
// Make newform 2001.2.a.h in Magma, downloaded from the LMFDB on 08 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2001_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_2001_2_a_h();".
// 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_2001_2_a_h();".
// 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 := [-3, -1, 11, -5, -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], [1, -5, 0, 1, 0], [1, 5, -5, -1, 1], [-3, -5, 6, 1, -1]];
    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_2001_a();"
function MakeCharacter_2001_a()
    N := 2001;
    order := 1;
    char_gens := [668, 1132, 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_2001_a_Hecke(Kf)
    return MakeCharacter_2001_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], [-1, 0, 0, 0, 0], [-1, 1, 1, 0, 0], [-1, 0, -1, 0, 0], [-2, 0, 0, 0, 1], [1, -1, -1, 0, 1], [0, 1, 0, 0, 0], [-1, 0, 0, 0, -2], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [-2, 0, -1, 2, 0], [2, -2, 1, 1, 1], [-1, 0, 2, -1, -2], [-1, -2, -2, 1, -1], [-5, -1, 2, 4, 3], [-1, 2, 1, 2, -1], [2, 4, 0, -2, -4], [3, 3, -1, 2, -1], [0, -1, 0, 0, -4], [-3, 1, 1, 2, -1], [7, -3, -3, -1, 2], [0, 2, 1, 0, 2], [5, -7, 0, -2, 0], [1, -4, 0, 2, 1], [1, -2, 2, -2, 3], [-2, -1, 2, 0, -2], [-12, -1, 0, 2, 2], [3, -2, 3, -3, -1], [0, 3, -2, 4, 5], [3, -3, 1, -3, -6], [-1, 5, 3, 3, -2], [-2, 0, -5, -4, 2], [-1, -4, -3, -6, 0], [-9, -1, -3, 0, 3], [-4, 1, 3, 0, 1], [-1, -5, -6, -6, 2], [2, 5, 1, -2, 1], [0, 5, 0, -5, -2], [-3, 2, 7, -1, -1], [-5, -1, 0, 4, 3], [-7, -6, 3, 2, 1], [-5, -8, -2, -1, 3], [-8, -1, -4, -8, 4], [3, -1, -1, 1, -5], [-5, 7, -1, 3, 0], [9, -3, -1, -5, 1], [7, -6, 0, -6, -2], [-3, 8, 3, 1, -8], [-6, 2, 0, -1, -7], [1, -5, -2, -5, 2], [-8, 7, 2, -1, 1], [4, -3, 4, -3, -6], [-10, -1, -3, 1, 6], [-1, -3, 2, 1, 6], [9, -2, -3, -9, -8], [5, -1, -6, 4, 3], [-1, 0, -6, 1, 2], [2, -4, -3, -8, -1], [-21, 10, 4, 3, 3], [-14, -3, 3, 5, 4], [5, -2, 0, -4, 7], [11, 5, 5, 3, -5], [-2, 8, 1, 2, -5], [6, -5, -4, -1, 4], [10, 9, 2, -5, -8], [2, 0, -7, -1, 3], [-8, 1, -7, 0, 2], [-5, 4, 9, 10, 4], [-5, 6, 3, 6, -5], [15, -7, -4, -4, -1], [-1, -1, 4, -3, 5], [3, 7, -1, 1, 8], [-10, 7, 6, 5, -4], [14, -6, -4, -12, 1], [-6, -3, -4, 5, 2], [-9, 1, -1, 4, -2], [-5, 5, 6, -6, -14], [6, -1, 2, 1, -2], [-7, -3, -4, 7, 1], [-4, 4, 7, 7, 1], [0, 0, 0, 6, -3], [15, -1, -1, -5, -3], [-2, 9, 14, -2, -9], [28, -11, -6, -12, -3], [-23, 1, 2, 4, 3], [-2, -3, -3, 5, 8], [-4, 8, 6, 11, -3], [-6, 2, -1, -1, 3], [11, -12, -3, 0, 7], [-5, -3, -6, -1, 10], [1, 7, 1, 1, -7], [-1, 2, -6, 5, 7], [-9, -5, 3, 4, 5], [-7, -8, -2, -6, 