
// Make newform 1470.2.n.k in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1470_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1470_n_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_1470_2_n_k();".
// 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_1470_2_n_k();".
// 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 := [4096, 0, -11264, 0, 24768, 0, -14768, 0, 6177, 0, -1394, 0, 227, 0, -18, 0, 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, 0, 0, 0, 0], [-716288, 0, 38912, 0, -85120, 0, 51859, 0, -18886, 0, 3705, 0, -326, 0, 19, 0], [0, 46691840, 0, 6760448, 0, -14788480, 0, 12013925, 0, -3281194, 0, 643695, 0, -53930, 0, 3301], [-2710528, 0, 5963712, 0, -3327060, 0, 1506549, 0, -344566, 0, 58575, 0, -4686, 0, 269, 0], [0, -33238528, 0, -4159488, 0, 9098880, 0, -7668591, 0, 2018814, 0, -396045, 0, 32894, 0, -2031], [0, 65518592, 0, -143948224, 0, 89641160, 0, -35273581, 0, 7811442, 0, -1194855, 0, 93154, 0, -4893], [1554176, 0, -16501984, 0, 8620820, 0, -3903653, 0, 854742, 0, -151775, 0, 12142, 0, -733, 0], [-87668736, 57592320, 192477056, 2836480, -126049360, -6204800, 46442378, 5905385, -10109828, -1376690, 1452510, 270075, -111332, -21810, 5482, 1385], [87668736, 57592320, -192477056, 2836480, 126049360, -6204800, -46442378, 5905385, 10109828, -1376690, -1452510, 270075, 111332, -21810, -5482, 1385], [0, -46285824, 0, 101725760, 0, -63272840, 0, 25102547, 0, -5601470, 0, 878105, 0, -69038, 0, 3715], [29806080, -9632000, 6760448, 85946592, -14788480, -53427500, 12013925, 24192875, -3281194, -5519626, 643695, 940625, -53930, -75250, 3301, 4339], [-29806080, -9632000, -6760448, 85946592, 14788480, -53427500, -12013925, 24192875, 3281194, -5519626, -643695, 940625, 53930, -75250, -3301, 4339], [-19785216, -14430464, -1558528, 133656288, 3409280, -80043980, -3323257, 36245267, 756434, -8276154, -148395, 1409225, 11858, -112738, -761, 6491], [-14535680, 120224768, 122123648, -264476096, -80627600, 153883160, 36509540, -66585149, -8295864, 15175618, 1419500, -2548375, -113560, 203666, 6596, -11597], [427520, 5463040, -3591872, -12012480, 2371400, 7162480, -1073810, -2995845, 243996, 676090, -41750, -110095, 3340, 8730, -194, -485], [19785216, -23843840, 1558528, 209010624, -3409280, -132258800, 3323257, 59889020, -756434, -13822472, 148395, 2328500, -11858, -186280, 761, 10588]];
    Rf_basisdens := [1, 496640, 33771520, 2110720, 16885760, 33771520, 4221440, 33771520, 33771520, 16885760, 16885760, 16885760, 16885760, 33771520, 993280, 16885760];
    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_1470_n();"
