
// Make newform 1003.2.a.h in Magma, downloaded from the LMFDB on 10 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1003_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_1003_2_a_h();".
// 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_1003_2_a_h();".
// 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 := [-1, -50, 179, 301, -1323, -151, 2641, -695, -2105, 951, 723, -456, -82, 90, -5, -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, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1629531, -12584729, 12947770, 58636299, -20768470, -111632131, 21890310, 96459971, -22572604, -38927099, 12562022, 6450296, -2846302, -210704, 206853, -21607], [-252191, -18751633, 6021412, 129950349, -6096132, -259608919, 17845342, 222583253, -38866840, -88715715, 25713044, 14620862, -6086430, -494240, 448281, -46479], [621523, -8034099, 13601600, 58146626, -81535535, -102838342, 140772231, 63824178, -102911265, -11490419, 35569812, -2256821, -5685137, 929938, 331382, -73855], [-1684399, 26785741, -1162256, -170098381, 109309352, 314110313, -258192220, -215179431, 216056280, 54087255, -80818546, 173262, 13610466, -1814188, -821071, 169225], [-1684399, 22006609, -1162256, -169301859, 109309352, 314110313, -258192220, -215179431, 216056280, 54087255, -80818546, 173262, 13610466, -1814188, -821071, 169225], [623575, 12942671, -17602844, -126473895, 112170862, 263860137, -220121182, -212042053, 188185360, 68975265, -76771974, -4762104, 14140202, -1423874, -912757, 171663], [-5345561, -39120777, 54040858, 327552561, -221528078, -648191089, 397855998, 491995215, -328549356, -156621049, 129619958, 14593232, -23191430, 1709236, 1466667, -252191], [-317590, 17324960, -5471927, -149988142, 76413493, 321400128, -185448483, -265461630, 177914468, 90954465, -77542723, -8664781, 14819972, -1206377, -976139, 174999], [-180195, 54032887, -38353282, -474483529, 280288612, 991791603, -580720748, -795429433, 518142928, 267361123, -216678478, -26534880, 40413834, -2947570, -2624923, 454985], [-1226399, -60979649, 68918706, 472849827, -351211644, -949139277, 642026498, 755318627, -548042556, -252272051, 226708120, 22964484, -42280306, 3520236, 2751753, -493455], [303563, -87539399, 66001372, 695675295, -466744348, -1385054993, 954414268, 1066670989, -822852954, -336831221, 332167660, 25543250, -60313288, 5665410, 3849819, -710147], [1585084, 47365022, -62155134, -364544169, 293307769, 724676789, -521963732, -560595280, 430148838, 179552242, -171588528, -14693575, 31115804, -2756504, -1988496, 360387], [-1545655, -59282695, 57412699, 461460405, -329011326, -927386318, 644160968, 728495661, -554950973, -236146200, 226475122, 19291176, -41578626, 3747450, 2675091, -488289]];
