LEVEL 17 RAMANUJAN-SATO SERIES. 1. Introduction Let τ be a complex number with positive imaginary part and set q = e 2πiτ. Define

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1 LEVEL 17 RAMANUJAN-SATO SERIES TIM HUBER, DANIEL SCHULTZ, AND DONGXI YE Abstract. Two level 17 modular functions r q 1 q n n 17, s q 1 q 17n 3 1 q n 3, are used to construct a new class of Ramanujan-Sato series for 1/. The expansions are induced by modular identities similar to those level of 5 and 13 appearing in Ramanujan s Notebooks. A complete list of rational and quadratic series corresponding to singular values of the parameters is derived. 1. Introduction Let τ be a complex number with positive imaginary part and set q e iτ. Define rτ q 1 q n 17 n, sτ q 1 q 17n 3 1 q n 3. In this paper, we derive level 17 Ramanujan-Sato expansions for 1/ of the form q d dq log s rr n s + 8rs r s 1 A n 8r 3 s 3r, s + r s s r r 4r 15, 1.1 r n0 where A n is defined recursively. These relations are analogous to those at level 13 and 5 [5, 1], q d dq log S R1 3R R n 1 An 1 + R, S 1 3 R, 1. R n0 q d dq log S R R 5 R 10 n 1 an 1 + R 10, S 1 R 5 11 R n0 Here an, An are recursively defined sequences induced from differential equations and Rτ q 1 q n 13 n, Rτ q 1/5 1 q n n Identity 1.3 and explicit evaluations for Rτ were used to formulate expansions for 1/ including an n n n n n n + j, an n j j n0 This expression is a generalization of 17 such formulas stated by Ramanujan [3, 18]. In each formula, the algebraic constants come from explicit evaluations for a modular function. The sequence are coefficients in a series solution to a differential equation satisfied by a relevant modular form. Generalizing such relations to higher levels requires finding differential equations for modular parameters and relevant identities. The series [5, 9, 10, 1] have common construction for primes p 1 4, where X 0 p has genus zero. More work remains to unify constructions for other levels. 1 j0

2 TIM HUBER, DANIEL SCHULTZ, AND DONGXI YE The purpose of this paper is to construct level 17 Ramanujan-Sato series as a prototype for levels such that X 0 N has positive genus. Central to the construction is the fact that r and s generate the field of functions invariant under action by an index two subgroup of Γ These constructions and singular value evaluations yield new Ramanujan-Sato expansions, including the rational series k k+ Following [5, 6], an expansion for 1/ is said to be rational or quadratic if C/ can be expressed as a series of algebraic numbers of degree 1 or, respectively, for some algebraic number C. We derive a complete list of series of rational and quadratic series from singular values of parameters in 1.1. In the next section, we given an overview of results at levels 5 and 13 that motivate the approach of the paper. Section 3 includes an analogous construction of level 17 modular functions. This construction is used to motivate the differential equation satisfied by with coefficients in the field Cx, where zτ θ q log s, θ q : q d dq, 1.7 xτ rr s + 8rs r s 8r 3 s 3r s + r s. 1.8 We conclude with Section 4 in which singular values are derived for x and used to construct a new class of series approximations for 1/ of level 17. We derive a complete list of values of xτ with [Qxτ : Q] within the radius of convergence for z as a powers series in x and therefore provide a complete list of linear and quadratic Ramanujan-Sato series corresponding to xτ.. Level 5 and 13 Series The product Rτ, defined by 1.4, is the Rogers-Ramanujan continued fraction [19]. Together, Rτ and Sτ, defined by q 1/5 1 q 5n 6 Rτ, Sτ q q n 6,.1 q q+ q 1+ q3 1+ generate the field of functions invariant under the congruence subgroup Γ 0d { a b c d SL Z c 0 mod p and χd 1}. This motivates Ramanujan s reciprocal identity [17], [1, p. 67] 1 R 5 11 R5 1 S.. Equation 1.3 expresses the logarithmic derivative Z θ q log Sτ in terms of a series solution to a third order linear differential equation [5] satisfied by S and the Γ 0 5 invariant function T T τ 16T + 44T 1Z T T T + 48T + 66T Z T T + 44T + 34T Z T + 1T + 6T Z 0,.3 Z T T d dt Z, T R5 1 11R 5 R R The form of the equation may be anticipated from a general theorem [] c.f. [4].

