A226347 Expansion of the unique normalized cusp form of Gamma_0(5) of weight 6 in powers of q.

1, 2, -4, -28, 25, -8, 192, -120, -227, 50, -148, 112, 286, 384, -100, 656, -1678, -454, 1060, -700, -768, -296, 2976, 480, 625, 572, 1880, -5376, -3410, -200, -2448, 5152, 592, -3356, 4800, 6356, 182, 2120, -1144, -3000, -9398, -1536, -1244, 4144, -5675

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..2500

Formula

Expansion of (eta(q) * eta(q^5))^7 / (eta(q^3) * eta(q^15)) + 9 * (eta(q) * eta(q^5))^5 * eta(q^3) * eta(q^15) + 27 * (eta(q) * eta(q^3) * eta(q^5) * eta(q^15))^3 in powers of q.

a(n) is multiplicative with a(p^e) = p^(2*e) if p=5, else a(p^e) = a(p) * a(p^(e-1)) - p^6 * a(p^(e-2)).

G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 125 (t/i)^6 f(t) where q = exp(2 Pi i t).

a(5*n) = 25 * a(n) for all n in Z.

Example

G.f. = q + 2*q^2 - 4*q^3 - 28*q^4 + 25*q^5 - 8*q^6 + 192*q^7 - 120*q^8 - 227*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ With[{A = QPochhammer[ q] QPochhammer[ q^5], B = QPochhammer[ q^3] QPochhammer[ q^15]}, With[{U = q A B, V = A^3/B}, U (V^2 + 9 U V + 27 U^2)]], {q, 0, n}];
a[ n_] := SeriesCoefficient[ With[{ V = q (QPochhammer[ q^5] / QPochhammer[ q])^6, U = q^3 (QPochhammer[ q] QPochhammer [q^5])^12}, Sqrt[ U (1/V + 22 + 125 V)]], {q, 0, n}]; (* Michael Somos, Dec 05 2014 *)

Prog

(PARI) {a(n) = my(A, u1, u3, u, v); if( n<1, 0, n--; A = x * O(x^n); u1 = eta(x + A) * eta(x^5 + A); u3 = eta(x^3 + A) * eta(x^15 + A); u = x * u1 * u3; v = u1^3 / u3; polcoeff( u * (v^2 + 9 * u*v + 27 * u^2), n))};
(Sage) CuspForms( Gamma0(5), 6, prec=45).0;
(Magma) Basis( CuspForms( Gamma0(5), 6), 46) [1]; /* Michael Somos, Nov 12 2014 */

Crossrefs

Sequence in context: A256451 A059719 A264930 * A305738 A323447 A227303

Adjacent sequences: A226344 A226345 A226346 * A226348 A226349 A226350

Keyword

sign,mult

Author

Michael Somos, Jun 04 2013

Status

approved