
// Make newform 448.3.o.b in Magma, downloaded from the LMFDB on 21 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_448_o();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_448_o_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_448_3_o_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_448_3_o_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 := [7311616, 0, -10729472, 0, 10423552, 0, -5640416, 0, 2189920, 0, -522048, 0, 89921, 0, -9698, 0, 755, 0, -34, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [21278031177915975424, 0, -49320243572313119232, 0, 24359618557475718448, 0, -10781175884092385424, 0, 2671067767894593040, 0, -492293261842127317, 0, 54580483450150554, 0, -4416640623007247, 0, 202194994815850, 0, -6178891093061, 0], [-632229630954928155136, 0, 350503147037072684672, 0, -821187645630119796624, 0, 235705441743286751296, 0, -51987628482883683752, 0, 4227458635693993999, 0, -234433062312013526, 0, -10978015213222019, 0, 1067983757829386, 0, -74793058692353, 0], [116537606365877382400, 0, -107567176715361278976, 0, 78115055174157948176, 0, -17425707316030040784, 0, 2908209572205776656, 0, -110649151533636599, 0, -7176487769216610, 0, 2625159897435883, 0, -160634970287730, 0, 7536626957017, 0], [0, -3274729407844104016, 0, -249477129907553184, 0, 235226423466409760, 0, -193402940515066221, 0, 42881297504398934, 0, -8647825122939002, 0, 951551210160756, 0, -83302075295446, 0, 3873553393682, 0, -131763721003], [1532180810549839400704, 0, -1856815766311165669024, 0, 1351536002054552727152, 0, -512774788700197275776, 0, 122713892798819153718, 0, -19159671770658851653, 0, 1970113303308170644, 0, -139909272161600103, 0, 6030031367115544, 0, -156849163360857, 0], [1395702621642671902976, 0, -425105100516737080000, 0, 1607936590867659792, 0, 333885858635670341776, 0, -123375241796011738556, 0, 30468859185364189369, 0, -3776929569241084022, 0, 338148454923425803, 0, -16199692497006958, 0, 530520981186385, 0], [38154005573433587968, 0, -41628914775537630400, 0, 26833486753116299216, 0, -21496998405021403424, 0, 6532569131054483988, 0, -1431496643706984907, 0, 170330401022527330, 0, -14739620565077505, 0, 695686864779370, 0, -22319680131651, 0], [429845974066435307264, 0, -594994276770647054208, 0, 491083925296069143728, 0, -216838348259813301264, 0, 55527581278339512248, 0, -9770648532838180637, 0, 1067438358262724994, 0, -83044881986249671, 0, 3745899429973250, 0, -109071631282621, 0], [0, 90236297134745464352, 0, -171963172249235811056, 0, 3924875485191293632, 0, -17704464274993342318, 0, 5187467924178384081, 0, -1625350691097557762, 0, 209485482305406983, 0, -20695054627309458, 0, 1018101072048017, 0, -36381780868686], [-8524654922546073382656, 0, 4213862590456357254016, 0, -1676700291424060982032, 0, -269418682644302592304, 0, 159814565750350547272, 0, -49034974172334106905, 0, 6353471410864236074, 0, -598477354455709499, 0, 29113576762981002, 0, -990559116329657, 0], [0, 395913599749630277952, 0, -99282454045618892800, 0, 251826577876038372208, 0, -142055676676975786604, 0, 41435500691128453400, 0, -7777053218347731981, 0, 879208334111140618, 0, -69875666598910231, 0, 3193444958825346, 0, -93940477944073], [0, 9824395700065024, 0, -20683811259392352, 0, 11255930628950992, 0, -4986055868667184, 0, 1224415269632070, 0, -228797968753211, 0, 