
// Make newform 4100.2.d.g in Magma, downloaded from the LMFDB on 08 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4100_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4100_d_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_4100_2_d_g();".
// 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_4100_2_d_g();".
// 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 := [9, 0, 549, 0, 1896, 0, 2289, 0, 1141, 0, 261, 0, 27, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1194, 0, -12474, 0, 3665, 0, 9926, 0, 3015, 0, 317, 0, 11, 0], [30909, 0, 23175, 0, -3665, 0, -9926, 0, -3015, 0, -317, 0, -11, 0], [-55269, 0, -238413, 0, -205077, 0, -84469, 0, -18561, 0, -1974, 0, -76, 0], [43749, 0, -117510, 0, -438054, 0, -284399, 0, -72225, 0, -7860, 0, -299, 0], [0, -255924, 0, -743613, 0, -881244, 0, -445073, 0, -102927, 0, -10713, 0, -398], [0, 1088289, 0, 2932104, 0, 2488857, 0, 945079, 0, 177792, 0, 16023, 0, 541], [21792, 0, 607209, 0, 1282252, 0, 795630, 0, 203067, 0, 22348, 0, 858, 0], [0, -244029, 0, -756087, 0, -877579, 0, -435147, 0, -99912, 0, -10396, 0, -387], [0, 1244169, 0, 4739757, 0, 5565507, 0, 2688422, 0, 600321, 0, 61029, 0, 2234], [59667, 0, 1705125, 0, 3473993, 0, 2083974, 0, 519654, 0, 56324, 0, 2142, 0], [0, -34740, 0, -322644, 0, -585784, 0, -337957, 0, -82459, 0, -8811, 0, -332], [0, 846729, 0, 8410254, 0, 15261117, 0, 8885411, 0, 2187345, 0, 235404, 0, 8915]];
    Rf_basisdens := [1, 1, 10701, 10701, 10701, 10701, 32103, 32103, 10701, 10701, 32103, 10701, 3567, 32103];
    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_4100_d();"
function MakeCharacter_4100_d()
    N := 4100;
    order := 2;
    char_gens := [2051, 1477, 3901];
    v := [2, 1, 2];
    // 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_4100_d_Hecke();"
function MakeCharacter_4100_d_Hecke(Kf)
    N := 4100;
    order := 2;
    char_gens := [2051, 1477, 3901];
    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], [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 := 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]];
