
// Make newform 1470.2.b.d in Magma, downloaded from the LMFDB on 25 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_d();".
// 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_d();".
// 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 := [6561, 0, 0, -3888, 162, 0, 864, -240, 2, -80, 96, 0, 2, -16, 0, 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], [37179, -114453, 101574, -8910, 35019, -49437, 6318, -3814, 9355, -4053, 294, -814, 971, -45, 30, -86], [0, -28269, 4212, 2385, 12096, -3125, -636, -1431, 1376, -125, -28, -271, 64, 11, 20, -7], [81648, -5589, 16200, -27459, 6696, -7809, 3232, -3055, 1392, -597, 544, -131, 168, -33, 8, -15], [-69984, 54675, -239598, -15471, -24624, 99387, -270, 3769, -14656, 7347, 354, 769, -2096, -117, -30, 185], [527067, 258066, 125874, -192537, -75141, -42822, 27498, -5393, -565, -4974, 3714, 1639, 1147, -342, -150, -97], [236925, 99387, 94122, -86238, -35595, -39573, 12754, -1094, 2797, -3429, 1642, 738, 917, -165, -94, -86], [778572, 438129, 244944, -337851, -158004, -90423, 55008, -2675, 716, -9231, 6816, 3253, 2188, -711, -336, -211], [710775, 587574, 306666, -282771, -215649, -104418, 47826, 9565, -2713, -10698, 5178, 4621, 2431, -594, -462, -211], [153090, 153090, 93069, -60102, -62424, -33534, 10803, 4714, 2, -3246, 1005, 1306, 796, -126, -141, -70], [-470934, -140211, -123363, 185058, 42786, 51597, -26715, 6922, -3910, 4701, -3555, -926, -1214, 333, 117, 122], [599238, 236925, 99387, -240246, -72306, -35595, 34731, -7886, -922, -4083, 4827, 1642, 910, -459, -165, -94], [-494262, -290142, -121581, 205929, 103698, 38922, -32133, 361, 3194, 4626, -4029, -2327, -974, 426, 219, 89], [-142884, -143127, -73872, 57537, 53964, 24501, -9976, -3211, 860, 2505, -1000, -1119, -596, 117, 112, 53], [-1692738, -919998, -415287, 699597, 313254, 137322, -106695, 8789, 6094, 15666, -13863, -6643, -3418, 1386, 633, 325], [1421550, 850743, 651240, -544077, -322434, -236961, 89928, 6251, 6958, -22521, 10104, 6803, 5438, -1089, -744, -485]];
    Rf_basisdens := [1, 34992, 3888, 5832, 34992, 34992, 11664, 34992, 34992, 8748, 11664, 11664, 11664, 5832, 34992, 34992];
    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, 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]];
