
// Make newform 2142.2.p.h in Magma, downloaded from the LMFDB on 18 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2142_p();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2142_p_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_2142_2_p_h();".
// 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_2142_2_p_h();".
// 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 := [388, -2792, 10964, -26632, 41270, -42696, 30950, -16208, 6149, -1804, 674, -424, 244, -108, 36, -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], [-76395493561780151130848, 506538614981225614494006, -1877104889236322930463630, 3453926376238674918082904, -3783929615224231476364634, 2790392699169146971029975, -1466608751795197602942023, 542602547851280460555612, -152654468589152385737773, 59079873652956449074792, -37733928354753219350747, 22035837556550890788974, -9703641051278112982219, 3208590268295068201125, -693869388216574127960, 83697067141096116888], [-3358102369452620884074, 121948972426168563545126, -426895847355200568983120, 864251874465104075410743, -1088755784388747813520602, 912204050648384912306555, -536773719072913944072475, 224369814277419183003060, -66933575683604009820954, 19963021516308940633542, -12074013251513553297532, 7777123945042826055521, -3701190392156518965305, 1378788799153837318231, -334014119879790158171, 49982728503937424302], [104272945613991252879888, -677319109984523107813940, 2314521422074226931653352, -4451463448985146208068866, 5349591721392553436443770, -4329905617177027515944238, 2481581723393047415060683, -1013053059204338560216780, 300802621739720484005801, -94563280990424275804432, 57640710718669544347397, -36279854323899924358840, 16963331200025317471493, -6183254630377521138700, 1469995026914423724846, -214394476059204155940], [-112635588667614036121246, 879310162154290044015786, -3407017707222247243882006, 7314818004510471842671810, -9479868369697540668383448, 8040513055437175312092331, -4748283775502786629048561, 1987425573139125473701044, -583930956806773207704023, 171439164869721140288402, -106706904868494847561699, 68310948748720342568346, -32866784842726203197479, 12142653177386855640965, -2950912076889156818026, 426781146651813411300], [-338839287028343294661310, 1897809886849165083524058, -6460912624634877129215666, 13068976208935164400574592, -16130025392627510442983300, 13158639538229019330316897, -7540143219452943033635444, 3061315430800084763253964, -882425253779165735346270, 277662823017572123894450, -175723073075201760226340, 109433055893938092396788, -51610969076436181762140, 18493479662593302633675, -4376817517934527811606, 599951581167756684496], [-319785014539055370317984, 1909857010530659729094760, -6897038323903945504989760, 14500398471852032178758804, -18232729043119896225507282, 15027463429723995299311800, -8665628249546759695697497, 3530675474976371657521072, -1014343518163621032119266, 314671876813684187040744, -200539417748269754278787, 125285764756570584473255, -59449440504393264097776, 