Testwiki:Reference desk/Archives/Mathematics/2007 December 24

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December 24

Computing with modular forms

Two questions regarding modular forms:

1. Using SAGE or MAGMA, is there a (relatively) painless way to work with modular forms associated with the principal congruence subgroup $\mathrm{\Gamma }(N)$ of the (full) modular group, $PS{L}_2(\mathbb{Z})$? Looking through the documentation (here, for example), it only seems possible to define "ModularForms" objects with respect to the congruence subgroups ${\mathrm{\Gamma }}_0(N)$, or ${\mathrm{\Gamma }}_1(N)$. But surely there must be some way around this?
2. How does one go about computing q-expansions for specific modular forms, in a practical manner? The definition (in the wiki article) is nice from a theoretical standpoint, but I find it unwieldy from an everyday-use point of view.

Help with either or both is appreciated! --PeruvianLlama^(spit) 00:51, 24 December 2007 (UTC)

You should look at the book Modular Forms, a Computational Approach by William Stein (the lead developer of SAGE). The book's ISBN is 0-8218-3960-8.

--Petekl (talk) 01:16, 26 December 2007 (UTC)

Attachment point formula

I want to weld a cyliner to a bar at point D such that when the cylinder is rolled 45 degrees the distance between A and B will equal 11.86. What formula would I use to determine the distance between point A and point D to achieve a distance of 11.86 between A and B? Multimillionaire (talk) 20:16, 24 December 2007 (UTC)

Denoting the angle C by α, point D is $r(1-\cos \alpha )$ above ground. Thus we have $\mathrm{B}\mathrm{D}=\frac{r(1-\cos \alpha )}{\sin \alpha }$, and $\mathrm{A}\mathrm{D}=\mathrm{A}\mathrm{B}-\mathrm{B}\mathrm{D}=11.86-\frac{r(1-\cos \alpha )}{\sin \alpha }$. For α=45° this is $11.86-0.4142r$. -- Meni Rosenfeld (talk) 21:27, 24 December 2007 (UTC)