
// Make newform 1470.2.n.j in Magma, downloaded from the LMFDB on 25 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_j();".
// 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_j();".
// 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 := [64, -192, 288, -112, 84, -216, 368, -108, 21, -14, 18, -6, 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], [11668096, -7541936, -2644000, 4101028, 25802216, -5356872, 192316, 26475, 1200782, -320538, 32246, 10967], [376704, 29588496, -10074032, 7241340, -5540672, 23734728, -7374100, 1522973, -559878, 1115946, -395262, 71537], [2629264, 4633520, -1196060, 2618720, 9778420, 8245842, -3125365, 552915, 334360, 367550, -170111, 38425], [21343872, -29339888, 11904288, -14480484, 15651488, -31186560, 9354400, -2126039, 1056954, -1490814, 504878, -87643], [-1612208, 15880448, -690028, -1987220, 801716, 13520382, -4079825, 510126, -106362, 629192, -237241, 44272], [-33288832, 32776656, -21185216, -17165628, -54894792, 52377004, -14110440, 1600117, -2969054, 2628966, -805130, 112417], [-20220800, 31227728, -6351360, 15014452, -24737320, 46490256, -13565876, 2890131, -1687650, 2298830, -773178, 131935], [-16768, 54704, -36720, 228, -20464, 75424, -23804, 3655, -1778, 3622, -1290, 227], [9604288, -12722696, 7880932, 5302254, 10669054, -16904415, 4941719, -534986, 642506, -829457, 276857, -43824], [-2470736, -17921792, 14559756, 1994316, -14186852, -34275032, 11871603, -1660469, -220844, -1597836, 645645, -128463], [782560, -13207560, 10439620, -1154946, -800286, -24050043, 7919495, -1234124, 259810, -1148873, 432381, -80946]];
    Rf_basisdens := [1, 6321728, 6321728, 3160864, 6321728, 3160864, 6321728, 6321728, 8768, 1580432, 3160864, 1580432];
    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], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 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, -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], [0, 0, 1, 0, -1, 0, -1, 1, -1, 0, 0, -1], [0, 1, 0, -1, 0, 1, 1, 1, 0, 0, -1, 0], [0, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2], [-1, 0, -1, 0, 1, 0, 0, 0, 1, 1, 0, 0], [0, 0, 0, -3, 0, -1, 0, 0, 0, 0, 1, 0], [-4, 0, 0, 0, 0, -1, -1, 1, 0, 0, -1, 