
// Make newform 5040.2.t.y in Magma, downloaded from the LMFDB on 16 April 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_y();".
// 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_y();".
// 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, -20, 25, -4, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [100, -121, 10, -25, -2, -5], [223, 0, -14, 35, 27, 7], [258, -605, 50, -4, -10, -25], [574, -1331, 372, 38, -26, -65]];
    Rf_basisdens := [1, 1, 121, 121, 242, 242];
    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+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, -1, 0, 0, 0, 1], [0, 0, 0, 0, -1, 0], [2, -1, -1, -1, 0, 0], [0, 2, -2, 0, -2, -1], [0, -1, 1, 0, 0, -1], [2, 1, 1, 2, 0, 0], [0, -2, 2, 0, 2, 0], [0, -3, -3, -3, 0, 0], [-4, 0, 0, -2, 0, 0], [0, 0, 0, 0, 6, 0], [-6, -2, -2, 0, 0, 0], [0, -2, 2, 0, 0, 0], [0, -3, 3, 0, -2, 3], [0, 2, -2, 0, 2, -4], [-2, 3, 3, 0, 0, 0], [8, -1, -1, 0, 0, 0], [0, 2, -2, 0, 4, 2], [2, 0, 0, 2, 0, 0], [0, -2, 2, 0, 2, 0], [4, -1, -1, -5, 0, 0], [0, -1, 1, 0, 6, 6], [2, 2, 2, 6, 0, 0], [0, 1, -1, 0, -4, -3], [0, -3, -3, -4, 0, 0], [0, 1, -1, 0, 2, -1], [0, 0, 0, 0, 0, 0], [-4, -3, -3, -7, 0, 0], [0, 0, 0, 0, 6, -4], [0, -6, 6, 0, 2, 8], [-2, -1, -1, -4, 0, 0], [0, 4, -4, 0, 0, 0], [-2, -5, -5, -8, 0, 0], [-10, 0, 0, -4, 0, 0], [-2, 1, 1, -1, 0, 0], [0, -3, 3, 0, 4, 2], [0, 0, 0, 0, 4, -2], [0, -3, 3, 0, -14, -1], [0, -2, 2, 0, 14, -1], [-4, -4, -4, 0, 0, 0], [-8, -3, -3, -2, 0, 0], [-4, 1, 1, -3, 0, 0], [0, -2, 2, 0, 10, 6], [0, -2, 2, 0, -14, -8], [12, 8, 8, 2, 0, 0], [2, 1, 1, 7, 0, 0], [0, -1, 1, 0, -6, -1], [0, -4, 4, 0, -8, 1], [-16, 1, 1, -2, 0, 0], [0, 0, 0, 0, -4, -4], [14, 1, 1, 3, 0, 0], [2, 2, 2, 0, 0, 0], [-10, 3, 3, -8, 0, 0], [0, -2, 2, 0, -6, 0], [0, -4, 4, 0, 8, 0], [-8, -3, -3, 2, 0, 0], [4, -8, -8, -8, 0, 0], [0, 2, -2, 0, -10, -4], [16, -3, -3, -5, 0, 0], [0, 2, -2, 0, 4, -9], [0, 4, -4, 0, 10, 3], [0, 2, -2, 0, 28, -3], [-12, -4, -4, -4, 0, 0], [0, 5, -5, 0, 8, 5], [0, 0, 0, 0, -22, -2], [8, 2, 2, -4, 0, 0], [0, -4, 4, 0, 2, 4], [0, 12, -12, 0, -8, -6], [-12, -7, -7, -8, 0, 0], [0, -5, 5, 0, -12, 9], [-8, -4, -4, -4, 0, 0], [0, -3, 3, 0, 2, 13], [0, -4, 4, 0, -6, -2], [-4, -2, -2, 0, 0, 0], [0, -2, 