3 LEVEL 17 RAMANUJAN-SATO SERIES 3 Theorem.1. Let Γ be subgroup of SL R commensurable with SL Z. If tτ is a nonconstant meromorphic modular function and F τ is a meromorphic modular form of weight k with respect to Γ, then F, τf,..., τ k F, as functions of t, are linearly independent solutions to a k + 1st order differential linear equation with coefficients that are algebraic functions of t. The coefficients are polynomials when Γ \ H has genus zero and t generates the field of modular functions on Γ. Therefore, from.3, Z ant n, T < ,.5 8 where an is recursively determined from.3, and expressible in closed form [5] in terms of the summand appearing in 1.5. The final ingredient needed for Ramanujan-Sato series at level 5 are explicit evaluations for the Rogers-Ramanujan continued fraction within the radius of convergence of the power series. Such singular values for Rτ were given by Ramanujan in his first letter to Hardy [] and can be derived from modular equations satisfied by T τ and T nτ. We provide a general approach in Section 4. To formulate the analogous construction at level 13, define R Rτ by 1.4 and Sτ q 1 q 13n 1 q n, T τ R1 3R R 1 + R..6 A third order linear differential equation [1] is satisfied by the Eisenstein series Zτ θ q log S with coefficients that are polynomials in the Γ 0 13 invariant function T τ. For both the level 5 and 13 cases, the weight zero functions T and T may be uniformly presented as the quotient of a weight 4 cusp form and the square of a weight Eisenstein series T UV Z, where Uτ θ q log R, Uτ θ q log R, n Vτ 13 q n 1 q n, V τ T UV Z,.7 n 5 q n 1 q n..8 Both levels require singular values for T, T [11]. Explicit evaluations for Z and Z follow from W θ q log T 1 1T 16T Z, W θ q log T 1 44T + 16T Z..9 The pairs T, W, T, W, respectively, generate the field of invariant functions for Γ 0 13, Γ 0 5, and r and s generate invariant function fields for the congruence subgroup Γ Proposition.. Let A 0 Γ denote the field of functions invariant under Γ and denote by χ the real quadratic character modulo p. Then 1 A 0 Γ 0 5+ CT and A 0 Γ 0 5 CT, W. A 0 Γ CT and A 0 Γ 0 13 CT, W. 3 For prime p 1 mod 4, A 0 Γ 0 p CR p, S p, where l p R p q lp p 1 nn p χn, p 1 q n χn, S p q ap 1 q np bp 1 q n,.10 bp p 1 4 a p b p, gcda p, b p 1..11

4 4 TIM HUBER, DANIEL SCHULTZ, AND DONGXI YE A proof of the first two parts of Proposition. may be given along the lines of the proof of Proposition 3.. The third part of the Proposition is a main result of [15]. The results of [15] explain Ramanujan s level 5 reciprocal relation. and his level 13 reciprocal relation [1, Equation 8.4] 1 R 3 R 1 S..1 For our present work at level 17, we apply a new identity proven in [15] r + 1 r 4 r 4r 15 1 s..13 Our next task is to construct functions analogous to T and W in terms of r, s and Eisenstein series. 3. Functions invariant under Γ 0 17 and a differential equation In this Section we prove an analogue to Proposition. and derive a second order linear differential equation for z defined by 1.7 with coefficents in Cx, where x is defined by 1.8. In order to construct functions that are invariant under Γ 0 17, we introduce sums of eight Eisenstein series considered in [14]. Set E 1 τ : 1 8 χ 1 1 E χ,k τ, E χ,k τ 1 + L1 k, χ χn nk 1 q n 1 q n, 3.1 where the sum in 3.1 is over the odd primitive Dirichlet characters modulo 17 and L1 k, χ is the analytic continuation of the associated Dirichlet L-series and χ 1 1 k. For a Z/17Z, apply the diamond operator [13] to define, for 1 k 8, a E 1 τ 1 χae χ,1 τ, E k τ ± 3 k 1 E 1 τ χ 1 1 The sign in Equation 3. is chosen so that the first coefficient in the q series expansion is 1. The parameters E k τ have the product