
// Make newform 847.2.n.i in Magma, downloaded from the LMFDB on 27 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_i();".
// 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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_847_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 20, 250, 3258, 24298, -50294, -106907, -197039, 610692, 416600, 1517223, -301274, -5792858, -2511863, 6053344, 1289830, 29144169, 16819433, -21361401, 3428771, 16948009, -2572439, -4492979, 327838, 27220, -167082, 271995, 79672, -76929, -25573, 16655, 4999, -1752, -817, -137, 90, 66, 2, -9, -2, 1]>],Snew);
    return Vf;
end function;