    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, 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, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, -1, 1, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0], [0, 1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 1, 0], [0, 1, 0, 0, 0, 0, 2, 0, 0, 0, -1, 0, 0, 1], [1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 1, -2, 0, -1, 0, 0, -3, -1], [0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 1, 0, 0], [0, 2, 0, 0, 0, 0, 3, -1, 0, 0, 0, 0, 1, -1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -2, 0, 1, 0], [0, -1, 0, 0, 0, 0, 3, 0, 0, 0, -1, 0, -2, 1], [0, 2, 0, 0, 0, 0, -1, 2, 0, -1, -2, 0, 1, 2], [4, 0, 5, -1, 0, 2, 0, 0, -3, 0, 0, 2, 0, 0], [1, 0, 0, -2, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0], [0, -2, 0, 0, 0, 0, 3, 1, 0, 0, -1, 0, -1, -2], [3, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0], [0, 2, 0, 0, 0, 0, -3, 2, 0, 1, -2, 0, 0, 1], [3, 0, 4, -1, -1, 2, 0, 0, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, -1], [3, 0, 3, -1, -1, -2, 0, 0, -1, 0, 0, 2, 0, 0], [0, 2, 0, 0, 0, 0, 4, 5, 0, -1, -2, 0, 1, 2], [3, 0, 3, 0, -2, -1, 0, 0, 0, 0, 0, -1, 0, 0], [0, 2, 0, 0, 0, 0, -3, -1, 0, 1, 3, 0, 1, 0], [0, 0, 0, 0, 0, 0, 5, 3, 0, 2, -1, 0, 3, 0], [3, 0, 0, -2, -3, -1, 0, 0, -1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, -1, -4, 0, -1, 1, 0, -5, -4], [0, 2, 0, 0, 0, 0, 1, -2, 0, 2, 4, 0, 1, 0], [7, 0, 1, 0, -4, 0, 0, 0, -2, 0, 0, -2, 0, 0], [0, 3, 0, 0, 0, 0, 6, -2, 0, 1, 1, 0, -2, -1], [1, 0, -3, 2, 1, -1, 0, 0, 6, 0, 0, -1, 0, 0], [1, 0, 1, -2, -2, -3, 0, 0, -2, 0, 0, 0, 0, 0], [3, 0, 3, 0, -2, -1, 0, 0, -4, 0, 0, -1, 0, 0], [0, 3, 0, 0, 0, 0, 1, 0, 0, 2, 4, 0, 3, 1], [0, -2, 0, 0, 0, 0, -4, 0, 0, 1, 2, 0, -4, -1], [0, -2, 0, 0, 0, 0, 6, 0, 0, 0, -1, 0, 0, 2], [0, 2, 0, 0, 0, 0, -1, -5, 0, -1, 1, 0, -3, -1], [1, 0, 1, 4, 4, 3, 0, 0, -2, 0, 0, 0, 0, 0], [-1, 0, 2, -2, -1, -5, 0, 0, 4, 0, 0, 2, 0, 0], [-1, 0, 1, -1, 4, -1, 0, 0, 0, 0, 0, 1, 0, 0], [0, -1, 0, 0, 0, 0, -7, 4, 0, -3, -3, 0, -2, 0], [0, -5, 0, 0, 0, 0, 5, -2, 0, -2, 1, 0, -6, -2], [-2, 0, -6, 2, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0], [1, 0, 0, 3, 3, -3, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, -3, 0, 3, -2, 0, 0, 2], [0, 4, 0, 0, 0, 0, 1, -2, 0, 4, 3, 0, 7, 2], [-4, 0, -6, 3, 0, 1, 0, 0, 2, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, -7, -2, 0, 2, 3, 0, 2, -3], [7, 0, -2, -2, -4, 0, 0, 0, -5, 