
// Make newform 5292.2.bm.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5292_bm();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_5292_bm_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_5292_2_bm_a();".
// 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_5292_2_bm_a();".
// 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 := [6561, -4374, 3645, -4131, 1782, -837, 558, -156, 52, -52, 62, -31, 22, -17, 5, -2, 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], [4788801, 2597427, 2064528, -63747, -231723, -123732, -17106, -26486, -19426, -23272, 14762, 10009, 8221, 1928, 139, 19], [-19166868, -4573017, -11094408, 2054484, -900801, 1174032, -84090, 181946, 7210, 113602, -77018, -2716, -49267, -824, -5068, 1307], [46300977, -33241671, 20620980, -18766647, 9488205, -2586978, 844320, -711356, 298928, -187876, 360098, -215537, 99223, -51544, 20725, -3695], [-5872095, -16161201, 15997176, -3514995, 7713063, -3624714, 202764, -214640, 277628, -101824, 11630, -145673, 91129, -16612, 21577, -6677], [-4533651, 11432178, 2423439, 6259194, -910737, 118692, -419040, 54754, -86836, -18340, -82192, 44857, 5248, 23477, 1930, 1669], [-35170605, -9423297, -19275408, 4418847, -1523241, 2194674, -278440, 332068, 28864, 195292, -147394, -8827, -88319, 1724, -9229, 2739], [-11234619, -15432201, -13165740, -11460447, -2209221, -981702, 1092156, 318940, 178940, 87524, 66614, -20561, -27863, -52096, -16535, -8405], [34596153, 614547, 17226513, -9141984, 1955178, -2581722, 605064, -305642, 38834, -186610, 193856, -24581, 86401, -21253, 6616, -4934], [-61443765, -7676370, -29851092, 18020475, -63018, 5698206, -1370658, 284726, -100070, 361210, -343040, -15775, -164782, 45760, -1333, 13862], [-47062053, 7598367, 5383908, 36196875, 10347075, 5542434, -3257256, -426532, -326936, 157996, -442826, -95755, -58111, 129088, 45263, 27959], [23648031, -21030192, 16524972, -20396205, 5124384, -5939262, 1677798, -282422, 398066, -166462, 247964, -128807, 123904, -73804, 9175, -16952], [66410442, -26451765, 21512304, -34358094, 3336147, -6290388, 2453682, -442766, 407870, -286834, 529214, -121322, 147133, -108688, -1430, -19793], [-10890531, 1887138, -4856112, 4880493, -523818, 1284126, -379710, 79862, -61412, 55828, -75428, 13505, -31234, 15076, -865, 3656], [92446677, 4219452, 56735154, -29717685, 5227686, -11690478, 2355330, -708026, 383294, -614170, 523952, -60809, 318436, -80794, 18943, -27554], [111152817, -24427089, 51803064, -41432283, 8671095, -9787698, 2760316, -995208, 458764, -571592, 740670, -185713, 283569, -116460, 19169, -23045]];