representations [14, Theorems ] q E 1 τ 8, q 9, q 17, q 17 q, q 3, q 14, q 15 ; q 17 q, E τ q 3, q 14, q 17, q 17 q, q 5, q 1, q 16 ; q 17, E 3 τ q 3 q, q 16, q 17, q 17 q 4, q 6, q 11, q 13 ; q 17 q, E 4 τ q 6, q 11, q 17, q 17 q, q 7, q 10, q 15 ; q 17, E 5 τ q 3 q, q 15, q 17, q 17 q 5, q 8, q 9, q 1 ; q 17 q, E 6 τ q 5, q 1, q 17, q q 3, q 4, q 13, q 14 ; q 17, q E 7 τ q 4, q 13, q 17, q 17 q, q 7, q 10, q 16 ; q 17, E 8 τ q q 7, q 10, q 17, q 17 q 6, q 8, q 9, q 11 ; q 17, a1,..., a m b 1,..., b n ; z a 1 ; z a m ; z b 1 ; z b n ; z, a; z 1 az n. A function Ω is now introduced as a level 17 analogue to the level 5 cusp form UV. Define Ωτ E 1 E E E 3 + E 3 E 4 E 4 E 5 + E 5 E 6 E 6 E 7 E 7 E 8 E 8 E Proposition 3.1 demonstrates that the weight two parameters z and Ω, respectively, play a role at level seventeen analogous to that played by the parameters Z and UV at level 5. Proposition 3.1. Let z zτ be defined by 1.7. Then 1 The Eisenstein space of weight two E Γ 0 17 is generated by z. n0

5 LEVEL 17 RAMANUJAN-SATO SERIES 5 The space of cusp forms of weight two S Γ 0 17 is generated by Ω. 3 Both z and Ω change sign under W17,, where f W17,kτ 17 k/ τ k f 1/17τ. 4 Both z and Ω have zeros at the elliptic points ρ ±, and in the case of Ω, the zeros are simple. Proof. From 3. and the definition of the E i, if arithmetic is performed modulo 8 on the subscripts, 3 E k ɛ k E k+1, ɛ 1,..., ɛ 8 +1, 1, +1, 1, +1, 1, 1, This, coupled with the transformation formula for Eisenstein series, aτ + b a b E k cτ + d a E k τ, Γ cτ + d c d implies that Ω and z are modular forms of weight two with respect to Γ From their q-expansions, we deduce that Ω and z are linearly independent over C. Therefore, from dimension formulas for the respective vector spaces [13], we see that these parameters generate the vector space of weight two forms for Γ Thus, we obtain the first two claims of Proposition 3.1. The third claim follows from the fact that W 17 normalizes Γ As fundamental domain for H/Γ 0 17 we take 8 k 8 F kd D, where D is the usual fundamental domain for the full modular group and F k τ 1 τ+k. The two elliptic points of order are ρ ± F ±4 i. Since H/Γ 0 17 has two elliptic points of order and two cusps, the valence formula for a weight k modular form f on Γ 0 17 reads as ord f + ord 0 f + ord ρ + f + ord ρ f + τ H {ρ ± } ord τ f k 1 18 From the q-expansion and the fact that Ω changes sign under W17,, we know that the two cusps are zeros of Ω, so the valence formula for f Ω reads as ord ρ + f + ord ρ f + ord τ f 3. τ H {ρ ± } Since Ω changes sign under W17, and the fixed point of W 17 is not a zero as one may check numerically, the zeros must come in pairs. Accordingly, the two other zeros must be the two elliptic points, and these are simple zeros. A similar argument gives the result for z. The cusp form and Eisenstein series from Proposition 3.1 can now be used in the construction of a Γ 0 17 invariant function of the same form as T, T given by.7. Although the representation for xτ given here appears to differ from that given in the introduction, we ultimately demonstrate agreement of the two representations in Proposition 3.3. The parameters xτ and wτ, defined in Proposition 3., play roles analogous to corresponding parameters T and W in Proposition.. Proposition 3.. If the Fricke involution is denoted W 17 W 17,0 and x and w are defined by xτ Ω z, wτ z θ q log x, x is invariant under Γ 0 17 as well as W 17 ; and x has two simple zeros on H/Γ 0 17 at the two cusps. 3 The field of functions invariant under Γ 0 17 and W 17 is A 0 Γ Cx. 4 The field of functions invariant under Γ 0 17 is given by A 0 Γ 0 17 Cx, w. 5 The relation w 17x 4 48x 3 66x 16x + 1 holds.

