
// Make newform 1008.2.r.n in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1008_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1008_r_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_1008_2_r_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_1008_2_r_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 := [243, -324, 162, -63, 75, -66, 25, -7, 6, -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], [-81, 81, -243, 156, 27, 31, 2, 3, -1, 1], [405, -180, 57, -39, 64, -33, 5, -4, 3, -2], [-3807, 3375, -441, 600, -939, 461, -86, 45, -59, 11], [-495, 657, -201, 54, -141, 101, -20, 3, -11, 5], [-8019, 6075, -2133, 912, -1935, 1129, -214, 129, -127, 55], [-4050, 3321, -702, 393, -1017, 566, -77, 51, -74, 29], [-9639, 6939, -2169, 1428, -2271, 1277, -290, 153, -155, 71], [-2457, 2241, -891, 300, -645, 439, -94, 27, -49, 25], [8667, -7965, 2853, -798, 2265, -1577, 308, -111, 167, -89]];
    Rf_basisdens := [1, 648, 108, 648, 72, 648, 324, 648, 216, 648];
    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_1008_r();"
function MakeCharacter_1008_r()
    N := 1008;
    order := 3;
    char_gens := [127, 757, 785, 577];
    v := [3, 3, 1, 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_1008_r_Hecke();"
function MakeCharacter_1008_r_Hecke(Kf)
    N := 1008;
    order := 3;
    char_gens := [127, 757, 785, 577];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 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 := true, 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+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, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, -1, 0, 0], [-1, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 0, -1, 1, 0], [0, -1, -1, 1, -1, -1, -1, -1, 0, -1], [0, -1, 0, -1, 0, -1, 0, 0, 1, 0], [1, 1, 0, 0, -1, 1, 2, 0, 1, 2], [0, -1, -2, 0, 1, -1, 0, 1, 0, 0], [-3, 0, 3, -1, 0, 0, 0, 1, -1, -1], [0, 1, 1, -1, 0, 1, 0, 0, 0, 1], [-1, 2, 0, 0, -1, -2, 0, 0, 0, -1], [0, -2, -4, 0, 1, -2, -1, -1, 0, 0], [3, -3, -3, 1, -1, -1, 0, -2, 2, 2], [-1, 3, 1, 0, 2, 2, -2, 0, 0, -1], [6, -2, 0, 0, 1, 1, -1, 0, 0, 0], [0, 1, -1, 2, -1, 1, 2, -1, 0, -2], [-2, 2, 2, 2, -2, -2, 3, -1, 1, -1], [0, -2, 0, -1, 1, -2, -2, -1, 0, 1], [1, -2, 0, 0, 1, -1, -3, 0, 