    Rf_basisdens := [1, 103518, 621108, 1242216, 1242216, 310554, 414072, 621108, 310554, 621108, 1242216, 621108, 621108, 103518, 621108, 414072];
    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_5292_bm();"
function MakeCharacter_5292_bm()
    N := 5292;
    order := 6;
    char_gens := [2647, 785, 1081];
    v := [6, 5, 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_5292_bm_Hecke();"
function MakeCharacter_5292_bm_Hecke(Kf)
    N := 5292;
    order := 6;
    char_gens := [2647, 785, 1081];
    char_values := [[1, 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], [1, 0, -1, 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, 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, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0], [0, 0, -2, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, 0, 0, 0, 2, 0, 2, 0, -1, -2, 0, 0, 1], [0, -1, 1, 1, 0, 0, 0, 1, 0, 0, 1, -1, 0, 0, 1, 0], [-1, 0, 1, -1, 1, 0, 0, -2, 0, -1, 0, 1, 2, 1, -1, 0], [-2, 0, -1, 2, -1, -1, 1, 0, 2, 0, 0, 0, 1, -1, 1, -2], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -2, 1, 0, 0, 0], [1, 0, 0, -1, 0, 0, 1, 2, 0, 2, -1, -1, -1, 0, 0, 1], [1, 0, 1, -2, 2, 0, -2, 0, 0, 0, 0, -3, 0, 0, 0, 1], [0, -1, -3, 1, -2, 0, 0, 0, 0, -1, 1, 0, 0, 1, 2, 0], [-2, 1, 1, 1, 0, -2, 0, -1, 1, -1, 0, 0, 2, 1, 2, -1], [1, 0, 1, -3, -1, 2, 0, 2, 0, 2, 1, -1, 0, 0, -2, 1], [0, 0, 0, -1, 2, 0, -1, 0, 0, 1, -1, 0, -2, 0, 0, 1], [-4, 0, 3, 0, -1, -3, -1, -2, 0, -3, -1, 1, 1, 0, 0, -1], [-1, -1, 0, 0, -3, 1, 2, 0, 1, 1, 1, 1, 0, 0, 1, 0], [-1, -1, 2, 0, 0, -2, -1, -1, 1, -3, -1, -1, 3, -1, 0, 0], [1, 0, -1, 0, 0, 0, 0, 2, 0, 3, 0, -1, -1, 0, 0, 0], [0, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 2, 0], [-5, 0, 3, 1, -2, -1, 1, -1, 0, -1, -1, 1, 0, 0, -1, -1], [-1, 0, 0, 1, -1, 0, 0, -1, 0, 1, 2, -1, 1, -1, -1, -2], [3, 0, 3, -2, 3, -1, -1, -2, 1, -1, -2, -1, 4, 0, 0, 2], [0, 1, 3, -1, 1, -1, -2, 1, -1, -1, 0, -2, 0, 0, 0, 0], [-5, 0, -4, 1, 0, -1, 1, 0, 2, -2, -2, 1, 1, -1, 2, -1], [2, 0, -1, -2, 0, 0, -2, 1, -3, -3, 1, -2, -2, 0, 0, 1], [6, 0, 0, -3, 2, 4, -1, 2, -2, 2, -1, 0, 0, 0, -4, 3], [-4, -1, 1, 0, 1, -3, 0, -1, 3, -5, -1, 2, 2, -2, 1, 0], [3, 0, -1, 3, -4, -2, 2, -1, 2, -2, 3, 1, 2, 0, 1, -4], [2, 0, 0, -2, -3, -1, 0, 4, -1, 0, 1, -4, 0, 0, 1, 0], [4, 1, -2, -2, -1, 2, 0, 1, -1, 1, 1, 4, -2, 1, -3, 1], [-3, 2, 5, 0, 2, 2, 0, -1, 2, 1, -2, 2, 0, 0, -2, 0], [-8, 0, -2, 7, -3, -3, 2, 0, 3, -3, -2, 3, 0, 0, 6, -4], [-4, 0, 0, 8, -6, -2, 2, -2, 4, -4, 1, 4, 4, 2, 5, -5], [-5, 0, 3, 0, -1, -3, -1, -3, 0, -3, 0, 0, 0, 0, 1, -1], [-1, 0, 7, -4, 3, 0, -3, -4, -2, -2, -2, 1, 1, 0, -3, -1], [0, 1, -4, -2, 2, 0, -1, -1, -1, 2, -1, 0, 