
// Make newform 1849.2.a.n in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1849_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_1849_2_a_n();".
// 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_1849_2_a_n();".
// 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 := [-27, 513, -2493, -27, 9945, -5118, -14483, 10400, 9909, -8851, -3223, 3855, 364, -897, 47, 106, -15, -5, 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, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-16390890, 140920983, 104938560, -580027476, -215589719, 916736762, 175925085, -726303448, -52495987, 314282184, -2151884, -76161915, 4293380, 10040797, -817698, -638270, 49216, 13275], [-7389, -32487966, 40814856, 64429719, -167553770, 27615381, 228076724, -120576325, -146365724, 90932936, 50308357, -31639730, -9559915, 5791002, 949666, -543260, -38599, 20710], [401490, -7993449, -48095238, -26028240, 150145347, 100861486, -253491883, -86897862, 233856473, 16877036, -112470798, 8945071, 28270866, -4677973, -3533264, 771312, 173548, -44003], [9437373, -73666089, -89122374, 221535819, 317741963, -228333277, -500960689, 127874509, 397852707, -64968220, -169440829, 28825423, 39149731, -7782617, -4620818, 1056380, 218229, -55583], [-4659615, 3726342, 218833080, 11271585, -746262274, -24846673, 1015190912, -19113311, -702822648, 55159664, 268538099, -35687486, -57120029, 10180162, 6326890, -1359184, -284265, 69430], [5508270, -41178123, -128625066, 157106100, 485295733, -255948658, -729037413, 248450834, 544218431, -155901156, -219749186, 60465153, 48709646, -13573619, -5570484, 1599640, 256828, -76293], [4297068, -43668981, 94679412, 170060541, -424393738, -230450510, 663208641, 123911218, -504355352, -6203235, 206093975, -18893697, -46037891, 7351847, 5288955, -1095040, -244357, 59046], [2059911, -43334415, 231376572, 130684977, -915839985, -94828159, 1360900177, -64409787, -1004438171, 131050912, 401793495, -74360329, -88235805, 19725175, 9987848, -2516574, -455245, 124637], [8750907, -109231326, 172094370, 426518757, -817237726, -608721671, 1347595896, 361330573, -1073642906, -48490122, 454974713, -36379716, -104501855, 16443476, 12268074, -2561026, -576685, 141246], [6009885, -99718389, 250284606, 411372357, -1061265551, -561275045, 1655953135, 277767581, -1276405047, 16191022, 529610989, -61900753, -119904103, 21852299, 13928240, -3148820, -649353, 167003], [9088893, -91873764, 175378248, 311539797, -742219478, -361418209, 1105999782, 147391919, -810758590, 25631112, 322516021, -40633854, -70696261, 13158388, 8017242, -1822472, -367211, 94434], [11305827, -126336348, 318068358, 526887669, -1383174472, -747127031, 