
// Make newform 6027.2.a.v in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6027_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_6027_2_a_v();".
// 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_6027_2_a_v();".
// 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 := [2, 10, 6, -6, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-3, -1, 1, 0, 0], [5, 6, -5, -2, 1], [5, 10, -4, -3, 1]];
    Rf_basisdens := [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_6027_a();"
function MakeCharacter_6027_a()
    N := 6027;
    order := 1;
    char_gens := [4019, 493, 2794];
    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_6027_a_Hecke(Kf)
    return MakeCharacter_6027_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, 1, 0, 0, 0], [1, 0, 0, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 0], [0, 1, -1, 1, 0], [0, -1, 0, 0, 1], [1, 0, 0, 1, -2], [-4, -1, -1, 0, -1], [-1, 1, 2, 0, 0], [2, 0, 1, 0, 0], [-2, -1, 3, 0, 1], [-4, 1, -1, -2, 0], [-1, 0, 0, 0, 0], [3, -4, 0, 1, 2], [-1, 0, 0, -1, 3], [-6, -1, -2, 2, 1], [-7, 0, -2, 1, -4], [-2, -2, 0, 1, 3], [5, -2, 0, -1, -1], [2, -2, 0, -1, -1], [2, -3, 3, 2, -1], [5, -3, -1, 2, 1], [-4, 0, 4, 0, 2], [-4, 3, -1, -1, -2], [-8, 2, -1, -2, 0], [0, 1, 3, 1, 2], [-5, 1, -2, 1, -1], [1, 0, 1, -2, 3], [3, -1, 3, -4, 2], [2, -7, 1, 1, 2], [-4, 3, -1, -3, 0], [-5, 7, 1, 0, 2], [-2, -1, -4, -3, -2], [-5, 2, -3, 3, -2], [-4, 1, 3, -2, 1], [2, 7, -5, 1, -2], [-4, -1, -7, 0, -3], [-5, 3, -5, 0, 2], [-4, 4, -6, -4, -1], [-3, 7, -5, 1, 2], [4, 7, -3, -2, -5], [4, -3, 7, 4, 3], [0, 1, 3, 0, -1], [-1, 6, -8, 1, 0], [-10, 2, 2, 2, 0], [-2, -2, -2, 1, -1], [13, -3, 5, 2, 1], [-1, -2, -3, -5, -2], [-7, 2, 0, 1, -3], [-8, -1, 4, 1, -4], [-6, 3, 5, 0, -1], [6, -1, 3, -5, -2], [-8, 2, 0, -2, -4], [4, 1, 3, 3, 0], [6, -2, -2, 4, -4], [4, -7, -1, -3, 0], [1, 6, 2, -2, 6], [7, 3, -2, 4, 2], [5, -5, 5, 1, 2], [-12, -2, -1, 3, -1], [-16, 0, -6, 1, -7], [-7, 8, -2, -7, 0], [-1, -6, -3, 0, -5], [-7, -11, 9, -2, 3], [-8, 0, -5, 5, -3], [-4, 6, 3, -3, 1], [6, -6, -6, 8, -7], [-2, 0, 0, 5, -8], [0, 1, -3, 2, -5], [2, -9, 3, 2, 3], [4, -1, 7, 1, 6], [3, 1, 2, -1, -1], [9, 4, 9, -6, 7], [-2, 2, 4, -7, 2], [3, 2, 2, -7, 0], [3, 1, 7, 0, 1], [-10, 3, -3, 2, -1], [-16, 1, -7, 2, -1], [-8, 3, 1, -5, -2], [4, -5, 3, -3, 2], [-9, -4, -4, 5, 2], [-3, -6, -2, 9, 0], [8, -7, 5, 0, 9], [-2, 10, -6, -4, 2], [8, -5, 5, -6, 5], [-14, 9, -5, -8, -3], [-2, 6, -10, 3, 1], [10, -2, 4, 2, 0], [9, -10, 8, -4, -6], [-13, 10, 6, -1, 3], [11, -3, 1, 10, 0], [-16, 11, 1, -5, 0], [-8, 15, 5, -5, 4], [-2, 0, 