
// Make newform 6027.2.a.u 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_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_6027_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 := [4, 3, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, 0, 1, 0], [-1, -4, 0, 1]];
    Rf_basisdens := [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 := [[0, 1, 0, 0], [1, 0, 0, 0], [1, 0, 1, 0], [0, 0, 0, 0], [-1, -1, 1, -1], [2, -1, 2, 0], [-2, 2, -1, 0], [1, 1, 0, -1], [2, -3, 2, 0], [0, 3, -1, 2], [2, 2, -1, 0], [2, -3, 2, -1], [1, 0, 0, 0], [-6, 2, -1, -2], [0, -1, -2, -3], [2, 1, 2, -2], [5, -3, 2, -1], [8, -2, 1, 2], [-3, 0, 3, -2], [-2, 4, -3, -2], [6, 2, 1, 0], [1, -4, 5, -4], [0, -2, -2, 0], [4, -4, 2, 0], [-3, -4, -2, 1], [3, -1, 5, -1], [2, -3, -3, -2], [-3, 2, -6, 5], [-5, 1, 0, 3], [10, -2, 2, -2], [-4, 4, -6, 2], [7, -3, 6, 1], [6, 1, -1, 2], [8, -3, -2, -2], [7, -5, 0, -3], [-2, 2, 2, 0], [-5, 7, 0, 1], [-6, 4, -7, 0], [4, 3, 3, -1], [-4, 7, -5, 9], [4, -6, -1, -4], [3, -9, 2, -7], [-1, -9, 6, -3], [-1, -3, -4, -1], [14, -4, 6, -2], [1, 9, 0, -1], [-1, -4, -3, -4], [0, -7, 2, -2], [4, 1, 4, -5], [-3, -2, -4, -5], [-1, 7, -4, 1], [-2, -2, -2, -4], [-3, 5, 1, 5], [14, -4, -2, -2], [-1, -11, 7, -3], [-1, 7, -1, 1], [10, 5, -9, 1], [-7, 4, -1, 1], [-13, -6, -4, 4], [-5, 12, -3, 1], [14, 6, 3, -2], [-12, 0, -7, -2], [20, -1, 1, 6], [-1, -2, -7, 8], [2, -7, -4, -2], [1, -6, -7, -1], [-12, 7, -5, 9], [8, 5, -6, 9], [14, -12, -3, 2], [4, 4, -9, 6], [-6, -8, -2, 2], [-5, -2, -6, 9], [4, -1, 7, -6], [-8, 11, 6, -1], [-1, 9, -8, 11], [-6, -9, 12, -1], [-15, 5, -8, -1], [-5, 7, -10, 13], [6, -2, -4, 2], [-3, -5, 5, 1], [19, -3, -2, 1], [15, 9, -8, 3], [5, -1, 6, 3], [1, 9, 1, -3], [-3, -1, 0, -9], [3, -3, 10, 1], [3, -7, 8, -1], [-12, 8, -4, 0], [-6, 1, -11, -7], [3, 10, 5, -4], [11, -7, -2, 3], [5, -1, -11, 3], [17, -1, 7, 5], [-13, 1, -14, -1], [-2, -10, 10, -12], [23, -2, 6, 4], [-2, 4, -1, -6], [5, 1, -1, 7], [9, 15, -6, 7], [-9, -10, 9, -2], [16, -5, 11, -9], [22, 4, -3, 0], [3, -6, 4, -2], [23, -3, 2, 9], [-14, 6, -6, 8], [-9, 11, -6, -3], [-9, 1, 9, -5], [-13, 5, -1, -1], [-10, 7, -10, -4], [1, -18, 18, -3], [5, -21, 6, -1], [0, -17, -1, 5], [24, -8, 2, 8], [6, 1, -3, 14], [3, -3, -7, 9], [10, 8, -3, 12], [-5, 5, 6, -11], [15, -3, 14, -7], [6, 0, -1, 10], [-18, -12, 2, 0], [0, -8, -6, 4], [1, -9, 6, -7], [14, 0, 6, 6], [4, 0, 8, 8], [-18, 10, -16, 6], [14, 6, 0, 0], [-8, 10, -4, -2], [5, 15, 1, 3], [-30, -2, 0, -8], [-3, -21, 17, -3], [-6, -2, 1, 6], [-11, 3, 0, -5], [-7, 6, -15, -4], [32, -2, 2, 0], [9, 2, -1, 0], [9, -9, 19, -5], [5, -13, 7, -7], [-13, 8, -3, 3], [-16, -2, -4, 2], [10, -11, 16, -4], [-12, 1, 5, 2], [25, -3, 16, -11], [-2, 10, 2, -10], [-15, -1, -17, 7], [-25, -1, 1, -11], [0, 5, 