
// Make newform 4100.2.a.h in Magma, downloaded from the LMFDB on 05 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4100_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_4100_2_a_h();".
// 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_a_h();".
// 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 := [-11, -37, -4, 49, 7, -13, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-4, -1, 1, 0, 0, 0, 0], [2, -6, -1, 1, 0, 0, 0], [5, 4, -7, -1, 1, 0, 0], [-19, -39, 31, 24, -10, -3, 1], [-38, -39, 74, 24, -23, -3, 2]];
    Rf_basisdens := [1, 1, 1, 1, 1, 3, 3];
    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_a();"
function MakeCharacter_4100_a()
    N := 4100;
    order := 1;
    char_gens := [2051, 1477, 3901];
    v := [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_4100_a_Hecke(Kf)
    return MakeCharacter_4100_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], [0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0], [0, 1, 1, 0, 1, 1, 0], [-1, 0, 0, 1, 0, 0, 0], [-1, 0, -1, 0, -1, -1, 1], [0, 1, -1, 0, 0, 1, 0], [-1, -1, -1, 0, -1, 0, 0], [0, -1, -1, -2, 2, 1, -1], [0, 0, 0, -1, 1, 1, 1], [-4, 2, 0, 0, 1, 1, -2], [-1, 0, 0, 0, 0, 0, 0], [-3, -1, -1, -1, 0, 0, -2], [-2, 2, 1, 0, -1, 1, -1], [-1, -1, 0, 1, -2, 0, 2], [2, 2, 0, 3, -2, -2, 0], [-1, -3, -1, -1, -1, -1, 2], [-5, 1, 1, 2, -4, -2, 1], [-3, 2, 1, 1, 1, -1, 0], [-2, -1, 2, -1, -1, 0, 0], [1, 0, -1, -1, -2, 1, 1], [0, 0, -1, -2, 1, 0, -2], [0, 2, 0, 0, 0, -1, -3], [-6, -1, 0, -3, 2, 3, 0], [-1, 0, 1, -1, 5, 3, -2], [2, 3, 4, 2, -2, -2, 1], [-3, 1, -1, 0, 0, 0, 1], [-3, -1, -1, -2, -2, 0, -3], [-2, -2, 2, 2, 0, -3, -1], [4, -1, 3, -1, 4, 2, 4], [-5, 3, 2, 4, -4, -2, 2], [-6, 2, 0, 0, -3, -1, -3], [2, -2, -1, 2, -1, 0, -1], [9, -1, 4, 2, 0, -1, 4], [-1, 0, -3, -1, -3, -1, 2], [-10, 3, 0, 2, -3, 0, 0], [-1, -3, 0, -3, 5, 2, 0], [1, -2, 0, -1, 0, -1, -1], [-6, 2, -1, 3, -3, 2, -1], [5, -5, -2, -4, 4, 3, -2], [0, 0, 2, 1, 0, 0, -3], [2, 0, 2, 1, -1, -3, 2], [-7, 1, -1, 3, -2, -3, 3], [-6, -5, -2, -3, 0, -1, -1], [3, 3, 3, -1, 4, 4, -2], [-8, -1, 0, 0, 2, 0, -5], [1, 4, -1, 0, 0, -1, 2], [-3, -5, 1, -2, 4, 1, -3], [-5, -3, -2, 1, 3, 3, 0], [-2, -5, 1, 0, -3, -3, 1], [4, 