
// Make newform 1450.2.e.f in Magma, downloaded from the LMFDB on 27 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1450_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1450_e_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_1450_2_e_f();".
// 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_1450_2_e_f();".
// 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 := [16, 0, 105, 0, 186, 0, 99, 0, 18, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-56, 0, -211, 0, -155, 0, -33, 0, -2, 0], [134, 0, 237, 0, 155, 0, 33, 0, 2, 0], [184, 318, 450, 687, 175, 330, 23, 56, 1, 3], [0, -313, 0, -992, 0, -627, 0, -122, 0, -7], [-308, -75, -1297, -755, -924, -472, -188, -89, -11, -5], [440, 70, 1747, 1060, 1099, 769, 211, 155, 12, 9], [0, 879, 0, 2554, 0, 1571, 0, 300, 0, 17], [-440, 70, -1747, 1060, -1099, 769, -211, 155, -12, 9]];
    Rf_basisdens := [1, 1, 26, 26, 52, 52, 52, 52, 52, 52];
    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_1450_e();"
function MakeCharacter_1450_e()
    N := 1450;
    order := 4;
    char_gens := [1277, 901];
    v := [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_1450_e_Hecke();"
function MakeCharacter_1450_e_Hecke(Kf)
    N := 1450;
    order := 4;
    char_gens := [1277, 901];
    char_values := [[0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 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 := [[-1, 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, 0, 0, 0], [0, -1, -1, 0, 0, 0, 1, 0, 0, 0], [0, -1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, -1, -1, -1, -1, -1, -1, 0, 0], [-1, 0, 0, -1, 0, -1, -2, -2, 1, 0], [-1, 0, 0, 0, 0, 1, 0, 0, 0, 1], [1, 1, 1, 1, -1, 1, 0, 1, 0, -1], [0, -1, 1, 0, -1, 0, 0, 0, 0, 1], [0, -2, 0, 0, 0, 0, 0, -1, 2, -1], [1, 2, 2, 0, 0, 1, 0, 2, 0, 0], [1, 1, 0, 0, -1, 2, 1, 2, 1, 1], [-1, 0, 0, 0, 1, -1, -1, -2, 2, -1], [-1, 0, 0, -1, 1, 1, 0, 0, -1, 1], [1, 1, 0, 0, -1, -3, 1, 0, 0, -1], [-3, 1, -1, 0, 2, 3, 0, 0, 0, 3], [1, 1, -1, -2, -4, -1, 0, 0, -2, -2], [-1, -2, 0, 0, 1, 5, -1, -1, 2, 0], [0, 0, 3, -1, -1, -1, -1, -1, 0, 0], [-2, -3, -3, 0, 0, -2, 2, -1, 0, 0], [0, 1, -1, -1, 2, 0, 0, 0, -1, 0], [-4, 0, 0, -1, 0, -4, -2, 0, 1, 0], [-1, 