21344021111792446830230, -5071379448529711928044, 692689064430707852081], [-44739747780844192361008, 562523049761601483792788, -2417106023645750296783048, 6744037342619549577042876, -10543131891130648496563216, 10030916787923567313498022, -6421973363776157130228789, 2879309021700076544121800, -867413741462952935486856, 219671291490941591218927, -130349547563871245450182, 90308990897787457030701, -45482637129521982927154, 17973349184141107650735, -4580444688927294085539, 723645995847453730375], [423336165377736737634650, -2447704630289010604398518, 8461906245978232853652098, -17280406551598649172145478, 21520172082653530119860628, -17687558752172205917254021, 10197104817369951715976142, -4166101430396557502356684, 1205548245167674917684122, -374005694949074405644954, 235930426855070890564053, -147728232943551047861635, 69956104174946912050426, -25225174361927438359239, 6003431659039223279389, -832400517928609161633], [-422065577481920090333574, 2576392729078516605630586, -8937641769845813698159810, 18343936806760614201704848, -22975480042736294227253610, 18968460506654124800578275, -10974047535600343184733319, 4501208738323750448582056, -1304261128870665682460173, 401251894759331730277362, -252895965114646748537750, 158789878035155524993667, -75400243820413954797775, 27275084409094928861139, -6512831813634688499321, 907008254631839283197], [413462689269289153891870, -2540522601928176694572610, 8769341787517303760630746, -17975063069851650046668138, 22578272400259646690276540, -18703517451337506158770157, 10855641158211228810990939, -4470341301584103965499194, 1299721602366510836063448, -397141741284622115719176, 249137932805835998166334, -157019431549438540573768, 74668341784381686276870, -27130322678742158158273, 6497943847961854870173, -913503855597074493028], [522456062716210459196306, -3086311069616311089107052, 10603157624481140679865168, -21661938449948699408700610, 27123031869892314729014682, -22441596058815660278651418, 13016036855322253320567143, -5355314534477784710327638, 1558989141201386380062768, -477181148550334417174384, 298921502515770130273068, -188356251438068684867318, 89483345105699397055364, -32513914381511823245486, 7782724746808995011409, -1096273854753039552366], [6562235677822830, -39019858127949324, 135202838889544168, -277066206541023404, 346616329436288712, -286112234910474506, 165538244736382493, -67897354507591046, 19696331787165103, -6060167566143162, 3813895431244811, -2395808023801618, 1137194504320201, -411624462299224, 98291343706372, -13722817214568], [1722045027702135056412422, -10307818488813526118399978, 36185353484313506173145914, -73941163951111855107256924, 91887577728360199214257924, -75403271963912992328542157, 43399798670659419298879977, -17705790448903327729003500, 5121405513732680188478711, -1593977841918984532826206, 1005893814649427170039772, -629143284163554839353161, 297610354736793191280555, -107178801384215486960287, 