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A140686
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Coefficients of L-series for elliptic curve "49a1": y^2 + x * y = x^3 - x^2 - 2 * x - 1.
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
| W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170. See p. 163.
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LINKS
| F. Brown and O. Schnetz, A K3 in phi^4 see Remark 60.
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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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 two-variable theta function. [from the Parry reference]
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 Fourier series of a weight 2 level 49 modular form. 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.
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EXAMPLE
| q + q^2 - q^4 - 3*q^8 - 3*q^9 + 4*q^11 - q^16 - 3*q^18 + 4*q^22 + ...
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PROG
| (PARI) {a(n) = ellak( ellinit( [1, -1, 0, -2, -1], 1), n)}
(PARI) {a(n) = if( n<1, 0, polcoeff( qfrep( [2, 1, 0, 0; 1, 4, 0, 0; 0, 0, 14, 7; 0, 0, 7, 28], n, 1) - qfrep( [4, 2, 2, -1; 2, 8, 1, 3; 2, 1, 8, 3; -1, 3, 3, 16], n, 1), n))}
(PARI) {a(n) = local(A, p, e, y, a0, a1); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; 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)))))))}
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CROSSREFS
| Sequence in context: A084103 A036477 A128164 * A116580 A096439 A128046
Adjacent sequences: A140683 A140684 A140685 * A140687 A140688 A140689
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, May 22 2008
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