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

%I A129666 #28 Aug 06 2025 17:33:06
%S A129666 1,-1,-2,-7,16,2,-7,15,-23,-16,-8,14,28,7,-32,41,54,23,-110,-112,14,8,
%T A129666 48,-30,131,-28,100,49,-110,32,12,-161,16,-54,-112,161,-246,110,-56,
%U A129666 240,182,-14,128,56,-368,-48,324,-82,49,-131,-108,-196,-162,-100,-128
%N A129666 Expansion of unique cusp form of weight 4 level 7 in powers of q.
%C A129666 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A129666 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.
%H A129666 G. C. Greubel, <a href="/A129666/b129666.txt">Table of n, a(n) for n = 1..1000</a>
%H A129666 Mathieu Lemire and Kenneth S. Williams, <a href="https://people.math.carleton.ca/~williams/papers/pdf/284.pdf">Evaluation of two convolution sums involving the sum of divisors function</a>, Bulletin of the Australian Mathematical Society, Volume 73, Issue 1 February 2006, pp. 107-115. See c7() p. 108.
%H A129666 LMFDB, <a href="https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/7/4/">Space of modular forms of level 7 and weight 4</a>.
%H A129666 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A129666 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A129666 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.
%F A129666 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.
%F A129666 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)).
%F A129666 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).
%F A129666 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.
%F A129666 Convolution of A002652 and A002656.
%e A129666 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 - ...
%t A129666 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 *)
%o A129666 (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))};
%o A129666 (Sage) CuspForms( Gamma0(7), 4, prec=55).0; # _Michael Somos_, May 28 2013
%o A129666 (Magma) Basis( CuspForms( Gamma0(7), 4), 56)[1]; /* _Michael Somos_, Nov 11 2015 */
%Y A129666 Cf. A002652, A002656.
%K A129666 sign,mult
%O A129666 1,3
%A A129666 _Michael Somos_, Apr 27 2007