25477699019184746464625, -3527049722754317143905], [1744054756938595731239022, -10690053763243833065877978, 38074537316596223823901570, -80056865567094840130254050, 102111029118157588201689822, -85465808303840115623311329, 49990352526667460439567806, -20710559672797892706348486, 6035110167835369906823324, -1817449080765909995003446, 1135698643478905653063717, -721381073169232292068817, 344517863649785672516338, -126070578540969894583909, 30341459025870070471317, -4315598932975937341439], [2463792384185489887992886, -14918900250227352071285144, 52738926801394581422131092, -109299771207163289985137836, 137545772308397081434586148, -113915049276630032759059780, 66061366418485975909690664, -27141293221876034883000134, 7871607887508144170182031, -2413103460291467735591443, 1516947736253662539724123, -955482360175181969651513, 454047630363642034367095, -164692431449799412159685, 39370424315931024658315, -5513804559118835924977]];
    Rf_basisdens := [1, 14524676816773762526606, 7262338408386881263303, 14524676816773762526606, 14524676816773762526606, 14524676816773762526606, 14524676816773762526606, 14524676816773762526606, 14524676816773762526606, 14524676816773762526606, 14524676816773762526606, 14524676816773762526606, 160550220964286, 14524676816773762526606, 14524676816773762526606, 14524676816773762526606];
    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_2142_p();"
function MakeCharacter_2142_p()
    N := 2142;
    order := 4;
    char_gens := [1667, 1837, 1261];
    v := [4, 4, 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_2142_p_Hecke();"
function MakeCharacter_2142_p_Hecke( : Kf := false)
    N := 2142;
    order := 4;
    char_gens := [1667, 1837, 1261];
    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], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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 := 2;
    raw_aps := [[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, -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], [-1, 1, 0, -1, -1, 0, 0, 1, 1, 1, -1, -1, 0, -1, 1, 1], [-1, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0], [0, 1, -1, -1, -1, 1, 1, 2, 1, 0, -1, -1, -1, 0, 1, 0], [-1, 1, 0, 0, -1, 1, 0, 1, 1, 0, -1, 0, 0, -1, 0, 2], [1, 0, -2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 1, 0, 0, 0], [1, -1, -2, -1, -2, -1, 0, 1, 0, 0, 0, -1, 0, 0, 0, 1], [2, -1, 0, -1, -1, -2, 0, 0, -1, 0, 0, 1, 2, -1, 0, 0], [-1, 0, 0, 0, 0, 1, 0, 0, -3, 0, 0, 0, -1, -1, 0, 0], [-1, 1, -1, -1, -1, 0, 2, 2, 1, -6, 0, 0, 0, -1, 2, 0], [-1, 1, 0, 0, -2, 1, -1, 1, 1, 1, -1, -3, -1, -2, -1, 3], [-4, 0, 0, 0, 0, 1, 0, 2, -2, -2, 1, 0, 0, -2, 2, 0], [-1, 1, 0, 0, 1, 1, 2, 1, 1, -2, -1, -1, -1, 0, 1, -1], [-1, 1, 0, 0, 0, 2, 1, 1, 0, 0, -2, -2, 0, -1, 0, 2], [0, -1, 2, 1, 1, 0, 0, -2, -1, 0, 0, -1, 0, 1, 0, 0], [-3, 0, 2, 1, 0, 0, 0, -2, -1, -1, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, -1, 0, 0, 2], [4, -3, 0, -3, 1, -5, 0, -2, -1, 0, 0, 3, 2, -1, 0, -2], [-1, 0, 0, 0, 0, 0, 3, 0, 0, -5, 0, 0, 1, 0, 2, 