Sorry to disturb. I have a question need some explanations from the experts MO-website.

As usual, we let $L(f,s)$ be the corresponding $L$-function associated to the newform $f$ on $SL_2(\mathbb{Z})$, of level $q$, $q\in \mathbb{N}^+$, say. It is known that, regarding the $s$-aspect subconvex bound, one has $L(f,1/2+it)\ll q^A (|t|+1)^{1/3+\varepsilon}$ for any $t \in \mathbb{R}$; see, e.g, A. Good's paper. Here, based on the functional equation, one sees that $A$ can be taken as $A=1/4+\varepsilon $; this corresponds to the convexity bound for $L(f,s)$ in the critical strip.

My question if we know a subconvex bound in $q$-aspect, whereby one can achieve a hybrid bound simultaneously in the s and q aspects? That is, whether or not the exponent $A$ here can be taken as an exponent which is less than 1/4?

If any expert knows some relevant knowledge upon this, please show some guides. Thanks in advance.

Many many thanks.

Sorry to disturb. I have a question need some explanations from the experts on the MO-website.

As usual, we let $L(f,s)$ be the corresponding $L$-function associated to the newform $f$ on $SL_2(\mathbb{Z})$, of level $q$, $q\in \mathbb{N}^+$, say. It is known that, regarding the $s$-aspect subconvex bound, one has $L(f,1/2+it)\ll q^A (|t|+1)^{1/3+\varepsilon}$ for any $t \in \mathbb{R}$; see, e.g, A. Good's paper. Here, based on the functional equation, one sees that $A$ can be taken as $A=1/4+\varepsilon $; this corresponds to the convexity bound for $L(f,s)$ in the critical strip.

My question if we know a subconvex bound in $q$-aspect, whereby one can achieve a hybrid bound simultaneously in the s and q aspects? That is, whether or not the exponent $A$ here can be taken as an exponent which is less than 1/4?

If any expert knows some relevant knowledge upon this, please show some guides. Thanks in advance.

Many many thanks.