A306935 -id:A306935 - cp's OEIS frontend

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

N. J. A. Sloane, Mar 16 2019

sign

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).

Cf. A259825, A306935, A306936, A306937.

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

Corrected and more terms from Robin Visser, Feb 24 2024

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

Original entry on oeis.org

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

N. J. A. Sloane, Mar 16 2019

nonn

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.

Cf. A259825, A306934, A306935, A306937.

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

More terms from Robin Visser, Feb 24 2024

cp's OEIS Frontend

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

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