
// Make newform 740.2.i.a in Magma, downloaded from the LMFDB on 09 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_740_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_740_i_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_740_2_i_a();".
// 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_740_2_i_a();".
// 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 := [3, 0, 64, 0, 426, 0, 1008, 0, 727, 0, 204, 0, 24, 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, 0, 0, 0], [60, 0, 863, 0, 1496, 0, 893, 0, 226, 0, 25, 0, 1, 0], [141, 0, 890, 0, 1496, 0, 893, 0, 226, 0, 25, 0, 1, 0], [30, 27, 361, 0, 954, 0, 713, 0, 203, 0, 24, 0, 1, 0], [-30, 0, -361, 0, -954, 0, -713, 0, -203, 0, -24, 0, -1, 0], [303, 0, 3680, 0, 11588, 0, 9017, 0, 2602, 0, 310, 0, 13, 0], [-303, 765, -3680, 5637, -11588, 10806, -9017, 7092, -2602, 1905, -310, 219, -13, 9], [60, -540, 863, -7101, 1496, -18009, 893, -13077, 226, -3672, 25, -432, 1, -18], [-354, 0, -6034, 0, -20287, 0, -15895, 0, -4595, 0, -548, 0, -23, 0], [279, 0, 3306, 0, 9367, 0, 7067, 0, 2016, 0, 239, 0, 10, 0], [9, -279, 0, -3306, 0, -9367, 0, -7067, 0, -2016, 0, -239, 0, -10], [-279, -576, -3306, -3834, -9367, -9072, -7067, -6543, -2016, -1836, -239, -216, -10, -9], [-141, -1881, -890, -21570, -1496, -63432, -893, -48387, -226, -13863, -25, -1647, -1, -69], [354, 2772, 6034, 38589, 20287, 121431, 15895, 93933, 4595, 27039, 548, 3219, 23, 135]];
    Rf_basisdens := [1, 27, 27, 18, 9, 27, 54, 54, 27, 9, 18, 18, 54, 54];
    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_740_i();"
function MakeCharacter_740_i()
    N := 740;
    order := 3;
    char_gens := [371, 297, 261];
    v := [3, 3, 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_740_i_Hecke();"
function MakeCharacter_740_i_Hecke( : Kf := false)
    N := 740;
    order := 3;
    char_gens := [371, 297, 261];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 1, 1, 0, 0, 0, -1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, -1, -1, 0, -1, 0, -1, -1], [0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, -1, 0], [1, 0, 0, -1, 0, 0, 0, -1, -1, 0, -1, 0, -1, -1], [1, 1, 0, 0, 1, 1, 0, 0, 0, -1, 0, 0, 0, 0], [0, 1, 1, 0, 0, -1, 0, 0, 1, 1, 0, 0, 0, 0], [-1, 1, 1, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0], [1, 0, 1, -1, 1, 1, 0, -1, -1, 0, 1, -1, 1, 0], [0, 0, 0, 2, 0, -1, -1, 1, 1, 0, 0, 1, 0, 1], [-1, -1, -2, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 2, 0, 0, -1, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 2, 0, 0, 1], [0, -1, -1, 1, 1, 0, 2, 1, 0, 2, -3, 2, -1, 1], [1, 0, 0, 0, 0, 0, 0, -1, -1, 0, -1, 1, 1, -1], [1, 0, 0, -3, 0, 2, 