    Rf_basisdens := [1, 1, 1, 796522, 796522, 398261, 796522, 796522, 796522, 796522, 398261, 796522, 796522, 796522, 398261, 398261];
    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_1003_a();"
function MakeCharacter_1003_a()
    N := 1003;
    order := 1;
    char_gens := [768, 120];
    v := [1, 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_1003_a_Hecke(Kf)
    return MakeCharacter_1003_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 := [[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, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 1, 0, 1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0], [-1, 2, 0, -1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 1, 0, 0], [-2, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, -1, 0, -1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, -1, 0, 0, -1, -1, 0, 0, 1, 1, -1, 2, 0, 0, 0], [0, 1, 0, 0, 0, 1, 0, 0, -1, 0, -1, 1, -2, -1, 0, 0], [-2, 0, -1, 0, 0, -1, -1, 0, 0, 1, 0, 0, 0, -1, 0, 1], [0, 0, 1, 0, 0, 0, 1, 1, 1, -1, 0, 1, -1, -1, 0, 0], [-2, 0, 0, 1, -1, -2, 0, -1, -1, -1, 0, 0, 0, 1, -1, 0], [-1, 0, 0, -1, 0, 0, 1, 0, 2, 0, -2, 0, -1, 0, 0, 0], [2, -3, 0, -1, 1, 0, 1, 0, 1, 2, -2, 0, 0, -1, 0, 1], [-5, 0, -2, 1, 1, 0, 0, 0, 1, 1, 2, -1, 1, 1, 2, -1], [-2, 0, 1, -1, 1, 1, -1, 2, 0, -1, 0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -1, -1, 2, -1, -1, 0, -3, -1, 1, 1, -2, 0, 1, 0, -1], [1, 1, 0, -3, 0, 1, 0, 1, 0, 1, -1, 1, -2, -2, -2, 1], [-1, 0, 0, 0, 2, -1, 0, 0, 0, -2, 1, 2, 0, 2, 1, 0], [-1, 1, 2, 0, 1, 0, 0, -1, 1, -2, -1, 3, 1, 1, -1, -1], [-2, -1, 0, 2, 0, -2, -1, -3, -1, -1, 1, -1, 1, 0, 1, 0], [-5, 2, -1, -1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 1, 1, -1], [2, -2, 1, 0, 1, 1, -1, -1, 1, 1, 1, -1, 2, 1, 3, -2], [-1, -3, -3, 1, 2, -1, 1, -4, -1, 2, -1, -1, 0, 3, 0, -2], [-1, -1, 0, -1, -1, -2, 1, -1, -3, -1, 0, -1, 0, 1, -1, 1], [-1, -2, -2, 2, 0, 0, 1, -1, 0, 3, 1, -1, 2, 1, 1, -2], [1, -3, 0, -1, 1, -1, -2, 0, -1, 1, 0, 0, 0, 0, 1, -1], [-4, 3, -2, -2, 5, 2, -2, 3, 1, -1, 0, 1, 0, 1, 3, 0], [-1, -3, -1, 0, 1, 0, 0, -1, 0, 1, -1, -1, 0, -1, -1, 2], [0, 2, 2, -1, -2, -2, 0, -1, -1, -1, -3, 5, -1, -1, -3, -1], [1, 1, 1, 0, -1, -3, 0, -2, 0, 0, -2, 1, 1, 1, -1, 0], [-3, -3, -2, 2, 3, 3, -3, 2, 1, 1, 4, -5, 1, -2, 2, 2], [0, -1, 1, -3, 0, 0, 0, 2, 1, -2, 1, -1, -1, -2, 0, 1], [-1, -3, 1, 2, -2, 4, 0, 3, 0, 1, 3, -2, 0, -3, 2, -1], [4, 4, -1, -2, -3, 1, -1, 2, 2, 2, 1, 1, 0, -1, -2, 1], [-2, 1, 1, -1, 0, -3, 1, -3, -1, 0, 0, 0, 2, -1, -1, -1], [-5, 2, -2, 1, 1, 2, -4, 1, -3, -1, 4, -2, 0, 1, 2, 0], [-2, 2, -1, 1, -3, 2, 0, -2, -5, 1, 4, -3, 0, -1, -1, 