2161286394, 404042737, -1657489664, -11491806, 683247641, -68256762, -153706883, 25899626, 17751510, -3821242, -823249, 204816], [24053769, -274838571, 497629530, 1061084985, -2117529921, -1460957353, 3175200109, 859608609, -2337456797, -152533502, 932572533, -52751011, -204864951, 27818161, 23271404, -4433508, -1067329, 245597], [-2620152, 59409549, -412418418, -290653932, 1642014991, 382011948, -2498571613, -84082444, 1898220181, -157879042, -780467606, 122991985, 175651034, -36421791, -20322290, 4923820, 944792, -253199], [-572175, -38129769, 327551883, 145462917, -1248308096, -138676475, 1826041525, -46035718, -1335019437, 146650858, 531312130, -89543356, -116487784, 24565100, 13198295, -3199685, -603273, 160886], [13820985, -160192611, 420139959, 651327492, -1791688513, -915641593, 2759880713, 495031120, -2090623248, -15885937, 853971125, -82343444, -190931687, 31393966, 21964198, -4644175, -1016292, 249517]];
    Rf_basisdens := [1, 1, 1312164, 1312164, 1312164, 1312164, 1312164, 1312164, 656082, 1312164, 1312164, 1312164, 656082, 1312164, 1312164, 1312164, 656082, 656082];
    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_1849_a();"
function MakeCharacter_1849_a()
    N := 1849;
    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_1849_a_Hecke(Kf)
    return MakeCharacter_1849_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, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 0, -1, -1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0], [-2, 1, 1, 0, -1, 0, -2, 1, -1, -1, 0, -1, 0, 0, 0, 1, 1, 0], [0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0], [-1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0], [1, -1, -2, 1, 1, 0, 1, -1, 0, 1, 0, 1, 0, 0, 0, 0, -1, 1], [0, -2, 1, 0, 0, -1, 0, 0, 1, 0, -1, 0, 1, 1, -1, 0, 0, 0], [-1, -1, 0, 1, 0, -1, 0, 3, 1, -1, 0, -1, 0, -1, 0, 0, 0, 0], [-2, 0, -2, -2, 1, 2, 2, -3, -1, 1, 2, 1, -1, 1, 1, 0, -1, 0], [2, 1, 1, 0, 2, 0, 2, -1, 1, -1, 0, 1, 1, 1, -1, 0, 1, -1], [2, 2, 0, 0, 0, 0, 1, -2, 1, -2, 1, -1, 0, 1, -1, 1, 0, 0], [1, 1, 0, -2, 1, 1, 5, -3, -1, 2, 2, 1, 0, 1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, 0, -1, 1, 0, 0, -1, 2, 1, 3, 0, -2, 0, 1, 0, 1], [0, 0, -3, -2, 1, 2, 2, -4, 1, 1, 0, 2, -3, 0, 2, -1, -3, -1], [1, 1, 0, -1, 1, 3, 0, -1, 0, 1, -1, 2, -1, 1, 1, -1, -1, -1], [-3, -1, -2, 0, -1, -1, -2, -1, -1, 1, 0, 0, -2, -1, -1, 2, 0, 1], [-1, 0, 1, -1, 2, 2, -4, 4, -1, -1, 0, -3, 0, 2, -1, 1, -1, 0], [-6, -1, -2, 0, 0, 0, -1, 1, 1, 1, -2, -1, -2, -2, 0, 0, -2, 1], [1, 0, -2, -1, 0, -2, 1, -2, 3, 1, -1, -2, 0, 2, 2, -2, -1, -2], [1, -1, 3, -1, 1, 1, -2, -1, 0, 1, 1, -1, -1, 2, -1, 