
// Make newform 1502.2.a.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1502_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1502_2_a_f();".
// 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_1502_2_a_f();".
// 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 := [-16, -72, 163, 341, -193, -354, 70, 134, -7, -20, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [65684, -43087, -563689, -248639, 734978, 232522, -324370, -66273, 56308, 5788, -3377], [-11908, 218599, -293791, -935697, 783622, 800590, -488494, -224575, 106732, 19580, -7719], [128476, 366731, -896739, -1070205, 1447470, 917910, -773302, -260419, 154204, 22668, -10331], [59296, -32459, -510145, -133691, 654830, 174802, -297982, -58357, 53496, 5548, -3309], [-1891, -66654, 48043, 180678, -90328, -130539, 55763, 33449, -12598, -2757, 944], [-286404, -10397, 1862053, 1055579, -2596834, -1152074, 1316490, 361301, -255572, -33284, 16813], [-54224, -218005, 687729, 967795, -1075122, -838742, 582246, 234113, -119748, -20172, 8321], [137972, 583519, -1492535, -2127889, 2247430, 1792686, -1204054, -498655, 248332, 43044, -17407], [194524, 543615, -3000471, -3061481, 4827926, 2870574, -2578286, -846295, 516508, 75532, -34863]];
    Rf_basisdens := [1, 1, 26056, 52112, 52112, 13028, 3257, 52112, 13028, 26056, 52112];
    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_1502_a();"
function MakeCharacter_1502_a()
    N := 1502;
    order := 1;
    char_gens := [3];
    v := [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;

function MakeCharacter_1502_a_Hecke(Kf)
    return MakeCharacter_1502_a();
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 := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0], [-1, 0, 0, 0, 1, 0, -1, 0, 0, -1, 0], [0, -2, -1, 1, -1, 0, 1, 0, -1, -1, -1], [-3, -1, 1, 1, 0, 0, 1, 1, 1, 1, 0], [-2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2], [-1, -1, -1, -1, -1, 0, 0, 1, -1, -1, -1], [-1, 1, 2, -1, 1, -1, 0, 1, 1, 0, 1], [0, 0, 1, 0, 0, 1, -1, 0, 0, 1, -1], [-2, -1, -2, 1, 0, -2, 0, -1, -1, 1, 1], [-1, 0, 0, 1, -2, -2, 2, -2, -1, 2, 1], [1, 1, 0, -2, 0, 0, 0, -2, 0, -1, -2], [0, 3, 1, -1, -1, 2, -2, 0, -1, -1, 0], [-4, 2, -1, 0, 3, 2, -3, 1, 0, -2, 0], [-2, -1, 1, -1, 0, -2, 3, 3, -1, 0, -1], [-2, 2, -1, -1, 1, 3, 0, -1, 2, 0, -1], [-1, 2, 2, -3, 1, -1, 0, -1, 0, 0, -2], [-2, 1, -1, 1, -1, 2, 3, 0, 3, 2, 1], [0, 0, -2, 0, 0, 3, -5, -2, -1, -2, -2], [-4, 1, 1, -1, 1, 1, 0, 0, 1, 1, 2], [0, -1, -1, 1, 1, 0, 1, -1, 2, 2, -1], [-2, 2, 0, -2, 0, 1, -1, 3, 2, -1, 0], [-1, -2, 2, 2, -1, -1, 3, 2, 0, 