
// Make newform 1710.2.t.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1710_t();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1710_t_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_1710_2_t_b();".
// 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_1710_2_t_b();".
// 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 := [4, -8, 8, -24, 20, 20, -8, 16, 4, -8, 2, -2, 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, 0], [748, 48, -48, 144, -146, -94, 22, -18, 28, -4, 1, -1], [612, 148, -148, 444, -454, -286, 64, -44, 94, -20, 5, -5], [-12, 12, -1370, -50, 110, -414, 264, 166, -103, 43, -43, 12], [-142, -154, 154, 1148, 524, 246, 560, -109, -201, 9, -31, 31], [12, -12, 1140, 50, -110, 414, -264, -166, 103, -43, 43, -12], [-32, 32, -1982, -118, 278, -1058, 658, 412, -267, 107, -107, 32], [-420, -484, 484, 3424, 1614, 806, 1208, -244, -566, 48, -81, 81], [-12, 32, -38, 86, 74, -12, 46, 4, -23, 5, -5, 3], [476, 524, -524, -3856, -1774, -846, -1734, 344, 666, -36, 101, -101], [-1060, 1060, -1294, 2790, 2510, -92, 1700, 250, -941, 195, -195, 117], [612, -612, 732, -1636, -1424, 184, -906, -94, 469, -100, 100, -60]];
    Rf_basisdens := [1, 460, 460, 460, 230, 230, 460, 460, 20, 460, 460, 230];
    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_1710_t();"
function MakeCharacter_1710_t()
    N := 1710;
    order := 6;
    char_gens := [191, 1027, 1351];
    v := [6, 3, 2];
    // 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_1710_t_Hecke();"
function MakeCharacter_1710_t_Hecke(Kf)
    N := 1710;
    order := 6;
    char_gens := [191, 1027, 1351];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1], [0, 0, 0, 0, 3, 0, 0, 0, 0, -1, 0, 0], [1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0], [0, 1, -1, 0, 0, 0, 0, 0, 1, 0, -2, 1], [0, 0, 0, 4, 4, 3, 5, 5, 0, 3, 0, 0], [0, 3, -2, 0, 0, 0, 0, 0, 2, 0, -2, -3], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, -2], [0, 0, 0, -4, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 7, 7, 3, 5, 5, 0, 3, 0, 0], [0, 0, 0, 6, 6, 0, -3, -3, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, -5], [0, 0, 5, 0, 0, 0, 0, 0, -3, 0, 5, 0], [0, 0, 0, 2, 2, -5, -2, -2, 0, -5, 0, 0], [-7, 0, 0, 0, 0, 0, 0, 0, 7, 0, -1, 2], [0, 0, 0, 7, 0, -2, 1, 0, 0, 0, 0, 0], [-6, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, -2], [0, 0, 0, 0, -4, 0, 0, -4, 0, 0, 0, 0], [0, 5, 6, 0, 0, 0, 0, 0, 1, 0, 6, -5], [0, 0, 0, -5, 0, 0, -5, 0, 0, 0, 0, 0], [0, -8, 2, 0, 0, 0, 0, 0, 0, 0, 2, 8], [0, 0, 0, 0, -4, 0, 0, -1, 0, -2, 0, 0], [0, 0, 0, 0, -1, 0, 0, -3, 0, -8, 0, 0], [3, 0, 0, 0, 0, 0, 0, 0, -3, 0, -5, 0], [0, 0, 0, 0, 2, 0, 0, -4, 0, 5, 0, 0], [0, 0, 0, 2, 2, 2, 9, 9, 0, 2, 0, 0], [-13, 0, 0, 0, 0, 0, 0, 0, 13, 0, 2, 2], [0, 0, 0, -6, -6, 2, 2, 2, 0, 2, 0, 0], [0, -2, -4, 0, 0, 0, 0, 0, -4, 0, -4, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 3], [-2, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 9, 0, 4, 8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, 0, 10, 0, 2, 0, 0], [0, 0, 0, -10, -10, 7, -1, -1, 0, 7, 0, 0], [0, 0, 0, -9, 0, -1, -8, 0, 0, 0, 0, 