3, -1], [-2, -16, 4, 0], [1, 6, -21, 4], [1, -2, 15, -9], [-11, 12, 2, -7], [-5, -5, 13, 1], [15, -14, 9, -4], [6, -5, -7, 5], [-15, -10, 0, -2], [2, 8, 8, -6], [9, 11, 0, 9], [-14, 5, 7, 9], [-14, 8, -1, 4], [11, -1, 14, -3], [6, 11, -11, 1], [27, -2, 16, -3], [3, -5, -2, -11], [-12, 5, -13, -11], [-2, 13, -14, 15], [14, -16, 1, -16], [9, 13, -10, 1], [4, -3, -11, -3], [-15, 12, -18, 15], [3, -26, 12, -2], [8, 0, -16, 2], [-4, 16, -10, 10], [11, -15, 12, 1], [-36, -2, -2, -6], [6, 5, -6, 1], [10, 2, 18, 0], [-14, -16, 13, -4], [-22, 3, 3, 3], [9, 11, 6, 1], [-10, 20, 2, 2], [16, 1, -10, -4], [13, -5, -3, 11], [-9, 5, -2, 11], [-8, -6, 12, -4], [24, 9, 1, 3], [9, 6, 0, -5], [10, 8, 4, -8], [-12, -6, -4, 10], [19, -2, 0, -1], [-5, 15, 0, 11], [11, 13, -3, 1], [34, 0, -11, -2], [7, 5, 12, -13], [-23, 3, 4, 5], [22, 8, 6, -2], [-27, 8, -22, 11], [-12, 6, -6, -6], [-9, 13, 8, -3], [1, -21, -4, 9], [-15, 21, 4, 11], [-7, -9, 1, -15], [0, -2, -6, 6], [-12, 5, -23, 2], [-11, 5, -10, 1], [-21, -4, -4, 5], [7, -7, -11, -3], [-11, -3, -15, 5], [-30, 3, 0, 6], [-24, 12, -5, -4], [3, -9, 22, -11], [-8, 16, -33, 4], [-17, -19, 2, 5], [-19, 19, 1, -9], [33, -13, 6, 19], [-6, 8, -14, 12], [4, 5, -18, 5], [19, 5, 5, 5], [31, -6, -1, 2], [0, 2, 18, -14], [-9, -9, -13, -1], [-47, 3, 2, -11], [-37, 5, -18, -1], [2, -5, -1, -4], [35, -6, -3, -3], [8, -5, 15, -11], [-12, 8, 14, -8], [-28, -6, 6, -4], [-15, 8, -3, 4], [-10, 17, -18, -6], [2, -30, 2, 4], [-14, -3, 18, -4], [14, 21, -7, 5], [25, -11, 18, -5], [6, -4, -21, -10], [10, 8, 14, -8], [18, -14, 18, 6], [-20, 24, 2, 6], [-1, -2, -4, -9], [-18, 11, -30, -2], [-20, -5, 8, 0], [11, 17, -7, 7], [-16, 18, -6, -2], [-6, -4, 14, -20], [22, 14, -2, 6], [22, -13, 18, -13], [4, -13, 27, -9], [2, 18, 6, 2], [-8, -5, 10, 6], [9, 34, -22, 4], [17, 9, 5, -5], [1, 30, -28, 15], [-23, 0, -8, 2], [26, 29, -21, 13], [13, 16, -24, 3], [6, 16, 5, 10], [18, -20, 24, -4], [-16, -3, 12, 6], [5, -3, -8, -9], [0, 22, 8, -10], [24, -2, 13, -2], [25, -24, 6, 1], [-16, 0, 3, -8], [1, 5, 7, -15], [-6, -11, 10, -6], [19, 10, -8, 11], [23, -4, 27, -16], [-30, 19, 17, 6], [-24, -14, 3, 2], [-16, 17, 3, 21], [-10, 6, 8, 2], [12, 3, 3, -2], [1, 6, 8, -7], [21, 11, -3, 5], [4, -4, -5, 14], [2, -4, -1, -8], [-29, -1, -4, -7], [-7, 23, 2, 15], [6, -20, -7, -8], [-38, 6, -12, 6], [19, -13, 28, -3], [3, 26, 6, -6], [9, -9, -23, -7], [-2, -5, 13, -13], [19, -17, 12, -11], [5, 21, -24, 15], [-16, 10, -20, 8], [18, -12, 25, -4], [35, 9, -10, 7], [9, 11, 14, -7], [-26, -18, 0, 6], [-16, -10, -1, -18], [-20, -25, 1, 6], [-28, 4, -4, 6], [4, 6, -7, 8], [-17, 13, -32, -5], [16, -6, -24, -6], [-10, 18, -3, 22], [-28, -6, 16, -8], [10, 19, -15, 5], [6, -9, -21, 13], [-29, 11, -30, 17], [22, -2, -22, -6], [-37, 20, 10, 7], [1, 3, -17, 