0], [1, -1, 0, 0, 1, -1, 0, -1, 1, -2, 1, -1, 0, 2, 1, 0], [-3, 0, 1, 2, 0, 1, 0, 2, -2, -2, 1, 0, 0, -1, 1, 0], [2, -1, -4, -1, -3, 1, 0, 2, 3, 0, 0, -3, 0, -3, 0, 2], [0, 0, 3, -1, 0, -1, 0, 1, 2, 2, -1, 0, 0, -2, 2, 0], [1, 0, 3, 2, 2, 0, -2, -6, 0, 2, 0, 0, 0, 1, -1, 0], [4, 0, -2, 0, 1, 1, 0, 0, -1, 0, 0, -2, 2, 3, 0, 0], [2, 0, 2, 0, 0, 0, -1, -2, 0, -2, -3, -2, -2, 0, 1, 2], [1, -1, 0, 1, 1, 0, -1, -3, -1, 3, 2, 1, 2, 1, -2, 1], [2, -2, 0, 0, 3, -2, 1, -2, -6, 3, 2, 4, 4, -2, -4, 0], [-8, 3, 0, 3, -5, 2, 0, 4, 5, 0, 0, -3, -4, -3, 0, 4], [6, 0, -2, 0, -1, 0, 1, 0, 2, 1, 0, 0, 0, 0, 0, 0], [4, -1, -2, -1, -1, -1, 0, 0, -5, 0, 0, -1, 2, -1, 0, 0], [-1, 0, 2, 1, 2, 1, -2, -4, -4, -2, 1, 0, 0, 3, -3, 0], [2, -2, 0, 0, 2, 2, 0, -2, 0, -2, -2, 2, 2, 2, 0, 0], [-4, 0, 3, -1, -1, 0, 1, 1, 5, 4, 0, 0, 0, 0, 0, 0], [-1, -2, 4, 2, 2, 0, -3, -4, -2, 8, -1, -2, 1, 2, -3, 0], [1, -1, -1, -1, -1, -1, 0, 2, -1, 0, 0, 0, 2, -1, 0, 2], [-2, -2, 2, -2, 2, -6, 0, -4, 2, 0, 0, 4, -4, 0, 0, -4], [0, 0, 0, 0, 2, 1, 2, 0, -7, 5, -1, 0, 0, 1, 1, -2], [-1, 1, 3, -1, -1, 0, -2, 2, 1, -2, 2, -4, -4, -1, 0, 0], [1, 0, -4, -3, -1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0], [3, -2, 2, 2, 2, 0, -8, -6, -2, 8, 2, 0, 1, 2, 0, 2], [3, 0, -4, 0, 0, 0, -3, -2, 0, -6, 3, 4, 3, 0, -3, 2], [2, 0, -4, 0, -4, -2, 0, 2, 10, 0, 0, -4, 0, 0, 0, 2], [3, 1, 0, -1, -1, 0, 1, 3, 1, 0, 1, -1, -6, -1, -1, -1], [2, -2, 0, 0, 5, -2, 3, -2, -7, 2, 2, 10, -2, 0, -2, -2], [3, 2, -5, -2, -2, 0, 2, 2, 2, 4, -2, 3, 0, -2, 2, 2], [-3, 3, 0, 0, 1, 4, 4, 3, -3, 2, -4, -3, -2, 2, 5, -4], [5, -3, 2, -3, 2, -2, 0, -1, 3, 0, 0, 5, 6, 1, 0, -1], [1, 0, 0, 1, 0, 3, 0, 0, -2, -2, 3, 0, 0, 1, -1, 0], [6, -2, -2, -2, 6, -4, 0, -4, 2, 0, 0, 0, 0, 2, 0, -4], [3, 0, -4, -1, -4, -1, 4, 5, -4, -8, -1, 0, 0, -1, 1, 0], [-6, 6, 0, 0, -6, 1, 0, 6, 10, -4, -1, -5, -4, -3, 3, 4], [-3, 3, 0, 0, -3, -1, 0, 3, 5, -2, 1, -7, 1, -2, 1, 3], [-1, 3, -1, 3, -5, -4, 0, 0, 5, 0, 0, -4, -2, 1, 0, 0], [-3, 0, 7, 6, 4, 0, -4, -6, -2, 2, 0, 0, 0, 5, -5, 0], [-6, 6, 0, 6, -3, 11, 0, 6, 1, 0, 0, -6, 0, -1, 0, 6], [2, -2, 0, 0, 3, 0, 1, -2, -10, 7, 0, 2, 2, 2, 0, -2], [-3, -2, 1, 2, 2, 0, 0, 0, -2, -2, 1, 1, 2, 2, -4, -4], [-2, 0, 5, 3, 0, 1, 0, -1, 6, 6, 1, 0, 0, 2, -2, 0], [-2, 0, 9, 1, 4, 1, -4, 3, -2, 2, 1, 0, 0, 0, 0, 0], [9, -3, -3, -3, 5, -1, 0, -2, -15, 0, 0, 0, 4, -1, 0, -2], [-8, 0, 0, 0, 0, 0, 2, 2, 0, -8, -2, 0, 6, 0, 4, -2], [1, -3, 2, 3, 3, 0, -1, -5, -3, 10, 1, 1, 2, 3, -7, -1], [-3, 3, 0, 0, -3, 2, 0, 3, -2, 5, -2, -3, 5, 1, 4, 3], [11, -5, 0, -5, 3, -10, 0, -3, -1, 0, 0, 5, 8, 3, 0, -3], [-3, -1, -2, 1, 1, 0, 2, 1, -1, -17, -1, 3, 4, 1, 1, -3], [-3, 3, 0, 0, 1, -2, 4, 3, 5, -6, 2, 3, 6, -2, 1, 4], [5, 0, 3, -4, 0, -5, 0, 0, -12, -12, -5, 0, 0, -3, 3, 0], [4, -4, 0, 0, 3, -1, -1, -4, 5, -8, 1, 4, 4, 1, -3, 4], [-6, 7, -4, -7, -7, 0, 6, 12, 7, 4, -6, -3, -2, -7, 6, 2], [-6, 0, 4, 2, 1, 0, -1, -2, -8, -7, 0, 0, 0, 2, -2, 0], [-3, 3, 0, 3, -2, 7, 0, 5, 2, 0, 0, -3, 2, -4, 0, 5], [-4, 4, 0, 0, -2, 