
// Make newform 2106.2.b.c in Magma, downloaded from the LMFDB on 11 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2106_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2106_b_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_2106_2_b_c();".
// 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_2106_2_b_c();".
// 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 := [729, 0, 5589, 0, 10260, 0, 7651, 0, 2617, 0, 435, 0, 34, 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], [0, 15714, 0, 113931, 0, 84463, 0, 25015, 0, 3630, 0, 256, 0, 7], [-3996, 0, -18795, 0, -19789, 0, -8236, 0, -1545, 0, -130, 0, -4, 0], [116478, 0, 618183, 0, 747619, 0, 318433, 0, 59250, 0, 4936, 0, 151, 0], [114129, 0, 484605, 0, 500504, 0, 196739, 0, 35070, 0, 2852, 0, 86, 0], [0, 424197, 0, 574866, 0, 82484, 0, -51001, 0, -16356, 0, -1645, 0, -55], [20250, 0, 248967, 0, 332563, 0, 143425, 0, 26634, 0, 2206, 0, 67, 0], [-163863, 0, -1111986, 0, -1365586, 0, -580945, 0, -108174, 0, -9031, 0, -277, 0], [0, 10773, 0, 49293, 0, 54901, 0, 22336, 0, 4044, 0, 331, 0, 10], [49815, 820773, -473634, 2771208, -932316, 2578730, -462417, 979343, -92628, 172284, -8007, 13931, -249, 419], [49815, 27621, -473634, -1621476, -932316, -2413762, -462417, -1081345, -92628, -204996, -8007, -17221, -249, -529], [0, 420795, 0, 1616553, 0, 1809329, 0, 754208, 0, 139614, 0, 11627, 0, 356], [0, -18630, 0, -465399, 0, -625807, 0, -269017, 0, -49980, 0, -4156, 0, -127], [0, 729, 0, -25620, 0, -36671, 0, -16301, 0, -3090, 0, -260, 0, -8]];
    Rf_basisdens := [1, 9234, 513, 9234, 4617, 27702, 3078, 9234, 1458, 55404, 55404, 27702, 9234, 513];
    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_2106_b();"
function MakeCharacter_2106_b()
    N := 2106;
    order := 2;
    char_gens := [1379, 1783];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2106_b_Hecke();"
function MakeCharacter_2106_b_Hecke( : Kf := false)
    N := 2106;
    order := 2;
    char_gens := [1379, 1783];
    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]];
