// Make newform 75.22.a.j in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_75_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_75_22_a_j();". // 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_75_22_a_j();". // 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 := [-684850637484000, 2888057920800, 59940607490, -23850203, -530373, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0], [-1, 4, 0, 0, 0, 0], [-2828475, -1100, 16, 0, 0, 0], [-909462022508, 36666498918, 37318593, -456377, -157, 1], [2940276712412, -112675632118, -66403649, 712633, 157, -1], [11600260895639, 48958029442, -249005991, -544465, 571, 1]]; Rf_basisdens := [1, 1, 1, 4004, 4004, 1001]; 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_75_a();" function MakeCharacter_75_a() N := 75; order := 1; char_gens := [26, 52]; v := [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; function MakeCharacter_75_a_Hecke(Kf) return MakeCharacter_75_a(); 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 := 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 := 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 := 22; raw_aps := [[15, -1, 0, 0, 0, 0], [-59049, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [221197120, 19683, 49, 1, 0, 0], [-13051322369, 1089973, 856, -7, -22, 5], [19343717437, 16269174, -10225, 192, 24, -41], [897755498349, 1817817035, 1581598, -2477, -38, 561], [1161770576808, 495339975, 3120522, -15105, 798, -2147], [49295643731423, 9502341945, 18751382, 19741, 38260, 6181], [-649494712225261, -672270729223, -285261600, -1093903, -252640, -15905], [82134241435154, 109726643871, 427740317, 3620509, 571692, -147244], [-4348206583309412, 6053235920406, -1353709002, -35939638, 3513864, 54740], [18776843987443269, -918470060407, -9494465230, 66201257, 7767938, 6182151], [-2230212264864090, -1710617680719, 24098195076, -120771947, -3067722, 3760761], [-40539911597465698, -16645600760068, -48020087986, 31728304, -59951504, -14866322], [92188076557950117, -457869494067319, -14170250238, 548552517, -240014172, -31693719], [-1167261537163076950, 187342105277152, -480592048664, 450027504, -736420800, 382429144], [-1705585442274439029, 768622219250436, -1987595684115, -1714908382, 36522072, 255238947], [1495388408654534528, -1753033449013857, -2330566481472, -4807105613, 2330873922, 60620699], [3000121344133000703, -4352407507149649, -12342344929028, 10905027519, 4282719048, -1873775339], [13529113804116431132, 1164975244939998, 2325143951242, 3988710098, 6099236292, -2615196796], [-13187168464942142112, 13268060385847560, 3293544994810, -4751197684, 27162572412, 337321682], [39777973966115832594, -23832973295969916, -22288259005224, 2877328860, -41057878548, 6876039408], [-118740197357584871088, -54924537499891992, 20582166427116, 84112477464, -47283516420, -17280743208], [249832862838690670887, 221943586458790080, 105090571152148, 430757643256, -149295927768, 9935314628], [377642773955322036863, 85878233961313875, 342383368703694, -1333846083889, 170760821936, -6906659777], [-9402371611911428862, 484810731053188542, -199417264290530, -284400998038, 246582167724, 49699750248], [-38132322252390791307, 440440507365598749, -114615107435332, -1988438977159, 129075116762, -107912174439], [-140595842690915621147, 1300674914334263160, 181450532040781, 939264851318, 129534370272, -43037796681], [-1100019923928857095960, 289850466580673612, -359414172187024, -933046489628, -900222309884, 116897806024], [-2872133229004937683226, 7025675179569849990, -1861377661120684, 1146847088446, -1808692405872, -353981711242], [-9496658385962319755467, -4616018814626155875, -3299969493672284, -22103699035935, 1893408250350, 457975309765], [-6769476295580973162028, 16631458429353421914, -6831692191912840, 27218939890, -4559822692748, 1289841405778], [-2649472612005665327202, 1520768107703018394, -7963008121834318, 5139963179670, -7837000933560, -974636182972], [9081878934981937352348, 27144612490590882890, 13163160772488582, 45539773746614, -1395292259308, -71777681636], [-13938349807558290316772, -8072549863862695893, 5946536474367773, -14692196738239, -14086362597216, -85565156356], [-1437153288002888970167, 29767451620372071876, 17626950730648995, -118821613078618, -2531689401984, -935854822435], [-2831204035921662472818, -28427269923167944185, -19252424637293996, -129949195894589, -23014018882446, -3270214416577], [45382584721449415427049, 12759581712324772533, -18347976862295672, 101639859256861, 3188281431976, 11654489203587], [-43688235351148547835785, -43452836891118906699, -130323118885374816, -92937844635803, 41183366900260, 22227804224807], [-1501104129541232833960, -54947613773947276946, 181647339700879612, 51294234638070, -37754031945864, 29324409941614], [83751266954438837824719, 205208094812329439694, 36522187048897055, -412461056856872, 78154032199392, -16598380779465], [-322606222609162200288297, 245778575245327787285, -331819871331291306, -116738243214255, -37944039860568, 35907236549373], [148610749434285143780799, -281313358076385694566, 320928502687846894, 495981921876070, 62707092073716, -36619048924060], [-23852045252597648694396, -472076764393106361654, -409127612639236454, -330539645218178, 112148987818588, 5171968681344], [-443510430516845722150502, 23183060933654565093, -29637808104194743, -68246361822397, 466453749751248, -42790706703994], [-667377629087024361123708, 844865932069405104291, 247497522794043986, 1708655090105627, -109326785424546, 12089431195129], [-2140214150417857361042224, -177967173490491226149, 1068042869238797657, 2023997376542025, 710210508722616, -195526118747464], [3079007450313926013355419, 581766396000866697127, 393991185306402632, -1073395886216677, -44828600520454, 392847728241699], [-3723907027327776821425563, -1724747412929589584604, 331772526716074789, 3134041985578066, -1226268838054776, -52830266060645], [-3356881237329377388176197, 483231222876050409711, -818546375553356062, 9463310665483279, 938886183757306, 180680438530681], [-5291823480624079977176037, 474181169723486510191, 2938489624674359900, -3377455568399017, 1266945009815564, 166276876188249], [-1941746475326054242469599, -673959516068581362000, 3379175377826067046, -332672349665100, -486044377063944, -179231596992982], [-3875053918569347859505930, -6109937601446846155854, 1766610099256052200, 4930823306528418, -889303135125960, 604493061879406], [10371676666450484137535702, 2069389592876472191202, -2778388815394392560, -15083654248346094, -2143989162686928, -481897758822098], [-4271493256251669140243437, 9510146482577497473547, 7501218263206847292, -17459475198113365, -2517975663498640, -1331952926861183], [-27256054053855701864639653, 8373971437736966351989, 2734395391349562960, -10105608118260211, -3920045905536688, -123812116893053], [-8833758269478223588153522, -3221987430469594242402, -8227235872976884046, -840811860480750, 6133623662053812, -1356480630075220], [18482437130083582727248773, 3252989958599503805994, -18803754082691349775, 15260019346947680, 647183831781576, -1431892139928727], [-7621757456854839349808053, -16173819424401445551951, 5016756761189297822, 13702782545532241, 4609923434061862, 2101226564704903], [105472270164383327139108, 7208219340836005496049, 6513842060403549290, 14890351766752609, 17228409506795058, -478880330284937], [-25058808260435881638799718, -3576986302665205099992, 22172839172208666258, -45046207206280388, -17736610263672500, -344732308553974], [38816934778441208381885092, 3231215841821459554305, 9841122866535751730, -166582818510402591, -6897141212740446, -7615986433277201], [-27642023976341979838385716, -8162229975748902652374, 50948984963279448870, -31166158562578498, -13314914661069376, -7690665315968600], [-72134098979549033410083577, -47301637114680223071222, -5803232011540100956, -129187053840677558, 11289526311169044, 5726229397835814], [78924460530983422734889162, -69728838555492487241654, 28408459869167424468, -8090303834401238, 25691447623263220, -525019267932970], [-34755877007052497549103538, 147661220156724691815474, -6430349436160534444, 142564662922248498, -8132009155505580, -17001645312776690], [119997508431247843522072457, 38966747795782291836810, 18660711224880720070, 164361916902373142, 24900757439176308, 35553746827293308], [99928941670758554860096534, -150348568167162823060852, -8983937617745813504, 251253195786580772, 8419942143259412, -5780558860529080], [49155542997663368754245662, 139930181745424535422836, -1683619401299161504, -107390464812853868, -150916773379777104, 39157728624431932], [307938523811169516664013453, -50717674832018379648681, -9832435275275006758, 270338305432805839, -41273434058010266, 25228851118577221], [217314685147258970997247019, 61341338330213163755571, -16711276211306579200, -228800704750751813, 131824009473938488, -3603920867156723], [-4854521659119248033008626, 388905560007936116322843, -43304021408689157863, -298502450457037567, -68603100360572940, -10044613264809964], [725548661096171072291790487, -363096945848667466878678, 48020484302783025623, 312314867545476372, 4519829162877432, 63325095638367931], [478962294115035617527401408, 