LMFDB - Embedded newform 2.82.a.b.1.1

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			1.1
Root			$2.03307e16$ 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)

$f(q)$	$=$	\(q+1.09951e12 q^{2} -4.08130e19 q^{3} +1.20893e24 q^{4} +2.59550e28 q^{5} -4.48743e31 q^{6} +1.64669e34 q^{7} +1.32923e36 q^{8} +1.22227e39 q^{9} +2.85378e40 q^{10} +2.10493e42 q^{11} -4.93398e43 q^{12} +6.02269e44 q^{13} +1.81056e46 q^{14} -1.05930e48 q^{15} +1.46150e48 q^{16} +6.31832e49 q^{17} +1.34390e51 q^{18} +2.57279e51 q^{19} +3.13776e52 q^{20} -6.72063e53 q^{21} +2.31439e54 q^{22} +8.80406e54 q^{23} -5.42497e55 q^{24} +2.60070e56 q^{25} +6.62202e56 q^{26} -3.17869e58 q^{27} +1.99073e58 q^{28} -2.59830e59 q^{29} -1.16471e60 q^{30} -7.51097e59 q^{31} +1.60694e60 q^{32} -8.59084e61 q^{33} +6.94706e61 q^{34} +4.27398e62 q^{35} +1.47763e63 q^{36} +3.27166e63 q^{37} +2.82881e63 q^{38} -2.45804e64 q^{39} +3.45001e64 q^{40} +5.29625e64 q^{41} -7.38941e65 q^{42} -7.79339e65 q^{43} +2.54470e66 q^{44} +3.17240e67 q^{45} +9.68016e66 q^{46} +1.36148e67 q^{47} -5.96482e67 q^{48} -1.25943e67 q^{49} +2.85950e68 q^{50} -2.57869e69 q^{51} +7.28099e68 q^{52} +9.04185e69 q^{53} -3.49501e70 q^{54} +5.46334e70 q^{55} +2.18883e70 q^{56} -1.05003e71 q^{57} -2.85686e71 q^{58} +2.63654e71 q^{59} -1.28061e72 q^{60} +2.00209e72 q^{61} -8.25840e71 q^{62} +2.01270e73 q^{63} +1.76685e72 q^{64} +1.56319e73 q^{65} -9.44573e73 q^{66} -1.04609e74 q^{67} +7.63838e73 q^{68} -3.59320e74 q^{69} +4.69929e74 q^{70} -6.00902e73 q^{71} +1.62468e75 q^{72} +1.61879e75 q^{73} +3.59723e75 q^{74} -1.06142e76 q^{75} +3.11031e75 q^{76} +3.46617e76 q^{77} -2.70264e76 q^{78} -7.95967e76 q^{79} +3.79332e76 q^{80} +7.55331e77 q^{81} +5.82329e76 q^{82} -8.38108e77 q^{83} -8.12475e77 q^{84} +1.63992e78 q^{85} -8.56892e77 q^{86} +1.06044e79 q^{87} +2.79793e78 q^{88} -1.88972e78 q^{89} +3.48809e79 q^{90} +9.91752e78 q^{91} +1.06435e79 q^{92} +3.06545e79 q^{93} +1.49696e79 q^{94} +6.67767e79 q^{95} -6.55839e79 q^{96} -3.96442e80 q^{97} -1.38475e79 q^{98} +2.57279e81 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−4.08130e19	−1.93815	−0.969075	−	0.246768i	$-0.920631\pi$
$3$	−4.08130e19	−1.93815	−0.969075	+	0.246768i	$0.920631\pi$
$4$	1.20893e24	0.500000
$5$	2.59550e28	1.27625	0.638124	−	0.769933i	$-0.279711\pi$
$5$	2.59550e28	1.27625	0.638124	+	0.769933i	$0.279711\pi$
$6$	−4.48743e31	−1.37048
$7$	1.64669e34	0.977556	0.488778	−	0.872408i	$-0.337443\pi$
$7$	1.64669e34	0.977556	0.488778	+	0.872408i	$0.337443\pi$
$8$	1.32923e36	0.353553
$9$	1.22227e39	2.75642
$10$	2.85378e40	0.902444
$11$	2.10493e42	1.40228	0.701138	−	0.713025i	$-0.252675\pi$
$11$	2.10493e42	1.40228	0.701138	+	0.713025i	$0.252675\pi$
$12$	−4.93398e43	−0.969075
$13$	6.02269e44	0.462472	0.231236	−	0.972898i	$-0.425723\pi$
$13$	6.02269e44	0.462472	0.231236	+	0.972898i	$0.425723\pi$
$14$	1.81056e46	0.691236
$15$	−1.05930e48	−2.47356
$16$	1.46150e48	0.250000
$17$	6.31832e49	0.927727	0.463863	−	0.885907i	$-0.346463\pi$
$17$	6.31832e49	0.927727	0.463863	+	0.885907i	$0.346463\pi$
$18$	1.34390e51	1.94908
$19$	2.57279e51	0.417719	0.208859	−	0.977946i	$-0.433025\pi$
$19$	2.57279e51	0.417719	0.208859	+	0.977946i	$0.433025\pi$
$20$	3.13776e52	0.638124
$21$	−6.72063e53	−1.89465
$22$	2.31439e54	0.991560
$23$	8.80406e54	0.623312	0.311656	−	0.950195i	$-0.399116\pi$
$23$	8.80406e54	0.623312	0.311656	+	0.950195i	$0.399116\pi$
$24$	−5.42497e55	−0.685239
$25$	2.60070e56	0.628811
$26$	6.62202e56	0.327017
$27$	−3.17869e58	−3.40421
$28$	1.99073e58	0.488778
$29$	−2.59830e59	−1.54018	−0.770089	−	0.637937i	$-0.779788\pi$
$29$	−2.59830e59	−1.54018	−0.770089	+	0.637937i	$0.779788\pi$
$30$	−1.16471e60	−1.74907
$31$	−7.51097e59	−0.298915	−0.149457	−	0.988768i	$-0.547753\pi$
$31$	−7.51097e59	−0.298915	−0.149457	+	0.988768i	$0.547753\pi$
$32$	1.60694e60	0.176777
$33$	−8.59084e61	−2.71782
$34$	6.94706e61	0.656002
$35$	4.27398e62	1.24760
$36$	1.47763e63	1.37821
$37$	3.27166e63	1.00600	0.503001	−	0.864286i	$-0.332229\pi$
$37$	3.27166e63	1.00600	0.503001	+	0.864286i	$0.332229\pi$
$38$	2.82881e63	0.295372
$39$	−2.45804e64	−0.896339
$40$	3.45001e64	0.451222
$41$	5.29625e64	0.254813	0.127407	−	0.991851i	$-0.459335\pi$
$41$	5.29625e64	0.254813	0.127407	+	0.991851i	$0.459335\pi$
$42$	−7.38941e65	−1.33972
$43$	−7.79339e65	−0.544821	−0.272411	−	0.962181i	$-0.587821\pi$
$43$	−7.79339e65	−0.544821	−0.272411	+	0.962181i	$0.587821\pi$
$44$	2.54470e66	0.701138
$45$	3.17240e67	3.51788
$46$	9.68016e66	0.440748
$47$	1.36148e67	0.259446	0.129723	−	0.991550i	$-0.458591\pi$
$47$	1.36148e67	0.259446	0.129723	+	0.991550i	$0.458591\pi$
$48$	−5.96482e67	−0.484537
$49$	−1.25943e67	−0.0443845
$50$	2.85950e68	0.444637
$51$	−2.57869e69	−1.79807
$52$	7.28099e68	0.231236
$53$	9.04185e69	1.32765	0.663827	−	0.747886i	$-0.268931\pi$
$53$	9.04185e69	1.32765	0.663827	+	0.747886i	$0.268931\pi$
$54$	−3.49501e70	−2.40714
$55$	5.46334e70	1.78965
$56$	2.18883e70	0.345618
$57$	−1.05003e71	−0.809601
$58$	−2.85686e71	−1.08907
$59$	2.63654e71	0.502953	0.251477	−	0.967863i	$-0.419084\pi$
$59$	2.63654e71	0.502953	0.251477	+	0.967863i	$0.419084\pi$
$60$	−1.28061e72	−1.23678
$61$	2.00209e72	0.989978	0.494989	−	0.868899i	$-0.335172\pi$
$61$	2.00209e72	0.989978	0.494989	+	0.868899i	$0.335172\pi$
$62$	−8.25840e71	−0.211365
$63$	2.01270e73	2.69456
$64$	1.76685e72	0.125000
$65$	1.56319e73	0.590229
$66$	−9.44573e73	−1.92179
$67$	−1.04609e74	−1.15755	−0.578777	−	0.815486i	$-0.696470\pi$
$67$	−1.04609e74	−1.15755	−0.578777	+	0.815486i	$0.696470\pi$
$68$	7.63838e73	0.463863
$69$	−3.59320e74	−1.20807
$70$	4.69929e74	0.882190
$71$	−6.00902e73	−0.0635097	−0.0317549	−	0.999496i	$-0.510110\pi$
$71$	−6.00902e73	−0.0635097	−0.0317549	+	0.999496i	$0.510110\pi$
$72$	1.62468e75	0.974542
$73$	1.61879e75	0.555412	0.277706	−	0.960666i	$-0.410426\pi$
$73$	1.61879e75	0.555412	0.277706	+	0.960666i	$0.410426\pi$
$74$	3.59723e75	0.711350
$75$	−1.06142e76	−1.21873
$76$	3.11031e75	0.208859
$77$	3.46617e76	1.37080
$78$	−2.70264e76	−0.633807
$79$	−7.95967e76	−1.11429	−0.557145	−	0.830415i	$-0.688103\pi$
