Hecke Operator- sum over divisors of a number

Question

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!!

Answer 1

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]