156949099880480452028709, -372663514483077272028, -709249537659954391, 55500844966709670, -59092550372696139], [152780918335818664462715130, -69164610134856264495234, -557093940332872942388, -352173630915040994, 45894319345601596, -53392195742441934], [752315544722937260735502577, 123520266662614605789777, -73320432101629425918, -652485217876338011, -273586673941466600, -9039237183408139], [-461263059565960679812364913, 763326211852604255340294, 415380729015088240239, -792550745299039904, -183067385213099736, -69743039680520649], [231348945172884179347561599, -1012365203180333992860585, 57057053716177821914, 1299378289283918063, 311531122474735382, 30354199552449141], [1923945556292129697519621263, 1232530177951494033266826, -549520151075812795896, 2297623977191375378, 24105115440597108, -68818813272784366], [2274478198884003436995861497, 14909307535843231767329, -145227890320098428976, -1140757259653630339, -20352341827177498, -187749505014250307], [-972800886747985169361576380, 82484825237051788719270, 62201593604561130698, -1513496915736350782, -641334649383585024, -277233689672191912], [466963510929467835475582965, -2501987272563247383948453, -811007827107987589274, -7293256410434297, 69601202214978136, 116586560357899815], [540105622148771598917385475, -2885106182927793801685938, -84314225217605437162, 1245812413730709674, -328316412837638484, -267169277227514496], [2678497574997923176374710910, 818387610576970729053555, -751256690302960687341, 935989504428689549, 640971653062555008, 149791418539638246], [-155534746902750610247086518, -1470100601277162137552406, 345148882797923557184, -3574051943216033230, 829144742473691588, -373297258994260950], [-347076467241550147547535330, 274479015755610153713362, 265296326249581714620, -6635860287352886910, -4685291781655548, 540915911077742742], [-1232823071141183949468726830, -2824590870961922798145072, -1167645138632768507878, 4723442194804949564, 384405017471274816, -583153009700268978], [-1723454633474157828449902575, -1357216141906402299955155, -277359689433539756470, -9994148884331660983, 187116045663694316, -187825783226894523], [1715918909456995831039028218, -1409250847134291809949846, 1186638214322480071894, -7048102556656676074, 615465915915677172, 1207913155710366412], [-59013289931709894679222771, -3245386284345911008240193, 1317464392728318946304, -11557073604373272149, 730466636423100886, 685938322941477871], [-9920217454206305919916597090, 1468591724785387783225668, -1010148766508587848882, -695429347200628960, 1082874925500434204, -53384904528508046], [6470914763914561421773539632, -5154587304347362041887499, -1336245661761075395053, -852955107224037633, 2635043816694877248, -92095549907844652], [5135411133192415609688000632, -342516293952965817962392, 3749061942566418904392, -9111660207442793792, -608146071071833796, 78725081324681612], [-5667841910554065512800419086, 4646871032179400309841201, 6966597611583814034360, 683217042884044045, 1580285920350125598, -633933268427040227], [-11838855677954094073265088969, -7873681326964499608942937, 3982132178646599566716, -9618905200726258161, -2508423634340152116, 295449596486655105], [-21144232357527320524727127604, 1818266314980593875421998, -541825570437300699862, -5006712929838063086, 1429326120052867252, -418066620969307580], [-17820468251807242408190749, -8981283495664735411561255, 3518731937649324621610, 24785832206144169993, 687648015520312914, -596927476411073873], [-16743661543876359307683182782, -10441970501846481471214089, 351167382635065562722, 28028218288129417599, 4178445282386355438, -608801353171182843], [-18203847008883867849168192497, -23382747856639887574237542, -833466448415711321541, -11995121974056299652, 13868006932886810448, -898596829396833937], [-10762160407262595785094952662, -3628324010749263981967638, -14329914703997538118278, -31490077421457357130, -10417259601781034232, -1696566955862574260], [-19539501079488439783286575882, 18296707431364047700926000, -3669949755382239832278, 58186522434768155612, -7488873384871197496, 3325670432737994278], [-39937536737768995451158396929, -20228757704845298287784211, 2816768802084037607840, -29599618760923301359, -23177759899655174110, 2214954593436665757], [-12744756817341573831259525449, 41140834557621501175765053, 23131691079512860741018, 68550354933946250197, 2879913894696434542, -4970061196385097609], [4155677187439667198496526860, -51881317556061456587440485, -737803338483334442680, 30218456901412163631, 3457953368279365386, 11901086729553695], [32909370823441453344173583953, -10608884571817039179918996, 8467811604574264456304, 49224049003936733372, -478251361035981840, 9725426805086129332], [-59389941062568661153667899395, -26155670038146861948112227, -12504498918292520669672, -50563960207132849991, 8996068786709888686, -10556574487157345535], [9792759715412775572466100085, -33670108439652038988650567, -26272419252276914491334, 73187819978924476113, 23033601592932337518, 7197459005268213499], [-35270075173341487860396670078, 2395329825279072896618798, 35934399501837026779152, -90983810549263441234, -5966837853812486296, 1511263131887256442], [21955947477595240063375026777, -53998483074969758593606650, 43612954225263512441360, -25643091450895402882, -2948249666496571644, -1800714628169217122], [8926658091104066922401534348, -59639035143075532066780884, 19888212082858600826662, -124740928415763039464, 40521812364756608460, -5379186109403669438], [68509895618705124291991168654, -42784732659790630558128024, -43885357091404121036278, 12973000432752076212, 11365825978968330768, -15589860929164553450], [-125978763288144302596077337283, -17573723599784850057046227, 40386741482212229225986, -26034793457810636875, -10637202932265704050, -2614661487992144737], [49253820617545961549833892136, -77208257410536925586303829, -10407019962154641454690, -76134060845131221397, -8544140991232433346, -8149375211405005355], [-30246751823718047469166237012, -112339534594319278907374539, -19914693688599198052055, 88133189969291668347, -5862137887310078076, 1729908766278829418], [38068558044250066807968127302, 16141070713849289658469986, -88214001454246906890160, -211542753951558751030, 22277573302056583268, -7480224353043538110], [33291223389917056983524180850, -22824712034292509993835882, -28569751400432090867626, -29906710967579406998, -58973344409094257352, -3947505189114069404], [-228827163582972277749694551508, -35226530694002001256586296, 55776999483286776982088, -241853953851011594744, 6885758062407342472, 6608445026610326104], [-39392681403519676669358843830, -29072720402604857426582156, 26693330241168261362862, 169800406975603269584, -112261613835978417892, 32912425205178707938], [16768365674918183999470767373, 9056177212387028893761985, -91645986153307045144840, 33124994506767273117, -66168965272979015154, 15887789318922837437], [-117892434636692072445095405470, 4607788882337796854341008, -4901657338527731953046, -338660821312467501572, 54839424449178944352, 573457703165304990], [132247300374564705781345813520, 163009082853509117191963542, 46969594117336117623246, -192293863952949730654, 48148835515916997204, 30052583119063116168], [-330360260725014066155915954741, 67451660569306584993222573, 121454075183181163924568, 59657735038947541149, 33581326566810651612, 18161339438056865615], [242144750307110179669669729972, 62751406775154956430196404, 130430650760060841654020, -21863199323467875564, 119774716605436115532, -48518489200517392252], [-217691993262571582103669908014, 102742245841170302920849986, 11575911014244132886512, 186382129588388195658, 143278881449089381476, 17609906009135390146], [227067971748657137826864360915, -199711109930716334677332331, -12229066901910997003072, -169186611220824997291, -159477536924156193556, -5472982956915534033], [181567924745291854286420595681, -40751062590146912613733392, -1728913981579240623939, -979390174645869180834, -112480180622973187608, -17986704509748543977], [-333761175887046794711324462339, -23528727483995703640220589, 6523963228499675865710, 757187560079585521503, -61608648460780077840, 95007755253703159967], [478915558383051153835347737208, 299080718467300736765807283, -144149668226705193863603, 724958597946161539689, -108226352194907204280, -13648945269400024724], [360532648175393180515822117238, -223137678775318312277227212, -325988187892706083572376, 980528427846602909220, -147579929961331566912, -55757154205376982172], [-72057971929808216013406969628, 207511657245381821627848320, 65090167805929167353876, -94691055922256633064, 13516530740954956800, -117518575497134119716], [30378102291633054752907710500, 487769128499525554729371776, 135500423248709618380952, 451389219798045800520, -211048614983964555996, 3791276954455387340], [-1216689014076958794600586938654, 235274016595230694154237346, -186241012371170131363872, -42474786338913088798, -378157572674370513360, -120753839774940979170], [826901010799728567015196252339, 656284092071420476719810252, -134126945566773453937855, 149139985620009049762, 106021891439513209272, -113593914131169404137], [273715972534607203951225862604, 