
// Make newform 561.2.m.d in Magma, downloaded from the LMFDB on 10 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_561_m();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_561_2_m_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_561_m();"
function MakeCharacter_561_m()
    N := 561;
    order := 5;
    char_gens := [188, 409, 496];
    v := [5, 1, 5];
    // 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_561_m_Hecke(Kf)
    return MakeCharacter_561_m();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_561_2_m_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_561_2_m_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_561_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -26, 277, -698, 2115, -4348, 9007, -7265, 12401, 2215, 21953, -14374, 19168, -10503, 9514, -3545, 2727, -667, 546, -146, 94, -27, 14, -3, 1]>],Snew);
    return Vf;
end function;