
// Make newform 1849.2.a.q 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_q();".
// 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_q();".
// 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, 0, -75, 0, 1500, 0, -6823, 0, 12809, 0, -11859, 0, 6037, 0, -1775, 0, 300, 0, -27, 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, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -26526277, 0, 275318380, 0, -747265236, 0, 818927614, 0, -424283337, 0, 98515291, 0, -2821968, 0, -3048022, 0, 533429, 0, -27659], [2418746, 0, -145773292, 0, 580699660, 0, -895324181, 0, 698608976, 0, -307062428, 0, 79319430, 0, -11941032, 0, 969056, 0, -32741, 0], [14278584, 0, -387773159, 0, 1697907062, 0, -2934073585, 0, 2450520491, 0, -1107993096, 0, 285881978, 0, -42032783, 0, 3273453, 0, -104687, 0], [0, 27579461, 0, -411757298, 0, 1510642955, 0, -2355356698, 0, 1920911813, 0, -900255456, 0, 250857133, 0, -40943432, 0, 3602040, 0, -131444], [2656569, 0, -232881548, 0, 976052115, 0, -1681080551, 0, 1517964126, 0, -785122399, 0, 239188604, 0, -42164100, 0, 3956200, 0, -152197, 0], [8548128, 0, -234845401, 0, 976052115, 0, -1681080551, 0, 1517964126, 0, -785122399, 0, 239188604, 0, -42164100, 0, 3956200, 0, -152197, 0], [1053184, 0, -136438918, 0, 763377719, 0, -1536429084, 0, 1496628476, 0, -801740165, 0, 248035165, 0, -43991454, 0, 4135469, 0, -159103, 0], [-9133342, 0, 329474479, 0, -1678562036, 0, 3302046534, 0, -3062673744, 0, 1522703970, 0, -431405640, 0, 69816397, 0, -6006257, 0, 212899, 0], [8669543, 0, -337531739, 0, 1813167422, 0, -3722103403, 0, 3565031955, 0, -1825452336, 0, 532470474, 0, -88705119, 0, 7849454, 0, -285767, 0], [0, -10092877, 0, 117336103, 0, -832529259, 0, 2103410868, 0, -2315478651, 0, 1327270034, 0, -426972760, 0, 77502090, 0, -7388254, 0, 286689], [12511623, 0, -263109395, 0, 1413228919, 0, -2998735049, 0, 3014087627, 0, -1634332462, 0, 506292190, 0, -89443122, 0, 8357310, 0, -319430, 0], [0, -42842185, 0, 830303886, 0, -3584557446, 0, 6443159602, 0, -5802940707, 0, 2901019397, 0, -842131947, 0, 140982897, 0, -12595441, 0, 463799], [14696711, 0, -612215946, 0, 3233121230, 0, -6535167537, 0, 6325178486, 0, -3325698762, 0, 1003467048, 0, -173276438, 0, 15875946, 0, -596780, 0], [0, 27774347, 0, -681655983, 0, 3487143410, 0, -7086537953, 0, 6896116521, 0, -3635096403, 0, 1097567004, 0, -189482587, 0, 17350711, 0, -651785], [0, -11460945, 0, 951458866, 0, -5025209785, 0, 9911337190, 0, -9367021555, 0, 4817550600, 0, -1424650828, 0, 241632621, 0, -21795831, 0, 808455], [0, -68776600, 0, 2439434596, 0, -12284174748, 0, 24019751632, 0, -22595687461, 0, 11561404040, 0, -3399374624, 0, 573187456, 0, -51413396, 0, 1897282], [0, 104904189, 0, -3086037295, 0, 14770869818, 0, -28056188881, 0, 26034563400, 0, -13246781895, 0, 3889420361, 0, -656294988, 0, 58971636, 0, -2180923], [0, 78017444, 0, -2907161545, 0, 14236278978, 0, -27381912734, 0, 25631615713, 0, -13131648838, 0, 3877751832, 0, -657515656, 0, 59325796, 0, -2201676]];
