Determinant-normalized slash operator ($f{|}_k\gamma =(\det \gamma {)}^{k/2}(cz+d{)}^{-k}f(\gamma z)$)

For a rational matrix $\gamma$ of positive determinant, use the positive real power of $\det \gamma$ in the displayed slash operator for modular forms. It satisfies $(f{|}_k\gamma ){|}_k\eta =f{|}_k(\gamma \eta )$ by the automorphy factor identity and multiplicativity of the determinant. It agrees with the usual operator on the modular group. The determinant factor is essential for the stated normalization of the Hecke operators.