LMFDB - Embedded newform 1.68.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1,68,Mod(1,1)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1, base_ring=CyclotomicField(1))

chi = DirichletCharacter(H, H._module([]))

N = Newforms(chi, 68, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1.1");

S:= CuspForms(chi, 68);

N := Newforms(S);

Level:	$ N $	$=$	$ 1 $
Weight:	$ k $	$=$	$ 68 $
Character orbit:	$[\chi]$	$=$	1.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$28.4290351930$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} + \cdots + 20\!\cdots\!00 $ x^5 - x^4 - 939384011925257456x^3 + 31046449413968483513911200x^2 + 156793504704482691874379743265203200*x + 20916736226052669578405116700517591609696000
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{15}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 13\cdot 17 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.64845e8$ of defining polynomial
Character	$\chi$	$=$	1.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.06726e9 q^{2} -7.82458e15 q^{3} -1.21897e20 q^{4} -8.33551e22 q^{5} -3.96491e25 q^{6} -1.15070e28 q^{7} -1.36548e30 q^{8} -3.14854e31 q^{9} +O(q^{10})$	\(q+5.06726e9 q^{2} -7.82458e15 q^{3} -1.21897e20 q^{4} -8.33551e22 q^{5} -3.96491e25 q^{6} -1.15070e28 q^{7} -1.36548e30 q^{8} -3.14854e31 q^{9} -4.22381e32 q^{10} -1.09259e35 q^{11} +9.53792e35 q^{12} +1.54273e37 q^{13} -5.83087e37 q^{14} +6.52218e38 q^{15} +1.10696e40 q^{16} +8.24456e40 q^{17} -1.59545e41 q^{18} -5.93328e42 q^{19} +1.01607e43 q^{20} +9.00371e43 q^{21} -5.53645e44 q^{22} +2.28489e45 q^{23} +1.06843e46 q^{24} -6.08146e46 q^{25} +7.81738e46 q^{26} +9.71773e47 q^{27} +1.40266e48 q^{28} +1.07438e49 q^{29} +3.30496e48 q^{30} +1.32534e50 q^{31} +2.57601e50 q^{32} +8.54908e50 q^{33} +4.17773e50 q^{34} +9.59164e50 q^{35} +3.83798e51 q^{36} -5.20056e52 q^{37} -3.00655e52 q^{38} -1.20712e53 q^{39} +1.13819e53 q^{40} -1.63973e54 q^{41} +4.56241e53 q^{42} +5.64177e54 q^{43} +1.33184e55 q^{44} +2.62447e54 q^{45} +1.15781e55 q^{46} +2.41027e55 q^{47} -8.66148e55 q^{48} -2.85968e56 q^{49} -3.08163e56 q^{50} -6.45102e56 q^{51} -1.88053e57 q^{52} +2.84898e57 q^{53} +4.92422e57 q^{54} +9.10731e57 q^{55} +1.57125e58 q^{56} +4.64254e58 q^{57} +5.44415e58 q^{58} -3.30680e59 q^{59} -7.95034e58 q^{60} -6.89383e59 q^{61} +6.71583e59 q^{62} +3.62302e59 q^{63} -3.28249e59 q^{64} -1.28594e60 q^{65} +4.33204e60 q^{66} +2.58683e61 q^{67} -1.00499e61 q^{68} -1.78783e61 q^{69} +4.86033e60 q^{70} -1.34953e62 q^{71} +4.29927e61 q^{72} -1.11285e62 q^{73} -2.63526e62 q^{74} +4.75848e62 q^{75} +7.23248e62 q^{76} +1.25724e63 q^{77} -6.11677e62 q^{78} +1.23916e63 q^{79} -9.22705e62 q^{80} -4.68471e63 q^{81} -8.30894e63 q^{82} +3.41100e63 q^{83} -1.09752e64 q^{84} -6.87226e63 q^{85} +2.85883e64 q^{86} -8.40656e64 q^{87} +1.49191e65 q^{88} +3.04246e65 q^{89} +1.32989e64 q^{90} -1.77521e65 q^{91} -2.78521e65 q^{92} -1.03702e66 q^{93} +1.22134e65 q^{94} +4.94569e65 q^{95} -2.01562e66 q^{96} +2.29986e66 q^{97} -1.44907e66 q^{98} +3.44008e66 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5554901256 q^{2} + 34\!\cdots\!72 q^{3}+ \cdots + 27\!\cdots\!85 q^{9}+O(q^{10}) $ 5 * q + 5554901256 * q^2 + 3443360269119372 * q^3 + 350472004382296600640 * q^4 + 330744836907160597528950 * q^5 + 146648871573834733762548960 * q^6 + 33632726520317603069152899256 * q^7 + 3261706912683890613885787783680 * q^8 + 27806018581845734541016911293385 * q^9	$ 5 q + 5554901256 q^{2} + 34\!\cdots\!72 q^{3}+ \cdots + 16\!\cdots\!20 q^{99}+O(q^{100}) $ 5 * q + 5554901256 * q^2 + 3443360269119372 * q^3 + 350472004382296600640 * q^4 + 330744836907160597528950 * q^5 + 146648871573834733762548960 * q^6 + 33632726520317603069152899256 * q^7 + 3261706912683890613885787783680 * q^8 + 27806018581845734541016911293385 * q^9 - 5287067006725590176865792208438800 * q^10 + 203006981551281631527218766884109060 * q^11 - 2089518810401181456110201520986913024 * q^12 + 177822884499813229570219072356766702 * q^13 + 290753425875732621001638220329334092480 * q^14 - 933024301033068664231979455289562936600 * q^15 + 18266362102387359184851236859301091348480 * q^16 + 75241497376497768433807587544150344102906 * q^17 - 1011568566626978308983328666525487111002968 * q^18 + 3998884102997801557241177083441346393295100 * q^19 - 15074831378020267075078660994426474705846400 * q^20 - 175719863949295325296418063144372762630915040 * q^21 + 1841247880573095918698534631505063585473750432 * q^22 - 419660874260151197180020017656265233426170968 * q^23 - 13019747846573945420584224979196286639928166400 * q^24 + 175279270003539476977813620578789764300129011875 * q^25 + 21416337629300654169124097666599354769371062960 * q^26 + 1886613800166950113417786970194782722207999659320 * q^27 + 14982728306233364463143494158089725946900023694848 * q^28 + 18733547497571939101380180097730717676662894233950 * q^29 + 183881202860964913168534714290880945912096094414400 * q^30 + 365849822471060800690376948081970588059264217288160 * q^31 + 1591165751988992347021834813546084975107380979204096 * q^32 + 2437038903031213839498705428654461423996034595356784 * q^33 + 7817592154883765712044327776251337623941567500782480 * q^34 + 4524513410554564733118320802153390591800073874285200 * q^35 - 38131548457864843751915538710226876689225094472585920 * q^36 - 56389392588768330454829645685931746748224606527380394 * q^37 - 311451303114725973135630000941921262878276604287141280 * q^38 - 717177850582388225418943791249053833224228521090828280 * q^39 - 1886768345048751997029226502271137564956948018338432000 * q^40 + 113555410342193387626817165052765483601006465327103010 * q^41 + 2805540126450529847390961864461873572829871059166350592 * q^42 + 6508304671085340439704487085091235272293711787089861092 * q^43 + 49369449025916528958230735645862721892607029455332199680 * q^44 + 99011086960300194550392336618799141939148872728728967150 * q^45 + 79552980744915855625545425031873854934595203714399466560 * q^46 - 128023594814179612806658270243785955024442659789917751344 * q^47 - 198512903890152994486195174416803543869354744423782432768 * q^48 - 771329931628098140389185999842408514636676231229111640435 * q^49 - 3919481096957516748709549462396523762064894577356951885000 * q^50 - 3397415009313164473667296477133432796692029796730856284840 * q^51 + 672997930141532412373441872733517069231734104886640838016 * q^52 + 1046765637663599376128164919371991945530174947374966490822 * q^53 + 19066352042639687223500180230679796188217876586398799185600 * q^54 + 66942753107037791618627925698973654692865254681839842449400 * q^55 + 164746697572073652525116376573879338548073684495562965299200 * q^56 - 9288557942524759644038647059781434529266663439727382990960 * q^57 - 61137956327686914952144218440883774026433838412764071303120 * q^58 - 305790738450157663862297559072875409310668010528073986627500 * q^59 - 1346157500908637758724100110370785768539984220742218248588800 * q^60 - 1137073938560830615075915586814815057732339483683661907261890 * q^61 - 2197846158587850633685459639420082896328263471188214086352128 * q^62 - 2062780018748959038039139080074136999529046449885010133153768 * q^63 + 6319522016046144185339524760669817745425888834305772325437440 * q^64 + 17054811715744261908524663200762497384422620874171150779332900 * q^65 + 27998862803339055552343423092993933514291039899414979026779520 * q^66 - 11903044989154510038048691316368615691989662189985228212145844 * q^67 + 81678078346863753465296539273887923556029130514802676363013248 * q^68 + 3593002077831059525793506029116145162632074080747869594484320 * q^69 - 262230520857255225386866306600227822969034558447869634844476800 * q^70 - 112169920751076079402369696382938772997794660792753351047772040 * q^71 - 747723581155501189674233444839980291686130221322311676193789440 * q^72 - 304061800126793465330511223999919410949872771539142074085150718 * q^73 + 921008183492866283853128598587548033201480125306857491530532080 * q^74 + 21623812027928499471467875193108522496460255319651250940492500 * q^75 + 2076567511595431334156222539483464646991367936386801535927366400 * q^76 + 5018131462902473316385236580394702266689060843204393245422638432 * q^77 + 6187393864780941106550100388692932558738785969110711253222726464 * q^78 + 2994504004186984867153587379277397704940492543518650040519815600 * q^79 - 15198313539268031787508043655897930060910106707913153957798092800 * q^80 - 11195187555119704986426983039082721262159900559267345597680888595 * q^81 - 17699192859543411700126655967680257958811789647196784631772687408 * q^82 - 47352447490969920981134910324833485175129837165394254819464203588 * q^83 - 52124143836539534782259645807914007868879894236203487647926097920 * q^84 - 39334683419450918052255554891934302881190270223971152032299949300 * q^85 + 49699285824187070765304255878405188987409992188766856866277888160 * q^86 + 357574798440063707158439877351651513807909832776407769090038489160 * q^87 + 430257938373980762031490456670953174964896424231159472833711400960 * q^88 - 11988583288945008097106276879882407358382120120051430459779195150 * q^89 + 477054381540878916877701150770415125576892082956059127548575220400 * q^90 + 232376792470844928685229521265495056030669651812230900159714321360 * q^91 - 986292139147147043544160128863480288571347084605459783959415954944 * q^92 - 1849280210034321990512094428557344731461785100166176155774311127936 * q^93 - 4927754861315037106505826725122767884474273389521981763161929704320 * q^94 - 2022377488008256909110920831467942087293430657450319294506545963000 * q^95 - 2242556165580376632567217570505477297218451346654758522676359331840 * q^96 + 2897556354128426205551279700626881901316694084191208635967319112106 * q^97 + 12426885478574475523720488645206508880013087100212747539199689691208 * q^98 + 16175146031966622867876651602334534091105243319832315048256094545620 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	5.06726e9	0.417127	0.208563	−	0.978009i	$-0.433121\pi$
$2$	5.06726e9	0.417127	0.208563	+	0.978009i	$0.433121\pi$
$3$	−7.82458e15	−0.812641	−0.406321	−	0.913731i	$-0.633188\pi$
$3$	−7.82458e15	−0.812641	−0.406321	+	0.913731i	$0.633188\pi$
$4$	−1.21897e20	−0.826005
$5$	−8.33551e22	−0.320211	−0.160106	−	0.987100i	$-0.551184\pi$
$5$	−8.33551e22	−0.320211	−0.160106	+	0.987100i	$0.551184\pi$
$6$	−3.96491e25	−0.338974
$7$	−1.15070e28	−0.562570	−0.281285	−	0.959624i	$-0.590761\pi$
$7$	−1.15070e28	−0.562570	−0.281285	+	0.959624i	$0.590761\pi$
$8$	−1.36548e30	−0.761676
$9$	−3.14854e31	−0.339614
$10$	−4.22381e32	−0.133569
$11$	−1.09259e35	−1.41842	−0.709208	−	0.705000i	$-0.750947\pi$
$11$	−1.09259e35	−1.41842	−0.709208	+	0.705000i	$0.750947\pi$
$12$	9.53792e35	0.671246
$13$	1.54273e37	0.743338	0.371669	−	0.928365i	$-0.378786\pi$
$13$	1.54273e37	0.743338	0.371669	+	0.928365i	$0.378786\pi$
$14$	−5.83087e37	−0.234663
$15$	6.52218e38	0.260217
$16$	1.10696e40	0.508290
$17$	8.24456e40	0.496739	0.248370	−	0.968665i	$-0.420105\pi$
$17$	8.24456e40	0.496739	0.248370	+	0.968665i	$0.420105\pi$
$18$	−1.59545e41	−0.141662
$19$	−5.93328e42	−0.861091	−0.430546	−	0.902569i	$-0.641679\pi$
