%I #18 May 14 2023 09:39:59
%S 1,0,2,0,0,2,2,0,4,0,0,2,0,2,0,6,0,0,10,8,14,12,0,0,20,0,6,0,16,0,0,8,
%T 18,18,12,0,0,12,0,8,0,6,0,0,30,22,20,10,0,0,22,0,14,0,38,0,0,22,30,
%U 18,48,0,0,30,0,12,0,22,0,0,38,16,50,30,0,0,46,0
%N Theta series of quadratic form with Gram matrix [ 4, 1, 0, 1; 1, 10, 0, 3; 0, 0, 26, 13; 1, 3, 13, 36].
%C G.f. is theta_6 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="/A107502/b107502.txt">Table of n, a(n) for n = 0..20000</a>
%H W. R. Parry, <a href="http://gdz.sub.unigoettingen.de/dms/resolveppn/?PPN=GDZPPN002196476">A negative result on the representation of modular forms by theta series</a>, J. Reine Angew. Math., 310 (1979), 151170.
%e G.f. = 1 + 2*q^4 + 2*q^10 + 2*q^12 + ...
%o (Magma)
%o prec := 90;
%o ls := [[4, 1, 0, 1], [1, 10, 0, 3], [0, 0, 26, 13], [1, 3, 13, 36]];
%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*i1] : 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, A107499, A107500, A107501, A107503, A107504, A107505.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, May 28 2005
%E Name clarified and more terms from _Andy Huchala_, May 13 2023