1, 0, 0, 1, -3, -1, -1, 4, 0], [1, -3, 3, 0, -2, -1, 0, 0, 0, -1], [-2, 2, -2, -1, 0, 2, 0, 0, -1, 0], [-3, 1, 1, 2, 0, -3, 2, 2, -2, 0], [-2, 0, -4, -2, -2, -2, -2, -2, 0, 0], [5, 0, 3, 2, 1, 1, 1, 1, 0, 0], [-5, 0, 0, -2, -1, -1, -1, -1, 0, 0], [-3, 1, 1, -1, 0, -3, 0, -4, 1, 0], [9, 0, 3, 0, -1, -1, -1, 1, 0, -2], [4, -1, 0, 0, -4, 1, 4, 1, -1, -3], [3, 0, 2, -2, -3, -3, -3, -3, 0, 0], [0, 6, 0, 0, 0, -2, 0, 1, 2, 1], [-2, 6, 0, 0, 2, 6, -2, 0, -2, 2], [-6, 0, -6, -2, 0, 0, 0, 0, 0, 0], [-1, 1, -1, 0, 3, 1, 0, 0, 0, 1], [-2, -4, -4, 0, 0, -2, -2, 1, 0, 0], [-7, 0, -1, -2, -1, -1, -1, -4, 0, 3], [4, 0, 0, -4, 0, 0, 0, -1, 0, 1], [-4, 1, -1, 2, -3, 4, 0, 0, 2, -3], [1, -3, 0, 0, -1, 1, 1, -1, 2, -2], [6, 0, 0, 3, 0, 6, 1, 0, -3, 0], [-1, 2, 0, 0, 1, 3, -1, 1, 4, 2], [1, 1, 1, -1, 0, 1, 2, 2, 1, 0], [3, 2, -2, 2, 0, -3, 0, 0, 2, -3], [2, 1, 1, 1, 0, 2, -2, 2, -1, 0], [6, 0, 0, 3, -1, -6, 0, 0, 3, -1], [9, -5, 5, 1, 0, -9, 0, 0, 1, 0], [-3, -2, 0, 0, 3, 7, -3, -1, 2, 2], [-2, 5, 0, 0, 2, -2, -2, 1, 0, 3], [6, 2, -2, -2, -4, -6, 0, 0, -2, -2], [3, 4, 4, -1, 0, 3, -2, 4, 1, 0], [2, -4, 0, 0, -2, 2, 2, 1, -4, -1], [10, 0, 0, -1, 1, -10, 0, 0, -1, -3], [6, -4, -4, 2, 0, 6, 2, 0, -2, 0], [8, 4, 4, 0, 0, 8, 0, 2, 0, 0], [11, 0, 1, -3, -2, -2, -2, -2, 0, 0], [-11, -3, -3, 0, 0, -11, -2, -2, 0, 0], [0, 0, 0, 0, 0, -12, 0, 2, -2, 2], [8, 0, -2, 2, 4, 4, 4, 4, 0, 0], [-4, -3, 3, 2, 4, 4, 0, 0, 2, 1], [-10, -4, 4, 4, 4, 10, 0, 0, 4, -2], [-2, -12, 0, 0, 2, 2, -2, -3, 2, -1], [0, 5, 5, -2, 0, 0, -4, -2, 2, 0], [3, 1, 0, 0, -3, 4, 3, 7, -3, 4], [-3, -3, 3, 2, 2, 3, 0, 0, 2, -2], [1, -2, 0, 0, -1, -3, 1, 0, 2, -1], [9, 7, 7, 1, 0, 9, 0, 6, -1, 0], [8, -3, -3, 4, 0, 8, 2, 3, -4, 0], [-2, 0, -2, 2, 4, 4, 4, 4, 0, 0], [3, -4, 4, 1, -5, -3, 0, 0, 1, -3], [-15, -3, -3, 1, 0, -15, 0, 0, -1, 0], [4, 1, 1, 4, 0, 4, 1, 4, -4, 0], [0, -4, 4, 3, 6, 0, 0, 0, 3, 4], [-6, 0, 0, -3, 0, -6, -5, -4, 3, 0], [-2, 0, -3, 6, 4, 4, 4, 3, 0, 1], [-11, -3, -3, 0, 0, -11, -4, -6, 0, 0], [-17, 0, -13, -3, 2, 2, 2, -1, 0, 3], [4, 10, 10, 3, 0, 4, 4, 10, -3, 0], [-9, 0, -4, 0, -1, -1, -1, 1, 0, -2], [10, 0, 2, -4, -4, -4, -4, 0, 0, -4], [10, 0, 6, 0, 2, 2, 2, 6, 0, -4], [2, 0, 3, -4, 0, 0, 0, 