
// Make newform 5040.2.t.z in Magma, downloaded from the LMFDB on 09 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_5040_t();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_5040_t_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_5040_2_t_z();".
// 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_5040_2_t_z();".
// 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 := [2, -4, 4, 2, 2, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [13, -32, -30, -5, 10, -7], [-3, -74, -32, 10, 3, -9], [-11, -72, -10, -17, 11, -10], [-65, 68, 12, 25, -27, 12], [55, -54, -42, -30, 37, -19]];
    Rf_basisdens := [1, 23, 23, 23, 23, 23];
    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_5040_t();"
function MakeCharacter_5040_t()
    N := 5040;
    order := 2;
    char_gens := [3151, 3781, 2801, 2017, 3601];
    v := [2, 2, 2, 1, 2];
    // 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_5040_t_Hecke();"
function MakeCharacter_5040_t_Hecke( : Kf := false)
    N := 5040;
    order := 2;
    char_gens := [3151, 3781, 2801, 2017, 3601];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 1, 0, 0, 0, 0], [0, 0, -1, -1, -1, 1], [0, -4, 0, -1, 1, 0], [0, -4, 1, -2, 2, 1], [0, 0, -1, 2, 2, 1], [0, -2, -2, 1, -1, -2], [-2, 0, 0, 0, 0, 0], [4, 0, 1, 0, 0, -1], [0, 4, 2, 0, 0, 2], [4, 0, -1, 0, 0, 1], [0, 0, 0, -2, 2, 0], [0, 0, 0, 0, 0, 0], [0, 4, 0, -1, 1, 0], [4, 0, -2, 2, 2, 2], [-2, 0, 4, 0, 0, -4], [0, 4, 2, -2, 2, 2], [-4, 0, -3, -1, -1, 3], [0, -8, 2, -1, 1, 2], [0, 0, 2, 0, 0, -2], [0, -4, 2, -2, 2, 2], [0, 0, -3, 2, 2, 3], [0, 0, 2, -1, 1, 2], [0, 0, 1, 2, 2, -1], [0, 8, -2, 0, 0, -2], [0, 6, -2, -1, 1, -2], [-6, 0, -2, 2, 2, 2], [0, 0, 0, -3, 3, 0], [0, -4, 2, 0, 0, 2], [4, 0, 0, 4, 4, 0], [0, 0, 2, 1, -1, 2], [12, 0, -3, 0, 0, 3], [6, 0, 2, 0, 0, -2], [0, 0, 0, 2, 2, 0], [0, 4, -2, -1, 1, -2], [0, 4, 0, 0, 0, 0], [0, -8, -2, -2, 2, -2], [0, 8, -1, 2, -2, -1], [-8, 0, 1, -3, -3, -1], [6, 0, 2, 4, 4, -2], [-12, 0, -1, 5, 5, 1], [0, 4, -4, -2, 2, -4], [0, 4, -4, 3, -3, -4], [0, 0, 1, -2, -2, -1], [4, 0, -2, 2, 2, 2], [0, 0, 0, -4, 4, 0], [0, 8, 2, 0, 0, 2], [-2, 0, -2, 4, 4, 2], [0, 8, 2, 5, -5, 2], [12, 0, -1, -3, -3, 1], [2, 0, -2, 0, 0, 2], [-4, 0, 6, -2, -2, -6], [0, -12, -3, 0, 0, -3], [0, -14, -2, 3, -3, -2], [-4, 0, 1, 6, 6, -1], [-4, 0, -5, 0, 0, 5], [0, -4, -2, -4, 4, -2], [6, 0, -6, 4, 4, 6], [0, -4, -2, 