Let $f=\sum a(n)q^n\in S_k(N,\chi)$ be a cusp form of integral weight $k$ and level $N$ with Dirichlet character $\chi\pmod N$, let $\alpha_p,\beta_p$ the local parameter of $f$, ie., $$1-a(p)X+\chi(p)p^{k-1}X^2=(1-\alpha_pX)(1-\beta_pX)$$ It's known that if $\chi$ is the trivial character then $\beta_p=\bar{\alpha_p}.$

My question is the following : If $\chi$ is not trivial, on what condition on $f$ we have $\beta_p=\bar{\alpha_p}$ ?

For $q = e^{2\pi iz}$, let $f(z) = \sum_{n = 1}^{\infty} a(n)q^n\in S_k(N,\chi)$ be a cusp form of integral weight $k \geq 2$ and level $N$ with Dirichlet character $\chi\pmod N$. Let $\alpha_p,\beta_p$ be the local Satake parameters of $f$, i.e. \[1-a(p)X+\chi(p)p^{k-1}X^2=(1-\alpha_p p^{\frac{k - 1}{2}} X)(1-\beta_p p^{\frac{k - 1}{2}} X).\] It is known that if $\chi$ is the trivial character then $\beta_p=\bar{\alpha_p}.$

My question is the following : If $\chi$ is not trivial, on what condition on $f$ we have $\beta_p=\bar{\alpha_p}$ ?