4, 2, 4, -10, 12, -4, -2, -2, -6, -2, 6], [4, -4, 0, 0, -1, -1, -5, -4, -3, 4, 1, 2, -2, -1, -5, 2], [2, -2, -8, 2, 2, 0, 8, 0, -2, 4, -2, 10, 6, 2, -2, -4], [4, -8, 4, 8, 8, 0, -4, -12, -8, -6, 8, 4, 4, 8, -8, -4], [0, 0, -2, -8, -2, -1, 2, 8, 6, 4, -1, 0, 0, -4, 4, 0], [-4, 1, 0, -1, -1, 0, 2, 0, 1, 2, -4, -1, 4, -1, 2, 2], [-2, 0, 2, 2, 1, -3, -1, 0, -7, -6, -3, 0, 0, 1, -1, 0], [3, 2, 0, -2, -2, 0, 2, 4, 2, 0, 2, -2, -7, -2, 8, 0], [-2, 2, 0, 0, -4, 0, -2, 2, 4, 0, 0, 0, -10, -4, -2, 0], [-1, 1, -3, 1, -3, 6, 0, 4, 1, 0, 0, -4, 0, 3, 0, 4], [-6, 0, -6, -2, -4, 3, 4, 8, -3, -7, 3, 0, 0, -5, 5, 0], [5, -1, -2, 1, 1, 0, 1, -1, -1, -9, 4, 3, -2, 1, -4, -1], [-3, 3, 0, 0, -2, -1, 1, 3, 11, -9, 1, -3, -5, 2, 5, 1], [-3, 3, 0, 0, -5, 5, -2, 3, 3, 2, -5, -2, 6, -5, -2, 6], [-4, 0, 0, 2, 1, 6, -1, 0, 0, 1, 6, 0, 0, 4, -4, 0], [-1, 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5, -2, 2, 10], [14, -14, 0, 0, 12, -6, -2, -14, -14, 2, 6, 11, 22, 9, -5, -8], [10, 4, -2, -4, -4, 0, 3, 6, 4, 10, -13, -2, -14, -4, 7, 2], [-2, 2, 0, 0, 0, -1, 2, 2, -9, 9, 1, 8, 4, -5, -3, 4], [0, -4, 4, -4, 8, -8, 0, 0, -8, 0, 0, 8, 4, -4, 0, 0], [8, -8, 0, 0, 11, -2, 3, -8, 0, -11, 2, 12, 18, 10, 2, -14], [6, -6, 0, 0, 9, -6, 3, -6, 10, -19, 6, 4, 8, 8, 2, -2], [-8, 0, 1, 11, 2, 0, -2, -7, -2, 0, 0, 0, 0, 8, -8, 0], [2, -2, 0, 0, 0, 8, -2, -2, 0, 0, -8, -12, -8, 4, 2, 4], [14, 1, -4, -1, -1, 0, 0, 0, 1, 16, -5, 3, -10, -1, 0, 2], [11, -3, -8, -3, 0, -3, 0, 1, 0, 0, 0, -5, 4, -4, 0, 1], [-4, -4, -6, -4, -3, -9, 0, 2, 29, 0, 0, -2, -8, 5, 0, 2], [8, -8, 0, 0, 14, -6, 6, -8, -30, 16, 6, 4, 6, 2, -6, -6], [-27, 5, -2, -5, -5, 0, 8, 11, 5, 11, -7, -3, 18, -5, 3, -1], [4, 0, 3, -7, -2, -4, 2, 7, -22, -24, -4, 0, 0, -2, 2, 0], [19, 0, 7, 4, 6, -4, -6, -4, -2, 4, -4, 0, 0, 3, -3, 0], [-6, 6, 0, 0, 4, -3, 10, 6, 3, -7, 3, 6, 16, -1, 5, -4], [-6, 12, -6, -12, -12, 0, 8, 22, 12, -6, -18, -6, -10, -12, 24, 2], [-27, 1, -3, 1, 1, 3, 0, 0, 1, 0, 0, -4, -30, -3, 0, 0], [-7, 12, 5, 12, -6, 26, 0, 10, -2, 0, 0, -7, 8, -2, 0, 10], [-3, -2, -1, 2, 2, 0, 0, -4, -2, -30, 10, 3, 8, 2, -6, 0], [3, -3, 0, 0, 8, 1, 5, -3, 1, -9, -1, 1, 11, 2, -1, -1], [5, -5, 0, 0, 4, -2, -1, -5, 18, -22, 2, 1, 9, 3, -2, 3], [-8, 0, -6, 0, -2, 16, 2, 4, 2, 0, 16, 0, 0, 6, -6, 0], [-10, -2, -2, -2, -6, -4, 0, 8, 30, 0, 0, 0, -4, -2, 0, 8], [-14, 14, 0, 0, -13, 7, 1, 14, 5, 8, -7, -14, 20, -7, 7, 14], [-8, 5, -10, -5, -5, 0, 8, 2, 5, 0, -1, 5, 16, -5, 4, 8], [11, -11, 0, 0, 9, -2, -2, -11, -23, 14, 2, 20, 6, 7, -4, -10], [-16, 0, -8, -6, -12, -11, 12, 8, 15, 3, -11, 0, 0, -11, 11, 0], [-13, 4, 2, -4, -4, 0, 2, 10, 4, 19, 1, -6, 1, -4, 1, -2], [-3, 5, 11, 5, 1, 10, 0, 4, 7, 0, 0, 6, 12, 1, 0, 4], [-10, -2, 0, -2, -8, -9, 0, 0, 8, 0, 0, 2, -10, -12, 0, 0], [12, -12, 0, 0, 10, -1, -2, -12, -6, -4, 1, 14, 14, 10, -2, -12], [-15, 0, 2, 0, -5, -12, 0, 0, 18, 0, 0, 2, -13, 2, 0, 0], [22, 0, 0, -2, 1, 3, -1, 4, 11, 12, 3, 0, 0, 1, -1, 0], [15, -6, 1, -6, 6, -12, 0, -4, 2, 0, 0, 7, 12, 4, 0, -4], [-5, -5, 