
// Make newform 4002.2.a.bb in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4002_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_4002_2_a_bb();".
// 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_4002_2_a_bb();".
// 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, 0, -6, -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, -1, 1, 0], [0, -5, -1, 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_4002_a();"
function MakeCharacter_4002_a()
    N := 4002;
    order := 1;
    char_gens := [2669, 3133, 553];
    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_4002_a_Hecke(Kf)
    return MakeCharacter_4002_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], [1, 0, 0, 0], [-1, 0, 0, 1], [0, -1, 0, -1], [-3, 1, 0, 0], [0, -1, -1, 0], [0, 1, 0, 1], [2, 1, 2, -1], [1, 0, 0, 0], [-1, 0, 0, 0], [-4, -1, 1, 0], [3, 3, -2, 0], [0, 3, 1, -2], [2, -6, 0, 2], [2, -1, 2, -1], [0, 3, -2, -1], [-4, 3, 1, 0], [-3, -1, 0, 0], [-3, 2, -2, -3], [-4, -3, -1, 4], [-6, -1, 0, 3], [-6, -3, 0, 1], [-4, -4, 2, -2], [-6, 1, -4, -1], [-2, 1, -4, -1], [-2, -1, -3, 0], [-5, 1, -2, 2], [-4, -1, 4, -1], [-2, -2, 4, -2], [2, -2, 0, -4], [-8, -1, -3, 0], [-6, -1, 0, -3], [4, 5, 0, 1], [0, 3, 0, -1], [-7, 1, 0, 0], [-8, 4, 4, 0], [6, -3, 0, 3], [-4, 1, -3, 6], [4, -4, 3, -1], [-2, 7, -2, -3], [-6, -2, 2, -2], [12, -2, 2, 6], [1, -4, 0, -1], [-8, -3, -2, 5], [-12, -6, -2, 6], [-5, 3, 2, 0], [8, -3, 3, 4], [-4, 3, -6, 1], [-8, -4, 6, -2], [-1, 0, -6, 1], [2, -4, -2, 6], [-4, -4, 5, 1], [-4, 7, 0, 5], [-7, -2, 0, 1], [-4, 1, -4, -1], [-2, 5, 2, -5], [2, 4, -3, 5], [6, -5, 1, -2], [-6, 9, 1, 2], [-8, -3, -2, 1], [9, 3, 2, -6], [2, -4, 2, 0], [14, -1, 1, -6], [6, 9, -6, 5], [-14, 5, 0, 1], [8, 7, 7, -8], [10, -4, 2, -4], [7, -1, 2, 8], [-8, 0, 4, 0], [-10, -4, -3, 5], [2, 1, 2, -1], [0, 13, 0, -9], [6, -2, -4, 2], [2, -1, 4, -1], [-2, 1, 0, 9], [-6, -7, -4, 3], [10, 3, 6, -13], [-4, 0, -2, 0], [5, 13, 0, -4], [14, -7, -2, -1], [-6, 8, -4, -6], [-7, -2, 2, -3], [6, -7, 0, 7], [20, -3, -6, 3], [-12, -10, 2, 4], [-2, 7, 2, -1], [-16, 3, 5, 2], [-4, 14, 0, -8], [-24, -4, -7, 7], [-4, -5, 6, -3], [-3, -1, -8, 2], [-15, 3, 2, -4], [-22, -5, 0, 5], [-6, -5, 2, 3], [0, -6, 6, 4], [2, -7, 6, -1], [-8, 1, 4, -1], [4, 3, 2, -1], [-1, -2, 0, 5], [-12, 2, -6, -2], [12, -2, 0, -10], [-11, -6, -8, 9], [11, -6, 0, -5], [24, -3, 0, 1], [15, 4, -6, 1], [-10, 1, 8, -7], [-16, -2, 0, -6], [-6, 8, 2, -10], [-14, 5, -4, 7], [-4, 1, -12, 3], [6, -10, 2, -6], [-6, 4, -4, 8], [12, -6, 2, 12], [-4, 9, -2, -3], [-17, 0, -4, 3], [18, 0, 6, -4], [16, -1, 0, -5], [-6, 2, 3, -5], [4, 4, 3, -7], [-4, 9, -4, 11], [4, 0, -12, 6], [6, -6, -4, -6], [20, 4, 0, 0], [-20, -11, -7, 10], [30, 2, 4, -8], [21, -3, 0, 0], [18, -17, 2, 5], [8, 7, 5, -8], [4, 10, -2, 2], [13, -6, -4, 3], [0, -12, -3, -3], [5, -8, 0, -1], [8, -15, 0, 3], [11, -7, 6, -6], [-4, -4, -2, 0], [-3, -6, 4, -1], [-6, -2, 