3], [-6, 7, 2, 0, 1], [-13, -12, -9, 0, 13], [16, -3, -4, -6, -1], [3, 11, 3, 0, -7], [3, -10, 9, -2, 3], [2, -1, -2, 5, 5], [2, -1, -2, -7, -9], [-2, -6, -2, 9, 3], [1, 10, 1, 1, -3], [6, -6, -6, 5, 0], [-20, 9, -4, 3, 1], [-13, 5, 0, 12, 14], [11, -8, -9, -3, 5], [-6, 1, 0, 1, 8], [-22, 12, 5, 9, -2], [6, -5, 7, -3, -1], [5, 10, 11, 1, -4], [14, -4, -6, -4, 1], [11, 3, -3, 4, -4], [-21, 13, 8, 14, 2], [9, 2, -3, -17, -8], [4, 4, 1, -4, -7], [13, -14, -5, -8, 6], [-20, 11, -4, 6, 1], [-5, -7, -4, 0, 9], [7, -1, -3, 2, 7], [1, -4, -1, 5, 10], [7, -1, -3, -10, -10], [-9, -6, -6, 6, 10], [-11, -9, -3, 0, -6], [-30, 15, 8, 6, 6], [-12, -2, 2, 0, -12], [-18, -15, -7, 2, 7], [-11, -4, 4, 4, -6], [20, -6, 0, -6, -15], [25, 18, -1, -4, -5], [-17, -1, 5, 3, -6], [16, -17, 1, -11, -6], [-8, 9, -1, 6, 10], [12, -1, -8, 12, 0], [-21, 3, -8, -7, -4], [9, -6, -1, -9, 7], [9, -3, -8, 2, 3], [1, -4, -4, 10, 11], [-12, -1, 1, 5, 0], [-18, 24, 8, 12, -8], [10, -4, -8, -5, 2], [-2, 0, -5, -3, 8], [-30, 11, 12, 2, 1], [-28, 17, 13, 3, -2], [-2, 3, -4, 15, -3], [-7, -4, -3, -3, -10], [-20, 14, 3, -3, 7], [-18, 4, 6, 9, 10], [-2, -1, 3, -6, -2], [-9, 13, 2, 0, 5], [-29, 11, 3, 6, -1], [1, -7, -11, -1, 10], [-3, -5, 14, 11, 7], [-21, -5, -7, -3, 9], [-9, 8, -5, 4, 18], [3, 8, 1, -11, -9], [-13, -5, 3, -6, -6], [11, 0, 12, 8, 2], [37, -16, -12, -8, -1], [-11, 6, -12, -5, 1], [17, 6, -2, 7, 4], [4, -13, -1, 3, 0], [-4, -3, 3, -5, 10], [20, 4, 12, -2, 3], [10, 11, 0, 2, 8], [1, 7, 1, 6, -13], [-25, 10, 4, 1, 4], [-21, 11, -1, 6, 14], [2, -4, -4, 12, 13], [13, -7, -4, -21, -2], [-11, -2, -7, -12, -6], [-8, -14, 2, 2, -9], [-23, -1, 2, 1, 2], [6, -1, -4, 11, -7], [-6, -7, 0, -6, -2], [-8, -8, -12, -14, 7], [-30, 0, 4, 21, 11], [-19, 5, 4, 12, 0], [13, -12, 11, 3, -8], [-21, 10, 11, 1, 6], [-5, 2, -7, 1, -2], [10, 4, 4, -8, -3], [3, 2, 12, 7, 13], [28, -14, 3, 3, -15], [-15, 8, 6, 4, 4], [34, -12, -3, -13, -8], [20, 5, 5, -17, -22], [2, 6, -16, -10, -4], [-3, 3, -6, 0, -11], [4, 6, 7, -10, -13], [7, -5, -3, -23, -2], [1, 6, 6, 11, -5], [-17, -15, -9, -2, 0], [6, -4, -10, -24, 0], [25, -8, 8, 1, -6], [14, 14, 0, -7, -16], [5, 7, 0, -17, -1], [23, -1, 0, 3, 4], [-3, -1, -10, -13, -11], [-24, 7, -5, 6, 4], [23, -17, -8, -5, -1], [8, 21, 12, 11, -11], [5, -9, 4, 5, -11], [24, 0, 2, -1, -9], [40, 7, 6, 1, -9], [12, 5, 5, 16, 1], [-18, 17, 7, 3, 0], [18, -11, -12, -15, 6], [-14, -9, -11, 0, 3], [-1, -2, 21, 8, 8], [-1, -5, -3, 19, 4], [-11, 4, -1, 5, 7], [4, 9, 23, 6, 3], [-31, -6, 3, -8, 4], [1, 16, 13, -6, -4], [7, -11, -11, -10, 10], [-39, 19, 7, 19, 4], [5, 7, 13, -10, -6], [13, -1, 