I am mainly thinking about the group $\Gamma(N)$. A weakly modular form of weight one is a holomorphic function $f: \mathfrak{H} \to \mathbb{C}$ satisfying
$$
f(\gamma \tau) = (c\tau+d)f(\tau), \qquad \gamma \in \Gamma(N),
$$
and which is also "meromorphic at the cusps". It corresponds to an algebraic section of the Hodge line bundle $\underline{\omega}$ over $Y(N) = \mathfrak{H}/\Gamma(N)$.


I believe that, at least for small values of $N$, the Hodge line bundle is actually trivial (in other words, the universal elliptic curve over $Y(N)$ admits a Weierstrass equation). In particular, the trivialization of the $\underline{\omega}$ should correspond to some explicit weakly modular form of weight 1 with no zeros.


Where can I find explicit formulas for such modular forms? For instance, can they be defined by a series on $\mathfrak{H}$ (much like Eisenstein or Poincaré series)?



Explicit expressions for "weakly holomorphic" modular forms of weight 1
Completing David's answer, it is clear that any eta quotient of level $N$ and weight 1 satisfies the required condition. If you ask in addition that the
form be holomorphic, it is not difficult to write a program giving all holomorphic eta quotients of a given level. For instance there are none in prime level
$N\equiv1\pmod4$ (and more generally 50 levels out of the first 100 have none), but otherwise some levels have hundreds. The smallest level is of course level $4$ with
the eta quotient $\eta(\tau)^{-4}\eta(2\tau)^{10}\eta(4\tau)^{-4}=\theta(\tau)^2$.


This begs for instance the additional question: do there exist weakly modular forms of weight $1$ and level $N\equiv1\pmod4$ prime? I assume this is known.


