
// Make newform 3800.2.a.z in Magma, downloaded from the LMFDB on 23 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_z();".
// 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_z();".
// 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, -32, 22, 20, -11, -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], [12, -26, -16, 11, 2, -1], [-28, 22, 20, -11, -2, 1], [-10, 24, 9, -11, -1, 1], [48, -98, -38, 37, 4, -3]];
    Rf_basisdens := [1, 1, 4, 4, 2, 4];
    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], [-1, -1, 0, -1, 0, 0], [0, 0, -1, 1, 1, 0], [-2, 1, -1, 0, -1, -1], [-3, 1, 1, -1, 1, 1], [1, 0, 0, 0, 0, 0], [0, -1, 0, -1, -1, 1], [0, 1, -1, -1, -2, 0], [2, 0, -1, 0, -1, 1], [-2, 2, 2, 1, 1, -1], [0, 0, 1, 0, 1, -1], [-1, 0, 1, 2, 2, 0], [-1, 0, 0, 0, 1, 1], [2, 3, 1, 2, -1, -1], [0, 1, 0, 1, -1, -3], [2, 0, -1, -1, -1, 2], [-6, -1, -2, 0, 0, 0], [-2, 2, 1, 0, -1, -1], [-1, 3, 1, 1, 1, 1], [-4, 2, 0, 0, 0, 0], [-4, 0, 2, 0, 2, 0], [8, 0, -3, 4, -1, -1], [-8, 0, 2, 1, 3, 1], [-6, 0, -1, 0, 1, -3], [-2, 2, 1, 1, -2, 0], [0, -3, 0, 4, 0, 0], [0, -1, -1, -3, -2, -2], [4, 2, 3, -1, 0, 0], [-4, 2, -2, -1, -3, 1], [0, -4, 2, -1, 0, 3], [-1, 1, 1, 3, -5, -1], [-2, 2, 4, 3, 2, 1], [0, -4, -2, -3, 4, 1], [0, -2, 4, 0, 2, 0], [-4, 2, -2, 4, 2, 2], [2, -2, -5, 2, -3, 1], [-6, 4, 1, 1, -2, 0], [2, 2, 2, -3, -3, -1], [-4, -2, 2, -2, 2, 0], [-10, -2, 4, -2, 2, 2], [-4, 3, 0, 0, -7, -2], [-8, -4, 2, -7, -1, 1], [-10, -2, -3, -4, 1, -1], [4, 3, 0, 6, -3, 0], [-4, -3, -4, 3, 5, -1], [-8, -2, -2, 1, 3, -1], [2, 1, -2, 0, -4, 4], [0, -2, 0, -1, 4, 1], [-1, -2, -2, 0, 3, -3], [-12, -3, 0, 0, 3, 2], [6, 6, 2, 4, -2, -6], [-6, 6, 5, -1, -1, 0], [-6, -4, -3, 1, -2, 2], [7, -2, 2, -4, 3, 3], [10, -8, 0, -4, 2, 0], [2, 1, 2, 1, 5, 3], [7, -6, -2, -6, -3, -1], [12, -8, -1, -4, -1, 1], [3, -2, -8, 4, 1, -3], [-10, 1, -1, -4, -5, 3], [-10, 0, 2, 3, 3, -1], [-2, -9, -4, 0, 3, 2], [-10, 3, 3, 1, 0, -2], [0, 1, -1, 4, 1, -3], [4, 1, -2, -3, -5, -1], [-2, -4, -3, -3, 6, 2], [-7, -2, -4, 4, -1, -1], [2, -8, 2, -7, -4, 1], [10, -1, -3, 7, 0, -2], [-6, -3, 2, -2, 3, 2], [0, 6, 4, 0, 4, -2], [-22, 3, 3, 2, 1, 1], [-4, 3, -2, 9, 1, 3], [4, 2, -3, -1, -2, -4], [0, -2, -7, -5, -5, 0], [9, -6, -2, -2, -1, 1], [12, 8, 