0, -1], [-6, -4, 0, 3, 0, -3, 3, 1, -3, -2, -2], [-1, -1, 0, -1, 2, 2, -2, 3, -2, -2, 0], [-1, 1, -1, 2, -3, -2, 0, 0, 0, 2, 5], [0, 3, 1, -1, 3, 2, 0, -1, 0, 0, -4], [0, -2, -3, 3, -2, -3, 3, -3, -2, 3, 1], [3, 4, 1, -2, 1, -3, -2, -1, 0, 2, 2], [-3, 2, 0, 2, 3, 6, -2, 0, 4, 0, 2], [-3, -2, 0, 2, -3, -5, 2, 0, -4, 2, 3], [3, 0, 0, 3, 0, -1, -1, -1, 3, -2, 1], [-3, -2, -1, 1, -2, 3, 4, 2, -5, -3, -4], [3, -1, 1, 0, -1, 2, -1, 0, -5, -4, 0], [7, 2, -3, -1, -2, 0, -4, -4, -1, -1, 3], [-5, -2, 1, 2, 0, 1, 0, 0, -3, -1, -4], [4, 0, -1, -2, -1, -2, -1, -1, 2, 2, -1], [-2, -3, 0, 2, -1, -5, 4, -1, 3, 1, 1], [-2, 3, 0, 2, 4, 5, -2, -1, 2, -4, 0], [0, 1, -4, 3, -4, -3, 2, 0, -4, 1, 4], [2, 2, 2, -6, 5, 3, -6, -1, 2, 0, -1], [6, 3, -4, 0, 1, -1, -7, 0, -1, -3, 5], [-3, -4, 1, 2, 0, -3, 3, 1, 3, -2, -1], [0, -6, 0, 2, 5, -2, 0, 1, 4, -1, -3], [-6, 4, 1, -1, 4, 8, -8, 2, -1, -6, -3], [1, -1, -3, 3, -1, -6, 3, -5, -1, 5, 3], [-7, -2, 3, 0, 1, 1, 2, 1, -2, -4, -4], [-1, 6, -1, -3, 4, 2, -3, -6, 4, 3, -1], [-3, 2, 5, -3, 2, 6, 0, 4, 1, 0, -1], [-1, -4, 4, 3, 3, -3, 6, 0, 2, 4, 4], [-1, 6, 0, -1, 1, -2, -6, -3, -1, 0, 2], [0, -6, -2, 3, -4, -2, 4, -1, -1, 3, -2], [-2, -6, 1, -1, 0, 1, 4, -3, 1, 2, -2], [6, 1, -2, -1, -4, -5, 4, 3, -3, 3, 1], [-3, -3, 1, 1, 0, -1, -2, 3, 0, -2, 2], [6, 1, -5, 3, -5, -2, 2, -9, -4, -1, -5], [-5, 2, 7, -2, 2, 2, -2, 2, 1, 2, -1], [-2, 6, 0, 0, 7, 6, -12, -2, 1, -1, 3], [7, -5, -2, 4, 0, -4, 4, 1, -1, 0, -1], [-9, 1, 0, -1, 0, -3, 5, 2, 2, 7, 5], [2, 3, 2, -4, -3, -3, 3, 3, 1, 6, 2], [-4, -3, 2, 0, 0, 0, -1, -3, 4, 4, 0], [2, -7, -2, 1, -6, -3, 1, -1, 0, 0, 2], [-5, -5, -1, 2, 0, -7, 0, -6, -4, 0, -6], [0, -5, 0, 4, -2, 7, 1, 3, 2, -6, -2], [7, -4, -1, -1, -6, -5, 10, -1, -9, 1, 0], [1, 0, 5, -1, -2, 2, 2, 1, 5, 6, 3], [5, 3, -3, -1, -5, 3, -3, 5, 1, 1, 7], [5, -2, -3, -2, -5, -2, 7, -1, -4, 1, 2], [10, 2, -3, 1, -8, -3, 1, -2, -9, -4, 2], [-1, -4, 1, 1, -1, -7, 1, -2, -4, 3, -6], [-3, 0, 6, -3, 2, -3, -5, 0, 0, -3, -4], [0, 6, -5, -6, 4, 1, -6, 1, 4, 1, 5], [1, -6, -4, 2, 4, 1, 0, -3, 4, 1, -2], [2, 0, -8, 4, -3, -2, 1, -1, 0, 5, 7], [-2, 8, 7, -1, 4, -1, 3, -1, 7, 9, 10], [-17, 3, 5, -2, 7, 1, 2, 8, 2, 0, 3], [4, -2, 6, 2, 0, -1, 4, 5, 1, -2, -3], [-3, 4, 3, 1, -2, -2, -2, 0, 4, 0, 3], [-7, -5, -2, 2, -4, -2, 1, 4, -3, -1, 3], [2, 12, 2, -2, 9, 6, -10, -4, 8, 2, 6], [6, 2, 0, -1, -3, 5, -7, -2, -4, -5, -4], [-4, 3, -4, -1, -1, 0, -5, 2, 7, 1, 11], [5, 1, -3, -7, -7, -4, 1, -1, -5, -3, 0], [9, -1, -5, -5, -5, 1, -1, 3, -4, -2, -3], [2, 0, 1, 3, 1, -2, 3, -4, 2, 5, -5], [-2, -3, -5, 8, 