A006571 Expansion of q*Product_{k>=1} (1-q^k)^2*(1-q^(11*k))^2. (Formerly M0092)

1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4, -2, 4, 0, 2, 2, -2, -1, 0, -4, -8, 5, -4, 0, 2, 7, 8, -1, 4, -2, -4, 3, 0, -4, 0, -8, -4, -6, 2, -2, 2, 8, 4, -3, 8, 2, 8, -6, -10, 1, 0, 0, 0, 5, -2, 12, -14, 4, -8, 4, 2, -7, -4, 1, 4, -3, 0, 4, -6, 4, 0, -2, 8, -10, -4, 1, 16, -6, 4, -2, 12, 0, 0, 15, 4, -8, -2, -7, -16, 0, -8, -7, 6, -2, -8

Offset

1,2

Comments

Number 23 of the 74 eta-quotients listed in Table I of Martin (1996).

Unique cusp form of weight 2 for congruence group Gamma_1(11). - Michael Somos, Aug 11 2011

For some elliptic curves with p-defects given by this sequence, and for more references, see A272196. See also the Michael Somos formula from May 23 2008 below. - Wolfdieter Lang, Apr 25 2016

References

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

M. du Sautoy, Review of "Love and Math: The Heart of Hidden Reality" by Edward Frenkel, Nature, 502 (Oct 03 2013), p. 36.

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

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.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1002 terms from T. D. Noe)

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

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.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.

A. W. Knapp, Review of "Love and Math: The Heart of Hidden Reality" by E. Frenkel, Notices Amer. Math. Soc., 61 (2014), pp. 1056-1060; see p. 1058, but beware typos.

LMFDB, Newform orbit 11.2.a.a

LMFDB, Elliptic curve with LMFDB label 11.a3 (Cremona label 11a3)

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

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

G. Shimura, A reciprocity law in non-solvable extensions, J. Reine Angew. Math. 221 1966 209-220. [Annotated scan of pages 218, 219 only]

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

Formula

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

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

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

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

G.f. A(q) satisfies 0 = f(A(q), A(q^2), A(q^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).

Convolution square of A030200.

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

Convolution inverse is A032442. - Michael Somos, Apr 21 2015

a(prime(n)) = prime(n) - A272196(n), n >= 3.

a(2) = -2 is not 2 - A272196(1) = 0. Modularity pattern of some elliptic curves. - Wolfdieter Lang, Apr 25 2016

Example

G.f.: q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 + q^11 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^11])^2, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)
a[ n_] := SeriesCoefficient[ q (Product[ (1 - q^k), {k, 11, n, 11}] Product[ 1 - q^k, {k, n}])^2, {q, 0, n}]; (* Michael Somos, May 27 2014 *)

Prog

(PARI) {a(n) = if( n<1, 0, ellak( ellinit( [0, -1, 1, 0, 0], 1), n))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^2, n))};
(PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; 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 */
(Sage) CuspForms( Gamma1(11), 2, prec = 101).0 # Michael Somos, Aug 11 2011
(Magma) [ Coefficient(qEigenform(EllipticCurve([0, -1, 1, 0, 0]), n+1), n) : n in [1..100] ]; /* Klaus Brockhaus, Jan 29 2007 */
(Magma) [ Coefficient(Basis(ModularForms(Gamma0(11), 2))[2], n) : n in [1..100] ]; /* Klaus Brockhaus, Jan 31 2007 */
(Magma) Basis( CuspForms( Gamma1(11), 2), 101) [1]; /* Michael Somos, Jul 14 2014 */

Crossrefs

Cf. A002070 (terms with prime indices), A032442, A030200.

Sequence in context: A208845 A232506 A133099 * A243906 A100889 A206828

Adjacent sequences: A006568 A006569 A006570 * A006572 A006573 A006574

Keyword

sign,easy,nice,mult

Author

N. J. A. Sloane

Status

approved