
// Make newform 819.2.et.c in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_819_et();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_819_2_et_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_819_et();"
function MakeCharacter_819_et()
    N := 819;
    order := 12;
    char_gens := [92, 703, 379];
    v := [12, 2, 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_et_Hecke(Kf)
    return MakeCharacter_819_et();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_819_2_et_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_819_2_et_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_819_et();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2304, 28800, 180000, 462168, 667593, 608310, 1802178, 4586184, 6662169, 3562446, 2124402, 5096652, 9588994, 3930512, 717164, 1452738, 5032807, 1284942, 87502, 87086, 1121398, 154620, 3732, -7712, 117684, 6572, 50, -616, 5749, 88, 0, -10, 126, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;