0, 0, 1, 0, 0], [-5, 0, 3, 2, 3, 0, 0, 0, 2, 0, 0, -1, 0, 0], [10, 0, -2, 4, 0, 3, 0, 0, 3, 0, 0, -3, 0, 0], [0, 8, 0, 0, 0, 0, -4, 1, 0, 4, 4, 0, 7, 0], [0, 5, 0, 0, 0, 0, 3, 4, 0, -2, -2, 0, 6, 6], [5, 0, -2, 1, -4, -1, 0, 0, -1, 0, 0, 0, 0, 0], [1, 0, 2, 0, 2, -1, 0, 0, 2, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0, 8, 1, 0, 4, -3, 0, 5, 5], [2, 0, 11, -5, -5, 0, 0, 0, -8, 0, 0, 3, 0, 0], [0, 3, 0, 0, 0, 0, -6, -7, 0, 5, 6, 0, 4, -2], [0, -5, 0, 0, 0, 0, 0, -7, 0, 2, 0, 0, -3, 2], [0, -9, 0, 0, 0, 0, -1, -4, 0, -4, 0, 0, -8, -7], [9, 0, 4, -3, -2, -1, 0, 0, -5, 0, 0, 2, 0, 0], [0, 3, 0, 0, 0, 0, -10, 5, 0, 1, 0, 0, 1, 0], [0, -4, 0, 0, 0, 0, 1, -8, 0, -5, 1, 0, -5, -4], [4, 0, 4, -1, -1, -1, 0, 0, 1, 0, 0, 1, 0, 0], [0, 2, 0, 0, 0, 0, 5, 2, 0, 1, -3, 0, -4, 2], [0, 0, 0, 0, 0, 0, 3, 3, 0, -1, 0, 0, 5, 5], [-5, 0, -3, 1, 1, 4, 0, 0, 0, 0, 0, 3, 0, 0], [0, 4, 0, 0, 0, 0, 0, 5, 0, -4, -2, 0, 7, 2], [1, 0, -11, 1, 1, -3, 0, 0, 4, 0, 0, -6, 0, 0], [0, 2, 0, 0, 0, 0, -4, -7, 0, 1, 5, 0, 2, -3], [0, 5, 0, 0, 0, 0, -10, -3, 0, -4, -1, 0, -3, -2], [-1, 0, -2, -4, -2, 0, 0, 0, -7, 0, 0, 2, 0, 0], [0, 2, 0, 0, 0, 0, 1, 3, 0, 2, 1, 0, 0, -1], [-10, 0, -6, -1, 4, 2, 0, 0, 5, 0, 0, 4, 0, 0], [0, -12, 0, 0, 0, 0, 8, 1, 0, -2, 0, 0, -2, -6], [-4, 0, -6, 3, 8, 4, 0, 0, 6, 0, 0, -1, 0, 0], [-11, 0, -1, 2, 0, -5, 0, 0, 6, 0, 0, 1, 0, 0], [8, 0, 0, -1, -3, -8, 0, 0, 5, 0, 0, -5, 0, 0], [-10, 0, 3, 4, 6, 7, 0, 0, 1, 0, 0, 4, 0, 0], [1, 0, 6, -1, -3, 1, 0, 0, -3, 0, 0, 1, 0, 0], [0, 8, 0, 0, 0, 0, -4, 5, 0, -5, 0, 0, 3, 1], [-4, 0, 1, -2, 0, -5, 0, 0, 1, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, 1, 4, 0, 0, -2], [2, 0, -2, -2, 3, 5, 0, 0, 0, 0, 0, 3, 0, 0], [0, 8, 0, 0, 0, 0, 9, 0, 0, 2, -1, 0, 1, 5], [0, 0, -2, -3, 0, -4, 0, 0, 1, 0, 0, -5, 0, 0], [0, 1, 0, 0, 0, 0, 5, 2, 0, -8, -1, 0, -6, -5], [0, 1, 0, 0, 0, 0, -5, 7, 0, 2, -2, 0, -1, 5], [2, 0, -3, 0, -6, -5, 0, 0, -2, 0, 0, 2, 0, 0], [0, 2, 0, 0, 0, 0, -7, 3, 0, -2, -1, 0, 3, 6], [1, 0, 6, -1, -2, 0, 0, 0, 5, 0, 0, 1, 0, 0], [-1, 0, -3, -2, -3, -7, 0, 0, 1, 0, 0, -4, 0, 0], [0, -4, 0, 0, 0, 0, 13, -1, 0, 6, -1, 0, -1, 1], [5, 0, 2, -6, -7, -1, 0, 0, -5, 0, 0, -1, 0, 0], [8, 0, 2, -1, -5, 2, 0, 0, -5, 0, 0, 1, 0, 0], [0, 3, 0, 0, 0, 0, -1, 3, 0, -2, -6, 0, 5, 6], [-7, 0, -2, 