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A028887 Theta series of 4-dimensional 5-modular lattice with det 25 and minimal norm 2.

%I A028887 #30 Sep 08 2022 08:44:50
%S A028887 1,6,18,24,42,6,72,48,90,78,18,72,168,84,144,24,186,108,234,120,42,
%T A028887 192,216,144,360,6,252,240,336,180,72,192,378,288,324,48,546,228,360,
%U A028887 336,90,252,576,264,504,78,432,288,744,342,18,432,588,324,720,72,720,480
%N A028887 Theta series of 4-dimensional 5-modular lattice with det 25 and minimal norm 2.
%D A028887 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 463 Entry 4(i).
%H A028887 John Cannon, <a href="/A028887/b028887.txt">Table of n, a(n) for n = 0..5000</a>
%H A028887 Shaun Cooper and Dongxi Ye, <a href="http://www.researchgate.net/profile/Shaun_Cooper/publication/276369530_Level_14_and_15_analogues_of_Ramanujans_elliptic_functions_to_alternative_bases/links/555fb1a108ae86c06b637117.pdf">Level 14 AND 15 Analogues of Ramanujan's Elliptic Functions to Alternative Bases</a>, preprint, 2015.
%H A028887 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/QQF.4.a.html">Home page for this lattice</a>
%F A028887 a(n) = 6*b(n) where b(n) is multiplicative with a(0) = 1,  b(5^e) = 1, b(p^e) = (p^(e+1) - 1) / (p - 1) otherwise. - _Michael Somos_, Feb 04 2006
%F A028887 G.f. 1 + 6 * (Sum_{k>0} k * x^k / (1 - x^k) - 5*k * x^(5*k) / (1 - x^(5*k))). - _Michael Somos_, Feb 04 2006
%e A028887 G.f. = 1 + 6*x + 18*x^2 + 24*x^3 + 42*x^4 + 6*x^5 + 72*x^6 + 48*x^7 + ...
%e A028887 G.f. = 1 + 6*q^2 + 18*q^4 + 24*q^6 + 42*q^8 + 6*q^10 + 72*q^12 + 48*q^14 + ...
%t A028887 a[ n_] := If[ n < 1, Boole[ n == 0], 6 Sum[ If[ Mod[ d, 5] > 0, d, 0], {d, Divisors @ n }]];  (* _Michael Somos_, Jun 12 2014 *)
%t A028887 a[ n_] := SeriesCoefficient[ 1 + 6 Sum[ k x^k / (1 - x^k) - 5 k x^(5 k) / (1 - x^(5 k)), {k, n}], {x, 0, n}]; (* _Michael Somos_, Jun 12 2014 *)
%o A028887 (PARI) {a(n) = if( n<1, n==0, 6 * sumdiv(n, d, (d%5>0) * d))}; /* _Michael Somos_, Feb 04 2006 */
%o A028887 (PARI) {a(n) = my(G); if( n<0, 0, G = [ 2, 1, 0, 0; 1, 2, 1, 0; 0, 1, 4, 5; 0, 0, 5, 10]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n, 1)), n))}; /* _Michael Somos_, Jun 12 2014 */
%o A028887 (Sage) ModularForms( Gamma0(5), 2, prec=70).0;  # _Michael Somos_, Jun 12 2014
%o A028887 (Magma) Basis( ModularForms( Gamma0(5), 2), 70) [1]; /* _Michael Somos_, Jun 12 2014 */
%K A028887 nonn
%O A028887 0,2
%A A028887 _N. J. A. Sloane_