I would like to know what is the latex command to include a double lower index using sigma notation (i.e: second index lies underneath the first index)?

I tried typing:

\displaystyle\sum_{m,n=-\infty, \; (m,n)\neq (0,0)}^{\infty}
\left[
  \frac{1}{(z-2 m\omega_{1} - 2 n\omega_{2})^{2}}
 -\frac{1}{   4(m\omega_{1} +   n\omega_{2})^{2}}
\right] 


I would appreciate the assistance. Thanks


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