L-series values for certain modular forms computed incorrectly

[arminstraub]

Using Sage 8.1 on CoCalc:

sage: f5 = Newforms(Gamma1(4), 5, names='a')[0]; f5
q - 4*q^2 + 16*q^4 - 14*q^5 + O(q^6)
sage: L5 = f5.lseries()
sage: L5.check_functional_equation()
0.148408065960889 - 9.08737314733255e-18*I

An example of an incorrect L-value (and the correct one, for comparison):

sage: L5(4)
0.379630585317869 + 2.49933520900079e-17*I
sage: sum(f5.coefficient(n)/n^4 for n in [1..200]).n()
0.787848673384282
sage: (gamma(1/4)^8/(3840*pi^2)).n()
0.787803000538474

Just in case it is helpful, here is another example with similar issues:

sage: f3 = Newforms(Gamma1(24), 3, names='a')[0]; f3
q - 2*q^2 + 3*q^3 + 4*q^4 + 2*q^5 + O(q^6)
sage: f3.lseries().check_functional_equation()
1.13933480080878 - 3.61460956971046e-16*I

[alexjbest]

It seems the behaviour here is changed by #24086, it still doesn't seem to work even after that for me though.

The LMFDB has this form ​http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/4/5/3/a/ and says the associated L-function is ​http://www.lmfdb.org/L/ModularForm/GL2/Q/holomorphic/4/5/3/a/0/, which has epsilon factor 1, rather than -1 as L5.eps reports. If you set the epsilon factor to 1 then the functional equation checks out (for both examples).

Conceivably this could just be an issue of embeddings? In the first case the atkin lehner eigenvalue is reported with #24086 to be zeta4 or simply I using the 0th embedding of the heceke_eigenvalue_field into QQbar, which if embedded as the same as I gives I^5 * I = -1, in the second case the eigenvalue is -I with the 0th embedding giving I^3 *(-I) = -1 again. So both examples would be fixed by a judicious conjugate embedding? But I think I have stretched my lack of the appropriate mathematics too far now, as I really don't know what all these embeddings etc. "should" be.

[davidloeffler]

This is an interesting case: it's a form of odd weight and rational Hecke eigenvalues -- this can only happen for CM forms. Since the Hecke eigenvalue field is QQ, the "embedding" parameter to atkin_lehner_eigenvalue() has no effect.

My best guess is that this is a discrepancy of normalisations between atkin_lehner_eigenvalue() and lseries(). The Atkin-Lehner eigenvalue code uses the normalisations from Atkin-Li's 1978 paper; but that defines W_N to be [0,1; -N,0], whereas a lot of subsequent literature uses [0,-1; N,0], and that causes a sign discrepancy in odd weights.

I will investigate more and get back to you.

[arminstraub]

Thank you, Alex! Manually adjusting the epsilon factor indeed makes the functional equation work out. The L-values I am interested in are then also computed correctly. This is very helpful as a temporary work-around. It is awesome to see you and David investigate this issue so swiftly!

[davidloeffler]

Yes, I was correct: this is the discrepancy between the Atkin--Li definition of $W_N$ and the other definition when the weight is odd. I'll to do a fix on top of my existing branch for #24086.

[davidloeffler]

Here's a fix -- it's literally a single-byte change! I also added your weight 5 example as a doctest.

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New commits:

​2a58074	Trac 24086: Atkin--Lehner operators - now works for any character, + normalisations fixed
​76e7262	Fix failing doctest in abvar/lseries.py
​4265032	Trac 25369: fix modular form L-series root numbers in odd weights

[alexjbest]

Just ran through a load of examples with this patch and all the functional equations check out, so positive review!

[arminstraub]

Fantastic! It's been a while, but I updated Sage on my machine to the newest beta, applied your patch, and computed lots of L-values (the ones I included in the report are part of an infinite family). It is working beautifully! Thank you!

[davidloeffler]

Here's a new branch which merges cleanly with 8.3.beta2.

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New commits:

​6756c1e	Merge branch 'public/24086_better' in 8.3.b2
​51266f9	Trac 24086: stamp out some fuzz accidentally re-introduced by auto-merge
​6e97ac5	Trac 25369: fix modular form L-series root numbers in odd weights