
// Make newform 1071.2.i.g in Magma, downloaded from the LMFDB on 26 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1071_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1071_i_Hecke();"
// 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_i_g();".
// 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_i_g();".
// 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 := [243, -324, 54, 90, -24, -12, -8, 10, 2, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [324, -54, -90, 24, 12, 8, -10, -2, 4, -1], [810, -594, -189, 219, 39, -26, -38, 11, 17, -5], [1863, -1701, -342, 552, 69, -71, -89, 38, 32, -17], [2916, -1728, -765, 483, 183, -22, -130, 19, 43, -13], [-567, 720, 78, -237, -13, 33, 32, -18, -11, 7], [-414, 330, 84, -103, -17, 12, 20, -6, -7, 3], [459, -423, -87, 132, 16, -15, -23, 9, 8, -4], [-6642, 4995, 1575, -1569, -306, 158, 317, -89, -107, 44]];
    Rf_basisdens := [1, 1, 81, 162, 162, 162, 54, 18, 18, 162];
    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_i();"
function MakeCharacter_1071_i()
    N := 1071;
    order := 3;
    char_gens := [596, 766, 190];
    v := [3, 2, 3];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1071_i_Hecke();"
function MakeCharacter_1071_i_Hecke( : Kf := false)
    N := 1071;
    order := 3;
    char_gens := [596, 766, 190];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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]];
    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, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, 0, 1, 0, 1, 0, -1, 0], [0, -1, 0, 1, 0, -1, 1, 1, 1, 0], [-2, 0, 0, 1, 2, 1, 0, 1, 0, 0], [1, 2, 0, 0, -1, 2, 0, -1, 1, 0], [1, 0, 0, 0, -1, 0, 0, 0, 0, 0], [1, -1, -2, 0, -1, -1, 2, 2, 0, 0], [0, 0, 1, 0, -5, 0, 0, 0, 1, 0], [4, 2, -3, -1, 0, 0, 4, 0, 4, 1], [-1, 1, -1, -3, 1, 0, 1, 0, 0, 0], [-1, 4, 2, 0, -2, 1, -5, -2, 1, 1], [-1, 2, 0, -1, -1, 2, -1, -1, 0, 1], [4, -1, -5, -3, 2, -4, 5, 2, 3, 3], [-1, 2, 2, 0, 1, 1, -3, -2, 0, 0], [-1, 0, 0, 3, 1, 0, -1, 0, 1, 0], [-4, 2, -2, -1, 4, 0, 3, 0, -1, 0], [1, -2, -1, 0, -5, -1, 3, 2, 0, -1], [-3, -1, 1, -5, 3, -1, 1, -1, -2, 0], [5, 0, -3, -2, 0, 0, 1, 0, 1, 2], [-3, 0, 0, 1, 3, 2, 2, 2, -2, 0], [1, 0, 1, 0, -1, -1, 1, 2, 4, 1], [-3, 4, 4, -1, -3, 6, -7, -3, -4, 1], [-2, 3, 3, 0, 