  We systematically study how the integrality of the conformal characters
shapes the space of fermionic rational conformal field theories in two
dimensions. The integrality suggests that conformal characters on torus with a
given choice of spin structures should be invariant under a principal
congruence subgroup of $\mathrm{PSL}(2,\mathbb{Z})$. The invariance strongly
constrains the possible values of the central charge as well as the conformal
weights in both Neveu-Schwarz and Ramond sectors, which improves the
conventional holomorphic modular bootstrap method in a significant manner. This
allows us to make much progress on the classification of fermionic rational
conformal field theories with the number of independent characters less than
five.