25502866317136, 0, -2091233989993, 0, 96267694836, 0, -2988257823], [-6465796365905380519808, 0, 8136449507789524414912, 0, -5375220038073089723472, 0, 2122934290676172565208, 0, -514418044875550539124, 0, 86936790867594699083, 0, -9310320541279217746, 0, 711300543268637825, 0, -31791203852696906, 0, 912490276567931, 0], [0, 29993395722959108476608, 0, -19967202659044399612016, 0, 11682726240455782291328, 0, -2519318944938424039204, 0, 397878740655691040581, 0, -11121229037171874972, 0, -1644918613584829765, 0, 426868954167226576, 0, -25404125874592791, 0, 1149081987071530], [0, -14759481736598933553024, 0, 6080498311118676477632, 0, -1842484234641363557360, 0, -2052627007259774918072, 0, 774230743814170314500, 0, -198443152695679083543, 0, 24619090686964719322, 0, -2232219399239715685, 0, 107087891990781810, 0, -3558483468007111], [0, 14093297948364485392672, 0, -17227181945929812473216, 0, 12352443708280474803120, 0, -4740827789170912527118, 0, 1141713444372033945584, 0, -187943249978958733909, 0, 19897521154364416862, 0, -1489732507877524111, 0, 65983179645813322, 0, -1847591803122019], [0, 13390332098534788189312, 0, 508597861003358733712, 0, -4593480822678658789872, 0, 4604863326413934097880, 0, -1436617094085253395427, 0, 309405830131469303141, 0, -36592576967270116039, 0, 3125666720204697347, 0, -146811829881845009, 0, 4647543482269847], [0, 15675961902170367559840, 0, -36564184481230787324288, 0, 27331650355530432771408, 0, -12616193571836466848614, 0, 3250645707535600403228, 0, -586435673371413528013, 0, 64665994016157714674, 0, -5114973818920810567, 0, 232292021608071190, 0, -6893548144068763], [0, -3641233076634698553424, 0, 14515954323121488713536, 0, -10754952510429423833312, 0, 5801696677979333302499, 0, -1573197916892454553518, 0, 305499488820758421163, 0, -34734340844283280018, 0, 2858209816537212057, 0, -132069536783915884, 0, 4068596376660450]];
    Rf_basisdens := [1, 29018519847896796672, 203129638935277576704, 9672839949298932224, 1953169605146899776, 101564819467638788352, 203129638935277576704, 7812678420587599104, 29018519847896796672, 101564819467638788352, 203129638935277576704, 203129638935277576704, 6222806573156992, 101564819467638788352, 1320342653079304248576, 1320342653079304248576, 660171326539652124288, 1320342653079304248576, 1320342653079304248576, 660171326539652124288];
    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_448_o();"
function MakeCharacter_448_o()
    N := 448;
    order := 6;
    char_gens := [127, 197, 129];
    v := [3, 3, 4];
    // 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_448_o_Hecke();"
function MakeCharacter_448_o_Hecke( : Kf := false)
    N := 448;
    order := 6;
    char_gens := [127, 197, 129];
    char_values := [[-1, 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], [-1, -1, 0, 0, 0, 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 := ISA(Type(Kf), Fld), 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]];
    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 := 3;
