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

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, 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, 6, 2, -1, -15, 9, 8, 12, 1, 0, 3, -16, -10, -6, 16, 15, 9, 14, 0, 4

Offset

0,3

Comments

Fourier expansion denoted by f_44(tau) on p. 80 of Umbral Moonshine.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

M. Cheng, J. Duncan, and J. Harvey, Umbral Moonshine, arXiv:1204.2779 [math.RT], 2012-2013.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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.

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.

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.

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).

Example

G.f. = 1 + x - 3*x^2 + 2*x^3 - 2*x^4 - x^5 - 4*x^6 - 3*x^7 + 6*x^8 + ...
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 + ...

Mathematica

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}];

Prog

(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))};
(Sage) A = CuspForms( Gamma0(44), 2, prec=146) . basis(); A[0] + A[2];
(Magma) A := Basis( CuspForms( Gamma0(44), 2), 146); A[1] + A[3];

Crossrefs

Cf. A030200.

Sequence in context: A183043 A050604 A262672 * A239066 A239067 A131015

Adjacent sequences: A252728 A252729 A252730 * A252732 A252733 A252734

Keyword

sign

Author

Michael Somos, Dec 21 2014

Status

approved