
// Make newform 201.4.j.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_201_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_201_j_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_201_4_j_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_201_4_j_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 := [1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 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, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 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, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3], [-3, 0, 3, -3, 0, 3, -3, 0, 3, -3, 6, 0, -3, 3, 0, -3, 3, 0, -3, 3]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1];
    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_201_j();"
function MakeCharacter_201_j()
    N := 201;
    order := 22;
    char_gens := [68, 136];
    v := [11, 5];
    // 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_201_j_Hecke();"
function MakeCharacter_201_j_Hecke(Kf)
    N := 201;
    order := 22;
    char_gens := [68, 136];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 0, 0, 0, 0, 0, 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 := 4;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -10, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, -3, 0, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 35, 35, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -28, 0, -28, 0, 0, 0, 0, 0, 0, -15, 15, -15, -30, 15, -15, -15, 15, -15, 15], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 154, 0, 0, 154, 0, 0, 0, 0, 0, 0, -15, 0, 0, 0, -15, 0, 0, 0], [0, 0, 0, 0, 0, 0, -55, 0, -55, 0, 0, 0, 0, 0, -42, 0, 0, 42, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 260, 0, -260, 0, 0, 0, 0, 0, -21, 0, 0, -21, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [91, 91, 91, 91, 91, -91, 91, -182, 91, -91, 0, 0, 0, 0, 0, 90, 0, 0, 0, -90], [0, 0, 0, 0, 0, 0, 0, 0, 440, 0, 0, 0, 0, 0, 0, 0, 0, -63, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-595, 0, 0, 0, 0, 0, -595, 0, 0, 0, 36, 0, 0, 0, 36, 0, 0, 0, 0, 0], [-442, 442, 0, 0, 0, 0, 0, 0, 0, 0, 105, 0, 0, 105, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 665, 0, 0, 0, 0, 0, 0, 0, 665, 0, 0, 0, 132, 0, 0, 0, 0, 132, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [910, 910, 910, 0, 910, -910, 910, -910, 910, -910, 99, -99, 99, 99, -99, 99, 99, -99, 99, -198], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -323, 0, 0, 0, -323, 0, 0, 210, -210, 210, 210, -210, 0, 210, -210, 210, -210], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -190, -190, 0, 0, 0, 0, 0, -273, 546, -273, -273, 273, -273, -273, 273, -273, 273], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1288, 0, 1288, 0, 0, 0, 0, 0, 0, 195, 0, 0, 0, 0, -195, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-874, 0, -874, -874, -874, 874, -874, 874, -874, 874, 0, 0, 0, -315, 0, 0, 0, 0, 0, -315], [0, -1925, 0, 0, 0, 1925, 0, 0, 0, 0, 0, 0, -78, 78, 0, 0, 0, 0, 0, 0], [1700, 1700, 1700, 1700, 0, -1700, 1700, -1700, 1700, -1700, 0, -231, 0, 0, 0, 0, 0, 0, 0, 231], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1729, 0, 0, 0, 0, 0, 0, -1729, 0, 0, 330, 0, 0, 0, 0, -330, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [575, 575, 575, 0, 575, -575, 575, -575, 575, -575, -504, 504, -504, -504, 504, -504, -504, 504, -504, 1008], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2618, 0, 0, 0, 2618, 0, 0, 0, 0, 0, 0, 0, 195, 0, 195, 0, 0, 0], [0, 0, 0, 