0, -2], [5, 0, 0, 0, -2, -1, -1, 0, 4, 1], [-1, 5, 1, 0, 1, 1, 2, 3, -3, -1], [-4, 3, 4, 2, -3, -3, 4, -1, 1, -3], [0, 3, 0, 1, -3, 0, 1, 0, 3, 2], [-7, 0, 7, 1, -2, -2, 4, 1, -1, 1], [1, -3, -1, 3, -3, -3, 1, -5, 5, 1], [0, 0, 6, 1, -2, 0, -1, 0, 0, -1], [1, 7, 0, 5, -2, 0, -3, 0, -3, -3], [5, 2, 0, 2, 0, -1, -3, 0, -2, -3], [0, 5, 4, -1, -1, 5, 3, 3, 0, 1], [-2, 4, 0, 2, -1, 2, 2, 0, -2, 1], [0, 2, -3, 1, -1, 2, 2, 2, 0, -1], [-5, 5, 5, 1, 1, 1, 2, 2, -2, 0], [0, -2, 5, 1, 0, -2, -1, -5, 0, -1], [0, 2, -6, 0, 3, 2, 5, 1, 0, 0], [7, -2, -7, 1, -3, -3, 0, -4, 4, -3], [0, -2, -5, 1, 2, -2, 1, -3, 0, -1], [-2, 5, 0, 5, -2, 1, -4, 0, -1, -2], [0, -3, -7, 1, -1, -3, -3, -5, 0, -1], [-7, 3, 7, -3, 1, 1, 0, 4, -4, -4], [9, -4, 0, 0, 2, 3, -1, 0, 0, 1], [9, -3, 0, -4, 2, -2, 3, 0, -1, 0], [1, -1, -1, -1, 0, 0, 2, 3, -3, 2], [0, -2, -9, 3, -4, -2, -3, 0, 0, -3], [-1, 3, 0, 3, 1, 1, -2, 0, -5, -3], [-6, 2, 0, 0, -1, -1, 1, 0, 0, 0], [0, 1, 9, -1, -1, 1, -1, 7, 0, 1], [13, 6, -13, 4, -1, -1, 4, -1, 1, 0], [-9, 8, 9, -2, 3, 3, 3, 8, -8, -1], [0, 2, 3, 2, -1, 2, 3, 2, 0, -2], [3, -5, 0, 3, 3, 6, -5, 0, -1, 0], [0, 4, -3, -4, 0, 4, 0, 4, 0, 4], [-8, -2, 8, 1, -1, -1, -2, -4, 4, -1], [2, 4, 0, -1, -3, -7, -1, 0, 2, -3], [0, 2, -4, 0, 0, 2, 2, 0, 0, 0], [-7, -7, 7, -2, 3, 3, -10, -5, 5, 1], [11, 10, 0, 1, -4, -3, 5, 0, -2, 0], [2, 7, 0, 5, -3, 2, -1, 0, -1, 0], [-2, 1, 2, 0, -1, -1, -2, -3, 3, -5], [6, 3, -6, 2, 0, 0, 2, 0, 0, 1], [0, 0, 9, 0, -2, 0, -2, -1, 0, 0], [0, 4, 11, 3, -1, 4, 6, 2, 0, -3], [-2, 2, 0, 1, -1, 3, 3, 0, 0, 3], [0, 0, -6, -2, 1, 0, -1, -3, 0, 2], [-3, 6, 3, -1, 2, 2, 4, 7, -7, 1], [-8, 7, 8, -1, 6, 6, -6, 1, -1, -2], [-2, 9, 2, 0, 1, 1, 2, 3, -3, -5], [0, 5, 15, -4, 1, 5, 2, 9, 0, 4], [0, -3, -13, 0, 4, -3, 1, 2, 0, 0], [-8, -12, 8, -4, 2, 2, -10, -4, 4, 2], [-10, 4, 10, -1, 1, 1, 0, 2, -2, -3], [6, 2, 0, -4, -2, -1, 9, 0, 2, 5], [-1, -3, 1, 1, -1, -1, -2, -4, 4, 0], [0, 3, 11, -3, -4, 3, -4, 2, 0, 3], [1, -1, 0, -5, 1, -1, 8, 0, -1, 3], [0, 7, 7, -5, 4, 7, 6, 2, 0, 5], [-13, 3, 13, 0, 0, 0, 4, 4, -4, 1], [-5, -2, 0, 2, 1, 4, -2, 0, 0, 1], [0, 7, 4, -5, 6, 7, 8, 4, 0, 5], [0, 4, -4, -2, 6, 4, 8, 2, 0, 2], [0, -7, 6, 2, 0, -7, -5, -3, 0, -2], [9, -13, -9, 2, -2, -2, -8, -12, 12, 3], [-9, -7, 0, -5, 7, 1, 4, 