
// Make newform 805.2.i.e in Magma, downloaded from the LMFDB on 16 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_805_i();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_805_2_i_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_805_i();"
function MakeCharacter_805_i()
    N := 805;
    order := 3;
    char_gens := [162, 346, 281];
    v := [3, 2, 3];
    // 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_805_i_Hecke(Kf)
    return MakeCharacter_805_i();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_805_2_i_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_805_2_i_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_805_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![36, -102, 1243, -2337, 27489, -48144, 259738, -109958, 991194, -58617, 2733760, 1055734, 3585055, 1879644, 3349761, 1841621, 1982069, 948903, 771987, 329432, 214256, 78419, 42138, 13320, 6028, 1570, 585, 120, 37, 5, 1]>],Snew);
    return Vf;
end function;