
// Make newform 738.2.e.j in Magma, downloaded from the LMFDB on 01 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_738_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_738_e_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_738_2_e_j();".
// 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_738_2_e_j();".
// 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 := [2187, -2916, 2916, -2511, 1971, -1350, 891, -549, 297, -150, 73, -31, 12, -4, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [250047, -114696, 172449, -124389, 82323, -57186, 38331, -19449, 10668, -5102, 1769, -573, 419, -8], [457812, -710775, 423792, -204552, 103410, -66960, 11376, -13005, 5601, -892, 331, -1044, 841, -130], [925830, -672867, 480897, -177822, 146907, -47358, 35271, -11304, -3192, 5345, -623, 1701, -914, 203], [446877, -672867, 1944, -177822, 146907, -47358, 35271, -11304, -3192, 5345, -623, 1701, -914, 203], [-1744497, 878445, -829602, 803250, -500121, 388665, -221931, 121833, -48174, 30548, -10166, 3204, -1340, -253], [17496, 155763, -61803, 2565, -4158, 4509, 9729, -4824, 3294, -3745, 2125, -231, 472, -124], [-2916, 2916, -2511, 1971, -1350, 891, -549, 297, -150, 73, -31, 12, -4, 1], [-289413, 52488, -274347, 242298, -54243, 147069, -62028, 25047, -16437, 7159, -838, -495, -313, -1100], [-1944, -25191, 10152, -16929, 12069, -7947, 5562, -3771, 1897, -1052, 502, -169, 53, -43], [207036, -998001, 805059, -740637, 511218, -260361, 195390, -136143, 65040, -31334, 15011, -6858, 1838, -1370], [1422279, -2042658, 1453869, -1249452, 1049409, -703134, 428688, -267102, 131502, -67969, 32308, -12546, 4522, -1381], [-25515, 29889, -26973, 22005, -15822, 10800, -7443, 4302, -2112, 1073, -509, 207, -80, 26]];
    Rf_basisdens := [1, 1, 159651, 478953, 478953, 478953, 478953, 159651, 729, 478953, 17739, 478953, 478953, 6561];
    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_738_e();"
function MakeCharacter_738_e()
    N := 738;
    order := 3;
    char_gens := [83, 703];
    v := [1, 3];
    // 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_738_e_Hecke();"
function MakeCharacter_738_e_Hecke( : Kf := false)
    N := 738;
    order := 3;
    char_gens := [83, 703];
    char_values := [[-1, 0, 1, 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]];
