A107499 Theta series of quadratic form with Gram matrix [ 6, 2, 2, 1; 2, 18, 5, 9; 2, 5, 18, 9; 1, 9, 9, 24].

1, 0, 0, 2, 0, 0, 0, 0, 0, 4, 4, 0, 8, 2, 10, 0, 10, 8, 0, 0, 0, 0, 8, 6, 0, 12, 6, 8, 0, 4, 22, 0, 0, 0, 0, 18, 32, 0, 10, 8, 22, 0, 26, 12, 0, 0, 0, 0, 36, 18, 0, 20, 14, 16, 0, 20, 34, 0, 0, 0, 0, 10, 22, 0, 42, 12, 42, 0, 44, 26, 0, 0, 0, 0, 38, 34, 0, 30

Offset: 0

Author

N. J. A. Sloane, May 28 2005

Keywords

• nonn

Comments

G.f. is theta_3 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

Examples

			G.f. = 1 + 2*q^6 + 4*q^18 + 4*q^20 + ...
		

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.

Crossrefs

Cf. A107498, A107500, A107501, A107502, A107503, A107504, A107505.

Programs

Magma

prec := 90;
ls := [[6, 2, 2, 1], [2, 18, 5, 9], [2, 5, 18, 9], [1, 9, 9, 24]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := 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

Extensions

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