4, -1, -5], [4, 5, -9, -1, 2], [0, -8, 4, 0, -9], [9, 1, -7, -2, 10], [0, 9, -7, 1, -4], [0, 7, -7, -4, 5], [16, -11, 1, -4, 6], [18, -1, 3, -9, 1], [2, 5, 11, -2, -1], [-16, -3, -3, 5, 5], [-12, 10, -12, -3, 3], [-2, -13, -5, 1, -6], [8, 3, -3, -6, -5], [-2, -4, 6, 8, 0], [-10, 10, 6, 0, 0], [-3, -1, 10, -8, 4], [-10, 6, -5, 10, -12], [0, -3, -3, -8, 1], [-5, -2, -8, 2, -6], [-8, -2, 4, 6, -6], [13, -7, -8, 7, -7], [-8, 0, 6, 0, -2], [6, -1, 19, -2, 7], [-18, 4, -8, 7, 5], [11, -5, 3, 4, 8], [-18, 2, -8, 1, 1], [6, -5, 1, -11, 4], [20, 1, 1, -1, 6], [-3, -12, 6, -1, -2], [-2, 1, -1, -1, 10], [-4, -9, -1, 1, -2], [-8, 13, -3, -3, 4], [8, 3, -1, -1, 4], [-4, 13, 5, -7, 4], [-4, -3, -1, 1, -8], [-2, 14, 10, -2, 6], [-16, 2, 2, -4, -2], [-23, 4, 2, -1, 4], [-10, 3, 3, -8, 9], [9, -4, 6, -3, -7], [2, 1, 3, 7, 6], [26, 1, 1, 0, 6], [10, -6, -6, 0, 8], [-2, 1, 1, -9, -6], [-5, 1, 8, 2, 4], [10, -2, 0, 8, 0], [-2, -10, 1, 3, 3], [-13, 6, -5, -6, 9], [2, 2, 0, 3, -3], [-6, 0, 2, -4, -8], [-4, -12, -2, 4, 6], [-2, 5, -3, -13, 2], [4, -1, -9, -1, -7], [-2, 11, -13, -5, 4], [14, 5, -1, 4, 0], [-19, 22, 3, -5, 0], [13, -18, 7, 4, -7], [4, -10, 6, -4, -4], [22, -15, -1, -2, 4], [20, 6, 2, -2, 3], [14, 1, 11, -7, -1], [10, 12, -4, 0, 0], [2, 0, -8, -9, -1], [8, 1, -1, 3, 11], [-17, -2, 2, 9, -4], [-10, 17, 1, 2, 1], [-21, 15, 1, -5, -2], [7, -5, 18, 1, 5], [-6, -1, 3, 0, -1], [-10, -3, -7, 11, -3], [-1, -16, 2, 5, -5], [-30, 5, -1, 10, -5], [-39, 3, -1, 0, -8], [16, -5, 15, -9, 5], [-14, -3, -2, 5, -2], [-3, 18, -8, -4, 4], [16, -6, 10, 6, 12], [-8, 12, -4, 0, -4], [2, 4, -2, 1, -9], [-8, 9, -1, -7, -4], [10, -2, 6, 7, 2], [-12, 11, -5, 3, -8], [13, -1, -5, 2, -4], [-4, -11, -9, 5, 3], [20, 4, 4, -3, -1], [8, 1, 1, -9, 16], [-20, -6, 1, -5, -5], [4, 1, 3, -13, 6], [13, -15, 7, 6, -4], [14, 2, -8, 0, 4], [-29, 13, -11, -5, 10], [5, -13, -6, 11, -5], [0, 8, 6, -4, 0], [12, 6, -16, 4, -10], [-9, 15, 4, -9, 9], [39, -3, -3, 4, -2], [-24, 13, -9, -9, 6], [-4, 7, -17, 0, -3], [12, 0, 6, 1, -1], [1, -2, 12, 3, 6], [4, -13, 15, 1, 2], [13, -1, 14, -3, -9], [-16, -4, 4, 12, 2], [-7, 5, -3, -2, -12], [-5, 14, -6, -1, -8], [23, 5, -1, -2, 2], [4, 6, -6, 4, -2], [16, 11, -3, 1, 6], [17, -2, 3, -18, -1], [-38, -2, 4, 3, -1], [-6, -7, -2, 13, 8], [-14, 1, -15, 3, -2], [-4, -13, -13, 13, -14], [21, -17, 6, 6, -10], [-2, -6, 2, 9, 5], [36, -2, 10, -5, -3], [0, -14, -10, 11, -5], [5, 3, 13, -6, -4], [16, -1, 15, -1, -6], [15, 9, -7, 4, 6], [10, -3, 11, -17, 6], [-7, 3, -1, -6, -3], [-32, 1, 3, -3, 6], [27, -11, -5, -2, -1], [10, -24, 12, 12, 4], [8, 4, -2, 0, 0], [-8, -15, 1, 