2, 0, 0, 0], [-20, -1, -1, -9, 0, 0], [0, -2, 2, 0, -2, -1], [6, -3, -3, -1, 0, 0], [-2, 4, 4, 2, 0, 0], [6, -9, -9, -12, 0, 0], [8, -7, -7, -13, 0, 0], [6, 9, 9, 11, 0, 0], [0, 4, -4, 0, 22, 0], [-12, 2, 2, -6, 0, 0], [0, 4, -4, 0, 0, -8], [0, -1, -1, -7, 0, 0], [0, 8, -8, 0, 6, -16], [4, 9, 9, 4, 0, 0], [0, 0, 0, 0, 0, -8], [0, 4, -4, 0, 0, -5], [0, -4, -4, 10, 0, 0], [0, 6, -6, 0, -18, -4], [18, 7, 7, 9, 0, 0], [14, -7, -7, 11, 0, 0], [0, 3, -3, 0, 10, -5], [16, -1, -1, 2, 0, 0], [-10, 4, 4, 6, 0, 0], [0, -9, 9, 0, 6, 14], [8, -5, -5, 5, 0, 0], [0, 10, -10, 0, -8, -12], [0, 4, -4, 0, -2, 2], [0, -5, 5, 0, 22, 2], [-4, 2, 2, 10, 0, 0], [-20, -2, -2, -8, 0, 0], [0, -3, 3, 0, -12, 7], [0, -7, 7, 0, 10, 2], [0, 5, -5, 0, 0, 7], [4, 15, 15, 7, 0, 0], [2, 2, 2, 10, 0, 0], [0, -5, 5, 0, 2, -9], [0, -6, 6, 0, -6, 6], [0, -16, 16, 0, 10, 8], [34, 1, 1, 4, 0, 0], [4, -1, -1, 3, 0, 0], [4, 10, 10, -6, 0, 0], [0, 8, -8, 0, -8, -7], [0, 2, -2, 0, 0, 8], [0, -2, 2, 0, 18, 10], [-6, 5, 5, 9, 0, 0], [8, 5, 5, -8, 0, 0], [0, 8, -8, 0, 24, -12], [0, 0, 0, 0, -10, 11], [0, 4, -4, 0, 16, -4], [-10, 11, 11, 2, 0, 0], [-8, -5, -5, -13, 0, 0], [-12, -1, -1, -9, 0, 0], [12, -8, -8, 2, 0, 0], [0, -6, 6, 0, -4, 8], [0, -10, 10, 0, 18, 11], [14, 15, 15, 11, 0, 0], [0, 2, -2, 0, -2, -4], [-2, -7, -7, 3, 0, 0], [0, 2, -2, 0, 6, 6], [-26, 2, 2, -4, 0, 0], [-30, 4, 4, 10, 0, 0], [0, -8, 8, 0, 14, 9], [0, 2, -2, 0, 4, -5], [0, -4, 4, 0, -6, -3], [6, 11, 11, 13, 0, 0], [2, -3, -3, -4, 0, 0], [0, -5, -5, 1, 0, 0], [0, 10, -10, 0, 20, -6], [0, 6, -6, 0, 16, -2], [12, 3, 3, -2, 0, 0], [16, -8, -8, -2, 0, 0], [0, 11, -11, 0, 4, -6], [0, -4, 4, 0, 2, -4], [6, 11, 11, -6, 0, 0], [0, -6, 6, 0, 36, -2], [0, 6, -6, 0, -2, 2], [10, 2, 2, 18, 0, 0], [0, 20, -20, 0, -16, -8], [0, 2, -2, 0, 16, 4], [0, 2, -2, 0, -4, 14], [-16, 12, 12, -4, 0, 0], [36, -5, -5, -5, 0, 0], [10, 6, 6, 20, 0, 0], [0, 11, -11, 0, -12, -15], [-24, -9, -9, -16, 0, 0], [0, 0, 0, 0, -24, -2], [0, 2, -2, 0, -8, 6], [0, 12, -12, 0, -18, -2], [10, 5, 5, 8, 0, 0], [0, 4, -4, 0, 8, -8], [0, 13, -13, 0, 6, 3], [-14, 12, 12, 10, 0, 0], [0, 6, -6, 0, 2, -9], [-22, 9, 9, 7, 0, 0], [0, 3, -3, 0, 28, -6], [-4, 4, 4, -16, 0, 0], [12, 17, 17, 4, 0, 0], [-32, -7, -7, -7, 0, 