
// Make newform 6027.2.a.r in Magma, downloaded from the LMFDB on 25 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_r();".
// 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_r();".
// 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 := [5, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, 0, 1]];
    Rf_basisdens := [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 := [[0, 1, 0], [-1, 0, 0], [-1, 0, 0], [0, 0, 0], [-1, -1, 1], [-3, 1, 0], [0, -1, -2], [2, 0, -1], [0, 1, -2], [1, -2, 1], [1, -3, 1], [5, 2, -1], [-1, 0, 0], [4, -2, -2], [-1, -1, 2], [5, -1, -1], [-4, 2, -1], [-4, 3, 1], [-5, -1, -1], [11, -2, -2], [-7, 0, -1], [-8, -1, 1], [-6, 5, 4], [2, -3, 1], [-3, -6, 0], [1, -2, 1], [-3, 3, -3], [4, 1, 0], [1, -1, 4], [4, 1, 0], [0, 4, 3], [-10, 4, 2], [3, -6, 0], [6, -6, -1], [8, -1, 6], [-9, 2, -4], [-6, 7, 1], [-10, 1, 0], [7, 5, 1], [10, -2, 1], [1, -1, -1], [1, 2, -4], [2, 3, -4], [-4, -8, 3], [1, 0, -3], [-2, 2, 6], [-6, 2, 3], [1, -3, -5], [-6, -5, 1], [-9, -2, 4], [17, -4, -2], [-2, -2, 3], [-12, 3, 6], [-4, -2, 1], [-2, -2, -7], [-9, -6, 0], [-8, 1, 0], [-7, -3, -1], [17, 5, -2], [-3, -4, 0], [8, -6, -4], [2, 5, -1], [14, -2, 1], [14, 2, 4], [10, 2, -1], [-17, -5, -2], [7, -9, -5], [-7, 1, -10], [-13, 10, 5], [-21, 5, 6], [-5, 7, 2], [13, -2, 3], [7, 8, 2], [-1, 4, -1], [-5, -4, -4], [-5, -4, 5], [1, 6, 4], [-9, -13, 2], [-5, 13, 0], [-8, 1, 8], [-13, 6, 3], [-10, -4, 5], [-10, 4, -5], [-1, 3, 5], [8, -2, 0], [4, 2, 5], [-9, 3, 9], [21, -5, 3], [-12, 4, 3], [15, -10, 1], [-5, -5, 0], [4, 4, -9], [6, -1, 1], [-5, 5, -5], [-22, 4, 4], [8, 7, -2], [-24, 5, 5], [-32, 2, 2], [5, -2, -11], [-24, 13, 8], [7, -7, 6], [-5, 2, -12], [-5, 13, 2], [3, -3, -10], [12, -15, -1], [-5, 9, -4], [-4, 11, -1], [-16, -3, -5], [12, 0, -3], [-13, 3, 11], [-14, 9, -4], [-26, 1, 0], [-3, 15, 2], [23, -4, 6], [4, 3, 12], [30, 9, -2], [2, -5, 2], [-6, 2, 5], [34, -6, 0], [1, 1, -9], [8, 4, 1], [1, 14, -6], [-25, -6, -1], [-18, -2, 6], [-26, 2, 3], [15, 0, 10], [-19, 9, -1], [0, -3, 8], [14, 4, -2], [-9, 15, 4], [1, 7, 9], [7, -4, -6], [-1, 13, -4], [-22, 6, 1], [40, -5, -2], [13, -11, -13], [-12, -9, 5], [0, -7, -4], [-4, -16, 0], [-22, 4, -3], [8, -5, -5], [2, 14, -4], [-16, 9, 12], [-16, -1, -9], [-14, 3, 0], [-8, 5, -12], [-24, 3, 5], [-4, 10, -8], [-23, -1, -3], [-14, 7, 3], [5, -11, 5], [2, 1, 13], [20, 10, -8], [7, 3, -14], [2, -9, -7], [18, 1, 2], [-12, -12, 10], [-14, 16, 4], [-20, -3, 10], [-14, -15, 2], [12, 9, 2], [29, -16, -8], [25, 0, -7], [-12, 1, -9], [-7, -15, -1], [6, -2, -1], [-11, 21, 2], [-15, 1, -12], [2, 10, -7], [13, 10, 1], [-8, 16, 2], [-22, 6, -1], [-25, 6, -6], [-23, -14, 5], [-12, 18, 5], [40, -16, -9], [13, 1, 9], [-23, 5, -7], [-8, 6, -5], [-19, 1, -13], [-21, 19, 13], [-3, -17, 3], [-27, -11, 0], [-8, -6, -8], [-2, -3, 2], [-1, -8, -11], [10, 2, -7], [-1, 8, -1], [1, -6, 0], [15, 5, 1], [4, -8, 7], [-21, -7, -6], [-8, 0, -14], [26, -1, 0], [-14, -2, 4], [-37, 0, -7], [-12, -11, -4], [-5, -18, 4], [-11, -4, 5], [-42, 2, -5], [-18, 9, 3], [-2, -7, -6], [-10, -5, 9], [-13, -7, 12], [-1, -13, -8], [5, -13, 10], [-9, -5, -15], [20, 6, 9], [16, 1, 11], [-3, 11, 5], [-28, 6, -7], [-15, -2, 7], [15, -2, 0], [-3, -10, -3], [30, 5, -1], [-2, 3, 7], [16, -20, 0], [36, -12, -7], [16, 2, -13], [-4, 5, -12], [-14, 8, 2], [3, -1, 4], [-24, -11, 0], [16, -7, 