
// Make newform 4004.2.a.f in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4004_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_4004_2_a_f();".
// 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_4004_2_a_f();".
// 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 := [1, 6, 6, -6, -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], [2, 5, -5, -2, 1], [5, 6, -6, -2, 1], [-6, -17, 10, 5, -2]];
    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_4004_a();"
function MakeCharacter_4004_a()
    N := 4004;
    order := 1;
    char_gens := [2003, 3433, 365, 925];
    v := [1, 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_4004_a_Hecke(Kf)
    return MakeCharacter_4004_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, 0, 0], [1, -1, 0, 0, 0], [0, 0, -1, 0, 0], [-1, 0, 0, 0, 0], [-1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [-3, 1, 0, 2, 1], [-1, 1, 1, 1, -1], [-1, 1, -1, 1, -1], [-1, 0, 2, -1, -1], [1, 2, 0, -2, 0], [-3, 2, 1, 1, 0], [-4, 0, 0, 0, 1], [-5, -1, -1, 1, 2], [3, -1, 1, -3, 1], [2, 1, 2, -1, -2], [0, 2, 0, 1, -3], [-5, -1, 0, 3, -1], [-4, 3, 1, 4, 1], [2, 0, 0, 0, 4], [-8, 4, 1, 3, 2], [0, -5, -2, -2, 3], [2, 2, -1, -3, 0], [-5, 1, 1, -1, 5], [-5, -2, 0, -2, -1], [1, -3, 2, -1, -1], [1, -6, -1, -4, 2], [-2, 2, 2, 0, 3], [-10, -1, -3, -2, 1], [2, 4, 1, 0, 2], [3, 3, 0, 2, -6], [4, -2, 4, -2, -5], [2, -5, -1, -3, -1], [-7, 4, 1, -1, 4], [-4, 2, -2, 3, -6], [-1, 1, 0, 0, -6], [-1, -8, -1, -4, 3], [-1, -2, -5, 2, -1], [-8, 7, 2, 4, 4], [-3, -8, 1, -5, -1], [-4, -5, -4, -5, 2], [-4, 1, -3, 9, 1], [1, -9, 0, -3, 5], [-4, 7, -1, -1, 2], [5, -6, 1, -2, -2], [2, 7, -3, 1, 6], [-1, -1, -3, 7, 1], [4, 4, 1, -1, -5], [-5, -1, 0, 1, -6], [-8, 0, 0, 5, 7], [-2, -1, -4, 5, -2], [-16, 0, 0, 3, 4], [-5, -2, -6, -4, -2], [4, 6, 6, 4, -4], [3, 4, 11, 2, -4], [-11, 5, -1, 12, 5], [15, 2, 5, -1, -6], [2, -5, 3, -6, -4], [2, -4, 5, -10, -4], [3, 2, -7, 9, 2], [8, -1, 1, -12, -4], [9, -8, -4, -6, 0], [0, 1, -6, -1, 4], [4, 5, -3, 8, 3], [9, -7, 2, 0, 3], [-3, -11, 2, 0, 3], [6, -6, -2, -9, 6], [0, -3, 4, -6, -5], [5, -1, 2, -10, 0], [-12, 6, 5, 4, -5], [-10, 4, 0, -4, 2], [7, 1, -2, 0, 1], [6, 5, 1, 10, 6], [1, 5, -4, 7, 1], [4, -10, 0, 4, -3], [-9, -6, 7, -4, -5], [-8, 8, -4, 4, 9], [9, -3, -3, 3, -12], [8, -9, -3, -7, 4], [-3, 0, 6, -3, 2], [3, 5, -2, 6, -10], [-10, 7, 5, 8, -6], [-8, -3, -3, -1, -4], [4, -11, 3, -7, 2], [-5, -3, 0, -2, 4], [5, 2, 2, 3, -6], [17, -10, -7, -2, 1], [-8, 8, 6, -1, 6], [9, -4, -9, -1, 4], [8, 3, 7, 8, 1], [10, -10, 2, 6, 5], [2, -2, -5, -1, 1], [-8, 7, 7, -9, -2], [-5, 