0], [0, -2, 2, 0, -24, -14], [0, -14, -14, -8, 0, 0], [10, 6, 6, 2, 0, 0], [2, -9, -9, 1, 0, 0], [-8, -3, -3, 5, 0, 0], [0, -4, 4, 0, 4, 0], [-40, 3, 3, 4, 0, 0], [0, 8, -8, 0, 26, -10], [-34, 3, 3, 6, 0, 0], [0, -6, 6, 0, 22, -6], [0, 22, -22, 0, -14, -12], [0, -8, 8, 0, 18, 18], [0, 1, 1, -6, 0, 0], [0, -10, 10, 0, 14, 0], [0, 0, 0, 0, 24, 5], [12, -13, -13, -7, 0, 0], [-12, 10, 10, 4, 0, 0], [0, -2, 2, 0, 38, -4], [0, 8, -8, 0, 28, 4], [-30, -15, -15, -13, 0, 0], [40, -5, -5, 2, 0, 0], [0, 12, -12, 0, -12, -8], [0, 1, -1, 0, 24, -13], [-30, -3, -3, 3, 0, 0], [0, 8, -8, 0, -6, -28], [0, 9, -9, 0, -20, -9], [0, -3, 3, 0, -6, 9], [-26, -12, -12, 0, 0, 0], [-12, 2, 2, 0, 0, 0], [0, 1, -1, 0, -40, 2], [26, -10, -10, 2, 0, 0], [6, 15, 15, 10, 0, 0], [0, -8, 8, 0, -18, 5], [-36, -4, -4, 2, 0, 0], [0, 14, -14, 0, 8, -12], [-36, -10, -10, 2, 0, 0], [14, 7, 7, -6, 0, 0], [0, 12, -12, 0, -2, -20], [12, -7, -7, 14, 0, 0], [0, -14, 14, 0, 10, 24], [0, -12, 12, 0, -8, 25], [-16, -8, -8, -20, 0, 0], [2, -8, -8, 10, 0, 0], [0, -10, 10, 0, 4, -2], [10, 6, 6, 24, 0, 0], [0, 16, -16, 0, -16, 0], [0, -4, 4, 0, -10, -4], [-2, 2, 2, -16, 0, 0], [-8, 8, 8, 8, 0, 0], [-8, -15, -15, -9, 0, 0], [0, -8, 8, 0, 30, -6], [0, -20, 20, 0, 16, 17], [6, -10, -10, 8, 0, 0], [0, 3, -3, 0, 24, 17], [-30, -5, -5, -7, 0, 0], [0, 1, -1, 0, 30, 9], [-16, -6, -6, -12, 0, 0], [0, 14, -14, 0, -10, 2], [-6, 13, 13, 14, 0, 0], [30, 4, 4, 6, 0, 0], [40, -9, -9, -15, 0, 0], [0, 0, 0, 0, 8, 17], [0, 3, -3, 0, -22, 3], [14, 6, 6, 16, 0, 0], [0, 12, -12, 0, 10, -18], [-2, -15, -15, -1, 0, 0], [48, 6, 6, 2, 0, 0], [0, -2, 2, 0, 40, 12], [18, 21, 21, 10, 0, 0], [0, -1, 1, 0, 22, 23], [18, -8, -8, 24, 0, 0], [0, -7, 7, 0, 12, -5], [-8, 14, 14, 4, 0, 0], [0, 12, -12, 0, 12, -28], [-2, 15, 15, -10, 0, 0], [-18, -15, -15, 1, 0, 0], [0, 4, -4, 0, 0, 12], [0, 0, 0, 0, -30, 6], [-34, 2, 2, 6, 0, 0], [0, -2, 2, 0, -4, -14], [10, -2, -2, 6, 0, 0], [0, 19, -19, 0, 0, -10], [-14, 7, 7, 21, 0, 0], [18, 10, 10, 4, 0, 0], [0, -13, 13, 0, -10, 2], [0, 0, 0, 0, -46, -5], [0, 11, -11, 0, 12, -1], [0, -12, 12, 0, 14, -2], [0, 8, -8, 0, -36, -14], [-28, 9, 9, -2, 0, 0], [0, -4, 4, 0, 2, -9], [0, -8, 8, 0, 2, 4], [-2, -1, -1, 4, 0, 0], [24, 9, 9, 3, 0, 0], [10, 