
// Make newform 4410.2.d.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4410_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4410_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_4410_2_d_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_4410_2_d_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 := [390625, -468750, 328125, -168750, 70625, -21000, 4650, -420, -6, -84, 186, -168, 113, -54, 21, -6, 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], [-198906250, 262743750, -114151250, 95015625, -32915700, 8780040, -870576, 415452, 259320, -27834, -101484, 103248, -49670, 25998, -7678, 189], [-346640625, 510212500, -263960625, 103996375, -48097950, 6932940, 199494, 88968, 790644, 488220, -129186, 157080, -86901, 37592, -5487, 1553], [-491250000, -69500000, 33712500, -64549375, 14222250, -15270150, -2041800, -12822, -736638, -29238, -120366, -19254, 46872, -26998, 4278, -3503], [-3281250, -606250, -1177500, -382375, 164850, -79170, -7452, -13746, -5610, -5118, -918, -354, -90, -104, -6, -47], [81562500, -1492703125, 1084443750, -557861875, 266151750, -79847700, 11659590, -6240882, 527442, 1249302, -162846, -463584, 336102, -195713, 58968, -28283], [31171875, -22375000, 16706250, -7088125, 3376500, -1110600, 246270, -28224, 28464, -7506, 12828, -9648, 5859, -2816, 1326, -221], [4389921875, -7280656250, 3728618750, -2250321875, 552840000, -315330600, 81074310, -10144404, -7849236, -3946746, 1295088, -2564568, 1097059, -816586, 152446, -72631], [-7956484375, 6112843750, -3710312500, 1666289375, -578292000, 74294100, -9287550, -5331624, -1678296, 1836654, -2848872, 2136732, -1227551, 538634, -180524, 28999], [3061796875, -2635218750, 1557531250, -737662500, 287925000, -65558250, 10333650, -43854, -1282116, -369666, 1301988, -982878, 546479, -238236, 79646, -22146], [-1730156250, 1293390625, -926287500, 435921250, -165279750, 35428200, -7484130, -206466, 35346, 628476, -749898, 483708, -303924, 129881, -45366, 10646], [-8881406250, 10084203125, -4732575000, 2373254375, -869065500, 222951450, -31295250, -993864, 2386644, 3230094, -3388392, 2476302, -1660236, 730399, -239664, 63739], [-22617265625, 19754156250, -12008093750, 5620186875, -1823016000, 612694800, -46368690, -9342132, 2114052, 8198442, -7941696, 8035536, -3465913, 1702962, -489382, 134847], [17355937500, -16017953125, 8548481250, -3833020000, 1225090500, -328546950, 23859630, -6017556, 