3, -4, -3, 1], [12, -4, -1, 2, -1, -1], [-2, -4, -1, -6, 1, 3], [0, -5, -3, 1, -4, 4], [-14, 2, 5, 4, 1, -1], [-12, -6, -1, 3, 0, 0], [2, 0, -5, 2, -5, 3], [-3, 6, 1, 2, -4, 2], [14, 0, 4, 0, 2, 4], [-5, 9, -3, 1, -7, -3], [-12, 4, 9, -3, 1, 2], [-1, 8, -2, 2, -3, 5], [11, -6, 3, -6, 4, -2], [-6, 6, -4, 2, 0, -2], [-4, 6, 6, -1, 1, 5], [-10, -4, 3, -10, 1, 3], [-8, 2, 1, -5, 9, 0], [12, -3, -4, 1, 1, 3], [-6, -2, 6, -4, 4, 2], [10, 2, 2, 0, 4, -2], [-8, -1, -2, -10, -4, 0], [-6, 8, -7, 3, -7, -4], [-6, -2, 0, -1, 1, -3], [-1, 2, 6, -2, 1, 3], [-4, 4, 2, 3, 5, -3], [14, 2, -6, 6, -2, -4], [4, 2, 5, 4, -1, -1], [-5, -5, 3, -9, 1, -3], [15, 0, -5, 4, -6, 0], [8, -6, 6, -2, 8, 4], [-2, 4, 1, 0, 9, -1], [-12, -4, 1, 4, 1, -1], [6, 10, 2, 7, 3, -5], [-1, 2, 1, -8, 0, 4], [5, -6, -4, 2, 1, -5], [-8, 8, -1, -2, -5, -1], [-2, 8, 2, -1, 2, -5], [12, 6, -3, 0, 3, 3], [-1, 6, 0, -4, 5, -3], [9, -3, -8, 3, 2, 2], [9, -2, -2, 10, 3, -3], [-4, -7, -4, -11, -1, -3], [4, -1, 5, -3, -4, -4], [-18, -6, 0, 3, 1, -1], [8, 7, -5, 6, 3, 3], [26, -8, -2, -5, -1, -1], [6, -8, 0, 1, 0, 3], [-12, 4, 4, 2, 12, 2], [-10, -4, 0, 4, 4, 4], [6, 5, -2, -2, -5, 6], [3, -7, -6, 5, -8, 0], [16, -8, -4, 0, -4, 4], [6, -4, 3, 1, -1, -4], [4, 4, -7, -5, -2, -4], [14, 0, -1, 14, -7, -1], [-1, -4, 0, -2, 7, 1], [18, -9, -5, -8, -2, -1], [-2, 1, 3, -2, 8, -3], [12, 11, -5, 2, -1, -5], [10, -1, -14, 4, -4, -4], [6, -9, -3, 6, 3, 3], [8, -7, 3, 2, 4, 1], [-12, -1, 0, -7, 1, 9], [-2, 14, 4, 1, 0, -5], [-19, 5, 0, 5, -8, -4], [6, 3, -4, -2, -4, -8], [-4, 1, -1, 7, 0, -4], [16, 0, 0, 12, -2, -4], [8, 6, 6, -6, 4, -4], [6, -6, -4, -3, -1, 1], [6, -6, -5, 3, -3, -2], [2, 2, -5, 9, -4, 2], [12, -7, -5, 8, -3, -7], [20, -12, -6, -1, 0, -5], [-27, 6, 0, 2, 5, 1], [-6, 2, 7, -1, 10, 0], [-14, -5, 2, -2, 2, 6], [-4, -12, -4, 6, 0, -2], [-16, -11, 2, -15, 3, 3], [0, -1, 3, -7, -8, 2], [3, -1, -1, 1, -1, -5], [-16, -9, 5, -11, -2, 4], [0, 10, -2, -2, 0, 6], [-2, 6, -1, 3, -10, 2], [0, 2, 4, -2, 10, 4], [12, -4, -12, 6, -10, -2], [10, -16, 3, -9, 2, 2], [18, -6, -2, -3, -15, 1], [-6, 4, 5, -4, -1, -1], [-9, -4, 5, -6, -2, -2], [-10, 2, 14, -8, 0, 2], [5, 8, -1, 10, -2, -6], [-20, -7, -6, 3, 3, 5], [-12, -7, 1, 5, 8, 0], [-2, 8, 0, -4, 6, -4], [-14, 5, 5, -9, 0, -2], [2, 