14, 14, 6, 0, 0], [0, 14, -14, 0, -4, -26], [0, -6, 6, 0, -10, 6], [8, 21, 21, 10, 0, 0], [0, -4, 4, 0, 16, 10], [0, -23, 23, 0, -8, 27], [-32, -8, -8, 0, 0, 0], [0, 6, -6, 0, 14, -16], [0, 7, -7, 0, -34, -17], [0, -14, 14, 0, 4, -8], [10, 8, 8, -8, 0, 0], [-22, 3, 3, 13, 0, 0], [-18, 3, 3, 22, 0, 0], [0, -3, 3, 0, -18, 13], [-4, 1, 1, -23, 0, 0], [0, 10, -10, 0, 32, 0], [-4, -19, -19, -10, 0, 0], [0, -6, 6, 0, -12, 3], [-18, -7, -7, 7, 0, 0], [0, 0, 0, 0, -36, -4], [0, 24, -24, 0, -10, -14], [-20, 0, 0, -10, 0, 0], [-18, 0, 0, 16, 0, 0], [12, 3, 3, -15, 0, 0], [0, 4, -4, 0, -8, -21], [0, -16, 16, 0, 18, 16], [-6, 5, 5, -20, 0, 0], [0, -8, 8, 0, 46, 6], [8, 13, 13, 10, 0, 0], [20, -14, -14, -30, 0, 0], [0, -2, 2, 0, 2, -15], [-46, -7, -7, -14, 0, 0], [0, -12, 12, 0, -32, 3], [0, -5, 5, 0, 28, -9], [0, 8, -8, 0, -2, 20], [-4, 1, 1, -15, 0, 0], [0, 18, -18, 0, -4, -24], [-46, -5, -5, -9, 0, 0], [0, 6, -6, 0, -12, -6], [0, -6, 6, 0, 44, 20], [0, -13, -13, -24, 0, 0], [24, 4, 4, 16, 0, 0], [0, 6, -6, 0, -2, -4], [0, 24, -24, 0, -16, -16], [28, 17, 17, 9, 0, 0], [18, 0, 0, -4, 0, 0], [0, 2, -2, 0, 44, 0], [0, 8, -8, 0, 16, 4], [-2, -14, -14, -30, 0, 0], [22, -13, -13, 6, 0, 0], [56, -7, -7, -11, 0, 0], [0, -11, 11, 0, -36, -9], [8, -7, -7, -17, 0, 0], [-14, -7, -7, 2, 0, 0], [0, -6, 6, 0, -12, -18], [-20, 15, 15, 18, 0, 0], [0, -6, 6, 0, -60, -2], [0, -2, 2, 0, -26, 6], [22, -2, -2, -16, 0, 0], [-6, -11, -11, -7, 0, 0], [0, 1, -1, 0, -6, 18], [0, 2, -2, 0, -34, -20], [0, 8, -8, 0, 34, 12], [22, -16, -16, 0, 0, 0], [0, 4, -4, 0, 10, 8], [16, -12, -12, -22, 0, 0], [0, 2, -2, 0, -20, -11], [64, 0, 0, 4, 0, 0], [0, 20, -20, 0, 16, 5], [20, 13, 13, -3, 0, 0], [0, -6, 6, 0, 30, -8], [6, -10, -10, -22, 0, 0], [0, 19, -19, 0, 10, -31], [0, -14, 14, 0, -22, 16], [0, 12, -12, 0, 22, -12], [16, 9, 9, 10, 0, 0], [48, -1, -1, 15, 0, 0], [0, -16, 16, 0, -38, 0], [-4, 2, 2, -28, 0, 0], [-8, -19, -19, -34, 0, 0], [0, -6, 6, 0, 8, -20], [28, -6, -6, 6, 0, 0], [0, -22, 22, 0, 22, 25], [-46, 1, 1, -16, 0, 0], [0, -16, 16, 0, 28, 12], [-4, 3, 3, 5, 0, 0], [0, -3, 3, 0, -14, -21], [20, 3, 3, -11, 0, 0], [0, -22, 22, 0, 22, 20], [0, -14, -14, -24, 0, 0], [-18, -5, -5, 8, 0, 0], [0, 0, 0, 0, 28, -4], [0, -18, 18, 0, 10, 