0, 0, 0], [11, -1, -1, 2, 0, 11, 2, 6, -2, 0], [4, -3, 3, -4, -6, -4, 0, 0, -4, 0], [-1, 0, 0, 5, 0, -1, 7, 10, -5, 0], [-14, -2, -2, -3, 0, -14, -6, -6, 3, 0], [-18, 0, -3, -1, -3, -3, -3, -4, 0, 1], [-4, -1, 1, -6, -3, 4, 0, 0, -6, -1], [11, 0, 0, 0, 0, 11, -4, -3, 0, 0], [-4, 3, 3, -4, 0, -4, -2, -8, 4, 0], [1, 0, 7, 2, -1, -1, -1, -4, 0, 3], [17, 0, 6, 0, 3, 3, 3, 3, 0, 0], [-4, -6, 0, 0, 4, 6, -4, -5, 2, -1], [-4, -15, 0, 0, 4, -9, -4, -2, 3, 2], [20, 0, 0, 1, -2, -20, 0, 0, 1, -4], [-10, 0, 0, 3, 7, 10, 0, 0, 3, 1], [-3, -3, -3, -4, 0, -3, 4, -6, 4, 0], [0, 6, -6, 2, 6, 0, 0, 0, 2, 5], [-12, 0, 1, 0, 4, 4, 4, 0, 0, 4], [7, -2, 2, 0, -2, -7, 0, 0, 0, 0], [9, 0, -7, 2, 7, 7, 7, 4, 0, 3], [-3, -1, 0, 0, 3, -24, -3, -5, 3, -2], [5, -4, -4, -4, 0, 5, -6, -4, 4, 0], [8, 7, 7, 4, 0, 8, 0, -2, -4, 0], [15, 7, 7, 2, 0, 15, 7, 8, -2, 0], [7, -7, 7, -4, -6, -7, 0, 0, -4, 2], [2, -6, 0, 0, -2, 10, 2, 2, -2, 0], [18, 2, 2, 3, 0, 18, 7, 4, -3, 0], [1, 5, 0, 0, -1, -12, 1, -3, -1, -4], [-11, 6, -6, 1, 4, 11, 0, 0, 1, 4], [4, -16, 0, 0, -4, -16, 4, -2, 4, -6], [-16, -1, 1, 3, 2, 16, 0, 0, 3, -4], [6, -1, -1, -3, 0, 6, -4, -8, 3, 0], [2, 0, 0, 5, -4, -2, 0, 0, 5, -6], [3, 0, -4, -2, 7, 7, 7, 1, 0, 6], [-4, 0, 0, -6, 0, -4, -4, -2, 6, 0], [24, 0, -4, 4, 0, 0, 0, 1, 0, -1], [8, 3, 3, -2, 0, 8, -8, -10, 2, 0], [7, -12, 0, 0, -7, -21, 7, -1, 2, -8], [-11, 3, -3, -2, 0, 11, 0, 0, -2, 6], [-15, 0, -7, 1, -4, -4, -4, -5, 0, 1], [-1, -4, 0, 0, 1, 1, -1, 2, 0, 3], [-8, 0, 6, 0, -6, -6, -6, 0, 0, -6], [-3, 10, 0, 0, 3, 9, -3, 2, 0, 5], [-6, 0, 0, 0, -6, -6, -6, -8, 0, 2], [-10, 0, -6, 6, 2, 2, 2, 5, 0, -3], [4, 5, 5, 2, 0, 4, -4, -6, -2, 0], [-10, 0, 8, -4, 0, 0, 0, 4, 0, -4], [-10, -8, -8, -4, 0, -10, -6, -4, 4, 0], [-24, 0, -6, 0, 2, 2, 2, 2, 0, 0], [-3, 0, 9, -3, 2, 2, 2, 6, 0, -4], [-8, -1, 1, -1, -2, 8, 0, 0, -1, 2], [3, 4, 0, 0, -3, 5, 3, 0, -8, -3], [9, 0, 0, 4, 10, -9, 0, 0, 4, 4], [-17, 0, -10, -6, -5, -5, -5, -5, 0, 0], [9, -6, 6, 6, 6, -9, 0, 0, 6, 0], [0, -1, 0, 0, 0, -5, 0, 3, -7, 3], [1, -2, 0, 0, -1, -9, 1, -2, 6, -3], [3, -14, 0, 0, -3, -31, 3, -2, 2, -5], [-5, 7, 0, 0, 5, 8, -5, -6, 9, -1], [-1, -2, -2, 3, 0, -1, 9, 6, -3, 0], [-8, 4, -4, -2, 8, 8, 0, 0, -2, 