A129666 Expansion of unique cusp form of weight 4 level 7 in powers of q.

1, -1, -2, -7, 16, 2, -7, 15, -23, -16, -8, 14, 28, 7, -32, 41, 54, 23, -110, -112, 14, 8, 48, -30, 131, -28, 100, 49, -110, 32, 12, -161, 16, -54, -112, 161, -246, 110, -56, 240, 182, -14, 128, 56, -368, -48, 324, -82, 49, -131, -108, -196, -162, -100, -128

Offset

1,3

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

H. Rosson and G. Tornaria, Central values of quadratic twists for a modular form of weight 4, pp. 315-321 of J. B. Conrey et al., ed., Ranks of Elliptic Curves and Random Matrix Theory, Cambridge University Press, 2007.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Mathieu Lemire and Kenneth S. Williams, Evaluation of two convolution sums involving the sum of divisors function, Bulletin of the Australian Mathematical Society, Volume 73, Issue 1 February 2006, pp. 107-115. See c7() p. 108.

LMFDB, Newforms of weight 4 on Gamma_0(7).

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q * phi(-q)^3 * psi(q) * phi(-q^7)^3 * psi(q^7) + 4*q^2 * (phi(-q) * psi(q) * phi(-q^7) * psi(q^7))^2 in powers of q.

Expansion of ((eta(q) * eta(q^7))^3 + 4 * (eta(q^2) * eta(q^14))^3) * (eta(q) * eta(q^7))^2 / (eta(q^2) * eta(q^14)) in powers of q.

a(n) is multiplicative with a(7^e) = (-7)^e, a(p^e) = a(p) * a(p^(e-1)) - p^3 * a(p^(e-2)).

G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 49 (t/i)^4 f(t) where q = exp(2 Pi i t).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 + 2*u*v + 16*u*w + 12*v^2 + 32*v*w + 256*w^2) * (-v^3 + 2*w*u*v + w*u^2 + 16*w^2*u) + 2*v^5.

Convolution of A002652 and A002656.

Example

G.f. = q - q^2 - 2*q^3 - 7*q^4 + 16*q^5 + 2*q^6 - 7*q^7 + 15*q^8 - 23*q^9 - ...

Mathematica

a[ n_] := With[ {A1 = QPochhammer[ q] QPochhammer[ q^7], A2 = QPochhammer[ q^2] QPochhammer[ q^14]}, SeriesCoefficient[  (A1^3 + 4 q A2^3) A1^2 / A2, {q, 0, n}]]; (* Michael Somos, Nov 11 2015 *)

Prog

(PARI) {a(n) = my(A, A1, A2); if( n<1, 0, n--; A = x * O(x^n); A1 = eta(x + A) * eta(x^7 + A); A2 = eta(x^2 + A) * eta(x^14 + A); polcoeff( (A1^3 + 4*x * A2^3) * A1^2 / A2, n))};
(Sage) CuspForms( Gamma0(7), 4, prec=55).0; # Michael Somos, May 28 2013
(Magma) Basis( CuspForms( Gamma0(7), 4), 56)[1]; /* Michael Somos, Nov 11 2015 */

Crossrefs

Cf. A002652, A002656.

Sequence in context: A047694 A338399 A262016 * A288675 A135781 A233580

Adjacent sequences: A129663 A129664 A129665 * A129667 A129668 A129669

Keyword

sign,mult

Author

Michael Somos, Apr 27 2007

Status

approved