1982616, -6953964, 6806232, -5451762, 2832846, -1192979, 333594, -82424], [-12730312500, 13369062500, -7391446875, 4053550625, -1325565750, 347872800, -45729360, 5982396, 6684504, 5948364, -4615362, 4791012, -2573376, 1157764, -393129, 113839], [63741953125, -51938718750, 30395350000, -13935140625, 4897207500, -1103936400, 199376490, 39547164, 3731676, -17319114, 22180692, -18306312, 10377581, -4250574, 1400564, -298929]];
    Rf_basisdens := [1, 113625000, 340875000, 568125000, 3375000, 1704375000, 11250000, 3408750000, 1136250000, 852187500, 340875000, 1704375000, 3408750000, 1704375000, 1704375000, 3408750000];
    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_4410_d();"
function MakeCharacter_4410_d()
    N := 4410;
    order := 2;
    char_gens := [3431, 2647, 1081];
    v := [1, 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_4410_d_Hecke();"
function MakeCharacter_4410_d_Hecke(Kf)
    N := 4410;
    order := 2;
    char_gens := [3431, 2647, 1081];
    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], [-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 := [[-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, 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, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, -1, -1, 0, 0, 0, -1, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, -1, 1, 0, 0, 1, 1, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0], [0, 0, -2, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 2, 0], [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, -1, 0, 1, 1, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0], [0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 1, 0, -1, 0, 1], [0, 0, 0, 0, 0, 1, 0, 0, 0, -3, -1, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, -2, -1, 0, -1], [0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, -2, 1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -3, 0, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0, -3], [0, 0, 2, 0, 0, -2, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 1, -2, 1, 0, 0], [0, 0, 1, 0, 0, 2, 0, 0, 0, -6, -2, 0, 0, 0, -1, 0], [0, 0, 1, 0, 1, -1, -2, 0, 0, 0, -1, 0, 0, 0, 1, 0], [0, 0, -1, 0, 0, -2, 0, 0, 0, 0, -2, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0], [0, 0, -1, 0, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0], [6, 0, 0, -1, 0, 0, 0, -2, 1, 0, 0, 2, 0, 2, 0, 1], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, -2, 