
// Make newform 1071.2.a.n in Magma, downloaded from the LMFDB on 29 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1071_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_1071_2_a_n();".
// 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_1071_2_a_n();".
// 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, 2, 8, -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], [-2, -1, 1, 0, 0], [4, 3, -6, -1, 1], [-2, -7, 5, 2, -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_1071_a();"
function MakeCharacter_1071_a()
    N := 1071;
    order := 1;
    char_gens := [596, 766, 190];
    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_1071_a_Hecke(Kf)
    return MakeCharacter_1071_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, 0, 0, -1, 0], [-1, 0, 0, 0, 0], [1, 0, 1, 0, -1], [1, 1, 0, -1, 1], [1, 0, 0, 0, 0], [1, 0, 0, -1, -2], [3, -1, -1, 0, 0], [4, -1, 0, 2, 1], [0, -1, -1, -1, 0], [-2, -2, 1, 1, 1], [7, -1, -2, -1, 1], [-3, 4, 1, 2, -1], [-2, -3, -1, 1, 2], [0, 0, 1, 1, 3], [2, 3, -3, -1, -2], [-2, -5, -1, 1, 2], [2, -5, 0, 2, 3], [2, -3, 0, 0, -3], [0, 0, -2, -2, 0], [2, -2, -1, 1, -1], [2, -4, -2, 4, 2], [0, 4, 4, 2, -4], [-2, -2, -2, 0, 2], [4, -5, 1, -1, 2], [3, -5, 2, 3, 3], [3, -1, 1, -2, -2], [-2, 0, 1, -3, -1], [11, -4, -3, -2, 1], [1, -1, -5, 0, 0], [-3, 2, 6, 3, 0], [0, 8, 1, -3, -1], [-4, 0, 4, 0, -4], [10, -2, 0, 2, 2], [-6, -1, -2, 4, 1], [5, 6, 2, -3, -4], [0, -2, 4, 4, -2], [-5, 5, 2, 1, 3], [-3, 0, 2, 1, -2], [-2, 2, -2, -4, -2], [6, -4, 2, 4, 0], [2, -2, -4, 2, 4], [-6, -6, -3, 1, 5], [-1, 2, 5, 2, 3], [-2, 5, -1, 5, 2], [-12, 4, 3, -1, -3], [-7, 3, 0, 7, 1], [-3, 5, 6, -1, -1], [6, -5, -6, 2, 1], [3, -9, -3, 2, 0], [0, 0, -3, 5, 5], [-8, 6, 6, 2, 2], [-6, 1, 3, -3, -2], [-6, -1, -1, -5, -4], [-4, 2, 0, 4, -4], [-9, 9, 0, 1, 1], [13, -4, -6, -3, 0], [4, 0, 4, -2, -8], [14, 4, 4, 2, -4], [10, -5, -7, -5, 2], [10, 0, -2, -6, -10], [0, 6, -4, -2, 0], [-8, 6, -2, -6, 0], [-10, 13, 5, -5, -2], [6, 6, 2, 2, 2], [9, -5, 5, 6, 2], [-6, 8, 3, 7, -3], [-6, -9, -2, 6, 5], [-1, 8, -2, -1, 6], [8, 3, -3, -1, -2], [10, -2, -1, -7, -3], [4, 12, -4, 0, 0], [8, 1, -8, -6, 5], [-6, 0, -2, 4, 10], [-12, 5, 3, 3, 2], [0, -6, -2, -6, 0], [-10, 1, 7, 5, -4], [17, -6, -3, 0, -1], [7, -7, 6, 7, 3], [8, -3, -4, -8, -3], [1, -1, 3, 2, 8], [4, 8, 0, -2, -8], [-3, 2, 0, 3, -2], [2, -1, 1, -9, -2], [6, 8, -5, -3, 3], [16, -1, 2, 0, -1], [3, 5, -3, -2, -4], [8, 1, -3, 3, 0], [14, -1, 0, 2, -1], [-10, 12, 0, 2, 4], [15, 1, 2, -3, 7], [8, -12, -8, 4, 0], [12, -8, -6, -2, 0], [-2, 12, -5, -3, 