0], [4, 13, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, -1, 0, 0, 0, 0, 0, 1, 0, -1, -4], [4, -8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 11, 0, 4, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 10, 0, 0, -5, 0, -6, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, 4, 0, 0, 5], [-1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -4, 5], [0, 0, 0, -10, 0, -4, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -11, 0, 0, 1, 0, 2, 0, 0], [-8, 4, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 4, 0, -6, -8, 0, 0, 0, 0, 0], [-2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, -13, 0, 0, 0, 0, 0, -1, 0, -13, 3], [0, -8, 2, 0, 0, 0, 0, 0, -2, 0, 2, 8], [0, 0, 0, -2, -2, -8, 10, 10, 0, -8, 0, 0], [0, 0, 0, -9, 0, 0, 10, 0, 0, 0, 0, 0], [6, 0, 0, 0, 0, 0, 0, 0, -6, 0, 12, -7], [-22, 0, 0, 0, 0, 0, 0, 0, 22, 0, 14, 6], [0, 0, 0, 0, -5, 0, 0, 11, 0, -5, 0, 0], [0, 4, -11, 0, 0, 0, 0, 0, -2, 0, -11, -4], [0, 0, 0, -16, 0, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, 0, -6, 0, 5, 0, 0], [0, 0, 0, -14, 0, 5, -6, 0, 0, 0, 0, 0], [4, 10, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -6, -6, -2, 6, 6, 0, -2, 0, 0], [0, 0, 0, -16, -16, -4, -3, -3, 0, -4, 0, 0], [9, -1, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 16, 0, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 6, 0, -4, 8, 0, 0, 0, 0, 0], [-5, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -16, 0, 0, -10, 0, -1, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 3, 0, 3, -8], [0, 0, 0, -22, -22, -6, -10, -10, 0, -6, 0, 0], [0, 0, 0, 0, 0, 0, 0, -13, 0, 5, 0, 0], [18, 8, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -5, 0, 1, -7, 0, 0, 0, 0, 0], [0, -1, -6, 0, 0, 0, 0, 0, 8, 0, -6, 1], [0, 0, 0, -2, 0, 9, -3, 0, 0, 0, 0, 0], [-6, 0, 0, 0, 0, 0, 0, 0, 6, 0, 8, -6], [0, 19, 1, 0, 0, 0, 0, 0, 4, 0, 1, -19], [1, -14, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [11, 0, 0, 0, 0, 0, 0, 0, -11, 0, -9, -2], [0, 6, -3, 0, 0, 0, 0, 0, -23, 0, -3, -6], [0, 0, 0, -11, -11, -4, 3, 3, 0, -4, 0, 0], [16, 0, 0, 0, 0, 0, 0, 0, -16, 0, 0, 9], [0, 0, 0, -19, -19, 8, 1, 1, 0, 8, 0, 0], [-11, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 7, 0, 0, 15, 0, 12, 0, 0], [12, 0, 0, 0, 0, 0, 0, 0, -12, 0, 12, 6], [0, 0, 0, 0, -13, 0, 0, -10, 0, 1, 0, 0], [0, 0, 0, 0, 7, 0, 0, -3, 0, -8, 0, 0], [0, -10, -9, 0, 0, 0, 0, 0, 9, 0, -9, 10], [0, 0, 0, 0, -19, 0, 0, -2, 0, -7, 0, 0], [-5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 12, 18], [-5, 0, 0, 0, 0, 0, 0, 0, 5, 0, 2, -11], [0, 0, 0, -8, -8, 13, -1, -1, 0, 13, 0, 0], [0, -4, -6, 0, 0, 0, 0, 0, 12, 0, -6, 4], [-12, 8, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 2, -7, -18, -18, 0, -7, 0, 0], [0, -10, -8, 0, 0, 0, 0, 0, -4, 0, -8, 10], [0, 0, 0, 4, 4, -4, 12, 12, 0, -4, 0, 0], [0, 0, 0, 16, 16, 6, 15, 15, 0, 6, 0, 0], [0, 0, 0, 0, -2, 0, 0, -12, 0, -16, 0, 0], [10, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7, -9, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -11, 0, 0, -9, 0, -12, 0, 0], [0, 0, 0, 8, 0, 8, 2, 0, 0, 0, 0, 0], [0, 0, 0, 24, 24, 7, -10, -10, 0, 7, 0, 