
// Make newform 1110.2.x.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1110_x();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1110_x_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_1110_2_x_e();".
// 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_1110_2_x_e();".
// 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 := [676, 0, 13976, 0, 43297, 0, 47024, 0, 23344, 0, 5750, 0, 724, 0, 44, 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], [-9055852, -88306866, 0, -431800511, 0, -559225583, 0, -298884555, 0, -75642377, 0, -9594515, 0, -582741, 0, -13206], [-77890999576, 115482059490, -1722644084824, 13243132119, -4574527274083, -183908713051, -3505456229509, -149605457411, -1066046293013, -45846209935, -150568927483, -6548710747, -9788447929, -431418313, -231600785, -10324114], [-122757597300, 144197912780, -1484648364030, 621706144390, -3219686942498, 1209580766261, -2297096731332, 856452813701, -682692894520, 254037921393, -95765609498, 35550593381, -6219207696, 2303097077, -147256174, 54424395], [980070, 46766, 19510371, 0, 51587280, 0, 39518805, 0, 12057906, 0, 1710274, 0, 111621, 0, 2649, 0], [-2258341020, -39312620648, 248883798774, -493899653872, 907543313781, -1348129378719, 754965960931, -1051583349562, 237377029149, -322877995643, 34160381775, -45886051749, 2247594011, -2995690422, 53593657, -71079673], [-17830174076, -9758281326, -336823640582, -145114756365, -873572804013, -371795294250, -664771352973, -282656815476, -202135749621, -85967283874, -28605027321, -12170127966, -1863894877, -793263294, -44185843, -18809805], [-17830174076, 9758281326, -336823640582, 145114756365, -873572804013, 371795294250, -664771352973, 282656815476, -202135749621, 85967283874, -28605027321, 12170127966, -1863894877, 793263294, -44185843, 18809805], [-152403200300, -116168789070, -3000525724378, -1147108653689, -7827014551031, -2427112866829, -5959533316776, -1727111569368, -1810543749131, -511794695604, -255907946385, -71545576365, -16655667834, -4631516922, -394487353, -109394411], [0, 36856818194, 0, -974556175625, 0, -2880167470489, 0, -2252772156535, 0, -691602201221, 0, -98320814245, 0, -6422799157, 0, -152483460], [-283340406336, -148807246794, -6033812610046, -1641008307561, -16096784942732, -3775242245548, -12354386014477, -2778694918930, -3768752775164, -834672691247, -534148371796, -117431628114, -34833141239, -7627207344, -826130396, -180474084], [-283340406336, 