8], [-20, 0, 2, -4, 2, 2, 2, -1, 0, 3], [9, -3, -3, 5, 0, 9, 2, -2, -5, 0], [6, -2, 2, 2, 2, -6, 0, 0, 2, -10], [-13, 2, 2, 0, 0, -13, 2, 5, 0, 0], [4, 4, 4, 2, 0, 4, 8, 13, -2, 0], [13, -11, 11, 2, 0, -13, 0, 0, 2, 0], [2, -7, -7, 3, 0, 2, 0, 2, -3, 0], [-5, 0, 2, -4, -7, -7, -7, -5, 0, -2], [-3, 3, 0, 0, 3, 14, -3, 0, 5, 3], [-10, -1, -1, 0, 0, -10, -4, -11, 0, 0], [-3, -6, 0, 0, 3, 3, -3, 2, -2, 5], [4, 5, 0, 0, -4, -5, 4, 6, -1, 2], [-13, 2, -2, 3, 4, 13, 0, 0, 3, 10], [8, -6, 0, 0, -8, 4, 8, 2, -12, -6], [-2, 0, -17, -6, -2, -2, -2, -4, 0, 2], [-3, -3, 3, 1, 8, 3, 0, 0, 1, 6], [-8, 8, 0, 0, 8, -2, -8, -1, 6, 7], [-10, 3, -3, -4, 6, 10, 0, 0, -4, 2], [-7, -8, -8, -1, 0, -7, 0, 4, 1, 0], [32, 0, -8, 4, 2, 2, 2, -1, 0, 3], [-3, -8, 0, 0, 3, -7, -3, -7, 0, -4], [-28, 0, 0, -2, -4, -4, -4, -7, 0, 3], [-6, 1, 0, 0, 6, -15, -6, -8, 3, -2], [1, 0, -2, 6, -3, -3, -3, 5, 0, -8], [-10, 0, -14, -2, 0, 0, 0, -8, 0, 8], [-5, 0, 0, 0, 5, -15, -5, 1, -2, 6], [6, 0, -8, 2, 8, 8, 8, 6, 0, 2], [-21, 0, -17, -6, 3, 3, 3, -3, 0, 6], [-5, 4, 0, 0, 5, -1, -5, 3, -4, 8], [-27, 0, 3, -1, 0, 0, 0, 0, 0, 0], [-1, 0, 5, 5, 6, 6, 6, 5, 0, 1], [-29, 9, -9, 0, 10, 29, 0, 0, 0, 9], [3, 0, 0, 0, -3, -7, 3, 3, -10, 0], [-1, -4, 0, 0, 1, -23, -1, -8, 0, -7], [2, 0, 0, -4, 0, 0, 0, 3, 0, -3], [9, -2, -2, -3, 0, 9, 6, 4, 3, 0], [11, 2, -2, 5, 7, -11, 0, 0, 5, -1], [-1, -12, -12, -7, 0, -1, -4, -14, 7, 0], [15, -3, 3, 0, -9, -15, 0, 0, 0, 1], [10, 2, 0, 0, -10, 8, 10, 6, -8, -4], [5, 2, 0, 0, -5, 1, 5, 6, -2, 1], [-6, 9, 0, 0, 6, 12, -6, -6, 4, 0], [-4, -8, 8, 5, 8, 4, 0, 0, 5, -10], [31, 0, 8, 4, -1, -1, -1, 4, 0, -5], [-19, -3, -3, -5, 0, -19, -12, -10, 5, 0], [-5, -15, 0, 0, 5, -32, -5, -6, 9, -1], [13, 1, -1, -5, -2, -13, 0, 0, -5, 6], [-16, 4, -4, -1, 3, 16, 0, 0, -1, -1], [6, -13, 0, 0, -6, -8, 6, 12, -10, 6], [31, 6, 6, -1, 0, 31, -2, -2, 1, 0], [9, 4, 4, 6, 0, 9, 14, 12, -6, 0], [-14, 8, -8, -6, 0, 14, 0, 0, -6, 3], [-13, -3, 3, -3, 2, 13, 0, 0, -3, 8], [3, 0, 0, 2, 0, 3, -4, 5, -2, 0], [-3, 7, 7, -8, 0, -3, -8, -5, 8, 0], [-8, 8, 0, 0, 8, 6, -8, -1, 2, 7], [11, 0, 12, 2, 3, 3, 3, 6, 0, -3], [-21, 0, 0, -6, 0, -21, -6, -6, 6, 0], [9, -9, 9, -5, -14, -9, 0, 0, -5, -12], [-8, 