
// Make newform 2001.2.a.i in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2001_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_2001_2_a_i();".
// 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_2001_2_a_i();".
// 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 := [8, 1, -26, 4, 18, -5, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-2, -1, 1, 0, 0, 0, 0], [1, -3, -1, 1, 0, 0, 0], [4, 3, -5, -1, 1, 0, 0], [-2, 7, 4, -6, -1, 1, 0], [-6, -5, 13, 5, -7, -1, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 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_2001_a();"
function MakeCharacter_2001_a()
    N := 2001;
    order := 1;
    char_gens := [668, 1132, 553];
    v := [1, 1, 1];
    // 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_2001_a_Hecke(Kf)
    return MakeCharacter_2001_a();
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, -1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0], [-1, 0, -1, -1, 0, -1, 1], [-1, 1, 0, 1, 1, 0, 0], [-3, 1, 0, 0, 0, 0, -1], [-1, 0, 1, 0, 1, 1, 0], [0, -1, 0, 1, 1, 1, -1], [1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0], [-3, -1, 1, 1, 1, 0, -1], [-5, 0, 3, 0, 1, -1, -1], [-1, 1, 0, 1, -2, 0, -1], [0, 0, 0, 1, 1, 2, 2], [-4, 2, 1, 2, 1, 2, 1], [-2, 1, 1, -2, -1, -1, -3], [0, -2, 0, -1, 4, -2, 3], [-3, -1, 0, 1, -2, 0, 0], [3, 0, -3, -1, -3, 1, 3], [-1, -1, -2, -4, -2, -2, 1], [3, 0, 2, 1, -2, 1, 1], [-1, -1, -1, 0, -1, 0, -2], [3, 1, 1, -1, -2, 2, -5], [3, -2, -2, 0, 0, 3, 1], [3, -2, 2, 1, 0, 3, 0], [-5, 4, -1, -5, -1, -1, 3], [-5, 2, 1, -1, 1, -5, 1], [1, -3, 1, 3, 4, -2, 2], [-2, -1, -1, -6, -4, -1, 1], [-3, 4, 4, 1, 0, 1, -5], [3, 0, 0, 5, 2, -1, 1], [3, 1, -5, -3, -3, -4, -3], [-1, -2, -7, -4, 2, -3, 2], [-3, 1, 6, 3, 0, 2, 0], [-1, 6, 6, 3, -1, 3, -4], [-5, 1, -1, 2, 2, 2, -4], [-9, 2, -4, -2, -3, 1, -1], [-6, 2, -3, 0, 4, 0, 6], [5, -5, -5, 2, 2, 0, 1], [6, 0, -1, 1, -3, 4, 4], [-4, -5, -7, -5, -1, -3, 4], [2, -6, -4, 2, -1, -2, 1], [-5, -2, -5, -2, 1, -3, 6], [-2, -1, 2, 4, -1, 1, -5], [3, -2, -4, -4, 0, -1, -4], [-6, 3, 0, -4, -1, -1, -3], [-4, -1, 0, -4, -1, -1, -2], [2, -4, -7, -1, 7, -4, 3], [0, -3, 2, -2, 0, -7, -1], [-10, -5, -1, -1, 5, -3, 3], [1, -3, 3, 6, -1, -2, -5], [-8, 4, -1, -5, 0, 0, 7], [7, -1, -4, -3, -5, -4, -1], [2, -3, -1, 