A187076 Coefficients of L-series for elliptic curve "144a1": y^2 = x^3 - 1.

1, 4, 2, -8, -5, 4, -10, -8, 9, 0, 14, 16, -10, 4, 0, 8, 14, -20, 2, 0, -11, -20, -32, 16, 0, 4, 14, -8, -9, -20, 26, 0, 2, 28, 0, 16, 16, 28, -22, 0, 14, -16, 0, -40, 0, 28, 26, -32, -17, 0, -32, 16, -22, 0, -10, -32, -34, 8, 14, 0, 45, 4, 38, -8, 0, 0, -34

Offset

0,2

Comments

Number 67 of the 74 eta-quotients listed in Table I of Martin (1996).

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(x)^4 in powers of x where f() is a Ramanujan theta function.

Expansion of q^(-1/6) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^4 in powers of q.

Euler transform of period 4 sequence [4, -8, 4, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = f(t) where q = exp(2 Pi i t).

a(n) = b(6*n + 1) and b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)), b(p) = 0 if p == 5 (mod 6), b(p) = 2 * x where p = x^2 + 3 * y^2 == 1 (mod 6) and x == 4, 5 (mod 6).

G.f.: Product_{k>0} (1 - (-x)^k)^4. a(n) = (-1)^n * A000727(n).

Convolution cube is A209676. - Michael Somos, Jun 10 2015

a(2*n) = A258779(n). a(2*n + 1) = 4 * A187150(n). - Michael Somos, Jun 10 2015

Example

G.f. = 1 + 4*x + 2*x^2 - 8*x^3 - 5*x^4 + 4*x^5 - 10*x^6 - 8*x^7 + 9*x^8 + ...
G.f. = q + 4*q^7 + 2*q^13 - 8*q^19 - 5*q^25 + 4*q^31 - 10*q^37 - 8*q^43 + 9*q^49 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^4, {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 0, 0, -1], 1), 6*n + 1))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A) * eta(x^4 + A)))^4, n))};
(PARI) {a(n) = my(m, A, p, e, x, y, a0, a1); if( n<0, 0, m = 6*n + 1; A = factor(m); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, p%6==5, if(e%2, 0, (-p)^(e/2)), for( y=1, sqrtint(p\3), if( issquare(p - 3*y^2, &x), break)); a0 = 1; if( x%6>3, x = -x); a1 = x = 2*x; for( i=2, e, y = x*a1 - p*a0; a0 = a1; a1 = y); a1)))};
(Magma) A := Basis( CuspForms( Gamma0(144), 2), 398); A[1] + 4*A[7] + 2*A[11] - 8*A[13]; /* Michael Somos, Jan 01 2017 */

Crossrefs

Cf. A000727, A187150, A209676, A258779, A280384.

Sequence in context: A194064 A194054 A191536 * A000727 A030181 A021879

Adjacent sequences: A187073 A187074 A187075 * A187077 A187078 A187079

Keyword

sign

Author

Michael Somos, Mar 05 2011

Status

approved