
// Make newform 3330.2.a.bk in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3330_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_3330_2_a_bk();".
// 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_3330_2_a_bk();".
// 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 := [-36, 9, 39, -20, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [-22, -19, 20, 0, -1], [-36, -21, 37, -1, -2], [120, 33, -94, 4, 5], [-96, -39, 77, -2, -4]];
    Rf_basisdens := [1, 2, 3, 6, 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_3330_a();"
function MakeCharacter_3330_a()
    N := 3330;
    order := 1;
    char_gens := [371, 667, 631];
    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_3330_a_Hecke(Kf)
    return MakeCharacter_3330_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 := [[-1, 0, 0, 0, 0], [0, 0, 0, 0, 0], [-1, 0, 0, 0, 0], [1, 0, 0, 0, 1], [-1, 0, 0, -1, 0], [1, 0, -1, 0, 1], [-1, 0, 0, 0, 1], [0, 1, 1, -1, 0], [-1, 0, -1, 0, 1], [0, 0, 1, 0, 0], [3, -1, -1, 0, 0], [1, 0, 0, 0, 0], [-1, 1, -1, 0, 0], [3, -1, 1, 0, 0], [-1, 1, 0, 1, 1], [0, -1, 0, 0, 1], [2, 0, 0, 2, 0], [2, 2, 1, -1, -3], [0, 0, 0, 2, -2], [2, -2, 0, 0, 0], [-1, 2, 1, -2, -1], [0, 2, 0, 0, 0], [1, 0, -1, 2, 1], [3, 0, 1, 0, 3], [2, 0, -1, -1, -1], [-1, 3, 0, 1, -1], [1, 1, 0, 1, 1], [-1, 0, -3, 0, 1], [3, -2, 0, -1, 0], [-2, 0, -1, -2, 0], [-3, 2, 3, 0, -3], [2, 0, 0, -2, -4], [1, -1, 0, 1, 1], [-2, 2, -1, -2, 0], [-1, 3, 2, -1, -3], [9, -2, -3, 0, 1], [-2, 0, -3, 2, 0], [4, -3, -2, 2, 3], [1, 0, 1, 0, -1], [0, 1, 4, 0, -1], [8, 2, -2, 0, 2], [10, 0, 0, -2, 2], [9, -2, 2, 0, -1], [3, 1, 2, 3, 1], [-5, 0, -1, -2, 3], [0, 0, -2, 2, 4], [-2, 2, 3, -2, 0], [2, 2, -1, 0, -2], [2, 0, -1, -2, 2], [-2, 2, 4, 0, 2], [-5, 1, -2, -1, 3], [6, 2, 1, -2, -4], [14, 0, 4, 0, 0], [0, 2, -2, 2, 0], [-3, 2, 1, 4, -1], [16, -2, 1, 3, 1], [8, -1, -3, -1, 4], [6, 2, 0, 2, 2], [3, 0, -1, -2, -3], [9, -2, 1, 0, -1], [5, -2, -3, 2, 1], [0, -1, -2, 2, 3], [-6, 0, 0, -2, 0], [10, 2, 1, -2, 4], [7, 1, -2, 3, -3], [-11, 3, 1, -2, -6], [1, 3, -2, -1, -3], [-3, 0, 3, -4, 1], [-2, 0, 2, 0, 2], [-2, -4, 2, 2, -2], [10, 0, -3, -2, 0], [16, 0, 2, 4, 0], [3, -2, -6, 2, 3], [-2, -2, -2, -2, 4], [-6, -2, 4, 2, -6], [7, 2, 1, 2, -7], [6, 4, 3, -4, -2], [-8, -2, -4, 2, 2], [13, -2, -3, 0, 7], [-8, 2, 0, 2, -2], [10, -3, -1, -1, 4], [13, 1, 0, -1, -1], [-1, 6, 2, -4, -3], [-9, 0, -1, 0, -1], [-7, -1, -1, -6, 4], [8, 4, -2, 0, 0], [6, 0, 2, -2, -2], [4, -2, -3, 3, 3], [4, -6, -3, 2, 4], [5, -1, 