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%I #20 May 14 2023 09:40:09
%S 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,
%T 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,
%U 10,48,0,0,24,0,12,0,24,0,0,46,22,60,12,0,0,32,0
%N Theta series of quadratic form with Gram matrix [ 10, 4, 4, 1; 4, 12, -1, 3; 4, -1, 12, 3; 1, 3, 3, 30].
%C 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
%H Andy Huchala, <a href="/A107500/b107500.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^10 + 4*q^12 + 4*q^14 + ...
%o (Magma)
%o prec := 90;
%o ls := [[10, 4, 4, 1], [4, 12, -1, 3], [4, -1, 12, 3], [1, 3, 3, 30]];
%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, A107499, A107501, A107502, A107503, A107504, A107505.
%K nonn
%O 0,6
%A _N. J. A. Sloane_, May 28 2005
%E Name clarified and more terms from _Andy Huchala_, May 13 2023
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