
// Make newform 3800.2.a.ba in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3800_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_3800_2_a_ba();".
// 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_3800_2_a_ba();".
// 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, -14, 15, 16, -10, -2, 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], [-4, -1, 1, 0, 0, 0], [5, 6, -9, -1, 1, 0], [-8, 6, 15, -9, -2, 1], [-7, 15, 15, -10, -2, 1]];
    Rf_basisdens := [1, 1, 1, 2, 2, 2];
    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_3800_a();"
function MakeCharacter_3800_a()
    N := 3800;
    order := 1;
    char_gens := [951, 1901, 1977, 401];
    v := [1, 1, 1, 1];
    // 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;

function MakeCharacter_3800_a_Hecke(Kf)
    return MakeCharacter_3800_a();
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, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, -1, 0, 0, -1, 0], [1, -1, 0, 1, -1, 1], [0, 0, 1, -1, -1, 0], [3, -1, 0, 0, 1, -1], [-1, 0, 0, 0, 0, 0], [-2, 1, -1, 0, 1, -2], [1, -1, 0, -1, 0, 1], [0, 2, -1, -1, 0, -2], [2, -3, 0, 0, 0, 1], [-1, 1, -1, -1, 0, 1], [-2, -2, 0, 1, -1, 2], [-1, 0, -1, 0, 1, 0], [-1, 0, 0, 3, 0, 1], [1, -4, 0, 0, 0, 1], [0, 4, 1, -1, 0, 0], [2, -1, 0, 1, 0, -2], [2, 2, -2, -1, 1, -2], [-2, 1, -1, -3, 0, -2], [-3, -1, -2, -2, 0, -3], [-5, -1, -1, 1, 0, 1], [3, -1, -1, 2, 1, 1], [2, -2, 1, 1, -2, 4], [1, 1, 2, -2, -2, 1], [-1, 1, 2, 0, 0, -1], [1, 3, 1, 0, 2, -1], [4, -5, -1, 1, -3, 2], [2, 0, -1, 0, 3, 0], [-5, 5, 1, -2, -1, 1], [9, -3, -3, 3, 2, -1], [12, 1, 1, 1, 1, 0], [-4, 0, -2, -1, 3, 0], [7, 3, 1, 1, -2, -1], [10, -2, 1, 2, -1, 2], [1, -1, -1, -2, -3, -1], [2, 2, 3, -1, -2, 2], [-1, 5, -1, -3, 2, -1], [-6, -1, 0, -1, -3, 5], [1, 0, 2, -2, 2, -2], [-1, -2, 0, -2, -2, 0], [4, 1, 2, 1, -5, 6], [-1, 5, 1, -1, 0, -1], [1, -5, -2, 3, -5, 3], [4, 1, 0, -1, 2, -2], [11, 2, 2, -3, 1, -3], [-6, 4, 2, -3, -3, 0], [2, -1, -1, 3, 5, 2], [10, 0, -1, 4, -1, 0], [6, 3, -2, 4, -2, 1], [3, -2, -1, 1, 0, 1], [5, -1, 1, -1, -2, -3], [4, 4, 1, 0, 1, -2], [11, -3, 0, -2, -2, 1], [-4, -4, -1, 3, -4, 2], [-1, 2, -1, -3, 2, -2], [2, 7, 0, 1, 1, 4], [0, 0, -2, -1, 1, -2], [4, 2, 4, 0, 0, -2], [13, -5, -1, 7, 0, 1], [-7, 7, 1, -2, 2, -1], [-4, -1, -3, 1, -2, -3], [3, -2, -1, -1, 1, -3], [7, 7, 2, -1, -2, 