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Following the comments, here is perhaps the simplest counterexample (once you know the Breuil-Conrad-Diamond-Taylor modularity theorem). The curves $y^2 + y = x^3$ and $y^2 + y = x^3 + 2x$ both reduc …

The map $f(z) \mapsto f(z)\cdot(dz)^k$ establishes a bijection between modular forms of weight $2k$ on $\Gamma \backslash \mathbb{H}^*$ and sections of $\Omega^{\otimes k}(\log \text{cusps})$, the $k$ …

Let $D(\tau) = \Delta(\tau)/q = \prod_{n=1}^\infty (1-q^n)^{24} = 1 - 24q + 252q^2 + \cdots$, where $q = e^{2 \pi i \tau}$. This is a holomorphic function on $\mathbb{H}$ that takes the value 0 at al …

Many "natural" examples of automorphic infinite products (also known as Borcherds products) can be explained using the singular theta lift of Harvey-Moore and Borcherds. These examples have the prope …

Since I'm not an expert, this will be just a minor note in addition to the excellent answers already given. The example modular forms $E_4$ and $\Delta$ you've written down are defined over the integ …

Characters of rational vertex operator algebras tend to yield modular functions. This is due to the space of torus partition functions in a chiral conformal field theory being a complex moduli invari …

There is an algebraic object naturally attached to $1/\Delta$, namely the fake monster Lie algebra. It was introduced in Borcherds's paper The monster Lie algebra, but the word "fake" was later attac …

I'm not an expert on black holes, but I can give you a couple pointers. From work of Bekenstein and Hawking in the 1970s, we are pretty sure that macroscopic black holes in our 3+1 dimensional univer …

Note that for any $\left(\begin{smallmatrix} a & b \\ cpN & d \end{smallmatrix} \right) \in \Gamma_0(pN)$, we have $$f\left(\frac{a\tau+bN}{cp\tau + d}\right) = \phi\left(\frac{a\tau+bN}{cpN\tau+Nd}\r …

Sorry, the first edition of this answer was shamefully incoherent. We'll see if this attempt is any better. Any double coset KgK (for G and K as given) has a unique representative in elementary divi …

Edit: Here's a rather silly method that should work if SAGE is just giving you cusp forms: $\Gamma_0(4)$ has a single normalized cusp form of weight 6, given by $\eta(2\tau)^{12} = q - 12q^3 + 54q^5 …

The free product $\mathbf{Z} * \mathbf{Z}$ is typically viewed as a discrete group, since I believe that is the coproduct in the category of topological groups. Even if you topologize the free produc …

Sorry about the one-sentence answer. Here is an expanded version that uses the following facts: The modular function $\lambda$ is holomorphic on the upper half-plane, invariant under the action of $ …

In addition to the power-series methods using Eisenstein series and $\Delta$, and the modular equation methods using, e.g., $h_5$ given in the other answers and comments, there are transcendental meth …

You can think of a topological modular form as a point on the tmf spectrum, but it's not clear that this is a useful view, unless you expand your notion of "point" to a large class of objects. The re …