#4062: [with patch, needs review] Problems with Eisenstein series code?
---------------------------+------------------------------------------------
Reporter: craigcitro | Owner: craigcitro
Type: defect | Status: assigned
Priority: major | Milestone: sage-3.2
Component: modular forms | Resolution:
Keywords: |
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Changes (by craigcitro):
* status: new => assigned
* summary: Problems with Eisenstein series code? => [with patch, needs
review] Problems with Eisenstein series code?
* milestone: sage-3.2.1 => sage-3.2
Comment:
This is a fix for the above problem. In fact, the fix was suggested by
Peter Bruin, who originally reported the fix. Here's what he had to say:
{{{
The other possibility is to define E_{k,chi,psi} as the unique modular
form whose L-series equals L(s,chi) L(s-k+1,psi); this is the form
which Miyake considers in Theorem 4.7.1 (cf. E. Hecke, Math. Ann. 114
(1937), 316--351 [= Mathematische Werke, 672--707]). Then the
formulas for the q-expansion as they are now in William Stein's book
and in Sage remain correct (i.e. without replacing psi by its
conjugate), and the change that should be made in this case (in the
book and in Sage) is to change the relation between chi, psi and the
character epsilon of E_{k,chi,psi} from
chi = epsilon * psi
to
chi * psi = epsilon.
This would mean that the comment (not the code!) in __find_eisen_chars
in eis_series.py should be changed (refer to Miyake's Theorem 4.7.1),
and that in the method called `character' of the class
EisensteinSeries in element.py, the line
self.__character = self.__chi * (~self.__psi)
should be replaced by
self.__character = self.__chi * self.__psi
}}}
The attached patch fixes this, and adds a few doctests to catch it in the
future. Credit for the patch should also go to Peter.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/4062#comment:1>
Sage <http://sagemath.org/>
Sage - Open Source Mathematical Software: Building the Car Instead of
Reinventing the Wheel
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