0, 0, -2], [0, -8, 1, -2, 2, 1], [0, 12, -2, -4, 4, -2], [-24, 0, 0, -2, -2, 0], [0, -8, 4, 5, -5, 4], [0, -12, 4, -7, 7, 4], [-12, 0, 2, 2, 2, -2], [0, 8, -4, 4, -4, -4], [0, -2, -2, -1, 1, -2], [-14, 0, 4, -4, -4, -4], [0, -12, 5, -6, 6, 5], [20, 0, 1, -1, -1, -1], [0, 8, 0, 4, -4, 0], [0, -8, -2, 4, -4, -2], [12, 0, 0, 2, 2, 0], [0, 4, 6, -4, 4, 6], [18, 0, -4, 4, 4, 4], [0, 12, -2, -3, 3, -2], [-14, 0, 2, 0, 0, -2], [-6, 0, -6, -4, -4, 6], [4, 0, 4, -2, -2, -4], [14, 0, -2, 6, 6, 2], [12, 0, 1, -1, -1, -1], [0, 0, -4, 5, -5, -4], [16, 0, -3, -2, -2, 3], [0, -14, 2, -7, 7, 2], [-6, 0, 0, 4, 4, 0], [0, 0, 4, 2, -2, 4], [12, 0, 5, 0, 0, -5], [0, 4, 6, 0, 0, 6], [0, -16, 2, -4, 4, 2], [24, 0, 0, 2, 2, 0], [0, -16, -2, -6, 6, -2], [-16, 0, 7, -5, -5, -7], [20, 0, 4, -2, -2, -4], [0, 4, 2, 4, -4, 2], [-4, 0, -5, -2, -2, 5], [-16, 0, -1, 4, 4, 1], [0, 20, 0, 4, -4, 0], [10, 0, 2, -6, -6, -2], [0, -20, -6, 2, -2, -6], [0, -4, 0, -7, 7, 0], [0, 0, 4, -2, 2, 4], [-2, 0, -2, -4, -4, 2], [-20, 0, 4, 0, 0, -4], [0, -16, -4, 3, -3, -4], [0, 0, 8, -2, 2, 8], [0, -20, -1, 6, -6, -1], [12, 0, 1, 3, 3, -1], [-10, 0, 8, -4, -4, -8], [0, 0, 8, -4, 4, 8], [0, -8, 2, -6, 6, 2], [0, 24, 2, -5, 5, 2], [12, 0, -7, 0, 0, 7], [8, 0, -2, -8, -8, 2], [6, 0, -6, -4, -4, 6], [0, -12, 2, 8, -8, 2], [0, 16, -2, 2, -2, -2], [0, -12, 6, -3, 3, 6], [0, 0, 3, -9, -9, -3], [-34, 0, 4, -4, -4, -4], [0, 4, 4, -4, 4, 4], [0, 0, -3, 6, -6, -3], [0, -6, 2, 5, -5, 2], [4, 0, -7, 0, 0, 7], [2, 0, 4, -4, -4, -4], [6, 0, 8, 0, 0, -8], [8, 0, -6, 4, 4, 6], [0, 16, 6, 4, -4, 6], [0, -20, -8, 3, -3, -8], [28, 0, 0, -4, -4, 0], [0, -2, -6, 5, -5, -6], [-8, 0, 10, -10, -10, -10], [0, -8, 6, -12, 12, 6], [-16, 0, 5, -6, -6, -5], [10, 0, -4, 0, 0, 4], [0, -16, -3, -2, 2, -3], [0, 4, 4, 4, -4, 4], [0, 8, 1, 6, -6, 1], [-10, 0, 4, -8, -8, -4], [4, 0, 3, 8, 8, -3], [14, 0, -10, 8, 8, 10], [0, -20, -2, 0, 0, -2], [0, 30, -2, 3, -3, -2], [-14, 0, 2, 8, 8, -2], [32, 0, -2, -2, -2, 2], [0, 4, 14, -9, 9, 14], [0, -4, -7, 4, -4, -7], [48, 0, 1, 2, 2, -1], [0, 6, 6, 1, -1, 6], [0, 16, 2, -2, 2, 2], [12, 0, 5, -10, -10, -5], [0, -16, 2, -8, 8, 2], [0, -8, -4, -8, 8, -4], [0, 16, -2, 8, -8, -2], [-4, 0, -1, -11, -11, 1], [-16, 0, 6, -2, -2, -6], [-8, 0, 3, 2, 2, -3], [0, 8, 4, 9, -9, 4], [16, 0, -5, 4, 4, 5], [0, -6, -2, 9, -9, -2], [0, -32, 4, -3, 3, 4], [0, 12, -8, 14, -14, -8], [20, 