Congruences for Siegel modular forms of nonquadratic nebentypus mod $p$

We prove that weights of two Siegel modular forms of nonquadratic nebentypus should satisfy some congruence relations if these modular forms are congruent to each other. Applying this result, we prove that there are no mod $p$ singular forms of nonquadratic nebentypus. Here we consider the case where the Fourier coefficients of the modular forms are algebraic integers, and we emphasize that $p$ is a rational prime. Moreover, we construct some examples of mod $\mathfrak{p}$ singular forms of nonquadratic nebentypus using the Eisenstein series studied by Takemori.