I'm working with a special case of Ichino's triple product formula, which for classical holomorphic newforms $f$, $g$ ,$h$ of weights $k$, $m-k$, $m$ (and central characters satisfying $\chi_f \chi_g = \chi_h$) gives me an identity of the form $$ |\langle fg,h \rangle|^2 = C \cdot L(f\times g\times \bar{h},m-1) $$ (where $m-1$ is the central value for this L-function). This is great in principle, but if you're familiar with Ichino's formula you probably know it requires a bunch of work to actually nail down the constant $C$ correctly (there's lots of normalizations you have to get right, and then you have to deal with some unpleasant local integrals). So I want to be able to computationally check the formula I have in a bunch of cases, to convince myself it's actually right. I'm particularly interested in cases where $g$ and $h$ are CM newforms from Hecke characters of an imaginary quadratic field, and the triple-product $L$-value splits up as a product of two Rankin-Selberg $L$-values (twists of $f$ by Hecke characters).

So, my question is: what's the best way to work with these sort of quantities computationally? I haven't used computer algebra systems for number theory of this sort, and I'm not entirely sure where to start. Googling around I found that Pari and Sage both have algorithms that should (?) be able to compute my $L$-value, and another MO question giving some ways to compute Petersson inner products numerically.

My thought is just to boot up Sage and start messing around with the things suggested in those links, but I figured I'd ask here first to see if those are actually the right tools for the job, or if I'm making things way harder on myself than I need to somehow. (I don't really need a ton of precision - just enough to convince me that my constant isn't off by powers of 2 or by Euler factors or something - but I do want to be able to try it for a bunch of modular forms).

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    $\begingroup$ The main thing: the number of coefficients you need is approximately the size of square root of the conductor (and you need to know the conductor of the triple product, and the bad Euler factors), and any of the methods (including Magma, maybe to a lesser extent) suffers in high weight, due to precision issues. Also, you should definitely split up into primitive $L$-functions. Another option is to use Hecke characters directly to twist $f$, but I think they only exist in Magma currently. $\endgroup$ – kantelope Aug 24 '15 at 18:13
  • $\begingroup$ Am I correct that $m-1$ is the center of the critical strip? If it were the edge, you could probably use an Euler product approximation (like is done in class number computations, approx the $hR$ to within a factor of 2, at least assuming GRH-ish stuff). (Sum of modular weights is $2m$, so motivic weights is $2m-3$, which is odd, and the integer $m-1$ is indeed the center.) $\endgroup$ – kantelope Aug 24 '15 at 20:54
  • $\begingroup$ Yes, $m-1$ is the central value, and accordingly if I put in two CM forms the L-function factors as central critical values of Rankin-Selberg L-functions. I should have mentioned that in my original question! $\endgroup$ – Dan Collins Aug 24 '15 at 20:58
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    $\begingroup$ For Petersson products, see also Cohen's ANTS-X paper: math.ucsd.edu/~kedlaya/ants10/paper-cohen.html $\endgroup$ – kantelope Aug 24 '15 at 21:11

This is a good question around an important issue. The first thing that came to my mind is Popa's article "Central values of Rankin $L$-series over real quadratic fields", which is about a related problem. In Remark 6.3.3 he writes:

Using the PeriodMapping algorithm implemented by William Stein in MAGMA, and an algorithm implemented by Dokchitser [Dok04] in PARI for computing the special values, we have checked that the formula in Theorem 6.3.1 is exact up to more than 10 decimal places for a range of forms $f$ of weight 2, 4, 6, and 8, taking for $χ$ the trivial character.

In short, I recommend that you contact Popa and Stein (and perhaps also Rubinstein, Farmer, Booker) who are familiar with these kinds of calculations.


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