// Make newform 198.10.a.n in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_198_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_198_10_a_n();". // 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_198_10_a_n();". // 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 := [-222, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; 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_198_a();" function MakeCharacter_198_a() N := 198; order := 1; char_gens := [155, 145]; 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_198_a_Hecke(Kf) return MakeCharacter_198_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 := 10; raw_aps := [[16, 0], [0, 0], [309, -97], [-3910, 330], [-14641, 0], [74912, 490], [-342030, -6412], [237200, 36812], [-459705, 44659], [1601652, -252246], [-3092959, 367213], [1531241, -403577], [-6828420, 228846], [-9471298, 1121834], [-29057640, 2071176], [-49710978, 2579204], [63684453, -8232723], [-39600544, -11223238], [-145292041, -5360809], [161673573, -393879], [-128786344, 1597174], [9488714, -19867086], [118728066, 40242910], [-674517477, -14637263], [679350203, 39733637], [-1419880530, -2378236], [-1697553808, -17013944], [-1039722606, 28043494], [782194130, -125246444], [-464287899, 46621163], [-1755216496, 105723280], [-27850482, 123730090], [-1405519707, -319175005], [-4209906250, -234544926], [-4437359502, 305482088], [2584061534, 599572622], [112399139, 230885601], [-4577614588, -550579168], [-5285158884, 418889652], [-15636658818, -224078020], [18865835985, 70170113], [9975071285, -1180197613], [3734216601, -1050535131], [-7685825728, 603209894], [-6989955246, -2160431816], [15527918288, 1082572872], [22410227036, -765776664], [-19237062907, -122044031], [12889259970, -490418898], [-51217917811, 1621049115], [1970224212, -3224785946], [27073709202, -41517862], [3142499708, -3418420030], [14182873545, 4771199105], [44043053130, 4250753756], [10043132502, -916991802], [26408839662, 3791253740], [103592630564, -2333007996], [155220277790, -2094776856], [89200587786, 2576620436], [63729742244, -2780817688], [42139676532, -2985410414], [164286428648, -7992824092], [9114836076, 17297687476], [201868022933, 7287906399], [-36840988047, -14364559293], [-16065778999, -2817702567], [96310783970, -9339440], [229854360876, 8834295656], [-394896899098, -2424801368], [209987957607, 10247528373], [118892138004, 445151052], [329605034777, 12331092981], [317343788846, 9598510424], [336387552101, 7295242005], [-316124011737, 2623281755], [-618204560235, -13723863297], [-103414771282, 16651455680], [-365885032722, 10411705216], [-559822331182, -6348179368], [777936298044, -25860903072], [313270443206, 39076752768], [-382033399338, -7038605682], [281018559071, -55052130231], [-67421812072, -70304663776], [-515686319691, 62311945021], [-459443036961, 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480235722693], [3240255903138, 352586506302], [712489913687, 263372931373], [-7385190829834, -21426310908], [-4479884241870, -302703105608], [1686089288904, -166763249820], [4469818623695, 137427367135], [-6767582194818, 12242805516], [-4759039324363, 365651495491], [-8440227925353, -1634112549], [-2237329399195, -152023244855], [4665387584144, -141224261622], [-4368970784698, -291357361542], [1421557547796, 572081092108], [4501531758335, -85022685405], [9552641713538, 456036426492], [137674242816, -577980880974], [14089666753880, -251353390690], [-2772327210450, 489983987844], [-974269683304, 56525667188], [350747023209, 747344913339], [-6322973228988, -444340028138], [2420136792254, -353969621718], [-10186244640246, -382617976256], [2626875294401, 464137806165], [-4488769420836, -487193565064], [4889054919875, 496496919137], [-10412227457229, -207509704161], [5476424482622, -736034227064], [127649170260, 69105679958], [3475834285631, 1050583130567], [-2966616724968, 1210185957176], [-24428051846272, 571174667050], [-13251123820623, 393234406855], [-1903859982604, 181599859712], [-18225238293222, -104406985302], [-18826874529052, 557026820216], [-821105979876, 551776800924], [8735291822630, 666117215094], [-11363197009650, -673939309200], [6632526578228, 2138853879362], [2417244491934, 2527813532124], [-12333992632251, 295107282597], [7190297937606, -1287664471296], [-591574142560, -384520384072], [-17457374996769, -204602633553], [19756888921377, 815483641559], [2826949269855, -2092078004181], [18399086370440, 1192226625000], [-20631266051458, 840798271160], [14562903520190, 405530476316], [1974213844779, 1029812314253], [16506333290766, 2236914555842], [-32639721071578, 1075057105320], [-18960379662696, -2104588337184], [-10167183536212, -246727446534], [9904412019011, -3298676681225], [20408253279621, -1442031652013], [-25261718472178, -761234718198], [8879300237319, 3299807877033], [-15336323072284, 1387703082892], [30417594609872, 899299121658], [15210526115192, 2478504774968], [-17000258014098, -1264321798518], [-11377453904641, 2696703367621], [-4089916085070, -3460035280884], [40861533560037, -2085759087855], [63549188455014, -1141537469332], [58232035622324, -773757563106], [-8544845312128, -5499364704212], [37327250645978, -2292664045136], [-12943959310062, 2262650239594], [90334707387683, -1288418403411], [-23607370638750, 1906038674646], [30019848238835, 5594883785299], [-37473005502990, 1414395653040], [68061373448196, 3126286856], [-65568702734889, -351806552755], [37921801559882, 4789049603512], [-57562566982699, 419681275659], [-34660356692382, 2500484459988], [-4292232650202, 605456463894], [115502349127992, -1601540856434], [-125118088758094, 2295778400762], [-53209461002854, 677663912468], [8942524499756, 8413983050434], [20137779634167, -1388926781433], [-59327836359915, -3259958550313], [-102030651000457, 