function MakeCharacter_1470_n()
    N := 1470;
    order := 6;
    char_gens := [491, 1177, 1081];
    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_1470_n_Hecke();"
function MakeCharacter_1470_n_Hecke(Kf)
    N := 1470;
    order := 6;
    char_gens := [491, 1177, 1081];
    char_values := [[1, 0, 0, 0, 0, 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, 1, 0, 0, 0, 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, 1, 0, 0, 0, 0, 0, 0, 0, 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, -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, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, -1, 1, 0], [0, 0, 0, 0, -2, 0, 0, 0, 0, 2, -1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, -1, -1, -1, 1, 1, 0, 1, 1, 0], [0, 2, -1, -4, 0, 0, -2, -1, 1, 0, 0, 0, -1, -1, 1, 1], [0, 0, -1, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1], [0, -1, -1, 0, 0, 1, 0, 0, 0, 0, -2, 2, -1, 0, 0, 1], [-4, 0, 0, -4, 0, -1, -1, 0, 0, 0, 0, 0, 0, -1, 1, 0], [0, 0, -2, 0, -3, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0], [4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0], [0, 0, 1, 0, 1, -1, 0, 0, 0, -1, -2, -2, -1, 0, 0, -1], [0, 0, 3, 0, -1, 0, 0, -2, -2, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 5, 0, -1, -1, -2, 1, 1, 0, -1, -1, 0], [4, 0, 0, 4, 0, 1, -4, -1, 1, 0, -1, 1, 0, 1, -1, 0], [0, 0, 0, 0, 0, 0, 0, -3, 3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, -1, -1, 0], [-8, -2, -1, 0, 0, 1, 0, 0, 0, 0, -3, 3, -1, 0, 0, 1], [0, 0, 0, 0, 0, -3, 0, -2, -2, 2, 2, 2, 0, -1, -1, 0], [0, 4, 2, 0, 0, 0, -4, -2, 2, 0, 0, 0, 2, 2, -2, -2], [0, 0, 2, 0, -2, -2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2], [0, 4, 0, -12, 0, 0, -4, -1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, -5, 0, 4, 5, 0, 0, 0, -4, -2, -2, 1, 0, 0, 1], [-8, 0, 0, -8, 0, 3, 4, -2, 2, 0, -2, 2, 0, 3, -3, 0], [0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2], [0, 0, 2, 0, -6, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2], [4, 0, 0, 4, 0, 1, -6, 1, -1, 0, 1, -1, 0, 1, -1, 0], [0, 0, 5, 0, 8, -5, 0, 0, 0, -8, 1, 1, 3, 0, 0, 3], [0, 0, 8, 0, -6, -8, 0, 0, 0, 6, -2, -2, 0, 0, 0, 0], [0, 0, -1, 4, 0, 0, 0, 1, -1, 0, 0, 0, -1, -1, 1, 1], [0, 0, 0, 0, 0, -5, 0, -1, -1, -6, 1, 1, 0, -1, -1, 0], [-4, 0, -3, 0, 0, 3, 0, 0, 0, 0, -3, 3, -3, 0, 0, 3], [0, 7, 0, 2, 0, 0, -7, 1, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 4, 1, -1, 0, 1, -1, 0, -1, 1, 0], [0, 0, 0, 0, 0, 3, 0, -4, -4, -4, 4, 4, 0, -3, -3, 0], [0, 0, 5, 0, 5, 0, 0, -2, -2, 0, 0, 0, 3, 3, 3, 3], [0, 0, -7, 0, 3, 7, 0, 0, 0, -3, 4, 4, 3, 0, 0, 3], [0, 0, 2, 0, -2, 