mixedmathart

A level 3, weight 32 traceform

When asked if I might contribute an image for MSRI program 332, I thought it would be fun to investigate a modular form with a label roughly formed from the program number, 332. We investigate the trace form with label 3.32.a.a.

Image of 3.32.a.a without contours
DLD 2021.

The space of weight \(32\) modular forms on \(\Gamma_0(3)\) with trivial central character is an \(11\)-dimensional vector space. The subspace of newforms is a \(5\)-dimensional vector space.

These newforms break down into two groups: the two embeddings of an abstract newform whose coefficients lie in a quadratic field, and the three embeddings of an abstract newform whose coefficients lie in a cubic field. The label 3.32.a.a is a label for the two newforms with coefficients in a quadratic field.

These images are for the trace form, made by summing the two conjugate newforms in 3.32.a.a. This trace form is a newform of weight \(32\) on \(\Gamma_1(3)\).

Each modular form is naturally defined on the upper half-plane. In these images, the upper half-plane has been mapped to the unit disk. This mapping is uniquely specified by the following pieces of information: the real line \(y = 0\) in the plane is mapped to the boundary of the disk, and the three points \((0, i, \infty)\) map to the (bottom, center, top) of the disk.

Image of 3.32.a.a with contours
DLD 2021.

I've changed how I indicate contours from my previous modular forms images. The effect is very pronounced, but the actual change is a minor difference in hue-saturation-lightness pathing.

Image of 3.32.a.a with contours
DLD 2021.
Image of 3.32.a.a with contours, in pastel
DLD 2021.
Image of 3.32.a.a without contours
DLD 2021.