
// Make newform 5520.2.be.d in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5520_be();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_5520_2_be_d();".
// 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_5520_be();"
function MakeCharacter_5520_be()
    N := 5520;
    order := 2;
    char_gens := [4831, 1381, 1841, 4417, 1201];
    v := [1, 2, 2, 2, 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_5520_be_Hecke(Kf)
    return MakeCharacter_5520_be();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_5520_2_be_d();".
// 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_5520_2_be_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5520_be();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-239616, 1790976, -485120, -2472448, 915168, 1335632, -515276, -367664, 137556, 56420, -19571, -4876, 1519, 220, -61, -4, 1]>],Snew);
    return Vf;
end function;