0, 0, -1, -1, 0, 0, 0, -2, -2, -2, -2], [-8, 0, 0, -8, 0, -1, -15, -2, 2, 0, -2, 2, 0, -1, 1, 0], [0, -6, -2, 0, 0, 2, 0, 0, 0, 0, -3, 3, -2, 0, 0, 2], [0, -4, 1, 0, 0, 0, 4, 3, -3, 0, 0, 0, 1, 1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 2, 2, 2, -2, -2, 0, 0, 0, 0], [0, 0, -7, 0, -2, 7, 0, 0, 0, 2, 3, 3, 3, 0, 0, 3], [12, 0, 0, 12, 0, 1, 3, -2, 2, 0, -2, 2, 0, 1, -1, 0], [4, 6, -2, 0, 0, 2, 0, 0, 0, 0, 4, -4, -2, 0, 0, 2], [0, 0, 0, 0, -2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, -6, 0, 2, 2, 0, -2, -2, 0, 2, 2, 0], [0, -6, 4, -16, 0, 0, 6, -1, 1, 0, 0, 0, 4, 4, -4, -4], [0, 0, -4, 0, 8, 0, 0, 2, 2, 0, 0, 0, -2, -2, -2, -2], [0, -8, -3, 0, 0, 3, 0, 0, 0, 0, 1, -1, -3, 0, 0, 3], [0, 0, 0, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0, 2, -2, 0], [-4, 8, 2, 0, 0, -2, 0, 0, 0, 0, -2, 2, 2, 0, 0, -2], [0, 0, -3, 0, -9, 0, 0, 1, 1, 0, 0, 0, -5, -5, -5, -5], [0, 0, 0, 0, 0, 3, 0, 1, 1, -6, -1, -1, 0, -3, -3, 0], [8, 0, 0, 8, 0, 6, 0, 1, -1, 0, 1, -1, 0, 6, -6, 0], [0, -9, 1, 12, 0, 0, 9, 4, -4, 0, 0, 0, 1, 1, -1, -1], [0, 0, 0, 0, 0, 5, 0, 0, 0, -1, 0, 0, 0, 3, 3, 0], [-16, 6, -1, 0, 0, 1, 0, 0, 0, 0, 1, -1, -1, 0, 0, 1], [0, 0, 0, 0, 0, -1, 0, 5, 5, 6, -5, -5, 0, -3, -3, 0], [0, 0, -16, 0, 4, 16, 0, 0, 0, -4, 3, 3, 0, 0, 0, 0], [0, 0, 0, 0, 8, 0, 0, 0, 0, -8, 0, 0, -4, 0, 0, -4], [8, 0, 0, 8, 0, 2, 8, 2, -2, 0, 2, -2, 0, 2, -2, 0], [0, 0, 3, 0, -6, 0, 0, 2, 2, 0, 0, 0, -7, -7, -7, -7], [0, 0, 5, 0, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, -1, -1], [0, 10, -2, -12, 0, 0, -10, -4, 4, 0, 0, 0, -2, -2, 2, 2], [0, 0, 19, 0, -2, -19, 0, 0, 0, 2, 3, 3, -1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, -4, -4, 0], [0, 10, 2, 0, 0, -2, 0, 0, 0, 0, -3, 3, 2, 0, 0, -2], [0, 0, 0, 0, 0, -9, 0, 3, 3, -1, -3, -3, 0, 1, 1, 0], [0, -2, 3, 0, 0, 0, 2, 3, -3, 0, 0, 0, 3, 3, -3, -3], [0, 0, 0, 0, 0, -2, 0, 6, 6, 12, -6, -6, 0, 2, 2, 0], [0, 0, 17, 0, 1, 0, 0, -2, -2, 0, 0, 0, -1, -1, -1, -1], [12, -4, -8, 0, 0, 8, 0, 0, 0, 0, 0, 0, -8, 0, 0, 8], [0, 0, -5, 0, 13, 0, 0, 4, 4, 0, 0, 0, 1, 1, 1, 1], [8, 0, 0, 8, 0, 3, -13, 0, 0, 0, 0, 0, 0, 3, -3, 0], [0, 0, -17, 0, 12, 0, 0, -2, -2, 0, 0, 0, 1, 1, 1, 1], [0, -4, -1, 0, 0, 0, 4, 1, -1, 0, 0, 0, -1, -1, 1, 1], [8, 0, 0, 8, 0, -4, 7, 4, -4, 0, 4, -4, 0, -4, 4, 0], [4, -16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, 6, 0, 0, -6, 0, 0, 0, 0, 0, 0, 6, 0, 0, -6], [16, 0, 0, 16, 0, 1, 2, -3, 3, 0, -3, 3, 0, 1, -1, 0], [0, 0, -7, 0, -8, 7, 0, 0, 0, 8, -6, -6, -5, 0, 0, -5], [0, -1, -3, -12, 0, 0, 1, 4, -4, 0, 0, 0, -3, -3, 3, 3], [0, 0, -6, 0, 0, 0, 0, -2, -2, 0, 0, 0, 2, 2, 2, 