    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, 1, 0, 0, 0, 0, 0], [0, 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, 1, 0, 0, 0, 0, 0, 0, 1, 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, -1, 0, 0, -1, 0, 0, 0, 0], [0, 0, 1, -1, 0, 1, -1, -1, 0, -1, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [1, 0, -1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 2, 0, 1, -1, 0, 0, -1, -1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -1], [0, -1, 0, 0, 0, 1, 0, 0, 1, 1, -1, 1, -1, 1], [0, 0, 0, 0, 0, -1, 0, 0, 1, 2, -2, 0, -1, 0], [-1, 0, 1, -1, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, -1, 1, -1, -1, -1, 1], [-3, 0, 0, 2, 0, 0, -1, 2, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 2, 0, 1], [-1, 0, 0, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, 0], [0, -1, 0, 0, 0, 1, 0, 0, -4, 2, -2, 0, 0, 1], [0, 1, 0, 0, 0, -2, 0, 0, -2, -1, 1, 0, 2, -2], [0, -1, 0, 0, 0, 1, 0, 0, 1, 1, -1, -2, 0, 1], [-1, 0, -1, 2, 1, -1, 1, 2, 0, 1, 1, 0, 0, 0], [0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 1], [0, 1, 0, 0, 0, 0, 0, 0, 2, 1, -1, 2, -1, 1], [0, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, -1, 0], [1, 0, 0, 0, 2, 1, -1, 0, 0, -1, -1, 0, 0, 0], [3, 0, 1, 0, 1, -1, 1, 2, 0, 1, 1, 0, 0, 0], [5, 0, 1, 2, 0, 0, -1, -2, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 1, 0, 0, -3, 1, -1, 1, -2, 2], [-3, 0, -1, -2, 2, 2, 0, 0, 0, -2, -2, 0, 0, 0], [-4, 0, 0, -2, -2, -1, -2, -4, 0, 1, 1, 0, 0, 0], [-6, 0, 1, 3, -1, 1, -1, -1, 0, -1, -1, 0, 0, 0], [0, 3, 0, 0, 0, -1, 0, 0, 1, -1, 1, 0, 0, -3], [0, 0, -1, 2, -2, -1, 1, -1, 0, 1, 1, 0, 0, 0], [0, -5, 0, 0, 0, 2, 0, 0, -3, 3, -3, 0, 0, 1], [0, 1, 0, 0, 0, 2, 0, 0, 2, -1, 1, -1, 3, -1], [1, 0, 0, 0, 0, -2, 1, 0, 0, 2, 2, 0, 0, 0], [0, -1, 0, 0, 0, -1, 0, 0, 4, 0, 0, -2, -1, -3], [0, -1, 0, 0, 0, -1, 0, 0, -1, -2, 2, 0, 0, -1], [0, 0, 0, 0, -2, 0, -2, -2, 0, 0, 0, 0, 0, 0], [1, 0, -2, 4, -2, -1, 2, 3, 0, 1, 1, 0, 0, 0], [4, 0, 0, 0, 0, 1, -2, 0, 0, -1, -1, 0, 0, 0], [-5, 0, 1, 0, -1, 2, -1, 0, 0, -2, -2, 0, 0, 0], [0, 3, 0, 0, 0, 1, 0, 0, 1, -3, 3, 2, 4, -1], [0, -2, 0, 0, 0, 1, 0, 0, -2, -1, 1, -4, 3, 1], [-4, 0, 2, 0, 2, 2, 0, -2, 0, -2, -2, 0, 0, 0], [1, 0, 1, 1, -1, 0, 1, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, -14, 0, 0, -1, -3, 1], [0, -2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -1, 4], [0, 