0, 7, -2, 5, 7], [16, 4, 0, 3, 2, -7], [10, -16, 1, -10, 7, 5], [-8, -10, 4, 1, 4, 3], [16, 0, -5, 7, -8, -10], [-10, 6, -3, -1, -5, 0], [1, 3, 0, -7, 0, 8], [8, 7, -8, 1, 1, 1], [-6, -2, -6, 7, 15, -3], [10, 6, 8, -5, 5, 3], [-9, -1, 4, -3, -6, -2], [-18, 0, 10, 5, -4, -3], [-24, -1, 3, -8, 9, 1], [0, -8, 8, -5, 7, 7], [-4, 2, 5, -2, -9, 1], [20, 9, -8, -3, -3, 1], [-4, -6, 3, -1, 0, 8], [-9, 8, 0, 2, 3, -9], [4, -24, -6, -6, 4, 0], [12, 3, -7, 11, -4, -10], [-4, 10, 2, 0, 10, 0], [-18, 0, 7, -11, 4, 12], [12, 5, -13, 2, -4, 1], [16, 0, -4, -8, -12, 4], [10, 7, 3, 5, 0, 2], [9, -5, -14, 11, -8, -4], [-6, 6, 6, -4, -8, 2], [-20, -12, 0, -2, 0, -2], [-14, -5, -7, -2, 3, -5], [-4, 0, -7, -14, -5, -3], [22, -16, -2, -8, -8, 2], [-11, -14, -14, -2, 7, 1], [6, -13, 12, -6, 7, 8], [10, 3, -8, 14, 0, 0], [4, -1, 5, 2, 4, -5], [-8, 5, 12, 1, 15, -7], [-30, -5, 1, -7, 6, -4], [-24, 0, 0, -6, 0, 6], [7, -8, -6, -2, 5, -3], [-18, -1, 2, -10, -6, -2], [2, -4, -9, 0, -7, 1], [16, -10, -1, 6, -3, 7], [-8, -1, 0, 5, 13, -5], [14, 4, 0, 10, -4, 6], [-2, 6, -6, -5, -5, -5], [-15, 0, 0, -6, -15, -1], [2, -6, 6, -4, 10, 2], [12, -2, -12, -8, -2, 4], [12, -6, 11, -4, 7, 5], [-1, -13, 10, -3, 4, 8], [4, -9, -16, -2, -2, -2], [-14, 0, 2, -5, -8, -1], [6, 4, -2, 13, -11, -9], [2, -2, 9, -4, -3, 7], [-22, 10, 7, 7, -6, 0], [-8, 18, 2, 11, 2, -5], [-17, -6, 9, 2, 10, 2], [0, 5, 12, -3, 9, 1], [8, -16, 4, -8, -4, 0], [18, -16, -4, -2, -2, -2], [-25, 6, 20, -8, 13, 1], [-5, -11, 14, 1, 8, 8], [4, -8, -2, 9, 12, -3], [11, -4, 9, -8, -4, 0], [-18, 0, -2, 0, -4, 2], [2, 0, 3, 6, -3, 7], [-8, -12, -2, -3, 3, 7], [2, 10, 1, -9, -9, -4], [-4, -7, -10, 3, 9, 7], [-4, 3, 1, 5, 2, 8], [-20, 4, 5, -5, -14, 2], [0, 16, 4, -3, -2, 3], [7, 2, 2, 10, -7, -7], [-32, -4, -6, -4, 2, -10], [24, -8, -10, -4, -2, -2], [-15, -11, 12, -7, 4, 8], [-22, 8, 19, -2, 9, 3], [-24, 5, -5, 0, 4, -5], [-8, -11, 6, -3, -9, 1], [16, -2, -7, 8, -5, -5], [53, -4, -5, -4, -4, -4], [-18, 10, 11, -8, 11, -3], [-26, -12, 6, -17, 0, -1], [-40, 0, 14, -3, 7, 3], [-6, 6, -2, -5, 3, 5], [-16, -4, 6, -6, 4, -4], [6, -20, 7, -13, -2, 8], [-16, 8, 18, -5, 5, 5], [-14, 18, 1, -1, -1, -4], [12, -7, -13, 4, -7, -11], [-41, 6, 0, 0, -7, -1], [28, 9, 0, 3, 7, -1], [-36, -7, -5, -7, 