
// Make newform 1386.2.k.w in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1386_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1386_k_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1386_2_k_w();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1386_2_k_w();".
// 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 ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [4, -18, 27, -19, 12, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [3, -1, 1, 0, 0, 0], [-18, 43, -28, 22, -5, 2], [22, -56, 37, -31, 7, -3], [50, -121, 82, -64, 15, -6], [24, -62, 44, -33, 8, -3]];
    Rf_basisdens := [1, 1, 2, 1, 2, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1386_k();"
function MakeCharacter_1386_k()
    N := 1386;
    order := 3;
    char_gens := [155, 199, 1135];
    v := [3, 1, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1386_k_Hecke();"
function MakeCharacter_1386_k_Hecke(Kf)
    N := 1386;
    order := 3;
    char_gens := [155, 199, 1135];
    char_values := [[1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 1, -1, 1], [1, 0, 1, 0, 0, 0], [0, 2, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0], [1, 1, 0, 1, 0, 2], [-1, -1, -4, -1, 1, -2], [-4, 0, 0, 1, 0, -1], [-2, 2, 0, 4, -4, 2], [1, 1, 4, 1, 0, 2], [-5, 0, 0, 1, 0, -1], [2, -2, 0, 1, 0, -1], [-1, -1, 4, -1, 1, -2], [0, 2, 0, 0, -2, 0], [3, -3, 4, 2, 2, 1], [-2, -2, -4, -2, 5, -4], [-2, 1, -1, 2, -2, 1], [-2, 0, 0, 1, 0, -1], [0, 0, -2, -4, 2, -2], [0, 0, 4, 0, -3, 0], [3, 4, 0, -1, 0, 1], [0, 0, 12, 0, 2, 0], [1, 0, 0, 2, 0, -2], [-5, 5, -6, -2, -4, -1], [2, 2, -2, 2, 0, 4], [-1, -1, -9, -1, 6, -2], [-2, -3, -4, -4, 5, -2], [6, 0, 0, -2, 0, 2], [2, -3, 0, 1, 0, -1], [2, 2, 8, 2, -8, 4], [0, 2, 0, 0, -2, 0], [12, 0, 0, -3, 0, 3], [1, 1, -8, 1, 0, 2], [-1, 4, 0, 2, -5, 1], [3, -3, 6, 6, 0, 3], [-1, -1, -5, -1, 6, -2], [-6, -6, 0, 2, 0, -2], [2, 2, 16, 2, -4, 4], [-9, -3, -10, -2, 4, -1], [12, -2, 0, -2, 0, 2], [0, 0, -8, 0, 4, 0], [2, 4, 0, -4, -2, -2], [-4, 2, 0, -3, 0, 3], [12, 2, 10, -4, 0, -2], [8, 2, 0, -4, 0, 4], [-2, 2, 0, 2, 0, -2], [4, 1, 5, 2, -2, 1], [2, 2, 0, 2, -2, 4], [-2, -2, -9, -2, 0, -4], [0, -2, 0, 4, 0, -4], [0, 6, -2, -4, -4, -2], [8, 6, 0, -1, 0, 1], [0, 0, 16, 0, 2, 0], [-10, 6, -14, -8, -2, -4], [-10, -5, -12, -4, 7, -2], [0, 0, 8, 0, -4, 0], [-8, -6, -8, 0, 6, 0], [11, 0, 0, -2, 0, 2], [-4, 6, 0, 8, -10, 4], [10, 0, 0, -3, 0, 3], [-2, -4, 0, 2, 0, -2], [-3, 4, -4, -2, -3, -1], [1, 1, 26, 1, -2, 2], [4, 4, 8, 4, -9, 8], [-5, -5, -5, -5, 10, -10], [0, 10, 0, -2, 0, 2], [-14, -5, -15, -2, 6, -1], [12, 5, 0, 0, 0, 0], [-16, -4, -18, -4, 6, -2], [0, 0, 10, 0, -8, 0], [-20, -6, -20, 0, 6, 0], [2, 2, 4, 2, 3, 4], [-1, 4, 0, 1, 0, -1], [5, 5, 0, 5, -6, 10], [-4, 1, -4, 0, -1, 0], [-3, -3, 10, -3, 2, -6], [-2, -2, 0, -2, 6, -4], [18, -4, 12, -12, 10, -6], [0, 6, 0, 1, 0, -1], [-20, -2, 0, 3, 0, -3], [4, -10, 8, 8, 6, 4], [11, -4, 0, -2, 0, 2], [-3, -3, -4, -3, -1, -6], [-1, -1, 18, -1, 