
// Make newform 847.2.n.e in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_847_n();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_847_2_n_e();".
// 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_847_n();"
function MakeCharacter_847_n()
    N := 847;
    order := 15;
    char_gens := [122, 365];
    v := [5, 9];
    // 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_847_n_Hecke(Kf)
    return MakeCharacter_847_n();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_847_2_n_e();".
// 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_847_2_n_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_847_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![6561, -10935, 0, 26001, -39690, 54000, 2187, -140415, 214000, -972, -53600, 21375, -5220, 1200, 4250, -1983, -445, 375, 152, -30, 0, -6, -5, 0, 1]>],Snew);
    return Vf;
end function;