3072047663979], [-65654562647388, -3739415017976], [-15143617068324, -1666878775698], [-25450345950736, 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[-67221235572105, 12944906073399], [76092247863837, 19560796260385], [25456239532503, -14837979019377], [51410650478264, -9637089997428], [-18657795178300, 2448268420460], [162650986025195, 13808988779865], [-26407317107580, 27742372933406], [-126020386697244, -12867205439668], [-159392914093084, -498446724332], [139949832697884, 312009447616], [263771510121830, -2778874793550], [-19140489346206, 27215830390442], [88945001751164, 16238773192934], [235215101496390, 6259924301160], [36462518049897, -29231547555875], [-278389052548882, -1376295589868], [26306518167174, -20021106169660], [131719585652532, -963374647128], [-15151392711082, -19138908253344], [-178406842499086, -6696039768418], [171079643691477, 3637742554143], [199311466287950, -10486544559164], [127210884551084, 279802037180], [443696991575520, 6312875785428], [489939672507986, -8128924141028], [399102118817066, 9451311085660], [-251253428100894, 4300969304244], [-329506696950064, 8767004152572], [37190434895511, -13684125799007], [146483138966457, 9079461094167], [-219165283074682, 11519968844866], [44671808543262, 29504120215124], [-590920973892901, -335199959991], [292144176632825, -11006721438815], [-91019072185726, 9031387917512], [-226099645078624, 6233838106304], [126906315834740, 22418337977882], [-273687866944483, -11434437950655], [-129943812760953, 14838180813415], [-46897539054622, 26136231029652], [-401357406898696, 22852599609686], [444110142082524, -1032425706448], [-254131441748640, 27097298357568], [-236093358204877, 19756464692207], [-84062587339980, -15070912558980], [-18448929564706, 15006051915576], [-105287021371348, 11847200007520], [-6728596548747, -20341591844895], [118967741276234, -56294585541736], [168170841460974, -31321849096908], [669845157041531, -7866646574873], [-87957622647126, 41183564900380], [-35606031647055, 48323308689395], [-1003000115492940, 299387912904], [321298591949160, -39047803831878], [369270319360878, -8955268594254], [-484153249693522, 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[-214350488976318, -50712371953760], [432755418413363, 75600321652439], [-324297519242622, -31252944383108], [-322058904819871, -13709669576925], [-504254715884371, 21060756073237], [32816860186817, 61613722128801], [-118558981712760, -33510568225464], [-1114731482144712, -4020477949178], [585146944889030, -79174483951496], [509691035331618, 41436117452736], [987803928894686, 12621697525254], [488749330257648, 72729009423908], [-334832065895368, -12297973772804], [-1448939591678931, 25253226474933], [-595588249000141, -81331618507151], [-496292916816000, 21433462264442], [-138726702487891, 53861738631463], [1396273659468242, 30909086957114], [666625609316783, 91500214725317], [-773203587235773, 43513307255385], [287502949127208, 94409338762390], [839927563606277, -29491146444295], [116851606065279, -93464755822695], [523594728182226, 71435381439358], [-854995015803271, -2108587104553], [-319710583516363, -61695366085899], [69273493302024, 23348701353944], [317617220027115, 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76624288003462], [-684964162789834, 60867896110596], [-1399058006193522, 156168401526016], [-1296264200220790, -5735467417584], [-1894862721137046, -168484514151832], [-413287347338253, 173892052187097], [1257423652473602, -144132591266526], [1660595065692762, 182444921152652], [-1989405130576480, 139239412794060], [1187252248108239, -5379078211605], [1972264937674856, -103143407430664], [-374521680850237, 73441485343817], [-123926201052304, 83954065853284], [-3480703132866801, 112570101024291], [2244371570291183, -63000719346679], [-337386887119941, -13094523292755], [-1339949293753863, -228139329573119], [1738872550444481, -14284511831927], [3021610468022991, 95149290007259], [-917707864272364, -13937080730574], [-1558340358651010, -138061679014970], [-534831040040742, -75774082459916], [1394392266217772, 220393200119488], [791758885576794, 183051944057232], [1468095794902130, -50954966165596], [-1232721641115555, 116552429026707], [2061521718100016, 112001995546208], [1152669402249522, -182419963288484], [163377914078736, 96528809813950], [1190182066075406, -82803331362258], [4335823827434630, -65587179190440], [-1973904994290936, 67760771485554], [-3190755024058948, -92328902915800], [177992266102359, 283016634944095], [-1391509637607238, 182022733424904], [2617724043668094, 34861010768260], [2250447776827500, -78662350483416], [689421722992238, -36597037485374], [268775700043070, -368843031457084], [3662874755395701, 70000135723559], [7254890514360648, -784900212952], [-364887609670111, 231963105646757], [-2094609803997918, 59755927118180], [1806471733060038, 11142348925830], [-4406910189104613, 150255950125509], [-1635599293942780, -375842449620026], [-4109499056308248, 170958808466648], [1269935768874030, -107371398857502], [-112142454513910, -36800582552256], [-264283434679305, 161761202596449], [-1269674988935745, 341805396858999], [43022049403578, -395298925074556], [-118773738158827, 329649706802789], [333710633756160, -272873131874280]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_198_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_198_10_a_n();". // 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_198_10_a_n(:prec:=2) chi := MakeCharacter_198_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 10)); 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_198_10_a_n();". // 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_198_10_a_n( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_198_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign))); Vf := Kernel([<5,R![-2023290, -521, 1]>],Snew); return Vf; end function;