1], [-15, -4, -4, 5, -4], [2, -8, -6, 6, 6], [3, 1, 2, -1, 3], [4, -6, -4, -2, -4], [-8, -10, 4, 2, -2], [-2, -4, 1, -1, -1], [12, -10, -3, -7, 5], [-2, -16, 0, 2, 0], [18, 0, -8, -6, 4], [16, -6, 0, -8, 6], [19, -3, -4, 1, -3], [-2, 7, -9, -3, -8], [2, -9, -1, -1, 4], [6, 2, -4, 2, -4], [-22, 11, 7, 5, -2], [6, -3, -1, -3, 2], [7, -2, 5, 2, 1], [16, -13, 2, 8, 7], [-20, 5, 3, 7, -2], [-1, -8, -1, 2, 9], [5, 3, 9, -4, 4], [8, 2, 0, -4, 2], [-20, 2, -4, 0, -6], [-3, -11, -1, 2, 4], [6, 0, 5, 3, 1], [-17, 0, 0, 1, 4], [-8, -2, 2, -4, -4], [-25, 5, 0, 5, -3], [-5, 2, 3, 12, 3], [6, 13, 11, -7, -10], [4, 2, -2, 2, 4], [10, 0, -8, 4, 4], [-3, -10, -6, -1, 0], [-4, 12, -6, -4, -4], [-10, -6, -8, 12, 2], [-1, 14, 1, -12, 1], [-6, -11, -8, 4, 1], [-2, 2, 1, -1, -15], [-25, 1, 5, 6, 0], [-6, 14, 8, -4, -2], [-1, 5, -6, 3, 7], [10, 7, -1, -5, -8], [-8, 7, 7, -1, 4], [-22, 6, 6, -6, -4], [21, 8, -5, -6, -5], [8, 0, 2, -14, -10], [-27, -1, -3, 8, -4], [-12, 0, -3, -3, -1], [-13, 14, 7, -4, 3], [14, 8, 0, 0, -4], [1, -7, -8, -3, 9], [-28, 9, 1, 1, 6], [10, -8, -8, 0, 4], [-24, 6, 0, -2, -8], [-12, 20, -1, -9, 3], [-14, 8, 13, 1, -5], [-2, 6, -2, 10, 16], [21, -19, -5, -4, 2], [-11, -11, 6, -1, -9], [-20, 2, 6, 2, 0], [-3, 0, 1, 12, -1], [7, -4, -7, -12, -5], [1, 3, 4, 7, -1], [-10, 3, 6, -2, 5], [14, -10, -8, 0, 6], [-34, 11, 4, 8, -1], [14, 8, 7, -3, -7], [11, 13, 5, -12, -4], [8, 4, 6, -6, 2], [0, -8, 6, -2, -2], [-3, 9, -4, 1, -3], [-14, -18, -2, 0, 8], [4, 4, -8, 4, 10], [-18, 0, 6, -2, -2], [-26, 6, 4, 0, 2], [8, 10, 3, -1, -7], [11, -4, -6, -9, -10], [-21, 3, -3, 2, 6], [7, -14, 1, 2, 1], [13, -5, 6, 5, 7], [-3, -2, -4, 15, 6], [6, 2, 7, -3, 7], [-1, -3, 1, -6, -2], [7, -14, -4, -3, 0], [5, 9, 12, -1, -7], [5, 10, 9, 2, -7], [-4, 10, -10, -14, -4], [0, 2, -7, -17, -11], [12, -4, -12, 2, -4], [0, 2, -4, -8, 4], [12, -2, 2, -8, -8], [14, -8, 0, -12, 0], [-1, -11, -2, -1, -7], [-20, 12, -3, -5, 3], [8, 15, -4, 4, 7], [-16, 3, 11, 7, 6], [2, -7, 8, 8, -7], [19, -16, -9, -2, 5], [-14, -11, 1, 5, -4], [21, -5, 11, 14, 4], [18, -5, -6, -10, 9], [2, -10, -14, -2, 4], [2, -12, 2, -2, -6], [10, 6, -2, -10, -4], [-14, -4, 0, -6, -16], [-5, -7, 5, 10, -6], [0, -6, -2, 10, 0], [15, -4, -2, 7, -2], [11, 2, 2, -17, -20], [-18, 13, 5, 3, -4], [4, -13, -5, 21, 12], [16, 18, -2, -8, -18], [-28, 4, 8, 0, -10], [23, -7, 3, 4, 0], [-4, 3, -8, 10, 9], [-8, 2, 15, 5, 3], [-26, -15, -3, 5, 8], [-4, -2, 7, -13, -1], [-2, -28, 4, 6, 4], [-11, 8, -6, -5, 2], [-8, 0, 0, 4, 6], [-9, -5, -1, 2, 8], [44, -2, 6, 0, 8], [5, 7, 3, -6, -16], [6, 6, 