0, -10, 0, 0, 10], [0, -16, -2, -5, 5, -2], [0, -36, 4, -10, 10, 4], [-24, 0, -6, 6, 6, 6], [0, -4, 2, 9, -9, 2], [22, 0, 4, -8, -8, -4], [0, -32, -3, 2, -2, -3], [0, 0, -11, -7, -7, 11], [2, 0, -4, 12, 12, 4], [20, 0, 5, -5, -5, -5], [0, 36, 4, 2, -2, 4], [-16, 0, -5, -6, -6, 5], [0, 0, -1, 8, 8, 1], [4, 0, 10, -8, -8, -10], [-10, 0, -8, 4, 4, 8], [0, 0, 2, 0, 0, 2], [-2, 0, -4, 4, 4, 4], [0, 12, 0, -10, 10, 0], [12, 0, -10, 8, 8, 10], [0, 4, -10, -2, 2, -10], [0, -20, 3, -10, 10, 3], [0, 2, 6, -1, 1, 6], [-16, 0, -7, 14, 14, 7], [0, -8, 6, 6, -6, 6], [0, 20, -8, 8, -8, -8], [2, 0, 8, -8, -8, -8], [-16, 0, 2, -4, -4, -2], [0, -16, -10, 9, -9, -10], [0, -6, -2, 5, -5, -2], [4, 0, -8, 2, 2, 8], [4, 0, -5, 0, 0, 5], [0, 10, -6, 5, -5, -6], [0, -4, -9, 2, -2, -9], [26, 0, -4, -8, -8, 4], [0, -20, 10, -2, 2, 10], [0, 4, 5, 12, -12, 5], [0, -24, -6, 2, -2, -6], [26, 0, 2, 4, 4, -2], [40, 0, -9, 2, 2, 9], [0, 44, -4, -1, 1, -4], [-2, 0, -2, -8, -8, 2], [-36, 0, -2, 0, 0, 2], [0, 0, 1, -6, 6, 1], [-16, 0, 3, 6, 6, -3], [0, -14, -6, 1, -1, -6], [-2, 0, 8, 0, 0, -8], [-16, 0, 3, 10, 10, -3], [0, -16, -4, 6, -6, -4], [20, 0, 1, 2, 2, -1], [0, 32, -4, 16, -16, -4], [0, -12, -10, -2, 2, -10], [32, 0, 2, -10, -10, -2], [6, 0, 8, -16, -16, -8], [0, 0, -10, -2, 2, -10], [20, 0, 11, 2, 2, -11], [0, -10, 6, 5, -5, 6], [0, -44, 2, 1, -1, 2], [-22, 0, 8, 0, 0, -8], [-12, 0, 3, -4, -4, -3], [6, 0, 0, 0, 0, 0], [0, 8, 4, -4, 4, 4], [0, -4, 8, -8, 8, 8], [6, 0, 8, 0, 0, -8], [0, -20, 7, -12, 12, 7], [36, 0, -7, -5, -5, 7], [0, -8, 6, -8, 8, 6], [-8, 0, -1, 7, 7, 1], [0, 24, -2, 12, -12, -2], [8, 0, 3, -10, -10, -3], [24, 0, -16, 14, 14, 16], [-30, 0, 0, 8, 8, 0], [0, -4, -2, -16, 16, -2], [0, 12, -4, -2, 2, -4], [2, 0, -18, 0, 0, 18], [0, 4, -12, 1, -1, -12], [-32, 0, 1, 1, 1, -1], [24, 0, -2, -4, -4, 2], [0, -18, -6, -1, 1, -6], [-60, 0, 3, -4, -4, -3], [0, -8, 0, -4, 4, 0], [-14, 0, 0, 0, 0, 0], [0, 12, 15, -8, 8, 15], [-24, 0, 10, 2, 2, -10], [0, 24, -6, 4, -4, -6], [-4, 0, 0, -6, -6, 0], [-4, 0, 6, 4, 4, -6], [0, -18, -6, -7, 7, -6], [0, 8, -6, -2, 2, -6], [-4, 0, -1, 2, 2, 1], [0, 54, -2, 5, -5, -2], [34, 0, 2, 0, 0, -2], [0, -16, 5, -14, 14, 5], [-16, 0, -3, 1, 1, 3], [26, 0, -8, 8, 8, 8], [0, -36, -2, -4, 4, -2], [0, 0, -1, 2, -2, -1], [0, 48, -4, -1, 1, -4], [0, 4, -14, 12, -12, -14], [0, -18, 6, -5, 5, 6], [38, 0, -2, -4, -4, 2], [0, -4, -2, 