148807246794, -6033812610046, 1641008307561, -16096784942732, 3775242245548, -12354386014477, 2778694918930, -3768752775164, 834672691247, -534148371796, 117431628114, -34833141239, 7627207344, -826130396, 180474084], [-246614842136, -228113543072, -5011488932190, -4803947370306, -13751840979565, -12918094576644, -10655274888141, -9931123320356, -3261632348195, -3031013989376, -462969734669, -429677117692, -30213515345, -28023362596, -716837485, -664665858], [-315674197904, 191256724878, -6236365419466, 5778503545931, -16511281626086, 15798262047133, -12650799195788, 12183895476891, -3852924582910, 3722616190597, -545217898602, 527997931937, -35506775252, 34446161753, -841250956, 817149318], [-111042129536, -411912946812, -3117800473762, -9254818590818, -8423226198985, -24366401654201, -6463613063453, -18615042339947, -1970674050267, -5668426141845, -279214096005, -802702515889, -18205033405, -52322147503, -431724501, -1240593227], [-111042129536, 411912946812, -3117800473762, 9254818590818, -8423226198985, 24366401654201, -6463613063453, 18615042339947, -1970674050267, 5668426141845, -279214096005, 802702515889, -18205033405, 52322147503, -431724501, 1240593227]];
    Rf_basisdens := [1, 18111704, 13348325848, 13348325848, 46766, 6674162924, 1213484168, 1213484168, 6674162924, 6674162924, 6674162924, 6674162924, 13348325848, 13348325848, 13348325848, 13348325848];
    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_1110_x();"
function MakeCharacter_1110_x()
    N := 1110;
    order := 6;
    char_gens := [371, 667, 631];
    v := [6, 6, 5];
    // 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_1110_x_Hecke();"
function MakeCharacter_1110_x_Hecke(Kf)
    N := 1110;
    order := 6;
    char_gens := [371, 667, 631];
    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], [0, -1, 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, 1, 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, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 1, 0, -1, 0, -1, 0, 0, 1, -1, 0, 0, 0], [0, 0, 1, 0, 0, 1, 0, 0, 0, -1, 1, 1, -1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, -1], [-1, 1, -1, 0, 1, -1, 1, 0, 0, 0, 1, 2, -1, -1, 0, -1], [0, 1, 0, 0, 1, 0, 1, 0, 0, 0, -2, 0, 0, 1, 1, -1], [-2, 0, -1, 0, 1, 0, 1, -1, -1, -1, 0, 1, -1, 1, 1, -1], [-1, 2, 0, 1, 1, -1, 1, -2, -1, 0, 0, 1, -1, 1, 0, 0], [-1, -4, -1, -1, -1, 0, 0, 1, 1, 1, -1, 0, 0, 0, 0, 0], [2, 0, 0, -1, -1, 0, -1, 1, 2, 1, 0, -1, 1, -1, -2, 