TITLE: On the local automorphic components of classical Siegel modular forms
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: Look at the tables of Ralf Schmidt and Brooks Roberts:
Tables for Representations of GSp(4)
Section 2 also tells you which representations you should find in Saito-Kurokawa versus non-Saito-Kurokawa spaces.  
Section 5 tells you about what representations can occur for prime level for various congruence subgroups.  They also do higher level for paramodular groups (see also their book).  I saw a similar table in a talk by Ralf 2 days ago for higher level Siegel congruence subgroups--I don't think it's public yet, but I expect it will appear in a future paper.