1, 3, 1], [13, -2, -9, -2, -1], [8, 4, -2, 2, -10], [20, 4, 4, 4, -4], [-2, -12, -1, 13, 13], [-14, 8, 10, 0, -6], [20, 3, -6, 4, 5], [-7, -3, -2, 9, 7], [31, -15, -3, 6, 8], [11, -17, -14, -5, 7], [-2, -23, 0, 0, 9], [9, 1, 9, 4, 12], [10, 27, -1, -3, -8], [-14, 7, -2, -8, 5], [18, -8, -5, -7, 1], [-12, 26, 8, -2, 0], [-6, -15, -7, 15, 8], [-24, 9, -3, 1, 2], [-21, 13, -1, -4, 12], [-31, 6, 5, 0, 1], [-10, -14, -6, 8, 0], [19, 3, -7, -10, 6], [-8, 14, -8, -6, 0], [5, -3, 10, 13, 1], [3, 16, 5, -14, -5], [-15, 28, 6, -3, -2], [19, 16, -12, -5, -8], [-30, 0, -8, 2, -4], [17, -10, 0, -3, -14], [-15, 12, -3, -6, -5], [16, -8, 0, 0, -6], [1, 28, 3, -10, 1], [30, -10, -8, 12, 6], [22, 0, -9, -7, 3], [-12, -2, 0, -12, -12], [22, -16, 0, 8, 4], [-16, -12, -7, 5, 5], [-10, 14, -2, -2, -8], [-22, -7, -11, -1, 4], [-11, -8, 0, 3, 4], [-14, -15, -6, 10, -5], [4, 2, -10, 10, 4], [38, 0, -18, -4, 4], [-30, -1, 5, 11, 12], [48, -20, -10, -6, 0], [11, 11, 6, 3, 1], [1, 7, 4, -1, -3], [2, 12, 5, -9, -13], [34, 7, -1, -1, 4], [-6, 18, -2, 8, -8], [-14, 6, -13, -11, -13], [-28, 11, 5, -3, -12], [11, -15, 1, -4, -4], [-15, 4, 10, 17, 2], [30, -5, -4, -10, -1], [38, 4, -8, 4, 0], [13, 3, 6, 5, -5], [-36, -14, 10, -2, -2], [-15, 22, 7, 2, -1], [1, -14, -1, 6, -5], [24, 4, 10, -2, 2], [0, -29, 3, 7, 16], [-32, 18, 14, 10, 8], [-24, 16, 6, -2, 2], [-18, 6, -4, 4, 2], [20, -15, -3, 5, 4], [22, 6, 0, -2, -4], [8, 6, -9, -19, -5], [-2, -4, 8, -12, -2], [-19, -2, -6, -3, 10], [6, -12, -2, 2, -14], [-35, 13, 5, 0, -6], [-1, 7, -4, 3, -13], [-12, 2, 12, -4, -2], [-15, 17, -2, 3, -5], [14, 8, -11, -7, -9], [-3, 11, -6, -15, 5], [-8, 4, 8, 14, 18], [-22, 0, 10, 2, -14], [15, 1, -8, 11, 17], [-12, 15, 4, -14, -7], [-1, -19, 10, 3, 15], [18, -14, 2, -16, 4], [52, 2, -4, 8, 6], [7, -9, 7, 0, 8], [-36, -12, -2, 6, -10], [-32, 6, 0, -2, -14], [-10, 8, 11, 13, -5], [6, -27, 5, 7, 10], [0, -4, 4, 12, 14], [48, 5, 6, -4, -5], [-8, 2, 12, 12, -10], [10, -28, -4, -4, 6], [-13, 18, 13, -10, -5], [33, 19, -11, -2, -6], [2, -8, 10, 8, 4], [-11, 20, -4, -1, 8], [26, -10, 12, 10, -2], [37, 15, 1, 0, -6], [22, -35, 3, 5, 16], [-4, -4, 7, 11, 9], [8, -4, -2, -2, 2], [0, -6, -8, -6, -6], [-16, 13, 7, 1, -2], [-33, -14, 3, -2, -1], [-24, 1, 12, 6, -5], [-15, 6, 2, 1, -12], [30, 6, -4, 6, -6], [14, -7, 9, 19, 4], [30, -11, 8, -4, 13], [17, 0, 1, 20, 13], [-40, 6, -4, 2, -6], [-29, 27, -7, -18, -4], [-38, 19, 3, 9, -12], [44, -22, -6, -14, 0], [18, 10, -3, 3, -3], [-21, -1, -4, 3, 11], [-21, -3, -7, 10, 16], [-7, 29, 0, -9, -15], [-18, 9, 15, -7, -8], [-41, 18, 