2], [-6, -6, 4, 0, 0, -4, 0, 0, 0, 0, -2, 2, 4, 0, 0, -4], [0, 0, -12, 0, -10, 0, 0, -6, -6, 0, 0, 0, -4, -4, -4, -4], [-8, 8, 2, 0, 0, -2, 0, 0, 0, 0, 3, -3, 2, 0, 0, -2], [0, 0, -8, 0, 20, 8, 0, 0, 0, -20, 4, 4, 0, 0, 0, 0], [0, 0, 10, 0, -6, 0, 0, 4, 4, 0, 0, 0, 2, 2, 2, 2], [8, 0, 0, 8, 0, -6, -16, 2, -2, 0, 2, -2, 0, -6, 6, 0], [0, 0, 0, 0, 0, 14, 0, 0, 0, 14, 0, 0, 0, 2, 2, 0], [0, 9, -3, 0, 0, 3, 0, 0, 0, 0, 2, -2, -3, 0, 0, 3], [0, 12, -2, -4, 0, 0, -12, -2, 2, 0, 0, 0, -2, -2, 2, 2], [0, 0, -3, 0, 15, 3, 0, 0, 0, -15, 0, 0, -1, 0, 0, -1], [0, 4, 1, -16, 0, 0, -4, -2, 2, 0, 0, 0, 1, 1, -1, -1], [-4, 0, 0, -4, 0, -4, 8, 3, -3, 0, 3, -3, 0, -4, 4, 0], [0, 0, -1, 0, 2, 0, 0, 1, 1, 0, 0, 0, 5, 5, 5, 5], [0, 2, -4, 10, 0, 0, -2, 2, -2, 0, 0, 0, -4, -4, 4, 4], [0, 0, 23, 0, -11, -23, 0, 0, 0, 11, -8, -8, 1, 0, 0, 1], [0, 0, 0, 0, 0, -1, 0, -1, -1, -4, 1, 1, 0, 5, 5, 0], [0, 0, 0, 0, 0, -6, 0, 0, 0, 10, 0, 0, 0, 2, 2, 0], [0, 8, -3, -16, 0, 0, -8, -1, 1, 0, 0, 0, -3, -3, 3, 3], [-4, 0, 0, -4, 0, 6, 8, -2, 2, 0, -2, 2, 0, 6, -6, 0], [0, 0, 0, 0, 0, 11, 0, 0, 0, 18, 0, 0, 0, 1, 1, 0], [0, 0, 20, 0, 6, -20, 0, 0, 0, -6, 2, 2, 0, 0, 0, 0], [0, 0, 5, 0, -7, 0, 0, -7, -7, 0, 0, 0, 3, 3, 3, 3], [8, 0, 0, 8, 0, 5, 4, 7, -7, 0, 7, -7, 0, 5, -5, 0], [8, -1, 6, 0, 0, -6, 0, 0, 0, 0, 0, 0, 6, 0, 0, -6], [0, 0, 12, 0, 26, 0, 0, 2, 2, 0, 0, 0, 4, 4, 4, 4], [0, 0, 0, 0, 0, -6, 0, -1, -1, -5, 1, 1, 0, 8, 8, 0], [0, 0, -13, 0, 6, 13, 0, 0, 0, -6, -9, -9, 3, 0, 0, 3], [4, 0, 0, 4, 0, -1, -6, -1, 1, 0, -1, 1, 0, -1, 1, 0], [0, 4, -7, 0, 0, 7, 0, 0, 0, 0, -3, 3, -7, 0, 0, 7], [0, 0, 0, 0, 0, -7, -4, 7, -7, 0, 7, -7, 0, -7, 7, 0], [0, 0, -17, 0, 6, 17, 0, 0, 0, -6, -3, -3, 5, 0, 0, 5], [0, 0, 0, 0, 0, 5, 0, -4, -4, 7, 4, 4, 0, -1, -1, 0], [0, 0, 6, 0, -4, 0, 0, 0, 0, 0, 0, 0, -4, -4, -4, -4], [0, -3, 3, 0, 0, -3, 0, 0, 0, 0, 0, 0, 3, 0, 0, -3], [0, 0, 0, 0, 0, 2, -16, -1, 1, 0, -1, 1, 0, 2, -2, 0], [0, 0, -19, 0, 4, 19, 0, 0, 0, -4, -5, -5, -3, 0, 0, -3], [0, 0, -20, 0, -2, 0, 0, -3, -3, 0, 0, 0, 4, 4, 4, 4], [0, 0, 0, 0, 0, -4, 0, -2, -2, -2, 2, 2, 0, 8, 8, 0], [0, 4, -11, 4, 0, 0, -4, -1, 1, 0, 0, 0, -11, -11, 11, 11], [0, -13, 11, 0, 0, -11, 0, 0, 0, 0, -2, 2, 11, 0, 0, -11], [0, -8, 1, 4, 0, 0, 8, -9, 9, 0, 0, 0, 1, 1, -1, -1], [0, 2, -6, 8, 0, 0, -2, 0, 0, 0, 0, 0, -6, -6, 6, 6], [0, 0, 6, 0, 6, -6, 0, 0, 0, -6, 0, 0, 10, 0, 0, 10], [0, 0, 19, 0, -4, 0, 0, 0, 0, 0, 0, 0, 5, 5, 5, 5], [4, 0, 0, 4, 0, 2, 4, 6, -6, 0, 6, -6, 0, 2, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12, -12, 0, 0, 0, 0], [0, -4, -7, -8, 0, 0, 4, -1, 1, 0, 0, 0, -7, -7, 7, 7], [0, 0, 24, 0, 13, -24, 