
// Make newform 1400.2.bh.j in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1400_bh();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1400_bh_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_1400_2_bh_j();".
// 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_1400_2_bh_j();".
// 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 := [1, 0, -18, 0, 269, 0, -960, 0, 2754, 0, -789, 0, 170, 0, -15, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1313504207, 0, 20064172, 0, -299619996, 0, 10229021874, 0, -3009625800, 0, 677333628, 0, -60216631, 0, 4102340, 0], [-98128630, 0, -27392539, 0, 409055127, 0, -13986921834, 0, 4108880850, 0, -924727311, 0, 82189681, 0, -5600705, 0], [99650, 0, -1489165, 0, 7255677, 0, -15233538, 0, 4324629, 0, -913449, 0, 80335, 0, -5297, 0], [-48212247, 0, 720514496, 0, -2197202760, 0, 7378916580, 0, -2121507168, 0, 460586700, 0, -40688940, 0, 2723695, 0], [0, 48212247, 0, -720514496, 0, 2197202760, 0, -7378916580, 0, 2121507168, 0, -460586700, 0, 40688940, 0, -2723695], [0, 3763758820, 0, 173068144, 0, -2584441392, 0, 88439791707, 0, -25960221600, 0, 5842497456, 0, -519187807, 0, 35385680], [-1014108521, 0, -193132316, 0, 2884061388, 0, -98668813581, 0, 28969847400, 0, -6519831084, 0, 579404438, 0, -39488020, 0], [0, -1373145044, 0, -193132316, 0, 2884061388, 0, -98668813581, 0, 28969847400, 0, -6519831084, 0, 579404438, 0, -39488020], [185133080, 0, -17010925465, 0, 48895329456, 0, -164206309848, 0, 47199500037, 0, -10249640520, 0, 905469064, 0, -60623903, 0], [1359797656, 0, -20321026133, 0, 88415581794, 0, -207944878650, 0, 59254651869, 0, -12618776481, 0, 1111329695, 0, -73615375, 0], [0, 1793707879, 0, -26805656597, 0, 108190406634, 0, -274355127870, 0, 78348216381, 0, -16764056781, 0, 1477530155, 0, -98128630], [0, 3662269979, 0, 552004409, 0, -8243129037, 0, 282019518909, 0, -82800661350, 0, 18634765941, 0, -1656023633, 0, 112863355], [0, 18, 0, -269, 0, 960, 0, -2754, 0, 789, 0, -170, 0, 15, 0, -1], [0, -6603304643, 0, 98681459059, 0, -402966711759, 0, 1009972746282, 0, -288326988363, 0, 61649473755, 0, -5432977525, 0, 360686108]];
    Rf_basisdens := [1, 1, 359036523, 359036523, 259983, 39892947, 39892947, 359036523, 359036523, 359036523, 359036523, 359036523, 359036523, 359036523, 1, 359036523];
    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_1400_bh();"
function MakeCharacter_1400_bh()
    N := 1400;
    order := 6;
    char_gens := [351, 701, 1177, 801];
    v := [6, 6, 3, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1400_bh_Hecke();"
function MakeCharacter_1400_bh_Hecke(Kf)
    N := 1400;
    order := 6;
    char_gens := [351, 701, 1177, 801];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, -1, -1, -1, -1], [0, -1, 0, 0, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, -3, 0, 0, 2, 0, -1, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, -3, 0], [-2, 0, -1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 0, -4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -1, 0, 1, 1], [-3, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -1, 2, 0, 2, 0, 0, 5, 1, -2, 1], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -4, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, 0, -1, -1, 5, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, -2, 0, -2, 0, 0, 0, 2, 0, 0], [-5, 0, 0, -4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 