0], [0, -6, 5, 0, 0, 0, 0, 0, 31, 0, 5, 6], [-6, 15, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -18, 0, 0, -13, 0, -10, 0, 0], [0, 0, 0, -10, 0, -7, -1, 0, 0, 0, 0, 0], [0, 0, 0, 6, 6, -2, -6, -6, 0, -2, 0, 0], [16, -2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 15, 10, 0, 0, 0, 0, 0, 2, 0, 10, -15], [-10, 0, 0, 0, 0, 0, 0, 0, 10, 0, 8, -2], [0, 0, 0, 6, 0, 11, -6, 0, 0, 0, 0, 0], [0, 0, 0, 0, -9, 0, 0, -2, 0, 6, 0, 0], [0, 0, 0, 0, -7, 0, 0, 8, 0, 23, 0, 0], [0, 11, 1, 0, 0, 0, 0, 0, 26, 0, 1, -11], [0, -14, 2, 0, 0, 0, 0, 0, 6, 0, 2, 14], [0, 0, 0, 0, 4, 0, 0, 2, 0, 16, 0, 0], [0, 0, 0, 0, 11, 0, 0, 12, 0, 11, 0, 0], [0, 0, 0, 0, 27, 0, 0, 9, 0, 0, 0, 0], [12, 14, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-28, 0, 0, 0, 0, 0, 0, 0, 28, 0, 4, -2], [0, 18, -1, 0, 0, 0, 0, 0, 17, 0, -1, -18], [32, 0, 0, 0, 0, 0, 0, 0, -32, 0, 2, 0], [0, 0, 0, -15, 0, -9, 9, 0, 0, 0, 0, 0], [0, 0, 0, 0, -7, 0, 0, -12, 0, -20, 0, 0], [-36, 0, 0, 0, 0, 0, 0, 0, 36, 0, -7, -4], [0, 0, 0, -18, 0, 11, -7, 0, 0, 0, 0, 0], [0, -2, 22, 0, 0, 0, 0, 0, 10, 0, 22, 2], [0, 0, 0, -1, 0, -16, -20, 0, 0, 0, 0, 0], [34, -6, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, -3, 0, 0, 0, 0, 0, 9, 0, -3, -1], [0, 0, 0, -1, -1, 9, -6, -6, 0, 9, 0, 0], [0, 0, 0, 0, 32, 0, 0, 8, 0, 1, 0, 0], [0, 0, 0, 0, -27, 0, 0, -4, 0, -13, 0, 0], [-5, 3, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 5, 0, 0, 0, 0, 0, 38, 0, 5, -2], [0, 12, -10, 0, 0, 0, 0, 0, 18, 0, -10, -12], [0, 0, 0, 30, 30, 12, 13, 13, 0, 12, 0, 0], [0, 0, 0, 13, 13, 6, 15, 15, 0, 6, 0, 0], [0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 0, 0, 0, 0, 0, 0, 0, 6, 0, -12, -18], [0, 0, 0, -22, 0, 0, 4, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 18, 16, 0, 0, 0, 0, 0], [0, -7, 26, 0, 0, 0, 0, 0, 21, 0, 26, 7], [0, 0, 0, 0, 7, 0, 0, 16, 0, 6, 0, 0], [0, 0, 0, 6, 0, 9, -5, 0, 0, 0, 0, 0], [-4, -10, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 2, 15, 10, 10, 0, 15, 0, 0], [0, 0, 0, 1, 1, 11, 3, 3, 0, 11, 0, 0], [0, 0, 0, 14, 0, -4, 14, 0, 0, 0, 0, 0], [-4, 2, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-20, 3, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [15, 0, 0, 0, 0, 0, 0, 0, -15, 0, -4, 5], [0, 0, 0, 15, 15, 16, 5, 5, 0, 16, 0, 0], [0, 1, -2, 0, 0, 0, 0, 0, -56, 0, -2, -1], [0, 0, 0, 3, 0, -16, -15, 0, 0, 0, 0, 0], [0, 0, 0, 10, 0, 6, -12, 0, 0, 0, 0, 0], [0, 0, 0, 32, 0, 12, 12, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -27, 5], [0, 0, 0, 0, 2, 0, 0, -4, 0, -11, 0, 0], [0, 0, 0, -12, 0, -11, -9, 0, 0, 0, 0, 0], [4, 0, 0, 0, 0, 0, 0, 0, -4, 0, -12, -7], [0, 0, 0, -20, -20, -9, -17, -17, 0, -9, 0, 0], [11, 0, 0, 0, 0, 0, 0, 0, -11, 0, 8, 12], [0, 0, 0, 4, 4, 10, 3, 3, 0, 10, 0, 0], [24, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 0, 0, 0, 0, 0, 0, 4, 0, -2, -8], [33, 0, 0, 0, 0, 0, 0, 0, -33, 0, -1, 8], [0, 0, 0, 0, -4, 0, 0, 14, 0, -16, 0, 0], [0, -8, -2, 0, 0, 0, 0, 0, -24, 0, -2, 8], [0, 9, -4, 0, 0, 0, 0, 0, -25, 0, -4, -9], [0, 18, -8, 0, 0, 0, 0, 0, 28, 0, -8, -18], [-8, 0, 0, 0, 0, 0, 0, 0, 8, 0, -6, -25], [0, 0, 0, 0, -32, 0, 0, -19, 