1, 3, 1, 3], [-4, -2, 1, 5, -1, -2, -1], [4, -4, -5, 0, -5, 6, 3], [1, -7, -4, -3, -4, -2, 3], [-4, -4, -1, -4, 2, -6, 3], [2, 0, 0, 1, -5, 4, -2], [5, -3, 6, 1, -3, 0, 1], [-11, -2, -2, -1, 4, -7, 6], [10, -3, 4, 5, 3, 1, -2], [-6, 0, -1, -4, 0, -8, 3], [-12, 4, -1, -2, 2, -4, 2], [-4, 8, -1, -4, 0, -6, 4], [7, -8, -3, 3, -3, 1, -2], [0, 5, 10, 4, -8, 7, -6], [1, 2, 1, -8, -8, -1, -4], [2, -5, 1, 3, 3, -1, 10], [-2, 4, 5, -1, -5, 4, 6], [-7, 0, -7, -7, 0, -11, 6], [-1, -2, 0, 0, 12, -5, 8], [-4, -8, -9, 2, 2, 2, 0], [-1, 7, 2, -1, -7, -2, 2], [8, 4, -5, 3, -4, 2, -5], [14, 2, 6, 4, 1, 2, -6], [7, 7, 1, 0, -2, 4, 0], [6, -4, -1, 6, -4, 4, -10], [5, -2, 9, 5, 2, 3, -2], [-7, 4, 3, -1, 5, 3, 6], [1, -1, -2, 3, 1, -2, 0], [4, 3, 6, 3, -1, 9, -2], [-4, -3, -5, 4, 0, 1, -5], [-10, -1, -3, 2, -6, 1, -5], [2, 10, 7, -3, 1, 4, 0], [-1, -9, 6, 5, 5, -2, -1], [-12, 11, -2, -4, 0, -3, -1], [17, -8, -5, -1, -1, 3, 0], [-2, 1, -5, -8, 5, 3, 7], [-8, 17, 7, 3, -5, 5, -3], [10, -5, -2, -5, -3, 5, -2], [-10, 14, 0, 0, 1, -2, 5], [-11, 5, 4, 3, -4, -2, -6], [-11, -8, 4, 3, 10, -5, 8], [-4, 11, -1, -5, -2, 3, -2], [-2, -1, 7, 7, 1, 1, -8], [8, 3, -3, -5, 4, -7, 2], [-5, 5, 10, 9, -2, 2, -6], [-2, 3, -9, -4, -9, -5, -1], [-11, -9, -3, 5, 3, 0, 0], [-3, 5, 5, -6, 3, -12, 3], [16, -5, 0, 6, 12, 1, 1], [1, 7, 1, -4, 4, -2, 7], [11, 0, 6, 11, 3, -3, -7], [-5, -7, 1, 1, 9, 2, 0], [-1, 5, 1, -9, -4, -4, -5], [11, 7, -7, -6, -18, 2, -9], [-4, -2, 15, 3, -2, 4, 1], [-10, -7, -13, -4, 6, -7, 2], [0, 4, 4, 6, -8, 0, -11], [-4, 0, -9, -2, 3, -2, 10], [-17, 13, 2, 3, 1, 4, -2], [-2, 6, 12, 8, 3, 10, -6], [-3, 15, 1, -6, 2, 0, -4], [-2, 6, 9, 14, 1, 10, -8], [0, -8, -5, 3, 2, -2, -6], [7, -8, 5, 7, 2, 3, -9], [10, 7, 3, 3, -6, 13, -8], [-12, 12, -3, -2, -7, 2, -5], [3, 11, 0, -8, -6, 4, -5], [4, 0, -7, -12, 7, -8, 12], [-16, 8, 8, 3, 11, 0, -1], [18, -5, -14, -2, 3, 5, 10], [18, -8, 4, 6, 1, 8, 4], [9, -8, 15, 9, -1, 3, -3], [-2, 0, 0, 4, -6, -4, 0], [-15, 0, 4, 3, 1, -1, 4], [-6, -7, 0, -3, 5, -1, -1], [3, -1, 4, 1, 7, -2, 6], [12, 1, 3, 0, -1, 11, -1], [-1, -6, -4, 3, 10, 1, 3], [-3, 7, 16, 12, -7, 6, -12], [8, -19, -4, 9, -4, -3, 3], [15, -12, -13, 8, 1, 1, 4], [8, -7, 13, 5, -1, 7, 3], [11, 11, -3, -4, 4, -2, 1], [4, 0, 3, -2, 7, 