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.

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.

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.