0, 0, 0, -13, 3, 3, 6, 0, 0, 6], [0, 4, -4, -20, 0, 0, -4, -1, 1, 0, 0, 0, -4, -4, 4, 4], [16, 3, 4, 0, 0, -4, 0, 0, 0, 0, 2, -2, 4, 0, 0, -4], [0, 0, 0, 0, 0, -10, 0, 1, 1, -4, -1, -1, 0, -6, -6, 0], [0, 0, 0, 0, 0, 8, 0, 0, 0, 8, 0, 0, 0, 4, 4, 0], [0, 0, 32, 0, -6, -32, 0, 0, 0, 6, 3, 3, 0, 0, 0, 0], [14, 0, 0, 14, 0, -4, 10, -2, 2, 0, -2, 2, 0, -4, 4, 0], [-4, 8, -11, 0, 0, 11, 0, 0, 0, 0, 1, -1, -11, 0, 0, 11], [0, 7, -8, -10, 0, 0, -7, -7, 7, 0, 0, 0, -8, -8, 8, 8], [0, 0, 0, 0, 0, -2, 0, 6, 6, -8, -6, -6, 0, -6, -6, 0], [0, 0, -18, 0, -12, 18, 0, 0, 0, 12, -2, -2, -10, 0, 0, -10], [0, 0, 0, 0, 0, 6, 12, 1, -1, 0, 1, -1, 0, 6, -6, 0], [0, -10, -4, 0, 0, 4, 0, 0, 0, 0, 2, -2, -4, 0, 0, 4], [0, 0, -23, 0, -4, 23, 0, 0, 0, 4, -2, -2, 7, 0, 0, 7], [0, 0, 0, 0, 0, 15, 0, 3, 3, -7, -3, -3, 0, 1, 1, 0], [0, -12, -1, 12, 0, 0, 12, -3, 3, 0, 0, 0, -1, -1, 1, 1], [0, 0, -20, 0, 0, 0, 0, 4, 4, 0, 0, 0, -4, -4, -4, -4], [0, 0, -2, 0, 7, 0, 0, 7, 7, 0, 0, 0, -4, -4, -4, -4], [20, 2, 2, 0, 0, -2, 0, 0, 0, 0, 5, -5, 2, 0, 0, -2], [0, 0, 7, 0, 5, -7, 0, 0, 0, -5, -4, -4, 9, 0, 0, 9], [0, 0, -1, 0, -17, 0, 0, -2, -2, 0, 0, 0, 5, 5, 5, 5], [0, 0, 0, 0, 0, -7, 0, 4, 4, 11, -4, -4, 0, -9, -9, 0], [16, -16, 5, 0, 0, -5, 0, 0, 0, 0, 1, -1, 5, 0, 0, -5], [0, 0, -3, 24, 0, 0, 0, 3, -3, 0, 0, 0, -3, -3, 3, 3], [0, 8, 6, 4, 0, 0, -8, -1, 1, 0, 0, 0, 6, 6, -6, -6], [0, 0, 11, 0, 8, -11, 0, 0, 0, -8, -6, -6, -7, 0, 0, -7], [-8, 0, 0, -8, 0, -9, 0, -6, 6, 0, -6, 6, 0, -9, 9, 0], [0, 0, 16, 0, 2, 0, 0, -6, -6, 0, 0, 0, -4, -4, -4, -4], [0, 0, 8, 0, -6, -8, 0, 0, 0, 6, 2, 2, -8, 0, 0, -8], [0, 0, 40, 0, 12, -40, 0, 0, 0, -12, 0, 0, 0, 0, 0, 0], [0, 0, 10, -4, 0, 0, 0, -2, 2, 0, 0, 0, 10, 10, -10, -10], [0, 0, 0, 0, 0, 4, 0, 6, 6, -8, -6, -6, 0, -6, -6, 0], [0, 0, 0, 0, 0, 1, 0, 2, 2, 17, -2, -2, 0, -5, -5, 0], [8, 0, 0, 8, 0, 3, 0, 5, -5, 0, 5, -5, 0, 3, -3, 0], [0, 0, 0, 0, 0, 23, 0, -2, -2, 12, 2, 2, 0, 1, 1, 0], [-22, 4, -9, 0, 0, 9, 0, 0, 0, 0, -9, 9, -9, 0, 0, 9], [0, 0, 16, 0, 10, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0], [-8, 0, 0, -8, 0, -7, 9, -4, 4, 0, -4, 4, 0, -7, 7, 0], [0, -2, -2, 0, 0, 2, 0, 0, 0, 0, -5, 5, -2, 0, 0, 2], [0, 4, 13, 8, 0, 0, -4, -1, 1, 0, 0, 0, 13, 13, -13, -13], [0, 0, 0, 0, 0, 21, 0, -7, -7, 2, 7, 7, 0, -11, -11, 0], [-12, 0, 0, -12, 0, 1, 17, 0, 0, 0, 0, 0, 0, 1, -1, 0], [12, 4, 6, 0, 0, -6, 0, 0, 0, 0, 7, -7, 6, 0, 0, -6], [4, -2, -4, 0, 0, 4, 0, 0, 0, 0, 10, -10, -4, 0, 0, 4], [0, 0, 0, 0, 0, 5, -7, -4, 4, 0, -4, 4, 0, 5, -5, 0], [0, 0, 6, 0, -6, -6, 0, 0, 0, 6, 8, 8, -6, 0, 0, -6], [0, 0, -2, 32, 0, 0, 0, -3, 3, 0, 0, 0, -2, -2, 2, 