# On the local automorphic components of classical Siegel modular forms

I am looking for a dictionary that relates the level of a classical genus 2 Siegel modular form and the local components of the corresponding automorphic representation of $Gsp_4(\mathbb{A}_{\mathbb{Q}})$ (Relative to the Sally-Tadic classification of the non-supercuspidals). The notes of J. Booher compile the relevant information for classical elliptic modular forms (http://stanford.edu/~jbooher/expos/adelic_mod_forms.pdf) and I would love to have a similar breakdown for Siegel if such a treatise exists.

For instance: If I start with a Siegel modular form $F$ of level $N$, and $p \vert N$, what representations (of type I-XI in S-T) should I expect to see and when?

• – Pig
Sep 2 '16 at 0:13
• Incidentally, there's a trivial but dangerously misleading typo in the concluding Theorem 5.6 of Booher's notes: in the third bullet point, "if the conductor of $\pi_{f, p}$ is $p^r$" should say "if the conductor of $\omega_{p}$ is $p^r$". Sep 2 '16 at 12:39