0], [-4, 3, 0, -2, -1, -2, 3, -1, 1, 2, -6, 3, 0, 2, -3, -1], [4, 2, 5, -1, 0, 0, -2, 1, 2, -3, 1, 4, -1, 1, -1, 1], [-1, 3, 1, -2, -5, -4, 5, -3, 1, 2, -6, 5, 0, 2, -5, -2], [5, 1, 2, 3, -4, 2, -2, 2, -1, 0, 1, -1, 2, -2, 0, 1], [-3, -4, -5, 3, 1, 3, 0, 0, -2, 4, 3, -7, 1, -1, 4, 0], [-5, 1, 1, -2, -3, 1, 3, 2, 2, -1, -1, 1, -2, 2, 1, 1], [2, -5, 2, 0, -1, 2, 4, -1, 3, 0, -2, -3, -1, 1, 0, 0], [0, -1, -3, 4, 2, -3, 0, -4, -2, 1, 0, -2, 3, 3, 4, -1], [-1, -1, -4, -2, 0, 0, 1, -3, 1, 6, -1, -1, 0, 0, -1, -1], [0, 5, 0, -2, 0, 1, 1, 3, 3, -2, -2, 2, -3, 2, -2, 4], [-1, -3, 4, 2, -4, 3, 4, 3, 0, 0, -2, 0, -2, 1, 1, 0], [0, 7, -1, -2, 3, -5, -3, 1, 2, -2, -3, 3, 0, 2, -1, 1], [-5, 3, -1, 3, -1, -1, 0, -1, -2, -1, 4, 1, 0, 0, 0, -2], [-1, -5, -8, 4, 2, 1, 1, -4, -1, 5, -2, -4, 2, 0, 4, -4], [1, -8, -4, 5, 4, 0, -1, -6, -5, 2, 6, -6, 4, 2, 3, 0], [-6, 6, 0, -2, -3, -2, -1, 0, -1, -1, -3, 1, 2, 1, -1, -1], [1, -3, -1, 1, 3, 1, 1, -5, 1, 3, -1, 2, 0, 1, 3, -4], [-4, 5, 3, -3, 1, 1, -2, 7, 6, -2, 0, 4, -1, -4, -1, 1], [10, -3, 0, -1, -3, 1, 1, 1, 0, -2, -4, 1, -1, -5, -4, 2], [0, 1, 2, -2, 0, -1, 1, 2, 1, -1, -5, 4, 0, -2, 0, 0], [0, -4, -1, 2, -1, 4, 2, 4, 1, 1, -2, -1, -2, 0, 1, 0], [3, 2, 2, 1, -1, -1, 1, 0, 2, 2, -6, 3, -1, -2, -1, 0], [-8, 4, 1, 1, -1, 0, 0, -3, -3, -3, 0, 3, -1, 1, -2, 1], [0, 2, -3, -2, 5, 1, -2, -2, 0, 3, -1, -3, 0, 0, 1, 2], [-5, -5, 2, 2, 0, -4, 2, -1, 5, 2, -2, 0, 3, 1, 2, -2], [1, 5, -3, -3, 1, -1, 2, 0, 0, -1, -7, 3, -3, 4, -1, 1], [-9, -3, -1, 3, 1, 1, 2, -2, -1, 0, -1, -6, 1, 0, 1, -1], [3, -2, 2, -2, 1, 0, 3, -1, 3, -3, -6, 2, 1, 1, -5, 1], [5, -5, 5, 3, -1, 3, 2, -1, -3, -3, -4, 0, -3, -2, 1, -2], [-2, 6, 3, -1, 5, 0, -3, 0, -2, -3, 3, 3, -1, 4, -1, 2], [0, 4, -2, -3, 3, 5, -4, 5, 2, -2, 0, 2, -2, -1, 1, 1], [-8, 2, -2, 0, 4, 0, -1, 0, -1, 4, -3, 2, -3, -1, -3, 1], [4, 0, 1, -2, 5, -1, -1, -3, -1, -2, -1, -1, 1, 3, -2, -1], [11, -3, 5, -3, -3, 1, 0, 3, 0, -1, -4, 4, -2, -2, -4, 3], [8, -1, 2, 3, 3, -3, 1, -5, -1, -3, -3, 6, -3, 3, -2, 0], [-4, -3, 3, 2, -1, -2, -1, -1, -3, -2, 2, 2, -2, 0, -3, 2], [-1, 4, 2, 1, 2, 0, -4, 4, 0, 0, 2, -1, -1, -5, -1, 3], [-5, 7, -2, -3, 3, -2, -2, 0, 1, -3, 0, 3, -1, 2, 3, -2], [-3, -7, -4, 3, -3, 1, -2, -3, -2, 3, 4, -7, 4, 0, 1, -2], [2, 2, -3, 1, 7, 3, -7, 3, 5, 2, 3, -1, 3, 0, 5, 2], [2, 2, 10, 0, -4, 4, 6, -1, -1, -5, 0, 6, -4, 2, -3, -1], [1, 0, 4, -2, -4, 1, 4, 2, -4, -4, -5, 4, -3, -4, -4, 3], [2, -4, 4, 4, -7, 1, 3, 2, -4, -2, 5, 3, -1, -1, 1, -1], [11, -5, 4, 1, -9, -2, 5, -4, -4, -1, -3, 2, -1, -3, -6, -2], [-5, -2, -2, 2, -5, -1, 2, 0, -6, 0, -5, 1, -4, -4, -3, 2], [-11, 7, -1, 4, 4, 0, 0, -3, -2, 0, 4, 2, 2, 9, 0, -2], [-8, -5, 4, -2, -4, 0, 3, 0, 2, 