// Make newform 384.8.a.l in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_384_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_384_8_a_l();". // 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_384_8_a_l();". // 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 := [-1680, -286, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [-763, -24, 4], [2288, 168, -12]]; Rf_basisdens := [1, 1, 1]; 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_384_a();" function MakeCharacter_384_a() N := 384; order := 1; char_gens := [127, 133, 257]; v := [1, 1, 1]; // chi(gens[i]) = zeta^v[i] assert UnitGenerators(DirichletGroup(N)) eq char_gens; F := CyclotomicField(order); chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]); return MinimalBaseRingCharacter(chi); end function; function MakeCharacter_384_a_Hecke(Kf) return MakeCharacter_384_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 := 8; raw_aps := [[0, 0, 0], [27, 0, 0], [103, 1, 0], [2, 0, 1], [492, 8, 2], [1025, 5, 3], [-1692, 50, 16], [4722, -2, 8], [19992, -4, -34], [23867, -383, 2], [59306, 448, -103], [-61527, -567, 173], [219132, 146, 36], [-39734, 1342, -284], [221860, 448, -258], [-38497, -3603, -814], [-345648, -3636, -1364], [1195645, 2629, -115], [-782296, -1988, -1380], [2104328, 1460, 2054], [-707288, -9690, -2938], [669046, -11692, -4627], [-2481312, -13452, 2838], [-4442546, -27228, 2720], [994426, 22864, -204], [1641491, -27767, -4174], [2213578, -23748, -10713], [5067212, -27128, 8860], [1977689, 19333, 3887], [-7127110, -41208, 2956], [-2846554, 50248, 17715], [12190916, 40728, -8680], [-21994064, 22406, -37804], [43788040, 50996, -4812], [15394583, -19871, -25000], [28688574, -85956, 10657], [7943229, -129563, -54015], [28927242, 5894, 34628], [36196292, 77468, 2020], [19133015, -84991, 58040], [16254768, -219132, -5148], [61829669, 188425, 20163], [29028976, -185416, -67312], [-37319708, -353166, 23346], [141234711, 217805, -7370], [17972610, -180260, 67403], [7875936, -551756, 79196], [143427906, 125836, -51873], [122942648, 824188, -7918], [194320813, -414903, -67473], [-113067046, -51376, -228148], [163568728, -407096, 89500], [15240300, 168362, -128410], [49234220, 479480, 65810], [165024930, 523568, 16344], [78133724, 312604, 17348], [53138063, 190285, -166134], [175570394, 1190420, 43807], [-73435507, -913391, 89387], [-168755778, 244052, 288060], [120146036, 1327600, 82856], [293582687, 132669, 161082], [49364872, -2291700, -248084], [-206599860, 343892, 57664], [-469490140, -511830, -63062], [-437784473, 1483501, 183310], [46132004, -996448, -10244], [136177296, -2039274, -154378], [-343326104, -919548, 272758], [-1040205251, -509131, -292299], [563128618, -2127016, -571532], [118550672, -329220, 230502], [-882189142, -844736, 226617], [-220219471, 3428033, -170687], [262381202, -839498, -969912], [-285833912, 530080, -191096], [1038060691, -946507, 31232], [-1706852523, -116123, 152061], [1517066520, -2536074, -231408], [-312902270, 2629952, -751044], [-1766645112, 1046412, 24778], [544416861, 683585, -519661], [-144534348, 6344296, 26546], [1077014830, 113396, 1558684], [-869601302, 2611788, 688783], [-535686496, 3054204, 1444718], [1932741308, -1760406, -1002872], [-499995614, 690400, 1890532], [-505929897, 576785, 2334448], [-1469733354, -2720256, -287465], [-2651581160, -1969428, 351114], [-222484740, 1684840, -101694], [-923776414, 2951312, -1168239], [207012664, -128380, 56912], [1268415628, 2949664, -2026200], [1009645064, -6081388, -1769370], [94746739, 6027873, -1763338], [-382645040, -5113738, -640704], [415091914, -8992146, 976152], [-3651838295, 7090317, 1423887], [1961006166, -3476430, -1590020], [1194663811, 8967645, -615324], [-495725052, 3281736, 4967466], [-836709388, -5967686, -5739860], [2434552584, -14159820, -3883576], [-2576241236, -9733550, -3446014], [908706600, 12969556, -4449848], [685098106, 7686888, 3421468], [1917114680, -14746924, 4591466], [-3782689084, 18933602, 1874762], [-1305337506, 12256640, -1062325], [-8528655, -10570479, 3864353], [355619438, 21566756, 4546524], [5020407708, 7879992, -1424096], [-1256091462, 7167016, -3480383], [-2046624248, 6913014, 1986340], [7790703198, -6476950, 5895828], [5111520504, 1375884, 193638], [6105239975, -856679, 3322268], [3121835272, 14638588, -2957552], [-3254132223, 1454729, 5840645], [-7059694578, 19603204, 5014580], [5327235471, 11342353, 7176140], [3243738032, -4035108, -8848438], [9539488402, 13906430, -1335284], [7761888191, -4354715, 2074078], [1451537389, 36421721, -6647545], [5637483068, -2949760, 6414662], [-7706891098, -44511468, 2399465], [3594941961, 11501093, -3186765], [1252926640, -48763388, -5644424], [11651861452, -10761892, -435880], [4760486758, -3472188, 7337853], [-3624595875, -1427255, -15868289], [1239606220, -32159438, 12899336], [-8026286340, 10949258, -1327982], [1445614747, 1610553, 4642030], [3779153606, -16848174, -2016152], [8981851915, 13422405, -6849788], [4976962832, -16906986, -6750772], [4743889750, 4900282, 11645152], [-7993159101, -29211503, 4491574], [-8712527222, -5763648, -3109179], [7740528876, -45634696, 5813540], [5077734769, -12529011, 6426951], [14709509424, 9835748, -4681470], [-8612054743, 29943697, 16448949], [2765570932, -30346902, 8157364], [-1409885626, 13436458, -12530724], [-21193382928, 12174952, -10910896], [2505687117, 30047205, -6801251], [-440914150, 69199096, 1357384], [-3207534214, -18502666, 9728388], [3494913028, 51749908, 17548348], [-1257344042, 74941546, 915300], [-17939053664, -37641056, -2003180], [-20392079734, -20296496, 925397], [-2649746048, -11836194, -17562824], [7016453754, -120106288, -22415104], [-7503602817, 31981577, -28101848], [-26386607328, -30279404, -13352572], [8129976496, -43628762, -2059504], [2061982798, -46237712, -3926809], [8083238152, -11660972, -1203046], [-1605659952, 26671134, 7649496], [-6564600844, -48303116, 11011188], [-7497769386, 133862932, 15396229], [-2101083495, 30850641, -2247855], [-2781413206, -124990688, 9723756], [3965461731, 62453581, -28546988], [6502216584, 46159684, 6151608], [1250850701, -78087787, -22348063], [-4395951084, -162116012, -7511464], [22343820890, -10336784, 22626816], [39599971446, -102778164, -17133623], [18569219494, -44275172, -23366348], [-23103953820, -1959232, 22362120], [-13144646081, 133750489, 21475584], [12661496722, 166933156, 14778087], [12710179157, 96646741, -13893759], [-10974585106, -184589940, 30151749], [-2200928080, 166528980, -10006834], [7746621389, 10305153, 5257655], [19002934014, 37549572, -8703956], [6760471152, 27184648, -766320], [16957359883, 31291237, 419452], [8946251897, -117538203, -9265917], [13135094420, 231431672, 15622936], [7805164364, -132389054, 38328910], [-70849771128, -73842056, 11504240], [46936131666, -32257376, 2984600], [25333408664, -91437660, -25242628], [11789198620, -128415888, -25120072], [-25283433601, 34499329, 2616196], [949447420, -33911472, -40924080], [-48139046418, 159851652, 10449788], [26851938380, -14173150, -41520654], [30870069297, 234367549, -12832045], [3879560334, 99925484, -6454324], [18285257564, -55491644, 66137648], [-27987974429, 74164693, 42029064], [74987964486, -85525024, 28112687], [83777867045, -26098367, 6987307], [35383433144, 276779694, -69170], [50820884580, 49847328, 62286030], [19388938379, 1212489, -10118890], [-12135745478, -218220292, -44608453], [-25696166012, -253018088, 28070514], [-60063296030, -275324984, 13382244], [-2908082618, -206211926, -57401956], [-5674318292, -63837494, 70303874], [-5375925017, -207545143, 15163268], [-22918526526, -272939420, 15361159], [66377369352, -242468380, -35515924], [70864274208, 27629892, -34669278], [-83199607914, -139525116, -29267908], [126715792122, 22763864, 30388653], [-36059454888, 71438278, 31393420], [42686380868, -153922740, -8901652], [-6847989589, 49780557, -39505744], [53040792637, 50752713, 53992475], [25751974502, 314475672, -34706209], [11933379676, -204900262, -49490276], [145657972362, 45501812, -8591097], [-31753751792, 169010532, -102598458], [104339340897, -8287735, 26976781], [-56824533458, 102426644, 91819008], [-82688766604, 323682320, 113860770], [119364278830, -183189096, -10376149], [81736003808, -191802644, 107965480], [6453385057, -78864707, -45977761], [-23624754002, 133920282, -41877708], [-63719101730, 62883800, 