5, -5, -2], [0, 12, 9, 2, -2, 9], [-4, 0, -5, -4, -4, 5], [-6, 0, -2, -4, -4, 2], [48, 0, -1, 4, 4, 1], [0, 52, -2, 6, -6, -2], [0, 44, -2, -3, 3, -2], [10, 0, -16, 12, 12, 16], [0, 16, -8, 18, -18, -8], [0, 16, 6, -3, 3, 6], [-16, 0, 6, 2, 2, -6], [0, 56, 2, -1, 1, 2], [0, 24, -2, 4, -4, -2], [0, -2, 10, -9, 9, 10], [2, 0, -6, -10, -10, 6], [-22, 0, 10, -6, -6, -10], [-20, 0, -2, -2, -2, 2], [0, -36, 4, 2, -2, 4], [0, 0, -6, -12, -12, 6], [0, -12, 12, -2, 2, 12], [-46, 0, 8, 0, 0, -8], [0, 52, -4, 4, -4, -4], [4, 0, 11, -15, -15, -11], [0, 24, -12, -2, 2, -12], [0, 12, -8, -15, 15, -8], [-44, 0, -5, 4, 4, 5], [40, 0, -13, 6, 6, 13], [-2, 0, 6, 8, 8, -6], [0, 4, 6, -14, 14, 6], [0, 32, 18, -3, 3, 18], [36, 0, 8, -6, -6, -8], [0, 32, -6, 20, -20, -6], [-12, 0, 11, -4, -4, -11], [-16, 0, 9, 2, 2, -9], [0, 8, -7, 8, -8, -7], [4, 0, 2, -14, -14, -2], [0, -28, 8, 8, -8, 8], [0, -48, 4, 7, -7, 4], [0, 44, -2, 7, -7, -2], [0, 0, 4, -20, -20, -4], [0, -26, 2, -17, 17, 2], [-20, 0, 16, -12, -12, -16], [0, -48, 0, 4, -4, 0], [0, 58, 2, 5, -5, 2], [38, 0, -10, 4, 4, 10], [4, 0, -1, 9, 9, 1], [0, 28, -6, 18, -18, -6], [0, 20, 0, 6, -6, 0], [-26, 0, 4, 4, 4, -4], [42, 0, 6, -12, -12, -6], [0, -8, 10, -8, 8, 10], [0, 50, -2, -5, 5, -2], [54, 0, -10, 4, 4, 10], [-28, 0, 0, 8, 8, 0], [36, 0, -5, 9, 9, 5], [0, -72, -2, 1, -1, -2], [-26, 0, 2, 12, 12, -2], [-16, 0, -5, 10, 10, 5], [0, -20, 4, -20, 20, 4], [4, 0, -13, 16, 16, 13], [0, 8, -8, -12, 12, -8], [0, -16, -20, 11, -11, -20], [-62, 0, 2, 4, 4, -2], [-44, 0, 8, 4, 4, -8], [0, 36, 2, -12, 12, 2], [0, -18, -10, -11, 11, -10], [0, -4, 2, 11, -11, 2], [-6, 0, -2, 6, 6, 2], [0, 20, -6, -9, 9, -6], [-36, 0, 1, 4, 4, -1], [0, 8, 4, 14, -14, 4], [-20, 0, -6, 24, 24, 6], [0, 0, -12, 14, -14, -12], [-22, 0, 8, -8, -8, -8], [0, 20, -11, -16, 16, -11], [42, 0, -22, 12, 12, 22], [0, 0, -14, 0, 0, -14], [0, -16, 6, 8, -8, 6], [0, 16, -10, -5, 5, -10], [-52, 0, 21, -10, -10, -21], [-40, 0, 8, -6, -6, -8], [0, 20, 2, -7, 7, 2], [-8, 0, 11, -9, -9, -11], [-46, 0, -10, -8, -8, 10], [0, 24, -10, 12, -12, -10], [-24, 0, 6, -8, -8, -6], [0, 8, -5, 10, -10, -5], [8, 0, -11, 14, 14, 11], [0, -4, -4, 0, 0, -4], [22, 0, -10, -12, -12, 10], [0, 16, -8, -4, 4, -8], [-2, 0, 10, 14, 14, -10], [0, 36, -7, -2, 2, -7], [-16, 0, 14, -16, -16, -14], [12, 0, -12, 4, 4, 12], [0, -24, 0, -21, 21, 0], [0, -2, -6, 5, -5, -6], [0, -48, 