0, 0, 0, -4, 2, 0, 0, 0, 0, 0, 0, 2], [0, 0, 0, 4, 0, 0, 0, 0, 1, 0, 0, 2, 0, -2, 0, 0], [0, 0, -2, 0, -1, -1, 1, 0, 0, 0, -1, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 4, 2, 0, 0], [0, 0, 2, 0, 0, 2, 0, 0, 0, -2, -2, 0, 0, 0, -2, 0], [0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [-10, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, -3, -4, -3, 0, -2], [0, 0, -3, 0, -7, -1, 1, 0, 0, 0, -1, 0, 0, 0, -3, 0], [0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0, -1], [0, 1, 1, 0, 0, 5, 0, 0, 0, -3, -5, 0, 0, 0, -1, 0], [0, 0, 1, 0, 0, 2, 0, 0, 0, 3, -2, 0, 0, 0, -1, 0], [0, 0, 0, 4, 0, 0, 0, 0, -3, 0, 0, 1, 0, -1, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 3], [-6, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [-3, 0, 0, -2, 0, 0, 0, -2, 2, 0, 0, 1, 2, 1, 0, 2], [0, 0, -3, 0, -3, 0, 4, 0, 0, 0, 0, 0, 0, 0, -3, 0], [0, -2, 1, 0, 0, 2, 0, 0, 0, -6, -2, 0, 0, 0, -1, 0], [0, 0, 2, 0, 0, 2, 0, 0, 0, -4, -2, 0, 0, 0, -2, 0], [-4, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 1, -2, 1, 0, -1], [0, 0, 0, 9, 0, 0, 0, 0, -1, 0, 0, -1, 0, 1, 0, -3], [0, 1, 2, 0, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0, -2, 0], [0, 0, -1, 0, 8, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, 2, 0, 0, 0, -2, 0, 0, 0, -6, 2, 0, 0, 0, 0, 0], [14, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 1], [0, 0, -4, 0, -6, -1, 2, 0, 0, 0, -1, 0, 0, 0, -4, 0], [0, 0, 4, 0, 0, 1, 0, 0, 0, -2, -1, 0, 0, 0, -4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -3, 0, 3, 0, -1], [0, 0, 0, -10, 0, 0, 0, 0, -1, 0, 0, -3, 0, 3, 0, 0], [0, 0, -3, 0, -2, 1, -2, 0, 0, 0, 1, 0, 0, 0, -3, 0], [0, 1, 5, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, -5, 0], [0, 0, 3, 0, 10, 1, -2, 0, 0, 0, 1, 0, 0, 0, 3, 0], [0, 0, 4, 0, 0, -4, -2, 0, 0, 0, -4, 0, 0, 0, 4, 0], [0, 0, 1, 0, 8, 4, 2, 0, 0, 0, 4, 0, 0, 0, 1, 0], [10, 0, 0, -4, 0, 0, 0, -2, 4, 0, 0, 1, 6, 1, 0, 4], [-4, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, -3, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 5], [10, 0, 0, 2, 0, 0, 0, -2, -2, 0, 0, -2, -6, -2, 0, -2], [0, 0, -4, 0, 0, 2, 0, 0, 0, -4, -2, 0, 0, 0, 4, 0], [0, 2, 0, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 8, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, -6], [0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 3, 0, -3, 0, 1], [7, 0, 0, -1, 0, 0, 0, -4, 1, 0, 0, 