
// Make newform 6042.2.a.u in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6042_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_6042_2_a_u();".
// 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_6042_2_a_u();".
// 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 := [6, 19, 5, -9, -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], [-4, -2, 1, 0, 0], [5, -3, -3, 1, 0], [9, 10, -6, -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_6042_a();"
function MakeCharacter_6042_a()
    N := 6042;
    order := 1;
    char_gens := [2015, 4771, 2281];
    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_6042_a_Hecke(Kf)
    return MakeCharacter_6042_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], [-1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, -1, 0, 0], [-1, -1, -1, 0, -1], [1, 0, 0, 0, -1], [1, 1, -1, 0, -1], [1, 0, 0, 0, 0], [2, -1, 1, 1, 1], [1, 0, 0, -1, 0], [1, 1, 0, 0, 1], [-1, -1, -1, 0, -1], [4, 1, -1, 0, 0], [0, 1, -1, 1, -1], [-5, 1, 0, 1, 0], [1, 0, 0, 0, 0], [3, -2, 0, 1, 0], [4, -3, 0, 1, -1], [2, -3, 0, 0, 0], [-2, 4, 2, 0, 2], [3, 1, 3, 0, 1], [-3, -2, -1, 1, 0], [3, 0, 1, -1, 2], [-1, 1, 1, 1, 0], [3, 2, 1, -1, 2], [3, -4, 2, 1, 0], [5, 1, 3, 1, 2], [6, -1, 0, 0, 2], [-2, 4, -1, 0, 0], [4, 0, 0, -1, -1], [9, 3, 1, -1, 0], [3, 2, 3, -1, 0], [3, -1, 3, -1, 2], [1, -1, 1, 0, 1], [5, 1, 0, -1, 2], [1, 1, 4, 2, -1], [-6, -1, 0, -1, 3], [-5, -1, 2, 0, -1], [5, -5, -2, -1, 0], [-4, -7, 0, 1, -3], [2, -2, -2, 4, -2], [7, -1, 1, 1, -2], [7, 1, 0, 1, 0], [6, 2, 0, 2, 2], [0, 2, -1, -1, 1], [1, 1, 5, 1, -2], [6, 0, -3, -1, 1], [-6, -1, -5, -2, 0], [-6, -2, -5, -2, -2], [2, -2, 0, -1, -5], [11, -2, 1, 1, -2], [5, 2, -4, 0, -1], [2, 3, 0, -1, 1], [4, 0, 1, 1, 3], [10, -1, 2, -1, 5], [3, -4, -4, -1, 0], [4, -2, -3, 4, -2], [-6, -3, -7, 1, -3], [0, 3, 3, 0, 2], [-2, -1, -1, 1, -3], [-13, -6, 0, 0, -3], [14, -5, 1, 0, -2], [1, -2, 2, 0, 3], [11, 2, 2, 0, 5], [2, 2, 1, 1, 5], [-1, 4, -2, 0, -5], [-15, 0, 1, -1, 2], [6, 2, 3, 2, 2], [0, -3, -1, 2, -4], [-3, 5, -3, -2, -5], [13, -2, -2, -4, -1], [-17, 1, -1, -3, 0], [15, -2, 7, 1, 2], [6, -1, 0, -2, 4], [-15, 0, -2, 0, -1], [3, 6, -1, -1, 0], [5, -4, -5, -2, 3], [5, 4, 6, 0, 3], [2, -1, 3, 2, 0], [-7, 0, 0, 4, 1], [4, -1, -4, -1, -1], [4, 4, 2, -5, 5], [-2, 7, 3, 1, 1], [9, -4, 2, 0, 1], [-8, 2, -5, 1, -1], [2, -1, 7, -1, 1], [-2, -2, 1, 3, 5], [1, 5, -2, -1, 2], [1, -4, -8, 0, -7], [2, 9, 1, -2, 0], [-12, 5, 4, 3, 1], [18, -6, 5, 4, 2], [3, -2, 2, 0, 7], [-7, 4, 4, 