6 6 TIM HUBER, DANIEL SCHULTZ, AND DONGXI YE Proof. The first two assertions follow directly from Proposition 3.1. The third assertion is then a direct consequence of the first two. For the fourth assertion, the functions xτ and wτ are invariant under Γ 0 17, so it suffices to show that they generate the whole field. Since x has order, we have [A 0 Γ 0 17 : Cx]. Since w Cx because it changes sign under W 17, we must have [A 0 Γ 0 17 : Cx, w] 1, that is, the second assertion holds. For the final assertion, the function w is fixed under W 17 and has the same set of poles as x, hence it is a polynomial in x. We bound the degree of this polynomial by 4 and find its coefficients by comparing q expansions. The parameter x is expressible as the rational function of r and s appearing in the Introduction and in terms of the McKay-Thompson series 17A [8, Table 4A]. Proposition 3.3. Define ητ q 1/4 q; q, and let x be defined as in Proposition 3.. Then x rr s + 8rs r s 8r 3 s 3r s + r s, x x 1 4ητ η17τ m,n e im e in q 1 4 n m. 3.9 Proof. From the product representation for r and those for the Eisenstein sums E i, from 3.3 r E 1E 3 E 5 E 7 E E 4 E 6 E Therefore, r is the quotient of weight four modular forms for Γ 1 17, and x Ω/z is the quotient of weight two modular forms for Γ Hence, the quadratic relation between x and r, 4 r 4r 15 xr 1 4r 1 x + r 3.11 may be transcribed as a relation between modular forms of weight 0 for Γ 1 17 and proved from the Sturm bound by verifying the q-expansion to order /1. Then 4 xr 14r 1 4r 15, 3.1 r x + r where the branch of the square root is determined using the definition of x and r. Therefore, 4r 1 + rβr 4 x 4r r βr, βr 4r r The first equation of 3.13 is seen to be equivalent to 3.8 by applying.13. Equation 3.9 may be derived from respective q-expansions since each side is a Hauptmodul for Γ It follows from the first part of Proposition 3. and Theorem.1 that z satisfies a third order linear homogeneous differential equation with coefficients in Cx. In order to formulate the differential equation, we state the following preliminary nonlinear differential equation in terms of the differential operator θ q : q q dq. This is written even more succinctly as f q : θ q f. Lemma 3.4. zz qq 3z q 3z 4 x 17x 5 x x 3 + 4x + 7x + 4x 1

7 LEVEL 17 RAMANUJAN-SATO SERIES 7 Proof. Let fτ denote the function on the left hand side of the proposed equality. If z satsifies the functional equation aτ + b cτ + d z ɛ cτ + d ad bc zτ one can compute that zz qq 3z q 3z 4 aτ + b 1 zz qq 3zq cτ + d ɛ 3z 4 τ. By Proposition 3.1, we have ɛ 1 for elements of Γ 0 17 and ɛ 1 for W 17. Thus we see that fτ is invariant under Γ 0 17 and W 17 in weight 0. According to Theorem 4.4, we see that x does not have a pole at the two elliptic points, i.e. xρ ± 1. This means that the two zeros of z at these elliptic points are both simple. Hence, z has two other simple zeros p 1 and p W 17 p 1, which are also the poles of x, modulo Γ 0 17, as observed in the proof of Proposition 3.1. Since all of the poles are z are simple, we can take the expansion zτ cτ r +... at the zeros r ρ +, ρ, p 1, p, where c is non-zero. quadruple pole to fτ since zz qq 3z q 3z 4 τ 3 c τ r 4 + Each of these zeros contributes a In the fundamental domain of H/Γ 0 17, the translate F 4 D is adjacent to itself. Thus xτ must identify the two halves of the corresponding side of F 4 D the side that contains F 4 i. Likewise for F 4. Therefore, at the elliptic point ρ ±, the