A138661 Expansion of a level 11 weight 7 multiplicative modular form in powers of q.

1, 0, 10, 64, 74, 0, 0, 0, -629, 0, -1331, 640, 0, 0, 740, 4096, 0, 0, 0, 4736, 0, 0, -12670, 0, -10149, 0, -13580, 0, 0, 0, 56018, 0, -13310, 0, 0, -40256, 87050, 0, 0, 0, 0, 0, 0, -85184, -46546, 0, -206350, 40960, 117649, 0, 0, 0, 246890, 0, -98494, 0, 0, 0, 107642, 47360, 0

Offset

1,3

Comments

This is a member of an infinite family of odd weight level 11 multiplicative modular forms. g_1 = A035179, g_3 = A129522, g_5 = A065099, g_7 = A138661. - Michael Somos, Jun 07 2015

Links

Table of n, a(n) for n=1..61.

Formula

a(4*n + 2) = a(11*n + 2) = a(11*n + 6) = a(11*n + 7) = a(11*n + 8) = a(11*n + 10) = 0.

a(n) is multiplicative with a(11^e) = (-1331)^e, a(p^e) = p^(3*e) * (1 + (-1)^e) / 2 if p == 2, 6, 7, 8, 10 (mod 11), a(p^e) = a(p) * a(p^(e-1)) - p^6 * a(p^(e-2)) if p == 1, 3, 4, 5, 9 (mod 11) where a(p) = y^6 - 6*p*y^4 + 9*p^2*y^2 - 2*p^3 and 4 * p = y^2 + 11 * x^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(7/2) (t/i)^7 f(t) where q = exp(2 Pi i t).

Example

G.f. = q + 10*q^3 + 64*q^4 + 74*q^5 - 629*q^9 - 1331*q^11 + 640*q^12 + 740*q^15 + ...

Prog

(PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==11, (-1331)^e, kronecker(-11, p)==-1, if(e%2, 0, (p^3)^e), for(x=1, sqrtint(4*p\11), if( issquare(4*p - 11*x^2, &y), break)); y = y^6 - 6*p*y^4 + 9*p^2*y^2 - 2*p^3; a0=1; a1=y; for(i=2, e, x = y * a1 - p^6 * a0; a0=a1; a1=x); a1)))};
(PARI) {a(n) = my(A, F1, F2, G1); if( n<1, 0, A = x * O(x^n); F1 = x * (eta(x + A) * eta(x^11 + A))^2; F2 = x * eta(x^2 + A) * eta(x^22 + A); G1 = (F1 + 4 * F2^2 + 8 * x^4 * (eta(x^4 + A) * eta(x^44 + A))^2) / F2; polcoeff( G1 * F1 * (G1^4 - 8*G1^2*F1 + 7*F1^2), n))};
(Magma) A := Basis( CuspForms( Gamma1(11), 7), 58); A[1] + 10*A[3] + 64*A[4] + 74*A[5] - 629*A[9] - 1331*A[11] + 640*A[12] + 740*A[15] + 4096*A[16] + 4736*A[20] - 12670*A[23]; /* Michael Somos, Jun 07 2015 */

Crossrefs

Cf. A129522, A065099.

Sequence in context: A046638 A101467 A162473 * A341681 A269538 A178256

Adjacent sequences: A138658 A138659 A138660 * A138662 A138663 A138664

Keyword

sign,mult

Author

Michael Somos, Mar 25 2008

Status

approved