3, -6, -3, -5, -4, -7, -3], [-2, -7, 5, 2, -4, -9, 8, 4, 0, 6, 2], [-7, 0, 1, -3, 8, 6, -3, 3, -2, -6, -6], [0, 3, 3, -3, 2, 4, -1, 1, -5, -1, 2], [-8, 10, -6, -1, 4, 7, -10, -9, 1, -1, 2], [-7, 4, -4, -2, -4, 1, -4, 2, 3, 3, 2], [-5, 0, 5, 3, 4, -3, 10, -2, 4, 7, -1], [5, -2, -7, 1, -15, -7, 4, -2, -8, -1, 4], [-4, 9, -4, -2, -5, 9, -2, -1, 4, 2, 8], [-5, 7, -2, 1, 5, 6, -10, 4, 7, -1, 10], [8, -1, 3, -6, 0, 13, -5, 3, 1, -5, -7], [0, -2, -3, 4, -4, -2, -6, -3, -2, -2, 1], [0, 1, 3, -8, 6, 7, -12, 4, -1, -4, -2], [-6, 2, 4, 1, 5, 0, 2, -3, 5, -3, -2], [0, -8, 7, 3, -3, -11, 12, 0, -6, 5, -4], [-11, 1, 1, -1, 6, 4, -2, -1, 14, 7, 12], [-5, 0, 4, 2, 2, -4, 8, 11, 7, 4, 3], [1, -1, 1, 0, 0, 6, -3, 8, 2, -8, 0], [-10, 4, 5, -4, -3, -5, 5, 2, -1, -1, 2], [-1, -11, 4, 5, -3, -8, 8, 2, -1, 3, 0], [0, 8, 8, -5, 2, 5, -5, 7, 0, -5, 1], [-6, -2, 5, -8, 2, -4, 7, 2, -6, 3, -7], [7, -6, 3, 4, 3, -10, 3, -2, -3, 0, -6], [-1, -9, 2, 1, 3, 2, 1, 0, -7, -12, -12], [-10, 2, 2, 3, -2, -4, -2, -2, -5, 1, 2], [15, -5, -10, -1, -11, 0, 2, -11, -8, 1, -3], [-17, 0, 3, -2, 5, 1, 4, 9, 8, 1, 1], [-8, 2, -4, 0, -2, 4, -6, -3, 0, 6, 3], [9, -1, 4, -6, -2, 0, -1, 2, 2, 4, -2], [-11, -6, 1, -1, 3, -3, 2, -7, -1, 3, -12], [-2, 5, 0, -4, 1, -2, -3, -12, -2, 3, 1], [-10, -3, -1, 2, 3, 0, 0, 4, 12, 4, 5], [-5, 4, -4, 3, 5, 13, -14, -5, -2, -11, -3], [-1, -3, -6, 4, -1, -4, -2, -3, 2, 3, 1], [2, -9, -3, 1, -8, -8, 7, 7, -7, -4, 6], [-5, -3, 8, 3, -2, -5, 9, 12, -4, -2, 0], [-16, 7, 3, 1, 10, 6, -9, 5, 8, -2, 6], [-7, -1, 1, 11, 6, -4, -1, -8, 6, 1, 1], [8, -6, -1, -3, -6, 10, 2, 4, -3, -4, -5], [2, 5, -1, -9, 2, 13, -9, 6, -1, -7, -6], [4, -5, -10, 7, -11, -1, 8, -3, -3, -2, 2], [0, -6, 2, -8, 0, 4, -1, 0, -5, -7, -10], [16, 5, -5, -6, -3, -6, 0, -3, -5, -4, 6], [-8, 6, 0, -1, 6, 7, -8, -1, 2, -9, -4], [5, -3, 3, -7, 1, -1, -4, 2, -7, -4, -7], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5, -2, 6, 6, 2, 2, 3, 3, 1, 3, -1], [19, 0, -4, -5, -9, -5, 0, -4, -8, -2, -7], [8, -10, -5, 4, -1, -2, 7, -4, 2, 3, -2], [14, -1, 0, -6, -11, 0, 3, 1, -4, 5, 0], [1, -4, 4, 6, 2, -2, -2, -3, -4, -2, -6], [4, -8, -1, 2, -2, -4, 2, 5, 7, -1, 5], [17, -7, -3, 0, -7, -8, 8, -5, -5, 9, -5], [-11, -3, 1, 5, -10, -6, 7, -3, 2, 6, 0], [-5, -12, -3, 9, -4, -4, 9, -8, -2, 1, -1], [8, -8, -3, 4, -1, -10, -2, -1, -4, -7, 8], [9, -4, -1, 3, -6, -6, 5, -9, -5, 0, -8], [-9, -11, -1, 9, -6, -6, 1, -1, -8, 1, -2], [14, -11, -1, 0, -4, -5, 8, 1, -8, 1, -7], [-15, -4, 5, -3, 4, 0, 6, 6, -2, -9, -9], [0, -3, -8, 2, 1, 4, -4, 0, 2, -10, -6], [-5, 11, 2, -5, -2, 10, -9, 11, 2, -1, 15], [9, -5, -3, 