1], [-20, 30, 11, 4], [-31, -9, 9, -9], [-22, 28, -16, 24], [-26, 21, 0, 2], [21, -23, -5, 1], [51, -5, -4, -5], [-5, 27, -26, -1], [5, 0, 24, -14], [39, -4, 2, 19], [-21, 1, -28, 3], [-22, 8, -1, -16], [6, 7, -14, 18], [-23, 19, -18, 5], [10, -6, 20, 8], [-5, -4, -8, 15], [8, 33, -2, 0], [-6, -12, 11, -16], [26, -27, 10, -12], [3, -18, -1, -4], [-14, 8, -22, 18], [22, -10, -12, 6], [15, -17, -2, 13], [28, 19, -4, -6], [16, 1, -3, 21], [-58, 10, -4, -2], [19, -5, 7, 3], [21, 9, 14, 5], [35, -5, 0, 19], [-33, 5, -20, -7], [39, 8, 0, -1], [-5, 2, -27, -8], [12, 0, -19, 24], [38, 13, 8, -1], [13, 11, -12, 5], [-31, -12, 13, -12], [-3, 8, -18, 3], [-26, -6, 21, -4], [49, -19, 12, 17], [9, 3, 6, 15], [-16, 24, -26, 20], [42, 2, -2, -6], [28, 8, 28, -20], [-4, -23, 12, -24], [-1, -3, -20, 9], [-37, -9, -19, 11], [-25, -4, -7, 12], [38, 4, -12, 6], [-15, -3, -6, 19], [-34, 3, 6, -27], [-25, -14, 3, 5], [-46, -14, -3, 0], [-21, 9, -6, -7], [33, -17, 6, 17], [-34, 26, 4, 16], [36, -16, -2, 6], [6, 6, -4, -8], [32, -11, -8, -8], [-15, 28, -14, -4], [30, -2, 4, -12], [-14, 12, 25, -6], [7, 18, 4, -1], [-33, 25, -22, -5], [28, -16, 36, -8], [43, 9, -12, -3], [0, 12, 25, -2], [-45, -6, -22, 12], [-3, 9, -10, -5], [-8, 4, 14, 6], [1, 19, -10, -9], [6, 12, 7, 24], [4, 9, 16, 16], [22, -16, 14, -10], [19, -35, -2, -7], [46, 13, -8, 0], [-10, -20, 2, 14], [0, 4, 27, -12], [68, -5, 4, 18], [10, -30, 1, 0], [-8, 14, 10, 10], [14, 6, -3, 0], [10, 4, 24, 14], [-5, 11, -28, -7], [19, -3, -2, 11], [1, -2, -31, 14], [22, -5, 0, 14], [33, -11, 25, 1], [-24, 3, -11, 13], [-49, -3, -4, 13], [-2, -2, -13, -12], [-3, -35, 38, -9], [-23, 14, 6, 11], [3, 2, -20, 7], [18, 6, 26, 0], [40, -22, 26, 16], [-21, 38, -9, -12], [-30, 16, -25, 16], [-19, 13, 10, -11], [57, -21, 7, 27], [25, -10, 14, -19], [15, 1, -23, 7], [-12, 8, -4, 22], [-3, -32, 19, -16], [-18, 6, 10, -6], [5, -39, -1, 7], [-50, 12, -11, -14], [-14, 22, -14, 22], [-29, 21, -6, 1], [34, 16, -13, -4], [48, 0, 24, -12], [49, 11, -20, 5], [13, 4, 1, -6], [2, -12, 4, 8], [29, -18, 3, 8], [-6, 11, 6, 14], [9, -3, -20, 27], [-38, -22, 1, -10], [-19, -7, 10, 9], [-44, 12, -12, -4], [9, -27, -6, 5], [75, -4, -2, 11], [14, 15, -2, -17], [54, 6, -11, -8], [38, -11, 23, -7], [8, 4, 11, 0], [-44, 5, -9, 2], [6, -8, 28, 12], [-44, -5, -18, 11]];
    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_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_6027_2_a_u(:prec:=4)
    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_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_6027_2_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6027_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 3, -5, -1, 1]>,<5,R![2, 7, -3, -3, 1]>,<13,R![88, 47, -13, -5, 1]>],Snew);
    return Vf;
end function;