LMFDB - Embedded newform 2.82.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2,82,Mod(1,2)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2.1"); S:= CuspForms(chi, 82); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 82, names="a")

Level:	$ N $	$=$	$ 2 $
Weight:	$ k $	$=$	$ 82 $
Character orbit:	$[\chi]$	$=$	2.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$83.1002571076$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2 x^{3} + \cdots + 15\!\cdots\!00 $ x^4 - 2x^3 - 376262508638451474994292856206343x^2 - 1502772308966385407243870196944706973427347215700*x + 15227343706090890463549797596060496062977217468321124839077527200
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{45}\cdot 3^{15}\cdot 5^{5}\cdot 7^{2}\cdot 11 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.77338e15$ of defining polynomial
Character	$\chi$	$=$	2.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.09951e12 q^{2} -1.09428e19 q^{3} +1.20893e24 q^{4} -1.35941e28 q^{5} -1.20318e31 q^{6} -2.01060e34 q^{7} +1.32923e36 q^{8} -3.23681e38 q^{9} -1.49469e40 q^{10} +1.34090e41 q^{11} -1.32291e43 q^{12} -1.41164e45 q^{13} -2.21068e46 q^{14} +1.48758e47 q^{15} +1.46150e48 q^{16} -6.93481e49 q^{17} -3.55891e50 q^{18} +2.85317e51 q^{19} -1.64343e52 q^{20} +2.20016e53 q^{21} +1.47434e53 q^{22} +1.73783e55 q^{23} -1.45455e55 q^{24} -2.28790e56 q^{25} -1.55212e57 q^{26} +8.39432e57 q^{27} -2.43067e58 q^{28} -1.94235e59 q^{29} +1.63561e59 q^{30} -8.81707e59 q^{31} +1.60694e60 q^{32} -1.46733e60 q^{33} -7.62490e61 q^{34} +2.73323e62 q^{35} -3.91306e62 q^{36} +2.77158e63 q^{37} +3.13710e63 q^{38} +1.54474e64 q^{39} -1.80697e64 q^{40} -2.02778e65 q^{41} +2.41911e65 q^{42} +2.10478e66 q^{43} +1.62105e65 q^{44} +4.40016e66 q^{45} +1.91077e67 q^{46} +2.81604e67 q^{47} -1.59930e67 q^{48} +1.20498e68 q^{49} -2.51558e68 q^{50} +7.58864e68 q^{51} -1.70657e69 q^{52} -7.63795e69 q^{53} +9.22966e69 q^{54} -1.82284e69 q^{55} -2.67255e70 q^{56} -3.12218e70 q^{57} -2.13564e71 q^{58} +4.52685e71 q^{59} +1.79837e71 q^{60} +4.55117e71 q^{61} -9.69447e71 q^{62} +6.50793e72 q^{63} +1.76685e72 q^{64} +1.91900e73 q^{65} -1.61334e72 q^{66} -1.17113e74 q^{67} -8.38367e73 q^{68} -1.90168e74 q^{69} +3.00522e74 q^{70} +7.88481e74 q^{71} -4.30246e74 q^{72} -5.80736e73 q^{73} +3.04738e75 q^{74} +2.50361e75 q^{75} +3.44927e75 q^{76} -2.69602e75 q^{77} +1.69846e76 q^{78} +7.16447e76 q^{79} -1.98678e76 q^{80} +5.16711e76 q^{81} -2.22957e77 q^{82} +4.36912e77 q^{83} +2.65984e77 q^{84} +9.42725e77 q^{85} +2.31423e78 q^{86} +2.12548e78 q^{87} +1.78236e77 q^{88} +1.73505e79 q^{89} +4.83802e78 q^{90} +2.83825e79 q^{91} +2.10091e79 q^{92} +9.64836e78 q^{93} +3.09627e79 q^{94} -3.87863e79 q^{95} -1.75844e79 q^{96} +4.37906e80 q^{97} +1.32489e80 q^{98} -4.34025e79 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4398046511104 q^{2} - 71\!\cdots\!04 q^{3} + 48\!\cdots\!04 q^{4} + 24\!\cdots\!80 q^{5} - 78\!\cdots\!04 q^{6} + 18\!\cdots\!92 q^{7} + 53\!\cdots\!04 q^{8} + 10\!\cdots\!92 q^{9} + 27\!\cdots\!80 q^{10}+ \cdots + 23\!\cdots\!24 q^{99}+O(q^{100}) $ 4 * q + 4398046511104 * q^2 - 7111737378636958704 * q^3 + 4835703278458516698824704 * q^4 + 24913167970238703572051264280 * q^5 - 7819437941500545712789131362304 * q^6 + 18392891597037634084943392123253792 * q^7 + 5316911983139663491615228241121378304 * q^8 + 1013046472853099475277764895915187498292 * q^9 + 27392317868014062887782031287821564641280 * q^10 + 3293502419914460072529071908820017291223088 * q^11 - 8597562939352679680661076005207963445755904 * q^12 + 567209646784673546391552295946811229381397816 * q^13 + 20223198179366361312267969326253833663484526592 * q^14 - 547217381485998970079962981893261936776292350880 * q^15 + 5846006549323611672814739330865132078623730171904 * q^16 + 17089897952366178968039636285810569503954644106312 * q^17 + 1113856376379446798989506750446739019487282139758592 * q^18 + 5756965931766410168329316740594195077449201960279120 * q^19 + 30118172007617752210204452443556450103768978324193280 * q^20 - 14643153298636475055051905666955183519768753981624192 * q^21 + 3621244206804343073026598875549252663285163724833292288 * q^22 + 19646604683226764943245317684559999335272488116648750176 * q^23 - 9453120422354276003431179546249863341640979885802389504 * q^24 + 733266156338164296167626938048487221981492621013192555900 * q^25 + 623653602026466415561742315971908501933736430204371337216 * q^26 - 40421324236200783167310017942871741694795540187528627261280 * q^27 + 22235641549031747542709906392215390663498217245358677819392 * q^28 + 89899184350048092395043102182944472414475457640075074740600 * q^29 - 601671873864991093346079619103283246391243600682126546042880 * q^30 + 4430148341240723314324773447291665259367527167357662548603008 * q^31 + 6427752177035961102167848369364650410088811975131171341205504 * q^32 - 69664471619294722934395229831718213827407646316847541036666688 * q^33 + 18790541516131866747958596372298574046878755610605871916122112 * q^34 - 110027170646399928174365046715945192018225940599726953771773760 * q^35 + 1224698037501642467387345239127034234507515770416141817601851392 * q^36 + 2868361968223519440594845730542456366299838282287085971112221592 * q^37 + 6329850982687462191179845211872610167062658437298665326504837120 * q^38 + 18942502886425029009441809968742163165248613621078317208744463584 * q^39 + 33115280329733352604636119223977533216139153469886213934840545280 * q^40 + 241785427475215901047307993827246250994683885604889744530507410088 * q^41 - 16100317319157294529066940012044692765461751765438896538220756992 * q^42 + 3906640790087972714399517126127070696761217293523540873168832368816 * q^43 + 3981600112397853227369825768600170041868973598762120557816067391488 * q^44 + 40553804958029668344694291747814466848881682300767140094162198787640 * q^45 + 21601670295526245166878191503440804171462931381646083872049726488576 * q^46 + 68150727659512952126173314586975326413685192716985190293893331241152 * q^47 - 10393815823145298626686591984088904319185224597226259875413417263104 * q^48 + 404071745256872837110945159389335319846982704413781054108834196201188 * q^49 + 806234665148425924790992853208798894255207199876348009508134872678400 * q^50 - 1064663054384373454693815044115707495986393228123917639685336221542112 * q^51 + 685714387132485780807687351264933240450496186063759741333121968111616 * q^52 - 12504487477710484326069573437834673828075644450184339810661130387062824 * q^53 - 44443716007806603006255058779598675686836853379094825254648741657313280 * q^54 - 42234307462751559870688815279523036389647114318141024835956467841276640 * q^55 + 24448346434219554857386857259465331683222677509918394630166844058632192 * q^56 - 108722241788231835680029530262218202922060026555257931870949646820720320 * q^57 + 98845198520456082653157487714849975809061439375041480958881966954905600 * q^58 + 489644385838630680453793035064910893924161250200777882813761286795049200 * q^59 - 661545221420332509504689964508349027705625929138948302588827600295034880 * q^60 + 1366702264988165160362277893397817161446550792366271584021176863925910008 * q^61 + 4870999613966734002792865351354081807964899028017325921601006040889950208 * q^62 + 25467416254050289220554969290497643614373758623650152422090650290214784416 * q^63 + 7067388259113537318333190002971674063309935587502475832486424805170479104 * q^64 - 2818715554123324305996683900688488859608094117415047465075683253256636080 * q^65 - 76596896588285695382683818410402127840320212324145930898048348624034725888 * q^66 - 138862674451263516298138468667610680340162456232851749705287510352340126448 * q^67 + 20660418889194655771113483822358173664967328313530555040079187502706982912 * q^68 - 672683900106157260734473471072801550870108283451440958413239964932071157376 * q^69 - 120976153497010911141285596469887181170360486608554468602744869660403957760 * q^70 + 530690635697814275660599829540627664439950625326921320686595878050597442848 * q^71 + 1346569732747503601590628957775849376430636774436242794777074912689171464192 * q^72 + 4820651523365553876354090014651981899402331637274186511812407488393695849256 * q^73 + 3153797336732213047136019762904340078787789299843033684241252475787534139392 * q^74 + 8602612673085854634235499843915898492279116054592658607479566152826055983600 * q^75 + 6959744757554204748890607318869773205936950878980155945417583978548947845120 * q^76 + 110102493064509063187502350556580136272265257076594423627460955165606056817024 * q^77 + 20827502182804762099559385351883359501065891638239106218565497936471794909184 * q^78 - 50648470438514104286474913909108065961634468975354377982407985787923842139840 * q^79 + 36410635779603672534361228812119143475730269065801269680998301939816233697280 * q^80 + 379041758338986105391656527861324532842247800229613612526137961094691354412004 * q^81 + 265845888935790629257564155475812485570758338183988897393746995357086843404288 * q^82 - 238520107352811898084467415485453163522726145612576880661027053586010982834864 * q^83 - 17702486103296761416238650559110605188614661675886162871804355919104853409792 * q^84 + 6847336985153689699318057514777445791814846059666211714926452668399553888133040 * q^85 + 4295396974245745605449286229736365000414556517589309608033131241106158141833216 * q^86 + 24810367340380739951538667183142703925070608394534355695353250835478847036268000 * q^87 + 4377815620735668160553332165979083271645745234885307873133897377221885126770688 * q^88 + 27428045100278388486412864000971428693338849280996141787710697205603108083187560 * q^89 + 44589380101913620003376240172734928535455487911745767018464513103965350149488640 * q^90 + 83746848785098964567359078917389054910460771603691882810793485825135745141227968 * q^91 + 23751287669314528793963493100263251299423658690678278368493215681069782428286976 * q^92 + 175965595874091058205048012635001521337190765247409917262039952747606802639176192 * q^93 + 74932517503029952683603981255418566273459934607080646663323772465470189917437952 * q^94 + 380998938988702403083396234132524561810537502861981189014224981328509785911813600 * q^95 - 11428121354510432629189792105819544680287663362944398970188201375906177906376704 * q^96 - 154675480473123296332819855790359897912276472938070448066911736856000222852914808 * q^97 + 444281582365673460383294613306035847659645552572115066650073420126860773464997888 * q^98 + 2329598594513713332190439448236254139876473757732194671418010179022886450464947824 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.09951e12	0.707107
$2$	1.09951e12	0.707107
$3$	−1.09428e19	−0.519659	−0.259830	−	0.965654i	$-0.583666\pi$
$3$	−1.09428e19	−0.519659	−0.259830	+	0.965654i	$0.583666\pi$
$4$	1.20893e24	0.500000
$5$	−1.35941e28	−0.668445	−0.334222	−	0.942494i	$-0.608474\pi$
$5$	−1.35941e28	−0.668445	−0.334222	+	0.942494i	$0.608474\pi$
$6$	−1.20318e31	−0.367455
$7$	−2.01060e34	−1.19359	−0.596795	−	0.802394i	$-0.703559\pi$
$7$	−2.01060e34	−1.19359	−0.596795	+	0.802394i	$0.703559\pi$
$8$	1.32923e36	0.353553
$9$	−3.23681e38	−0.729954
$10$	−1.49469e40	−0.472662
$11$	1.34090e41	0.0893292	0.0446646	−	0.999002i	$-0.485778\pi$
$11$	1.34090e41	0.0893292	0.0446646	+	0.999002i	$0.485778\pi$
$12$	−1.32291e43	−0.259830
$13$	−1.41164e45	−1.08398	−0.541988	−	0.840386i	$-0.682328\pi$
$13$	−1.41164e45	−1.08398	−0.541988	+	0.840386i	$0.682328\pi$
$14$	−2.21068e46	−0.843996
$15$	1.48758e47	0.347364
$16$	1.46150e48	0.250000
$17$	−6.93481e49	−1.01825	−0.509123	−	0.860694i	$-0.670030\pi$
$17$	−6.93481e49	−1.01825	−0.509123	+	0.860694i	$0.670030\pi$
$18$	−3.55891e50	−0.516156
$19$	2.85317e51	0.463241	0.231621	−	0.972806i	$-0.425597\pi$
$19$	2.85317e51	0.463241	0.231621	+	0.972806i	$0.425597\pi$
$20$	−1.64343e52	−0.334222
$21$	2.20016e53	0.620260
$22$	1.47434e53	0.0631653
$23$	1.73783e55	1.23036	0.615178	−	0.788388i	$-0.289084\pi$
$23$	1.73783e55	1.23036	0.615178	+	0.788388i	$0.289084\pi$
$24$	−1.45455e55	−0.183727
$25$	−2.28790e56	−0.553181
$26$	−1.55212e57	−0.766486
$27$	8.39432e57	0.898987
$28$	−2.43067e58	−0.596795
$29$	−1.94235e59	−1.15135	−0.575676	−	0.817678i	$-0.695261\pi$
$29$	−1.94235e59	−1.15135	−0.575676	+	0.817678i	$0.695261\pi$
$30$	1.63561e59	0.245623
$31$	−8.81707e59	−0.350894	−0.175447	−	0.984489i	$-0.556137\pi$
$31$	−8.81707e59	−0.350894	−0.175447	+	0.984489i	$0.556137\pi$
$32$	1.60694e60	0.176777
$33$	−1.46733e60	−0.0464207
$34$	−7.62490e61	−0.720009
$35$	2.73323e62	0.797849
$36$	−3.91306e62	−0.364977
$37$	2.77158e63	0.852231	0.426115	−	0.904669i	$-0.359882\pi$
$37$	2.77158e63	0.852231	0.426115	+	0.904669i	$0.359882\pi$
$38$	3.13710e63	0.327561
$39$	1.54474e64	0.563298
$40$	−1.80697e64	−0.236331
$41$	−2.02778e65	−0.975608	−0.487804	−	0.872953i	$-0.662202\pi$
$41$	−2.02778e65	−0.975608	−0.487804	+	0.872953i	$0.662202\pi$
$42$	2.41911e65	0.438590
$43$	2.10478e66	1.47141	0.735706	−	0.677301i	$-0.236851\pi$
$43$	2.10478e66	1.47141	0.735706	+	0.677301i	$0.236851\pi$
$44$	1.62105e65	0.0446646
$45$	4.40016e66	0.487934
$46$	1.91077e67	0.869993
$47$	2.81604e67	0.536631	0.268315	−	0.963331i	$-0.413533\pi$
$47$	2.81604e67	0.536631	0.268315	+	0.963331i	$0.413533\pi$
$48$	−1.59930e67	−0.129915
$49$	1.20498e68	0.424657
$50$	−2.51558e68	−0.391158
$51$	7.58864e68	0.529141
$52$	−1.70657e69	−0.541988
$53$	−7.63795e69	−1.12151	−0.560756	−	0.827981i	$-0.689490\pi$
$53$	−7.63795e69	−1.12151	−0.560756	+	0.827981i	$0.689490\pi$
$54$	9.22966e69	0.635680
$55$	−1.82284e69	−0.0597116
$56$	−2.67255e70	−0.421998
$57$	−3.12218e70	−0.240728
$58$	−2.13564e71	−0.814129
$59$	4.52685e71	0.863555	0.431778	−	0.901980i	$-0.357887\pi$
$59$	4.52685e71	0.863555	0.431778	+	0.901980i	$0.357887\pi$
$60$	1.79837e71	0.173682
$61$	4.55117e71	0.225043	0.112521	−	0.993649i	$-0.464107\pi$
$61$	4.55117e71	0.225043	0.112521	+	0.993649i	$0.464107\pi$
$62$	−9.69447e71	−0.248119
$63$	6.50793e72	0.871266
$64$	1.76685e72	0.125000
$65$	1.91900e73	0.724578
$66$	−1.61334e72	−0.0328244
$67$	−1.17113e74	−1.29592	−0.647958	−	0.761676i	$-0.724377\pi$
$67$	−1.17113e74	−1.29592	−0.647958	+	0.761676i	$0.724377\pi$
$68$	−8.38367e73	−0.509123
$69$	−1.90168e74	−0.639366
$70$	3.00522e74	0.564165
$71$	7.88481e74	0.833351	0.416676	−	0.909055i	$-0.363195\pi$
$71$	7.88481e74	0.833351	0.416676	+	0.909055i	$0.363195\pi$
$72$	−4.30246e74	−0.258078
$73$	−5.80736e73	−0.0199252	−0.00996261	−	0.999950i	$-0.503171\pi$
$73$	−5.80736e73	−0.0199252	−0.00996261	+	0.999950i	$0.503171\pi$
$74$	3.04738e75	0.602618
$75$	2.50361e75	0.287466
$76$	3.44927e75	0.231621
$77$	−2.69602e75	−0.106622
$78$	1.69846e76	0.398312
$79$	7.16447e76	1.00297	0.501483	−	0.865167i	$-0.332788\pi$
$79$	7.16447e76	1.00297	0.501483	+	0.865167i	$0.332788\pi$
$80$	−1.98678e76	−0.167111
$81$	5.16711e76	0.262788
$82$	−2.22957e77	−0.689859
$83$	4.36912e77	0.827430	0.413715	−	0.910406i	$-0.364231\pi$
$83$	4.36912e77	0.827430	0.413715	+	0.910406i	$0.364231\pi$
$84$	2.65984e77	0.310130
$85$	9.42725e77	0.680642
$86$	2.31423e78	1.04044
$87$	2.12548e78	0.598311
$88$	1.78236e77	0.0315826
$89$	1.73505e79	1.94544	0.972720	−	0.231983i	$-0.0745213\pi$
$89$	1.73505e79	1.94544	0.972720	+	0.231983i	$0.0745213\pi$
$90$	4.83802e78	0.345022
$91$	2.83825e79	1.29382
$92$	2.10091e79	0.615178
$93$	9.64836e78	0.182345
$94$	3.09627e79	0.379455
$95$	−3.87863e79	−0.309651
$96$	−1.75844e79	−0.0918636
$97$	4.37906e80	1.50358	0.751788	−	0.659405i	$-0.229192\pi$
$97$	4.37906e80	1.50358	0.751788	+	0.659405i	$0.229192\pi$
$98$	1.32489e80	0.300278
$99$	−4.34025e79	−0.0652062
$100$	−2.76591e80	−0.276591
$101$	2.54574e81	1.70137	0.850683	−	0.525678i	$-0.176188\pi$
$101$	2.54574e81	1.70137	0.850683	+	0.525678i	$0.176188\pi$
$102$	8.34380e80	0.374159
$103$	−4.16422e81	−1.25784	−0.628921	−	0.777469i	$-0.716503\pi$
$103$	−4.16422e81	−1.25784	−0.628921	+	0.777469i	$0.716503\pi$
$104$	−1.87640e81	−0.383243
$105$	−2.99093e81	−0.414610
$106$	−8.39802e81	−0.793029
$107$	−1.44636e82	−0.933753	−0.466877	−	0.884323i	$-0.654621\pi$
$107$	−1.44636e82	−0.933753	−0.466877	+	0.884323i	$0.654621\pi$
$108$	1.01481e82	0.449493
$109$	−5.76944e82	−1.75938	−0.879692	−	0.475544i	$-0.842251\pi$
$109$	−5.76944e82	−1.75938	−0.879692	+	0.475544i	$0.842251\pi$
$110$	−2.00423e81	−0.0422225
$111$	−3.03289e82	−0.442869
$112$	−2.93850e82	−0.298397
