
// Make newform 5040.2.t.x in Magma, downloaded from the LMFDB on 15 November 2024.
// 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_x();".
// 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_x();".
// 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)
    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 := 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]];
    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, -1, 0, 0], [0, 1, 0, 0, 0, 0], [2, 0, -1, 1, 1, 1], [0, 2, 0, -1, 1, 0], [0, 0, 0, -1, 1, 0], [2, 0, -1, 0, 0, 1], [0, 4, -1, 0, 0, -1], [0, 0, 0, 0, 0, 0], [2, 0, 1, -2, -2, -1], [0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 0], [0, 0, 2, 0, 0, 2], [0, -4, 0, 0, 0, 0], [0, -2, -3, 2, -2, -3], [-8, 0, 0, 0, 0, 0], [2, 0, 0, 2, 2, 0], [0, -4, 0, 2, -2, 0], [2, 0, -1, -1, -1, 1], [0, -2, 2, -3, 3, 2], [4, 0, -2, 0, 0, 2], [0, 0, -2, 0, 0, -2], [8, 0, 0, -1, -1, 0], [0, -2, 2, -3, 3, 2], [0, 0, -4, -1, -1, 4], [0, 4, -2, 2, -2, -2], [0, 0, -3, 0, 0, -3], [2, 0, 2, 2, 2, -2], [0, 6, -1, 2, -2, -1], [0, 4, 2, 2, -2, 2], [4, 0, 2, -2, -2, -2], [0, 6, -1, 0, 0, -1], [-2, 0, 1, -4, -4, -1], [8, 0, -4, 2, 2, 4], [4, 0, 2, 0, 0, -2], [0, 2, -4, -1, 1, -4], [0, 8, -2, 4, -4, -2], [0, -4, 2, -4, 4, 2], [0, 4, -2, -1, 1, -2], [-10, 0, -3, -1, -1, 3], [-14, 0, 4, -2, -2, -4], [-10, 0, -3, 1, 1, 3], [0, 8, -4, 0, 0, -4], [0, -10, -3, -2, 2, -3], [-6, 0, 5, -2, -2, -5], [-4, 0, 4, -4, -4, -4], [0, 0, 0, -4, 4, 0], [0, 8, -4, 4, -4, -4], [2, 0, 4, 2, 2, -4], [0, -2, -1, -4, 4, -1], [-10, 0, -3, 1, 1, 3], [2, 0, 0, -4, -4, 0], [8, 0, -8, 4, 4, 8], [0, -12, 2, 3, -3, 2], [0, 4, 1, 0, 0, 1], [16, 0, -4, -1, -1, 4], [-2, 0, -5, 2, 2, 5], [0, -20, 0, 2, -2, 0], [-8, 0, 4, -2, -2, -4], [0, 20, -2, 6, -6, -2], [0, 8, -4, -3, 3, -4], [0, -4, -2, -6, 6, -2], [12, 0, 2, -2, -2, -2], [0, 10, 6, -1, 1, 6], [0, 10, -5, -2, 2, -5], [-4, 0, -4, 4, 4, 4], [0, 4, -4, -2, 2, -4], [0, 8, -3, 0, 0, -3], [6, 0, 4, -6, -6, -4], [0, -12, 2, 3, -3, 2], [-14, 0, -1, 7, 7, 1], [0, -8, 4, -8, 8, 4], [0, 12, 8, -2, 2, 8], [8, 0, 6, 4, 4, -6], [0, -12, -4, 0, 0, -4], [16, 0, 0, -4, -4, 0], [0, -10, 4, 5, -5, 4], [0, 0, 8, 2, 2, -8], [-14, 0, -4, 0, 0, 4], [-24, 0, 