4, 1, -1], [-10, -4, -1, 2, -2], [25, 0, 1, -2, -3], [-5, 1, 0, 3, -3], [2, 3, 5, -3, 4], [-8, 3, 1, 1, -4], [6, 2, 4, 2, -6], [-2, -1, -1, -5, -2], [-1, 3, 5, 2, 0], [6, -2, -6, 0, 10], [0, 1, 3, 1, 2], [4, -3, 4, 0, 1], [-20, 2, -4, 0, 0], [14, 0, -1, 2, 6], [-7, 4, 3, 0, -3], [-12, 0, 1, 0, -10], [-3, 3, 2, 1, 3], [-4, 4, 5, 2, -4], [-5, -5, 2, 1, -1], [4, 0, 0, 2, -2], [3, 0, -2, -2, 3], [-23, 3, 4, 1, -1], [8, 2, 1, 0, -2], [-25, 3, -2, -3, -5], [-16, 0, -5, -4, 2], [7, -3, -3, -2, 8], [-7, 3, 1, -4, 0], [-4, -5, -4, 4, 3], [-3, -2, 3, -2, 3], [0, -2, 6, 0, -4], [15, -3, -4, 1, 1], [-5, -4, -4, 3, 2], [1, 0, 1, 4, 5], [-25, -4, -1, 2, -5], [-10, 0, -5, -2, 2], [-12, -2, 1, 2, -6], [-12, 0, -2, 4, -2], [-1, 0, -4, 3, 6], [12, 2, 4, -6, -8], [-5, 5, 0, -3, -1], [-14, -2, -3, 4, 4], [-14, 0, 1, -4, 0], [12, -2, -3, -1, 1], [-20, 4, 0, -2, -2], [-13, 4, -3, -2, -3], [-5, -3, 1, 2, -2], [-16, 0, 2, 4, -2], [-6, 3, 6, 0, -9], [-4, 0, -4, -2, -2], [-10, 0, -2, -4, 0], [-1, 0, -1, -8, -1], [-12, 2, 6, 2, -4], [-30, -1, 1, -1, -6], [-7, 2, -1, -4, -3], [-12, 0, -1, -2, 4], [-23, -4, 2, -1, 4], [4, 2, -4, 0, 6], [15, 2, 3, -2, -5], [-19, 0, 2, -4, 3], [-16, -1, -5, 1, 8], [-4, 4, 3, 1, 1], [-28, -2, -5, 4, 6], [-17, 3, -3, 2, 4], [-6, 3, 2, 0, -7], [-6, -2, 7, -1, -5], [25, 2, 5, -2, -3], [-4, -4, 0, 0, -4], [-2, 0, -4, -2, -10], [19, -1, 1, 2, 8], [-6, 0, 6, -4, 4], [11, -1, 2, 5, -5], [-10, 0, -5, -2, 2], [-23, 1, 0, 1, -1], [1, -1, 2, -1, 1], [-24, -2, -1, 3, 5], [1, -2, 2, 2, -9], [30, -2, -5, -1, 7], [1, 3, 1, 4, 2], [13, 2, 5, 4, -9], [2, 0, 7, 0, -6], [-20, 2, -4, 0, 0], [12, 4, 6, -2, 6], [4, 0, -1, -5, -7], [10, -6, -5, -2, 0], [3, 2, -3, 2, -1], [1, 2, 5, 0, -7], [-8, -2, 4, 2, -6], [-21, 1, -4, 5, 3], [-6, -5, -11, 5, 4], [8, 0, 9, 2, -4], [8, -2, 2, 4, 0], [31, 3, 4, -1, -9], [0, 6, 2, -2, -8], [-15, -4, -3, 6, 3], [-8, -6, -3, 4, 6], [8, 2, 5, -1, 1], [-10, -3, -5, -3, -2], [12, -8, -5, 2, 6], [22, 1, 2, 0, 3], [14, -4, -8, -6, 10], [4, 0, 8, -4, -8], [19, -1, 8, -3, -7], [20, 4, 4, -2, -2], [14, -1, 9, -1, 0], [-22, 5, 5, 5, -2], [-20, 4, -2, -2, -10], [-3, -5, 4, 3, -1], [18, 0, 2, 2, 10], [8, -2, -10, -2, 6], [-5, -1, -2, 5, -1], [-9, -2, -7, 4, 3], [6, -2, 2, 2, -8], [-1, 0, -3, -10, -1], [-8, 2, -1, -4, -2], [14, 0, -3, 0, 6], [10, 0, 1, 7, 9], [-12, 8, 6, 0, -2], [31, 0, 5, -6, -5], [-21, 0, 9, -2, -5], [17, 2, -1, 8, -5], [2, 0, 1, 12, 0], [2, 2, -3, -3, 7], [2, -4, 1, 0, 2], [27, -3, 4, 1, 3], [-15, -2, 1, 4, -3], [21, 6, 0, 0, 3], [-3, -4, 