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Eisenstein series: Small documentation improvement #38484

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Nov 16, 2024
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7 changes: 5 additions & 2 deletions src/sage/lfunctions/dokchitser.py
Original file line number Diff line number Diff line change
Expand Up @@ -43,6 +43,9 @@ class Dokchitser(SageObject):
r"""
Dokchitser's `L`-functions Calculator.

PARI code can be found on
`Dokchitser's homepage <https://people.maths.bris.ac.uk/~matyd/computel>`_.

Create a Dokchitser `L`-series with

Dokchitser(conductor, gammaV, weight, eps, poles, residues, init,
Expand Down Expand Up @@ -153,7 +156,7 @@ class Dokchitser(SageObject):

We redefine the default bound on the coefficients: Deligne's
estimate on tau(n) is better than the default
coefgrow(n)=`(4n)^{11/2}` (by a factor 1024), so
coefgrow(n)= `(4n)^{11/2}` (by a factor 1024), so
re-defining coefgrow() improves efficiency (slightly faster). ::

sage: L.num_coeffs()
Expand Down Expand Up @@ -571,7 +574,7 @@ def taylor_series(self, a=0, k=6, var='z'):

- ``a`` -- complex number (default: 0); point about which to expand

- ``k`` -- integer (default: 6); series is `O(``var``^k)`
- ``k`` -- integer (default: 6); series is `O(\texttt{var}^k)`

- ``var`` -- string (default: ``'z'``); variable of power series

Expand Down
23 changes: 14 additions & 9 deletions src/sage/modular/modform/eis_series.py
Original file line number Diff line number Diff line change
Expand Up @@ -384,11 +384,12 @@ def eisenstein_series_lseries(weight, prec=53,
max_imaginary_part=0,
max_asymp_coeffs=40):
r"""
Return the `L`-series of the weight `2k` Eisenstein series
Return the `L`-series of the weight `2k` Eisenstein series `E_{2k}`
on `\SL_2(\ZZ)`.

This actually returns an interface to Tim Dokchitser's program
for computing with the `L`-series of the Eisenstein series
for computing with the `L`-series of the Eisenstein series.
See :class:`~sage.lfunctions.dokchitser.Dokchitser`.

INPUT:

Expand All @@ -400,18 +401,22 @@ def eisenstein_series_lseries(weight, prec=53,

- ``max_asymp_coeffs`` -- integer

OUTPUT: the `L`-series of the Eisenstein series
OUTPUT: the `L`-series of the Eisenstein series. This can be
evaluated at argument `s`, or have
:meth:`~sage.lfunctions.dokchitser.Dokchitser.derivative` called, etc.

EXAMPLES:

We compute with the `L`-series of `E_{16}` and then `E_{20}`::

sage: L = eisenstein_series_lseries(16)
sage: L(1)
-0.291657724743874
sage: L = eisenstein_series_lseries(20)
sage: L(2)
-5.02355351645998
sage: L = eisenstein_series_lseries(16)
sage: L(1)
-0.291657724743874
sage: L.derivative(1)
0.0756072194360656
sage: L = eisenstein_series_lseries(20)
sage: L(2)
-5.02355351645998

Now with higher precision::

Expand Down
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