6, -2], [-4, 2, 0, 4, 0, -4], [-4, -4, -14, -4, 0, -8], [28, 4, 0, -5, 0, 5], [-22, -2, 0, 2, 0, -2], [-3, -3, -24, -3, 2, -6], [-2, 6, -2, 0, -6, 0], [28, -4, 30, 4, 2, 2], [3, 12, 0, -5, 0, 5], [-2, -2, 0, -2, -8, -4], [-4, -6, 0, 2, 0, -2], [-4, -4, 0, -4, 2, -8], [-2, -8, -2, 0, 8, 0], [-4, -4, 16, -4, 10, -8], [-4, -4, 12, -4, 3, -8], [2, -2, 0, 3, 0, -3], [-23, -1, -26, -6, 4, -3], [0, 0, 0, 0, 0, 0], [-3, -3, -18, -3, 10, -6], [-9, -1, -10, -2, 2, -1], [-12, 7, -9, 6, -10, 3], [30, 2, 0, 2, 0, -2], [-5, -5, 6, -5, 10, -10], [-4, -1, -8, -8, 5, -4], [24, 0, 0, -2, 0, 2], [-4, -4, 8, -4, 3, -8], [0, 5, 4, 8, -9, 4], [-4, -2, 0, 6, 0, -6], [-16, 9, -19, -6, -6, -3], [2, 2, 0, 6, 0, -6], [-10, 2, -12, -4, 0, -2], [-16, 0, 0, 6, 0, -6], [12, -5, 16, 8, 1, 4], [-4, -4, -4, -4, 7, -8], [-35, -4, 0, -1, 0, 1], [11, 5, 10, -2, -4, -1], [-2, -4, 0, 6, 0, -6], [3, 3, 28, 3, 0, 6], [-5, -7, -12, -14, 14, -7], [-3, -3, -3, -3, 10, -6], [-2, -2, 0, -3, 0, 3], [-1, -1, -16, -1, 2, -2], [-1, -1, 8, -1, -3, -2], [2, -8, 0, 4, 0, -4], [2, 2, 8, 2, -1, 4], [16, -6, 10, -12, 12, -6], [32, -2, 0, 2, 0, -2], [2, 2, -8, 2, 6, 4], [4, 4, 0, 3, 0, -3], [-1, -1, -21, -1, 2, -2], [28, 6, 0, -10, 0, 10], [-26, -9, -28, -4, 11, -2], [-7, 1, -16, -18, 8, -9], [0, 7, 0, 2, 0, -2], [-16, 1, -15, 2, -2, 1], [-42, 0, 0, 2, 0, -2], [2, 2, -8, 2, -8, 4], [6, 0, 12, 12, -6, 6], [-21, 0, 0, 5, 0, -5], [-47, 1, -48, -2, 0, -1], [0, -9, 0, 8, 0, -8], [12, 3, 0, 0, 0, 0], [5, 0, 9, 8, -4, 4], [1, 1, -1, 1, 2, 2], [4, 4, 32, 4, -5, 8], [-4, -4, -4, -4, 11, -8], [2, 0, 0, -2, 0, 2], [17, 0, 0, -9, 0, 9], [6, 6, -6, 6, -6, 12], [6, -9, 9, 6, 6, 3], [-14, 15, 0, -1, 0, 1], [-5, -5, -24, -5, -1, -10], [6, 6, -4, 6, -14, 12], [-14, 0, 0, 6, 0, -6], [-33, 3, -30, 6, -6, 3], [-5, -5, 6, -5, 14, -10], [-15, 12, 0, -5, 0, 5], [30, -5, 0, 1, 0, -1], [6, 6, 18, 6, -8, 12], [-12, 4, -4, 16, -12, 8], [-33, 4, -36, -6, -1, -3], [-8, -10, -8, 0, 10, 0], [-40, -2, -40, 0, 2, 0], [20, 6, 0, 0, 0, 0], [-6, -6, -8, -6, 10, -12], [-8, 7, -13, -10, -2, -5], [4, 18, 0, 2, 0, -2], [12, 12, 10, 12, -16, 24], [-20, 10, -28, -16, -2, -8], [28, -12, 34, 12, 6, 6], [0, -18, 0, 2, 0, -2], [12, 8, 0, -1, 0, 1], [26, 8, 28, 4, -10, 2], [-20, 5, 0, 8, 0, -8], [-4, -4, -20, -4, 14, -8], [-4, -4, -12, -4, -2, -8], [-9, -12, 0, 1, 0, -1], [10, 10, 0, 5, 0, -5], [2, 2, 1, 2, -8, 4], [7, 5, 16, 18, -14, 9], [-20, 10, -20, 0, -10, 0], [-4, 9, 0, 8, -13, 4], [22, -3, 23, 2, 2, 1], [4, 4, 40, 4, -8, 8], [-10, 8, -12, -4, -6, -2], [5, 5, 34, 5, -6, 10], [7, -12, 0, -3, 0, 3], [-1, -1, -11, -1, -6, -2], [-2, -2, 52, -2, 2, -4], [18, 2, 20, 4, -4, 2], [-38, 0, -38, 0, 0, 0], [-40, -6, -38, 4, 4, 2], [3, 3, 20, 3, -6, 6], [-2, 16, 0, 0, 0, 0], [-6, -6, -44, -6, 4, -12], [2, -12, -6, -16, 20, -8], [-6, 12, 0, -5, 0, 5], [6, 6, 20, 6, -2, 12], [16, 8, 20, 8, -12, 4], [12, -4, 0, -11, 0, 11], [-12, 10, -12, 0, -10, 0], [8, 8, -4, 8, -12, 16], [5, 5, 9, 5, -22, 10], [-13, -16, 