    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, -1, 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, 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, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, -2, 0, -1], [0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0], [-3, 1, 2, 0, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, -1, 1, 0, 1, -2], [0, 1, 0, 0, 0, 1, 1, 2, 1, -2, -2, 0, -2, 1, 0, 0], [0, -1, 0, 0, 0, -1, 1, -1, 1, 2, 1, 0, 1, 0, 0, 0], [-2, -1, 0, 0, 0, -1, 1, -1, 1, 0, -1, 2, -1, 0, -2, 0], [0, 1, -3, 0, -2, -1, 0, 0, 0, 0, -2, 0, 2, 0, 0, 0], [-1, -2, -2, 2, 2, 2, 2, 0, -2, 0, 0, 1, 0, 0, 1, 0], [1, -1, 1, -2, 2, 1, -2, 0, 2, 0, 1, 1, -1, 0, 1, 0], [-2, -1, -1, 0, -2, 1, 2, 0, -2, 0, -2, 0, 2, 0, 0, 0], [0, 2, 0, 0, 0, 2, 0, 0, 0, -2, 2, 0, 2, -2, 0, 2], [-2, 0, 0, -3, 0, 0, -1, 0, 1, 0, 0, 2, 0, 0, 2, 0], [-1, 1, 0, 0, 0, 1, 1, 0, 1, 0, -1, 1, -1, 6, -1, 0], [1, 2, 4, 0, -3, -2, -2, 0, 2, 0, 0, 1, 0, 0, 1, 0], [1, 1, 0, 0, 0, 1, 1, 0, 1, -2, 1, -1, 1, -2, 1, 6], [-2, 1, 0, 0, 0, 1, -3, -2, -3, -4, 1, 2, 1, 2, -2, 0], [-3, 1, 0, -2, -4, -1, -1, 0, 1, 0, -1, 1, 1, 0, 1, 0], [-3, 2, -4, 4, 2, -2, 1, 0, -1, 0, 3, 1, -3, 0, 1, 0], [-1, 0, 0, 2, -2, 0, 2, 0, -2, 0, 0, 1, 0, 0, 1, 0], [-2, -1, 0, 0, 0, -1, 0, 4, 0, 8, -2, 2, -2, 0, -2, -2], [-1, -4, -2, 5, 4, 4, -2, 0, 2, 0, 1, 1, -1, 0, 1, 0], [-2, 0, 0, 0, 0, 0, 2, -2, 2, 4, 2, 2, 2, 0, -2, 2], [1, -1, 0, 0, 0, -1, -3, 4, -3, 0, 1, -1, 1, 4, 1, 2], [2, 2, 2, 2, 0, -2, 2, 0, -2, 0, 4, 0, -4, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 4, 0, 0, -3, -1, -3, -4, 1, 2], [-2, -2, 2, -6, 0, 2, -2, 0, 2, 0, -2, -2, 2, 0, -2, 0], [2, 2, -4, -1, 4, -2, 0, 0, 0, 0, -3, 2, 3, 0, 2, 0], [0, 0, 0, 0, 0, 0, -2, 6, -2, 0, 0, 0, 0, -1, 0, 4], [-1, -2, 0, 0, 0, -2, 2, -2, 2, 4, -2, 1, -2, 2, -1, -2], [-4, 1, 0, 0, 0, 1, -2, 0, -2, 2, 0, 4, 0, -2, -4, 3], [-1, 0, 8, 3, 0, 0, -2, 0, 2, 0, 1, 3, -1, 0, 3, 0], [2, 0, 0, 0, 0, 0, -2, 4, -2, 6, 2, -2, 2, 4, 2, 4], [1, -3, -10, 0, 1, 3, 3, 0, -3, 0, 4, 1, -4, 0, 1, 0], [4, 3, 0, 2, -1, -3, 3, 0, -3, 0, -1, -2, 1, 0, -2, 0], [-8, 0, 0, 2, -2, 0, -4, 0, 4, 0, 2, -2, -2, 0, -2, 0], [3, -2, 0, 0, 0, -2, 0, -1, 0, 4, -2, -3, -2, 6, 3, 0], [5, -1, 0, 0, 0, -1, -1, 4, -1, 4, -3, -5, -3, 0, 5, -2], [3, -3, 0, 0, 0, -3, 1, 4, 1, -2, 1, -3, 1, -2, 3, -4], [-1, -6, -2, -2, -10, 6, 2, 0, -2, 0, -1, -1, 1, 0, -1, 0], [2, 0, 0, 0, 0, 0, 2, 4, 2, 6, -2, -2, -2, 2, 2, 0], [2, -1, 0, 0, 0, -1, -1, -5, -1, -4, 1, -2, 1, 4, 2, 0], [5, -1, 4, -4, -6, 1, -1, 0, 1, 0, -5, -1, 5, 0, -1, 0], [2, 0, 0, 0, 0, 0, 2, -2, 2, -4, 6, -2, 6, -6, 2, 0], [-1, 0, -2, 6, -2, 0, 2, 0, -2, 0, 0, -1, 0, 0, -1, 0], [1, -1, 0, 0, 0, -1, -3, 2, -3, 8, -1, -1, -1, 4, 1, -4], [-3, 2, 0, 0, 0, 2, 6, 4, 