    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, -1, 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], [1, -1, -1, 0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, -1, 0, 0, 1, -1, 0, -1, 0, 0, 0], [0, 0, 0, 0, -1, -1, 1, -1, 0, 0, 1, -1, 0, -1], [-2, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, -2], [2, 0, -1, -1, 0, 1, 1, 1, -1, -2, -1, 2, 2, 0], [4, 0, -3, 0, -1, 0, 2, 0, 1, -1, -1, 0, 0, -1], [0, -1, 4, 0, 1, 0, -1, 0, 0, 0, 0, 0, -1, 0], [0, 0, 1, 0, -3, 2, 2, 2, -3, -1, 1, 0, 0, -3], [2, 3, 0, 3, 0, 3, 0, 2, 0, 0, 2, 0, 0, 0], [-1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 1, 0, 0, -1, 1, 0, 1, 0, 0, 0], [0, 1, 0, -1, -1, 0, 2, 0, -2, -1, -2, 0, 3, 0], [-2, -1, 0, 0, 0, 0, 0, -3, 1, 0, -3, 0, 0, 0], [0, -3, -2, 0, 0, -1, -2, 2, 0, 2, -2, -2, 0, 0], [0, -2, 9, -1, -1, 0, 0, -3, 1, 0, 1, 0, 0, 0], [2, 1, -2, 0, 1, 0, -1, -1, 2, 0, -1, 1, 0, 1], [1, -3, -1, 0, 0, 0, 1, -1, 3, -2, -1, 0, 0, -3], [-4, -4, 2, -3, 0, -4, -2, -1, 1, 4, 0, -1, -1, 0], [0, 0, -3, 0, -2, 0, -1, 3, -2, 0, -2, 0, -1, 0], [0, -1, 5, 2, 1, 0, -1, 1, 0, 1, 0, 0, -2, 0], [-5, -3, 2, 0, 0, -2, -2, -2, 3, 4, 0, -2, -2, 3], [0, 4, 1, 1, -2, 0, 1, -1, 2, -2, 2, 0, 3, 0], [0, -2, 2, 2, -1, 0, -1, 3, 0, 4, 0, 0, -5, 0], [3, 0, -2, 0, 4, 3, 0, 3, -1, -1, -2, 2, 0, 4], [-4, 5, -2, 2, 0, 2, 2, 1, -3, -4, 1, 0, 0, -2], [0, -1, -2, -2, 0, -3, 2, -1, -1, -4, 0, -1, -1, 1], [6, 1, -4, 0, -2, 2, 1, 1, 1, -2, -2, 3, 0, -2], [1, -1, 0, -3, 0, -3, 0, -4, -2, 0, -4, 0, 0, -2], [-7, 0, 4, 0, 0, -2, -5, -2, 2, 3, 0, -1, 0, 0], [0, 0, -1, -4, 1, 0, 2, 3, -7, -2, -7, 0, 4, 0], [1, -3, -1, 0, 2, -6, 1, -3, 4, 0, -1, -1, 0, 2], [9, -5, -8, 0, -1, 3, 0, 8, -4, -1, -4, 2, 0, -1], [0, 0, -3, 2, 2, 0, 0, 2, 1, 3, 1, 0, -3, 0], [7, -3, -4, 0, -2, 4, -1, 7, -4, -3, -3, 7, 0, -2], [-4, 3, 0, 2, 0, 1, 0, -2, -1, 0, -1, -1, -1, 3], [4, -3, -3, 0, 1, -2, -3, 1, 4, -1, -5, 5, 0, 1], [0, -2, -2, -1, 2, 0, 0, 2, -2, 2, -2, 0, -2, 0], [-8, -1, 1, -3, 0, -1, -1, -5, -2, 2, -7, 2, 2, 6], [4, 3, -2, 3, 0, 1, 2, 5, 0, -4, 7, -2, -2, -4], [0, -1, -1, -4, -2, 0, 2, 3, -5, 1, -5, 0, 1, 0], [-8, -3, 5, 0, -1, -4, -1, -1, 2, 3, -1, -5, 0, -1], [-4, -3, 1, -6, 0, -4, -1, 2, -3, 2, 0, 2, 2, 2], [-6, -5, 4, 1, 0, -4, -4, -5, 6, 8, 0, -5, -5, 