We study
![$ p$](img1.png)
-divisibility of discriminant of Hecke algebras associated
to spaces of cusp forms of prime level. By considering cusp forms of
weight bigger than
![$ 2$](img2.png)
, we are are led to make a conjecture about
indexes of Hecke algebras in their normalization which, if true,
implies that there are no mod
![$ p$](img1.png)
congruences between
non-conjugate newforms in
![$ S_2(\Gamma_0(p))$](img3.png)
.