6, -4, -3, 3, -3, 0, -3, -2], [1, -1, 0, 0, 0, -1, 1, 0, 1, -6, -1, -1, -1, 10, 1, 0], [0, 0, 0, 0, 0, 0, 1, -4, 1, 0, 3, 0, 3, 2, 0, 7], [4, 1, 6, 6, 2, -1, 3, 0, -3, 0, -3, 0, 3, 0, 0, 0], [-1, 3, 5, 0, 2, -3, 3, 0, -3, 0, 3, -1, -3, 0, -1, 0], [4, 3, 0, 0, 0, 3, 1, -2, 1, -6, -2, -4, -2, -1, 4, 8], [1, -3, -2, 0, -10, 3, -1, 0, 1, 0, -3, -1, 3, 0, -1, 0], [0, -1, 0, 0, 0, -1, -2, -6, -2, 4, -4, 0, -4, 4, 0, -5], [0, -3, 6, 8, 1, 3, 1, 0, -1, 0, 1, 0, -1, 0, 0, 0], [3, 4, 0, 0, 0, 4, -2, 6, -2, -4, 3, -3, 3, 0, 3, -2], [2, 4, 0, 0, 0, 4, 1, 2, 1, -8, 0, -2, 0, 11, 2, -4], [8, 0, 0, -4, 0, 0, -8, 0, 8, 0, 2, -2, -2, 0, -2, 0], [-4, -3, 0, 0, 0, -3, -3, -6, -3, 6, 1, 4, 1, -2, -4, -6], [0, 2, 0, -6, -2, -2, 2, 0, -2, 0, -2, -4, 2, 0, -4, 0], [2, -5, 0, 0, 0, -5, 5, 10, 5, 4, -1, -2, -1, -4, 2, -6], [-2, -2, 0, 0, 0, -2, 0, -4, 0, 2, 2, 2, 2, -4, -2, 4], [7, 0, 0, -8, -12, 0, -2, 0, 2, 0, -3, 1, 3, 0, 1, 0], [1, 1, 2, 2, -8, -1, 5, 0, -5, 0, 1, 1, -1, 0, 1, 0], [2, 1, 0, 0, 0, 1, 1, -2, 1, 8, -8, -2, -8, 0, 2, 0], [3, 1, 0, 0, 0, 1, -3, 2, -3, 0, -1, -3, -1, 2, 3, 8], [7, 1, -2, 6, 5, -1, 3, 0, -3, 0, -7, 3, 7, 0, 3, 0], [3, -1, 0, 0, 0, -1, 1, -2, 1, 14, 3, -3, 3, 0, 3, -8], [0, -4, 0, 0, 0, -4, 4, 0, 4, 4, 2, 0, 2, -6, 0, -2], [0, -3, -1, -4, 2, 3, 2, 0, -2, 0, 0, 4, 0, 0, 4, 0], [9, 4, -2, -4, 6, -4, 4, 0, -4, 0, -3, -1, 3, 0, -1, 0], [0, -3, -4, -10, -5, 3, -3, 0, 3, 0, 3, 2, -3, 0, 2, 0], [-2, -3, 0, 0, 0, -3, -2, 4, -2, 14, 0, 2, 0, 6, -2, 7], [-4, 2, 0, 0, 0, 2, -2, 4, -2, -6, -2, 4, -2, 4, -4, 8], [4, -10, 0, 0, 0, -10, -2, 0, -2, 8, 0, -4, 0, 2, 4, -8], [-2, 2, 0, 0, 0, 2, 2, -8, 2, 0, 2, 2, 2, 4, -2, -5], [5, -2, 2, -2, -10, 2, -1, 0, 1, 0, -7, 5, 7, 0, 5, 0], [-2, -4, -4, 2, -6, 4, 2, 0, -2, 0, -4, -4, 4, 0, -4, 0], [-7, -5, 0, 0, 0, -5, 1, -8, 1, -2, -3, 7, -3, 0, -7, -2], [0, 3, 0, 0, 0, 3, 0, -2, 0, -4, 2, 0, 2, -12, 0, 12], [-4, 1, 0, 0, 0, 1, -2, 2, -2, -4, -2, 4, -2, 8, -4, -1], [-2, 4, 0, 0, 0, 4, -2, -6, -2, -4, 0, 2, 0, 2, -2, 2], [3, 1, 0, 0, 0, 1, -5, 6, -5, -4, 3, -3, 3, 6, 3, 4], [-9, 2, 4, 2, 0, -2, -4, 0, 4, 0, 1, -1, -1, 0, -1, 0], [3, 1, -6, -4, 2, -1, 1, 0, -1, 0, 1, 1, -1, 0, 1, 0], [-12, 0, 10, 4, -2, 0, -2, 0, 2, 0, 2, 0, -2, 0, 0, 0], [3, 0, 0, -2, -6, 0, -3, 0, 3, 0, 1, -1, -1, 0, -1, 0], [10, 0, -6, 14, -8, 0, 4, 0, -4, 0, 6, -2, -6, 0, -2, 0], [16, -8, -2, -4, 8, 8, -6, 0, 6, 0, 2, 0, -2, 0, 0, 0], [0, -1, 0, 0, 0, -1, -1, -11, -1, 12, 5, 0, 5, -6, 0, 2], [5, 1, 10, -2, 4, -1, 3, 0, -3, 0, 1, 3, -1, 0, 3, 0], [2, 3, -11, 8, 