118977687], [-35057531106, 272965380, -80028908], [19772298390, 260196570, -124643428], [45433082684, -117583688, 127098034], [42895611042, -48254392, -97058588], [47185223807, -73881463, -83271992], [83997142440, 72137092, -92071446], [133582556984, 486993116, -23099962], [8455569060, 526461480, -7814454], [-37763905068, 175160048, 117496820], [55500515438, -181819076, 4621929], [31148506157, -288032427, 77580409], [33100781862, 317585572, -131586908], [-36298360768, -176506940, -10745294], [-39478851746, 472649620, 54539841], [-26397074308, -197088592, -98935372], [-29908679932, 589247152, 91365684], [21128498984, -554867072, -109390708], [-25860026359, 373349445, -74647585], [60029805674, 273322664, 20867052], [44236088632, -609205988, -86284434], [119895388202, 320164880, 27031960], [-32402280941, 272062137, -132414934], [-52455421300, 547224080, 107586324], [-125708418027, 189297, -167410301], [-57208141688, -216943804, 112407568], [10672987015, -65242531, -8586434], [22328657870, -605091236, -3121228], [-204187395386, -37257080, -25789797], [-94629007616, 245199012, 75138884], [-26519732795, 24312945, 192262115], [-123986693643, -385549139, 9232609], [75300598876, 359618602, -126161156], [-16882757884, -734920856, -60964252], [40368197307, 674174325, 111891052], [56450677498, 212604304, -6758948], [-56486408698, -414627494, -189877944], [-95578771233, -304471751, 23409536], [-35075510163, 588467637, -204067835], [8147568680, 244339740, 48219568], [14607256546, -790565608, -112036152], [-117423733878, 69710456, -120442099], [91046899778, 912661280, 4134272], [-217591484106, -362470120, 31050167], [-101948617316, 797817416, -4073074], [-113668382195, -431409091, -163348883], [106949318332, 100351658, -143498030], [114154633532, 140701440, -137245468], [-65801032076, 973996304, 148333730], [-105540489158, 172547868, -86124249], [93213441936, 1100673356, -88230366], [-234848991423, -711992431, 110482697], [752675518, 151261314, 178362488], [283690205344, -205512920, 192843396], [-41003223084, 1076218258, 77672342], [-230378314513, -175398655, -84862132], [-30484767166, 216870672, -333374799], [41779587402, 690045176, 221791156], [-32781552617, -752931207, 90419084], [-253381215344, 229988100, 19056220], [240161155318, -425402116, -178144616], [62789690512, 874244220, 108263140], [32687220493, -259182035, 55545465], [201380204255, 182003625, 144742280], [-271130241686, 43657180, -70919221], [-202290200573, -784154951, -12887686], [-201728664780, -647727232, 91025982], [99069844046, 854886570, 103156920], [223814473932, 36380162, 27166858], [371275338327, -253552631, -142215204], [85506571086, -828440684, -224662623], [-77502207932, -1231581016, 55396240], [294998838874, 224264574, 111767252], [-359819974908, -556219150, 84809070], [-87773898572, -442341584, -233924454], [196049513113, -84180715, 154846755], [-59958391248, -1523752140, 7678986], [-51747165543, 1732887193, 86905769], [271992888912, -870413832, -7229752], [76904207531, -344198751, -390001174], [-366532933384, -490786970, -156395140], [145694849802, 1068980246, 103921480], [171997618532, -1615815508, -479491848], [-99852047222, 1304245792, -18669032], [226826845928, -16197028, -156557880], [58745610796, 1435144912, -167836170], [-419320810128, -267522650, 30219990], [-352744002634, 737424260, 241528792], [29994200292, -1003592968, 118895768], [-108619138948, -869963326, 41923914], [418049133651, -26522503, -69009486], [-427111333310, -47369064, -11854935], [-127074869940, 216400498, 195826284], [-224787349662, 739791960, 246839020], [373087096962, 691574286, -208174416], [217838393410, 21914854, 112120260], [51720545384, -1794422584, -81378000], [-12414363749, 960782717, 526220384], [110946351329, -381266131, 256153327], [144004380459, -577346259, 174368520], [-23076207662, 2279076524, 111857391], [453664678116, 1675929528, -207959340], [595706063360, 307404044, 160246116], [272272455844, 1796212304, 188372110], [-215584999771, 750436669, 272969457], [22631855366, 2090461780, 116777104], [-466618585240, 1985541382, 252714974], [-111269412974, -288167948, 310023291], [-250019305303, -2102302655, 150437965], [-122453832488, 123923838, 