$113$	5.44775e82	0.385958	0.192979	−	0.981203i	$-0.438185\pi$
$113$	5.44775e82	0.385958	0.192979	+	0.981203i	$0.438185\pi$
$114$	−3.43287e82	−0.170220
$115$	−2.36243e83	−0.822426
$116$	−2.34816e83	−0.575676
$117$	4.56922e83	0.791252
$118$	4.97733e83	0.610626
$119$	1.39431e84	1.21537
$120$	1.97733e83	0.122812
$121$	−2.23526e84	−0.992020
$122$	5.00406e83	0.159129
$123$	2.21897e84	0.506984
$124$	−1.06592e84	−0.175447
$125$	8.73260e84	1.03822
$126$	7.15555e84	0.616078
$127$	2.28875e85	1.43070	0.715350	−	0.698767i	$-0.246267\pi$
$127$	2.28875e85	1.43070	0.715350	+	0.698767i	$0.246267\pi$
$128$	1.94267e84	0.0883883
$129$	−2.30322e85	−0.764632
$130$	2.10997e85	0.512354
$131$	4.18237e84	0.0744621	0.0372311	−	0.999307i	$-0.488146\pi$
$131$	4.18237e84	0.0744621	0.0372311	+	0.999307i	$0.488146\pi$
$132$	−1.77389e84	−0.0232104
$133$	−5.73659e85	−0.552920
$134$	−1.28767e86	−0.916351
$135$	−1.14113e86	−0.600923
$136$	−9.21794e85	−0.360005
$137$	−3.09640e86	−0.898824	−0.449412	−	0.893325i	$-0.648366\pi$
$137$	−3.09640e86	−0.898824	−0.449412	+	0.893325i	$0.648366\pi$
$138$	−2.09092e86	−0.452100
$139$	−2.04547e86	−0.330136	−0.165068	−	0.986282i	$-0.552784\pi$
$139$	−2.04547e86	−0.330136	−0.165068	+	0.986282i	$0.552784\pi$
$140$	3.30428e86	0.398925
$141$	−3.08154e86	−0.278865
$142$	8.66944e86	0.589268
$143$	−1.89288e86	−0.0968306
$144$	−4.73060e86	−0.182489
$145$	2.64045e87	0.769616
$146$	−6.38526e85	−0.0140893
$147$	−1.31859e87	−0.220677
$148$	3.35063e87	0.426115
$149$	−6.65863e87	−0.644675	−0.322338	−	0.946625i	$-0.604469\pi$
$149$	−6.65863e87	−0.644675	−0.322338	+	0.946625i	$0.604469\pi$
$150$	2.75275e87	0.203269
$151$	2.85938e88	1.61327	0.806633	−	0.591052i	$-0.201287\pi$
$151$	2.85938e88	1.61327	0.806633	+	0.591052i	$0.201287\pi$
$152$	3.79252e87	0.163780
$153$	2.24467e88	0.743274
$154$	−2.96430e87	−0.0753934
$155$	1.19860e88	0.234553
$156$	1.86747e88	0.281649
$157$	−1.25466e89	−1.46080	−0.730398	−	0.683022i	$-0.760665\pi$
$157$	−1.25466e89	−1.46080	−0.730398	+	0.683022i	$0.760665\pi$
$158$	7.87741e88	0.709205
$159$	8.35808e88	0.582804
$160$	−2.18449e88	−0.118165
$161$	−3.49409e89	−1.46854
$162$	5.68130e88	0.185819
$163$	−4.72225e89	−1.20379	−0.601896	−	0.798575i	$-0.705588\pi$
$163$	−4.72225e89	−1.20379	−0.601896	+	0.798575i	$0.705588\pi$
$164$	−2.45144e89	−0.487804
$165$	1.99470e88	0.0310297
$166$	4.80390e89	0.585081
$167$	−1.94133e90	−1.85388	−0.926939	−	0.375211i	$-0.877570\pi$
$167$	−1.94133e90	−1.85388	−0.926939	+	0.375211i	$0.877570\pi$
$168$	2.92452e89	0.219295
$169$	2.96794e89	0.175002
$170$	1.03654e90	0.481287
$171$	−9.23518e89	−0.338145
$172$	2.54452e90	0.735706
$173$	4.83196e90	1.10473	0.552365	−	0.833603i	$-0.313726\pi$
$173$	4.83196e90	1.10473	0.552365	+	0.833603i	$0.313726\pi$
$174$	2.33699e90	0.423070
$175$	4.60006e90	0.660272
$176$	1.95973e89	0.0223323
$177$	−4.95366e90	−0.448754
$178$	1.90771e91	1.37563
$179$	−2.64540e91	−1.52035	−0.760175	−	0.649719i	$-0.774887\pi$
$179$	−2.64540e91	−1.52035	−0.760175	+	0.649719i	$0.774887\pi$
$180$	5.31946e90	0.243967
$181$	−1.53949e91	−0.564151	−0.282076	−	0.959392i	$-0.591023\pi$
$181$	−1.53949e91	−0.564151	−0.282076	+	0.959392i	$0.591023\pi$
$182$	3.12069e91	0.914870
$183$	−4.98027e90	−0.116945
$184$	2.30998e91	0.434997
$185$	−3.76771e91	−0.569669
$186$	1.06085e91	0.128937
$187$	−9.29890e90	−0.0909591
$188$	3.40438e91	0.268315
$189$	−1.68776e92	−1.07302
$190$	−4.26460e91	−0.218956
$191$	−2.24535e92	−0.932037	−0.466018	−	0.884775i	$-0.654312\pi$
$191$	−2.24535e92	−0.932037	−0.466018	+	0.884775i	$0.654312\pi$
$192$	−1.93343e91	−0.0649574
$193$	−4.62320e92	−1.25855	−0.629277	−	0.777181i	$-0.716649\pi$
$193$	−4.62320e92	−1.25855	−0.629277	+	0.777181i	$0.716649\pi$
$194$	4.81483e92	1.06319
$195$	−2.09993e92	−0.376534
$196$	1.45673e92	0.212328
$197$	3.38177e92	0.401108	0.200554	−	0.979683i	$-0.435726\pi$
$197$	3.38177e92	0.401108	0.200554	+	0.979683i	$0.435726\pi$
$198$	−4.77215e91	−0.0461078
$199$	−3.93082e91	−0.0309694	−0.0154847	−	0.999880i	$-0.504929\pi$
$199$	−3.93082e91	−0.0309694	−0.0154847	+	0.999880i	$0.504929\pi$
$200$	−3.04115e92	−0.195579
$201$	1.28155e93	0.673435
$202$	2.79907e93	1.20305
$203$	3.90529e93	1.37424
$204$	9.17410e92	0.264571
$205$	2.75659e93	0.652140
$206$	−4.57861e93	−0.889428
$207$	−5.62504e93	−0.898104
$208$	−2.06312e93	−0.270994
$209$	3.82582e92	0.0413810
$210$	−3.28856e93	−0.293173
$211$	1.42673e94	1.04931	0.524654	−	0.851315i	$-0.324195\pi$
$211$	1.42673e94	1.04931	0.524654	+	0.851315i	$0.324195\pi$
$212$	−9.23372e93	−0.560756
$213$	−8.62821e93	−0.433059
$214$	−1.59028e94	−0.660263
$215$	−2.86126e94	−0.983557
$216$	1.11580e94	0.317840
$217$	1.77276e94	0.418823
$218$	−6.34357e94	−1.24407
$219$	6.35490e92	0.0103543
$220$	−2.20367e93	−0.0298558
$221$	9.78948e94	1.10375
$222$	−3.33470e94	−0.313156
$223$	7.54258e93	0.0590436	0.0295218	−	0.999564i	$-0.490602\pi$
$223$	7.54258e93	0.0590436	0.0295218	+	0.999564i	$0.490602\pi$
$224$	−3.23091e94	−0.210999
$225$	7.40551e94	0.403797
$226$	5.98987e94	0.272914
$227$	1.56153e95	0.594981	0.297491	−	0.954725i	$-0.403850\pi$
$227$	1.56153e95	0.594981	0.297491	+	0.954725i	$0.403850\pi$
$228$	−3.77448e94	−0.120364
$229$	1.76125e95	0.470419	0.235209	−	0.971945i	$-0.424422\pi$
$229$	1.76125e95	0.470419	0.235209	+	0.971945i	$0.424422\pi$
$230$	−2.59752e95	−0.581543
$231$	2.95021e94	0.0554073
$232$	−2.58182e95	−0.407065
$233$	1.18481e96	1.56940	0.784702	−	0.619873i	$-0.212816\pi$
$233$	1.18481e96	1.56940	0.784702	+	0.619873i	$0.212816\pi$
$234$	5.02391e95	0.559500
$235$	−3.82816e95	−0.358708
$236$	5.47263e95	0.431778
$237$	−7.83995e95	−0.521201
$238$	1.53306e96	0.859396
$239$	−1.86426e96	−0.881844	−0.440922	−	0.897546i	$-0.645348\pi$
$239$	−1.86426e96	−0.881844	−0.440922	+	0.897546i	$0.645348\pi$
$240$	2.17410e95	0.0868409
$241$	−3.70273e96	−1.24977	−0.624887	−	0.780715i	$-0.714855\pi$
$241$	−3.70273e96	−1.24977	−0.624887	+	0.780715i	$0.714855\pi$
$242$	−2.45769e96	−0.701464
$243$	−4.28769e96	−1.03555
$244$	5.50203e95	0.112521
$245$	−1.63806e96	−0.283860
$246$	2.43978e96	0.358492
$247$	−4.02766e96	−0.502142
$248$	−1.17199e96	−0.124060
$249$	−4.78105e96	−0.429982
$250$	9.60159e96	0.734130
$251$	1.76418e97	1.14751	0.573756	−	0.819026i	$-0.305486\pi$
$251$	1.76418e97	1.14751	0.573756	+	0.819026i	$0.305486\pi$
$252$	7.86761e96	0.435633
$253$	2.33027e96	0.109907
$254$	2.51651e97	1.01166
$255$	−1.03161e97	−0.353702
$256$	2.13599e96	0.0625000
$257$	3.89420e97	0.973030	0.486515	−	0.873672i	$-0.338268\pi$
$257$	3.89420e97	0.973030	0.486515	+	0.873672i	$0.338268\pi$
$258$	−2.53242e97	−0.540677
$259$	−5.57253e97	−1.01721
$260$	2.31993e97	0.362289
$261$	6.28702e97	0.840435
$262$	4.59857e96	0.0526527
$263$	−5.39423e97	−0.529324	−0.264662	−	0.964341i	$-0.585260\pi$
$263$	−5.39423e97	−0.529324	−0.264662	+	0.964341i	$0.585260\pi$
$264$	−1.95041e96	−0.0164122
$265$	1.03831e98	0.749670
$266$	−6.30745e97	−0.390974
$267$	−1.89864e98	−1.01097
$268$	−1.41581e98	−0.647958
$269$	−1.52947e98	−0.601969	−0.300984	−	0.953629i	$-0.597315\pi$
$269$	−1.52947e98	−0.601969	−0.300984	+	0.953629i	$0.597315\pi$
$270$	−1.25469e98	−0.424917
$271$	6.58423e98	1.91977	0.959885	−	0.280396i	$-0.0904657\pi$
$271$	6.58423e98	1.91977	0.959885	+	0.280396i	$0.0904657\pi$
$272$	−1.01352e98	−0.254562
$273$	−3.10585e98	−0.672347
$274$	−3.40453e98	−0.635564
$275$	−3.06786e97	−0.0494152
$276$	−2.29899e98	−0.319683
$277$	3.36599e98	0.404280	0.202140	−	0.979357i	$-0.435210\pi$
$277$	3.36599e98	0.404280	0.202140	+	0.979357i	$0.435210\pi$
$278$	−2.24901e98	−0.233441
$279$	2.85392e98	0.256136
$280$	3.63309e98	0.282082
$281$	−3.06754e98	−0.206151	−0.103075	−	0.994674i	$-0.532868\pi$
$281$	−3.06754e98	−0.206151	−0.103075	+	0.994674i	$0.532868\pi$
$282$	−3.38819e98	−0.197187
$283$	2.40864e99	1.21457	0.607284	−	0.794485i	$-0.292259\pi$
$283$	2.40864e99	1.21457	0.607284	+	0.794485i	$0.292259\pi$
$284$	9.53215e98	0.416676
$285$	4.24432e98	0.160913
$286$	−2.08124e98	−0.0684696
$287$	4.07706e99	1.16448
$288$	−5.20135e98	−0.129039
$289$	1.70814e98	0.0368266
$290$	2.90321e99	0.544201
$291$	−4.79193e99	−0.781347
$292$	−7.02067e97	−0.00996261
$293$	−2.20050e99	−0.271883	−0.135942	−	0.990717i	$-0.543406\pi$
