
// Make newform 1143.2.a.h in Magma, downloaded from the LMFDB on 08 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1143_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_1143_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_1143_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 := [-2, 5, 3, -5, -1, 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, 0, 1, 0, 0], [1, -3, -1, 1, 0], [2, 2, -4, -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_1143_a();"
function MakeCharacter_1143_a()
    N := 1143;
    order := 1;
    char_gens := [128, 892];
    v := [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_1143_a_Hecke(Kf)
    return MakeCharacter_1143_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, 0], [0, 0, 0, 0, 0], [1, -1, 1, 0, 1], [0, 0, 0, 0, 2], [3, -1, 0, -1, 0], [0, 0, 1, -1, -1], [1, 0, -1, -1, 1], [-3, 0, 1, -3, -1], [1, 1, -1, -2, -1], [3, 3, -1, 0, 1], [0, -1, -2, 3, 2], [0, 3, -2, 1, -2], [-1, 3, 2, -1, -4], [0, -2, -2, 0, 0], [3, 3, -4, 3, 0], [5, -3, 1, 4, 1], [5, 1, -3, 2, -1], [0, 1, -2, -1, 2], [0, 4, -2, 2, -4], [1, 2, -1, 3, 5], [4, 0, 3, 5, -3], [-7, 5, 0, -1, -2], [-3, 3, 1, -6, -1], [1, -5, -1, 0, -3], [4, 0, -4, -6, 2], [1, 3, -5, 0, 1], [3, 3, -4, 5, 2], [5, -6, 7, 3, 1], [-2, -2, 2, 2, 4], [3, -5, 2, 3, 0], [-1, 0, 0, 0, 0], [0, 0, 4, -4, -4], [-3, 1, -1, -2, -5], [-2, 2, -4, -4, -4], [3, 6, 1, -1, -9], [-12, 6, -2, -6, -6], [5, 1, 0, -3, -4], [-5, -5, 1, -6, -1], [-6, 4, 0, 2, 0], [2, 2, 4, 8, -2], [9, -6, -1, -1, 1], [4, 0, 2, 8, 2], [-4, 4, 4, 0, 0], [6, -2, 6, -4, 0], [-5, -1, -2, -9, -4], [-6, 2, 1, 3, 9], [-8, 4, -3, 5, 1], [-4, -6, 2, 6, -6], [-4, 0, 4, 4, -4], [8, -2, 6, -2, -2], [5, 5, -9, 4, -7], [-5, 1, -5, 0, 7], [10, -4, 2, 2, -4], [7, -5, -2, 5, 10], [-1, -7, -3, -8, 9], [1, 0, -1, 1, 5], [3, -8, 5, -3, -1], [5, 1, -1, 6, -3], [6, 8, -4, -4, -2], [-8, 8, 2, 6, 4], [-6, -10, 2, -4, -6], [-4, -4, 6, 6, 6], [-12, 14, 4, -6, -8], [9, 13, -7, 6, 1], [8, 0, -2, -2, 10], [2, 4, 6, -4, -2], [-4, -6, 2, 6, -8], [8, 10, 0, 0, -8], [-13, -3, 1, -8, -3], [14, -2, 8, 4, -6], [7, -6, 5, 3, -5], [-12, -6, -2, 4, 0], [12, -9, 10, 3, 6], [12, -12, 4, 6, 4], [-16, -4, 8, -4, 4], [-16, 4, 0, -4, 4], [-1, -9, -2, 7, 0], [4, -5, 10, -3, 6], [9, -1, 5, -4, 7], [12, 6, 2, 12, -4], [9, 4, 3, 9, -7], [-2, -12, 12, 0, 14], [7, -9, 4, -1, 0], [17, -16, -7, 7, 7], [-2, 14, -4, -4, -6], [9, 2, 7, -1, -3], [3, 3, 4, 1, 2], [-8, 8, 3, 5, -7], [-2, 10, 2, -12, 2], [4, -7, 10, 1, 10], [9, -7, -1, 0, -5], [3, -13, 0, -1, 8], [-10, -12, 4, 4, 0], [4, -2, 6, 0, 0], [0, 0, -6, 4, 12], [-3, -3, -4, -7, -8], [0, -6, 