A140686 Coefficients of L-series for elliptic curve "49a1": y^2 + x * y = x^3 - x^2 - 2 * x - 1.

1, 1, 0, -1, 0, 0, 0, -3, -3, 0, 4, 0, 0, 0, 0, -1, 0, -3, 0, 0, 0, 4, 8, 0, -5, 0, 0, 0, 2, 0, 0, 5, 0, 0, 0, 3, -6, 0, 0, 0, 0, 0, -12, -4, 0, 8, 0, 0, 0, -5, 0, 0, -10, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 7, 0, 0, 4, 0, 0, 0, 16, 9, 0, -6, 0, 0, 0, 0, 8, 0, 9, 0, 0, 0, 0, -12, 0, -12, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, -12, 5, 0, 0

Offset

1,8

Comments

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

The g.f. is half the difference of the theta series of the Gram matrices denoted by B_1 and B_2 in the Parry 1979 reference on page 163.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

F. Brown and O. Schnetz, A K3 in phi^4, arXiv:1006.4064 [math.AG], 2010-2011 and Duke Math. J. 161, No. 10, 1817-1862 (2012)

W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (q * f(-q^21, -q^28) + q^2 * f(-q^14, -q^35) - q^4 * f(-q^7, -q^42)) * f(-q^7)^3 in powers of q where f() is Ramanujan's general theta function. [see Remark 60 from the Brown and Schnetz link on page 30]

a(n) is multiplicative with a(7^e) = 0^e, a(p^e) = (1 + (-1)^e) / 2 * (-p)^(e/2) if p == 3, 5, 6 (mod 7)

G.f. is a period 1 Fourier series which satisfies f(-1 / (49 t)) = 49 (t/i)^2 f(t) where q = exp(2 Pi i t).

a(7*n) = a(7*n + 3) = a(7*n + 5) = a(7*n + 6) = 0. a(9*n) = -3 * a(n), a(9*n + 3) = a(9*n + 6) = 0.

Example

G.f. = q + q^2 - q^4 - 3*q^8 - 3*q^9 + 4*q^11 - q^16 - 3*q^18 + 4*q^22 + ...

Mathematica

a[ n_] := SeriesCoefficient[ With[{Q = q^7}, Sum[ {-1, 1, 1}[[k]] q^{4, 2, 1}[[k]] QPochhammer[ Q^k, Q^7] QPochhammer[ Q^(7 - k), Q^7], {k, 3}] QPochhammer[ Q^7] QPochhammer[ Q]^3], {q, 0, n}]; (* Michael Somos, May 09 2015 *)

Prog

(PARI) {a(n) = ellak( ellinit( [1, -1, 0, -2, -1], 1), n)};
(PARI) {a(n) = my(G1, G2); if( n<1, 0, G1 = [ 2, 1, 0, 0; 1, 4, 0, 0; 0, 0, 14, 7; 0, 0, 7, 28]; G2 = [ 4, 2, 2, -1; 2, 8, 1, 3; 2, 1, 8, 3; -1, 3, 3, 16]; qfrep( G1, n, 1)[n] - qfrep( G2, n, 1)[n])}; /* (theta(B_1) - theta(B_2))/2 */
(PARI) {a(n) = my(A, p, e, y, a0, a1); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==7, 0, if( kronecker(-7, p)==1, y = if( p==2, 1, for( x=1, sqrtint(p\7), if( issquare( p - 7 * x^2, &y), break)); 2 * y * kronecker(-7, y)); a0 = 1; a1 = y; for(i=2, e, x = y * a1 - p * a0; a0=a1; a1=x); a1, if( e%2==0, (-p)^(e/2) )))))};
(Magma) Basis( CuspForms( Gamma0(49), 2), 103)[1]; /* Michael Somos, May 09 2015 */

Crossrefs

Sequence in context: A260636 A245256 A347149 * A116580 A096439 A256119

Adjacent sequences: A140683 A140684 A140685 * A140687 A140688 A140689

Keyword

sign,mult

Author

Michael Somos, May 22 2008

Status

approved