0], [0, 1, 0, 2, -1, -2, 1, -3, 0, -2, 1, -1, 0, 1, 0, -1], [-2, 0, -1, 0, 1, 0, 1, -1, -1, 0, 1, 0, -1, 1, 0, 0], [-1, 0, -1, 2, 0, -1, 1, -1, 0, -1, 2, 2, -1, -1, -1, -1], [1, -3, 2, -1, -2, 1, -1, 0, 1, -1, 0, -1, 1, -2, -1, 0], [3, -3, 1, -2, 0, 2, -1, 2, 0, 1, -2, -1, 0, 0, 1, 2], [-2, -1, -4, -2, 1, -2, 1, 1, 2, 2, 1, 1, -2, -1, 0, 1], [0, -1, 0, 0, 0, 0, -2, 1, 0, 0, 1, -1, 0, 0, -1, 0], [-3, 3, 0, 0, 4, 1, 2, -2, -3, -1, 0, 5, -3, -1, 1, -1], [-4, 0, -3, -2, 1, -2, 1, 3, 1, 2, 1, 2, -1, -1, 0, 0], [6, 4, -2, 0, 1, -1, 1, 2, 0, 1, -2, 2, 0, 1, -1, -3], [3, 5, 2, 3, 1, 0, 0, -2, -1, 0, 0, 1, 0, 1, 0, 0], [1, -1, 1, 1, -2, -1, -3, -2, -1, 0, -2, -3, 1, 2, 1, 2], [-2, 4, 1, 2, 3, 0, -7, 5, -3, 0, 1, 2, -1, 1, 0, 0], [0, 0, 1, -1, 1, 2, -1, 0, 1, -1, 0, 1, -2, -2, -1, -1], [-3, -2, 1, 2, 1, -2, 1, -3, -1, 0, 0, 1, -1, 1, 1, -1], [1, 1, 0, 1, -2, -2, 6, -4, 1, 0, 1, -2, 1, 1, 0, -1], [3, -3, -1, -3, 2, 0, -1, 3, 0, 2, -1, 1, -2, -2, 0, 1], [-1, 1, 1, 0, 0, 2, 0, 0, 0, -1, -1, 0, 0, 0, 1, 1], [-1, -5, 1, -2, -2, 1, -4, 7, 1, -1, -1, -3, 1, -2, 2, -1], [2, -2, 0, -1, 0, -1, 0, 0, -1, 1, 0, 0, -1, 0, 0, 0], [-4, 0, -2, 0, 1, -1, 6, 6, 1, 1, -2, -1, 0, 0, 2, 2], [5, 5, -3, -1, 0, -1, -1, 4, 1, 2, -2, 1, -1, 0, 1, -2], [8, 0, 1, -2, -2, -1, -3, -1, -2, -1, 3, 1, -3, -3, -2, -2], [2, -2, 0, 0, -2, -2, 2, 0, 2, 2, -2, -2, 2, 0, 0, 2], [0, 2, -1, 0, 1, 1, 3, -6, -2, 0, 0, -1, 1, 1, 2, 0], [1, -1, 0, 0, 1, 2, -9, 1, 1, -1, -1, 2, 0, -1, 2, 1], [2, 0, -2, 0, 0, -1, 0, 0, 2, 1, 2, -1, -2, -2, 1, 3], [-1, 5, 0, 2, 3, -1, 5, -8, -2, -1, 0, 3, -1, 3, 0, 0], [2, -2, 0, -1, -2, 0, 2, -1, 2, -2, -2, 0, 1, -1, 1, -1], [2, 6, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, -2, 0, 0], [-6, -2, 0, 2, 3, -1, 5, -7, -3, 0, -1, 4, -2, 2, 1, -1], [-3, 2, 2, 3, 3, -1, 8, -11, -3, -1, 1, 2, -1, 1, -1, 1], [-2, 2, 5, 4, 2, 1, -2, 1, -3, -3, 3, -2, 1, 2, 1, 3], [-1, 2, -1, 0, 1, 0, -6, 6, -1, 4, -1, 2, -1, 1, 1, -1], [-3, 0, 2, 5, -3, -1, 1, -4, -3, -1, -3, -6, 5, 5, 4, 4], [-1, 0, -3, -3, 3, 0, 4, 7, 3, 2, 0, 3, -2, -2, 1, 1], [9, 3, 0, -6, -1, 0, -1, 12, 1, 0, -2, -5, -1, -2, 4, 2], [0, 5, 0, 2, -2, -5, -4, -1, 0, 1, -1, -3, 0, 2, -1, 2], [2, 0, -3, -5, 4, 1, 5, 10, 4, 2, 2, 6, -4, -4, -1, -1], [-4, 4, -1, 0, 0, -2, -5, 0, 0, 1, -1, 1, 0, 0, 2, 1], [-2, -3, 4, -1, -3, 2, -3, 0, -2, -2, -1, -3, 2, -1, 0, -1], [3, -2, 0, -3, -1, 5, -2, 5, 1, -4, 2, -3, 3, -3, -1, 1], [3, -3, -1, 2, 0, 0, 0, 0, 2, -1, 1, -2, 2, 0, -3, -1], [0, -2, 