2, -2, 4, -7, -4, 0], [4, -2, 1, 0, 1, -3, 1], [-3, -4, -2, -2, 1, -1, 6], [1, 1, 1, -3, 4, 1, 4], [-1, -1, -2, 0, 2, 0, -4], [1, -7, -4, 3, -2, -3, 0], [5, 3, 5, 1, 1, 5, 0], [-2, 3, 6, -1, 5, 4, 1], [0, 0, 2, 2, 3, 3, -1], [-1, 1, -1, 0, -2, -3, 0], [3, 6, 5, -1, 2, 3, 0], [-9, -2, -1, -1, 5, 4, -2], [-9, -3, -4, -3, -2, -3, -2], [-8, 0, 4, 3, -4, -5, 2], [0, 1, -2, 0, -2, -3, 5], [1, 2, -1, 3, 1, 4, -1], [-11, 0, -3, -1, -4, -1, -1], [1, 4, 3, 2, 5, 1, 2], [1, 2, 6, 0, 1, 3, -1], [1, -3, -1, -1, 6, 5, -4], [7, -3, 1, -3, -2, -6, 1], [6, 5, 2, 2, 1, 0, -3], [-4, 6, 5, 4, 4, 0, 1], [-2, -1, -3, 5, -2, -6, 2], [10, 5, 5, -2, -1, 0, -1], [2, -3, -4, 1, 2, 0, 2], [-8, 0, -2, -4, -2, 3, -4], [-1, -9, -6, -3, 5, 4, -2], [-7, 6, 3, 1, 1, 3, -4], [15, 0, 1, 1, 7, 1, 1], [3, -2, 3, 1, 0, -1, 2], [-4, -2, -7, 1, 3, -2, 1], [-10, 0, -3, 0, 3, 3, 0], [-5, -10, -3, -1, -6, -5, 4], [-2, 2, -7, 2, -4, 2, -2], [-1, 5, -1, -2, -3, -2, 3], [-5, 10, 0, 5, -5, 3, 3], [-4, 6, -1, 2, -3, 0, -2], [0, -6, -3, 2, -1, 1, -3], [-4, -3, 4, -2, 5, 1, 2], [-2, 3, 0, -6, 2, 3, 2], [5, 5, 1, -2, 2, 4, 1], [-3, -4, -1, -2, -5, 2, 4], [-9, 4, 1, 0, -2, 2, 5], [7, 4, 2, 5, 1, 0, 1], [-7, -6, -2, -9, 5, 4, -1], [11, 2, 5, -3, -3, -5, 3], [0, -8, 1, -3, 4, 5, 2], [6, -6, -5, -9, 5, 8, -1], [16, -9, 2, -2, 2, -5, 3], [-3, 6, 0, 5, -4, 1, -4], [17, -9, 0, -1, 1, -3, 7], [-13, 10, -4, -2, 0, 2, -7], [4, -1, 0, 2, 3, 3, 3], [8, 0, -1, 3, 0, -5, 3], [17, -2, -6, 2, -1, -4, 1], [3, -7, 4, -6, 3, 0, -1], [-18, 7, -2, 1, 1, 0, -7], [14, 1, -2, -5, 7, -2, -6], [5, 7, -4, 4, -5, -4, -4], [-21, -2, -4, 3, 2, -1, -4], [-12, 3, -3, -2, -7, 2, 0], [-6, -15, -8, -6, -1, -6, -3], [8, 0, 8, 4, 11, 5, -1], [0, -1, -2, 3, -8, 0, 1], [0, 0, 0, 2, 0, -5, 2], [14, 4, 4, 3, -6, 2, -2], [1, 0, 4, -3, 4, 9, 1], [-3, 3, 1, -2, -2, 6, 0], [3, 10, 0, 1, 0, 4, -5], [-9, -13, -2, 1, 3, 2, -1], [10, 4, 5, 2, -6, -4, 4], [4, -1, 3, 6, -2, -6, -5], [2, 2, -5, 0, -6, 0, -1], [3, -1, 0, 3, -3, -3, 1], [-15, 3, 3, 3, 4, 2, -1], [7, -10, -4, -1, 3, 4, 2], [3, -11, -5, -7, 4, 0, -3], [-19, 3, 2, 3, -7, -3, -6], [-19, -4, 0, 3, 4, -1, -7], [16, 3, 4, 1, 0, 7, -3], [13, -8, -3, -6, 2, 3, -7], [-2, 12, 6, 3, -1, -2, 1], [5, 5, 6, -1, 7, 10, 0], [-5, 