0, 2, -5, -4, -2, -1], [3, -4, 5, 7, 1, -2, 0, 1, -1, -7], [1, -5, 5, -1, -1, -3, -1, -3, -4, 0], [0, 2, -4, 0, 0, 0, -2, 0, 0, 4], [0, 0, -1, 0, 3, 0, 0, 0, -2, -1], [-2, 0, 0, 0, 2, 2, -3, 2, 3, 0], [9, -2, -2, -3, 0, 0, -3, 0, -3, 3], [-4, 7, 3, -2, -4, 8, -6, -4, -2, 2], [2, 1, -8, 0, -3, -2, 1, 4, -1, 3], [2, 4, -4, -3, -2, 3, 1, 3, 3, 0], [1, -1, 4, 2, -4, 8, -11, -4, -7, -2], [2, -8, -1, 0, 3, -2, 10, 4, 5, 2], [-1, 4, -4, -3, 1, 2, 3, 2, 1, 0], [-3, 3, -3, -1, 3, -2, 3, -2, 0, 0], [1, -2, -2, 0, 0, -1, 3, 2, 3, 3], [-4, 6, 4, -3, 0, 0, -1, 0, -1, 3], [4, -6, -7, 0, 6, -4, 10, 8, 7, 6], [-2, -2, 2, 1, 2, -4, 0, -4, -2, 0], [3, -2, 3, -1, 3, -6, 0, 3, -3, 1], [3, -5, -10, 0, 8, -3, 8, 6, 3, 7], [-5, 5, -5, -3, 5, -1, 3, -1, 2, 0], [2, 2, 5, 3, -2, 4, -4, -2, -2, -3], [6, -1, 1, -2, -6, 2, 1, 2, -2, 0], [-1, 4, -2, -5, -1, 2, -1, -1, 0, 5], [7, 6, 1, -5, -4, 8, -8, -4, -4, 5], [-2, -4, 4, 2, 2, 2, -3, 2, -1, 0], [-2, 1, 0, 0, -16, 2, -3, -4, -3, 1], [-4, 3, 10, 0, -6, 4, -7, -8, -2, -4], [12, -10, -11, 1, 5, -10, 12, 5, 7, -1], [6, 1, -1, 4, -6, 3, -2, 3, 3, 0], [-10, -1, 2, -3, 1, -2, -4, 1, -5, 3], [2, 2, -2, 0, -2, -2, 0, 4, 2, 0], [-3, 1, -1, -4, 3, -6, 2, -6, -1, 0], [11, -3, 3, -1, -11, 2, 3, 2, -6, 0], [2, -5, 2, 0, 3, -2, 7, 4, 8, 2], [6, -7, 7, 6, -6, 3, -8, 3, 1, 0], [-8, 1, -3, 0, 0, 0, 4, 0, 4, 0], [2, -1, 1, 1, -2, 1, -4, 1, 3, 0], [5, 1, 2, -2, -4, 8, -11, -4, -7, 2], [-2, 8, 8, -2, -1, 2, -4, -1, -3, 2], [3, -4, 4, -3, -3, 0, 4, 0, -8, 0], [-1, -1, 0, 0, -1, 1, 0, -2, 1, 3], [-4, 17, 4, 0, -2, 4, -21, -8, -4, 0], [-3, 8, 1, 0, -13, 3, -11, -6, -3, 2], [0, 3, -9, 0, -1, 2, 10, -1, 11, 0], [-15, -9, 9, 0, 15, 0, -4, 0, -5, 0], [10, 1, 4, -4, -4, 8, -15, -4, -11, 4], [6, -6, 6, 5, -6, 1, -4, 1, -2, 0], [-3, 3, 7, 0, 7, 3, -6, -6, -2, -3], [8, 7, -7, -2, -8, 7, 4, 7, 3, 0], [-1, 4, -3, 0, -6, 1, -5, -2, -5, 0], [-4, -6, 8, 4, 1, -2, -8, 1, -9, -4], [-3, -2, 9, 0, 5, 3, -1, -6, 0, -3], [9, 0, 0, -3, -9, 6, -2, 6, 2, 0], [2, -1, -4, 0, -13, -2, 3, 4, 4, 4], [2, -2, 4, 0, -6, -2, 4, 4, 5, -3], [12, -1, 1, -5, -12, -2, -4, -2, 3, 0], [-7, -6, -3, 5, 5, -10, 12, 5, 7, -5], [-4, -6, 9, 8, 4, -8, 1, 4, -3, -8], [9, 9, -9, -1, -9, 9, 4, 9, 5, 0], [-2, -5, -8, 1, 3, -6, 10, 3, 7, -1], [8, -5, -2, 0, -8, -8, 13, 16, 15, 1], [4, 6, -5, 