
// Make newform 819.2.fm.g in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_819_fm();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_819_2_fm_g();".
// 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_819_fm();"
function MakeCharacter_819_fm()
    N := 819;
    order := 12;
    char_gens := [92, 703, 379];
    v := [12, 6, 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_819_fm_Hecke(Kf)
    return MakeCharacter_819_fm();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_819_2_fm_g();".
// 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_819_2_fm_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_819_fm();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, -162, 891, -1218, 1423, -1834, 1076, -138, -178, 394, -68, -10, 31, -28, 11, -4, 1]>,<5,R![187388721, 0, 0, 0, 7550441730, 0, 0, 0, 4376614033, 0, 0, 0, 922081398, 0, 0, 0, 87250612, 0, 0, 0, 3728042, 0, 0, 0, 64945, 0, 0, 0, 454, 0, 0, 0, 1]>,<19,R![5551952306507776, 0, -21444855162851328, 0, 33199642062013440, 0, -21587280125838336, 0, 4518047221302976, 0, 975055014734976, 0, -84921500605744, 0, -22059856200240, 0, 2649756264513, 0, 110678036082, 0, -5208544805, 0, -245274306, 0, 7775888, 0, 670548, 0, 17159, 0, 204, 0, 1]>],Snew);
    return Vf;
end function;