Coefficients in a small quadratic field, but not shown explicitly

[davidfarmer]

The arithmetically normalized coefficients of this L-function http://beta.lmfdb.org/L/ModularForm/GL2/Q/holomorphic/1041/2/a/a/1/1/ lie in Qsqrt5. But on the L-function home page those are only shown as decimals. It would be nice to see those coefficients explicitly.

This is an interesting example, because it has rank 2, but not rational integer coefficients. And its Galois conjugate also has rank 2: http://www.lmfdb.org/L/ModularForm/GL2/Q/holomorphic/1041/2/a/a/1/2/ which apparently follows from some conjecture.

[AndrewVSutherland]

Hmm, currently I don't think the L-function "knows" the (embedded) number field in which its coefficients lie, if any (although we could and perhaps should add this information to the database in order to make the L-functions fully independent of their origins).

Are you saying you would like to see arithmetic L-functions store their coefficients as exact values (which in general would mean storing them as algebraic expressions along with a choice of embedding) that would be displayed when the arithmetic normalization is chosen?

In the real quadratic case there is an unambiguous way to display the image of a generator for the coefficient field under a chosen embedding (e.g. \sqrt{5} denotes the positive square root of 5), but in general this is going to be a bit tricky -- if the coefficients lie in, say, one of the embeddings of the cubic field Q[x]/(x^3+x+1), how would you want them displayed?

Or are you suggesting we treat real (and possibly also imaginary) quadratic fields as a special case?

[edgarcosta]

I think it would be tricky and overwhelming to display the coefficients as algebraic integers. The first question is, what do we gain with it? Second, will it be distracting? We went through a lot of hoops to get the q-expansions for classical modular forms to display in some reasonable manner by picking an optimal basis. Doing something like that for L-fcns might be a bit distracting from some of the key content, e.g., the zeros and the euler factors.

[AndrewVSutherland]

@edgarcosta I agree with you, although I can imagine making an exception for quadratic fields (which is perhaps all that @davidfarmer is suggesting).

However, this issue does raise a point that I think we should consider. An arithmetic L-function really ought to "know" the embedded number field in which its Dirichlet coefficients lie, even if we don't use this information when displaying them. This is an important invariant of the L-function.

[davidfarmer]

I agree that the field of coefficients is an important invariant of an arithmetic L-function and it should (eventually) be somewhere on the home page.

Note that Maass type L-functions don't have an associated number field, and in fact should not have the option of toggling between arithmetic and analytic normalizations. We can deal with that when the GL(2) Maass form L-functions are computed rigorously.

I was suggesting that the case of coefficients in a quadratic field be given special treatment. (Although I am not sure why seeing q-expansion coefficients explicitly is important but seeing Dirichlet coefficients is not.)

On Tue, 30 Apr 2019, Andrew Sutherland wrote:

@edgarcosta I agree with you, although I can imagine making an exception for quadratic fields (which is perhaps all that @davidfarmer is suggesting).

However, this issue does raise a point that I think we should consider. An arithmetic L-function really ought to "know" the embedded number field in which its Dirichlet coefficients lie, even if we don't use this information when displaying them. This is an important invariant of the L-function.

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[edgarcosta]

I agree that we should store the field as an invariant.

However, I would like to point out that this L-fcn corresponds to the embedded modular form: http://beta.lmfdb.org/ModularForm/GL2/Q/holomorphic/1041/2/a/a/1/1/ and for that one, we also only display the coefficients as floating point numbers, and one needs to navigate to the Galois orbit to see them as algebraic integers.

[AndrewVSutherland]

@davidfarmer Regarding your last point, on embedded newform pages (which are the objects that give rise to the L-functions we are considering) we do not display algebraic q-expansions. We only display algebraic q-expansions on the page for the Galois orbit of the newform, and there we are not picking an embedding. For example, if you go to

http://www.lmfdb.org/L/ModularForm/GL2/Q/holomorphic/53/2/b/a/52/1/

you will see that the q-expansion coefficients are the same (but with more precision) as those listed in the Dirihlet series of its L-function

http://www.lmfdb.org/L/ModularForm/GL2/Q/holomorphic/53/2/b/a/52/1/

It is only the the home page for the Galois orbit of the modular form that we display an algebraic q-expansion

http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/53/2/b/a/

This q-expansion does not depend on a choice of embedding, where as the embedded modular forms and their L-functions do.

(EDIT: it looks like @edgarcosta made exactly the same point while I was typing this comment)

[davidfarmer]

Right. Those pages might also be improved by showing the quadratic case explicitly.

I'm not suggesting this is a 1.1 priority. And I think it is more relevant to L-functions, where one could toggle between arithmetic (explicit, showing the radicals) and analytic (decimals).