0, -1, 3, 5, 3, 1, 0, -3, 3, 0, 0, -3, -3, 0], [0, 0, 2, 5, -1, 1, 6, 1, -1, -2, -1, -2, 3, 3, 3, 3], [3, -5, 2, -2, -4, 0, -3, 4, 4, 0, 1, -5, 0, -4, 0, 1], [-9, -4, 2, 2, 0, 1, 0, -3, -1, -1, 3, 0, 1, 1, 0, 3], [-1, 1, -2, -1, 0, -4, 1, 1, 0, 3, -3, -3, 0, 0, 0, 3], [5, 5, 2, -1, 0, 1, 0, -8, -5, -1, -3, -6, 5, 5, 6, 3], [-1, 10, 0, 2, -1, -6, 5, -6, -1, 2, 3, -5, -1, 2, -4, 1], [-5, 0, 2, -1, -2, 0, -3, -2, -2, 1, 0, -2, 3, 3, 4, 4], [-1, 1, -2, 1, -1, -3, 5, 1, 2, 2, -2, 0, 3, 1, 3, 2], [-8, 8, 0, -2, 4, 2, 0, 0, -2, -2, 4, 8, -6, -4, 0, -4], [9, -5, 1, -4, -7, 2, -9, 13, 5, 2, -1, -3, 2, -7, -3, -1], [0, 0, 4, 0, 1, 6, -5, -3, -3, -5, 1, 6, -4, -1, 4, -1], [0, 11, 0, 4, 0, -3, -2, -3, 0, -5, 1, 3, 0, 0, 3, -4], [6, -4, 2, -1, -5, -1, -6, 7, 5, 0, -3, -2, 3, -3, -1, 1], [-20, -12, -4, 2, 4, -2, 0, 4, -2, 2, -2, 2, -2, 4, 4, -2], [6, -6, -1, 6, -2, -4, 3, -6, 0, -3, 3, 1, 2, 2, 0, -3], [-10, 0, 3, 3, 0, 3, 7, 4, 0, -2, 0, 0, 2, 2, 1, 1], [-3, 8, 0, -1, 1, 5, -3, 0, -3, -3, -2, 4, -3, 2, 3, -1], [0, 2, 0, 4, 2, 1, -4, -1, 0, -9, 0, 5, 0, -2, 3, -3], [5, 7, -5, -3, 1, -1, 4, -3, 0, 1, -5, 1, -1, 1, 5, -5], [-5, 0, 0, -2, 2, 0, 2, -4, -1, 0, -3, -2, 1, 3, 4, 1], [4, -2, 0, -1, -3, 0, -3, 2, 2, 0, 3, 1, -2, -5, -4, -1], [-7, 0, 0, 0, 1, 1, -8, -8, 1, -1, 2, 3, -2, -2, -1, -1], [-4, -16, -2, -1, -5, -5, -3, 4, 5, 6, -1, -4, -1, 1, 0, 0], [-2, 2, 1, 3, -4, -1, -1, -2, 1, 0, -4, -3, 5, 4, 5, 4], [-7, -5, -5, -1, 1, 1, -3, 7, -2, 3, 0, 2, 1, 1, -1, 0], [-7, 6, 1, 3, 8, 3, 9, -12, -8, -7, 4, 4, -2, 2, 3, -3], [5, 1, 6, -4, -2, 3, -2, -10, -1, -3, 0, -3, 1, -1, 1, 1], [10, 0, -3, -4, 3, 0, 3, 7, 3, 4, -3, 0, 1, 1, 0, 0], [4, 5, -2, 0, 3, -1, 3, -1, 0, 1, -5, 1, 0, 3, 2, -3], [0, 0, 6, 3, -4, 2, -5, -8, -4, -2, -2, -6, 5, 5, -2, -2], [2, 4, 4, 6, 0, 2, 0, -6, -6, -2, -4, -2, 6, 6, 2, -2], [-2, -2, 0, -4, 0, 0, 0, -12, 2, 0, -2, 0, -2, -2, 0, -2], [-1, 1, 1, 0, -3, 0, -6, 1, 1, 2, -3, -7, 4, 3, -1, 3], [-6, -6, 0, -5, -1, 0, -1, 10, 4, 0, 5, 3, -4, -5, -4, 1], [-4, 4, 0, 1, 5, 3, 6, -7, -5, -2, -1, 6, -1, 1, 3, -3], [8, -8, 0, 0, -2, 2, -10, 4, 4, 2, -2, -2, 6, 2, 2, 2], [4, 12, 2, 3, 1, -3, 16, -19, -1, -8, 1, 0, -1, 1, 3, -3], [-2, 0, -2, -1, 2, 0, 2, 3, 2, 2, 0, 2, 1, 1, -1, -1], [-5, -3, 2, 0, -2, 1, -2, 6, -3, -1, 0, 1, 3, 1, -3, -3], [0, 0, -3, -5, 3, 0, -4, 8, 3, 5, -3, 1, 0, -3, 1, 3], [8, 2, 1, 0, -3, 0, 5, -13, 3, 2, 1, 0, 0, -3, -4, 1], [0, -3, 0, -1, -2, -2, -4, 2, 0, 4, 3, -3, 0, 