3, -1, -1, 4, 2, -1, 2], [-5, 3, 1, -3, 0, -3, 0, -4, -3, -1, -3, 1, 1, 2, -2, 1], [5, 3, 7, -2, 0, 3, 1, 1, 0, -4, -2, 4, -4, -6, -4, 0], [-2, -5, -6, 5, 4, -4, -4, -2, 1, 0, 2, -4, 4, 0, 3, -1], [1, -2, -4, -3, 2, 2, 3, 2, 1, 3, 0, -2, -1, 2, 2, -1], [2, -2, 0, -1, -1, -3, -3, -2, 1, 1, -3, 0, 3, -6, -1, -2], [-4, 3, 1, -2, -1, 0, -3, 1, 0, 0, 5, -4, 1, -4, -2, 2], [-2, -1, -2, 2, -1, 0, 1, 1, 0, 1, 0, 0, 0, 4, 4, -4], [12, 8, 9, -8, -5, -5, 5, 4, 1, -4, -11, 10, -5, -1, -7, 2], [0, -1, 1, 2, -5, 1, -3, 0, 1, 1, 9, -4, 6, 2, 2, -1], [2, -1, 4, 1, -4, -3, 6, -3, 0, -3, -2, 5, 0, 8, -2, -2], [-5, -3, 2, 3, -4, 1, 1, -1, 4, 1, 7, -4, 6, 5, 2, -1], [-2, 2, -2, 1, 2, 2, 1, 0, 1, 2, 3, 0, 3, 0, 5, 0], [-7, 6, 3, -3, -1, 0, 1, 5, 5, 1, -3, 5, -4, 2, 2, -1], [5, -5, 6, 3, -1, 5, -1, -2, 0, 2, 1, 2, 1, -1, -2, 1], [8, 1, 3, 1, -3, 3, 3, -1, -1, -4, -2, -1, -5, -1, -3, -1], [-11, -3, 1, -1, -2, -2, -1, 3, 1, 2, 2, -3, 2, 1, -3, 0], [5, -3, 4, -2, 4, 5, 3, 4, 3, 0, 0, -4, -6, -2, 2, 1], [-12, 3, 1, -3, -2, -6, 2, 1, -2, -1, 5, 5, 2, 4, -4, 2], [7, -7, -6, 5, 5, -2, -3, -5, -4, -1, -1, 0, 0, -1, -1, 0], [1, -9, 1, 2, 0, 0, 3, 3, 6, -5, 1, -5, 0, 4, 7, 0], [-7, 5, -8, -4, 1, -1, 1, -1, 5, 4, -7, -1, -2, -2, -2, 0], [-2, 1, 3, -3, -6, -1, 4, 4, 0, -3, -9, 4, -6, -4, -3, 3], [-1, -4, -5, 3, -4, -2, 1, -1, -4, 0, 3, -5, 1, 2, 2, -3], [4, -1, -6, -1, 2, -3, 0, -6, -1, 3, 7, -6, 3, 2, 1, 1], [-2, 7, 1, -1, -2, -1, 2, 4, 3, 3, 5, -1, 4, 2, 0, 1], [-1, 4, 0, -2, 4, 3, -8, 4, -2, 2, 7, -3, 8, 4, 2, 0], [-6, 15, 7, -2, -5, 1, -1, 5, -4, -3, 3, 9, 0, -2, 1, 1], [-9, -3, -3, 1, 0, -4, 4, -5, -2, 3, 4, -3, 4, 0, -3, -2], [7, 3, 0, 0, -5, -6, 3, -5, 5, 0, -6, 3, 1, 0, -9, 1], [-2, 1, 6, 3, -1, 3, 3, 2, 0, 2, 1, 4, -4, 5, 2, 2], [0, 4, 9, -4, 0, -1, 0, 4, 5, -4, -3, 3, 2, -2, -6, 1], [3, -6, 2, 0, 2, 8, -3, 5, -1, -1, 5, 0, -6, -7, 1, 1], [-13, -3, 5, 0, -2, 2, 2, 1, 2, -3, 3, -1, 1, 1, 3, 2], [6, 7, 3, -5, -6, 0, 1, 5, 7, -3, -6, 2, 0, 1, -1, -2], [-8, 9, 9, -6, -1, -2, 3, 7, 4, -9, 3, 9, -3, 4, -1, 0], [4, -3, -1, 2, -3, 1, -4, -1, 0, 2, 3, 0, 0, -3, 1, -1], [-8, 1, 2, 1, 5, -2, -2, -5, -2, -1, 3, 4, 0, -2, -1, 1], [-7, 5, -8, -1, 7, -3, -2, -3, 1, -3, 0, -1, -3, -2, 4, 0], [8, 9, 8, -8, 0, -3, 0, -1, 2, -7, -9, 6, -8, -1, -8, 4], [8, -5, -7, 0, 3, 0, 0, 2, 5, 7, 1, -8, 1, 1, 2, -1], [-11, 0, 5, -4, 1, 1, 9, 2, 0, -5, -2, 6, -5, -1, -2, -1], [-5, 4, 1, -3, 5, 3, 3, 4, 9, -4, 1, 5, 2, 2, 3, -2], [2, -7, -10, 3, -1, 1, -1, -2, 0, 5, -2, -6, 1, 1, 7, -2], [1, 3, 4, -5, 0, 0, -1, 6, 1, -3, -2, 5, 0, -1, 5, 0], [11, -4, -2, -1, -3, 0, -2, -2, -1, -2, 2, -7, 4, -5, 0, -1], [-2, -2, -9, 3, 4, -5, 1, -2, -2, 4, 0, -5, 1, 4, 4, -2], [11, 6, 2, -2, 