
// Make newform 605.2.m.c in Magma, downloaded from the LMFDB on 10 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_605_m();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_605_2_m_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_605_m();"
function MakeCharacter_605_m()
    N := 605;
    order := 20;
    char_gens := [122, 486];
    v := [5, 2];
    // 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_605_m_Hecke( : Kf := Kf)
    return MakeCharacter_605_m();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_605_2_m_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_605_2_m_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_605_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![625, -5000, 20000, -53750, 108375, -171000, 211250, -181750, 78275, -14500, 3750, 42150, 2425, -53750, 13950, -18150, 12800, 8450, -7550, 11800, -4615, -1750, 2400, -1420, 555, 130, 200, 0, -25, -20, 0, 0, 1]>],Snew);
    return Vf;
end function;