
// Make newform 475.2.l.f in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_475_l();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_475_2_l_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
// To make the character of type GrpDrchElt, type "MakeCharacter_475_l();"
function MakeCharacter_475_l()
    N := 475;
    order := 9;
    char_gens := [77, 401];
    v := [9, 7];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_475_l_Hecke(Kf)
    return MakeCharacter_475_l();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_475_2_l_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_475_2_l_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_475_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![361, 0, 1995, 0, -79860, 0, 612550, 0, 23407380, 0, -25832994, 0, 615475776, 0, 252074172, 0, -249968517, 0, 154333422, 0, 329153946, 0, 122725170, 0, 55689493, 0, 31702815, 0, 5285235, 0, 527040, 0, 653217, 0, -24393, 0, -5028, 0, 10179, 0, -255, 0, -181, 0, 78, 0, -9, 0, 1]>],Snew);
    return Vf;
end function;