13, -3, -4, 1, 0, -3], [7, -5, 4, 1, 5, 2, 4], [3, 3, 4, -8, 10, 3, 1], [-2, -14, -4, 4, -1, -2, -2], [-7, 9, -2, -1, -4, -4, 0], [3, -2, 2, 0, 9, 8, 1], [12, -9, -5, 0, -1, -6, 2], [7, -4, -1, -2, -4, 4, 4], [12, -1, -2, -9, 3, 3, -4], [-14, -7, -5, -2, -7, -2, 3], [6, 10, 4, 2, 4, 2, 1], [-6, 10, 2, 5, -4, 2, 5], [29, 5, 4, -2, 5, 5, 4], [-11, -4, -8, -9, 7, 4, -5], [-6, -9, -9, 0, -2, -8, -3], [-16, -5, -2, 5, 0, 0, 0], [-12, 4, 1, 0, -3, -3, -3], [-6, 1, -7, -2, 8, -1, -4], [3, -3, 3, -2, -4, -8, 2], [1, 5, -1, -4, 5, -1, -4], [1, -4, 5, 3, 4, 2, 5], [-11, 6, 2, 6, -11, -5, -5], [-13, 10, 5, -4, 4, 8, -5], [-2, 11, 9, 8, -8, -9, 2], [10, 0, -7, -3, 5, 3, 1], [0, 3, -1, 2, 0, 6, -10], [0, -5, -5, 7, -11, -10, 7], [15, -3, -7, 5, -3, -7, 2], [-9, -1, -3, 1, 1, -2, -5], [-2, 8, 3, 5, 6, 2, 4], [23, 11, 7, 5, -7, 1, 6], [29, -1, 2, 2, -5, -2, 0], [15, -16, -7, -6, -2, -1, -1], [-15, 7, -5, -1, -4, 0, 5], [4, -2, 2, -10, 13, 6, 4], [18, 2, 7, 4, -8, -3, 7], [0, -6, -8, -3, 1, -8, -7], [11, -10, -2, 0, -1, -5, 3], [-28, 1, 0, -5, 2, 5, 2], [-7, -16, 0, -3, 4, 2, 2], [-20, 7, -2, 6, -8, 0, -9], [3, -3, 7, 1, 7, 1, -1], [-4, 11, 3, -1, 7, 1, -3], [1, 3, 0, 5, -9, -1, 1], [-14, 4, 3, 4, 0, -7, 3], [-1, 1, -4, -2, 1, 3, -9], [6, 7, 0, 4, 1, 1, -4], [-14, -1, -7, 2, -7, -7, 2], [-3, 0, 4, 1, -3, 6, 1], [-1, 6, -1, 0, -9, -5, -7], [7, -6, 3, 9, -2, -9, 3], [-13, -5, -7, -9, 9, 4, -1], [-22, 7, 0, 4, -9, 1, -3], [-14, -12, 5, 1, 1, -6, -3], [3, -13, -5, -2, -2, -1, 3], [4, -3, 9, -3, 13, 3, 1], [11, -1, 3, 5, -7, -8, 11], [7, -3, 8, 0, -1, 0, 5], [14, -13, 0, -6, 2, 2, 0], [-3, -8, -5, -10, -4, 3, 1], [3, 4, -9, -4, 2, 1, -5], [9, 11, 1, 9, -5, -3, -4], [-16, -13, -11, -2, -12, -14, -3], [20, 4, -4, -1, 7, -2, 0], [-18, -8, -4, -1, -1, -1, -11], [-1, 10, 7, 9, -2, -3, 7], [28, -7, 0, -4, 3, -5, -5], [4, 5, -2, 11, -6, -9, 1], [-7, 2, -7, -2, -3, -7, -1], [14, -16, -1, 0, 6, 3, -1], [16, -7, -2, -7, 10, 10, 3], [-8, -14, -5, 7, -10, -10, 5], [-3, -8, -3, -3, 1, 2, 3], [26, -13, 4, -4, 14, -4, 1], [-7, 4, 1, 3, -9, 0, -9], [-20, -5, 1, 1, 6, 3, 5], [-7, 9, 2, 5, -11, -9, 9], [1, 6, -7, -11, 1, 7, 1], [-9, 14, 1, 0, 3, 2, 1], [-13, 8, -8, 2, 1, 8, -4], [9, 1, 0, -5, 2, 0, 