Newform and Galois representation (Shimura-Deligne Reciprocity Law)

Question

Shimura-Deligne Reciprocity Law implies that to a newform $f \colon =f(z) \in S^{\mathrm{new}}_k(\Gamma_0(N))$ of weight $k$, one can associate Galois representation $\rho_{f,\lambda} \colon {\mathrm{Gal}}(\overline{\Bbb{Q}}/{\Bbb Q}) \to \mathrm{GL}_2({\cal O}_{\lambda})$.

I am awkward of the definition of a newform $f$.

I.e., a newform $f$ of weight $k$ with respect to $\Gamma_0(N)$ is the orthogonal complement to ``old" forms coming from $S_k(\Gamma_0(M))$ for every $M | N$ ($1 \leq M < N$) by degeneracy map (natural inclusion) with respect to the Peterson inner product; viz.

$(f, g) \colon= \int_{{\Bbb H}/\Gamma_0(N)}f(z)\overline{g(z)}y^{k}\frac{dx \wedge dy}{y^2}$.

Q: Is a newform $f $ defined also in the following manner?

1: $f(\sigma(z)) = (cz + d)^kf(z)$ for all $\sigma \in \Gamma_0(N)$

2: When $f(z) = \Sigma_{n \geq 1} a_nq^n$ with $q := {\mathrm{exp}}(2\pi iz)$, $a_1 = 1$, $a_n \in {\overline{\Bbb Q}}$.

3: $T_nf(z) = a_nf(z)$ for all Hecke operators $T_n$'s with $n \geq 1$

4: If another $g(z)$ satisfies $T_ng(z) = a_ng(z)$ for all $n$ s.t. $(n,N)=1$, then $g(z) = cf(z)$ with some $c \in {\Bbb C}$.

---

In some article, it was written that roughly" newform is given by conditions 1. ～ 4. Please help me with the reason why this isroughly". Is anything wrong if we define newform $f$ as satisfying 1. - 4. ?

Answer 1

Your alternate definition is ok if you restrict property 4 to level $N$ forms $g(z)$, where $N$ is the level of the newform $f(z)$. Actually it is sufficient to restrict property 4 to forms $g(z)$ of level at most $N$.

The restriction is necessary, because for any $d\in\mathbb{N}$ the oldform $f(dz)$ of level $dN$ has the same Hecke eigenvalues $a_n$ as $f(z)$ for $(n,dN)=1$. In particular, $f(Nz)$ has the same Hecke eigenvalues as $f(z)$ for $(n,N)=1$, rendering property 4 invalid in its original version.

In fact the name "newform of level $N$" comes from the fact that its Hecke eigenvalues $(a_n)$ are "new" for level $N$, i.e. they do not occur for smaller levels. In addition, these Hecke eigenvalues $(a_n)$ determine a 1-dimensional subspace within the space of level $N$ forms.

Remark 1. Note that the same concept makes sense for Maass forms where the Hecke eigenvalues $a_n$ are not expected to be algebraic numbers. So I suggest that in property 3 you allow $a_n\in\mathbb{C}$.

Remark 2. You quote the original definition of a newform incorrectly. The correct definition is that $f(z)$ is orthogonal to any oldform $g(dz)$, where $g(z)$ is a Hecke eigenform of level $M\mid N$ such that $M<N$ and $dM\mid N$. See Section 4 in Atkin-Lehner's paper "Hecke operators on $\Gamma_0(m)$" (Math. Ann. 185 (1970), 134-160).