1, -5, 1, 0, 0, 3, 0, 0, -5, 0, 0], [0, 2, 0, 0, 0, 0, 7, 8, 0, -5, -5, 0, -1, 0], [0, -7, 0, 0, 0, 0, 3, -1, 0, -5, -4, 0, -5, -4], [0, 7, 0, 0, 0, 0, 11, -2, 0, 4, 1, 0, 3, 1], [-6, 0, -1, 5, -3, 1, 0, 0, -1, 0, 0, -4, 0, 0], [-4, 0, 2, 5, -1, 8, 0, 0, 0, 0, 0, -1, 0, 0], [0, 4, 0, 0, 0, 0, 10, 2, 0, -1, 1, 0, 4, -2], [0, -5, 0, 0, 0, 0, 7, -7, 0, -3, -1, 0, -14, -7], [0, -2, 0, 0, 0, 0, -1, -3, 0, 7, 7, 0, 0, -1], [8, 0, -5, -2, -1, 1, 0, 0, -3, 0, 0, -3, 0, 0], [-13, 0, -4, 1, 0, -8, 0, 0, 6, 0, 0, 1, 0, 0], [0, 15, 0, 0, 0, 0, -16, 6, 0, 0, 2, 0, 7, -1], [0, -1, 0, 0, 0, 0, -13, -5, 0, 1, 2, 0, -5, -2], [0, 15, 0, 0, 0, 0, -16, 0, 0, 1, 0, 0, 7, 3], [8, 0, 10, 1, 3, 3, 0, 0, 0, 0, 0, 3, 0, 0], [-12, 0, -7, -3, -1, 2, 0, 0, 2, 0, 0, -1, 0, 0], [-13, 0, 3, 0, -5, 3, 0, 0, -6, 0, 0, 2, 0, 0], [0, -4, 0, 0, 0, 0, -6, 5, 0, 4, 2, 0, 8, 8], [0, 9, 0, 0, 0, 0, -10, -4, 0, 3, 6, 0, 1, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 8, -1, 0, 0, -2], [6, 0, -5, 1, 2, 0, 0, 0, 4, 0, 0, 2, 0, 0], [4, 0, 0, 2, -3, 3, 0, 0, 0, 0, 0, -8, 0, 0], [0, 5, 0, 0, 0, 0, -7, -13, 0, 0, 5, 0, -5, -9], [0, -12, 0, 0, 0, 0, -9, -8, 0, -7, -2, 0, -12, -8], [0, 0, 0, 0, 0, 0, 4, 5, 0, 0, -7, 0, 1, 4], [-18, 0, 1, 6, 7, 7, 0, 0, 5, 0, 0, 4, 0, 0], [-5, 0, -5, 0, 3, -4, 0, 0, 3, 0, 0, -5, 0, 0], [-5, 0, -1, -7, -3, -1, 0, 0, -8, 0, 0, 2, 0, 0], [0, 0, 13, -3, -2, -6, 0, 0, -2, 0, 0, -5, 0, 0], [0, -7, 0, 0, 0, 0, -19, -9, 0, -5, 1, 0, -4, -6], [0, -12, 0, 0, 0, 0, -6, -7, 0, -3, 2, 0, -6, -5], [4, 0, 14, 0, -1, 4, 0, 0, 2, 0, 0, 4, 0, 0], [0, -5, 0, 0, 0, 0, 5, -4, 0, -2, 9, 0, 0, -2], [-1, 0, 13, -4, -1, 4, 0, 0, -3, 0, 0, 8, 0, 0], [0, 0, 0, 0, 0, 0, 2, -1, 0, -4, 3, 0, -3, 1], [-6, 0, 0, -1, -2, -8, 0, 0, 0, 0, 0, -3, 0, 0], [-6, 0, -2, 0, -3, -11, 0, 0, 8, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, -4, 2, 0, 9, 4, 0, 4, 3], [0, 0, 0, 0, 0, 0, -8, 4, 0, -1, -1, 0, -5, 0], [0, -2, 0, 0, 0, 0, -9, 2, 0, -2, -6, 0, 5, 1], [-13, 0, 8, -7, 0, 4, 0, 0, -9, 0, 0, 8, 0, 0], [1, 0, 9, -1, -1, 3, 0, 0, -12, 0, 0, 9, 0, 0], [-2, 0, -7, 3, 2, -10, 0, 0, 5, 0, 0, -1, 0, 0], [0, 2, 0, 0, 0, 0, -4, -2, 0, 3, -2, 0, 2, 0], [0, -4, 0, 0, 0, 0, 15, 1, 0, -1, -6, 0, 4, -1], [-13, 0, 2, -2, -1, 1, 0, 0, 0, 0, 0, -1, 0, 0], [-3, 0, -7, 5, -5, -9, 0, 0, 5, 