0, -7, -1], [19, -7, 0, -5, 2, 2, 5, 0, 3, 5], [3, -2, -3, 1, -2, -2, 2, -1, 1, 1], [-14, -2, 14, 2, -6, -6, 6, -2, 2, -2], [-4, 2, 0, 0, -4, 1, 3, 0, 6, 5], [11, -9, -11, -2, 3, 3, -6, -1, 1, 7], [-18, -5, 18, 5, -8, -8, 5, -8, 8, -1], [0, 1, -10, 2, 1, 1, 4, 3, 0, -2], [-7, 2, 0, 4, 1, 2, -4, 0, -4, -3], [-3, -9, 3, 2, 0, 0, -6, -8, 8, 5], [-1, 10, 0, 6, 1, 3, 1, 0, -12, -4], [0, -9, -9, 5, -6, -9, -10, -6, 0, -5], [0, 6, -2, -2, 0, 6, 4, 4, 0, 2], [18, -3, 0, -5, 1, -4, 3, 0, 1, 0], [0, -2, 6, -1, -4, -2, -7, 0, 0, 1], [-3, 9, 0, 1, -4, -6, 1, 0, -1, -3], [3, -4, 0, -3, 0, -5, -3, 0, 4, -2], [-5, 7, 0, -1, -6, 2, 11, 0, 5, 9], [-2, -8, 2, -1, 3, 3, -12, -8, 8, 1], [15, 3, 0, -1, -5, 5, 10, 0, 7, 11], [0, -5, -18, 1, -1, -5, -5, 0, 0, -1], [-3, -14, 3, -4, -1, -1, -2, 1, -1, 6], [0, -5, -5, 2, -1, -5, -4, -5, 0, -2], [-6, 8, 0, 4, 1, 3, 3, 0, -10, -2], [6, -7, -6, 2, -7, -7, 5, -4, 4, 0], [-9, 2, 0, 2, 4, 3, 1, 0, -10, -3], [0, 10, -6, -4, 6, 10, 12, 2, 0, 4], [-14, -13, 14, -7, 2, 2, -10, -1, 1, 0], [0, -1, -11, -1, 5, -1, 3, 1, 0, 1], [-1, 7, 0, 7, -2, -4, -11, 0, -3, -9], [0, -4, 24, -2, 1, -4, -5, 7, 0, 2], [0, -18, 0, -6, 4, 4, -11, -1, 1, 9], [8, -8, 0, 0, 5, 2, -6, 0, -2, -3], [2, -10, -2, -3, 0, 0, -8, -5, 5, -1], [0, 3, 13, -6, 7, 3, 4, 7, 0, 6], [1, -2, 0, -6, -2, -1, 9, 0, 6, 7], [0, 6, -24, -3, 5, 6, 8, 1, 0, 3], [-9, 7, 9, -2, 4, 4, 0, 6, -6, -1], [0, 6, 13, 1, 1, 6, 8, 4, 0, -1], [10, -5, -10, -3, -1, -1, -2, 0, 0, -2], [-5, -4, 5, -1, 4, 4, -4, 1, -1, 7], [8, -8, 0, 0, 4, 2, -6, 0, 0, -2], [-12, -3, 12, -6, 7, 7, -9, 4, -4, 2], [10, -2, 0, -6, 0, -6, 4, 0, 2, 0], [18, 8, -18, 2, -4, -4, 12, 6, -6, -2], [-1, -15, 0, -1, 3, 5, -8, 0, 9, 3], [-7, 7, 7, -7, 9, 9, -4, 12, -12, 0], [0, 4, 1, 2, -5, 4, 1, 0, 0, -2], [11, 3, 0, 1, -7, 3, 4, 0, 11, 9], [0, 0, 14, 1, -1, 0, 0, -1, 0, -1], [0, 6, -3, 3, 3, 6, 12, -5, 0, -3], [-22, 9, 0, 1, -4, -1, 6, 0, -1, 2], [0, -4, 18, 2, -4, -4, -6, 0, 0, -2], [0, 4, -10, 3, 0, 4, 7, -8, 0, -3], [-1, 0, 0, -3, 1, 0, 6, 0, -2, 2], [0, 11, -1, -7, -4, 11, 0, 12, 0, 7], [0, -3, -12, 0, -5, -3, -8, 4, 0, 0], [-8, 1, 0, -3, 4, -7, 0, 0, -9, -8], [-8, -16, 0, -8, 9, 3, 3, 0, -2, 2], [-15, 7, 15, -4, 8, 8, -10, 2, -2, -5], [0, 0, 8, 8, -4, 0, 4, 0, 0, -8], [13, -7, 0, -3, -1, -7, -8, 