
// Make newform 8820.2.d.c in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8820_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_8820_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_8820_2_d_c();".
// 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_8820_2_d_c();".
// 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 := [68, -864, 4848, -15904, 34228, -51840, 58384, -51008, 35421, -19960, 9316, -3528, 1134, -280, 60, -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], [150044, -1317740, 4940444, -10553790, 14780744, -15072848, 11804248, -7200631, 3666175, -1449893, 503911, -126937, 29695, -3941, 563, 0], [-8164, 72096, -271104, 580008, -812840, 829298, -649602, 396339, -201813, 79821, -27743, 6989, -1635, 217, -31, 0], [-17398, 151536, -565228, 1204216, -1684475, 1716224, -1343486, 819204, -417015, 164884, -57300, 14432, -3376, 448, -64, 0], [251108, -2195268, 8210868, -17518682, 24522096, -24996888, 19572612, -11937273, 6077301, -2403203, 835197, -210375, 49213, -6531, 933, 0], [-130804, 1135602, -4225470, 8990894, -12569384, 12800879, -10018741, 6107891, -3108941, 1229122, -427122, 107571, -25163, 3339, -477, 0], [325876, -2842120, 10611548, -22619510, 31647880, -32249910, 25247728, -15396207, 7837701, -3099083, 1077003, -271269, 63457, -8421, 1203, 0], [245380, -2145226, 8021234, -17111062, 23949172, -24411051, 19113169, -11656637, 5934323, -2346616, 815524, -205417, 48053, -6377, 911, 0], [1295576, -13901100, 64854694, -174504810, 306047064, -380329884, 357257717, -262297961, 153778332, -74737703, 28961262, -9654653, 2412384, -538751, 71835, -9578], [-5590392, 59901234, -279252788, 751070018, -1316922472, 1636354831, -1536952540, 1128373599, -661508118, 321491944, -124576947, 41529045, -10376600, 2317365, -308985, 41198], [6148892, -65738314, 306021290, -822338834, 1441171232, -1790248231, 1681174263, -1234125131, 723438657, -351572654, 136226024, -45411237, 11346231, -2533879, 337845, -45046], [-525468, 5621528, -26179718, 70367120, -123337072, 153223032, -143895227, 105634416, -61923975, 30093874, -11660825, 3887188, -971243, 216902, -28920, 3856], [7973036, -85350718, 397656120, -1069135048, 1874234912, -2328586171, 2186961412, -1605516550, 941197611, -457411359, 177241460, -59084708, 14762917, -3296928, 439590, -58612], [588636, -6306008, 29394806, -79054960, 138610316, -172228752, 161764661, -118760880, 69623097, -33836622, 13111527, -4370860, 1092117, -243898, 32520, -4336], [2510464, -26878490, 125240438, -336737754, 590330982, -733450047, 688848917, -505708399, 296461545, -144077334, 55828410, -18610825, 4650113, -1038487, 138465, -18462], [26936464, -288572112, 1345157438, -3617702120, 6343074326, -7881522940, 7402668495, -5434732004, 3186090657, -1548430882, 600009509, -200019240, 49977447, -11161258, 1488180, -198424]];
    Rf_basisdens := [1, 56, 2, 4, 56, 28, 56, 28, 56, 56, 56, 4, 56, 4, 8, 56];
    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_8820_d();"
function MakeCharacter_8820_d()
    N := 8820;
    order := 2;
    char_gens := [4411, 7841, 7057, 1081];
    v := [2, 1, 2, 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_8820_d_Hecke();"
function MakeCharacter_8820_d_Hecke(Kf)
    N := 8820;
    order := 2;
    char_gens := [4411, 7841, 7057, 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], [-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, 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, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, -1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 1, -1, -1, 0], [0, 2, 1, -1, 0, -1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 1, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 1, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, -1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, -1, 0, -1, 1], [2, 0, 1, 2, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 1, 2, 0, -2, 1, -1], [-2, -2, -1, -1, 0, 1, -3, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 3, 1, -3, 0, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, -4, 1, -1, 2, 0], [0, 1, 0, -2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 2, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [4, 2, 1, -2, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -3, -3, 