0, 0, -2, 0, 0], [0, -6, 0, 0, 0, 0, -3, 11, 0, -6, -11, 0, 2, 4], [0, -1, 0, 0, 0, 0, -10, -7, 0, 0, 0, 0, -11, -3], [-5, 0, -6, 0, 3, 3, 0, 0, -9, 0, 0, -3, 0, 0], [0, 10, 0, 0, 0, 0, 12, 1, 0, 2, -3, 0, -1, 2], [0, 10, 0, 0, 0, 0, -6, 4, 0, -3, -2, 0, 3, 6], [-9, 0, -11, -6, -5, -4, 0, 0, -2, 0, 0, -1, 0, 0], [0, 5, 0, 0, 0, 0, -3, 1, 0, -3, 0, 0, -6, 0], [0, 3, 0, 0, 0, 0, -22, -4, 0, -1, 0, 0, -4, -4], [0, -5, 0, 0, 0, 0, 11, -5, 0, 7, 2, 0, -1, 3], [-5, 0, 0, -7, 1, -5, 0, 0, 4, 0, 0, -2, 0, 0], [-3, 0, -7, -3, 3, 4, 0, 0, -2, 0, 0, 7, 0, 0], [-9, 0, -3, 0, -1, 5, 0, 0, -7, 0, 0, -2, 0, 0], [0, -7, 0, 0, 0, 0, -7, -3, 0, -5, 2, 0, -2, -8], [5, 0, 12, -7, -9, -10, 0, 0, -7, 0, 0, 3, 0, 0], [0, 3, 0, 0, 0, 0, 3, 4, 0, 1, -2, 0, 0, 4], [0, -5, 0, 0, 0, 0, 17, -1, 0, -5, -9, 0, -3, 1], [0, 6, 0, 0, 0, 0, -17, -1, 0, 6, 2, 0, 10, 5], [16, 0, 7, -1, 7, 9, 0, 0, 3, 0, 0, 1, 0, 0], [0, -13, 0, 0, 0, 0, -9, -4, 0, -2, -6, 0, -11, -2], [0, -8, 0, 0, 0, 0, 21, 5, 0, -4, -4, 0, -8, 1], [-6, 0, 7, 0, -6, 2, 0, 0, 1, 0, 0, -3, 0, 0], [0, -11, 0, 0, 0, 0, 11, -2, 0, -2, 5, 0, -1, 0], [-1, 0, -7, -3, 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-22, -5, 0, -13, 2, 0, -14, -12], [8, 0, -23, 12, 8, 8, 0, 0, 12, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 14, -5, 0, -5, -5, 0, -11, 5], [-10, 0, -2, -1, 4, 3, 0, 0, -7, 0, 0, -3, 0, 0], [0, -19, 0, 0, 0, 0, 39, -2, 0, -4, 5, 0, -14, -6], [40, 0, 17, 5, -1, 5, 0, 0, 12, 0, 0, -1, 0, 0], [0, 4, 0, 0, 0, 0, 8, 21, 0, -4, -8, 0, 3, 5], [-27, 0, -13, -3, 5, 6, 0, 0, -13, 0, 0, 7, 0, 0], [0, -4, 0, 0, 0, 0, 2, 7, 0, 6, -3, 0, -2, -7], [0, -17, 0, 0, 0, 0, -17, -8, 0, -13, -1, 0, -17, -10], [0, 9, 0, 0, 0, 0, 25, -2, 0, 3, -7, 0, -4, 3], [16, 0, 16, 11, 10, 10, 0, 0, -5, 0, 0, 2, 0, 0], [7, 0, 3, -8, -4, 11, 0, 0, -18, 0, 0, 10, 0, 0], [0, 3, 0, 0, 0, 0, 7, -7, 0, 2, 13, 0, -2, -7], [-15, 0, -10, 16, 2, -6, 0, 0, 12, 0, 0, -8, 0, 0], [13, 0, 11, -7, -1, 16, 0, 0, -23, 0, 0, 18, 0, 0], [0, 5, 0, 0, 0, 0, 2, -5, 0, -3, -5, 0, -5, 7], [13, 0, -7, -3, -10, -5, 0, 0, -10, 0, 0, 3, 0, 0], [0, 4, 0, 0, 0, 0, -6, -1, 0, 0, -2, 0, 7, -4], [-20, 0, 2, -10, -7, -11, 0, 0, 6, 0, 0, -2, 0, 0], [0, 3, 0, 0, 0, 0, -7, 10, 0, 14, 7, 0, 17, 6], [26, 0, 3, 13, -4, -2, 0, 0, 10, 0, 0, -8, 0, 0], [0, -1, 0, 