A107500 Theta series of quadratic form with Gram matrix [ 10, 4, 4, 1; 4, 12, -1, 3; 4, -1, 12, 3; 1, 3, 3, 30].

1, 0, 0, 0, 0, 2, 4, 4, 2, 0, 0, 2, 0, 2, 0, 8, 0, 0, 14, 2, 8, 10, 0, 0, 18, 0, 6, 0, 18, 0, 0, 4, 24, 16, 22, 0, 0, 14, 0, 8, 0, 10, 0, 0, 20, 22, 22, 14, 0, 0, 30, 0, 14, 0, 30, 0, 0, 24, 22, 10, 48, 0, 0, 24, 0, 12, 0, 24, 0, 0, 46, 22, 60, 12, 0, 0, 32, 0

Offset

0,6

Comments

G.f. is theta_4 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 13 2023

Links

Andy Huchala, Table of n, a(n) for n = 0..20000

W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.

Example

G.f. = 1 + 2*q^10 + 4*q^12 + 4*q^14 + ...

Prog

(Magma)
prec := 90;
ls := [[10, 4, 4, 1], [4, 12, -1, 3], [4, -1, 12, 3], [1, 3, 3, 30]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T<q> := ThetaSeries(L, 48);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 13 2023

Crossrefs

Cf. A107498, A107499, A107501, A107502, A107503, A107504, A107505.

Sequence in context: A328412 A079536 A058923 * A108620 A070512 A156346

Adjacent sequences: A107497 A107498 A107499 * A107501 A107502 A107503

Keyword

nonn

Author

N. J. A. Sloane, May 28 2005

Extensions

Name clarified and more terms from Andy Huchala, May 13 2023

Status

approved