476012379735516742934086464, -146370678402636313704008, 692041946385076483504, 574061296710368028496, 74825400113902350456], [-440997105922635359922979353595, -289475226376683689470061862, -148638852783814452447300, 1870555402457166121738, -236252166461073409836, 84639318923906350366], [-608478211501674625185318923553, 753227485632426298246088007, 381814084509310378974718, 1858373576013827400883, 418152984516949351624, -46142765479705674093], [820250435588580426858608882254, 706946984888696872582048731, -486542748578418126472540, -1862424659943551858665, -214596172866089117790, 238799372610816652011], [-1726681732194655616213841605266, -975782960556130444596610286, -161014368706094265902524, -850435749273069483878, 599288619271616881456, -95477387136725442758], [-2833166580167618414245646263607, -1821322302155008061633671783, 794357619925313729641358, -691945703618228949663, -584572149723296585310, -11428342242436435573], [-2480439847236727992742331283334, 1365254883271241435154729825, 17896150053375911364182, -517562896953378869799, 127750425883439234154, -278255533709261492381], [-785080191073631886564553421170, -368028605692227596136840504, -813076896531991491398350, 1698853362777660179356, -131844294952142640116, -247937714016048418550], [345841605834673224730570338712, 195888999663497094746887647, 632873421482019802316223, 2776623814377050017241, 414318579723861227940, 108569162604208540610], [-2234624103809330470859015540815, -1408327333488890333565376183, 485009027293434101257608, 4584403574802679624229, 28648491761513753342, -230266191277799113067], [6971496750653914691907037334, 429958596628611581068770732, 798733411810527683423666, -1282828436385353739512, 1230481721137394947584, 158940375354626786138], [-2078637448984745792619433278995, -58206808567892696072408329, 1879098750165831145235302, 1965788009733639135755, 592589716105055167796, 67192005654363875687], [1709993183612678464561036017637, 742493662632202626273869064, -475442241133641241477823, 1315086016229892458254, 1826567521841748731760, -577683999517457179149], [-5401561562517813568782961861526, 1155137680462922297233749230, 1327774250269406863857668, -1000116847744645307658, -66697643085807748368, 401193318398979707774], [-972166310168615870041299786234, -664513537461488524810304154, -105221065215659968369164, 1346292812210627323190, -1526437811197806918516, -229946769434759832558], [128823708330716783152355547430, 2187739423021547745662941678, -659288500160456561402596, -1184157295109508740178, 1187666108657886993228, 522820709780685207746], [-3140320038990290486647290964083, 833834733727559418523681848, -1138261111739781962795551, -12071111937457252787426, 817992879733135824144, -562031844798039581117], [1721666768882003149543832698662, -2412443932947766323661347890, -1019670518875832884574628, 8269812721191712613790, -2396566300096720382172, -231642864172630967238], [4240922095150437020739709232878, 1595226107605612365855945009, -749230535632485581872254, -297031471597450287759, -1160683771678384695462, -64823966556484693785], [-5182007477915875372800658420306, 4297957560179514175737403800, 1751535035114647461625534, 6001075159351740595620, 329867917650476556612, -155192640242012120522], [-2010411571296668664732110947736, 4319109245569916274306223788, -256496908318935652616792, 10450063884014457828476, -739075610043679898352, 358706608601836145596], [1190833197960602778006710614718, -86891543475163661952184340, -2192606827144967340063466, -2824699529327305712456, 1146701910527316934204, 463710746129132694850], [3043743474999362926556773889982, -2282947363917923198021005203, -2363271282408634678635393, 16548188334051112634135, -1797145741929886533252, 40459884905720927004], [-2845757704657670882450896443411, 1445485047439584444351661523, -4452544059276482884482706, -9285316087718154976333, 3018883406432693736638, -1144809100467748599843], [-9125998740771330458661303256705, 1537797182846522964000458310, -1118843831392281447409008, 8555234983765682848318, 130518289995995388396, 184210753956531447878], [-1620661017949549394098692330298, 4672471417693772371484636552, 678564504045122645568990, -1685608355666930624524, 6954898484879978461412, -1270743933695819233466], [-9900404528215446551697962155523, -2009701503435443649478045749, 708490569165047506216660, 18527350919830939945151, 2849425579740674571806, -495641274263339137369], [8257225647678591506777892983180, 1667126241901605833588097860, 5140628374297433703516648, 726362933627017981412, 2840345829042545663816, -10136981291001743588], [-7290218880164334043053036397388, -8885610484622756431280767476, 1434763594786735693295718, -11439255270485828911560, 493506614373095461068, -833893820090347878366], [-5173617946104127526027340809944, 1322750008987483456873234434, 2667193858762117664731832, -4400854239600149527550, -8799625376125367034272, 87171911447162921206], [-10291967850703694997835517106564, -2372694222872953360119490, 8497759419690769816000576, -22909646280077633609858, 3705379966218203826424, 772881363866091723786], [1465504271016989807765975990668, -8686591049799555666519790800, -2510595477134592128218432, -627462947388398674296, -5823345695813639387100, 2108748266300416989140], [-9517656514624404266290776128068, -6364629503836659187578686547, 1068293023888595438232729, -7832477593795899843629, 2568633791189230857876, -1013284024399883912942], [-4197298448846036943100986649062, -15888869031660013658459875500, 3605921492737743579905168, 9305885665858275305412, 4269958681204914779160, -500824034991452087532], [-2481768342558891293263732775919, -11514029609895652856659091220, -3433578339779557254919474, -3799845353406202875536, 10141675791786011248920, -2248700494921399488282], [9994603477824371116049000762128, -19200390904277845301694517646, -198400528703290699384890, -17369174257361840195162, 2390352556513870624000, -3235134230286333808960], [2452192221428408229092476276993, -1785242760656432707636998945, 4156729423886439380462880, 47361835778591296414843, 1078946163106081538014, 28145618043699177335], [-20834527423633052734522558001195, -12233915623514973234351573270, 2650195169588360584270247, 23553149930262403392388, -3994840906694156751888, 1895663639558207115427], [-22985167542095424257185624488672, -3720983328271724556431236194, -8046399331009412246110578, 41206449092058935378898, 4105131615071870730216, 182333651899239268980], [6367853514157020448225601222265, 27787878635785135748331750, 1226664460303207296904736, -26512592869802771003746, 1822647938177963347500, 3053522964340211794838], [-12427285075448429983487325279224, -5365022972985051743519597493, 2331997973651284441764905, 59927462255410968544025, -2477538706426842716376, 3412782736322183615528], [-7407405419984287909648112314010, 8687410009248584868002919054, 6070850072552319093779188, 37524494353466700387230, -7286066618890890297412, -1049222339481113670694], [-24272248399707944418720148659594, -11887478007908776557408438795, -3808992160052188447253708, -55893697038921120814855, 8887461602583925247358, -1064652617664973180435], [-48172090747302033659044755131983, -11086645236112653963913469421, -11734305310399597420354192, 16636085285426065652059, -3751898581406647621856, 2611975945548894439141], [15355253579070539980761277653402, -16730023455474828172712327691, 10461377941174175051989285, -17080988600219291957973, -7774173193534338886752, -1715519424786319515398], [-36725397300943701912731182242173, 744373569320837325446510520, -293163680243905638906505, -37351809733787288122382, -4106214999685278175656, -4037629679195197928387], [-7434639132019949958225542433744, 22756201201087186314744085887, 7532993438433685155728299, -1675276289201047241199, -13234592959006955807604, 2504962114070683180270], [-66441223410791789686753735305108, -15406088684133972844069776004, 3407258151089129257046172, -3589930128497365207668, -10659697866711055516692, 3122737654599725654820], [-7930408184551973448019697925079, -3044271970959534514353424950, 16054219165776572931299761, 24063957366764187674576, -18094811476224209334072, -3965887987560340921015], [3462588856111315219728450330810, -13250454396301443841006277012, -19489987451523731087430664, -52471172388741920279500, -14547035643772159461676, 4842098659990406712912], [18351353671262050769791460674962, 5957964864032252399054467200, -17685006057346285672151170, -10682517071362014878260, -5000713184736001257340, 5031884684188170504846], [9209983502079370965314162330234, 28362007324575950879789993020, 6234902846129069699947882, 32595499053373906120224, 5035016735078095918476, 3832010017297658233054], [911573937698077291057765759757, 31077990332898603272850861486, -739668008584520383242069, 77312117540550613204800, 21715721735060713672752, -8113834938887324598101], [17062719329377887560469795768658, -24243219821995379347281529077, 10095497432971563674249234, -48944317214961452885005, 3370350086793467238294, 7119464918891429876745], [-25544022491798312853749345805193, 13356633448894083217322060970, 18308533958918720182181976, 39876124114604225854946, -12771261700303486912908, 8745838423992698748578], [-44696999934183861092635773982675, 