6, -3, -2, 0, 2, 0, 2, 0, -2, 0, 0, 0], [-1, -2, 14, -1, 4, 2, -2, 0, 2, 0, 1, -3, -1, 0, -3, 0], [-11, -2, 4, -4, -2, 2, -7, 0, 7, 0, 1, -1, -1, 0, -1, 0], [0, 0, 0, 0, 0, 0, -4, 6, -4, 0, 7, 0, 7, -3, 0, 0], [-8, 10, -6, 6, -2, -10, 4, 0, -4, 0, 0, -6, 0, 0, -6, 0], [-1, 3, -5, -2, 2, -3, 4, 0, -4, 0, 3, -5, -3, 0, -5, 0], [0, 4, 0, 0, 0, 4, -6, 2, -6, -6, 4, 0, 4, -4, 0, 6], [-5, 2, 6, -6, 8, -2, 0, 0, 0, 0, 2, -1, -2, 0, -1, 0], [5, 5, 0, 0, 0, 5, -1, 0, -1, 0, -5, -5, -5, -2, 5, 12], [3, 6, -4, -2, -6, -6, 0, 0, 0, 0, -1, 5, 1, 0, 5, 0], [-2, 1, 0, 0, 0, 1, 0, 6, 0, -16, -4, 2, -4, -6, -2, 10], [-5, -8, -4, -10, -2, 8, 0, 0, 0, 0, -6, -1, 6, 0, -1, 0], [1, 3, -9, 6, -2, -3, 4, 0, -4, 0, 2, -3, -2, 0, -3, 0], [-1, 7, 0, 0, 0, 7, -3, -14, -3, 2, 3, 1, 3, 6, -1, 6], [0, 2, 0, 0, 0, 2, -5, -1, -5, 12, -3, 0, -3, 6, 0, 4], [-2, 4, 0, 0, 0, 4, 0, -4, 0, 4, 0, 2, 0, -6, -2, 6], [-14, 1, -2, 8, 7, -1, 1, 0, -1, 0, 3, -2, -3, 0, -2, 0], [-5, -6, 0, 0, 0, -6, -2, 0, -2, 12, 5, 5, 5, 1, -5, 0], [-1, -4, 0, 0, 0, -4, -4, 0, -4, 4, -2, 1, -2, -8, -1, -6], [3, -2, -16, -1, 12, 2, -2, 0, 2, 0, 1, 3, -1, 0, 3, 0], [-5, -3, 0, 0, 0, -3, 3, 0, 3, 0, 1, 5, 1, -8, -5, 12], [-4, 4, 0, 0, 0, 4, 8, -6, 8, 0, -1, 4, -1, 5, -4, 0], [4, -1, -7, 0, 6, 1, 6, 0, -6, 0, 2, 2, -2, 0, 2, 0], [-4, 4, 0, 0, 0, 4, 2, -2, 2, -14, 4, 4, 4, 6, -4, -8], [-4, 0, 0, 0, 0, 0, 5, 12, 5, 0, -4, 4, -4, -2, -4, -5], [3, 1, 0, 0, 0, 1, -3, 2, -3, 12, -9, -3, -9, 10, 3, 10], [1, 4, -12, 6, 2, -4, 2, 0, -2, 0, 1, 5, -1, 0, 5, 0], [-4, -6, -12, 14, 10, 6, 0, 0, 0, 0, 4, -2, -4, 0, -2, 0], [6, -4, 0, 0, 0, -4, 4, 0, 4, -4, 2, -6, 2, -14, 6, -12], [7, 8, 0, 0, 0, 8, 2, 0, 2, -8, -2, -7, -2, -2, 7, 4], [-8, -1, 0, 0, 0, -1, 2, -8, 2, -2, 6, 8, 6, -10, -8, -3], [4, -4, -2, 8, 4, 4, 4, 0, -4, 0, 6, -6, -6, 0, -6, 0], [4, -6, 4, -8, 2, 6, -6, 0, 6, 0, 0, -4, 0, 0, -4, 0], [-4, 4, 0, 0, 0, 4, 6, 8, 6, -8, -2, 4, -2, 0, -4, -6], [3, 5, 0, 0, 0, 5, 1, 0, 1, 12, -1, -3, -1, -8, 3, 8], [1, -6, -2, -2, 14, 6, -6, 0, 6, 0, -3, 7, 3, 0, 7, 0], [-2, 2, 0, 0, 0, 2, 6, -8, 6, 4, 0, 2, 0, -12, -2, 0], [5, 4, 12, 3, 16, -4, 4, 0, -4, 0, 1, 5, -1, 0, 5, 0], [-8, 3, -5, -4, 2, -3, -2, 0, 2, 0, 4, -8, -4, 0, -8, 0], [7, -6, -12, -12, -4, 6, -3, 0, 3, 0, 1, -3, -1, 0, -3, 0], [-4, 4, 0, 0, 0, 4, -4, -9, -4, -12, 4, 4, 4, 8, -4, -4], [8, 4, 6, 2, 18, -4, 0, 0, 0, 0, 6, 2, -6, 0, 2, 0], [0, -1, 0, 0, 0, -1, -2, 14, -2, 10, -2, 0, -2, 8, 0, -2], [-4, -2, -6, 8, 8, 2, -2, 0, 2, 0, 10, -6, -10, 0, -6, 0], [1, 8, 0, 0, 0, 8, 4, 16, 