4], [-12, -8, 1, 5, -15, -9, 7], [-1, 5, 11, 3, 11, 10, 7], [16, -6, -2, 2, 10, 11, -3], [11, -3, 0, 7, 1, -1, -3], [24, 2, 9, -5, 14, 4, -2], [-20, 13, 4, 3, -8, -5, 5], [10, -2, -2, -3, -5, -1, -10], [-3, 1, -7, 5, 5, 1, 1], [-9, -14, 0, -6, -3, -5, -3], [-10, -3, -6, -9, 6, 6, -12], [-10, -4, 3, -2, 12, 12, 0], [5, -17, 5, 2, -7, -5, 8], [21, -6, -2, -7, 18, 5, -5], [-13, -16, 2, 2, -9, -5, 3], [-14, -3, -5, 0, -2, -15, 5], [0, -8, -3, -6, 0, -6, 6], [-4, 2, -6, 2, -7, -5, 0], [9, 23, 3, 13, -12, -1, 3], [7, 9, 10, 11, -2, -5, 3], [-11, 13, -1, 0, -3, 4, -3], [-9, -10, 4, -8, 12, 6, -3], [-2, 10, 2, 1, -3, 4, 9], [23, -20, -3, -7, 5, -5, -6], [-13, 11, -5, 0, -4, -1, 10], [14, 0, 8, -2, 8, -1, 13], [22, -13, 2, 4, 2, -2, 4], [-14, 13, -1, 3, 3, 5, -1], [5, -8, 0, 3, -10, -15, 9], [3, -12, -8, -4, -18, -8, 8], [-14, -13, 0, 3, -9, -3, 11], [7, -11, 2, -2, 12, 7, 4], [-17, 1, -14, -3, -5, 2, 8], [-3, 9, 8, -7, 2, 8, -9], [8, -6, -5, 0, 2, 2, -6], [-12, -7, -7, 8, -15, -12, 10], [-17, 5, 5, -1, 6, 3, 1], [-4, 3, 4, 4, -11, -20, 9], [3, 15, -3, 5, -8, -9, 6], [20, 18, -4, 4, -4, -4, -2], [48, 6, 5, -4, 8, 8, 2], [14, -3, 4, -6, 10, 4, -2], [11, -1, 0, -3, 5, 1, 12], [-6, -2, 10, 6, -3, -8, 10], [4, 18, 18, -1, 11, 9, -3], [-10, 1, 8, 1, 7, 5, -7], [-3, 8, 5, 0, -9, -2, -2], [0, -12, -2, -4, 12, -2, -3], [23, 11, 6, -1, 14, 11, 1], [-23, -16, 3, 3, -11, -9, 1], [-6, -7, 5, 5, 0, 4, -1], [0, 26, 1, 7, -5, -1, 0], [-26, 6, -3, -2, 0, 11, 0], [-19, 17, 1, -5, 3, 10, -12], [-5, 11, 3, 5, -5, -1, 6], [-2, -9, -5, -8, -4, 1, 8], [-6, 9, -6, 3, -5, 0, -9], [-6, -2, 9, 0, 16, 3, -1], [-23, -10, 1, -5, 13, 11, -7], [12, -9, 2, 3, 3, -4, 2], [-30, 10, 0, 10, -9, 1, 10], [-21, 0, -3, -3, 6, 0, 3], [27, -3, 2, 1, 8, 9, -5], [6, -8, -1, 0, 7, 13, 2], [3, 10, 10, 3, 14, 13, -4], [7, -14, -4, 4, 3, -4, 4], [13, 5, -8, 4, -15, -12, 1], [-9, -21, -1, -11, 21, 9, -7], [-40, 14, -4, -5, -9, 1, -2], [-21, -6, -2, 6, -11, -7, -3], [-23, 4, 9, 12, -5, -10, 7], [-3, -17, 2, 2, 3, 0, -4], [-13, -2, -6, -1, -10, -1, -5], [-12, 18, -3, -3, 3, 0, -5], [23, 16, -1, -2, 7, 11, 4], [-6, 16, -2, -3, 0, -5, 4], [-31, 2, -2, 4, -3, -2, 14], [14, -12, -4, -4, -5, -6, 3], [-5, -4, 7, 5, -4, 1, -7], [9, -8, -4, 