9571416], [77244918967, -1938905035, -348214502], [341901023686, 955596048, -310745221], [534311483915, 1616159561, -259056610], [294473774172, 1581185936, -338419142], [5674666397, -1590956239, 51223023], [358225324780, -1224683816, 408543724], [275389500412, -1465627584, -448399466], [364868877567, -1852787275, -20342930], [245015788366, -515229014, 109018184], [993591018, 647375576, -21921620], [-180483917685, 1147125401, -585734970], [115000128970, 2751940524, 216809251], [441896238345, 633596593, -194247779], [244889598194, 639648184, 715751560], [463581783680, -677459152, 1002999740], [-607247085632, 499664588, -195571424], [119975829096, -138436506, -71944500], [-776742963836, -766414020, -323872616], [-194094818175, 859749137, 258789125], [-194112070922, -1148477988, -59840408], [208682342896, 980659608, 539348200], [-720330489484, -1214274688, 287227902], [-693338261424, 364843124, 232156816], [9597054916, -1561691726, -622777654], [-425640092489, 832724341, 78510114], [-424471969646, -485069656, -698636203], [-75648553170, -1086450484, -13300388], [332062351332, 837607032, -83047690], [-286328748094, 2792546550, 124912968], [31340132128, -387956496, -70063760], [8972970279, 1062061189, -129211110], [85613519494, -2260780204, -226481623], [201445789385, 1522240213, -367284365], [526834183516, 89383456, 586564562], [291382140928, -2027476280, -133570160], [861401727778, -469788224, -66678000], [983757920612, -372488254, 572353628], [-40886879252, 972173234, 769382502], [-39395992441, -2278931231, 621307704], [78466309220, 1047715002, 637009904], [322362596094, 768673500, 532935665], [-139541910070, -924200344, 438851288], [-505822275418, -1565188398, 310767156], [-577358856844, 1141491900, -683222356], [-82934365690, 1589912900, -189764931], [669363098277, -1539169543, -683001493], [-789754714944, 1748056828, 494037020], [141354466076, 4392895584, -166418630], [654011596162, -1512027888, -360750168], [-557777486397, -2254942947, 651078516], [-775248263748, 82513304, -303671032], [-255743287534, -147111810, 387571584], [108188615412, 3031999028, -126173264], [379262839074, 3860126640, 142904036], [337431987022, -1402121120, -390800317], [-574083735832, 607632542, -1417632552], [192309290274, 1811760742, -76421952], [-137851997309, 1666809749, -172584296], [-221727150651, 4712355781, 712152189], [809122421228, -2139093478, -645367300], [-611700506214, 639368252, -229873629], [-1006821035212, -1449851238, -227273788], [418661151515, 2088741793, -263109238], [-136436123074, -3660182572, 61564829], [-294390573159, -629006059, -796083161], [228775522584, 2342321462, -419939664], [-1061297319590, -585276442, -887803776], [416503475684, -271260392, -243000846], [956781461428, 1620329450, 808019578], [13388861607, 1293762393, 1402593564], [-600084877052, 426386192, 714136482], [-1148733404878, -2958095074, -34911476], [-688257151072, -2860994850, 873504102], [-314084576569, 1622823717, -825126950], [1052559725296, -1610934824, -487969440], [754289274578, -2204852936, -819502115], [-100507456464, 3106986270, -44731512], [-784254824712, -1450887060, -671365734], [278698549843, 2019542101, 1308943064], [-182482509055, -2692429159, 153882557], [-1381724099592, -2064537696, 28857460], [80216314632, 759547044, 264871218], [30295419364, -5903981254, 899279938], [-528560929913, 3424216181, 805206010], [752689267640, 3623638476, -766096598], [-279817349402, -2827156740, -719512160], [-150035617402, -4574787654, -122905924], [51843988672, -4918801740, -1547961074]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_384_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_384_8_a_l();". // 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_384_8_a_l(:prec:=3) chi := MakeCharacter_384_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 8)); 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_384_8_a_l();". // 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_384_8_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_384_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign))); Vf := Kernel([<5,R![23304000, -85520, -308, 1]>,<7,R![765563000, -1837620, -6, 1]>],Snew); return Vf; end function;