2, -2, 1, 0, -1, -2, 1, 1], [-1, 0, 0, 0, 0, 1, 1, -2, -1, 0, 1, 2, -1, -1], [-3, -1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 1, 1, 1, -1, 0, 0, 0, 2, -1], [0, -1, -1, 0, 0, 0, -3, 1, 0, -2, 1, -2, -1, -1], [0, 3, -1, 0, 0, 0, -1, -3, 0, -3, 2, -3, -1, -1], [-5, 1, 0, 0, 2, -2, 0, 0, -1, 0, 0, 0, 0, 0], [-1, 3, 3, 0, -3, -1, 0, 0, 1, -3, 0, 0, 0, 0], [4, -1, -3, 0, -1, 1, 0, 0, 0, 1, 0, 0, 0, 0], [4, 0, 0, -1, 0, 2, 2, 2, -2, 0, -4, -1, 0, -2], [0, 0, -1, -1, -1, 0, 1, 0, 0, -2, 2, -2, -1, 1], [0, 0, 4, -1, -1, 0, 1, 0, 0, 1, 5, 1, 4, -1], [0, 1, -3, 0, 0, 0, 1, -1, 0, -1, 3, -1, -3, 1], [0, 2, 2, -1, -1, 0, 1, -2, 0, -2, 5, -2, 2, 0], [-1, 0, -3, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0], [0, -3, -1, -2, -2, 0, 0, 3, 0, 0, 2, 0, -1, -2], [-1, -5, 0, 0, -1, 0, 0, 0, -4, 1, 0, 0, 0, 0], [-7, 0, 0, 0, 0, -1, -1, -1, -2, 0, 7, -1, 3, -2], [0, 2, -2, -3, -3, 0, 1, -2, 0, 1, 0, 1, -2, -3], [0, -4, -1, -2, -2, 0, 2, 4, 0, 0, -5, 0, -1, 1], [4, 0, 0, 2, 0, -1, -1, 1, 0, 0, -4, 0, 1, 0], [0, -2, 3, 0, 0, 0, 0, 2, 0, -2, -5, -2, 3, 3], [-4, -1, -1, 0, 4, 2, 0, 0, 0, 1, 0, 0, 0, 0], [7, 0, 0, -1, 0, 1, 1, 2, -1, 0, -7, 3, 2, -1], [-1, 0, 2, 0, 2, 1, 0, 0, -2, -3, 0, 0, 0, 0], [-2, -1, -1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, -1, -1, -1, 0, 2, -1, 0, 1, 7, 1, -1, 0], [-3, 0, -1, 0, -1, 0, 0, 0, 2, 1, 0, 0, 0, 0], [-3, -3, -3, 0, -1, 1, 0, 0, -1, -3, 0, 0, 0, 0], [-7, 1, -3, 0, 0, 0, 0, 0, 4, -2, 0, 0, 0, 0], [-1, 0, 0, -1, 0, -1, -1, 3, 1, 0, 1, -1, -2, 1], [-6, 0, 0, -2, 0, 2, 2, 2, 3, 0, 6, 0, 4, 3], [0, 2, -1, 0, 1, -1, 0, 0, -4, -1, 0, 0, 0, 0], [0, 4, 0, 6, 6, 0, -2, -4, 0, -1, -1, -1, 0, 4], [1, 0, 0, 3, 0, -2, -2, -2, 4, 0, -1, -3, -2, 4], [3, -1, 0, 0, -2, -4, 0, 0, 2, 3, 0, 0, 0, 0], [0, -3, 1, 1, 1, 0, -1, 3, 0, 2, -3, 2, 1, -3], [-3, 0, 0, -3, 0, -2, -2, 2, 2, 0, 3, 1, 0, 2], [0, -2, -3, 0, 5, 6, 0, 0, -2, 0, 0, 0, 0, 0], [0, 2, 1, -2, -2, 0, 3, -2, 0, 1, 9, 1, 1, 0], [4, 0, 0, -1, 0, 0, 0, -2, -5, 0, -4, -2, 0, -5], [0, 6, 2, 1, 1, 0, 2, -6, 0, 0, -4, 0, 2, -1], [-4, 0, 0, -2, 0, -2, -2, 0, 2, 0, 4, -2, 2, 2], [0, 0, 0, 3, 0, 2, 2, -2, 1, 0, 0, 3, 1, 1], [-2, 5, 0, 0, 1, -3, 0, 0, 3, 1, 0, 0, 0, 0], [0, -9, 0, 2, 2, 0, 0, 9, 0, 5, 0, 5, 0, 3], [0, -2, -1, 2, 2, 0, 1, 2, 0, 3, 4, 3, -1, 2], [0, -1, -3, 2, 2, 0, 3, 1, 0, 2, 4, 2, -3, 2], [0, -3, 4, -2, -2, 0, 4, 3, 0, -4, 0, -4, 4, 2], [3, 0, 0, 0, 0, -1, -1, -4, -3, 0, -3, 0, -2, -3], [12, -1, -1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0], [0, 2, -4, 1, 1, 0, 2, -2, 0, 2, -1, 2, -4, -1], [0, 1, -3, 1, 1, 0, -3, -1, 0, 0, 6, 0, -3, -1], [2, 2, 4, 0, 0, -1, 0, 0, 0, 3, 0, 0, 0, 0], [-13, 0, 0, 4, 