$79$	−7.95967e76	−1.11429	−0.557145	+	0.830415i	$0.688103\pi$
$80$	3.79332e76	0.319062
$81$	7.55331e77	3.84144
$82$	5.82329e76	0.180180
$83$	−8.38108e77	−1.58722	−0.793610	−	0.608427i	$-0.791801\pi$
$83$	−8.38108e77	−1.58722	−0.793610	+	0.608427i	$0.791801\pi$
$84$	−8.12475e77	−0.947325
$85$	1.63992e78	1.18401
$86$	−8.56892e77	−0.385247
$87$	1.06044e79	2.98509
$88$	2.79793e78	0.495780
$89$	−1.88972e78	−0.211886	−0.105943	−	0.994372i	$-0.533786\pi$
$89$	−1.88972e78	−0.211886	−0.105943	+	0.994372i	$0.533786\pi$
$90$	3.48809e79	2.48752
$91$	9.91752e78	0.452092
$92$	1.06435e79	0.311656
$93$	3.06545e79	0.579342
$94$	1.49696e79	0.183456
$95$	6.67767e79	0.533113
$96$	−6.55839e79	−0.342620
$97$	−3.96442e80	−1.36121	−0.680603	−	0.732652i	$-0.738282\pi$
$97$	−3.96442e80	−1.36121	−0.680603	+	0.732652i	$0.738282\pi$
$98$	−1.38475e79	−0.0313846
$99$	2.57279e81	3.86527
$100$	3.14406e80	0.314406
$101$	1.29489e81	0.865403	0.432702	−	0.901537i	$-0.357560\pi$
$101$	1.29489e81	0.865403	0.432702	+	0.901537i	$0.357560\pi$
$102$	−2.83530e81	−1.27143
$103$	−2.75262e81	−0.831454	−0.415727	−	0.909489i	$-0.636473\pi$
$103$	−2.75262e81	−0.831454	−0.415727	+	0.909489i	$0.636473\pi$
$104$	8.00553e80	0.163508
$105$	−1.74434e82	−2.41804
$106$	9.94162e81	0.938793
$107$	−1.54658e82	−0.998454	−0.499227	−	0.866471i	$-0.666383\pi$
$107$	−1.54658e82	−0.998454	−0.499227	+	0.866471i	$0.666383\pi$
$108$	−3.84280e82	−1.70210
$109$	3.17791e82	0.969098	0.484549	−	0.874764i	$-0.338984\pi$
$109$	3.17791e82	0.969098	0.484549	+	0.874764i	$0.338984\pi$
$110$	6.00700e82	1.26548
$111$	−1.33526e83	−1.94978
$112$	2.40664e82	0.244389
$113$	−2.45745e83	−1.74103	−0.870517	−	0.492139i	$-0.836215\pi$
$113$	−2.45745e83	−1.74103	−0.870517	+	0.492139i	$0.836215\pi$
$114$	−1.15452e83	−0.572474
$115$	2.28509e83	0.795501
$116$	−3.14115e83	−0.770089
$117$	7.36136e83	1.27477
$118$	2.89890e83	0.355642
$119$	1.04043e84	0.906905
$120$	−1.40805e84	−0.874536
$121$	2.17749e84	0.966381
$122$	2.20132e84	0.700020
$123$	−2.16156e84	−0.493866
$124$	−9.08021e83	−0.149457
$125$	−3.98461e84	−0.473729
$126$	2.21299e85	1.90534
$127$	4.33079e83	0.0270718	0.0135359	−	0.999908i	$-0.495691\pi$
$127$	4.33079e83	0.0270718	0.0135359	+	0.999908i	$0.495691\pi$
$128$	1.94267e84	0.0883883
$129$	3.18071e85	1.05594
$130$	1.71874e85	0.417355
$131$	6.00252e85	1.06868	0.534338	−	0.845271i	$-0.320561\pi$
$131$	6.00252e85	1.06868	0.534338	+	0.845271i	$0.320561\pi$
$132$	−1.03857e86	−1.35891
$133$	4.23659e85	0.408343
$134$	−1.15019e86	−0.818514
$135$	−8.25029e86	−4.34462
$136$	8.39849e85	0.328001
$137$	4.60370e86	1.33636	0.668180	−	0.743999i	$-0.267073\pi$
$137$	4.60370e86	1.33636	0.668180	+	0.743999i	$0.267073\pi$
$138$	−3.95076e86	−0.854236
$139$	8.24008e85	0.132994	0.0664970	−	0.997787i	$-0.478818\pi$
$139$	8.24008e85	0.132994	0.0664970	+	0.997787i	$0.478818\pi$
$140$	5.16693e86	0.623802
$141$	−5.55660e86	−0.502846
$142$	−6.60698e85	−0.0449082
$143$	1.26773e87	0.648513
$144$	1.78635e87	0.689106
$145$	−6.74389e87	−1.96565
$146$	1.77988e87	0.392736
$147$	5.14009e86	0.0860239
$148$	3.95520e87	0.503001
$149$	1.68699e88	1.63331	0.816654	−	0.577127i	$-0.195826\pi$
$149$	1.68699e88	1.63331	0.816654	+	0.577127i	$0.195826\pi$
$150$	−1.16705e88	−0.861773
$151$	−7.71751e87	−0.435422	−0.217711	−	0.976013i	$-0.569859\pi$
$151$	−7.71751e87	−0.435422	−0.217711	+	0.976013i	$0.569859\pi$
$152$	3.41983e87	0.147686
$153$	7.72269e88	2.55721
$154$	3.81109e88	0.969305
$155$	−1.94947e88	−0.381490
$156$	−2.97159e88	−0.448169
$157$	−5.25995e87	−0.0612415	−0.0306207	−	0.999531i	$-0.509748\pi$
$157$	−5.25995e87	−0.0612415	−0.0306207	+	0.999531i	$0.509748\pi$
$158$	−8.75175e88	−0.787922
$159$	−3.69025e89	−2.57319
$160$	4.17080e88	0.225611
$161$	1.44976e89	0.609322
$162$	8.30495e89	2.71631
$163$	−3.03527e89	−0.773748	−0.386874	−	0.922133i	$-0.626445\pi$
$163$	−3.03527e89	−0.773748	−0.386874	+	0.922133i	$0.626445\pi$
$164$	6.40277e88	0.127407
$165$	−2.22975e90	−3.46862
$166$	−9.21509e89	−1.12233
$167$	1.32489e90	1.26521	0.632606	−	0.774474i	$-0.281985\pi$
$167$	1.32489e90	1.26521	0.632606	+	0.774474i	$0.281985\pi$
$168$	−8.93325e89	−0.669860
$169$	−1.33322e90	−0.786120
$170$	1.80311e90	0.837222
$171$	3.14465e90	1.15141
$172$	−9.42163e89	−0.272411
$173$	1.71035e90	0.391037	0.195518	−	0.980700i	$-0.437361\pi$
$173$	1.71035e90	0.391037	0.195518	+	0.980700i	$0.437361\pi$
$174$	1.16597e91	2.11078
$175$	4.28256e90	0.614698
$176$	3.07636e90	0.350569
$177$	−1.07605e91	−0.974798
$178$	−2.07777e90	−0.149826
$179$	−2.31211e91	−1.32881	−0.664403	−	0.747375i	$-0.731314\pi$
$179$	−2.31211e91	−1.32881	−0.664403	+	0.747375i	$0.731314\pi$
$180$	3.83520e91	1.75894
$181$	−1.17841e91	−0.431833	−0.215917	−	0.976412i	$-0.569274\pi$
$181$	−1.17841e91	−0.431833	−0.215917	+	0.976412i	$0.569274\pi$
$182$	1.09044e91	0.319677
$183$	−8.17112e91	−1.91872
$184$	1.17026e91	0.220374
$185$	8.49159e91	1.28391
$186$	3.37050e91	0.409656
$187$	1.32996e92	1.30093
$188$	1.64593e91	0.129723
$189$	−5.23433e92	−3.32780
$190$	7.34218e91	0.376968
$191$	−3.22369e92	−1.33814	−0.669071	−	0.743198i	$-0.733308\pi$
$191$	−3.22369e92	−1.33814	−0.669071	+	0.743198i	$0.733308\pi$
$192$	−7.21102e91	−0.242269
$193$	4.30383e92	1.17161	0.585807	−	0.810451i	$-0.300778\pi$
$193$	4.30383e92	1.17161	0.585807	+	0.810451i	$0.300778\pi$
$194$	−4.35892e92	−0.962518
$195$	−6.37983e92	−1.14395
$196$	−1.52255e91	−0.0221923
$197$	−3.88537e92	−0.460840	−0.230420	−	0.973091i	$-0.574010\pi$
$197$	−3.88537e92	−0.460840	−0.230420	+	0.973091i	$0.574010\pi$
$198$	2.82882e93	2.73316
$199$	6.02699e89	0.000474844 0	0.000237422	−	1.00000i	$-0.499924\pi$
$199$	6.02699e89	0.000474844 0	0.000237422		1.00000i	$0.499924\pi$
$200$	3.45693e92	0.222318
$201$	4.26942e93	2.24351
$202$	1.42375e93	0.611932
$203$	−4.27860e93	−1.50561
$204$	−3.11745e93	−0.899037
$205$	1.37464e93	0.325205
$206$	−3.02654e93	−0.587927
$207$	1.07609e94	1.71811
$208$	8.80218e92	0.115618
$209$	5.41554e93	0.585757
$210$	−1.91792e94	−1.70982
$211$	−3.12974e93	−0.230181	−0.115090	−	0.993355i	$-0.536716\pi$
$211$	−3.12974e93	−0.230181	−0.115090	+	0.993355i	$0.536716\pi$
$212$	1.09309e94	0.663827
