Expanding wavefunctions in terms of $L_z$ eigenkets

The angular momentum operator along the $z$-axis $L_z$ satisfies the secular equation $$ L_z|m,lrangle = hbar m |m,lrangle,$$ where $l$ is the corresponding (integer-valued) eigenvalue of the simultaneously diagonalizable operator $|mathbf L|^2$ and $m$ is an integer between $-l$ and $+l$. Dropping the number $l$ from the state, we can write in angular coordinates the associated wavefunction $$chi_m(varphi):=langle varphi|mrangle =frac{1}{sqrt{2pi}}e^{imvarphi}.$$
Consider the (angular) position eigenket $|varphirangle$. Since the eigenvectors of $L_z$ form a complete set of the Hilbert space, we should be able to write $$ |varphirangle=sum_m langle m|varphirangle|mrangle=sum_m chi_m^*(varphi)|mrangle,$$ or more generally any wavefunction $Psi(varphi)$ as $$Psi(varphi)=langle varphi|Psirangle=sum_m langlevarphi|mranglelangle m|Psirangle=sum_m c_mchi_m(varphi)=frac{1}{sqrt{2pi}}sum_m c_m e^{imvarphi} $$
which is just a Fourier expansion with coefficients given by $$c_m=int dvarphilangle m|varphiranglelangle varphi|Psirangle=int dvarphi chi_m^*(varphi)Psi(varphi).$$ My question, coming back to the original eigenket $|m,lrangle$, is the following: on all of these sums indexed by $m$, does $m$ vary freely in $mathbb Z$ or is it bound by the quantum numer $l$?