
// Make newform 6027.2.a.q in Magma, downloaded from the LMFDB on 23 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_q();".
// 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_q();".
// 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 := [-14, -12, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-8, -2, 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, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 0], [0, -1, 1], [1, 1, 1], [2, 0, 0], [3, 1, 0], [2, 1, 0], [0, 2, 0], [-1, -2, 1], [-1, 0, 0], [1, 0, 0], [-7, 0, -1], [0, 1, 3], [-2, -2, -2], [-6, -2, -2], [-2, -2, -3], [1, -1, -1], [0, 2, 2], [7, 0, 1], [7, -1, 2], [2, 1, 4], [-4, -1, -3], [-6, 4, 0], [2, -4, 0], [3, -2, 0], [0, -1, 0], [5, 3, 2], [2, 1, -4], [5, 0, -2], [14, -1, 0], [2, -2, -2], [-1, 2, 4], [4, 1, -1], [2, 0, -5], [10, 0, 0], [0, -4, -1], [6, -2, -2], [8, -2, -2], [8, 4, -2], [7, 3, -2], [2, 3, 3], [5, 1, 3], [8, 4, 6], [0, -2, -2], [-8, -4, 4], [0, 2, -3], [6, 2, 2], [7, -3, -4], [-10, 1, -1], [8, -4, 2], [10, 0, 1], [-2, -3, -2], [2, -3, -5], [-6, -1, 3], [12, 2, 4], [0, -6, -6], [9, 0, 1], [-10, -3, -5], [11, 2, -2], [8, -5, 1], [9, -4, 0], [-2, -3, 7], [11, 5, 5], [-8, -1, -5], [-13, -1, -4], [-27, 0, 0], [16, 2, 4], [-14, 2, -3], [10, -2, 0], [26, -3, -2], [1, -4, -1], [7, -4, 0], [3, 4, 9], [10, -6, -4], [-10, -1, 6], [-13, -5, 5], [-6, 0, 6], [-5, 0, -5], [-10, 2, -2], [-1, 9, -3], [8, -9, -4], [1, -6, 1], [18, 4, 5], [12, -2, 4], [4, -3, -6], [5, 3, -4], [8, 3, -2], [-19, 1, -1], [-10, -3, -2], [16, 3, 1], [-1, -4, -8], [12, 0, -2], [-11, -5, -2], [4, 7, 3], [-10, 0, -6], [12, -9, -3], [9, -6, 3], [13, 0, 1], [8, 10, 10], [-12, -3, -7], [-10, -3, 1], [2, 5, 6], [-9, -1, 11], [13, 7, -6], [18, -7, -1], [-6, -1, -7], [14, -1, -4], [-9, -5, 1], [-15, -6, 4], [-2, -2, 8], [26, 2, 6], [-3, 2, 7], [-20, -2, -2], [-18, 3, 3], [-11, -5, -13], [0, 0, 6], [-24, -3, -9], [-16, 8, 0], [17, -4, -7], [-11, 9, -3], [-20, -3, 4], [20, 3, 5], [-1, 11, 3], [4, 5, -4], [-10, 4, -4], [0, -5, -13], [11, 5, -3], [-21, -6, -8], [-23, 4, -1], [16, -2, 2], [-13, 3, 5], [0, -2, 11], [8, -4, -2], [-3, 6, -3], [16, 5, 11], [-20, -2, 11], [-22, -3, 4], [20, 3, -3], [-8, 0, -5], [18, 4, -4], [0, -10, -4], [0, -4, 4], [-5, -8, 2], [-4, -11, 5], [13, -2, 3], [-4, 0, -14], [-24, 4, 1], [6, -13, -4], [14, -2, -7], [-2, -6, 8], [-9, -9, 0], [-4, -2, 8], [3, -12, -1], [8, 6, 6], [-25, -3, 6], [-12, 6, 6], [17, -1, 4], [6, 10, 6], [12, -7, -6], [30, 6, -10], [31, -11, -2], [-8, 2, 2], [12, -8, 8], [-26, 3, 8], [-13, -1, 5], [-15, 1, -4], [25, -6, 1], [26, 7, -7], [-8, -6, 8], [14, -14, 3], [42, -5, -3], [-23, 0, -10], [-33, -1, 6], [22, -8, 0], [5, 9, -7], [32, 0, -2], [14, -8, 7], [-31, 1, 2], [-31, -8, -9], [20, 3, -8], [7, -5, 1], [-32, -3, 6], [-26, 4, 2], [30, 3, -6], [-2, 2, -7], [5, 0, -8], [-39, -1, -4], [16, 5, -7], [-29, 10, 3], [20, -5, -7], [32, -6, -3], [28, 3, 8], [8, -12, 4], [6, 6, 2], [-17, 5, 6], [-24, 2, 11], [6, -7, -3], [-30, 9, 3], [0, -9, -16], [1, 8, -1], [-16, 2, 5], [8, 2, 1], [4, 4, 6], [28, 1, -9], [4, -14, -11], [14, 9, 11], [-18, 7, -2], [12, 0, -9], [15, -1, 6], [6, -5, -3], [-24, 4, 5], [10, 15, 6], [-16, -10, 12], [9, 16, 5], [12, 4, 10], [-8, -1, 10], [-16, 9, -5], [4, -10, -10], [-13, -3, -1], [11, 6, 21], [-42, 2, 6], [4, 12, 4], [4, -1, -8], [-2, -10, 