$213$	2.45246e93	0.123091
$214$	−1.70048e94	−0.706014
$215$	−2.02277e94	−0.695327
$216$	−4.22521e94	−1.20357
$217$	−1.23683e94	−0.292206
$218$	3.49414e94	0.685256
$219$	−6.60677e94	−1.07647
$220$	6.60477e94	0.894827
$221$	3.80533e94	0.429047
$222$	−1.46814e95	−1.37870
$223$	1.46041e95	1.14321	0.571607	−	0.820528i	$-0.306320\pi$
$223$	1.46041e95	1.14321	0.571607	+	0.820528i	$0.306320\pi$
$224$	2.64613e94	0.172809
$225$	3.17876e95	1.73327
$226$	−2.70199e95	−1.23110
$227$	1.89896e95	0.723550	0.361775	−	0.932265i	$-0.382171\pi$
$227$	1.89896e95	0.723550	0.361775	+	0.932265i	$0.382171\pi$
$228$	−1.26941e95	−0.404800
$229$	5.51265e95	1.47239	0.736196	−	0.676769i	$-0.236620\pi$
$229$	5.51265e95	1.47239	0.736196	+	0.676769i	$0.236620\pi$
$230$	2.51248e95	0.562504
$231$	−1.41465e96	−2.65682
$232$	−3.45374e95	−0.544535
$233$	5.31175e95	0.703595	0.351797	−	0.936076i	$-0.385571\pi$
$233$	5.31175e95	0.703595	0.351797	+	0.936076i	$0.385571\pi$
$234$	8.09390e95	0.901396
$235$	3.53371e95	0.331118
$236$	3.18738e95	0.251477
$237$	3.24858e96	2.15966
$238$	1.14397e96	0.641279
$239$	−2.63809e96	−1.24789	−0.623943	−	0.781470i	$-0.714470\pi$
$239$	−2.63809e96	−1.24789	−0.623943	+	0.781470i	$0.714470\pi$
$240$	−1.54817e96	−0.618390
$241$	−4.78055e96	−1.61357	−0.806785	−	0.590846i	$-0.798794\pi$
$241$	−4.78055e96	−1.61357	−0.806785	+	0.590846i	$0.798794\pi$
$242$	2.39417e96	0.683334
$243$	−1.67321e97	−4.04108
$244$	2.42038e96	0.494989
$245$	−3.26884e95	−0.0566457
$246$	−2.37666e96	−0.349216
$247$	1.54951e96	0.193183
$248$	−9.98379e95	−0.105682
$249$	3.42056e97	3.07627
$250$	−4.38112e96	−0.334977
$251$	−1.30986e97	−0.852001	−0.426001	−	0.904723i	$-0.640078\pi$
$251$	−1.30986e97	−0.852001	−0.426001	+	0.904723i	$0.640078\pi$
$252$	2.43321e97	1.34728
$253$	1.85319e97	0.874056
$254$	4.76176e95	0.0191426
$255$	−6.69299e97	−2.29479
$256$	2.13599e96	0.0625000
$257$	7.62294e97	1.90472	0.952360	−	0.304977i	$-0.0986488\pi$
$257$	7.62294e97	1.90472	0.952360	+	0.304977i	$0.0986488\pi$
$258$	3.49723e97	0.746666
$259$	5.38742e97	0.983423
$260$	1.88978e97	0.295114
$261$	−3.17583e98	−4.24538
$262$	6.59984e97	0.755668
$263$	1.06189e98	1.04201	0.521007	−	0.853553i	$-0.325557\pi$
$263$	1.06189e98	1.04201	0.521007	+	0.853553i	$0.325557\pi$
$264$	−1.14192e98	−0.960895
$265$	2.34681e98	1.69442
$266$	4.65818e97	0.288742
$267$	7.71251e97	0.410667
$268$	−1.26465e98	−0.578777
$269$	1.68072e98	0.661497	0.330749	−	0.943719i	$-0.392699\pi$
$269$	1.68072e98	0.661497	0.330749	+	0.943719i	$0.392699\pi$
$270$	−9.07129e98	−3.07211
$271$	−2.84348e98	−0.829075	−0.414538	−	0.910032i	$-0.636057\pi$
$271$	−2.84348e98	−0.829075	−0.414538	+	0.910032i	$0.636057\pi$
$272$	9.23423e97	0.231932
$273$	−4.04763e98	−0.876221
$274$	5.06182e98	0.944950
$275$	5.47430e98	0.881768
$276$	−4.34391e98	−0.604036
$277$	3.39519e98	0.407787	0.203893	−	0.978993i	$-0.434640\pi$
$277$	3.39519e98	0.407787	0.203893	+	0.978993i	$0.434640\pi$
$278$	9.06007e97	0.0940410
$279$	−9.18044e98	−0.823935
$280$	5.68110e98	0.441095
$281$	2.44452e99	1.64281	0.821406	−	0.570343i	$-0.193190\pi$
$281$	2.44452e99	1.64281	0.821406	+	0.570343i	$0.193190\pi$
$282$	−6.10954e98	−0.355566
$283$	1.53972e99	0.776410	0.388205	−	0.921573i	$-0.373095\pi$
$283$	1.53972e99	0.776410	0.388205	+	0.921573i	$0.373095\pi$
$284$	−7.26445e97	−0.0317549
$285$	−2.72536e99	−1.03325
$286$	1.39389e99	0.458568
$287$	8.72129e98	0.249094
$288$	1.96411e99	0.487271
$289$	−6.46226e98	−0.139323
$290$	−7.41498e99	−1.38992
$291$	1.61799e100	2.63822
$292$	1.95700e99	0.277706
$293$	−5.68348e99	−0.702224	−0.351112	−	0.936333i	$-0.614196\pi$
$293$	−5.68348e99	−0.702224	−0.351112	+	0.936333i	$0.614196\pi$
$294$	5.65159e98	0.0608281
$295$	6.84313e99	0.641893
$296$	4.34878e99	0.355675
$297$	−6.69092e100	−4.77364
$298$	1.85486e100	1.15492
$299$	5.30241e99	0.288264
$300$	−1.28318e100	−0.609365
$301$	−1.28333e100	−0.532593
$302$	−8.48549e99	−0.307890
$303$	−5.28484e100	−1.67728
$304$	3.76014e99	0.104430
$305$	5.19642e100	1.26346
$306$	8.49119e100	1.80822
$307$	−4.38014e99	−0.0817304	−0.0408652	−	0.999165i	$-0.513011\pi$
$307$	−4.38014e99	−0.0817304	−0.0408652	+	0.999165i	$0.513011\pi$
$308$	4.19034e100	0.685402
$309$	1.12342e101	1.61148
$310$	−2.14347e100	−0.269754
$311$	−4.42777e100	−0.489091	−0.244545	−	0.969638i	$-0.578639\pi$
$311$	−4.42777e100	−0.489091	−0.244545	+	0.969638i	$0.578639\pi$
$312$	−3.26729e100	−0.316904
$313$	−1.09871e101	−0.936134	−0.468067	−	0.883693i	$-0.655049\pi$
$313$	−1.09871e101	−0.936134	−0.468067	+	0.883693i	$0.655049\pi$
$314$	−5.78338e99	−0.0433043
$315$	5.22396e101	3.43893
$316$	−9.62265e100	−0.557145
$317$	−1.43398e101	−0.730538	−0.365269	−	0.930902i	$-0.619023\pi$
$317$	−1.43398e101	−0.730538	−0.365269	+	0.930902i	$0.619023\pi$
$318$	−4.05747e101	−1.81952
$319$	−5.46924e101	−2.15975
$320$	4.58585e100	0.159531
$321$	6.31203e101	1.93515
$322$	1.59402e101	0.430856
$323$	1.62557e101	0.387529
$324$	9.13139e101	1.92072
$325$	1.56632e101	0.290807
$326$	−3.33731e101	−0.547122
$327$	−1.29700e102	−1.87826
$328$	7.03993e100	0.0900902
$329$	2.24194e101	0.253623
$330$	−2.45164e102	−2.45268
$331$	−1.98886e102	−1.76024	−0.880118	−	0.474755i	$-0.842537\pi$
$331$	−1.98886e102	−1.76024	−0.880118	+	0.474755i	$0.842537\pi$
$332$	−1.01321e102	−0.793610
$333$	3.99886e102	2.77296
$334$	1.45673e102	0.894639
$335$	−2.71513e102	−1.47733
$336$	−9.82222e101	−0.473662
$337$	1.38879e102	0.593779	0.296890	−	0.954912i	$-0.404051\pi$
$337$	1.38879e102	0.593779	0.296890	+	0.954912i	$0.404051\pi$
$338$	−1.46589e102	−0.555871
$339$	1.00296e103	3.37438
$340$	1.98254e102	0.592005
$341$	−1.58101e102	−0.419161
$342$	3.45758e102	0.814169
$343$	−4.87993e102	−1.02094
$344$	−1.03592e102	−0.192623
$345$	−9.32613e102	−1.54180
$346$	1.88055e102	0.276505
$347$	8.41632e102	1.10098	0.550489	−	0.834842i	$-0.314441\pi$
$347$	8.41632e102	1.10098	0.550489	+	0.834842i	$0.314441\pi$
$348$	1.28200e103	1.49255
$349$	6.69368e101	0.0693802	0.0346901	−	0.999398i	$-0.488956\pi$
$349$	6.69368e101	0.0693802	0.0346901	+	0.999398i	$0.488956\pi$
$350$	4.70872e102	0.434657
$351$	−1.91443e103	−1.57435