4, -8, -5, -1, -5, 10, 1, 0], [-3, 6, 0, 0, 0, 6, 3, 0, 3, -16, 1, 3, 1, -10, -3, -12], [-4, 1, 0, 0, 0, 1, -3, -15, -3, -22, 3, 4, 3, 12, -4, 8], [-4, -2, -8, -6, -4, 2, -2, 0, 2, 0, 4, -4, -4, 0, -4, 0], [3, -1, 0, 0, 0, -1, 3, -10, 3, 0, 1, -3, 1, -10, 3, 12], [-3, 5, 0, 0, 0, 5, -1, 8, -1, -8, -1, 3, -1, -14, -3, 4], [10, 2, 12, -10, 0, -2, 6, 0, -6, 0, -4, 6, 4, 0, 6, 0], [-5, 1, 0, 0, 0, 1, -1, -8, -1, -4, -9, 5, -9, 8, -5, 12], [-1, -1, -3, 2, -6, 1, -2, 0, 2, 0, 0, 3, 0, 0, 3, 0], [-7, -2, 0, 0, 0, -2, 1, -4, 1, 8, 1, 7, 1, -16, -7, -8], [6, 0, 0, 0, 0, 0, -10, 4, -10, -12, 0, -6, 0, 4, 6, 4], [-14, -3, 11, 4, 6, 3, 0, 0, 0, 0, 2, -6, -2, 0, -6, 0], [-7, -6, -10, 6, 0, 6, -7, 0, 7, 0, 3, 3, -3, 0, 3, 0], [0, -7, -14, 8, -9, 7, -1, 0, 1, 0, -7, 2, 7, 0, 2, 0], [-10, 5, 4, -6, -7, -5, 7, 0, -7, 0, 1, -4, -1, 0, -4, 0], [0, 5, -2, 0, -1, -5, -1, 0, 1, 0, 1, -6, -1, 0, -6, 0], [-5, -3, 0, 0, 0, -3, -1, 2, -1, 14, 1, 5, 1, 8, -5, 2], [-1, 8, -4, -5, 8, -8, 0, 0, 0, 0, 1, 7, -1, 0, 7, 0], [-9, 6, 0, 14, 0, -6, 7, 0, -7, 0, 5, -7, -5, 0, -7, 0], [0, 3, 0, 0, 0, 3, 1, -2, 1, -4, -1, 0, -1, 0, 0, 10], [3, -6, -6, -3, 0, 6, -6, 0, 6, 0, -1, 1, 1, 0, 1, 0], [-2, 5, 0, 0, 0, 5, -3, -2, -3, 4, 0, 2, 0, -10, -2, 8], [8, -1, 0, 0, 0, -1, 3, -10, 3, -4, 10, -8, 10, -5, 8, -2], [-6, 0, 6, -4, -6, 0, -4, 0, 4, 0, 0, 6, 0, 0, 6, 0], [12, 5, 2, 4, 2, -5, -2, 0, 2, 0, 0, 4, 0, 0, 4, 0], [4, -5, 0, 0, 0, -5, 1, 4, 1, 6, -2, -4, -2, 8, 4, 10], [-8, 1, 0, 2, 15, -1, -7, 0, 7, 0, 3, -2, -3, 0, -2, 0], [5, 4, 0, -15, 8, -4, -2, 0, 2, 0, -3, -3, 3, 0, -3, 0], [5, 3, 0, 0, 0, 3, -7, -12, -7, -4, 7, -5, 7, -4, 5, 12], [-18, -10, -12, -1, 0, 10, 2, 0, -2, 0, 2, 2, -2, 0, 2, 0], [-4, 0, -18, 6, -8, 0, 2, 0, -2, 0, -4, 2, 4, 0, 2, 0], [-4, 3, 0, 0, 0, 3, 0, 0, 0, 12, 2, 4, 2, 18, -4, 9], [9, -1, 0, 0, 0, -1, 7, 8, 7, -16, -3, -9, -3, 8, 9, -6], [-1, 11, 0, 0, 0, 11, 11, 6, 11, -2, -7, 1, -7, 4, -1, -10], [-7, 0, 4, -20, 4, 0, -6, 0, 6, 0, -5, 5, 5, 0, 5, 0], [6, -5, 0, 0, 0, -5, 0, -2, 0, 32, 2, -6, 2, 4, 6, -5], [-1, 2, 8, 6, -20, -2, -1, 0, 1, 0, -3, -3, 3, 0, -3, 0], [-8, 2, 0, -6, 0, -2, -2, 0, 2, 0, -8, 4, 8, 0, 4, 0], [10, -1, 0, 0, 0, -1, 4, 8, 4, 10, 10, -10, 10, -2, 10, -11], [-2, -6, 0, 0, 0, -6, -6, 10, -6, 4, 0, 2, 0, 8, -2, 8], [-1, -5, 0, 0, 0, -5, -1, 0, -1, -8, 7, 1, 7, -14, -1, -4], [-12, -6, -4, -14, -14, 6, -2, 0, 2, 0, -4, 4, 4, 0, 4, 0], [-8, 0, -16, 5, -12, 0, 1, 0, -1, 0, 2, 4, -2, 0, 4, 0], [0, 7, 6, 0, -15, -7, 9, 0, -9, 0, -3, -8, 3, 0, -8, 0], [3, 3, 6, 