1], [-4, 3, -3, -1, 2, 4], [8, -8, -5, 2, 4, -2], [8, 6, 3, -3, 4, -2], [-3, 5, 2, 1, 1, -3], [18, 0, -1, 0, 3, -4], [11, 4, -3, 0, 4, -3], [6, 0, -1, -3, -3, -4], [3, -3, 1, 1, 0, 1], [1, -6, 1, 3, 2, 1], [3, 4, 1, -5, 1, -1], [-4, -2, 1, -1, -2, 4], [-6, -2, 0, -1, 3, 4], [-12, 4, -1, -4, 3, -4], [-1, 1, 0, -2, -4, 5], [-1, 3, -3, -4, -1, -7], [-8, 0, -1, -2, -3, 0], [-11, 2, -3, 5, 4, 1], [-3, 1, -2, 1, -5, 3], [5, -7, -1, 4, -1, 5], [13, -1, -2, 1, 5, -1], [-9, -1, 1, -1, 0, -1], [-16, -2, 1, 5, 0, 6], [6, -4, 3, 5, 1, 4], [-2, -8, 3, 0, -3, 8], [11, -2, -1, 1, 0, -4], [-15, 1, -1, -5, 2, 1], [-17, 4, 4, -1, -1, -2], [-9, 7, 3, -5, 0, 1], [-1, 9, 5, 1, 0, 5], [5, -3, 2, 0, -8, 3], [-5, 3, 2, -4, 1, 3], [-6, -5, -1, -4, 3, -5], [-6, 12, 2, -4, 6, -2], [-2, 9, 4, -2, 5, 0], [-11, -3, -4, 3, -1, 7], [2, -10, -1, 6, 1, 2], [3, 3, 4, -2, 2, 3], [-10, -9, 0, 0, 2, -1], [-4, 8, -2, 0, 8, 0], [13, 7, 5, -5, 0, -5], [-6, -5, -2, 0, 0, -4], [-11, -1, 0, 1, 5, -3], [9, 4, 1, -4, 1, 0], [-5, 7, 2, -5, -3, 3], [-5, 9, -3, -1, 8, -9], [-11, 1, 1, -1, 8, -3], [10, 8, -1, 4, 1, 4], [19, -4, 4, 3, -3, 6], [13, -5, 1, -3, -4, 7], [0, 11, 3, -2, -5, 1], [-8, -2, -2, -5, -1, -2], [20, -4, 0, 0, -2, 2], [-4, -1, 1, 0, -8, 0], [-3, -1, 1, -2, 3, -5], [9, 0, 5, 0, 6, -1], [-24, -4, 3, 0, -4, 2], [7, -7, -1, -2, -1, -1], [10, -13, -8, 1, 4, -4], [-8, 1, -6, -3, -1, -5], [-6, -4, 4, -4, -6, 4], [6, 0, 5, -2, 5, -4], [6, 0, 5, 1, -6, 6], [11, -2, -1, -1, 2, 1], [-6, 9, -2, -6, 2, -2], [14, -8, 0, 3, -11, 8], [22, -2, -3, 8, 3, 0], [1, 11, -3, 4, 7, -1], [-3, -5, -1, 0, 5, 1], [-13, 5, 7, 2, 1, -1], [1, -7, -5, 1, 5, -5], [-2, 8, -2, -4, 11, -14], [-14, -3, -3, 2, -1, -4], [6, 2, 1, -3, 3, -10], [-7, -1, 9, 5, -3, 5], [-3, -12, -1, -1, -3, -3], [1, 4, -1, -2, 2, -5], [-11, 7, 3, 1, -2, -3], [-5, -2, -7, 4, 3, -1], [-8, 0, 7, -6, -4, 2], [-9, 6, 0, -4, 4, -3], [-4, -10, -3, 5, 4, 2], [-7, 13, 2, -9, 1, -1], [-6, -2, -3, 6, 1, 4], [-7, -9, -1, 5, 0, -3], [8, -2, 6, 1, -7, 8], [-9, 4, 6, -8, 2, -3], [0, -7, 1, 11, -2, 9], [7, 5, -6, 2, -2, -1], [5, 5, 0, -2, -4, 1], [2, 5, 5, 0, -8, 4], [-6, 2, 1, -1, 2, -10], [20, 2, -5, 2, 4, 0], [10, -1, -3, 3, 8, 2], [-1, 3, -4, 4, 7, -1], [-4, 1, -3, -3, 5, -2], [-5, -3, -1, 4, 3, -1], [-13, 5, -1, -1, -6, 7], [2, 6, 3, 4, 3, 6], [-10, -6, -2, -2, 0, 6], [22, 2, 3, -1, -4, 