2, -2, 2, 0, 1], [0, 0, -3, 0, 0, 4, 0, 0, 0, 3, -4, 0, 0, 0, 3, 0], [0, 0, 0, 12, 0, 0, 0, 0, 2, 0, 0, 2, 0, -2, 0, 0], [0, 0, -5, 0, -11, 3, 2, 0, 0, 0, 3, 0, 0, 0, -5, 0], [0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3], [0, -2, 1, 0, 0, 4, 0, 0, 0, -12, -4, 0, 0, 0, -1, 0], [0, 0, -3, 0, -5, -3, 3, 0, 0, 0, -3, 0, 0, 0, -3, 0], [-14, 0, 0, 1, 0, 0, 0, 2, -1, 0, 0, -1, 0, -1, 0, -1], [0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6], [0, 0, -2, 0, -10, -2, -4, 0, 0, 0, -2, 0, 0, 0, -2, 0], [0, -2, 2, 0, 0, -1, 0, 0, 0, -6, 1, 0, 0, 0, -2, 0], [-8, 0, 0, 1, 0, 0, 0, -2, -1, 0, 0, -2, -4, -2, 0, -1], [0, 0, 0, 1, 0, 0, 0, 0, 7, 0, 0, 1, 0, -1, 0, 6], [0, 0, 0, 17, 0, 0, 0, 0, -3, 0, 0, -2, 0, 2, 0, -3], [0, 0, -2, 0, 4, 4, 4, 0, 0, 0, 4, 0, 0, 0, -2, 0], [0, 0, 0, -4, 0, 0, 0, 0, 4, 0, 0, -2, 0, 2, 0, -1], [0, 2, 4, 0, 0, 2, 0, 0, 0, -6, -2, 0, 0, 0, -4, 0], [0, 0, 5, 0, 2, -1, -4, 0, 0, 0, -1, 0, 0, 0, 5, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, -9, 0, 0, 0, 0, 4, 0, 0, 5, 0, -5, 0, 6], [12, 0, 0, -2, 0, 0, 0, -2, 2, 0, 0, 4, 2, 4, 0, 2], [0, -4, -1, 0, 0, 3, 0, 0, 0, 0, -3, 0, 0, 0, 1, 0], [0, 0, -4, 0, -8, -3, 2, 0, 0, 0, -3, 0, 0, 0, -4, 0], [0, 0, -6, 0, -1, 0, 3, 0, 0, 0, 0, 0, 0, 0, -6, 0], [0, 0, -5, 0, 0, 5, -4, 0, 0, 0, 5, 0, 0, 0, -5, 0], [-8, 0, 0, 1, 0, 0, 0, -4, -1, 0, 0, -7, -6, -7, 0, -1], [0, 0, 0, 9, 0, 0, 0, 0, -1, 0, 0, -3, 0, 3, 0, -5], [7, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 5, 2, 5, 0, 1], [0, -2, 1, 0, 0, 2, 0, 0, 0, 6, -2, 0, 0, 0, -1, 0], [0, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0, 6], [10, 0, 0, -3, 0, 0, 0, -4, 3, 0, 0, 5, 2, 5, 0, 3], [0, 0, -1, 0, -8, 2, 0, 0, 0, 0, 2, 0, 0, 0, -1, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 1, 0], [0, -4, 5, 0, 0, 3, 0, 0, 0, 0, -3, 0, 0, 0, -5, 0], [0, 0, 0, 10, 0, 0, 0, 0, 4, 0, 0, -2, 0, 2, 0, 0], [0, 0, -1, 0, 0, -4, 0, 0, 0, 14, 4, 0, 0, 0, 1, 0], [0, 0, 0, 0, -12, -3, -5, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0, 0, -2, 0, 0, -2, 0, 2, 0, 1], [18, 0, 0, 3, 0, 0, 0, 4, -3, 0, 0, -1, -2, -1, 0, -3], [0, 2, 2, 0, 0, -1, 0, 0, 0, -11, 1, 0, 0, 0, -2, 0], [-18, 0, 0, 4, 0, 0, 0, 4, -4, 0, 0, 1, -4, 1, 0, -4], [0, 0, 0, 5, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, -9], [0, 0, 0, 0, 2, 2, -2, 0, 0, 0, 2, 0, 0, 0, 0, 0], [0, 4, -4, 0, 0, 3, 0, 0, 0, -9, -3, 0, 0, 0, 4, 0], [-8, 0, 0, 2, 0, 0, 0, 9, -2, 0, 0, 5, 