22, -1, -3, 5, 0, -5, 0, 3, -5, -3, 0, -5, 0], [-4, -7, 0, 0, 0, -7, -1, -6, -1, -10, -4, 4, -4, -1, -4, -8], [1, 1, -14, 4, -18, -1, -1, 0, 1, 0, 1, -5, -1, 0, -5, 0], [4, -1, -6, 12, 15, 1, 7, 0, -7, 0, -1, 0, 1, 0, 0, 0], [0, -4, 0, 0, 0, -4, 5, 14, 5, 8, 2, 0, 2, -3, 0, -8], [0, 1, 10, -16, 8, -1, -7, 0, 7, 0, -4, -4, 4, 0, -4, 0], [-6, 2, -2, 0, 12, -2, 4, 0, -4, 0, -8, -2, 8, 0, -2, 0], [-4, -7, 0, 0, 0, -7, 2, 10, 2, -12, -14, 4, -14, 10, -4, 0], [4, 8, 0, 0, 0, 8, -2, -8, -2, 0, 0, -4, 0, -6, 4, 2], [-6, -8, 4, -4, 2, 8, 0, 0, 0, 0, 4, -6, -4, 0, -6, 0], [-2, 4, 20, 12, -4, -4, -8, 0, 8, 0, 2, 8, -2, 0, 8, 0], [-3, 1, 0, 0, 0, 1, -5, 2, -5, 8, -5, 3, -5, 0, -3, -10], [7, -1, 13, 6, -6, 1, 6, 0, -6, 0, -4, 7, 4, 0, 7, 0], [8, -9, -14, 11, -8, 9, -1, 0, 1, 0, -4, 0, 4, 0, 0, 0], [18, 7, 6, 16, 1, -7, -5, 0, 5, 0, -3, 2, 3, 0, 2, 0], [-23, 5, 7, -16, 6, -5, -7, 0, 7, 0, -3, 1, 3, 0, 1, 0], [-4, 5, 8, -2, -1, -5, 3, 0, -3, 0, 3, 2, -3, 0, 2, 0], [6, -7, 0, 0, 0, -7, -2, 12, -2, 22, -2, -6, -2, 2, 6, -5], [0, -1, 0, 0, 0, -1, 1, 7, 1, -12, 9, 0, 9, -20, 0, 8], [0, -5, 0, 0, 0, -5, -7, -4, -7, 16, 8, 0, 8, -11, 0, 0], [2, 2, 0, 0, 0, 2, 3, -3, 3, 0, 5, -2, 5, 2, 2, -4], [8, 9, 14, 10, -10, -9, -9, 0, 9, 0, -3, -4, 3, 0, -4, 0], [-10, -4, -2, -12, -10, 4, 2, 0, -2, 0, -2, -4, 2, 0, -4, 0], [4, 5, 0, 0, 0, 5, 3, 3, 3, -16, -3, -4, -3, 4, 4, 0], [6, -15, 0, 0, 0, -15, -5, 6, -5, 20, 0, -6, 0, 6, 6, 2], [-5, 6, 0, 0, 0, 6, -6, -6, -6, -4, -1, 5, -1, 8, -5, -8], [-9, 12, 0, 0, 0, 12, 6, -12, 6, -8, 0, 9, 0, 8, -9, 16], [-3, -13, -14, -10, -2, 13, -5, 0, 5, 0, -1, 5, 1, 0, 5, 0], [-6, 14, 12, -4, 0, -14, 0, 0, 0, 0, -6, -8, 6, 0, -8, 0], [-6, -12, -6, 2, -6, 12, -2, 0, 2, 0, -6, -6, 6, 0, -6, 0], [-9, -2, -14, 2, 8, 2, -5, 0, 5, 0, -1, 3, 1, 0, 3, 0], [4, 6, -10, -10, -14, -6, 6, 0, -6, 0, -8, 0, 8, 0, 0, 0], [-10, -4, 0, 0, 0, -4, 6, 4, 6, -8, -2, 10, -2, 0, -10, -15], [2, -4, -16, 0, -18, 4, 14, 0, -14, 0, -6, 2, 6, 0, 2, 0], [17, 2, 16, 18, 8, -2, 9, 0, -9, 0, -5, -1, 5, 0, -1, 0], [-10, -5, 0, 0, 0, -5, 7, 14, 7, 2, -10, 10, -10, 3, -10, -4], [6, -4, 0, 0, 0, -4, 2, 16, 2, 18, -2, -6, -2, 0, 6, -2], [2, -18, -8, -6, 10, 18, 4, 0, -4, 0, 6, 4, -6, 0, 4, 0], [8, -3, 0, 0, 0, -3, 0, 10, 0, -16, -4, -8, -4, 16, 8, 11], [5, 9, 0, 0, 0, 9, 1, 0, 1, -8, -5, -5, -5, -8, 5, 10], [0, 6, -2, 6, -16, -6, -6, 0, 6, 0, 0, -12, 0, 0, -12, 0], [23, 6, 18, 4, -4, -6, -7, 0, 7, 0, 11, -5, -11, 0, -5, 0], [2, -4, -12, -16, 14, 4, -8, 0, 8, 0, -8, 4, 8, 0, 4, 0], [-15, 15, 