9, -7, -12], [-8, -10, -7, 17, 15], [4, 5, -6, 9, 7], [-9, 2, -10, -4, 13], [9, 17, 3, -6, -6], [-9, 19, 2, 2, -2], [12, -16, 14, -5, -3], [17, 12, 13, 10, -19], [-6, -10, -10, -15, 12], [-24, -7, 5, 14, 15], [19, 18, 11, -6, -19], [26, -11, 10, 3, 14], [-19, -2, -3, -3, -7], [-26, 12, -3, -3, 6], [5, -5, 16, -1, -8], [-5, -2, 17, 21, 12], [-3, 14, 3, -9, -16], [-8, -4, 16, -4, 6], [26, 12, -8, -12, -5], [6, 2, 15, 2, 9], [-37, 20, 2, 11, -3], [-23, 22, -6, 10, -11], [-3, 22, 15, -2, -25], [26, -3, -4, -11, 9], [9, 1, 0, -5, -2], [-25, 13, 21, -5, -8], [-43, 16, 4, 3, -4], [-13, 4, -11, 5, 9], [-27, 27, 0, 8, -4], [8, 21, 8, -9, 1], [-6, 6, 2, 10, 4], [-9, -1, 8, -9, -4], [-11, 28, 3, 16, -6], [-9, 12, -1, 8, 13], [12, 18, 2, -17, -4], [38, -14, -11, -5, 0], [28, -5, -2, -17, -16], [-17, -16, -4, 15, 16], [-25, 26, 22, 8, -6], [5, 2, 6, 15, 11], [-1, 4, -4, -10, -5], [22, -12, -17, 9, 5], [-47, 1, -2, 2, 7], [-4, -9, -1, -4, 14], [17, 4, 10, 16, 3], [-16, 4, 2, -8, 5], [16, -31, -8, -6, 0], [29, -29, -8, -8, 10], [-5, -15, -19, 1, 2], [19, -5, -13, -24, -13], [6, 1, 1, -8, -5], [5, 10, 0, 4, 4], [42, 6, 10, -4, -18], [13, 0, 0, -6, 1], [23, -10, -6, 13, -7], [-34, -6, -7, 5, 0], [19, -11, 4, 8, -12], [10, -22, -24, -3, 13], [11, 17, 4, -8, -17], [2, -6, -8, 11, 32], [36, -5, -4, -3, 6], [26, 25, 3, -7, -6], [5, -6, 1, -5, 12], [30, 3, 12, 0, -11], [-17, -8, 4, 13, 16], [24, -20, 7, -3, -7], [26, -1, 7, -2, -13], [-1, -7, 2, -13, -4], [22, -4, 6, 6, -7], [15, 0, -8, -23, 0], [0, 4, 6, 22, 4], [22, 3, 2, -15, 3], [-43, -2, -15, -14, 10], [4, -2, -11, 0, 1], [-29, 20, 2, 21, -9], [-6, 25, 18, -8, -12], [-25, -3, -9, -5, -5], [-10, 2, -1, 14, 10], [1, 0, 13, 25, 1], [-5, -7, 13, -8, -5], [20, -26, 2, 4, -9], [23, -23, 14, -2, 8], [20, -2, 1, -9, -5], [-3, 2, -4, -19, -13], [-29, 14, 0, 15, 15], [-14, 32, -3, 4, 6], [14, 8, 20, -3, -17], [-12, 4, 4, 11, -11], [32, -19, -19, -12, 1], [-11, 5, 3, 4, 14], [-20, 15, 21, -13, -9], [17, -13, 5, 1, 10], [4, 4, -2, -10, 8], [-2, -17, 7, -8, 7], [-33, -4, -9, 7, 24], [1, 5, -16, 0, 12], [24, -14, -16, 6, 19], [0, -6, -9, 22, 3], [-9, -26, -4, 8, 9], [12, -10, -6, -15, 4], [16, 8, 15, 9, -20], [23, -33, 7, -5, 0], [-2, 10, 11, -13, -25], [44, -4, 1, -3, -14], [-24, 39, 11, 24, -16], [4, -17, -8, -22, 10], [5, -3, 15, -9, 11], [-10, 25, 14, 3, 3], [-8, -9, -27, -6, 5], [12, -22, -27, 6, 15], [-1, 3, 18, 0, -14], [-53, 21, 2, 17, 4], [38, -9, 1, -4, 13], [-17, -12, 15, 4, 6], [-24, -19, -12, 20, 21], [48, -4, 4, -7, -15], [-17, -34, -24, 2, 25], [2, -8, -8, -12, -17], [39, 8, -4, -32, -27], [-10, -3, 