$19$	−5.93328e42	−0.861091	−0.430546	+	0.902569i	$0.641679\pi$
$20$	1.01607e43	0.264496
$21$	9.00371e43	0.457167
$22$	−5.53645e44	−0.591659
$23$	2.28489e45	0.550785	0.275393	−	0.961332i	$-0.411192\pi$
$23$	2.28489e45	0.550785	0.275393	+	0.961332i	$0.411192\pi$
$24$	1.06843e46	0.618969
$25$	−6.08146e46	−0.897465
$26$	7.81738e46	0.310066
$27$	9.71773e47	1.08863
$28$	1.40266e48	0.464686
$29$	1.07438e49	1.09856	0.549281	−	0.835638i	$-0.314902\pi$
$29$	1.07438e49	1.09856	0.549281	+	0.835638i	$0.314902\pi$
$30$	3.30496e48	0.108543
$31$	1.32534e50	1.45114	0.725572	−	0.688147i	$-0.241575\pi$
$31$	1.32534e50	1.45114	0.725572	+	0.688147i	$0.241575\pi$
$32$	2.57601e50	0.973697
$33$	8.54908e50	1.15266
$34$	4.17773e50	0.207203
$35$	9.59164e50	0.180141
$36$	3.83798e51	0.280523
$37$	−5.20056e52	−1.51807	−0.759037	−	0.651048i	$-0.774330\pi$
$37$	−5.20056e52	−1.51807	−0.759037	+	0.651048i	$0.774330\pi$
$38$	−3.00655e52	−0.359184
$39$	−1.20712e53	−0.604067
$40$	1.13819e53	0.243897
$41$	−1.63973e54	−1.53643	−0.768217	−	0.640190i	$-0.778856\pi$
$41$	−1.63973e54	−1.53643	−0.768217	+	0.640190i	$0.778856\pi$
$42$	4.56241e53	0.190697
$43$	5.64177e54	1.07207	0.536034	−	0.844197i	$-0.319922\pi$
$43$	5.64177e54	1.07207	0.536034	+	0.844197i	$0.319922\pi$
$44$	1.33184e55	1.17162
$45$	2.62447e54	0.108748
$46$	1.15781e55	0.229747
$47$	2.41027e55	0.232695	0.116347	−	0.993209i	$-0.462881\pi$
$47$	2.41027e55	0.232695	0.116347	+	0.993209i	$0.462881\pi$
$48$	−8.66148e55	−0.413057
$49$	−2.85968e56	−0.683515
$50$	−3.08163e56	−0.374356
$51$	−6.45102e56	−0.403671
$52$	−1.88053e57	−0.614001
$53$	2.84898e57	0.491414	0.245707	−	0.969344i	$-0.420980\pi$
$53$	2.84898e57	0.491414	0.245707	+	0.969344i	$0.420980\pi$
$54$	4.92422e57	0.454095
$55$	9.10731e57	0.454193
$56$	1.57125e58	0.428496
$57$	4.64254e58	0.699759
$58$	5.44415e58	0.458239
$59$	−3.30680e59	−1.56987	−0.784937	−	0.619575i	$-0.787305\pi$
$59$	−3.30680e59	−1.56987	−0.784937	+	0.619575i	$0.787305\pi$
$60$	−7.95034e58	−0.214941
$61$	−6.89383e59	−1.07130	−0.535650	−	0.844440i	$-0.679933\pi$
$61$	−6.89383e59	−1.07130	−0.535650	+	0.844440i	$0.679933\pi$
$62$	6.71583e59	0.605311
$63$	3.62302e59	0.191057
$64$	−3.28249e59	−0.102135
$65$	−1.28594e60	−0.238025
$66$	4.33204e60	0.480806
$67$	2.58683e61	1.73485	0.867424	−	0.497569i	$-0.165774\pi$
$67$	2.58683e61	1.73485	0.867424	+	0.497569i	$0.165774\pi$
$68$	−1.00499e61	−0.410309
$69$	−1.78783e61	−0.447591
$70$	4.86033e60	0.0751418
$71$	−1.34953e62	−1.29726	−0.648630	−	0.761104i	$-0.724658\pi$
$71$	−1.34953e62	−1.29726	−0.648630	+	0.761104i	$0.724658\pi$
$72$	4.29927e61	0.258676
$73$	−1.11285e62	−0.421814	−0.210907	−	0.977506i	$-0.567642\pi$
$73$	−1.11285e62	−0.421814	−0.210907	+	0.977506i	$0.567642\pi$
$74$	−2.63526e62	−0.633229
$75$	4.75848e62	0.729317
$76$	7.23248e62	0.711266
$77$	1.25724e63	0.797958
$78$	−6.11677e62	−0.251973
$79$	1.23916e63	0.333135	0.166568	−	0.986030i	$-0.446732\pi$
$79$	1.23916e63	0.333135	0.166568	+	0.986030i	$0.446732\pi$
$80$	−9.22705e62	−0.162760
$81$	−4.68471e63	−0.545048
$82$	−8.30894e63	−0.640887
$83$	3.41100e63	0.175294	0.0876468	−	0.996152i	$-0.472065\pi$
$83$	3.41100e63	0.175294	0.0876468	+	0.996152i	$0.472065\pi$
$84$	−1.09752e64	−0.377623
$85$	−6.87226e63	−0.159062
$86$	2.85883e64	0.447188
$87$	−8.40656e64	−0.892736
$88$	1.49191e65	1.08037
$89$	3.04246e65	1.50890	0.754448	−	0.656360i	$-0.227905\pi$
$89$	3.04246e65	1.50890	0.754448	+	0.656360i	$0.227905\pi$
$90$	1.32989e64	0.0453618
$91$	−1.77521e65	−0.418180
$92$	−2.78521e65	−0.454951
$93$	−1.03702e66	−1.17926
$94$	1.22134e65	0.0970633
$95$	4.94569e65	0.275731
$96$	−2.01562e66	−0.791266
$97$	2.29986e66	0.638041	0.319021	−	0.947748i	$-0.396646\pi$
$97$	2.29986e66	0.638041	0.319021	+	0.947748i	$0.396646\pi$
$98$	−1.44907e66	−0.285112
$99$	3.44008e66	0.481714
$100$	7.41311e66	0.741311
$101$	8.83103e66	0.632769	0.316385	−	0.948631i	$-0.397531\pi$
$101$	8.83103e66	0.632769	0.316385	+	0.948631i	$0.397531\pi$
$102$	−3.26890e66	−0.168382
$103$	−1.08466e67	−0.402947	−0.201473	−	0.979494i	$-0.564573\pi$
$103$	−1.08466e67	−0.402947	−0.201473	+	0.979494i	$0.564573\pi$
$104$	−2.10656e67	−0.566182
$105$	−7.50505e66	−0.146390
$106$	1.44365e67	0.204982
$107$	−2.87086e67	−0.297616	−0.148808	−	0.988866i	$-0.547544\pi$
$107$	−2.87086e67	−0.297616	−0.148808	+	0.988866i	$0.547544\pi$
$108$	−1.18456e68	−0.899211
$109$	5.52911e66	0.0308225	0.0154112	−	0.999881i	$-0.495094\pi$
$109$	5.52911e66	0.0308225	0.0154112	+	0.999881i	$0.495094\pi$
$110$	4.61491e67	0.189456
$111$	4.06922e68	1.23365
$112$	−1.27377e68	−0.285949
$113$	5.97000e68	0.995054	0.497527	−	0.867449i	$-0.334242\pi$
$113$	5.97000e68	0.995054	0.497527	+	0.867449i	$0.334242\pi$
$114$	2.35250e68	0.291888
$115$	−1.90457e68	−0.176368
$116$	−1.30963e69	−0.907418
$117$	−4.85734e68	−0.252448
$118$	−1.67564e69	−0.654837
$119$	−9.48698e68	−0.279451
$120$	−8.90589e68	−0.198201
$121$	6.00410e69	1.01190
$122$	−3.49328e69	−0.446868
$123$	1.28302e70	1.24857
$124$	−1.61555e70	−1.19865
$125$	1.07176e70	0.607590
$126$	1.83588e69	0.0796948
$127$	5.53532e70	1.84382	0.921909	−	0.387406i	$-0.126629\pi$
$127$	5.53532e70	1.84382	0.921909	+	0.387406i	$0.126629\pi$
$128$	−3.96786e70	−1.01630
$129$	−4.41445e70	−0.871206
$130$	−6.51619e69	−0.0992867
$131$	−3.42212e70	−0.403373	−0.201687	−	0.979450i	$-0.564642\pi$
$131$	−3.42212e70	−0.403373	−0.201687	+	0.979450i	$0.564642\pi$
$132$	−1.04211e71	−0.952106
$133$	6.82741e70	0.484424
$134$	1.31081e71	0.723652
$135$	−8.10022e70	−0.348590
$136$	−1.12578e71	−0.378354
$137$	4.32164e71	1.13634	0.568171	−	0.822910i	$-0.307651\pi$