5, 5, 0, -2], [0, 0, 0, 4, 0, 0, 0, 0, -7, 0, 0, 1, 0, -1, 0, -3], [0, 2, 1, 0, 0, -5, 0, 0, 0, 6, 5, 0, 0, 0, -1, 0], [0, 0, 0, -15, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1], [0, -3, -3, 0, 0, -2, 0, 0, 0, -15, 2, 0, 0, 0, 3, 0], [-6, 0, 0, -3, 0, 0, 0, 2, 3, 0, 0, 4, 8, 4, 0, 3], [0, 4, -2, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0], [0, 0, 0, -14, 0, 0, 0, 0, -2, 0, 0, 1, 0, -1, 0, 0], [14, 0, 0, 4, 0, 0, 0, 6, -4, 0, 0, 0, -2, 0, 0, -4], [0, 0, 6, 0, 6, -4, -6, 0, 0, 0, -4, 0, 0, 0, 6, 0], [0, 0, -4, 0, 0, 7, -3, 0, 0, 0, 7, 0, 0, 0, -4, 0], [0, 0, -4, 0, -3, -8, 3, 0, 0, 0, -8, 0, 0, 0, -4, 0], [0, 0, 0, 4, 0, 0, 0, 1, -4, 0, 0, -2, -7, -2, 0, -4], [-7, 0, 0, 3, 0, 0, 0, 6, -3, 0, 0, -2, 0, -2, 0, -3], [2, 0, 0, -4, 0, 0, 0, -6, 4, 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12, -6, 0, 0, 3, 0, 3, 0, -6], [0, 4, 2, 0, 0, 3, 0, 0, 0, 15, -3, 0, 0, 0, -2, 0], [0, -8, -2, 0, 0, -2, 0, 0, 0, -4, 2, 0, 0, 0, 2, 0], [0, 0, 0, 0, -2, -14, 4, 0, 0, 0, -14, 0, 0, 0, 0, 0], [0, 0, 0, 19, 0, 0, 0, 0, -9, 0, 0, -5, 0, 5, 0, -3], [0, 0, 0, -17, 0, 0, 0, 0, 9, 0, 0, -4, 0, 4, 0, 7], [28, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, -3, -7, -3, 0, 0], [0, 0, 2, 0, 0, -1, 0, 0, 0, -14, 1, 0, 0, 0, -2, 0], [0, 0, -3, 0, 3, 15, -3, 0, 0, 0, 15, 0, 0, 0, -3, 0], [0, 0, 0, -14, 0, 0, 0, 0, 4, 0, 0, 4, 0, -4, 0, 6], [0, -4, 7, 0, 0, 14, 0, 0, 0, -18, -14, 0, 0, 0, -7, 0], [-2, 0, 0, -8, 0, 0, 0, -6, 8, 0, 0, 2, 10, 2, 0, 8], [0, 0, -16, 0, -10, 2, 5, 0, 0, 0, 2, 0, 0, 0, -16, 0], [0, 0, 0, 9, 0, 0, 0, 0, 2, 0, 0, 6, 0, -6, 0, 1], [0, 0, 0, 0, -20, -5, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0], [0, 4, 2, 0, 0, -10, 0, 0, 0, 0, 10, 0, 0, 0, -2, 0], [0, 0, 2, 0, 0, 0, 0, 0, 0, -24, 0, 0, 0, 0, -2, 0], [0, 0, 3, 0, 3, -1, -9, 0, 0, 0, -1, 0, 0, 0, 3, 0], [0, 0, -1, 0, 2, 3, 6, 0, 0, 0, 3, 0, 0, 0, -1, 0], [0, 0, -6, 0, 6, -6, 4, 0, 0, 0, -6, 0, 0, 0, -6, 0], [0, 0, 0, -4, 0, 0, 0, 1, 4, 0, 0, 5, 9, 5, 0, 4], [4, 0, 0, 6, 0, 0, 0, 8, -6, 0, 0, -7, -4, -7, 0, -6], [-12, 0, 0, 3, 0, 0, 0, -2, -3, 0, 0, 8, -8, 8, 0, -3], [22, 0, 0, 7, 0, 0, 0, 8, -7, 0, 0, -1, -6, -1, 0, -7], [0, 0, 0, 14, 0, 0, 0, 0, -6, 0, 0, 4, 0, -4, 0, 12], [-8, 0, 0, -1, 0, 0, 0, -4, 1, 0, 0, -6, -2, -6, 0, 1], [0, 2, 6, 0, 0, 12, 0, 0, 0, 10, -12, 0, 0, 0, -6, 0], [0, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 10, 0, 0, 0, 0, -12, 0, 0, 2, 0, -2, 0, -1], [0, -5, 