
// Make newform 690.2.i.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_690_i();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_690_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_690_i();"
function MakeCharacter_690_i()
    N := 690;
    order := 4;
    char_gens := [461, 277, 511];
    v := [2, 1, 4];
    // 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_690_i_Hecke(Kf)
    return MakeCharacter_690_i();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_690_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_690_2_i_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_690_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![21566736, -18055872, 7558272, 4866048, 3819672, -1606848, 555584, 286144, 94081, -14196, 4936, 2504, 671, -24, 8, 4, 1]>],Snew);
    return Vf;
end function;