  Let $p\geq 6$ be any integer, where $p\equiv 2$ mod $4$. Let $f(x)=\int
e^{i\pi \tau |x|^2}d\mu(\tau)$ and $\mathcal{F}(f)$ be the Fourier transform of
$f$, where $x\in \mathbf{R}^2$ and $\mu$ is a measure with bounded variation
and supported on a compact subset of $\tau \in\mathbf{C}$, where
$\Im(\tau),\Im(-\frac{1}{\tau})>\sin(\frac{\pi}{p}).$ For every integer $k\geq
0$ and $x\in \mathbf{R}^2,$ we express $f(x)$ by the values of $\frac{d^k
f}{du^k}$ and $\frac{d^k \mathcal{F}f}{du^k}$ at $u=\frac{2n}{\lambda},$ where
$u=|x|^2$ and $\lambda=2\cos(\frac{\pi}{p}).$ We show that the condition
$\Im(\tau),\Im(-\frac{1}{\tau})>\sin(\frac{\pi}{p})$ is optimal. We also
identify the cokernel to these values with a specific space of holomorphic
modular forms of weight $2k+1$ associated to the Hecke triangle group
$(2,p,\infty)$. Using our explicit formulas for $p=6$ and developing new
methods, we prove a conjecture of Cohn, Kumar, Miller, Radchenko and
Viazovska~\cite[Conjecture 7.5]{Maryna3} motivated by the universal optimality
of the hexagonal lattice.