0], [0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 3, 0, 1, 0, 0, 0, 0, -1, 0], [3, 0, 0, 0, 0, -2, -2, 2, 0, -1, -2, 1], [0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, -3, 0, 0, 0, 0, 3, 0], [0, -3, 2, 0, 2, 0, 2, 2, 0, 0, 0, 2], [0, 0, 2, 0, -2, 0, 1, -1, 4, 0, 0, -2], [2, 0, -2, 0, 2, 2, 0, 0, -2, 2, 2, 0], [0, 2, -2, 0, -2, 0, 2, 2, 0, 0, 0, -2], [6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, 0, -1, 0, 1, 0, 0, 0, -8, 1, 0, 0], [0, 4, 0, -4, 0, 1, 1, 1, 0, -2, -1, -2], [2, 0, 2, 0, -2, 2, 0, 0, -2, -2, 2, 0], [0, 8, 0, -8, 0, 2, 2, 2, 0, 1, -2, 1], [0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0], [0, 0, -2, -8, -2, 0, 0, 0, 0, 2, 0, 0], [0, 0, -2, 6, -2, -1, 0, 0, 0, 2, 1, 0], [0, 0, 0, 0, 0, 0, -4, 4, 6, 0, 0, 0], [0, 2, 0, -2, 0, 2, 2, 2, 0, 0, -2, 0], [0, 9, 0, -9, 0, -1, -1, -1, 0, 3, 1, 3], [-3, 0, 1, 0, -1, -1, 0, 0, 3, -1, -1, 0], [0, 0, 4, 0, 4, 0, -2, -2, 0, 0, 0, 4], [12, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 2], [14, 0, -2, 0, 2, 0, 0, 0, -14, 2, 0, 0], [0, 0, -3, 0, 3, 0, 0, 0, -4, 0, 0, 3], [0, 9, -2, 0, -2, 0, -1, -1, 0, 0, 0, -2], [0, 0, -2, -8, -2, 2, 0, 0, 0, 2, -2, 0], [0, -5, 0, 5, 0, -1, -1, -1, 0, -2, 1, -2], [0, 0, 2, -1, 2, 1, 0, 0, 0, -2, -1, 0], [0, 0, 3, 0, -3, 0, 0, 0, 7, 0, 0, -3], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, -6], [0, 0, -4, 0, 4, 4, 0, 0, 0, 4, 4, 0], [0, -10, -1, 0, -1, 0, 0, 0, 0, 0, 0, -1], [0, 5, 0, -5, 0, 1, 1, 1, 0, -2, -1, -2], [0, 0, 2, 0, -2, 0, 0, 0, 10, 0, 0, -2], [-7, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1], [0, -4, 0, 4, 0, -4, -4, -4, 0, 1, 4, 1], [0, 12, 2, 0, 2, 0, 1, 1, 0, 0, 0, 2], [12, 0, 0, 0, 0, -4, 0, 0, -12, 0, -4, 0], [0, 0, -2, 4, -2, 2, 0, 0, 0, 2, -2, 0], [-8, 0, 0, 0, 0, -2, -2, 2, 0, -2, -2, 2], [0, 0, -2, 0, 2, 0, 0, 0, 3, 0, 0, 2], [1, 0, 0, 0, 0, 5, 5, -5, 0, -5, 5, 5], [0, 0, 4, -2, 4, 4, 0, 0, 0, -4, -4, 0], [0, -8, 2, 0, 2, 0, -4, -4, 0, 0, 0, 2], [0, 0, -2, 0, 2, 0, -1, 1, 10, 0, 0, 2], [-10, 0, -1, 0, 1, -4, 0, 0, 10, 1, -4, 0], [0, 14, 4, 0, 4, 0, -2, -2, 0, 0, 0, 4], [7, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1], [0, 2, 4, 0, 4, 0, -4, -4, 0, 0, 0, 4], [0, 7, 0, -7, 0, 0, 0, 0, 0, 2, 0, 2], [0, 10, 0, -10, 0, -4, -4, -4, 0, -2, 4, -2], [0, 0, 2, 0, -2, 0, 0, 0, -14, 0, 0, -2], [0, 0, 1, -18, 1, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, -8, 0, 3, 0, 0, 0, 0, -3, 0], [-11, 0, -5, 0, 5, -2, 0, 0, 11, 5, -2, 0], [0, 2, 0, -2, 0, -4, -4, -4, 0, 1, 4, 1], [0, -8, -4, 0, -4, 0, 0, 0, 0, 0, 0, -4], [12, 0, 0, 0, 0, 4, 4, -4, 0, 2, 4, -2], [0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 4], [4, 0, 0, 0, 0, -2, 0, 0, -4, 0, -2, 0], [0, -13, -1, 0, -1, 0, -3, -3, 0, 0, 0, -1], [0, 0, 8, 2, 8, 2, 0, 0, 0, -8, -2, 0], [-5, 0, 0, 0, 0, 6, 6, -6, 0, 3, 6, -3], [0, 0, -2, -9, -2, 1, 0, 0, 0, 2, -1, 0], [0, 0, 2, 0, -2, 0, 4, -4, 10, 0, 0, -2], [0, 0, 4, 18, 4, -2, 0, 0, 0, -4, 2, 0], [1, 0, 7, 0, -7, -2, 0, 0, -1, -7, -2, 0], [0, 0, 1, 0, -1, 0, 2, -2, -4, 0, 0, -1], [9, 0, 0, 0, 0, 0, 0, 0, 0, -7, 0, 7], [-4, 0, 0, 0, 0, 6, 6, -6, 0, -4, 6, 4], [0, 0, -6, 0, 6, 0, -4, 4, -6, 0, 0, 6], [0, -12, 0, 12, 0, -4, -4, -4, 0, 0, 4, 0], [18, 0, -3, 0, 3, 0, 0, 0, -18, 3, 0, 0], [0, 0, 6, 4, 6, -1, 0, 0, 0, -6, 1, 0], [-7, 0, 0, 0, 0, 4, 4, -4, 0, -1, 4, 1], [0, 0, -1, 2, -1, 0, 0, 0, 0, 1, 0, 0], [4, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, -8], [0, 3, 0, -3, 0, 5, 5, 5, 0, -4, -5, -4], [0, 0, 4, -10, 4, -6, 0, 0, 0, -4, 6, 0], [0, 0, -2, 0, 2, 0, -8, 8, 4, 0, 0, 2], [0, 20, -3, 0, -3, 0, -4, -4, 0, 0, 0, -3], [8, 0, 0, 0, 0, -3, -3, 3, 0, -2, -3, 2], [8, 0, 2, 0, -2, 0, 0, 0, -8, -2, 0, 0], [0, 4, 0, -4, 0, 2, 2, 2, 0, -2, -2, -2], [-14, 0, 6, 0, -6, -5, 0, 0, 14, -6, -5, 0], [0, 0, 1, 0, -1, 0, -6, 6, -9, 0, 0, -1], [0, 0, 2, -24, 2, 6, 0, 0, 0, -2, -6, 0], [-12, 0, 4, 0, -4, 6, 0, 0, 12, -4, 6, 0], [0, -2, 0, 2, 0, -4, -4, -4, 0, -10, 4, -10], [0, 5, -6, 0, -6, 0, 2, 2, 0, 0, 0, -6], [0, 8, -6, 0, -6, 0, 1, 1, 0, 0, 0, -6], [-5, 0, 7, 0, -7, 4, 0, 0, 5, -7, 4, 0], [0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0], [0, -8, 7, 0, 7, 0, -2, -2, 0, 0, 0, 7], [0, -28, 0, 28, 0, 1, 1, 1, 0, 2, -1, 2], [0, 0, 2, 2, 2, -4, 0, 0, 0, -2, 4, 0], [0, 0, -4, 0, 4, 0, -6, 6, 4, 0, 0, 4], [-16, 0, 0, 0, 0, 2, 2, -2, 0, -5, 2, 5], [0, 0, 7, 19, 7, -1, 0, 0, 0, -7, 1, 0], [0, -29, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0], [0, -26, 0, 26, 0, 2, 2, 2, 0, 2, -2, 2], [0, 0, -9, 0, 9, 0, 4, -4, 11, 0, 0, 9], [12, 0, 0, 0, 0, -4, -4, 4, 0, 7, -4, -7], [0, 0, -3, 0, 3, 0, -2, 2, -1, 0, 0, 3], [0, 6, 0, -6, 0, 10, 10, 10, 0, -2, -10, -2], [0, -13, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0], [0, 0, -6, 21, -6, -4, 0, 0, 0, 6, 4, 0], [-20, 0, 0, 0, 0, 2, 2, -2, 0, 0, 2, 0], [0, 0, -8, 0, 8, 0, 2, -2, 0, 0, 0, 8], [0, 4, 0, -4, 0, -6, -6, -6, 0, -7, 6, -7], [0, 0, -2, -23, -2, 0, 0, 0, 0, 2, 0, 0], [0, -2, -6, 0, -6, 0, 9, 9, 0, 0, 0, -6], [-4, 0, 0, 0, 0, -8, 0, 0, 4, 0, -8, 0], [2, 0, 0, 0, 0, -1, -1, 1, 0, 2, -1, -2], [4, 0, -10, 0, 10, -4, 0, 0, -4, 10, -4, 0], [-18, 0, -2, 0, 2, 10, 0, 0, 18, 2, 10, 0], [0, 11, 0, -11, 0, 9, 9, 9, 0, -5, -9, -5], [0, 0, 2, -5, 2, -7, 0, 0, 0, -2, 7, 0], [0, 0, -5, 0, 5, 0, -6, 6, 9, 0, 0, 5], [0, 31, 0, -31, 0, -1, -1, -1, 0, -2, 1, -2], [6, 0, -3, 0, 3, 8, 0, 0, -6, 3, 8, 0], [0, -14, 0, 14, 0, 4, 4, 4, 0, 4, -4, 4], [19, 0, -5, 0, 5, 0, 0, 0, -19, 5, 0, 0], [15, 0, 0, 0, 0, -4, -4, 4, 0, 2, -4, -2], [0, -3, -6, 0, -6, 0, -3, -3, 0, 0, 0, -6], [0, -22, -4, 0, -4, 0, 0, 0, 0, 0, 0, -4], [0, -12, 0, 12, 0, 9, 9, 9, 0, 8, -9, 8], [0, 0, -3, 0, 3, 0, -2, 2, -11, 0, 0, 3], [-5, 0, 0, 0, 0, -10, -10, 10, 0, -3, -10, 3], [-6, 0, -6, 0, 6, 1, 0, 0, 6, 6, 1, 0], [0, -16, 2, 0, 2, 0, -8, -8, 0, 0, 0, 2], [0, 0, 0, 0, 0, -1, -1, -1, 0, 6, 1, 6], [0, 0, 0, 0, 0, 0, -2, 2, -2, 0, 0, 0], [10, 0, 0, 0, 0, -2, -2, 2, 0, 4, -2, -4], [0, -13, 0, 13, 0, 1, 1, 1, 0, 4, -1, 4], [0, 16, -6, 0, -6, 0, -8, -8, 0, 0, 0, -6], [-4, 0, 4, 0, -4, -6, 0, 0, 4, -4, -6, 0], [0, 0, 2, 9, 2, -5, 0, 0, 0, -2, 5, 0], [0, 0, -2, -23, -2, -3, 0, 0, 0, 2, 3, 0], [25, 0, 0, 0, 0, 4, 4, -4, 0, -3, 4, 3], [0, -4, 0, 4, 0, 4, 4, 4, 0, 8, -4, 8], [0, 0, -10, 16, -10, 6, 0, 0, 0, 10, -6, 0], [0, 12, 6, 0, 6, 0, -6, -6, 0, 0, 0, 6], [-6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-24, 0, 4, 0, -4, -4, 0, 0, 24, -4, -4, 0], [-15, 0, -11, 0, 11, 2, 0, 0, 15, 11, 2, 0], [0, 34, 0, -34, 0, 0, 0, 0, 0, 3, 0, 3], [0, 0, -8, 0, 8, 0, -1, 1, 12, 0, 0, 8], [0, 0, 0, 12, 0, -12, 0, 0, 0, 0, 12, 0], [0, -16, 0, 16, 0, 8, 8, 8, 0, -2, -8, -2], [0, 4, 0, -4, 0, 4, 4, 4, 0, 5, -4, 5], [21, 0, -7, 0, 7, 7, 0, 0, -21, 7, 7, 0], [0, -2, 6, 0, 6, 0, -2, -2, 0, 0, 0, 6], [0, 3, 2, 0, 2, 0, 1, 1, 0, 0, 0, 2], [0, 0, 7, 0, -7, 0, 0, 0, -2, 0, 0, -7], [0, 6, -10, 0, -10, 0, 2, 2, 0, 0, 0, -10], [1, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, -6], [0, 0, 6, -27, 6, 6, 0, 0, 0, -6, -6, 0], [0, 0, -6, 0, 6, 0, 7, -7, -8, 0, 0, 6], [-14, 0, 0, 0, 0, -2, -2, 2, 0, 6, -2, -6], [6, 0, 4, 0, -4, -2, 0, 0, -6, -4, -2, 0], [0, -8, 5, 0, 5, 0, 10, 10, 0, 0, 0, 5], [0, 0, 5, 0, -5, 0, 4, -4, 16, 0, 0, -5], [-18, 0, 0, 0, 0, 4, 4, -4, 0, -2, 4, 2], [1, 0, 0, 0, 0, -8, -8, 8, 0, -7, -8, 7], [0, 0, -4, 0, 4, 0, 5, -5, 12, 0, 0, 4], [0, 21, 0, -21, 0, -5, -5, -5, 0, -4, 5, -4], [0, 0, -8, 0, 8, -2, 0, 0, 0, 8, -2, 0], [0, 0, 5, 27, 5, -1, 0, 0, 0, -5, 1, 0], [-17, 0, 0, 0, 0, 9, 9, -9, 0, 1, 9, -1], [0, 21, 0, -21, 0, -5, -5, -5, 0, -4, 5, -4], [0, 0, -12, -4, -12, -2, 0, 0, 0, 12, 2, 0], [0, 15, -6, 0, -6, 0, -5, -5, 0, 0, 0, -6], [0, 0, -2, 0, 2, 0, -1, 1, -38, 0, 0, 2], [0, 27, -6, 0, -6, 0, 7, 7, 0, 0, 0, -6], [0, 0, -6, 0, -6, 0, 8, 8, 0, 0, 0, -6], [-15, 0, 2, 0, -2, -4, 0, 0, 15, -2, -4, 0], [0, 0, -8, 0, 8, 0, 2, -2, 2, 0, 0, 8], [0, 0, 3, 24, 3, 6, 0, 0, 0, -3, -6, 0], [0, 22, 0, -22, 0, -7, -7, -7, 0, 6, 7, 6], [-11, 0, 5, 0, -5, -12, 0, 0, 11, -5, -12, 0], [-12, 0, -4, 0, 4, -8, 0, 0, 12, 4, -8, 0], [0, -16, 10, 0, 10, 0, 3, 3, 0, 0, 0, 10], [0, -14, 8, 0, 8, 0, 0, 0, 0, 0, 0, 8], [0, 0, -5, 0, 5, 0, -2, 2, -27, 0, 0, 5], [0, 0, -12, -10, -12, 4, 0, 0, 0, 12, -4, 0], [0, -2, 0, 2, 0, -6, -6, -6, 0, -10, 6, -10], [0, 0, -14, 0, -14, 4, 0, 0, 0, 14, -4, 0], [0, 0, -8, 0, 8, 0, -5, 5, -36, 0, 0, 8], [50, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 2], [0, 0, -2, 26, -2, -6, 0, 0, 0, 2, 6, 0], [0, 0, 3, 0, -3, 0, -6, 6, 5, 0, 0, -3], [-36, 0, 0, 0, 0, 10, 10, -10, 0, 0, 10, 0], [0, 11, -2, 0, -2, 0, 0, 0, 0, 0, 0, -2], [30, 0, 6, 0, -6, 0, 0, 0, -30, -6, 0, 0], [0, 0, 6, -32, 6, 13, 0, 0, 0, -6, -13, 0], [-15, 0, 0, 0, 0, 4, 4, -4, 0, -9, 4, 9], [0, 0, 9, 0, -9, 0, -10, 10, 3, 0, 0, -9], [0, 0, 8, 2, 8, 10, 0, 0, 0, -8, -10, 0], [-10, 0, 0, 0, 0, -1, -1, 1, 0, 6, -1, -6], [0, -24, 0, 24, 0, 0, 0, 0, 0, -5, 0, -5], [0, 0, -2, 6, -2, -1, 0, 0, 0, 2, 1, 0], [0, 0, 10, 0, -10, 0, 2, -2, 