Let $K \supseteq F$ be a finite extension of number fields, not necessarily Galois. Let $f$ be a modular form on $\GL(2)$ over $F$ of level $\mathfrak{N}$.

For a (nonzero) prime $\mathfrak{p} \nmid \mathfrak{N}$ of $F$, let $L_{\mathfrak{p}}(f,T)$ be the (good, degree 2) Euler factor at $\mathfrak{p}$. For $r \in \Z_{\geq 1}$ and a polynomial $h(T) \in \C[T]$, let $h^{(r)}(T)$ be the polynomial whose roots are the $r$-th powers of the roots of $h(T)$ (with $\deg h^{(r)}(T)=\deg h(T)$).

We say that a modular form $g$ for $\GL(2)$ over $K$ is a **base change** of $f$ to $K$ if
\[ L_{\mathfrak{q}}(g,T) = L_{\mathfrak{p}}^{(r)}(f,T) \]
for all but finitely many primes $\mathfrak{q}$ of $K$, where $\mathfrak{q}$ lies over $\mathfrak{p} \nmid \mathfrak{N}$ and $r$ is the degree of $\mathfrak{q}$ over $\mathfrak{p}$.

In general, we say that a form $g$ for $\GL(2)$ over $K$ is a **base change** if there exists a proper subfield $K_0 \subsetneq K$ and a modular form $g_0$ on $\GL(2)$ over $K_0$ such that $g$ is the base change of $g_0$ to $K$.

In particular, if $g$ is a base change from $\Q$ to $K$, then $a_{\mathfrak{q}}(g) = a_{\sigma(\mathfrak{q})}(g)$ for all $\sigma \in \mathrm{Aut}(K)$.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Voight on 2019-03-21 14:11:23

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- 2019-03-21 14:11:23 by John Voight (Reviewed)
- 2019-03-21 14:11:09 by John Voight
- 2019-03-21 14:10:58 by John Voight
- 2019-03-21 14:10:50 by John Voight
- 2019-03-21 14:10:08 by John Voight
- 2019-03-21 14:09:01 by John Voight
- 2019-03-21 13:43:47 by John Voight
- 2019-03-21 13:42:24 by John Voight
- 2019-03-21 13:42:12 by John Voight
- 2019-03-21 13:29:45 by John Voight
- 2019-03-21 13:19:13 by John Voight
- 2019-03-21 13:18:41 by John Voight
- 2019-03-21 13:18:00 by John Voight
- 2019-03-21 13:17:32 by John Voight
- 2019-03-21 12:31:59 by John Voight
- 2019-03-21 12:31:31 by John Voight
- 2019-03-21 12:31:08 by John Voight
- 2019-03-21 12:30:54 by John Voight
- 2019-03-21 12:29:00 by John Voight

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