0, -1, -2, 0], [-3, -2, 1, 2, -3, -6, 0, -3, 3, -8, 1, 1, 5, -2, 3, -3], [0, 1, 2, 0, 4, 1, -2, 0, 1, 0, -2, 0, 0, 0, -2, 0], [7, 0, -4, 3, -4, -2, 1, -1, 1, -4, 0, 3, 3, 0, 0, -2], [0, -2, -4, 7, -2, -3, 3, -1, 0, -4, 1, 2, 2, -1, 5, -3], [2, 1, -5, 4, 1, 3, 3, 2, 3, 4, 0, 2, 0, 0, 0, 0], [-1, 2, 2, -3, 1, -4, -2, -2, 2, -2, 1, 1, 3, 2, 1, 0], [-1, -1, -2, 2, -1, 0, 1, 1, 3, 4, -1, 1, 1, 1, 1, -2], [2, 0, 2, -3, 2, 1, -1, -1, -2, 3, -2, -1, 0, 2, 0, 3], [8, 1, 3, -3, 2, 0, -1, -2, 0, -1, 0, 1, 4, 2, -2, 2], [-3, 0, -1, 2, 0, 0, 2, -5, 0, -1, 0, 4, 0, 0, 0, 0], [-9, 0, -4, 6, -4, -2, 2, -3, 4, 1, 1, 1, 5, 1, 3, -3], [11, 1, -3, -5, 5, 2, -2, 1, -1, 4, 0, -3, -1, 1, -1, 3], [4, -3, -5, -2, -1, 1, -1, 2, 1, 1, 0, -1, 0, 0, 0, 0], [3, -1, -2, 4, -4, 2, 1, 1, -2, -1, 1, 0, -2, -2, 1, -2], [8, 0, 0, 0, 3, 1, 0, 0, -1, 4, -3, -4, 0, 0, 3, 0], [2, 1, 2, -3, 4, 1, -5, 1, 1, -1, -2, -2, 0, 0, -2, 0], [-3, 2, -10, 3, -2, 0, 2, -1, 5, 0, -2, 3, 3, -2, 2, -3], [-1, 0, -3, 3, -3, 0, 0, 2, 0, 2, 1, -1, -2, 0, 2, -2], [1, -1, -3, 2, -3, 0, 0, 3, 0, 4, 1, -3, -6, -2, -1, 0], [2, 0, 4, -4, 3, 1, -1, -3, -2, 1, -1, -2, 2, -2, -2, 1], [-4, 0, -2, 8, -5, -3, 3, 1, 6, 1, -1, 1, 2, 0, 6, -1], [-1, 0, 5, 4, -2, 0, 2, -1, 1, -1, 3, 1, 1, 0, 2, 1], [-2, 1, 3, -2, -4, 0, 2, 1, 0, 2, 0, 1, 0, 0, 0, 0], [-1, 0, 4, -5, 3, 4, -2, 1, 0, 4, -3, 1, -3, 0, -6, 3], [-1, -1, 2, -2, -2, 2, -1, 1, -1, 0, -1, 1, -1, -1, -6, 2], [19, -1, -5, -5, 8, 4, -2, 2, -2, 1, -1, -1, 1, -1, 0, 4], [0, 0, 3, -4, 4, 0, -4, -3, -3, -6, -1, 0, 0, 0, -4, -1], [-4, -1, -2, 0, 0, 0, 3, -5, 3, -5, -3, 4, 4, 1, 0, 0], [-10, 0, -5, 8, -5, -3, 3, -2, 6, -4, 0, 4, 5, 1, 5, -6], [2, 0, 1, -1, 2, -1, 1, 2, 2, 2, -3, -1, -1, -1, 1, 5], [12, -1, -1, 0, -3, -1, 0, 1, 1, 2, 3, -3, -2, -2, -3, 0], [-6, 0, -6, 0, -1, 1, -1, -1, -2, -2, 2, 2, 0, 3, -1, -1], [-2, 2, 10, -5, 5, 1, -4, 2, 1, 1, -6, -1, 0, 0, -6, 0], [-2, 1, 1, 2, -3, -3, 1, -2, 3, -1, 2, 1, 4, 2, 0, -2], [0, 2, 1, 2, -3, 0, 1, -2, 0, -2, 2, 4, 4, 4, 0, -2], [11, 0, 2, 3, -1, -4, -1, -1, 4, -6, -3, 5, 2, 0, 1, 2], [-4, -1, 8, -2, -1, -2, -3, -3, -2, -1, 2, 2, 0, 0, 2, 0], [2, 0, 0, 0, -1, 1, -1, 1, -2, 0, 4, 1, -2, 3, -3, 4], [-4, -1, 12, 0, -1, 2, 1, 5, 2, 3, 0, -2, 0, 0, 0, 0], [-10, 0, -6, -1, 3, -2, 2, -2, 4, -8, -2, 4, 4, -2, -1, -2], [1, 1, -6, 4, -1, 0, 1, 2, 0, -3, 4, -1, -1, -1, 1, 1], [1, 0, 1, -2, -2, 0, 0, 2, 0, 0, 6, 2, 2, 0, 4, 0], [3, 0, -1, -2, -5, -2, 0, -1, 1, 0, 3, -5, 3, 0, -3, -1], [3, 1, -6, 0, -1, -1, 1, 1, -1, 2, 0, 1, 0, 0, 0, 0], [-8, -2, 1, 4, 0, -4, 0, -2, 2, -5, -2, 4, -2, -2, 4, -2], [7, 2, -5, 4, 0, -6, -1, -3, 3, 1, -3, -5, 1, 2, 4, -2], [-4, 2, 3, 6, 2, -1, 