A306934 Coefficients of q-expansion of Eisenstein series G_{5/2}(tau) multiplied by 120.

1, -10, 0, 0, -70, -48, 0, 0, -120, -250, 0, 0, -240, -240, 0, 0, -550, -480, 0, 0, -528, -480, 0, 0, -720, -1210, 0, 0, -960, -720, 0, 0, -1080, -1440, 0, 0, -1750, -1200, 0, 0, -1680, -1920, 0, 0, -1680, -1488, 0, 0, -2160, -3370, 0, 0, -2640, -1680, 0, 0, -2400, -3360, 0, 0, -2880, -2640

Offset: 0

Author

N. J. A. Sloane, Mar 16 2019

Keywords

• sign

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 (erroneously gives a(5) = -72).

Crossrefs

Cf. A259825, A306935, A306936, 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*sigma(f/d, 3) for d in f.divisors()])
    return round((120*X.lfunction(100)(-1)*s).real()) # Robin Visser, Feb 24 2024

Extensions

Corrected and more terms from Robin Visser, Feb 24 2024