0, 9, -3], [5, -6, 0, 6, 0, 9, -9, 0, 6, 3], [-3, 14, 3, -2, 5, 5, 4, 11, -11, -2], [-17, 5, 17, 3, -1, -1, 7, 3, -3, 3], [14, -8, -14, -3, -7, -7, 2, -2, 2, -7], [0, -1, -11, 1, -1, -1, -1, -11, 0, -1], [-14, -6, 0, -2, 2, -5, -7, 0, 2, -5], [0, 2, 26, -1, -5, 2, -4, -3, 0, 1], [-21, -14, 0, -8, 7, -8, -6, 0, 0, -7], [-1, 6, 0, 4, 3, 6, 4, 0, -12, -1], [0, -8, -12, 5, 4, -8, 1, -10, 0, -5], [-25, 16, 25, -2, 1, 1, 12, 15, -15, -4], [9, -4, -9, -5, -1, -1, 0, 4, -4, -3], [22, 8, 0, 0, -7, 2, 10, 0, 6, 9], [39, -4, -39, 1, -2, -2, 0, -3, 3, 1], [-8, 9, 0, 1, -4, 9, 16, 0, -1, 12], [0, 0, -1, -4, -4, 0, -8, 3, 0, 4], [6, 16, -6, -1, 5, 5, 6, 12, -12, -1], [2, 4, 0, 0, 0, 8, 12, 0, -4, 8], [0, 7, -6, 0, -4, 7, 3, 6, 0, 0], [-2, 4, 0, 5, -1, 3, -3, 0, -2, -1], [0, 9, -1, -7, 9, 9, 11, 11, 0, 7], [-20, -6, 20, 3, -7, -7, 4, -6, 6, -1], [-21, 12, 0, 6, -4, -5, -5, 0, -4, -7], [-4, 8, 4, 1, 5, 5, -1, 3, -3, 2], [-18, 1, 0, 1, 2, -3, -4, 0, -5, -6], [0, 4, -15, -9, 14, 4, 9, 13, 0, 9], [-35, 8, 35, -5, 4, 4, -4, 5, -5, -9], [0, 3, 6, -5, 3, 3, 1, 2, 0, 5], [0, -14, -19, 8, -2, -14, -8, -20, 0, -8], [3, 0, -3, 4, -8, -8, 12, 0, 0, 0], [0, 5, 1, -9, 3, 5, -1, 1, 0, 9], [0, -4, 26, -4, -7, -4, -15, 7, 0, 4], [-7, -17, 7, -1, -3, -3, -8, -10, 10, 2], [9, -2, 0, 2, 3, 8, 2, 0, -4, 3], [-24, 4, 0, 4, 1, -5, -9, 0, -6, -10], [-3, 7, 3, -6, 8, 8, -7, 7, -7, -4], [-28, 10, 28, 1, 0, 0, 6, 5, -5, -3], [9, -3, -9, 2, -6, -6, 6, -2, 2, -1], [0, -10, 8, 4, -8, -10, -14, -2, 0, -4], [-29, -6, 0, 6, 0, 5, -13, 0, 6, -1], [0, 9, -20, -2, -1, 9, 6, 3, 0, 2], [-29, -3, 29, -4, 2, 2, -4, 2, -2, -1], [13, -2, 0, 0, -7, 6, 4, 0, 16, 13], [-6, 15, 6, 1, 4, 4, -1, 2, -2, -7], [0, 1, 4, -3, 6, 1, 4, 6, 0, 3], [7, 4, 0, -4, -4, -11, 1, 0, 4, -3], [-37, 11, 0, 1, -1, -13, -4, 0, -9, -13], [21, 1, -21, 7, -7, -7, 6, -8, 8, -2], [11, -3, 0, -3, 9, -3, 0, 0, -15, -9], [22, -19, -22, -2, -5, -5, -7, -10, 10, 0], [-29, -2, 0, -2, -5, 0, 2, 0, 12, 7], [0, 2, -6, 1, 6, 2, 9, -2, 0, -1], [0, 2, 5, -3, -7, 2, -8, 11, 0, 3], [-9, -9, 9, 2, -12, -12, 6, -8, 8, -7], [2, 17, -2, 2, 7, 7, 6, 11, -11, 5], [-13, -9, 0, -5, 4, -2, -1, 0, 1, -1], [0, 3, -7, 1, -5, 3, -1, 7, 0, -1], [13, 8, -13, 0, 3, 3, 4, 7, -7, 2], [0, 6, 33, -5, -1, 6, 0, 14, 0, 