There are several confusions in your post.

1. Automorphic forms for the group $\mathrm{SL}_2(\mathbb{Z})$ have level $q=1$ by definition. You probably wanted to talk about newforms for the Hecke congruence subgroup $\Gamma_0(q)$.

2. Good only treated the case of $q=1$. The bound you claim was only proved recently (with an unspecified $A>0$) by Booker-Milinovich-Ng (2009) and Aggarwal (2020).

3. It is not known that $A=1/4+\varepsilon$ is admissible in the bound you claim. One can take $A=1/4+\varepsilon$ if at the same time the exponent of $q$ is enlarged to $1/2+\varepsilon$; this is the convexity bound.

4. As far as I know, that best hybrid subconvexity bound for the family at hand is due to Wu (2022): $$L(f,1/2+it)\ll_\varepsilon (q(|t|+1)^2)^{1/4-1/224+\varepsilon}.$$ If the nebentypus of $f$ is fixed, then $1/224$ can be improved to $25/1536$.

There are several confusions in your post.

1. Automorphic forms for the group $\mathrm{SL}_2(\mathbb{Z})$ have level $q=1$ by definition. You probably wanted to talk about newforms for the Hecke congruence subgroup $\Gamma_0(q)$.

2. Good only treated the case of $q=1$. The bound you claim was only proved recently (with an unspecified $A>0$) by Booker-Milinovich-Ng (2009) and Aggarwal (2020).

3. It is not known that $A=1/4+\varepsilon$ is admissible in the bound you claim. One can take $A=1/4+\varepsilon$ if at the same time the exponent of $|t|+1$ is enlarged to $1/2+\varepsilon$; this is the convexity bound.

4. As far as I know, that best hybrid subconvexity bound for the family at hand is due to Wu (2022): $$L(f,1/2+it)\ll_\varepsilon (q(|t|+1)^2)^{1/4-1/224+\varepsilon}.$$ If the nebentypus of $f$ is fixed, then $1/224$ can be improved to $25/1536$.

There are several confusions in your post.

1. Automorphic forms for the group $\mathrm{SL}_2(\mathbb{Z})$ have level $q=1$ by definition. You probably wanted to talk about newforms for the group $\Gamma_0(q)$.

2. Good only treated the case of $q=1$. The bound you claim was only proved recently (with an unspecified $A>0$) by Booker-Milinovich-Ng (2009) and Aggarwal (2020).

3. It is not known that $A=1/4+\varepsilon$ is admissible in the bound you claim. One can take $A=1/4+\varepsilon$ if at the same time the exponent of $q$ is enlarged to $1/2+\varepsilon$; this is the convexity bound.

4. As far as I know, that best hybrid subconvexity bound for the family at hand is due to Wu (2022): $$L(f,1/2+it)\ll_\varepsilon (q(|t|+1)^2)^{1/4-1/224+\varepsilon}.$$ If the nebentypus of $f$ is fixed, then $1/224$ can be improved to $25/1536$.

There are several confusions in your post.

1. Automorphic forms for the group $\mathrm{SL}_2(\mathbb{Z})$ have level $q=1$ by definition. You probably wanted to talk about newforms for the Hecke congruence subgroup $\Gamma_0(q)$.

2. Good only treated the case of $q=1$. The bound you claim was only proved recently (with an unspecified $A>0$) by Booker-Milinovich-Ng (2009) and Aggarwal (2020).

3. It is not known that $A=1/4+\varepsilon$ is admissible in the bound you claim. One can take $A=1/4+\varepsilon$ if at the same time the exponent of $q$ is enlarged to $1/2+\varepsilon$; this is the convexity bound.

4. As far as I know, that best hybrid subconvexity bound for the family at hand is due to Wu (2022): $$L(f,1/2+it)\ll_\varepsilon (q(|t|+1)^2)^{1/4-1/224+\varepsilon}.$$ If the nebentypus of $f$ is fixed, then $1/224$ can be improved to $25/1536$.