7, -4, -1, 12], [-5, -1, 6, 7, -4], [10, 7, 1, 2, -4], [2, 6, -5, 0, -8], [-1, 8, 1, 10, -1], [7, -2, -11, 3, 1], [9, -9, 2, -2, 1], [-5, 2, 8, 0, 5], [-15, 8, -1, -3, -7], [0, -7, -1, 10, 6], [-11, 12, 4, 0, -7], [-5, 3, -1, 0, -10], [-6, -5, -11, -6, 0], [14, -8, -3, -16, -4], [10, -8, -3, -13, -8], [6, 7, 6, 12, 4], [-1, 3, 10, 3, 4], [13, 9, 0, 5, -2], [-8, -3, -7, -3, 5], [21, -9, -4, -10, -2], [9, -6, -3, 3, -11], [14, -1, 3, 1, -1], [-10, -7, -2, -7, -5], [15, 4, 3, -3, -12], [-16, 16, 2, 9, -1], [10, 3, 11, 5, 1], [18, 2, 7, 4, -9], [-8, 12, -3, 4, 0], [0, -9, -6, -3, -7], [4, 1, -3, -7, 11], [-3, -11, -4, -6, 6], [-12, 0, -3, 0, 9], [17, 7, -3, 0, 6], [-4, 2, -8, 4, 3], [11, 6, -3, 4, -4], [24, -8, 0, -8, 9], [-6, -5, 6, -6, 5], [-10, -2, -2, -7, 3], [10, 7, -1, 13, 2], [6, -10, 1, -11, -3], [-1, 4, -2, 11, -8], [-19, -11, 3, -5, 4], [-11, 4, -10, -8, -5], [-14, 8, -2, 9, 13], [22, -10, 1, 0, -10], [-20, 3, -7, 12, 7], [4, -4, 1, -6, 1], [-6, 8, 2, 5, 4], [-15, 13, 12, 7, -1], [-10, -20, 8, -12, -3], [-7, -7, -2, 4, 7], [-7, -10, -7, -7, 7], [4, 7, 9, -5, 0], [8, 8, -5, 11, 2], [-7, 15, 2, -6, -1], [-24, 3, -3, 20, -5], [8, 10, 0, 9, 5], [-8, -13, 1, -9, 3], [22, 7, -1, -3, 1], [-4, 1, -5, -7, 8], [24, -4, -2, 0, -13], [8, 7, 8, -10, -4], [-7, 3, -9, 2, 4], [19, 5, 6, 11, 8], [-11, 8, -2, 8, -15], [-12, 11, -7, 8, -9], [22, -11, 0, -14, -7], [9, -3, 2, -6, -6], [-1, 5, -7, 9, -14], [-5, 11, 2, 1, 13], [7, -20, -3, 2, 8], [-4, 1, -3, -3, -11], [42, -9, -6, 4, 1], [22, -8, 2, -14, -2], [0, -7, -9, -12, 2], [4, -13, 1, -13, 14], [13, 3, 2, -2, -1], [-9, 3, -6, -9, 3], [10, -17, 1, 7, 2], [-14, 8, -1, 20, -6], [-12, -10, 4, 0, -10], [-4, 4, -6, 15, 7], [1, 9, -16, 16, 0], [-19, 13, 2, 11, 5], [-6, -19, 5, 1, 3], [5, 3, -10, -3, -2], [-22, -3, -9, -4, 9], [21, -8, 1, -18, -16], [5, -11, -4, 3, -6], [-12, 11, -11, 8, 1], [-5, -11, -1, 4, -5], [1, 10, -5, -5, -6], [-13, 9, 8, -10, 9], [-8, 2, 15, 8, -8], [1, 12, 4, 4, -9], [-23, 24, 5, 9, 1], [2, -21, 4, -12, 14], [7, -3, 0, -1, -10], [-2, 3, 9, 1, -2], [-29, 0, -8, -1, 5], [0, 3, 4, 1, 4], [5, -14, -14, -3, 9], [8, -5, 17, 0, -1], [-17, -16, 1, -9, 11], [-18, 20, 9, 13, -2], [3, 22, -6, 10, 8], [-1, -23, -6, -11, 7], [4, -4, -8, 8, -19], [25, -23, -10, -7, 2], [4, -20, -9, -21, 1], [-14, -3, -3, -1, -14], [-21, 1, 3, 12, 1], [-15, 8, 9, 8, 4], [25, 5, 1, -3, -16], [-10, 5, -6, 8, -3], [16, 3, 3, -21, 4], [17, -12, -11, -7, 9], [-30, 11, -2, 1, -1], [-4, -10, 4, 12, 4], [23, -1, -10, 1, 5], [-14, 3, 5, 8, -9], [19, -7, 11, 4, -9], [-28, 13, 14, 1, 2], [-28, 3, -8, -2, 8], [4, 