5], [20, -5, -20, 11, -4, -4, 2, -13, 13, 10], [-2, 6, 0, 1, -2, 6, 10, 0, -2, 7], [18, 0, 0, 3, -2, -2, -8, 0, 4, -3], [6, 4, 0, -2, 0, -2, 6, 0, -4, 0], [0, -2, -6, -1, 10, -2, 7, -4, 0, 1], [20, 12, -20, 6, -8, -8, 20, 6, -6, -2], [0, -5, 2, 7, -5, -5, -3, -2, 0, -7], [-14, -20, 0, -8, 13, -4, -8, 0, -6, -9], [0, -1, -7, 4, 6, -1, 9, -10, 0, -4], [-4, 2, 4, 2, -4, -4, 0, -6, 6, -8], [-22, 3, 22, 2, -5, -5, 3, -4, 4, -8], [0, 10, 8, -5, 3, 10, 8, 9, 0, 5], [3, -34, -3, -3, -12, -12, -6, -15, 15, 1], [0, -1, -2, 0, -3, -1, -4, 15, 0, 0], [11, 25, -11, 7, 5, 5, 6, 4, -4, -2], [24, 3, 0, 3, -6, 3, 0, 0, 9, 6], [0, 7, 19, -2, 5, 7, 10, -1, 0, 2], [0, -23, -23, 10, -1, -23, -14, -11, 0, -10], [0, 0, 26, -4, -6, 0, -10, -4, 0, 4], [0, -11, -8, -3, 5, -11, -9, -1, 0, 3], [-7, 10, 0, 0, -5, -10, 0, 0, 0, -5], [0, 0, 18, 2, -4, 0, -2, 12, 0, -2], [-28, 23, 28, -4, 3, 3, 14, 21, -21, -7], [-4, 17, 4, 1, 6, 6, -4, 1, -1, -8], [-2, -7, 0, 7, -1, 5, -16, 0, 9, -1], [12, -13, -12, 7, -6, -6, -8, -21, 21, 0], [-33, -3, 33, -7, 1, 1, 0, 8, -8, -2], [0, -1, -3, -7, 10, -1, 2, 0, 0, 7], [-28, 13, 0, 6, -3, -4, -3, 0, -7, -7], [0, -3, -12, 1, -3, -3, -5, -19, 0, -1], [0, 3, 15, 8, -6, 3, 5, 8, 0, -8], [16, 18, -16, 9, 3, 3, 7, 1, -1, 4], [39, -19, -39, 1, -5, -5, -4, -10, 10, 6], [0, -10, -2, 6, -4, -10, -8, 0, 0, -6], [6, -12, 0, 0, 15, -1, -13, 0, -18, -16], [0, -9, 5, 1, 6, -9, -2, -8, 0, -1], [-7, 10, 0, 4, -4, -3, -1, 0, -2, -3], [0, -3, 0, 5, -4, -4, 6, -3, 3, 6], [5, -5, 0, 9, -1, 3, -20, 0, 7, -5], [0, 0, 14, 6, -3, 0, 3, 9, 0, -6], [5, 7, 0, 1, -5, -5, 0, 0, 3, -1], [0, 2, -12, -1, 0, 2, 1, 4, 0, 1], [36, 3, -36, 6, -3, -3, 2, -7, 7, -1], [35, 1, 0, 1, -7, 15, 14, 0, 13, 21], [0, -3, 18, -1, -5, -3, -9, 8, 0, 1], [0, 5, 23, -6, 2, 5, 1, 2, 0, 6], [-19, 14, 0, 8, -8, -6, -8, 0, 2, -6], [0, 14, 4, -8, 1, 14, 7, 3, 0, 8], [-18, -20, 0, -4, 12, -2, -14, 0, -4, -10], [0, -3, -36, -4, -1, -3, -8, -3, 0, 4], [-12, -5, 12, 0, 5, 5, -8, -3, 3, 7], [7, -13, -7, 0, 4, 4, -12, -8, 8, 9], [0, -19, 0, 3, 6, 11, -14, 0, 7, 2], [9, 2, 0, 4, 0, -3, -9, 0, -2, -7], [29, 3, -29, -3, 3, 3, 1, 7, -7, 1], [-7, 15, 0, 8, -6, 12, 11, 0, -3, 10], [11, 9, 0, 1, -14, 10, 17, 0, 19, 23], [4, 26, -4, 7, -1, -1, 16, 8, -8, -5], [18, -15, -18, 6, -3, -3, -6, -15, 