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A252731 Fourier expansion of the unique newform on Gamma_0(44).

%I A252731 #23 Feb 16 2025 08:33:24
%S A252731 1,1,-3,2,-2,-1,-4,-3,6,8,2,-3,4,-5,0,5,-1,-6,-1,-4,0,-10,6,0,-3,6,-6,
%T A252731 3,8,3,-4,-4,12,-1,-3,15,-4,4,-2,2,1,6,-18,0,-9,-8,5,-24,-7,2,18,8,-6,
%U A252731 6,2,-1,-15,9,8,12,1,0,3,-16,-10,-6,16,15,9,14,0,4
%N A252731 Fourier expansion of the unique newform on Gamma_0(44).
%C A252731 Fourier expansion denoted by f_44(tau) on p. 80 of Umbral Moonshine.
%C A252731 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A252731 G. C. Greubel, <a href="/A252731/b252731.txt">Table of n, a(n) for n = 0..1000</a>
%H A252731 M. Cheng, J. Duncan, and J. Harvey, <a href="https://arxiv.org/abs/1204.2779">Umbral Moonshine</a>, arXiv:1204.2779 [math.RT], 2012-2013.
%H A252731 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A252731 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A252731 Expansion of 2*F(x^2)^5 / (F(x) * F(x^4)^2) - F(x)^3 / F(x^2) - 4*x * F(x) * F(x^2) in powers of x where F() is the g.f. for A030200.
%F A252731 Expansion of f(-q^22) * phi(-q^22)^3 / chi(-q^2) - 2 * q^4 * f(-q^22) * (chi(-q) * psi(q^11)^3 + chi(q) * psi(-q^11)^3) in powers of q^2 where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F A252731 a(n) = b(2*n + 1) where b() is multiplicative with b(p^e) = (-1)^e if p = 11, b(p^e) = b(p)*b(p^(e-1)) - p*b(p^(e-2)) if p != 11.
%F A252731 G.f. is a period 1 Fourier series which satisfies f(-1 / (44 t)) = 44 (t/i)^2 f(t) where q = exp(2 Pi i t).
%e A252731 G.f. = 1 + x - 3*x^2 + 2*x^3 - 2*x^4 - x^5 - 4*x^6 - 3*x^7 + 6*x^8 + ...
%e A252731 G.f. = q + q^3 - 3*q^5 + 2*q^7 - 2*q^9 - q^11 - 4*q^13 - 3*q^15 + 6*q^17 + ...
%t A252731 a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^11] (2 EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^11] - EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^11] - 4 x QPochhammer[ x^2] QPochhammer[ x^22]), {x, 0, n}];
%o A252731 (PARI) {a(n) = local(A, F1, F2, F4); if( n<0, 0, A = x * O(x^n); F1 = eta(x + A) * eta(x^11 + A); F2 = subst(F1, x, x^2); F4 = subst(F1, x, x^4); polcoeff( 2*F2^5 / (F1 * F4^2) - F1^3 / F2 - 4*x * F1*F2, n))};
%o A252731 (Sage) A = CuspForms( Gamma0(44), 2, prec=146) . basis(); A[0] + A[2];
%o A252731 (Magma) A := Basis( CuspForms( Gamma0(44), 2), 146); A[1] + A[3];
%Y A252731 Cf. A030200.
%K A252731 sign
%O A252731 0,3
%A A252731 _Michael Somos_, Dec 21 2014