3, 0, 0, -5, 0, 0, 0, -9, 5, 0, 0, 0, -3, 0], [0, 0, 0, 18, 0, 0, 0, 0, -8, 0, 0, 8, 0, -8, 0, 0], [0, 0, 0, -9, 0, 0, 0, 0, -6, 0, 0, -4, 0, 4, 0, -6], [0, 0, 0, 26, 0, 0, 0, 0, 0, 0, 0, 4, 0, -4, 0, 12], [-3, 0, 0, -4, 0, 0, 0, -12, 4, 0, 0, 3, -4, 3, 0, 4], [0, -7, 3, 0, 0, 3, 0, 0, 0, -1, -3, 0, 0, 0, -3, 0], [0, 0, 5, 0, 0, -4, -13, 0, 0, 0, -4, 0, 0, 0, 5, 0], [0, 0, 0, 5, 0, 0, 0, 0, -3, 0, 0, -3, 0, 3, 0, -9], [0, 0, -1, 0, -6, -14, 0, 0, 0, 0, -14, 0, 0, 0, -1, 0], [0, 0, 0, 6, 0, 0, 0, 0, -5, 0, 0, -1, 0, 1, 0, -3], [0, 8, 4, 0, 0, 4, 0, 0, 0, 4, -4, 0, 0, 0, -4, 0], [0, 0, 0, -4, 0, 0, 0, 0, 4, 0, 0, 10, 0, -10, 0, 10], [0, 0, 2, 0, -18, -5, -4, 0, 0, 0, -5, 0, 0, 0, 2, 0], [0, 0, 0, -10, 0, 0, 0, 0, 3, 0, 0, 1, 0, -1, 0, -12], [42, 0, 0, 4, 0, 0, 0, 10, -4, 0, 0, 2, 2, 2, 0, -4], [0, 7, -12, 0, 0, -3, 0, 0, 0, 19, 3, 0, 0, 0, 12, 0], [-27, 0, 0, 4, 0, 0, 0, -2, -4, 0, 0, -5, -10, -5, 0, -4], [0, 0, 0, 0, -10, 2, -8, 0, 0, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 6, 0, 0], [16, 0, 0, -4, 0, 0, 0, -14, 4, 0, 0, 1, -6, 1, 0, 4], [-12, 0, 0, -7, 0, 0, 0, -2, 7, 0, 0, 1, 12, 1, 0, 7], [16, 0, 0, 6, 0, 0, 0, 10, -6, 0, 0, 1, -2, 1, 0, -6], [0, -2, 8, 0, 0, -4, 0, 0, 0, -16, 4, 0, 0, 0, -8, 0], [0, 2, -4, 0, 0, 6, 0, 0, 0, -24, -6, 0, 0, 0, 4, 0], [-37, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1], [0, -4, 6, 0, 0, 4, 0, 0, 0, -12, -4, 0, 0, 0, -6, 0], [-2, 0, 0, 3, 0, 0, 0, 14, -3, 0, 0, 11, 8, 11, 0, -3], [0, 6, -1, 0, 0, -15, 0, 0, 0, 12, 15, 0, 0, 0, 1, 0], [0, 0, -8, 0, 0, 4, 0, 0, 0, -5, -4, 0, 0, 0, 8, 0], [0, 0, 1, 0, 5, 4, -1, 0, 0, 0, 4, 0, 0, 0, 1, 0], [0, 0, 0, -3, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, -10], [-14, 0, 0, -9, 0, 0, 0, -6, 9, 0, 0, 5, 12, 5, 0, 9], [0, 0, -2, 0, 0, 13, -2, 0, 0, 0, 13, 0, 0, 0, -2, 0], [10, 0, 0, 2, 0, 0, 0, 14, -2, 0, 0, 11, 10, 11, 0, -2], [28, 0, 0, 2, 0, 0, 0, -2, -2, 0, 0, 1, -6, 1, 0, -2], [0, 0, 0, -3, 0, 0, 0, 0, 14, 0, 0, 5, 0, -5, 0, 0], [0, -4, 4, 0, 0, -8, 0, 0, 0, -10, 8, 0, 0, 0, -4, 0], [0, 0, 0, -2, 0, 0, 0, 0, -14, 0, 0, -4, 0, 4, 0, 0], [0, 0, 2, 0, 24, 8, 2, 0, 0, 0, 8, 0, 0, 0, 2, 0], [0, 10, 11, 0, 0, -3, 0, 0, 0, 9, 3, 0, 0, 0, -11, 0], [0, -2, 8, 0, 0, -1, 0, 0, 0, -31, 1, 0, 0, 0, -8, 0], [0, 0, 0, -11, 0, 0, 0, 0, -5, 0, 0, -4, 0, 4, 0, -11], [0, 0, 0, 6, 0, 0, 0, 0, 8, 0, 0, -4, 0, 4, 0, 6], [0, 0, 18, 0, 16, -5, -5, 0, 0, 0, -5, 0, 0, 0, 18, 0], [30, 0, 