$352$	3.38249e102	0.247890
$353$	1.61588e100	0.00105568	0.000527840	−	1.00000i	$-0.499832\pi$
$353$	1.61588e100	0.00105568	0.000527840		1.00000i	$0.499832\pi$
$354$	−1.18313e103	−0.689286
$355$	−1.55964e102	−0.0810542
$356$	−2.28453e102	−0.105943
$357$	−4.24631e103	−1.75772
$358$	−2.54220e103	−0.939607
$359$	4.71907e103	1.55787	0.778936	−	0.627104i	$-0.215760\pi$
$359$	4.71907e103	1.55787	0.778936	+	0.627104i	$0.215760\pi$
$360$	4.21684e103	1.24376
$361$	−3.13159e103	−0.825511
$362$	−1.29568e103	−0.305352
$363$	−8.88697e103	−1.87299
$364$	1.19895e103	0.226046
$365$	4.20157e103	0.708844
$366$	−8.98424e103	−1.35674
$367$	1.01865e104	1.37736	0.688680	−	0.725065i	$-0.258190\pi$
$367$	1.01865e104	1.37736	0.688680	+	0.725065i	$0.258190\pi$
$368$	1.28671e103	0.155828
$369$	6.47345e103	0.702373
$370$	9.33660e103	0.907860
$371$	1.48891e104	1.29786
$372$	3.70590e103	0.289671
$373$	−1.39478e104	−0.977911	−0.488956	−	0.872309i	$-0.662622\pi$
$373$	−1.39478e104	−0.977911	−0.488956	+	0.872309i	$0.662622\pi$
$374$	1.46231e104	0.919896
$375$	1.62624e104	0.918157
$376$	1.80972e103	0.0917281
$377$	−1.56488e104	−0.712288
$378$	−5.75520e104	−2.35311
$379$	−5.89533e103	−0.216581	−0.108290	−	0.994119i	$-0.534538\pi$
$379$	−5.89533e103	−0.216581	−0.108290	+	0.994119i	$0.534538\pi$
$380$	8.07281e103	0.266556
$381$	−1.76752e103	−0.0524691
$382$	−3.54449e104	−0.946209
$383$	4.88721e104	1.17357	0.586787	−	0.809741i	$-0.300393\pi$
$383$	4.88721e104	1.17357	0.586787	+	0.809741i	$0.300393\pi$
$384$	−7.92861e103	−0.171310
$385$	8.99643e104	1.74949
$386$	4.73211e104	0.828456
$387$	−9.52563e104	−1.50176
$388$	−4.79268e104	−0.680603
$389$	5.76769e104	0.737978	0.368989	−	0.929434i	$-0.379704\pi$
$389$	5.76769e104	0.737978	0.368989	+	0.929434i	$0.379704\pi$
$390$	−7.01470e104	−0.808896
$391$	5.56268e104	0.578263
$392$	−1.67407e103	−0.0156923
$393$	−2.44981e105	−2.07125
$394$	−4.27201e104	−0.325863
$395$	−2.06593e105	−1.42211
$396$	3.11032e105	1.93263
$397$	1.32360e104	0.0742578	0.0371289	−	0.999310i	$-0.488179\pi$
$397$	1.32360e104	0.0742578	0.0371289	+	0.999310i	$0.488179\pi$
$398$	6.62675e101	0.000335766 0
$399$	−1.72908e105	−0.791430
$400$	3.80093e104	0.157203
$401$	−3.56169e105	−1.33140	−0.665701	−	0.746219i	$-0.731867\pi$
$401$	−3.56169e105	−1.33140	−0.665701	+	0.746219i	$0.731867\pi$
$402$	4.69427e105	1.58640
$403$	−4.52363e104	−0.138240
$404$	1.56543e105	0.432702
$405$	1.96046e106	4.90263
$406$	−4.70437e105	−1.06463
$407$	6.88662e105	1.41069
$408$	−3.42767e105	−0.635715
$409$	−3.39272e105	−0.569842	−0.284921	−	0.958551i	$-0.591967\pi$
$409$	−3.39272e105	−0.569842	−0.284921	+	0.958551i	$0.591967\pi$
$410$	1.51143e105	0.229955
$411$	−1.87890e106	−2.59007
$412$	−3.32771e105	−0.415727
$413$	4.34156e105	0.491665
$414$	1.18318e106	1.21489
$415$	−2.17531e106	−2.02569
$416$	9.67810e104	0.0817542
$417$	−3.36302e105	−0.257762
$418$	5.95445e105	0.414193
$419$	1.55456e106	0.981614	0.490807	−	0.871268i	$-0.336702\pi$
$419$	1.55456e106	0.981614	0.490807	+	0.871268i	$0.336702\pi$
$420$	−2.10878e106	−1.20902
$421$	−3.11467e106	−1.62176	−0.810882	−	0.585209i	$-0.801012\pi$
$421$	−3.11467e106	−1.62176	−0.810882	+	0.585209i	$0.801012\pi$
$422$	−3.44119e105	−0.162762
$423$	1.66410e106	0.715144
$424$	1.20187e106	0.469396
$425$	1.64321e106	0.583365
$426$	2.69650e105	0.0870387
$427$	3.29682e106	0.967758
$428$	−1.86970e106	−0.499227
$429$	−5.17400e106	−1.25692
$430$	−2.22406e106	−0.491671
$431$	6.65230e105	0.133857	0.0669287	−	0.997758i	$-0.478680\pi$
$431$	6.65230e105	0.133857	0.0669287	+	0.997758i	$0.478680\pi$
$432$	−4.64566e106	−0.851052
$433$	−2.93728e106	−0.489989	−0.244994	−	0.969525i	$-0.578786\pi$
$433$	−2.93728e106	−0.489989	−0.244994	+	0.969525i	$0.578786\pi$
$434$	−1.35990e106	−0.206621
$435$	2.75238e107	3.80972
$436$	3.84185e106	0.484549
$437$	2.26510e106	0.260369
$438$	−7.26422e106	−0.761180
$439$	6.07029e106	0.579957	0.289979	−	0.957033i	$-0.406352\pi$
$439$	6.07029e106	0.579957	0.289979	+	0.957033i	$0.406352\pi$
$440$	7.26202e106	0.632738
$441$	−1.53936e106	−0.122343
$442$	4.18400e106	0.303382
$443$	2.89211e106	0.191365	0.0956827	−	0.995412i	$-0.469497\pi$
$443$	2.89211e106	0.191365	0.0956827	+	0.995412i	$0.469497\pi$
$444$	−1.61423e107	−0.974890
$445$	−4.90477e106	−0.270420
$446$	1.60574e107	0.808374
$447$	−6.88510e107	−3.16560
$448$	2.90945e106	0.122194
$449$	−1.57018e107	−0.602520	−0.301260	−	0.953542i	$-0.597407\pi$
$449$	−1.57018e107	−0.602520	−0.301260	+	0.953542i	$0.597407\pi$
$450$	3.49509e107	1.22561
$451$	1.11482e107	0.357319
$452$	−2.97087e107	−0.870517
$453$	3.14974e107	0.843913
$454$	2.08793e107	0.511627
$455$	2.57409e107	0.576982
$456$	−1.39573e107	−0.286237
$457$	−2.91315e107	−0.546709	−0.273354	−	0.961913i	$-0.588133\pi$
$457$	−2.91315e107	−0.546709	−0.273354	+	0.961913i	$0.588133\pi$
$458$	6.06122e107	1.04114
$459$	−2.00840e108	−3.15818
$460$	2.76250e107	0.397751
$461$	5.81060e107	0.766185	0.383092	−	0.923710i	$-0.374859\pi$
$461$	5.81060e107	0.766185	0.383092	+	0.923710i	$0.374859\pi$
$462$	−1.55542e108	−1.87866
$463$	6.33079e107	0.700532	0.350266	−	0.936650i	$-0.386091\pi$
$463$	6.33079e107	0.700532	0.350266	+	0.936650i	$0.386091\pi$
$464$	−3.79742e107	−0.385044
$465$	7.95636e107	0.739384
$466$	5.84033e107	0.497517
$467$	1.45465e108	1.13612	0.568062	−	0.822986i	$-0.307693\pi$
$467$	1.45465e108	1.13612	0.568062	+	0.822986i	$0.307693\pi$
$468$	8.89934e107	0.637383
$469$	−1.72259e108	−1.13157
$470$	3.88536e107	0.234136
$471$	2.14674e107	0.118695
$472$	3.50456e107	0.177821
$473$	−1.64045e108	−0.763990
$474$	3.57185e108	1.52711
$475$	6.69107e107	0.262666
$476$	1.25781e108	0.453452
$477$	1.10516e109	3.65957
$478$	−2.90061e108	−0.882389
$479$	−6.17811e108	−1.72690	−0.863450	−	0.504435i	$-0.831701\pi$
$479$	−6.17811e108	−1.72690	−0.863450	+	0.504435i	$0.831701\pi$
$480$	−1.70223e108	−0.437268
$481$	1.97042e108	0.465247
$482$	−5.25627e108	−1.14097
$483$	−5.91688e108	−1.18096
$484$	2.63242e108	0.483190
$485$	−1.02896e109	−1.73724
$486$	−1.83972e109	−2.85747
$487$	7.70426e108	1.10105	0.550526	−	0.834818i	$-0.314427\pi$