6, 2], [22, -5, -5, 10, -10, -11], [-16, 3, 4, 6, -10, -14], [-7, -14, 7, -2, 6, -2], [-32, 11, 7, 7, 8, -4], [16, 7, -4, 6, -6, -6], [-23, 11, 3, -7, -3, -3], [2, 3, 4, -8, -5, -6], [-6, -6, -9, 9, 8, 0], [32, 1, -14, 7, -3, -1], [-27, 18, -3, 8, -2, -8], [-8, 11, -15, 11, -2, -10], [18, 2, -18, 0, -6, -2], [-14, -2, -8, -7, -6, -1], [8, 6, 6, 5, 11, -1], [-22, -5, 0, 3, 5, 7], [-51, 1, -2, 3, 0, 0], [-28, -1, 1, -3, 2, 0], [-11, -16, -4, -10, -3, -3], [-22, 2, 2, -11, 12, 9], [1, 9, -3, -3, -1, -5], [-16, -4, -2, -7, -9, 1], [16, -11, 6, -12, 13, 8], [-6, 12, 8, 0, 2, -8], [4, -10, 1, -12, -3, -5], [19, 12, 4, 12, 5, -3], [-10, -10, -11, -3, -2, -2], [0, -4, 20, -14, 8, 2], [-22, 10, 4, -7, -1, 5], [-30, 2, 11, -18, 3, 7], [-12, 30, 12, 12, -4, -4], [-13, 16, 0, -6, 7, -3], [-24, 3, 12, 7, 3, 3], [29, -6, -21, 16, -16, -6], [15, -6, 0, -2, 7, 9], [4, -9, 1, 12, 11, 7], [-26, 21, 10, 2, 3, 6], [-22, 23, 6, 12, -4, -4], [26, -4, -27, -3, -11, -2], [4, -20, -2, -1, 5, 3], [18, -3, -6, 8, 16, -4], [20, -3, -3, 11, -4, -2], [0, 2, -8, -3, 8, -5], [-16, -12, -8, 0, -8, 8], [33, -8, -20, -8, 3, 3], [22, -12, 7, 0, 13, 5], [4, 10, 1, -10, 13, 1], [-36, -4, -14, 3, -3, -3], [16, -6, -4, -14, -18, -8], [-20, 0, 21, -4, 7, 7], [2, -4, 8, -17, 4, -3], [-18, -6, 1, -4, -3, -17], [-11, -1, -1, -5, -1, 7], [-10, 0, 6, -7, -4, 5], [-8, 19, 2, 7, -9, -3], [-19, 7, 9, 5, 7, 3], [16, 15, -7, 3, -12, -14], [-32, 6, 3, 3, -8, 2], [-38, -5, -5, -13, 6, -8], [12, 2, -7, 8, -13, -13], [0, -7, 8, -9, 11, -1], [24, 11, 0, -2, -2, 2], [-22, -4, 7, -1, 6, -16], [-11, 12, 3, 2, 10, 6], [28, -8, 10, -3, 8, 5], [-4, 19, 3, 12, -7, 1], [-10, 5, -8, 5, 7, -7], [1, 14, -8, 14, -3, 9], [30, -8, -4, -1, -12, 1], [9, 16, -3, 22, -8, -2], [0, -3, -9, 7, 4, 8], [-22, -10, 4, 13, 7, 5], [-8, 1, 1, -12, -6, 1], [-8, 5, -2, 1, 3, 1], [22, -1, -13, 4, 3, 3], [-7, 1, 19, -13, 7, -5], [-28, -1, 11, 7, 2, -8], [28, 6, -16, 12, -12, -12], [-16, -7, 13, -4, 3, -5], [0, -4, -10, -18, -16, 4], [34, 4, 0, -12, -12, -4], [0, 19, 12, 2, 6, -2], [-26, 6, -7, 6, 7, -7], [17, -22, -6, -10, 11, 5], [-24, -4, 6, -12, -4, 2], [2, -14, 1, -11, 6, 6], [12, 16, 6, 9, -12, 5], [-12, 8, 18, -5, 1, 5], [6, 26, 18, -4, -8, 2], [20, -10, -12, -9, -13, -3], [-20, 11, 0, 