0, 0, 0], [-3, -1, 2, 0, -2, -4, -3, 3, 1, -1, -3, -1, 1, 0, 1, -1, 2, -2], [-4, -2, -5, 0, 0, 0, -1, 0, 2, 0, 0, 1, -1, -2, 2, -1, -1, 1], [0, -2, 0, 1, -1, -6, -2, 3, 2, -1, -4, 0, 3, -1, -3, -1, 0, -1], [4, -1, -3, -2, 2, 0, 1, -6, 0, 1, -3, 2, -1, -2, 1, -1, 1, -3], [-3, 1, -2, 0, -1, -1, -1, 0, 3, 0, 1, -1, -1, -1, 0, -1, 1, 1], [2, -4, 0, 0, 0, -1, 0, 0, 0, 1, -1, 4, 1, 2, 0, 0, -2, 1], [-2, -1, 1, 0, -2, 2, -1, 5, -1, 0, 2, 1, 0, 2, 2, 1, -2, 1], [2, -1, -1, 0, 2, -2, 0, 5, 2, -2, 0, 0, 2, 2, 0, 0, 0, -1], [2, 0, -2, 1, 0, 3, 0, -4, -3, 2, 1, 0, -2, 0, -1, 3, -2, 1], [-8, 3, 4, 0, -1, 0, -2, 2, -3, -3, 0, -2, 2, -1, 2, 2, 4, 0], [0, 2, -1, 1, 0, 3, 4, -3, -1, 1, 4, 1, -3, 1, 3, -2, -2, 0], [-3, 2, 4, 4, -2, 1, -5, 7, 0, -2, 4, -5, 1, 0, -2, 2, 4, 1], [-1, -1, 3, 1, -3, -3, 1, 0, 1, 1, -1, -1, 4, 0, -3, -1, -1, 0], [-3, 1, 3, 1, 0, 0, 0, 4, 2, -4, 2, -3, 1, 1, 0, 2, 3, 0], [5, 1, 3, 1, 2, 0, 2, -2, -1, 1, 2, 2, 3, 0, -3, 2, 0, -2], [1, 1, -2, 1, -1, 4, -4, -1, -4, 3, 0, -2, -2, -2, 1, -2, -1, 1], [-4, -2, 6, -1, 0, 3, -3, 1, -1, 1, 2, 1, -2, 3, 1, -1, -1, 0], [2, -6, 2, 0, 1, -5, -1, 6, 6, -4, -4, -1, 6, 2, 0, -1, 0, -2], [-2, 3, -2, -4, 3, 1, 5, -6, 1, -1, -4, 2, -1, 2, -2, -1, -1, -3], [-5, -5, -4, 2, -3, -4, -9, 5, 3, 0, -3, -1, 1, -2, 3, -2, 0, 0], [3, 0, 0, 1, -1, -3, 2, -4, 4, -4, 0, -2, 5, 4, -4, 1, -2, -3], [1, -2, -3, -3, 5, 5, 3, -7, -1, 4, -2, 2, 1, 1, 2, -2, 0, 0], [-6, -2, -1, 3, -2, -4, 2, 1, -1, 1, 2, -1, -2, -5, 3, -2, 1, 1], [-12, 0, 2, 2, -7, -2, -5, 3, -1, -1, 0, -4, -2, -2, 4, 1, 1, 1], [6, 1, 3, 3, 2, 1, 8, -1, 5, 2, 5, -1, 3, 4, -2, -2, -1, 0], [-1, 1, 1, -4, 2, 1, 1, -1, 3, 1, 2, -3, 2, 1, -3, -2, 1, -2], [6, -5, -7, 1, 2, 0, 4, -7, 3, -2, -2, 1, 2, 4, -6, 3, -4, 1], [-1, -1, 2, 3, -4, -1, -6, 3, -4, 1, -4, 1, 0, -2, 4, 1, 2, 3], [-1, 1, 3, -3, 1, 1, -3, 2, -6, 1, 0, -2, 2, 2, 0, 0, 1, 0], [-11, 5, 1, 0, -3, -2, -1, 4, -3, -5, 4, -2, -3, 0, 2, 2, 3, -1], [1, 0, 0, 2, -4, -1, 0, -6, -5, 1, 2, 2, 2, -1, -4, 3, 0, 2], [-5, 0, -1, 3, -2, -6, -1, 4, 2, -3, 1, -1, 4, -2, -1, 2, 5, -2], [-2, 3, 8, 1, 0, -2, -1, 5, 1, -5, -2, -4, 5, -2, -3, -1, 3, 0], [-8, 5, -4, -4, -1, 1, -2, -4, -1, -1, -2, 3, -2, 0, 2, 0, 3, -1], [-5, -2, 0, -1, -5, -3, -7, 8, 1, 0, 3, -1, 0, -2, 2, 3, 0, 2], [2, 8, 3, -5, 1, 8, 3, -6, -7, 2, 2, 2, -3, -3, -1, 1, 2, -1], [7, 1, 2, -5, 3, 8, 9, -5, 0, 3, 7, 4, -3, 1, 6, -4, -1, 1], [11, -1, -1, -1, 2, 0, 1, -7, -1, 2, -3, 4, 0, -4, -3, -3, 2, -1], [-2, -1, 0, 5, -3, -1, -7, 3, 2, -1, 3, 1, 3, 2, 0, 2, -2, -1], [-3, 2, 6, 1, -3, -4, 8, 7, 2, -2, -1, -1, 2, 1, 0, 1, -2, 0], [-3, 2, -2, -1, 4, 4, 0, 4, -5, 3, 0, 0, -3, 1, 1, -2, -1, 3], [2, -4, 2, 2, -4, -10, -2, 