2, -1, 2], [4, 0, -1, 1, -2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -3, -2, 0, 3, -1, 2, 1], [0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 2, 1, -2, -1], [2, 0, 2, -2, 0, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 2, -5, 1, -1, 0], [-2, 6, 0, 1, 1, -2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 3, 2, -1, 2, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, -3, -3, 1, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -1, -2, 0, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 5, 0, -1, 3, 2, 0, -1], [2, 0, -1, 1, -1, 0, -5, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, -3, 0, -5, -1, 1], [4, 1, 1, -1, -1, 1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, 0, 4, 1, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [2, -3, -2, -3, 1, 0, 4, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 1, -2, -7, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -3, 0, -4, -2, 4, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, -4, -1, 0, 4, 0], [-4, -4, 0, 1, 2, 1, -2, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 2, -1, -2, -1, -1, -1, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -4, -3, -5, 0, -1], [2, 5, 0, 1, 1, 1, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, -5, 4, 1, 1, 0], [4, 4, 2, 5, 1, -1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 4, -2, -4, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -2, 2, 6, 0, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 5, -1, -1, -4, 1, 2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 4, -1, 0, -3, -6, -1, -1], [-4, 1, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 2, 1, -3, 0, 1, -4, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 1, -2, -2, -4, 7, 0, -1], [2, 3, 0, 5, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 2, -2, -4, -9, -1, 0], [6, -4, 1, 2, -2, 0, -5, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 2, -5, -7, -6, 2, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 1, -9, 2, 1, 1, 0], [-2, 4, 2, 5, 1, -1, 5, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, -1, 0, 2, 3, 1, 0, -3], [6, 4, 1, 3, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -4, 0, 4, 2, -2, -2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 3, 7, 2, -2, 4, -2, 0, 0], [4, 3, 1, 1, 0, -1, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0], [10, 1, 1, 1, 1, 3, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, -6, 2, -2, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, -4, 0, 6, -2, -2], [8, 1, 0, 2, 3, -3, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 1, 1, 1, -6, -2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -2, -6, -1, -3, 0, 1], [0, 7, 1, -4, 3, 0, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0], [6, 4, -1, -3, -3, 2, -3, -1, 0, 0, 0, 0, 0, 0, 0, 0], [6, -6, 0, 5, -2, -1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, -4, -2, 5, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 7, -1, 3, -3, -2, 3, -2], [0, 0, 0, 0, 0, 0, 0, 0, -2, 4, 0, -6, 4, 1, -6, 2], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, -4, 7, -4, -1, 3], [-2, 6, 1, 2, 2, -3, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, 0, 9, 0, -2, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -3, 1, -1, -10, -7, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -3, 1, 2, -8, 9, 3, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 6, -1, -4, 2, -5, 6, -3], [0, 0, 0, 0, 0, 0, 0, 0, -3, 4, 1, 6, -5, -5, -5, 2], [0, 0, 0, 0, 0, 0, 0, 0, -5, 3, 2, -2, -1, -6, -4, -1], [6, 5, 0, 1, 1, 1, 7, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, -1, 10, 2, 1, -4, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 10, 4, -7, 0, -3, -4, 1, 0, 0, 0, 0, 0, 0, 0, 0], [12, 2, 2, -7, 3, -3, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, -3, 3, 2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [4, 8, 3, -4, 3, -3, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 5, -1, -2, -2, -9, -6, 1], [-4, 0, -2, 1, -2, -2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 6, -2, -1, -2, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 2, 0, -5, 2, -1, -2, 3, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 9, 3, 5, 2, -2, 5, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -11, 4, -4, 7, -9, 0, -1], [-2, 4, 3, 5, 1, 2, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0], [2, 6, -1, 1, -3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, -11, -4, 4, -4, -3, 0, -1], [2, -10, 0, 5, -2, 3, -6, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 10, -2, -1, 0, 5, 2, -2], [6, 4, 4, -6, 0, -4, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 5, -1, 2, -4], [8, -6, 1, -1, -3, -1, -1, 5, 0, 0, 0, 0, 0, 0, 0, 0], [2, -12, 2, -12, -2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 9, 1, -2, -4, -7, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 3, 1, -1, -2, -10, 6, 3, 1], [0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 3, -5, -5, -4, -1, 5], [-8, -7, -4, 0, -6, -1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, -10, -5, 4, 7, 1], [0, 0, 0, 0, 0, 0, 0, 0, -2, -1, 5, -12, -3, 1, 2, -1], [6, 9, 1, 8, 0, -3, -3, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 2, -3, 6, -5, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 5, -3, -3, 5, 11, 4, 3, 1], [-4, -1, -5, 8, -2, 1, 5, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 8, 3, 4, 3, -7, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 3, -1, -4, -2, 5, 0, 7], [0, 0, 0, 0, 0, 0, 0, 0, -3, 7, -1, -2, 8, 0, 3, -1], [-8, 0, -1, -9, 1, -2, 2, -4, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 10, -1, 3, 0, -3, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 1, 0, -13, 6, 3, -3], [0, 0, 0, 0, 0, 0, 0, 0, -4, 9, -3, -2, -5, -1, -4, 3], [0, 0, 0, 0, 0, 0, 0, 0, 4, 2, -6, -5, 6, 1, 6, -2], [-2, -4, -4, 3, -6, 4, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0], [8, 6, -2, -2, 4, 0, 9, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, -8, -11, 3, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, -10, -1, 11, 6, -3, -5, -3], [-2, 0, 3, 2, 1, 2, -11, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 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-4, 21, -10, -6, -1], [-14, 10, 2, 32, -2, 4, -4, 2, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, 0, 8, 0, 6, -19, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 7, 7, -1, 9, 16, 7, 6], [0, 0, 0, 0, 0, 0, 0, 0, -6, -14, 7, -6, 15, -12, -5, 7], [6, 0, 3, 29, -1, 0, -15, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 16, 9, 0, -1, 0, -9, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -15, 1, 1, -15, -8, -19, 0, 1], [0, -6, 6, 0, -4, 4, -8, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 36, 1, -7, -4, 15, -3, -4], [-8, 6, 8, 9, 3, -6, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0], [6, 9, 6, 5, -5, -5, -14, 8, 0, 0, 0, 0, 0, 0, 0, 0], [26, -7, 0, 6, -15, 1, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, -11, 4, 14, -9, 12, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -3, 24, 12, 6, -4, -5, -11, -2], [-8, 7, -4, 25, -3, 5, -3, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -8, 6, -3, -4, 9, 6, 1, 15], [-12, 5, 2, 33, 0, 5, -7, -3, 0, 0, 0, 0, 0, 0, 0, 0], [8, 4, 1, 12, -7, 4, -13, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, -9, 9, -5, 10, 7, -12, 7], [0, 0, 0, 0, 0, 0, 0, 0, 12, 0, -9, 15, 14, -7, 3, -7], [0, 0, 0, 0, 0, 0, 0, 0, 12, 12, -2, 2, -24, -5, 10, -2], [0, 0, 0, 0, 0, 0, 0, 0, -5, -5, 0, -8, 5, -27, 7, 2], [-8, 0, 3, 27, 9, -1, -3, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 1, -4, 6, 8, 1, -10, -3], [0, 0, 0, 0, 0, 0, 0, 0, 8, -12, -2, 0, 6, -16, 4, 3], [2, -9, -4, 7, -14, -5, -3, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 11, 4, -2, -15, 1, 2, -6], [-12, 11, 7, 20, 6, 1, 6, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 9, -15, 9, -1, -7, -6, -1, -7], [0, 0, 0, 0, 0, 0, 0, 0, -1, 3, -4, -12, 2, -8, -4, 8], [24, -8, 2, -6, -12, 0, -22, 0, 0, 0, 0, 0, 0, 0, 0, 0], [50, 3, -1, -29, 6, 6, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 13, 12, 2, 7, -3, -3, -8], [-6, -6, -8, -18, -6, 6, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -10, -6, -14, -10, 8, 2, -3], [-20, -5, 3, 0, 10, -5, 11, 4, 0, 0, 0, 0, 0, 0, 0, 0], [24, 14, -1, -17, 5, -8, 5, 3, 0, 0, 0, 0, 0, 0, 0, 0], [12, -6, 4, -2, 3, -2, -7, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 2, -9, -20, -27, -24, -4, -2], [0, 0, 0, 0, 0, 0, 0, 0, 5, 14, -7, 0, -19, -10, 15, -11], [0, 