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I'm trying to compute special values of Hecke L-function for the field $K=\mathbb{Q}(\sqrt[5]{1})$ using Magma (more exactly, I need $L(k, \chi^k)$, $k$ - integer, $\chi$ - Hecke character for the field $K$). However, I'm very confused, because the text http://magma.maths.usyd.edu.au/~watkins/papers/hecke.pdf says (as far as I understood) that it is possible to do it as we're dealing with a CM field. On the other hand, (again, as far as I got it), one needs to specify real places to create a HeckeCharacterGroup in Magma, but there are no real places for this field? (While looking for the answer found this question Special values of Hecke L-function, but couldn't understand how to do similarly in my case). I feel that I don't realize something simple here.

And one more question on the subject: there are many Dirichlet characters of given modulus, but when we compute Dirichlet L-function for a quadratic field, we take one specific character which is a Kronecker symbol. Is there something similar for Hecke characters in my case?

I'm sorry if my question is somehow very silly but I couldn't find much clear information on the subject. Thank you in advance!

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    $\begingroup$ There seems to be an example for exactly this field $K$ in the Magma documentation at magma.maths.usyd.edu.au/magma/handbook/text/422#4502. Does that help at all? $\endgroup$ Jan 4, 2021 at 21:11
  • $\begingroup$ thank you! I saw this example, but couldn't understand it :( I think my problem is that I don't really understand which character do I need. But I guess this link will be helpful when I understand the subject better. $\endgroup$
    – tyazko
    Jan 8, 2021 at 16:59

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The linked Magma documentation notes that HeckeCharacterGroup can have the real infinite places ("oo") omitted, corresponding to no real infinite places being ramified (as is trivially the case in your example).

HeckeCharacterGroup(I) : RngOrdIdl -> GrpHecke
HeckeCharacterGroup(I, oo) : RngOrdIdl, SeqEnum -> GrpHecke

Alternatively, an empty sequence of places ("[]") should give the same result. However, you will first need to know where your prospective Hecke character is ramified (the ideal $I$).

For the second question, the Kronecker character is a specific character corresponding to a quadratic extension of the rationals. Are you dealing with a specific quadratic extension of the 5th cyclotomic field in your case? If so, there is a QuadraticCharacter command in Magma that might help (same documentation), with an example of extending ${\bf Q}(\sqrt{-7})$ by $\sqrt{-118-18\sqrt{-7}}$.

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  • $\begingroup$ Thank you! As for the second question, I'm considering the field $\mathbb{Q}(\sqrt[5]{1})$ as a totally imaginary quadratic extension of a quadratic field $\mathbb{Q}(\sqrt{5}) $ if that helps. I'll look for this command, thank you very much for your answer! $\endgroup$
    – tyazko
    Jan 8, 2021 at 16:50

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