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A107497 Theta series of quadratic form with Gram matrix [ 2, 1, 1, 1; 1, 20, 7, 7; 1, 7, 20, 7; 1, 7, 7, 46].

%I A107497 #32 May 14 2023 09:39:51
%S A107497 1,2,0,0,2,0,0,0,0,2,8,0,8,2,4,0,10,4,0,0,0,0,12,4,0,6,6,12,0,8,20,0,
%T A107497 0,0,0,16,22,0,24,8,24,0,32,20,0,0,0,0,36,14,0,20,14,16,0,24,32,0,0,0,
%U A107497 0,20,28,0,30,12,44,0,24,24,0,0,0,0,28,44,0,32
%N A107497 Theta series of quadratic form with Gram matrix [ 2, 1, 1, 1; 1, 20, 7, 7; 1, 7, 20, 7; 1, 7, 7, 46].
%C A107497 G.f. is theta_1 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 A107497 Andy Huchala, <a href="/A107497/b107497.txt">Table of n, a(n) for n = 0..20000</a>
%H A107497 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 A107497 G.f. = 1 + 2*q^2 + 2*q^8 + 2*q^18 + ...
%o A107497 (Magma)
%o A107497 prec := 60;
%o A107497 ls := [[2,1,1,1],[1,20,7,7],[1,7,20,7],[1,7,7,46]];
%o A107497 S := Matrix(ls);
%o A107497 L := LatticeWithGram(S);
%o A107497 M := ThetaSeriesModularFormSpace(L);
%o A107497 B := Basis(M, prec);
%o A107497 T<q> := ThetaSeries(L,44);
%o A107497 coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
%o A107497 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 A107497 Cf. A107498-A107505, A140686.
%K A107497 nonn
%O A107497 0,2
%A A107497 _N. J. A. Sloane_, May 28 2005
%E A107497 Name clarified and more terms from _Andy Huchala_, May 13 2023