1, -2], [0, -3, -9, 0, -2], [-3, -3, -3, -3, 6], [0, -7, -1, 2, 2], [19, 6, 2, -4, 1], [4, -2, -3, 1, 1], [-16, -5, -1, 1, -3], [-9, -5, 0, -1, -6], [9, 1, 0, -6, -3], [-15, 6, 0, 2, 5], [9, -4, -1, 3, 0], [0, 4, 9, -1, 3], [-19, -3, -3, 0, -1], [-1, -8, 2, 0, 1], [-12, 0, -7, -3, 3], [1, 5, -1, -2, 3], [1, -6, 2, -1, 0], [-14, 2, -6, 0, -8], [-15, 7, 3, 4, 5], [-6, -5, 1, -2, -2], [-12, 10, 4, -3, 1], [-15, -5, 2, -3, 4], [5, 1, 3, 2, -3], [12, 5, -4, 2, 0], [-2, 5, 1, 2, -2], [-12, 7, 5, 4, -4], [-1, 10, 0, -4, 5], [2, -4, 1, 3, -3], [11, -3, -1, -6, -5], [3, 3, -3, -6, -5], [-16, 7, 3, 4, -8], [-2, -7, 3, 0, 8], [-9, -4, -2, 3, -2], [-28, 2, -9, -3, -3], [-1, -4, -3, 2, -5], [-3, -4, 0, 2, 1], [-19, 2, -4, 1, -2], [-8, 0, -3, -5, 7], [-8, -13, -8, 1, -9], [-17, 4, -7, -3, -8], [-6, -11, 0, -2, -8], [-8, 1, 6, -5, 5], [14, -1, 1, -2, -4], [26, -4, 8, 2, 2], [4, 2, 5, 4, 4], [-17, 1, 5, -2, -1], [1, 2, 5, 0, 11], [-6, -4, -4, 0, -2], [-5, -5, 1, -5, 0], [10, -2, 4, -2, -4], [-12, 5, 1, -4, 2], [12, -7, 2, -2, 6], [12, 2, 2, -1, 3], [-12, -4, -6, 2, -6], [0, 1, 6, 2, -4], [-14, 2, -5, -1, -7], [8, -3, 3, -1, -3], [-17, 13, 6, -3, 4], [2, -2, 5, 0, -2], [-12, 13, 4, 1, -5], [22, -4, -7, -1, -3], [0, -9, -2, 3, -3], [-17, -2, -5, -8, -3], [-18, 0, 6, -2, -2], [25, 1, 0, -5, 6], [20, -11, 2, 4, 0], [-8, 13, 3, -7, 5], [10, -9, 0, 9, -3], [-6, 0, -15, 0, -10], [7, 9, 12, 1, 8], [8, 8, -3, -4, 2], [1, 11, 2, -5, 2], [-1, -2, -2, -1, 8], [-13, -1, 1, 2, 7], [-15, 8, -5, -4, 1], [1, -10, -3, -1, -4], [16, -6, 5, -1, 1], [14, 13, 0, -1, 7], [-21, 11, -6, -3, -2], [-7, 0, 11, -3, 4], [-13, -11, 0, 3, 0], [5, -2, 2, 5, -4], [19, -2, 1, -1, 2], [1, -7, 5, 4, 5], [11, -4, -4, -6, -3], [6, -15, -3, 3, -5], [2, -17, 1, 5, -1], [-3, 0, -4, 2, -1], [37, -3, 8, 1, 2], [9, -6, 2, 5, -2], [1, -9, -15, 0, -7], [2, 11, 3, 0, 4], [-9, 0, -1, -1, -6], [-13, -8, 0, -2, 3], [-15, -5, 4, 1, -2], [-17, -8, 0, 4, -5], [-7, 0, -6, -6, -3], [-3, -7, -12, -5, -2], [-6, -5, 3, 1, 3], [-22, -4, -7, 4, -6], [28, -12, -6, 1, -1], [-17, -2, -5, -6, -3], [-4, -2, -2, -2, 4], [-5, 12, 10, 2, 5], [10, -15, 4, -1, -3], [-7, 7, -3, 1, 2], [-2, -8, -4, 2, 4], [-6, -13, -3, 1, -1], [-14, 2, -5, 0, 10], [-13, 1, 2, -6, 3], [6, 4, -10, 3, -5], [17, -6, 11, 3, 2], [-20, -2, -4, -1, -3], [0, -4, 2, -3, 5], [-8, 13, 5, 1, 5], [35, -7, -8, -1, 0], [-17, 7, 5, -3, 4], [10, 1, 2, 4, -2], [0, -17, -9, 10, -6], [13, -7, 7, 0, 7], [9, 1, 5, 6, 7], [-13, 3, -4, 0, -1], [-16, -1, -5, -2, -6], [5, 0, 12, 4, 13], [-22, -12, -2, 