0, 0, 0, 17, 3, 0, -5, -11, 0, 7, 6], [-1, 0, 22, -7, -4, 14, 0, 0, -21, 0, 0, 12, 0, 0], [0, 9, 0, 0, 0, 0, 7, 8, 0, -2, 5, 0, 9, -3], [-13, 0, -12, 5, 10, 8, 0, 0, 0, 0, 0, 9, 0, 0], [17, 0, -11, 11, 1, -8, 0, 0, 7, 0, 0, 1, 0, 0], [0, 11, 0, 0, 0, 0, 1, 8, 0, 12, -1, 0, 13, 12], [0, -7, 0, 0, 0, 0, -4, 9, 0, 4, 4, 0, 3, 12], [0, 1, 0, 0, 0, 0, 4, -12, 0, 10, 16, 0, -7, -11], [-15, 0, 6, -3, -11, -8, 0, 0, 7, 0, 0, -7, 0, 0], [0, -5, 0, 0, 0, 0, 25, -2, 0, 3, 3, 0, 16, -4], [-1, 0, 3, -19, -12, -16, 0, 0, -16, 0, 0, -3, 0, 0], [0, -17, 0, 0, 0, 0, 44, -3, 0, -1, 4, 0, -10, 2], [0, 6, 0, 0, 0, 0, -37, -4, 0, 1, 8, 0, 2, -10], [0, 0, 0, 0, 0, 0, -2, -7, 0, -6, 13, 0, -5, -7], [0, -11, 0, 0, 0, 0, 1, 2, 0, -18, 0, 0, -12, -10], [-22, 0, -23, 10, 13, -8, 0, 0, 20, 0, 0, -4, 0, 0], [-14, 0, 0, 11, 1, -3, 0, 0, 4, 0, 0, -7, 0, 0], [-23, 0, -8, 1, 3, -1, 0, 0, 6, 0, 0, -4, 0, 0], [0, 20, 0, 0, 0, 0, -28, 5, 0, -4, -4, 0, 1, -2], [6, 0, -4, 6, -13, -18, 0, 0, 9, 0, 0, -17, 0, 0], [-22, 0, -3, 5, 2, 11, 0, 0, -10, 0, 0, 4, 0, 0], [0, 6, 0, 0, 0, 0, -4, 10, 0, -11, -12, 0, -10, -8], [12, 0, -5, -8, 1, -4, 0, 0, 6, 0, 0, -3, 0, 0], [-13, 0, 13, -20, -7, -4, 0, 0, -15, 0, 0, 5, 0, 0], [0, -9, 0, 0, 0, 0, 1, -1, 0, 14, 13, 0, 3, 0], [-21, 0, 6, 1, 7, 4, 0, 0, -13, 0, 0, 10, 0, 0], [0, -5, 0, 0, 0, 0, -22, -14, 0, -8, 3, 0, -14, -5], [8, 0, -32, 9, 2, 4, 0, 0, 14, 0, 0, -11, 0, 0], [0, 7, 0, 0, 0, 0, -7, -10, 0, 8, 6, 0, -10, -9], [0, -9, 0, 0, 0, 0, -20, -12, 0, -7, -1, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, -3, 0, 7, -4, 0, 4, 7], [-12, 0, -15, 8, 4, 9, 0, 0, -4, 0, 0, -15, 0, 0], [0, 5, 0, 0, 0, 0, -16, -4, 0, 1, 8, 0, -10, 8], [13, 0, -3, -13, -11, 8, 0, 0, -19, 0, 0, 10, 0, 0], [0, 9, 0, 0, 0, 0, 16, 21, 0, 5, 1, 0, 13, 2], [0, 3, 0, 0, 0, 0, -9, -2, 0, 4, -3, 0, -7, -9], [0, 5, 0, 0, 0, 0, 6, -8, 0, -9, -1, 0, 4, -6], [17, 0, -20, 16, 10, -17, 0, 0, 15, 0, 0, -9, 0, 0], [0, -8, 0, 0, 0, 0, 8, -6, 0, -12, -5, 0, -3, 5], [-4, 0, -15, 1, 0, 13, 0, 0, -6, 0, 0, 3, 0, 0], [0, 19, 0, 0, 0, 0, 33, 15, 0, 12, 2, 0, 7, 10], [-16, 0, -5, 5, 17, -1, 0, 0, 9, 0, 0, 2, 0, 0], [0, -13, 0, 0, 0, 0, -33, 1, 0, -12, -2, 0, -3, -1], [43, 0, 7, -5, -13, -2, 0, 0, -13, 0, 0, -6, 0, 0], [-2, 0, 3, 0, -12, 2, 0, 0, 5, 0, 0, -14, 0, 0], [5, 0, -3, 