    Rf_basisdens := [1, 1, 1963853, 1963853, 1963853, 1963853, 1963853, 1963853, 1963853, 1963853, 1963853, 1963853, 1963853, 1963853, 1963853, 1963853, 1963853, 1963853, 1963853, 1963853];
    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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, -1], [0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 1, 0, 1, 0], [-1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 1, 0, 1, 0, -1, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, -1, -1, -1, 1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 2, 0, -1], [-2, 0, 0, -1, 1, 0, 0, 1, 0, -1, 0, 0, -2, 0, 1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 1, 0, -2, -1, 0], [-3, 0, 0, -2, 0, 0, 0, 0, -1, 1, 1, 0, 1, 0, -2, 0, 0, 0, 0, 0], [0, 1, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, -1, -1, -2, -1], [-1, 0, 0, 4, 0, 0, -1, -1, 0, 0, -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, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 1, -3, -2, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 0, -4, 2, 0, 2, 3, 2, -3, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 0, -1, 0, 0, -1, 1, 0, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, -1, 0, -2, 2, 0, -3, -2], [-3, 0, 0, -2, -1, 0, 1, 2, -2, 0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, -3, -1, 0, 1], [0, -2, 3, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 2, -1, 0, 3], [-2, 0, 0, -2, -2, 0, 3, 0, 3, 0, 1, 0, 2, 0, -1, 0, 0, 0, 0, 0], [-2, 0, 0, 2, -2, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 1, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 2, 4, 4, -1], [-1, 0, 0, 4, -1, 0, -1, 0, 0, 3, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0], [-1, 0, 0, 6, -1, 0, -2, -7, -7, 4, 1, 0, 1, 0, -1, 0, 0, 0, 0, 0], [-3, 0, 0, 4, 0, 0, 3, -2, -5, 0, -2, 0, 3, 0, 3, 0, 0, 0, 0, 0], [-3, 0, 0, 2, -2, 0, -1, -4, -2, 3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, -1, 0, 0, 4, -1, -2, 0, -2, 0, 5, 0, -2, 0, 0, 0, 0, 0], [0, -3, 3, 0, 0, 4, 0, 0, 0, 0, 0, 5, 0, 2, 0, 1, -6, -1, -1, 1], [-6, 0, 0, -6, 2, 0, -1, 2, 9, -1, 5, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, -5, 5, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, -1, 0, -1, 0, 0, -3, 2], [0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 4, -2, 3, 3, 3], [4, 0, 0, 5, -2, 0, 0, -2, -4, 