3, -11], [18, 5, -3, -3, 1], [4, -17, 2, 7, -1], [2, -13, 3, 1, 7], [-13, -2, 9, 0, 1], [12, -3, -4, -9, -5], [9, 2, 11, -1, 4], [20, -11, 9, 5, -3], [2, -6, -16, 4, -2], [13, 6, -1, -5, 0], [-3, -8, 4, 4, 3], [3, -1, 0, -1, -4], [-19, 20, 5, 0, -3], [9, 9, 7, -4, 5], [5, -1, 1, -8, 1], [4, 6, 8, -2, 0], [-25, 4, 2, 2, 9], [-26, 6, -3, 2, 8], [-6, 1, -9, -1, -9], [-15, -1, -3, -6, 5], [14, 3, 4, 3, 9], [9, -10, 8, -2, 17], [-23, 0, 2, 6, -15], [-14, 15, 5, 2, -2], [-56, 9, 0, -2, 0], [6, 5, 0, -10, 2], [-12, 14, 2, 0, -2], [17, 8, 2, -7, 4], [-18, 8, 3, 6, 0], [3, 0, -5, -1, -6], [16, -6, 10, 4, -4], [-2, 5, 6, 3, 7], [24, 14, 2, -2, 2], [25, 6, -4, -4, 1], [-12, 0, 15, -3, 7], [1, 16, -10, 2, 5], [17, 24, 5, -8, 9], [27, 2, -13, 5, -4], [-42, -1, -5, 2, -2], [9, 5, 2, -3, -2], [4, 9, -6, -1, -11], [11, -13, -9, -7, -10], [1, 3, -10, 1, -4], [-13, -12, -9, 3, -4], [20, 9, 12, 2, 6], [32, -1, 7, -2, 16], [-28, -8, -5, 5, -5], [20, -18, -7, 5, -1], [-5, 16, -1, -5, -4], [-2, 7, 13, 1, 11], [13, -9, -5, -10, -3], [-7, 10, 1, 6, -11], [8, -1, 6, 3, -9], [15, 8, 3, -1, 6], [11, -8, 7, -4, 5], [-3, -1, -8, -6, -11], [-38, -6, -4, -6, -4], [11, 9, 1, 2, -5], [21, 5, -3, -6, 1], [-26, -8, -8, 0, -6], [11, 4, 0, 10, -9], [-22, -10, 3, 3, 3], [10, -7, -11, 8, -8], [7, -24, 5, 1, -6], [-40, -2, 0, 7, -1], [23, 10, 4, -8, 9], [0, -5, 9, 4, 2], [-33, -3, 0, 5, 0], [-5, 12, 2, -10, -1], [5, -2, 9, 9, -6], [-12, 10, 7, 11, 5], [6, -7, 6, 5, 9], [26, 3, -3, 2, 8], [39, -11, 2, 1, 2], [23, 2, -3, -1, 0], [1, -4, 2, 2, -1], [10, 21, 2, -7, 9], [30, 1, 15, 0, 0], [0, -4, 11, -4, -6], [8, -10, -5, -1, 5], [22, -11, -8, -2, -4], [-4, -3, -7, 0, 2], [-34, 14, 2, 2, 4], [10, 6, 3, 1, 5], [-6, 19, -1, -6, -8], [-13, 14, -3, -5, 0], [0, 5, -8, -4, -4], [3, -7, -12, 1, -8], [9, 1, -7, 0, 3], [15, 1, -2, -10, 7], [2, -5, 11, 2, 6], [43, 5, -1, -2, 1], [5, -17, 7, 2, 3], [-1, 17, 19, -2, 5], [-9, 3, 9, 6, -17], [-5, 3, 11, 2, 11], [15, -12, -5, -5, 8], [4, 14, -2, -5, 7], [-11, 5, -7, 1, 0], [-5, 11, 1, 3, 8], [46, 6, 9, -3, 7], [4, -10, -3, 8, 2], [-5, -2, -4, -6, 1], [10, 23, 6, 1, 9], [5, -21, -23, 1, -12], [23, 11, 0, 5, 6], [1, -10, -6, -5, 2], [39, -5, 4, -3, 0], [5, 15, 5, 7, 10], [34, -12, 6, 0, -2], [33, -17, -2, 6, -1], [-33, -12, -1, -1, 0], [21, 13, 13, 2, 17], [0, 10, 7, -7, -3], [-36, 17, 7, 0, 0], [6, -30, 9, 0, -2], [-9, -7, -5, -2, 5], [-9, 15, 12, -12, 7], [-39, 11, 2, 4, -1], [9, 19, 2, 5, 10], [11, 5, 3, 11, -10], [2, -4, 4, -10, 2], [10, -5, -6, -7, 