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Not an answer to your question, but a traceback

sage: hecke_series(191,1,4,3)
4742530*x^3 + 2222858*x^2 + 2482*x + 1
sage: parent(_)
Univariate Polynomial Ring in x over Ring of integers modulo 6967871
sage: __.factor()
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-4-0070f8eac88e> in <module>
----> 1 __.factor()

~/sage/local/var/lib/sage/venv-python3.10/lib/python3.10/site-packages/sage/rings/polynomial/polynomial_zmod_flint.pyx in sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint.factor (build/cythonized/sage/rings/polynomial/polynomial_zmod_flint.cpp:19015)()
    749         """
    750         if not self.base_ring().is_field():
--> 751             raise NotImplementedError("factorization of polynomials over rings with composite characteristic is not implemented")
    752 
    753         return factor_helper(self)

NotImplementedError: factorization of polynomials over rings with composite characteristic is not implemented