10, -5], [9, 3, -15, -14, -4], [9, -10, -13, -6, -5], [0, -16, 11, 7, 5], [14, 6, -8, 0, -1], [-8, 3, -9, -15, -4], [-6, 0, -6, 12, -18], [-16, 2, -4, 5, -6], [-9, 0, -3, 1, -14], [-2, -7, 3, -9, -10], [13, 1, -6, 0, -2], [-13, -3, -1, -3, 8], [-8, 6, 12, -11, 7], [1, 16, 14, -9, 2], [8, -10, 18, -8, 16], [-28, 11, -11, -1, -14], [14, -15, -7, -3, 8], [-10, -14, -5, 14, -2], [-18, -13, -10, 2, -9], [-9, 8, -7, -3, -2], [2, -4, 6, 12, 8], [-6, 1, -7, 7, -2], [10, -20, 16, -6, -6], [0, 5, -9, 13, -2], [0, 2, -18, 1, -12], [25, 0, 16, 2, 16], [-2, -5, -1, -9, 0], [5, 0, 1, 11, 10], [14, 6, -6, 2, 3], [32, -19, -3, -7, 0], [-27, 7, -4, 9, -17], [17, -19, 3, -3, 16], [11, -20, 14, -4, -10], [-13, 19, -6, -5, 11], [20, -6, 10, 15, -3], [-30, 7, 5, 5, -2], [51, -1, 4, -6, -2], [3, -7, -1, -10, -2], [-28, 14, -4, 10, 0], [-37, 13, -3, 2, -4], [-2, -4, 7, -10, -2], [-3, -3, 9, 4, 4], [-2, 14, -10, 2, -12], [26, -6, 11, -1, 1], [-8, -18, -11, 4, -16], [-14, -10, -18, 7, -18], [37, -3, 2, 9, 7], [-6, -1, 19, 2, 3], [-19, -4, -6, 0, -10], [-32, -9, 3, 5, -4], [0, -6, 9, -8, -10], [-18, 6, -9, -8, 6], [-4, -14, -2, 8, -6], [-16, 5, -11, 0, -11], [-9, -3, -9, 4, -18], [-24, -1, 5, 8, -3], [46, -3, 13, 0, 7], [-2, 26, -14, -1, -7], [-4, 1, 7, -3, 0], [31, -2, 10, 11, 6], [16, -23, -7, 3, 11], [-4, -10, 10, 16, 2], [-16, -1, -15, 13, -19], [21, 5, 13, -6, -4], [-28, 26, -2, -1, -3], [-2, 9, 7, -15, 14], [10, -10, 10, 3, 9], [-17, -11, -1, 8, 6], [-4, 3, -25, 8, -9], [20, 18, -12, 2, -6], [1, 8, 0, 17, 8], [26, -4, -7, -10, -6], [0, -5, -1, 11, -6], [-4, 2, -4, 1, -9], [-41, 8, -4, -9, 8], [2, 25, -5, -11, -8], [-13, -8, -10, 11, 4], [32, -13, 9, -5, 4], [37, 1, -7, -7, -6], [-22, -11, -25, 7, -9], [38, 6, 6, -11, 7], [-18, 11, -3, -15, 14], [-26, 15, -12, -7, -14], [-18, -17, -11, 11, 2], [-4, 7, 9, -10, 15], [20, 6, 6, 0, 10], [0, 2, -10, 2, -14], [19, -2, 3, -11, 24], [-6, 1, 1, -19, 4], [28, -15, 13, 2, -9], [3, -11, -9, 6, -10], [18, 0, 16, -8, -3], [28, -23, 4, 1, 4], [-8, -10, -12, 1, -13], [-31, 1, 15, 0, 8], [7, -12, 29, 5, 8], [0, 7, -5, -24, 7], [0, -3, -3, 11, 0], [25, -10, 17, -6, -1], [-6, -22, -3, 3, 5], [6, 11, 5, -6, -11], [-11, 2, -11, 17, -8], [-20, 14, -18, 7, -2], [-6, 1, -11, 3, -8], [-8, -10, -20, 4, 0], [10, 11, 1, -2, -13], [12, 19, -2, 2, -1], [-30, -1, -7, 15, -3], [12, -11, 9, 5, 8], [-24, -2, -16, 10, 8], [18, -15, 1, 6, 7], [1, -8, 8, -15, 16], [10, -6, 12, -17, 7], [35, -11, 0, -9, 7], [-1, -1, -7, 8, -1], [-37, 7, 5, 0, 6], [-15, 12, 0, -21, 9], [-7, 7, 9, -14, 8], [-24, -8, 2, -9, 6], [-24, 15, -10, -11, 12], [6, -1, -5, 