0, -3], [9, -2, 12, 15, 8, 1, -2], [-5, 7, 12, 1, 5, -2, 3], [2, -8, -4, 0, 0, 4, -4], [-9, -4, -2, 1, 9, 1, -1], [-16, 15, 5, 1, -4, -5, 3], [2, -5, -3, -4, -8, 1, -3], [-3, 3, 6, 0, 1, -10, 0], [5, 13, -1, -3, -9, 10, -4], [-10, 0, -1, 8, 3, 2, -2], [-5, 14, 13, 7, 7, 3, -8], [13, 6, 8, 1, 1, 3, -13], [-22, -3, -10, -6, 0, -7, 8], [-20, 10, 15, 4, 5, 2, -3], [-11, 9, 4, -16, -6, -8, 5], [1, 2, -4, 0, -4, 3, -6], [-7, 6, -5, -4, 8, 5, 3], [-8, -13, -6, 4, 7, 1, -3], [1, -8, 7, 6, 8, -9, 4], [-3, 7, 7, 8, -10, 6, -7], [-2, 2, 4, 12, 1, 18, 4], [-4, 9, 16, 1, 1, 3, -5], [13, 8, -4, -9, -4, 5, -2], [-4, 14, -4, -10, -1, -4, -9], [-3, 3, 2, 1, -10, 6, 1], [-7, 3, -4, -13, -5, -6, -1], [1, 3, -10, -5, -3, 4, -9], [-1, 1, 16, 9, 3, 12, -2], [9, -4, -3, 5, -7, 9, -3], [-21, 13, 12, -4, 4, -2, 7], [3, 5, 2, 11, 8, 6, 1], [14, -18, -12, -6, 5, 0, 0], [17, -5, 24, 13, -1, 0, -4], [-12, 1, -7, -8, 7, -9, 12], [-5, -2, 3, -7, -10, -7, -1], [-29, 17, 2, -2, -5, -8, -3], [4, -7, 11, 6, -3, 1, -4], [-15, 13, -9, -2, 9, -4, 9], [-37, 0, 5, -1, 1, 5, -5], [25, -5, -10, -1, 1, 8, 14], [6, -9, -16, 1, -4, -7, 8], [-5, -15, 3, 1, 12, -4, -3], [-6, -10, 5, 0, 5, 4, 2], [-2, 10, -5, 4, 11, 4, 0], [-12, 2, -7, -18, -19, 0, 4], [5, 1, -4, -14, -15, -10, 5], [10, -10, -16, -14, -3, -6, 9], [-13, 12, -9, -6, 3, -7, -1], [-12, 3, 8, 2, -3, 11, -10], [2, 7, 9, -2, -8, -3, -2], [-7, -1, 8, 1, 5, -2, -7], [-14, 4, 10, -2, 12, 4, 6], [-10, 4, 9, -6, -1, -8, 3], [-2, 6, -3, -2, -18, 2, -7], [5, 0, -14, -6, 4, -9, 14], [-8, 2, -8, -6, -13, -2, 9], [10, -14, -8, -6, 7, -4, -2], [0, 6, 10, 0, 2, 0, -8], [11, -19, -16, 6, 6, 4, -4], [1, 8, 8, -1, 1, -7, 11], [11, -2, -9, -9, -4, -9, 4], [-6, -1, -9, -2, 9, -9, -4], [-3, 12, 9, -13, -14, -7, 0], [-16, -15, 2, 1, -4, 3, -3], [3, -10, 3, 10, 0, 3, -9], [9, 7, 13, 22, -3, 18, -19], [5, -14, 1, 9, -2, -5, -4], [16, -21, -2, 7, 14, -1, 0], [15, -5, -9, 4, -5, 2, 3], [-17, -4, 6, -2, 13, 1, 9], [1, -9, 6, 8, 7, 2, 6], [16, -10, -9, 9, -6, 6, 1], [-25, 12, 8, 0, 3, 1, -7], [-11, 0, 7, 11, 14, 17, -3], [-29, 30, 20, 7, -10, 3, -7], [3, -7, 1, -6, -8, 10, -11], [-19, 2, -2, -6, 9, 1, 9], [7, 12, -3, -4, 0, -7, 6], [-21, 14, 17, 5, 10, 9, -23], [32, -14, -20, -10, 3, -6, 2], [33, 0, -4, 4, -6, 9, -6], [-12, -10, 2, -8, 13, -10, 