Let $f(z)$ be a newform of even weight $k$. We can numerically
approximate the integral along the path from 0 to infinity of
$f(z)z^j dz$, using a change of variable trick. In this talk I will
describe some of the interesting relations which arise, and then give
examples of $j$-invariants of elliptic curves attached to higher weight
rational newforms, e.g., Ramanujan's $\Delta$ function.
REFERENCES:
- Lang, "Introduction to Modular Forms", Chapter 5,
- Zagier, "Periods of modular forms and Jacobi theta functions",
Invent. math. 104, 449-465 (1991).
ERRATA: Last week I incorrectly asserted that the period ratios for j
odd are 0. In fact only their real parts must be 0, as they are pure
imaginary.