0, 0, -4, -2, 8, 3, 0], [-15, 0, 1, 2, -3, 6, -5, -3, -2, -2], [-5, 11, 14, 0, -14, 5, -16, -10, -8, -12], [-1, -1, -6, -7, 2, -4, 2, 2, 0, 7], [7, -2, -19, -7, 5, -10, 20, 5, 15, 7], [1, -9, -2, 0, -4, -1, 10, 2, -2, -2], [-10, 1, -1, 1, 10, 4, -1, 4, 2, 0], [5, 12, -12, -6, -5, -4, 1, -4, 11, 0], [2, 1, -15, -7, 1, -2, 11, 1, 10, 7], [-5, 6, -2, 0, 7, 5, -11, -10, -5, 7], [-5, 0, 19, 12, -1, 2, -9, -1, -8, -12], [-1, 0, 11, 0, 15, 1, -1, -2, -3, -12], [20, 11, -11, -5, -20, -2, 6, -2, 5, 0], [0, 5, 8, 0, -13, 0, -5, 0, 3, -5], [1, 0, -16, 0, 2, -1, 1, 2, 0, 14], [9, -3, 9, 12, 1, -2, 2, 1, 1, -12], [-8, -2, 2, 0, 8, 2, 7, 2, -9, 0], [-5, 10, -10, -5, 5, -2, 1, -2, 9, 0], [2, -3, 3, 0, -10, -2, 5, 4, 2, -5], [3, -10, 10, 1, -3, 0, -6, 0, -4, 0], [-9, 3, -3, -6, 9, -9, 6, -9, -3, 0], [6, -5, -5, 2, 3, -6, 8, 3, 5, -2], [0, -1, 0, 0, -19, 0, 1, 0, 6, 6], [-13, 3, -3, -1, 13, -4, -2, -4, 5, 0], [-2, -9, 7, 4, 6, -12, 0, 6, -6, -4], [4, -7, -5, 0, 29, -4, 11, 8, 3, 0], [-10, 9, -9, 5, 10, 0, 4, 0, 5, 0], [-15, 5, 1, 7, 1, -2, 13, 1, 12, -7], [-1, -1, 1, -5, 1, 5, 1, 5, -2, 0], [0, 4, -8, -2, -5, 10, 0, -5, 5, 2], [7, -9, 9, -4, -7, 0, -4, 0, -5, 0], [11, 6, -20, -9, 4, -8, 25, 4, 21, 9], [2, 1, -1, 9, -2, -5, -2, -5, 3, 0], [3, -1, -3, 0, 0, -3, 4, 6, -6, -9], [9, 1, -7, -7, 4, -8, 9, 4, 5, 7], [-17, 2, -2, 2, 17, 2, 0, 2, 2, 0], [-2, 5, -23, -15, 2, -4, 17, 2, 15, 15], [0, 0, 0, 0, 6, 0, 0, 0, 0, 0], [25, -7, 7, 1, -25, 7, -6, 7, -1, 0], [-6, 11, -1, 0, 15, 6, -17, -12, -13, 0], [8, 10, -8, -5, 1, -2, 15, 1, 14, 5], [-2, -2, 6, 0, 10, 2, 0, -4, 0, -2], [-4, 2, -10, 0, 8, 4, -6, -8, -9, 9], [1, 3, -12, 0, 3, -6, 21, 3, 18, 0], [2, 8, -4, 0, -4, -2, -6, 4, -1, -1], [-21, -2, 2, 0, 21, 1, 7, 1, -9, 0], [0, -1, 11, 6, -2, 4, -10, -2, -8, -6], [3, -6, -7, 0, -14, -3, 9, 6, -5, -4], [5, 1, 13, 4, -3, 6, -14, -3, -11, -4], [2, -11, -10, 0, 23, -2, 13, 4, -4, 2], [1, 8, 1, -4, -1, 2, 1, -1, 2, 4], [-4, 10, -10, 4, 4, 4, 3, 4, 7, 0], [-14, -12, 12, 4, 14, -1, -13, -1, 1, 0], [-8, -7, 1, 1, 4, -8, 1, 4, -3, -1], [0, 2, -2, -12, 0, -12, 4, -12, -2, 0], [16, -8, 5, 12, 2, -4, 3, 2, 1, -12], [-11, 7, 4, 0, -1, 11, -18, -22, -8, 10], [-7, -14, 14, 1, 7, -5, 1, -5, -15, 0], [1, 9, 4, -4, 2, -4, 5, 2, 3, 4], [-15, 4, -4, 7, 15, 4, 4, 4, 0, 0], [-11, 10, 