This question is a follow-up of an [old question][1] 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][2]).

**Experiment:** According to the Proposition 2.17 of Pizer's [paper][2], 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][3] 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 $p=227$. 

  [1]: https://mathoverflow.net/questions/231770/integral-quaternary-forms-and-theta-functions
  [2]: https://www.sciencedirect.com/science/article/pii/0021869380901519
  [3]: https://eudml.org/doc/151386