    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, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -1, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 1, 0, 0, -1, 0, -1], [-1, -1, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [1, -3, 2, 0, 0, 2, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, -4, 0, 1, 0, 1, -2, 1, 2], [0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 2, -1, 0, 1, -2, -1, -1, -1, -1], [-1, -3, 1, -1, 0, -3, 0, 0, -1, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, -8, 0, -4, -1, 0, -1, -1, 1, -1, -1, 0], [2, -6, -4, 2, 0, 2, -2, 4, -4, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0], [-10, 0, 0, 2, 0, 3, -6, 0, -2, 0, 2, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, 0, 0, -4, 0, -4, -3, 0, -2, 3, 0, -4, -2, -2], [0, 0, 0, 0, 2, 0, 0, 0, 0, -6, 0, -12, -3, 0, 4, 4, 2, -3, -4, -3], [2, 0, 0, 2, 0, 0, 2, 2, -1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 13, 5, 0, 2, -1, -1, -2, 2, -4], [2, -5, -3, 1, 0, -2, 2, 3, -2, 0, 2, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, -1, -9, 0, 2, -2, -2, -2, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, -14, -3, 0, 0, 0, 0, 0, 0, 0], [5, -25, 10, 0, 0, 4, -2, -5, 0, 0, 2, 0, 0, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -15, 0, 0, 0, 0, 7, 0, 14, 3, 0, -1, -6, -3, 3, 1, 3], [0, 0, 0, 0, -22, 0, 0, 0, 0, 0, 0, 0, 14, 0, 2, 3, 0, 6, 4, 4], [-15, -17, 2, 8, 0, 0, 0, 2, 0, 0, -4, 0, 0, -4, 0, 0, 0, 0, 0, 0], [-12, 8, 8, -9, 0, 3, -6, -16, 9, 0, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0], [-69, -28, 0, -14, 0, 0, -9, -13, 7, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -19, 0, 0, 0, 0, -5, 0, -10, 9, 0, -9, -8, -4, 9, 9, 9], [0, 0, 0, 0, 16, 0, 0, 0, 0, -8, 0, 0, -40, 0, 5, 0, 4, 8, 5, -8], [39, 26, 0, 0, 0, 0, 5, -13, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [41, -1, -1, -8, 0, 3, -6, 2, 8, 0, -10, 0, 0, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, 0, 0, 12, 0, -12, -46, 0, 10, 3, 6, -4, 6, -6], [0, 0, 0, 0, -6, 0, 0, 0, 0, -23, 0, 0, -2, 0, -2, 0, -1, 14, -2, -14], [-7, 17, -14, 0, 0, 10, -5, 7, -5, 0, 6, 0, 0, -12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -42, 0, 0, 0, 0, 4, 0, 4, 23, 0, 4, -10, 0, 4, 0, 0], [64, -48, 16, -2, 0, -9, 9, -16, 4, 0, -12, 0, 0, 12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -29, 0, 0, 0, 0, -8, 0, -4, 29, 0, 7, 3, -3, 7, -1, -8], [-152, -78, 0, -8, 0, 0, -4, 4, 4, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 27, 0, 0, 0, 0, -25, 0, 0, -60, 0, -6, 0, -15, 6, -6, -6], [0, 0, 0, 0, 11, 0, 0, 0, 0, -9, 0, 9, 2, 0, 24, 0, 0, -11, 13, -13], [32, -38, -6, -5, 0, 4, -4, 6, 10, 0, -10, 0, 0, 10, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -21, 0, 0, 0, 0, 0, 0, 32, 9, 0, -15, 7, 7, 15, 10, 5], [60, 102, 18, 7, 0, 18, 0, 0, 7, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -70, 0, 0, 0, 0, 0, 0, 0, 1, 0, -14, 8, 4, 1, 14, 1], [-1, 51, -2, 0, 0, 2, -1, 1, 6, 0, -5, 0, 0, 10, 0, 0, 0, 0, 0, 0], [-44, -58, -30, -12, 0, -19, 0, 0, -12, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 82, 0, 0, 0, 0, 20, 0, 10, -82, 0, -5, 1, -1, -5, -5, 0], [0, 0, 0, 0, 42, 0, 0, 0, 0, 8, 0, 8, -30, 0, 6, -3, 0, -18, -24, -24], [0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, -4, 15, 0, 4, -10, -20, -2, 2, -2], [0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 0, 30, 68, 0, -7, 5, 5, 7, 18, -11], [14, -7, 7, 0, 0, 39, -39, -7, 0, 0, 6, 0, 0, -6, 0, 0, 0, 0, 0, 0], [73, 58, 15, 16, 0, -7, -7, 15, 0, 0, -5, 0, 0, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, 0, 0, -13, 0, 13, -61, 0, -14, -5, -10, 3, -11, 11], [0, -177, 0, 0, 0, 16, -8, 0, 20, 