18, -6, -1, -15, -3, 0, 3, 0, -13, -3, 13, 0, -3, 0], [-7, 1, 0, 0, 0, 1, -5, -4, -5, 6, -1, 7, -1, 12, -7, -16], [-4, 2, 0, 0, 0, 2, 6, -8, 6, -8, 4, 4, 4, -16, -4, -20], [4, 2, 0, 0, 0, 2, 6, 9, 6, 0, -5, -4, -5, 10, 4, -4], [4, 5, 7, -12, -6, -5, 4, 0, -4, 0, -8, 16, 8, 0, 16, 0], [3, 4, 0, 0, 0, 4, 0, 0, 0, -20, 6, -3, 6, -8, 3, 10], [8, -4, 16, 4, -4, 4, 0, 0, 0, 0, 2, 4, -2, 0, 4, 0], [-8, 10, 0, 0, 0, 10, -2, -8, -2, -12, 0, 8, 0, -2, -8, 8], [8, -9, 0, 0, 0, -9, -10, -6, -10, 18, -2, -8, -2, -4, 8, -6], [24, 1, 1, -8, -10, -1, -2, 0, 2, 0, 6, -2, -6, 0, -2, 0], [-2, -2, 0, 0, 0, -2, -7, -17, -7, 12, 3, 2, 3, -2, -2, 0], [-6, -8, 0, 0, 0, -8, -4, -12, -4, -2, 12, 6, 12, -6, -6, 4], [-2, -9, 0, 0, 0, -9, 6, -12, 6, -8, -4, 2, -4, 14, -2, -9], [8, 6, -6, 4, 2, -6, 4, 0, -4, 0, 6, -4, -6, 0, -4, 0], [8, -7, 0, 0, 0, -7, -7, 8, -7, 4, 0, -8, 0, -8, 8, -8], [-1, -2, 0, 0, 0, -2, 8, -14, 8, -8, 6, 1, 6, 4, -1, -8], [4, -2, 0, 0, 0, -2, -2, 12, -2, 4, 8, -4, 8, 10, 4, -4], [-2, -2, 2, 26, -6, 2, 0, 0, 0, 0, 14, -4, -14, 0, -4, 0], [17, -1, -5, -6, 18, 1, 4, 0, -4, 0, -6, 13, 6, 0, 13, 0], [-8, 2, 6, -16, 4, -2, -6, 0, 6, 0, -4, 0, 4, 0, 0, 0], [10, -2, 0, 0, 0, -2, 0, -14, 0, 12, 2, -10, 2, -16, 10, -8], [0, -4, -12, -9, -8, 4, 3, 0, -3, 0, 2, -4, -2, 0, -4, 0], [-9, 12, 12, 20, -4, -12, 6, 0, -6, 0, 1, 1, -1, 0, 1, 0], [-5, 11, 0, 0, 0, 11, -5, -6, -5, -18, 3, 5, 3, 2, -5, 18], [-4, -5, 18, -4, 1, 5, -5, 0, 5, 0, -3, -12, 3, 0, -12, 0], [3, -6, -2, 4, 10, 6, -6, 0, 6, 0, 2, -3, -2, 0, -3, 0], [-1, -3, 0, 0, 0, -3, -1, -10, -1, 18, 9, 1, 9, -2, -1, -10], [-6, -4, 0, 0, 0, -4, 9, -4, 9, 8, 11, 6, 11, -18, -6, -5], [20, -10, 0, 12, 0, 10, 0, 0, 0, 0, 4, -2, -4, 0, -2, 0], [-11, 6, 0, 0, 0, 6, 8, 5, 8, -12, 2, 11, 2, -2, -11, -4], [20, -4, 4, 2, 6, 4, 6, 0, -6, 0, 4, 2, -4, 0, 2, 0], [6, 8, 0, 0, 0, 8, 4, 6, 4, -16, -5, -6, -5, -7, 6, -8], [-10, 4, -12, -2, -4, -4, 8, 0, -8, 0, 0, 0, 0, 0, 0, 0], [2, -3, 0, 0, 0, -3, 9, 6, 9, -20, 3, -2, 3, 4, 2, -24], [2, -2, 14, 10, 6, 2, -2, 0, 2, 0, 12, -2, -12, 0, -2, 0], [0, -7, 0, 0, 0, -7, -5, -16, -5, 4, 4, 0, 4, -16, 0, 10], [-3, -1, 0, 0, 0, -1, -7, 0, -7, 4, -13, 3, -13, 24, -3, 0], [0, -17, 0, 0, 0, -17, 7, 8, 7, 24, -6, 0, -6, 1, 0, -12], [-3, 7, -2, 14, 13, -7, 5, 0, -5, 0, -1, 5, 1, 0, 5, 0], [-3, -10, 0, 0, 0, -10, -3, 2, -3, 24, -5, 3, -5, 6, -3, 14], [-12, 9, 0, 0, 0, 9, 5, -13, 5, -10, -7, 12, -7, 0, -12, 4], [-9, 2, 0, -8, -14, -2, -4, 0, 4, 0, 4, 1, -4, 0, 1, 0], [-4, -5, 10, 4, -23, 5, -7, 0, 7, 0, -5, 2, 5, 