0, 0, 0, 0], [2, 0, -10, -2, -2, 10], [18, 0, -1, -1, -1, 1], [0, 2, 2, 3, -3, 2], [-6, 0, -3, 2, 2, 3], [0, 0, 3, -8, 8, 3], [24, 0, -4, 4, 4, 4], [0, 0, 0, 8, -8, 0], [16, 0, -8, 7, 7, 8], [0, 12, 10, -2, 2, 10], [0, 0, 0, 4, -4, 0], [-12, 0, -2, -6, -6, 2], [0, -12, 2, 2, -2, 2], [-6, 0, 3, 1, 1, -3], [8, 0, -10, 4, 4, 10], [0, -4, 4, -8, 8, 4], [16, 0, -4, 1, 1, 4], [-16, 0, 0, 1, 1, 0], [0, 24, -4, -4, 4, -4], [-10, 0, 2, -6, -6, -2], [0, -4, 8, -6, 6, 8], [0, 2, -5, 2, -2, -5], [0, 8, 10, -4, 4, 10], [8, 0, 4, 6, 6, -4], [-20, 0, 4, -4, -4, -4], [0, 6, -2, 1, -1, -2], [0, 8, -6, 8, -8, -6], [0, 36, 2, 1, -1, 2], [-14, 0, 7, -5, -5, -7], [14, 0, 4, 0, 0, -4], [0, -8, -4, 0, 0, -4], [0, 24, 0, 0, 0, 0], [0, 2, 9, -4, 4, 9], [-2, 0, 1, 8, 8, -1], [4, 0, 6, 8, 8, -6], [16, 0, 0, -2, -2, 0], [0, -4, -6, 2, -2, -6], [0, -12, 6, -4, 4, 6], [0, -30, 3, -8, 8, 3], [-6, 0, 3, 5, 5, -3], [26, 0, -8, 6, 6, 8], [0, 12, -8, 6, -6, -8], [0, -8, -4, 5, -5, -4], [0, 8, -5, 8, -8, -5], [14, 0, -7, 4, 4, 7], [-8, 0, -4, 0, 0, 4], [-6, 0, 0, -4, -4, 0], [-24, 0, -4, 0, 0, 4], [0, 4, -10, 2, -2, -10], [0, 18, 0, -1, 1, 0], [-4, 0, 8, -4, -4, -8], [0, -4, 3, 8, -8, 3], [16, 0, 0, -4, -4, 0], [0, -24, -8, 0, 0, -8], [-8, 0, 8, -5, -5, -8], [-6, 0, 4, 8, 8, -4], [0, 4, -10, 9, -9, -10], [0, 8, 8, 4, -4, 8], [0, -12, -2, -3, 3, -2], [24, 0, 4, 0, 0, -4], [-22, 0, -1, -4, -4, 1], [-24, 0, 4, 2, 2, -4], [0, 28, 2, 6, -6, 2], [0, 8, 1, -8, 8, 1], [6, 0, 4, -2, -2, -4], [-20, 0, 10, 2, 2, -10], [0, 26, 4, 3, -3, 4], [0, 16, 0, 11, -11, 0], [-18, 0, -3, -4, -4, 3], [0, 12, -5, -8, 8, -5], [0, -16, 8, -4, 4, 8], [-8, 0, 0, 11, 11, 0], [0, 4, 8, -2, 2, 8], [0, 20, 8, 4, -4, 8], [0, -20, 0, 2, -2, 0], [-10, 0, 5, -7, -7, -5], [-24, 0, 4, -4, -4, -4], [-8, 0, 8, -7, -7, -8], [0, 18, -2, 3, -3, -2], [0, 0, 8, -7, -7, -8], [0, 0, 7, -8, 8, 7], [0, 14, 7, -6, 6, 7], [0, 0, 4, 4, -4, 4], [4, 0, 2, -14, -14, -2], [0, 10, 9, 4, -4, 9], [0, 12, 2, 12, -12, 2], [-8, 0, -8, 4, 4, 8], [0, -26, 0, -3, 3, 0], [14, 0, -8, 4, 4, 8], [0, 32, 0, 9, -9, 0], [-10, 0, 5, 3, 3, -5], [6, 0, 4, 6, 6, -4], [10, 0, 11, 7, 7, -11], [0, 12, 0, -2, 2, 0], [-18, 0, 7, -10, -10, -7], [16, 0, 8, 5, 5, -8], [-16, 0, 10, 4, 