5], [-11, -2, 8, 0, 1, -9, -4, -7, 5, 3, 2, -2, 0, 1], [0, -1, 5, -2, 5, 0, 0, -4, 0, -3, 0, 0, 3, 0], [0, 1, -1, 0, 4, 0, -2, 4, 0, 1, 0, 0, -3, 0], [9, 1, -9, 0, 1, -1, 2, -2, 4, 0, -2, -2, 0, 1], [-16, 2, 0, -2, 0, 0, 0, 2, -4, 0, 0, 2, 2, 0], [8, 2, -6, 0, -3, 10, 3, 8, -6, -2, -2, 1, 0, -3], [0, -2, -2, 2, 1, 0, -1, -4, 2, -1, 2, 0, 0, 0], [-4, 3, -1, -1, 0, -2, 1, 0, -4, -2, 1, -1, -1, -1], [8, 2, -8, 0, -3, 6, 1, 4, -3, 0, -1, -1, 0, -3], [0, 3, 6, 4, 1, 0, -2, 4, 4, 3, 4, 0, -5, 0], [4, -2, 1, 1, 0, 2, -1, 3, 3, 2, 2, 1, 1, 1], [2, -3, 0, 0, 0, -3, 0, -3, 3, 0, 0, -3, -3, 0], [0, 0, -1, 3, -2, 0, -2, -2, 3, -1, 3, 0, -1, 0], [0, 1, 1, 6, -6, 0, 0, 1, 1, 1, 1, 0, -1, 0], [-2, -4, 3, 0, 0, -3, 2, 1, 5, -1, -6, 0, 0, 0], [2, 6, -2, 0, 3, 0, 3, -6, 1, 0, 5, -3, 0, 3], [2, 2, 0, 3, 0, 5, 0, 1, 1, 0, -1, 2, 2, 0], [6, 7, -4, 0, 4, 5, 2, -2, -1, -2, 3, 2, 0, 4], [0, -7, 8, -10, -2, 0, 3, -2, -7, 1, -7, 0, 2, 0], [0, -2, 1, 0, -7, 0, -3, 1, 1, 1, 1, 0, -4, 0], [0, 0, 3, 5, 4, 0, 1, -5, 2, -2, 2, 0, 3, 0], [3, -5, -4, 0, 4, 0, -8, 5, -3, 1, -2, 6, 0, 4], [7, 6, -3, 0, 0, 4, 3, -2, 0, -4, 2, 5, 0, 0], [0, -3, -11, -3, -8, 0, 0, -1, -1, 1, -1, 0, -1, 0], [0, 3, 7, 0, 1, 0, 1, -1, 3, 0, 3, 0, 1, 0], [12, 7, -1, 1, 0, 3, 1, 4, -6, -2, 2, 2, 2, 5], [0, -2, -5, -9, -5, 0, 6, -5, -2, 1, -2, 0, 5, 0], [-14, 2, 12, 0, 3, 0, -7, -2, 5, 2, -3, 3, 0, 3], [-3, 4, -2, 7, 0, 4, 2, -1, 3, -4, 2, -3, -3, -6], [-2, -11, -3, 0, 5, -8, -11, 3, 3, 5, -6, 1, 0, 5], [0, 1, -5, 0, -7, 0, -4, 2, 3, 0, 3, 0, -4, 0], [2, 5, -1, 0, 0, 3, 1, 5, -5, -2, 2, 3, 3, 1], [0, 6, 1, 0, -1, -1, 7, -7, -2, -1, 9, -5, 0, -1], [-2, 4, 2, 0, -3, 7, 2, 3, -12, 0, 9, -2, 0, -3], [4, -1, -1, 0, -1, -4, 7, -3, 3, -3, 0, -1, 0, -1], [0, 0, 8, 4, -6, 0, 3, -4, 2, 1, 2, 0, 2, 0], [-2, 9, -1, 4, 0, 2, 1, 1, -5, -2, 3, -2, -2, -3], [-5, 0, 4, 3, 0, -2, -4, -3, 3, 8, 2, -5, -5, 8], [0, -6, -6, 1, 0, 0, 0, 0, -4, 2, -4, 0, -2, 0], [0, 3, -4, 0, 3, 0, -5, 1, 3, -4, 3, 0, -1, 0], [-4, -8, 2, 3, 0, -2, -2, -5, 11, 4, 0, -5, -5, -2], [0, -9, 0, -8, 1, 0, -3, -3, -1, 2, -1, 0, -5, 0], [0, 1, -4, 9, -4, 0, -6, 5, 2, 0, 2, 0, -6, 0], [-15, 3, 12, 0, 6, -7, -7, -10, 10, 3, 0, 1, 0, 6], [-5, -15, 6, -4, 0, -12, -6, -4, 11, 12, 4, -8, -8, 3], [0, -1, 6, -1, -4, 0, -2, 3, 0, 2, 0, 0, -4, 0], [1, -7, 