# SageMath code for working with modular form 39710.2.a.e


# Compute space of new eigenforms: 
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39710, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
B = ModularForms(chi, 1).cuspidal_submodule().basis()
N = [B[i] for i in range(len(B))]

# Compute space of new eigenforms: 
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39710, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a") 

# select newform: 
traces = [1,-1,0,1,-1,0,-2,-1,-3,1,-1,0,-3,2,0,1,0,3,0,-1,0,1,0,0,1,3,
0,-2,1,0,-4,-1,0,0,2,-3,0,0,0,1,-12,0,1,-1,3,0,-1,0,-3,-1,0,-3,0,0,1,2,
0,-1,12,0,-11,4,6,1,3,0,14,0,0,-2,7,3,-12,0,0,0,2,0,-10,-1,9,12,3,0,0,
-1,0,1,9,-3,6,0,0,1,0,0,-17,3,3,1]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(None)] == traces)

# q-expansion: 
f.q_expansion() # note that sage often uses an isomorphic number field