function xτ is locally a holomorphic function of τ ρ ± /τ ρ ± so that xτ 1 + c ± τ ρ ± + We see now that x 1 f has poles only at p 1 and p, each of order six. It is therefore a polynomial of degree six in x, and we can compute that 4x 1 f x17x 5 x x 3 + 4x + 7x + Oq 7. The left hand side has poles of order 6 at p 1, p and zeros at least order 7 at 0 and. This contradicts the valence formula unless the left hand side is constant. We now give the third order linear differential equation for z with rational coefficients in x. The concise formulation of the differential equation in 3.14 is motivated by the general form of such differential equations from [, 4]. Theorem 3.5. With respect to the function x, the form f z satisfies the differential equation. 0 3x54x 6 714x x 4 50x 3 6x 8x 1f + xx 11397x 5 48x x 3 8x + 197x + 14f x + 6xx x x + 33x + 4f xx + x x x x + 16x 1f xxx. Proof. The differential equation satisfied by f z is given as f f x f xx f xxx det z z x z xx z xxx z log q z log q x z log q xx z log q xxx z log q z log q x z log q xx z log q xxx

8 8 TIM HUBER, DANIEL SCHULTZ, AND DONGXI YE When expanding this determinant, we make the following substitutions: 1 For the diffential with respect to x, use the definition 3.7 in the form θ x x x wz θ q. When the first derivative x q appears, use the definition 3.7 in the form x q 1 xwz. 3 When the first derivative w q appears, use the relation between w and x to obtain w q x17x x + 33x + 4z. 4 When the second derivative z qq appears, use Lemma 3.4 in the form z qq 3z q z + 3x17x5 x x 3 + 4x + 7x + 8x 1 z 3 When these substitutions are made in 3.14, the claimed differential equation results after clearing denominators by multiplying by 1 x 3 w 5 /16 and using Proposition The linear differential equation in Theorem 3.5 induces a series expansion for z in terms of x with coefficients A n. Corollary 3.6. z where A 0, A 1,..., 6 0 and A n x n x < , 3.15 n0 0 n A n n 3 4n 14n 3A n 3 5n 3 + 7n 8n + 4 A n n 3 300n + 13n 5 A n 3 55n 3 67n + 491n 305 A n 3 + 3n 3 101n 97n + 53 A n 4 9n 4n 337n 66A n n 5n 4n 3A n 6. The radius of convergence is the positive root of 17x x x + 16x 1. To make use of the series appearing in Corollary 3.6, we require explicit evaluations for the xτ within the domain of validity. In the next section, we prove that the number of singular values is finite and compile a complete list of quadratic evaluations and expansions. 4. Singular Values and Series for 1/ In this Section, singular values for xτ are derived and used to formulate Ramanujan- Sato expansions. The work culminates in a proof that there are precisely 11 singular values for xτ of degree at most two over Q within the radius of convergence of Corollary 3.6. The series given by 1.6 is the only such expansion with a rational singular value for xτ. The main challenge in proving the expansions lies in rigorously determining exact evaluations for xτ for given τ and deriving constants appearing in the Ramanujan-Sato series. To do this, we formulate modular equations for xτ and provide an explicit relation between the modular equations and constants appearing in the series. We demonstrate in the proof of Theorem 4.7 that the following table is a complete list of singular values for xτ in a fundamental domain for Γ 0 17 with [Qxτ : Q] within the radius of convergence of Corollary 3.6. Each value τ is listed by the coefficients a, b, c of its minimal polynomial, and the values are ordered by discriminant.

