# SageMath code for working with modular form 17850.2.a.ba


# Compute space of new eigenforms: 
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(17850, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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(17850, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 2, names="a") 

# select newform: 
traces = [1,-1,1,1,0,-1,1,-1,1,0,4,1,-6,-1,0,1,-1,-1,4,0,1,-4,0,-1,0,6,
1,1,-2,0,-8,-1,4,1,0,1,10,-4,-6,0,2,-1,-4,4,0,0,0,1,1,0,-1,-6,-6,-1,0,
-1,4,2,-12,0,-2,8,1,1,0,-4,-12,-1,0,0,-8,-1,-2,-10,0,4,4,6,8,0,1,-2,-4,
1,0,4,-2,-4,10,0,-6,0,-8,0,0,-1,-10,-1,4,0]
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