0], [0, -4, 4, 0, -58, 10], [2, -12, -12, -16, 0, 0], [0, -10, 10, 0, 30, -8], [6, 7, 7, -15, 0, 0], [0, 3, -3, 0, -50, 6], [0, 10, -10, 0, -58, -6], [0, 12, -12, 0, -22, -11], [0, 8, -8, 0, 30, 26], [6, 11, 11, -3, 0, 0], [-62, 5, 5, -5, 0, 0], [-38, -19, -19, -10, 0, 0], [0, -14, 14, 0, -4, -14], [-32, -9, -9, 1, 0, 0], [-20, -24, -24, -12, 0, 0], [0, 24, -24, 0, -6, -26], [-6, -7, -7, 2, 0, 0], [62, 5, 5, -1, 0, 0], [0, 3, -3, 0, -52, -7], [-2, 18, 18, 0, 0, 0], [0, 11, -11, 0, 44, -13], [12, -9, -9, 0, 0, 0], [0, 0, 0, 0, 2, 16], [0, -18, 18, 0, -4, 14], [0, -10, 10, 0, 12, 30], [-22, 13, 13, 12, 0, 0], [0, 1, -1, 0, 38, 1], [0, 15, 15, -1, 0, 0], [0, 16, -16, 0, -34, -11], [0, -8, 8, 0, 36, 8], [0, 15, -15, 0, 26, 1], [-14, -11, -11, -21, 0, 0], [0, -6, 6, 0, 26, -1], [-38, -35, -35, -17, 0, 0], [0, 5, -5, 0, 74, 10], [-74, 1, 1, 7, 0, 0], [0, -12, 12, 0, -48, 8], [12, 10, 10, -8, 0, 0], [-30, -10, -10, -4, 0, 0], [-18, -9, -9, -23, 0, 0], [-6, 10, 10, 12, 0, 0], [40, 9, 9, 24, 0, 0], [0, 10, -10, 0, -56, -18], [0, 14, -14, 0, -36, -10], [0, -9, 9, 0, 52, 19], [36, 17, 17, 8, 0, 0], [-12, 22, 22, 48, 0, 0], [0, -10, 10, 0, 22, 0], [54, -5, -5, 29, 0, 0], [0, -11, 11, 0, 18, 10], [18, 29, 29, 18, 0, 0], [0, 21, -21, 0, 4, -17], [0, -2, 2, 0, 42, -10], [0, 8, -8, 0, -4, 14], [8, 24, 24, 20, 0, 0], [0, 2, -2, 0, -16, -18], [-4, 3, 3, 0, 0, 0], [8, -25, -25, -9, 0, 0], [0, -14, 14, 0, 0, 16], [0, -8, 8, 0, 18, 28], [0, 13, -13, 0, -34, -23], [12, -5, -5, -21, 0, 0], [0, -5, 5, 0, -32, 10], [0, -4, 4, 0, 0, -20], [-2, -5, -5, -2, 0, 0], [0, 5, -5, 0, 16, 1], [0, 4, -4, 0, -58, -3], [0, 8, -8, 0, 40, 6], [-18, 12, 12, 20, 0, 0], [2, -19, -19, 4, 0, 0], [48, 22, 22, 16, 0, 0]];
    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_y();".
// 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_y(: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_y();".
// 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_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5040_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![8, 8, -7, 1]>,<13,R![11236, 0, 1544, 0, 69, 0, 1]>,<17,R![400, 0, 536, 0, 49, 0, 1]>,<19,R![-8, -14, -4, 1]>,<29,R![108, -72, -3, 1]>],Snew);
    return Vf;
end function;