2, 8, 2], [25, 8, -9, 3, -7], [-4, -8, 1, -7, 1], [2, 0, 4, -4, 4], [2, -6, 2, -2, -6], [2, 0, 0, 0, 12], [10, -6, 8, -10, -18], [13, -7, -10, 1, 8], [-16, 12, -2, -6, 0], [-8, 3, -2, 5, -6], [-3, -10, -5, -1, -7], [-1, 7, -5, -12, 7], [-9, -11, 13, 2, 1], [6, 8, 2, 4, 2], [6, 15, -4, 5, -4], [16, -12, -6, 4, 4], [7, 13, -7, 8, -11], [-8, -4, -2, 8, -4], [-8, -2, 6, -4, -10], [22, 8, -4, 4, -4], [7, -1, 8, 5, 6], [-7, -3, 19, -6, 7], [21, -3, 2, 13, 4], [3, -9, -7, -2, -3], [6, 10, -9, -1, -11], [-12, 9, -2, -5, -6], [20, 2, -6, 4, -6], [8, -10, -6, -8, 2], [6, 0, -10, 0, 0], [3, -9, -3, -4, -17], [-1, -9, 0, -17, 8], [5, 2, -1, -5, 17], [20, 20, -6, 6, -4], [5, 7, -3, -10, -1], [-10, 2, 7, 5, 3], [5, -7, 5, 0, 15], [0, 0, 4, 22, -4], [-2, 3, 12, -5, 0], [-3, -5, -1, -6, -3], [2, 14, -18, 0, -10], [3, 0, 1, -11, 7], [2, -14, -1, -3, 11], [-8, 6, 10, -8, -2], [-9, 2, -3, -17, 3], [-22, -6, 15, 1, 3], [-9, 1, -13, -2, 3], [-7, -16, 1, 1, 19], [-7, 5, 11, 16, 3], [-20, -12, 12, -18, -6], [-11, 9, 3, 6, -17], [14, -2, 0, 16, 16], [3, -13, 7, 8, -1], [-16, 22, -2, -8, -8], [7, 3, -13, 18, 3], [1, 4, -19, 3, -9], [5, 3, 15, -6, -15], [2, -27, 8, 7, 0], [17, 5, 5, 6, -9], [-12, -4, 1, 5, 9], [-10, 2, 2, 4, 6], [10, -7, -8, -1, -8], [-14, 6, -4, -4, 2], [2, -6, -4, -6, -4], [-9, -1, -4, 1, 2], [-18, 12, 0, -14, -4], [9, -7, -4, -5, -18], [-8, 4, 6, -14, -4], [17, -3, 6, 23, 14], [15, -1, -16, 1, -6], [-11, 1, -12, 17, 12], [-24, -6, -6, -2, -4], [16, 4, 4, 4, 14], [-14, -7, 0, 5, -20], [-9, 19, -16, -3, -2], [5, -13, -1, 8, 3], [-20, -6, 8, 6, 2], [7, -2, 5, -7, -1], [0, -20, 15, 5, -3], [16, -8, 0, -12, -16], [-21, 3, -3, -14, 3], [-16, -11, 14, 5, 2], [-4, -26, 14, 6, -10], [-17, -4, -5, -1, -11], [-20, 12, -4, 12, 8], [8, -16, 1, 9, 17], [-3, -17, 15, 8, -3], [-16, -4, 4, 14, 2], [49, -2, -5, 5, 1], [-27, -14, 3, -1, 21], [15, 1, -11, 2, -11], [10, -24, -2, 10, 8], [2, -2, -3, -17, -7], [11, -24, -1, 9, 9], [-14, -14, 14, 0, 10], [38, -2, 2, 12, 6], [26, 0, -16, 10, 2], [-20, -14, 6, -12, 2], [17, 5, -18, 13, 4], [3, -1, -16, -5, 4], [33, -3, 4, -5, -14], [-16, 4, -2, 0, 8], [-2, 1, 12, -19, 8], [23, -10, 9, 19, 23], [-1, 3, 1, -6, -3], [13, -23, 5, 2, 17], [-14, -6, -9, -3, 7], [-5, 1, 3, 2, 5], [10, -20, 10, -4, 0], [-3, 7, -7, -2, 19], [16, 20, -10, 6, 4], [2, 18, -3, -13, 9], [0, 12, -8, -8, -16], [19, 0, 5, -3, -21], [-3, -12, -11, 17, 15], [20, 8, -28, 6, -8], [-19, 14, -5, -7, -19], [30, 8, -4, 16, 6], [9, -6, 15, -13, -15], [-31, 20, 7, -23, -15], [-18, -10, 0, 4, -8], [8, 0, 10, 14, -2], [-11, -3, -7, 14, -17], [8, 8, -20, 0, -16], [-17, -2, -3, 