0, -4, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 21, 28], [0, 0, 0, 10, 10, 0, -7, -7, 0, 0, 0, 0], [-30, 13, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 3, 8, 12, 12, 0, 8, 0, 0], [0, 0, 0, -26, 0, -18, -6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 11, 0, 0, 0, 0, 12, 0, 0], [-18, 8, -12, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 17, 17, 17, 25, 25, 0, 17, 0, 0], [0, 0, 0, 24, 0, 11, -8, 0, 0, 0, 0, 0], [5, -7, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [23, 24, -13, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 26, 26, 2, -6, -6, 0, 2, 0, 0], [0, 0, 0, -7, -7, 2, 1, 1, 0, 2, 0, 0], [-22, -6, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-12, 0, 0, 0, 0, 0, 0, 0, 12, 0, 6, -3], [0, 0, 0, 12, 12, 14, 12, 12, 0, 14, 0, 0], [0, 0, 0, -34, -34, 20, 10, 10, 0, 20, 0, 0], [0, 6, -12, 0, 0, 0, 0, 0, -37, 0, -12, -6], [0, 0, 0, -23, 0, -10, 10, 0, 0, 0, 0, 0], [0, 0, 0, 0, 16, 0, 0, 8, 0, 18, 0, 0], [0, 0, 0, 0, -22, 0, 0, -1, 0, -11, 0, 0], [0, -15, 14, 0, 0, 0, 0, 0, 14, 0, 14, 15], [0, 6, 18, 0, 0, 0, 0, 0, 12, 0, 18, -6], [0, 0, 0, -15, 0, -8, -12, 0, 0, 0, 0, 0], [29, 0, 0, 0, 0, 0, 0, 0, -29, 0, -8, -4], [28, 0, 0, 0, 0, 0, 0, 0, -28, 0, 7, 3], [0, 0, 0, -22, -22, -10, -15, -15, 0, -10, 0, 0], [0, -15, -6, 0, 0, 0, 0, 0, -12, 0, -6, 15], [0, 0, 0, -33, -33, -14, -11, -11, 0, -14, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 20, -4], [-18, -12, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, -24, -22, 0, 0, 0, 0, 0], [0, -9, -4, 0, 0, 0, 0, 0, 12, 0, -4, 9], [0, 0, 0, 0, -28, 0, 0, 8, 0, 16, 0, 0], [0, 0, 0, -23, -23, -18, -27, -27, 0, -18, 0, 0], [3, 10, -13, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 7, 0, 0, 0, 0, 0, 22, 0, 7, 3], [0, 0, 0, 12, 0, 20, -8, 0, 0, 0, 0, 0], [11, -9, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, -13, 0, 9, 0, 0], [0, 0, 0, -2, 0, 8, -20, 0, 0, 0, 0, 0], [0, 8, -12, 0, 0, 0, 0, 0, 24, 0, -12, -8], [42, 1, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [45, 0, 0, 0, 0, 0, 0, 0, -45, 0, -4, -5], [0, 0, 0, -30, 0, -14, -22, 0, 0, 0, 0, 0], [0, 0, 0, 24, 0, 4, 24, 0, 0, 0, 0, 0], [0, 6, -35, 0, 0, 0, 0, 0, -7, 0, -35, -6], [0, 0, 0, 0, -10, 0, 0, 12, 0, -20, 0, 0], [27, 0, 0, 0, 0, 0, 0, 0, -27, 0, 13, 2], [0, 0, 0, -17, 0, 11, -6, 0, 0, 0, 0, 0], [-16, 15, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 6, 6, -7, -5, -5, 0, -7, 0, 0], [0, -11, 6, 0, 0, 0, 0, 0, -7, 0, 6, 11], [14, 29, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [16, -18, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, 34, 0, -2, 0, 0], [0, 0, 0, -11, 0, -14, -24, 0, 0, 0, 0, 0], [-35, 8, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -37, 0, 0, 8, 0, -7, 0, 0], [-32, 0, 0, 0, 0, 0, 0, 0, 32, 0, 27, -1], [0, 14, 16, 0, 0, 0, 0, 0, -26, 0, 16, -14], [0, 0, 0, 18, 0, -14, 4, 0, 0, 0, 0, 0], [-11, 31, -26, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -16, -16, -18, -20, -20, 0, -18, 0, 0], [0, -28, -6, 0, 0, 0, 0, 0, -16, 0, -6, 28], [0, 0, 0, 28, 0, -27, 2, 0, 0, 0, 0, 0], [-3, 10, -21, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, -2, 