2, 2, 3, 0, -8, 3, 10, -3, 0], [16, 5, 5, 6, 0, 16, -4, 5, -6, 0], [-12, -5, 5, -1, 4, 12, 0, 0, -1, -6], [-2, 0, -3, 0, 0, 0, 0, 5, 0, -5], [-9, 7, 7, 1, 0, -9, -14, -6, -1, 0], [24, 0, 0, 2, -4, -4, -4, 6, 0, -10], [-6, 0, 2, -8, -4, -4, -4, -8, 0, 4], [11, 0, -20, -2, 1, 1, 1, -7, 0, 8], [34, 0, -2, 0, -2, -2, -2, 2, 0, -4], [16, -5, -5, -2, 0, 16, 6, -3, 2, 0], [0, 0, 7, 6, 0, 0, 0, 3, 0, -3], [-13, -2, 2, -6, 8, 13, 0, 0, -6, 11], [-8, -6, -6, -7, 0, -8, -6, -8, 7, 0], [14, -6, 6, 5, 12, -14, 0, 0, 5, 0], [37, 0, 16, 4, -5, -5, -5, 8, 0, -13], [-11, -14, -14, -7, 0, -11, 3, -10, 7, 0], [1, 8, 0, 0, -1, 13, 1, 8, -8, 7], [0, -11, -11, -5, 0, 0, 4, -10, 5, 0], [-4, 12, 0, 0, 4, 36, -4, 0, 0, 4], [14, 6, -6, -1, 0, -14, 0, 0, -1, 0], [1, -1, 1, 4, 4, -1, 0, 0, 4, 7], [-22, 0, 6, 0, -12, -12, -12, -4, 0, -8], [2, -11, 11, 6, 2, -2, 0, 0, 6, 1], [6, -12, 0, 0, -6, -26, 6, 3, 6, -3], [-15, 0, -27, -7, 0, 0, 0, -13, 0, 13], [1, 22, 0, 0, -1, -17, 1, 6, 0, 5], [1, 0, -3, 0, -3, -3, -3, 2, 0, -5], [14, -16, 16, 0, -6, -14, 0, 0, 0, -18], [-6, 8, -8, -1, -6, 6, 0, 0, -1, 12], [12, -3, -3, 5, 0, 12, 0, -2, -5, 0], [-6, 8, 0, 0, 6, 10, -6, 3, -4, 9], [6, 5, -5, -1, -4, -6, 0, 0, -1, 6], [-9, 8, 0, 0, 9, -11, -9, -5, 4, 4], [-2, -11, 0, 0, 2, 11, -2, -9, 11, -7], [10, -6, 6, -6, -10, -10, 0, 0, -6, -4], [2, 0, -9, -1, 1, 1, 1, 2, 0, -1], [37, 0, 25, 6, -5, -5, -5, -2, 0, -3], [20, -5, -5, -4, 0, 20, -1, -6, 4, 0], [21, -9, 9, 6, -6, -21, 0, 0, 6, -13], [-17, 4, -4, -5, -4, 17, 0, 0, -5, 0], [14, 0, -12, -6, -4, -4, -4, -8, 0, 4], [-36, 3, 3, -2, 0, -36, 1, -4, 2, 0], [2, -3, 0, 0, -2, -19, 2, 1, -3, -1], [3, 0, 5, -9, -8, -8, -8, -5, 0, -3], [1, 6, 0, 0, -1, -25, 1, -9, 6, -10], [-3, 0, -6, -4, 3, 3, 3, 2, 0, 1], [4, -4, 0, 0, -4, -32, 4, -2, 0, -6], [-1, 24, 0, 0, 1, 21, -1, 10, -6, 11], [8, -3, -3, -5, 0, 8, 8, 0, 5, 0], [-16, 0, -2, 0, -2, -2, -2, -5, 0, 3], [-2, -17, 0, 0, 2, 5, -2, -11, 13, -9], [-5, 3, 3, 4, 0, -5, 4, -5, -4, 0], [-34, 0, -7, 4, 2, 2, 2, -4, 0, 6], [26, -6, 6, -4, -12, -26, 0, 0, -4, 1], [37, -10, 10, -1, -10, -37, 0, 0, -1, -4], [38, 0, 10, -2, 0, 0, 0, 0, 0, 0], [-18, 0, -7, 4, 2, 2, 2, 4, 0, -2], [6, -2, 0, 0, -6, 2, 6, 3, 6, -3], [0, -2, 