1, -7, -2, -1, -1, -2, -1, 9, 0, 0, -2, -2], [-9, 10, -2, -4, -4, 0, 3, 7, -1, 1, 6, 3, -4, -1, -1, 3], [5, 2, 5, -5, 3, 3, 9, 2, 4, 0, -4, 3, -6, 0, -3, 1], [-1, 4, -4, 0, 3, -4, 2, -7, -4, -4, -2, 2, 1, 6, -2, 0], [-18, 9, -2, 4, -4, -1, -5, 0, -5, -10, 8, 1, -1, 2, 1, -3], [-3, 4, 7, 1, 2, 0, 4, 3, -3, -4, 3, 7, -5, 1, -6, 4], [-2, -7, -4, 9, -9, -4, 4, -10, -6, 1, 1, -6, 8, 4, 1, -3], [2, -3, -2, -1, 0, -2, -1, -2, 1, 4, -4, -1, -1, -7, -4, 1], [-6, 2, 2, -1, -2, 4, -1, 3, -3, -4, 4, -2, -4, 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2], [-1, 15, 9, 7, -12, 2, 5, -1, -9, -6, 0, 14, -1, 5, 0, -7], [16, 23, 10, -10, 4, -7, -3, 8, 1, 0, -14, 20, -4, 0, -8, 5], [10, -15, -6, 1, -2, 4, -2, -4, -5, -6, 22, -17, 7, -6, 3, 3], [-2, -13, -4, 10, 0, 6, -5, -4, -3, 4, 5, -11, 0, -2, 9, -3], [8, -12, -3, 4, 12, 1, -7, 3, 2, 6, 5, -14, 3, 1, 14, -2], [-8, 2, -11, 6, 8, 7, 1, 2, -6, -3, 15, -15, 3, 1, 6, 5], [13, -9, 4, -5, 6, 13, -2, 1, 5, 2, -10, -2, -5, 2, 8, -2], [10, 1, -7, 8, -6, 2, 1, -1, -2, -3, 4, -9, 4, 0, 0, -5], [-17, -5, 5, -1, 9, 4, -2, 10, 13, -4, 7, 1, -2, -2, 0, 5], [-7, -13, -6, 1, 6, 4, 4, 4, 1, 2, 1, -5, 4, 0, 14, -7], [6, 4, 3, 0, 2, -5, -1, -12, -3, -11, -7, 4, 5, 3, -11, 3], [-8, 6, 3, -12, -5, 10, -8, 15, -3, -3, 5, -4, -5, -10, -4, 7], [-5, 14, -3, 0, 0, 2, -12, 6, 7, 2, 10, -7, 3, -4, 1, 1], [11, -2, -2, 0, 16, 0, 1, -4, 10, 7, -13, -1, -2, 7, 4, -3], [5, -2, 2, -10, 4, 2, 1, 0, -1, 10, -1, 2, -3, -9, -6, 1], [-15, 4, -9, 3, 19, -5, -3, -8, 3, 1, 14, -11, 12, 8, 1, -1], [6, 3, 16, 2, -9, 4, 8, 4, 2, -12, -5, 14, -4, 3, -6, -5], [-1, -2, -16, 4, 2, -1, -3, -2, -6, 8, -10, -4, -1, -1, 8, -7], [-5, 7, -17, 8, 2, -2, -7, -15, -13, 2, 8, -7, 9, 4, 2, -1], [1, -12, -15, 7, 3, -2, -10, -2, -4, 10, 4, -16, 6, -7, -1, 3], [0, 13, 2, 6, -7, 1, -3, 0, -1, -1, 7, -6, 4, -8, -5, 1], [-9, -11, 13, -5, 0, 4, -1, 1, -5, -4, -6, 0, -5, -5, -5, 4], [5, -10, -13, 6, 2, 15, -3, 1, -2, 9, 11, -7, -3, -9, 7, -1], [-7, -8, 3, 2, 1, -6, 8, -6, 7, 0, -5, 1, -2, 6, -3, 1], [7, 20, 8, -11, 4, -5, 8, -1, 2, -4, -21, 21, -8, 3, -5, 0], [3, 16, -3, -13, 10, 4, -12, 11, 5, 1, -3, 14, -2, -3, -1, 2], [7, -16, -11, 5, 6, 1, 8, -5, -7, 3, -6, -4, -11, -3, 10, 0], [11, -8, -1, 4, 4, 2, 0, -4, -7, -5, 3, -16, 0, -2, -1, 3], [-2, -15, -6, 7, 6, -1, -1, -5, 1, 4, 1, -13, 0, 5, 2, 3], [11, 13, 5, -6, 3, -3, -8, 5, 6, 0, -10, 13, -3, 3, 1, 3], [-9, 5, 4, 4, -9, 9, 1, 14, -1, 2, 14, -5, 7, 2, 1, 2], [-4, 15, 3, 1, -7, -7, -4, -2, 5, -4, 1, 13, 6, 10, 0, -11], [7, 12, -2, -8, -3, -5, 0, -7, 0, 1, -7, 4, -9, -4, -17, 0], [-16, 17, -9, -4, 4, 10, -5, 4, -8, 2, 4, -3, -3, -6, 2, -2], [-4, -2, 1, 2, 10, -6, 4, -7, -7, -10, 1, 9, 5, 12, 0, 4], [-11, 