$137$	4.32164e71	1.13634	0.568171	+	0.822910i	$0.307651\pi$
$138$	−9.05938e70	−0.186702
$139$	8.25204e71	1.33526	0.667629	−	0.744494i	$-0.267309\pi$
$139$	8.25204e71	1.33526	0.667629	+	0.744494i	$0.267309\pi$
$140$	−1.16919e71	−0.148798
$141$	−1.88593e71	−0.189098
$142$	−6.83840e71	−0.541122
$143$	−1.68557e72	−1.05436
$144$	−3.48530e71	−0.172622
$145$	−8.95549e71	−0.351772
$146$	−5.63910e71	−0.175950
$147$	2.23758e72	0.555453
$148$	6.33932e72	1.25394
$149$	−6.68758e72	−1.05567	−0.527836	−	0.849346i	$-0.676997\pi$
$149$	−6.68758e72	−1.05567	−0.527836	+	0.849346i	$0.676997\pi$
$150$	2.41125e72	0.304218
$151$	2.60779e72	0.263357	0.131679	−	0.991292i	$-0.457963\pi$
$151$	2.60779e72	0.263357	0.131679	+	0.991292i	$0.457963\pi$
$152$	8.10176e72	0.655872
$153$	−2.59584e72	−0.168700
$154$	6.37077e72	0.332849
$155$	−1.10474e73	−0.464673
$156$	1.47144e73	0.498963
$157$	−4.90233e73	−1.34204	−0.671018	−	0.741441i	$-0.734143\pi$
$157$	−4.90233e73	−1.34204	−0.671018	+	0.741441i	$0.734143\pi$
$158$	6.27915e72	0.138960
$159$	−2.22921e73	−0.399344
$160$	−2.14724e73	−0.311789
$161$	−2.62921e73	−0.309855
$162$	−2.37386e73	−0.227354
$163$	1.63101e74	1.27108	0.635538	−	0.772069i	$-0.280778\pi$
$163$	1.63101e74	1.27108	0.635538	+	0.772069i	$0.280778\pi$
$164$	1.99878e74	1.26910
$165$	−7.12609e73	−0.369096
$166$	1.72844e73	0.0731196
$167$	8.02398e73	0.277580	0.138790	−	0.990322i	$-0.455679\pi$
$167$	8.02398e73	0.277580	0.138790	+	0.990322i	$0.455679\pi$
$168$	−1.22944e74	−0.348213
$169$	−1.92729e74	−0.447449
$170$	−3.48235e73	−0.0663489
$171$	1.86812e74	0.292439
$172$	−6.87714e74	−0.885533
$173$	−4.19257e74	−0.444566	−0.222283	−	0.974982i	$-0.571351\pi$
$173$	−4.19257e74	−0.444566	−0.222283	+	0.974982i	$0.571351\pi$
$174$	−4.25982e74	−0.372384
$175$	6.99791e74	0.504887
$176$	−1.20945e75	−0.720966
$177$	2.58743e75	1.27574
$178$	1.54169e75	0.629401
$179$	2.46277e75	0.833389	0.416695	−	0.909046i	$-0.363188\pi$
$179$	2.46277e75	0.833389	0.416695	+	0.909046i	$0.363188\pi$
$180$	−3.19915e74	−0.0898267
$181$	−5.01874e75	−1.17048	−0.585238	−	0.810862i	$-0.698999\pi$
$181$	−5.01874e75	−1.17048	−0.585238	+	0.810862i	$0.698999\pi$
$182$	−8.99544e74	−0.174434
$183$	5.39413e75	0.870583
$184$	−3.11996e75	−0.419520
$185$	4.33493e75	0.486104
$186$	−5.25486e75	−0.491900
$187$	−9.00795e75	−0.704583
$188$	−2.93804e75	−0.192207
$189$	−1.11822e76	−0.612428
$190$	2.50611e75	0.115015
$191$	−1.28809e75	−0.0495825	−0.0247912	−	0.999693i	$-0.507892\pi$
$191$	−1.28809e75	−0.0495825	−0.0247912	+	0.999693i	$0.507892\pi$
$192$	2.56841e75	0.0829991
$193$	5.85943e76	1.59106	0.795528	−	0.605916i	$-0.207193\pi$
$193$	5.85943e76	1.59106	0.795528	+	0.605916i	$0.207193\pi$
$194$	1.16540e76	0.266144
$195$	1.00619e76	0.193429
$196$	3.48586e76	0.564587
$197$	−6.37062e76	−0.870087	−0.435044	−	0.900409i	$-0.643267\pi$
$197$	−6.37062e76	−0.870087	−0.435044	+	0.900409i	$0.643267\pi$
$198$	1.74317e76	0.200936
$199$	−9.92421e76	−0.966313	−0.483157	−	0.875534i	$-0.660510\pi$
$199$	−9.92421e76	−0.966313	−0.483157	+	0.875534i	$0.660510\pi$
$200$	8.30409e76	0.683577
$201$	−2.02408e77	−1.40981
$202$	4.47491e76	0.263945
$203$	−1.23628e77	−0.618018
$204$	7.86359e76	0.333434
$205$	1.36680e77	0.491984
$206$	−5.49626e76	−0.168080
$207$	−7.19407e76	−0.187054
$208$	1.70773e77	0.377831
$209$	6.48266e77	1.22139
$210$	−3.80300e76	−0.0610633
$211$	−2.29628e77	−0.314459	−0.157229	−	0.987562i	$-0.550256\pi$
$211$	−2.29628e77	−0.314459	−0.157229	+	0.987562i	$0.550256\pi$
$212$	−3.47282e77	−0.405911
$213$	1.05595e78	1.05421
$214$	−1.45474e77	−0.124144
$215$	−4.70270e77	−0.343288
$216$	−1.32693e78	−0.829180
$217$	−1.52506e78	−0.816369
$218$	2.80174e76	0.0128569
$219$	8.70759e77	0.342784
$220$	−1.11015e78	−0.375166
$221$	1.27191e78	0.369245
$222$	2.06198e78	0.514588
$223$	−3.32066e77	−0.0712871	−0.0356436	−	0.999365i	$-0.511348\pi$
$223$	−3.32066e77	−0.0712871	−0.0356436	+	0.999365i	$0.511348\pi$
$224$	−2.96421e78	−0.547773
$225$	1.91477e78	0.304792
$226$	3.02515e78	0.415064
$227$	1.36540e79	1.61582	0.807909	−	0.589307i	$-0.200599\pi$
$227$	1.36540e79	1.61582	0.807909	+	0.589307i	$0.200599\pi$
$228$	−5.65911e78	−0.578004
$229$	4.64291e78	0.409545	0.204773	−	0.978810i	$-0.434354\pi$
$229$	4.64291e78	0.409545	0.204773	+	0.978810i	$0.434354\pi$
$230$	−9.65094e77	−0.0735677
$231$	−9.83739e78	−0.648453
$232$	−1.46704e79	−0.836747
$233$	−2.73417e79	−1.35021	−0.675105	−	0.737721i	$-0.735902\pi$
$233$	−2.73417e79	−1.35021	−0.675105	+	0.737721i	$0.735902\pi$
$234$	−2.46134e78	−0.105303
$235$	−2.00908e78	−0.0745116
$236$	4.03088e79	1.29672
$237$	−9.69591e78	−0.270719
$238$	−4.80730e78	−0.116566
$239$	5.49826e79	1.15850	0.579249	−	0.815150i	$-0.303346\pi$
$239$	5.49826e79	1.15850	0.579249	+	0.815150i	$0.303346\pi$
$240$	7.21978e78	0.132266
$241$	−3.38224e79	−0.539056	−0.269528	−	0.962993i	$-0.586868\pi$
$241$	−3.38224e79	−0.539056	−0.269528	+	0.962993i	$0.586868\pi$
$242$	3.04243e79	0.422091
$243$	−5.34366e79	−0.645697
$244$	8.40336e79	0.884900
$245$	2.38368e79	0.218869
$246$	6.50140e79	0.520812
$247$	−9.15342e79	−0.640082
$248$	−1.80972e80	−1.10530
$249$	−2.66897e79	−0.142451
$250$	5.43086e79	0.253442
$251$	−8.54919e79	−0.349024	−0.174512	−	0.984655i	$-0.555835\pi$
$251$	−8.54919e79	−0.349024	−0.174512	+	0.984655i	$0.555835\pi$
$252$	−4.41635e79	−0.157814
$253$	−2.49645e80	−0.781242
$254$	2.80489e80	0.769106
$255$	5.37725e79	0.129260
$256$	−1.52620e80	−0.321791
$257$	7.14436e80	1.32191	0.660957	−	0.750423i	$-0.270150\pi$
$257$	7.14436e80	1.32191	0.660957	+	0.750423i	$0.270150\pi$