16], [-23, 12, -4, -13, -8, -3, 7], [11, 14, 7, -5, 7, -1, -2], [-11, -5, 15, 9, 5, 8, -2], [-23, 4, -4, -3, 10, -1, -2], [8, -4, -14, -4, 9, -4, 8], [29, -11, 9, 6, 2, -2, 0], [-4, 11, -8, -1, 6, -9, -2], [-8, 17, -3, -10, -15, -11, -5], [25, 0, 10, 0, -5, 15, -14], [-29, -10, -14, -3, -1, -7, 4], [8, -17, -9, -9, -5, -13, 16], [2, 4, 9, 8, -2, 2, -12], [-15, 5, 9, 7, 10, 8, -14], [14, -3, -13, -2, 12, -11, 12], [-4, 5, -3, -11, -17, -1, 11], [2, 0, -11, -18, 1, -6, -4], [-4, 8, -13, -6, -1, -14, -12], [24, -12, -16, 8, 4, 2, 14], [13, 21, 2, 4, -7, 4, -5], [10, -8, -13, 5, -6, 12, 12], [-42, 12, 4, -1, -1, -6, -2], [-11, -2, 10, -11, -6, -3, 8], [-12, -5, -19, -7, 13, -17, 24], [11, -11, 17, 14, -7, 2, -7], [18, 1, 9, 11, -9, -1, 5], [-7, -2, -14, -5, -12, -7, 5], [5, -5, -6, -5, 8, -8, 3], [-27, 17, 11, -6, 4, 2, 7], [-25, -1, -3, 9, 6, -4, -3], [5, -17, 3, 3, 13, -2, 12], [10, -12, 0, 10, -4, 2, -14], [-10, -3, 11, 9, 7, 5, -7], [17, -4, -5, 3, 0, 13, 5], [18, 1, -7, -4, -17, -17, 1], [1, 0, 10, -2, -7, -11, -8], [-2, -9, -3, 4, -6, 7, 2], [24, 6, 5, 2, 4, 16, 2], [11, -5, -14, -1, -12, 8, 11], [-20, 7, 16, 4, 7, 11, -6], [20, 2, -4, -10, -19, -12, 3], [25, -16, 4, 10, -4, -3, -9], [-7, 12, 13, 12, -1, -5, -13], [-14, 0, -3, -14, -5, -8, 13], [8, 9, 0, 1, -10, 9, -9], [1, -20, 6, 8, 0, 9, -3], [-9, 7, 12, 15, 15, 16, -12], [27, -10, -5, 9, -3, 17, -9], [-23, 1, -1, -8, 12, -6, 10], [11, -24, -12, -6, 0, -5, 4], [13, 11, 6, 10, -8, 4, -15], [-17, 14, 12, 3, 15, -1, -2], [10, 21, -2, -2, 3, -7, 2], [-10, -12, 0, 3, 18, -4, 1], [31, -8, 0, -3, -10, -3, 10], [-8, 4, -6, -14, -7, -6, 7], [-24, 13, 1, -17, -10, -7, 1], [9, -1, -15, -7, 14, -12, 23], [12, -25, -20, 8, 0, 3, -3], [12, 4, 9, 8, -11, 12, -21], [13, -8, -12, -15, 7, -11, 21], [6, -20, 5, 10, 10, -6, 2], [-11, 1, 6, 13, 1, 6, 3], [-8, 8, -7, -14, 1, -18, 8], [-6, -7, 5, -10, -18, -1, 1], [3, -5, -4, 3, -8, -8, 3], [35, -16, -11, -9, -13, -11, 10], [15, -22, -2, 9, 3, 5, -18], [-18, 1, 7, -12, 15, -9, 14], [31, -3, -12, 2, -9, -2, -1], [-1, 15, 4, 4, 12, -16, 4], [4, 6, 2, -12, -16, -2, -4], [25, -3, -5, 6, -3, 10, 13], [31, -28, -1, 11, 6, 17, -3], [14, 18, -5, 4, 4, -2, -7], [-8, -8, 0, 7, 9, 14, -4], [15, -2, 23, 18, 3, 29, -18], [6, -13, 8, 6, -13, 17, -11], [39, 