3, -4, -7, 14, -11, -7, -4, 4], [-3, -15, 6, 3, 2, -4, -14, 2, -16, -3], [7, 5, -5, -6, -7, -7, 11, -7, -6, 0], [0, 1, 6, 0, -21, 0, -1, 0, 4, -2], [8, -4, -5, 0, -19, -8, 12, 16, 13, 2], [-8, 24, 15, 0, -27, 8, -32, -16, -4, -3], [-19, -3, 17, 12, 1, -2, -6, 1, -7, -12], [-5, 10, -15, -10, 2, -4, 19, 2, 17, 10], [3, 1, 0, 0, -9, -3, 2, 6, 5, -1], [5, 17, -17, -1, -5, 2, 7, 2, 10, 0], [3, -13, -8, -7, 6, -12, 0, 6, -6, 7], [-4, -1, 6, 0, 1, 4, -3, -8, 3, 5], [-2, 6, 5, 0, 26, 2, -8, -4, -2, -3], [15, -6, 9, 6, -1, 2, -11, -1, -10, -6], [-1, -12, 12, 15, 1, -2, -1, -2, -11, 0], [2, 3, -7, 0, 16, -2, -1, 4, -1, 2], [-25, -4, 11, 10, 0, 0, -5, 0, -5, -10], [12, -6, -14, 1, 0, 0, 9, 0, 9, -1], [5, -22, -6, 0, -8, -5, 27, 10, 1, -3], [18, 6, -6, 5, -18, 1, 2, 1, 4, 0], [5, -2, 2, 5, -5, 2, -9, 2, 7, 0], [14, 5, -5, -7, -14, 1, 7, 1, -2, 0], [12, -18, 18, 2, -12, 4, -3, 4, -15, 0], [22, -4, -2, -3, -4, 8, -13, -4, -9, 3], [0, 5, 17, 0, -11, 0, -5, 0, 2, -15], [-7, 15, -15, -9, 7, -4, 9, -4, 6, 0], [29, 3, 2, 7, -4, 8, 0, -4, 4, -7], [-7, 13, -4, 0, 23, 7, -20, -14, -14, 4], [7, -7, 7, -8, -7, 9, 4, 9, -11, 0], [-7, 13, -13, -2, 7, -4, 11, -4, 2, 0], [21, -1, -33, -9, 6, -12, 35, 6, 29, 9], [-36, 13, 27, 11, -13, 26, -29, -13, -16, -11], [11, 0, 0, 7, -11, 7, -9, 7, 9, 0], [-26, 17, 7, -9, -6, 12, -11, -6, -5, 9], [1, -11, -12, 0, -21, -1, 12, 2, -9, 1], [-11, 2, 9, 0, 11, 11, -13, -22, -20, -7], [-10, 24, 12, -10, -7, 14, -12, -7, -5, 10], [-16, 8, 2, -7, -1, 2, -3, -1, -2, 7], [-8, -5, 20, 0, 13, 8, -3, -16, -3, -7], [8, 9, -9, 5, -8, 7, 9, 7, 0, 0], [14, 0, 0, -13, -14, 9, 8, 9, -8, 0], [-6, -3, 3, 13, 6, -4, -10, -4, 7, 0], [1, 26, -26, 3, -1, -2, 6, -2, 20, 0], [-5, -5, 6, 0, -1, 5, 0, -10, -3, 1], [-6, 9, -9, 1, 6, 0, 10, 0, -1, 0], [3, 9, 12, 0, -5, -3, -6, 6, 18, 0], [21, -7, 2, -2, 8, -16, 5, 8, -3, 2], [-3, -6, -2, 0, 0, 3, 3, -6, -6, 2], [-1, 14, 8, 0, 8, 1, -15, -2, 6, 0], [-4, 7, -7, -9, 4, 6, 6, 6, 1, 0], [-13, 1, -1, -6, 13, -5, 0, -5, 1, 0], [-6, -12, 12, 7, 6, 5, -9, 5, -3, 0], [4, -5, -4, 0, 23, -4, 9, 8, 13, 9], [14, -17, -7, -4, -3, 6, -20, -3, -17, 4], [7, 5, -9, 0, -7, -7, 2, 14, 11, 6], [-13, -5, 5, -4, 13, 1, 7, 1, -12, 0], [20, -3, -15, -12, 0, 0, 0, 0, 0, 12], [2, 1, -23, 0, 10, -2, 1, 4, 1, 20], [-1, 2, -2, -14, 1, -5, 5, -5, -3, 