3, -2, -4, 3, -5, 3, -3, 8, 3, 11, -8, -1, 4, 6, -5], [-1, 6, -3, 7, -2, -4, -5, 3, 0, 3, -1, -1, -2, 0, 4, 4], [17, 5, 12, 3, -9, 6, 7, 0, -6, -9, -4, 13, -4, 5, 1, -4], [15, -2, 15, -4, -8, 3, 6, 13, 7, -8, -6, 21, -10, -10, -8, 3], [-12, -15, 1, 6, 10, 1, 5, 0, 11, 2, 0, -13, 6, 6, 14, -4], [-6, -20, -15, 8, -1, -9, -1, -13, -6, 10, 3, -13, 4, -5, 2, 3], [0, -9, -13, -5, 3, -15, -2, -10, -2, 5, -13, -1, 2, 2, 0, 0], [18, 4, 4, -4, -9, -6, 5, 2, 0, -5, -6, 13, 3, -3, -8, 5], [16, -5, -11, 8, -5, -16, 0, -16, 1, 7, -4, 3, 11, 11, 4, -2], [25, -3, 9, -12, -6, -4, 7, 9, 9, -4, -7, 0, 1, 0, -3, 0], [6, 3, -2, -3, 0, -11, 3, -13, 10, -2, -12, 13, 4, 19, -2, -9], [2, 11, 2, -3, 9, -6, 1, -4, -3, -8, -3, 6, 3, 1, -3, 3], [1, -3, 4, -1, -6, 4, 13, -3, 3, 2, -9, 9, -15, -7, -9, -1], [-9, -7, -7, 9, 14, 8, -13, 1, -4, 1, 10, -6, 8, 1, 15, 4], [5, 17, -5, -3, -5, -6, 3, 6, 14, 0, -4, 3, 8, 1, 0, -9], [-9, 27, -2, -2, 10, 2, 0, -2, -2, -6, 2, 7, -3, 5, 0, -5], [-21, -5, 2, -6, -6, 5, 11, 0, -7, -6, 5, -1, -5, 0, -3, 2], [-5, 2, 10, -7, -4, -2, 6, -9, 4, -8, -10, 12, -6, 3, -1, -10], [-15, 16, 4, -6, -7, 1, 6, 8, 1, -8, -13, 8, -6, 2, 1, -2], [9, 13, 8, -6, -8, -22, 6, -18, -10, -11, -8, 14, 1, 7, -14, -6], [-3, -4, 0, 14, -2, -2, 0, -2, -8, 7, 14, -4, 5, 6, 11, -1], [21, -16, -7, 13, -16, 3, 0, 0, -10, 6, -5, -13, -2, -5, 6, 0], [-4, -3, -1, 8, 14, -2, -11, 1, 0, -9, 1, -6, 4, 0, 7, 4], [-4, -15, 1, 2, 8, 11, 0, 7, -1, -4, 5, -8, -1, -9, 10, 2], [5, -24, -11, 15, 10, 10, 0, -9, -3, 19, 8, -19, 2, -1, 2, -1], [5, 1, 1, -12, 0, -9, 7, -3, -1, -8, -9, 7, -12, 0, -17, 7], [15, 1, 10, -1, -14, -14, 12, -5, 9, -7, -16, 13, 1, -2, -17, 3], [-17, 20, 14, 4, -9, -8, -8, 8, -4, -8, 2, 9, 6, 2, -6, 2], [-10, -17, 3, 6, -1, -5, 15, -5, 2, 6, -1, -10, -2, 2, -3, 0], [-10, 8, -9, -2, -5, 3, -2, 6, 2, 11, 11, -6, 6, -4, 1, 2], [-10, 14, -6, -5, 7, -3, -7, -2, 3, 2, 8, -4, 3, 2, 1, -2], [-6, 11, 1, -7, 10, 9, 7, 8, 8, -2, 6, 4, 2, 5, -1, -1], [-4, 8, -14, -5, 5, -2, -8, 12, 3, 4, 4, -8, 2, -7, 6, 2], [-9, -13, -11, 5, 8, 11, -11, 7, 0, 5, 5, -13, -5, -13, 12, 4], [6, -15, 5, 11, 5, -8, 1, -14, -8, -6, -8, -2, -4, -3, 0, 1], [-20, 1, -14, 0, -3, -1, 8, -2, 8, 9, -1, -10, 2, 3, 2, -7], [3, 0, 9, 5, 5, 9, -7, -4, -9, -4, 13, 0, 5, -12, 0, 1], [10, -5, -4, 5, 0, -3, -6, 7, 1, -8, 8, -7, 3, 8, 4, 1], [-13, 8, -1, -4, -6, 1, 3, 11, 4, -11, 9, -8, 1, 5, 4, 4], [-4, -12, 6, 15, -1, 13, -5, 10, -4, 3, 21, -18, 4, -6, 8, 7], [12, 4, -1, -3, -2, 13, -9, 9, -1, 9, 3, -3, -9, -18, -1, 5], [1, 5, 8, 0, 2, 9, 8, 10, 9, -3, -13, 18, -13, -5, -5, -1], [29, -1, 10, -4, -9, -6, 8, -10, -3, 1, -15, 14, -5, 1, -21, 1], [6, -10, 2, 5, 8, 5, -3, -15, -6, 6, -8, 