0, -2, -2, -3, -1, 0, 13, -3, 3, -1], [-7, 0, 0, -2, 0, 0, 0, 2, -2, 0, 7, -1, -5, -2], [0, -3, 0, -3, -3, 0, -4, 3, 0, 0, 3, 0, 0, 2], [-6, 0, 0, 4, 0, -2, -2, -5, 1, 0, 6, -2, -1, 1], [7, 0, 0, -5, 0, 1, 1, -2, 3, 0, -7, 0, 0, 3], [-3, -4, -1, 0, 5, -2, 0, 0, -8, 0, 0, 0, 0, 0], [-1, 10, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 8, 0, 0, -1, 2, -1, 2, -2], [0, -2, 3, 6, 6, 0, -5, 2, 0, -1, 11, -1, 3, 4], [2, -1, 3, 0, -1, -6, 0, 0, -1, 4, 0, 0, 0, 0], [12, 0, 0, -4, 0, 4, 4, -3, -3, 0, -12, -1, -2, -3], [9, -6, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 0, 0], [10, 0, 0, 2, 0, -1, -1, 8, 5, 0, -10, 0, 2, 5], [-3, 1, -5, 0, -2, 0, 0, 0, -2, -1, 0, 0, 0, 0], [-4, 0, 0, 0, 0, 0, 0, 6, -5, 0, 4, 1, -5, -5], [5, 0, 0, -4, 0, 4, 4, -4, 4, 0, -5, 0, 0, 4], [0, 1, 1, 0, 0, 0, -4, -1, 0, 0, 4, 0, 1, -6], [11, 0, 0, -4, 0, -2, -2, -7, -2, 0, -11, 0, 0, -2], [-10, 1, -1, 0, 2, 1, 0, 0, -4, -3, 0, 0, 0, 0], [0, -2, -2, 5, 5, 0, -5, 2, 0, 6, 2, 6, -2, 1], [-5, 1, -4, 0, -1, 3, 0, 0, 2, -5, 0, 0, 0, 0], [-17, -5, -1, 0, 0, -4, 0, 0, -3, -2, 0, 0, 0, 0], [3, 0, 0, -1, 0, 0, 0, -8, 2, 0, -3, 1, 7, 2], [0, 5, -1, 3, 3, 0, -5, -5, 0, -2, 5, -2, -1, 6], [0, -12, 1, -4, -4, 0, 1, 12, 0, 3, 11, 3, 1, -4], [-9, 0, 0, -1, 0, -1, -1, 6, 3, 0, 9, 1, 2, 3], [-7, 0, 0, 0, 0, -4, -4, -4, 1, 0, 7, -4, 3, 1], [-10, -2, -3, 0, 5, 3, 0, 0, 1, 6, 0, 0, 0, 0], [0, -2, 5, 0, -1, 3, 0, 0, 0, -1, 0, 0, 0, 0], [0, -2, 2, 1, 1, 0, 4, 2, 0, -5, 6, -5, 2, 2], [-1, 4, -2, 0, 0, -1, 0, 0, 2, -4, 0, 0, 0, 0], [2, 7, -4, 0, 0, -4, 0, 0, -1, 1, 0, 0, 0, 0], [0, 0, -1, -2, -2, 0, 0, 0, 0, -5, 5, -5, -1, -2], [0, -3, 3, 6, 6, 0, -3, 3, 0, 5, -7, 5, 3, 4], [5, 0, 0, 2, 0, -3, -3, 1, 5, 0, -5, 1, 3, 5], [-10, -1, -4, 0, -4, -2, 0, 0, 3, 5, 0, 0, 0, 0], [3, 0, 0, 5, 0, 1, 1, -1, 5, 0, -3, 3, -4, 5], [0, 0, -1, -3, -3, 0, 8, 0, 0, 2, -11, 2, -1, -3], [0, -2, 5, -3, -3, 0, 6, 2, 0, -3, -18, -3, 5, 1], [0, 2, -3, 5, 5, 0, -6, -2, 0, -7, -3, -7, -3, 0], [0, 7, 6, -2, -2, 0, -2, -7, 0, -7, -5, -7, 6, 6], [0, 6, 6, 0, 0, -2, 0, 0, 2, -7, 0, 0, 0, 0], [0, 2, -3, -5, -5, 0, -2, -2, 0, 4, 1, 4, -3, -5], [0, -4, -1, 3, 3, 0, 2, 4, 0, -7, 5, -7, -1, 3], [7, -9, 4, 0, 0, -2, 0, 0, -2, -2, 0, 0, 0, 0], [5, 0, 0, 1, 0, -2, -2, -4, -9, 0, -5, 6, -1, -9], [2, 0, 0, 6, 0, -4, -4, 4, 8, 0, -2, -1, -4, 8], [-5, 0, 0, 4, 0, 1, 1, 1, -6, 0, 5, 4, -1, -6], [14, 0, 0, 3, 0, 3, 3, 7, 9, 0, -14, -3, 2, 9], [-4, 0, 0, 2, 0, 0, 0, 14, 1, 0, 4, 3, -2, 1], [-9, 3, 2, 0, -3, -2, 0, 0, 3, 1, 0, 0, 0, 0], [0, 4, 2, -9, -9, 0, 1, -4, 0, -8, -7, -8, 2, -5], [0, 5, 1, -9, -9, 0, 0, -5, 0, -2, -14, -2, 1, -8], [0, 3, -8, -1, -1, 0, 5, -3, 0, 2, -6, 2, -8, -3], [-10, -8, -1, 