3], [25, -2, -4, -8, -3], [30, 5, 2, -1, 3], [-15, 8, 5, -1, 8], [-5, 27, 0, -6, 5], [44, -14, -9, -2, 0], [-2, 6, -10, -3, -15], [17, -24, -8, 3, -8], [23, -19, 12, 7, -6], [17, 7, 14, -5, 0], [-20, 4, -16, -4, -6], [-1, 10, -6, 1, 14], [-19, -5, -12, -6, -7], [25, -7, -15, 0, -1], [39, 1, 2, -6, 9], [4, -12, -4, 2, -2], [26, -2, 20, 0, 4], [-24, 7, -9, -13, -3], [-6, 10, -11, 0, -12], [-23, 15, 7, -2, -1], [-9, -4, -21, 2, -21], [-4, -8, -1, -3, -13], [25, 9, 15, -4, 17], [-4, 22, 9, 2, 2], [-11, -10, 4, 6, 1], [20, 8, -2, 3, 3], [-1, -4, 0, -2, 13], [22, 15, 11, -7, 7], [20, 7, -3, -7, 7], [1, -2, 2, -3, 2], [-63, -2, -9, -5, -6], [-4, 20, 2, 2, -6], [18, -20, -11, 3, -3], [-20, -5, -3, 0, -4], [-28, -6, -1, -5, -1], [-12, 11, 6, -1, 11], [-22, 0, -8, 4, 2], [25, -9, 6, 5, -6], [-3, -10, -1, 6, -5], [24, -13, 10, 0, 2], [-36, 19, 1, -8, 0], [-3, -18, 12, 0, 5], [-18, 28, -2, -4, 4], [-17, 5, 2, 5, 10], [52, -11, 7, 10, -2], [15, 10, -5, -7, -6], [1, 13, -13, 1, -14], [-20, -5, -4, -7, 9], [15, -21, 5, 0, 11], [-35, 18, 8, 10, 5], [18, -25, 4, 2, 10], [-6, -7, -7, -9, -3], [38, 5, 8, 8, 2], [17, 6, 6, -8, 3], [0, -10, -5, -2, 2], [22, 17, 1, -12, 16], [29, 21, 16, -7, 16], [-3, -6, 15, -1, 4], [4, -6, 12, -2, -8], [-6, 6, 13, 7, -3], [-21, -8, 3, -3, 4], [18, 3, -4, -3, -11], [20, 6, 0, -7, 5], [-21, 21, 0, -3, 10], [-29, -9, -3, 4, 5], [-41, 9, -2, 1, -2], [12, 9, 17, 4, 12], [-5, -1, -12, -1, 2], [29, 9, -6, -3, 12], [-32, -6, -11, -3, 1], [-16, 1, -15, -2, -4], [31, 9, 1, -2, -1], [0, -15, -13, 0, -8], [-23, 0, -16, 0, -17], [9, -6, -10, -9, 10], [9, -2, 0, 0, 5], [-24, -8, -5, -3, 7], [7, -12, 1, -7, 14], [-9, 34, 1, -7, 8], [-19, -10, -2, -1, 10], [-25, 9, -11, 4, 11], [10, -6, -9, -2, -2], [-19, -3, 8, -7, 4], [-12, 1, 0, 3, 5], [-42, -1, -11, 1, -9], [-6, 2, -11, -5, -17], [-44, 6, -10, 0, 6], [-24, -6, 0, 0, -4], [-17, 5, 3, 4, 17], [-9, 10, 3, 5, 6], [34, 19, -9, 1, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6042_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6042_2_a_u();".
// 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_6042_2_a_u(:prec:=5)
    chi := MakeCharacter_6042_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_6042_2_a_u();".
// 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_6042_2_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6042_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![6, 19, 5, -9, -2, 1]>,<7,R![20, 91, 17, -19, -2, 1]>,<11,R![192, 44, -77, -24, 3, 1]>],Snew);
    return Vf;
end function;