1, -4, -1, -1, 3, 3, 0, 0, 3, 0], [2, -1, -7, 5, -2, 0, 2, 7, 3, 4, 2, -2, -2, 1, 2, -1], [12, -2, -16, -1, -7, 2, 4, 0, 0, 1, 3, 0, -2, -1, -2, -1], [1, 0, -1, 4, -5, 0, 0, 2, 0, -3, 4, -1, -1, 0, 5, 0], [-13, 2, 16, -6, 3, -1, -5, -2, 0, 0, 1, -3, -1, 1, 0, 2], [2, -2, 1, -3, 0, -1, 0, -1, 1, -4, 3, -1, 2, -4, -3, 0], [1, 1, -13, 1, -1, 0, 2, 5, 3, 8, -2, -1, -1, -1, 1, -1], [15, 0, -3, -6, 9, 0, -6, 0, 0, 1, -6, 1, -3, 0, -3, 6], [3, -1, -1, -3, -2, 3, 2, -1, -3, 4, 7, -7, 2, -2, -3, -4], [-3, -2, 0, 0, 2, -1, -1, 1, 1, -1, -3, -2, -2, -4, 1, 2], [19, 1, -4, 0, 1, 5, -2, 6, -5, 8, -3, 0, -12, 2, -1, 4], [4, 0, -13, -2, 0, 0, 0, 7, -1, 5, -1, -4, -4, 0, 0, 1], [-5, 0, -2, 0, 2, -2, 2, -1, 4, -6, -6, 5, 3, 2, 4, -3], [3, -2, 2, -3, 2, 3, -1, 4, 0, 2, 0, -2, 1, -1, -2, 2], [-7, 2, 4, -1, -2, -3, -1, -4, 0, -2, -4, 5, -1, 1, -4, -1], [1, 3, 0, 2, 0, 4, 0, 2, -2, 1, -4, 11, -3, 3, -4, 2], [-10, -4, 4, 9, -6, -1, 7, 1, 0, -3, 4, -3, 2, -2, 6, -4], [20, 0, -6, -11, 10, 4, -5, 2, -2, 5, -1, 0, -6, 0, -4, 7], [13, -2, -6, 1, 9, 0, -4, 3, 0, -1, -2, -6, 3, -1, 3, 2], [11, 1, -7, -1, -2, 4, 3, 2, -2, 7, 5, -7, -1, 1, 0, -1], [2, 0, 4, -5, 6, 1, -5, 1, 0, 1, -6, -1, 0, 0, -6, 3], [-8, 2, 10, -7, 2, 3, -3, -1, 0, 4, -1, -3, -4, 1, -5, 3], [-2, 0, 2, 1, -1, 0, 0, -4, 0, -12, 0, 4, 4, -4, 1, -4], [0, 0, 0, 3, -3, 0, 0, 0, 0, -8, -2, 3, 0, -2, 5, -6], [-2, -1, 11, -3, 4, -2, -9, -2, -2, -7, 2, -5, 0, 0, 2, 0], [-3, -3, 1, 1, -2, 0, 5, -3, 3, 0, -2, 3, 3, 3, 2, 2], [-9, 0, -3, 6, -12, 0, 6, 0, 0, -2, 6, 7, -3, 0, 0, -6], [5, -3, -14, 5, -3, -1, 3, -1, -1, -2, 5, -1, 0, 0, 5, 0], [-3, 0, -4, 10, -9, 0, 2, 2, 0, 3, 1, -1, -4, 0, 3, -4], [-1, -1, -4, 4, -5, -1, 8, 2, -1, 2, 1, 0, 0, 0, 1, 0], [-8, -2, 1, 6, -5, 2, 3, 1, -1, 1, -1, -2, -2, -2, -1, -2], [2, -2, -1, -6, 3, 0, -4, -1, -5, 2, -1, -2, -2, 2, -3, 1], [-8, -3, 11, 0, -2, 2, 3, -2, 2, 4, -1, 6, 0, 0, -1, 0], [-8, 0, -10, 4, -5, 1, -1, 1, -2, 4, 2, 0, -2, 5, 3, -3], [7, 0, 5, 0, -3, 3, -3, -1, -6, -7, 4, 2, -2, 0, -1, -2], [-1, -2, 2, -2, 7, 3, -3, -2, 0, 2, -7, 10, -5, -1, -6, 3], [-2, -3, 5, -9, 8, 0, -4, -10, -6, -9, -3, 2, 2, 3, -8, 2], [21, -2, -6, -9, 7, 4, -3, 2, -2, 5, 1, -9, -1, -2, -3, 5], 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5, 11, -6, -6, 2, -7, 6, -6, -3, 5, 14, 6, 7, -4], [0, 0, -11, 6, -5, -1, 1, 12, 2, -1, -7, -1, -11, -4, 12, -2], [17, 0, 12, -1, -2, 3, -3, 2, -6, -4, -4, -1, -5, -3, 6, -2], [2, -1, -3, 2, 1, 5, 8, 9, 5, 9, -7, 0, 0, 0, -7, 0], [-13, -4, 15, -3, 6, -5, -7, -5, 0, -7, -1, -4, 0, -2, 3, 1], [19, -1, -11, -7, 0, 12, 0, 6, -6, 17, 1, -9, -15, -1, -12, 6], [4, 10, 4, -4, 11, 1, -11, 1, 1, 2, -4, 