2, 1, 0, -2, -2, -1, 8, 4, 2, -3, -3, 7], [2, -2, -2, 0, -4, -4, 0, -2, 3, 0, -1, 0, 0, -4], [1, -9, 2, 0, 2, -3, -1, 6, -2, -3, -4, 7, 0, 2], [3, -4, -1, -3, 0, 3, 1, -1, 1, -2, -7, 6, 6, 7], [-16, 8, 14, 0, -1, -3, -1, -11, 0, 2, 11, -3, 0, -1], [4, -10, 0, -10, 0, -3, 0, -3, 0, 0, -10, 7, 7, 1], [2, -8, 3, -2, 0, -3, -3, -8, 6, 6, -7, -1, -1, 1], [-12, 6, 3, 9, 0, 2, -3, 3, 3, 6, 10, -7, -7, -5], [0, 4, -3, 1, 7, 0, 5, 2, -6, -3, -6, 0, 8, 0], [-27, 1, 2, -2, 0, 0, -2, 0, -3, 4, -2, 2, 2, 1], [0, 0, 2, 0, 1, 2, 7, 2, -7, -2, 5, -3, 0, 1], [0, -10, 7, -2, 3, 0, 4, -9, -4, 1, -4, 0, 3, 0], [10, -2, -5, 0, 1, -1, 3, 1, 1, -5, -2, 7, 0, 1], [16, -3, -6, -5, 0, 3, 6, 4, -2, -12, -4, 8, 8, -4], [0, 7, 15, 6, 7, 0, 2, -3, 4, -4, 4, 0, 6, 0], [14, 8, -5, 5, 0, 10, 5, 11, -3, -10, 6, 5, 5, -1], [13, -3, -10, 0, 6, -4, -4, -1, 4, -3, -3, 10, 0, 6], [0, 5, 0, 11, 1, 0, -3, 3, 5, 0, 5, 0, -3, 0], [-2, -2, 0, 0, 6, -8, 2, -6, 10, 2, -4, -6, 0, 6], [9, 12, -1, 11, 0, 7, 1, 3, -1, -2, 7, -4, -4, -1], [15, -12, -17, 0, 9, -2, -5, 10, 4, 2, -14, 1, 0, 9], [0, 4, -1, 5, 12, 0, -1, 1, 2, -2, 2, 0, 1, 0], [29, -3, -4, 0, 0, 2, 4, -1, 3, -8, -3, 2, 2, 0], [0, -4, 18, -9, -2, 0, 9, -11, -1, 1, -1, 0, 8, 0], [0, 9, 6, 0, 1, 8, 5, -1, 2, -6, -1, 7, 0, 1], [-7, 4, 3, 7, 0, 3, -3, 1, 3, 6, 5, -4, -4, -5], [4, 10, -2, 0, -2, -1, 10, -11, 0, -2, 11, -6, 0, -2], [0, 2, -2, 2, 12, 0, 0, -1, 3, 0, 3, 0, 0, 0], [-2, -2, 6, 0, -3, -6, 9, -4, -7, -4, 11, -1, 0, -3], [0, 5, -1, 11, 0, 0, 1, -6, 6, -4, 6, 0, 5, 0], [0, -1, 5, -9, -10, 0, 4, -1, -5, -1, -5, 0, 5, 0], [-8, 2, -5, -2, 0, 1, 5, 5, -4, -10, 2, 3, 3, -3], [0, -6, 5, 8, -4, 0, -2, -5, 0, -1, 0, 0, -1, 0], [7, 12, 1, 5, 0, 9, -1, 7, -7, 2, 3, 4, 4, 3], [0, 3, 10, -5, 0, 0, -3, 4, 3, 1, 3, 0, -4, 0], [7, -7, -2, -6, 0, 3, 2, 3, 1, -4, -6, 9, 9, 12], [0, -3, -14, -3, -9, 0, -5, 9, -3, 4, -3, 0, -9, 0], [-3, -5, 1, 0, -9, -2, 4, 3, -1, 2, -2, -8, 0, -9], [0, 9, -4, 1, 0, 4, 4, 15, -8, -8, 12, 3, 3, -11], [3, 6, -5, 0, 2, 4, -4, -2, 1, 2, 1, 0, 0, 2], [-24, -2, 18, 0, 1, -10, -11, -8, 8, 6, 0, -1, 0, 1], [-2, 6, -1, -1, 0, 2, 1, -2, -7, -2, -5, 3, 3, -1], [14, 7, -12, 0, 2, 6, 5, -1, 1, -2, 0, -1, 0, 2], [-5, 6, 5, 0, -1, 4, -1, -2, -6, 0, 8, 1, 0, -1], [5, -16, 0, -9, 0, -5, 0, -1, 7, 0, -5, 4, 4, -5], [-14, -2, 