2], [0, 0, -30, 0, -6, 0, 0, -8, -8, 0, 0, 0, -2, -2, -2, -2], [20, 12, -3, 0, 0, 3, 0, 0, 0, 0, -3, 3, -3, 0, 0, 3], [0, 0, -10, 0, 9, 10, 0, 0, 0, -9, 9, 9, 4, 0, 0, 4], [0, 0, -23, 0, -11, 0, 0, -3, -3, 0, 0, 0, -1, -1, -1, -1], [0, 0, 0, 0, 0, 5, 0, 7, 7, -6, -7, -7, 0, -5, -5, 0], [-8, 0, 0, -8, 0, 0, -4, -9, 9, 0, -9, 9, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, -19, 0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, -11, 0, 3, 3, 2, -3, -3, 0, 7, 7, 0], [0, -20, 7, 16, 0, 0, 20, 9, -9, 0, 0, 0, 7, 7, -7, -7], [0, 0, 0, 0, 0, -2, -18, 4, -4, 0, 4, -4, 0, -2, 2, 0], [0, 0, -19, 0, -28, 0, 0, -2, -2, 0, 0, 0, -1, -1, -1, -1], [0, 0, 16, 0, 6, -16, 0, 0, 0, -6, 2, 2, -12, 0, 0, -12], [0, 4, -4, 28, 0, 0, -4, 8, -8, 0, 0, 0, -4, -4, 4, 4], [0, -16, 2, 8, 0, 0, 16, 7, -7, 0, 0, 0, 2, 2, -2, -2], [0, 0, 0, 0, 0, 14, 0, -4, -4, 18, 4, 4, 0, 6, 6, 0], [0, 0, 0, 0, 0, -21, 0, 3, 3, 23, -3, -3, 0, -3, -3, 0], [0, 0, 0, 0, 0, -6, 0, -2, 2, 0, -2, 2, 0, -6, 6, 0], [0, 0, -18, 0, -19, 0, 0, -3, -3, 0, 0, 0, -12, -12, -12, -12], [0, 0, -9, 0, 9, 9, 0, 0, 0, -9, 5, 5, -7, 0, 0, -7], [0, 0, -17, 0, -7, 0, 0, 4, 4, 0, 0, 0, -3, -3, -3, -3], [26, 0, 0, 26, 0, 0, 17, 9, -9, 0, 9, -9, 0, 0, 0, 0], [4, -5, 15, 0, 0, -15, 0, 0, 0, 0, 2, -2, 15, 0, 0, -15], [0, 0, -18, 0, 22, 0, 0, 11, 11, 0, 0, 0, -6, -6, -6, -6], [-24, 0, 0, -24, 0, 0, 5, 6, -6, 0, 6, -6, 0, 0, 0, 0], [-12, -16, -2, 0, 0, 2, 0, 0, 0, 0, -10, 10, -2, 0, 0, 2], [0, 0, 0, 0, 0, 4, 0, -5, -5, 4, 5, 5, 0, -4, -4, 0], [0, -13, 9, 20, 0, 0, 13, 4, -4, 0, 0, 0, 9, 9, -9, -9], [0, 0, 26, 0, -4, 0, 0, -2, -2, 0, 0, 0, 2, 2, 2, 2], [6, 0, 2, 0, 0, -2, 0, 0, 0, 0, 10, -10, 2, 0, 0, -2], [12, 0, 0, 12, 0, 9, 8, -1, 1, 0, -1, 1, 0, 9, -9, 0], [0, 0, -11, 0, -4, 0, 0, 11, 11, 0, 0, 0, 5, 5, 5, 5], [-16, -16, -4, 0, 0, 4, 0, 0, 0, 0, -3, 3, -4, 0, 0, 4], [0, 0, 4, 0, -12, -4, 0, 0, 0, 12, -4, -4, -4, 0, 0, -4], [0, 0, 28, 0, -6, 0, 0, 2, 2, 0, 0, 0, -8, -8, -8, -8], [-24, 0, 0, -24, 0, 4, -6, 2, -2, 0, 2, -2, 0, 4, -4, 0], [0, -3, 2, -8, 0, 0, 3, 2, -2, 0, 0, 0, 2, 2, -2, -2], [0, 0, 0, 0, 0, -18, 0, -4, -4, 22, 4, 4, 0, 2, 2, 0], [36, 0, 0, 36, 0, 4, 18, 1, -1, 0, 1, -1, 0, 4, -4, 0], [0, 0, 9, 0, 18, 0, 0, -1, -1, 0, 0, 0, -1, -1, -1, -1], [0, 0, 8, 0, 0, -8, 0, 0, 0, 0, 2, 2, 6, 0, 0, 6], [0, 12, 5, 12, 0, 0, -12, 7, -7, 0, 0, 0, 5, 5, -5, -5], [12, -9, -5, 0, 0, 5, 0, 0, 0, 0, 2, -2, -5, 0, 0, 5], [0, -34, 4, -6, 0, 0, 34, -2, 2, 0, 0, 0, 4, 4, -4, -4], [0, 0, 20, 0, 10, 0, 0, 10, 10, 0, 0, 0, 4, 4, 4, 4], [0, 0, -20, 0, -2, 20, 0, 0, 0, 2, -6, -6, 0, 0, 0, 0], [28, -8, 11, 0, 0, -11, 0, 0, 0, 0, -5, 5, 11, 