$487$	7.70426e108	1.10105	0.550526	+	0.834818i	$0.314427\pi$
$488$	2.66123e108	0.350010
$489$	1.23878e109	1.49964
$490$	−3.59413e107	−0.0400546
$491$	−1.13884e109	−1.16859	−0.584294	−	0.811542i	$-0.698628\pi$
$491$	−1.13884e109	−1.16859	−0.584294	+	0.811542i	$0.698628\pi$
$492$	−2.61316e108	−0.246933
$493$	−1.64169e109	−1.42886
$494$	1.70371e108	0.136601
$495$	6.67768e109	4.93304
$496$	−1.09773e108	−0.0747287
$497$	−9.89499e107	−0.0620843
$498$	3.76095e109	2.17525
$499$	−3.33949e109	−1.78077	−0.890386	−	0.455207i	$-0.849565\pi$
$499$	−3.33949e109	−1.78077	−0.890386	+	0.455207i	$0.849565\pi$
$500$	−4.81709e108	−0.236864
$501$	−5.40727e109	−2.45217
$502$	−1.44021e109	−0.602456
$503$	−2.78394e109	−1.07438	−0.537188	−	0.843463i	$-0.680513\pi$
$503$	−2.78394e109	−1.07438	−0.537188	+	0.843463i	$0.680513\pi$
$504$	2.67534e109	0.952670
$505$	3.36089e109	1.10447
$506$	2.03761e109	0.618051
$507$	5.44125e109	1.52362
$508$	5.23561e107	0.0135359
$509$	4.84639e109	1.15704	0.578519	−	0.815669i	$-0.303631\pi$
$509$	4.84639e109	1.15704	0.578519	+	0.815669i	$0.303631\pi$
$510$	−7.35902e109	−1.62266
$511$	2.66565e109	0.542946
$512$	2.34854e108	0.0441942
$513$	−8.17811e109	−1.42200
$514$	8.38151e109	1.34684
$515$	−7.14441e109	−1.06114
$516$	3.84525e109	0.527972
$517$	2.86582e109	0.363816
$518$	5.92353e109	0.695385
$519$	−6.98044e109	−0.757888
$520$	2.07783e109	0.208677
$521$	−3.47017e109	−0.322421	−0.161210	−	0.986920i	$-0.551540\pi$
$521$	−3.47017e109	−0.322421	−0.161210	+	0.986920i	$0.551540\pi$
$522$	−3.49186e110	−3.00194
$523$	5.25290e109	0.417908	0.208954	−	0.977925i	$-0.432994\pi$
$523$	5.25290e109	0.417908	0.208954	+	0.977925i	$0.432994\pi$
$524$	7.25660e109	0.534338
$525$	−1.74784e110	−1.19138
$526$	1.16756e110	0.736815
$527$	−4.74567e109	−0.277311
$528$	−1.25555e110	−0.679455
$529$	−1.21994e110	−0.611482
$530$	2.58035e110	1.19813
$531$	3.22256e110	1.38635
$532$	5.12173e109	0.204172
$533$	3.18977e109	0.117844
$534$	8.48000e109	0.290385
$535$	−4.01413e110	−1.27428
$536$	−1.39050e110	−0.409257
$537$	9.43642e110	2.57542
$538$	1.84797e110	0.467749
$539$	−2.65101e109	−0.0622394
$540$	−9.97399e110	−2.17231
$541$	1.36770e110	0.276377	0.138189	−	0.990406i	$-0.455872\pi$
$541$	1.36770e110	0.276377	0.138189	+	0.990406i	$0.455872\pi$
$542$	−3.12644e110	−0.586245
$543$	4.80946e110	0.836957
$544$	1.01531e110	0.164001
$545$	8.24825e110	1.23681
$546$	−4.45042e110	−0.619582
$547$	−6.41351e110	−0.829103	−0.414552	−	0.910026i	$-0.636062\pi$
$547$	−6.41351e110	−0.829103	−0.414552	+	0.910026i	$0.636062\pi$
$548$	5.56553e110	0.668180
$549$	2.44710e111	2.72880
$550$	6.01905e110	0.623504
$551$	−6.68489e110	−0.643361
$552$	−4.77618e110	−0.427118
$553$	−1.31071e111	−1.08928
$554$	3.73305e110	0.288349
$555$	−3.46567e111	−2.48841
$556$	9.96165e109	0.0664970
$557$	5.37985e109	0.0333914	0.0166957	−	0.999861i	$-0.494685\pi$
$557$	5.37985e109	0.0333914	0.0166957	+	0.999861i	$0.494685\pi$
$558$	−1.00940e111	−0.582610
$559$	−4.69372e110	−0.251964
$560$	6.24643e110	0.311901
$561$	−5.42796e111	−2.52140
$562$	2.68778e111	1.16164
$563$	5.81626e110	0.233913	0.116956	−	0.993137i	$-0.462686\pi$
$563$	5.81626e110	0.233913	0.116956	+	0.993137i	$0.462686\pi$
$564$	−6.71751e110	−0.251423
$565$	−6.37830e111	−2.22199
$566$	1.69294e111	0.549005
$567$	1.24380e112	3.75522
$568$	−7.98735e109	−0.0224541
$569$	−4.76895e109	−0.0124846	−0.00624231	−	0.999981i	$-0.501987\pi$
$569$	−4.76895e109	−0.0124846	−0.00624231	+	0.999981i	$0.501987\pi$
$570$	−2.99656e111	−0.730620
$571$	2.30933e111	0.524474	0.262237	−	0.965003i	$-0.415540\pi$
$571$	2.30933e111	0.524474	0.262237	+	0.965003i	$0.415540\pi$
$572$	1.53260e111	0.324257
$573$	1.31568e112	2.59352
$574$	9.58916e110	0.176136
$575$	2.28967e111	0.391946
$576$	2.15957e111	0.344553
$577$	−9.91954e111	−1.47527	−0.737635	−	0.675200i	$-0.764057\pi$
$577$	−9.91954e111	−1.47527	−0.737635	+	0.675200i	$0.764057\pi$
$578$	−7.10533e110	−0.0985160
$579$	−1.75652e112	−2.27076
$580$	−8.15286e111	−0.982825
$581$	−1.38010e112	−1.55160
$582$	1.77900e112	1.86550
$583$	1.90325e112	1.86174
$584$	2.15174e111	0.196368
$585$	1.91064e112	1.62692
$586$	−6.24905e111	−0.496547
$587$	−2.55604e112	−1.89550	−0.947751	−	0.319011i	$-0.896649\pi$
$587$	−2.55604e112	−1.89550	−0.947751	+	0.319011i	$0.896649\pi$
$588$	6.21399e110	0.0430119
$589$	−1.93242e111	−0.124862
$590$	7.52410e111	0.453887
$591$	1.58573e112	0.893177
$592$	4.78154e111	0.251500
$593$	−1.12864e112	−0.554424	−0.277212	−	0.960809i	$-0.589410\pi$
$593$	−1.12864e112	−0.554424	−0.277212	+	0.960809i	$0.589410\pi$
$594$	−7.35675e112	−3.37547
$595$	2.70044e112	1.15744
$596$	2.03944e112	0.816654
$597$	−2.45979e109	−0.000920319 0
$598$	5.83006e111	0.203833
$599$	1.96290e112	0.641374	0.320687	−	0.947185i	$-0.396086\pi$
$599$	1.96290e112	0.641374	0.320687	+	0.947185i	$0.396086\pi$
$600$	−1.41087e112	−0.430886
$601$	2.12630e112	0.607027	0.303513	−	0.952827i	$-0.401840\pi$
$601$	2.12630e112	0.607027	0.303513	+	0.952827i	$0.401840\pi$
$602$	−1.41104e112	−0.376600
$603$	−1.27861e113	−3.19071
$604$	−9.32989e111	−0.217711
$605$	5.65166e112	1.23334
$606$	−5.81075e112	−1.18602
$607$	−5.99961e112	−1.14546	−0.572731	−	0.819743i	$-0.694116\pi$
$607$	−5.99961e112	−1.14546	−0.572731	+	0.819743i	$0.694116\pi$
$608$	4.13432e111	0.0738429
$609$	1.74622e113	2.91810
$610$	5.71352e112	0.893400
$611$	8.19977e111	0.119987
$612$	9.33616e112	1.27860
$613$	4.42008e112	0.566606	0.283303	−	0.959030i	$-0.408570\pi$
$613$	4.42008e112	0.566606	0.283303	+	0.959030i	$0.408570\pi$
$614$	−4.81601e111	−0.0577921
$615$	−5.61031e112	−0.630297
$616$	4.60733e112	0.484652
$617$	−6.56554e111	−0.0646728	−0.0323364	−	0.999477i	$-0.510295\pi$
$617$	−6.56554e111	−0.0646728	−0.0323364	+	0.999477i	$0.510295\pi$
$618$	1.23522e113	1.13949
$619$	1.64373e113	1.42023	0.710117	−	0.704084i	$-0.248642\pi$
$619$	1.64373e113	1.42023	0.710117	+	0.704084i	$0.248642\pi$
$620$	−2.35677e112	−0.190745
$621$	−2.79854e113	−2.12188
$622$	−4.86839e112	−0.345840
$623$	−3.11179e112	−0.207131
$624$	−3.59243e112	−0.224085
$625$	−2.10983e113	−1.23341
$626$	−1.20805e113	−0.661947
$627$	−2.21024e113	−1.13528