6, 11], [-45, -5, -5, 4, 8], [0, -3, 13, -20, 17], [-50, -2, -10, 8, -4], [14, 22, -4, -14, 0], [-16, -29, 3, 11, 8], [-5, -14, -1, 3, -16], [20, -17, 17, 8, 5], [4, 7, 13, -3, 8], [1, 0, 18, -7, 21], [-2, 17, -5, -15, 0], [26, 12, 0, -3, 9], [10, -7, -3, -8, -4], [58, -10, -3, 15, -11], [-18, -9, 17, 4, 3], [-13, -13, 11, -2, 12], [49, 12, 10, -9, 2], [0, -28, 20, 10, 18], [-28, 14, 12, 4, 0], [-32, -11, 1, 1, 0], [5, -12, -17, 15, -14], [17, -14, 0, 12, -16], [-32, -1, -25, -1, -10], [2, -5, 15, 0, 9], [-5, -8, 11, 8, -1], [26, -1, 13, -12, 13], [10, 6, 2, -2, 6], [-29, -5, 13, -4, 12], [-27, 37, -5, -14, 0], [-19, 27, -1, -5, -10], [36, -6, -12, -7, 7], [14, -20, 2, 2, 6], [-2, -5, -13, 26, -17], [-38, -9, -21, 16, -11], [-11, -3, -2, 16, -8], [-8, -15, -21, 1, -2], [-37, 8, 4, 5, 2], [23, 1, -6, 2, 14], [10, -6, 2, -6, -8], [-33, 5, 13, 2, -6], [-40, 1, -22, -2, -1], [-15, 9, -25, 10, -10], [12, 3, -11, -7, 22], [-3, 0, -12, 3, -2], [12, -14, -4, -2, -8], [-4, -15, 19, -14, 1], [7, 29, -3, -2, -8], [21, -19, 9, 4, 7], [8, 20, 7, -11, 11], [32, -10, 8, -18, -4], [-18, 9, -9, 1, -17], [9, 4, 18, -13, -6], [-27, -5, -9, 18, 10], [18, 4, -8, -15, 5], [9, 0, 15, -4, -21], [-13, -30, -9, 16, 7], [6, -37, 5, 5, -8], [16, -14, -12, 26, -16], [28, 11, -11, 10, -8], [9, -16, 0, 15, 8], [-4, 26, -14, 9, -3], [34, 7, -15, -13, -4], [-22, -1, 20, -5, 0], [0, -21, 11, -5, 14], [-10, -10, 14, 6, 10], [20, 16, -8, -11, -6], [16, -11, -11, 3, 6], [14, -26, 4, -12, 0], [-14, -12, -8, 15, -17], [-10, -18, 10, 6, -6], [69, 9, 7, -2, 8], [14, 4, -2, -1, 3], [-22, 0, 12, 4, -10], [-10, -11, 9, -8, 5], [19, -8, -6, 19, -9], [0, 1, -15, -5, -10], [28, -19, 1, 24, -8], [-7, -2, 9, 21, -6], [-6, -15, 21, -22, 21], [19, -27, -3, 2, -10], [-1, 10, -20, 3, -6], [18, 4, -16, 12, -6], [-54, 4, -4, 19, -1], [-23, -4, 1, -8, 1], [-49, 21, -3, -2, 5], [-14, -22, 16, 7, 3], [-61, 2, -8, 2, -6], [-14, 7, -21, 28, -21], [-6, -12, -3, 0, -6], [-46, 11, -9, 5, -26], [19, 9, -9, -2, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6027_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6027_2_a_v();".
// 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_6027_2_a_v(:prec:=5)
    chi := MakeCharacter_6027_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_6027_2_a_v();".
// 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_6027_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6027_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![11, 1, -14, -4, 3, 1]>,<5,R![-35, 53, -2, -14, 1, 1]>,<13,R![-1, 91, -26, -18, 3, 1]>],Snew);
    return Vf;
end function;