Category Archives: sagemath

Using lcalc to compute half-integral weight L-functions

This is a brief note intended primarily for my collaborators interested in using Rubinstein’s lcalc to compute the values of half-integral weight $L$-functions.

We will be using lcalc through sage. Unfortunately, we are going to be using some functionality which sage doesn’t expose particularly nicely, so it will feel a bit silly. Nonetheless, using sage’s distribution will prevent us from needing to compile it on our own (and there are a few bugfixes present in sage’s version).

Some $L$-functions are inbuilt into lcalc, but not half-integral weight $L$-functions. So it will be necessary to create a datafile containing the data that lcalc will use to generate its approximations. In short, this datafile will describe the shape of the functional equation and give a list of coefficients for lcalc to use.

Building the datafile

It is assumed that the $L$-function is normalized in such a way that
$$\begin{equation}
\Lambda(s) = Q^s L(s) \prod_{j = 1}^{A} \Gamma(\gamma_j s + \lambda_j) = \omega \overline{\Lambda(1 – \overline{s})}.
\end{equation}$$
This involves normalizing the functional equation to be of shape $s \mapsto 1-s$. Also note that $Q$ will be given as a real number.

An annotated version of the datafile you should create looks like this

2                  # 2 means the Dirichlet coefficients are reals
0                  # 0 means the L-function isn't a "nice" one
10000              # 10000 coefficients will be provided
0                  # 0 means the coefficients are not periodic
1                  # num Gamma factors of form \Gamma(\gamma s + \lambda)
1                  # the \gamma in the Gamma factor
1.75 0             # \lambda in Gamma factor; complex valued, space delimited
0.318309886183790  # Q. In this case, 1/pi
1 0                # real and imaginary parts of omega, sign of func. eq.
0                  # number of poles
1.000000000000000  # a(1)
-1.78381067250408  # a(2)
...                # ...
-0.622124724090625 # a(10000)

If there is an error, lcalc will usually fail silently. (Bummer). Note that in practice, datafiles should only contain numbers and should not contain comments. This annotated version is for convenience, not for use.

You can find a copy of the datafile for the unique half-integral weight cusp form of weight $9/2$ on $\Gamma_0(4)$ here. This uses the first 10000 coefficients — it’s surely possible to use more, but this was the test-setup that I first set up.

Generating the coefficients for this example

In order to create datafiles for other cuspforms, it is necessary to compute the coefficients (presumably using magma or sage) and then to populate a datafile. A good exercise would be to recreate this datafile using sage-like methods.

One way to create this datafile is to explicitly create the q-expansion of the modular form, if we happen to know a convenient expression. For us, we happen to know that $f = \eta(2z)^{12} \theta(z)^{-3}$. Thus one way to create the coefficients is to do something like the following.

num_coeffs = 10**5 + 1
weight     = 9.0 / 2.0

R.<q> = PowerSeriesRing(ZZ)
theta_expansion = theta_qexp(num_coeffs)
# Note that qexp_eta omits the q^(1/24) factor
eta_expansion = qexp_eta(ZZ[['q']], num_coeffs + 1)
eta2_coeffs = []
for i in range(num_coeffs):
    if i % 2 == 1:
        eta2_coeffs.append(0)
    else:
        eta2_coeffs.append(eta_expansion[i//2])
eta2 = R(eta2_coeffs)
g = q * ( (eta2)**4 / (theta_expansion) )**3

coefficients = g.list()[1:] # skip the 0 coeff
print(coefficients[:10])

normalized_coefficients = []
for idx, elem in enumerate(coefficients, 1):
    normalized_coeff = 1.0 * elem / (idx ** (.5 * (weight - 1)))
    normalized_coefficients.append(normalized_coeff)
print(normalized_coefficients[:10])

Using lcalc now

Suppose that you have a datafile, called g1_lcalcfile.txt (for example). Then to use this from sage, you point lcalc within sage to this file. This can be done through calls such as

# Computes L(0.5 + 0i, f)
lcalc('-v -x0.5 -y0 -Fg1_lcalcfile.txt')

# Computes L(s, f) from 0.5 to (2 + 7i) at 1000 equally spaced samples
lcalc('--value-line-segment -x0.5 -y0 -X2 -Y7 --number-samples=1000 -Fg1_lcalcfile.txt')

# See lcalc.help() for more on calling lcalc.

The key in these is to pass along the datafile through the -F argument.

Posted in Data, Mathematics, Programming, sage, sagemath, sagemath | Tagged | Leave a comment

A Jupyter Notebook from a SageMath tutorial

I gave an introduction to sage tutorial at the University of Warwick Computational Group seminar today, 2 November 2017. Below is a conversion of the sage/jupyter notebook I based the rest of the tutorial on. I said many things which are not included in the notebook, and during the seminar we added a few additional examples and took extra consideration to a few different calls. But for reference, the notebook is here.

The notebook itself (as a jupyter notebook) can be found and viewed on my github (link to jupyter notebook). When written, this notebook used a Sage 8.0.0.rc1 backend kernel and ran fine on the standard Sage 8.0 release , though I expect it to work fine with any recent official version of sage. The last cell requires an active notebook to be seen (or some way to export jupyter widgets to standalone javascript or something; this either doesn’t yet exist, or I am not aware of it).

I will also note that I converted the notebook for display on this website using jupyter’s nbconvert package. I have some CSS and syntax coloring set up that affects the display.

Good luck learning sage, and happy hacking.

Sage

Sage (also known as SageMath) is a general purpose computer algebra system written on top of the python language. In Mathematica, Magma, and Maple, one writes code in the mathematica-language, the magma-language, or the maple-language. Sage is python.

But no python background is necessary for the rest of today’s guided tutorial. The purpose of today’s tutorial is to give an indication about how one really uses sage, and what might be available to you if you want to try it out.

I will spoil the surprise by telling you upfront the two main points I hope you’ll take away from this tutorial.

  1. With tab-completion and documentation, you can do many things in sage without ever having done them before.
  2. The ecosystem of libraries and functionality available in sage is tremendous, and (usually) pretty easy to use.

Lightning Preview

Let’s first get a small feel for sage by seeing some standard operations and what typical use looks like through a series of trivial, mostly unconnected examples.

In [1]:
# Fundamental manipulations work as you hope

2+3
Out[1]:
5

You can also subtract, multiply, divide, exponentiate…

>>> 3-2
1
>>> 2*3
6
>>> 2^3
8
>>> 2**3 # (also exponentiation)
8

There is an order of operations, but these things work pretty much as you want them to work. You might try out several different operations.

Sage includes a lot of functionality, too. For instance,

In [2]:
factor(-1008)
Out[2]:
-1 * 2^4 * 3^2 * 7
In [3]:
list(factor(1008))
Out[3]:
[(2, 4), (3, 2), (7, 1)]

In the background, Sage is actually calling on pari/GP to do this factorization. Sage bundles lots of free and open source math software within it (which is why it’s so large), and provides a common access point. The great thing here is that you can often use sage without needing to know much pari/GP (or other software).

Sage knows many functions and constants, and these are accessible.

(more…)

Posted in Expository, Mathematics, sage, sagemath | Tagged , , , , , | Leave a comment