0, -9, -3, 0, 4, -7, -4, 5], [9, -7, 1, -3, -6, -9, 14, -7, 0, 2, -5], [-8, 7, -8, 6, 0, 4, -5, 2, 3, 1, 10], [-13, 1, 2, -6, 13, 4, -3, -6, 3, -2, -7], [4, 13, -3, -2, 7, 16, -22, -6, 2, -6, 6], [-7, 10, -1, 6, 10, 19, -13, -6, 11, -5, 2], [2, -10, -1, 1, -6, -2, 15, -5, -1, 3, -7], [3, 11, 6, -10, -5, 2, 5, 5, -1, 7, 3], [4, 0, -1, -2, 0, 0, -1, -11, 7, 3, -5], [0, 6, 4, 4, 5, -3, -2, -4, 14, 7, 13], [4, 11, -4, -1, -2, 2, -10, 1, -6, 1, 0], [0, 9, -2, -2, 7, 6, -13, 3, 5, 11, 7], [15, -12, -2, 2, -7, -5, 10, 11, -5, 4, 4], [-13, 2, 11, -4, -1, -6, 3, 14, 2, 11, 11], [-17, -2, 6, 4, 7, 8, 5, 15, 0, -8, -4], [10, 8, 1, 3, 2, -7, 0, -9, -2, 7, -4], [-1, -5, 5, 4, -9, -2, 4, 9, 0, -1, 7], [-5, 9, 7, -10, 1, 1, 4, 4, 10, 14, -1], [-8, -10, -7, -3, -9, 4, 4, 14, -1, -9, -1], [-14, 6, -2, -2, 3, 3, 8, 3, -2, -3, -5], [-7, -1, 5, -10, 7, 3, -7, 2, -5, -1, -9], [-3, -9, 15, -6, 2, -1, 7, 8, 1, 2, -6], [-7, -16, 4, 8, 4, -6, -3, -1, 3, 1, -3], [-2, 4, 3, -4, 6, 15, 1, 3, 7, 0, -5], [-13, 4, -3, 4, -1, 8, -9, -2, -6, -8, 5], [-8, -2, -5, -1, 7, 5, -4, -9, 1, -3, -7], [13, 7, -6, 2, 0, 11, -5, -1, 9, 2, 8], [16, -7, 2, 1, 5, -6, 1, -6, -2, 2, 0], [-17, -5, 4, -4, 4, -5, 9, 15, -2, -1, 3], [1, 1, -1, -9, 1, 15, -12, 1, 7, -5, -1], [1, 2, 10, -2, 6, -2, 10, 0, 3, -3, -6], [-5, -2, -4, 2, -4, -6, 15, 0, 2, 0, -2], [-12, 1, -5, 2, -4, -3, -3, -3, 3, 1, 7], [-10, -4, 8, 2, -2, -4, 11, 5, 0, 5, -9], [15, 3, -10, -6, -8, 2, -1, 5, -8, 1, 5], [2, 8, 3, -2, 5, 9, -1, -5, -4, -1, -16], [-1, 3, -3, 0, 12, 13, -5, 9, 2, -6, 7], [-3, -6, -7, -4, -10, 9, 7, 8, 2, -1, 1], [-14, -6, 6, -1, 6, 4, 1, 7, 2, -11, -4], [0, -3, -5, 2, -10, 3, 9, -6, -4, 12, 4], [-13, 8, 0, -3, 7, 2, -1, 10, 6, 2, 0], [-18, 7, 6, -2, 2, 7, 1, 4, -1, -1, -8], [-1, -3, -11, 2, -6, 3, -9, -13, -5, -2, -10], [-11, -9, -1, 7, 8, 2, -1, -7, 11, 9, -7], [-18, -21, 0, 5, -6, -8, 12, 1, -4, 0, -6], [15, -2, -13, 3, 4, 7, -5, -10, 2, 2, -3], [-15, -7, -6, 13, 5, -17, 5, -15, -1, 4, -1], [4, 10, -1, -6, 3, 4, 0, 14, -2, -13, 9], [9, -9, 5, 5, 1, -6, 3, 5, -8, -10, -15], [13, 1, -7, -7, -3, 2, -11, -3, -12, -23, -10], [-13, -13, 1, 7, -2, -11, 11, -4, -5, 0, 1], [-11, 1, 10, 2, 4, -11, 17, 5, 13, 18, 1], [7, 8, 4, 1, -5, 0, 1, 7, -5, -3, 3], [-26, 3, 7, 3, 12, 0, -1, 1, 12, 5, -5], [4, 2, -7, 7, 4, 11, -1, -5, 6, 1, -8], [10, -10, -4, 3, -2, 17, -11, -7, -6, -12, -18], [12, 6, -2, -4, -7, -1, 8, 15, -4, 0, 10], [17, 0, -3, 7, -8, 5, -11, -4, -4, -11, -1], [-2, 4, 7, 4, 0, -10, 0, -5, 12, 15, 13], [12, -2, -1, 2, 9, -6, -5, 2, -1, -12, -5], [4, 6, -5, 7, 1, 9, -4, -8, 0, -7, 