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Theta series of extremal even unimodular lattice in dimension 56.
2

%I #17 Jun 05 2021 17:02:24

%S 1,0,0,15590400,36957286800,15284192071680,2099603881267200,

%T 134803322124134400,4960017097962973200,119289241340847513600,

%U 2051414989573311774720,26894038407511734144000,281804009505443595441600

%N Theta series of extremal even unimodular lattice in dimension 56.

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 195.

%H Andy Huchala, <a href="/A004673/b004673.txt">Table of n, a(n) for n = 0..20000</a>

%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

%F E4^7 - 1680 * E4^4 * Delta + 347760 * E4 * Delta^2 with E4(q) as in A004009 and Delta(q) as in A000594. - _Andy Huchala_, Jun 05 2021

%e G.f.: 1 + 15590400*q^3 + 36957286800*q^4 + ...

%o (Sage)

%o e4 = eisenstein_series_qexp(4,20,normalization = "integral");

%o delta = CuspForms(1,12).0.q_expansion(20);

%o e4^7 - 1680*e4^4*delta + 347760*e4*delta^2 # _Andy Huchala_, Jun 05 2021

%Y Cf. A000594, A004009.

%K nonn

%O 0,4

%A _N. J. A. Sloane_