  Let $f$ be a cuspidal eigenform (holomorphic or Maass) on the full modular
group $SL(2, \mathbb{Z})$ . Let $\chi$ be a primitive character of modulus $P$.
We shall prove the following results:
  1. Suppose $P = p^r$, where $p$ is a prime and $r\equiv 0 (\textrm{mod} \
3)$. Then we have \[ L\left( f \otimes \chi, \frac{1}{2}\right) \ll_{f,
\epsilon} P^{1/3 +\epsilon}, \] where $\epsilon > 0$ is any positive real
number.
  2. Suppose $\chi$ factorizes as $\chi= \chi_1 \chi_2$, where $ \chi_i$'s are
primitive character modulo $P_i$, where $P_i$ are primes, $P^{1/2 -\epsilon}
\ll P_i \ll P^{1/2 + \epsilon}$ for $i=1,2$ and $P=P_1 P_2$. We have the
Burgess bound \[ L\left( f \otimes \chi, \frac{1}{2}\right) \ll_{f, \epsilon}
P^{3/8 +\epsilon}, \] where $\epsilon > 0$ is any positive real number.