8, 6, -1, -7, -5, 4, -4, -4, -3, -4, -3], [7, 6, 2, -2, -2, -1, 2, -4, 3, -2, 2, 1, 0, -2, -1, 1, 1, 0], [0, 0, 5, 6, -5, -5, -1, 10, 1, -2, 0, 5, 4, -4, 2, 1, 3, 2], [-4, -1, -8, 2, -1, 3, -1, 0, 0, 3, 5, 0, -1, -3, -1, -1, -3, 3], [-2, -4, -1, -3, -2, -2, 1, -2, 4, 2, -2, -2, 2, 3, 0, -1, -5, -5], [-2, 4, 0, 8, -1, -1, 2, 4, 0, -4, 6, -6, 1, 1, -5, 5, 1, -1], [0, -6, -4, -2, 2, 3, -3, -2, -5, 3, 2, 2, 1, -2, -5, 4, -3, 1], [-2, 3, -4, -3, 4, 1, 9, -5, 0, 2, -3, 3, -4, -3, 0, -5, -1, 1], [-6, -2, 1, -1, 0, -2, -3, 3, -2, 1, -4, 3, -1, -5, 0, 0, 0, 5], [6, 0, 0, 2, -4, -2, 7, -7, 2, 0, -4, 5, 1, -1, -1, 1, 1, 4], [7, 0, 2, -7, 8, 7, 9, -14, -6, 4, -2, 6, -4, 2, -3, -3, -3, 3], [12, -6, 1, 5, 0, -7, 7, -1, 8, -3, 2, -1, 5, 7, -4, 0, 0, -2], [-11, -5, 3, 3, -5, -8, -2, 5, 5, -7, 2, 2, 1, 2, -1, 3, -2, -1], [-5, -2, 5, -4, -1, 3, -7, 1, -2, 3, -10, 3, 1, 0, 3, -3, 0, 2], [-7, 6, 5, 1, 0, -1, -4, 5, -1, -4, -5, -8, 2, 0, 2, -3, 4, -1], [14, -1, -11, -1, 1, 0, 9, -4, 7, 2, -3, 5, -3, 0, -2, -1, -7, -2], [-5, 5, 5, -8, -1, 6, 3, -4, -7, 4, 6, 1, -5, 2, 5, -1, 3, 1], [13, -2, -5, -1, 6, 1, 15, -16, -2, 1, 1, 8, 0, -1, -5, 0, -2, 3], [3, -2, 2, 0, -3, -3, -9, 4, -3, -4, -3, -6, 6, 8, -8, 6, -3, 1], [-9, 2, 0, 1, -3, -1, -6, 10, 6, -4, -3, -9, -3, 2, -2, 1, -3, -2], [-6, 2, 1, -1, -3, 5, -3, 1, -10, 6, 3, 2, -1, -1, 2, -3, 1, 4], [0, -1, -1, -1, -3, 0, 4, -6, -5, 1, 0, 1, 0, -1, -2, -1, -3, -3], [9, 2, 3, 1, 0, -1, 8, 0, 3, -4, 0, -7, 2, 5, -5, 2, 2, -4], [1, -5, -1, 8, -1, -2, -9, 10, -4, 1, 2, -1, 3, -8, 1, 4, 5, 5], [10, -1, -3, 2, 4, 0, 1, -4, -3, -1, -5, -4, 4, 2, -3, 0, 2, 0], [0, 8, 0, -8, 5, 8, 4, -12, -1, -1, -2, 1, -4, 3, -2, 1, -1, -6], [4, -5, 8, 4, -3, -4, -7, 4, -1, 1, 2, -7, 2, -1, 2, 1, 2, 0], [-2, -3, 0, 4, 4, -3, -3, 6, 6, -4, 0, -3, 4, 6, -1, -4, -1, -3], [11, -11, -1, -1, -2, -1, -4, -1, 5, 2, -5, 4, 2, 5, -5, 2, -5, -1], [-3, 4, -5, -3, 5, 5, 5, -6, 4, 0, 1, 3, -1, 2, 1, -4, 0, -8], [-17, 6, 4, 1, -2, 3, -9, 3, -7, -2, 3, 1, -6, -5, -1, 2, 1, 4], [-6, 6, 11, -2, -5, -1, -12, 11, -2, -11, 4, -2, 3, 0, 4, 2, 6, 0], [-5, 2, -3, 1, 3, 0, 13, -7, -2, 7, 5, 3, -3, -5, 0, -4, -3, 4], [-8, 4, 2, 1, -4, -1, -4, 15, -5, -2, 4, -2, -4, -3, 5, -3, 0, 5], [-1, -1, -5, -4, 4, 5, 9, -3, 12, -2, -5, 1, -1, 5, 7, -7, -5, -4], [1, -9, -4, -3, -4, -2, -1, -9, 3, 4, -12, 1, -2, -2, 2, -1, -1, -1], [0, 0, 4, -2, 1, 1, 7, -3, 1, 5, 7, 4, 3, 4, -2, 0, -3, -4], [-1, 3, -9, -5, 7, 14, 1, -7, -7, 4, 4, 4, -8, -2, 4, -2, 3, 4], [4, 3, 0, 4, 4, 7, -1, 8, 0, -1, 5, 2, -3, 1, 6, 1, 2, 5], [-13, 1, -3, -1, 0, -3, -7, -2, 0, 3, -1, -9, 4, 2, -7, -2, 1, -2], [3, -2, 2, -5, 4, 5, 0, -8, -5, 3, -1, 10, 3, 0, 