3, -2, -4, -5, 0], [5, -3, 9, -3, -18, -1, 6, 4, -3, -5, 9, 4, 6, -8, -1, 0], [-9, 3, -12, 3, -5, -7, -5, -16, -11, 4, 3, -10, 0, 1, 1, -8], [-17, 7, -1, -4, -2, 9, 3, 8, 7, 9, 5, -2, -1, -4, -4, -1], [10, -14, 6, -1, -20, -3, 12, 4, 5, -7, -8, 2, 1, 5, 4, -3], [7, -9, 10, -4, -13, -1, 12, -2, -14, -4, -9, 14, -7, -5, -8, -4], [1, 20, 8, -9, 7, -10, -3, 7, 3, -21, 0, 14, -12, 6, -7, 8], [-34, 4, -6, 1, -3, 4, 3, 8, 10, 9, 5, -15, 2, 0, -1, 2], [7, -10, 20, -5, -8, 6, 5, 4, -5, -9, -4, 10, 5, 0, -5, 9], [-6, -6, -1, 5, 0, 11, -11, 19, 0, 4, 10, -7, 4, -16, 11, 8], [5, -10, -2, 4, -5, -5, -5, -19, -2, 9, -8, 2, 1, 3, -3, -3], [14, 11, -12, -5, -2, -5, 7, -4, -4, 6, -4, -1, 4, -1, 4, -5], [-32, 5, -2, -2, -1, 6, -11, 8, -7, 6, 8, -6, -1, -10, 3, 4], [0, -9, 7, 11, 6, -4, 6, -16, -6, 2, -5, 11, 1, 10, 3, -3], [9, -18, -7, -4, 2, -5, 17, -6, 6, -3, -2, -1, -8, 7, -3, 2], [1, 13, 17, -2, -8, 9, 2, 6, -8, -7, 3, 13, -11, -13, -7, 3], [-8, 8, -6, 0, 9, 12, -10, 17, 9, 5, 12, -8, -3, 0, 10, 1], [-5, 3, 13, -3, 4, -4, 6, 7, 18, 2, 5, 6, 12, 8, 3, 0], [0, -11, 6, -4, -6, 5, -1, 2, 3, 6, -3, -6, 4, -12, -4, 13], [-17, 7, -12, 0, -5, -7, -1, 0, -8, 0, 7, -4, 9, 0, 4, -1], [-17, 4, 0, -6, 3, 7, 16, 5, 3, 1, -16, 9, -12, 2, -5, 2], [42, -9, 10, 4, 4, -7, 9, -7, -1, -11, -22, 20, -9, 2, 1, 0], [-9, 6, -4, 10, 3, 1, 4, -2, -4, -7, -9, -3, -4, 2, 11, 0], [1, 7, 14, -13, -15, 2, 6, 1, -1, -6, 4, 11, 3, -2, -18, 1], [6, -19, -6, 4, 1, 1, -6, -7, -3, 1, 8, -13, 4, -5, 10, -3], [3, -23, -2, 8, -2, 14, 4, 7, -13, 10, -4, -12, -7, -15, 0, -2], [-15, 4, 0, -2, 9, 18, -6, 22, 9, 8, -1, 0, -3, -16, 4, 5], [-9, 13, -15, 0, 16, 0, -12, 8, -3, -10, 4, 0, -7, -7, 5, -1], [16, -11, 4, -3, -1, -11, 7, -7, 5, -3, -12, 12, 3, 5, 7, -11], [14, -18, -15, 8, 6, 3, -3, -12, -7, 10, -4, -7, -2, 1, 5, -1], [-17, 13, 2, -7, -19, 3, 3, 21, 5, -4, 7, -9, -7, -7, -6, 5], [-8, -11, -8, 10, 1, 7, -11, -9, -5, 8, 13, -20, 15, -4, -1, -2], [-3, -3, -2, -7, 6, -1, 2, 3, 15, -9, -3, -1, -8, 10, -1, 7], [0, 1, -4, 10, -9, -4, 8, -3, 4, 7, 7, 5, -3, 5, 5, -5], [-12, 15, 2, -3, 2, 6, 2, 16, 11, 0, 8, 7, -7, 0, -1, 5], [-16, 13, 15, -5, -7, -7, 3, 11, -4, -11, -1, 11, 0, 3, -6, 5], [7, -13, -6, 10, 9, 4, -12, 5, -5, -2, 15, -19, 9, -3, 9, 10], [-6, 0, 3, 12, -11, -6, -11, -11, 0, 11, 6, -7, 13, -1, 2, -7], [0, -21, 4, 0, 6, 12, -3, 8, 6, 3, 2, 2, -5, -4, 0, 2], [15, -12, -5, 10, 10, 1, -17, -6, 1, 9, 21, -16, 12, -1, 11, -4], [17, -13, 3, 12, 4, 1, -9, -10, -16, -1, -5, 1, 1, -4, 8, -2], [-10, -11, 16, 5, -1, 16, -5, 13, -11, -9, 19, -1, -4, -7, 3, 8], [0, 20, 3, -4, 5, -1, -14, 1, -1, -12, 13, 1, 3, -6, 1, 6], [-6, -14, 1, 5, -8, -1, 3, -11, -7, -1, -3, -1, 0, 0, 0, -4], [-39, -17, -12, 