0, 0, 0, 0, 0, 0, 0, -1, -3, 1, 11, 11, 2, -1, -11], [0, 0, 0, 0, 0, 0, 0, 0, -3, -1, 10, -12, -2, -4, -8, -4], [0, 0, 0, 0, 0, 0, 0, 0, 6, -36, -1, -2, -2, -5, -2, -3], [0, 6, -9, -18, -5, 4, 3, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 17, -13, 4, -4, 3, 8, 8, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 27, 4, -14, 5, 15, 2, 3], [0, 0, 0, 0, 0, 0, 0, 0, 5, 15, -2, 8, -14, -14, 8, -8], [-16, 18, 5, 8, -3, 1, -10, -7, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 5, -1, -5, 7, 9, -16, -4, 0, 0, 0, 0, 0, 0, 0, 0], [-14, 0, -11, -20, -6, 4, -4, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, 16, 5, -14, -5, 1, -15, 2], [0, 0, 0, 0, 0, 0, 0, 0, 7, 6, -8, 14, 8, 13, -9, 2], [34, 6, 2, -6, 4, -10, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 12, 2, -2, -2, 0, -8, 6, -4], [-18, 15, 7, -21, 10, -4, 11, -3, 0, 0, 0, 0, 0, 0, 0, 0], [-6, -5, -5, 7, 10, 0, 17, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 3, 3, 1, -34, 3, -10], [0, 0, 0, 0, 0, 0, 0, 0, -1, 22, 1, 4, -7, -5, -5, 2], [38, -5, -1, 12, -2, 5, 9, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-8, -4, -10, 1, 5, 4, 5, 2, 0, 0, 0, 0, 0, 0, 0, 0], [16, -7, -5, 27, 0, 8, -13, -11, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -5, -1, -12, 20, 18, 1, -3], [0, 0, 0, 0, 0, 0, 0, 0, 5, 35, 1, 17, -8, -27, 2, -3], [0, 0, 0, 0, 0, 0, 0, 0, 0, -6, -4, 0, 26, -32, -4, 0], [4, 7, 3, -16, 4, -15, 17, 10, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 12, 10, 31, -4, 0, -8, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, -29, 0, -5, 5, 4, -10, 5], [0, 0, 0, 0, 0, 0, 0, 0, -3, -11, -8, 14, -24, -20, -2, 6], [0, 0, 0, 0, 0, 0, 0, 0, -3, 17, 5, -24, 8, -2, -3, 5], [-36, 17, 8, -7, -6, -5, -8, 9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, -13, -8, -16, -9, -11, 0, 10], [-32, 0, -5, 4, -1, -4, 1, -7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, -4, 7, 14, 11, 2, -11, -1], [-10, -3, 2, 23, -2, 1, 8, -1, 0, 0, 0, 0, 0, 0, 0, 0], [28, 2, -5, 6, 2, -6, -1, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, -19, 0, 4, -16, -7, -2, 9], [-2, 3, -1, 12, 8, -3, 7, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 27, 7, 14, 4, -19, -10, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 15, -2, -2, -13, -5, -10, 4], [30, 6, -9, 18, -7, 0, 13, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -4, 11, -7, -6, 17, -1, -7], [46, -9, -8, 4, 2, 2, 10, -2, 0, 0, 0, 0, 0, 0, 0, 0], [2, 4, 2, 6, -4, -6, -2, 12, 0, 0, 0, 0, 0, 0, 0, 0], [2, 22, 6, 3, -5, -4, -3, 4, 0, 0, 0, 0, 0, 0, 0, 0], [-24, -13, -14, 11, -2, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 15, 0, -14, -18, -20, -4, 8], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 2, -22, -24, -19, 1, -4], [-2, -1, 8, 41, -4, -8, -11, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 26, 2, 2, 12, 7, -10, -2], [-12, -7, 0, 10, 9, -11, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [10, 19, 1, 6, 0, -1, -1, 10, 0, 0, 0, 0, 0, 0, 0, 0], [-26, -3, -6, -1, -2, 0, 11, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 1, -7, 5, 11, -6, 11, -9], [12, -4, -7, 4, -2, -5, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, -1, -4, -6, -33, -8, -6, -5], [0, 0, 0, 0, 0, 0, 0, 0, 5, -1, -8, 8, 3, 27, -4, 1], [30, 6, 0, -3, 10, 5, 20, -14, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 8, -14, 12, -10, 2, -4], [0, 0, 0, 0, 0, 0, 0, 0, -2, -4, 7, -21, 15, -10, -4, 2], [0, 0, 0, 0, 0, 0, 0, 0, -9, 15, -1, -16, 12, -16, -3, 9], [4, -4, 2, 46, -4, 6, 8, -2, 0, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8820_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8820_2_d_c();".
// 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_8820_2_d_c(:prec:=16)
    chi := MakeCharacter_8820_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(4027) - 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_8820_2_d_c();".
// 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_8820_2_d_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8820_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![73984, 0, 1339392, 0, 5820672, 0, 6144000, 0, 1442400, 0, 137984, 0, 6160, 0, 128, 0, 1]>,<17,R![1552, -2176, -5088, -128, 1288, 32, -72, 0, 1]>],Snew);
    return Vf;
end function;