6], [-4, -10, -8, 7, 5, 0], [19, 1, 1, 3, 6, -1], [-13, 7, 7, 0, 1, 3], [19, -13, -2, 7, -1, 3], [-2, 0, 4, 4, 4, 4], [-17, 16, -1, -3, 2, -7], [-13, -12, -2, 5, 1, -1], [-22, 18, 3, -4, -1, -6], [19, -6, 2, 0, -4, 1], [13, 7, 5, -2, 9, -1], [-19, 2, -1, 2, 1, 2], [0, 4, 4, 4, -2, 2], [10, 2, -5, 4, 5, -6], [15, -5, 0, -7, -3, 3], [-4, -12, 4, 0, -2, 4], [-13, 6, -2, -3, -1, 5], [-6, 3, 1, 3, 7, -10], [-5, 4, -5, 4, 7, -2], [-4, 2, -1, 1, 8, 6], [-19, -3, 0, 2, 5, 5], [5, 1, 2, 8, -6, 9], [-12, -1, -4, -5, -3, 8], [23, 1, 4, -3, -1, 5], [1, 9, 0, 8, 0, -1], [-22, -5, -2, 1, -2, 12], [12, 6, 2, 9, 3, 6], [7, 7, 0, 3, -7, 1], [-22, 5, -5, -2, 7, -11], [-16, 0, 2, -3, -10, 4], [0, 16, -4, -5, 3, -12], [-14, 8, 6, 0, -2, 4], [10, 12, -6, -3, 6, -4], [25, -5, -1, 6, -1, 3], [10, 3, 0, 7, -6, 2], [1, 0, 4, 0, 5, -3], [-5, -6, -5, 4, -5, 0], [4, -10, -1, -8, -5, 2], [-33, 4, -2, -7, 0, -3], [-4, 8, 11, -6, -3, 0], [4, 4, 5, -4, 7, -6], [9, -13, 2, 3, -9, 9], [-11, 8, 3, 9, 4, 7], [3, -12, -1, 6, 5, 3], [-7, 17, -4, -2, -1, -11], [19, -3, -3, 2, 8, -11], [1, -10, 7, 10, -4, 15], [26, 8, 0, 1, 3, -4], [24, -3, -3, -5, 4, -11], [4, 9, -6, -10, 4, -8], [3, -9, 9, 3, -14, 17], [22, -10, -3, 0, -1, -2], [-23, 5, -1, -3, 5, -1], [9, -17, 0, -4, -2, -5], [-13, -7, 0, -7, -9, -1], [-17, 11, 0, -6, -4, 9], [-1, 7, 4, -3, 7, -1], [-18, -2, 1, -8, -11, 2], [6, -14, -3, 3, 0, 4], [14, -8, 1, -6, -2, -6], [12, -5, 1, 0, -8, 2], [18, 4, -6, 1, -1, 2], [-2, -6, -8, 3, -5, 2], [-31, 11, 7, -6, -3, 1], [18, 0, 8, -12, 0, 0], [18, -4, -3, 8, -1, -4], [-12, -2, -3, -5, -4, -6], [-10, 3, 5, 2, 3, -2], [7, -5, 7, 1, -10, -3], [23, 7, 3, 0, 1, 9], [2, 14, 2, -7, 7, -4], [-3, -8, -1, 0, -7, 13], [-5, 8, -2, -2, 8, 4], [27, 1, 2, 7, -11, 11], [18, 6, -1, -4, -3, 4], [-17, -19, -1, 1, -4, -7], [-9, -14, 5, 1, -18, 18], [20, 10, 0, -7, -3, -6], [10, 12, 6, -8, -5, -2], [25, 8, -1, 3, -3, -1], [-14, 4, -3, 4, -1, -8], [-1, -2, -3, 0, 7, -2], [10, -2, -2, 2, 8, -10], [23, -6, 1, 2, 5, 4], [12, -1, -2, -6, -6, -1], [-25, -8, 0, 0, 2, 7], [6, -8, 1, -1, 12, -4], [25, -10, -4, 3, 8, 1], [-13, 11, 7, -3, -3, 5], [14, 18, 0, -4, -2, -4], [8, -16, -3, -1, 6, -12], [-4, -14, -7, 7, 2, 10], [9, -19, 1, 0, -9, -3], [35, -7, -2, -4, -6, 3], [-22, 11, 3, 3, 4, 5], [5, 0, -5, 0, 11, -20], [8, 2, -8, 11, 9, -4], [26, 17, -1, -5, -2, -5], [32, -8, 