Quantum Phase estimation with $2pi$ replaced with $2e$

The QPE on IBM platform finds the eigenvalue of a unitary operator, i.e $$U|phirangle=e^{2pi itheta}|phirangle$$
and uses the rotation operators as $$U(theta)=begin{bmatrix}0 & 1
1 &e^{itheta}end{bmatrix}$$
My question is we can write $U|phirangle=e^{2pi itheta}|phirangle$ as $$U|phirangle=e^{2e itheta}|phirangle$$ where, since even then the magnitude of the eigenvalue remains same. Of course that would require more number of measurement qubits. For instance for $theta=0.5$, we had $4$, so this time we can increase the $n$ and get closer to $theta=0.5$. My question is can this be done.