2, 0, 0, 0, -3, 0, 0, 6, 1, -1, -1, 0, -2], [3, 0, 3, -2, 0, 3, -2, 1, 0, -3, -3, 0, 0, 0], [0, -4, 0, 0, 0, 3, 0, 0, 4, -1, 1, -3, 1, -1], [0, 3, 0, 0, 0, -3, 0, 0, 7, 1, -1, 2, -2, 1], [4, 0, 0, 0, 2, 1, -1, 0, 0, -1, -1, 0, 0, 0], [-3, 0, 0, 1, -1, 0, -3, -6, 0, 0, 0, 0, 0, 0], [11, 0, -1, 4, -1, -2, 3, 2, 0, 2, 2, 0, 0, 0], [-8, 0, 0, -6, -2, -2, -2, -4, 0, 2, 2, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 1, -2, 2, 5, 3, -1], [-3, 0, -4, -2, 0, -1, -1, 0, 0, 1, 1, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0, 6, 0, 0, 2, 4, 0], [-3, 0, 2, -2, 3, 4, -2, 0, 0, -4, -4, 0, 0, 0], [0, 3, 0, 0, 0, -4, 0, 0, 1, 2, -2, 0, -2, 0], [0, 0, 0, 0, 0, -2, 0, 0, 10, 0, 0, 2, 1, 0], [-3, 0, -1, -4, -1, -4, 1, -2, 0, 4, 4, 0, 0, 0], [1, 0, 0, -7, -1, 0, 1, -2, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 2, 0, 0, -1, 1, -1, 0, 0, 5], [0, -1, 0, 0, 0, 1, 0, 0, 10, -2, 2, -8, 1, -1], [-3, 0, -1, -2, 0, 1, -2, 3, 0, -1, -1, 0, 0, 0], [-14, 0, -2, -1, -1, -4, 4, 2, 0, 4, 4, 0, 0, 0], [0, 3, 0, 0, 0, -1, 0, 0, -1, -3, 3, 1, 3, 1], [0, -2, 0, 0, 0, -3, 0, 0, -1, -2, 2, 4, 3, -2], [0, 9, 0, 0, 0, -1, 0, 0, 1, -4, 4, 0, 2, -3], [2, 0, -6, 4, -2, -3, 2, 2, 0, 3, 3, 0, 0, 0], [-8, 0, -2, 2, 0, 3, 0, 0, 0, -3, -3, 0, 0, 0], [0, -3, 0, 0, 0, -1, 0, 0, -18, -2, 2, 0, -2, -1], [0, 1, 0, 0, 0, 1, 0, 0, 4, -4, 4, -4, 0, -2], [2, 0, -4, -2, 2, 1, 0, 2, 0, -1, -1, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 6, -1, 1, 4, 3, -4], [0, 1, 0, 0, 0, -1, 0, 0, -3, 3, -3, -2, 0, 1], [0, 6, 0, 0, 0, -3, 0, 0, 11, -4, 4, -2, -1, -4], [11, 0, 0, -1, 0, -1, -1, -2, 0, 1, 1, 0, 0, 0], [0, -1, 0, 0, 0, 3, 0, 0, -7, -3, 3, 0, -2, -2], [0, -3, 0, 0, 0, 5, 0, 0, -2, 2, -2, 3, -3, 3], [3, 0, -4, 6, -1, 1, 2, 1, 0, -1, -1, 0, 0, 0], [6, 0, 0, 4, -2, 2, -2, 2, 0, -2, -2, 0, 0, 0], [-6, 0, -1, 2, -2, -1, -3, -3, 0, 1, 1, 0, 0, 0], [0, -7, 0, 0, 0, 1, 0, 0, 3, 3, -3, -4, -4, 5], [0, 4, 0, 0, 0, -3, 0, 0, -17, 0, 0, 0, 5, 0], [0, -6, 0, 0, 0, -4, 0, 0, -2, 2, -2, -4, -1, 3], [0, 1, 0, 0, 0, 0, 0, 0, 11, -5, 5, 8, 2, 1], [-4, 0, -4, 0, -2, 1, 1, 0, 0, -1, -1, 0, 0, 0], [0, 3, 0, 0, 0, 2, 0, 0, -8, -1, 1, 5, 1, -3], [0, 4, 0, 0, 0, 3, 0, 0, 14, 1, -1, -4, 0, 4], [-5, 0, 3, -1, 0, 4, -4, -3, 