0, 2, 0], [4, 8, 0, 0, 0, 8, -8, -14, -8, -30, 4, -4, 4, 8, 4, 14], [-1, 7, 0, 0, 0, 7, -3, -16, -3, 12, 9, 1, 9, 2, -1, 2], [3, 3, 17, -22, 2, -3, -2, 0, 2, 0, 11, -1, -11, 0, -1, 0], [0, -3, 0, 0, 0, -3, 5, 12, 5, -24, 5, 0, 5, -10, 0, -20], [-16, -12, -12, 4, 4, 12, 6, 0, -6, 0, -6, 0, 6, 0, 0, 0], [-16, -3, 0, 0, 0, -3, -1, 2, -1, 8, -5, 16, -5, 10, -16, 6], [4, -8, 0, 0, 0, -8, -4, 18, -4, 0, 2, -4, 2, 18, 4, -8], [0, 3, 0, 0, 0, 3, -3, -2, -3, -16, -2, 0, -2, 22, 0, 10], [16, 18, 22, -4, 0, -18, 4, 0, -4, 0, 6, -2, -6, 0, -2, 0], [16, 6, -12, 1, -8, -6, 6, 0, -6, 0, -4, 8, 4, 0, 8, 0], [-4, -2, 8, 3, -16, 2, 4, 0, -4, 0, 13, -12, -13, 0, -12, 0], [6, -5, 25, -8, 6, 5, -12, 0, 12, 0, 4, 0, -4, 0, 0, 0], [-10, 6, 0, 12, -16, -6, 4, 0, -4, 0, 10, -2, -10, 0, -2, 0], [20, 3, 4, -6, 9, -3, -1, 0, 1, 0, 5, -10, -5, 0, -10, 0], [-7, 3, 0, 0, 0, 3, 11, 10, 11, 4, -3, 7, -3, -12, -7, -14], [4, -8, 0, 0, 0, -8, 3, 2, 3, 40, -2, -4, -2, 7, 4, 4], [-3, -5, 0, 0, 0, -5, -9, -2, -9, 18, -1, 3, -1, -6, -3, -4], [-7, -5, 12, 4, 14, 5, -19, 0, 19, 0, -5, 5, 5, 0, 5, 0], [2, 12, 0, 0, 0, 12, 10, 10, 10, 6, 0, -2, 0, -4, 2, -4], [2, 3, 0, 0, 0, 3, 3, -1, 3, -16, -3, -2, -3, -8, 2, 20], [-27, -8, 2, -6, -10, 8, 3, 0, -3, 0, -7, 3, 7, 0, 3, 0], [-4, -1, 0, 0, 0, -1, 3, 7, 3, 30, 5, 4, 5, -12, -4, 4], [-23, -2, -8, 4, 16, 2, -2, 0, 2, 0, 2, 1, -2, 0, 1, 0], [4, 4, 0, 0, 0, 4, -2, -10, -2, -12, 2, -4, 2, 8, 4, -10], [6, 2, 4, -11, 24, -2, -4, 0, 4, 0, 19, -2, -19, 0, -2, 0], [-26, -9, -6, 0, 1, 9, 7, 0, -7, 0, -9, -6, 9, 0, -6, 0], [15, -10, 6, -1, 16, 10, -10, 0, 10, 0, 5, -11, -5, 0, -11, 0], [12, -7, 0, 0, 0, -7, -2, 18, -2, -12, 6, -12, 6, 0, 12, -5], [-16, -6, -4, -20, -10, 6, 6, 0, -6, 0, -6, 2, 6, 0, 2, 0], [22, 11, -6, 4, -6, -11, 8, 0, -8, 0, -6, -2, 6, 0, -2, 0], [5, 8, 0, 0, 0, 8, -9, -8, -9, -32, -1, -5, -1, -18, 5, 8], [-4, 5, 0, 0, 0, 5, 16, 14, 16, 12, -10, 4, -10, -8, -4, -10], [12, 10, 0, 0, 0, 10, -4, -2, -4, -10, 4, -12, 4, 0, 12, 8], [-14, -2, 12, 16, -16, 2, -2, 0, 2, 0, 4, -14, -4, 0, -14, 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_d();".
// 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_d(:prec:=16)
    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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1470_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![64, 0, 2688, 0, 26016, 0, 83808, 0, 74484, 0, 20144, 0, 2092, 0, 80, 0, 1]>,<17,R![8452, 7456, -25728, -18832, -3596, 304, 192, 24, 1]>],Snew);
    return Vf;
end function;