0, -1, 0, 0, 0, 10, 1, 0, 0, 13, 12, 13, 0, 1], [0, 8, -1, 0, 0, 3, 0, 0, 0, 0, -3, 0, 0, 0, 1, 0], [0, 0, 3, 0, -8, -7, -12, 0, 0, 0, -7, 0, 0, 0, 3, 0], [0, 0, -2, 0, -5, -11, -8, 0, 0, 0, -11, 0, 0, 0, -2, 0], [-50, 0, 0, 4, 0, 0, 0, 0, -4, 0, 0, -10, -8, -10, 0, -4], [0, 0, 0, -2, 0, 0, 0, -10, 2, 0, 0, -8, -6, -8, 0, 2], [0, 0, 0, -1, 0, 0, 0, 0, 7, 0, 0, 1, 0, -1, 0, -2], [0, 0, 11, 0, 9, -5, -10, 0, 0, 0, -5, 0, 0, 0, 11, 0], [-12, 0, 0, 4, 0, 0, 0, 8, -4, 0, 0, -2, 0, -2, 0, -4], [0, 0, 0, -12, 0, 0, 0, 0, 10, 0, 0, 2, 0, -2, 0, 9], [0, 0, -9, 0, 4, 13, -8, 0, 0, 0, 13, 0, 0, 0, -9, 0], [0, 0, 0, 3, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 3], [0, 4, 3, 0, 0, -8, 0, 0, 0, 3, 8, 0, 0, 0, -3, 0], [0, 0, 0, -14, 0, 0, 0, 0, -10, 0, 0, -4, 0, 4, 0, 7], [-4, 0, 0, 3, 0, 0, 0, -3, -3, 0, 0, -6, -9, -6, 0, -3], [0, 0, 0, -1, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 0, -6], [0, -2, 7, 0, 0, -14, 0, 0, 0, 31, 14, 0, 0, 0, -7, 0], [12, 0, 0, 4, 0, 0, 0, -4, -4, 0, 0, -2, -12, -2, 0, -4], [0, 0, 0, -4, 0, 0, 0, 0, 4, 0, 0, -9, 0, 9, 0, -12], [0, -6, -2, 0, 0, -8, 0, 0, 0, 18, 8, 0, 0, 0, 2, 0], [0, 0, 0, -13, 0, 0, 0, 0, -8, 0, 0, 4, 0, -4, 0, -4], [0, 0, 9, 0, 7, 1, -9, 0, 0, 0, 1, 0, 0, 0, 9, 0], [0, 0, 0, 9, 0, 0, 0, 0, 1, 0, 0, 5, 0, -5, 0, 9], [0, 0, 1, 0, -10, -14, 2, 0, 0, 0, -14, 0, 0, 0, 1, 0], [0, 0, 1, 0, -5, -13, -2, 0, 0, 0, -13, 0, 0, 0, 1, 0], [0, 0, -2, 0, 4, 0, 8, 0, 0, 0, 0, 0, 0, 0, -2, 0], [0, 0, 8, 0, 24, 6, 2, 0, 0, 0, 6, 0, 0, 0, 8, 0], [2, 0, 0, -9, 0, 0, 0, -18, 9, 0, 0, -5, 0, -5, 0, 9], [0, 0, 0, -26, 0, 0, 0, 0, -6, 0, 0, 8, 0, -8, 0, 3], [-30, 0, 0, 3, 0, 0, 0, 4, -3, 0, 0, 5, -2, 5, 0, -3], [0, 2, -8, 0, 0, 13, 0, 0, 0, 1, -13, 0, 0, 0, 8, 0], [0, -7, -6, 0, 0, 4, 0, 0, 0, 3, -4, 0, 0, 0, 6, 0], [8, 0, 0, 4, 0, 0, 0, 12, -4, 0, 0, 0, 4, 0, 0, -4], [0, 0, 10, 0, 2, -20, -7, 0, 0, 0, -20, 0, 0, 0, 10, 0], [0, 0, 0, -19, 0, 0, 0, 0, 7, 0, 0, 1, 0, -1, 0, 5], [0, 0, -3, 0, 0, -5, 0, 0, 0, -18, 5, 0, 0, 0, 3, 0], [8, 0, 0, -5, 0, 0, 0, 0, 5, 0, 0, 0, 10, 0, 0, 5], [0, -4, -8, 0, 0, 8, 0, 0, 0, 12, -8, 0, 0, 0, 8, 0], [0, 0, 0, 11, 0, 0, 0, 0, 8, 0, 0, 3, 0, -3, 0, 6], [0, 0, -1, 0, 6, 7, -12, 0, 0, 0, 7, 0, 0, 0, -1, 0], [0, 0, 0, 22, 0, 0, 0, 0, 12, 0, 0, -2, 0, 2, 0, -6], [0, 0, 0, 25, 0, 0, 0, 0, 3, 0, 0, -4, 0, 4, 0, -3], [0, 4, -16, 0, 0, -13, 0, 0, 0, 32, 13, 0, 0, 0, 16, 0], [0, 0, 5, 0, 1, -15, -1, 0, 0, 0, -15, 0, 0, 