0, 0, -11], [0, 0, -15, 0, 7, 0, 0, 1, 1, 0, 0, 0, -9, -9, -9, -9], [8, 0, 0, 8, 0, -5, 12, -3, 3, 0, -3, 3, 0, -5, 5, 0], [0, 0, 36, 0, 16, 0, 0, -4, -4, 0, 0, 0, -12, -12, -12, -12], [0, 3, -1, 16, 0, 0, -3, 6, -6, 0, 0, 0, -1, -1, 1, 1], [0, 0, 0, 0, 0, 6, 0, 5, 5, 13, -5, -5, 0, 12, 12, 0], [-20, 0, 0, -20, 0, -3, -2, -3, 3, 0, -3, 3, 0, -3, 3, 0], [-8, 4, 4, 0, 0, -4, 0, 0, 0, 0, 4, -4, 4, 0, 0, -4], [6, 0, 0, 6, 0, 2, -16, 6, -6, 0, 6, -6, 0, 2, -2, 0], [0, 0, 8, 0, -8, -8, 0, 0, 0, 8, 12, 12, 4, 0, 0, 4], [0, 0, 0, 0, 0, -1, 0, -2, -2, -25, 2, 2, 0, -3, -3, 0], [0, -39, 0, 8, 0, 0, 39, 4, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, -27, 0, 6, 0, 0, 7, 7, 0, 0, 0, 7, 7, 7, 7], [0, 17, 5, 0, 0, -5, 0, 0, 0, 0, 2, -2, 5, 0, 0, -5], [-32, 2, -2, 0, 0, 2, 0, 0, 0, 0, -4, 4, -2, 0, 0, 2], [0, 0, 0, 0, 0, 6, 0, -12, -12, -10, 12, 12, 0, 6, 6, 0], [0, -18, 9, 4, 0, 0, 18, -3, 3, 0, 0, 0, 9, 9, -9, -9], [0, 0, 0, 0, 0, -6, 0, 6, 6, -20, -6, -6, 0, -10, -10, 0], [0, -6, -2, 10, 0, 0, 6, 16, -16, 0, 0, 0, -2, -2, 2, 2], [0, 0, 43, 0, -7, -43, 0, 0, 0, 7, -1, -1, -3, 0, 0, -3], [0, 6, 2, 24, 0, 0, -6, 0, 0, 0, 0, 0, 2, 2, -2, -2], [0, 0, -14, 0, 10, 14, 0, 0, 0, -10, 8, 8, -2, 0, 0, -2], [-20, 0, 0, -20, 0, 5, 8, 3, -3, 0, 3, -3, 0, 5, -5, 0], [4, 0, 0, 4, 0, 6, 26, 8, -8, 0, 8, -8, 0, 6, -6, 0], [0, 0, -20, 0, 0, 20, 0, 0, 0, 0, 4, 4, 12, 0, 0, 12], [0, 0, 9, 0, -17, -9, 0, 0, 0, 17, 4, 4, -9, 0, 0, -9], [0, -14, -4, -28, 0, 0, 14, -5, 5, 0, 0, 0, -4, -4, 4, 4], [0, 0, 0, 0, 0, -4, 0, 0, 0, -4, 0, 0, 0, -4, -4, 0], [0, 15, 2, 0, 0, -2, 0, 0, 0, 0, -2, 2, 2, 0, 0, -2], [0, 0, 0, 0, 0, -16, 0, -7, -7, -4, 7, 7, 0, 0, 0, 0], [0, 5, -3, -24, 0, 0, -5, 2, -2, 0, 0, 0, -3, -3, 3, 3], [-12, 0, 0, -12, 0, -1, 2, 3, -3, 0, 3, -3, 0, -1, 1, 0], [0, 0, 0, 0, 0, -31, 0, -5, -5, 6, 5, 5, 0, -5, -5, 0], [0, 0, 2, 0, 4, -2, 0, 0, 0, -4, 5, 5, 14, 0, 0, 14], [0, 0, -5, 0, 12, 0, 0, -4, -4, 0, 0, 0, 9, 9, 9, 9], [0, 0, 0, 0, 0, -12, 0, -8, -8, -24, 8, 8, 0, 4, 4, 0], [0, 0, -32, 0, 12, 32, 0, 0, 0, -12, 4, 4, 4, 0, 0, 4], [-16, 0, 0, -16, 0, 4, 16, -3, 3, 0, -3, 3, 0, 4, -4, 0], [0, 0, -2, 0, 22, 2, 0, 0, 0, -22, 7, 7, -6, 0, 0, -6], [0, 0, 0, 0, 0, 5, 0, 9, 9, 5, -9, -9, 0, -5, -5, 0], [0, 16, 3, 36, 0, 0, -16, 1, -1, 0, 0, 0, 3, 3, -3, -3], [-18, 9, -6, 0, 0, 6, 0, 0, 0, 0, 3, -3, -6, 0, 0, 6], [-12, -20, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0], [0, 0, -27, 0, -1, 27, 0, 0, 0, 1, -12, -12, 3, 0, 0, 3], [0, 0, 0, 0, 0, -36, 0, 12, 12, 18, -12, -12, 0, -10, -10, 0], [0, 22, -2, 0, 0, 0, -22, 5, -5, 0, 0, 0, -2, -2, 2, 2], [0, 0, 0, 0, 0, -3, 0, 14, 14, 