$258$	−2.23691e80	−0.363403
$259$	5.98427e80	0.854022
$260$	1.56752e80	0.196610
$261$	−3.38273e80	−0.373087
$262$	−1.73408e80	−0.168258
$263$	8.76506e80	0.748578	0.374289	−	0.927312i	$-0.377887\pi$
$263$	8.76506e80	0.748578	0.374289	+	0.927312i	$0.377887\pi$
$264$	−1.16736e81	−0.877955
$265$	−2.37477e80	−0.157357
$266$	3.45962e80	0.202066
$267$	−2.38059e81	−1.22619
$268$	−3.15326e81	−1.43299
$269$	4.76354e81	1.91086	0.955428	−	0.295223i	$-0.0953940\pi$
$269$	4.76354e81	1.91086	0.955428	+	0.295223i	$0.0953940\pi$
$270$	−4.10459e80	−0.145406
$271$	−3.02040e81	−0.945352	−0.472676	−	0.881236i	$-0.656712\pi$
$271$	−3.02040e81	−0.945352	−0.472676	+	0.881236i	$0.656712\pi$
$272$	9.12638e80	0.252488
$273$	1.38903e81	0.339830
$274$	2.18988e81	0.473999
$275$	6.64456e81	1.27298
$276$	2.17931e81	0.369712
$277$	−3.30476e81	−0.496670	−0.248335	−	0.968674i	$-0.579883\pi$
$277$	−3.30476e81	−0.496670	−0.248335	+	0.968674i	$0.579883\pi$
$278$	4.18152e81	0.556972
$279$	−4.17289e81	−0.492829
$280$	−1.30972e81	−0.137209
$281$	−9.67683e81	−0.899645	−0.449823	−	0.893118i	$-0.648513\pi$
$281$	−9.67683e81	−0.899645	−0.449823	+	0.893118i	$0.648513\pi$
$282$	−9.55650e80	−0.0788776
$283$	−1.18626e82	−0.869630	−0.434815	−	0.900520i	$-0.643186\pi$
$283$	−1.18626e82	−0.869630	−0.434815	+	0.900520i	$0.643186\pi$
$284$	1.64503e82	1.07154
$285$	−3.86979e81	−0.224071
$286$	−8.54122e81	−0.439802
$287$	1.88683e82	0.864351
$288$	−8.11069e81	−0.330681
$289$	−2.07499e82	−0.753250
$290$	−4.53798e81	−0.146733
$291$	−1.79954e82	−0.518499
$292$	1.35653e82	0.348421
$293$	7.50510e82	1.71906	0.859530	−	0.511085i	$-0.170756\pi$
$293$	7.50510e82	1.71906	0.859530	+	0.511085i	$0.170756\pi$
$294$	1.13384e82	0.231694
$295$	2.75638e82	0.502692
$296$	7.10125e82	1.15628
$297$	−1.06175e83	−1.54412
$298$	−3.38877e82	−0.440349
$299$	3.52495e82	0.409419
$300$	−5.80044e82	−0.602420
$301$	−6.49196e82	−0.603113
$302$	1.32144e82	0.109853
$303$	−6.90991e82	−0.514214
$304$	−6.56789e82	−0.437684
$305$	5.74636e82	0.343043
$306$	−1.31538e82	−0.0703691
$307$	1.38135e83	0.662475	0.331237	−	0.943547i	$-0.392534\pi$
$307$	1.38135e83	0.662475	0.331237	+	0.943547i	$0.392534\pi$
$308$	−1.53254e83	−0.659117
$309$	8.48703e82	0.327451
$310$	−5.59799e82	−0.193827
$311$	3.94064e83	1.22488	0.612441	−	0.790516i	$-0.290188\pi$
$311$	3.94064e83	1.22488	0.612441	+	0.790516i	$0.290188\pi$
$312$	1.64829e83	0.460103
$313$	2.68725e83	0.673864	0.336932	−	0.941529i	$-0.390611\pi$
$313$	2.68725e83	0.673864	0.336932	+	0.941529i	$0.390611\pi$
$314$	−2.48414e83	−0.559799
$315$	−3.01997e82	−0.0611785
$316$	−1.51050e83	−0.275171
$317$	2.06813e83	0.338917	0.169458	−	0.985537i	$-0.445798\pi$
$317$	2.06813e83	0.338917	0.169458	+	0.985537i	$0.445798\pi$
$318$	−1.12960e83	−0.166577
$319$	−1.17386e84	−1.55822
$320$	2.73612e82	0.0327048
$321$	2.24633e83	0.241855
$322$	−1.33229e83	−0.129249
$323$	−4.89173e83	−0.427738
$324$	5.71052e83	0.450213
$325$	−9.38202e83	−0.667120
$326$	8.26474e83	0.530200
$327$	−4.32629e82	−0.0250476
$328$	2.23902e84	1.17026
$329$	−2.77349e83	−0.130907
$330$	−3.61097e83	−0.153960
$331$	5.77944e83	0.222663	0.111332	−	0.993783i	$-0.464488\pi$
$331$	5.77944e83	0.222663	0.111332	+	0.993783i	$0.464488\pi$
$332$	−4.15791e83	−0.144793
$333$	1.63742e84	0.515559
$334$	4.06596e83	0.115786
$335$	−2.15625e84	−0.555518
$336$	9.96673e83	0.232374
$337$	3.23409e84	0.682576	0.341288	−	0.939959i	$-0.389137\pi$
$337$	3.23409e84	0.682576	0.341288	+	0.939959i	$0.389137\pi$
$338$	−9.76609e83	−0.186643
$339$	−4.67127e84	−0.808622
$340$	8.37707e83	0.131386
$341$	−1.44806e85	−2.05832
$342$	9.46624e83	0.121984
$343$	8.10488e84	0.947095
$344$	−7.70371e84	−0.816568
$345$	1.49024e84	0.143324
$346$	−2.12448e84	−0.185440
$347$	7.38268e84	0.585028	0.292514	−	0.956261i	$-0.405508\pi$
$347$	7.38268e84	0.585028	0.292514	+	0.956261i	$0.405508\pi$
$348$	1.02473e85	0.737405
$349$	2.25397e85	1.47331	0.736657	−	0.676266i	$-0.236403\pi$
$349$	2.25397e85	1.47331	0.736657	+	0.676266i	$0.236403\pi$
$350$	3.54602e84	0.210602
$351$	1.49918e85	0.809217
$352$	−2.81453e85	−1.38111
$353$	−2.28605e85	−1.02008	−0.510038	−	0.860152i	$-0.670369\pi$
$353$	−2.28605e85	−1.02008	−0.510038	+	0.860152i	$0.670369\pi$
$354$	1.31112e85	0.532147
$355$	1.12490e85	0.415397
$356$	−3.70866e85	−1.24636
$357$	7.42317e84	0.227093
$358$	1.24795e85	0.347629
$359$	−7.82216e84	−0.198455	−0.0992276	−	0.995065i	$-0.531637\pi$
$359$	−7.82216e84	−0.198455	−0.0992276	+	0.995065i	$0.531637\pi$
$360$	−3.58366e84	−0.0828309
$361$	−1.22740e85	−0.258521
$362$	−2.54312e85	−0.488237
$363$	−4.69796e85	−0.822313
$364$	2.16392e85	0.345418
$365$	9.27618e84	0.135070
$366$	2.73335e85	0.363143
$367$	6.12079e85	0.742155	0.371077	−	0.928602i	$-0.378988\pi$
$367$	6.12079e85	0.742155	0.371077	+	0.928602i	$0.378988\pi$
$368$	2.52927e85	0.279959
$369$	5.16277e85	0.521794
$370$	2.19662e85	0.202767
$371$	−3.27831e85	−0.276455
$372$	1.26410e86	0.974074
$373$	−1.28610e86	−0.905791	−0.452895	−	0.891564i	$-0.649609\pi$
$373$	−1.28610e86	−0.905791	−0.452895	+	0.891564i	$0.649609\pi$
$374$	−4.56456e85	−0.293900
$375$	−8.38604e85	−0.493753
$376$	−3.29117e85	−0.177238
$377$	1.65747e86	0.816602
$378$	−5.66628e85	−0.255460
$379$	5.21083e85	0.215027	0.107514	−	0.994204i	$-0.465711\pi$
$379$	5.21083e85	0.215027	0.107514	+	0.994204i	$0.465711\pi$
$380$	−6.02864e85	−0.227756
$381$	−4.33115e86	−1.49836
$382$	−6.52708e84	−0.0206822
$383$	−1.33278e86	−0.386901	−0.193451	−	0.981110i	$-0.561968\pi$
$383$	−1.33278e86	−0.386901	−0.193451	+	0.981110i	$0.561968\pi$
$384$	3.10468e86	0.825888
$385$	−1.04798e86	−0.255515