0], [-24, -6, 17, 6, 1, -2, -15, 1, -16, -6], [2, 12, -12, 8, -2, -2, 9, -2, 3, 0], [1, -13, -8, 0, -5, -1, 14, 2, 6, 12], [-1, -2, -11, 0, -10, 1, 1, -2, 6, 19], [-23, 10, 10, 6, 1, -2, 8, 1, 7, -6], [-3, -2, 2, -12, 3, -5, 8, -5, -10, 0], [6, -14, -6, 0, 12, -6, 20, 12, 19, 13], [-8, -5, 21, 6, -1, 2, -22, -1, -21, -6], [4, -14, 1, 10, 2, -4, -1, 2, -3, -10], [-7, 0, 11, 0, -14, 7, -7, -14, -13, -10], [-8, 19, -19, 9, 8, 2, 12, 2, 7, 0], [-6, 15, 0, 0, -9, 6, -21, -12, -17, -5], [-6, -5, 5, 18, 6, 7, 2, 7, -7, 0], [-5, -13, 13, 13, 5, 1, 0, 1, -13, 0], [-6, 9, 10, 0, -13, 6, -15, -12, -17, -15], [-10, 13, 13, -5, -5, 10, -15, -5, -10, 5], [-3, 13, 8, 0, 28, 3, -16, -6, 2, 0], [-18, -5, -18, 5, 8, -16, 34, 8, 26, -5], [-3, -7, 10, 0, 19, 3, 4, -6, 6, 2], [-1, -11, 6, 0, -12, 1, 10, -2, 2, -2], [22, 14, -1, -5, -3, 6, 4, -3, 7, 5], [3, 13, 3, -8, -5, 10, -8, -5, -3, 8], [-13, 5, 24, 0, -8, 13, -18, -26, -9, -7], [17, 8, -8, 8, -17, 9, 0, 9, 8, 0], [-5, 2, 20, 0, 6, 5, -7, -10, 0, -10], [9, -4, -33, 0, 21, -9, 13, 18, -2, 13], [12, 1, -1, -5, -12, 0, -8, 0, 9, 0], [24, -3, 3, 1, -24, 9, -12, 9, 9, 0], [27, -3, -22, -10, 6, -12, 21, 6, 15, 10], [-14, 3, -3, -7, 14, 2, 6, 2, -3, 0], [22, -7, 9, 11, -1, 2, -7, -1, -6, -11], [-42, -13, 13, 6, 42, 0, -8, 0, -5, 0], [6, -18, 9, 0, -13, -6, 24, 12, 12, -9], [15, -14, -17, 0, 6, -15, 29, 30, 34, 21], [-12, 4, 8, 6, -6, 12, -10, -6, -4, -6], [-18, 8, -1, 0, -4, 8, 1, -4, 5, 0], [-3, 13, -13, -1, 3, 15, 6, 15, 7, 0], [7, -10, 1, 0, 17, -7, 17, 14, 4, -11], [-1, -9, 9, 0, 1, -2, 2, -2, -11, 0], [-1, 0, -7, 0, 2, 1, -1, -2, -16, -7], [3, -10, 19, 12, 6, -12, -5, 6, -11, -12], [-9, 22, 17, 0, -27, 9, -31, -18, -1, 0], [-14, -3, -6, -8, 11, -22, 17, 11, 6, 8], [23, 7, -7, 8, -23, 6, -10, 6, 17, 0], [3, 15, -15, -7, -3, 9, 17, 9, -2, 0], [16, 12, -9, -15, -1, 2, 4, -1, 5, 15], [-4, 9, -1, -11, -2, 4, -5, -2, -3, 11], [-3, -10, -3, 0, 12, 3, 7, -6, -3, 6], [-1, -4, 4, 14, 1, -6, -5, -6, 1, 0], [-17, 9, 2, 1, -5, 10, -2, -5, 3, -1], [-27, -5, 5, -20, 27, 4, 14, 4, -19, 0], [11, -13, -7, 0, -11, -11, 24, 22, 14, -1], [22, 11, -11, 7, -22, -4, -3, -4, 14, 0], [20, -6, 6, -10, -20, -5, 3, -5, -9, 0], [-12, -22, 6, 14, 9, -18, 4, 9, -5, -14], [-10, 6, 24, 0, -17, 10, -16, -20, 2, -2], [-26, -7, 7, 5, 26, 5, 