4, -10], [-8, 0, -4, -4, -4, 4], [0, 44, -2, 6, -6, -2], [2, 0, -8, 10, 10, 8], [0, -32, 0, -4, 4, 0], [-4, 0, 6, 6, 6, -6], [0, 8, 0, -12, 12, 0], [0, -16, 8, -3, 3, 8], [0, 36, 5, 0, 0, 5], [0, 0, -12, 7, 7, 12], [0, -20, -8, 6, -6, -8], [0, 24, 8, -4, 4, 8], [-6, 0, 8, 4, 4, -8], [0, 0, -8, -4, -4, 8], [0, 10, 10, -1, 1, 10], [0, -16, -1, -8, 8, -1], [24, 0, 2, -4, -4, -2], [16, 0, 8, -3, -3, -8], [0, -24, 7, -8, 8, 7], [0, -12, 10, 5, -5, 10], [-18, 0, 0, 12, 12, 0], [0, 24, 0, -4, 4, 0], [0, -4, 6, -9, 9, 6], [0, -36, 6, 0, 0, 6], [-32, 0, 0, -6, -6, 0], [6, 0, -9, 2, 2, 9], [0, -38, 4, -13, 13, 4], [-10, 0, -4, 4, 4, 4], [12, 0, -2, -2, -2, 2], [0, -12, -2, 5, -5, -2], [30, 0, -1, -2, -2, 1], [0, 40, 7, 0, 0, 7], [0, 0, 8, 4, 4, -8], [26, 0, -9, 4, 4, 9], [0, -4, 4, -14, 14, 4], [0, 0, -4, 7, 7, 4], [0, 32, -4, 12, -12, -4], [0, 32, -2, 4, -4, -2], [-4, 0, 10, -10, -10, -10], [14, 0, 4, -4, -4, -4], [0, 20, 2, 10, -10, 2], [40, 0, 0, -3, -3, 0], [0, -20, -1, 0, 0, -1], [0, 10, 3, -4, 4, 3], [22, 0, 8, 4, 4, -8], [-2, 0, -13, -2, -2, 13], [-8, 0, 12, 0, 0, -12], [0, 0, 8, -12, 12, 8], [0, 8, 16, -4, 4, 16], [-6, 0, 8, -8, -8, -8], [0, 36, 2, 5, -5, 2], [-6, 0, -13, -9, -9, 13], [0, -12, 2, -6, 6, 2], [-22, 0, 3, 9, 9, -3], [0, 16, 0, -4, 4, 0], [18, 0, -13, 0, 0, 13], [-36, 0, -6, -8, -8, 6], [-8, 0, 12, 0, 0, -12], [0, 28, 2, 2, -2, 2], [0, 28, -6, 12, -12, -6], [26, 0, 4, 4, 4, -4], [0, -22, -9, -6, 6, -9], [-2, 0, -7, 11, 11, 7], [24, 0, -4, 8, 8, 4], [0, 56, 5, 0, 0, 5], [-6, 0, -5, -8, -8, 5], [0, -16, 12, -4, 4, 12], [-18, 0, 0, 4, 4, 0], [0, 0, 8, 1, -1, 8], [28, 0, 10, -6, -6, -10], [0, -12, 14, 2, -2, 14], [-8, 0, 8, -4, -4, -8], [48, 0, -2, 4, 4, 2], [0, -36, -1, 0, 0, -1], [0, -12, 0, -6, 6, 0], [8, 0, 0, -7, -7, 0], [0, 4, 3, 16, -16, 3], [10, 0, 0, 4, 4, 0], [0, -36, 10, -7, 7, 10], [-26, 0, -3, 7, 7, 3], [-6, 0, -8, 8, 8, 8], [0, 4, -6, 14, -14, -6], [0, 4, -2, -5, 5, -2], [0, 6, 6, -7, 7, 6], [0, 4, -2, 10, -10, -2], [0, 0, 5, -8, 8, 5], [-22, 0, -8, -2, -2, 8], [0, 22, 4, -3, 3, 4], [0, 4, 18, -5, 5, 18], [-14, 0, -5, 12, 12, 5], [-48, 0, 0, -6, -6, 0], [16, 0, 0, -7, -7, 0], [0, 4, -4, -2, 2, -4], [0, -14, -9, 12, -12, -9], [-10, 0, 4, -2, -2, -4], [0, -16, 