I have three questions related to the theory of modular forms and it was frequently asked to me by my collegues and even my invited teacher in our seminars of the number theory at the faculty of sciences of Monastir (Tunisia) and as I was not expert in this theory, I couldn't answer it:

The first one is: Are there examples of usual functions (functions known by everyone) which are modular forms?

The second one is: Given a modular form $f$ of an even weight $k$ and a level $N\geq 1$ over the congruence subgroup $\Gamma_{0}(N),$ we construct its $r-$th symmetric power $Sym^{r}f.$ Is the form $Sym^{r}f$ also a modular form and if yes what are its weight and level?

The last one is: Are there any applications or connections of the theory of modular forms in other field of mathematics? I will be grateful if you reply to my questions.

## closed as off-topic by Olivier, Chris Godsil, Marco Golla, Johannes Hahn, Yoav KallusJul 28 '15 at 20:21

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Olivier, Chris Godsil, Marco Golla, Johannes Hahn, Yoav Kallus
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• Crossposted: math.stackexchange.com/questions/1369276/… – Qiaochu Yuan Jul 21 '15 at 22:55
• The zero function is a modular form, and is known by pretty much everyone. – Gerry Myerson Jul 21 '15 at 23:46
• Please don't crosspost, without waiting an appropriate amount of time to receive answers on the first site (a week, as a rule of thumb?). – Scott Morrison Jul 22 '15 at 2:14
• @GerryMyerson With all due respect, I think your comment is a good example of the worst kind of MO response. I can't imagine how it got two upvotes. – David Loeffler Jul 23 '15 at 0:52
• @David, probably two people meant to flag it for moderator attention, but accidentally clicked in the wrong place. – Gerry Myerson Jul 23 '15 at 2:33

• Second question: No, the symmetric power $\operatorname{Sym}^r f$ is not a modular form for $r > 1$. It should be something called an "automorphic form for $\operatorname{GL}_{r + 1}$", but this is only known for a few small values of $r$ (this is a topic of ongoing research).