How to determine whether a power of eta function is a eigenform? I find that it is complicated to do this from the definition. In fact, I know that $\eta^k(m z)$ is a eigenform for $k=1,2,3,4,6,8,12,24$ and $m=\frac{24}{gcd(k,24)}$. What I want to know is the cases that $k=5,7,11$. In addition, how to choose proper $N$ and find the weight $k$ as well as the $\chi(p)$. Thank you so much.
 A: The weight is $k/2$, so $\eta^{5}(24z)$, $\eta^{7}(24z)$ and $\eta^{11}(24z)$ are half-integer weight modular forms. The level $N = 576$,
because $\eta(24z) = \sum_{n=1}^{\infty} \chi_{12}(n) q^{n^{2}}$
is a single-variable theta series with level $576$ (this is stated as Corollary 1.62 in Ono's book "The Web of Modularity"). When you ask for $\chi(p)$, I presume you're asking for the character in the transformation law. This is 
$$
\chi_{12}(n) = \begin{cases} 1 & \text{ if } n \equiv 1 \text{ or } 11 \pmod{12} \\ -1 & \text{ if } n \equiv 5 \text{ or } 7 \pmod{12}. \end{cases}
$$
For clarify, the transformation law for $f(z) = \eta^{r}(24z)$ (if $r$ is odd) is
$$
  f\left(\frac{az+b}{cz+d}\right) = \chi_{12}(d) \left(\frac{c}{d}\right)^{r} \epsilon_{d}^{-r} (cz+d)^{r+1/2} f(z),
$$
where $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is any matrix in $\Gamma_{0}(576)$, $\epsilon_{d} = \begin{cases} 1 & \text{ if } d \equiv 1 \pmod{4}\\ i & \text{ if } d \equiv 3 \pmod{4} \end{cases}$. If $d > 0$,
$\left(\frac{c}{d}\right)$ is the usual Jacobi symbol, and
$$
  \left(\frac{c}{d}\right) = \begin{cases}
    \left(\frac{c}{|d|}\right) & \text{ if } d < 0 \text{ and } c > 0 \\
    -\left(\frac{c}{|d|}\right) & \text{ if } d < 0 \text{ and } c < 0.
\end{cases}
$$
