A027364 Coefficients of unique normalized cusp form Delta_16 of weight 16 for full modular group.

1, 216, -3348, 13888, 52110, -723168, 2822456, -4078080, -3139803, 11255760, 20586852, -46497024, -190073338, 609650496, -174464280, -1335947264, 1646527986, -678197448, 1563257180, 723703680, -9449582688, 4446760032, 9451116072, 13653411840, -27802126025, -41055841008

Offset: 1

Author

Paolo Dominici (pl.dm(AT)libero.it), N. J. A. Sloane

Keywords

• sign
• easy

Examples

			G.f. = q + 216*q^2 - 3348*q^3 + 13888*q^4 + 52110*q^5 - 723168*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]
• Author?, Table of coefficients c16(n) of the weight 16 cusp form on Gamma_0(1) for n up to 1000
• F. Q. Gouvea, Non-ordinary primes, Experimental Mathematics 6 195, 1997.
• LMFDB, Newform orbit 1.16.a.a.
• 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. A013963, A126763, A290152, A290178.

The unique normalized cusp form: A000594 (k=12), this sequence (k=16), A037944 (k=18), A037945 (k=20), A037946 (k=22), A037947 (k=26).

Programs

Maple

with(numtheory): DO := qs -> q*diff(qs,q)/2: E2:=1-24*add(sigma(n)*q^(2*n),n=1..100): delta16:=(-1/24)*(DO@@6)(E2)*E2+(9/8)*(DO@@5)(E2)*(DO@@1)(E2)-(45/8)*(DO@@4)(E2)*(DO@@2)(E2)+(55/12)*(DO@@3)(E2)*(DO@@3)(E2):seq(coeff(delta16,q,2*i),i=1..40); with(numtheory): E2n:=n->1-(4*n/bernoulli(2*n))*add(sigma[2*n-1](k)*q^(2*k),k=1..100): qs:=(E2n(2)^4-E2n(3)^2*E2n(2))/1728: seq(coeff(qs,q,2*i),i=1..40); # C. Ronaldo

Mathematica

terms = 26;
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}];
(E4[x]^4 - E6[x]^2*E4[x])/1728 + O[x]^(terms+1) // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Feb 27 2018, after Seiichi Manyama *)

PARI

N=66; q='q+O('q^N); Vec(q*(1+240*sum(n=1,N,sigma(n,3)*q^n))*eta(q)^24) \\ Joerg Arndt, Nov 23 2015

Formula

G.f.: q*(1 + 240*Sum_{n>=1} sigma_3(n)q^n) Product_{k>=1} (1-q^k)^24, where sigma_3(n) is the sum of the cubes of the divisors of n (A001158).

(E_4(q)^4 - E_6(q)^2*E_4(q))/1728.

a(n) == A013963(n) mod 3617. - Seiichi Manyama, Feb 01 2017

G.f.: -691/(1728*250) * (E_4(q)*E_12(q) - E_8(q)^2). - Seiichi Manyama, Jul 25 2017

Extensions

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 17 2005