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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].

%I A107499 #23 May 14 2023 09:40:22
%S A107499 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 A107499 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 A107499 0,10,22,0,42,12,42,0,44,26,0,0,0,0,38,34,0,30
%N 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].
%C A107499 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 A107499 Andy Huchala, <a href="/A107499/b107499.txt">Table of n, a(n) for n = 0..20000</a>
%H A107499 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 A107499 G.f. = 1 + 2*q^6 + 4*q^18 + 4*q^20 + ...
%o A107499 (Magma)
%o A107499 prec := 90;
%o A107499 ls := [[6, 2, 2, 1], [2, 18, 5, 9], [2, 5, 18, 9], [1, 9, 9, 24]];
%o A107499 S := Matrix(ls);
%o A107499 L := LatticeWithGram(S);
%o A107499 M := ThetaSeriesModularFormSpace(L);
%o A107499 B := Basis(M, prec);
%o A107499 T<q> := ThetaSeries(L, 48);
%o A107499 coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
%o A107499 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 A107499 Cf. A107498, A107500, A107501, A107502, A107503, A107504, A107505.
%K A107499 nonn
%O A107499 0,4
%A A107499 _N. J. A. Sloane_, May 28 2005
%E A107499 Name clarified and more terms from _Andy Huchala_, May 13 2023