29, -14, -14, 0, 3, 3, 0], [0, 0, 0, 0, 0, 35, 0, -4, -4, -14, 4, 4, 0, -7, -7, 0], [0, -24, -10, -24, 0, 0, 24, 2, -2, 0, 0, 0, -10, -10, 10, 10], [0, 0, -19, 0, -26, 19, 0, 0, 0, 26, 0, 0, -9, 0, 0, -9], [0, 0, -2, 0, 0, 0, 0, -10, -10, 0, 0, 0, 2, 2, 2, 2], [0, 0, -36, 0, -2, 0, 0, 10, 10, 0, 0, 0, 8, 8, 8, 8], [-30, 0, 0, -30, 0, 8, 16, -8, 8, 0, -8, 8, 0, 8, -8, 0], [0, -16, 3, 34, 0, 0, 16, 1, -1, 0, 0, 0, 3, 3, -3, -3], [0, 4, -1, -52, 0, 0, -4, 5, -5, 0, 0, 0, -1, -1, 1, 1], [0, 0, 0, 0, 0, 7, 0, -2, -2, 31, 2, 2, 0, 5, 5, 0], [24, 0, 0, 24, 0, -10, -28, 2, -2, 0, 2, -2, 0, -10, 10, 0], [0, 0, -35, 0, 5, 35, 0, 0, 0, -5, -6, -6, -1, 0, 0, -1], [-32, 24, -4, 0, 0, 4, 0, 0, 0, 0, -5, 5, -4, 0, 0, 4], [0, 0, 15, 0, -6, 0, 0, -15, -15, 0, 0, 0, 5, 5, 5, 5], [0, 14, -13, 8, 0, 0, -14, -5, 5, 0, 0, 0, -13, -13, 13, 13], [0, 0, 0, 0, 0, -18, 0, -8, -8, 18, 8, 8, 0, 2, 2, 0], [0, 0, 17, 0, 14, -17, 0, 0, 0, -14, -5, -5, 11, 0, 0, 11], [-12, 0, 0, -12, 0, 9, -7, -20, 20, 0, -20, 20, 0, 9, -9, 0], [12, 0, 10, 0, 0, -10, 0, 0, 0, 0, 11, -11, 10, 0, 0, -10], [-34, 0, 0, -34, 0, 2, 5, -13, 13, 0, -13, 13, 0, 2, -2, 0], [0, 0, 0, 0, 0, -14, 0, 6, 6, 28, -6, -6, 0, 2, 2, 0], [0, 0, 34, 0, 24, 0, 0, 4, 4, 0, 0, 0, 4, 4, 4, 4], [-4, 0, 11, 0, 0, -11, 0, 0, 0, 0, -1, 1, 11, 0, 0, -11], [0, 0, -5, 0, -29, 5, 0, 0, 0, 29, 2, 2, -11, 0, 0, -11], [-32, 0, 0, -32, 0, -3, -16, 6, -6, 0, 6, -6, 0, -3, 3, 0], [0, -5, 1, -28, 0, 0, 5, 8, -8, 0, 0, 0, 1, 1, -1, -1], [0, 0, -12, 0, 2, 12, 0, 0, 0, -2, 1, 1, 12, 0, 0, 12], [0, 12, 2, -28, 0, 0, -12, -10, 10, 0, 0, 0, 2, 2, -2, -2], [0, 0, -16, 0, 8, 16, 0, 0, 0, -8, 12, 12, 4, 0, 0, 4], [0, 0, 25, 0, 12, 0, 0, -8, -8, 0, 0, 0, 3, 3, 3, 3], [0, 0, 42, 0, -12, 0, 0, 12, 12, 0, 0, 0, -8, -8, -8, -8], [-8, 0, 0, -8, 0, -8, -12, 0, 0, 0, 0, 0, 0, -8, 8, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1470_n_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1470_2_n_k();".
// 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_1470_2_n_k(:prec:=16)
    chi := MakeCharacter_1470_n();
    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_1470_2_n_k();".
// 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_1470_2_n_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1470_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![1024, 512, 832, -288, 292, -32, 18, 0, 1]>,<17,R![236421376, 0, -123008000, 0, 43457664, 0, -8227840, 0, 1129520, 0, -90880, 0, 5064, 0, -80, 0, 1]>,<19,R![200704, -100352, 55552, -8064, 3280, -592, 132, -12, 1]>,<31,R![4096, 8192, 11392, 7936, 3972, 992, 178, 16, 1]>],Snew);
    return Vf;
end function;