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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.

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    $\begingroup$ Crossposted: math.stackexchange.com/questions/1369276/… $\endgroup$
    – Qiaochu Yuan
    Commented Jul 21, 2015 at 22:55
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    $\begingroup$ The zero function is a modular form, and is known by pretty much everyone. $\endgroup$
    – Gerry Myerson
    Commented Jul 21, 2015 at 23:46
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    $\begingroup$ 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?). $\endgroup$
    – Kim Morrison
    Commented Jul 22, 2015 at 2:14
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    $\begingroup$ @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. $\endgroup$
    – David Loeffler
    Commented Jul 23, 2015 at 0:52
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    $\begingroup$ @David, probably two people meant to flag it for moderator attention, but accidentally clicked in the wrong place. $\endgroup$
    – Gerry Myerson
    Commented Jul 23, 2015 at 2:33

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  • First question: There are plenty of "standard" examples, for instance Eisenstein series, theta series, and eta products (doing a web search for any of these will bring up a lot of information).

  • 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).

  • Third question: yes, lots of them! Do a web search for "modular elliptic curves", or "expander graphs", or "Witten genus", or "monstrous moonshine" to see connections to (respectively) algebraic geometry, combinatorics, differential geometry, and finite group theory.

To answer your (implied) zero-th question, which is how you can learn more about modular forms with the limited resources available to you: there are many excellent online resources, some of them free, such as William Stein's e-book "Modular Forms: A Computational Perspective" or James Milne's online notes.

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  • $\begingroup$ Thank you @DavidLoeffler for your answers. It is so helpful for me! $\endgroup$
    – Khadija Mbarki
    Commented Jul 22, 2015 at 6:31

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