0, -4, 0, 0, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -111, 0, 0, 0, 0, -11, 0, -11, 62, 0, -4, -10, 0, 13, 17, 17], [7, 22, -15, 1, 0, -9, -9, -15, 0, 0, -6, 0, 0, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -35, 0, 0, 0, 0, 82, 0, 41, 35, 0, -18, -9, 9, -18, -1, 17], [-283, -146, 0, -6, 0, 0, 4, 9, 3, 0, -16, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 87, 0, 0, 0, 0, 1, 0, 2, 13, 0, 13, -12, -6, 13, -13, 13], [100, 24, 0, 22, 0, 0, -6, 52, -11, 0, -12, 0, 0, 0, 0, 0, 0, 0, 0, 0], [40, 16, 16, 17, 0, -11, 22, -32, -17, 0, 4, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 76, 0, 0, 0, 0, 0, 0, -54, 22, 0, 27, -8, -8, -27, -8, -19], [172, 340, 4, -14, 0, 0, 0, 0, -14, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -90, 0, 0, 0, 0, 0, 0, 0, 36, 0, -28, -4, 0, -18, 10, 10], [0, 0, 0, 0, 123, 0, 0, 0, 0, 22, 0, 11, -123, 0, 6, 6, -6, 6, -3, -9], [31, 31, 0, 5, 0, 12, 12, 0, 0, 0, 5, 0, 0, 5, 0, 0, 0, 0, 0, 0], [20, -11, 9, 18, 0, 6, -6, -9, -36, 0, -23, 0, 0, 23, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 62, 0, 0, 0, 0, -33, 0, 0, -102, 0, -20, 0, 15, -22, -20, 22], [47, 15, 15, -3, 0, 27, -54, -30, 3, 0, 16, 0, 0, -8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -36, 0, 0, 0, 0, 0, 0, 49, 0, 0, -18, -12, -12, 18, 36, -18], [-11, 7, -29, 9, 0, 15, 0, 0, 9, 0, 0, 0, 0, -18, 0, 0, 0, 0, 0, 0], [-16, -79, -32, 0, 0, -50, 25, 16, -41, 0, 4, 0, 0, -8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -15, 0, 0, 0, 0, -18, 0, -36, -11, 0, -18, -28, -14, -11, 18, -11], [0, 0, 0, 0, -58, 0, 0, 0, 0, -26, 0, -13, 58, 0, 0, -2, 2, 0, -43, -43], [124, -95, 29, 4, 0, 10, -10, -29, -8, 0, 11, 0, 0, -11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 41, 0, 0, 0, 0, 21, 0, 21, -22, 0, -8, 16, 0, -3, 5, 5], [0, 0, 0, 0, 56, 0, 0, 0, 0, 42, 0, 84, -13, 0, -32, 2, 1, -13, 32, -13], [42, 4, 0, -16, 0, 0, 57, 34, 8, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-30, -13, -43, 20, 0, -31, 31, 43, -40, 0, -8, 0, 0, 8, 0, 0, 0, 0, 0, 0], [101, 87, 14, -26, 0, 3, 3, 14, 0, 0, -9, 0, 0, -9, 0, 0, 0, 0, 0, 0], [3, -19, 6, 0, 0, -2, 1, -3, -23, 0, -14, 0, 0, 28, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -100, 0, 0, 0, 0, 98, 0, 98, 67, 0, -2, -1, 0, 34, 36, 36], [-113, -197, -29, 22, 0, -37, 0, 0, 22, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 69, 0, 0, 0, 0, 20, 0, 10, -69, 0, 19, 12, -12, 19, -23, -42], [-128, -8, -8, 37, 0, -19, 38, 16, -37, 0, 12, 0, 0, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -127, 0, 0, 0, 0, 50, 0, 100, 13, 0, -6, -20, -10, 13, 6, 13], [0, 0, 0, 0, 0, 0, 0, 0, 0, -52, 0, 0, -34, 0, 17, 0, 11, 34, 17, -34], [-63, -7, -7, 41, 0, -11, 22, 14, -41, 0, -20, 0, 0, 10, 0, 0, 0, 0, 0, 0], [229, 220, 9, -34, 0, 24, 24, 9, 0, 0, 6, 0, 0, 6, 0, 0, 0, 0, 0, 0], [110, 236, -16, 14, 0, -1, 0, 0, 14, 0, 0, 0, 0, 29, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -36, 0, 0, 0, 0, -38, 0, 38, -53, 0, -50, 9, 18, 36, -14, 14], [0, 0, 0, 0, 45, 0, 0, 0, 0, -59, 0, 0, -132, 0, 36, 0, -21, 42, 36, -42], [0, 0, 0, 0, -35, 0, 0, 0, 0, 114, 0, 57, 35, 0, -20, 4, -4, -20, 33, 53], [0, 0, 0, 0, -100, 0, 0, 0, 0, 78, 0, 39, 100, 0, -4, 14, -14, -4, 29, 33], [0, 0, 0, 0, 81, 0, 0, 0, 0, -27, 0, -27, -28, 0, 44, -2, 0, 25, -19, -19], [0, 0, 0, 0, 8, 0, 0, 0, 0, 97, 0, 0, -16, 0, -30, 0, 30, 0, -30, 0], [0, 