The analytic conductor of a newform $f \in S_k^{\mathrm{new}}(N,\chi)$ is the positive real number \[ N\left(\frac{\exp(\psi(k/2))}{2\pi}\right)^2, \] where $\psi(x):=\Gamma'(x)/\Gamma(x)$ is the logarithmic derivative of the Gamma function.