4, -2, -2, 14], [16, 7, 0, 5, 6, -2], [17, 6, -1, -4, -1, -2], [-3, 10, -3, 9, 3, 1], [29, -10, 1, 8, -4, -3], [5, -6, -5, -3, -15, 5], [25, 0, 7, 5, -2, 11], [-23, 9, 0, 1, 5, -1], [0, -5, -4, 9, 11, 0], [8, -24, 0, -1, 2, 0], [-5, 8, 1, -2, 2, 1], [53, -6, -4, 2, 4, 1], [-3, 21, -2, -3, 13, -3], [0, 2, -3, -10, 4, 0], [-14, 8, -4, 3, -3, -6], [18, -1, -3, 1, 13, -16], [-4, -8, -7, 15, -6, 2], [-20, 6, 6, 3, -3, 8], [-7, -3, 7, 5, -10, 5], [3, 3, 4, 3, -2, -7], [15, -16, -11, 5, 6, -3], [22, 15, 2, 2, -3, 6], [20, -10, 0, 12, 4, 4], [11, -8, -4, 7, 7, -2], [17, -7, -2, -5, 2, -5], [12, -23, -5, 8, 7, 9], [7, 3, -4, -1, -16, 1], [-15, 1, 8, 7, 1, 5], [19, 1, 8, 5, -3, 9], [-1, -19, 3, 5, -6, 23], [0, 8, -2, -1, -3, 2], [13, -2, 2, 6, 10, 8], [-19, -3, 8, -4, -8, 7], [8, 2, 1, 3, -6, 8], [8, 0, -6, 1, 7, 0], [14, -18, 4, -2, -10, 6], [-12, 16, 13, -11, -1, 0], [13, -23, -1, -3, -8, -1], [-5, 3, -10, 0, -2, 5], [-52, 15, 3, -11, -6, 0], [-8, -6, 1, -12, 2, -8], [-37, 8, -4, -3, 11, -3], [11, 3, 2, -8, 6, -7], [19, -1, 8, 0, -6, 9], [2, 10, -1, 8, -6, 4], [10, -1, -10, 7, 11, -6], [25, -9, 3, -8, -5, -7], [-8, 2, 6, 6, 2, 8], [-28, -8, -5, 11, 0, 14], [-7, -5, -7, -6, 1, -5], [5, 1, -7, -4, 7, -7], [-33, 8, 5, -2, -3, 6], [-22, 2, 4, 6, 6, 6], [26, -8, 4, 3, -9, 0], [5, 23, -5, -1, 4, -7], [11, -17, -16, 11, 7, -1], [-20, -15, -1, 3, 1, 4], [14, -17, 0, 0, 4, -15], [4, 1, -1, 5, -1, 0], [-36, 5, -1, -10, -13, -6], [18, -2, 2, -2, 3, -20], [28, -8, -3, 2, -9, 8], [-2, 8, -3, -8, -2, -14], [23, -17, 2, 5, 13, 5], [-5, -3, -8, 6, -13, 9], [-8, 1, -9, 1, 14, -14], [-6, -4, -2, 0, 8, -8], [16, 4, -6, 1, 11, -12], [20, -18, -3, 7, 4, -6], [34, 2, -2, 1, 8, -18], [39, -1, 4, 3, -2, -3], [-29, -11, 2, -8, 2, -5], [-8, 8, -6, 5, 13, 2], [-13, 15, 0, -11, 9, -9], [-12, 4, 10, -7, 2, 0], [-11, -7, -5, 10, 11, 7], [-4, -1, 0, -1, -8, -8], [-4, 6, 0, -1, 14, -2], [-4, 7, 4, -2, 2, -2], [-13, 6, 10, 5, -2, -5], [4, 12, 1, -1, -3, 8], [-14, -6, -2, 0, 8, 10], [-20, -16, 11, 4, -12, 14], [23, 5, -10, 2, 4, 3], [-27, 9, 3, 2, -3, 5], [-44, -1, 0, 5, -2, 0], [11, -9, 3, -9, 6, -5], [-1, -3, 10, 7, 5, 3], [22, -13, -4, 6, -4, -5], [1, -11, 1, 5, -10, 9], [5, -5, -2, 4, 2, 7], [-17, -3, -11, 5, -2, -9], [35, -14, 2, 15, -5, 6], [25, -13, -6, -6, -2, -1], [-14, 24, 3, -9, -3, -4], [19, 7, -11, 8, 13, -3], [26, 