0, 0, 0, -38, 0, 0, 0, 0, -28, 0, 28, 107, 0, -16, -12, -24, 38, 22, -22], [40, -48, -8, -8, 0, -1, 1, 8, 16, 0, -30, 0, 0, 30, 0, 0, 0, 0, 0, 0], [30, 103, 60, 0, 0, 20, -10, -30, 4, 0, -22, 0, 0, 44, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -10, 0, 0, 0, 0, -46, 0, 0, 16, 0, 27, 0, 38, 4, 27, -4], [120, -97, 23, -29, 0, 31, -31, -23, 58, 0, 11, 0, 0, -11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -35, 0, 0, 0, 0, -7, 0, -7, 22, 0, 16, 52, 0, 9, -7, -7], [-143, -42, 0, -60, 0, 0, -85, -59, 30, 0, -18, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -42, 0, 0, 0, 0, 0, 0, 81, 62, 0, -52, -37, -37, 52, 26, 26], [-163, -164, 1, 6, 0, 10, 10, 1, 0, 0, 4, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 37, 0, 0, 0, 0, 0, 0, -78, 23, 0, 7, 21, 21, -7, -62, 55], [-2, 11, -4, 0, 0, -68, 34, 2, -10, 0, 32, 0, 0, -64, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -82, 0, 0, 0, 0, 152, 0, 152, 50, 0, -16, -2, 0, 18, 34, 34], [157, 142, 15, -38, 0, -51, -51, 15, 0, 0, -19, 0, 0, -19, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -37, 0, 0, 0, 0, -66, 0, -33, 37, 0, -6, -14, 14, -6, -11, -5], [35, 37, 37, -62, 0, -29, 58, -74, 62, 0, -22, 0, 0, 11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 218, 0, 0, 0, 0, -20, 0, -40, -57, 0, 14, -2, -1, -57, -14, -57], [11, -18, 0, 28, 0, 0, 119, 47, -14, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-49, -37, -37, -5, 0, -21, 42, 74, 5, 0, 28, 0, 0, -14, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -97, 0, 0, 0, 0, 0, 0, 43, -69, 0, -14, 2, 2, 14, 12, 2], [0, 0, 0, 0, -32, 0, 0, 0, 0, 14, 0, -14, 317, 0, -32, 12, 24, 32, 0, 0], [-19, -79, -38, 0, 0, 76, -38, 19, -29, 0, -9, 0, 0, 18, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 172, 0, 0, 0, 0, 154, 0, 154, -100, 0, 16, 18, 0, -28, -44, -44], [0, 0, 0, 0, 92, 0, 0, 0, 0, -130, 0, -65, -92, 0, -42, 11, -11, -42, -17, 25], [-260, 228, -32, 23, 0, -116, 116, 32, -46, 0, 10, 0, 0, -10, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 36, 0, 0, 0, 0, -34, 0, -34, -36, 0, -18, 45, 0, -36, -18, -18], [360, 174, 0, 20, 0, 0, 66, 12, -10, 0, 45, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-162, -32, -32, 42, 0, 13, -26, 64, -42, 0, 22, 0, 0, -11, 0, 0, 0, 0, 0, 0], [70, -54, 16, 21, 0, 99, -99, -16, -42, 0, -31, 0, 0, 31, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 81, 0, 0, 0, 0, 0, 0, -118, 71, 0, 5, 12, 12, -5, 12, -17], [0, 0, 0, 0, -99, 0, 0, 0, 0, -151, 0, 0, 202, 0, -40, 0, -22, -4, -40, 4], [-51, -167, 65, 29, 0, -18, 0, 0, 29, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 0], [-170, 176, 6, 30, 0, 10, -10, -6, -60, 0, -19, 0, 0, 19, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 331, 0, 0, 0, 0, -3, 0, -6, -27, 0, 65, 12, 6, -27, -65, -27], [0, 0, 0, 0, 44, 0, 0, 0, 0, -22, 0, 22, -27, 0, -8, -12, -24, -44, -52, 52], [-128, 144, 16, -1, 0, 114, -114, -16, 2, 0, 12, 0, 0, -12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -19, 0, 0, 0, 0, 0, 0, 4, -41, 0, 11, -48, -48, -11, 56, -67], [0, 0, 0, 0, 34, 0, 0, 0, 0, 26, 0, -26, 166, 0, 54, -3, -6, -34, 20, -20], [0, 0, 0, 0, 56, 0, 0, 0, 0, -50, 0, -100, 69, 0, 12, 12, 6, 69, -12, 69], [239, 545, -67, -39, 0, 9, 0, 0, -39, 0, 0, 0, 0, -48, 0, 0, 0, 0, 0, 0], [281, 251, 30, 44, 0, 33, 33, 30, 0, 0, 27, 0, 0, 27, 0, 0, 0, 0, 0, 0], [553, 17, 