1, -8, -30], [-16, -12, -13, -10, -6], [25, -26, -15, -1, -1], [-13, 5, -5, -6, 5], [-21, 26, -6, -3, -2], [-17, 31, 17, -4, 7], [-26, 11, 0, 1, -12], [-4, -22, -5, 13, 6], [32, -14, 9, 3, 14], [-9, 20, 18, -4, 8], [-2, -1, 3, -4, 7], [-37, 0, 4, 4, -9], [-9, 11, 2, 1, -18], [-16, -10, -14, -14, -2], [-23, -1, -2, -9, 7], [1, -27, 3, -13, -5], [23, 4, 19, 2, -11], [14, -12, 11, -14, -16], [21, -29, 1, -21, 0], [-49, -4, -11, -7, 11], [10, -18, -13, 5, 15], [19, -39, -2, -16, 1], [21, -10, -9, -27, 6], [29, 5, 4, 2, 2], [8, -16, 7, 3, 16], [-31, 24, 0, -3, -1], [23, -22, -23, -17, -2], [-8, -6, 2, 23, -12], [4, -13, -24, -22, 6], [-4, 7, 7, -9, 14], [-15, -3, 12, -15, -5], [-64, 24, 12, 18, 12], [-22, 0, -5, -12, -18], [7, -20, 9, 13, -8], [-48, 18, 8, 11, -14], [-12, 28, 16, 25, -8], [-27, 10, 5, -5, -13], [-39, -24, -10, 9, 15], [-46, 17, -10, 8, 12], [-50, 8, 4, 1, -3], [5, -15, -10, 11, 19], [9, 28, 25, 5, -28], [-19, 25, -5, 15, 13], [-5, -13, -4, -3, -23], [-31, -14, 5, 24, 7], [9, -6, -20, -10, -8], [-23, 15, 13, -8, -10], [18, -2, -10, -12, -24], [6, 9, 6, -9, -23], [9, -7, 8, -32, -18], [15, -17, 0, -2, 22], [-1, 0, -10, -32, -3], [-43, 1, -11, 5, 1], [-18, -4, -13, 21, 31], [-26, 7, 5, -6, -15], [-10, -9, 12, 1, 4], [69, -3, -6, 1, -1], [11, -6, -17, 6, 27], [3, 0, -7, -6, -11], [-48, 24, 8, 30, -9], [-26, 21, -8, 4, -5], [17, 12, -4, -6, -19], [1, 0, -8, 5, -11], [10, 21, 2, 11, -2], [5, -23, -3, -2, -5], [38, 3, -12, 22, 13], [-46, 26, 8, 20, 28], [-11, -37, -2, 4, 3], [-12, -18, -11, 18, -6], [13, 6, -19, 30, 29], [-6, 10, -23, -15, -5], [-11, 1, -9, 18, 19], [-5, -14, 1, 10, -9], [-12, -26, -5, -14, 0], [17, -2, -3, -14, 1], [16, 26, 7, -15, -9], [-32, 5, 25, 7, 8], [-15, 9, 5, 9, 8], [-21, 16, 0, 35, 19], [-26, 13, 9, 7, 8], [1, 8, -7, -6, -8], [-10, 25, -3, 13, 2], [4, -17, -2, 7, 14], [-2, -20, -10, -6, 10], [12, 28, 26, -6, -31], [14, 5, -20, 6, 16], [45, -34, -2, -5, -5], [27, 1, -15, -13, -11], [-19, 14, 18, 11, 10], [-23, -28, -4, -8, 3], [13, -21, 15, -1, -17], [58, -20, 2, -18, -30], [66, -32, -18, -30, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2001_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2001_2_a_h();".
// 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_2001_2_a_h(:prec:=5)
    chi := MakeCharacter_2001_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_2001_2_a_h();".
// 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_2001_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2001_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![19, 11, -13, -7, 2, 1]>,<5,R![57, 17, -28, -9, 3, 1]>],Snew);
    return Vf;
end function;