3, -9, 7], [-20, 20, 4, 4, -8, 0], [5, -17, -17, 1, -11, 9], [-12, 10, 10, 15, 1, 1], [-9, 20, 5, 10, 6, 6], [4, 20, -8, 10, 4, -2], [18, -22, -5, 1, -12, 10], [16, 1, 8, 9, 1, 1], [3, 18, 11, -10, 8, -2], [-20, 8, -3, 5, -12, 0], [-4, 6, 2, 0, 10, -2], [-10, -14, -3, -1, 8, -2], [22, 12, -8, -8, -6, -4], [-38, -4, 6, 12, 8, 6], [-44, 9, 14, 5, -3, -1], [21, -7, -2, -7, -10, -6], [-16, 11, 1, -1, 4, -6], [48, 0, -8, -3, 10, 7], [-11, -18, 6, -18, 3, 9], [-8, -3, -2, -19, -9, 3], [-8, -8, 0, -4, 4, 12], [21, -22, -4, -12, -5, 5], [8, 5, -3, 2, 6, -1], [-18, 12, -9, -1, -2, -6], [14, -2, 8, -2, -2, -2], [23, 18, 4, -16, -3, -1], [11, 13, 6, 21, 4, -4], [3, 12, -12, 14, -17, -11], [28, -3, -12, 17, -13, -3], [14, 10, 2, 7, 3, 7], [-46, -5, 4, -17, -7, 7], [6, 30, 7, 14, 1, -17], [-35, 14, 12, 10, -7, -3], [-32, 4, 14, -7, 3, 3], [6, 12, -8, 18, 2, 6], [0, 7, 9, 20, 15, -5], [4, 2, 8, -2, 2, -10], [-13, 14, -13, 2, -20, -6], [-2, 0, 1, 12, -1, 7], [-2, 0, -2, 9, 1, -5], [-12, 2, 12, 12, 8, 8], [22, -18, -10, -10, -4, 4], [-56, 6, -2, 2, 14, -8], [12, -32, -5, 6, 1, 3], [6, 20, 4, -4, 2, 12], [11, 16, -8, 18, 7, -7], [6, -2, -18, 9, -7, -7], [8, 8, -1, 3, 16, 0], [34, -10, -22, -10, -10, 8], [12, -11, -12, 12, -1, -14], [9, -6, 11, 8, 4, 0], [10, 2, 10, 6, 8, 8], [-4, 11, 10, -2, -8, -8], [6, -12, -13, 15, -9, -2], [18, -16, -3, -19, -10, -10], [-11, -8, -12, 14, -11, 3], [-16, 5, -22, 8, -6, -14], [24, 4, 3, -9, -11, -2], [34, -7, -3, 5, -18, 4], [0, 12, 1, 18, -9, 1], [-18, 6, -11, -14, -21, 1], [24, -8, -13, -9, -4, 10], [6, -1, -7, 23, 0, 2], [-24, -20, -6, -17, 1, 5], [-10, -10, -22, 6, 2, -8], [0, 5, 7, -20, -3, -3], [7, -1, 8, 19, 12, 4], [-24, -1, -6, -7, -11, 3], [-30, -10, 3, -21, -6, 10], [14, 9, -13, 10, -3, -7], [4, -1, 6, -14, 16, 8], [14, -21, 1, 8, 8, -7], [-24, -4, 10, -19, 4, 5], [30, -1, -8, -2, 3, -8]];
    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_z();".
// 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_z(: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_z();".
// 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_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![8, 32, 22, -20, -11, 2, 1]>,<7,R![-4, 36, -49, -62, -4, 6, 1]>],Snew);
    return Vf;
end function;