$293$	−2.20050e99	−0.271883	−0.135942	+	0.990717i	$0.543406\pi$
$294$	−1.44980e99	−0.156042
$295$	−6.15386e99	−0.577239
$296$	3.68406e99	0.301309
$297$	1.12560e99	0.0803057
$298$	−7.32124e99	−0.455854
$299$	−2.45320e100	−1.33368
$300$	3.02668e99	0.143733
$301$	−4.23187e100	−1.75626
$302$	3.14393e100	1.14075
$303$	−2.78576e100	−0.884131
$304$	4.16992e99	0.115810
$305$	−6.18691e99	−0.150429
$306$	2.46804e100	0.525574
$307$	9.90809e100	1.84878	0.924391	−	0.381446i	$-0.124574\pi$
$307$	9.90809e100	1.84878	0.924391	+	0.381446i	$0.124574\pi$
$308$	−3.25929e99	−0.0533112
$309$	4.55683e100	0.653649
$310$	1.31788e100	0.165854
$311$	6.14070e100	0.678301	0.339150	−	0.940732i	$-0.389860\pi$
$311$	6.14070e100	0.678301	0.339150	+	0.940732i	$0.389860\pi$
$312$	2.05331e100	0.199156
$313$	−4.63035e100	−0.394520	−0.197260	−	0.980351i	$-0.563204\pi$
$313$	−4.63035e100	−0.394520	−0.197260	+	0.980351i	$0.563204\pi$
$314$	−1.37951e101	−1.03294
$315$	−8.84696e100	−0.582393
$316$	8.66131e100	0.501483
$317$	1.51283e101	0.770710	0.385355	−	0.922768i	$-0.374079\pi$
$317$	1.51283e101	0.770710	0.385355	+	0.922768i	$0.374079\pi$
$318$	9.18980e100	0.412105
$319$	−2.60450e100	−0.102849
$320$	−2.40187e100	−0.0835556
$321$	1.58272e101	0.485233
$322$	−3.84179e101	−1.03842
$323$	−1.97862e101	−0.471694
$324$	6.24666e100	0.131394
$325$	3.22971e101	0.599635
$326$	−5.19217e101	−0.851209
$327$	6.31340e101	0.914280
$328$	−2.69539e101	−0.344930
$329$	−5.66193e101	−0.640517
$330$	2.19319e100	0.0219413
$331$	−1.44712e102	−1.28077	−0.640384	−	0.768055i	$-0.721225\pi$
$331$	−1.44712e102	−1.28077	−0.640384	+	0.768055i	$0.721225\pi$
$332$	5.28194e101	0.413715
$333$	−8.97107e101	−0.622089
$334$	−2.13451e102	−1.31089
$335$	1.59205e102	0.866248
$336$	3.21554e101	0.155065
$337$	3.45224e102	1.47601	0.738007	−	0.674793i	$-0.235767\pi$
$337$	3.45224e102	1.47601	0.738007	+	0.674793i	$0.235767\pi$
$338$	3.26329e101	0.123745
$339$	−5.96138e101	−0.200567
$340$	1.13969e102	0.340321
$341$	−1.18228e101	−0.0313450
$342$	−1.01542e102	−0.239105
$343$	3.28242e102	0.686724
$344$	2.79773e102	0.520222
$345$	2.58517e102	0.427381
$346$	5.31279e102	0.781162
$347$	−2.75746e102	−0.360716	−0.180358	−	0.983601i	$-0.557726\pi$
$347$	−2.75746e102	−0.360716	−0.180358	+	0.983601i	$0.557726\pi$
$348$	2.56955e102	0.299156
$349$	−2.00163e102	−0.207469	−0.103735	−	0.994605i	$-0.533079\pi$
$349$	−2.00163e102	−0.207469	−0.103735	+	0.994605i	$0.533079\pi$
$350$	5.05782e102	0.466883
$351$	−1.18498e103	−0.974479
$352$	2.15475e101	0.0157913
$353$	−9.19073e102	−0.600447	−0.300223	−	0.953869i	$-0.597061\pi$
$353$	−9.19073e102	−0.600447	−0.300223	+	0.953869i	$0.597061\pi$
$354$	−5.44660e102	−0.317317
$355$	−1.07187e103	−0.557049
$356$	2.09755e103	0.972720
$357$	−1.52577e103	−0.631578
$358$	−2.90865e103	−1.07505
$359$	−1.16554e103	−0.384769	−0.192385	−	0.981320i	$-0.561622\pi$
$359$	−1.16554e103	−0.384769	−0.192385	+	0.981320i	$0.561622\pi$
$360$	5.84881e102	0.172511
$361$	−2.97945e103	−0.785408
$362$	−1.69269e103	−0.398915
$363$	2.44601e103	0.515513
$364$	3.43124e103	0.646911
$365$	7.89459e101	0.0133189
$366$	−5.47586e102	−0.0826929
$367$	4.33470e103	0.586116	0.293058	−	0.956095i	$-0.405327\pi$
$367$	4.33470e103	0.586116	0.293058	+	0.956095i	$0.405327\pi$
$368$	2.53985e103	0.307589
$369$	6.56355e103	0.712149
$370$	−4.14264e103	−0.402817
$371$	1.53569e104	1.33863
$372$	1.16642e103	0.0911726
$373$	−1.07564e104	−0.754151	−0.377075	−	0.926183i	$-0.623070\pi$
$373$	−1.07564e104	−0.754151	−0.377075	+	0.926183i	$0.623070\pi$
$374$	−1.02242e103	−0.0643178
$375$	−9.55593e103	−0.539519
$376$	3.74316e103	0.189728
$377$	2.74191e104	1.24804
$378$	−1.85572e104	−0.758741
$379$	3.00956e104	1.10564	0.552822	−	0.833299i	$-0.313551\pi$
$379$	3.00956e104	1.10564	0.552822	+	0.833299i	$0.313551\pi$
$380$	−4.68898e103	−0.154826
$381$	−2.50454e104	−0.743476
$382$	−2.46879e104	−0.659050
$383$	2.43967e104	0.585843	0.292922	−	0.956136i	$-0.405372\pi$
$383$	2.43967e104	0.585843	0.292922	+	0.956136i	$0.405372\pi$
$384$	−2.12583e103	−0.0459318
$385$	3.66500e103	0.0712712
$386$	−5.08327e104	−0.889932
$387$	−6.81277e104	−1.07406
$388$	5.29396e104	0.751788
$389$	9.77624e104	1.25087	0.625436	−	0.780275i	$-0.284921\pi$
$389$	9.77624e104	1.25087	0.625436	+	0.780275i	$0.284921\pi$
$390$	−2.30890e104	−0.266249
$391$	−1.20515e105	−1.25281
$392$	1.60169e104	0.150139
$393$	−4.57670e103	−0.0386949
$394$	3.71829e104	0.283626
$395$	−9.73945e104	−0.670428
$396$	−5.24704e103	−0.0326031
$397$	−2.87083e105	−1.61061	−0.805307	−	0.592858i	$-0.797999\pi$
$397$	−2.87083e105	−1.61061	−0.805307	+	0.592858i	$0.797999\pi$
$398$	−4.32198e103	−0.0218987
$399$	6.27745e104	0.287330
$400$	−3.34378e104	−0.138295
$401$	4.08222e105	1.52598	0.762992	−	0.646408i	$-0.223730\pi$
$401$	4.08222e105	1.52598	0.762992	+	0.646408i	$0.223730\pi$
$402$	1.40908e105	0.476190
$403$	1.24466e105	0.380360
$404$	3.07761e105	0.850683
$405$	−7.02423e104	−0.175659
$406$	4.29391e105	0.971737
$407$	3.71641e104	0.0761290
$408$	1.00870e105	0.187080
$409$	5.01221e104	0.0841852	0.0420926	−	0.999114i	$-0.486598\pi$
$409$	5.01221e104	0.0841852	0.0420926	+	0.999114i	$0.486598\pi$
$410$	3.03090e105	0.461133
$411$	3.38834e105	0.467082
$412$	−5.03423e105	−0.628921
$413$	−9.10170e105	−1.03073
$414$	−6.18480e105	−0.635055
$415$	−5.93943e105	−0.553091
$416$	−2.26842e105	−0.191622
$417$	2.23832e105	0.171558
$418$	4.20654e104	0.0292608
$419$	−3.00211e105	−0.189565	−0.0947825	−	0.995498i	$-0.530216\pi$
$419$	−3.00211e105	−0.189565	−0.0947825	+	0.995498i	$0.530216\pi$
$420$	−3.61581e105	−0.207305
$421$	−1.76425e106	−0.918621	−0.459310	−	0.888276i	$-0.651903\pi$
$421$	−1.76425e106	−0.918621	−0.459310	+	0.888276i	$0.651903\pi$
$422$	1.56871e106	0.741973
$423$	−9.11499e105	−0.391716
$424$	−1.01526e106	−0.396515
$425$	1.58662e106	0.563275
$426$	−9.48681e105	−0.306219
$427$	−9.15058e105	−0.268609
$428$	−1.74854e106	−0.466877
$429$	2.07134e105	0.0503189
$430$	−3.14599e106	−0.695480
$431$	8.04943e106	1.61970	0.809852	−	0.586635i	$-0.199548\pi$
$431$	8.04943e106	1.61970	0.809852	+	0.586635i	$0.199548\pi$
$432$	1.22683e106	0.224747
$433$	1.01833e107	1.69875	0.849376	−	0.527788i	$-0.176978\pi$
$433$	1.01833e107	1.69875	0.849376	+	0.527788i	$0.176978\pi$
$434$	1.94917e106	0.296153
$435$	−2.88940e106	−0.399938
$436$	−6.97483e106	−0.879692
$437$	4.95834e106	0.569952
$438$	6.98728e104	0.00732161
$439$	−7.01037e105	−0.0669773	−0.0334887	−	0.999439i	$-0.510662\pi$
$439$	−7.01037e105	−0.0669773	−0.0334887	+	0.999439i	$0.510662\pi$
$440$	−2.42297e105	−0.0211112
$441$	−3.90029e106	−0.309980
$442$	1.07636e107	0.780472
$443$	1.59443e107	1.05500	0.527501	−	0.849554i	$-0.323129\pi$
$443$	1.59443e107	1.05500	0.527501	+	0.849554i	$0.323129\pi$
$444$	−3.66654e106	−0.221435
$445$	−2.35865e107	−1.30042
$446$	8.29316e105	0.0417502
$447$	7.28642e106	0.335012
$448$	−3.55242e106	−0.149199
$449$	−2.12994e107	−0.817316	−0.408658	−	0.912688i	$-0.634003\pi$
$449$	−2.12994e107	−0.817316	−0.408658	+	0.912688i	$0.634003\pi$
$450$	8.14245e106	0.285528
$451$	−2.71906e106	−0.0871503
$452$	6.58593e106	0.192979
$453$	−3.12897e107	−0.838349
$454$	1.71692e107	0.420715
$455$	−3.85835e107	−0.864849
$456$	−4.15008e106	−0.0851100
$457$	4.95978e107	0.930800	0.465400	−	0.885100i	$-0.345910\pi$
$457$	4.95978e107	0.930800	0.465400	+	0.885100i	$0.345910\pi$
$458$	1.93652e107	0.332636
$459$	−5.82130e107	−0.915390
$460$	−2.85600e107	−0.411213
$461$	−3.15964e107	−0.416629	−0.208315	−	0.978062i	$-0.566798\pi$
$461$	−3.15964e107	−0.416629	−0.208315	+	0.978062i	$0.566798\pi$
$462$	3.24379e106	0.0391789
$463$	−3.81465e107	−0.422109	−0.211054	−	0.977474i	$-0.567690\pi$
$463$	−3.81465e107	−0.422109	−0.211054	+	0.977474i	$0.567690\pi$
$464$	−2.83875e107	−0.287838
$465$	−1.31161e107	−0.121888
$466$	1.30272e108	1.10974
$467$	−1.99161e107	−0.155550	−0.0777749	−	0.996971i	$-0.524782\pi$
$467$	−1.99161e107	−0.155550	−0.0777749	+	0.996971i	$0.524782\pi$
$468$	5.52385e107	0.395626
$469$	2.35468e108	1.54679
$470$	−4.20910e107	−0.253645
$471$	1.37295e108	0.759116
$472$	6.01722e107	0.305313
$473$	2.82230e107	0.131440
$474$	−8.62012e107	−0.368545