1, -3, 6, -1], [-28, -10, -4, -1, 11], [20, -19, 7, 5, 5], [5, 5, 3, -1, -6], [13, -2, 15, -4, -12], [-12, 14, 4, 0, 3], [9, -4, -17, 6, 2], [-32, -3, -11, 6, 2], [-22, -10, 11, -7, -5], [-4, 6, -10, 6, 6], [2, 12, 3, 8, 10], [-8, -9, -5, 8, -3], [-3, 13, 4, 4, -8], [-25, 17, -2, 9, -3], [-4, 7, 3, 5, 0], [-28, -9, 9, 2, -12], [-24, -9, -7, 4, 9], [-7, 7, -4, 13, 0], [2, -13, -2, 9, -4], [-8, -3, 11, -13, 17], [3, 16, -7, -4, 5], [-10, 5, -12, -14, -1], [19, -7, -16, 13, -2], [17, -23, 2, -10, -5], [-16, -6, -8, 11, -2], [-22, 20, 2, 25, 16], [-46, 17, 7, 1, 4], [-9, 8, -4, -10, 18], [14, -15, -15, 1, 3], [4, -5, 14, -4, -13], [2, -7, 2, -12, -10], [-27, 3, 8, 1, 6], [24, 1, 4, 9, -10], [-46, 11, 7, -5, 6], [-36, 0, 4, 9, -1], [6, 1, -13, 1, 6], [-1, -1, -2, -7, -12], [-17, 9, 4, -6, -6], [-19, 12, -9, -3, -13], [13, 9, -9, -6, -1], [-11, -10, 3, 12, -4], [18, -15, 9, -9, 10], [19, -11, 1, -21, -5], [-11, 6, -2, 6, 16], [12, 2, -5, -10, -2], [24, 11, 12, 4, -3], [35, -4, 3, 0, 2], [25, -11, 6, 1, -6], [9, -15, 2, -6, -4], [-27, 7, 0, 23, 0], [28, -7, 12, 2, -2], [5, 8, 3, 1, -17], [-1, 11, 6, -5, -6], [10, -7, 11, 5, -3], [-35, 16, 5, -1, 5], [-14, 16, 8, 7, -4], [-3, 3, 11, -14, -14], [-4, -6, -8, -3, 8], [-4, -5, 1, -10, -14], [-24, 10, -10, -8, 1], [-28, 15, -5, 21, -14], [0, -24, 4, -13, -4], [-32, -5, -17, -2, 13], [41, -14, -7, -14, 6], [-33, 6, -7, 28, 7], [-11, -7, -6, -9, 3], [-21, -3, -17, -5, -8], [-2, 0, 4, -2, -1], [-7, 10, -18, 20, -10], [-5, 10, 16, -7, -11], [-33, 6, -11, 1, 16], [3, 0, -13, -7, 4], [7, -15, -18, 4, 5], [-31, 12, -6, 10, 6], [10, -3, 13, 4, -8], [5, 11, 14, -3, -3], [-12, 7, -14, 5, -4], [10, 3, -4, -6, -1], [-5, 7, 12, -8, 4], [14, 28, 12, 7, -20], [44, -8, -10, -4, 3], [-6, -11, -3, 2, 19], [-29, 4, -6, -10, 1], [-35, -3, -4, 8, 6], [20, 6, -10, 25, 7], [-9, -8, 10, -8, -14], [22, 0, -9, -19, 9], [-14, -12, 13, -25, -7], [-15, 15, -10, 11, -10], [-26, 13, -1, 17, -13], [-3, -19, 16, -15, 11], [19, -9, 4, -13, 13], [-4, 10, -17, -2, 2], [-11, 6, 5, 14, -10], [-15, -2, -7, 11, -3], [-34, 15, 11, 19, -3], [36, -5, -8, 1, 3], [-21, 4, 15, -16, 3], [-36, 20, -2, 0, 3], [29, -30, -10, -31, -1], [-10, 10, -2, 5, -21], [-14, -9, 2, 5, 14], [33, -18, -6, 6, 0], [-24, -9, 9, -2, -10], [-19, -15, 16, -3, -8], [13, 17, -4, 24, 4], [-17, 16, 15, -6, -2], [37, -23, -2, -22, -1], [-16, 15, -7, -3, -5], [-17, 15, -10, 9, 25], [5, 11, 17, -16, -10], [-12, 4, 8, -9, 18], [-19, 29, 10, 6, -4], [-51, 10, 3, 15, 4], [-26, -10, -3, -6, 22], [29, 16, -5, 7, -15], [-33, 6, 12, -10, 6], [-7, -4, 19, -6, -11], [-33, 0, 