2, 6, 6, 0, 2, 0, 0], [0, -1, -5, 0, 0, 0, 0, 0, -15, 0, -5, 1], [4, 0, 0, 0, 0, 0, 0, 0, -4, 0, 5, -2], [0, 0, 0, 9, 9, 25, 3, 3, 0, 25, 0, 0], [0, 0, 0, 0, 9, 0, 0, -14, 0, 6, 0, 0], [-7, 0, 0, 0, 0, 0, 0, 0, 7, 0, -14, -19], [0, 0, 0, 0, 22, 0, 0, 21, 0, 3, 0, 0], [0, -7, -27, 0, 0, 0, 0, 0, -18, 0, -27, 7], [0, 0, 0, -60, 0, 0, -3, 0, 0, 0, 0, 0], [0, 20, -20, 0, 0, 0, 0, 0, -16, 0, -20, -20], [0, -16, 17, 0, 0, 0, 0, 0, 7, 0, 17, 16], [0, 0, 0, 0, -16, 0, 0, -2, 0, 15, 0, 0], [0, 0, 0, 1, 0, -3, -4, 0, 0, 0, 0, 0], [0, 0, 0, 9, 9, 14, -29, -29, 0, 14, 0, 0], [0, 0, 0, 15, 15, -33, -6, -6, 0, -33, 0, 0], [0, 0, 0, 6, 0, -2, 2, 0, 0, 0, 0, 0], [12, 0, 0, 0, 0, 0, 0, 0, -12, 0, 22, 38], [0, 0, 0, 21, 0, 19, 25, 0, 0, 0, 0, 0], [0, 0, 0, -10, -10, 21, -2, -2, 0, 21, 0, 0], [-35, 17, -36, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-12, 0, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0], [11, -25, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 12, 12, -7, -20, -20, 0, -7, 0, 0], [0, 0, 0, 40, 40, 6, 18, 18, 0, 6, 0, 0], [-41, 10, -15, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 26, 0, 0, -20, 0, -10, 0, 0], [0, 0, 0, -1, 0, -14, -1, 0, 0, 0, 0, 0], [-18, 7, -20, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, -2, -12, 26, 26, 0, -12, 0, 0], [0, 0, 0, -52, 0, 8, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0, 0], [-36, 0, 0, 0, 0, 0, 0, 0, 36, 0, -8, -2], [0, -7, -7, 0, 0, 0, 0, 0, 25, 0, -7, 7], [0, 12, -26, 0, 0, 0, 0, 0, -37, 0, -26, -12], [0, 0, 0, 0, -1, 0, 0, -31, 0, -21, 0, 0], [40, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -11, -11, 27, 3, 3, 0, 27, 0, 0], [8, 30, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -54, 0, 0, -12, 0, 0, 0, 0, 0], [0, -24, 18, 0, 0, 0, 0, 0, -10, 0, 18, 24], [0, 0, 0, 0, -53, 0, 0, -7, 0, 2, 0, 0], [0, 0, 0, 1, 1, 11, -9, -9, 0, 11, 0, 0], [4, 0, 0, 0, 0, 0, 0, 0, -4, 0, 14, 1], [30, -12, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 20, -34, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, -1, 0, -12, 0, 0], [0, 0, 0, -42, -42, 8, 14, 14, 0, 8, 0, 0], [5, 9, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 18, 0, -8, 0, 0, 0, 0, 0, 0], [22, -3, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 12, -10, 0, 0, 0, 0, 0, -2, 0, -10, -12], [0, 0, 0, -5, 0, -7, 2, 0, 0, 0, 0, 0], [-19, 0, 0, 0, 0, 0, 0, 0, 19, 0, -14, 26], [0, 0, 0, 0, 18, 0, 0, -34, 0, 7, 0, 0], [0, 0, 0, -41, 0, 8, -15, 0, 0, 0, 0, 0], [0, 0, 0, -38, 0, -12, -33, 0, 0, 0, 0, 0], [0, 10, -28, 0, 0, 0, 0, 0, -10, 0, -28, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1710_t_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1710_2_t_b();".
// 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_1710_2_t_b(:prec:=12)
    chi := MakeCharacter_1710_t();
    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_1710_2_t_b();".
// 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_1710_2_t_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1710_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![169, 0, 295, 0, 35, 0, 1]>],Snew);
    return Vf;
end function;