2 Added 2 more questions after a comment

I am not familiar with newforms, so this may not make any sense.

OEIS sequence A116418 is Expansion of a newform level 18 weight 3 and character [3]

Numerical evidence suggest that up to $10^5$ $$\text{A116418}[n] \equiv \sigma(3n+1) \pmod 3$$

What is the complexity of computing A116418[n], possibly assuming $n$ is factored (for modular form coefficients after $n$ is factored the coefficient is efficiently computable).

Gjergji Zaimi proved a similar congruence involving eta and A116418 is expansion of an eta formula.

Added My main interest is computing $\sigma(3n+1) \mod 3$ and a comment by Dror Speiser suggests the coefficient of the newform is computable in polynomial time assuming $n$ is factored.

The factorization of $n$ is not related related to the factorization of $3n+1$ and for numbers of form $3 \cdot 2^n + 1$ the factorization is trivial.

Is A116418 really the expansion of the newform or is it a typo in OEIS?

Is the congruence $\text{A116418}[n] \equiv \sigma(3n+1) \pmod 3$ identity or just coincidence for the the first $10^5$ terms?

1

# Complexity of computing expansion of a newform level 18 weight 3 and character [3] - OEIS A116418

I am not familiar with newforms, so this may not make any sense.

OEIS sequence A116418 is Expansion of a newform level 18 weight 3 and character [3]

Numerical evidence suggest that up to $10^5$ $$\text{A116418}[n] \equiv \sigma(3n+1) \pmod 3$$

What is the complexity of computing A116418[n], possibly assuming $n$ is factored (for modular form coefficients after $n$ is factored the coefficient is efficiently computable).

Gjergji Zaimi proved a similar congruence involving eta and A116418 is expansion of an eta formula.