6, -2, 1], [9, -3, -2, 1, 9], [-10, -4, -10, 2, -2], [15, -4, 5, -8, -7], [8, 1, -2, -6, 9], [6, 0, 2, -6, 2], [-33, 4, -3, -6, -5], [-9, 5, 3, 0, 4], [19, 1, 6, 3, -3], [2, -2, 12, -4, -4], [-8, 5, -1, 9, 2], [1, 1, 0, -9, 1], [20, -2, -2, -2, 4], [38, -2, -7, 0, 2], [-6, 4, 4, 2, 4], [14, 2, -1, -8, -4], [-8, 4, 0, 8, 8], [7, -2, -5, 2, 5], [4, 2, 2, 2, 8], [4, 4, 6, -4, 8], [10, 8, 1, -3, 3], [7, -2, -9, 4, 1], [-18, 0, 4, 8, -4], [22, -10, -10, 0, 12], [24, 4, -4, 0, -4], [2, 8, -1, -6, 2], [4, 1, 11, -3, -6], [-13, -8, 3, -6, -1], [-19, 2, 2, -1, 6], [-3, -1, -16, 1, 5], [13, 2, 0, 4, -5], [21, 1, -4, 3, -5], [-12, -4, -8, 8, 0], [-4, -10, 1, 6, -2], [-1, -2, -1, -4, -1], [-19, 0, -3, -6, -9], [6, 0, -2, -4, 2], [52, -4, 3, 5, 5], [20, 0, 2, 8, 2], [26, -6, 2, 4, 2], [-22, -8, 1, 3, -3], [8, 8, 2, 4, 2], [-14, 2, 2, 4, -12], [-8, -1, 4, -8, 1], [8, 4, 5, -1, 1], [43, -1, 10, -1, -5], [10, -12, -2, 4, 6], [4, -2, -6, 2, 14], [6, 6, 3, 8, -8], [33, 6, 1, -4, 11], [2, 0, 3, -8, -8], [-18, 5, 5, 1, -10], [10, -4, -2, -2, -2], [38, 4, 2, -4, -2], [4, 8, 2, 4, -14], [12, 1, 3, 1, 2], [40, 5, -4, 0, 5], [1, 5, 8, -9, -1], [-23, -3, -5, 2, -6], [-29, -4, -3, 6, 5], [-11, 6, -2, -2, -7], [-13, -2, -9, 4, -5], [15, 7, 8, -11, -7], [3, -8, -2, -10, 3], [-8, 0, -10, 8, 4], [17, -5, -8, 3, 3], [11, 5, 7, -8, -10], [-6, -4, 5, 5, -9], [12, -6, 2, 2, -2], [-4, 8, -6, 6, 4], [2, -2, -2, 10, 10], [28, -10, -11, -5, 3], [9, 3, 3, 0, 12], [0, 2, 0, 4, -8], [-13, -4, -5, 8, -5], [-30, -4, -7, 4, 2], [0, 8, 4, 0, -20], [3, 1, 0, -7, 1], [-9, -10, 2, 1, 8], [8, 4, 1, 2, 10], [-1, -5, 6, -1, 5], [-51, 2, 1, 2, -1], [-11, -4, -3, 2, 5], [-6, 6, 8, -2, -2], [-18, 8, -6, 0, -6], [-20, -2, 3, 5, -7], [2, 0, 2, 8, 6], [-16, 4, 8, -2, 6], [-28, -4, 0, 8, -8], [6, 1, 13, 5, -12], [18, 4, -3, -3, 1], [-12, -8, -3, 2, 4], [-5, 3, 4, 9, 1], [12, -2, 0, -8, 6], [27, 2, -3, -4, 5], [21, -1, 16, -5, -13], [-4, -3, 3, -5, 14], [-43, -1, -5, -6, 2], [2, -4, 4, -10, 0], [11, -6, -13, 4, -1], [-16, 4, -4, -10, -8], [-55, 4, -2, -5, -2], [-11, 12, 5, 6, -7], [20, -6, -4, 2, 2], [19, 6, 1, 4, 9], [-2, -3, -11, 9, 4], [9, 0, 11, 8, -11], [6, 4, -9, 0, 2], [17, 4, 2, 8, 1], [-1, -1, -4, -5, -3], [-20, -4, -4, -4, -2], [-14, 6, 3, -10, 0], [1, 1, 5, 6, -6], [-5, 2, -5, -8, -5], [-25, -3, 3, 2, 6], [-34, 8, 8, 2, -6], [10, 0, -8, 4, -12], [33, -1, -7, -8, 10], [-10, 0, 2, 8, -6], [22, -3, 3, -1, 6], [2, 6, 6, -4, -24], [22, -2, -2, 10, 4], [1, 5, 2, -13, -7], [0, 4, 2, 2, 0], [14, -14, -4, -6, 