A306936 Coefficients of q-expansion of Eisenstein series G_{9/2}(tau) multiplied by 240.

1, 2, 0, 0, 242, 480, 0, 0, 2640, 4322, 0, 0, 11040, 13920, 0, 0, 30962, 39360, 0, 0, 65760, 73920, 0, 0, 125280, 156002, 0, 0, 216960, 226080, 0, 0, 340560, 406080, 0, 0, 522962, 541920, 0, 0, 756960, 860160, 0, 0, 1033440, 1063200, 0, 0, 1424160, 1646402, 0, 0, 1907040, 1860000, 0, 0

Offset: 0

Author

N. J. A. Sloane, Mar 16 2019

Keywords

• nonn

Links

• H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271-285.
• X. Wang and D. Pei, Modular Forms with Integral and Half-Integral Weights, Science Press Beijing, Springer Berlin Heidelberg, 2012. x+432 pp.
• D. Zagier, Modular Forms of One Variable, Notes based on a course given in Utrecht, 1991. See page 50.

Crossrefs

Cf. A259825, A306934, A306935, A306937.

Programs

Sage

def a(n):
    if n==0: return 1
    if (n%4) not in [0, 1]: return 0
    D = Integer(n).squarefree_part()
    f = Integer(sqrt(n/D))
    if (D%4) not in [0, 1]: D, f = 4*D, f//2
    X = kronecker_character(D)
    s = sum([moebius(d)*X(d)*d^3*sigma(f/d, 7) for d in f.divisors()])
    return round((240*X.lfunction(100)(-3)*s).real()) # Robin Visser, Feb 24 2024

Extensions

More terms from Robin Visser, Feb 24 2024