1, 5, 5, 0, -12, -2, -5, 4, 16, 4, 6, 5, 0, -8], [-7, 1, -7, 1, 3, 6, 0, 6, -15, 0, 0, -8, -8, 1, 0, 6], [-4, 0, -15, -1, 2, 5, -2, 7, -2, 0, 5, 0, 0, -2, 2, 0], [-4, 4, 0, 0, 1, -1, 5, 4, -1, 0, 1, 6, -22, -7, -3, 6], [9, 0, -9, -8, -6, -7, 6, 0, 10, 4, -7, 0, 0, -11, 11, 0], [19, -8, 12, 8, 8, 0, -29, -26, -8, 16, 17, -4, -5, 8, -21, 10], [-5, 0, -4, 15, 2, 15, -2, -2, 8, 10, 15, 0, 0, 15, -15, 0], [-8, -2, 6, 2, 2, 0, -6, 4, -2, 12, 8, -4, -2, 2, 4, -8], [-17, 0, 0, 7, 1, 1, -1, 4, 2, 3, 1, 0, 0, 5, -5, 0], [-10, 10, 0, 0, -6, 5, 4, 10, -10, 16, -5, 4, -18, -8, 2, -4], [-4, 4, 0, 0, 4, 14, 8, 4, -18, 14, -14, 2, 8, -4, 0, 0], [-8, 8, 0, 0, -8, 1, 0, 8, 26, -18, -1, -10, -8, 0, 8, -2], [0, 0, 0, 0, 6, -6, 6, 0, -12, 6, 6, 5, 2, 1, 1, -8], [-16, 16, 0, 0, -11, 7, 5, 16, 25, -14, -7, -16, -22, -7, 9, 6], [12, 0, -20, -12, -7, 2, 7, 26, 12, 5, 2, 0, 0, -8, 8, 0], [-1, 0, -4, 1, -1, -7, 1, 2, 6, 5, -7, 0, 0, -3, 3, 0], [3, -3, 0, 0, 9, -11, 6, -3, -7, -2, 11, 3, -24, 2, -1, -12], [-26, 0, -2, 8, 2, 8, -2, 6, 14, 16, 8, 0, 0, 4, -4, 0], [-7, -6, 8, -6, 15, 2, 0, -2, -2, 0, 0, 14, -1, -2, 0, -2], [17, -9, -9, 9, 9, 0, -2, -14, -9, 14, 7, 18, 6, 9, -6, -4], [13, -13, 0, 0, 21, 4, 8, -13, -23, 2, -4, 23, -6, 12, -1, -20], [-12, 9, 8, 9, 11, 15, 0, -2, 3, 0, 0, -1, -6, -5, 0, -2], [-15, 0, 8, -1, -9, 2, 9, 8, 3, -6, 2, 0, 0, -6, 6, 0], [15, 0, 6, 0, 2, -10, 0, 0, 12, 0, 0, 6, 21, 0, 0, 0], [-5, 9, 0, -9, -9, 0, 3, 13, 9, -18, -5, -9, -8, -9, 9, 5], [6, -6, 0, 0, -3, 0, -9, -6, 11, -8, 0, -10, 2, 4, -2, 2], [-10, 10, 0, 0, -6, 5, 4, 10, 14, -8, -5, -21, -4, -3, 7, -4], [17, -16, -14, -16, -5, -11, 0, 0, -9, 0, 0, 2, 3, -1, 0, 0], [37, -11, -7, -11, 5, -14, 0, -14, 31, 0, 0, 4, 16, 15, 0, -14], [-10, 10, 0, 0, -2, 10, 8, 10, 0, 2, -10, -2, 10, 0, 10, 2], [-12, 6, -6, -6, -6, 0, 12, 14, 6, -14, 0, 0, 4, -6, 14, -2], [-11, 0, 2, 0, -8, 5, 0, 2, 1, 0, 0, 2, -7, -9, 0, 2], [-11, 3, -3, -3, -3, 0, -4, 4, 3, 8, 7, 0, 10, -3, -8, 2], [9, 0, -9, 2, -8, 6, 8, 6, -2, -10, 6, 0, 0, 9, -9, 0], [4, 0, 7, 3, 0, -3, 0, 3, 14, 14, -3, 0, 0, -6, 6, 0], [3, 2, 9, -2, -2, 0, 2, 0, 2, -18, -14, -11, -12, -2, 12, 4], [5, -5, 0, 0, 1, -3, -4, -5, -7, 6, 3, -6, 26, 7, 2, -4], [-8, 0, 8, -2, 2, -16, -2, -2, 18, 20, -16, 0, 0, -4, 4, 0], [11, -11, 0, 0, 9, -13, -2, -11, -3, -6, 13, 15, -5, 4, -7, -17], [-15, 6, -6, -6, -6, 0, 7, 8, 6, -27, 2, 0, 13, -6, 0, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2142_p_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2142_2_p_h();".
// 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_2142_2_p_h(:prec:=16)
    chi := MakeCharacter_2142_p();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_2142_2_p_h();".
// 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_2142_2_p_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2142_p();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![12544, 3584, 512, -8576, 41136, -4384, 0, 704, 7201, 32, 0, 0, 210, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;