2, -1, -2], [-10, -6, -2, -2, -1, -1, -1, 2, 0, 1, -2, 0, 0, -1, -1, -3], [-5, -2, 0, 3, 1, -5, 8, -11, -1, -3, 5, -4, -3, 3, -1, 1], [2, -12, 0, -4, -4, 4, -2, 6, 4, -2, -6, 2, 4, -4, 0, 0], [-8, 8, -1, 5, 1, -3, 0, -7, -2, -5, 7, 6, -3, -1, -2, -7], [-4, -2, -6, -3, 0, 0, 1, 2, 0, 0, 2, -2, -3, 3, -2, 2], [2, 0, -3, -3, -2, 1, -10, 13, 2, 2, 0, -2, 0, 0, 0, 0], [1, 1, 0, -1, 0, 3, -12, 22, -3, -1, 1, -2, 3, 0, 1, 1], [3, -3, 2, -3, 0, 5, -3, 4, 1, 1, -4, 1, 1, 0, 5, 4], [9, 5, 0, -2, 1, 0, 1, 20, 1, 0, -2, -1, -1, 0, 2, 0], [9, 0, -2, 0, 2, 0, -7, -7, 2, 0, 1, 3, -2, -2, -3, -3], [-3, 9, 0, 4, 2, -5, -20, 4, -3, -3, 2, 1, -3, 1, -1, -1], [-11, -5, 1, 6, 3, -2, 3, -7, -1, 2, 1, 4, -2, 3, -2, 1], [2, -2, 2, 4, -2, 1, -11, -5, -1, -4, 2, -1, 1, 2, -1, -2], [2, 2, -4, 0, 0, -2, 1, 0, 2, 4, 5, 2, -2, 0, -7, 5], [-13, -9, -7, -2, 2, -6, 10, -10, 4, 3, -1, 6, -6, 2, -3, -1], [14, 0, 0, -2, 1, 1, -5, -3, 1, 2, -1, 0, 2, 2, 3, 3], [-7, 7, -4, 4, 1, -8, -13, -5, -1, -1, 3, -2, -2, -1, -6, -3], [-2, 6, -9, -2, 4, -2, -2, 12, -2, 3, -6, 8, -2, 4, 2, -6], [9, 6, -3, -2, -7, -6, -6, 8, 7, 10, -1, -6, -1, 1, -5, 5], [-4, -2, 2, -3, 2, 1, 2, 1, 4, -1, -3, -1, -4, -2, 3, 0], [7, 8, -4, -2, 5, -2, 5, 2, 1, 2, -5, 5, -1, 4, 0, -5], [-6, 11, 0, -1, 6, 3, 16, -10, -6, -1, 5, 6, -6, 0, 0, -5], [10, 0, 2, -4, -5, 1, -5, 8, 0, -1, 2, -4, 0, -5, -1, 1], [-2, -5, 0, 0, 3, -4, -7, 3, -2, 4, 1, 3, -2, -1, 0, -1], [10, 0, -5, 4, 2, -3, 6, 2, 2, 3, 1, 3, 5, 5, 0, 0], [-10, 10, 0, -2, 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-16, -2, 7, -5, 3, 8, -2, 7, 6, 1, -5, 7, 0, 6], [-4, 4, -3, 0, 3, 0, 5, -5, -3, -6, 1, 2, -3, 3, 4, -4], [-6, 6, -7, -5, 9, 3, -12, 13, 8, 3, -3, 8, -1, -9, 2, 3], [7, -7, 6, 7, -7, 0, -4, -12, -5, -6, 0, -9, 2, 7, -2, 0], [2, -5, 0, -4, 2, 9, 44, -15, 2, -1, 3, -1, 2, -4, -3, 0], [-2, 2, 11, 3, 2, 8, -4, -1, -10, -10, 0, -4, 8, 2, 6, 0], [8, 3, -6, -5, 4, -3, 4, 21, 10, 3, 0, 11, -10, -6, -7, -7], [3, 4, 0, -1, -1, -8, 35, -14, 3, 10, -5, 0, 3, -2, 1, 4], [-1, -17, 0, 0, 5, 5, 7, -2, -1, -5, -8, 16, -1, -4, 11, -3], [-12, -12, 4, -4, -6, 2, -6, 0, 0, -2, 0, -6, 0, -6, 0, 0], [-7, -11, 9, 4, -2, 7, -1, -11, -5, -3, 4, -3, 7, -2, -3, 4], [-6, 6, -2, 0, 7, 5, 11, -11, -7, 3, -4, 11, -2, 2, 0, 0], [-19, 19, 1, -1, 2, 2, -17, -1, -2, -2, 3, 11, -4, -2, 6, -3], [-14, -14, 3, 5, 5, -2, 11, -16, -5, 3, 3, 2, -2, 2, -1, 1], [31, 26, 1, -5, -12, -1, -3, 8, 12, -3, -4, -8, 6, -6, -1, 1], [1, -1, 2, 6, -7, 2, -3, -1, 5, -1, -5, -6, 