
// Make newform 5472.2.k.d in Magma, downloaded from the LMFDB on 07 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5472_k();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_5472_2_k_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_5472_k();"
function MakeCharacter_5472_k()
    N := 5472;
    order := 2;
    char_gens := [4447, 2053, 1217, 3745];
    v := [1, 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_5472_k_Hecke(Kf)
    return MakeCharacter_5472_k();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_5472_2_k_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_5472_2_k_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5472_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1024, 0, 1752, 0, -1072, 0, 281, 0, -30, 0, 1]>,<31,R![-131072, 0, 327680, 0, -76320, 0, 5400, 0, -140, 0, 1]>],Snew);
    return Vf;
end function;