= Prime stabilization of an oldform
{title2=$f_\alpha=f-\beta f(p\tau),\quad\alpha\beta=\chi(p)p^{k-1}$}
= p-stabilization
{c}
{synonym}
For a nonzero positive-weight eigenform at a good prime, the two oldforms $f(\tau),f(p\tau)$ are independent. At level multiplied by $p$, the <Hecke operator> is the bad-prime operator, with <matrix> $\begin{pmatrix}\lambda&1\\-\chi(p)p^{k-1}&0\end{pmatrix}$. Distinct roots $\alpha,\beta$ of its <characteristic polynomial> give eigenforms $f-\beta f(p\tau)$ and $f-\alpha f(p\tau)$. Weight-zero constants are an exception to the independence assertion.
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