2, 8, -8, 2], [-12, 0, -3, 0, 0, 3], [0, 10, 6, -5, 5, 6], [0, 0, -13, -1, -1, 13], [0, -36, 8, -16, 16, 8], [0, 28, -8, 0, 0, -8], [0, 8, 1, -4, 4, 1], [0, -4, 8, 6, -6, 8], [58, 0, 2, -6, -6, -2], [56, 0, -13, -1, -1, 13], [8, 0, -15, -10, -10, 15], [0, 30, -10, 21, -21, -10], [22, 0, -18, 8, 8, 18], [36, 0, -25, 8, 8, 25], [0, 20, -10, 12, -12, -10], [12, 0, -8, 8, 8, 8], [28, 0, 11, 5, 5, -11], [0, -8, -16, -11, 11, -16], [24, 0, -15, 16, 16, 15], [0, -24, -6, -7, 7, -6], [-40, 0, -7, -6, -6, 7], [0, 32, -4, -11, 11, -4], [0, -8, -6, 10, -10, -6], [0, 0, -8, 1, -1, -8], [-12, 0, -7, -8, -8, 7], [0, -44, -14, 4, -4, -14], [32, 0, 6, 2, 2, -6], [0, -4, -4, 9, -9, -4], [0, -28, -2, -22, 22, -2], [0, 16, 4, -12, 12, 4], [10, 0, 8, -8, -8, -8], [0, -40, 5, -16, 16, 5], [-16, 0, 7, -9, -9, -7], [0, -20, -10, -4, 4, -10], [-20, 0, 13, -1, -1, -13], [0, -8, 8, -8, 8, 8], [56, 0, -13, 10, 10, 13], [-28, 0, 7, 10, 10, -7], [44, 0, -20, 4, 4, 20], [42, 0, -16, 4, 4, 16], [6, 0, 12, 12, 12, -12], [0, -64, 0, -7, 7, 0], [0, 20, 28, -10, 10, 28], [0, -12, -13, -4, 4, -13], [48, 0, 1, -8, -8, -1], [56, 0, -11, 2, 2, 11], [0, -36, 6, 0, 0, 6], [46, 0, -16, 4, 4, 16], [0, -28, -4, 4, -4, -4], [28, 0, -2, 6, 6, 2], [0, -24, -8, -5, 5, -8], [0, 36, -6, 1, -1, -6], [0, 58, 10, -3, 3, 10], [36, 0, 6, 2, 2, -6], [0, 4, -4, 6, -6, -4], [-52, 0, 3, -6, -6, -3], [20, 0, -5, -7, -7, 5], [0, -40, 2, -4, 4, 2], [0, -28, -1, -2, 2, -1], [0, -8, 8, -16, 16, 8], [-14, 0, -12, -8, -8, 12], [0, 12, -6, -19, 19, -6], [0, -6, -2, -21, 21, -2], [28, 0, -18, 0, 0, 18], [0, -8, 8, -15, 15, 8], [0, 24, -5, 18, -18, -5], [0, 14, 10, -1, 1, 10], [30, 0, 10, -16, -16, -10], [-44, 0, 9, -8, -8, -9], [56, 0, -14, 2, 2, 14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5040_t_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5040_2_t_z();".
// 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_5040_2_t_z(:prec:=6)
    chi := MakeCharacter_5040_t();
    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_5040_2_t_z();".
// 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_5040_2_t_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5040_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-104, -36, 2, 1]>,<13,R![64, 0, 560, 0, 60, 0, 1]>,<17,R![73984, 0, 5376, 0, 128, 0, 1]>,<19,R![-200, -60, 2, 1]>,<29,R![2, 1]>],Snew);
    return Vf;
end function;