I don't think pi^*(c) can be 0. Suppose pi^*(c) = div(f); then f would be a map from X_1(N) to P^1 of degree about N. But in fact any such map is of degree at least ~N^2, i.e. the gonality of X_1(N) is bounded below by a constant multiple of N^2. This was proved independently by Zograf ("Small eigenvalues of automorphic Laplacians in spaces of cusp forms") and Abramovich ("A linear lower bound on the gonality of modular curves.")
Update: As Kevin points out in comments I should say "for N large enough." But "large enough" is effective here since the constants in the gonality bounds are effective (albeit small.)