6, -6], [13, 2, 4, -5], [-28, -3, -2, 11], [4, 0, 9, 3], [-8, 11, 6, -3], [-16, -1, 6, 3], [-14, 10, -3, -7], [5, -10, -2, -9], [28, 5, -2, -3], [5, 6, 2, 3], [-14, 5, -8, 1], [-12, -18, -2, 10], [10, 6, -6, -2], [18, 11, -9, 8], [10, 4, 4, -16], [32, 5, 10, -9], [30, -1, -4, -3], [-10, -16, -2, 4], [12, -12, 2, -8], [-7, -21, 0, 6], [11, -2, -2, 11], [4, 3, -16, 1], [-2, 11, 0, -1], [1, -4, 6, -11], [8, 3, -6, 1], [-8, 5, 2, 13], [6, -16, 2, 4], [5, 0, 2, -3], [23, 8, -10, 3], [3, -12, 10, 7], [-38, -2, -6, 10], [-10, 5, -2, 7], [-4, -12, 12, -2], [6, 16, -4, 4], [-4, -11, -12, 13], [8, -20, 6, 8], [16, 11, -10, 5], [10, 11, 8, -11], [-8, -10, 4, 2], [19, -6, -6, 1], [10, -13, 0, 9], [26, -6, -4, 2], [8, -4, 6, 4], [28, 1, 10, -7], [4, 5, -1, -4], [1, -9, 6, -16], [8, 2, 7, -13], [-6, 15, -2, -3], [10, -16, -2, 8], [-3, -15, -2, 2], [0, 19, 0, -11], [24, -16, -6, 8], [10, 6, 6, -16], [-12, -14, 8, 6], [28, -7, 7, -6], [22, 9, -6, -3], [12, -20, 4, -2], [12, -12, 6, 0], [-9, -20, 2, 3], [34, -13, 0, -1], [11, -6, -10, -1], [30, 4, -10, 2], [-2, -5, 0, -5], [18, -6, -5, 3], [24, 7, 10, -19], [-16, 17, -4, -11], [-26, 8, 10, -2], [4, 1, -2, 15], [6, 25, -6, -5], [-32, 14, 4, 8], [-12, -7, -14, 5], [12, -9, 5, -16], [12, 9, -18, 9], [4, 17, 9, -14], [-2, -8, -4, 16], [16, -1, -6, -1], [-18, 8, 2, 14], [-38, -3, -6, -1], [14, 15, 4, -3], [-42, 12, 0, -4], [-4, 32, -2, -14], [4, 5, 17, -8], [4, 10, 8, 6], [-8, -13, 17, -6], [8, -4, -10, 4], [-23, 0, -2, 11], [38, 9, -3, 2], [-32, -8, -14, 16], [18, 14, -2, -14], [6, 7, 10, -7], [-2, 8, 2, 16], [6, -12, 0, -14], [-18, 6, 10, -8], [0, -5, -1, -14], [28, 3, 6, -21], [28, -3, 0, 9], [20, 11, 14, -9], [-32, -6, -14, 6], [25, 1, -8, 10], [-44, -5, -12, 13], [1, -22, 6, 5], [-44, -5, -12, 11], [-14, -5, -7, -10], [-14, -12, -2, 20], [-22, -5, -6, 13], [-6, 20, -6, -12], [-14, 16, 0, 2], [12, -4, -2, 8], [-18, -1, 6, 5], [-4, 0, 24, -4], [2, -3, -8, 7], [2, -6, 4, 4], [15, -10, 16, -3], [-17, 18, 10, -7], [-38, -5, 8, -3], [-27, -13, 8, -6], [10, 21, -6, 3], [25, -4, 6, -9], [10, -8, -8, 4], [10, -12, 7, 5], [-26, -7, 0, 11], [-12, 11, 4, 9], [-2, 9, -10, 11], [-18, -1, -19, 20], [-30, 25, 10, -5], [6, -7, 14, -5], [-8, 8, -9, -7], [-11, 15, 2, 6], [10, -16, -8, 16], [20, 17, 2, -11], [-36, 12, -6, -2], [32, -22, -8, 6], [10, 8, -2, -6], [8, -7, -2, 5], [-10, -7, -13, 0], [-8, 11, 4, 19], [18, 6, 20, -10], [-16, -19, 18, -3], [-4, 7, 6, 11], [28, -14, 8, 18], [22, 11, 4, -19], [-12, 18, -6, -10], [41, -1, 8, -4], [12, -15, -8, 17], [-6, 1, -13, 10], [-10, 15, 2, 15], [12, 11, -20, 7], [-26, 22, -2, -12], [-34, 10, -6, -6], [-16, 1, 8, -11], [-2, 27, -8, -7], [-4, -13, 4, -1], [6, -6, 14, -2], [-4, -8, 10, -12], [-28, 23, 6, 3], [8, 7, -10, 21], [6, -12, -6, 22], [-30, 3, -2, -15], [-16, 15, 14, -5], [25, 8, -8, -9], [6, 7, 12, -13], [26, 0, 0, 4], [-6, 16, 1, -13], [-20, 0, -16, -4], [-23, 4, 6, -13], [-1, 25, 