5, 13, -6, -3, -5, 1, -4, -2, -8, 10, 7, 8, -2], [-4, 10, -13, 1, 9, -5, -13, -4, 5, 4, 12, -9, 7, -6, -1, 2], [-5, 16, 1, -7, 2, -3, 1, -2, -4, 4, -18, 2, -2, 8, -5, 3], [-13, -1, 12, 1, -17, -6, 14, -8, -4, -2, -15, 7, -1, -4, -11, -1], [-1, -5, 3, -7, 1, 10, 7, 12, 2, -6, 6, 1, 0, 1, 7, 1], [-5, 10, -4, 3, 2, -6, -8, 3, -5, 1, 4, -7, -2, -11, -9, 9], [-8, -2, -7, -10, 4, -2, -6, -7, 0, 5, 11, -8, 6, 10, -3, -1], [12, 18, -3, -10, 3, -20, -5, -8, -4, -2, -26, 26, -1, -1, -6, -5], [-3, 3, 9, -2, 8, 0, -6, 5, -1, 1, -9, 13, 0, -7, -11, 5], [16, -2, 6, 0, 1, -12, -6, -8, -2, -5, 0, 4, 12, 14, 4, -5], [-31, 18, -13, -10, 4, -8, -4, -1, 7, 2, -2, 0, 8, 0, 5, -7], [24, -5, 18, 1, -11, -3, 8, -6, -6, -14, -3, 18, -2, 0, -11, -3], [-5, -4, 0, 1, 2, -7, 3, 0, 1, -16, 1, 5, 7, 10, 11, 1], [39, -11, 11, 4, -7, 2, -2, -3, -7, 14, -5, 0, -4, -15, -3, 0], [5, 3, 17, 0, -19, 0, 12, -10, -13, -7, 2, 15, -8, -1, -20, -3], [20, 15, -1, -6, -1, -2, -12, 6, 6, 0, 1, 4, -6, -13, -3, 3], [-4, 23, -1, -22, 9, 2, -3, 7, 8, 9, -1, 10, 4, -6, -1, -2], [-17, -5, -6, -5, 19, 0, -3, -3, 5, -9, 5, 3, -7, 1, 2, -1], [-5, -3, 2, 2, 19, 0, -12, 5, -3, -7, 11, 1, -2, 1, 5, 6], [-12, 1, 4, 3, -6, 1, -1, -6, -9, 8, -4, 3, 2, -3, 2, 2], [3, -9, 6, 2, -2, 9, -6, 12, 3, 2, 4, 3, -7, -7, -1, 10], [-12, 3, 11, -7, 0, 15, 4, 8, 9, -2, -3, 3, 4, -2, -7, 3], [0, 14, -1, -5, -8, 2, -6, 13, 4, 2, 0, 0, 2, -13, -2, 2], [-10, 2, 7, -3, 6, 12, 8, 7, 2, 2, 9, 4, 1, 3, -4, -1], [-18, 23, 0, 0, 4, -1, 2, 5, -8, -5, 0, 7, 2, 11, -4, -1], [11, 8, -7, -11, 11, -14, -7, -18, 4, -3, -15, 9, -4, -6, -6, 0], [-2, -14, 10, 2, -8, 0, 5, -1, -4, -5, 5, 0, -3, 11, 9, -4], [-35, 34, -2, -12, 10, -13, -3, -8, 0, -6, -21, 15, -3, 11, -3, -2], [-6, -9, -15, 7, 18, 8, -7, -1, 5, 7, 12, -2, -4, 7, 13, -3], [22, -37, -12, 8, 1, 4, -4, -6, -12, 6, 1, -18, 0, 5, 10, 1], [-12, -4, -9, 4, 5, -2, 1, 5, -1, 7, 1, -6, -1, 1, 15, -4], [7, -4, -15, 5, 7, -9, 2, -13, 8, 12, -6, -6, 2, 13, 8, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1003_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1003_2_a_h();".
// 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_1003_2_a_h(:prec:=16)
    chi := MakeCharacter_1003_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_1003_2_a_h();".
// 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_1003_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1003_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 50, 179, -301, -1323, 151, 2641, 695, -2105, -951, 723, 456, -82, -90, -5, 6, 1]>,<3,R![540, 4266, -1953, -27478, -25563, 17216, 30581, 1539, -13179, -3944, 2482, 1228, -157, -154, -8, 7, 1]>],Snew);
    return Vf;
end function;