0, 0, 0, -22, 0, 4, 12, 4], [0, 6, 0, 0, 0, -2, 0, -2, 0, -2], [2, -8, -8, 0, 0, 2, 2, -2, 0, 0], [14, 6, -6, -11, -15, -14, 0, 0, -11, -5], [-16, 4, 4, -7, 0, -16, -14, -14, 7, 0], [-20, 0, 11, 16, 6, 6, 6, 12, 0, -6], [-12, -6, -6, 2, 0, -12, 12, -5, -2, 0], [7, 0, 22, -2, -9, -9, -9, -1, 0, -8], [-17, -9, -9, 2, 0, -17, 4, 0, -2, 0], [10, 5, 5, -4, 0, 10, 8, 8, 4, 0], [5, 7, 7, -2, 0, 5, -11, 6, 2, 0], [-1, 9, 0, 0, 1, 15, -1, -3, -6, -2], [-5, -12, 0, 0, 5, -25, -5, -9, 0, -4], [3, 11, 0, 0, -3, 26, 3, 10, -7, 7], [-1, -4, 0, 0, 1, 7, -1, 7, 0, 8], [-24, 7, -7, 3, 8, 24, 0, 0, 3, 6], [8, 8, 0, 0, -8, 38, 8, 4, 2, -4], [-19, 6, -6, 2, 6, 19, 0, 0, 2, 7], [-6, 14, 0, 0, 6, 34, -6, 5, 0, 11], [-11, -18, -18, -4, 0, -11, 0, -13, 4, 0], [6, -18, 0, 0, -6, 4, 6, 0, 4, -6], [-20, 0, -9, -10, 0, 0, 0, 2, 0, -2], [3, -5, -5, 4, 0, 3, -1, 10, -4, 0], [-11, -1, 0, 0, 11, 9, -11, -3, 10, 8], [-18, 0, -6, -8, -12, -12, -12, -15, 0, 3], [-4, -12, 0, 0, 4, -14, -4, -6, -2, -2], [6, 0, -2, 4, 10, 10, 10, 4, 0, 6], [3, 0, 0, 10, 3, 3, 3, 0, 0, 3], [-18, 5, -5, -11, 0, 18, 0, 0, -11, 6], [-14, -6, -6, -10, 0, -14, -2, -8, 10, 0], [-25, 5, -5, -9, 4, 25, 0, 0, -9, 6], [-9, 16, 0, 0, 9, 5, -9, 0, 8, 9], [-23, 0, -18, 6, 9, 9, 9, 6, 0, 3], [6, -10, -10, 2, 0, 6, 4, 6, -2, 0], [-1, -14, 14, 7, 15, 1, 0, 0, 7, 5], [-5, -19, 0, 0, 5, -5, -5, -8, 16, -3], [7, 9, 0, 0, -7, 2, 7, 0, -1, -7], [5, -9, 0, 0, -5, 8, 5, 3, -13, -2], [4, -2, -2, -13, 0, 4, -11, -12, 13, 0], [0, -1, 1, -12, -1, 0, 0, 0, -12, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1450_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1450_2_e_f();".
// 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_1450_2_e_f(:prec:=10)
    chi := MakeCharacter_1450_e();
    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_1450_2_e_f();".
// 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_1450_2_e_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1450_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 0, 105, 0, 186, 0, 99, 0, 18, 0, 1]>,<7,R![4050, 14130, 24649, 4448, 278, -126, 311, 36, 2, -2, 1]>,<11,R![512, 640, 400, -352, 672, 448, 156, -16, 8, 4, 1]>],Snew);
    return Vf;
end function;