8, 0, 2, -4, -4, -2, 3, 6, -1, -8, 0, 2], [10, 4, -4, 0, 0, 10, 6, 6, -6, -6, 0, 6, 0, 0], [6, -9, 3, 1, 0, 2, -3, -2, 10, 6, -3, 1, 1, 5], [3, 10, -1, 3, 0, 9, 1, 5, -7, -2, -1, 6, 6, -7], [0, 9, -6, 14, 7, 0, -2, 8, 0, -3, 0, 0, 1, 0], [-18, 8, 19, 0, 4, -5, 10, -13, 6, -1, 7, -8, 0, 4], [1, 1, -4, -13, 0, -8, 4, 3, -14, -8, -2, 5, 5, -4], [-6, -6, 1, -4, 0, -6, -1, -1, 2, 2, 1, -2, -2, -4], [0, 3, -5, -3, -1, 0, 5, -15, 8, -5, 8, 0, 10, 0], [0, 6, 9, -7, 4, 0, 0, 6, -3, -3, -3, 0, 3, 0], [0, 0, 4, 6, 5, 0, -7, 0, 6, -1, 6, 0, -6, 0], [-12, -2, 7, 0, 8, 3, -16, 5, -2, 5, -3, 6, 0, 8], [19, -8, -1, -2, 0, 3, 1, 3, 6, -2, -2, 5, 5, 12], [14, 10, -15, 0, 0, 3, 6, -7, 7, 1, 0, -8, 0, 0], [14, 14, -2, 14, 0, 7, 2, 6, 0, -4, 13, -7, -7, 4], [3, -3, 4, -3, 0, -4, -4, 1, 0, 8, 2, -1, -1, 8], [8, 16, -7, 0, 2, 5, 10, -11, 0, -1, 11, -8, 0, 2], [0, 6, 5, 7, 3, 0, -6, 1, 11, 0, 11, 0, -6, 0], [0, 5, -1, 6, -4, 0, 2, 7, -7, -3, -7, 0, 5, 0], [6, -5, -2, 1, 0, 1, 2, -4, 6, -4, -4, 0, 0, 6], [0, -2, -7, -9, -3, 0, 3, -14, 8, -1, 8, 0, 4, 0], [7, 1, -4, -8, 0, -5, 4, 5, -9, -8, 2, 3, 3, 0], [-20, -4, 13, 0, 2, -5, -12, -1, 2, 7, -1, -2, 0, 2], [0, -3, -5, 7, 4, 0, -5, -2, 10, 6, 10, 0, -11, 0], [-6, -22, -1, -16, 0, -8, 1, -2, 6, -2, -10, 8, 8, -3], [-13, -6, 17, 0, -1, -1, 4, 5, 4, -4, -9, 4, 0, -1], [15, 7, 2, 7, 0, 0, -2, 4, 0, 4, 11, -7, -7, -10], [1, -3, -3, 0, -2, -4, 3, -1, 8, 2, -7, -7, 0, -2], [0, -3, -12, -2, -7, 0, -7, 1, 2, -1, 2, 0, -6, 0], [17, 7, 0, 4, 0, -6, 0, 4, -3, 0, 14, -10, -10, -7], [0, 10, -9, 5, 13, 0, 0, 14, -2, 2, -2, 0, -2, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_738_e_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_738_2_e_j();".
// 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_738_2_e_j(:prec:=14)
    chi := MakeCharacter_738_e();
    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_738_2_e_j();".
// 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_738_2_e_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_738_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1936, 1408, 10000, -4944, 38628, -2040, 16792, -6470, 5389, -1356, 567, -102, 37, -5, 1]>,<7,R![64, -128, 448, -288, 1176, -864, 1708, -494, 829, -270, 273, -36, 19, 1, 1]>],Snew);
    return Vf;
end function;