1, 0, 0, -4, 0], [-6, -4, -5, 9, -16, -15, 5, -6, 0, -17, 12, 0, 9, -2, 16, -10], [-2, 0, -6, 3, -7, 0, 3, -1, 3, 0, 0, 2, 2, 0, 7, 0], [2, 0, 9, -4, 6, -2, 2, -5, 4, -15, -8, 5, 7, -7, 2, 4], [0, -2, -1, 4, 1, 2, 3, 2, 2, 1, 0, -1, 0, 0, 0, 0], [0, 6, 4, 3, 1, 0, 2, 3, 3, -9, 0, 0, 0, -6, -1, -1], [16, 9, -27, 4, -5, 3, 5, 10, 3, 11, 4, 1, 0, 0, 4, 0], [6, 0, 4, -11, 11, 3, -2, -2, -3, -3, -2, 1, 4, 0, -2, 4], [-31, 8, 20, 5, 6, -6, -3, -3, 3, -1, -8, 3, 9, 8, 6, 0], [-18, 0, -11, 19, -13, -6, 6, -1, 12, 4, 4, 2, 7, 2, 9, -11], [10, 2, -10, -1, 0, 2, 4, 14, 2, 14, -5, 0, 0, 0, -5, 0], [2, 4, 0, -2, 3, 3, 0, 0, -3, 8, -1, 0, 0, 8, 1, 0], [7, 4, -13, 8, -7, -9, 3, -3, 0, -7, 4, -9, 6, 2, 9, -6], [-2, -1, 4, 4, -2, 0, -1, 0, 4, -7, 0, 2, 2, 1, 2, -5], [38, -4, -33, 4, 11, 5, -1, 4, 0, 3, -1, -7, -1, -2, 4, 3], [-16, -4, 20, -6, 14, 0, -10, -6, 0, -2, -5, 0, -6, -2, -1, 5], [8, 4, -9, 3, -4, 3, 5, 4, 0, 5, -6, -1, 1, 2, -8, -1], [6, 0, 13, 6, -1, -5, 5, -2, 10, 9, -3, -2, 7, 0, 4, -5], [13, -1, -18, -11, 1, -3, -7, 1, -3, 2, -5, 1, 0, 0, -5, 0], [-7, 0, -6, 5, -1, -4, 4, 3, 8, -2, 3, 1, 1, -4, -2, 0], [5, -4, 11, -6, -4, 0, 5, 7, -3, 10, -3, -5, -5, 4, 4, 8], [-23, -3, 20, -12, 0, 6, -1, 3, -3, 1, 7, -11, 9, -3, -8, 4], [-14, -4, 27, 5, -12, 3, 8, 7, 3, 8, 9, 1, 0, 0, 9, 0], [-5, -1, -4, 8, -9, -1, 4, 0, 1, -7, 5, 5, 0, -2, 3, -8], [7, -9, -8, 10, 5, 3, 3, 9, 3, 5, 2, -4, 0, 0, 2, 0], [0, 0, -5, -8, 3, 0, 4, 4, 4, 11, -12, 0, 0, 0, -3, 0], [13, 0, -7, -18, 8, -2, -20, -1, -2, -7, -6, -6, 0, 0, -6, 0], [-3, -1, -18, -10, 5, 0, -5, -13, -7, -16, -3, 3, 3, 1, -5, 2], [10, 2, 2, -11, -1, -6, -8, 4, 0, -5, 5, -14, 10, 1, 2, 1], [16, 0, -9, -1, 6, 9, 1, 8, 0, 9, -4, 1, -1, 0, -6, 4], [-2, -7, -2, 4, -2, -2, 2, -1, 1, -9, 1, 2, 0, -7, 4, -3], [19, 8, -1, -16, 11, 13, -7, 11, 0, 17, -2, -5, -2, 4, -11, 10], [-10, -4, -3, 14, -17, -9, 11, -2, 0, -11, 11, -3, 7, -2, 14, -10], [-6, 0, -7, -4, 3, 1, -1, 0, -2, -7, -3, 3, -1, 0, 0, -4], [0, 0, 1, -4, -3, 4, 0, 1, 4, 8, -1, 7, 0, 0, -1, 0], [14, 0, 16, -7, 4, 3, -3, -5, -6, -7, 6, 4, 2, 4, -10, 6], [7, -2, -20, -17, 11, 0, -11, -1, -15, -4, -2, -7, -7, 2, -11, 4], [12, 5, -3, -15, -1, -4, -3, -2, 2, 2, 11, -2, 4, 5, -3, 1], [-12, 5, -5, 19, -23, -3, 6, -1, 3, -5, 10, 14, 2, 10, 2, -12], [-3, 4, 3, 0, -1, 3, 2, -5, -3, -1, 3, 4, 10, 8, 1, -4], [2, 2, 4, -7, -11, 1, 1, 9, 1, 8, 3, -1, 0, 0, 3, 0], [39, -2, -6, -17, 14, 10, -6, 5, -5, 4, 0, -7, 5, -2, -8, 11], [-6, 1, -3, -5, 8, 2, -3, -18, 2, -9, -10, 9, 0, 0, -10, 0], [-12, 2, 14, -1, 6, -3, -5, -3, 0, -2, -3, 3, 0, 1, 1, 1], [-2, -1, -25, 15, -4, 0, 10, 9, 13, 10, 2, 2, 2, 1, 4, -3], [-10, 1, -3, 