3, -13, -1, 7, -4, 9], [5, 9, 17, 12, 6, 20, -9], [-9, -1, 26, 8, 0, 4, -3], [-7, 3, 16, 19, 7, 14, -8], [3, 3, -1, -13, -4, 12, 5], [-17, -8, -5, -16, 3, -1, 9], [0, 20, 8, -14, -1, 6, 5], [-18, -10, -10, -12, 9, -22, 5], [-7, -13, -3, 21, 7, 4, -7], [4, -27, 4, 4, 2, -9, -3], [24, -15, -4, 7, 0, 11, 0], [-29, 8, -14, -13, -5, -5, -4], [-44, 14, -2, -10, -7, -2, 2], [10, 0, -20, -12, -20, 12, -9], [-3, 6, 6, 4, -2, 13, -12], [8, -6, -10, 16, 16, -2, 4], [1, 2, -8, 3, -5, -9, 8], [-10, 18, -3, -8, -15, 8, 12], [-9, -25, -1, 0, 16, -4, 8], [-7, -19, 4, 3, 5, -4, -6], [-8, 4, -7, -24, 4, -4, 1], [29, -6, 0, 21, 4, 17, -8], [-25, -8, -2, 5, 1, 5, -7], [12, 9, -3, -14, -8, 5, -18], [-15, -6, -30, -11, 6, -15, 21], [25, -26, -15, 3, -3, -3, 8], [-20, -11, -5, 4, 12, -3, 2], [4, -13, -2, -9, 0, -21, 11], [11, 2, -7, 1, 13, -1, 5], [26, -19, -14, -10, -7, -9, -7], [21, -19, -5, -5, -9, -12, 8], [-25, 22, 6, -1, 1, -7, 2], [-20, -4, -1, 7, 0, 6, 12], [-29, 25, 15, -7, -4, -4, 13], [6, -9, -31, -14, -11, -13, 6], [31, -16, 4, 17, 7, 17, -8], [-7, 10, 3, -2, 4, -11, -2], [8, -21, -29, -11, 0, -17, 16], [30, 7, 12, 10, 2, -7, -3], [3, 18, 4, 0, -28, 5, -7], [-23, -13, 0, -13, 1, -6, 8], [15, -16, -16, -1, 12, -1, 0], [4, 15, -4, -16, -2, -7, 18], [13, 5, 1, 9, 15, 0, -11], [-17, 19, -3, -5, -18, -2, 10], [31, -3, 6, 7, 0, 18, 2], [31, 9, 12, 17, 0, 24, -13], [35, -15, 8, 9, -4, -2, 6], [16, 0, -9, -6, 0, -4, 18], [14, -7, 1, 8, 10, -9, -10], [-10, 7, 1, -22, -4, -25, 8], [-14, 13, 12, 18, 1, 9, -10], [-19, -10, 10, 3, 5, 7, 8], [-49, 8, 6, -1, 12, -1, 6], [14, 7, 5, -7, -9, -11, -5], [18, -16, -8, -7, 0, 18, 7], [15, -12, 11, -4, -14, 15, -13], [-38, 23, 10, 0, -5, 1, -9], [12, -9, -5, 12, -19, 9, -11], [-37, 7, 19, -5, -1, -10, -17], [23, -6, -8, -11, -10, 19, 5], [-35, 3, -9, 6, 14, 18, -5], [23, 0, -17, -5, -11, 7, 2], [-6, 10, -9, -4, 15, 14, 9], [12, 4, 7, 3, -9, 6, -29], [-19, -3, -13, -7, 10, 6, -3], [12, 1, 5, -2, -8, -15, -8], [-18, 20, 20, 14, 1, -6, -11], [-14, -8, -19, -16, -3, -4, 10], [15, -2, 6, 1, -5, -13, -1], [13, -12, 13, 5, -9, 9, -15], [21, -9, -2, 5, 5, 26, 5], [5, -8, 1, -5, 2, -13, 10], [0, 4, -8, 5, 2, 8, 5], [-37, 37, -1, -7, -7, -8, -11], [20, -8, -13, 1, -21, -2, 5], [-5, 8, 24, 8, -13, 1, -14], [-3, 4, 6, 8, 1, 