3, 0, 4, 2], [-3, 2, -7, 10, -1, -6, -1, 6, 6, -2, -1, -5, -2, -2, -1, 0, 2, -3], [5, -6, 2, 1, -3, -8, 2, 9, 8, -4, -5, 0, 5, 1, 1, 0, 3, 1], [-3, -3, 3, 12, -7, -12, -11, 19, -2, -4, -5, -1, 5, -9, -3, 1, 4, 0], [-5, 8, 6, -1, -2, 3, 3, -5, -5, 3, 3, 7, -8, 0, 6, -3, -2, 3], [0, 5, 8, 1, 1, -3, -3, 10, -1, -4, 1, -9, 3, -3, -5, 3, 7, 2], [-1, -5, -1, -1, 0, 4, -3, 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-10, -15, -1, 3, 4, 11], [-1, -4, 10, 13, -2, 2, -32, 17, -8, 14, 11, 6, 1, -4, 12, 0, 7, -2], [-5, 0, 1, 9, 4, 1, 6, 31, 9, 1, 19, -5, 2, 7, 9, -13, -8, 4], [3, 10, 0, 12, -10, -10, -18, 19, 1, -10, -4, -16, 1, -8, 2, 7, 6, 0], [18, -15, -24, -1, 9, -9, 4, -19, 16, 8, -15, -9, 3, -8, -2, -4, 2, -13], [-22, 27, -6, 1, -2, 10, -12, -16, -12, -6, 1, -6, -6, -5, 16, -2, 21, -1], [-15, 15, -5, -20, 13, 23, 2, -23, 2, 5, -12, 7, -12, -2, 13, -3, 2, -11], [12, -5, 8, 17, -6, -23, 17, 11, 22, -16, 10, -11, 16, -3, -18, 6, -2, -6], [-13, 7, -8, 6, -9, 5, 16, -2, -1, 6, 12, 17, -14, -9, 4, -8, -12, 12], [8, 11, -1, 20, 4, 4, -2, -24, -24, 12, 1, 6, 0, -9, -11, 5, 5, 11], [7, -8, -21, 6, 12, 5, -12, -16, -13, 8, -6, -3, -6, -12, -6, 5, -6, 2], [-21, 14, -21, -7, 8, 6, 22, -18, -1, 10, 6, 6, -3, -11, -1, -2, -1, -1], [-18, 10, -3, -2, 5, 11, -12, 12, -14, 14, 6, 11, -11, -7, 3, -6, 5, -6], [13, -17, 1, -14, 1, 13, 2, -18, -13, 13, -19, 5, -6, -10, -1, 4, -9, 15], [0, 6, -12, 1, 12, 7, -3, -3, 2, 8, 5, 2, -6, 1, 13, -10, -1, -2], [-16, 0, -13, -12, 12, 13, -20, 1, -3, -3, -15, 18, -12, -18, 14, -7, -2, -11], [14, 2, 3, -5, 6, -2, 16, -2, 12, -3, -17, 3, -9, 3, -5, 0, -9, -12], [0, 6, 3, -3, -6, -3, 4, -16, -10, 14, -20, 10, -5, -14, -2, -1, -4, 1], [9, 0, 0, -13, 11, 6, 23, -30, -7, 12, 14, 17, -9, 0, 13, -9, 0, -6], [-5, 21, 0, -25, 12, 10, 12, -9, 13, -9, 10, 4, -4, 12, 4, -7, 11, -15], [-12, -12, 14, 12, -17, -34, -8, 24, 11, -12, -13, -12, 24, -2, -25, 10, 7, 1], [14, -7, 8, 15, 0, -10, 21, 27, 6, 6, 11, -5, 4, 2, -13, 8, -10, 1], [2, 7, 19, -4, 14, 16, 3, -2, -7, 1, 3, 7, 5, 18, 0, -7, -2, 1], [27, 19, 12, 4, -9, -11, 29, -19, -4, -1, 11, 5, 17, -1, -3, 2, 5, 4], [-25, -6, 10, 16, -14, -30, -33, 38, 19, -19, -8, -5, 25, 7, 5, -5, 9, -4], [10, 2, 4, 9, -3, -2, 5, -8, -11, -4, 10, 9, -1, 4, -9, 10, -3, -4], [-13, 17, 15, 8, -14, -11, 11, 15, -7, -4, 21, -12, 4, 3, -13, 5, -1, 12], [1, -15, -13, -10, 4, -4, 24, -12, 11, 4, 1, 28, -2, 5, 3, 8, -7, 2], [7, 11, 5, 13, -3, -13, 8, 1, 2, -11, 6, -27, 15, 7, -19, 12, 4, -6], [-11, 13, -13, 17, -2, -11, -7, 16, -2, -7, 4, -11, 5, -10, 1, 6, 1, -7], [3, 3, -10, -19, 6, 5, 11, -32, 7, 0, -1, 6, 0, 6, -3, -5, -4, -15], [2, 2, -4, 2, -10, 5, 12, -2, 