2, -1, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, -10, 0, 0, 0, 0, 0, -5, 0, -3, 0, -3, 0, 2, 7, -1], [0, -3, -1, 0, 0, 8, 0, 0, 0, 0, 0, 1, 0, 2, 0, 4, -2, -7, -8, 3], [0, 1, 2, 0, 0, -4, 0, 0, 0, 0, 0, 2, 0, 4, 0, -1, 1, 5, 7, 1], [0, 3, -5, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 7, 0, -2, 5, 2, 4, 3], [5, 0, 0, 7, 2, 0, -2, -2, -3, 0, -2, 0, -1, 0, 4, 0, 0, 0, 0, 0], [-4, 0, 0, -2, 1, 0, 0, 2, 4, -2, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 8, -5, 0, 0, -8, 0, 0, 0, 0, 0, -4, 0, -3, 0, -1, 1, 2, 7, 0], [3, 0, 0, -4, -3, 0, -2, 2, 1, -2, 4, 0, -2, 0, -2, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, -4, 0, 0, 0, 0, 0, -2, 0, 0, 0, 3, 3, -2, 4, 0], [-7, 0, 0, -1, 2, 0, -1, 1, 7, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -4, 0, 0, 0, 0, 6, 3, 0, 0, 3, 0, -3, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, -1, 6], [0, 0, -4, 0, 0, 5, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 2, -6, -5, 8], [0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 3, 0, -1, 1, -1, 4, -4], [0, 7, -6, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 2, 0, 0, 6, -1, -3, 2], [-1, 0, 0, -5, 6, 0, -1, 4, -2, -3, -1, 0, -1, 0, -2, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, -1, 0, 0, 0, 10, 0, -1, -4, -2], [-1, 0, 0, 0, -3, 0, 0, -2, -4, -2, 0, 0, 1, 0, -2, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, -5, 0, 0, 0, 0, 0, 1, 0, 3, 0, -7, 2, 1, 1, 1], [7, 0, 0, 9, 1, 0, -5, -6, -4, 4, -6, 0, 1, 0, 4, 0, 0, 0, 0, 0], [0, 5, -2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, -3, 0, 2, -2, -1, -4, -10], [0, 0, 2, 0, 0, 8, 0, 0, 0, 0, 0, 2, 0, 5, 0, -4, -5, -2, 0, 1], [-4, 0, 0, -5, 2, 0, -1, 7, -5, -2, 3, 0, -4, 0, -1, 0, 0, 0, 0, 0], [-6, 0, 0, -8, -1, 0, 3, -1, 2, -2, 1, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, -1, 6, 0, 0, 7, 0, 0, 0, 0, 0, 5, 0, 9, 0, -3, -5, 2, 4, 3], [-6, 0, 0, 3, 4, 0, 1, 9, 5, -6, -6, 0, -2, 0, 4, 0, 0, 0, 0, 0], [-6, 0, 0, -3, 1, 0, -1, -1, -1, 2, -1, 0, 2, 0, 3, 0, 0, 0, 0, 0], [-3, 0, 0, 0, -6, 0, -5, 3, 7, -4, 4, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-6, 0, 0, 0, 0, 0, -3, 4, 12, -5, 5, 0, -1, 0, 1, 0, 0, 0, 0, 0], [-7, 0, 0, -6, 3, 0, -1, 7, 0, -5, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 6, 0, 0, -2, 0, 0, 0, 0, 0, -1, 0, -7, 0, 2, -7, 1, 3, 0], [7, 0, 0, 4, 4, 0, 5, -3, -5, -2, -6, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, -9, 11, 0, 0, 10, 0, 0, 0, 0, 0, -1, 0, -2, 0, -9, -8, 3, -1, 3], [14, 0, 0, 6, -2, 0, 5, -5, -2, 8, 2, 0, -2, 0, -4, 0, 0, 0, 0, 0], [0, 0, 6, 0, 0, -3, 0, 0, 0, 0, 0, -3, 0, -9, 0, 2, 3, 2, -2, -10], [0, -4, -1, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, -4, 0, 7, -1, -2, -7, -8], [14, 0, 0, 