7, -2], [-1, -5, 9, 1, 10, -14, 24], [4, 8, -4, -19, 7, -18, 6], [-3, 0, -3, -3, -23, -13, -5], [6, 9, -10, 2, 2, 5, 19], [9, -4, 3, 6, 0, 19, -1], [2, -10, 7, 4, 7, 0, 24], [4, -7, 8, -1, -1, 9, -6], [11, 16, 11, -12, 4, 1, -5], [4, -21, -15, -6, -21, -1, 13], [6, 15, 12, 6, -9, 17, -10], [30, -14, -18, 0, -5, -12, 16], [20, -4, 4, 26, -14, 12, -34], [21, -11, -3, 8, 1, -12, -15], [5, -23, 5, 9, 0, 0, 17], [-15, 9, 13, 13, 2, 10, -29], [29, -12, -28, -8, -14, 1, 1], [32, -17, -8, 19, -1, 19, -10], [-7, 4, 27, 8, -17, 9, -19], [5, -6, -4, 11, -1, 9, 22], [-6, 8, -5, 14, 16, -10, 14], [1, 8, -27, -11, 6, -5, 14], [-8, 9, -7, -7, 3, 9, 4], [20, 3, -19, -2, 1, -9, -17], [-13, 9, 12, -9, 15, -6, 8], [-10, 1, -11, -19, -2, -1, 10], [19, 20, 16, 7, -6, 9, -12], [2, 12, -16, -12, 1, -12, 17], [-10, 0, 5, -2, 12, 12, 10], [-5, 7, -14, -7, -6, -12, -6], [-36, 1, -11, -3, -4, 7, 2], [-12, -4, 6, -4, 2, -26, -4], [34, 0, -3, -8, -19, 8, -1], [21, -15, 9, 4, -1, -10, -20], [6, 5, -1, -6, 17, -5, 15], [-13, 8, -13, -15, 5, -7, 4], [-37, -1, -6, 2, 24, -4, 21], [-12, -7, 23, 7, 17, 15, 2], [7, 3, 35, 12, 1, 20, -30], [-10, 27, -21, -14, -13, -3, -5], [23, -18, 7, 13, 11, 29, 4], [23, 5, 2, -13, 13, -24, 3], [-31, 26, 20, -16, -18, -9, -10], [18, 4, 0, 14, -9, 0, 1], [-39, -5, -22, -1, 11, 8, 1], [19, -18, -7, 18, 32, 1, 18], [-3, 19, 18, 18, -9, 24, -30], [10, 18, 19, 12, -4, 14, -8], [28, 0, -6, -8, -18, 18, 10], [-5, 11, 12, 6, 5, 20, -1], [1, -6, 23, 15, 29, 9, 1], [-44, 10, 12, 4, 11, 6, -25], [-22, -3, -22, -7, -9, -9, 12], [19, 3, 6, 5, -18, 14, -21], [-5, -6, -16, -19, 4, 3, 12], [-20, 7, -8, -26, -3, -17, -11], [-18, 6, 6, 18, 24, -8, 8], [-9, 11, 24, -3, 3, 0, -20]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2001_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2001_2_a_i();".
// 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_2001_2_a_i(:prec:=7)
    chi := MakeCharacter_2001_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_2001_2_a_i();".
// 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_2001_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2001_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-8, 1, 26, 4, -18, -5, 3, 1]>,<5,R![-8, 6, 29, -5, -30, -9, 3, 1]>],Snew);
    return Vf;
end function;