13, -18, -8, 5, -4, 4, -5, 1, 2, 2, 1, 0, -9], [-28, 1, 16, 12, -6, -16, 0, -8, 8, -16, -4, -2, 20, 1, 10, -8], [29, 6, -22, -11, -3, 0, -4, 3, 0, 3, -2, 12, 3, 3, -9, 2], [-6, -7, 8, 0, 8, -1, -6, -10, -1, -10, -2, 0, 0, 0, -2, 0], [-6, -5, 3, 1, 6, -12, -9, -6, 6, -11, -13, 6, -6, -5, 0, 3], [-45, 0, 35, 3, -9, -4, 4, -6, 0, -4, -5, 1, -2, 0, -6, -4], [-4, 0, 5, 2, 1, -3, 3, -6, 6, 0, -8, 6, 9, 9, 7, -2], [15, 0, 2, -18, 9, 9, -9, 4, -18, 1, -2, -5, -13, 0, -7, 8], [1, -7, -7, 3, -8, 6, 4, 3, -3, -1, 2, 5, -11, -7, -6, -1], [18, -1, -46, 5, 1, 2, 3, -7, 2, -2, 1, 5, 0, 0, 1, 0], [-22, 8, -3, 12, -11, 0, 1, 2, 0, 1, 0, 17, -4, 16, 2, -2], [-11, -4, 5, 0, 3, 1, 0, -5, -1, -15, -3, 2, 10, -8, 3, 0], [-10, 0, 1, 16, -14, -9, 6, -7, 9, 0, 4, -7, 14, 0, 8, -12], [-1, 4, 11, 0, 7, 1, -11, -2, 1, -3, 4, -1, 0, 0, 4, 0], [-16, 0, -10, 16, -14, -2, 2, -4, 4, -17, 2, 8, 6, -3, 12, -18], [13, -2, 0, -8, 4, 12, 0, 13, 0, 11, -1, 4, 1, -1, -9, 6], [-27, 6, 21, 2, 8, -10, -5, -5, 5, -7, -7, 1, 15, 6, 8, 0], [-29, -1, 21, -13, 20, -6, -12, -3, 3, 4, -8, -11, -5, -1, 2, 9], [18, 0, -36, 6, -4, 10, 14, 14, 10, 18, -4, 4, 0, 0, -4, 0], [4, -1, 10, 14, -14, 0, -3, 5, -3, -8, 17, -4, -4, 1, 14, 0], [1, 0, 2, 4, 0, -4, 4, 3, 8, 10, -2, 1, 1, 8, 2, -2], [-1, 0, 9, 15, -10, -5, 5, -5, 10, -8, 4, 7, 10, 1, 6, -5], [-5, -2, 2, -11, 1, 4, -6, 4, -4, -1, 4, 1, -8, -4, -16, 12], [3, 2, 26, -3, -1, 0, -6, 3, -3, -11, 0, -3, -3, -2, 1, -3], [17, -1, 12, -14, 16, -2, -6, -6, 2, -6, -8, -2, 12, -2, -4, 12], [7, 0, 8, 12, 1, -6, 2, -10, 6, -15, -11, 5, 20, 0, 15, -4], [-21, -1, 2, 10, -9, -5, 5, -7, 5, -9, 4, 0, 14, -2, 6, -10], [-21, -9, 0, 14, -12, -12, 2, -6, 6, -11, -4, -3, -5, -9, 4, -8], [4, 0, 5, -9, 9, 0, 0, -1, 0, -9, -3, -1, 1, -11, -6, 10], [-6, -5, 4, 20, -16, -15, 9, -13, 15, -34, 5, 11, 26, -10, 13, -18], [24, 0, 32, -24, 20, 4, -4, -12, -8, -2, -11, 1, 8, 4, -9, 13], [6, 3, 5, -9, 10, 0, 0, 4, -8, 16, -1, -6, -6, -3, -10, 8], [14, 0, -2, -9, 19, 0, -14, -2, 0, 0, -14, 4, -2, 0, -9, 7], [-6, 0, 10, -1, 16, 0, -10, 0, 0, 2, -19, 2, -6, 0, -4, 10], [-12, -2, 16, 6, 7, 1, 6, -2, 1, 5, -7, 7, 0, 0, -7, 0], [23, 1, -7, -11, -2, -8, -8, -4, 4, 0, -1, -7, 1, 1, -10, 4], [-4, 1, -11, 7, -8, 0, -7, -2, 6, -16, 1, 4, 4, -1, 8, -13], [-3, 4, 9, -12, 1, 7, -3, 3, 0, 9, -4, 12, -4, 2, -13, 5], [41, -8, -37, -7, -6, -5, -3, 1, 0, -9, 1, -7, 6, -4, -3, -1], [-9, 4, 32, -7, 11, 0, -10, 4, 0, 5, -8, 1, 0, 0, -8, 0], [6, -1, -29, -7, 2, 0, -6, 3, -9, 1, 2, -6, -6, 1, -2, 3], [-8, -3, 4, -2, -2, -2, 5, -1, 1, -5, 13, -10, 12, -3, 10, -6], [11, 3, -7, -7, 1, 0, -3, 12, 0, 12, -5, 0, 0, 0, -5, 0], [26, -3, 6, -22, 23, 