2, -1, 0, 0, 2, -10, 5, -2, 0, -2, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, -1, 4, 14, -10, 4, 0, 1, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, -4, -4, 0, -1, 4, 6, -3, 5, 0, 0, 0, 2, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, -2, 0, 0, 0, 0, 0, -1, 0, 0, 0, -9, -1, -1, -3, -7], [5, 0, 0, 0, 8, 0, -1, -3, 0, -4, -4, 0, -2, 0, 8, 0, 0, 0, 0, 0], [0, -5, 3, 0, 0, 9, 0, 0, 0, 0, 0, 5, 0, 7, 0, 2, 0, 3, -7, 1], [0, 1, -6, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 11, 0, 6, 0, -7, 2, 3], [6, 0, 0, 3, -1, 0, 6, -12, 1, 7, -2, 0, 1, 0, -1, 0, 0, 0, 0, 0], [15, 0, 0, 11, 1, 0, -5, -4, -9, 10, 1, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, -7, 9, 0, 0, 6, 0, 0, 0, 0, 0, -2, 0, 1, 0, -5, -2, 4, 6, 11], [0, -11, 10, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0, 3, 0, 8, -2, -2, -4, -4], [0, -2, 2, 0, 0, 6, 0, 0, 0, 0, 0, 3, 0, 8, 0, 8, 3, -7, -4, 2], [-4, 0, 0, 8, -4, 0, -4, -9, -2, 12, 3, 0, 5, 0, -4, 0, 0, 0, 0, 0], [0, 1, -8, 0, 0, -4, 0, 0, 0, 0, 0, -4, 0, 1, 0, 1, -1, 2, -6, -3], [-18, 0, 0, -3, 3, 0, -5, 3, 6, -12, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0], [5, 0, 0, 6, 1, 0, -1, -2, 0, 0, -5, 0, -4, 0, -2, 0, 0, 0, 0, 0], [0, -1, -3, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 2, 0, 9, -1, -3, -5, 1], [0, -1, 6, 0, 0, -6, 0, 0, 0, 0, 0, -3, 0, -9, 0, -2, -2, 3, 0, -6], [2, 0, 0, 0, -1, 0, 0, -8, 4, -4, -6, 0, 0, 0, 5, 0, 0, 0, 0, 0], [0, -19, 8, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, -2, 0, 6, -4, -3, -16, 5], [0, -3, -4, 0, 0, -12, 0, 0, 0, 0, 0, 2, 0, 2, 0, 6, -2, -4, 4, 4], [-10, 0, 0, 4, 1, 0, -3, -5, 0, 0, 0, 0, -2, 0, -1, 0, 0, 0, 0, 0], [-3, 0, 0, -9, 5, 0, 8, -1, 0, -4, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0], [0, -8, -1, 0, 0, 7, 0, 0, 0, 0, 0, 2, 0, 14, 0, 1, 4, -4, 8, 16], [0, 4, -2, 0, 0, -3, 0, 0, 0, 0, 0, -3, 0, -5, 0, -4, -7, 4, 4, 7], [0, 1, -6, 0, 0, -11, 0, 0, 0, 0, 0, -7, 0, -4, 0, -2, 6, 6, 14, -2], [-6, 0, 0, 1, 4, 0, -1, 3, -3, 11, 10, 0, 1, 0, -5, 0, 0, 0, 0, 0], [0, -4, -1, 0, 0, -7, 0, 0, 0, 0, 0, -4, 0, -9, 0, 1, 5, 1, 0, -4], [0, -6, 0, 0, 0, 1, 0, 0, 0, 0, 0, -8, 0, -2, 0, -7, 2, -3, -1, 2], [6, 0, 0, 4, 5, 0, 3, 8, -4, 1, -8, 0, -3, 0, 3, 0, 0, 0, 0, 0], [2, 0, 0, 14, -4, 0, -1, -8, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, -9, 4, 0, 7, 9, 6, -2, -5, 0, 1, 0, 4, 0, 0, 0, 0, 0], [-7, 0, 0, -6, 2, 0, 0, 14, -1, 3, 8, 0, -1, 0, -5, 0, 0, 0, 0, 0], [-7, 0, 0, -4, -10, 0, -1, 4, 6, -3, 5, 0, 6, 0, -1, 0, 0, 0, 0, 0], [0, -9, 4, 0, 0, -6, 0, 0, 0, 0, 0, 4, 0, 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0, -4, 3, -3, -26, 6], [0, -3, -15, 0, 0, -3, 0, 0, 0, 0, 0, -14, 0, -1, 0, 1, 19, 