0, -4, -4, 0, 0, 0], [0, 9, 0, 0, 0, -7, 0, 0, -10, 0, 0, 4, 2, -3], [-7, 0, 5, 0, 3, 1, 3, 2, 0, -1, -1, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, -10, 1, -1, 4, 3, 0], [-7, 0, 2, -3, -1, 4, -7, -2, 0, -4, -4, 0, 0, 0], [-3, 0, 2, 2, 1, 2, -2, -4, 0, -2, -2, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 11, -4, 4, 4, 2, 2], [6, 0, 2, -3, 5, 2, 0, -2, 0, -2, -2, 0, 0, 0], [0, 1, 0, 0, 0, -2, 0, 0, 7, -4, 4, -3, 4, -2], [0, 0, -1, -2, 2, -1, 3, -3, 0, 1, 1, 0, 0, 0], [-17, 0, 2, -1, 3, 6, -1, -6, 0, -6, -6, 0, 0, 0], [-1, 0, 0, 5, 5, -1, 6, 1, 0, 1, 1, 0, 0, 0], [0, -2, 0, 0, 0, 1, 0, 0, -19, 0, 0, 2, -1, 2], [0, 2, 0, 0, 0, 0, 0, 0, 4, 2, -2, -4, 3, -4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 6, -6, 2, 0, 0], [-3, 0, 3, 4, 3, 3, 7, 4, 0, -3, -3, 0, 0, 0], [-8, 0, 6, 1, 1, 3, -1, -1, 0, -3, -3, 0, 0, 0], [11, 0, 1, -8, -1, 0, -5, -2, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, -2, 0, 0, -13, 5, -5, 0, -6, 1], [0, -3, 0, 0, 0, 3, 0, 0, -3, -3, 3, -4, 2, -3], [0, -2, 0, 0, 0, 0, 0, 0, 16, 0, 0, 6, 1, 0], [0, 0, 0, 0, 0, -1, 0, 0, 6, 1, -1, -3, -5, -1], [5, 0, -4, -4, -2, -2, -6, -4, 0, 2, 2, 0, 0, 0], [0, -2, 0, 0, 0, -2, 0, 0, 4, -4, 4, -6, 5, -2], [-12, 0, -2, -2, -4, -8, 6, -6, 0, 8, 8, 0, 0, 0], [8, 0, 0, 0, -4, 2, -4, -2, 0, -2, -2, 0, 0, 0], [-15, 0, -1, -6, 0, -1, 0, -6, 0, 1, 1, 0, 0, 0], [0, -7, 0, 0, 0, -1, 0, 0, -9, 5, -5, -4, -5, 3], [3, 0, -6, -1, -4, 2, -3, 4, 0, -2, -2, 0, 0, 0], [-1, 0, -4, 0, -4, 4, -1, 4, 0, -4, -4, 0, 0, 0], [0, 3, 0, 0, 0, 3, 0, 0, 6, -6, 6, 6, 4, 3], [0, 3, 0, 0, 0, -1, 0, 0, -22, 0, 0, 4, 3, -1], [0, 0, -2, 8, -2, -2, 0, -6, 0, 2, 2, 0, 0, 0], [0, -2, 0, 0, 0, -4, 0, 0, 0, -3, 3, 4, 7, -2], [-1, 0, 1, -2, -3, 0, 1, -6, 0, 0, 0, 0, 0, 0], [-6, 0, 4, 2, 0, 1, -2, 0, 0, -1, -1, 0, 0, 0], [0, -5, 0, 0, 0, -2, 0, 0, -15, 4, -4, 4, -6, 0], [0, 3, 0, 0, 0, -1, 0, 0, -12, -2, 2, 6, 0, -1], [0, 4, 0, 0, 0, 6, 0, 0, 5, -5, 5, -3, -1, 1], [9, 0, -5, 0, 1, 2, 1, 2, 0, -2, -2, 0, 0, 0], [-8, 0, 2, 2, -8, -3, -4, -4, 0, 3, 3, 0, 0, 0], [0, 0, 0, 0, 0, -4, 0, 0, -8, -4, 4, 2, 2, 0], [0, -5, 0, 0, 0, -2, 0, 0, -14, -1, 1, -6, 1, 1], [0, 3, 0, 0, 0, 2, 0, 0, -11, -4, 4, 5, 2, 0], [0, 3, 0, 0, 0, 5, 0, 0, 2, 