%I M0092 #88 Sep 14 2024 06:54:16

%S 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,

%T -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,

%U 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

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

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

%C Unique cusp form of weight 2 for congruence group Gamma_1(11). - _Michael Somos_, Aug 11 2011

%C 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

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

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

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

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

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

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

%H Seiichi Manyama, <a href="/A006571/b006571.txt">Table of n, a(n) for n = 1..10000</a> (first 1002 terms from T. D. Noe)

%H J. Cowles, <a href="http://dx.doi.org/10.1016/0022-314X(80)90076-1">Some congruence properties of three well-known sequences: Two notes</a>, J. Num. Theory 12(1) (1980) 84.

%H H. Darmon, <a href="http://www.ams.org/notices/199911/comm-darmon.pdf">A proof of the full Shimura-Taniyama-Weil conjecture is announced</a>, Notices Amer. Math. Soc., Dec. 1999, pp. 1397-1401.

%H F. Diamond, <a href="http://www.pnas.org/content/94/21/11115.full">Congruences between modular forms: raising the level and dropping Euler factors</a>, in Elliptic curves and modular forms (Washington, DC, 1996). Proc. Nat. Acad. Sci. U.S.A. 94 (1997), 11143-11146.

%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.

%H A. W. Knapp, <a href="http://www.ams.org/notices/201409/rnoti-p1056.pdf">Review of "Love and Math: The Heart of Hidden Reality" by E. Frenkel</a>, Notices Amer. Math. Soc., 61 (2014), pp. 1056-1060; see p. 1058, but beware typos.

%H LMFDB, <a href="http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/11/2/a/a/">Newform orbit 11.2.a.a</a>

%H LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/11/a/3">Elliptic curve with LMFDB label 11.a3 (Cremona label 11a3)</a>

%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

%H Shimura, Goro, <a href="http://dx.doi.org/10.1515/crll.1966.221.209">A reciprocity law in non-solvable extensions</a>, J. Reine Angew. Math. 221 1966 209-220.

%H G. Shimura, <a href="/A002070/a002070.pdf">A reciprocity law in non-solvable extensions</a>, J. Reine Angew. Math. 221 1966 209-220. [Annotated scan of pages 218, 219 only]

%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>

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

%F a(n) == A000594(n) (mod 11). [Cowles]. - _R. J. Mathar_, Feb 13 2007

%F Euler transform of period 11 sequence [ -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -4, ...]. - _Michael Somos_, Feb 12 2006

%F 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

%F 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

%F 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).

%F Convolution square of A030200.

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

%F Convolution inverse is A032442. - _Michael Somos_, Apr 21 2015

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

%F a(2) = -2 is not 2 - A272196(1) = 0. Modularity pattern of some elliptic curves. - _Wolfdieter Lang_, Apr 25 2016

%e 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 + ...

%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^11])^2, {q, 0, n}]; (* _Michael Somos_, Aug 11 2011 *)

%t 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 *)

%o (PARI) {a(n) = if( n<1, 0, ellak( ellinit( [0, -1, 1, 0, 0], 1), n))};

%o (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))};

%o (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 */

%o (Sage) CuspForms( Gamma1(11), 2, prec = 101).0 # _Michael Somos_, Aug 11 2011

%o (Magma) [ Coefficient(qEigenform(EllipticCurve([0, -1, 1, 0, 0]), n+1),n) : n in [1..100] ]; /* _Klaus Brockhaus_, Jan 29 2007 */

%o (Magma) [ Coefficient(Basis(ModularForms(Gamma0(11), 2))[2], n) : n in [1..100] ]; /* _Klaus Brockhaus_, Jan 31 2007 */

%o (Magma) Basis( CuspForms( Gamma1(11), 2), 101) [1]; /* _Michael Somos_, Jul 14 2014 */

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

%K sign,easy,nice,mult

%O 1,2

%A _N. J. A. Sloane_