0, 5, 0], [42, 0, 0, 3, 0, 0, 0, -5, -3, 0, 0, 1, -11, 1, 0, -3], [0, 0, 0, -18, 0, 0, 0, 0, -6, 0, 0, -7, 0, 7, 0, -6], [0, 2, 7, 0, 0, 1, 0, 0, 0, 8, -1, 0, 0, 0, -7, 0], [-18, 0, 0, -1, 0, 0, 0, 10, 1, 0, 0, 5, 12, 5, 0, 1], [0, 0, 0, -16, 0, 0, 0, 0, 5, 0, 0, 1, 0, -1, 0, -14], [0, 4, -8, 0, 0, 14, 0, 0, 0, -24, -14, 0, 0, 0, 8, 0], [0, 0, -20, 0, -22, 5, 4, 0, 0, 0, 5, 0, 0, 0, -20, 0], [0, -2, -8, 0, 0, -11, 0, 0, 0, -14, 11, 0, 0, 0, 8, 0], [0, 0, 0, 8, 0, 0, 0, 0, 3, 0, 0, 1, 0, -1, 0, 8], [0, 0, 0, 7, 0, 0, 0, 0, -10, 0, 0, -2, 0, 2, 0, 6], [0, 0, 3, 0, -2, -5, -2, 0, 0, 0, -5, 0, 0, 0, 3, 0], [0, 0, -8, 0, 17, 1, 7, 0, 0, 0, 1, 0, 0, 0, -8, 0], [0, 0, 7, 0, -2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 7, 0], [-2, 0, 0, -10, 0, 0, 0, -14, 10, 0, 0, 7, 6, 7, 0, 10], [18, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 6, 2, 6, 0, 1], [26, 0, 0, -7, 0, 0, 0, -9, 7, 0, 0, -3, 5, -3, 0, 7], [0, 0, 0, -28, 0, 0, 0, 0, -12, 0, 0, -2, 0, 2, 0, -4], [0, 0, 24, 0, 36, -7, -6, 0, 0, 0, -7, 0, 0, 0, 24, 0], [0, 0, 0, -16, 0, 0, 0, 0, 10, 0, 0, 6, 0, -6, 0, 12], [0, 0, -9, 0, -3, -15, 7, 0, 0, 0, -15, 0, 0, 0, -9, 0], [0, -2, 13, 0, 0, -1, 0, 0, 0, 6, 1, 0, 0, 0, -13, 0], [0, -1, 8, 0, 0, -6, 0, 0, 0, -15, 6, 0, 0, 0, -8, 0], [0, 0, 0, 28, 0, 0, 0, 0, 2, 0, 0, -5, 0, 5, 0, 0], [0, 0, -3, 0, 19, 1, -5, 0, 0, 0, 1, 0, 0, 0, -3, 0], [-26, 0, 0, 11, 0, 0, 0, 6, -11, 0, 0, -7, -16, -7, 0, -11], [0, 0, -8, 0, 3, 1, -8, 0, 0, 0, 1, 0, 0, 0, -8, 0], [0, 0, 6, 0, 2, 2, 4, 0, 0, 0, 2, 0, 0, 0, 6, 0], [0, -9, 3, 0, 0, -11, 0, 0, 0, 15, 11, 0, 0, 0, -3, 0], [48, 0, 0, 3, 0, 0, 0, 6, -3, 0, 0, 6, 0, 6, 0, -3], [0, 0, 0, 3, 0, 0, 0, 0, -10, 0, 0, -4, 0, 4, 0, -15], [0, -3, 5, 0, 0, -1, 0, 0, 0, -1, 1, 0, 0, 0, -5, 0], [0, 0, 6, 0, 6, 14, 6, 0, 0, 0, 14, 0, 0, 0, 6, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4410_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4410_2_d_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_4410_2_d_a(:prec:=16)
    chi := MakeCharacter_4410_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(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_4410_2_d_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_4410_2_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4410_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![961, 0, 1154, 0, 426, 0, 50, 0, 1]>,<23,R![-294, -264, -53, 4, 1]>],Snew);
    return Vf;
end function;