16, 4, 1, 8, 8], [-26, 4, -2, -4, -18, 8], [13, -13, -10, 11, 9, 3], [34, -13, 3, -3, -10, -1], [-16, 6, 9, 1, -6, 10], [-10, 19, -1, -12, 11, -10], [16, 6, 15, -6, -7, 4], [-20, 0, 6, -1, 11, -2], [12, 6, 3, 9, -10, 8], [-15, 9, 7, 2, -7, -7], [-36, 8, -1, -2, 5, -4], [3, 3, -4, 4, 16, -13], [-37, 10, 3, -1, -11, 11], [-38, 5, -1, -10, 0, -6], [6, 5, -1, 6, 17, 4], [-24, -1, 0, 5, -3, 7], [-18, -18, 7, -3, -10, 10], [-4, 21, -5, 5, 7, -8], [2, -21, -6, 8, -12, 13], [9, -15, -1, 6, -3, 3], [-36, -3, 1, 8, 6, 2], [-18, 16, -5, -5, -2, 2], [-26, -8, -9, 4, -13, -4], [64, -2, 1, 7, 6, 4], [-6, 3, -10, -8, 0, -2], [13, -11, -6, 7, -1, 9], [18, -10, 6, -14, -15, -4], [10, -2, -6, 11, -7, -2], [25, -22, 0, 18, -7, 5], [-8, -14, -3, 0, 17, -2], [30, -14, -9, 4, 9, -4], [5, 3, -3, -5, 4, -3], [-10, -10, -17, 8, 7, -4], [-2, -5, -1, -5, 1, 16], [-12, 2, 1, 13, 6, 6], [-5, -5, 5, -3, -22, 17], [-9, -21, 1, 1, -14, 15], [-13, 7, 10, -5, 7, -9], [7, 5, 11, -3, -8, 7], [-37, 7, 3, -11, -12, -3], [-6, -10, -1, -2, -23, 10], [17, -7, -6, -2, 0, -3], [-4, -10, 5, -3, -20, 16], [31, -20, -5, 7, -16, 10], [-24, -16, 0, 2, -12, 6], [19, -25, -1, -1, -16, 13], [-39, 5, 0, -2, -6, -1], [8, 9, 5, 14, 0, 4], [9, 11, 5, 7, -6, 3], [-6, 14, -9, -6, 7, -20], [3, 9, 12, 7, -6, 19], [13, -3, 3, 4, 7, 11], [-3, -5, -18, -9, 1, -9], [2, 2, 11, -3, -10, 8], [-43, -8, -8, 3, 2, -7], [30, 10, 10, -9, 3, 2], [-39, -10, -7, 1, 11, -7], [-4, 2, 4, -18, 2, -8], [-14, 22, -3, -2, 15, -16], [8, 10, 7, -8, 1, -14], [14, -13, -10, 11, -3, -2], [-18, -4, -11, 9, 10, 8], [18, 11, 13, -6, -1, -9], [44, 0, 5, -3, -5, 12], [3, 7, 3, -11, -5, 5], [-9, 9, -7, -5, 1, -9], [14, -16, 8, 6, -12, 6], [14, 1, -2, 3, 3, 14], [17, -20, -8, 1, -8, 5], [-5, -13, 3, 6, 0, 11], [19, -19, -14, 2, 8, -7], [31, 11, 2, -3, 2, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3800_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3800_2_a_ba();".
// 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_3800_2_a_ba(:prec:=6)
    chi := MakeCharacter_3800_a();
    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_3800_2_a_ba();".
// 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_3800_2_a_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-3, 14, 15, -16, -10, 2, 1]>,<7,R![-383, 154, 223, -48, -30, 2, 1]>],Snew);
    return Vf;
end function;