
// Make newform 624.4.n.c in Magma, downloaded from the LMFDB on 16 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_624_n();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_624_4_n_c();".
// 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_624_n();"
function MakeCharacter_624_n()
    N := 624;
    order := 2;
    char_gens := [79, 469, 209, 145];
    v := [1, 2, 1, 1];
    // 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_624_n_Hecke(Kf)
    return MakeCharacter_624_n();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_624_4_n_c();".
// 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_624_4_n_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_624_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![-425520, 0, 72546, 0, -549, 0, 1]>,<7,R![-9542880, 0, 215178, 0, -885, 0, 1]>],Snew);
    return Vf;
end function;