0, -3016, 0, 0, 3016, 0, 0, 0, 105, 0, 0, 0, 105, 0, 0, 0, 0], [-1610, 0, -1610, 0, 0, 0, 0, 0, 0, 0, -561, 0, 0, 0, 0, 0, 561, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2233, 0, 0, 0, 0, 0, 0, 0, 0, 2233, 510, 0, 0, 0, 0, 0, 0, 0, 510, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3689, -3689, 0, 0, 0, 0, 0, 0, 120, -120, 120, 120, -120, 120, 240, -120, 120, -120], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -406, 0, 0, 0, -406, 0, 0, 0, 0, 0, 0, 855, 855, 0, 0, 0, 0, 0, 0], [0, 0, 2015, 0, 0, 0, 2015, 0, 0, 0, 0, 0, 0, 0, 798, 0, 798, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2800, 0, 0, -2800, 0, 0, 0, 0, -741, 0, 0, 0, -741, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 5320, 0, 0, 0, 0, 0, -5320, 0, 0, 0, 0, 0, 153, 0, -153, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -5005, 0, 0, 0, 0, 0, -5005, 456, -456, 456, 456, -456, 456, 456, -456, 0, -456], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 496, 0, 0, 496, 0, 0, 0, 1155, 0, 0, 0, 0, -1155, 0, 0], [0, 0, 0, 2465, -2465, 0, 0, 0, 0, 0, -1092, 0, -1092, -1092, 1092, -1092, -1092, 1092, -1092, 1092], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5957, -5957, 0, 0, 0, 0, 0, 0, 510, 510, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2170, 0, 0, 2170, 0, 0, 0, 1287, 0, 0, 0, -1287, 0, 0, 0, 0], [0, 0, 6175, 0, 0, 0, 0, 6175, 0, 0, 0, 0, 0, 0, 0, 714, 714, 0, 0, 0], [0, -4292, 0, 0, 0, -4292, 0, 0, 0, 0, 0, 0, -1155, -1155, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [595, 595, 595, 595, 595, -595, 595, -595, 595, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1518, -1518], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4123, 0, 0, 0, 0, 0, 0, -4123, 0, 0, 0, 0, -1380, 0, 0, 0, 1380, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8569, 8569, 8569, 8569, 8569, 0, 8569, -8569, 8569, -8569, 0, 0, 210, 0, 0, 0, 0, 0, 0, 210], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1295, 0, 0, 0, 0, 0, 0, 0, 0, 1295, 1716, 0, 0, 0, 0, 0, 0, 0, 1716, 0], [7462, 7462, 7462, 7462, 0, -7462, 7462, -7462, 7462, -7462, 0, -1035, 0, 0, 0, 0, 0, 0, 0, 1035], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6355, -6355, 0, 0, 0, 0, 0, -1428, -1428, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9890, -9890, 0, 0, 0, 0, 0, 0, 0, -231, 231, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-10450, 0, 0, 10450, 0, 0, 0, 0, 0, 0, 0, -483, 483, 483, -483, 483, 483, -483, 483, -483], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -7568, 0, 7568, 0, 0, 0, 0, 0, -1575, 0, 0, -1575, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6020, -6020, 0, 0, 0, 0, 0, 1989, 1989, 0, 0, 0, 0, 0, 0, 0], [11339, 11339, 11339, 11339, 11339, -11339, 11339, -11339, 11339, -22678, 0, 0, 0, 0, 0, 0, 0, 0, 1050, -1050], [820, 820, 820, 820, 820, -820, 820, -820, 820, -1640, 0, 0, 0, 0, 0, 0, 0, 0, 2457, -2457], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [11656, 0, 11656, 0, 0, 0, 0, 0, 0, 0, -1365, 0, 0, 0, 0, 0, 1365, 0, 0, 0], [8855, 17710, 8855, 8855, 8855, -8855, 8855, -8855, 8855, -8855, 0, 0, 0, 2052, 0, 0, 0, 0, 0, -2052], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 14651, 0, 0, 0, 0, 0, 0, 14651, 0, 0, 0, 0, -300, 0, 0, 0, -300, 0, 0], [0, 0, 0, 0, 0, 0, -14210, 14210, 0, 0, 0, 0, 0, 0, -897, -897, 0, 0, 0, 0], [0, 0, 0, 0, 0, -8695, 8695, 0, 0, 0, 0, 0, 2394, 0, 2394, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-13328, 0, 0, 0, 0, 0, 0, 0, -13328, 0, 1485, 0, 0, 0, 0, 0, 0, 1485, 0, 0], [0, 0, 0, 