5], [15, 16, -5, -7, -7, 9, -1, -4, 1, 7, 10], [13, 1, -5, 6, -7, -4, 5, -7, -6, -6, -4], [-2, -1, -3, -3, 6, 14, -13, 7, 8, -3, 8], [-6, 1, 5, 8, 10, -1, -3, -9, 10, 2, -6], [-5, 3, 15, -7, 2, -2, 3, 5, 2, -7, 3], [4, -11, 3, 5, -11, -25, 11, -1, 0, 11, -1], [13, -1, 3, 0, 7, 5, 2, 7, 14, -3, -4], [-7, 4, 0, 5, 8, -11, -3, -7, 4, 11, 11], [-1, 5, 6, 1, 4, -2, -2, -2, -1, 1, -8], [-5, -12, 3, -4, 10, -2, -1, 15, 11, -1, -4], [10, 8, -14, 0, -8, 1, 0, -17, -7, 0, 5], [4, -4, -3, 9, 4, -5, 3, -5, -1, 5, 3], [19, -12, -1, 15, -5, -8, 14, -4, -2, 11, 5], [-11, 7, -10, 2, -4, 5, -7, -11, -5, -3, 0], [-7, 14, 8, -2, 15, 13, -1, 10, 11, -6, -5], [10, -3, -4, 18, 2, -9, -2, -5, 6, 7, 8], [12, -3, -6, 3, -9, -3, 10, 7, -1, 3, 7], [9, 20, 2, -5, 10, 4, -19, -8, 8, 0, 15], [-4, 4, 3, 5, 8, 7, -1, 8, 6, 4, 0], [-17, -1, 9, -2, -4, -1, 15, 8, 9, 13, 3], [4, 10, 5, -2, 3, 5, -9, 7, 7, -6, 7], [17, -13, -6, 4, -7, 5, 12, -4, 1, 12, 4], [25, 1, 1, 1, -6, -5, 21, 9, -3, 0, 0], [13, -5, 5, -2, 5, 10, -4, 1, -3, -7, -21], [-1, 7, 10, -7, 7, 13, 2, 19, 12, -6, 0], [14, 9, -2, 4, 7, 3, -1, -10, 2, 1, -8], [18, -3, -14, 5, -14, -14, -1, -6, -16, -1, -6], [14, 20, -2, -3, -8, -4, -4, -7, -11, 5, 5], [2, 4, 10, -5, 3, 1, -5, 3, -4, 5, 3], [8, 0, 7, -14, -2, -7, 12, 0, 0, 6, -16], [11, 2, -4, -13, -8, 0, 1, 7, -11, -8, -5], [6, -2, -6, 9, -1, 4, -2, -2, -14, -6, -14], [-13, -15, 4, 15, 1, -15, 13, 1, 8, 4, 11], [1, 0, -6, -3, 2, 5, -17, -3, -20, -13, -7], [22, -8, -3, 7, -12, -12, 16, -12, 1, 15, 2], [-21, -4, -1, 7, 2, -6, 0, 0, -1, 11, -11], [-22, 1, 9, 3, 11, 4, -1, 11, 1, 0, -12], [0, 15, -2, -10, 7, 20, -26, 11, 3, -21, 5], [6, 17, 4, -9, 10, 1, -14, 11, 7, -4, 6], [-2, -3, 1, 12, 4, -1, 7, 10, -2, 4, 17], [2, -1, 2, 2, -12, -7, 13, -4, -5, 8, 2], [12, 4, 7, -6, 17, 1, -10, -5, 7, -9, 2], [24, 1, 0, -3, 1, 4, 1, 4, 2, 9, 5], [1, -3, -5, -4, 11, -7, -4, -1, -8, -2, -5], [-23, 9, 8, -4, 3, 3, -14, -3, 5, 10, 14], [-5, 3, 3, -6, -4, -22, -1, -2, -15, 5, 2], [14, 13, -5, -6, -7, 11, -9, 4, -1, -9, 20], [-13, 8, -14, 7, 0, 6, -12, -3, 1, -4, 14], [8, -6, -2, -1, -6, -10, -10, -6, -16, -11, -1], [10, -3, -12, 4, 3, 1, -9, -3, 6, 0, 9], [3, 27, -15, -11, 1, 4, -16, -4, -1, -3, 16], [19, -9, 4, -6, -12, 3, 10, 5, -3, 1, -4], [-10, -3, 3, -5, -2, -2, 10, 8, -3, 10, -8], [6, -3, -11, 9, -8, 2, 6, -9, -7, 2, 8], [8, -10, 13, 11, -11, -13, 22, 3, 8, 6, 2], [-1, 0, 9, 3, -2, -3, 3, 10, -11, -1, -3], [22, 6, -3, 0, 6, -2, -3, -5, -4, -5, -1], [-5, -9, -7, 4, -12, -20, 7, -2, -5, 4, 13], [-11, -9, 2, 3, 1, -1, -4, -3, -9, -12, 3], [2, 11, 3, 5, -1, 6, -2, 