6, -9, 3, -6, 15, -10, -18, -6, -6, -8, 18], [-7, 3, -6, 14, -3, -7, 7, -1, 7, 10, -4, -5, 2, 6, 18, -14], [-6, 2, -11, 27, -22, -3, 8, 3, 3, -2, 3, 9, -6, 4, 13, -16], [-7, -1, 1, 13, -7, 0, 10, -2, 12, 11, 1, 7, 7, 1, 7, -2], [12, 2, -18, -16, 4, -2, -12, -6, -2, -5, -8, 1, 0, 0, -8, 0], [-17, 0, -6, -2, 3, -1, 1, -10, 2, -22, -1, 15, 11, 7, -2, -4], [10, 4, 7, -14, 20, -6, -20, -3, 0, -4, -11, 3, 3, 2, -5, 7], [-7, 0, 20, 1, -8, 0, 5, -10, 4, -7, -3, 7, 7, 0, 8, 1], [-34, -8, 10, 21, -13, -18, 5, -9, 9, -24, -2, 0, 14, -8, 15, -14], [35, -1, -5, -21, 20, 20, -8, 10, -10, 15, -5, -1, -11, -1, -16, 18], [41, -9, -16, -10, 1, 0, -2, 0, 0, 2, 6, -27, 5, -9, -3, 2], [9, -2, 5, -20, 4, 0, -12, 3, -15, 2, -5, -9, -9, 2, -4, 3], [19, -6, -39, 17, -21, 6, 22, 2, 0, 3, 14, 9, -4, -3, 9, -8], [-3, 0, 5, 10, -9, 0, 2, -3, 3, -12, 7, 3, 3, 0, 9, -1], [21, -4, 2, -28, 29, 6, -6, 4, -6, 13, -7, -17, -8, -8, -5, 12], [-8, 1, 18, 3, -2, 1, 5, 7, 1, -1, 0, -8, 0, 0, 0, 0], [14, 0, 2, -30, 19, 11, -11, 1, -22, 11, -3, -11, -12, 0, -16, 18], [14, -6, -18, -11, -1, 3, -4, 6, 3, 1, -6, -5, 0, 0, -6, 0], [-13, 0, -3, 8, -2, -6, 6, -4, 12, 6, -4, 4, 10, 8, 6, 4], [6, 0, -5, 8, -8, 0, 0, 11, 0, -16, 1, 8, -11, 0, 7, 1], [21, 12, -12, -4, 14, 6, -6, 3, 0, 12, -14, 6, -3, 6, -12, 6], [6, -6, 2, -7, 10, -5, -11, -3, 0, -8, -11, -2, 2, -3, -7, 3], [5, -3, -9, 10, -9, 8, 12, 13, 8, 15, 7, 2, 0, 0, 7, 0], [-15, 6, 21, -16, 2, 8, -4, 4, -4, 4, 4, 1, 11, 6, -14, 8], [-1, 4, 9, -9, 2, -15, -13, -5, 0, -13, -9, 1, 10, 2, -5, -1], [24, 0, 11, -20, 13, 7, -7, 6, -14, -10, 0, -3, -13, -9, -13, 8], [-5, 6, 4, -16, 19, -4, -7, 3, 4, 21, -10, -6, -6, 12, -4, 14], [-28, 1, 18, 4, -4, 2, 1, 1, -1, 1, -3, -6, 8, 1, -4, 0], [12, -1, -2, -6, -6, 3, -1, 3, -3, 19, 11, -18, -6, -2, -13, 2], [11, 2, -18, -18, 1, 0, -12, -8, 0, -2, -7, 6, 0, 0, -7, 0], [-25, 2, 20, -15, 5, -8, -7, -4, 4, 3, 5, -19, 9, 2, -1, 3], [-7, -1, 32, 8, -1, 0, 5, -10, 4, -11, 4, 7, 7, 1, 1, 1], [34, -1, 3, 1, 10, 0, -3, 0, 0, -5, -14, 3, 0, -2, 8, 6], [-32, 0, -30, 17, -15, -2, 2, 0, 4, -8, 2, 6, 2, 2, 13, -10], [19, 8, -24, 12, -7, -9, 5, 0, 0, -5, 12, 6, 9, 4, 19, -7], [13, 0, -3, 2, -9, 0, 1, 20, -6, 6, 8, -13, -13, 0, 9, 7], [-3, -12, 2, -11, -5, 6, 0, -1, 0, 0, -1, 14, -7, -6, -12, 3], [-8, -3, 25, 11, -4, 0, 1, -2, 14, -6, -3, 8, 8, 3, 4, -13], [6, 7, -5, 23, -9, -18, 3, -9, 9, -19, -8, 14, 20, 7, 15, -12], [-13, 0, -1, 2, -12, -6, 3, -3, 3, 9, 7, -21, -3, 0, 0, -6], [-44, 5, 19, 1, -20, -4, 3, -2, 2, -4, 7, 10, 4, 5, -10, -5], [-26, -6, 10, 0, -12, -12, 0, -16, 0, -15, 10, -1, -4, -3, 10, -6], [5, 2, 2, 7, 5, -10, 2, -5, 5, -9, 2, -11, 23, 2, 