
// Make newform 1134.2.d.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1134_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1134_d_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_1134_2_d_b();".
// 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_1134_2_d_b();".
// 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 := [457, -3160, 10748, -23676, 38144, -47396, 47500, -38796, 26533, -15036, 7240, -2860, 962, -252, 56, -8, 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], [1872, -31549, 140708, -336644, 520258, -592287, 520549, -361020, 205329, -92699, 35295, -10122, 2506, -378, 54, 0], [-76759, 481367, -1488709, 2902165, -4047300, 4253522, -3566011, 2370641, -1315127, 578736, -217429, 61305, -15086, 2247, -321, 0], [62645, -408925, 1265008, -2459279, 3411739, -3565294, 2973244, -1965161, 1084910, -474754, 177662, -49829, 12233, -1813, 259, 0], [-164484, 1008213, -3034651, 5784968, -7943631, 8232039, -6833518, 4497900, -2477718, 1081803, -404410, 113274, -27797, 4116, -588, 0], [96267, -636174, 1957354, -3787767, 5237615, -5458815, 4545742, -3000797, 1655774, -724205, 270978, -75991, 18656, -2765, 395, 0], [-30378190, 362310557, -1498869805, 3648052981, -6025679763, 7417334981, -7112671903, 5486496404, -3438485903, 1787650380, -755616592, 267766992, -73571231, 16986696, -2510469, 325218], [-49140353, 300070321, -1068108878, 2514291521, -4290519843, 5483567995, -5464826120, 4380742786, -2827132387, 1520242992, -655964315, 239986362, -66777328, 15988020, -2367801, 325218], [4929077, -71812067, 366902317, -1028814578, 1890068654, -2493491829, 2528025602, -2031718950, 1316620605, -704002373, 304140298, -110181896, 30712461, -7240864, 1078230, -143764], [11072049, 8683691, -254206816, 884700954, -1770414957, 2413011717, -2495224381, 2023238010, -1319456100, 707350209, -306216779, 110985188, -30949178, 7295622, -1086090, 144812], [3067531, -51282013, 261301701, -717486533, 1303324972, -1709227692, 1726154823, -1384310135, 895321773, -477998494, 206151407, -74578887, 20752290, -4887955, 726555, -96874], [-3363404, 31438212, -127013004, 307360314, -516105789, 645331338, -628987746, 493311054, -313213977, 165381778, -70590828, 25400574, -7024134, 1650786, -244350, 32580], [117878701, -938083265, 3527197972, -8310030777, 13736874985, -17049504748, 16530696510, -12928279637, 8189368116, -4318389740, 1840904054, -661951717, 182902889, -42971171, 6356685, -847558], [-347093285, 2498153638, -8561202083, 18622064190, -28885056971, 34027921247, -31646865318, 23833677907, -14664774819, 7501129528, -3131539819, 1094893661, -298263571, 67896127, -9990348, 1265468], [-99619787, 898877062, -3639199676, 9082243404, -15662214209, 20218823969, -20134441467, 16227690157, -10467672483, 5665618522, -2445049144, 902505863, -251052352, 60905395, -8991672, 1265468], [64404140, -455572548, 1575990199, -3521575313, 5601651077, -6780921079, 6443817363, -4972565887, 3112931382, -1628974479, 689195683, -246733441, 67817711, -15898989, 2342355, -312314]];
    Rf_basisdens := [1, 655, 1965, 393, 655, 393, 2595765, 2595765, 865255, 865255, 519153, 173051, 2595765, 2595765, 2595765, 519153];
    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_1134_d();"
function MakeCharacter_1134_d()
    N := 1134;
    order := 2;
    char_gens := [407, 325];
    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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1134_d_Hecke();"
function MakeCharacter_1134_d_Hecke(Kf)
    N := 1134;
    order := 2;
    char_gens := [407, 325];
    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]];
    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, 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, 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, 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, 1, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, -3, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1], [1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0], [0, 0, 1, 1, 0, -2, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [4, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, -1, 0, -1, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, -1, -1, 1, 0, -4, 0, 0, 0, 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, -2, 0, 1, 1, 1, -1], [-1, -1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0], [0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 0, -2, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, -1, 0], [2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 2, 0, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0], [0, 0, -2, 2, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2], [0, 0, 0, 0, 0, 0, 3, 3, 0, -2, 0, 0, 0, 0, 0, 0], [1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0], [0, 0, 0, 0, 0, 0, 1, 1, -3, -2, 0, 1, 0, 0, 0, 0], [-1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 0], [0, 0, 2, -2, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 1, -3, 2, 0, 1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 1, 1, -1], [0, 0, 0, 0, 0, 0, -3, -3, 2, -4, 0, 3, 0, 0, 0, 0], [1, -4, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -4, 0, 0, -1], [-1, -2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 0], [0, 0, 1, -1, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 