2, 0, -2, 0, 0, 0, 2, 0, 0], [-1, 0, 0, 0, 0, -1, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, 5, 0, 0, 8, 1, -2, 1], [-5, 0, 0, 3, 3, -5, 0, 0, -2, 0, 1, -2, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 2, 1, -1, 1], [0, 0, 0, 0, -3, -3, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, -3, -1], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 5, 0, 0, -1], [-1, 0, -1, 0, -6, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 2, 4, 0, 4, 0, 0, 7, -1, -4, -1], [0, 3, 0, 0, 0, 0, -3, 0, 0, 10, 0, 0, 9, -1, 0, -1], [7, 0, 0, 5, 5, 7, 0, 0, -3, 0, 0, -3, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 3, 0, -1, 0, 0, 0, -2, 0, 0], [-6, 0, -1, 6, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -2, -2, -1, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0], [-1, 0, -1, 0, -2, 13, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, -2, 0, -9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 9, 0, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 5, 5, 1, 5], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -8, 0, 7, 0], [4, 0, 4, 0, 0, 2, 0, 0, 0, 0, -4, -2, 0, 0, 0, 0], [4, 0, 1, 6, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1, 0, 0, -4, 0, 4, -4, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, -2, 0, 0], [0, -9, 0, 0, 0, 0, 9, 0, 0, 7, 0, 0, 5, -2, 0, -2], [-1, 0, -1, 0, -5, 3, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0], [3, 0, 4, -1, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 2, -1, 0, -2, 0, 0, -4, -1, 1, -1], [0, -3, 0, 0, 0, 0, 0, -1, 0, 9, 0, 0, 0, -2, 0, 0], [-15, 0, 0, 1, 1, -15, 0, 0, 4, 0, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 2, 0, -1, 1], [6, 0, -2, -11, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0], [3, 0, 3, 0, -3, -4, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0], [-1, 0, 2, -1, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -10, 0, -2, 0], [0, -4, 0, 0, 0, 0, 0, -1, 0, -2, 0, 0, 0, -3, 0, 0], [1, 0, 1, 0, 5, -2, 0, 0, 0, 0, -1, 5, 0, 0, 0, 0], [19, 0, 0, 5, 5, 19, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, 9, 0, 0, 0, 0, 0, -4, 0, 3, 0, 0, 0, 2, 0, 0], [-6, 0, 3, -9, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, 0, 0, 2, 0, -11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3, 0, -6, 0, 0, -3, 6, 3, 6], [0, 3, 0, 0, 0, 0, -3, 6, 0, -12, 0, 0, -9, -3, -6, -3], [-3, 0, -3, 0, 3, -19, 0, 0, 0, 0, 3, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -8, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -8, 0, 3, 3], [0, 0, 0, 3, 3, 0, 0, 0, 3, 0, -3, 3, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, -6, 4, 0, 6, 0, 0, 9, -1, -4, -1], [0, -5, 0, 0, 0, 0, 0, -4, 0, 11, 0, 0, 0, 0, 0, 0], [6, 0, -2, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 2, 0, -10, 0, 0, 0, 3, 0, 0], [7, 0, 0, 7, 7, 7, 0, 0, -5, 0, 0, -5, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, -1, 0, 11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0, -4, 0, 9, -2], [15, 0, -2, 1, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 4, 6], [3, 0, 3, 0, -3, -2, 0, 0, 0, 0, -3, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, -5, 6], [-13, 0, 0, 1, 1, -13, 0, 0, -6, 0, 2, -6, 0, 0, 0, 0], [2, 0, 2, 0, 2, 27, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0], [-8, 0, -3, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-17, 0, 0, 1, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 10, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, -3, -1, 0, 2, 0, 0, -3, -4, 1, -4], [9, 0, 0, -4, -4, 9, 0, 0, -1, 0, -4, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, -2, 0, -1, 0], [5, 0, 6, 6, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 13, 0, -9, 2], [-3, 0, 0, -11, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0], [0, -9, 0, 0, 0, 0, 9, 7, 0, -3, 0, 0, 7, 3, -7, 3], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -8, 0, 11, -2], [0, 0, 0, 0, -3, -2, 0, 0, 0, 0, 0, -12, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, -5, 0, 10, 0, 0, 0, -2, 0, 0], [-20, 0, -6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [-8, 0, 0, -1, -1, -8, 0, 0, 9, 0, 1, 9, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, -8, -2, 0, 7, 0, 0, 2, -3, 2, -3], [-8, 0, 0, 0, 0, -8, 0, 0, 8, 0, -2, 8, 0, 0, 0, 0], [-5, 0, -5, 0, 1, 9, 0, 0, 0, 0, 5, -11, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -12, 0, 5, 3], [2, 0, 0, -8, -8, 2, 0, 0, -5, 0, 0, -5, 0, 0, 0, 0], [0, 9, 0, 0, 0, 0, -9, 2, 0, 5, 0, 0, 5, -2, -2, -2], [0, -9, 0, 0, 0, 0, 0, 6, 0, 4, 0, 0, 0, -5, 0, 0], [0, -1, 0, 0, 0, 0, 0, 1, 0, -28, 0, 0, 0, 3, 0, 0], [2, 0, 0, 11, 11, 2, 0, 0, -5, 0, 2, -5, 0, 0, 0, 0], [0, 0, 0, 0, -7, 3, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0], [0, 15, 0, 0, 0, 0, 0, -13, 0, -9, 0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, 0, -16, 0, 0, -27, -4, 7, -4], [0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 7, 0, -11, -5], [-5, 0, -5, 0, 3, 13, 0, 0, 0, 0, 5, -13, 0, 0, 0, 0], [5, 0, -1, 9, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, -1, 0, -7, 2], [0, 14, 0, 0, 0, 0, 0, -2, 0, -17, 0, 0, 0, 1, 0, 0], [0, 6, 0, 0, 0, 0, -6, 6, 0, -20, 0, 0, -13, 1, -6, 1], [1, 0, 1, 0, -11, 16, 0, 0, 0, 0, -1, 11, 0, 0, 0, 0], [1, 0, 7, 4, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 4, -10, 0, 0, 0, 0, -1, -14, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 5, 11, 0, -14, 0, 0, -4, -1, -11, -1], [0, 0, 0, 0, 0, 0, 0, 3, 0, -8, 0, 0, 0, -6, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -5, 0, -1, -2], [7, 0, -2, -5, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0], [2, 0, 2, 0, -8, 8, 0, 0, 0, 0, -2, -4, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 2, 5, 0, 14, 0, 0, 20, 1, -5, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -24, 0, 6, -3], [0, 12, 0, 0, 0, 0, 0, -9, 0, -2, 0, 0, 0, -2, 0, 0], [9, 0, 0, -12, -12, 9, 0, 0, 11, 0, -1, 11, 0, 0, 0, 0], [0, 0, 7, 4, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0], [-19, 0, 0, -5, -5, -19, 0, 0, 12, 0, -2, 12, 0, 0, 0, 0], [8, 0, 0, -10, -10, 8, 0, 0, 4, 0, 3, 4, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -1, -8, 0, -8, 0, 0, -12, 4, 8, 4], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -8, 0, -7, 4], [-3, 0, -3, 0, -3, 2, 0, 0, 0, 0, 3, -7, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 2, -7, 0, 0, 0, 0, -8, -1, 7, -1], [3, 0, 0, 7, 7, 3, 0, 0, 8, 0, 2, 8, 0, 0, 0, 0], [0, 12, 0, 0, 0, 0, -12, -12, 0, -3, 0, 0, -12, 3, 12, 3], [-1, 0, 0, -2, -2, -1, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0], [23, 0, 4, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, -2, 0, 10, 0, 0, 0, -1, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 0, 28, 0, 0, 0, 1, 0, 0], [0, -9, 0, 0, 0, 0, 9, 4, 0, 3, 0, 0, 11, 4, -4, 4], [1, 0, 