6], [4, 16, 4], [32, 7, 10], [-7, 5, -6], [20, 4, -4], [-59, 2, -1], [-4, 0, -4], [7, -8, 3], [28, -13, -8], [7, -5, -4], [-52, -5, 5], [-23, 3, -9], [-8, -10, -6], [0, -1, -16], [10, 5, -5], [-6, 5, 11], [25, 7, -7], [-15, -15, -5], [-18, 8, 0], [20, 11, -12], [10, 0, 6], [-7, 18, 5], [38, 8, 10], [20, 10, 1], [16, -1, 12], [-3, 6, -13], [-26, 1, -10], [30, 6, 6], [-37, 2, 0], [-8, -3, -1], [24, 0, 12], [-31, -11, -8], [-19, -1, -2], [4, -21, -1], [-2, 6, 12], [-9, 12, 5], [-2, -2, 6], [9, 13, 15], [33, 5, 16], [-34, -9, -10], [-5, -18, -12], [50, 1, -1], [14, 9, -14], [-8, 4, 9], [-30, -4, -16], [23, -1, -3], [4, 16, -7], [51, -2, 2], [-37, -12, -9], [-26, 6, -10], [4, 10, -15], [-26, -12, -4], [-39, -1, 3], [-7, -19, -17], [0, -10, -16], [32, 7, 5], [-5, -3, -12], [8, -7, 14], [-31, 4, -9], [44, -12, -2], [30, 2, -8], [-5, 5, -1], [-20, -15, 0], [9, 5, -1], [-2, 19, 3], [-4, -12, -14], [2, 6, -2], [-34, -6, 4], [-14, -12, -10], [-17, -15, -6], [-18, -9, 9], [29, 3, -13], [38, 10, 0], [-34, 4, -8], [35, -1, -9], [-10, -8, -1], [-22, 8, 10], [28, 8, -9], [24, -5, -5], [-11, -4, 5], [-27, -16, -15], [-62, 3, 2], [7, -16, -17], [-40, 17, 3], [13, -13, -11], [18, -21, -3], [40, -2, -14], [30, 5, -10], [-52, 10, 1], [-20, -15, -16], [-5, -10, 8], [2, -1, 2], [-14, 17, 10], [-9, 17, 6], [-24, 9, 15], [-33, 2, 7], [39, 2, -9], [-28, 5, 8], [-15, 5, 11], [-32, -6, 16], [43, -11, 4], [37, -15, 3], [-21, -11, -4], [34, -4, -12], [-6, 6, 6], [33, -13, 6], [-30, 17, 2], [-53, -10, -6], [24, -5, -5], [3, 2, 10], [-62, 6, 6], [-40, 2, -11], [-6, 12, -4], [21, -7, 9], [-21, -16, -12], [13, -9, 5], [-58, 10, 0], [38, 11, 1], [-27, -1, 6], [10, -4, 0], [2, 2, -18], [-13, -8, -16], [60, -4, 1], [-6, -4, -6], [-32, -4, -10], [-48, 2, -2], [10, -4, 8], [-56, 4, -6], [0, -16, -12], [21, 5, 7], [-30, 14, -8], [-40, 14, 0], [-2, 5, -16], [34, 11, 16], [-52, 9, 4], [-10, 12, 21], [60, -4, -4], [46, 1, -9], [-34, -7, -16], [-53, 1, -10], [-32, -14, -3], [-6, -5, -21], [-27, 10, 16], [30, -24, -8], [-14, -3, -15], [5, -1, 25], [10, 3, 0], [16, -9, -7], [45, 8, 3], [41, -5, -10], [16, 19, 18], [-32, -2, 2], [6, 4, -15], [-54, -2, 14], [-13, 1, 2], [20, 15, -5], [60, -13, 2], [20, 2, 21], [-26, -11, -10], [46, -12, -9], [6, -1, -8], [-7, -21, -13], [2, -14, -10], [31, 12, -15], [-18, 3, 17], [-3, -1, 8], [-10, 15, -9], [52, -9, 5], [1, 6, -6], [0, -5, -10], [-14, 10, -2], [-28, 2, 10], [-22, -2, 12], [43, -10, -13], [-4, 12, -8], [54, -18, -1], [20, -3, -21], [24, 16, -3], [8, 12, 26], [40, 14, 16], [-5, 19, -2], [10, -24, -9], [-70, -1, 3], [-11, -9, -5], [-10, 17, 2], [10, -10, -13], [10, 11, -2], [-44, -7, 7], [36, -14, 4], [-15, -17, -6], [64, -4, 0], [8, 8, 18], [19, -11, 12], [10, 11, -9], [-22, 26, 12], [38, -13, 6], [27, -9, 5], [18, -2, -4], [20, -2, 4], [-2, 10, 11], [48, -16, -8], [12, 18, 0], [-24, -18, -22], [0, 16, -10], [14, -15, -15]];
    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_q();".
// 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_q(: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_q();".
// 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_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6027_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<5,R![-14, -12, 0, 1]>,<13,R![35, -15, -3, 1]>],Snew);
    return Vf;
end function;