This question is a follow-up of an [old question](https://mathoverflow.net/questions/231770/integral-quaternary-forms-and-theta-functions) posted on MathOverflow.


**Motivation:** The exact equations of modular curves $X\_0^{+}(p)=X\_0(p)/w\_p$($p>13$ is a prime number,$w\_p$ is the Fricke involution) are intriguing objects. One can construct exact equations from modular forms of weight 2 on $\Gamma\_0(p)$ from a quaternion algebra $A(p)$ over $\mathbb{Q}$ ramified at $p$ and $\infty$. Let $O$ be a maximal order of $A(p)$ and $I$ be a left $O$-ideal. The theta function determined by $I$ $$\theta\_I(\tau)=\sum\_{x\in I}e^{2\pi i\tau\frac{N(x)}{N(I)}}$$ is always a modular form of weight 2 on $\Gamma\_0(p)$ with Fricke eigenvalue $-1$(see A. Pizer's [paper](https://www.sciencedirect.com/science/article/pii/0021869380901519)).


**Experiment:** According to the Proposition 2.17 of Pizer's [paper](https://www.sciencedirect.com/science/article/pii/0021869380901519), the number of different theta functions determined by left ideals of $A(p)$ is bounded by the *type number* $T(p)$, which can be computed explicitly in [another paper](https://eudml.org/doc/151386) of Pizer (p. 94). A numerical computation with MAGMA seems to suggest that the number of distinct theta functions associated to the left $O$-ideals of $A(p)$ always reaches the upper bound, and all these theta functions are linear independent.


**Question:** Is there any example that the linear independence of theta functions fails for a certain prime number $p>13$? If not, is there a reference for the proof of this fact? 


**Update:** The conjecture is likely to be false for $p=227$. 



Basis problem of modular forms arising from quaternion algebra
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.