$628$	−6.35889e111	−0.0306207
$629$	2.06714e113	0.933295
$630$	5.74381e113	2.43169
$631$	−1.89895e112	−0.0753920	−0.0376960	−	0.999289i	$-0.512002\pi$
$631$	−1.89895e112	−0.0753920	−0.0376960	+	0.999289i	$0.512002\pi$
$632$	−1.05802e113	−0.393961
$633$	1.27734e113	0.446125
$634$	−1.57668e113	−0.516568
$635$	1.12406e112	0.0345503
$636$	−4.46124e113	−1.28660
$637$	−7.58514e111	−0.0205266
$638$	−6.01350e113	−1.52718
$639$	−7.34464e112	−0.175060
$640$	5.04219e112	0.112806
$641$	−7.44903e113	−1.56441	−0.782203	−	0.623024i	$-0.785904\pi$
$641$	−7.44903e113	−1.56441	−0.782203	+	0.623024i	$0.785904\pi$
$642$	6.94015e113	1.36836
$643$	5.48077e112	0.101461	0.0507303	−	0.998712i	$-0.483845\pi$
$643$	5.48077e112	0.101461	0.0507303	+	0.998712i	$0.483845\pi$
$644$	1.75265e113	0.304661
$645$	8.25553e113	1.34765
$646$	1.78733e113	0.274024
$647$	−7.43938e113	−1.07131	−0.535653	−	0.844438i	$-0.679934\pi$
$647$	−7.43938e113	−1.07131	−0.535653	+	0.844438i	$0.679934\pi$
$648$	1.00401e114	1.35815
$649$	5.54972e113	0.705279
$650$	1.72219e113	0.205632
$651$	5.04785e113	0.566339
$652$	−3.66942e113	−0.386874
$653$	5.05344e113	0.500730	0.250365	−	0.968152i	$-0.419449\pi$
$653$	5.05344e113	0.500730	0.250365	+	0.968152i	$0.419449\pi$
$654$	−1.42606e114	−1.32813
$655$	1.55795e114	1.36390
$656$	7.74048e112	0.0637034
$657$	1.97860e114	1.53095
$658$	2.46503e113	0.179339
$659$	8.48064e113	0.580189	0.290094	−	0.956998i	$-0.406313\pi$
$659$	8.48064e113	0.580189	0.290094	+	0.956998i	$0.406313\pi$
$660$	−2.69560e114	−1.73431
$661$	−1.78496e114	−1.08011	−0.540057	−	0.841628i	$-0.681597\pi$
$661$	−1.78496e114	−1.08011	−0.540057	+	0.841628i	$0.681597\pi$
$662$	−2.18678e114	−1.24467
$663$	−1.55307e114	−0.831558
$664$	−1.11404e114	−0.561167
$665$	1.09961e114	0.521148
$666$	4.39679e114	1.96078
$667$	−2.28756e114	−0.960011
$668$	1.60169e114	0.632606
$669$	−5.96036e114	−2.21572
$670$	−2.98532e114	−1.04463
$671$	4.21426e114	1.38822
$672$	−1.07996e114	−0.334930
$673$	6.12589e114	1.78879	0.894394	−	0.447279i	$-0.147607\pi$
$673$	6.12589e114	1.78879	0.894394	+	0.447279i	$0.147607\pi$
$674$	1.52699e114	0.419865
$675$	−8.26684e114	−2.14061
$676$	−1.61176e114	−0.393060
$677$	7.41723e114	1.70373	0.851865	−	0.523761i	$-0.175471\pi$
$677$	7.41723e114	1.70373	0.851865	+	0.523761i	$0.175471\pi$
$678$	1.10276e115	2.38605
$679$	−6.52817e114	−1.33065
$680$	2.17982e114	0.418611
$681$	−7.75022e114	−1.40235
$682$	−1.73833e114	−0.296392
$683$	−8.33710e114	−1.33960	−0.669802	−	0.742540i	$-0.733621\pi$
$683$	−8.33710e114	−1.33960	−0.669802	+	0.742540i	$0.733621\pi$
$684$	3.80165e114	0.575704
$685$	1.19489e115	1.70553
$686$	−5.36554e114	−0.721917
$687$	−2.24987e115	−2.85371
$688$	−1.13901e114	−0.136205
$689$	5.44563e114	0.614002
$690$	−1.02542e115	−1.09022
$691$	1.43633e115	1.44010	0.720050	−	0.693922i	$-0.244119\pi$
$691$	1.43633e115	1.44010	0.720050	+	0.693922i	$0.244119\pi$
$692$	2.06769e114	0.195518
$693$	4.23660e115	3.77851
$694$	9.25384e114	0.778509
$695$	2.13871e114	0.169734
$696$	1.40957e115	1.05539
$697$	3.34634e114	0.236397
$698$	7.35978e113	0.0490592
$699$	−2.16788e115	−1.36367
$700$	5.17729e114	0.307349
$701$	−2.33684e115	−1.30933	−0.654666	−	0.755918i	$-0.727191\pi$
$701$	−2.33684e115	−1.30933	−0.654666	+	0.755918i	$0.727191\pi$
$702$	−2.10494e115	−1.11323
$703$	8.41730e114	0.420225
$704$	3.71909e114	0.175285
$705$	−1.44221e115	−0.641756
$706$	1.77667e112	0.000746479 0
$707$	2.13229e115	0.845980
$708$	−1.30086e115	−0.487399
$709$	−5.49353e114	−0.194392	−0.0971962	−	0.995265i	$-0.530987\pi$
$709$	−5.49353e114	−0.194392	−0.0971962	+	0.995265i	$0.530987\pi$
$710$	−1.71484e114	−0.0573140
$711$	−9.72887e115	−3.07145
$712$	−2.51187e114	−0.0749131
$713$	−6.61270e114	−0.186317
$714$	−4.66887e115	−1.24289
$715$	3.29040e115	0.827664
$716$	−2.79517e115	−0.664403
$717$	1.07668e116	2.41859
$718$	5.18868e115	1.10158
$719$	−3.78793e115	−0.760120	−0.380060	−	0.924962i	$-0.624097\pi$
$719$	−3.78793e115	−0.760120	−0.380060	+	0.924962i	$0.624097\pi$
$720$	4.63647e115	0.879470
$721$	−4.53271e115	−0.812793
$722$	−3.44322e115	−0.583725
$723$	1.95108e116	3.12734
$724$	−1.42462e115	−0.215917
$725$	−6.75741e115	−0.968481
$726$	−9.77133e115	−1.32440
$727$	−1.14460e116	−1.46726	−0.733632	−	0.679547i	$-0.762176\pi$
$727$	−1.14460e116	−1.46726	−0.733632	+	0.679547i	$0.762176\pi$
$728$	1.31826e115	0.159839
$729$	3.47954e116	3.99077
$730$	4.61968e115	0.501228
$731$	−4.92411e115	−0.505445
$732$	−9.87828e115	−0.959362
$733$	−4.61315e115	−0.423923	−0.211961	−	0.977278i	$-0.567985\pi$
$733$	−4.61315e115	−0.423923	−0.211961	+	0.977278i	$0.567985\pi$
$734$	1.12001e116	0.973941
$735$	1.33411e115	0.109788
$736$	1.41476e115	0.110187
$737$	−2.20195e116	−1.62321
$738$	7.11764e115	0.496653
$739$	−1.51407e115	−0.100011	−0.0500053	−	0.998749i	$-0.515924\pi$
$739$	−1.51407e115	−0.100011	−0.0500053	+	0.998749i	$0.515924\pi$
$740$	1.02657e116	0.641954
$741$	−6.32402e115	−0.374417
$742$	1.63708e116	0.917722
$743$	−4.53258e114	−0.0240602	−0.0120301	−	0.999928i	$-0.503829\pi$
$743$	−4.53258e114	−0.0240602	−0.0120301	+	0.999928i	$0.503829\pi$
$744$	4.07468e115	0.204828
$745$	4.37857e116	2.08451
$746$	−1.53358e116	−0.691488
$747$	−1.02439e117	−4.37505
$748$	1.60782e116	0.650465
$749$	−2.54673e116	−0.976045
$750$	1.78807e116	0.649235
$751$	−3.53238e115	−0.121521	−0.0607603	−	0.998152i	$-0.519353\pi$
$751$	−3.53238e115	−0.121521	−0.0607603	+	0.998152i	$0.519353\pi$
$752$	1.98980e115	0.0648616
$753$	5.34594e116	1.65131
$754$	−1.72060e116	−0.503664
$755$	−2.00308e116	−0.555707
$756$	−6.32791e116	−1.66390
$757$	4.81739e116	1.20068	0.600342	−	0.799744i	$-0.295031\pi$
$757$	4.81739e116	1.20068	0.600342	+	0.799744i	$0.295031\pi$
$758$	−6.48198e115	−0.153146
$759$	−7.56342e116	−1.69405
$760$	8.87615e115	0.188484
$761$	−2.40796e116	−0.484808	−0.242404	−	0.970175i	$-0.577936\pi$
$761$	−2.40796e116	−0.484808	−0.242404	+	0.970175i	$0.577936\pi$
$762$	−1.94341e115	−0.0371013
$763$	5.23303e116	0.947348
$764$	−3.89721e116	−0.669071
$765$	2.00442e117	3.26363
$766$	5.37354e116	0.829842
$767$	1.58791e116	0.232601
$768$	−8.71759e115	−0.121134
