The modular bootstrap program is devoted to exploiting the modular properties of CFTs in order to constrain their spectrum. A number of bounds on the lightest part of the spectrum have been successfully obtained for various CFTs with or without charge, with potential connections to the weak gravity conjecture. I will show that for chiral CFTs with a U(1) charge,  states with conformal dimension as low as c/24 +1 are needed in order for the theory to be consistent. While previous work relied more or less on numerical techniques, this argument will be completely analytical.