Congruence Subgroup (awaiting review)

Let $F$ be a real quadratic field with ring of integers $\Z_F$. Let $\frak{b}$ be a fractional ideal in $F$ (the component ideal ).

A principal congruence subgroup $\Gamma_{\frak{b}}(\frak{N})$ is the kernel of reduction-modulo-$\frak{N}$ within $\GL^+(\Z_F \oplus \frak{b})$ or $\SL(\Z_F \oplus \frak{b})$. A congruence subgroup $\Gamma$ is a subgroup of either $\GL^+(\Z_F \oplus \frak{b})$ or $\SL(\Z_F \oplus \frak{b})$ which contains a principal congruence subgroup $\Gamma_{\frak{b}}(\frak{N})$. The minimal such $\frak{N}$ is called the level.