10, -11, 6, 9, 9, 7, 13, 3, 7, 1], [-7, 10, -15, 2, 6, 21, -12, -18, 1, -6, 4, 1, -17, 9, 2, 9, -8, 7], [24, -12, -6, -5, 8, -19, 10, -16, 21, 12, -19, 19, 9, -11, 2, -18, 7, -12], [8, -13, 8, 2, 0, -11, -10, 8, -2, -3, 7, -16, 3, -17, -13, -4, 9, 3], [-17, 10, -2, 1, 9, 13, 24, -12, -2, 10, 18, 14, -5, 10, 9, -8, 3, -7], [-21, 6, 17, 12, -5, -4, -14, 31, -10, -9, 0, -24, -4, -9, 2, 7, 13, 14], [17, 7, 9, 11, -1, -7, 37, 10, 16, -11, 25, -13, 15, 7, -6, 7, 8, 6], [9, -3, -1, -2, 10, -9, -1, -7, 5, 0, 2, 4, 6, 5, -4, -9, 7, -5], [20, -17, 2, -1, 4, -11, 15, -3, 18, 3, -10, -3, 4, 2, 7, -5, -6, -17], [15, -7, -7, -17, 11, 23, -8, -12, 4, 9, -15, 15, -20, -9, 16, -12, -12, -1], [-7, 4, -8, 11, -1, -3, 21, 0, -4, -4, 22, -4, 7, -7, -9, 7, -2, 6], [4, 1, 4, 9, -11, -15, 3, 25, 10, -9, -2, -19, -6, -5, -7, 9, 5, 11], [36, -18, -2, -4, 21, 4, 7, -12, 16, 13, -15, 6, 4, 12, 0, -11, -6, 0], [-3, -7, 12, 5, 2, -28, 3, 16, 9, -4, -17, -8, 18, 6, -8, -11, 3, -12], [-13, -9, -10, -22, 17, 24, -1, -5, -12, 9, -14, 19, -8, 0, 17, -2, -11, -2], [-10, -14, -3, 13, -1, -9, -4, 26, 8, -3, 12, 5, 9, 15, 5, 5, 6, 5], [-12, -14, 8, 2, -14, -17, 5, 1, -3, -5, 7, 2, 3, -4, -9, 14, -4, -2], [3, -23, -27, -4, 16, 13, 6, -24, -6, 15, 5, 8, -1, 15, 2, -4, -10, -5], [-24, 11, 3, 5, 6, 9, 5, -20, -22, 10, 15, 24, -4, -7, 10, -1, -7, 10], [-10, 9, -8, 2, -3, 10, -7, -10, -8, -14, 6, -11, -6, 2, 4, 12, 6, 13], [8, 6, 5, 15, -12, -9, 8, 9, -5, 0, 17, 1, -2, 4, -7, 7, -4, 5], [13, 7, 15, -4, 6, 5, 0, -10, 15, -2, 10, -4, 3, 11, -7, -6, -3, -13], [35, 2, 2, 1, 2, 12, 18, -5, -20, 0, 8, 5, -11, -2, -7, 10, -9, 2], [-14, 1, -7, -4, 9, 23, -11, -6, -6, 8, 20, 5, -7, 4, 2, 0, -4, 12], [1, -12, -5, -3, 8, 3, -8, 18, 5, 16, -10, 5, -3, -5, 10, -13, 5, -1], [35, -20, 2, -11, 2, 1, -2, -19, 4, 0, 1, 11, 8, 17, -11, 8, -5, -5], [-24, -2, 4, 9, -11, 2, -21, 15, -5, 9, 13, -8, -13, -8, 1, 7, -3, 2], [-12, 24, 2, -5, 7, 1, 9, 0, -1, -17, -10, 9, -11, -14, 14, 3, 22, 0], [-26, 2, 3, 13, 1, 7, 4, 14, -2, -4, 12, 0, 5, 11, -4, 16, -4, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1849_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1849_2_a_n();".
// 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_1849_2_a_n(:prec:=18)
    chi := MakeCharacter_1849_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_1849_2_a_n();".
// 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_1849_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1849_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-27, -513, -2493, 27, 9945, 5118, -14483, -10400, 9909, 8851, -3223, -3855, 364, 897, 47, -106, -15, 5, 1]>],Snew);
    return Vf;
end function;