For i=0,1,…,2d−1, let
The distribution ΛE,d
is the distribution of real numbers αm(i), where
we vary over all m and i>0.
More concretely, for each integer m coprime to NE such
that d∣φ(m), compute the d real numbers
αm(i) and add them to our set of values. The
distribution ΛE,d is then the result of doing
this as m goes to ∞.
NOTE: The term αm(0) is the sensitive theta coefficient.
Regarding complexity the work in doing this computation
is the work of computing the rational numbers
[a/m]E± for all a. It's the same bottlekneck
that goes into approximating p-adic L-series using
the classical Riemann sums algorithm. The code in Sage
for this is fairly slow, but I have some fast code in psage,
which I used for some papers on p-adic L-series.
In [ ]:
In terms of Sage, the rational_period_mapping
method on a modular symbols space computes
a choice of [a/m]E±: