
// Make newform 1470.2.b.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1470_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1470_b_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_b_a();".
// 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_b_a();".
// 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 := [729, -972, 891, -864, 576, -360, 237, -120, 64, -32, 11, -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, 0], [1701, -243, 351, -630, 63, -183, 123, -10, 35, -11, 1, -4], [1620, -837, 1521, -1413, 678, -651, 337, -153, 118, -39, 17, -7], [-4455, 2403, -2457, 2502, -1017, 891, -513, 170, -145, 43, -11, 8], [17010, -12879, 12879, -11925, 5976, -4533, 2631, -1145, 736, -265, 83, -35], [6885, -4644, 4878, -4635, 2157, -1734, 998, -415, 289, -98, 32, -15], [-3564, 2673, -2610, 2538, -1212, 927, -560, 228, -158, 57, -16, 8], [-7857, 5967, -6309, 5670, -2859, 2139, -1241, 542, -351, 127, -39, 16], [-6075, 4212, -4455, 4302, -2070, 1668, -969, 400, -284, 98, -31, 16], [-3888, 3078, -3348, 3123, -1638, 1230, -708, 317, -208, 76, -26, 11], [15633, -12771, 13005, -11916, 6135, -4599, 2657, -1176, 767, -279, 91, -38], [51516, -40743, 42147, -38385, 19782, -14865, 8475, -3809, 2458, -877, 287, -119]];
    Rf_basisdens := [1, 324, 324, 324, 972, 324, 162, 324, 243, 162, 324, 972];
    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_b();"
function MakeCharacter_1470_b()
    N := 1470;
    order := 2;
    char_gens := [491, 1177, 1081];
    v := [1, 2, 1];
    // 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_b_Hecke();"
function MakeCharacter_1470_b_Hecke(Kf)
    N := 1470;
    order := 2;
    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, 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, 1, 0, 0, 0, 0, 0, 0, 0], [0, -1, 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], [-1, 0, 1, 0, 1, -1, -1, 0, 0, -1, -1, 1], [0, 0, 0, 0, 0, 1, 1, 0, -1, 1, -1, -1], [3, -1, -1, 0, 0, 1, -1, 0, 0, 1, 0, 0], [2, -2, -2, 0, -2, 0, 1, 0, -1, 1, 2, -2], [0, -1, 0, 0, 0, -1, 1, 0, 1, 1, 0, 0], [-1, 0, 1, 0, 1, -2, 1, 0, -1, 1, 0, 2], [2, -1, -2, 0, 1, 1, -1, 0, -1, -1, 2, -2], [1, 0, 0, 1, 0, 0, 1, 1, 0, -2, 0, 0], [0, 0, 1, -3, 3, 0, -2, 3, 3, -1, 0, 0], [-1, 1, -3, 0, -1, -1, 2, -2, -1, 0, 0, 0], [2, 1, 1, 0, -2, -1, 2, -4, -2, 2, 0, 0], [1, 0, -1, 0, -1, 0, 3, 0, -4, 3, 2, 0], [0, 2, -2, -2, 0, -2, 2, -2, 0, 0, 0, 0], [-2, 3, 2, 0, -1, -1, 0, 0, -1, 0, 0, 4], [-1, 2, 1, 0, 0, -2, 1, 0, 0, -1, 0, 0], [1, -3, -1, 0, 2, 1, -1, 0, 0, -1, -2, -4], [1, 1, -1, 0, -4, 1, 5, 0, -2, 5, 2, 0], [-1, 1, -1, -2, 0, -1, 3, -2, 0, -1, 0, 0], [3, 0, 1, 0, 0, 0, 1, 0, 0, -1, 0, 0], [4, 1, -4, 2, -1, -1, 0, 0, -1, 0, 0, 0], [3, -1, -3, 0, 6, 3, -1, 0, -2, -1, 2, -4], [4, 1, 2, 0, -1, -1, -2, -2, -1, 4, 0, 0], [1, 2, -1, 0, 2, 2, 1, 0, 2, 1, 2, 0], [-3, 3, 3, 0, -3, -1, 4, 0, -1, 4, -2, 4], [-1, 2, 5, 0, -1, -2, 1, -2, -1, 1, 0, 0], [2, 1, -2, 0, -5, 1, 3, 0, -3, 3, 4, 0], [1, 1, -2, 1, 1, -1, 0, 3, 1, -3, 0, 0], [2, -2, -1, -1, 1, 2, -1, 1, 1, 0, 0, 0], [-2, -1, 2, 0, -1, -3, -1, 0, 1, -1, -2, 2], [0, 0, 0, 0, 6, 0, 2, 0, 2, 2, 0, 0], [3, -4, -3, 0, 9, 0, -3, 0, -1, -3, 2, -4], [0, 0, 2, 0, 4, 0, -6, 8, 4, -2, 0, 0], [-2, 2, 2, 0, 2, 1, -1, 0, -1, -1, -3, 1], [-2, 2, -2, 4, -2, -2, 0, 0, -2, 0, 0, 0], [-7, 1, -2, 2, -2, -1, 1, -2, -2, 1, 0, 0], [11, -6, 2, 4, -2, 6, 1, 0, -2, -1, 0, 0], [1, -2, -1, 0, 5, 1, -1, 0, -8, -1, -1, -3], [-2, -2, 2, 0, 8, -4, -3, 0, 2, -3, -2, 2], [3, -3, -3, 0, -6, -1, -1, 0, -4, -1, 4, -2], [4, 1, -4, 0, -3, -1, 5, -6, -3, 1, 0, 0], [-3, 6, 3, 0, 3, 2, -1, 0, 2, -1, -2, 4], [2, -6, -2, 0, 4, 0, -6, 0, 0, -6, -2, -6], [0, 0, 1, -6, 7, 0, -7, 8, 7, -1, 0, 0], [2, -2, -2, 0, 16, 0, -4, 0, 2, -4, 2, -2], [4, -4, 0, -2, 2, 4, -6, 2, 2, 4, 0, 0], [0, -3, 0, 0, 15, -3, -5, 0, 7, -5, 0, 0], [-2, 4, 2, 0, -4, 2, -6, 0, 2, -6, -2, 2], [-4, 4, 4, 0, -2, 0, -2, 0, 8, -2, -4, 4], [-2, -1, 2, 0, 5, -3, -6, 0, 2, -6, -2, 2], [-4, 4, 1, 3, -5, -4, 5, -7, -5, 2, 0, 0], [0, 0, 4, 2, 0, 0, 0, 2, 0, -2, 0, 0], [3, -2, -3, 0, 0, -4, -3, 0, 0, -3, 8, 2], [2, -3, -4, 2, 1, 3, -4, 4, 1, 0, 0, 0], [0, -4, 0, 0, 0, 0, -6, 0, 4, -6, -4, -4], [0, -4, 0, 0, 2, 4, -4, 4, 2, 0, 0, 0], [-2, -3, 2, 0, -1, -2, -6, 0, 0, -6, -5, -1], [0, 0, 0, 0, 12, 2, 2, 0, -2, 2, -2, -2], [-1, -2, -6, 0, -2, 2, 1, -4, -2, 3, 0, 0], [-2, 5, 2, 0, -4, 3, 2, 0, -4, 2, -2, 2], [-8, 2, 8, -2, 2, -2, -4, 2, 2, 2, 0, 0], [-4, 2, 4, 0, -2, 0, 2, 0, 6, 2, -6, 2], [2, 2, -2, 0, -2, 2, 2, 0, -10, 2, 4, 0], [2, 6, 5, -2, -1, -6, -1, -4, -1, 5, 0, 0], [16, -3, -2, 4, -3, 3, -1, -2, -3, 3, 0, 0], [3, -5, -3, 0, 9, 3, 2, 0, -5, 2, -2, -8], [-2, 6, 2, 0, -4, 2, 1, 0, 2, 1, 0, 4], [0, -2, 0, 4, 0, 2, 4, 4, 0, -8, 0, 0], [-1, -1, 1, 0, 16, -3, -3, 0, 0, -3, 0, 2], [2, -1, -2, 0, -2, 2, 1, 0, 5, 1, 1, -3], [-4, -6, -6, 2, 0, 6, -4, 2, 0, 2, 0, 0], [-8, 4, 1, -2, 3, -4, -7, 4, 3, 3, 0, 0], [-5, 3, 8, -10, 8, -3, -3, 6, 8, -3, 0, 0], [1, 1, -1, 0, 2, 3, 3, 0, 6, 3, 0, -2], [-4, 6, 4, 0, -14, 0, 2, 0, 0, 2, -2, 6], [2, 2, -2, 0, -8, 5, 2, 0, -7, 2, 1, -3], [-2, 9, 2, 0, -1, 3, 4, 0, 1, 4, 2, 6], [-12, -4, 1, 3, -3, 4, 1, -3, -3, 2, 0, 0], [-20, 9, 4, -2, 1, -9, 6, 0, 1, -6, 0, 0], [2, -2, -2, 0, -2, 0, 0, 0, 2, 0, 2, -2], [2, -4, -2, 0, -8, 0, 6, 0, -4, 6, 0, -4], [0, -1, 0, 0, -3, 1, 1, 0, -3, 1, -2, -2], [-5, -4, 5, 0, -10, -8, -5, 0, -4, -5, -6, 4], [2, -4, -2, 0, 22, -1, -2, 0, -1, -2, 1, -3], [10, -6, 4, 6, -6, 6, 4, -6, -6, 2, 0, 0], [14, 7, -2, 2, 1, -7, -5, 4, 1, 1, 0, 0], [-2, 3, -1, 9, -11, -3, 9, -13, -11, 4, 0, 0], [10, -5, 2, -2, 2, 5, -4, 2, 2, 2, 0, 0], [-8, 7, -4, -4, 5, -7, -3, 6, 5, -3, 0, 0], [0, 2, 12, -4, 4, -2, 0, 4, 4, -4, 0, 0], [-4, 8, 4, 0, -14, 4, 4, 0, 10, 4, -4, 4], [-2, -2, -6, 8, -6, 2, 8, -4, -6, -4, 0, 0], [3, -4, -5, -2, 6, 4, -7, 10, 6, -3, 0, 0], [12, -7, -8, 2, -3, 7, 6, -4, -3, -2, 0, 0], [6, -5, -3, 5, 0, 5, -10, 5, 0, 5, 0, 0], [1, -1, -1, 0, -16, 1, -3, 0, 0, -3, 0, -2], [-17, 0, 5, -6, 1, 0, 3, -4, 1, 1, 0, 0], [15, -10, -7, 8, -4, 10, -3, 0, -4, 3, 0, 0], [-1, 4, 1, 0, 7, 4, -9, 0, 0, -9, -2, 0], [3, -10, 3, 6, -6, 10, 3, -6, -6, 3, 0, 0], [-8, 13, 8, 0, 17, 4, 4, 0, -8, 4, -7, 9], [-14, -4, -6, 4, -2, 4, 2, 0, -2, -2, 0, 0], [-3, 3, 3, 0, 0, -5, -3, 0, 6, -3, 2, 8], [19, -11, 3, 2, 2, 11, -5, 6, 2, -1, 0, 0], [-10, 8, 6, 6, -8, -8, 10, -10, -8, 0, 0, 0], [-4, 0, 4, 0, 4, -4, -10, 0, 6, -10, -4, 4], [4, -4, -4, 0, -16, 4, -2, 0, 6, -2, 0, -8], [2, -3, -2, 0, 16, -2, -3, 0, -7, -3, 3, -1], [13, -6, 4, -7, 4, 6, -5, 1, 4, 4, 0, 0], [0, -9, 0, 0, 9, -7, 1, 0, 5, 1, -2, -2], [0, -2, 0, 0, -6, 0, 3, 0, -7, 3, -2, -2], [-7, -5, 3, 0, 0, 5, 3, 0, 0, -3, 0, 0], [-2, 4, 2, 0, 14, 1, 0, 0, 1, 0, -1, 3], [-6, 10, 6, 0, -12, 0, 0, 0, 6, 0, -2, 10], [-1, -7, -6, -2, -2, 7, -1, -6, -2, 7, 0, 0], [1, 2, -1, 0, -19, 6, 9, 0, -8, 9, -2, -4], [6, -6, -6, 0, 12, 2, 0, 0, -4, 0, 4, -8], [2, -9, -2, 0, 7, -7, 0, 0, 3, 0, 2, -2], [10, -3, 4, -6, 9, 3, -11, 12, 9, -1, 0, 0], [5, -8, -14, 4, -2, 8, -3, 0, -2, 3, 0, 0], [-9, 8, 9, 0, -6, -2, 11, 0, -12, 11, -8, 10], [2, 2, -2, 0, -26, 2, 10, 0, -8, 10, 4, 0], [2, 5, -2, 0, 1, 9, -2, 0, -9, -2, 0, -4], [9, -6, 5, 8, -5, 6, 5, -2, -5, -3, 0, 0], [8, -3, -12, -2, -5, 3, 5, -12, -5, 7, 0, 0], [-10, 9, 10, 0, -14, -6, -1, 0, 3, -1, -5, 15], [-4, 2, 4, 0, 16, -7, 7, 0, 1, 7, 1, 9], [0, -6, 5, 2, -1, 6, 3, 0, -1, -3, 0, 0], [-10, 9, 10, 0, 16, -7, -7, 0, 15, -7, -4, 16], [-2, 6, 0, -6, 6, -6, 0, 6, 6, -6, 0, 0], [4, 0, 4, -4, 6, 0, 0, 8, 6, -8, 0, 0], [14, -3, -3, 1, -2, 3, -4, -3, -2, 7, 0, 0], [2, -3, -2, 0, 7, 3, -4, 0, 4, -4, -2, -6], [21, 8, 12, -4, 0, -8, -1, -4, 0, 5, 0, 0], [11, -8, -11, 0, 4, 4, -5, 0, 8, -5, 10, -12], [8, -8, 0, 0, 4, 8, -8, 8, 4, 0, 0, 0], [2, 2, -2, 0, 4, -3, 6, 0, -3, 6, 9, 5], [-2, -10, 2, 0, 8, -12, -5, 0, 1, -5, -2, 2], [4, 2, -4, 0, 2, 8, -4, 0, -12, -4, 2, -6], [-3, 8, -9, 2, 2, -8, 7, 6, 2, -13, 0, 0], [-5, 6, 5, 0, 2, 0, 1, 0, 2, 1, -4, 6], [0, 5, 0, 0, -9, 1, -3, 0, 9, -3, 4, 4], [0, -1, 6, -2, 3, 1, 1, 4, 3, -5, 0, 0], [-6, -2, 6, 0, 0, -11, -1, 0, -7, -1, -3, 9], [8, 0, 6, -2, -2, 0, 8, -6, -2, -2, 0, 0], [-2, 6, 2, 0, 8, 6, 8, 0, -2, 8, -4, 0], [12, -5, -12, 0, -18, 11, 3, 0, 3, 3, 8, -16], [1, 2, -6, 3, -2, -2, 7, -1, -2, -6, 0, 0], [-2, 8, 1, -5, -1, -8, 10, -7, -1, -3, 0, 0], [-6, -3, 0, 4, -5, 3, 11, -6, -5, -5, 0, 0], [-19, 8, 1, -6, 4, -8, 5, 2, 4, -7, 0, 0], [-15, 6, -3, 6, -8, -6, 7, -10, -8, 3, 0, 0], [10, -10, -10, 0, -10, 0, 14, 0, -8, 14, 10, -10], [9, -5, 5, 4, 4, 5, -11, 12, 4, -1, 0, 0], [-14, 2, -13, -3, -1, -2, 4, -5, -1, 1, 0, 0], [11, -1, -11, 0, -26, 13, 5, 0, -6, 5, 8, -14], [4, 7, 6, 2, -7, -7, 13, -12, -7, -1, 0, 0], [3, -4, -3, 0, 0, 0, -5, 0, 8, -5, 2, -4], [0, -14, 0, 0, -18, -10, -2, 0, -6, -2, -4, -4], [-1, 8, 17, 2, 0, -8, -5, 2, 0, 3, 0, 0], [8, -2, 9, -9, 7, 2, -5, 5, 7, 0, 0, 0], [6, -10, -6, 0, 12, -2, 2, 0, -6, 2, 4, -8], [-23, 1, 6, 2, 2, -1, -11, 6, 2, 5, 0, 0], [-5, -1, -7, 6, -6, 1, -7, -6, -6, 13, 0, 0], [-10, 18, 10, 0, -14, 2, 4, 0, 2, 4, -4, 16], [3, 12, 4, 8, -4, -12, 8, 0, -4, -8, 0, 0], [13, -2, 6, 4, -4, 2, -3, -4, -4, 7, 0, 0], [-2, -2, 2, 0, -16, -2, 10, 0, -8, 10, -4, 0], [-6, 8, 6, 0, -18, -4, 3, 0, 10, 3, 0, 12], [3, -3, -3, 0, -6, 1, -11, 0, 4, -11, 2, -4], [-12, 13, 10, -6, 5, -13, 3, 4, 5, -7, 0, 0], [0, -8, 0, 0, 6, -10, 0, 0, -2, 0, 2, 2], [8, 3, 18, -12, 7, -3, -2, 2, 7, 0, 0, 0], [32, -2, 7, -2, 3, 2, -7, 4, 3, 3, 0, 0], [12, -6, -12, 0, -6, 6, -4, 0, 0, -4, 12, -12], [8, -1, -8, 0, -8, 12, -5, 0, -7, -5, 3, -13], [-2, 0, 2, 0, 2, -2, -5, 0, 10, -5, -2, 2], [-6, 3, 7, 1, 5, -3, 1, 11, 5, -12, 0, 0], [14, -14, -19, -3, 1, 14, -3, -1, 1, 4, 0, 0], [-13, 0, 10, 1, 6, 0, -13, 13, 6, 0, 0, 0], [12, 8, 10, -6, 6, -8, -8, 6, 6, 2, 0, 0], [1, -13, -1, 0, 26, -7, -20, 0, 9, -20, -4, -6], [8, -8, 2, 0, -2, 8, 9, -4, -2, -5, 0, 0], [-13, 12, 0, 3, -4, -12, 11, -5, -4, -6, 0, 0], [5, 7, -5, 0, -20, 13, 3, 0, -6, 3, 4, -6], [-19, 8, -8, -4, 0, -8, -6, -4, 0, 10, 0, 0], [12, -5, 0, 6, -3, 5, 3, 0, -3, -3, 0, 0], [5, -11, -5, 0, -14, 1, 5, 0, -12, 5, -2, -12], [-6, 2, 6, 0, 12, -4, -12, 0, -2, -12, -6, 6], [-14, 6, 19, 2, -3, -6, 3, -4, -3, 1, 0, 0], [-16, 0, 4, 2, 4, 0, 7, 10, 4, -17, 0, 0], [0, -5, 0, 0, -3, -11, -5, 0, -13, -5, 6, 6], [-5, 5, 7, -4, 8, -5, -13, 12, 8, 1, 0, 0], [7, 4, -12, 0, 0, -4, -6, 0, 0, 6, 0, 0], [-10, -4, -2, 8, -6, 4, -6, -4, -6, 10, 0, 0], [-7, 21, 15, -2, -2, -21, 5, -6, -2, 1, 0, 0], [-20, -12, 2, 8, -4, 12, 0, 0, -4, 0, 0, 0], [-6, 9, 6, 0, 9, -1, -2, 0, 5, -2, -2, 10], [-6, 4, 6, 0, -24, -4, 14, 0, 2, 14, -4, 8], [-6, 14, 6, 0, 30, 8, 2, 0, -4, 2, -6, 6], [-6, 10, 6, 0, 12, 6, -4, 0, -3, -4, -8, 4], [-6, 8, 6, 0, 0, -8, 0, 0, 0, 0, 0, 0], [9, -6, -2, -8, 12, 6, -7, 16, 12, -9, 0, 0], [-2, -2, 2, 0, -16, -4, 4, 0, -12, 4, -2, 2], [0, 0, 0, 0, -12, -2, -4, 0, 16, -4, 2, 2], [-16, 8, 16, 0, 4, -11, -2, 0, -5, -2, -13, 19], [-6, 4, 6, 0, -12, 2, 5, 0, 13, 5, -10, 2], [-10, 10, 6, 6, -6, -10, 0, -6, -6, 6, 0, 0], [14, -6, -10, 0, 4, 6, -9, 8, 4, 1, 0, 0], [7, 0, -3, -2, 6, 0, -11, 10, 6, 1, 0, 0], [1, -2, 1, 8, -4, 2, -1, 0, -4, 1, 0, 0], [-16, 11, -10, -12, 7, -11, -3, 2, 7, 1, 0, 0], [-2, 9, 2, 0, -13, 7, -2, 0, -10, -2, -2, 2], [28, -15, -11, 1, 3, 15, -7, 7, 3, 0, 0, 0], [-4, 0, -11, 5, -5, 0, 16, -5, -5, -11, 0, 0], [-3, -5, 3, 0, -6, -7, 4, 0, -3, 4, -4, 2], [-3, 8, 3, 0, 9, 12, 7, 0, -4, 7, -10, -4], [8, -1, -10, -8, 5, 1, 2, 2, 5, -4, 0, 0], [8, -11, -8, 0, 13, 1, -9, 0, 7, -9, 4, -12], [12, -4, -12, 0, 0, 13, 0, 0, -9, 0, 7, -17], [31, 1, 14, -1, -3, -1, -2, -7, -3, 9, 0, 0], [-13, -6, -15, 2, -2, 6, 3, -2, -2, -1, 0, 0], [21, -18, -1, 2, 1, 18, -7, 4, 1, 3, 0, 0], [-24, 6, 10, -16, 12, -6, -12, 8, 12, 4, 0, 0], [13, -11, -13, 0, 8, 13, -7, 0, -8, -7, 2, -24], [-2, 1, 2, 0, -4, 0, -9, 0, 15, -9, -3, 1], [2, 0, -2, 0, 16, 10, -2, 0, 2, -2, -6, -10], [13, -10, -6, -1, 6, 10, -7, 11, 6, -4, 0, 0], [-4, -6, 4, 0, 4, -12, 15, 0, -5, 15, -2, 6], [2, -16, -8, -2, 2, 16, 4, 2, 2, -6, 0, 0], [8, -6, -8, 0, -14, -7, 6, 0, -9, 6, 17, 1], [7, -2, -7, 0, -40, 2, 7, 0, -12, 7, 10, -4], [-5, -8, 11, 10, -10, 8, 5, -10, -10, 5, 0, 0], [-4, 1, 4, 0, 1, -1, 8, 0, 1, 8, -6, 2], [5, 4, -5, 0, 7, 0, -7, 0, 8, -7, 14, 4], [-9, 0, 9, 0, -9, -9, 7, 0, -6, 7, -9, 9], [-34, 10, 16, -10, 4, -10, 10, -2, 4, -8, 0, 0], [-16, 23, 16, 0, -11, 1, -1, 0, 19, -1, -10, 22], [-6, -2, 6, 0, 6, -8, 1, 0, 17, 1, -6, 6], [-6, 2, 6, 0, 0, 2, -22, 0, 4, -22, -12, 0], [-24, 13, 28, -2, 1, -13, 6, 0, 1, -6, 0, 0], [18, 0, -8, -8, 0, 0, 2, -8, 0, 6, 0, 0], [6, 2, -16, 18, -12, -2, 14, -6, -12, -8, 0, 0], [0, -3, 0, 0, -6, -2, 9, 0, 11, 9, -1, -1], [-10, 0, 17, -7, 7, 0, -1, 7, 7, -6, 0, 0], [-8, 16, 0, -20, 16, -16, -4, 12, 16, -8, 0, 0], [-5, 5, 5, 0, -34, 3, -4, 0, 19, -4, -8, 2], [4, -8, 0, 8, -12, 8, 20, -16, -12, -4, 0, 0], [0, -2, 11, -5, 3, 2, 0, 1, 3, -1, 0, 0], [4, -5, -4, 0, -4, -5, 9, 0, 13, 9, 8, 0], [2, -4, -2, 0, 10, 4, 0, 0, -15, 0, -4, -8], [-8, 0, 6, 4, -4, 0, -6, -4, -4, 10, 0, 0], [-3, 8, 3, 0, 9, 9, -7, 0, 14, -7, -7, -1], [-14, 9, 22, -2, -5, -9, 12, -12, -5, 0, 0, 0], [-2, 1, 2, 0, -4, 5, -8, 0, 6, -8, -8, -4], [9, 2, -4, 16, -12, -2, 15, -8, -12, -7, 0, 0], [3, -15, -3, 0, 6, -5, -3, 0, -2, -3, -4, -10], [24, -15, 5, 17, -7, 15, 3, 3, -7, -6, 0, 0], [11, -5, -11, 0, 1, 7, 2, 0, 3, 2, 10, -12], [4, 8, -4, 0, -4, 20, 6, 0, -2, 6, -4, -12], [-14, 20, 14, 0, 8, 2, 6, 0, -2, 6, -10, 18], [-5, -9, -9, -10, 14, 9, -17, 18, 14, -1, 0, 0], [4, -8, -4, 0, -16, -6, -7, 0, -5, -7, 6, -2], [-8, 8, 8, 0, 20, -6, 0, 0, -8, 0, -2, 14], [-2, 7, -5, -5, 14, -7, -8, 23, 