There are no newforms of weight $2$ for levels $1$ thru $10,12,13,16,18,22,25,28,60$.

Looking at the list given in Sloane's integer sequence A127788 it appears that the number of newforms increases with the level without bound!

My question is two-fold:

1) Is there an upper bound to the number of weight $2$ newforms for any level $N$?

2) Is there an upper bound to the level of usable weight $2$ newforms corresponding to the discriminant, $\Delta$, arising from a Weierstrass equation resulting from a Frey curve?

The Darmon-Granville Theorem 2 states that there is a finite number of solutions to $x^p+y^q=z^r$. Doesn't that imply an upper bound for question 2) above?



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