Title: Fourier Coefficients of Triangle Functions

Abstract: Triangle functions $J_m$ are generalizations of the j modular function that map the interior of a hyperbolic triangle with vertices $(i, -\exp(-\pi i/m), i \infty)$ to the upper half place, where $m \ge 3$ is an integer. The corresponding groups, generalizing the modular group, are known as Hecke Groups and are generated by $S(z) = -1/z$ and $T_m(z) = 2\cos(\pi/m)$.

Fourier coefficients of the $J_m$ were first studied by Lehner in 1954, but were shown to be transcendental (except in the cases $m=3,4,6,\infty$) by Wolfart in 1981 and research on them mostly stopped. However the transcendental part is easily factored out, and the remaining part, a rational integer, has very interesting properties, especially with respect to which primes divide the denominator. Experimental evidence, a conjecture, and a proof of part of the conjecture will be presented.