A248395 q-Expansion of the modular form of weight 3/2, g*theta(4) in Tunnell's notation (see Comments).

0, 1, 0, 0, 0, 2, 0, 0, 0, -1, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, 2, 0, 0, 0, -4, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0

Offset: 0

Author

N. J. A. Sloane, Oct 18 2014

Keywords

• sign

Comments

g = Product_{m>=1} (1-q^(8*m))*(1-q^(16*m)),

theta(t) = Sum_{n=-oo..oo} (q^(t*n^2)).

Although the OEIS does not normally include sequences in which only every fourth term is nonzero, this one is important enough to warrant an exception.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Crossrefs

The nonzero quadrisection is A080966, which has further information and references.

Cf. A248394.

Used in A248397-A248406.

Programs

Maple

# This produces a list of the first 100 terms:
g:=q*mul((1-q^(8*m))*(1-q^(16*m)),m=1..30);
g:=series(g,q,100);
th:=t->series( add(q^(t*n^2),n=-50..50), q, 100);
series(g*th(4),q,100);
seriestolist(%);

Mathematica

QP := QPochhammer; a:= CoefficientList[Series[QP[q^8]*QP[q^16]* EllipticTheta[3, 0, q^4], {q, 0, 60}], q]; Join[{0}, Table[a[[n]], {n, 1, 50}]] (* G. C. Greubel, Jul 02 2018 *)

PARI

my(q='q+O('q^80)); A = eta(q^8)^6/(q*eta(q^4)^2*eta(q^16)); concat([0], Vec(A)) \\ G. C. Greubel, Jul 02 2018

Formula

From G. C. Greubel, Jul 02 2018: (Start)

Expansion of eta(q^8)*eta(q^16)*theta_{3}(0, q^4)/q in powers of q.

Expansion of eta(q^8)^6/(q*eta(q^4)^2*eta(q^16)). (End)