I am trying to write out the Hecke Operator; however, I don't know how to sum over all divisors of an integer. Could someone please give me some advice how to do that. Below is the Hecke Operator definition I use:

$$T_NZ(\tau)= \sum_{d|N}\sum_{\kappa=0}^{d-1}Z\left(\frac{N\tau/d +\kappa}{d}\right)$$

Many thanks!!

  • 1
    $\begingroup$ E.g. Sum[f[d], {d, Divisors[122]] $\endgroup$ – Coolwater Jul 13 '17 at 9:02
  • $\begingroup$ Thank you!! I got it.. $\endgroup$ – user404302 Jul 13 '17 at 11:45

DivisorSum[] can be used for this:

hecke[f_, n_Integer?Positive, τ_] := 
      DivisorSum[n, Sum[Function[τ, f][(n τ/# + b)/#], {b, 0, # - 1}] &]

For example:

hecke[KleinInvariantJ[τ], 3, τ]
   KleinInvariantJ[τ/3] + KleinInvariantJ[3 τ] +
   KleinInvariantJ[(1 + τ)/3] + KleinInvariantJ[(2 + τ)/3]

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