11, -5, -23, 0, 0, 13, 0, 0, -16, 0, 0], [-28, 0, -8, -15, 1, -6, 0, 0, -12, 0, 0, 9, 0, 0], [-30, 0, 5, -10, 5, 10, 0, 0, -1, 0, 0, 8, 0, 0], [0, -13, 0, 0, 0, 0, 11, 18, 0, -13, -16, 0, -11, -6], [0, 6, 0, 0, 0, 0, -62, 9, 0, 0, 1, 0, 4, -1], [0, -9, 0, 0, 0, 0, -4, 4, 0, -13, -3, 0, -7, 15], [45, 0, 11, 2, 6, -14, 0, 0, 1, 0, 0, -3, 0, 0], [-5, 0, 11, -9, -11, -16, 0, 0, -7, 0, 0, -6, 0, 0], [0, -15, 0, 0, 0, 0, 22, -9, 0, -11, -10, 0, -17, -5], [25, 0, -11, 15, -7, -18, 0, 0, 8, 0, 0, -15, 0, 0], [0, 10, 0, 0, 0, 0, 15, 14, 0, -9, -10, 0, 14, 13], [3, 0, -14, 10, 3, -6, 0, 0, 19, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 4, 0, 0, -17, -16, 0, -14, -3], [0, 3, 0, 0, 0, 0, -42, -8, 0, 8, -2, 0, 4, 2], [0, 22, 0, 0, 0, 0, -14, 0, 0, -1, 5, 0, 17, 11], [-7, 0, -3, 1, -4, 1, 0, 0, -15, 0, 0, 13, 0, 0], [0, -10, 0, 0, 0, 0, 22, -8, 0, 5, 9, 0, 0, 3], [42, 0, -4, -4, -7, -2, 0, 0, -4, 0, 0, 4, 0, 0], [-50, 0, -19, 3, 1, 0, 0, 0, 13, 0, 0, -14, 0, 0], [0, -2, 0, 0, 0, 0, -5, -8, 0, 5, 14, 0, 8, 5], [0, 1, 0, 0, 0, 0, 0, -12, 0, 16, 11, 0, 18, 5], [0, -9, 0, 0, 0, 0, -1, -5, 0, 5, 4, 0, -3, 3], [2, 0, -4, 3, -3, 2, 0, 0, 5, 0, 0, 5, 0, 0], [0, -10, 0, 0, 0, 0, 25, 1, 0, -5, -9, 0, -12, 0], [0, -6, 0, 0, 0, 0, -60, -12, 0, -4, 0, 0, -11, -8], [29, 0, -19, -5, -4, -17, 0, 0, 0, 0, 0, -11, 0, 0], [0, 10, 0, 0, 0, 0, -14, -6, 0, 8, 11, 0, 4, -3], [0, -17, 0, 0, 0, 0, 13, 0, 0, -19, -4, 0, -22, -10], [0, 8, 0, 0, 0, 0, 26, -2, 0, 11, 0, 0, 9, -3], [5, 0, -18, 3, 14, 8, 0, 0, 2, 0, 0, 11, 0, 0], [-9, 0, -5, 0, -8, -6, 0, 0, 10, 0, 0, 6, 0, 0], [13, 0, 9, -15, -15, 8, 0, 0, -22, 0, 0, 10, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4100_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4100_2_d_g();".
// 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_4100_2_d_g(:prec:=14)
    chi := MakeCharacter_4100_d();
    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_4100_2_d_g();".
// 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_4100_2_d_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4100_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![9, 0, 549, 0, 1896, 0, 2289, 0, 1141, 0, 261, 0, 27, 0, 1]>,<7,R![1, 0, 145, 0, 4590, 0, 11090, 0, 4655, 0, 729, 0, 46, 0, 1]>],Snew);
    return Vf;
end function;