0, -4, -2, 0, 0, -2, 1, 0, 0, 0, 0], [0, -1, -2, 0, 0, 0, -2, 1, 0, -1, -3, -1, -2, -2], [-4, 0, 0, 1, 0, -7, -7, -1, 4, 0, 4, -2, 1, 4], [11, 0, 0, 0, 0, 2, 2, 4, -3, 0, -11, 6, 3, -3], [3, -6, 1, 0, -6, -1, 0, 0, -1, 1, 0, 0, 0, 0], [-23, 0, 0, -2, 0, -2, -2, 4, 0, 0, 23, 0, 6, 0], [5, 1, 2, 0, 0, -4, 0, 0, 1, 6, 0, 0, 0, 0], [0, -1, 1, 1, 1, 0, 0, 1, 0, 2, 31, 2, 1, -2], [0, 2, 5, -7, -7, 0, 3, -2, 0, 0, 2, 0, 5, -3], [8, 8, -2, 0, -1, -8, 0, 0, -1, 6, 0, 0, 0, 0], [10, 0, 0, -3, 0, -4, -4, -7, 1, 0, -10, -1, 5, 1], [-9, -2, 0, 0, 0, 5, 0, 0, -3, -7, 0, 0, 0, 0], [0, 4, 1, 2, 2, 0, -5, -4, 0, 0, 6, 0, 1, -1], [12, 0, 0, -6, 0, 4, 4, 8, -4, 0, -12, -3, 0, -4], [-12, 0, 0, 4, 0, -2, -2, 8, 9, 0, 12, 2, 3, 9], [-11, 0, 0, -4, 0, -2, -2, 0, -1, 0, 11, -3, 3, -1], [0, -3, -1, -2, -2, 0, -6, 3, 0, -6, -2, -6, -1, 3], [7, 0, 0, 5, 0, 2, 2, -8, 0, 0, -7, -4, -3, 0], [0, -5, -1, 2, 2, 0, -4, 5, 0, 6, -12, 6, -1, 1], [0, -4, 2, 11, 11, 0, -2, 4, 0, 4, -4, 4, 2, 6], [0, -5, -7, 4, 4, 0, -1, 5, 0, 8, 2, 8, -7, 3], [-21, -3, -1, 0, 3, 1, 0, 0, 0, -3, 0, 0, 0, 0], [-4, 6, -1, 0, 0, -6, 0, 0, -3, -4, 0, 0, 0, 0], [0, 2, 0, -4, -4, 0, 2, -2, 0, 4, 14, 4, 0, -9], [17, 6, 7, 0, 1, 7, 0, 0, 0, 1, 0, 0, 0, 0], [-14, 0, 0, -6, 0, -3, -3, -2, -6, 0, 14, 1, -2, -6], [19, 0, 0, 1, 0, 3, 3, -10, -1, 0, -19, 0, -2, -1], [-1, 3, 2, 0, -6, -9, 0, 0, 1, -2, 0, 0, 0, 0], [9, 0, 0, 5, 0, -9, -9, 5, 0, 0, -9, 1, -5, 0], [-1, -5, 1, 0, 7, 6, 0, 0, 2, 6, 0, 0, 0, 0], [-10, -4, 2, 0, -3, 5, 0, 0, -1, -8, 0, 0, 0, 0], [-6, 0, 0, 2, 0, 6, 6, -8, 7, 0, 6, 1, -5, 7], [0, -5, 5, 1, 1, 0, -8, 5, 0, -2, 25, -2, 5, -1], [0, 3, 0, 3, 3, 0, -9, -3, 0, 3, 0, 3, 0, 6], [0, 13, -4, -5, -5, 0, -5, -13, 0, -3, -22, -3, -4, -5], [0, 3, 5, 2, 2, 0, 2, -3, 0, 0, 19, 0, 5, -2], [14, 0, 0, 3, 0, -4, -4, 8, 4, 0, -14, 2, -1, 4], [0, 4, -9, -7, -7, 0, 6, -4, 0, 1, -15, 1, -9, -7], [0, 10, -5, -1, -1, 0, -1, -10, 0, -3, -10, -3, -5, 1], [0, 2, 3, 1, 1, 0, -7, -2, 0, -5, -4, -5, 3, 12], [8, 3, -4, 0, 3, 8, 0, 0, 3, -6, 0, 0, 0, 0], [0, -2, 7, 2, 2, 0, 5, 2, 0, 4, 4, 4, 7, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_740_i_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_740_2_i_a();".
// 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_740_2_i_a(:prec:=14)
    chi := MakeCharacter_740_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_740_2_i_a();".
// 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_740_2_i_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_740_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![361, 1539, 6884, 1397, 6183, 1402, 3969, 486, 1080, 34, 216, 2, 17, 0, 1]>],Snew);
    return Vf;
end function;