16, 1, -1, 7], [-27, -4, 13, 0, 12, 6, 1, 22, 13, -6, 6], [-19, 7, 12, 2, 5, -17, 8, 7, 9, 1, 9], [-20, 5, 5, -6, -1, 1, -9, -7, 6, 6, 10], [22, 0, -3, 8, -4, -6, 14, 2, -2, 0, -6], [4, 20, 0, -13, -2, 6, -6, -6, 7, 13, -5], [-7, 2, 14, -7, 5, 8, 2, 16, 20, 0, 5], [5, -4, -4, 7, -16, 8, -5, 3, -7, -1, 10], [3, 15, 7, -7, 9, 3, -13, 3, 1, -6, 5], [-18, 16, 20, 0, 14, -5, -5, 1, 15, 11, -3], [8, -3, -1, 9, -12, -8, 15, -1, -8, 7, 5], [-2, -10, -7, -5, 1, 3, 3, -14, -1, 10, -10], [-6, 11, 13, -8, 5, 21, -3, 12, -8, -10, -17], [-24, 4, 10, -13, 3, 2, 1, 22, 6, 0, 5], [10, 6, 1, -16, -9, -2, 0, -6, -13, -3, -5], [7, 4, -9, -9, 2, 12, -11, -13, -7, -2, -10], [-23, 3, -1, -7, 13, 5, -4, -2, 2, -9, -17], [-10, 0, -21, 8, 5, 3, -15, -11, 4, 0, 6], [2, 4, -2, 9, 8, -19, -12, -17, -1, -3, 9], [8, -11, 11, -7, -14, -18, 26, 5, -15, 6, -4], [1, 15, -4, -17, 9, 16, -25, 11, -11, -12, 3], [12, 3, 2, -3, -2, 2, 2, -2, 13, 0, 2], [-11, 0, 8, -9, 0, -1, 16, 4, 5, 18, -10], [-9, -11, 3, -2, 2, 1, 4, -5, 6, 3, -6], [-21, -1, 7, 4, 12, 2, -7, -11, 10, 5, -4], [-13, 18, 15, -7, 14, -7, -10, -5, 12, 12, 4], [-13, 22, -1, -2, -1, -6, -10, 1, 7, 8, 29], [20, 8, -12, -8, 0, 5, -16, 2, -2, 1, 6], [3, -6, -3, 6, 7, -5, -13, -1, 12, -6, 10], [-21, -15, -11, 18, -16, -19, 17, -5, -1, 10, 6], [9, 8, -2, -2, 6, 17, -6, 5, 4, 0, 9], [24, -17, -1, 9, -1, -13, 8, 4, 5, -1, 2], [-27, 17, 4, -10, -1, -2, 7, 9, 12, 5, 13], [-31, 15, 1, 5, 10, 9, 4, -8, 17, 7, -2], [2, 18, 11, -10, 7, 19, -2, 8, -7, -9, -19], [8, 10, 2, -11, 4, -1, -18, 1, 1, -11, -4], [2, -17, 1, 2, -7, -6, 0, -3, -4, -1, 1], [-18, -4, -1, 11, 6, -17, 2, -10, 9, 10, 3], [1, 9, -6, -3, -6, 1, -10, 6, -21, 0, -3], [3, -20, -1, 4, -16, -18, 23, 2, -5, 14, 1], [-6, 8, 4, -1, 2, -5, -2, -1, 5, -11, 7], [-10, -20, -4, 16, 2, -5, 10, 0, 1, 5, -8], [-22, 2, 2, -1, 8, 0, 4, 5, 9, -7, 2], [11, 5, -11, -4, -5, 19, -5, -6, -1, -7, -4], [13, 7, -11, 3, 1, 2, -14, -14, -14, -8, -11], [-5, -18, -16, 19, -9, -1, -6, -3, -8, -13, -5], [2, -24, 2, 11, -18, -13, 27, 4, -14, 13, -7], [12, -15, -13, -1, -12, -9, 10, -13, -5, 0, -9], [-9, -6, -15, -6, -9, -2, 0, 1, -22, -11, 2], [-4, 8, -13, 11, 9, 13, -14, -8, 9, -3, -6], [8, -1, -8, -10, 4, 6, 0, 1, -15, -13, -9], [-20, -1, 9, 14, 11, 5, 2, 5, 15, 15, 9], [-9, -2, -3, 7, -7, -6, -1, -5, -6, -10, 8], [-21, 14, 4, -15, -1, 14, -9, 15, -1, -24, 5], [-8, -8, 12, -4, -2, 2, 13, 10, 1, -3, -5], [2, 15, -3, -12, 4, 6, -4, -3, -7, 0, 2], [-17, 1, -1, -1, 4, 3, 12, 29, 9, 3, 10], [-17, 6, 6, -10, 8, 2, 10, -7, 8, 13, -12], [1, -14, -19, 8, -11, -8, 9, -9, -12, -3, -2], [-14, 