9 LEVEL 17 RAMANUJAN-SATO SERIES 9 b 4ac τa, b, c xτ , 17, , 17, , 17, , 7, i , 41, i , 34, , 68, , 34, , 17, , 17, 7 1/ , 34, Table 1. Complete list of singular values of xτ of degree at most within the radius of convergence of Corollary 3.6, ordered by discriminant. 4.1 By Proposition 3.3, finding singular values for xτ is equivalent to finding singular values for the normalized Thompson series 17A. The fundamental results needed for such evaluations are presented in [7]. Our work below is a detailed rendition of the general presentation in [7] tailored to the modular function xτ. We begin with the set of matrices n17 { α β γ δ Z gcdα, β, γ, δ 1 and αδ βγ n and γ 0 mod 17}. Lemma 4.1. If gcdn, 17 1, then n17 has the coset decomposition α β n17 Γ 0 17, 0 δ and the double coset representation αδn 0 β<δ gcdα,β,δ1 n17 Γ n Γ Proof. Any α β γ δ n can be converted to an upper triangular matrix by multiplying on the left by a matrix of the form γ gcdα,γ α gcdα,γ Γ It is then easy to see that the claimed representatives are distinct modulo Γ This proves the decomposition formula. Next, by performing elementary row and column operations on the matrix m n17, we find matrices γ 1, γ Γ1 such that m γ n 0 γ. a b a bn Since n nc d n we may find appropriate a, b, c, and d such that γ 1 γ 1 nc a d b Γ 017. This results in an equality of the form m γ n 0 γ where γ 1 Γ 017, which forces γ Γ 017 as well. This establishes the double coset representation. We now establish modular equations central to our explicit evaluations for xτ. c d,

10 10 TIM HUBER, DANIEL SCHULTZ, AND DONGXI YE Proposition 4.. For any integer n with gcdn, 17 1, there is a polynomial Ψ n X, Y of degree ψn n q n q in X and Y such that: q prime 1 Ψ n X, Y is irreducible and has degree ψn in X and Y. Ψ n X, Y is symmetric in X and Y. 3 The roots of Ψ n xτ, Y 0 are precisely the numbers Y xατ + β/δ for integers α, β and δ such that αδ n, 0 β < δ, and gcdα, β, δ 1. Proof. The polynomial Ψ n satisfies XY ψn Ψ n X, Y αδn 0 β<δ α,β,δ1 ατ + β 1 Y 1 x, 4. δ where the coefficients of Y k on the right hand side should expressed as polynomials in 1/X for X xτ as demonstrated in the proof of Corollary 4.3. This relies on the fact that Γ 0 17 and W 17 permute the set of functions x ατ + β/δ where αδ n, 0 β < δ, and gcdα, β, δ 1. The double coset representation in Lemma 4.1 shows that every orbit contains xτ/n, and hence the action of Γ 0 17 on the roots must be transitive. Since α β 0 δ δ 0 17β α n17, it is clear that W 17 permutes these functions as well by the decomposition in Lemma 4.1. The coefficient of X ψn Y ψn in Ψ n X, Y is the constant term of the product on the right hand side of 4., which is clearly non-zero because the function xτ does not have poles at the cusps of H/Γ Therefore, Ψ n X, X has the claimed degree ψn. The symmetry can be proven by noting that τ 1/17nτ interchanges xτ and xnτ. In Corollary 4.3, modular equations Ψ n X, Y 0 are derived for X xτ and y xατ +β/δ satisfying the conditions of Proposition 4.. The proof indicates how modular equations for larger n may be derived and involves techniques analogous to those used to deduce classical modular equations of level n satisfied by the j invariant [16, 1]. Corollary 4.3. We have Ψ X, Y 9X 3 Y 3 1X 3 Y + X 3 Y + X 3 1X Y 3 + 8X Y + 10X Y + XY XY XY + Y 3, Ψ 3 X, Y 435X 4 Y X 4 Y X 3 Y X 4 Y 385X 3 Y X Y 4 39X 4 Y 63X 3 Y 63X Y 3 39XY 4 + 4X 4 + 9X 3 Y + 13X Y + 9XY 3 + 4Y X Y + 15XY XY. Proof. The level n result is representative of n 3 and other cases. For n, we have {α, β, δ gcdα, β, δ 1, 0 β < δ, αδ n} {1, 0,,, 0, 1, 1, 1, }. Let xτ be defined as in Proposition 3.. Then τ + 1 x 1 xτ/, x xτ, x 3 x. By 4., Ψ 1/X, 1/Y Y 3 x1 x + x 1 x 3 + x x 3 Y + x 1 x x 3 x1 + x + x 3 x 1 x x 3 Y 1 1. x 1 x x 3

11 LEVEL 17 RAMANUJAN-SATO SERIES 11 By Theorem 3., we know that 1/xτ is analytic on X 0 17 except for simple poles at the cusps. Therefore, the only poles in H/Γ 0 17 of the coefficients of Y k are at points equivalent to the cusps 0 and. We can explicitly compute the q-expansion for each of the coefficients and deduce that each has a pole of order at most 3 at q 0. Since each coefficient is invariant under Γ 0 17 and W 17, the coefficients may be expressed as polynomials of degree at most 3 in 1/xτ. Explicitly, x 1x + x 1 x 3 + x x 3 x 1 x x Oq, q x 1 + x + x 3 0 x 1 x x 3 q Oq, q 1 8 x 1 x x 3 q q Oq. q Therefore, with x xτ, we may determine polynomials in 1/x such that c 1 τ 1 9 x 1 x x 6x x + x 3 Oq, 3 c τ x 1 + x + x 3 x 1 x x 3 c 3 τ x 1x + x 1 x 3 + x x 3 x 1 x x x 1 + 5x Oq, 1 + 5x 1 1 x Oq. Since the functions c j τ, j 1,, 3, are analytic on the upper half plane and at the cusps of X 0 17, each c j is constant, and c j i 0. In Theorem 4.4, we prove each evaluation from Table 1. The evaluations are proven by showing that xτ satisfies a modular equation of degree n for some n. Theorem 4.4. Each evaluation for xτa, b, c from Table 1 holds, and xτ17, 8, 1 1. Proof. First note that τ17, i Therefore, xτ x i 17 Observe that with τ i 17, 1 17τ τ. Since xτ is invariant under the Fricke involution W 17, we have xτ xτ/. That is, X Y in the degree modular equation above. Setting Y X and simplifying the equation, we get X X 1X + 19X + 4X Now that we have proven xτ satisfies 4.3, we may numerically deduce xτis a root of 9X + 4X 1, and x i We may similarly prove xτ17, 8, 1 1 and the remaining evaluations in Table 1 from modular equations of degree n if we can determine, for each given value of τ τa, b, c, an upper upper triangular matrix α, β; 0, γ such that xτ xα, β; 0, γτ and αδ n, 0 β < δ, gcdα, β, δ 1, with gcdn, For each τ, Table provides a γ n17 such that γτ τ or W 17 τ and a Γ 0 17 equivalent upper triangular matrix α, β; 0, γ.