2, 7, -2], [1, -12, -9, -2, 5], [-24, 22, -3, 9, 1], [13, -3, -12, -17, -19], [-11, -11, -5, -8, 2], [-6, 2, -6, -20, -16], [-9, 23, -9, 0, -16], [-4, -4, -2, -18, 4], [-23, -3, 9, 2, -8], [27, -29, -4, 15, 7], [2, 18, -10, -14, -8], [35, -2, -12, 13, 8], [4, -14, -3, 7, 13], [0, -2, 3, -1, -5], [22, 10, 10, -16, -16], [-33, 2, 3, 0, 7], [-14, 17, -9, -5, 10], [44, -29, -9, 7, 4], [20, 11, -12, -8, -5], [-5, 18, -1, 0, 5], [68, -12, -10, -6, 4], [0, -22, -16, 18, 2], [-17, -7, 12, -23, -15], [54, -10, -5, 1, -7], [-8, 8, 14, 4, -2], [7, 5, 20, 5, -5], [21, 24, -11, -8, 3], [7, 20, 8, -7, -16], [4, 0, 10, -10, -10], [-2, 12, -10, 10, 10], [38, 4, 7, -15, -5], [-10, -6, -4, -2, 14], [3, 12, 11, 12, -15], [-22, 32, -1, 1, -3], [-32, 27, 0, -26, -7], [-18, -2, 0, -4, -6], [-35, 12, 18, -9, -10], [32, 8, -18, -8, 14], [21, -10, -2, -15, -12], [8, 5, 7, -3, 2], [19, -4, 10, 7, -18], [-11, 7, 22, 5, -3], [16, -14, -1, 7, 3], [42, 10, -9, -7, 11], [-23, -15, -1, -2, 0], [38, -21, -15, -13, 2], [-17, 8, 10, -7, -14], [-37, -7, -7, 20, 8], [-10, 20, 4, 4, -20], [-12, 28, -1, -5, 1], [24, -14, -6, -6, 12], [14, -2, -5, -5, -17], [-13, 16, -2, -21, -2], [18, 8, 17, -5, -11], [13, -2, 4, -1, 0], [18, 10, -3, -13, 7], [20, 6, -11, -5, -7], [8, 0, 4, -10, -10], [2, 11, 15, 7, 8], [-50, 2, 3, 9, 3], [-62, 6, 13, -5, -3], [-8, 3, -1, -5, -6], [-22, -26, -3, 15, 9], [25, -2, -5, -4, 15], [-30, 14, -14, -2, -12], [-4, 5, -7, 7, 16], [40, -10, -14, -6, -4], [16, -33, 2, 20, 1], [32, 0, -15, -7, 7], [-1, -7, -2, -13, -15], [15, 2, 16, 3, 14], [-38, -1, 11, -3, -14], [-8, -30, 4, -8, 2], [-34, 8, -4, -14, -10], [8, 2, 0, -16, -10], [17, -26, -1, 2, 7], [10, -19, -6, -2, -17], [57, 6, 3, -2, -5], [-4, -42, 4, 12, 2], [37, 12, -4, -13, -16], [-40, 29, 3, -1, 6], [-38, 6, 2, 20, -12], [-18, -15, 15, 13, 4], [37, -9, -23, -8, -10], [-18, 11, -17, -11, 0], [-52, -15, -9, -5, 8], [-38, -16, 10, 2, 6], [42, 15, 6, 2, -15], [-7, -18, 5, 16, -13], [27, -5, 4, 5, 5], [-31, -11, -8, 5, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1071_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1071_2_a_n();".
// 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_1071_2_a_n(:prec:=5)
    chi := MakeCharacter_1071_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_1071_2_a_n();".
// 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_1071_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1071_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 2, 8, -5, -2, 1]>,<11,R![-36, -56, 71, -5, -7, 1]>],Snew);
    return Vf;
end function;