12, 7, -9, 6, 2, -15, 5, -3, 4, -5], [-1, -5, -7, 1, -9, 2, 7, -5, -11, 1, -10], [8, -9, 4, -6, -6, -12, 8, -9, 4, 6, 4], [-4, -10, 7, 0, -3, 7, 9, 2, 3, 15, -9], [3, -11, -5, -5, -12, -1, 10, -7, -7, 9, -5], [-20, 10, 3, 3, -2, -6, 9, 3, 16, 9, 19], [-7, 5, 3, -12, -5, -3, 16, 3, -9, 4, 4], [6, -3, -21, 3, -7, 2, -9, -9, -18, -14, -2], [-1, 11, -2, -4, -4, 10, -5, -3, 0, 1, -7], [-2, -24, 10, 14, 12, -6, 20, -1, 11, 7, -6], [-15, -2, 8, 5, 16, 8, -5, 0, 12, 5, 8], [-11, 14, 19, -7, 11, -8, -9, 17, 23, 18, 16], [-35, -15, -4, 18, 0, -15, 25, 6, -6, 7, -4], [10, -4, -6, 18, -1, -7, 0, -12, 8, -3, 3], [-6, 8, 15, -2, 9, 4, 8, 1, 15, 1, 3], [9, 5, 10, 5, -15, -10, 12, -11, -7, 9, -6], [10, -10, 14, -10, -6, -9, 4, 10, -6, -3, -5], [-5, -2, 0, -6, -8, -5, 13, 12, -18, -9, -6], [23, -1, -2, 1, -18, 1, 3, 8, -4, 5, 14], [-15, 10, 16, 0, 31, 8, -13, -4, 17, -3, -11], [-15, 0, 2, 5, -4, 2, 11, -1, -16, 2, -21], [-9, -10, -2, -1, 5, -6, 10, 1, 8, 10, -8], [4, -21, 1, -1, -2, -1, 12, -2, -6, 4, -6], [-13, 9, -3, -2, -3, 15, -4, 11, -4, -8, -6], [18, -4, 0, -10, 13, 7, -7, -9, 8, -5, -14], [8, 7, -12, -20, 2, 10, -21, 1, -14, -16, -1], [4, -10, -4, 4, 6, -1, 6, 20, 4, -7, -8], [5, 12, 11, -5, -4, 5, 4, 5, 0, 0, 14], [8, -15, 2, 6, -6, -9, -10, -13, 0, 11, 5], [18, -12, -11, 2, -4, -6, -1, -7, 7, 1, 10], [-5, -11, -16, 0, 0, 7, 0, 5, -7, -17, -19], [0, 4, 6, -9, 6, 1, -6, 14, 14, 4, 15], [9, -4, -5, 0, -10, -17, 10, -4, -9, 4, 0], [4, 8, -1, -13, -12, -2, -9, -1, -2, -1, 9], [-15, -9, -4, -3, 11, 14, -14, 11, 4, -21, 0], [16, 0, -12, 12, -9, -7, 8, -18, -10, -11, -11], [6, 6, 4, -7, -5, 13, 11, 6, -1, 10, -9], [19, 6, 5, -15, 4, -4, -5, 9, 12, 6, 8], [-2, -21, -13, 8, -16, -4, -8, -4, -22, -17, 5], [-7, -1, 3, -13, 2, 21, -12, 5, 1, -5, 6], [-6, -17, 2, 3, -2, -3, 12, -12, -14, 11, -26], [-31, -29, 1, 9, -6, -12, 15, 13, 5, -4, -4], [8, -17, -11, 9, -2, -19, 13, 6, -10, -21, -5], [25, -12, -13, 12, -3, 1, 8, -11, -12, -19, -22], [7, 16, -1, -21, -3, 16, -4, 6, -2, -10, -19], [9, 8, 5, -5, -4, -3, 12, 4, -3, -9, 1], [1, 2, -10, -10, -4, 11, -11, -20, 9, 6, -8], [0, 0, 11, -2, 4, -7, -9, 4, 0, 10, 6], [9, 8, -10, 5, -16, -5, 7, -6, -11, -2, -4], [-1, -10, -1, -1, -18, -8, 5, -10, 0, 9, 5], [8, -8, -2, 8, -7, -1, 3, -22, -6, 18, -11], [3, -1, -16, 12, -1, -2, -3, -17, -18, -7, -15], [3, 7, -1, -1, 12, 3, -17, 12, -3, -19, 19], [-22, 4, 7, 9, 5, -10, 15, -5, 26, 26, 17], [17, -4, -27, 3, -6, -10, -4, -13, -15, -6, 5], [-24, 6, 7, -6, -2, -6, 2, -6, 16, 8, 5], [-5, 11, 5, 6, 4, -14, -2, -17, 6, 11, 18], [2, -29, -11, 