Wrong q-expansion for 37.2.b.a

[Maarten Derickx]

Hi All,

It seems that there is something wrong with the q-expansion listed on https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/37/2/b/a/ . For example it show that the second coefficient in front of q^2 is i,  however I believe it should be 2i.
Reasons I believe it should be 2i. 
1) 2i is a root of the Hecke characteristic polynomial T^2 + 4 at 2 listed lower on the page.

2) downloading the q-expansion from https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/download_qexp/37.2.b.a and then the output of make_data() is:

    q + 2*a*q^2 - q^3 - 2*q^4 - 2*a*q^5 - 2*a*q^6 + 3*q^7 - 2*q^9 + 4*q^10 - 3*q^11 + 2*q^12 - 6*a*q^13 + 6*a*q^14 + 2*a*q^15 - 4*q^16 ....

and make_data().parent() outputs:

   Power Series Ring in q over Number Field in a with defining polynomial x^2 + 1

Actually all coefficients with i in them seem to be off by the same factor of 2. So maybe the problem is actually a mismatch between the generator i, and that the coefficients as displayed are actually assuming they are with respect to 2i instead of i.

I didn't do an extensive check so there might be similar issues with other modular forms.

[Andrew Sutherland]

Thank you for reporting this!  It appears to be related to an old bug I thought had been fixed.  It is causesed by the website arbitrarily deciding it likes i better than beta=2i, I switched it back to beta=2i, which is what is actually stored in the data, which is why the code downloads work correctly.  Looking at the code, I think this is specific to newforms with Hecke field Q(i) (there was a line that unilaterally set the Hecke ring to [1,i] whenever the field poly is Q(i), no matter what the Hecke ring basis says -- I just removed that line, which should fix any similar instances of the same problem in other newforms).  In any case, it is now fixed, as can be seen at https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/37/2/b/a/