0, 26, 8], [-16, -13, 11, -13, -2], [-4, -13, -7, 9, 15], [7, -12, -7, 0, 12], [-43, 14, 9, 6, 10], [-21, -6, -17, 7, 6], [4, -10, -10, 5, -17], [-10, -6, -8, 13, -1], [-2, -32, -8, -3, 11], [-22, 16, -8, -9, -14], [-12, -6, 14, -11, 6], [-28, 8, 16, -22, -4], [-20, 34, -4, 2, -4], [-14, -5, 4, -22, 2], [6, -5, -3, -5, -7], [28, -16, 3, -7, 17], [38, -4, -4, -4, -27], [17, -17, -2, -30, -15], [10, 9, 1, 2, -13], [-20, 4, -16, -12, 1], [22, 7, -11, 15, 3], [12, -12, 13, -28, -1], [-4, -8, 3, -4, -14], [22, 5, 7, 1, -8], [-31, -9, 2, -2, -10], [7, 15, 2, 6, 16], [4, 14, 4, 7, 4], [21, 2, -4, -19, -7], [-6, -5, 1, 24, -10], [10, -22, -4, 10, -8], [38, -3, 12, -5, 15], [-30, 21, -4, -11, -4], [-10, -16, 2, 11, 10], [-5, 18, -16, -7, 21], [-17, 9, 12, 11, 3], [-18, 7, 21, 10, -3], [1, -8, -3, -9, 7], [-4, 6, 11, 14, -2], [30, -21, -5, -28, -17], [34, 4, -1, -13, -18], [16, -3, 21, -15, 7], [-25, -10, 7, -15, 15], [5, -5, 2, 23, 5], [-5, 16, -5, -18, -8], [7, -1, -14, 1, -10], [60, -6, -7, -5, -2], [-25, 11, 0, 44, 11], [9, -16, 5, 9, -1], [-26, 5, -13, 29, 2], [-24, 35, 4, 13, 12], [26, 17, -2, 14, -21], [-7, -22, 1, 5, 8], [-5, 18, -11, 7, -18], [-12, 4, 12, -2, -11], [-20, 9, 13, 4, 10], [-28, -29, 1, -6, 9], [20, -9, 10, -7, -8], [21, 12, -9, 9, -13], [48, -22, -4, -16, -11], [-16, -8, 18, -5, -5], [-26, -21, -8, 10, 21], [36, -30, -12, -20, 11], [54, -6, 12, 6, -11], [-48, -8, 7, 1, -11], [3, -2, 9, -26, -6], [-15, -2, 2, 11, 20], [42, 0, -1, -4, 4], [12, -14, -4, 12, -3], [19, 27, -9, 11, -9], [-18, -18, -11, -3, 13], [18, -15, -5, 23, 8], [-18, 2, -4, -10, 14], [14, -22, -6, 3, -2], [17, 12, -10, -9, 5], [3, 5, 1, 25, 18], [24, -4, 18, 11, 1], [21, -12, 12, -38, -14], [-1, -6, -16, -10, -3], [21, -27, 0, -17, 18], [-2, 4, -3, 27, 13], [-19, 21, -6, 15, -4], [-18, -16, 2, -19, -2], [-8, 31, -1, 4, -3], [5, -22, 3, -18, -3], [12, 18, -18, 24, 1], [-12, 15, 18, 16, 0], [2, 25, -15, 23, 17], [24, -9, 17, 12, 1], [0, -6, 24, -23, 0], [-28, 19, -7, 6, 15], [-15, 6, 4, 10, -1], [-34, 4, -11, 6, 16], [18, -5, -7, -21, 15], [-27, 5, -15, -13, -1], [-5, -1, -2, -22, -16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4004_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4004_2_a_f();".
// 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_4004_2_a_f(:prec:=5)
    chi := MakeCharacter_4004_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_4004_2_a_f();".
// 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_4004_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-6, -3, 14, -4, -3, 1]>],Snew);
    return Vf;
end function;