  The Hodge bundle $\omega$ over a modular curve is a square-root of the
canonical bundle twisted by the cuspidal divisor, or a theta characteristic,
due to the Kodaira--Spencer isomorphism. We prove that, in most cases, a
section of a theta characteristic $\nu$ (or any odd power of it) different from
$\omega$ is a noncongruence modular form. On the other hand, we show how
$\nu\ne\omega$ gives rise to a ``twisted'' analogue of the diagonal period map
to a Siegel threefold, whose difference attributes to the stackiness of the
moduli of abelian surfaces $\mathcal{A}_{2}$. Some questions on the
Brill--Noether theory of the modular curves are answered.