0, 0, -2, -1, -1], [1, 0, 2, 2, 0, -3, 5, 6, 0, 3, 3, 0, 0, 0], [4, 0, 2, -5, 1, -3, 5, -3, 0, 3, 3, 0, 0, 0], [0, -2, 0, 0, 0, -2, 0, 0, -12, 4, -4, -8, 3, 0], [0, -9, 0, 0, 0, -1, 0, 0, -11, 5, -5, -7, -2, 0], [6, 0, 0, 12, 0, 2, -4, 6, 0, -2, -2, 0, 0, 0], [0, -5, 0, 0, 0, 3, 0, 0, 2, -2, 2, 2, -3, 1], [-7, 0, 8, -2, 6, 2, -1, 6, 0, -2, -2, 0, 0, 0], [0, -1, 0, 0, 0, -2, 0, 0, -3, -9, 9, -8, 2, -1], [0, 3, 0, 0, 0, -2, 0, 0, 21, -4, 4, -3, -2, -4], [9, 0, 2, 6, 3, -4, 6, 6, 0, 4, 4, 0, 0, 0], [2, 0, 1, 0, -5, 2, -5, -10, 0, -2, -2, 0, 0, 0], [0, -1, 0, 0, 0, 4, 0, 0, 15, 0, 0, 3, -1, -1], [0, 9, 0, 0, 0, 2, 0, 0, 5, 0, 0, 1, 1, -1], [-1, 0, 2, 3, 0, 4, -3, 8, 0, -4, -4, 0, 0, 0], [4, 0, -5, 3, -2, -3, 2, -4, 0, 3, 3, 0, 0, 0], [20, 0, -4, -8, -4, -8, 6, 2, 0, 8, 8, 0, 0, 0], [-8, 0, 6, -2, -5, -6, -1, -1, 0, 6, 6, 0, 0, 0], [0, 0, 6, -6, -2, -6, 4, -8, 0, 6, 6, 0, 0, 0], [4, 0, 4, 0, -2, 9, -10, -4, 0, -9, -9, 0, 0, 0], [0, -3, 0, 0, 0, -2, 0, 0, 2, 1, -1, 4, 1, -3], [-3, 0, 5, 10, 6, 4, -2, 0, 0, -4, -4, 0, 0, 0], [0, 10, 0, 0, 0, 0, 0, 0, 26, -2, 2, 4, 3, 1], [0, -3, 0, 0, 0, 3, 0, 0, 0, -6, 6, -4, 4, 3], [-7, 0, 2, -6, 5, 1, 8, 1, 0, -1, -1, 0, 0, 0], [0, -6, 0, 0, 0, 5, 0, 0, -19, 0, 0, -5, 5, 1], [-15, 0, 3, 7, 7, 0, 5, 8, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 1, 0, 0, 1, -4, 4, 0, 1, -2], [0, -11, 0, 0, 0, 5, 0, 0, 9, -3, 3, 1, 3, 1], [13, 0, -3, -6, 3, -5, 1, 0, 0, 5, 5, 0, 0, 0], [14, 0, -4, 12, -4, -6, -2, 4, 0, 6, 6, 0, 0, 0], [0, 2, 0, 0, 0, 6, 0, 0, 20, -6, 6, -6, 2, 0], [0, 0, 4, 0, 0, -2, 0, -12, 0, 2, 2, 0, 0, 0], [0, -4, 0, 0, 0, 2, 0, 0, -4, 4, -4, 0, -1, -4], [0, 2, 0, 0, 0, 0, 0, 0, -2, 2, -2, 4, 5, -1], [-2, 0, -4, 2, -2, -3, 6, 4, 0, 3, 3, 0, 0, 0], [0, 5, 0, 0, 0, 1, 0, 0, 7, 1, -1, 4, -4, 1], [-16, 0, -4, -10, -2, 0, -4, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 5, 2, 3, -5, -5, 0, -3, -3, 0, 0, 0], [0, 10, 0, 0, 0, -2, 0, 0, -10, 0, 0, 8, 1, -4], [0, -7, 0, 0, 0, -2, 0, 0, -9, 0, 0, 0, 1, -1], [22, 0, -2, -2, 2, -2, 2, 4, 0, 2, 2, 0, 0, 0], [5, 0, -6, 8, 2, -4, 9, 4, 0, 4, 4, 0, 0, 0], [0, 3, 0, 0, 0, 2, 0, 0, 30, 4, -4, 0, 0, 1], [12, 0, 2, 8, 0, -1, -1, 8, 0, 1, 1, 0, 0, 0], [-7, 0, 2, -8, 0, -1, 5, 0, 0, 1, 1, 0, 