9238057465295893704738500767, 36443725714793286791614782, 41035906390192149602675, 47285745610302339316316, -3196849647717505861073], [-11967150677561319351199358722371, 1335178707300962817714327438, 23393374586707772494484790, 4057297451656636477994, 31290800314651459451988, 2236855469668467551176], [77860889136654683784186605073307, -16350763931474524270606365097, 14143014634156166857328696, 68931461379110236057011, -929801160555838329306, 928922041410439322831], [-3431114783228021833668130709422, 55510098957291262447439313435, -13764655729528893201248506, 92388923005871956303707, 7975518230004047488662, 9745135852954242734197], [-109147173770303511266586038377931, -25327301858741136413750048705, 341802617440464534633028, 79316956819453226582159, -3153915105381794436424, -12897982622938923126067], [32066049825282246007495390405595, -26933160693534985694939588859, -23701108041713519259307228, -27273815673599751790847, 44656941440527194313090, -5717858864601815566895], [79105113053792788988849401623426, 35455626266184702775417935952, 29751943490489972961633396, 69503583080490850444200, -50627393304531415269456, 5312897894185294464348], [-37373072162197693414535983905904, 92586040190117789266048547754, -49395718595298328325206298, -79316413078153341743146, -22545261406020339812100, -6760295446030181804732], [-7180064751449057653315722130234, -116493590712591470721508707924, -30047640206596035509158320, 79972155086979581405804, 17928097812970978843152, -9464473218297214879500], [46554533311060356125831439673320, -35366478991548047935552035750, 6212019261842961995985696, -66440241748705137839262, 11069654736548407122756, 18829332224788188487770], [280716063962348788954320198967645, 18042125651732834536438320775, 51649307271439038516733094, -175854986153176858799901, -1759103177513354538816, -5726689646484889996285], [133540013295815506212491307192119, -65498801686381537378170612849, -58147004396992723728957646, 91965142313835002710411, -4995879241289243076548, -12773270768017518850085], [33843600179281858889525214082074, 147694386625824704195970491742, -62934355238070689623725362, -161566101661845162315846, 25232496584824851071508, 10208290958472410294400], [-158259998109724032046046394527342, 97844424376127439372662643444, 54735768026130772307347498, 170386796636266641375712, 41979202883263867641408, -7536306603118222248390], [143348575034959274942439356985360, -80299511546456603365318906074, 40412049117973910934849954, -43443337380089915689254, 1367780747074943357580, 1925583906985389688588], [174648411554201648276663518318456, -25724496030215205575597403948, -26881978164297538055332152, 243932386628611214120548, 69401259781866558189184, 19792822174365431735588], [-33631541154385148488951214263447, -51213108356809060441612288989, 31808723389914255516295472, 304008995529561816013195, -17717756191603933635632, -34342941166005454871099], [-1387726855945732906783910626402, -157606755774109403643423466767, -76750838871642573310611627, 11398462317518887412039, -79839086295313357045368, 9585469338304217844374], [40852700702700647537191323450233, 88306464719689212315916443347, -153732254141911046720962460, -149081035762437394463073, 16392199359086480021442, -19111458843281050662701], [228327442762080012164376629729238, 17477960875957545448467945594, -36929520132654627573696916, -310382606187257970717886, -48486355842889957164952, 17877167902754634438978], [160944312512062134566893848951178, -60642180946200175852453737453, 57628897972037761806872056, 57793627083091900776599, -63758050212260652856038, 3879185542063672849095], [-284859472408814070521207983244795, 142459936111237360536489774384, -44232173320907151622003374, 51349873954472780091260, 73403072898977625044088, 13449911334714168601934], [-168721144121756935681183385578526, 59690230363842510471271957496, 37891852117215594707569762, 61875457616778768958916, -18759771722873959955248, 37609654456129561897550], [-17637226932655324713838131087308, 58375806985282810222832869539, -22657024218154890234840977, -206998631701731224482323, -171475022021941271640444, 3595745323090024796942], [-117158911312985749444309779167788, 74190115602524135197045728248, -68705173261782534633545004, -324664535300671009557112, -32404803062801509016476, -6196056368500274887448], [67189857838054149249060008879066, 125442634781186909243069342196, -38217648293735689907720726, -720645365516469931009944, -39923024012776495466652, -8718411187679120272514], [-129302624613504158474455185830936, -202371409086536885584087113966, -50175933078530309921882522, 46494769896271912683230, -75270151505534228555508, 23712710108390984593068], [-17855076229539137524178979564818, 199500961588932232041388053993, 157357774893347146643648663, -185531463798996516930045, 57909161396854627946196, -19259522922712877345504], [-392741599460444253389115020529367, -28914551000676765197219072949, -229771146424665915619324466, -517318265151895000767213, 74318869752744002259162, -10001320553058263092799], [526003566898328788503470336988465, 224757442698485641769153045329, -68209107859797781846285280, 571255670931839687544601, 216089108992718551704352, 26070687172148724146631], [-248845294777468508926587824764307, 162797896396524356806043070555, 61931437045807270935693684, 869318433114363392349667, 67107319328971699533916, -8211231825910643232859], [-84611955318963326843069448131745, -130317027495086320412506221294, 158163376098128862239801775, -123238390868138816844500, 54484127520621609298800, 66875203444380087320915], [-700119343406985527812540392157590, -328604787968281998304128682671, 103421926499507436907576985, -1062583243213894841464337, 23512087802845236513600, 38405381232353450523162], [-383208597222783741622861746090846, 47536156973978779545383708424, 278580226647748203010505496, -764345375876504760013920, 6425290409800947616572, 24330960451507351592796], [15177990407335594694209272975622, -7712880446220826479435866178, 97172590109609331595857962, 673332316042886372475650, -214332072548192018296092, 75545557034247668602596], [881588091634161021280016473948853, -243861965919558852893061845399, 303479768055448510527986036, 500229594395422577573677, -187153883416625764094690, -36853257136863844065479], [210820985761603382160184173217269, -348944954816231465311883833182, 512624073600179775612902561, 577326064957590899763400, 71756074076991488925912, 6914958265215129383697], [-635945255061604277207680865108811, -145069166921103366631542720477, -273666521579802520427842762, -65772585048245275082701, 72625078769110752115790, 39938575627590197591349], [-464182184705298264272990153559720, 403845600412003600856578482794, 335783777853518989733830746, 256703635448175115068942, 345440007172665054295860, 4246122193606367118648], [1237947118517096685494918461803870, -205709830631960197714771676733, 85228168795722490475771997, -26387677371191863158255, 314585180324244424014132, -32072321629433772694888], [953919277950678045754010954751977, 578584220938209123538975487031, 268514444825444639608681772, 1510338846650774976668227, -281584636228274778352214, 16259766573542106303631], [-989628038323233414587117182704491, -148690506598473212440052235966, -276207773241522548127148751, -1152909729642586601037176, 104314802714274397870224, -71362820454473582034471], [-179933425453739501060380354325462, -787938737338524367301025981186, -92943934973077673011623160, 999129707050201698862438, -242570943069863208158676, -10384376271046119682634], [146960649753724937532881430996080, -58173771294950486558206029633, 204532953004625397162583767, -2601491092254143938383367, 81444970433113252218300, -39824348219940198273214], [209918526816559640122518610513922, 192640753654003603920266591692, 80201861711955504152557108, -527570443961861204333868, -189636916373932729264632, -115822945698855947922728], [-27637435041580164491031359942082, -403189559123454766669655267910, -250610961440519183010787232, 1251833789793179130457634, 125210818906114269700548, 128214108217951603749306], [622968900199476016313043056511172, -714896622648349528898616702084, 345660507896814416717071188, 2121115187016550063384276, -287531168205786600196928, 32808048951370642140032], [615463835194871414532459919351592, -354375479810297457332687542554, 150756324823376220619686086, -2601341722778153632382014, -155077682110204077684108, 113225278633362603180672], [-1132241643388032532539193352804951, 46580760174771008613249658909, 1182521568599076229184712606, -572592598881174825074183, -24681539835476433752264, 96244275713516407209245], [-507213575574613875545480512982170, 634440854842417204062830232168, 35024007883394835078061792, 1507727521673466287891784, -295364581968289327311264, 50750851468139042309848], [1270049722420846797108878402369989, 105741808359453521137716464913, -79953841500882605691620932, 1349533690899602615773497, 472457918680410169839540, -136474844139047944650001], [39517194269386726135246961220333, 235270773237496747277897412951, 1206684310845619083051824412, -325715348603110350374085, -190840399779586988790786, -62627578303030879305549], [-274740279702706912623907758061592, -957527688078909675136305294183, -289458202094668519110553452, -2285704185421992196630659, -569030012235004824732354, 