1, 0, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, -3, -4, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 5, -1, -1, -2], [0, 0, 0, 0, 0, 0, 3, 3, 1, 0, 0, 6, 0, 0, 0, 0], [4, 1, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 5, -5, 0], [0, 0, 0, 0, 0, 0, -2, -2, 1, 3, 0, -5, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 2, 2, -1], [7, -1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4], [0, 0, 0, 0, 0, 0, 3, -3, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 2, -2, -2, -5], [0, 0, 0, 0, 0, 0, 1, 1, -3, -4, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 3, -5, 0, 0, 6, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -6, -3, -3, 2], [0, 0, 0, 4, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 3, 0, -1, -3, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, -4, -5, 2, 0, -4, 0, 0, 0, 0], [0, 0, 2, -1, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 8, 2, 2, 1], [-4, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4, 0], [0, 0, 0, 0, 0, 0, -3, -3, 0, -1, 0, -15, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, 2, 2, 2], [0, 0, -6, -1, 0, 4, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 3, 2], [0, 0, 3, -3, 0, 1, -3, 3, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, 3, 3, 0], [0, 0, 0, 0, 0, 0, -1, -1, 4, 0, 0, -7, 0, 0, 0, 0], [7, 4, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 0], [7, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 0], [0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, -16, 0, 0, 0, 0], [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, -5, -5, -4], [0, 0, 6, -2, 0, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 12, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 1], [-10, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0], [1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 0], [0, 0, 8, -2, 0, 2, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 3, -5, 5, 0, 6, 0, 0, 0, 0], [4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, -4, -4, -4, -3], [0, 0, 0, 0, 0, 0, 5, 5, 2, -3, 0, 5, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 8, -3, -3, -2], [0, 0, 2, 2, 0, 2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 6, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0], [0, 0, 0, 0, 0, 0, 5, 5, -2, -2, 0, -10, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, -3, 1, 1, -4], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, -1, -1, -4], [0, 0, 0, 0, 0, 0, -2, -2, -2, -7, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, -2, -2, 4, 0, -8, 0, 0, 0, 0], [1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, 0], [0, 0, -6, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-10, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4, 0], [0, 0, -4, 0, 0, -1, -6, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, -1, 0, -3, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [10, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, 0], [0, 0, 0, 0, 0, 0, -2, -2, -2, 2, 0, -20, 0, 0, 0, 0], [-2, -1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -7, 3, 0, 1, -3, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, 4, 0, -4, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, -2, 0, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, -6, 0, 0, -2], [-19, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4, 0], [-1, 1, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0], [0, 0, 0, 0, 0, 0, -5, -5, 2, 4, 0, 7, 0, 0, 0, 0], [0, 0, 2, -6, 0, 2, 3, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 1, 4, 1, 0, -11, 0, 0, 0, 0], [2, -3, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 2, -1, -1, 2], [0, 0, 4, 0, 0, -5, -3, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 6, 0, -3, 6, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, -5, 3, 2, 0, -2, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -6, -1, -1, -2], [-8, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 4, 8, 8, 3], [14, 6, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 5, 2, -5, 0, 5, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -4, 1, 1, -4], [-10, 4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4, 0], [0, 0, 0, 0, 0, 0, -1, -1, -1, -4, 0, -31, 0, 0, 0, 0], [4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 1, -4, -4, 5], [0, 0, -4, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 1, -5, -3, 0, -14, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 6, 0, -12, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, -1, -1, -5], [4, -1, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 0], [0, 0, -3, 1, 0, -2, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0, 12, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12, -2, -2, 2], [0, 0, 0, 0, 0, 0, 4, 4, 3, 1, 0, 13, 0, 0, 0, 0], [7, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -6, 6, 0], [0, 0, 8, 2, 0, -6, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -10, 2, 2, 6], [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, -5, -5, 0], [-16, -5, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0], [0, 0, 0, 0, 0, 0, -4, -4, -2, -8, 0, -4, 0, 0, 0, 0], [4, 2, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, 0], [-2, 5, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0], [0, 0, -6, 5, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, -4, 7, 7, 0], [0, 0, 5, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 4, 1, 1, 4], [0, 0, 2, -1, 0, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -11, -11, 1, -4, 0, -29, 0, 0, 0, 0], [-4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 1, 4, 4, 3], [0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 0, 15, 0, 0, 0, 0], [-19, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 0], [0, 0, 0, 0, 0, 0, 1, 1, -2, -4, 0, -20, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 2, 2, -4], [0, 0, -6, -4, 0, 6, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 8, 0, 0, 0], [0, 0, 8, -5, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -2, -3, -3, 2], [0, 0, 0, 0, 0, 0, -2, -2, -1, 2, 0, 4, 0, 0, 0, 0], [11, -7, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0], [0, 0, -3, -1, 0, -4, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [22, -3, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, 0], [0, 0, -11, -1, 0, 7, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [25, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 0], [0, 0, 0, 0, 0, 0, -2, -2, 4, -6, 0, 28, 0, 0, 0, 0], [-17, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0], [0, 0, 4, -3, 0, 5, -3, 3, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 5, 7, 7, 6], [0, 0, 12, -5, 0, -6, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -7, -7, -4, -6, 0, -16, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 3, -6, -3, 0, 9, 0, 0, 0, 0], [26, 2, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, 0], [0, 0, 8, 0, 0, 1, -3, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 8, 2, 0, 0, 20, 0, 0, 0, 0], [0, 0, 0, 2, 0, 4, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-11, -6, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, -7, 7, 0], [4, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -10, -4, -4, 3], [-7, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0], [0, 0, 12, 0, 0, 4, 6, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 3, 2, 4, 0, -12, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -7, 1, 1, -5], [0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 0, -24, 0, 0, 0, 0], [-13, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -10, 3, 3, 0], [0, 0, -9, 5, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 8, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 7, -7, 0], [0, 0, 0, 0, 0, 0, 3, 3, 8, -6, 0, -15, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, -2, -2, -4], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -9, 1, 1, 5], [14, 4, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 6, -6, 0], [0, 0, -8, 8, 0, 2, -5, 5, 0, 0, 0, 0, 0, 0, 0, 0], [5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, -10, 0], [0, 0, -7, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, -1, -2, -10, 0, 2, 0, 0, 0, 0], [0, 0, -9, 4, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-5, -6, 0, 0, -11, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, -3, 2, 2, 9], [-7, -7, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0], [0, 0, 3, -5, 0, 5, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -7, 5, 5, 4], [0, 0, 0, 0, 0, 0, 2, 2, 6, 0, 0, -16, 0, 0, 0, 0], [28, 3, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -10, 10, 0], [0, 0, -4, 11, 0, -10, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 5, -2, 2, 0, -16, 0, 0, 0, 0], [0, 0, -9, -3, 0, 4, -3, 3, 0, 0, 0, 0, 0, 0, 0, 0], [-11, -6, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0], [2, -8, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, -6, 6, 0], [0, 0, -4, -5, 0, 9, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -5, -7, 0, 11, 7, -7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, -15, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -10, 0, 0, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 1, 1, 1, -4], [-5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 5, 5, 8], [0, 0, 4, 8, 0, -5, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -12, 2, 0, 4, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, -2, -2, -12, 5, 0, 16, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 4, 2, 2, 0, 46, 0, 0, 0, 0], [4, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, -2, -4, -4, -8], [-10, 2, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0], [0, 0, 19, -2, 0, -4, 8, -8, 0, 0, 0, 0, 0, 0, 0, 0], [-26, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 10, 0], [0, 0, 4, 0, 0, 6, 3, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, 4, 0, 2, -7, 7, 0, 0, 0, 0, 0, 0, 0, 0], [4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -4, -4, 0], [-2, -8, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, -10, 10, 0], [0, 0, -6, 3, 0, -3, 3, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, -10, -6, -2, 0, -40, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, -3, -4, -2, 0, 21, 0, 0, 0, 0], [19, 9, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0], [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, -3, -3, -3], [0, 0, 0, 0, 0, 0, -3, -3, 3, 6, 0, -15, 0, 0, 0, 0], [1, 8, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0], [0, 0, -10, 10, 0, -4, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [7, 3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0], [0, 