$475$	−6.52779e107	−0.256256
$476$	1.68562e108	0.607685
$477$	2.47226e108	0.818653
$478$	−2.04978e108	−0.623558
$479$	1.93862e108	0.541880	0.270940	−	0.962596i	$-0.412665\pi$
$479$	1.93862e108	0.541880	0.270940	+	0.962596i	$0.412665\pi$
$480$	2.39045e107	0.0614058
$481$	−3.91248e108	−0.923797
$482$	−4.07119e108	−0.883723
$483$	3.82352e108	0.763141
$484$	−2.70226e108	−0.496010
$485$	−5.95294e108	−1.00506
$486$	−4.71437e108	−0.732242
$487$	−9.56980e108	−1.36766	−0.683832	−	0.729639i	$-0.739688\pi$
$487$	−9.56980e108	−1.36766	−0.683832	+	0.729639i	$0.739688\pi$
$488$	6.04954e107	0.0795646
$489$	5.16748e108	0.625561
$490$	−1.80107e108	−0.200719
$491$	−5.92115e107	−0.0607583	−0.0303791	−	0.999538i	$-0.509671\pi$
$491$	−5.92115e107	−0.0607583	−0.0303791	+	0.999538i	$0.509671\pi$
$492$	2.68257e108	0.253492
$493$	1.34698e109	1.17236
$494$	−4.42846e108	−0.355068
$495$	5.90018e107	0.0435868
$496$	−1.28862e108	−0.0877234
$497$	−1.58532e109	−0.994679
$498$	−5.25682e108	−0.304043
$499$	2.91358e109	1.55366	0.776828	−	0.629713i	$-0.216828\pi$
$499$	2.91358e109	1.55366	0.776828	+	0.629713i	$0.216828\pi$
$500$	1.05571e109	0.519108
$501$	2.12436e109	0.963385
$502$	1.93974e109	0.811414
$503$	−1.57775e109	−0.608885	−0.304442	−	0.952531i	$-0.598470\pi$
$503$	−1.57775e109	−0.608885	−0.304442	+	0.952531i	$0.598470\pi$
$504$	8.65053e108	0.308039
$505$	−3.46070e109	−1.13727
$506$	2.56215e108	0.0777158
$507$	−3.24777e108	−0.0909417
$508$	2.76693e109	0.715350
$509$	−2.64024e109	−0.630336	−0.315168	−	0.949036i	$-0.602061\pi$
$509$	−2.64024e109	−0.630336	−0.315168	+	0.949036i	$0.602061\pi$
$510$	−1.13426e109	−0.250105
$511$	1.16763e108	0.0237825
$512$	2.34854e108	0.0441942
$513$	2.39505e109	0.416448
$514$	4.28171e109	0.688036
$515$	5.66088e109	0.840798
$516$	−2.78443e109	−0.382316
$517$	3.77603e108	0.0479368
$518$	−6.12707e109	−0.719279
$519$	−5.28753e109	−0.574083
$520$	2.55079e109	0.256177
$521$	−1.27624e110	−1.18579	−0.592893	−	0.805282i	$-0.702014\pi$
$521$	−1.27624e110	−1.18579	−0.592893	+	0.805282i	$0.702014\pi$
$522$	6.91265e109	0.594277
$523$	−1.70175e110	−1.35387	−0.676934	−	0.736044i	$-0.736692\pi$
$523$	−1.70175e110	−1.35387	−0.676934	+	0.736044i	$0.736692\pi$
$524$	5.05618e108	0.0372311
$525$	−5.03377e109	−0.343116
$526$	−5.93101e109	−0.374288
$527$	6.11447e109	0.357296
$528$	−2.14450e108	−0.0116052
$529$	1.02501e110	0.513777
$530$	1.14164e110	0.530096
$531$	−1.46526e110	−0.630356
$532$	−6.93511e109	−0.276460
$533$	2.86251e110	1.05754
$534$	−2.08758e110	−0.714861
$535$	1.96619e110	0.624162
$536$	−1.55670e110	−0.458175
$537$	2.89481e110	0.790064
$538$	−1.68167e110	−0.425656
$539$	1.61576e109	0.0379343
$540$	−1.37955e110	−0.300462
$541$	1.50504e110	0.304129	0.152064	−	0.988371i	$-0.451408\pi$
$541$	1.50504e110	0.304129	0.152064	+	0.988371i	$0.451408\pi$
$542$	7.23944e110	1.35748
$543$	1.68464e110	0.293166
$544$	−1.11438e110	−0.180002
$545$	7.84304e110	1.17605
$546$	−3.41492e110	−0.475421
$547$	8.07682e110	1.04413	0.522064	−	0.852907i	$-0.325162\pi$
$547$	8.07682e110	1.04413	0.522064	+	0.852907i	$0.325162\pi$
$548$	−3.74332e110	−0.449412
$549$	−1.47313e110	−0.164271
$550$	−3.37314e109	−0.0349418
$551$	−5.54186e110	−0.533354
$552$	−2.52777e110	−0.226050
$553$	−1.44049e111	−1.19713
$554$	3.70095e110	0.285869
$555$	4.12294e110	0.296034
$556$	−2.47282e110	−0.165068
$557$	−2.30434e111	−1.43024	−0.715122	−	0.698999i	$-0.753629\pi$
$557$	−2.30434e111	−1.43024	−0.715122	+	0.698999i	$0.753629\pi$
$558$	3.13792e110	0.181116
$559$	−2.97120e111	−1.59497
$560$	3.99462e110	0.199462
$561$	1.01756e110	0.0472678
$562$	−3.37280e110	−0.145770
$563$	1.46911e111	0.590832	0.295416	−	0.955369i	$-0.404542\pi$
$563$	1.46911e111	0.590832	0.295416	+	0.955369i	$0.404542\pi$
$564$	−3.72536e110	−0.139433
$565$	−7.40573e110	−0.257992
$566$	2.64833e111	0.858830
$567$	−1.03890e111	−0.313661
$568$	1.04807e111	0.294634
$569$	3.59914e111	0.942219	0.471109	−	0.882075i	$-0.343854\pi$
$569$	3.59914e111	0.942219	0.471109	+	0.882075i	$0.343854\pi$
$570$	4.66668e110	0.113783
$571$	−1.14530e110	−0.0260110	−0.0130055	−	0.999915i	$-0.504140\pi$
$571$	−1.14530e110	−0.0260110	−0.0130055	+	0.999915i	$0.504140\pi$
$572$	−2.28835e110	−0.0484153
$573$	2.45705e111	0.484342
$574$	4.48278e111	0.823409
$575$	−3.97600e111	−0.680610
$576$	−5.71895e110	−0.0912443
$577$	5.30900e111	0.789573	0.394787	−	0.918773i	$-0.370819\pi$
$577$	5.30900e111	0.789573	0.394787	+	0.918773i	$0.370819\pi$
$578$	1.87812e110	0.0260403
$579$	5.05909e111	0.654019
$580$	3.19211e111	0.384808
$581$	−8.78455e111	−0.987612
$582$	−5.26878e111	−0.552496
$583$	−1.02417e111	−0.100184
$584$	−7.71931e109	−0.00704463
$585$	−6.21145e111	−0.528909
$586$	−2.41947e111	−0.192250
$587$	1.25086e112	0.927614	0.463807	−	0.885936i	$-0.346483\pi$
$587$	1.25086e112	0.927614	0.463807	+	0.885936i	$0.346483\pi$
$588$	−1.59407e111	−0.110338
$589$	−2.51566e111	−0.162548
$590$	−6.76624e111	−0.408170
$591$	−3.70061e111	−0.208440
$592$	4.05066e111	0.213058
$593$	2.05837e112	1.01113	0.505566	−	0.862788i	$-0.331284\pi$
$593$	2.05837e112	1.01113	0.505566	+	0.862788i	$0.331284\pi$
$594$	1.23761e111	0.0567847
$595$	−1.89544e112	−0.812407
$596$	−8.04979e111	−0.322338
$597$	4.30143e110	0.0160936
$598$	−2.69733e112	−0.943051
$599$	−1.41395e112	−0.462005	−0.231002	−	0.972953i	$-0.574200\pi$
$599$	−1.41395e112	−0.462005	−0.231002	+	0.972953i	$0.574200\pi$
$600$	3.32787e111	0.101635
$601$	−1.18747e112	−0.339007	−0.169503	−	0.985530i	$-0.554216\pi$
$601$	−1.18747e112	−0.339007	−0.169503	+	0.985530i	$0.554216\pi$
$602$	−4.65299e112	−1.24186
$603$	3.79074e112	0.945959
$604$	3.45678e112	0.806633
$605$	3.03864e112	0.663111
$606$	−3.06297e112	−0.625175
$607$	3.10494e112	0.592804	0.296402	−	0.955063i	$-0.404213\pi$
$607$	3.10494e112	0.592804	0.296402	+	0.955063i	$0.404213\pi$
$608$	4.58487e111	0.0818902
$609$	−4.27349e112	−0.714138
$610$	−6.80258e111	−0.106369
$611$	−3.97525e112	−0.581694
$612$	2.71363e112	0.371637
$613$	1.07051e113	1.37228	0.686139	−	0.727471i	$-0.259304\pi$
$613$	1.07051e113	1.37228	0.686139	+	0.727471i	$0.259304\pi$
$614$	1.08941e113	1.30729
$615$	−3.01649e112	−0.338891
$616$	−3.58362e111	−0.0376967
$617$	4.81319e112	0.474115	0.237057	−	0.971496i	$-0.423817\pi$
$617$	4.81319e112	0.474115	0.237057	+	0.971496i	$0.423817\pi$
$618$	5.01029e112	0.462200
$619$	−2.44135e111	−0.0210940	−0.0105470	−	0.999944i	$-0.503357\pi$
$619$	−2.44135e111	−0.0210940	−0.0105470	+	0.999944i	$0.503357\pi$
$620$	1.44902e112	0.117277
$621$	1.45879e113	1.10607
$622$	6.75177e112	0.479631
$623$	−3.48850e113	−2.32206
$624$	2.25764e112	0.140824
$625$	−2.40864e112	−0.140809
$626$	−5.09113e112	−0.278968
$627$	−4.18653e111	−0.0215040
$628$	−1.51679e113	−0.730398
$629$	−1.92204e113	−0.867781
$630$	−9.72733e112	−0.411814
$631$	1.17093e113	0.464882	0.232441	−	0.972610i	$-0.425329\pi$
$631$	1.17093e113	0.464882	0.232441	+	0.972610i	$0.425329\pi$
$632$	9.52321e112	0.354602
$633$	−1.56125e113	−0.545283
$634$	1.66338e113	0.544974
$635$	−3.11136e113	−0.956344
$636$	1.01043e113	0.291402
$637$	−1.70100e113	−0.460318
$638$	−2.86368e112	−0.0727255
$639$	−2.55216e113	−0.608308
$640$	−2.64089e112	−0.0590827
$641$	9.00250e112	0.189066	0.0945329	−	0.995522i	$-0.469864\pi$
$641$	9.00250e112	0.189066	0.0945329	+	0.995522i	$0.469864\pi$
$642$	1.74022e113	0.343112
$643$	5.32112e113	0.985050	0.492525	−	0.870298i	$-0.336074\pi$
$643$	5.32112e113	0.985050	0.492525	+	0.870298i	$0.336074\pi$
$644$	−4.22410e113	−0.734271
$645$	3.13103e113	0.511115
$646$	−2.17552e113	−0.333538
$647$	8.89947e113	1.28157	0.640783	−	0.767722i	$-0.278610\pi$
$647$	8.89947e113	1.28157	0.640783	+	0.767722i	$0.278610\pi$
$648$	6.86827e112	0.0929094
$649$	6.07007e112	0.0771407
$650$	3.55110e113	0.424006
$651$	−1.93990e113	−0.217645
$652$	−5.70885e113	−0.601896
$653$	1.74265e114	1.72673	0.863367	−	0.504577i	$-0.168351\pi$
$653$	1.74265e114	1.72673	0.863367	+	0.504577i	$0.168351\pi$
$654$	6.94166e113	0.646494
$655$	−5.68557e112	−0.0497738
$656$	−2.96361e113	−0.243902
$657$	1.87973e112	0.0145445
$658$	−6.22536e113	−0.452914
$659$	−4.23044e113	−0.289418	−0.144709	−	0.989474i	$-0.546225\pi$
$659$	−4.23044e113	−0.289418	−0.144709	+	0.989474i	$0.546225\pi$