189798057010113392729993], [552892825767111254878939924003938, -1693397195057694046859905571577, 243716670815026844329705797, -864358384675806828437643, -214141511051818286907372, -227121251504533098573364], [441140161847687176437377527198603, -214148012231097084122636336388, -647094150013698823258664787, -728722610493008206211046, 98035675039096271632920, -39364195614800276338069], [-1329496096182925081851883231113322, -660766849819406521768963971300, -128754901496763221954210300, -161873828246678248589916, -191816768102563121648280, -55165368944140643171624], [364042087159899217898921454654210, 699701848374975399640193405584, -242116944843559949403248714, 810656750462344614219292, 476500076196272122349212, -342925613112172808754498], [8298316433229676126231037492930, 873203772709684603557510461379, -1248772779774074554264464001, -821301926637931476100427, -508468577013687464175528, 26090685423900635064402], [3679210344033622187775367093304753, -1052640262349304600556330471937, -86650730402464768318856912, 589747230361533469417947, 326183684628478370841726, 226874140212707556881959], [1353307372351305677488153538248074, 790077300115000930433502981003, -648023434451621679659826592, -574760344170503230003665, 689067912281488455497754, 178271554771780317987527], [-434399532364721631864931816731210, -46990704406030430441624799232, -1372622449325227580389655194, 1871763146224830950948972, -234406246872849077362708, 197019560708056136422830], [568942871131719798925105986523934, 1250287181373491622894140432892, -823037170350327430703337758, 1568233914316264305750808, -1061661685407621781723104, -13962940558989329232326], [3202601877514504631626196486751707, -2015181700234842608394219276827, -848001767199257139040667062, -1165754975588003318407355, -856351076263031651929286, 163326618551832224904139], [-438267392559692744169402430720526, 44655327431695709720541341742, 557754294831996461052547136, 5138241055300220217361542, 1487932575549804231008124, -17099921920894814967490], [2160566757308152141127169315915695, -874156002322151718388731108222, -98121831635625711076956322, 2493665833681970474655646, -1900613286339477938710836, -115617189277651220715484], [-1208953453302927141920256622565243, 1249294826918139957147891239935, -751738459069425536418355804, 5006511119192845640346903, -516725599177784246868732, 1549018391731953347801], [663558196706330289176850097930816, 1415233668516383729772485203884, 771582323508549598051660480, 1265274399302881730629796, -542838085898419275063772, -113249814796862730085384], [3225505278288957933775203409419973, -239397232180953923661310676640, 1160345490475253680273973321, 379823815783262310609078, -728038781961506177135184, -17783388635309843619597], [2390461695369699150127999276883858, 1018706045424598532024014841766, 2919584209731768770864010872, 67464655535548495541006, -1000794461652270232786740, -359187643168423499113538], [-2057610601877480297252324973725989, -1701183517618415811946193799753, 1227847915075938647333994362, -3639454514827476161999909, -707523522046328894657696, 124412445405572878454095], [-1551259934354841909865630421630126, 111044436133562208145779969312, -2682655972960455723928632864, 1991994780694568261928368, 506802104205054103233912, 287108609744260426710968], [3248701719012417001914549757004684, -2252365417103535408850935262029, 1462599202621122821136809587, 5038334396457738529623781, 374900367397214457850812, 385558095324076816455586], [-3093863544780637619636961594445435, -1928443939326041113517781368477, 483381447248428885144732172, 900863039992417521282607, -1713562300925123351867702, 857470958065125132324547], [1579176081405293316019075505013626, -2611333352710129407010511722884, -1683215409287315627296258854, -4912714553604225853421000, 22975975304493379748952, 42769097835660939405658], [-2602539957239682182670926392827449, -4702247433200481782394816869526, 1127374695967944555835998193, 3401211258399270213670448, -991484228448068386618128, 25000966539957490188049], [-3980471787885763831353510667925754, -2522656452923601203401277014344, -1624267973078196326637331412, 107791120550052297519016, 728784849589469628955276, 267466809759458293178328], [1440961269267041532522767568237450, -2046387626263581459133408737462, 1767154285235255219959159464, -5547721190338265608972558, 104780835688341530009772, -125723065117007805600918], [-1605738726062109612032161780970692, -2237795975921419940018726712542, -1117339072361401296856063322, 327415293841054603139142, -841154302309476548250456, -286624626835332317150696], [3529693142047338055426625679226742, 1706316723071959479110298793670, -1540525036242452195760041168, 2583805149835833932152830, 1458660958007545444830972, 365539019382166234792630], [118587875243675000317555021380132, -1326247306241799420410676124392, 2494079107823981654475373000, -2763149849637833072748696, 1070860597120085567994240, -412275179723344227665984], [-5747140956584000688777117758039400, -2448923005012746252815580191734, -1321886731897601689960450834, 9889405638365880935003078, 933734924891762339756420, -462979076403801127714620], [-390856474800356466401442777513259, 266463356276772155262467384730, -534817359768079746388741701, 704728855506158581912572, 518700181099105509132816, -77706379231903128857169], [-2015062034319675919307540887968046, 3368192228121109370655443526738, -3995551171557709304939715176, -4722311277959462328454182, 865520887157760526449804, 643327532821571480799570], [-2666950132367259904075020782332602, -171711764971574960487811910040, 630133410128621723290466230, 972425452127951646962220, -1419699883710739202308776, 407046198288094304781826], [-4466618179441407378238937328440358, -2648838970927533253361568551397, -393726065863535858736207335, -3272315083515164139915503, -239534810947408759220892, 285581774093480673385156], [1915020252627502639065317444941271, 2988129750223272381052424373240, 482685459217297798518148350, 6409046856126069544770876, -4446195106015225691217984, 283750313657521186161970], [-5745938676118117879113867505955634, 362255424284134700979684467517, -5165302894404832833363534065, 5915121033759646617561751, -143024646166410824361564, 140166318321565676745956], [5073451512089371390368346603828646, 1844234266875111506213267511456, 1242102280644870282668185776, -566746902091460325089848, 1139990947369981793137100, -360679860291930687469028], [-2830710965548786367059667817245411, -326530437586946227304393989446, 6033020484421529711967406899, 13908415310763785714335708, 2114992359872549797999608, -586434983260042873553697], [7330097937784072360111355186372149, -3567316640882617260630118647666, 3516931990554771194320410246, -4191519338843152754762806, 1408218781559362067415660, -859382666217217555183664], [-4359904339494978528670878425946022, -3077780623245491651888026010550, 1761923622613786766188837932, 10554489727291631432764946, -905669954156412663725656, -1283658072821827101237902], [-1289577861215129873818209818341573, 1007603442680811910231682637907, 1663750138189014651458649620, -1797398846882781978427949, -3283849670458100345542472, -467822951252649626244023], [3205745189906788008019366762274448, -846129717022861351465552649376, 2268675609725546144599633278, 14026594197869649469078828, -1949494259952303237097116, 1270347962220329028054926], [6174486118385214811866460571801681, 6791930840316719033117551627975, -3248508481094252752443994458, -11549824027996328573105845, -2656654653783620416286884, -238161515174006525333441], [-5673718907364454242559469305350085, -4092412360468726338735763401804, -4004588100188083906782832747, 4022258576741195509955586, 5612116807794054356747520, 246868485210097465561171], [1404262215019801428129260639197084, 4852999858213625112893512338345, -1519010981455709555288641682, -13553830467555986389383599, 6636829054939526117695338, -43912729405949571087261], [-539852521501000074240723950296386, 3953124529872820240921973648808, 1646855997229939981208415950, 21909318934816312573144292, -192892553148269746289332, 963274457040949549645086], [7925152758614094428225445809420543, 6271083857135169097181541868464, 444846236673822152901932944, 14577085253032356528578576, 5204382840801085126797840, -816957148302868598728432], [11814330066060432926700914117868917, 5501106410175228783692716915965, 322104794974565667994938446, 9514641604578209154640273, -2675821370890454468864468, 1372102109820184368046561], [-5713705639386986904183058735997538, -9469659256763120885815424819154, 1711815719646478330121713028, -28392301560744316065463002, -3070219906151069590994784, 134574876673165991686190], [-654302277366230449290336036706515, -1159937495492705012597286701871, -4456974744700587000949562530, -995950988146213499531167, -652182229674841485999058, -3007866262694048610326289], [-18613511819348675336957492004958068, 4298813569899871422669356171882, -4674830294187497782940159782, -12247973161171895381713882, -322020329687653479881560, -1825123007321131132423092], [8664333957101961736161816068936714, -7515405286690912328158027537874, -7320277929666316577204758248, 