Let $f$ be a cusp form of integral weight $k$ for $\text{SL}_2(\mathbb{Z})$.
Its Mellin integral transform is a fundamental concept which
associates to $f$ the (completed) Dirichlet series $\Lambda(f; s)$.

The special values $\left\{\Lambda(f ; j+1) \mid 0 \leq j \leq k-2 \right\}$
define the odd and even period polynomials $p_f^{\text{odd}}(X)$ and
$p_f^{\text{even}}(X)$. Two important results in this context are the
Eichler-Shimura isomorphism theorem and Haberland's formula. The
former establishes the precise relations between the space of such
cusp forms and both sets of period polynomials, while the latter gives
the value of the Petersson inner product of two cusp forms $f$ and $g$
in terms of the special values $\Lambda(f; j+1)$ and $\Lambda(g; j+1)$.

This talk is about the integral transform
\[
D(f; s) = (-i) \int_{\rho}^{\rho^2} f(\tau) \left(\frac{\tau}{i}\right)^{s-1} d\tau,
\quad \text{where} \quad \rho = e^{\pi i/3},
\]
defined for any weakly holomorphic cusp form $f$ of weight $k$ for
$\text{SL}_2(\mathbb{Z})$. We will discuss the set of special values
$\left\{D(f ; j+1) \mid 1 \leq j \leq k-3 \right\}$ and certain polynomials
$Q_f^{\text{odd}}(X)$, $Q_f^{\text{even}}(X)$, defined by these values.
We will show a partial analogue of the Eichler-Shimura theorem for
the polynomials $Q_f^{\text{odd}}(X)$ and $Q_f^{\text{even}}(X)$ as
well as a formula similar to Haberland's involving the regularized Petersson
inner product of weakly holomorphic cusp forms and the special values
$D(f ; j+1)$. We will also discuss the relation between $Q_f^{\text{odd}}(X)$,
$Q_f^{\text{even}}(X)$ and analogues of period polynomials for weakly holomorphic
cusp forms studied in the literature.

The talk corresponds to joint work with \"O. Imamo\=glu and A. T\'oth.

The Frobenius Density Theorem is an early version of the Chebotarev Density Theorem concerning the distribution of the Frobenius substitutions among permutations of the roots of an integral polynomials.

We will survey the history of this result and the ingenious argument used by Frobenius to deduce it from an earlier statement of Kronecker. We will then present recent work with Forey and Fresán that can be interpreted as a variant of the Frobenius Density Theorem for the distribution of exponential sums.