$386$	2.96912e86	0.663672
$387$	−1.77634e86	−0.364089
$388$	−2.80346e86	−0.527025
$389$	9.94670e86	1.71541	0.857704	−	0.514144i	$-0.171890\pi$
$389$	9.94670e86	1.71541	0.857704	+	0.514144i	$0.171890\pi$
$390$	5.09864e85	0.0806845
$391$	1.88379e86	0.273597
$392$	3.90482e86	0.520617
$393$	2.67766e86	0.327798
$394$	−3.22815e86	−0.362937
$395$	−1.03290e86	−0.106674
$396$	−4.19335e86	−0.397898
$397$	−5.37440e86	−0.468649	−0.234325	−	0.972158i	$-0.575288\pi$
$397$	−5.37440e86	−0.468649	−0.234325	+	0.972158i	$0.575288\pi$
$398$	−5.02885e86	−0.403075
$399$	−5.34216e86	−0.393663
$400$	−6.73192e86	−0.456172
$401$	−4.13165e86	−0.257506	−0.128753	−	0.991677i	$-0.541097\pi$
$401$	−4.13165e86	−0.257506	−0.128753	+	0.991677i	$0.541097\pi$
$402$	−1.02565e87	−0.588069
$403$	2.04463e87	1.07869
$404$	−1.07647e87	−0.522671
$405$	3.90495e86	0.174531
$406$	−6.26456e86	−0.257792
$407$	5.68210e87	2.15326
$408$	8.80872e86	0.307466
$409$	−2.75139e87	−0.884753	−0.442376	−	0.896830i	$-0.645864\pi$
$409$	−2.75139e87	−0.884753	−0.442376	+	0.896830i	$0.645864\pi$
$410$	6.92592e86	0.205219
$411$	−3.38150e87	−0.923439
$412$	1.32217e87	0.332836
$413$	3.80512e87	0.883164
$414$	−3.64542e86	−0.0780254
$415$	−2.84324e86	−0.0561310
$416$	3.97408e87	0.723786
$417$	−6.45687e87	−1.08509
$418$	3.28493e87	0.509472
$419$	−1.09982e87	−0.157454	−0.0787268	−	0.996896i	$-0.525085\pi$
$419$	−1.09982e87	−0.157454	−0.0787268	+	0.996896i	$0.525085\pi$
$420$	9.14842e86	0.120919
$421$	−1.24851e88	−1.52385	−0.761923	−	0.647668i	$-0.775744\pi$
$421$	−1.24851e88	−1.52385	−0.761923	+	0.647668i	$0.775744\pi$
$422$	−1.16359e87	−0.131169
$423$	−7.58883e86	−0.0790265
$424$	−3.89022e87	−0.374298
$425$	−5.01389e87	−0.445806
$426$	5.35076e87	0.439738
$427$	7.93271e87	0.602681
$428$	3.49949e87	0.245832
$429$	1.31889e88	0.856818
$430$	−2.38298e87	−0.143195
$431$	−2.68649e87	−0.149347	−0.0746736	−	0.997208i	$-0.523791\pi$
$431$	−2.68649e87	−0.149347	−0.0746736	+	0.997208i	$0.523791\pi$
$432$	1.07571e88	0.553338
$433$	6.74157e87	0.320934	0.160467	−	0.987041i	$-0.448700\pi$
$433$	6.74157e87	0.320934	0.160467	+	0.987041i	$0.448700\pi$
$434$	−7.72789e87	−0.340530
$435$	7.00729e87	0.285864
$436$	−6.73981e86	−0.0254595
$437$	−1.35569e88	−0.474276
$438$	4.41236e87	0.142984
$439$	1.55673e88	0.467360	0.233680	−	0.972314i	$-0.424923\pi$
$439$	1.55673e88	0.467360	0.233680	+	0.972314i	$0.424923\pi$
$440$	−1.24358e88	−0.345948
$441$	9.00382e87	0.232131
$442$	6.44509e87	0.154022
$443$	−7.64486e88	−1.69373	−0.846866	−	0.531806i	$-0.821513\pi$
$443$	−7.64486e88	−1.69373	−0.846866	+	0.531806i	$0.821513\pi$
$444$	−4.96025e88	−1.01900
$445$	−2.53604e88	−0.483166
$446$	−1.68266e87	−0.0297358
$447$	5.23275e88	0.857883
$448$	3.77715e87	0.0574581
$449$	3.05563e88	0.431368	0.215684	−	0.976463i	$-0.430802\pi$
$449$	3.05563e88	0.431368	0.215684	+	0.976463i	$0.430802\pi$
$450$	9.70265e87	0.127137
$451$	1.79156e89	2.17930
$452$	−7.27724e88	−0.821920
$453$	−2.04049e88	−0.214015
$454$	6.91882e88	0.674001
$455$	1.47973e88	0.133906
$456$	−6.33929e88	−0.532989
$457$	−4.84706e88	−0.378692	−0.189346	−	0.981910i	$-0.560637\pi$
$457$	−4.84706e88	−0.378692	−0.189346	+	0.981910i	$0.560637\pi$
$458$	2.35268e88	0.170832
$459$	8.01184e88	0.540763
$460$	2.32161e88	0.145681
$461$	−2.17512e89	−1.26912	−0.634560	−	0.772874i	$-0.718818\pi$
$461$	−2.17512e89	−1.26912	−0.634560	+	0.772874i	$0.718818\pi$
$462$	−4.98486e88	−0.270487
$463$	−7.57996e88	−0.382564	−0.191282	−	0.981535i	$-0.561264\pi$
$463$	−7.57996e88	−0.382564	−0.191282	+	0.981535i	$0.561264\pi$
$464$	1.18929e89	0.558388
$465$	8.64410e88	0.377612
$466$	−1.38547e89	−0.563209
$467$	−1.00599e89	−0.380607	−0.190304	−	0.981725i	$-0.560947\pi$
$467$	−1.00599e89	−0.380607	−0.190304	+	0.981725i	$0.560947\pi$
$468$	5.92094e88	0.208523
$469$	−2.97665e89	−0.975973
$470$	−1.01805e88	−0.0310808
$471$	3.83587e89	1.09059
$472$	4.51536e89	1.19574
$473$	−6.16416e89	−1.52064
$474$	−4.91317e88	−0.112924
$475$	3.60830e89	0.772799
$476$	1.15643e89	0.230828
$477$	−8.97014e88	−0.166891
$478$	2.78611e89	0.483241
$479$	9.96571e89	1.61164	0.805821	−	0.592159i	$-0.201724\pi$
$479$	9.96571e89	1.61164	0.805821	+	0.592159i	$0.201724\pi$
$480$	1.68012e89	0.253373
$481$	−8.02304e89	−1.12844
$482$	−1.71387e89	−0.224855
$483$	2.05725e89	0.251801
$484$	−7.31881e89	−0.835836
$485$	−1.91705e89	−0.204308
$486$	−2.70777e89	−0.269337
$487$	−1.58587e90	−1.47247	−0.736236	−	0.676725i	$-0.763399\pi$
$487$	−1.58587e90	−1.47247	−0.736236	+	0.676725i	$0.763399\pi$
$488$	9.41337e89	0.815983
$489$	−1.27620e90	−1.03293
$490$	1.20787e89	0.0912963
$491$	2.42050e90	1.70874	0.854369	−	0.519668i	$-0.173944\pi$
$491$	2.42050e90	1.70874	0.854369	+	0.519668i	$0.173944\pi$
$492$	−1.56396e90	−1.03132
$493$	8.85778e89	0.545699
$494$	−4.63827e89	−0.266995
$495$	−2.86748e89	−0.154250
$496$	1.46709e90	0.737602
$497$	1.55290e90	0.729799
$498$	−1.35243e89	−0.0594200
$499$	1.18042e90	0.484918	0.242459	−	0.970162i	$-0.422046\pi$
$499$	1.18042e90	0.484918	0.242459	+	0.970162i	$0.422046\pi$
$500$	−1.30644e90	−0.501872
$501$	−6.27843e89	−0.225573
$502$	−4.33210e89	−0.145587
$503$	5.26757e90	1.65609	0.828043	−	0.560665i	$-0.189455\pi$
$503$	5.26757e90	1.65609	0.828043	+	0.560665i	$0.189455\pi$
$504$	−4.94715e89	−0.145523
$505$	−7.36111e89	−0.202620
$506$	−1.26502e90	−0.325877
$507$	1.50803e90	0.363615
$508$	−6.74738e90	−1.52300
$509$	−7.88037e90	−1.66533	−0.832666	−	0.553776i	$-0.813186\pi$
$509$	−7.88037e90	−1.66533	−0.832666	+	0.553776i	$0.813186\pi$
$510$	2.72479e89	0.0539178
$511$	1.28055e90	0.237300
$512$	5.08216e90	0.882073