0, 0], [0, -3, 0, 0, 0, 5, 0, 0, -23, -1, 1, -6, -2, 3], [0, -6, 0, 0, 0, 3, 0, 0, -15, -6, 6, -12, 2, 0], [-1, 0, 8, 2, 2, 4, -5, 6, 0, -4, -4, 0, 0, 0], [-3, 0, -4, -6, -4, -10, 3, -2, 0, 10, 10, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -16, 2, -2, -8, 4, 4], [0, 5, 0, 0, 0, 1, 0, 0, -15, -3, 3, -4, -4, -1], [0, 6, 0, 0, 0, -2, 0, 0, 2, -2, 2, 0, -2, -6], [6, 0, -4, 9, -1, -1, -3, -5, 0, 1, 1, 0, 0, 0], [0, 10, 0, 0, 0, 0, 0, 0, 8, -8, 8, 0, 3, -2], [-2, 0, 2, -3, -1, -2, 8, -2, 0, 2, 2, 0, 0, 0], [0, 2, 0, 0, 0, 2, 0, 0, -10, 4, -4, 4, 4, 2], [-3, 0, 9, 1, -2, 0, -4, -1, 0, 0, 0, 0, 0, 0], [-13, 0, -3, 0, -1, -4, 2, -2, 0, 4, 4, 0, 0, 0], [0, -1, 0, 0, 0, 7, 0, 0, -7, -1, 1, 0, 2, 5], [-3, 0, 0, -12, 6, 4, -5, 0, 0, -4, -4, 0, 0, 0], [0, 3, 0, 0, 0, -3, 0, 0, -3, -3, 3, 4, 0, -3], [3, 0, -3, -12, 3, -3, 9, 0, 0, 3, 3, 0, 0, 0], [0, -8, 0, 0, 0, 4, 0, 0, 16, 2, -2, 3, -7, 1], [0, 0, -4, -12, 2, -1, 6, 4, 0, 1, 1, 0, 0, 0], [0, 9, 0, 0, 0, 1, 0, 0, -13, 7, -7, 7, 0, 0], [21, 0, -2, 2, -3, -1, -4, -3, 0, 1, 1, 0, 0, 0], [0, -3, 0, 0, 0, -1, 0, 0, 22, 8, -8, 2, -6, 3], [0, 0, -4, 0, 4, -1, 10, 4, 0, 1, 1, 0, 0, 0], [0, -1, 0, 0, 0, 1, 0, 0, -18, 0, 0, -4, -2, -4], [-8, 0, -5, 4, -7, -4, -3, -2, 0, 4, 4, 0, 0, 0], [0, 0, 0, 0, 0, 5, 0, 0, 7, 2, -2, 2, -7, 6], [-7, 0, -11, -3, -2, -4, 4, -1, 0, 4, 4, 0, 0, 0], [-5, 0, -7, 0, -1, 0, -8, -2, 0, 0, 0, 0, 0, 0], [-10, 0, 8, -8, 4, 4, -4, 4, 0, -4, -4, 0, 0, 0], [-21, 0, 1, -2, -1, -2, 5, -2, 0, 2, 2, 0, 0, 0], [0, 5, 0, 0, 0, -1, 0, 0, -16, 4, -4, 2, -1, -1], [0, 5, 0, 0, 0, 0, 0, 0, 19, 2, -2, 5, 1, 5], [0, -4, 0, 0, 0, -1, 0, 0, 9, 2, -2, -10, 1, 0], [6, 0, 2, 12, 2, -5, 2, -2, 0, 5, 5, 0, 0, 0], [-5, 0, 4, -2, -1, 5, -6, 1, 0, -5, -5, 0, 0, 0], [0, 5, 0, 0, 0, -3, 0, 0, 16, -7, 7, 4, 6, -1], [0, 4, 0, 0, 0, -2, 0, 0, -8, -4, 4, 13, 1, -5], [9, 0, 0, -10, 2, 5, -3, 4, 0, -5, -5, 0, 0, 0], [0, 1, 0, 0, 0, -4, 0, 0, 30, -3, 3, 4, 0, 4], [0, -2, 0, 0, 0, -9, 0, 0, -7, -2, 2, 2, 3, -2], [0, 1, 0, 0, 0, -5, 0, 0, -6, -8, 8, 1, 9, -9], [14, 0, 8, 2, 7, 10, -9, 5, 0, -10, -10, 0, 0, 0], [-4, 0, 6, 8, 2, 12, -8, -2, 0, -12, -12, 0, 0, 0], [-3, 0, -2, -8, -4, -5, 0, -3, 0, 5, 5, 0, 0, 0], [5, 0, 1, 4, 7, 3, 5, 4, 0, -3, -3, 