4], [-7, -19, 2], [12, -15, 2], [30, -3, 6], [6, 16, -2], [0, 0, -17], [53, -4, 2], [-2, -3, 14], [-8, 3, 16], [31, 5, 1], [-21, 8, -12], [-27, 15, 9], [-8, -14, 10], [-22, 2, -14], [7, -17, -11], [12, 1, 1], [18, -4, 12], [29, -3, -9], [-21, -19, 11], [11, 1, -1], [-10, -2, 15], [15, 8, -4], [30, 0, 9], [4, 15, 5], [18, 12, -4], [-43, 0, 4], [7, 22, 0], [-26, 8, -4], [-32, -11, 7], [-15, -16, 12], [40, -13, 0], [9, -8, -7], [-14, -8, 15], [-3, 20, 8], [47, 10, 0], [-45, 6, 3], [-12, 15, -14], [-45, 5, -7], [9, -1, 8], [3, -3, -10], [-16, -7, 17], [-2, 14, -15], [-2, -6, 7], [14, 10, 1], [11, -14, -20], [-41, 19, 11], [-21, 9, -9], [-28, -4, 4], [-36, 1, -16], [2, -22, 7], [-27, 8, -6], [-14, 19, -6], [7, -6, 19], [5, -2, 8], [-19, -3, -3], [-24, -24, 11], [28, -20, -7], [3, -3, -2], [-69, 1, -6], [3, -12, 7], [-39, -1, 5], [15, 10, 15], [-42, -2, 6], [-23, 12, -8], [-7, 15, 2], [-11, -19, -2], [-19, -2, 19], [20, 9, 15], [-50, -2, -6], [-35, 6, 4], [0, 29, -9], [-30, -12, 2], [-57, 0, 0], [-20, 20, 1], [-18, -3, 0], [16, 4, 7], [8, 1, -18], [6, -7, -17], [32, -4, 4], [-55, -6, 7], [-11, 5, 5], [-7, 21, -3], [-4, -20, -9], [14, 1, 19], [-24, -3, -15], [22, 16, 3], [-22, -25, 1], [35, -23, -9], [-2, 10, -14], [-23, -8, -2], [-18, 4, 7], [11, 7, 18], [-9, -9, -14], [3, -12, -1], [-11, -14, 9], [14, 5, -3], [-1, -17, 1], [40, -22, -6], [7, -3, -15], [9, 29, -4], [14, -14, -23], [48, -14, -5], [-3, 13, 2], [20, -11, -20], [-20, -4, 1], [-19, -7, 2], [-34, 10, -5], [21, -6, 0], [-4, -1, 8], [14, -3, 15], [-21, -12, -3], [7, -1, 13], [-20, 19, -5], [33, -3, 3], [39, -28, -11], [10, -25, 7], [-4, -23, 13], [28, 14, 11], [18, 0, 14], [-43, 16, 8], [63, -13, -5], [-35, 18, 6], [-45, 9, -4], [-4, -6, 1], [2, 15, 0], [-24, -2, -10], [-14, 18, -10], [-41, 27, 9], [-14, 6, -2], [29, 19, -7], [-15, -22, -2], [7, 16, -4], [32, -11, -8], [26, 2, -10], [43, -25, -10], [-13, -19, -1], [-6, -8, 12], [-39, 17, 7], [-20, -20, 13], [-17, -10, 5], [21, 3, 22], [4, 13, 3], [35, -12, 11], [-16, 20, -2], [-15, 16, -11], [16, -11, 27], [6, 8, 7], [3, -11, 13], [44, -10, 9], [-43, 23, 18], [66, 2, -2], [-33, 1, 19], [40, 4, 17], [-16, 23, -16], [3, -15, 7], [1, 10, 6], [5, 19, -5], [-1, -8, 11], [21, 14, -2], [-22, 9, 5], [55, -19, -2], [-8, -13, 18], [35, 8, 3], [-9, 5, -2], [15, -23, -18], [-13, -14, 6], [1, -21, -3], [10, 3, -11], [-68, -11, 2], [46, -1, -8], [-25, -2, 9], [38, -4, -3], [-26, 25, 15], [7, -21, -16], [-8, -21, -10], [-39, 8, -16], [60, -5, -5], [17, 34, -2], [15, 24, -4], [8, -5, 4], [-37, -1, 1], [-47, -11, 7], [10, -11, -2], [-9, -16, -4], [25, 18, 12], [20, -19, 5], [-7, -26, 3], [19, 9, 4], [-13, 14, -8], [-48, -11, 9], [-34, -24, 5], [25, 26, -13], [46, -1, -1], [71, -7, -7], [0, 18, -15], [-52, 7, 15], [-11, -11, 5], [-38, 24, 9], [-12, 20, -11], [47, 5, 0], [-33, 12, 1], [-13, -2, 4], [2, 10, 4], [-7, 28, -18], [15, -13, 1], [5, 4, 14], [64, -16, -3]];
    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_r();".
// 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_r(:prec:=3)
    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_r();".
// 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_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6027_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![5, -6, -1, 1]>,<5,R![1, 1]>,<13,R![5, 15, 8, 1]>],Snew);
    return Vf;
end function;