$513$	−5.76580e90	−0.937406
$514$	3.62023e90	0.551406
$515$	9.04121e89	0.129028
$516$	5.38107e90	0.719621
$517$	−2.63344e90	−0.330058
$518$	3.03238e90	0.356235
$519$	3.28051e90	0.361273
$520$	1.75592e90	0.181298
$521$	−1.48169e90	−0.143448	−0.0717239	−	0.997425i	$-0.522850\pi$
$521$	−1.48169e90	−0.143448	−0.0717239	+	0.997425i	$0.522850\pi$
$522$	−1.71411e90	−0.155625
$523$	−1.36105e90	−0.115896	−0.0579481	−	0.998320i	$-0.518456\pi$
$523$	−1.36105e90	−0.115896	−0.0579481	+	0.998320i	$0.518456\pi$
$524$	4.17146e90	0.333188
$525$	−5.47557e90	−0.410292
$526$	4.44148e90	0.312252
$527$	1.09268e91	0.720840
$528$	9.46347e90	0.585887
$529$	−1.19887e91	−0.696636
$530$	−1.20336e90	−0.0656376
$531$	1.04116e91	0.533151
$532$	−8.32239e90	−0.400137
$533$	−2.52966e91	−1.14209
$534$	−1.20631e91	−0.511477
$535$	2.39301e90	0.0953000
$536$	−3.53226e91	−1.32139
$537$	−1.92702e91	−0.677247
$538$	2.41381e91	0.797069
$539$	3.12446e91	0.969508
$540$	9.87391e90	0.287938
$541$	−2.07523e90	−0.0568797	−0.0284399	−	0.999596i	$-0.509054\pi$
$541$	−2.07523e90	−0.0568797	−0.0284399	+	0.999596i	$0.509054\pi$
$542$	−1.53052e91	−0.394332
$543$	3.92695e91	0.951177
$544$	2.12381e91	0.483674
$545$	−4.60879e89	−0.00986971
$546$	7.03855e90	0.141752
$547$	5.89842e91	1.11728	0.558638	−	0.829412i	$-0.311324\pi$
$547$	5.89842e91	1.11728	0.558638	+	0.829412i	$0.311324\pi$
$548$	−5.26794e91	−0.938625
$549$	2.17055e91	0.363829
$550$	3.36697e91	0.530993
$551$	−6.37459e91	−0.945962
$552$	2.44124e91	0.340919
$553$	−1.42590e91	−0.187412
$554$	−1.67461e91	−0.207174
$555$	−3.39190e91	−0.395029
$556$	−1.00590e92	−1.10293
$557$	−5.36319e91	−0.553700	−0.276850	−	0.960913i	$-0.589290\pi$
$557$	−5.36319e91	−0.553700	−0.276850	+	0.960913i	$0.589290\pi$
$558$	−2.11451e91	−0.205572
$559$	8.70370e91	0.796908
$560$	1.06175e91	0.0915640
$561$	7.04834e91	0.572573
$562$	−4.90350e91	−0.375266
$563$	−2.64660e91	−0.190835	−0.0954174	−	0.995437i	$-0.530419\pi$
$563$	−2.64660e91	−0.190835	−0.0954174	+	0.995437i	$0.530419\pi$
$564$	2.29889e91	0.156196
$565$	−4.97630e91	−0.318628
$566$	−6.01108e91	−0.362746
$567$	5.39068e91	0.306628
$568$	1.84275e92	0.988091
$569$	1.56640e92	0.791848	0.395924	−	0.918283i	$-0.370424\pi$
$569$	1.56640e92	0.791848	0.395924	+	0.918283i	$0.370424\pi$
$570$	−1.96092e91	−0.0934659
$571$	−1.15568e92	−0.519433	−0.259717	−	0.965685i	$-0.583629\pi$
$571$	−1.15568e92	−0.519433	−0.259717	+	0.965685i	$0.583629\pi$
$572$	2.05466e92	0.870908
$573$	1.00788e91	0.0402928
$574$	9.56107e91	0.360544
$575$	−1.38954e92	−0.494310
$576$	1.03351e91	0.0346865
$577$	−2.87114e90	−0.00909210	−0.00454605	−	0.999990i	$-0.501447\pi$
$577$	−2.87114e90	−0.00909210	−0.00454605	+	0.999990i	$0.501447\pi$
$578$	−1.05145e92	−0.314201
$579$	−4.58476e92	−1.29296
$580$	1.09165e92	0.290565
$581$	−3.92503e91	−0.0986149
$582$	−9.11875e91	−0.216280
$583$	−3.11278e92	−0.697030
$584$	1.51957e92	0.321286
$585$	4.04884e91	0.0808367
$586$	3.80303e92	0.717066
$587$	2.33816e92	0.416387	0.208194	−	0.978088i	$-0.433242\pi$
$587$	2.33816e92	0.416387	0.208194	+	0.978088i	$0.433242\pi$
$588$	−2.72753e92	−0.458807
$589$	−7.86361e92	−1.24957
$590$	1.39673e92	0.209686
$591$	4.98474e92	0.707069
$592$	−5.75680e92	−0.771621
$593$	1.19409e93	1.51253	0.756267	−	0.654263i	$-0.227021\pi$
$593$	1.19409e93	1.51253	0.756267	+	0.654263i	$0.227021\pi$
$594$	−5.38017e92	−0.644095
$595$	7.90788e91	0.0894833
$596$	8.15196e92	0.871991
$597$	7.76528e92	0.785266
$598$	1.78618e92	0.170780
$599$	1.02253e93	0.924434	0.462217	−	0.886767i	$-0.347054\pi$
$599$	1.02253e93	0.924434	0.462217	+	0.886767i	$0.347054\pi$
$600$	−6.49760e92	−0.555503
$601$	−3.89386e92	−0.314837	−0.157418	−	0.987532i	$-0.550317\pi$
$601$	−3.89386e92	−0.314837	−0.157418	+	0.987532i	$0.550317\pi$
$602$	−3.28964e92	−0.251574
$603$	−8.14474e92	−0.589179
$604$	−3.17882e92	−0.217535
$605$	−5.00472e92	−0.324022
$606$	−3.50143e92	−0.214493
$607$	−2.57597e93	−1.49321	−0.746603	−	0.665270i	$-0.768317\pi$
$607$	−2.57597e93	−1.49321	−0.746603	+	0.665270i	$0.768317\pi$
$608$	−1.52842e93	−0.838442
$609$	9.67340e92	0.502227
$610$	2.91183e92	0.143092
$611$	3.71838e92	0.172971
$612$	3.16424e92	0.139347
$613$	4.23652e93	1.76638	0.883189	−	0.469017i	$-0.155392\pi$
$613$	4.23652e93	1.76638	0.883189	+	0.469017i	$0.155392\pi$
$614$	6.99967e92	0.276336
$615$	−1.06946e93	−0.399806
$616$	−1.71674e93	−0.607785
$617$	8.17102e92	0.273983	0.136991	−	0.990572i	$-0.456257\pi$
$617$	8.17102e92	0.273983	0.136991	+	0.990572i	$0.456257\pi$
$618$	4.30060e92	0.136589
$619$	−2.52814e92	−0.0760613	−0.0380306	−	0.999277i	$-0.512108\pi$
$619$	−2.52814e92	−0.0760613	−0.0380306	+	0.999277i	$0.512108\pi$
$620$	1.34664e93	0.383822
$621$	2.22039e93	0.599599
$622$	1.99683e93	0.510931
$623$	−3.50095e93	−0.848859
$624$	−1.33623e93	−0.307041
$625$	3.22759e93	0.702907
$626$	1.36170e93	0.281087
$627$	−5.07241e93	−0.992548
$628$	5.97579e93	1.10853
$629$	−4.28764e93	−0.754087
$630$	−1.53030e92	−0.0255192
$631$	−7.14064e93	−1.12915	−0.564577	−	0.825380i	$-0.690961\pi$
$631$	−7.14064e93	−1.12915	−0.564577	+	0.825380i	$0.690961\pi$
$632$	−1.69205e93	−0.253741
$633$	1.79675e93	0.255542
$634$	1.04798e93	0.141371
$635$	−4.61397e93	−0.590412
$636$	2.71733e93	0.329860
$637$	−4.41169e93	−0.508083
$638$	−5.94824e93	−0.649974
$639$	4.24905e93	0.440568
$640$	3.30741e93	0.325431
$641$	1.19515e94	1.11603	0.558017	−	0.829830i	$-0.311563\pi$
$641$	1.19515e94	1.11603	0.558017	+	0.829830i	$0.311563\pi$
$642$	1.13827e93	0.100884
$643$	−3.95521e93	−0.332738	−0.166369	−	0.986064i	$-0.553204\pi$