$660$	2.41144e112	0.0155148
$661$	7.44154e113	0.450302	0.225151	−	0.974324i	$-0.427712\pi$
$661$	7.44154e113	0.450302	0.225151	+	0.974324i	$0.427712\pi$
$662$	−1.59112e114	−0.905640
$663$	−1.07125e114	−0.573576
$664$	5.80756e113	0.292541
$665$	7.79838e113	0.369597
$666$	−9.86380e113	−0.439884
$667$	−3.37548e114	−1.41657
$668$	−2.34692e114	−0.926939
$669$	−8.25372e112	−0.0306826
$670$	1.75048e114	0.612530
$671$	6.10267e112	0.0201029
$672$	3.53553e113	0.109648
$673$	6.37304e114	1.86096	0.930478	−	0.366347i	$-0.119392\pi$
$673$	6.37304e114	1.86096	0.930478	+	0.366347i	$0.119392\pi$
$674$	3.79578e114	1.04370
$675$	−1.92054e114	−0.497303
$676$	3.58802e113	0.0875012
$677$	−5.60315e114	−1.28704	−0.643519	−	0.765430i	$-0.722526\pi$
$677$	−5.60315e114	−1.28704	−0.643519	+	0.765430i	$0.722526\pi$
$678$	−6.55461e113	−0.141822
$679$	−8.80454e114	−1.79465
$680$	1.25310e114	0.240643
$681$	−1.70876e114	−0.309188
$682$	−1.29993e113	−0.0221643
$683$	1.37058e114	0.220224	0.110112	−	0.993919i	$-0.464879\pi$
$683$	1.37058e114	0.220224	0.110112	+	0.993919i	$0.464879\pi$
$684$	−1.11646e114	−0.169072
$685$	4.20928e114	0.600814
$686$	3.60906e114	0.485587
$687$	−1.92731e114	−0.244458
$688$	3.07614e114	0.367853
$689$	1.07821e115	1.21569
$690$	2.84242e114	0.302204
$691$	−5.10818e114	−0.512159	−0.256080	−	0.966656i	$-0.582431\pi$
$691$	−5.10818e114	−0.512159	−0.256080	+	0.966656i	$0.582431\pi$
$692$	5.84148e114	0.552365
$693$	8.72650e113	0.0778295
$694$	−3.03186e114	−0.255065
$695$	2.78063e114	0.220678
$696$	2.82525e114	0.211535
$697$	1.40623e115	0.993410
$698$	−2.20082e114	−0.146703
$699$	−1.29652e115	−0.815556
$700$	5.56113e114	0.330136
$701$	1.73691e115	0.973189	0.486595	−	0.873628i	$-0.338239\pi$
$701$	1.73691e115	0.973189	0.486595	+	0.873628i	$0.338239\pi$
$702$	−1.30290e115	−0.689061
$703$	7.90779e114	0.394788
$704$	2.36917e113	0.0111661
$705$	4.18908e114	0.186406
$706$	−1.01053e115	−0.424580
$707$	−5.11846e115	−2.03073
$708$	−5.98860e114	−0.224377
$709$	6.35739e112	0.00224961	0.00112480	−	0.999999i	$-0.499642\pi$
$709$	6.35739e112	0.00224961	0.00112480	+	0.999999i	$0.499642\pi$
$710$	−1.17853e115	−0.393893
$711$	−2.31900e115	−0.732120
$712$	2.30628e115	0.687817
$713$	−1.53226e115	−0.431724
$714$	−1.67760e115	−0.446593
$715$	2.57320e114	0.0647259
$716$	−3.19809e115	−0.760175
$717$	2.04003e115	0.458258
$718$	−1.28152e115	−0.272073
$719$	−1.34275e115	−0.269447	−0.134724	−	0.990883i	$-0.543015\pi$
$719$	−1.34275e115	−0.269447	−0.134724	+	0.990883i	$0.543015\pi$
$720$	6.43084e114	0.121984
$721$	8.37258e115	1.50135
$722$	−3.27594e115	−0.555367
$723$	4.05183e115	0.649456
$724$	−1.86113e115	−0.282076
$725$	4.44391e115	0.636907
$726$	2.68941e115	0.364522
$727$	6.40980e115	0.821677	0.410838	−	0.911708i	$-0.365236\pi$
$727$	6.40980e115	0.821677	0.410838	+	0.911708i	$0.365236\pi$
$728$	3.77268e115	0.457435
$729$	2.40071e115	0.275344
$730$	8.68020e113	0.00941789
$731$	−1.45962e116	−1.49826
$732$	−6.02077e114	−0.0584727
$733$	1.13204e116	1.04028	0.520142	−	0.854080i	$-0.325879\pi$
$733$	1.13204e116	1.04028	0.520142	+	0.854080i	$0.325879\pi$
$734$	4.76605e115	0.414447
$735$	1.79250e115	0.147510
$736$	2.79259e115	0.217498
$737$	−1.57037e115	−0.115763
$738$	7.21670e115	0.503566
$739$	−1.93579e116	−1.27867	−0.639334	−	0.768929i	$-0.720790\pi$
$739$	−1.93579e116	−1.27867	−0.639334	+	0.768929i	$0.720790\pi$
$740$	−4.55488e115	−0.284835
$741$	4.40740e115	0.260943
$742$	1.68851e116	0.946552
$743$	−2.79056e116	−1.48130	−0.740652	−	0.671889i	$-0.765483\pi$
$743$	−2.79056e116	−1.48130	−0.740652	+	0.671889i	$0.765483\pi$
$744$	1.28249e115	0.0644687
$745$	9.05182e115	0.430930
$746$	−1.18268e116	−0.533265
$747$	−1.41420e116	−0.603986
$748$	−1.12417e115	−0.0454796
$749$	2.90804e116	1.11452
$750$	−1.05069e116	−0.381497
$751$	−3.90954e116	−1.34496	−0.672479	−	0.740117i	$-0.734770\pi$
$751$	−3.90954e116	−1.34496	−0.672479	+	0.740117i	$0.734770\pi$
$752$	4.11565e115	0.134158
$753$	−1.93051e116	−0.596315
$754$	3.01476e116	0.882496
$755$	−3.88708e116	−1.07838
$756$	−2.04038e116	−0.536511
$757$	3.79330e116	0.945441	0.472720	−	0.881212i	$-0.343272\pi$
$757$	3.79330e116	0.945441	0.472720	+	0.881212i	$0.343272\pi$
$758$	3.30905e116	0.781809
$759$	−2.54997e115	−0.0571140
$760$	−5.15559e115	−0.109478
$761$	−4.01135e116	−0.807628	−0.403814	−	0.914841i	$-0.632316\pi$
$761$	−4.01135e116	−0.807628	−0.403814	+	0.914841i	$0.632316\pi$
$762$	−2.75377e116	−0.525717
$763$	1.16000e117	2.09998
$764$	−2.71446e116	−0.466018
$765$	−3.05142e116	−0.496837
$766$	2.68245e116	0.414254
$767$	−6.39031e116	−0.936072
$768$	−2.33737e115	−0.0324787
$769$	1.43282e117	1.88875	0.944377	−	0.328865i	$-0.106666\pi$
$769$	1.43282e117	1.88875	0.944377	+	0.328865i	$0.106666\pi$
$770$	4.02971e115	0.0503963
$771$	−4.26135e116	−0.505644
$772$	−5.58911e116	−0.629277
$773$	1.34053e117	1.43222	0.716108	−	0.697989i	$-0.245922\pi$
$773$	1.34053e117	1.43222	0.716108	+	0.697989i	$0.245922\pi$
$774$	−7.49072e116	−0.759477
$775$	2.01726e116	0.194108
$776$	5.82077e116	0.531595
$777$	6.09793e116	0.528605
$778$	1.07491e117	0.884500
$779$	−5.78562e116	−0.451942
$780$	−2.53866e116	−0.188267
$781$	1.05728e116	0.0744426
$782$	−1.32508e117	−0.885868
$783$	−1.63047e117	−1.03505
$784$	1.76108e116	0.106164
$785$	1.70560e117	0.976461
$786$	−5.03213e115	−0.0273614
$787$	1.37146e117	0.708282	0.354141	−	0.935192i	$-0.384773\pi$
$787$	1.37146e117	0.708282	0.354141	+	0.935192i	$0.384773\pi$
$788$	4.08831e116	0.200554
$789$	5.90281e116	0.275068
$790$	−1.07086e117	−0.474064
$791$	−1.09533e117	−0.460676
$792$	−5.76918e115	−0.0230539
$793$	−6.42463e116	−0.243941
$794$	−3.15651e117	−1.13888
$795$	−1.13621e117	−0.389573
$796$	−4.75207e115	−0.0154847
$797$	−4.66086e117	−1.44346	−0.721729	−	0.692176i	$-0.756652\pi$
$797$	−4.66086e117	−1.44346	−0.721729	+	0.692176i	$0.756652\pi$
$798$	6.90213e116	0.203173
$799$	−1.95287e117	−0.546423
$800$	−3.67652e116	−0.0977896
$801$	−5.61604e117	−1.42008
$802$	4.48845e117	1.07903
$803$	−7.78711e114	−0.00177990
$804$	1.54930e117	0.336717
$805$	4.74990e117	0.981639
$806$	1.36851e117	0.268955
$807$	1.67367e117	0.312819
$808$	3.38387e117	0.601524
$809$	3.96055e117	0.669638	0.334819	−	0.942282i	$-0.391325\pi$
$809$	3.96055e117	0.669638	0.334819	+	0.942282i	$0.391325\pi$
$810$	−7.72322e116	−0.124210
$811$	6.26152e117	0.957933	0.478967	−	0.877833i	$-0.341012\pi$
$811$	6.26152e117	0.957933	0.478967	+	0.877833i	$0.341012\pi$
$812$	4.72120e117	0.687122
$813$	−7.20501e117	−0.997626
$814$	4.08624e116	0.0538314
$815$	6.41948e117	0.804668
$816$	1.10908e117	0.132285
$817$	6.00530e117	0.681618
$818$	5.51099e116	0.0595279
$819$	−9.18688e117	−0.944431
$820$	3.33252e117	0.326070
$821$	−1.02610e118	−0.955630	−0.477815	−	0.878460i	$-0.658571\pi$
$821$	−1.02610e118	−0.955630	−0.477815	+	0.878460i	$0.658571\pi$
$822$	3.72552e117	0.330277
$823$	9.96741e117	0.841180	0.420590	−	0.907251i	$-0.361823\pi$
$823$	9.96741e117	0.841180	0.420590	+	0.907251i	$0.361823\pi$
$824$	−5.53520e117	−0.444714
$825$	3.35710e116	0.0256791
$826$	−1.00074e118	−0.728837
$827$	−1.74802e118	−1.21220	−0.606100	−	0.795388i	$-0.707267\pi$
$827$	−1.74802e118	−1.21220	−0.606100	+	0.795388i	$0.707267\pi$
$828$	−6.80026e117	−0.449052
$829$	1.57176e118	0.988386	0.494193	−	0.869352i	$-0.335464\pi$
$829$	1.57176e118	0.988386	0.494193	+	0.869352i	$0.335464\pi$
$830$	−6.53047e117	−0.391095
$831$	−3.68334e117	−0.210088
$832$	−2.49416e117	−0.135497
$833$	−8.35630e117	−0.432406
$834$	2.46106e117	0.121310
$835$	2.63906e118	1.23922
$836$	4.62514e116	0.0206905
$837$	−7.40133e117	−0.315449
$838$	−3.30085e117	−0.134043
$839$	−3.67456e118	−1.42183	−0.710913	−	0.703280i	$-0.751718\pi$
$839$	−3.67456e118	−1.42183	−0.710913	+	0.703280i	$0.751718\pi$
$840$	−3.97563e117	−0.146587
$841$	9.26701e117	0.325613
$842$	−1.93982e118	−0.649563
$843$	3.35676e117	0.107128
$844$	1.72481e118	0.524654
$845$	−4.03466e117	−0.116980
$846$	−1.00220e118	−0.276985
$847$	4.49421e118	1.18407
$848$	−1.11629e118	−0.280378
$849$	−2.63574e118	−0.631162
$850$	1.74450e118	0.398296
$851$	4.81654e118	1.04855
$852$	−1.04309e118	−0.216529
$853$	6.01295e118	1.19029	0.595144	−	0.803619i	$-0.297095\pi$