0, -3, 0, 3], [-42, 2, -40, 4, -4, 2], [0, 0, 22, 0, -8, 0], [-12, 9, 0, -8, 0, 8], [-18, -2, 0, -2, 0, 2], [-6, -6, -14, -6, 6, -12], [32, 2, 28, -8, 2, -4], [1, 1, 22, 1, -14, 2], [8, 4, 4, -8, 0, -4], [12, 15, 11, -2, -14, -1], [-4, -4, 20, -4, 5, -8], [-16, -10, 0, 10, 0, -10], [6, 6, 28, 6, -14, 12], [-6, -3, 0, -5, 0, 5], [6, 6, -8, 6, 6, 12], [-2, -2, 36, -2, 0, -4], [-19, 4, 0, 5, 0, -5], [20, 1, 0, -8, 0, 8], [-4, -4, -22, -4, 2, -8], [-5, -8, -16, -22, 19, -11], [7, 7, 28, 7, -1, 14], [-4, -4, -40, -4, 4, -8], [-23, 3, -20, 6, -6, 3], [-15, 7, -22, -14, 0, -7], [18, -4, 0, -1, 0, 1], [25, -2, 23, -4, 4, -2], [-3, 4, 0, -5, 0, 5], [-8, -4, -2, 12, -2, 6], [-3, -3, 2, -3, -18, -6], [-4, -4, -10, -4, 14, -8], [8, 2, 0, -4, 0, 4], [-16, 5, 0, -4, 0, 4], [19, 13, 24, 10, -18, 5], [8, 8, 0, 8, -16, 16], [50, -2, 48, -4, 4, -2], [1, 1, 6, 1, -2, 2], [-44, 12, 0, -5, 0, 5], [-4, -4, -4, -4, 2, -8], [-10, 28, 0, -10, 0, 10], [9, 9, 6, -6, -6, -3], [21, -13, 14, -14, 20, -7], [-36, -12, 0, 4, 0, -4], [-4, -6, 0, 0, 0, 0], [3, 3, 26, 3, -10, 6], [31, -6, 40, 18, -3, 9], [11, -12, 0, 9, 0, -9], [-14, 16, -14, 0, -16, 0], [-6, -6, -4, -6, 4, -12], [2, -18, -2, -8, 22, -4], [-31, -11, -38, -14, 18, -7], [-8, -2, 0, 6, 0, -6], [-6, -6, 18, -6, 10, -12], [4, 17, 12, 16, -25, 8], [-4, 4, 0, 6, 0, -6], [12, 3, 16, 8, -7, 4], [32, -3, 0, 2, 0, -2], [-14, 10, -18, -8, -6, -4], [-2, -2, -12, -2, -12, -4], [24, 9, 0, -14, 0, 14], [-16, -4, 0, -2, 0, 2], [-50, 6, 0, 1, 0, -1], [22, 2, 30, 16, -10, 8], [-11, -11, -22, -11, 16, -22], [30, -1, 25, -10, 6, -5], [-38, -8, -38, 0, 8, 0], [-3, -3, 20, -3, 3, -6], [14, 2, 0, 8, 0, -8], [-2, -2, 36, -2, -6, -4], [9, 9, 20, 9, -8, 18], [-26, -9, -28, -4, 11, -2], [-4, -4, -34, -4, -6, -8], [-7, -7, 13, -7, 6, -14], [10, -4, 22, 24, -8, 12], [12, 18, 14, 4, -20, 2], [8, 8, 0, -6, 0, 6], [-24, 13, 0, 2, 0, -2], [-5, -5, -26, -5, -4, -10], [9, 9, -24, 9, -9, 18], [2, -4, 6, 8, 0, 4], [-48, -2, 0, 5, 0, -5], [-4, 2, 6, 20, -12, 10], [8, -16, 0, -1, 0, 1], [6, -9, 16, 20, -1, 10], [21, 25, 24, 6, -28, 3], [-5, -5, -13, -5, 18, -10], [-31, -8, 0, 7, 0, -7], [8, -19, 0, 2, 0, -2], [42, -12, 44, 4, 10, 2], [6, 6, -34, 6, -4, 12], [-4, -13, 0, 10, 0, -10], [-5, -5, -5, -5, 6, -10], [8, 0, 0, -13, 0, 13], [-8, -8, -26, -8, 26, -16], [3, 3, -26, 3, 14, 6], [9, 3, 8, -2, -2, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1386_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1386_2_k_w();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_1386_2_k_w(:prec:=6)
    chi := MakeCharacter_1386_k();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1386_2_k_w();".
// 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_1386_2_k_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1386_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16, -36, 81, -8, 9, 0, 1]>,<13,R![32, -36, 0, 1]>,<17,R![1, -1, 1]>,<23,R![441, 63, 198, -69, 78, -9, 1]>],Snew);
    return Vf;
end function;