$769$	8.46732e116	1.11617	0.558084	−	0.829785i	$-0.311537\pi$
$769$	8.46732e116	1.11617	0.558084	+	0.829785i	$0.311537\pi$
$770$	9.89168e116	1.23707
$771$	−3.11115e117	−3.69163
$772$	5.20301e116	0.585807
$773$	1.03355e117	1.10424	0.552118	−	0.833766i	$-0.313820\pi$
$773$	1.03355e117	1.10424	0.552118	+	0.833766i	$0.313820\pi$
$774$	−1.04735e117	−1.06190
$775$	−1.95338e116	−0.187961
$776$	−5.26961e116	−0.481259
$777$	−2.19876e117	−1.90602
$778$	6.34164e116	0.521829
$779$	1.36261e116	0.106440
$780$	−7.71275e116	−0.571976
$781$	−1.26486e116	−0.0890582
$782$	6.11623e116	0.408894
$783$	8.25920e117	5.24308
$784$	−1.84065e115	−0.0110961
$785$	−1.36522e116	−0.0781594
$786$	−2.69359e117	−1.46460
$787$	3.76130e117	1.94250	0.971251	−	0.238058i	$-0.0765108\pi$
$787$	3.76130e117	1.94250	0.971251	+	0.238058i	$0.0765108\pi$
$788$	−4.69712e116	−0.230420
$789$	−4.33390e117	−2.01958
$790$	−2.27151e117	−1.00558
$791$	−4.04666e117	−1.70196
$792$	3.41983e117	1.36658
$793$	1.20580e117	0.457836
$794$	1.45532e116	0.0525082
$795$	−9.57803e117	−3.28403
$796$	7.28619e113	0.000237422 0
$797$	5.62034e116	0.174061	0.0870303	−	0.996206i	$-0.472262\pi$
$797$	5.62034e116	0.174061	0.0870303	+	0.996206i	$0.472262\pi$
$798$	−1.90114e117	−0.559626
$799$	8.60226e116	0.240695
$800$	4.17917e116	0.111159
$801$	−2.30975e117	−0.584048
$802$	−3.91612e117	−0.941443
$803$	3.40744e117	0.778841
$804$	5.16141e117	1.12176
$805$	3.76284e117	0.777647
$806$	−4.97378e116	−0.0977502
$807$	−6.85951e117	−1.28208
$808$	1.72121e117	0.305966
$809$	−2.65150e117	−0.448308	−0.224154	−	0.974554i	$-0.571962\pi$
$809$	−2.65150e117	−0.448308	−0.224154	+	0.974554i	$0.571962\pi$
$810$	2.15555e118	3.46669
$811$	5.32527e117	0.814699	0.407350	−	0.913272i	$-0.366453\pi$
$811$	5.32527e117	0.814699	0.407350	+	0.913272i	$0.366453\pi$
$812$	−5.17251e117	−0.752805
$813$	1.16051e118	1.60687
$814$	7.57192e117	0.997510
$815$	−7.87804e117	−0.987495
$816$	−3.76876e117	−0.449518
$817$	−2.00508e117	−0.227582
$818$	−3.73034e117	−0.402939
$819$	1.21219e118	1.24616
$820$	1.66184e117	0.162603
$821$	−1.04278e117	−0.0971168	−0.0485584	−	0.998820i	$-0.515463\pi$
$821$	−1.04278e117	−0.0971168	−0.0485584	+	0.998820i	$0.515463\pi$
$822$	−2.06588e118	−1.83145
$823$	7.64299e117	0.645016	0.322508	−	0.946567i	$-0.395474\pi$
$823$	7.64299e117	0.645016	0.322508	+	0.946567i	$0.395474\pi$
$824$	−3.65886e117	−0.293963
$825$	−2.23422e118	−1.70900
$826$	4.77360e117	0.347659
$827$	−4.72687e117	−0.327793	−0.163897	−	0.986478i	$-0.552406\pi$
$827$	−4.72687e117	−0.327793	−0.163897	+	0.986478i	$0.552406\pi$
$828$	1.30092e118	0.859055
$829$	−1.40077e117	−0.0880861	−0.0440430	−	0.999030i	$-0.514024\pi$
$829$	−1.40077e117	−0.0880861	−0.0440430	+	0.999030i	$0.514024\pi$
$830$	−2.39177e118	−1.43238
$831$	−1.38568e118	−0.790352
$832$	1.06412e117	0.0578089
$833$	−7.95746e116	−0.0411767
$834$	−3.69768e117	−0.182265
$835$	3.43875e118	1.61472
$836$	6.54699e117	0.292879
$837$	2.38751e118	1.01757
$838$	1.70926e118	0.694106
$839$	6.03228e117	0.233411	0.116706	−	0.993167i	$-0.462767\pi$
$839$	6.03228e117	0.233411	0.116706	+	0.993167i	$0.462767\pi$
$840$	−2.31862e118	−0.854908
$841$	3.90515e118	1.37215
$842$	−3.42462e118	−1.14676
$843$	−9.97682e118	−3.18402
$844$	−3.78362e117	−0.115090
$845$	−3.46036e118	−1.00328
$846$	1.82969e118	0.505683
$847$	3.58565e118	0.944691
$848$	1.32147e118	0.331913
$849$	−6.28405e118	−1.50480
$850$	1.80673e118	0.412502
$851$	2.88039e118	0.627053
$852$	2.96484e117	0.0615457
$853$	5.47136e118	1.08308	0.541539	−	0.840676i	$-0.317842\pi$
$853$	5.47136e118	1.08308	0.541539	+	0.840676i	$0.317842\pi$
$854$	3.62490e118	0.684308
$855$	8.16192e118	1.46948
$856$	−2.05575e118	−0.353007
$857$	−7.07092e117	−0.115812	−0.0579058	−	0.998322i	$-0.518442\pi$
$857$	−7.07092e117	−0.115812	−0.0579058	+	0.998322i	$0.518442\pi$
$858$	−5.68887e118	−0.888773
$859$	1.15112e119	1.71553	0.857763	−	0.514045i	$-0.171853\pi$
$859$	1.15112e119	1.71553	0.857763	+	0.514045i	$0.171853\pi$
$860$	−2.44538e118	−0.347664
$861$	−3.55942e118	−0.482782
$862$	7.31428e117	0.0946514
$863$	6.53380e118	0.806730	0.403365	−	0.915039i	$-0.367840\pi$
$863$	6.53380e118	0.806730	0.403365	+	0.915039i	$0.367840\pi$
$864$	−5.10796e118	−0.601785
$865$	4.43921e118	0.499060
$866$	−3.22958e118	−0.346474
$867$	2.63744e118	0.270028
$868$	−1.49523e118	−0.146103
$869$	−1.67545e119	−1.56254
$870$	3.02627e119	2.69388
$871$	−6.30030e118	−0.535336
$872$	4.22416e118	0.342628
$873$	−4.84559e119	−3.75206
$874$	2.49050e118	0.184109
$875$	−6.56142e118	−0.463097
$876$	−7.98709e118	−0.538236
$877$	2.33914e119	1.50512	0.752562	−	0.658521i	$-0.228818\pi$
$877$	2.33914e119	1.50512	0.752562	+	0.658521i	$0.228818\pi$
$878$	6.67435e118	0.410092
$879$	2.31960e119	1.36101
$880$	7.98468e118	0.447414
$881$	−2.96116e119	−1.58467	−0.792333	−	0.610089i	$-0.791134\pi$
$881$	−2.96116e119	−1.58467	−0.792333	+	0.610089i	$0.791134\pi$
$882$	−1.69254e118	−0.0865092
$883$	−1.55880e119	−0.760994	−0.380497	−	0.924782i	$-0.624247\pi$
$883$	−1.55880e119	−0.760994	−0.380497	+	0.924782i	$0.624247\pi$
$884$	4.60036e118	0.214524
$885$	−2.79288e119	−1.24408
$886$	3.17990e118	0.135316
$887$	−3.08151e119	−1.25273	−0.626365	−	0.779530i	$-0.715458\pi$
$887$	−3.08151e119	−1.25273	−0.626365	+	0.779530i	$0.715458\pi$
$888$	−1.77487e119	−0.689352
$889$	7.13148e117	0.0264642
$890$	−5.39285e118	−0.191216
$891$	1.58992e120	5.38676
$892$	1.76553e119	0.571607
$893$	3.50280e118	0.108376
$894$	−7.57024e119	−2.23841
$895$	−6.00108e119	−1.69589
$896$	3.19898e118	0.0864045
$897$	−2.16407e119	−0.558699
$898$	−1.72643e119	−0.426046
$899$	1.95158e119	0.460382
$900$	3.84289e119	0.866635
$901$	5.71293e119	1.23170
$902$	1.22576e119	0.252663
$903$	5.23765e119	1.03224
$904$	−3.26651e119	−0.615548
$905$	−3.05857e119	−0.551127
$906$	3.46318e119	0.596737
$907$	2.76532e119	0.455669	0.227835	−	0.973700i	$-0.426835\pi$
$907$	2.76532e119	0.455669	0.227835	+	0.973700i	$0.426835\pi$
$908$	2.29570e119	0.361775
$909$	1.58271e120	2.38542
$910$	2.83024e119	0.407988
$911$	7.11634e119	0.981211	0.490606	−	0.871382i	$-0.336776\pi$
$911$	7.11634e119	0.981211	0.490606	+	0.871382i	$0.336776\pi$