2, -8, 8, 2], [0, 26, -10, 15, -15, -10], [-8, 0, 0, 4, 4, 0], [0, 46, 2, 5, -5, 2], [0, -20, 6, -2, 2, 6], [0, -40, -3, -8, 8, -3], [10, 0, -2, 6, 6, 2], [26, 0, -14, -2, -2, 14], [-24, 0, -8, 4, 4, 8], [0, 12, -2, 4, -4, -2], [-4, 0, -2, 8, 8, 2], [0, 28, 2, -12, 12, 2], [-34, 0, -4, -2, -2, 4], [0, 0, 0, 0, 0, 0], [30, 0, 1, 9, 9, -1], [0, -24, 8, -4, 4, 8], [0, -14, -5, -2, 2, -5], [-26, 0, -1, 2, 2, 1], [-24, 0, -8, 13, 13, 8], [48, 0, 0, 6, 6, 0], [0, 8, 2, 0, 0, 2], [0, -18, 11, -20, 20, 11], [0, 0, 4, -12, -12, -4], [0, -4, 8, 6, -6, 8], [32, 0, -8, 5, 5, 8], [2, 0, -3, 10, 10, 3], [0, 40, -4, -3, 3, -4], [8, 0, -8, -12, -12, 8], [0, 8, -12, 12, -12, -12], [0, -34, -10, -3, 3, -10], [0, -10, -3, -4, 4, -3], [8, 0, -12, 4, 4, 12], [0, -32, 9, -24, 24, 9], [-4, 0, -12, 4, 4, 12], [0, -12, 12, 6, -6, 12], [0, -32, -1, -16, 16, -1], [18, 0, 0, 2, 2, 0], [6, 0, 13, 5, 5, -13], [0, 20, -16, 14, -14, -16], [0, -68, 2, -4, 4, 2], [-8, 0, 12, -16, -16, -12], [24, 0, -4, 14, 14, 4], [0, 4, -8, 6, -6, -8], [0, 44, -7, 16, -16, -7], [22, 0, 0, 4, 4, 0], [28, 0, 6, -10, -10, -6], [-10, 0, -11, -3, -3, 11], [0, 18, 2, 3, -3, 2], [-16, 0, -8, 18, 18, 8], [34, 0, -1, 4, 4, 1], [0, 60, 0, 6, -6, 0], [-32, 0, 0, -3, -3, 0], [0, 24, -8, -8, 8, -8], [0, -46, 5, -18, 18, 5], [34, 0, -16, 12, 12, 16], [-12, 0, 8, 12, 12, -8], [0, 60, 2, -6, 6, 2], [0, 4, -13, -8, 8, -13], [0, 14, 1, -12, 12, 1], [10, 0, 6, 2, 2, -6], [0, 14, -11, 8, -8, -11], [10, 0, 17, 2, 2, -17], [0, -16, 2, 4, -4, 2], [-8, 0, -2, 4, 4, 2], [0, -16, -6, -4, 4, -6], [2, 0, 0, 4, 4, 0], [0, 60, -2, 7, -7, -2], [-30, 0, -4, -8, -8, 4], [0, 20, -14, 2, -2, -14], [0, 4, 0, 10, -10, 0], [0, -22, -3, 0, 0, -3], [-32, 0, 4, 7, 7, -4], [-12, 0, -6, 8, 8, 6], [0, -46, 7, 0, 0, 7], [2, 0, -1, 13, 13, 1], [22, 0, 8, -2, -2, -8], [0, -20, 4, -14, 14, 4], [-80, 0, 0, 0, 0, 0], [0, -16, -16, 7, -7, -16], [14, 0, 17, 12, 12, -17], [0, 56, 4, 0, 0, 4], [-16, 0, -16, 2, 2, 16], [0, -40, 0, 0, 0, 0], [2, 0, -2, 10, 10, 2], [0, -28, -6, -1, 1, -6], [24, 0, -12, 16, 16, 12], [12, 0, -10, 6, 6, 10], [0, 26, 5, -10, 10, 5], [0, -28, 3, 0, 0, 3], [0, -44, 8, 10, -10, 8], [32, 0, -16, 11, 11, 16], [0, -12, -7, 16, -16, -7], [66, 0, -9, 