15, 9], [0, -2, 0, 0, 1, -2, -1, 5, 0, 0], [-7, 6, 0, -5, 3, -6, 10, 0, -12, -4], [0, 4, 20, 0, 6, 4, 10, -4, 0, 0], [-10, -14, 10, -4, -6, -6, -4, -6, 6, -6], [-7, 8, 0, 6, 2, -7, -11, 0, -12, -15], [9, -15, -9, -9, -5, -5, 2, 6, -6, -2], [0, -8, -10, 0, 7, -8, -1, -11, 0, 0], [0, 10, -14, 1, 6, 10, 17, 2, 0, -1], [0, -12, -18, 3, -1, -12, -10, -23, 0, -3], [-5, 0, 0, 12, -4, 5, -19, 0, 8, -3], [-43, 0, 43, 2, -1, -1, 8, 5, -5, 8], [0, -2, -12, -9, -1, -2, -12, -1, 0, 9], [0, 7, -11, -2, 1, 7, 6, 11, 0, 2], [-13, 17, 0, 3, -2, -8, 3, 0, -13, -9], [0, 12, -8, 3, -2, 12, 13, -4, 0, -3], [-11, 7, 11, -8, 8, 8, -4, 12, -12, -3], [-18, 10, 0, 6, 0, 14, 12, 0, -10, 8], [17, 9, -17, -2, -8, -8, 11, 5, -5, -16], [0, -19, -11, 11, -7, -19, -15, -1, 0, -11], [0, -2, 16, -8, 7, -2, -3, 9, 0, 8], [-24, 6, 24, -3, 1, 1, -2, 2, -2, -9], [-21, 8, 21, -2, 3, 3, 1, 6, -6, -3], [0, 5, -10, 8, 1, 5, 14, -12, 0, -8], [7, -15, 0, -3, 8, 3, -6, 0, -1, -2], [13, 31, 0, 7, -12, -7, 10, 0, -7, -2], [0, 7, -9, -2, -1, 7, 4, 15, 0, 2], [5, 16, -5, 3, 4, 4, 12, 13, -13, 7], [6, 13, 0, -1, -6, 11, 26, 0, -1, 18], [-20, 6, 20, -1, 5, 5, 10, 16, -16, 13], [20, -1, 0, 6, 9, 6, -7, 0, -17, -9], [0, -12, 27, -3, -5, -12, -20, 9, 0, 3], [0, 5, 25, -10, 8, 5, 3, 14, 0, 10], [-10, -6, 10, 5, -15, -15, 8, -12, 12, -11], [0, -6, -3, -2, 2, -6, -6, 7, 0, 2], [11, 4, -11, -5, 6, 6, -4, 7, -7, -1], [-20, 4, 20, -3, 3, 3, -7, -1, 1, -8], [-9, -6, 0, -8, 1, 4, 14, 0, 4, 11], [31, -32, -31, 0, -1, -1, -14, -15, 15, 16], [0, -3, 9, -3, -5, -3, -11, 13, 0, 3], [46, 4, 0, 6, -7, -11, -19, 0, 10, -10], [-25, -2, 0, -4, 9, -3, 3, 0, -16, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1008_r_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1008_2_r_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_1008_2_r_n(:prec:=10)
    chi := MakeCharacter_1008_r();
    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_1008_2_r_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_1008_2_r_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1008_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3721, 2196, 3370, 362, 1441, 287, 235, 29, 22, 3, 1]>,<11,R![11664, 1728, 15052, 4936, 18865, 4501, 1621, 142, 49, 4, 1]>],Snew);
    return Vf;
end function;