22973139136613780540603174, -2748124439676343734351148, -112601627618519765323106], [10596845823146847740328187428329370, 5132415193883639858610297689334, 531732936301647155864007384, 16900798752435891267936918, -3674512632554477999691120, 346337214278172663524466], [-12126387298817137891495516258371154, 11365642938179407444413267965022, 3530634624473164201220873660, 26231878882831859189716030, 157434576261286318902796, 1837638046221126286113778], [3431389895985401579760453114192665, 1826580856161607465510461388032, 780649262944950698137045847, 6123802672388817997274218, -2622077630310206532981456, -1489700617712981578069395], [8888064402431624471035907268027371, -3172338657604449221732060463693, 1475677787885398089821913486, -11229787118405306966109097, 6749314608974466405026396, 2619290076623971346905771], [2468326067240754025494982331368714, -3334525940096350105210064918721, -6815301791851528794750931337, -10922355410759888393683263, 5198296904887870487629464, 369836332570760147260126], [17183003331173455803147644831838293, -9409160204516955509326696717203, -10088178218210450078075309282, 19949836520576133515778249, 1998719326697446270781352, -1488393548948881305014723], [32191432493447118909804033217945949, -18383250418820868573955264961661, -5059282998992223390815673536, -8232473902525814994400761, 1253110404069168687090522, -976524704510124203629457], [-8604077735831531880865481268754496, 2389259091709215108062063605116, 21617536347193686464880107928, 2628493692816481181171676, 1850924757066894447202872, -138846452185821021365724], [33498126200618114420932879981443550, 4119124699723376554977984292116, 4663776798648476495611782784, 10753197327937504820945028, 2579814009317750904235704, 1567494355623310729547972], [4573411902969705957082530243950668, -3056250966335139241414565324008, -2856137615361212186404178676, -14619897109883683847062520, 9048367349682708823126060, 3295051885304388407904440], [-7403707370376439092274733071499728, -11607077344469928732053682000039, -8064326546085638816096781161, -14205018859312881661463877, 5958966590190642400522104, -4701944969230457012666956], [2784348688653973835391270824826748, -6325186696487159652105293650734, -16194699314029584407458831988, -14574064815205869053807542, 4597860216971575164131696, 1596201897706904036113106], [-4179491166048329787964024403568924, -2399301961933524524300792007576, 4285359317729884948694219376, -7487421819891691421414840, 6489936815290773001969008, -1837823373490934815877992], [-22359571511983110321369108959983759, 4107930251478138779089587834582, 5139298032696233638243751798, -1051056117306348362864942, 3927859185504073087937124, -2884258554962465787017328], [-3802972652857577350833326800209998, -9979852894195032680367846862050, -6050808312050758112508898520, 33856136688545556035913734, 4209049937132465928415964, -630020621487561784089946], [2028076977490835956525543897407953, 4112836326746719432005863602836, 1148279039441219293632001851, 16715320299341860616363558, -2203615621907125890345168, 6085742816924126846107973], [25829871829833389461156290715726099, 1325604885170511581394680519895, 5864384349293278690115568292, -30976842717848777275081177, -4273581167944057630058944, -668680445116623774801763], [30141553037713210281023908879230402, -4181183353453108526220118319856, -5682440877454153853688773638, -16559789132366690578651764, -3239000913951409360700520, -2622061128192739069932090], [24206099655111729539300763126675529, -843697436546360852924438015073, 2463586820994991981517563298, 47192940066046541805478107, -3751079761999954130301996, 359116162789243204012979], [4065340163408319838397287682221882, 8373623495356290273811831903422, -12480410228291660304754573084, 27951560222145390529987446, -5156429690598986239154232, -441497792667401477313562], [-18341107079823027114494581185103068, -2143776880570014453222140736432, -1862584611518148395981356988, -55038299644331227422327968, 7083533052538979146934212, 7104123564237557425696896], [2964585026424984632315453592385856, -13617376217315387836875664400025, -249878466222538855427082808, 5042864588018774984596427, -2071162695380549467990134, 1439296294259167580953651], [-9157262720063377753376931676682563, 34112503981763594831996255478007, 3748035126320808287874900010, -27815236171834481654379357, -3232690169734461456649500, 3808153972320380527855747], [-46622311338311899011514604091335089, 23677181206781858148324397800006, -11932497554356841655103405478, -7666065503701900567742502, -1772391220324314008643852, -1446739958372426049130756], [8011805175936022255380909051547281, 16024041847525179327897936282807, -2695056498695281107339641584, 54295010967565777298843339, 2577589848578696425184258, 4738745880503930371354227], [-62716220281451545355978825141992656, -28963608282962306910605256885804, 2698744539707061858030122844, 38714280631970913839860356, -8314178691788153018909412, 124305183228053495756028], [22147599521858136111704947116751127, -7850107790490977806348474758144, 14918880379112873205286068294, -106207472316190688641501836, 466252318413228116973192, 4239114946412004011625034], [-20971091050458305915694457035249651, -10104435748017316709388411435309, -16369007921219437239229019602, -55924868758865319766636285, -9780398365643917432872394, 1503173300019678209888853], [-63042766443410000308143074355171934, -10014721374082866422199089299542, -18780732942340905338111965548, -43662443244415407322844838, 2487515640775135994814876, -69075890388210028048098], [-34430446167781968574761729423346135, 32437161290261360047490179671636, -12756063059184712849491504634, -31085310617121131327328104, 1442050284185799655311576, 311242019720000785219110], [-36003477664022076943036854134974081, -20102577234991409972546478624269, 10307719832088780675028371810, 66408975450506706727240319, 5192097169109972720397572, -4181909304202388000224145], [-25166014414940548514462118242840296, -14231075158525396217920841869749, -8745118632511334688201602069, 35577521468890414240300477, 687357605181042960444348, 403900822186272911332402], [-10315340041509360108263599283670624, -16971905887233903332050373555262, 10098401884466734577309404288, 14750680619621866782482722, 4850162393922920783016856, -3812779326467788520260794], [-38207351693041883814318471837782060, 17135920827975866245753128797034, 30080214524433927533526038162, -60032887431901145542571634, -10840426501632277592990988, 1057394567960099537116704], [-5676404791687147030808174395441538, 11714700119769659502443058933573, 5839892710612517871405559324, 70632115420571882610693481, -16200208382815794468861762, 1178549722589606795843861], [-47517435971223576812922566250300500, -22332261106428680808444117178539, 4699769573511514326968919902, -74189796728929867744195795, -33092481659403200852698398, 4995813396471361848854751], [-13492076474034936046942320032505787, -45491001273991743478544559478173, 19320193988067607870926716726, 173039487210534202834033015, -10392025020487839786799004, 1563796022896837004359483], [-7795582438552533348523203062315347, -18095394140135061964090926027691, 20916206818572114061175884686, 3323427230066607005002385, -16905072637823226951885784, 11972138331435179834125669], [10832485571677387820707746532888665, 12712648961797175536244596699884, 9475525754288684077593060907, -59986047505804937639238210, -13316593120523570755109016, -146481031701284848990539], [-30890233288707244566754475554560206, -35444582872434316410763875170248, -14269002194993171335349709650, -87526988507561210722173868, -38441205256334224825639876, -159133799795647206615778], [-66976482420752771017946814323462772, 40175177602762358548032883357845, 13760315663685270426207222449, -72648416812267184885164869, -1340302791131294451196236, -2941231951054957177695022], [-84367508389852661096464872722142283, 9684682603691019674986671504619, -11867203083366749934923024136, 24195525838425908233119551, -14041473357443026053099718, -5497099401919266501764465], [-22131479380437101681039392356552342, 11610113533870672022055897654369, -25575660280702816756338709636, 60681839325520731107672165, 17744722379661360070452534, -8894801835259912337736527], [-50118198002988473548226663404446276, -12903378494859274083324806245928, -3609519637427770307413827112, 43913342535086511848873288, 55945373905396261825817600, -7404866180943924069251280], [-47445527035697078889913017158255091, 50225137304880520211962132922814, -19623283837373224075194994075, -141386303044620303031682644, 30347847099373885215847800, -8579642881730896293707039], [10779742640725729320067504247739109, -40269338828410894547536328255171, 27010792492022020707478373990, -75905515556452457036257539, 26042571688933557592846398, -444721726889370026212981], [-29353356608769477740078597109430651, 36858348264731161914182659770504, -21248737025395255755222045840, 40668766776348423379275288, -15677954070203546004375360, 12828951721122552485358216], [25383758450911487553804139157602696, -58810016706838010226624680441424, -32729082000105144868679692118, 