1, 1, 9], [0, -16, -12, 0, 0, -12], [0, 4, 8, -6, 6, 8], [0, -8, 4, 1, -1, 4], [0, 0, 0, -4, 4, 0], [-6, 0, 18, -10, -10, -18], [-38, 0, 19, -3, -3, -19], [-26, 0, -7, 4, 4, 7], [0, 52, 3, -8, 8, 3], [-32, 0, 16, -14, -14, -16], [70, 0, 3, -6, -6, -3], [0, -60, -8, -2, 2, -8], [36, 0, 2, -2, -2, -2], [-10, 0, -19, -11, -11, 19], [0, -38, 14, -1, 1, 14], [-32, 0, 0, -9, -9, 0], [0, -38, 6, -1, 1, 6], [0, 0, 12, -21, -21, -12], [0, 22, -5, -2, 2, -5], [0, -20, -10, 14, -14, -10], [0, 6, -5, -6, 6, -5], [-10, 0, 9, 4, 4, -9], [0, 52, 12, -8, 8, 12], [-48, 0, 8, -4, -4, -8], [0, -26, 0, 13, -13, 0], [0, -12, 16, 14, -14, 16], [0, 4, -20, -8, 8, -20], [34, 0, 0, 4, 4, 0], [0, 76, 2, 5, -5, 2], [-14, 0, -17, -7, -7, 17], [0, 4, 14, -2, 2, 14], [-38, 0, -5, 11, 11, 5], [0, -12, -4, 14, -14, -4], [-18, 0, -5, 2, 2, 5], [-24, 0, -16, 1, 1, 16], [12, 0, -20, 4, 4, 20], [48, 0, 0, -8, -8, 0], [10, 0, -12, -6, -6, 12], [0, -34, -13, 6, -6, -13], [0, 24, -12, 8, -8, -12], [0, 8, 4, 1, -1, 4], [0, 0, -16, 19, 19, 16], [-6, 0, -3, -2, -2, 3], [0, -52, 16, -30, 30, 16], [48, 0, -16, 0, 0, 16], [0, 0, 0, -8, 8, 0], [-24, 0, 0, -16, -16, 0], [0, 14, -14, 13, -13, -14], [0, 18, 7, -4, 4, 7], [0, -8, -9, -16, 16, -9], [44, 0, 8, -4, -4, -8], [0, 52, 16, -6, 6, 16], [-16, 0, -12, 9, 9, 12], [38, 0, -3, -7, -7, 3], [0, -36, -6, -2, 2, -6], [0, 0, -24, 1, -1, -24], [0, -68, 0, -8, 8, 0], [-48, 0, 0, 16, 16, 0], [0, -34, 12, -23, 23, 12], [0, -12, 5, 16, -16, 5], [28, 0, -2, 2, 2, 2], [0, -46, -10, -5, 5, -10], [0, 40, -4, 15, -15, -4], [0, -16, 1, 0, 0, 1], [24, 0, -28, 2, 2, 28], [86, 0, 1, 0, 0, -1], [60, 0, -14, 2, 2, 14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5040_t_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5040_2_t_x();".
// 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_x(: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_x();".
// 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_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5040_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![32, -16, -4, 1]>,<13,R![256, 0, 320, 0, 48, 0, 1]>,<17,R![64, 0, 176, 0, 28, 0, 1]>,<19,R![16, -8, -4, 1]>,<29,R![0, 1]>],Snew);
    return Vf;
end function;