1, 0, 8, -9, 0, 0, 0, 0, -1, 12, 0, 0, 0, 0], [-15, 0, -5, -9, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0], [-35, 0, 0, -7, -7, -35, 0, 0, 11, 0, -4, 11, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 3, 0, -1, 0, 0, 0, 4, 0, 0], [0, 9, 0, 0, 0, 0, -9, -11, 0, 7, 0, 0, -4, 0, 11, 0], [4, 0, 4, 0, -14, 34, 0, 0, 0, 0, -4, 1, 0, 0, 0, 0], [-2, 0, -1, -23, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, 0, -3, 0, 0, -5, 5, 7, 5], [0, -5, 0, 0, 0, 0, 0, -2, 0, -20, 0, 0, 0, 3, 0, 0], [-7, 0, 0, 6, 6, -7, 0, 0, 3, 0, 1, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, -12, 0, 6, 3], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -14, 0, -10, -4], [-17, 0, -1, 10, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, -16, 0, 0, 0, 0, 16, 3, 0, 27, 0, 0, 27, -3, -3, -3], [0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 5, 0, 10, 3], [0, 7, 0, 0, 0, 0, 0, -1, 0, -22, 0, 0, 0, -1, 0, 0], [0, 0, -1, -1, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0], [-10, 0, 0, 16, 16, -10, 0, 0, 2, 0, 1, 2, 0, 0, 0, 0], [19, 0, 0, -3, -3, 19, 0, 0, -6, 0, 2, -6, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 6, 10, 0, -1, 0, 0, 10, 1, -10, 1], [3, 0, 3, 0, -8, 25, 0, 0, 0, 0, -3, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 2, -4], [0, 7, 0, 0, 0, 0, -7, -4, 0, 4, 0, 0, -5, -5, 4, -5], [0, 9, 0, 0, 0, 0, -9, 9, 0, -2, 0, 0, 7, 0, -9, 0], [-24, 0, 0, -7, -7, -24, 0, 0, 3, 0, -1, 3, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, 0, 28, 0, 0, 0, 0, 0, 0], [0, 9, 0, 0, 0, 0, 0, -2, 0, -9, 0, 0, 0, 0, 0, 0], [6, 0, 6, 0, 4, 3, 0, 0, 0, 0, -6, 2, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, -3, 0, 6, 0, 0, 0, -2, 0, 0], [2, 0, -1, -7, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 8, 0, 2, 1], [4, 0, 4, 0, 7, -15, 0, 0, 0, 0, -4, 9, 0, 0, 0, 0], [-17, 0, -8, 1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 7, 7, -1, 0, 0, -3, 0, 0, -3, 0, 0, 0, 0], [0, -14, 0, 0, 0, 0, 0, 6, 0, -10, 0, 0, 0, -1, 0, 0], [-4, 0, -4, 0, -1, 0, 0, 0, 0, 0, 4, -7, 0, 0, 0, 0], [-17, 0, -2, 8, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [12, 0, -8, -8, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0], [-8, 0, -8, 0, 1, -15, 0, 0, 0, 0, 8, 4, 0, 0, 0, 0], [0, 15, 0, 0, 0, 0, -15, -10, 0, -15, 0, 0, -21, 4, 10, 4], [12, 0, 0, -2, -2, 12, 0, 0, -4, 0, 8, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 27, 0, 4, 2], [-11, 0, -3, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, -2, -2, 0, -2, 0, 0, -14, -10, 2, -10], [0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -3, 0, -7, 1], [0, 1, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 2, 0, 0], [2, 0, 2, 0, -1, 17, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0], [0, 11, 0, 0, 0, 0, 0, 0, 0, -18, 0, 0, 0, 5, 0, 0], [0, 15, 0, 0, 0, 0, 0, -4, 0, -8, 0, 0, 0, 3, 0, 0], [-18, 0, 0, 4, 4, -18, 0, 0, -5, 0, 1, -5, 0, 0, 0, 0], [0, 0, 0, 0, 2, -4, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 14, 0, 13, 12], [0, -7, 0, 0, 0, 0, 7, -7, 0, 1, 0, 0, 2, 8, 7, 8], [16, 0, 0, 15, 15, 16, 0, 0, -14, 0, 7, -14, 0, 0, 0, 0], [8, 0, 0, -3, -3, 8, 0, 0, 4, 0, 1, 4, 0, 0, 0, 0], [0, 10, 0, 0, 0, 0, 0, 0, 0, -12, 0, 0, 0, -4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -12, 0, 0], [4, 0, 4, 0, -19, -5, 0, 0, 0, 0, -4, 17, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 0, -2, 10], [0, -9, 0, 0, 0, 