6, 5, -13, 0], [25, 8, -6, 2, 0, 0, 16, 0, 16, -2], [0, 26, -26, 0, 0, 0, 6, 0, 20, 0], [20, 10, -8, -12, 5, -10, 16, 5, 11, 12], [21, 4, -4, -13, -21, -13, -4, -13, 8, 0], [7, -14, -4, 0, 22, -7, 21, 14, 9, -1], [-26, 2, -24, -22, 0, 0, 4, 0, 4, 22], [34, 1, -11, -11, 2, -4, 5, 2, 3, 11], [28, -1, 18, 7, 0, 0, -12, 0, -12, -7], [8, 8, -8, -7, -8, -13, -2, -13, 10, 0], [-13, 19, 22, 0, -23, 13, -32, -26, -11, -7], [-14, 4, -4, 1, 14, -9, -10, -9, 14, 0], [-1, -11, 11, -15, 1, 2, 0, 2, -11, 0], [8, -2, -3, 0, 19, -8, 10, 16, 8, -5], [17, -16, 4, 0, 15, -30, 10, 15, -5, 0], [-10, 4, 31, 0, -15, 10, -14, -20, -5, -16], [6, -10, -4, 0, 4, -6, 16, 12, -5, -13], [11, -8, 8, 13, -11, -1, -10, -1, 2, 0], [-6, -1, 1, 0, 21, 6, -5, -12, -19, -8], [2, 8, 10, 0, 37, -2, -6, 4, -1, -15], [0, 6, -6, -16, 0, 2, -3, 2, 9, 0], [-17, -18, 18, 4, 17, 4, 6, 4, -24, 0], [-12, -5, 11, 13, -5, 10, -13, -5, -8, -13], [-12, 14, 20, 16, -8, 16, -6, -8, 2, -16], [-5, 10, 21, 0, 19, 5, -15, -10, 9, -2], [-7, 7, 22, 0, -5, 7, -14, -14, -2, -10], [9, 19, -19, -16, -9, -10, 7, -10, 12, 0], [21, -22, -18, 0, 6, -12, 8, 6, 2, 0], [25, 7, -7, -4, -25, 14, 3, 14, 4, 0], [-13, 4, 29, 17, -12, 24, -32, -12, -20, -17], [-27, 1, -1, -8, 27, -5, 8, -5, -7, 0], [-20, -9, 9, 9, 20, 10, 0, 10, -9, 0], [-3, 0, 18, 0, 42, 3, -3, -6, 3, -9], [-3, -3, 2, -1, 7, -14, 8, 7, 1, 1], [-23, 22, 31, 1, -7, 14, -22, -7, -15, -1], [12, -1, 1, 14, -12, -4, -5, -4, 4, 0], [-7, 21, 16, 0, -15, 7, -28, -14, -10, -12], [13, 17, 12, -5, -16, 32, -32, -16, -16, 5], [-1, 11, -15, 0, 0, 1, -12, -2, -8, 9], [-16, -10, 0, -14, 10, -20, -4, 10, -14, 14], [5, 1, 10, 0, 4, -5, 4, 10, 10, -10], [-5, 0, 36, 0, -30, 5, -5, -10, 7, -19], [-12, -15, 15, 12, 12, 2, -20, 2, 5, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1071_i_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1071_2_i_g();".
// 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_i_g(:prec:=10)
    chi := MakeCharacter_1071_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_i_g();".
// 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_i_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1071_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![225, 105, 259, 112, 215, 85, 70, 14, 11, 2, 1]>],Snew);
    return Vf;
end function;