$853$	6.01295e118	1.19029	0.595144	+	0.803619i	$0.297095\pi$
$854$	−1.00612e118	−0.189935
$855$	1.25544e118	0.226031
$856$	−1.92254e118	−0.330132
$857$	−1.07846e119	−1.76636	−0.883180	−	0.469034i	$-0.844602\pi$
$857$	−1.07846e119	−1.76636	−0.883180	+	0.469034i	$0.844602\pi$
$858$	2.27746e117	0.0355809
$859$	1.18938e119	1.77254	0.886272	−	0.463165i	$-0.153286\pi$
$859$	1.18938e119	1.77254	0.886272	+	0.463165i	$0.153286\pi$
$860$	−3.45905e118	−0.491779
$861$	−4.46146e118	−0.605131
$862$	8.85044e118	1.14530
$863$	7.58328e117	0.0936310	0.0468155	−	0.998904i	$-0.485093\pi$
$863$	7.58328e117	0.0936310	0.0468155	+	0.998904i	$0.485093\pi$
$864$	1.34892e118	0.158920
$865$	−6.56862e118	−0.738451
$866$	1.11967e119	1.20120
$867$	−1.86919e117	−0.0191373
$868$	2.14313e118	0.209412
$869$	9.60685e117	0.0895942
$870$	−3.17693e118	−0.282799
$871$	1.65322e119	1.40474
$872$	−7.66890e118	−0.622036
$873$	−1.41742e119	−1.09754
$874$	5.45175e118	0.403017
$875$	−1.75578e119	−1.23920
$876$	7.68260e116	0.00517716
$877$	−2.19337e119	−1.41133	−0.705665	−	0.708546i	$-0.749352\pi$
$877$	−2.19337e119	−1.41133	−0.705665	+	0.708546i	$0.749352\pi$
$878$	−7.70798e117	−0.0473601
$879$	2.40797e118	0.141287
$880$	−2.66408e117	−0.0149279
$881$	−3.17174e118	−0.169736	−0.0848680	−	0.996392i	$-0.527047\pi$
$881$	−3.17174e118	−0.169736	−0.0848680	+	0.996392i	$0.527047\pi$
$882$	−4.28841e118	−0.219189
$883$	−1.42439e119	−0.695375	−0.347688	−	0.937610i	$-0.613033\pi$
$883$	−1.42439e119	−0.695375	−0.347688	+	0.937610i	$0.613033\pi$
$884$	1.18348e119	0.551877
$885$	6.73406e118	0.299968
$886$	1.75309e119	0.745999
$887$	1.77457e119	0.721417	0.360709	−	0.932679i	$-0.382535\pi$
$887$	1.77457e119	0.721417	0.360709	+	0.932679i	$0.382535\pi$
$888$	−4.03140e118	−0.156578
$889$	−4.60177e119	−1.70767
$890$	−2.59337e119	−0.919535
$891$	6.92859e117	0.0234746
$892$	9.11842e117	0.0295218
$893$	8.03465e118	0.248589
$894$	8.01151e118	0.236889
$895$	3.59619e119	1.01627
$896$	−3.90593e118	−0.105499
$897$	2.68450e119	0.693057
$898$	−2.34189e119	−0.577930
$899$	1.71258e119	0.404002
$900$	8.95272e118	0.201899
$901$	5.29677e119	1.14198
$902$	−2.98964e118	−0.0616246
$903$	4.63086e119	0.912658
$904$	7.24130e118	0.136457
$905$	2.09280e119	0.377104
$906$	−3.44034e119	−0.592802
$907$	9.48113e119	1.56230	0.781151	−	0.624342i	$-0.214633\pi$
$907$	9.48113e119	1.56230	0.781151	+	0.624342i	$0.214633\pi$
$908$	1.88777e119	0.297491
$909$	−8.24007e119	−1.24192
$910$	−4.24230e119	−0.611540
$911$	−4.96720e119	−0.684885	−0.342443	−	0.939539i	$-0.611254\pi$
$911$	−4.96720e119	−0.684885	−0.342443	+	0.939539i	$0.611254\pi$
$912$	−4.56307e118	−0.0601819
$913$	5.85856e118	0.0739136
$914$	5.45334e119	0.658175
$915$	6.77023e118	0.0781716
$916$	2.12922e119	0.235209
$917$	−8.40908e118	−0.0888772
$918$	−6.40059e119	−0.647279
$919$	−8.81023e119	−0.852529	−0.426265	−	0.904599i	$-0.640171\pi$
$919$	−8.81023e119	−0.852529	−0.426265	+	0.904599i	$0.640171\pi$
$920$	−3.14021e119	−0.290771
$921$	−1.08422e120	−0.960737
$922$	−3.47406e119	−0.294601
$923$	−1.11305e120	−0.903332
$924$	3.56658e118	0.0277037
$925$	−6.34110e119	−0.471438
$926$	−4.19425e119	−0.298476
$927$	1.34788e120	0.918167
$928$	−3.12123e119	−0.203532
$929$	1.56217e120	0.975195	0.487597	−	0.873069i	$-0.337873\pi$
$929$	1.56217e120	0.975195	0.487597	+	0.873069i	$0.337873\pi$
$930$	−1.44213e119	−0.0861876
$931$	3.43801e119	0.196719
$932$	1.43235e120	0.784702
$933$	−6.71966e119	−0.352485
$934$	−2.18979e119	−0.109990
$935$	1.26410e119	0.0608012
$936$	6.07354e119	0.279750
$937$	−2.78813e120	−1.22987	−0.614936	−	0.788577i	$-0.710818\pi$
$937$	−2.78813e120	−1.22987	−0.614936	+	0.788577i	$0.710818\pi$
$938$	2.58900e120	1.09375
$939$	5.06692e119	0.205016
$940$	−4.62796e119	−0.179354
$941$	3.84851e120	1.42861	0.714303	−	0.699836i	$-0.246744\pi$
$941$	3.84851e120	1.42861	0.714303	+	0.699836i	$0.246744\pi$
$942$	1.50957e120	0.536776
$943$	−3.52395e120	−1.20035
$944$	6.61601e119	0.215889
$945$	2.29436e120	0.717256
$946$	3.10315e119	0.0929421
$947$	−2.79617e120	−0.802397	−0.401199	−	0.915991i	$-0.631406\pi$
$947$	−2.79617e120	−0.802397	−0.401199	+	0.915991i	$0.631406\pi$
$948$	−9.47792e119	−0.260601
$949$	8.19793e118	0.0215985
$950$	−7.17738e119	−0.181201
$951$	−1.65547e120	−0.400507
$952$	1.85336e120	0.429698
$953$	−7.39437e119	−0.164301	−0.0821503	−	0.996620i	$-0.526179\pi$
$953$	−7.39437e119	−0.164301	−0.0821503	+	0.996620i	$0.526179\pi$
$954$	2.71828e120	0.578875
$955$	3.05236e120	0.623015
$956$	−2.25375e120	−0.440922
$957$	2.85006e119	0.0534466
$958$	2.13153e120	0.383167
$959$	6.22563e120	1.07283
$960$	2.62833e119	0.0434204
$961$	−5.53649e120	−0.876874
$962$	−4.30182e120	−0.653223
$963$	4.68158e120	0.681597
$964$	−4.47632e120	−0.624887
$965$	6.28483e120	0.841274
$966$	4.20401e120	0.539622
$967$	−1.12670e121	−1.38687	−0.693436	−	0.720518i	$-0.743904\pi$
$967$	−1.12670e121	−1.38687	−0.693436	+	0.720518i	$0.743904\pi$
$968$	−2.97117e120	−0.350732
$969$	2.16517e120	0.245120
$970$	−6.54533e120	−0.710683
$971$	−9.85982e120	−1.02681	−0.513405	−	0.858146i	$-0.671616\pi$
$971$	−9.85982e120	−1.02681	−0.513405	+	0.858146i	$0.671616\pi$
$972$	−5.18350e120	−0.517773
$973$	4.11262e120	0.394047
$974$	−1.05221e121	−0.967085
$975$	−3.53421e120	−0.311606
$976$	6.65154e119	0.0562606
$977$	1.04037e120	0.0844225	0.0422112	−	0.999109i	$-0.486560\pi$
$977$	1.04037e120	0.0844225	0.0422112	+	0.999109i	$0.486560\pi$
$978$	5.68170e120	0.442339
$979$	2.32654e120	0.173785
$980$	−1.98030e120	−0.141930
$981$	1.86746e121	1.28427
$982$	−6.51037e119	−0.0429626
$983$	−1.54435e121	−0.977975	−0.488987	−	0.872291i	$-0.662634\pi$
$983$	−1.54435e121	−0.977975	−0.488987	+	0.872291i	$0.662634\pi$
$984$	2.94952e120	0.179246
$985$	−4.59721e120	−0.268119
$986$	1.48102e121	0.828985
$987$	6.19575e120	0.332851
$988$	−4.86915e120	−0.251071
$989$	3.65776e121	1.81036
$990$	6.48731e119	0.0308205
$991$	−1.83844e121	−0.838431	−0.419215	−	0.907887i	$-0.637695\pi$
$991$	−1.83844e121	−0.838431	−0.419215	+	0.907887i	$0.637695\pi$
$992$	−1.41685e120	−0.0620298
$993$	1.58356e121	0.665563
$994$	−1.74308e121	−0.703345
$995$	5.34360e119	0.0207014
$996$	−5.77994e120	−0.214991
$997$	9.76494e120	0.348751	0.174376	−	0.984679i	$-0.444209\pi$
$997$	9.76494e120	0.348751	0.174376	+	0.984679i	$0.444209\pi$
$998$	3.20351e121	1.09860
$999$	2.32655e121	0.766144

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2.82.a.b.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.82.a.b.1.2	✓	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2 \)
Weight:	\( k \)	\(=\)	\( 82 \)
Character orbit:	\([\chi]\)	\(=\)	2.a (trivial)

\(f(q)\)	\(=\)	\(q+1.09951e12 q^{2} -1.09428e19 q^{3} +1.20893e24 q^{4} -1.35941e28 q^{5} -1.20318e31 q^{6} -2.01060e34 q^{7} +1.32923e36 q^{8} -3.23681e38 q^{9} -1.49469e40 q^{10} +1.34090e41 q^{11} -1.32291e43 q^{12} -1.41164e45 q^{13} -2.21068e46 q^{14} +1.48758e47 q^{15} +1.46150e48 q^{16} -6.93481e49 q^{17} -3.55891e50 q^{18} +2.85317e51 q^{19} -1.64343e52 q^{20} +2.20016e53 q^{21} +1.47434e53 q^{22} +1.73783e55 q^{23} -1.45455e55 q^{24} -2.28790e56 q^{25} -1.55212e57 q^{26} +8.39432e57 q^{27} -2.43067e58 q^{28} -1.94235e59 q^{29} +1.63561e59 q^{30} -8.81707e59 q^{31} +1.60694e60 q^{32} -1.46733e60 q^{33} -7.62490e61 q^{34} +2.73323e62 q^{35} -3.91306e62 q^{36} +2.77158e63 q^{37} +3.13710e63 q^{38} +1.54474e64 q^{39} -1.80697e64 q^{40} -2.02778e65 q^{41} +2.41911e65 q^{42} +2.10478e66 q^{43} +1.62105e65 q^{44} +4.40016e66 q^{45} +1.91077e67 q^{46} +2.81604e67 q^{47} -1.59930e67 q^{48} +1.20498e68 q^{49} -2.51558e68 q^{50} +7.58864e68 q^{51} -1.70657e69 q^{52} -7.63795e69 q^{53} +9.22966e69 q^{54} -1.82284e69 q^{55} -2.67255e70 q^{56} -3.12218e70 q^{57} -2.13564e71 q^{58} +4.52685e71 q^{59} +1.79837e71 q^{60} +4.55117e71 q^{61} -9.69447e71 q^{62} +6.50793e72 q^{63} +1.76685e72 q^{64} +1.91900e73 q^{65} -1.61334e72 q^{66} -1.17113e74 q^{67} -8.38367e73 q^{68} -1.90168e74 q^{69} +3.00522e74 q^{70} +7.88481e74 q^{71} -4.30246e74 q^{72} -5.80736e73 q^{73} +3.04738e75 q^{74} +2.50361e75 q^{75} +3.44927e75 q^{76} -2.69602e75 q^{77} +1.69846e76 q^{78} +7.16447e76 q^{79} -1.98678e76 q^{80} +5.16711e76 q^{81} -2.22957e77 q^{82} +4.36912e77 q^{83} +2.65984e77 q^{84} +9.42725e77 q^{85} +2.31423e78 q^{86} +2.12548e78 q^{87} +1.78236e77 q^{88} +1.73505e79 q^{89} +4.83802e78 q^{90} +2.83825e79 q^{91} +2.10091e79 q^{92} +9.64836e78 q^{93} +3.09627e79 q^{94} -3.87863e79 q^{95} -1.75844e79 q^{96} +4.37906e80 q^{97} +1.32489e80 q^{98} -4.34025e79 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4398046511104 q^{2} - 71\!