17, -23, 0, -21, 42, -34, 23, 0, 56, 0, 0, -28, 0, 0, 0, 0, 0, 0], [366, 176, 0, 36, 0, 0, 25, 14, -18, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -150, 0, 0, 0, 0, 0, 0, -27, -186, 0, 18, -16, -16, -18, 32, -50], [27, 113, -59, -13, 0, 89, 0, 0, -13, 0, 0, 0, 0, -42, 0, 0, 0, 0, 0, 0], [28, -243, 56, 0, 0, 36, -18, -28, 49, 0, 29, 0, 0, -58, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 282, 0, 0, 0, 0, 108, 0, 108, -153, 0, -30, -63, 0, -24, 6, 6], [-238, 219, -19, -42, 0, 69, -69, 19, 84, 0, 58, 0, 0, -58, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -149, 0, 0, 0, 0, 166, 0, 83, 149, 0, 2, 4, -4, 2, -15, -17], [0, 0, 0, 0, -180, 0, 0, 0, 0, 2, 0, 2, 90, 0, -72, -62, 0, 0, 72, 72], [472, 262, 0, 38, 0, 0, -29, -52, -19, 0, -71, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 80, 0, 0, 0, 0, 0, 0, 0, -16, 0, 36, 26, 52, -80, -44, 44], [-326, -606, -46, -35, 0, -113, 0, 0, -35, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [41, -181, 82, 0, 0, -16, 8, -41, 82, 0, 30, 0, 0, -60, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -141, 0, 0, 0, 0, 76, 0, 0, 262, 0, -5, 0, 0, 20, -5, -20], [0, 0, 0, 0, -30, 0, 0, 0, 0, -95, 0, -190, -19, 0, 15, -12, -6, -19, -15, -19], [-268, 254, -14, -14, 0, 121, -121, 14, 28, 0, -10, 0, 0, 10, 0, 0, 0, 0, 0, 0], [-233, 65, 65, -42, 0, 2, -4, -130, 42, 0, -56, 0, 0, 28, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -148, 0, 0, 0, 0, 6, 0, 6, 101, 0, -48, 24, 0, 54, 102, 102], [0, 0, 0, 0, -472, 0, 0, 0, 0, -60, 0, -120, 89, 0, -50, -40, -20, 89, 50, 89], [0, 0, 0, 0, -77, 0, 0, 0, 0, 0, 0, -53, -21, 0, -28, 15, 15, 28, 50, -22], [0, 0, 0, 0, -42, 0, 0, 0, 0, -96, 0, 96, -175, 0, -50, 21, 42, 42, -8, 8], [0, 0, 0, 0, 213, 0, 0, 0, 0, -66, 0, -132, -119, 0, 94, 8, 4, -119, -94, -119], [-73, 19, -92, -39, 0, 4, 4, -92, 0, 0, 5, 0, 0, 5, 0, 0, 0, 0, 0, 0], [505, -11, -11, 79, 0, 30, -60, 22, -79, 0, -30, 0, 0, 15, 0, 0, 0, 0, 0, 0], [606, 320, 0, 26, 0, 0, -8, -34, -13, 0, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -229, 0, 0, 0, 0, 237, 0, 0, 462, 0, 24, 0, 22, -4, 24, 4], [271, 29, 29, -24, 0, 45, -90, -58, 24, 0, 74, 0, 0, -37, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 12, 0, 0, 0, 0, -44, 0, 44, 458, 0, 14, 9, 18, -12, 2, -2], [0, 0, 0, 0, -30, 0, 0, 0, 0, -33, 0, 0, 66, 0, -108, 0, -17, -6, -108, 6], [-23, -205, -46, 0, 0, -94, 47, 23, -78, 0, -33, 0, 0, 66, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -54, 0, 0, 0, 0, -200, 0, -100, 54, 0, 31, 4, -4, 31, -47, -78], [0, 0, 0, 0, 132, 0, 0, 0, 0, 222, 0, 111, -132, 0, -34, 13, -13, -34, -21, 13], [648, 326, 0, -24, 0, 0, 36, -4, 12, 0, -27, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_448_o_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_448_3_o_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_448_3_o_b(:prec:=20)
    chi := MakeCharacter_448_o();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_448_3_o_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_448_3_o_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_448_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![20820969, 0, 45543303, 0, 70955595, 0, 52972326, 0, 28554561, 0, 5523417, 0, 755737, 0, 50330, 0, 2415, 0, 59, 0, 1]>,<5,R![1982907, -490239, -354717, 97686, 62223, -31581, 4213, 210, -67, -3, 1]>],Snew);
    return Vf;
end function;