A037944 Coefficients of unique normalized cusp form Delta_18 of weight 18 for full modular group.

1, -528, -4284, 147712, -1025850, 2261952, 3225992, -8785920, -110787507, 541648800, -753618228, -632798208, 2541064526, -1703323776, 4394741400, -14721941504, -5429742318, 58495803696, 1487499860, -151530355200

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign

Examples

			G.f. = q - 528*q^2 - 4284*q^3 + 147712*q^4 - 1025850*q^5 + 2261952*q^6 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000
• Steven R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]
• Fernando Q. Gouvêa, Non-ordinary primes: a story, Experimental Mathematics, Volume 6, Issue 3 (1997), 195-205.
• S. C. Milne, Hankel determinants of Eisenstein series, preprint, arXiv:0009130 [math.NT], 2000.
• H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
• Index entries for sequences related to modular groups

Crossrefs

Cf. A000594, A013965, A013973, A027364, A037945, A037946, A037947, A290048.

Programs

Mathematica

terms = 20;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms+1}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms+1}];
E12[x_] = 1 + (65520/691)*Sum[k^11*x^k/(1 - x^k), {k, 1, terms}];
E14[x_] = 1 - 24*Sum[k^13*x^k/(1 - x^k), {k, 1, terms}];
(691/(1728*250))*(E4[x]*E14[x] - E6[x]*E12[x]) + O[x]^(terms+1) // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Feb 27 2018, after Seiichi Manyama *)

PARI

{a(n) = if( n<0, 0, polcoeff( x * eta(x + x * O(x^n))^24 * (1 - 504 * sum( k=1, n, sigma( k, 5) * x^k)), n))}; /* Michael Somos, Mar 18 2012 */

Formula

Convolution product of A000594 and A013973. - Michael Somos, Mar 18 2012

a(n) == A013965(n) mod 43867. - Seiichi Manyama, Feb 02 2017

G.f.: 691/(1728*250) * (E_4(q)*E_14(q) - E_6(q)*E_12(q)). - Seiichi Manyama, Jul 25 2017