7, -9], [4, 2, -6, -6, -18], [9, -7, 15, 14, 17], [5, -25, 15, 12, 11], [20, 8, -24, -4, -12], [-13, -1, 0, 7, 8], [14, 10, -17, -5, 9], [7, 11, 2, -21, 8], [-17, -14, 17, 5, -9], [9, -4, 5, -15, -25], [9, -1, -13, 4, -1], [14, -14, 2, 18, 10], [6, -18, 1, 23, 1], [23, 11, 4, 9, 6], [29, 12, -9, 3, 9], [12, 6, 4, -2, -22], [7, -13, 3, 2, -7], [22, 12, -2, 2, 10], [-36, 8, -6, -2, -12], [16, -10, 8, -4, 22], [-17, 5, -13, -2, -3], [2, -4, 16, 14, -8], [2, 0, 6, 16, 14], [-19, 5, 3, -30, -7], [-2, -18, 7, -5, 17], [9, -7, 1, 4, 5], [9, -3, 14, -9, 22], [24, -2, -4, 4, 14], [-8, 4, -20, 24, 12], [-20, 32, 0, -18, -14], [-12, 2, -12, -8, 16], [-11, 7, 5, -22, 3], [-2, 10, 6, 12, -12], [6, -14, 4, -14, 0], [6, -22, 4, 6, -8], [-7, -5, -13, 6, -1], [-10, 12, 20, -6, -12], [22, -26, 20, 8, -8], [33, 9, -1, -2, 9], [5, 6, -13, 9, 5], [-24, 2, -18, 10, 14], [2, -6, 6, 14, 14], [15, 39, -14, -3, 6], [-25, 8, -9, -7, -11], [-18, 22, -17, -9, 9], [17, 17, 3, -4, -9], [8, -6, -4, 4, -18], [-9, -23, 21, 2, 9], [-9, 6, 17, -9, -9], [-23, 20, -7, -11, 3], [-26, 12, 16, 4, 8], [-16, -8, 12, 6, -12], [-4, -26, 4, 12, -10], [28, -6, -14, 18, 6], [22, 14, 0, -10, 0], [40, 18, -10, 6, -8], [20, 4, 18, 0, -12], [-1, -1, -2, -7, 14], [-3, 5, -8, -11, -16], [4, -26, 16, 6, 14], [-23, 17, -23, -6, -3], [15, -13, -5, 10, -7], [-30, 2, -10, 2, -4], [5, 22, -5, 3, 13], [22, 12, -14, -8, 10], [-8, -16, 32, -8, -2], [44, -6, -2, 20, 12], [8, 4, 12, -20, 6], [-30, -28, 12, -20, 4], [12, 16, -8, -14, 8], [14, -16, 10, -24, -20], [28, -16, 16, 2, 2], [19, 0, -19, 27, 11], [7, 6, 9, -5, -25], [-5, 4, 13, 19, 11], [-34, 2, 4, -10, -2], [17, 2, -21, 25, 5], [6, 0, 12, -24, 10], [5, 0, 11, 19, -3], [17, 21, -32, 1, -4], [20, 25, -14, 21, 6], [-24, -9, 22, -11, 2], [3, 25, -1, -6, -3], [6, -21, 0, 5, -16], [-1, 14, 5, 21, -1], [-28, -14, 2, -14, -2], [-2, 34, -4, -14, -8], [-15, 8, -17, -27, -11], [-42, 4, -8, 12, 8], [31, 27, -13, 10, -11], [7, 16, -13, -11, -19], [-16, 4, -16, -4, 4], [19, 23, -4, -7, -14], [-4, 6, -10, -4, -6], [8, -20, -8, 8, 2], [-1, -5, 1, 28, -7], [-12, 24, -22, 6, 2], [-35, 15, -11, 0, 5], [-9, -4, -15, -13, -1], [-7, -9, -7, 2, 27], [-2, -2, 2, 12, 12], [-10, -21, 12, -23, 8], [6, 16, -16, 16, -18], [1, -21, -7, -12, 21], [-2, 2, 6, 18, -14], [20, 34, -14, 16, -18], [-18, -17, 16, 3, -8], [-52, -4, -6, -10, -2], [5, -19, -12, -17, 18], [33, -5, 13, -8, -9], [-31, -13, 13, -28, 5], [38, -27, -4, 21, 24], [-3, -13, 7, -4, 17], [36, -6, -18, -12, 0], [38, 18, -30, 8, -10], [-12, 2, -14, -34, 0], [6, -22, 18, 10, 16], [18, 6, 4, -2, -34], [-26, 0, 32, -12, 6], [0, 