12, 7, 6, 5], [10, 0, -4, -4, 2, -2, 6, 10, 2, 8, -12, -10, 8, 8, 8, 8], [-8, -8, 0, 4, 6, -4, -6, -6, -8, -4, 12, 2, -8, 2, -4, -8], [24, 0, 4, -1, -3, 1, -16, -15, -3, -4, -5, -8, -5, -5, 3, 3], [-50, -30, 4, 4, -6, 2, -6, 10, -2, -2, 2, -4, 2, -4, -2, 0], [6, 2, -5, -7, -2, -1, 11, -16, 3, 0, -6, 5, -1, -2, -1, -6], [-4, 4, 8, 9, -1, 3, 24, -21, -12, -16, 8, 3, -11, 1, -4, -8], [-7, 0, 3, -4, -2, 1, 5, 9, -2, 1, -4, -6, 1, 1, -3, -3], [-4, 1, -2, 0, -1, -1, -1, 13, -8, 1, -3, -5, 8, 7, 4, 1], [7, -6, 0, -1, -4, 2, 14, -1, 7, 0, 0, -8, 7, -3, -4, 4], [-26, 0, 3, -8, -1, 2, -13, -5, -1, 0, 3, 2, -5, -5, 0, 0], [-11, 11, 0, 14, -5, -14, -13, -23, -9, -9, 7, 2, -4, 5, 0, -7], [14, 22, 6, 6, 4, 4, 20, -46, -8, -4, -2, 10, 4, 4, -4, -2], [13, -18, 2, -5, -15, -3, -34, 39, 15, 13, 7, -22, 7, -7, -15, 15], [-12, 0, 2, -8, 3, 5, 1, 9, 3, -3, 12, 15, -12, -12, -6, -6], [-12, 2, -9, -1, 5, -2, -13, 14, -5, 7, 1, 4, -8, 8, 5, -5], [2, -2, 12, 0, -10, 13, -2, -1, -1, -2, 1, -8, 9, 10, 1, -1], [4, 4, -4, -5, -1, 5, -24, 29, 1, 10, -5, 4, 1, -1, -1, 1], [11, -11, -4, 5, 3, -8, -12, -8, -3, -4, 4, 7, -6, -3, 0, -4], [-8, 0, 7, 13, -1, 6, -19, -32, -1, -7, -2, -3, 6, 6, -1, -1], [13, -27, 0, 0, -13, -6, -9, 11, 13, 6, -3, -20, 13, 0, -7, 10], [-16, -10, -3, 5, 3, -7, 9, -13, 4, 5, 2, 6, -7, 3, -5, 2], [-23, -45, -3, -12, -11, 4, -16, 29, 7, 2, 5, -7, 4, -11, -9, 5], [2, 0, 11, 8, -9, 2, -11, -19, -9, -8, 9, 0, 3, 3, -8, -8], [-13, -13, -1, -5, 0, 2, 5, -7, -2, -4, 9, -7, 2, 0, -2, 9], [-26, -15, -2, 1, 1, -1, 1, -12, 6, 1, 1, 9, -6, -5, -8, -7], [-4, 0, 3, -1, 1, 4, 23, 24, 1, 1, -13, -12, 4, 4, 5, 5], [-8, 1, -6, -2, 6, -3, 6, -7, 2, 3, -1, -3, -2, 4, 9, 8], [11, 0, 2, 3, -3, -1, 8, 5, -3, 3, -9, -12, 11, 11, 1, 1], [8, 0, -19, -11, 5, -14, 13, 24, 5, 15, -15, -10, 0, 0, 7, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1110_x_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1110_2_x_e();".
// 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_1110_2_x_e(:prec:=16)
    chi := MakeCharacter_1110_x();
    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_1110_2_x_e();".
// 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_1110_2_x_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1110_x();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![3748096, -6473984, 15216960, 1563584, 8039408, 165456, 2348008, 237112, 341688, 29500, 32146, 3366, 1763, 128, 57, 4, 1]>],Snew);
    return Vf;
end function;