A107499 - OEIS

%I #23 May 14 2023 09:40:22

%S 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,

%T 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,

%U 0,10,22,0,42,12,42,0,44,26,0,0,0,0,38,34,0,30

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

%C 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

%H Andy Huchala, <a href="/A107499/b107499.txt">Table of n, a(n) for n = 0..20000</a>

%H W. R. Parry, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002196476">A negative result on the representation of modular forms by theta series</a>, J. Reine Angew. Math., 310 (1979), 151-170.

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

%o (Magma)

%o prec := 90;

%o ls := [[6, 2, 2, 1], [2, 18, 5, 9], [2, 5, 18, 9], [1, 9, 9, 24]];

%o S := Matrix(ls);

%o L := LatticeWithGram(S);

%o M := ThetaSeriesModularFormSpace(L);

%o B := Basis(M, prec);

%o T<q> := ThetaSeries(L, 48);

%o coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];

%o 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

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

%K nonn

%O 0,4

%A _N. J. A. Sloane_, May 28 2005

%E Name clarified and more terms from _Andy Huchala_, May 13 2023