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Published online by Cambridge University Press: 20 November 2018
Let
$f$ be a square-free integer and denote by
${{\Gamma }_{0}}{{\left( f \right)}^{+}}$ the normalizer of
${{\Gamma }_{0}}\left( f \right)$ in
$\text{SL}\left( 2,\,\mathbb{R} \right)$ . We find the analogues of the cusp form
$\Delta$ for the groups
${{\Gamma }_{0}}{{\left( f \right)}^{+}}$ .