1, -5, 24], [0, -9, 0, 0, 0, 6, 0, 0, 0, 0, 0, -4, 0, 12, 0, 17, -9, -8, -5, -11], [2, 0, 0, 23, -2, 0, -13, -39, -19, 7, 6, 0, -9, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 4, 0, 0, 0, 0, 0, 16, 0, 0, 0, 17, -14, 6, -16, 3], [0, 8, 8, 0, 0, 7, 0, 0, 0, 0, 0, 17, 0, -6, 0, 6, 28, 7, -4, -8], [-18, 0, 0, 3, 15, 0, -12, 25, 9, 0, -8, 0, -4, 0, 2, 0, 0, 0, 0, 0], [0, -8, 4, 0, 0, 5, 0, 0, 0, 0, 0, 13, 0, 13, 0, 21, 0, 9, -1, 10], [0, -21, 15, 0, 0, -1, 0, 0, 0, 0, 0, 21, 0, 9, 0, 23, -6, -12, 3, -13], [0, -10, 0, 0, 0, 26, 0, 0, 0, 0, 0, 6, 0, 17, 0, 11, -2, -12, -21, 7], [-15, 0, 0, -9, 6, 0, 5, 11, -9, -22, -16, 0, 0, 0, 5, 0, 0, 0, 0, 0], [0, 10, -4, 0, 0, -17, 0, 0, 0, 0, 0, -5, 0, -10, 0, 14, -15, 5, 10, -16], [-2, 0, 0, 32, 6, 0, -8, -19, -24, 21, 10, 0, -14, 0, 3, 0, 0, 0, 0, 0], [0, -9, 19, 0, 0, -18, 0, 0, 0, 0, 0, -2, 0, -18, 0, 1, -17, 3, 16, -6], [0, -10, 24, 0, 0, 34, 0, 0, 0, 0, 0, 3, 0, 4, 0, -7, -16, 2, 4, 31], [-17, 0, 0, 11, 8, 0, 2, 1, 10, -2, -4, 0, -8, 0, -3, 0, 0, 0, 0, 0], [-20, 0, 0, -31, 6, 0, 11, 34, 4, -2, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 21, -28, 0, 0, -7, 0, 0, 0, 0, 0, -21, 0, -2, 0, -6, 8, -9, 9, -8], [0, 25, -12, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, -8, 0, 5, 4, -2, -17, -1], [5, 0, 0, -4, 3, 0, -6, -11, 1, 2, -6, 0, -16, 0, -3, 0, 0, 0, 0, 0], [-5, 0, 0, -18, -11, 0, 2, 25, 15, 14, 3, 0, -2, 0, -12, 0, 0, 0, 0, 0], [-28, 0, 0, -1, 0, 0, -4, 14, 1, 5, -8, 0, 17, 0, -2, 0, 0, 0, 0, 0], [0, -12, 6, 0, 0, 16, 0, 0, 0, 0, 0, 14, 0, 2, 0, -6, 3, 3, 2, 14], [6, 0, 0, 26, 3, 0, 10, -30, -29, 10, -15, 0, 12, 0, -10, 0, 0, 0, 0, 0], [-10, 0, 0, -8, -25, 0, -9, 8, -5, 8, 13, 0, 15, 0, -14, 0, 0, 0, 0, 0], [0, -8, 8, 0, 0, 5, 0, 0, 0, 0, 0, 12, 0, -3, 0, 17, 7, -21, -25, -21], [28, 0, 0, 4, 22, 0, 11, -17, 3, -6, -25, 0, -3, 0, 21, 0, 0, 0, 0, 0], [-3, 0, 0, 1, -20, 0, -18, -15, -15, 9, 0, 0, -4, 0, 2, 0, 0, 0, 0, 0], [-21, 0, 0, -11, -1, 0, -12, 45, 31, -5, 7, 0, -11, 0, 4, 0, 0, 0, 0, 0], [-14, 0, 0, -16, 3, 0, 8, 31, 20, -21, -1, 0, -5, 0, 4, 0, 0, 0, 0, 0], [0, -11, 26, 0, 0, 7, 0, 0, 0, 0, 0, -4, 0, 0, 0, 5, 0, -6, -4, -4], [0, -25, 5, 0, 0, 8, 0, 0, 0, 0, 0, 11, 0, 7, 0, 31, 14, -10, 3, 23], [-6, 0, 0, -8, -2, 0, 6, -30, 7, -10, -16, 0, 14, 0, 7, 0, 0, 0, 0, 0], [0, 13, -11, 0, 0, -1, 0, 0, 0, 0, 0, 7, 0, -3, 0, -6, 11, -7, -22, -20], [-22, 0, 0, -4, -2, 0, 13, 21, -6, 0, 7, 0, 5, 0, 2, 0, 0, 0, 0, 0], [-20, 0, 0, -19, 1, 0, -3, 16, 8, -12, 9, 0, -26, 0, 1, 0, 0, 0, 0, 0], [0, -23, 2, 0, 0, 12, 0, 0, 0, 0, 0, 9, 0, 5, 0, 13, -1, -12, -1, 6], [0, 32, -8, 0, 0, 15, 0, 0, 0, 0, 0, 2, 0, 23, 0, -10, 0, 15, -2, -17], [0, -24, -2, 0, 0, 12, 0, 0, 0, 0, 0, 5, 0, -17, 0, 22, 