A107503 Theta series of quadratic form with Gram matrix [ 8, 1, 1, -1; 1, 18, 5, 8; 1, 5, 18, 8; -1, 8, 8, 18].

1, 0, 0, 0, 2, 0, 0, 0, 0, 6, 4, 0, 10, 2, 6, 0, 8, 6, 0, 0, 0, 0, 8, 10, 0, 6, 6, 14, 0, 14, 20, 0, 0, 0, 0, 10, 24, 0, 16, 8, 28, 0, 28, 8, 0, 0, 0, 0, 34, 16, 0, 18, 14, 8, 0, 18, 40, 0, 0, 0, 0, 22, 26, 0, 40, 12, 44, 0, 38, 28, 0, 0, 0, 0, 36, 38, 0, 30

Offset

0,5

Comments

G.f. is theta_7 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^8 + 6*q^18 + 4*q^20 + ...

Prog

(Magma)
prec := 90;
ls := [[8, 1, 1, -1], [1, 18, 5, 8], [1, 5, 18, 8], [-1, 8, 8, 18]];
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, A107500, A107501, A107502, A107504, A107505.

Sequence in context: A006792 A011992 A318329 * A350731 A364055 A184366

Adjacent sequences: A107500 A107501 A107502 * A107504 A107505 A107506

Keyword

nonn

Author

N. J. A. Sloane, May 28 2005

Extensions

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

Status

approved