90174348737607339060247780, 1943750325871396194383124, 8816138363760861186303954], [39413305006036240895712524928797457, -16854117250543817273283327214464, -25476901911176448534380079237, 11532174449487080831547282, 9785042163988896883354392, -6319525948446572002498783], [-38975854805772161569025813319385679, 23992956364484607452064799466557, 44162666528713900203338016566, -39439354516894999456918419, 7777525872982606067524662, 9395647577335981784794163], [-50673764465912746959420094508674453, -49439994442634512709773967594103, 19867651432249606216700801656, -38193858604438024009070367, -22714141553725675826725032, 18065154617580770468253007], [-50483166228785478574319857257320336, -4499525864014135546085824138173, 18349363312179859795148285361, 150440115121375332610541121, 10285048195510880201530944, -6333140606900684097325600], [-28580403242446501560486292909361771, 26245538410642184066479844890403, 22716794489668060795672978480, -44165439046655253767934061, -41114104767026027083830100, 22026729599227954061891585], [12264507692585536000430102262567416, 36230450695057822160296230102942, 4823510207269855962151159004, 26382151010064079986308494, 30821350222011331690585924, -1656574264991980401618422], [-16381481804401590196983926855281949, -5319814100188160134295074380474, 21940468803797324060361137575, -81299256022204550946735824, 10293702946732600726107864, 17581641854900019290323727], [-14856207638720772356699695352498990, -93111872998777704688589896680477, -16811552811702196155754611784, -59710104884147248843047049, -53718548387982406815724470, -3721601769681308036298969], [3090584371102483351397919178122345, -17664526628662382844833470163115, -56981088485926833940293868236, 44783736324704170161033405, -69221698823772670663877892, 3093954919487198601483579], [-56684085683748773867882478980040298, 41856885678469507776026561349874, 16101543172482668018561393280, 230911114063083203836658258, -8760710291230203102713128, -2207884709943460801232426], [62115457792382308286590168059118738, -21283372375834985920227624546807, 59676851471033747671898014542, -309894523822714842552026415, 35679613870752780919907946, 9664230766644383708646947], [52428593592773984551925488986312410, -77231644264558931914958078597364, -20306628049190904933237499590, 22620370274047164318346552, -80807392527683638317606384, 14837312126870403139919714], [-32513503701558691760642621944892535, 2161283140560797810773250581859, -47847990810040402587751500644, -299160281205954649638714461, -20144862887885862139371560, -15576945931507970896872207], [92258604342865519320133224831155782, 147000805875131181069843302810751, 30320327727348063895095156635, 80184960260514028638147049, 9333793942548403509911352, -14960459524694917608017398], [57062692803092252646911725628383259, 84304121053891475927322036652536, -1769818775264654963577320641, -125899875309969733980929502, -33182972318268465879177048, -12383610413162039982086171], [-155014922371431102736854545088156612, -21236902810718978849430866446686, -77877221847546201695633571496, 74968613803425838609755122, 27158884683466130894993288, -27978824609528751219519714], [34655594610037674646933133881221873, -63969607543397141815270989311119, 40971292617093691407675086628, -220910199245689642819269567, -45364050622021225849481832, -26545801357611233039592309], [-33892681786849364116361889049267213, 9122709798353124266294075413569, 19405157943827705055606953908, -240972657589956001894657851, 25118923665420714922215582, 16127535814062393975446497], [-150944326989966068159251630886241145, 107346803585428138025979755555181, 56848999485091042575849124250, 19051942333242423228210109, -43393633212620106640237646, -15070056794409553019032661], [-169286988543814320874205945538123501, 27662907588302695591584031062725, -107066234228088523961712071078, -83914607298863691043877615, 7340687171813575301928700, 10199859138428809008451821], [77802237650418527938791604181858221, 43231871426231604835988799436800, -11842567825970332715390785069, 33689746101258973795619186, 116400212158748095432272600, -17904447886526626639067799], [77792242748940438919124496213469420, 148870476999605019290267252855518, 25945548380289138294555083024, 213434459811912967066278966, 40184657817075523695342876, 11747847827400548834453534], [-160222450649619539750632017263709365, 80104742501883457431525369044865, 80892444721229031660933038134, -81500948354957585110148587, -38496801016118767894237108, -14281741919650328883733099], [127901741059426160558356924179423610, -21054897329640889671449701856222, -16185976702969910673841531688, -67257934031761889758940998, -70730722825375579557166060, -8010840143127411684681718], [69878549738604126025839306759097338, -56082327373396007242386567727914, -12316806021811573413087263204, -211653149774897447460423546, 120207349563607334206293828, -32230430232367296858654254], [172322177709428327547781167334663687, 100672745794911725124212102837610, -76010809096925321960929360198, 335489846008684660893683710, 80069133439336951921326780, -13539303577633494787860448], [-126419878124912014464260121131908232, 176498260395387191046857342471688, 458685545394993391431029812, 83308615690722544330446824, -13303835434421194509125908, 28829771264036300490430400], [159815743647957615614082735747956898, 4026984246476813534714459453502, 32308568364727020020654804438, 276705917319576281528460474, -126138616445406040071084612, -11059456164102790309137696], [-34058564706675543880030176171439143, 23586998233826745925224065231706, -219051772417314943593926774776, -336520430581520012579745246, -14572319594301032378518428, -25824134876747921055387982], [141791214116818798289549486640200064, 95105891586567738645924037358892, -63237923304401675267317194008, -165111676228085998882253108, 12909708449316744774530728, -32476343076519356355027372], [139438449331091947155308799742976794, 144053673431035191728242730180643, 177654927026588498796131212452, 165015905925006510525725743, 42110316892244129682240666, -40470107473562334883446501], [46800911325496901818186749339001991, 26769559100063520592613916345365, 82752726127813341424180562634, 700129200729104969919764297, 20480654761916908604363048, 19555033190928431658727801], [-161680816759020712387378619933192994, -97210530167532281489532223933998, -103379881666142124903050965008, -19397806872803874532415358, -112349823093648593459261856, 35108282358516154441477518], [299252461029196004408306375005074068, 97699563949336728038487506259684, -56116360818917234183360459002, 59099205903293247981706240, -77156980054515080759274828, 22828708653037627390299026], [274285420601590188215928112787837785, -167386598113831250349942768976830, 135257827283178974694322796855, -134926478498380273068362116, 33569500501885158585229680, -44733257660118846176732613], [131944930583022008585592386449584791, -365901224129586121367484033477821, -27297107398703065117285354900, -32486306737024769933877929, 139313938504846477365167254, -13105671761870949432861137], [52861963228883100419617241224151954, 259436316356606777133160257185820, 70569279620896505360173565434, -316813842345596272765956024, -124918679309897061555598296, 4760035156940835953067946], [-233128475121129947524198180008007857, 89301192088697044319058939106974, -80159475942972866944858467398, 133239205168683347162884562, 55573321277823500687230116, 23557885680621336378485556], [9785286181492382155638386333003748, -155599502959967809526367302742240, 174222386040593019794886417260, -372967216437905089374546664, -143743994780195963480942536, -25229869906481761408655364], [16437458939868939884887665074128979, -20194533105429120391008437623920, -284868847379357311193806156526, -48092300995903983952482932, 62228258773560018276802944, 57269090805902814004946038], [92185382284439484069104184003761212, -15554848293991356985951248993586, 183903361971557842664812568466, 169982235783608012918086882, 67554431146570412117910436, -21168752493765026273588980], [-916359805717218825613708958563150929, -106069305985896927285803546030199, 40555279206723061225385613738, -318900205238387610206955855, -135230038910783306358928122, 21831937805989428226946643], [334437846164914914712796594700344340, 132951532086490064292415768765284, 415487426800299849086280483426, 135941109403020702898759976, -166312119023609051412397740, -26893750943222581221044570], [625149859566515280816212200782577951, -77948784224532826209311844136181, 148585936819411102418635229230, 306935112409680698298440363, -30396663381635278194813250, 1662829110216751773861413], [-22081909898916138008186146217940556, 75300459214107684462595857432864, 242066917555834939460440182932, 199188406061021811853006216, 125002478328531395349889992, -14201183895931855639103164], [-280843289818395777805135620740916460, 244047597179660996676928470618578, 42715811465985908560438101638, 1100455832856081671785682998, 77106874779340679977059064, 19037211491641984367255512], [634312228774570389025415697835467454, -7624103166908465751402719886909, 230923320013110952254298282543, 576431932685737667914932485, -122601167929228993119357912, 