1, 10, 0, 10], [6, -5, -2, -2, 16, 14, -5], [19, 3, -1, -5, 10, 5, -1], [20, -2, 4, 2, -3, 3, -7], [9, -4, 3, 10, -2, -17, 1], [4, 9, 3, 3, -2, -4, 12], [3, -10, -6, -6, -8, -10, -3], [2, -7, 13, 4, -3, -8, 5], [-7, 10, 1, 4, -8, -9, 14], [4, 21, 9, 8, 0, 4, -4], [-30, -4, -4, -1, -3, -5, -1], [-39, 5, -8, 0, 0, -2, 1], [-10, -6, 18, 3, 4, -2, 5], [28, 7, -10, 3, -6, -4, 2], [7, 8, -10, -9, 13, 0, -9], [10, -2, 7, 1, 11, -2, 10], [16, -4, 5, 0, 5, 2, 1], [3, 22, -7, -7, 3, 4, 2], [11, 10, -1, 4, -10, -4, 12], [-32, 14, 8, 9, 5, 2, 5], [4, -2, -7, 9, -5, -5, 8], [-1, 15, 2, 6, -7, -3, 6], [22, 6, -7, -2, 5, 3, -2], [-4, -2, 6, -3, 11, -1, 7], [-23, 8, 0, 1, -3, 7, -2], [7, -3, -5, 1, 2, 1, 10], [-8, 27, 10, 4, -9, -7, -4], [-7, -4, -10, -17, 4, 10, 3], [-20, 6, -5, 7, -1, -7, 2], [-8, -5, 3, 1, -16, -12, 8], [-20, 3, -2, 11, 1, 1, -1], [-32, 6, 1, 4, -15, -2, -11], [-29, 13, -5, 5, -13, -3, -6], [-39, -6, -10, -9, 4, -5, 0], [-8, 10, 9, 1, -14, -11, 3], [-14, -24, -3, -9, 13, 8, -7], [-7, 1, 3, 10, -9, -10, -8], [-11, 1, -1, 8, -8, -11, -4], [-18, 1, 2, -8, 2, -2, -2], [5, 11, 3, 7, -7, -14, 5], [26, -2, 11, 0, 7, 5, 8], [-18, -5, -2, 1, -8, 5, 7], [4, -10, -4, 10, -7, -4, 5], [-3, 8, -4, -4, 5, 9, -2], [-16, 7, 5, -1, -5, 7, -5], [9, -1, -1, 6, -9, 4, -13], [-44, 0, -4, -8, 4, 1, 1], [43, -22, -6, 2, -3, -9, 5], [-12, -7, -2, 5, -18, -26, 12], [-11, 1, 4, 18, -10, -11, 10], [8, 2, 7, 2, 15, 12, -5], [37, -14, -4, -6, 0, 4, -6], [-10, 4, -12, 4, -2, -7, -9], [-7, 7, -1, -3, -11, -18, 9], [38, 0, 4, -9, -7, -1, 12], [-35, 17, 3, -1, 0, -3, 8], [-11, 12, 4, 15, -4, -10, -3], [-37, 10, 5, 4, 0, 1, 5], [3, 1, -1, -8, -8, 2, -10], [6, -22, 0, 0, 2, -10, -7], [21, 7, -2, 11, -8, 2, -4], [5, 7, -12, -7, -15, -5, -1], [-4, -9, -12, -7, -3, -2, 0], [-22, 19, 9, -1, 9, 5, -3], [-7, 20, -7, -2, 6, 7, -15], [5, 18, -1, 11, -14, -11, 4], [-25, -7, 6, -6, 13, 16, -2], [8, -16, -9, 1, -3, -3, -9], [-27, 1, 1, -15, 6, 15, -9], [8, -2, -6, -5, 3, -6, -8], [23, 5, 11, 6, -6, 7, 10], [-4, 0, 15, 0, 6, 2, 8], [4, 10, -6, -1, -1, 6, -5], [-5, 21, 6, 10, -16, -16, 14], [-31, -21, 2, 1, -6, -3, 11], [-2, 11, -4, -14, 5, 10, -3], [-13, -2, -9, 13, -10, -4, -7], [1, 23, 5, 15, -10, 3, 2], [-20, 