I am trying to solve a problem in modular form, I want to show that $\Gamma_1(N) \left( \begin{array}{cc} 1 & 0 \\ 0 & p \end{array} \right) \Gamma_1(N) $ $= \{ \gamma \in M_2(Z) \mid det \gamma$ $=p , \gamma\equiv \left( \begin{array}{cc} 1 & * \\ 0 & p \end{array} \right)(modN) $}
One part of it is clear , I can show every element im left side is in the form right hand side. but I could not show the other part.
Any help would be great thanks.