Hecke conjectured that $\theta_I$ form a basis for the space $S_2(p)$, but this was found to fail for $p=37$ by Eichler. In fact, Gross realized that whenever you get vanishing central $L$-values you get a linear relation among theta series. This happens in $S_2(37)$ since there is an elliptic curve with root number $-1$ of conductor 37.

However, Eichler showed you can find a basis for the space of modular forms among the theta series attached to lattices $IJ^{-1}$ as $I, J$ vary over left $O$-ideals. The precise linear relations are mysterious however. See for instance the introduction to Hijikata, Pizer and Shemanske's Memoirs article on the basis problem.

Hecke conjectured that $\theta_I$ form a basis for the space $S_2(p)$, but this was found to fail for $p=37$ by Eichler. In fact, Gross realized that whenever you get vanishing central $L$-values you get a linear relation among theta series. This happens in $S_2(37)$ since there is an elliptic curve with root number $-1$ of conductor 37.

However, Eichler showed you can find a basis for the space of modular forms among the theta series attached to lattices $J^{-1}I$ as $I, J$ vary over left $O$-ideals (so you range over left $O'$-ideals where $O'$ ranges over all maximal orders). The precise linear relations are mysterious however. See for instance the introduction to Hijikata, Pizer and Shemanske's Memoirs article on the basis problem.