12, 4, 13, -15, -15, -1], [23, 7, 5, -4, 16, 11, 8], [-19, -4, 3, 0, 0, 6, 10], [-1, 0, 3, 16, -15, -11, 9], [-9, -15, 8, 5, 7, -1, -9], [-17, 0, -13, 2, -2, 0, -9], [22, -5, 6, 8, -8, 13, -3], [-15, 2, -9, 0, -7, -3, 9], [-15, -5, 6, 3, 3, -14, -6], [3, 6, 1, -1, 5, 24, -4], [12, -13, 1, -5, -12, -7, 11], [-5, -5, -2, 13, -9, -23, -4], [-10, 2, -11, -5, -11, 3, -2], [3, 1, -1, -14, 16, 20, -4], [30, -10, -2, -8, -12, 1, 1], [43, -6, 0, -6, 8, 2, 11], [-19, 31, 8, 3, 1, -1, 5], [6, -21, -1, 1, 4, -1, 9], [26, -26, -6, -8, -1, -1, -10], [12, -4, 6, 5, 9, 13, -1], [5, 30, 8, 17, -5, -1, -4], [26, -10, 3, -7, -4, -1, -2], [25, -14, -8, -14, 2, 4, -9], [6, -15, -2, -12, 13, 11, 6], [12, 14, 24, 0, 0, 7, -3], [7, 0, 15, -5, -4, 0, 2], [18, -6, -12, 6, -8, -12, 9], [-10, 15, -2, 10, -11, 3, 5], [8, -18, 3, -6, 18, 13, 1], [1, -20, -20, -5, -10, -10, -8], [-11, 9, 17, -2, 21, 10, -1], [-21, 11, 11, 1, 5, 3, 1], [-10, -15, -12, -6, -11, -8, -16], [32, 0, -18, 4, -20, -7, 1], [-22, -17, 1, 8, -15, -9, 12], [17, -22, -12, -8, 9, 9, -9], [-34, 3, 2, 1, 15, -2, -12], [-1, 3, 1, -8, 24, 13, -11], [25, 23, 18, 15, -13, -4, 6], [-10, 5, 12, 4, -4, 0, -8], [19, 8, 1, -9, 11, 5, 15], [5, 11, 7, 16, -5, 1, 2], [-4, 17, -10, 6, -15, -3, -3], [1, 8, 6, -3, 12, 5, 1], [-7, 2, 14, 6, 2, 9, 3], [18, -21, -2, 1, 1, -4, -12], [15, -2, -7, 6, -13, -5, -3], [-7, 9, 1, -4, -5, 3, 2], [-19, 6, 15, 3, 9, -9, 5], [-3, -10, -19, -3, -2, -8, -5], [1, -1, 17, -5, 12, 0, 4], [-20, 8, -14, -1, 1, 11, -2], [11, 15, -7, 10, -7, -1, 5], [34, -1, -5, 0, 2, -12, -8], [-16, -22, -11, -20, 15, 4, -4], [-20, 3, 6, 5, -1, -8, -12], [-11, -7, -12, -2, 2, 0, -2], [1, -5, 13, 1, 20, 9, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4100_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4100_2_a_h();".
// 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_a_h(:prec:=7)
    chi := MakeCharacter_4100_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_4100_2_a_h();".
// 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_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4100_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![11, -37, 4, 49, -7, -13, 1, 1]>],Snew);
    return Vf;
end function;