0, 0, 0], [8, 0, 4, 14, -6, 2, -10, -6, 0, -2, -2, 0, 0, 0], [0, 2, 0, 0, 0, -4, 0, 0, -22, 4, -4, -6, 2, -4], [17, 0, 0, -2, -2, 0, -1, 2, 0, 0, 0, 0, 0, 0], [-1, 0, 4, 2, 3, 12, -12, 0, 0, -12, -12, 0, 0, 0], [0, 0, 0, 0, 0, -5, 0, 0, -17, 6, -6, 8, -4, 2], [31, 0, -5, -1, 2, -4, 1, 8, 0, 4, 4, 0, 0, 0], [-7, 0, 9, 4, 7, 6, 2, 5, 0, -6, -6, 0, 0, 0], [0, -5, 0, 0, 0, -4, 0, 0, -15, -6, 6, 7, 1, -5], [0, 10, 0, 0, 0, -6, 0, 0, 8, -2, 2, 8, -6, -2], [0, -5, 0, 0, 0, -3, 0, 0, -7, 7, -7, -4, -12, -2], [-16, 0, 4, -12, 6, 3, -3, -4, 0, -3, -3, 0, 0, 0], [21, 0, -3, -2, -7, 0, -9, 0, 0, 0, 0, 0, 0, 0], [0, 9, 0, 0, 0, -1, 0, 0, -8, -10, 10, 0, 6, 3], [-23, 0, 2, -4, -2, 3, 2, -13, 0, -3, -3, 0, 0, 0], [-10, 0, -2, 4, -4, -4, 0, -8, 0, 4, 4, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, -8, -12, 12, -1, 1, 1], [0, 12, 0, 0, 0, -5, 0, 0, -25, -2, 2, 10, 11, 0], [6, 0, 2, 4, 6, 2, -2, 4, 0, -2, -2, 0, 0, 0], [0, 3, 0, 0, 0, -6, 0, 0, -13, 2, -2, -5, -1, 3], [0, -2, 0, 0, 0, -6, 0, 0, 2, -2, 2, 4, -3, -2], [0, -10, 0, 0, 0, 4, 0, 0, 16, -2, 2, -8, 2, 6], [14, 0, -2, 6, 0, -3, 6, 14, 0, 3, 3, 0, 0, 0], [0, -1, 0, 0, 0, -2, 0, 0, -3, 0, 0, -13, 0, 2], [0, -7, 0, 0, 0, -7, 0, 0, -7, 1, -1, -6, -2, -1], [0, -7, 0, 0, 0, 6, 0, 0, -14, -1, 1, 3, -7, 11], [-20, 0, -4, 3, -4, 3, 1, 1, 0, -3, -3, 0, 0, 0], [-11, 0, -2, 0, 6, 2, 3, 4, 0, -2, -2, 0, 0, 0], [0, 0, 0, 0, 0, -10, 0, 0, -20, 8, -8, 2, -7, 0], [-4, 0, -6, -6, 0, 2, -4, 6, 0, -2, -2, 0, 0, 0], [0, 1, 0, 0, 0, 5, 0, 0, 4, -2, 2, 6, 1, -3], [31, 0, -6, 0, -4, -2, -1, 6, 0, 2, 2, 0, 0, 0], [0, -2, 0, 0, 0, 5, 0, 0, 1, -2, 2, 0, 7, -6], [15, 0, 3, -6, 3, 10, -5, 0, 0, -10, -10, 0, 0, 0], [20, 0, 5, 3, -2, 1, -2, 0, 0, -1, -1, 0, 0, 0], [0, -2, 0, 0, 0, 2, 0, 0, -20, -2, 2, -9, -3, 3], [15, 0, -2, -6, -8, 5, -5, -4, 0, -5, -5, 0, 0, 0], [0, -14, 0, 0, 0, 6, 0, 0, -4, 4, -4, -10, -4, 4], [14, 0, -3, 3, 6, -1, 8, 0, 0, 1, 1, 0, 0, 0], [0, 3, 0, 0, 0, 4, 0, 0, -1, -2, 2, -1, 11, -3], [0, 4, 0, 0, 0, -3, 0, 0, -7, -2, 2, 4, 3, -8], [0, 9, 0, 0, 0, -13, 0, 0, -10, 0, 0, 10, 6, -7], [-24, 0, 8, -2, 4, 8, 2, -4, 0, -8, -8, 0, 0, 0], [-6, 0, -2, 4, 8, -5, 14, 10, 0, 5, 5, 0, 0, 0], [0, 3, 0, 0, 0, 3, 0, 0, -8, 4, -4, -10, -1, -7], [0, -8, 