1,2

Coefficients of level 11 weight 2 cusp form with trivial character.

Barry Cipra, What's Happening in the Mathematical Sciences, Vol. 5, Amer. Math. Soc., 2002; see p. 5.

H. Darmon, A proof of the full Shimura-Taniyama-Weil conjecture is announced, Notices Amer. Math. Soc., Dec. 1999, pp. 1397-1401.

F. Diamond, Congruences between modular forms: raising the level and dropping Euler factors, in Elliptic curves and modular forms (Washington, DC, 1996). Proc. Nat. Acad. Sci. U.S.A. 94 (1997), 11143-11146.

N. D. Elkies, Elliptic and modular curves..., in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 42.

Shimura, Goro; A reciprocity law in non-solvable extensions. J. Reine Angew. Math. 221 1966 209-220.

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 412.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. Wiles, Modular forms, elliptic curves and Fermat's last theorem, pp. 243-245 of Proc. Intern. Congr. Math. (Zurich), Vol. 1, 1994.

T. D. Noe, Table of n, a(n) for n=1..1002

J. Cowles, Some congruence properties of three well-known sequences: Two notes, J. Num. Theory 12(1) (1980) 84.

W. A. Stein, The Modular Forms Database

a(n) is multiplicative with a(11^e) = 1, a(p^e) = a(p)*a(p^(e-1)) - p*a(p^(e-2)) for p != 11. - Michael Somos Feb 12 2006

a(n) = A000594(n) (mod 11). [Cowles]. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2007

Expansion of (eta(q)* eta(q^11))^2 in powers of q.

Euler transform of period 11 sequence [ -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -4, ...] . - Michael Somos Feb 12 2006

G.f. A(x) satisfies 0= f(A(x), A(x^2), A(x^4)) where f(u, v, w)= u*w*(u+ 4*v+ 4*w)- v^3 . - Michael Somos Mar 21 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11 (t/i)^2 f(t) where q = exp(2 pi i t).

Coefficients of L-series for elliptic curve "11a3": y^2 + y = x^3 - x^2 . - Michael Somos May 23 2008

q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 + ...

(MAGMA) [ Coefficient(Basis(ModularForms(Gamma0(11), 2))[2], n) : n in [1..100] ]; /* Klaus Brockhaus, Jan 31 2007 */

(PARI) {a(n)=local(A, p, e, x, 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==11, 1, a0=1; a1=y=-sum(x=0, p-1, kronecker(4*x^3-4*x^2+1, p)); for(i=2, e, x=y*a1-p*a0; a0=a1; a1=x); a1))))} /* Michael Somos Aug 13 2006 */

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^2, n))}

(MAGMA) [ Coefficient(qEigenform(EllipticCurve([0, -1, 1, 0, 0]), n+1), n) : n in [1..100] ]; /* Klaus Brockhaus, Jan 29 2007 */

(PARI) {a(n) = ellak( ellinit( [0, -1, 1, 0, 0], 1), n)}

Cf. A002070 (terms with prime indices). Convolution square of A030200.

Sequence in context: A144757 A002107 A133099 this_sequence A094781 A023582 A023518

Adjacent sequences: A006568 A006569 A006570 this_sequence A006572 A006573 A006574

sign,easy,nice,mult

N. J. A. Sloane (njas(AT)research.att.com).