0, 0, 0, -10], [-15, 0, -7, 0, 7, 6, 0, 0, 15, 7, 6, 0], [0, -9, -3, 0, -3, 0, -7, -7, 0, 0, 0, -3], [0, 0, 9, 0, -9, 0, 12, -12, 7, 0, 0, -9], [0, 0, -6, 17, -6, 3, 0, 0, 0, 6, -3, 0], [0, 31, 0, -31, 0, -1, -1, -1, 0, 6, 1, 6], [12, 0, -4, 0, 4, -8, 0, 0, -12, 4, -8, 0], [-4, 0, 0, 0, 0, 8, 8, -8, 0, 4, 8, -4], [-11, 0, -11, 0, 11, -6, 0, 0, 11, 11, -6, 0], [0, 0, 3, -25, 3, -13, 0, 0, 0, -3, 13, 0], [0, -22, 0, 22, 0, -5, -5, -5, 0, 2, 5, 2], [4, 0, 0, 0, 0, 12, 12, -12, 0, -4, 12, 4], [0, 0, 4, -14, 4, -4, 0, 0, 0, -4, 4, 0], [0, 0, 2, 0, -2, 0, -4, 4, -26, 0, 0, -2], [0, 0, -5, 13, -5, 5, 0, 0, 0, 5, -5, 0], [20, 0, 0, 0, 0, 10, 0, 0, -20, 0, 10, 0], [0, -11, 6, 0, 6, 0, -7, -7, 0, 0, 0, 6], [0, 0, 7, 0, -7, 0, 2, -2, -33, 0, 0, -7], [-30, 0, 0, 0, 0, -8, -8, 8, 0, -3, -8, 3], [0, 0, 5, 0, -5, 0, -4, 4, 45, 0, 0, -5], [0, 10, 0, -10, 0, -6, -6, -6, 0, -6, 6, -6], [0, 12, -6, 0, -6, 0, 2, 2, 0, 0, 0, -6], [-28, 0, -1, 0, 1, 6, 0, 0, 28, 1, 6, 0], [0, 0, -10, 27, -10, 8, 0, 0, 0, 10, -8, 0], [9, 0, 0, 0, 0, 12, 12, -12, 0, 9, 12, -9], [-22, 0, 0, 0, 0, 6, 6, -6, 0, -8, 6, 8], [0, 34, 4, 0, 4, 0, 2, 2, 0, 0, 0, 4], [11, 0, -11, 0, 11, 2, 0, 0, -11, 11, 2, 0], [0, -12, 1, 0, 1, 0, -4, -4, 0, 0, 0, 1], [-32, 0, 2, 0, -2, 4, 0, 0, 32, -2, 4, 0], [0, -2, 0, 2, 0, 8, 8, 8, 0, -10, -8, -10], [-18, 0, -6, 0, 6, 12, 0, 0, 18, 6, 12, 0], [0, 23, 0, -23, 0, -3, -3, -3, 0, 13, 3, 13], [0, 0, -3, 0, 3, 0, -12, 12, 33, 0, 0, 3], [0, 0, 12, 0, -12, 0, -4, 4, 4, 0, 0, -12], [0, -35, 0, 35, 0, -9, -9, -9, 0, -2, 9, -2], [0, -2, 0, 2, 0, -6, -6, -6, 0, 14, 6, 14], [-11, 0, 1, 0, -1, -14, 0, 0, 11, -1, -14, 0], [0, -43, 2, 0, 2, 0, 9, 9, 0, 0, 0, 2], [-59, 0, 0, 0, 0, -2, -2, 2, 0, -5, -2, 5], [0, 0, 0, 0, 0, 0, -7, -7, 0, 0, 0, 0], [-61, 0, -1, 0, 1, 3, 0, 0, 61, 1, 3, 0], [0, 0, -16, 0, 16, 0, -6, 6, 0, 0, 0, 16], [0, -8, -8, 0, -8, 0, 0, 0, 0, 0, 0, -8], [0, 7, 0, -7, 0, 2, 2, 2, 0, 14, -2, 14], [0, 0, -3, -14, -3, -12, 0, 0, 0, 3, 12, 0], [0, -11, 0, 0, 0, 0, -5, -5, 0, 0, 0, 0], [0, 18, 0, -18, 0, 13, 13, 13, 0, 2, -13, 2], [0, 0, -8, 0, 8, 0, -8, 8, -22, 0, 0, 8], [0, -14, 0, 14, 0, 2, 2, 2, 0, 0, -2, 0], [0, 36, 6, 0, 6, 0, 0, 0, 0, 0, 0, 6], [5, 0, 1, 0, -1, -6, 0, 0, -5, -1, -6, 