0, 9, -6, 0, 33, 0, 0, 25, -2, 6, -2], [0, 0, 0, 0, 0, 0, -21, 0, 0, 0, 0, 0, 16, 0, -3, -5], [-11, 0, -11, 0, -2, 9, 0, 0, 0, 0, 11, 3, 0, 0, 0, 0], [-8, 0, 4, 9, 0, 0, 0, 0, -17, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 6, 0, -11, 3], [1, 0, 1, 0, 0, 28, 0, 0, 0, 0, -1, -9, 0, 0, 0, 0], [-2, 0, 8, 16, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 17, 0, -17, 0, 0, 0, -9, 0, 0], [10, 0, 0, 14, 14, 10, 0, 0, -1, 0, 5, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, -1, -9], [3, 0, -6, -18, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0], [-4, 0, -4, 0, 9, 5, 0, 0, 0, 0, 4, -22, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -9, 0, 2, 3], [23, 0, -1, 19, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 5, -6, 0, -13, 0, 0, -16, 3, 6, 3], [0, 0, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 8, 0, -21, -4], [-1, 0, -1, 0, 10, -32, 0, 0, 0, 0, 1, 5, 0, 0, 0, 0], [-21, 0, 0, 16, 16, -21, 0, 0, -1, 0, 10, -1, 0, 0, 0, 0], [0, 10, 0, 0, 0, 0, 0, 4, 0, -17, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 21, -1, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -19, -2], [0, -17, 0, 0, 0, 0, 17, 1, 0, -9, 0, 0, 0, 8, -1, 8], [38, 0, 0, 0, 0, 38, 0, 0, -7, 0, -2, -7, 0, 0, 0, 0], [14, 0, 3, 1, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 11, 11, 0, 0, 0, 4, 0, 10, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -15, 0, -6, 7], [0, 10, 0, 0, 0, 0, -10, 1, 0, -38, 0, 0, -33, 4, -1, 4], [-25, 0, 6, -21, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 17, 0, 1, -2], [-7, 0, -7, 0, -5, 27, 0, 0, 0, 0, 7, -7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 11, 0, 2, -2], [1, 0, 0, 14, 14, 1, 0, 0, -13, 0, -1, -13, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, -19, 0, -8, 0, 0, 0, -2, 0, 0], [-2, 0, -2, 0, 7, 3, 0, 0, 0, 0, 2, 17, 0, 0, 0, 0], [29, 0, -3, -16, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0], [11, 0, 11, 0, -3, 3, 0, 0, 0, 0, -11, 3, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, -6, -8, 0, 9, 0, 0, 2, 1, 8, 1], [0, -6, 0, 0, 0, 0, 0, 3, 0, -7, 0, 0, 0, 8, 0, 0], [-7, 0, 0, -18, -18, -7, 0, 0, 6, 0, -1, 6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, -35, 0, -5, 0], [5, 0, -10, 3, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [-3, 0, -4, -4, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0], [0, -11, 0, 0, 0, 0, 0, 20, 0, 5, 0, 0, 0, -2, 0, 0], [-37, 0, 0, -2, -2, -37, 0, 0, -9, 0, -1, -9, 0, 0, 0, 0], [0, 19, 0, 0, 0, 0, 0, -10, 0, 28, 0, 0, 0, -7, 0, 0], [-23, 0, 0, -1, -1, -23, 0, 0, -9, 0, 4, -9, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, -3, -2, 0, -26, 0, 0, -32, -4, 2, -4], [21, 0, 0, 9, 9, 21, 0, 0, -4, 0, -9, -4, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 2, 20, 0, -16, 0, 0, 3, -1, -20, -1], [4, 0, 4, 0, 7, 9, 0, 0, 0, 0, -4, 13, 0, 0, 0, 0], [-3, 0, -3, 0, 3, -32, 0, 0, 0, 0, 3, 5, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 4, -3, 0, -43, 0, 0, -50, -4, 3, -4], [0, -15, 0, 0, 0, 0, 15, -1, 0, 17, 0, 0, 20, 4, 1, 4], [-18, 0, 0, 18, 18, -18, 0, 0, -7, 0, 12, -7, 0, 0, 0, 0], [0, 9, 0, 0, 0, 0, 0, -11, 0, -3, 0, 0, 0, 15, 0, 0], [36, 0, 4, 4, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 7, 0, 0, 0, 0, 0, 12, 0, 4, 0, 0, 0, 5, 0, 0], [-5, 0, 0, 11, 11, -5, 0, 0, 16, 0, 2, 16, 0, 0, 0, 0], [9, 0, 9, 0, 10, 3, 0, 0, 0, 0, -9, -8, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, -3, 