19, -7], [-21, 6, 30, -5, 8, -5, -9, 0, 0, -2, -6, -6, 5, 3, -2, 2], [-8, 2, -14, 9, -5, 0, 7, 2, 18, -5, -9, 8, 8, -2, 5, -11], [-15, 0, -7, 15, -8, -7, 7, -1, 14, -7, -9, 6, 8, -2, 17, -5], [-6, -8, 3, 10, -11, -10, 7, -2, -10, -7, 14, -5, 0, 0, 14, 0], [-22, 0, -19, 35, -22, -13, 13, 10, 26, -9, -4, 8, 3, -6, 26, -20], [-31, -4, 7, 21, -17, -15, 9, -16, 15, -26, 5, 2, 32, -8, 13, -18], [-40, 9, 4, -2, 4, 5, -3, -1, -5, 6, -5, 11, 2, 18, -1, 6], [-5, -6, 28, 9, 0, 0, 5, 0, 10, 6, -1, 5, 5, 6, 0, -5], [4, 0, 7, 2, 0, -2, 2, -1, 4, 11, -2, -7, 3, -5, 2, -3], [-44, -6, -4, 24, -21, -9, 12, -8, 9, -24, 9, 4, 16, -12, 15, -24], [17, 0, 9, -7, 3, 4, -4, 4, -8, 8, 7, -8, -8, -4, -10, -1], [18, 0, 12, -11, 11, 0, 0, 6, 0, 10, -2, -6, -6, -2, -9, 6], [5, -5, 24, -28, 16, 0, -17, -16, -26, -13, -2, -5, -5, 5, -16, 9], [12, 2, -6, -3, -18, -8, 4, -4, 4, -8, 15, -4, 16, 2, -2, -8], [-33, -4, 1, 8, -14, -2, 9, -1, 1, 3, 11, -5, -11, -4, 6, -10], [-5, 2, 16, 6, -14, 9, 10, 16, 9, 12, 10, -4, 0, 0, 10, 0], [4, 5, 12, -2, 10, 0, 3, 8, 0, -5, -2, -4, -4, -5, -10, 3], [1, 6, 10, 7, 2, 0, -2, 5, 3, 6, 4, -1, -1, -6, -2, -5], [22, -6, -17, 4, 7, -3, -3, 2, 0, -6, 4, -7, 5, -3, 11, 0], [14, 0, 15, -13, 14, -1, 1, 0, 2, 4, 3, -2, 1, -1, -17, 8], [26, 6, -39, 5, 6, 1, -6, 7, 1, 2, 5, -5, 0, 0, 5, 0], [-28, 3, -12, 14, -18, -1, 4, 8, 1, -2, 8, 16, -16, 6, 0, -8], [-14, -5, 4, -1, 0, -11, 4, -8, 11, -18, 5, 0, 16, -10, 3, -8], [19, 6, -32, -2, -12, -1, 10, 16, -1, 15, 0, -1, 0, 0, 0, 0], [-2, 0, -10, -2, 3, -1, 1, 9, 2, 7, 5, -6, -8, -6, -8, -5], [12, 0, 16, -3, 2, 1, -1, -5, -2, -2, -3, 0, 4, -1, 1, 8], [-6, 0, -40, 1, 5, 0, 1, -6, 6, 7, -5, 6, 6, 0, -5, -5], [-6, -10, 20, 6, 9, -5, -8, 0, -5, -9, 5, -9, 0, 0, 5, 0], [19, -2, -9, -10, 10, 16, -2, 8, 0, 15, 7, -10, -8, -1, -1, 9], [-11, -8, 4, 1, -16, -4, 11, -2, 2, -9, 23, -25, 23, -8, 10, -13], [-13, -6, -1, 8, -10, -4, 10, -17, -4, -10, 8, 7, 0, 0, 8, 0], [-56, -3, 19, 19, -7, -12, 9, -6, 6, -6, 5, -18, 12, -3, 23, -15], [-45, -2, 2, 15, -5, -6, 4, -6, 6, -7, -6, -3, 12, -4, 14, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5292_bm_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5292_2_bm_a();".
// 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_5292_2_bm_a(:prec:=16)
    chi := MakeCharacter_5292_bm();
    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_5292_2_bm_a();".
// 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_5292_2_bm_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5292_bm();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-18, 261, -324, -135, 171, 12, -24, 0, 1]>],Snew);
    return Vf;
end function;