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A136549 Expansion of (eta(q^3) * eta(q^5))^3 - (eta(q) * eta(q^15))^3 in powers of q.

%I A136549 #16 Jun 14 2023 09:35:01
%S A136549 1,-1,3,-3,-5,-3,0,7,9,5,0,-9,0,0,-15,5,14,-9,-22,15,0,0,-34,21,25,0,
%T A136549 27,0,0,15,2,-33,0,-14,0,-27,0,22,0,-35,0,0,0,0,-45,34,14,15,49,-25,
%U A136549 42,0,86,-27,0,0,-66,0,0,45,-118,-2,0,13,0,0,0,-42,-102,0,0,63,0,0,75,66,0,0,98,-25,81,0,-154,0
%N A136549 Expansion of (eta(q^3) * eta(q^5))^3 - (eta(q) * eta(q^15))^3 in powers of q.
%C A136549 G.f. is a newform level 15 weight 3 and nontrivial character.
%C A136549 The terms of A115155 differ only in sign from this sequence. - _Michael Somos_, Jun 14 2023
%F A136549 a(n) is multiplicative with a(3^e) = 3^e, a(5^e) = (-5)^e, a(p^e) = p^e * (1 + (-1)^e) / 2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = a(p) * a(p^(e-1)) - p^2 * a(p^(e-2)) if p == 1, 2, 4, 8 (mod 15).
%F A136549 G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Aug 29 2013
%e A136549 G.f. = q - q^2 + 3*q^3 - 3*q^4 - 5*q^5 - 3*q^6 + 7*q^8 + 9*q^9 + 5*q^10 + ...
%t A136549 QP = QPochhammer; s = (QP[q^3]*QP[q^5])^3-q*(QP[q]*QP[q^15])^3 + O[q]^100; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015 *)
%o A136549 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^5 + A))^3 - x * (eta(x + A) * eta(x^15 + A))^3, n))};
%o A136549 (Magma) A := Basis( CuspForms( Gamma1(15), 3), 80); A[1] - A[2] + 3*A[3] - 3*A[4] - 5*A[5] - 3*A[6] + 7*A[8]; /* _Michael Somos_, Oct 13 2015 */
%Y A136549 Cf. A115155.
%K A136549 sign,mult
%O A136549 1,3
%A A136549 _Michael Somos_, Jan 05 2008