1, -12, -1, 0, 8, -9, -1, -1], [17, -1, -8, -1, 3, -6, -3, -5, -2, 14, 3], [-26, 9, -17, -2, -10, 22, -12, -21, -5, 2, -3], [-29, 14, 11, -19, 19, 2, -9, 7, 9, -2, 0], [1, 23, 5, -9, 12, 23, -21, 7, 18, -6, 6], [12, 4, 2, 7, -6, 10, 9, 11, 3, -5, 6], [4, -13, 0, 24, -16, -7, 23, -9, -17, 13, -10], [-8, 5, -21, -2, 1, 13, -18, -14, -18, -13, -6], [41, -19, -14, 4, -5, -2, -5, -5, -22, -18, -20], [-4, 21, 2, -10, 8, 8, 1, -12, 9, 9, 1], [-1, -11, 2, 1, 0, -2, 14, -7, -4, 4, -9], [-17, 19, 7, -8, 11, 2, -13, 0, 15, 11, 3], [28, -18, -4, -1, -4, 14, 3, 8, -23, -16, -17], [35, -8, -4, -4, -7, 1, -6, -12, -26, -13, -26], [-19, 30, -8, -14, -11, 9, -7, 5, -12, 10, 13], [-8, 8, 20, -18, 16, 9, -5, 2, 7, -1, -16], [-9, -14, 2, 4, -5, -9, 2, -8, -7, 7, -9], [37, -12, -5, 5, -7, -6, 9, -2, 3, 8, 6], [-20, -4, 5, -3, 10, 15, -13, -12, 0, -4, -11], [-19, 20, 11, -21, 10, 14, -16, -3, 8, -3, 0], [-12, 11, -16, 5, 1, 0, -10, 3, -10, -19, 1], [10, 3, -11, -1, -8, -11, 2, 1, -9, 3, 12], [-19, -15, -6, 4, 6, -10, 13, 12, -12, -9, -22], [7, 21, 12, -10, -1, -1, -2, -6, 2, 14, -2], [1, 16, 15, -17, 14, 10, 1, 9, 14, 5, -10], [0, -12, 11, -8, 5, 9, 4, 12, 8, -8, -22], [0, 6, -3, -12, 9, 14, -7, -8, 27, 15, 15], [31, -15, 3, 2, 9, 6, -6, 14, 5, -10, 1], [14, 2, 3, -7, 9, 6, -3, 7, -3, -4, -11], [-10, 21, 3, -8, 2, 20, -15, 14, -13, -3, 2], [-13, 27, -3, -6, 8, 9, -19, -8, 15, 0, 4], [-25, -1, 2, 9, -2, 6, 7, 7, 9, 12, 0], [-8, 4, -11, 8, -2, 8, -10, -7, -1, 3, 11], [6, -16, -1, 7, 11, 3, -16, 4, 9, 0, 2], [-13, -6, 11, 4, -5, -1, 6, -1, 4, 15, -6], [-18, 1, 21, 0, 13, 7, 16, 3, 16, 8, -14], [7, 5, 9, -17, 0, -5, 7, -1, -8, 0, 2], [-44, 15, 4, 0, -1, -7, 4, 0, -3, 14, 14], [17, 0, 4, -10, -2, -13, 8, 10, -1, 2, 4], [-21, 14, -6, -14, -15, 13, -1, 0, -8, -5, 7], [12, 9, -2, -4, 18, 20, -2, 13, 12, -17, -15], [-6, 5, 2, 0, -4, -7, -14, -14, 4, 6, -7], [-12, 6, 14, 5, 6, -13, 10, 2, -3, -2, -9], [-9, 14, -3, -10, 11, 2, -11, -1, 20, -2, 0], [-10, 22, 12, -7, 8, -7, 7, -13, -1, 17, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1502_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1502_2_a_f();".
// 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_1502_2_a_f(:prec:=11)
    chi := MakeCharacter_1502_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_1502_2_a_f();".
// 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_1502_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1502_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![47, -132, -509, 212, 1097, 430, -406, -312, -22, 35, 11, 1]>],Snew);
    return Vf;
end function;