6, -16], [8, 3, 2, -13], [-14, -26, 4, 0], [-4, 1, -12, 19], [-72, -2, 6, -2], [-10, 8, -2, 4], [-16, -7, -16, 11], [16, 1, 18, -13], [-20, 28, -2, -16], [12, 12, -7, -11], [38, 9, 4, 3], [47, 3, 10, -6], [10, -14, 0, 8], [-32, 3, 2, 1], [10, -27, 4, 9], [-6, -28, 10, -4], [-28, 3, 1, 12], [-14, 14, 14, 0], [16, -16, 9, -21], [-6, -12, 10, -22], [-10, -8, -6, 18], [-14, 5, 8, 5], [-12, 8, -8, 0], [-8, 17, -6, -15], [-19, -4, -8, 17], [17, 7, -16, -4], [-4, 19, 0, 11], [21, -5, 2, 16], [3, 3, -10, 6], [-16, -25, 18, -1], [4, 24, 2, 0], [-6, -2, 8, -8], [-18, -10, 8, 4], [-28, 1, -6, 5], [-2, 2, 6, -22], [-14, -2, -26, 8], [-30, 22, -12, -12], [-32, -19, -8, 11], [9, 14, 2, -15], [-42, 24, 14, 6], [-3, -11, 6, -10], [42, -3, 5, -6], [3, 8, -4, 11], [0, -11, 22, -3], [10, -13, -4, -11], [-18, -26, -2, 20], [24, 2, -3, 15], [7, 11, 26, -10], [-18, -4, -10, 16], [28, 24, 2, -6], [-30, 20, -5, -17], [16, -31, 1, 12], [-59, 6, 4, -5], [0, 5, 10, -1], [18, -17, 7, -2], [35, -9, 4, -8], [12, -21, 10, 13], [30, -13, 1, 16], [22, 29, -11, 4], [-22, 37, 2, -5], [38, 17, 14, -25], [-34, 4, 8, -14], [-20, -3, -20, 9], [18, -3, 9, 0], [36, -17, -12, 1], [-28, 24, 2, 8], [13, 18, 2, -7], [-30, 5, -4, 11], [-2, 24, 8, -6], [16, -10, 6, 0], [-60, 8, 0, -12], [-2, -23, -4, -1], [25, 13, 2, 4], [-40, 5, -7, -4], [-50, 7, -2, -13], [-23, 23, -6, -24], [29, -9, 6, 10], [25, -21, 6, 20], [-18, -14, 8, -2], [-18, -3, -12, -9], [43, 0, 2, -21], [-6, -14, -15, 13], [4, -10, -3, 1], [64, -11, -4, 1], [-8, 5, -15, 26], [-40, 20, 0, -8], [-4, -8, 6, 20], [10, 19, -4, 7], [17, -14, -8, 1], [1, 22, -14, 5], [4, 12, -18, 0], [-36, 12, -12, 4], [-4, -38, -7, 23], [-7, -19, 8, 8], [-2, 4, -2, -6], [24, 1, 12, 1], [32, -6, 0, -14], [-7, 1, 16, -2], [-30, 10, -4, 2], [22, 7, 1, -20], [-44, 19, -7, -18], [2, 17, -12, -3], [-21, -2, -4, 11], [4, 3, -2, -1], [-32, -5, 16, -5], [-30, -14, -10, 16], [-14, 16, 5, 13], [26, 2, 12, -14], [46, -10, -3, -7], [-4, -21, 1, -6], [34, -14, 20, 18], [-80, 10, 0, 6], [-26, -15, 2, 21], [43, -27, -8, 4], [-14, 18, -4, 22], [0, 30, -14, 0], [-32, -13, 4, -1], [12, 14, 0, -22], [33, -8, 2, -23], [11, 2, -6, 25], [-6, -25, 8, -11], [-24, 6, -8, -14], [52, -1, -16, 3], [32, -9, -7, 14], [-46, 13, 16, 1], [42, 10, 6, -22], [-2, 0, -20, 8], [-12, 12, -8, 4], [12, 7, 6, -19]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4002_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4002_2_a_bb();".
// 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_4002_2_a_bb(:prec:=4)
    chi := MakeCharacter_4002_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_4002_2_a_bb();".
// 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_4002_2_a_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4002_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![14, -9, -7, 3, 1]>,<7,R![64, -4, -20, 2, 1]>,<11,R![2, 45, 39, 11, 1]>],Snew);
    return Vf;
end function;