\cdots\!04 q^{3} + 48\!\cdots\!04 q^{4} + 24\!\cdots\!80 q^{5} - 78\!\cdots\!04 q^{6} + 18\!\cdots\!92 q^{7} + 53\!\cdots\!04 q^{8} + 10\!\cdots\!92 q^{9} + 27\!\cdots\!80 q^{10}+ \cdots + 23\!\cdots\!24 q^{99}+O(q^{100}) \) 4 * q + 4398046511104 * q^2 - 7111737378636958704 * q^3 + 4835703278458516698824704 * q^4 + 24913167970238703572051264280 * q^5 - 7819437941500545712789131362304 * q^6 + 18392891597037634084943392123253792 * q^7 + 5316911983139663491615228241121378304 * q^8 + 1013046472853099475277764895915187498292 * q^9 + 27392317868014062887782031287821564641280 * q^10 + 3293502419914460072529071908820017291223088 * q^11 - 8597562939352679680661076005207963445755904 * q^12 + 567209646784673546391552295946811229381397816 * q^13 + 20223198179366361312267969326253833663484526592 * q^14 - 547217381485998970079962981893261936776292350880 * q^15 + 5846006549323611672814739330865132078623730171904 * q^16 + 17089897952366178968039636285810569503954644106312 * q^17 + 1113856376379446798989506750446739019487282139758592 * q^18 + 5756965931766410168329316740594195077449201960279120 * q^19 + 30118172007617752210204452443556450103768978324193280 * q^20 - 14643153298636475055051905666955183519768753981624192 * q^21 + 3621244206804343073026598875549252663285163724833292288 * q^22 + 19646604683226764943245317684559999335272488116648750176 * q^23 - 9453120422354276003431179546249863341640979885802389504 * q^24 + 733266156338164296167626938048487221981492621013192555900 * q^25 + 623653602026466415561742315971908501933736430204371337216 * q^26 - 40421324236200783167310017942871741694795540187528627261280 * q^27 + 22235641549031747542709906392215390663498217245358677819392 * q^28 + 89899184350048092395043102182944472414475457640075074740600 * q^29 - 601671873864991093346079619103283246391243600682126546042880 * q^30 + 4430148341240723314324773447291665259367527167357662548603008 * q^31 + 6427752177035961102167848369364650410088811975131171341205504 * q^32 - 69664471619294722934395229831718213827407646316847541036666688 * q^33 + 18790541516131866747958596372298574046878755610605871916122112 * q^34 - 110027170646399928174365046715945192018225940599726953771773760 * q^35 + 1224698037501642467387345239127034234507515770416141817601851392 * q^36 + 2868361968223519440594845730542456366299838282287085971112221592 * q^37 + 6329850982687462191179845211872610167062658437298665326504837120 * q^38 + 18942502886425029009441809968742163165248613621078317208744463584 * q^39 + 33115280329733352604636119223977533216139153469886213934840545280 * q^40 + 241785427475215901047307993827246250994683885604889744530507410088 * q^41 - 16100317319157294529066940012044692765461751765438896538220756992 * q^42 + 3906640790087972714399517126127070696761217293523540873168832368816 * q^43 + 3981600112397853227369825768600170041868973598762120557816067391488 * q^44 + 40553804958029668344694291747814466848881682300767140094162198787640 * q^45 + 21601670295526245166878191503440804171462931381646083872049726488576 * q^46 + 68150727659512952126173314586975326413685192716985190293893331241152 * q^47 - 10393815823145298626686591984088904319185224597226259875413417263104 * q^48 + 404071745256872837110945159389335319846982704413781054108834196201188 * q^49 + 806234665148425924790992853208798894255207199876348009508134872678400 * q^50 - 1064663054384373454693815044115707495986393228123917639685336221542112 * q^51 + 685714387132485780807687351264933240450496186063759741333121968111616 * q^52 - 12504487477710484326069573437834673828075644450184339810661130387062824 * q^53 - 44443716007806603006255058779598675686836853379094825254648741657313280 * q^54 - 42234307462751559870688815279523036389647114318141024835956467841276640 * q^55 + 24448346434219554857386857259465331683222677509918394630166844058632192 * q^56 - 108722241788231835680029530262218202922060026555257931870949646820720320 * q^57 + 98845198520456082653157487714849975809061439375041480958881966954905600 * q^58 + 489644385838630680453793035064910893924161250200777882813761286795049200 * q^59 - 661545221420332509504689964508349027705625929138948302588827600295034880 * q^60 + 1366702264988165160362277893397817161446550792366271584021176863925910008 * q^61 + 4870999613966734002792865351354081807964899028017325921601006040889950208 * q^62 + 25467416254050289220554969290497643614373758623650152422090650290214784416 * q^63 + 7067388259113537318333190002971674063309935587502475832486424805170479104 * q^64 - 2818715554123324305996683900688488859608094117415047465075683253256636080 * q^65 - 76596896588285695382683818410402127840320212324145930898048348624034725888 * q^66 - 138862674451263516298138468667610680340162456232851749705287510352340126448 * q^67 + 20660418889194655771113483822358173664967328313530555040079187502706982912 * q^68 - 672683900106157260734473471072801550870108283451440958413239964932071157376 * q^69 - 120976153497010911141285596469887181170360486608554468602744869660403957760 * q^70 + 530690635697814275660599829540627664439950625326921320686595878050597442848 * q^71 + 1346569732747503601590628957775849376430636774436242794777074912689171464192 * q^72 + 4820651523365553876354090014651981899402331637274186511812407488393695849256 * q^73 + 3153797336732213047136019762904340078787789299843033684241252475787534139392 * q^74 + 8602612673085854634235499843915898492279116054592658607479566152826055983600 * q^75 + 6959744757554204748890607318869773205936950878980155945417583978548947845120 * q^76 + 110102493064509063187502350556580136272265257076594423627460955165606056817024 * q^77 + 20827502182804762099559385351883359501065891638239106218565497936471794909184 * q^78 - 50648470438514104286474913909108065961634468975354377982407985787923842139840 * q^79 + 36410635779603672534361228812119143475730269065801269680998301939816233697280 * q^80 + 379041758338986105391656527861324532842247800229613612526137961094691354412004 * q^81 + 265845888935790629257564155475812485570758338183988897393746995357086843404288 * q^82 - 238520107352811898084467415485453163522726145612576880661027053586010982834864 * q^83 - 17702486103296761416238650559110605188614661675886162871804355919104853409792 * q^84 + 6847336985153689699318057514777445791814846059666211714926452668399553888133040 * q^85 + 4295396974245745605449286229736365000414556517589309608033131241106158141833216 * q^86 + 24810367340380739951538667183142703925070608394534355695353250835478847036268000 * q^87 + 4377815620735668160553332165979083271645745234885307873133897377221885126770688 * q^88 + 27428045100278388486412864000971428693338849280996141787710697205603108083187560 * q^89 + 44589380101913620003376240172734928535455487911745767018464513103965350149488640 * q^90 + 83746848785098964567359078917389054910460771603691882810793485825135745141227968 * q^91 + 23751287669314528793963493100263251299423658690678278368493215681069782428286976 * q^92 + 175965595874091058205048012635001521337190765247409917262039952747606802639176192 * q^93 + 74932517503029952683603981255418566273459934607080646663323772465470189917437952 * q^94 + 380998938988702403083396234132524561810537502861981189014224981328509785911813600 * q^95 - 11428121354510432629189792105819544680287663362944398970188201375906177906376704 * q^96 - 154675480473123296332819855790359897912276472938070448066911736856000222852914808 * q^97 + 444281582365673460383294613306035847659645552572115066650073420126860773464997888 * q^98 + 2329598594513713332190439448236254139876473757732194671418010179022886450464947824 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more