2], [28, -4, 2, -4, -2], [-27, 7, 8, 1, -15], [6, 2, -10, -2, 8], [-19, 3, -5, 6, 0], [-26, 2, -6, 8, 2], [-18, 0, -12, 6, 0], [18, -2, 16, -2, -4], [8, 4, 6, -4, 4], [2, -2, 5, 2, 8], [11, 9, 9, -8, -4], [24, -8, -1, 0, -2], [-15, 10, 7, -10, -3], [12, -6, 5, -10, -4], [-37, -9, 4, -5, 7], [42, 4, 0, 6, -2], [-31, -9, -8, 9, 1], [-3, -10, 0, 0, -5], [20, 4, 6, 0, -12], [-1, 1, -4, 3, -9], [20, 8, 4, 4, 0], [-4, 2, 8, 2, -6], [-18, -4, 2, -8, 2], [41, 4, 7, -2, 1], [-18, -6, 2, 2, 14], [-4, 18, 0, 2, -4], [12, -4, 5, -1, 5], [-6, 16, 4, -4, -8], [39, 3, 2, -1, -5], [20, -6, 0, 8, 8], [-16, 0, 4, -6, -2], [-31, -5, 0, 1, 15], [2, 6, -1, -1, -21], [-26, 1, -5, -5, 8], [-5, -5, 3, -8, 4], [0, 6, 9, 0, -10], [40, 1, -7, -1, 6], [38, 4, 2, 8, 2], [-45, 6, 1, -2, -9], [29, 0, 5, 8, -3], [-51, -8, -3, 4, -7], [14, 2, 3, 2, -2], [-3, -6, 1, 8, -1], [-27, -3, 8, 3, -13], [13, -3, 0, 1, -9], [-37, 5, 4, -7, -9], [14, -10, -7, 5, 13], [-15, -10, 1, 0, -3], [-26, -4, -4, -4, 4], [17, 5, 1, 6, -2], [-5, -2, -3, 0, -5], [-39, 0, -6, -2, 9], [-14, 8, 10, 4, -20], [0, 10, 0, -6, -4], [-12, 1, 4, 0, -11], [-10, 6, 0, 10, 2], [17, 0, -11, -4, 5], [22, -4, 4, 6, 14], [-39, 9, 1, -4, -10], [-36, -8, -9, 8, 18], [32, -3, 15, 3, -2], [-4, 2, 4, 4, -4], [1, -6, 8, 10, -9], [1, 7, -4, -3, 3], [24, -6, 0, 6, 18], [-34, 2, -2, -6, -16], [-27, 6, 11, -6, -1], [17, 2, 3, -10, -15], [5, -1, 3, -2, -6], [22, 2, -12, 8, -8], [-8, -6, -5, -5, 9], [35, -12, -9, 6, 9], [-24, -9, -4, 4, 1], [-8, 4, -3, -10, 0], [-2, 1, -5, 1, 4], [-6, -8, 1, -7, 9], [30, -5, 7, 5, 16], [4, 12, 6, -6, 2], [-17, -8, -4, 4, 19], [23, -1, 4, 9, -1], [-4, -8, 14, -4, -4], [-20, -4, 2, -12, 6], [-17, 2, 5, -4, 1], [21, -5, -8, -1, -5], [-21, 4, -2, 5, -2], [7, 2, 5, 8, 13], [49, -1, 3, -8, -8], [11, 0, -13, -2, 11], [-11, -1, -7, 0, 0], [-20, 6, 2, -2, -4], [14, -6, 7, -6, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3330_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3330_2_a_bk();".
// 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_3330_2_a_bk(:prec:=5)
    chi := MakeCharacter_3330_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_3330_2_a_bk();".
// 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_3330_2_a_bk( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3330_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-48, 48, 55, -21, -3, 1]>,<11,R![588, 210, -111, -29, 5, 1]>,<13,R![-368, 304, 112, -32, -5, 1]>,<17,R![84, 0, -63, -5, 7, 1]>],Snew);
    return Vf;
end function;