%I #11 Sep 08 2022 08:45:33

%S 1,0,10,64,74,0,0,0,-629,0,-1331,640,0,0,740,4096,0,0,0,4736,0,0,

%T -12670,0,-10149,0,-13580,0,0,0,56018,0,-13310,0,0,-40256,87050,0,0,0,

%U 0,0,0,-85184,-46546,0,-206350,40960,117649,0,0,0,246890,0,-98494,0,0,0,107642,47360,0

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

%C 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

%F 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.

%F 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.

%F 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).

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

%o (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)))};

%o (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))};

%o (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 */

%Y Cf. A129522, A065099.

%K sign,mult

%O 1,3

%A _Michael Somos_, Mar 25 2008