0, 31, 0, 0, 0, -7, 0, 0], [0, -2, 0, 0, 0, 0, 2, 8, 0, 10, 0, 0, 12, -6, -8, -6], [0, 0, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, -6, 0, 22, 6], [0, -6, 0, 0, 0, 0, 0, 6, 0, -27, 0, 0, 0, 5, 0, 0], [0, 2, 0, 0, 0, 0, -2, -3, 0, -23, 0, 0, -17, 9, 3, 9], [-4, 0, -4, 0, 10, 22, 0, 0, 0, 0, 4, -3, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, -5, 14, 0, -19, 0, 0, -7, -2, -14, -2], [0, -8, 0, 0, 0, 0, 0, 4, 0, 7, 0, 0, 0, 0, 0, 0], [-22, 0, 0, 3, 3, -22, 0, 0, -11, 0, -5, -11, 0, 0, 0, 0], [11, 0, -4, 1, 0, 0, 0, 0, -19, 0, 0, 0, 0, 0, 0, 0], [-20, 0, -11, -12, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, -5, -5, 0, 64, 0, 0, 58, -1, 5, -1], [0, -11, 0, 0, 0, 0, 0, 16, 0, 3, 0, 0, 0, 6, 0, 0], [-35, 0, 0, -14, -14, -35, 0, 0, 9, 0, 5, 9, 0, 0, 0, 0], [0, -11, 0, 0, 0, 0, 0, 8, 0, 1, 0, 0, 0, -4, 0, 0], [0, -2, 0, 0, 0, 0, 0, -3, 0, 28, 0, 0, 0, -17, 0, 0], [-2, 0, 0, -9, -9, -2, 0, 0, -11, 0, -8, -11, 0, 0, 0, 0], [0, -7, 0, 0, 0, 0, 7, -6, 0, -33, 0, 0, -33, 6, 6, 6], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 41, 0, -19, 1], [0, 0, 0, 0, 0, 0, -14, 0, 0, 0, 0, 0, -17, 0, 3, 4], [8, 0, 8, 0, -7, 15, 0, 0, 0, 0, -8, -11, 0, 0, 0, 0], [5, 0, 0, 6, 6, 5, 0, 0, 12, 0, -6, 12, 0, 0, 0, 0], [-50, 0, 0, -3, -3, -50, 0, 0, 4, 0, 3, 4, 0, 0, 0, 0], [0, 14, 0, 0, 0, 0, 0, 3, 0, 15, 0, 0, 0, -2, 0, 0], [1, 0, 1, 0, 6, -2, 0, 0, 0, 0, -1, -2, 0, 0, 0, 0], [0, 22, 0, 0, 0, 0, -22, -11, 0, 3, 0, 0, -4, 4, 11, 4], [19, 0, -5, -13, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -15, 0, 0, 0, 0, 0, 7, 0, 10, -4], [-8, 0, 0, 26, 26, -8, 0, 0, 7, 0, -1, 7, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, -2, 0, -12, 0, 0, 0, 1, 0, 0], [0, -7, 0, 0, 0, 0, 7, 21, 0, -37, 0, 0, -28, -12, -21, -12], [-10, 0, -10, 0, -2, 16, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0], [-44, 0, -4, 1, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0], [-7, 0, -7, 0, -11, -32, 0, 0, 0, 0, 7, 4, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 4, 0, 15, 0, 0, 0, -8, 0, 0], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, -6, 0, -1, -1], [41, 0, -4, -5, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, 0, 7, 13, 0, -2, 0, 0, 18, 7, -13, 7], [1, 0, 1, 0, 9, -46, 0, 0, 0, 0, -1, 6, 0, 0, 0, 0], [8, 0, 0, -3, -3, 8, 0, 0, 15, 0, 13, 15, 0, 0, 0, 0], [0, -34, 0, 0, 0, 0, 34, 22, 0, -15, 0, 0, 8, 1, -22, 1], [-30, 0, 0, 16, 16, -30, 0, 0, -3, 0, 0, -3, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 8, -1, 0, -37, 0, 0, -27, 11, 1, 11], [0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 39, 0, -4, -8], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -39, 0, 9, -4], [0, 0, 0, 0, 14, -47, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1400_bh_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1400_2_bh_j();".
// 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_1400_2_bh_j(:prec:=16)
    chi := MakeCharacter_1400_bh();
    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_1400_2_bh_j();".
// 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_1400_2_bh_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1400_bh();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![256, 0, -1424, 0, 6737, 0, -6042, 0, 3947, 0, -1080, 0, 215, 0, -17, 0, 1]>,<11,R![17956, 6566, 7359, -1813, 1235, -98, 37, 0, 1]>],Snew);
    return Vf;
end function;