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

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

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice
• mult

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

Examples

			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 + ...
		

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

Crossrefs

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

Programs

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 */

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 *)

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

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