8, -18, -20, 15], [-55, 0, 0, -4, -1, 0, 9, 13, 30, 7, 12, 0, 12, 0, -3, 0, 0, 0, 0, 0], [13, 0, 0, -3, -18, 0, -8, 18, -8, 25, 19, 0, -2, 0, -3, 0, 0, 0, 0, 0], [-7, 0, 0, -6, -6, 0, 14, -20, -20, 18, 7, 0, 10, 0, -25, 0, 0, 0, 0, 0], [2, 0, 0, 11, 0, 0, -3, 10, 15, -28, -1, 0, 9, 0, 3, 0, 0, 0, 0, 0], [0, 15, -29, 0, 0, -9, 0, 0, 0, 0, 0, -11, 0, 5, 0, -32, 30, -8, -4, 1], [0, 9, 32, 0, 0, -13, 0, 0, 0, 0, 0, -10, 0, 13, 0, -31, -21, 11, 30, -11], [0, 0, 1, 0, 0, 6, 0, 0, 0, 0, 0, 16, 0, 20, 0, 25, -4, -3, 9, 2], [-9, 0, 0, -19, 10, 0, 8, 12, 3, -38, -11, 0, 2, 0, 5, 0, 0, 0, 0, 0], [0, -14, 27, 0, 0, 19, 0, 0, 0, 0, 0, 17, 0, 10, 0, 3, -14, 13, 16, 19], [9, 0, 0, 22, -1, 0, 3, -19, -20, 7, 7, 0, -17, 0, -5, 0, 0, 0, 0, 0], [-24, 0, 0, 6, -4, 0, 15, -13, 13, 4, 9, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, 5, 11, 0, 0, 13, 0, 0, 0, 0, 0, -6, 0, 19, 0, 2, -25, 8, 15, -4], [0, -22, 9, 0, 0, -2, 0, 0, 0, 0, 0, -6, 0, 1, 0, 3, -15, -18, -11, 14], [-8, 0, 0, -12, -10, 0, -4, 5, 3, 1, 10, 0, 1, 0, 1, 0, 0, 0, 0, 0], [0, -1, 4, 0, 0, 19, 0, 0, 0, 0, 0, 4, 0, -30, 0, 16, 0, -18, -27, -24], [0, 0, 0, -7, 2, 0, -15, 28, 31, -24, 7, 0, -10, 0, 2, 0, 0, 0, 0, 0], [41, 0, 0, 16, -4, 0, 2, -27, -23, 16, 2, 0, -1, 0, 2, 0, 0, 0, 0, 0], [-4, 0, 0, 3, 2, 0, 3, -28, -16, 21, 7, 0, 4, 0, -11, 0, 0, 0, 0, 0], [19, 0, 0, -11, -11, 0, 2, -22, -13, -21, -11, 0, 13, 0, 1, 0, 0, 0, 0, 0], [0, 2, -15, 0, 0, -4, 0, 0, 0, 0, 0, -21, 0, -9, 0, -22, 2, -5, 1, 20], [0, -33, 24, 0, 0, 1, 0, 0, 0, 0, 0, 13, 0, -15, 0, 3, -27, 8, 6, 14], [-26, 0, 0, -28, -15, 0, 1, 35, 13, -9, 16, 0, 19, 0, -23, 0, 0, 0, 0, 0], [0, 10, -12, 0, 0, -2, 0, 0, 0, 0, 0, 6, 0, 24, 0, 17, 9, -4, 2, -23], [4, 0, 0, 21, -14, 0, 2, -25, -18, 20, -12, 0, 12, 0, 2, 0, 0, 0, 0, 0], [0, -14, 10, 0, 0, -7, 0, 0, 0, 0, 0, 11, 0, -10, 0, 26, -8, -10, -18, -17], [0, -16, 0, 0, 0, 20, 0, 0, 0, 0, 0, 8, 0, 13, 0, 13, 2, -9, -3, 9], [0, 12, -3, 0, 0, 1, 0, 0, 0, 0, 0, -3, 0, -22, 0, 10, -9, -2, 8, -24], [0, 2, -1, 0, 0, 24, 0, 0, 0, 0, 0, -1, 0, 19, 0, 21, -2, 6, -1, -6], [-3, 0, 0, 1, -1, 0, 18, 20, -2, -31, -4, 0, 9, 0, 8, 0, 0, 0, 0, 0], [0, 14, -8, 0, 0, -10, 0, 0, 0, 0, 0, 2, 0, 5, 0, 18, 10, 5, 4, 9]];
    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_q();".
// 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_q(:prec:=20)
    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_q();".
// 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_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1849_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 0, -75, 0, 1500, 0, -6823, 0, 12809, 0, -11859, 0, 6037, 0, -1775, 0, 300, 0, -27, 0, 1]>],Snew);
    return Vf;
end function;