0, 0, 0, -5, 0, 0, 3, 10, -10, -8, -9, 6], [-8, 0, 8, -2, 0, 7, 4, 0, 0, -7, -7, 0, 0, 0], [-21, 0, 2, 12, 2, 4, 2, 4, 0, -4, -4, 0, 0, 0], [0, -2, 0, 0, 0, -7, 0, 0, -22, 5, -5, 8, 0, 2], [0, 7, 0, 0, 0, -5, 0, 0, 16, 8, -8, 4, -5, 1], [0, 8, 0, 0, 0, 2, 0, 0, 24, -7, 7, -4, 3, 0], [0, -8, 0, 0, 0, -1, 0, 0, -11, -2, 2, 0, -5, 6], [-6, 0, -3, 2, 2, 9, -5, 7, 0, -9, -9, 0, 0, 0], [25, 0, -3, 2, 1, -5, 3, -2, 0, 5, 5, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, -22, -3, 3, 1, 1, -5], [10, 0, -10, -4, -2, -3, -6, -8, 0, 3, 3, 0, 0, 0], [0, -10, 0, 0, 0, -7, 0, 0, -18, 2, -2, -4, -5, -2], [0, -9, 0, 0, 0, 7, 0, 0, 12, 2, -2, -12, -2, -5], [0, 0, 8, -4, 6, 2, 2, -10, 0, -2, -2, 0, 0, 0], [12, 0, 0, -13, 2, -5, 5, 1, 0, 5, 5, 0, 0, 0], [0, 4, 0, 0, 0, -7, 0, 0, -24, -3, 3, 5, 0, 4], [0, 2, 0, 0, 0, -1, 0, 0, 3, -4, 4, -13, 7, -11], [-21, 0, -1, 0, 1, 2, -2, 10, 0, -2, -2, 0, 0, 0], [6, 0, 0, 6, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0], [-18, 0, 4, -10, 1, 4, -9, -3, 0, -4, -4, 0, 0, 0], [0, -11, 0, 0, 0, 7, 0, 0, -17, -7, 7, -6, -8, 7], [0, -11, 0, 0, 0, 1, 0, 0, 28, 8, -8, -6, -8, 1], [4, 0, 4, -10, -6, 1, -8, -18, 0, -1, -1, 0, 0, 0], [34, 0, 0, 2, -4, -5, -2, 2, 0, 5, 5, 0, 0, 0], [9, 0, 3, 6, 3, -7, 17, 0, 0, 7, 7, 0, 0, 0], [-17, 0, 4, 4, 2, 5, -3, -14, 0, -5, -5, 0, 0, 0], [9, 0, 2, 4, 0, 5, -14, -5, 0, -5, -5, 0, 0, 0], [0, 7, 0, 0, 0, -3, 0, 0, 14, 4, -4, 2, 3, -5], [-19, 0, 5, -10, 4, 9, -10, -13, 0, -9, -9, 0, 0, 0], [0, -10, 0, 0, 0, 7, 0, 0, 6, -3, 3, -13, -5, -1], [-15, 0, 1, 12, 1, 2, -5, 2, 0, -2, -2, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2106_b_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2106_2_b_c();".
// 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_2106_2_b_c(:prec:=14)
    chi := MakeCharacter_2106_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2106_2_b_c();".
// 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_2106_2_b_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2106_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![324, 0, 3240, 0, 10944, 0, 13932, 0, 4657, 0, 657, 0, 42, 0, 1]>,<17,R![8937, -2655, -2886, 867, 217, -61, -4, 1]>],Snew);
    return Vf;
end function;