On the maps from X(4p) to X(4)

We study pullbacks of modular forms of weight 1 from the modular curve X(4) to the modular curve X(4p), where p is an odd prime. We find the extent to which such modular forms separate points on X(4p). Our main result is that these modular forms give rise to a morphism F from the quotient of X(4p) by a certain involution i to projective space, such that F is a projective embedding of X(4p)/i away from the cusps. We also report on computer calculations regarding products of such modular forms, going up to weight 4 for p <= 13, and up to weight 3 for p <= 23, and make a conjecture about these products and about the nature of the singularities at the cusps of the image F(X(4p)/i).