Non-hyperelliptic modular curves of genus 3

A curve C defined over Q is modular of level N if there exists a non-constant morphism from X1(N) onto C defined over Q for some positive integer N. We provide a sufficient and necessary condition for the existence of a modular non-hyperelliptic curve C of genus 3 and level N such that Jac C is Q-isogenous to a given three dimensional Q-quotient of J1(N). Using this criterion, we present an algorithm to compute explicitly equations for modular non-hyperelliptic curves of genus 3. Let C be a modular curve of level N, we say that C is new if the corresponding morphism between J1(N) and Jac C factors through the new part of J1(N). We compute equations of 44 non-hyperelliptic new modular curves of genus 3, that we conjecture to be the complete list of this kind of curves. Furthermore, we describe some aspects of non-new modular curves and we present some examples that show the ambiguity of the non-new modular case.