12 1 TIM HUBER, DANIEL SCHULTZ, AND DONGXI YE τa, b, c Element of n17 α, β; 0, γ 17, 17, 5 1, ; 17, 5 1, ; 0, 59 17, 17, 19 1, 0; 17, 19 1, 0; 0, 19 17, 17, 13 1, 0; 17, 13 1, 0; 0, 13 17, 7, 17 13, 17; 17, 14 1, 81; 0, , 41, 31 0, 31; 17, 1 1, 68; 0, , 34, 3 1, 0; 34, 3 1, 0; 0, 3 34, 68, 37, 1; 51, 37 1, 11; 0, 3 17, 34, 1, 1; 17, 1, 4; 0, 5 17, 17, 9, 1; 17, 18 1, 9; 0, 19 17, 17, 7 1, 0; 17, 7 1, 0; 0, 7 17, 34, 19 1, 1; 17, 19 1, 1; 0, 17, 8, 1 3, 1; 17, 5 1, 1; 0, Table. Elements of n17 mapping τ to its image under W 17 or fixing τ and a corresponding Γ 0 17 equivalent upper triangular matrix α, β; 0, γ. 4.4 For values xτ in the domain of validity for series from Corollary 3.6, we may construct Ramanujan-Sato expansions via Theorem 4.5, a specialization of [4, Theorem.1]. Theorem 4.5 Series for 1/. Suppose there is a matrix a, b; c, d Γ 0 17, W 17 such that aτ + b cτ + d ατ + β δ for αδ n and 0 β < δ. Set X xτ, which is determined from Ψ n X, X 0, and further set W wτ, Ψ X Ψ p X, X, X Ψ and let ɛ Q and η ±1 satsify for all τ aτ + b z ɛcτ + d zτ, w cτ + d If is the sequence defined in Corollary 3.6 and then Y Ψ p X, X, Y aτ + b ηwτ. cτ + d B iw δ Ψ X ad bc + α ηɛψ Y cτ + d 4 α cηɛψ Y cτ + d 3, C Proof. Differentiate the relation iδ bc adw α cηɛψ 3 Y cτ + d3 Ψ XΨ Y Ψ X + Ψ Y 1 + θ X log W + Ψ XΨ YY Ψ X Ψ XY Ψ Y + Ψ XX Ψ Y X, z 1 Bk + CX k. aτ + b ɛcτ + d zτ cτ + d once and the relation ατ + β Ψ n xτ, x 0 δ twice and then set τ to the value in the hypothesis of the theorem.

13 LEVEL 17 RAMANUJAN-SATO SERIES 13 Corollary 4.6. If is the sequence defined in Corollary 3.6, k 1 k+, i 7 and / k 7 17 k+, k k+, k k+, k k+, ± i k ± 33i 7 k, k k+, k k+, k 55 ± 4 k+. Proof. The first five series may be derived by setting τ n 17τ 9 17i and using 34τ 17 τ+n 1/ n with ɛ 1/17 and η 1 in Theorem 4.5 for the values n 11, 19, 35, 59, 83. The subsequent pair may be derived from Theorem 4.5 by setting τ ±7 + 47i/34, using 11τ /17τ 3 τ +69/107 and 13τ +3/17τ +4 τ +8/107, respectively. The next three arise from setting τ n 1 17i and using 17τ τ n for n, 5, 6. The final series comes from setting τ 3/34i and using 1 17τ τ 3. Theorem 4.7. There are precisely eleven Γ 0 17 inequivalent algebraic τ in the upper half plane such that [Qxτ : Q] with xτ in the radius of convergence of Corollary 3.6. Proof. We formulate a complete list of algebraic τ such that [Qxτ : Q] using well known facts about the j invariant [0]. First, for algebraic τ, the only algebraic values of jτ occur at Iτ > 0 satisfying aτ + bτ + c 0 for a, b, c Z, with d b 4ac < 0. Moreover, [Qjτ : Q] hd, where hd is the class number. Since there is a polynomial relation P x, j between x and j of degree [7, Remark 1.5.3], we have [Qjτ : Q] [Qxτ : Q], and so values τ with [Qxτ : Q] satisfy [Qjτ : Q] hd 4. Therefore, the bound d 1555 for hd 4 from [3] implies that the following algorithm results in a complete list of algebraic τ, xτ with [Qxτ : Q] : For each discriminant 1555 d 1, 1 List all primitive reduced τ τa, b, c of discriminant d in a fundamental domain for P SL Z. Translate these values via a set of coset representatives for Γ 0 17 to a fundamental domain for Γ 0 17.

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