14, -15, 0, 0], [-19, 6, 13, 0, 6, -6, 1, 12, 6, -13, 0, 0], [-13, 16, 13, 0, -5, 0, 1, 0, 8, 1, -10, 16], [6, -5, -6, 0, 15, -3, 8, 0, -5, 8, 10, -2], [-10, -12, -4, -4, 2, 12, 0, 0, 2, 0, 0, 0], [-12, 4, 12, 0, -6, -8, -12, 0, 4, -12, -12, 12], [21, 1, -27, 0, -2, -1, 1, -4, -2, 3, 0, 0], [-4, -14, 4, 0, 10, -12, -6, 0, 8, -6, -10, -2], [-5, 19, 5, 0, -16, 6, 6, 0, -17, 6, 3, 13], [0, 2, 0, 0, -6, 0, 6, 0, 14, 6, 2, 2], [-17, 8, 3, 12, -9, -8, 3, -6, -9, 3, 0, 0], [1, -12, 8, 12, 0, 12, -10, 12, 0, -2, 0, 0], [-10, -4, 7, 7, -5, 4, -9, -3, -5, 12, 0, 0], [-33, -3, 6, 6, -6, 3, 3, -6, -6, 3, 0, 0], [-10, 10, -14, 4, -12, -10, 2, -20, -12, 18, 0, 0], [-10, -9, 9, -8, 4, 9, 8, 0, 4, -8, 0, 0], [2, -11, -2, 0, 13, -5, -12, 0, 25, -12, -2, -6], [-5, 2, 5, 0, 8, -16, 15, 0, 4, 15, 8, 18], [-9, 7, 9, 0, 6, -3, 9, 0, 10, 9, -8, 10], [10, 12, 10, 4, 0, -12, -10, 4, 0, 6, 0, 0], [9, -8, -9, 0, 21, 4, -15, 0, -10, -15, 6, -12], [10, -7, -10, 0, -13, -1, -1, 0, -1, -1, 14, -6], [0, 5, 3, -19, 12, -5, -4, 5, 12, -1, 0, 0], [3, -12, -3, 0, 21, -4, 9, 0, 11, 9, -2, -8], [5, 12, -11, 10, -2, -12, 9, 6, -2, -15, 0, 0], [-4, -11, 4, 0, 1, -19, -9, 0, 1, -9, 0, 8], [-8, 2, 13, 3, -7, -2, 9, -11, -7, 2, 0, 0], [-17, 10, 10, -7, -2, -10, 13, -11, -2, -2, 0, 0], [-12, -1, -10, 0, -3, 1, 8, -6, -3, -2, 0, 0], [-2, 14, 2, 0, -4, 8, 6, 0, -14, 6, 2, 6], [-14, 0, -4, 8, -10, 0, 0, -12, -10, 12, 0, 0], [-2, 12, -11, -3, -1, -12, 9, -5, -1, -4, 0, 0], [7, 1, -7, 0, 32, 9, -5, 0, 4, -5, 6, -8], [10, -14, -10, 0, 14, 2, -10, 0, 10, -10, 4, -16], [-11, 4, 11, 0, 11, -8, -3, 0, 10, -3, -10, 12], [-13, 13, 21, -10, 0, -13, 17, -10, 0, -7, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1470_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1470_2_b_a();".
// 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_b_a(:prec:=12)
    chi := MakeCharacter_1470_b();
    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_b_a();".
// 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_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1470_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![1296, 0, 36360, 0, 50113, 0, 14828, 0, 1590, 0, 68, 0, 1]>,<17,R![-288, 816, -572, 96, 32, -12, 1]>],Snew);
    return Vf;
end function;