$912$	−1.53462e119	−0.202400
$913$	−1.76416e120	−2.22572
$914$	−3.20304e119	−0.386581
$915$	−2.12081e120	−2.44877
$916$	6.66438e119	0.736196
$917$	9.88430e119	1.04469
$918$	−2.20826e120	−2.23317
$919$	−1.43235e120	−1.38603	−0.693013	−	0.720925i	$-0.743717\pi$
$919$	−1.43235e120	−1.38603	−0.693013	+	0.720925i	$0.743717\pi$
$920$	3.03741e119	0.281252
$921$	1.78766e119	0.158406
$922$	6.38882e119	0.541775
$923$	−3.61905e118	−0.0293714
$924$	−1.71020e120	−1.32841
$925$	8.50862e119	0.632585
$926$	6.96078e119	0.495351
$927$	−3.36444e120	−2.29184
$928$	−4.17531e119	−0.272267
$929$	3.35947e119	0.209718	0.104859	−	0.994487i	$-0.466561\pi$
$929$	3.35947e119	0.209718	0.104859	+	0.994487i	$0.466561\pi$
$930$	8.74812e119	0.522823
$931$	−3.24024e118	−0.0185402
$932$	6.42151e119	0.351797
$933$	1.80711e120	0.947931
$934$	1.59941e120	0.803361
$935$	3.45191e120	1.66031
$936$	9.78493e119	0.450698
$937$	6.82808e119	0.301193	0.150597	−	0.988595i	$-0.451881\pi$
$937$	6.82808e119	0.301193	0.150597	+	0.988595i	$0.451881\pi$
$938$	−1.89401e120	−0.800143
$939$	4.48417e120	1.81437
$940$	4.27200e119	0.165559
$941$	−3.27233e120	−1.21472	−0.607361	−	0.794426i	$-0.707772\pi$
$941$	−3.27233e120	−1.21472	−0.607361	+	0.794426i	$0.707772\pi$
$942$	2.36037e119	0.0839301
$943$	4.66285e119	0.158828
$944$	3.85330e119	0.125738
$945$	−1.35857e121	−4.24711
$946$	−1.80370e120	−0.540223
$947$	−3.43114e119	−0.0984610	−0.0492305	−	0.998787i	$-0.515677\pi$
$947$	−3.43114e119	−0.0984610	−0.0492305	+	0.998787i	$0.515677\pi$
$948$	3.92729e120	1.07983
$949$	9.74949e119	0.256862
$950$	7.35691e119	0.185733
$951$	5.85249e120	1.41589
$952$	1.38297e120	0.320639
$953$	−4.65198e120	−1.03365	−0.516827	−	0.856090i	$-0.672887\pi$
$953$	−4.65198e120	−1.03365	−0.516827	+	0.856090i	$0.672887\pi$
$954$	1.21514e121	2.58771
$955$	−8.36709e120	−1.70780
$956$	−3.18926e120	−0.623943
$957$	2.23216e121	4.18593
$958$	−6.79290e120	−1.22110
$959$	7.58087e120	1.30637
$960$	−1.87162e120	−0.309195
$961$	−5.74975e120	−0.910650
$962$	2.16650e120	0.328979
$963$	−1.89033e121	−2.75216
$964$	−5.77933e120	−0.806785
$965$	1.11706e121	1.49527
$966$	−6.50568e120	−0.835063
$967$	9.62670e120	1.18496	0.592482	−	0.805584i	$-0.298148\pi$
$967$	9.62670e120	1.18496	0.592482	+	0.805584i	$0.298148\pi$
$968$	2.89438e120	0.341667
$969$	−6.63444e120	−0.751088
$970$	−1.13136e121	−1.22841
$971$	−4.18917e120	−0.436264	−0.218132	−	0.975919i	$-0.569996\pi$
$971$	−4.18917e120	−0.436264	−0.218132	+	0.975919i	$0.569996\pi$
$972$	−2.02279e121	−2.02054
$973$	1.35689e120	0.130009
$974$	8.47093e120	0.778562
$975$	−6.39263e120	−0.563628
$976$	2.92606e120	0.247494
$977$	−9.07316e120	−0.736256	−0.368128	−	0.929775i	$-0.620001\pi$
$977$	−9.07316e120	−0.736256	−0.368128	+	0.929775i	$0.620001\pi$
$978$	1.36206e121	1.06040
$979$	−3.97773e120	−0.297123
$980$	−3.95178e119	−0.0283229
$981$	3.88426e121	2.67124
$982$	−1.25216e121	−0.826316
$983$	2.21239e120	0.140102	0.0700510	−	0.997543i	$-0.477684\pi$
$983$	2.21239e120	0.140102	0.0700510	+	0.997543i	$0.477684\pi$
$984$	−2.87320e120	−0.174608
$985$	−1.00845e121	−0.588146
$986$	−1.80506e121	−1.01036
$987$	−9.15000e120	−0.491560
$988$	1.87325e120	0.0965915
$989$	−6.86134e120	−0.339593
$990$	7.34218e121	3.48819
$991$	−1.01414e121	−0.462503	−0.231251	−	0.972894i	$-0.574282\pi$
$991$	−1.01414e121	−0.462503	−0.231251	+	0.972894i	$0.574282\pi$
$992$	−1.20697e120	−0.0528412
$993$	8.11713e121	3.41160
$994$	−1.08797e120	−0.0439002
$995$	1.56430e118	0.000606020 0
$996$	4.13521e121	1.53813
$997$	−2.18601e121	−0.780725	−0.390363	−	0.920661i	$-0.627650\pi$
$997$	−2.18601e121	−0.780725	−0.390363	+	0.920661i	$0.627650\pi$
$998$	−3.67181e121	−1.25920
$999$	−1.03996e122	−3.42464

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.82.a.b.1.1	✓	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)

\(f(q)\)	\(=\)	\(q+1.09951e12 q^{2} -4.08130e19 q^{3} +1.20893e24 q^{4} +2.59550e28 q^{5} -4.48743e31 q^{6} +1.64669e34 q^{7} +1.32923e36 q^{8} +1.22227e39 q^{9} +2.85378e40 q^{10} +2.10493e42 q^{11} -4.93398e43 q^{12} +6.02269e44 q^{13} +1.81056e46 q^{14} -1.05930e48 q^{15} +1.46150e48 q^{16} +6.31832e49 q^{17} +1.34390e51 q^{18} +2.57279e51 q^{19} +3.13776e52 q^{20} -6.72063e53 q^{21} +2.31439e54 q^{22} +8.80406e54 q^{23} -5.42497e55 q^{24} +2.60070e56 q^{25} +6.62202e56 q^{26} -3.17869e58 q^{27} +1.99073e58 q^{28} -2.59830e59 q^{29} -1.16471e60 q^{30} -7.51097e59 q^{31} +1.60694e60 q^{32} -8.59084e61 q^{33} +6.94706e61 q^{34} +4.27398e62 q^{35} +1.47763e63 q^{36} +3.27166e63 q^{37} +2.82881e63 q^{38} -2.45804e64 q^{39} +3.45001e64 q^{40} +5.29625e64 q^{41} -7.38941e65 q^{42} -7.79339e65 q^{43} +2.54470e66 q^{44} +3.17240e67 q^{45} +9.68016e66 q^{46} +1.36148e67 q^{47} -5.96482e67 q^{48} -1.25943e67 q^{49} +2.85950e68 q^{50} -2.57869e69 q^{51} +7.28099e68 q^{52} +9.04185e69 q^{53} -3.49501e70 q^{54} +5.46334e70 q^{55} +2.18883e70 q^{56} -1.05003e71 q^{57} -2.85686e71 q^{58} +2.63654e71 q^{59} -1.28061e72 q^{60} +2.00209e72 q^{61} -8.25840e71 q^{62} +2.01270e73 q^{63} +1.76685e72 q^{64} +1.56319e73 q^{65} -9.44573e73 q^{66} -1.04609e74 q^{67} +7.63838e73 q^{68} -3.59320e74 q^{69} +4.69929e74 q^{70} -6.00902e73 q^{71} +1.62468e75 q^{72} +1.61879e75 q^{73} +3.59723e75 q^{74} -1.06142e76 q^{75} +3.11031e75 q^{76} +3.46617e76 q^{77} -2.70264e76 q^{78} -7.95967e76 q^{79} +3.79332e76 q^{80} +7.55331e77 q^{81} +5.82329e76 q^{82} -8.38108e77 q^{83} -8.12475e77 q^{84} +1.63992e78 q^{85} -8.56892e77 q^{86} +1.06044e79 q^{87} +2.79793e78 q^{88} -1.88972e78 q^{89} +3.48809e79 q^{90} +9.91752e78 q^{91} +1.06435e79 q^{92} +3.06545e79 q^{93} +1.49696e79 q^{94} +6.67767e79 q^{95} -6.55839e79 q^{96} -3.96442e80 q^{97} -1.38475e79 q^{98} +2.57279e81 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4398046511104 q^{2} - 71\!\cdots\!04 q^{3} + 48\!\cdots\!04 q^{4} + 24\!\cdots\!80 q^{5} - 78\!\cdots\!04 q^{6} + 18\!\cdots\!92 q^{7} + 53\!\cdots\!04 q^{8} + 10\!\cdots\!92 q^{9} + 27\!\cdots\!80 q^{10}+ \cdots + 23\!\cdots\!24 q^{99}+O(q^{100}) \) 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