64470767889240491649491058], [179054386437492040481916072043698946, 76128093572147790984415564301952, -3979790710445489399757962838, 424766618557740835592949244, -151688858635015682619566856, -50995705518150731588834234], [-229110969358101490626334957561835774, -149524484312011092024869689062186, 299200291108943784866938517212, -728442909772313107483694650, -104689268247552202350601500, -34665641202781538022400334], [-32350398116201554107407718017507853, 94387296019262535827109724325529, -427883246419820106677817101006, -312003125179339082771766995, -53949914091205338493747520, -576718794629548243292379], [-813357285293554705302181100486698171, 223143458359277291270152600094400, -288601814230120259399961882838, 191520483732904398010123868, -168798795633037394029910232, 38532534908431531029530166], [-928757513566578869104230162840591235, 91371627458748889187815217468597, -156576982303345170750870697990, 174504532763143247804299297, -23339555980981395226997132, 89438813628699699755522005], [292659088688730007723349382244393157, 30659447422182765357635842880653, -8087148864899678379123461352, 692355817723906199070319889, 24915360268991570122262330, 9029346644705486906642173], [-126479139936649884716869167208712770, 52067499984889401541292366816925, -382850891726228475155903986990, -162713888862130972265096611, 95487205625569339803374010, -18810389691289256228819093], [-6864435162220402319796855412659578, -235081276334855430622793504656496, 73889888892586488623227577686, -522960560374462681536649876, 380811813928598602308602420, -3688321988365769076715018], [-598718282047247118977668863339641627, -18368662391949726769469413388274, 193890733725053400897736007574, -673620453799144055163852918, -206289830238676371038587716, -3042189165861617178253104], [-560785519302746904853633595212345346, 184076578830561301311370033469067, -190897694208547481447996052635, 863049015364141666658015881, -222851239582701749201802852, -58646923566987382092241456], [173695474078279696360594156514024417, 195330788117769656562491757132609, 191817754626363125800882339354, -42905141947092853591532631, -230423647956649312578622554, 27343246618461287571751627], [370598942922067875860972625194565578, 436220508260025045004973940525195, -44980914196200010272613094528, -1049281392679374827229287345, -115904394329028511466122110, 37455041837886408730240143], [-458766564591211999826408261481866526, 18521405496040330762616661715158, -72253233081833230810213887944, 510621305736954621610976278, 125032622316454826997006064, -5334273472148797604807022], [-351571101309697726977936396552173376, 410927198793451103819195427556926, 198627546990541500394201338186, -579025779870015485136946454, -61311459347648332957301352, 38277642830221539391219928], [-1030423420742263353793053592608592231, 240292096272896591636357636173617, -56381093386875788628394919250, -1031891901512384812663354783, 278873609856047782299575966, -7826239705682558686963601], [93571023891821268512976372489321594, 262976348483229782371252308113655, -2326282528904630571841860915, 228273755968734175291869685, 210499670382057243330265548, -66840074756744035449260052], [242328230939734373438732252962585355, 389319195465401609622932338914211, -350377793620501637134866999262, -1361128322139016954416490589, 137756823349308723081732682, -38616526717386694943497547], [889436608367769729925579802348010889, 599780641630436254104021851242901, -202914743457797249079547005594, 360266946122718608599490185, -70566773185895329808754764, -764348597972380913714983], [601206893246540162299420530740653264, 172408135333678488381418460726688, -375417111490157602653230140594, -555102585692768339256129412, -44630564685053902004347620, -56056061724175440146688970], [-551758607989684047336025621135730843, 83922721831277404751108311358742, -282905403932596905775409547399, -918954961268708840439163668, -261863601277716839821235400, -23545988781163397285088987], [-344440160379573510497190478543029375, 68461535707113952273473980265605, -175136915586842202727755591998, 1300884602421084931032762613, 143372458675270599198680650, 51195084750330098233434723], [680887450137793329429842294305589686, -597005905743707769137839016012481, -123377032185084097053927734186, -375543969669757976296733169, 84815402774010933471786678, -49644549603923263576405375], [1299974459872151047683512490654489322, 612484735243097411975904824531622, 148830681342825094359253877100, 1706465083763512007793578638, -325087900588483001541437720, -6714658204961998431139402], [-212507438708508031396214778407008425, -503348025840994294143012808035426, -126722650202876879618474771368, 92947239119388879192072454, 64166766393913132493830596, 115153007487769332022395934], [-59360504689099818647293915885158164, -696809329326191037837249966050304, -230644829696344205913767191672, -869868264938994051385387080, -91010222395089631285320324, 152559361262822498432596916], [150017032495449740532759018036912626, -552348175060217220548753080525120, -570205681508400603802315754204, 813621211555206331742619264, 208635714887072494930461372, -71458367592476846766718616], [844699995470215152889720676438754856, 292493995631504668696093860410487, 472058314460325441740107130622, -810864879305672139731455993, -52400395010434108235375298, -57777038680426820979682295], [-215369063676785277829864624185672291, -213503586410851707525162220736622, 610105890488343480500658969730, -1227038843222859764596002330, -341719962530510105382756540, 34475131294033328571180184], [-314358339576304970877467880462275432, -331723202761419420888526844120396, 452287585275406512950031155572, 277192841082532660215009716, -264022551741487973341195972, -15662883506657126189918908], [1540639394892661977035537939725420638, -203205040089578385929707096349800, 57212388295207650926999574782, -1540639637751560860775022964, 26040700306745654132300672, 99812697567608098194216978], [163612601644793225872198255160935684, -15674461313942002611006017901783, 330974114304063462319442871413, 2908918162988519977151638855, 363331277102706428452250244, 101747483999098433049351002], [677727471781999624033849051575626821, -371612953030890621960807599171583, 55069215883365269009933837522, -1037276754837431254336272279, 225381055279507080605624430, 53595887863097398082582267], [823054065132655510139798542675657715, -713485691022406938943356046498553, 438065114562182340990143714764, 8906642363129716806936231, 203383535126229494318414160, 146654532750276401223362869], [1615683362535770737573394035124654314, 575432407294476694282681700700026, -645950195268466858817422519408, -694220398334215734182168574, 35882628582528423286633956, -12743075547214594540545238], [-753518803851440430282115735348173839, -339301058366708126987936433584616, 414503176025799903552476564309, 95785444908001808433160246, 183103933305071273045823120, 14549564100404087871858911], [-160551797595793115475231689533593298, -613674695117458261535868695383344, 559866104869632859884725707462, -1934269060967853026067671436, 26342922965054098238993472, -108264319626412115118650510], [1676550221166823283421870518040625140, -423342827951291032048250626419266, -394696075589496239577545426056, -484179040143118616285469538, -472416269637412690752368224, 76511774884941849281862234], [-164409006694026934173329541186339810, -103799072691848605125393495105732, 111885227400277542696604115056, 49682331635173680639220060, -361839841391862493432055088, -59043395715045394223877244], [-337471867867892038894808751391326559, 21786556428585736914783548693489, -282873660839834401346053451126, 382033499317640362047746805, -315831603852961711606268376, -68646963557286762332469443], [387446937681413930640526006922385313, 315556236734802919384575847140325, 595931073839275505824843145576, 1786005297734866394192057145, -573925963296297772357681422, -2310464614025186682401203], [-320480454940591620964473033206611363, 1215741200043371049753882970026549, -988554832119033480168223458294, -1329100332156272277747634387, -26342507626160368298890498, 9188606285048279661388063], [339944393719969843991218630579658514, -946932055409034720604285663831677, -249078049825328457994272261970, 282239758914792253010023827, 46418079526578597228045294, 187662786756002442793226333], [1626521241129157325015393585167125583, 1778900731865489729922067785790609, -500722905767237631914414305590, -1071815981148720530059506083, 94009229629512888099175132, 158043820903494757962740465]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_75_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_75_22_a_j();". // 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_75_22_a_j(:prec:=6) chi := MakeCharacter_75_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 22)); 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_75_22_a_j();". // 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_75_22_a_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_75_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,22,sign))); Vf := Kernel([<2,R![-2753908810819043328, -3447093453160448, 15258494066176, 2069443264, -8482448, -92, 1]>],Snew); return Vf; end function;