0, 3, -5, 0, 2, 5, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 6, 6, -8, 0, 0, 12, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 6, 3, 3, -4], [0, 0, 0, 0, 0, 0, 6, 6, -6, -1, 0, 0, 0, 0, 0, 0], [13, 8, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0], [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 2, -5, -5, -2], [-11, -6, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, -7, 7, 0], [0, 0, 0, 0, 0, 0, 1, 1, 14, 1, 0, -29, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 11, -2, -2, -7], [0, 0, 6, 6, 0, 0, -6, 6, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, -5, -1, -1, 10], [0, 0, 0, 0, 0, 0, 7, 7, -5, 1, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 7, 7, 4, -11, 0, 16, 0, 0, 0, 0], [0, 0, 5, -3, 0, -11, -6, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 7, 7, 12, -5, 0, 4, 0, 0, 0, 0], [9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, -5, -9, -9, 5], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, -6, -2, -2, 10], [22, 7, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, -6, 6, 0], [0, 0, -13, -3, 0, 8, 3, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 11, -5, 0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, -12, -3, -3, 4], [0, 0, -12, 0, 0, -8, -3, 3, 0, 0, 0, 0, 0, 0, 0, 0], [-34, -2, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -6, 6, 0], [0, 0, 0, 0, 0, 0, 3, 3, -1, 0, 0, -42, 0, 0, 0, 0], [-11, -5, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, -6, 6, 0], [0, 0, -5, -6, 0, -11, -6, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 7, 7, 10, -12, 0, -2, 0, 0, 0, 0], [-7, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -7, 7, 7, 12], [0, 0, 12, 7, 0, -4, 8, -8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -18, -18, 2, 1, 0, -12, 0, 0, 0, 0], [11, -2, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 8, -8, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, -11, -1, -1, -1], [0, 0, -5, 8, 0, -6, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -2, -2, 8], [41, 4, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, -7, 7, 0], [0, 0, 0, 0, 0, 0, 7, 7, 2, 3, 0, 28, 0, 0, 0, 0], [-8, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 8, 8, 8, 7], [6, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -20, -6, -6, 2], [0, 0, -5, 5, 0, 0, -5, 5, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, -3, 5, 5, 1], [0, 0, 0, 0, 0, 0, 6, 6, 9, 5, 0, -9, 0, 0, 0, 0], [0, 0, 0, -7, 0, -7, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-25, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, -9, 9, 0], [0, 0, 0, 0, 0, 0, 11, 11, 8, 10, 0, 35, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 5, 1, 1, -3], [7, -11, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 5, -5, 0], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 8, -2, -2, 4], [4, -2, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 16, -16, 0], [-5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 15, 5, 5, 4], [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -10, -5, -5, -6], [0, 0, 0, 0, 0, 0, 8, 8, 0, 7, 0, -4, 0, 0, 0, 0], [-5, 12, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 10, -10, 0], [-17, -2, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, 0], [0, 0, -8, 0, 0, -4, -9, 9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 7, 9, 0, -7, 9, -9, 0, 0, 0, 0, 0, 0, 0, 0], [11, 4, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -13, 13, 0], [0, 0, 10, -12, 0, 0, 3, -3, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, -10, -6, -6, -9], [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 11, -5, -5, -9], [0, 0, 0, 0, 0, 0, 3, 3, -4, -8, 0, -18, 0, 0, 0, 0], [-5, 6, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 10, -10, 0], [0, 0, 0, 0, 0, 0, 11, 11, 1, 11, 0, 8, 0, 0, 0, 0], [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 8, -5, -5, -15], [0, 0, -9, 2, 0, -3, -8, 8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 1, 3, -11, 0, 40, 0, 0, 0, 0], [0, 0, 12, 0, 0, -7, 3, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -9, -9, 4, -1, 0, -3, 0, 0, 0, 0], [0, 0, 0, 8, 0, -3, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [5, -6, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -10, 10, 0], [0, 0, 17, -2, 0, 4, 8, -8, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -2, -2, 3], [0, 0, -7, 4, 0, 2, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -10, 10, 0, -8, 5, -5, 0, 0, 0, 0, 0, 0, 0, 0], [8, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 4, -8, -8, -8], [-7, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -5, 7, 7, 4], [0, 0, 0, 0, 0, 0, 0, 0, -14, 2, 0, 24, 0, 0, 0, 0], [2, -8, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 6, -6, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1134_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1134_2_d_b();".
// 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_1134_2_d_b(:prec:=16)
    chi := MakeCharacter_1134_d();
    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_1134_2_d_b();".
// 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_1134_2_d_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1134_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![225, 0, -360, 0, 162, 0, -24, 0, 1]>],Snew);
    return Vf;
end function;