0, 946, 0, 0, 0, -946, 0, 0, 3045, 0, 0, 0, 0, 0, 3045, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -6580, 0, 0, -6580, 0, 0, 0, 0, 0, 0, 2871, 0, 2871, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 10241, 0, 0, 0, -10241, 0, 0, 0, 0, 0, 0, 0, 2610, 0, -2610, 0, 0, 0], [-12025, -12025, -24050, -12025, -12025, 12025, -12025, 12025, -12025, 12025, 0, 0, 0, 0, 0, 0, -2436, 0, 0, 2436], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 8036, 8036, 0, 0, 0, 0, 0, 0, 3135, -3135, 3135, 3135, -3135, 3135, 6270, -3135, 3135, -3135], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -18073, 0, 0, 0, -18073, 0, 0, -1050, 0, 0, 0, 0, 0, 1050, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5390, 5390, 0, 0, 0, 0, 0, -3627, 0, 3627, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 7525, 7525, 0, 0, 0, 0, 0, 0, -3534, 3534, 0, 0, 0, 0], [0, 0, -15688, 0, 0, 0, 15688, 0, 0, 0, 0, 0, 0, 0, -2415, 0, 2415, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-11726, 0, 0, -11726, 0, 0, 0, 0, 0, 0, -6510, 3255, -3255, -3255, 3255, -3255, -3255, 3255, -3255, 3255], [-20735, -20735, -41470, -20735, -20735, 20735, -20735, 20735, -20735, 20735, 0, 0, 0, 0, 0, 0, 378, 0, 0, -378], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2303, -2303, -2303, -2303, -2303, 2303, -2303, 2303, -4606, 2303, 0, 0, 0, 0, 0, 0, 0, 4080, 0, 4080], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 21700, 0, -21700, 0, 0, 0, 0, 0, 783, 0, 0, 783, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 19684, -19684, 0, 0, 0, 0, 0, 2325, 0, 2325, 2325, -2325, 2325, 2325, -2325, 2325, -2325], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6110, 0, 0, 0, 6110, 0, 0, 0, 0, 0, -4389, -4389, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -8533, 0, 0, 0, 0, 0, 0, 8533, 0, 0, 0, 0, 0, 0, -4290, 0, 4290, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -23345, -23345, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1674, 0, 0, -1674, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [15652, 15652, 0, 0, 0, 0, 0, 0, 0, 0, 3795, 0, 0, -3795, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -25075, 25075, 0, 0, 0, 0, 0, 0, 0, -1302, -1302, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-10340, -10340, -10340, -10340, -10340, 10340, -20680, 10340, -10340, 10340, 0, 0, 0, 0, 4641, 0, 0, 0, 0, 4641], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 22420, 0, 0, -22420, 0, 0, 0, 0, 0, 0, 3003, 0, 0, 0, -3003, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1378, 0, 0, -1378, 0, 0, 0, 0, 5355, 0, 0, 0, -5355, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-27755, -27755, -27755, -27755, -55510, 27755, -27755, 27755, -27755, 27755, 0, -1392, 0, 0, 0, 0, 0, 0, 0, -1392], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 25010, 0, 0, 0, 0, 0, 0, 0, -25010, 0, 0, 0, 3213, 0, 0, 0, 0, -3213, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 22814, 0, 0, 0, 0, 22814, 0, -4095, 0, 0, 0, 0, 0, 0, 4095, 0], [0, 0, 0, 0, -14455, 0, 0, 0, -14455, 0, 0, 5382, 0, 0, 0, 0, 0, -5382, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_201_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_201_4_j_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_201_4_j_a(:prec:=20)
    chi := MakeCharacter_201_j();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_201_4_j_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_201_4_j_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_201_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;