0], [-30, 0, 0, 0, 0, -1, -1, 1, 0, 2, -1, -2], [15, 0, 0, 0, 0, -4, -4, 4, 0, -15, -4, 15], [0, 6, 0, -6, 0, 2, 2, 2, 0, 4, -2, 4], [0, 23, -6, 0, -6, 0, 8, 8, 0, 0, 0, -6], [-8, 0, 0, 0, 0, 8, 0, 0, 8, 0, 8, 0], [0, 8, 2, 0, 2, 0, -12, -12, 0, 0, 0, 2], [0, -24, -5, 0, -5, 0, 6, 6, 0, 0, 0, -5], [-26, 0, 3, 0, -3, 12, 0, 0, 26, -3, 12, 0], [0, 38, 0, -38, 0, -4, -4, -4, 0, -7, 4, -7], [0, 0, 8, -1, 8, -5, 0, 0, 0, -8, 5, 0], [0, 0, 10, 12, 10, 7, 0, 0, 0, -10, -7, 0], [0, 0, -14, 0, 14, 0, -2, 2, -4, 0, 0, 14], [29, 0, -3, 0, 3, -2, 0, 0, -29, 3, -2, 0], [51, 0, 5, 0, -5, 4, 0, 0, -51, -5, 4, 0], [0, 39, 4, 0, 4, 0, 1, 1, 0, 0, 0, 4], [0, 0, 3, 0, -3, 0, 16, -16, 2, 0, 0, -3], [0, -47, 0, 47, 0, -3, -3, -3, 0, 0, 3, 0], [20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -6, 24, -6, -8, 0, 0, 0, 6, 8, 0], [-20, 0, 6, 0, -6, -6, 0, 0, 20, -6, -6, 0], [0, 12, 3, 0, 3, 0, 6, 6, 0, 0, 0, 3], [0, 15, 0, -15, 0, 8, 8, 8, 0, -14, -8, -14], [0, 0, -7, 0, 7, 0, 0, 0, 14, 0, 0, 7], [9, 0, 0, 0, 0, -10, -10, 10, 0, -11, -10, 11], [0, 0, 2, 0, -2, 0, -8, 8, 30, 0, 0, -2], [0, -8, 2, 0, 2, 0, -9, -9, 0, 0, 0, 2], [0, 0, 8, -8, 8, 12, 0, 0, 0, -8, -12, 0], [11, 0, 0, 0, 0, -8, -8, 8, 0, 11, -8, -11], [0, 9, 0, -9, 0, 5, 5, 5, 0, -8, -5, -8], [0, 0, -12, 0, 12, 0, 3, -3, -12, 0, 0, 12], [-32, 0, 4, 0, -4, 12, 0, 0, 32, -4, 12, 0], [0, -6, 0, 6, 0, 4, 4, 4, 0, -8, -4, -8], [-51, 0, -1, 0, 1, 0, 0, 0, 51, 1, 0, 0], [0, 2, 0, -2, 0, 12, 12, 12, 0, -8, -12, -8], [0, 0, -16, -20, -16, -4, 0, 0, 0, 16, 4, 0], [0, 0, 2, -45, 2, 4, 0, 0, 0, -2, -4, 0], [0, 0, -5, 0, 5, 0, 16, -16, -34, 0, 0, 5]];
    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_j();".
// 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_j(:prec:=12)
    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_j();".
// 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_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1470_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![2401, 1323, 876, 17, 36, 3, 1]>,<17,R![5473632256, 0, -411942912, 0, 21236736, 0, -587008, 0, 11856, 0, -132, 0, 1]>,<19,R![576, 288, 216, 12, 21, 3, 1]>,<31,R![100, 150, 225, 20, 15, 0, 1]>],Snew);
    return Vf;
end function;