0, -17, -15, -3], [-2, 18, -31, 7, -11], [-10, 12, -4, 0, -6], [-11, 21, -4, -1, 2], [-47, 5, -7, -30, -3], [6, -18, 5, -1, -15], [-35, -27, 28, -13, 6], [-31, 21, 1, 16, 1], [8, 0, -16, 14, 4], [-33, 7, 16, 5, -10], [-25, -9, 9, -8, 5], [-28, 14, -2, -16, -14], [-20, 0, -14, 8, -2], [-34, -10, 10, 0, 16], [-15, 9, 19, -4, 13], [18, -10, -4, 4, 22], [-14, 2, -5, 11, 17], [-40, 18, -6, -8, -18], [-31, -7, 14, -1, -2], [43, -13, -7, -4, 3], [-11, 4, -21, 19, 9], [14, 22, 2, 18, -2], [18, 26, -26, 20, -6], [49, -19, 3, 8, 11], [-17, 7, 6, -27, -30], [17, -21, 35, 0, 15], [-6, 14, 14, -6, 14], [31, 13, -13, 14, -11], [11, -38, -3, 15, 19], [-38, 2, -2, 18, -8], [52, 2, 14, -4, 0], [12, 6, -8, -4, -12], [-2, 10, -6, 0, -12], [7, 21, -5, 20, -3], [-29, 11, 2, -13, -28], [2, 1, 8, 31, -12], [-28, -8, 1, 3, 23], [7, -21, -5, 10, 21], [27, -1, 11, 36, -7], [4, 34, -10, -10, -24], [-15, 1, -18, -3, 16], [-6, 14, -24, -12, 6], [-41, 3, 8, -15, -2], [-4, 26, 16, 6, -18], [-11, 25, -28, -5, 2], [8, -26, 26, 8, 6], [26, -16, 0, -2, 12], [-11, 15, 1, 2, -9], [26, 6, 1, 3, 5], [28, -10, 6, -20, 4], [-24, 25, -10, 5, 14], [23, -21, 7, 0, -23], [-30, 45, -4, -11, -4], [-9, -14, -3, -25, 15], [-29, 2, -3, -23, -17], [-12, 28, -10, -14, 12], [49, 4, 3, 17, 1], [6, 28, 0, 18, 10], [-34, 7, 8, 5, -12], [31, 7, -2, 23, 28], [12, 27, 6, 11, -22], [11, -30, -11, -13, 31], [-64, 11, 10, 13, -2], [-42, -20, 4, -20, 4], [-20, 32, -7, 13, 21], [51, -9, 9, 34, 23], [-61, 23, 12, -11, -14], [-8, -11, 6, -3, -22], [-24, 20, 10, -8, 12], [-2, -16, 26, -4, 34], [24, 35, -6, -5, -2], [9, -20, -1, -27, 9], [16, 6, 22, -8, -12], [2, 6, -18, -4, 14], [-1, -1, 4, 15, 0], [-6, -18, -8, -6, -26], [-38, 44, 4, -14, -26], [23, -21, 19, 2, 15], [-7, 23, -5, 12, -3], [-10, 8, 10, 18, -20], [-25, -5, -28, 21, 10], [67, 3, -15, 14, 1], [-4, 2, -18, 16, -26], [24, 6, -24, -4, -10], [1, 19, -29, 8, 11], [3, 22, -15, 1, -49], [-17, 16, -9, -15, -35], [-5, -25, 28, 13, 2], [41, 5, 20, -19, -12], [-21, -11, 11, 16, 19], [33, -31, 24, 7, 2], [-17, -35, 21, 10, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1143_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1143_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_1143_2_a_h(:prec:=5)
    chi := MakeCharacter_1143_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_1143_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_1143_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1143_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 5, 3, -5, -1, 1]>,<5,R![-16, 0, 24, -2, -5, 1]>,<7,R![-64, 80, 8, -24, 0, 1]>],Snew);
    return Vf;
end function;