$643$	−3.95521e93	−0.332738	−0.166369	+	0.986064i	$0.553204\pi$
$644$	3.20493e93	0.255942
$645$	3.67966e93	0.278970
$646$	−2.47876e93	−0.178421
$647$	−9.68870e93	−0.662174	−0.331087	−	0.943600i	$-0.607415\pi$
$647$	−9.68870e93	−0.662174	−0.331087	+	0.943600i	$0.607415\pi$
$648$	6.39687e93	0.415150
$649$	3.61298e94	2.22673
$650$	−4.75411e93	−0.278273
$651$	1.19330e94	0.663416
$652$	−1.98815e94	−1.04992
$653$	−2.58186e94	−1.29522	−0.647608	−	0.761974i	$-0.724230\pi$
$653$	−2.58186e94	−1.29522	−0.647608	+	0.761974i	$0.724230\pi$
$654$	−2.19224e92	−0.0104480
$655$	2.85251e93	0.129165
$656$	−1.81511e94	−0.780954
$657$	3.50386e93	0.143254
$658$	−1.40540e93	−0.0546049
$659$	−5.02240e92	−0.0185460	−0.00927300	−	0.999957i	$-0.502952\pi$
$659$	−5.02240e92	−0.0185460	−0.00927300	+	0.999957i	$0.502952\pi$
$660$	8.68648e93	0.304875
$661$	−2.75448e94	−0.918948	−0.459474	−	0.888191i	$-0.651962\pi$
$661$	−2.75448e94	−0.918948	−0.459474	+	0.888191i	$0.651962\pi$
$662$	2.92859e93	0.0928787
$663$	−9.95215e93	−0.300064
$664$	−4.65765e93	−0.133517
$665$	−5.69099e93	−0.155118
$666$	8.29723e93	0.215053
$667$	2.45483e94	0.605071
$668$	−9.78098e93	−0.229282
$669$	2.59827e93	0.0579309
$670$	−1.09263e94	−0.231722
$671$	7.53215e94	1.51955
$672$	2.31937e94	0.445143
$673$	7.54513e94	1.37772	0.688862	−	0.724893i	$-0.258111\pi$
$673$	7.54513e94	1.37772	0.688862	+	0.724893i	$0.258111\pi$
$674$	1.63880e94	0.284721
$675$	−5.90979e94	−0.977003
$676$	2.34931e94	0.369595
$677$	−4.57696e94	−0.685262	−0.342631	−	0.939470i	$-0.611318\pi$
$677$	−4.57696e94	−0.685262	−0.342631	+	0.939470i	$0.611318\pi$
$678$	−2.36705e94	−0.337298
$679$	−2.64644e94	−0.358943
$680$	9.38391e93	0.121153
$681$	−1.06837e95	−1.31308
$682$	−7.33767e94	−0.858582
$683$	9.77094e94	1.08854	0.544268	−	0.838911i	$-0.316807\pi$
$683$	9.77094e94	1.08854	0.544268	+	0.838911i	$0.316807\pi$
$684$	−2.27718e94	−0.241556
$685$	−3.60230e94	−0.363870
$686$	4.10695e94	0.395059
$687$	−3.63288e94	−0.332813
$688$	6.24520e94	0.544921
$689$	4.39519e94	0.365287
$690$	7.55145e93	0.0597841
$691$	−1.14117e95	−0.860669	−0.430334	−	0.902670i	$-0.641604\pi$
$691$	−1.14117e95	−0.860669	−0.430334	+	0.902670i	$0.641604\pi$
$692$	5.11061e94	0.367214
$693$	−3.95848e94	−0.270998
$694$	3.74099e94	0.244031
$695$	−6.87849e94	−0.427565
$696$	1.14790e95	0.679976
$697$	−1.35189e95	−0.763207
$698$	1.14214e95	0.614559
$699$	2.13937e95	1.09724
$700$	−8.53023e94	−0.417039
$701$	−7.36200e92	−0.00343117	−0.00171558	−	0.999999i	$-0.500546\pi$
$701$	−7.36200e92	−0.00343117	−0.00171558	+	0.999999i	$0.500546\pi$
$702$	7.59672e94	0.337546
$703$	3.08564e95	1.30720
$704$	3.58643e94	0.144870
$705$	1.57202e94	0.0605512
$706$	−1.15840e95	−0.425501
$707$	−1.01618e95	−0.355977
$708$	−3.15399e95	−1.05377
$709$	−2.42896e95	−0.774053	−0.387027	−	0.922069i	$-0.626498\pi$
$709$	−2.42896e95	−0.774053	−0.387027	+	0.922069i	$0.626498\pi$
$710$	5.70015e94	0.173273
$711$	−3.90155e94	−0.113137
$712$	−4.15441e95	−1.14929
$713$	3.02825e95	0.799268
$714$	3.76151e94	0.0947266
$715$	1.40501e95	0.337619
$716$	−3.00204e95	−0.688384
$717$	−4.30216e95	−0.941444
$718$	−3.96369e94	−0.0827810
$719$	6.97731e95	1.39082	0.695409	−	0.718614i	$-0.255223\pi$
$719$	6.97731e95	1.39082	0.695409	+	0.718614i	$0.255223\pi$
$720$	2.90518e94	0.0552757
$721$	1.24812e95	0.226686
$722$	−6.21957e94	−0.107836
$723$	2.64646e95	0.438059
$724$	6.11769e95	0.966819
$725$	−6.53379e95	−0.985920
$726$	−2.38058e95	−0.343009
$727$	5.00262e95	0.688327	0.344164	−	0.938910i	$-0.388162\pi$
$727$	5.00262e95	0.688327	0.344164	+	0.938910i	$0.388162\pi$
$728$	2.42401e95	0.318517
$729$	8.52436e95	1.06977
$730$	4.70048e94	0.0563412
$731$	4.65139e95	0.532538
$732$	−6.57528e95	−0.719106
$733$	−8.92678e95	−0.932634	−0.466317	−	0.884618i	$-0.654419\pi$
$733$	−8.92678e95	−0.932634	−0.466317	+	0.884618i	$0.654419\pi$
$734$	3.10156e95	0.309573
$735$	−1.86513e95	−0.177862
$736$	5.88590e95	0.536298
$737$	−2.82635e96	−2.46074
$738$	2.61611e95	0.217654
$739$	3.28768e95	0.261397	0.130699	−	0.991422i	$-0.458278\pi$
$739$	3.28768e95	0.261397	0.130699	+	0.991422i	$0.458278\pi$
$740$	−5.28415e95	−0.401525
$741$	7.16217e95	0.520157
$742$	−1.66120e95	−0.115317
$743$	−1.93595e96	−1.28460	−0.642301	−	0.766453i	$-0.722020\pi$
$743$	−1.93595e96	−1.28460	−0.642301	+	0.766453i	$0.722020\pi$
$744$	1.41603e96	0.898213
$745$	5.57444e95	0.338038
$746$	−6.51698e95	−0.377830
$747$	−1.07397e95	−0.0595322
$748$	1.09804e96	0.581989
$749$	3.30349e95	0.167430
$750$	−4.24942e95	−0.205957
$751$	2.33056e96	1.08025	0.540123	−	0.841586i	$-0.318378\pi$
$751$	2.33056e96	1.08025	0.540123	+	0.841586i	$0.318378\pi$
$752$	2.66806e95	0.118277
$753$	6.68938e95	0.283631
$754$	8.39883e95	0.340627
$755$	−2.17373e95	−0.0843301
$756$	1.36307e96	0.505869
$757$	−3.90449e96	−1.38628	−0.693141	−	0.720802i	$-0.743774\pi$
$757$	−3.90449e96	−1.38628	−0.693141	+	0.720802i	$0.743774\pi$
$758$	2.64046e95	0.0896936
$759$	1.95337e96	0.634869
$760$	−6.75323e95	−0.210018
$761$	4.58353e95	0.136400	0.0681999	−	0.997672i	$-0.478274\pi$
$761$	4.58353e95	0.136400	0.0681999	+	0.997672i	$0.478274\pi$
$762$	−2.19471e96	−0.625007
$763$	−6.36232e94	−0.0173398
$764$	1.57014e95	0.0409554
$765$	2.16376e95	0.0540196
$766$	−6.75355e95	−0.161387
$767$	−5.10148e96	−1.16695
$768$	1.19419e96	0.261501
$769$	6.37251e96	1.33591	0.667956	−	0.744201i	$-0.267169\pi$
$769$	6.37251e96	1.33591	0.667956	+	0.744201i	$0.267169\pi$
$770$	−5.31036e95	−0.106582
$771$	−5.59016e96	−1.07424
$772$	−7.14246e96	−1.31422
$773$	1.01535e97	1.78897	0.894486	−	0.447096i	$-0.147542\pi$
$773$	1.01535e97	1.78897	0.894486	+	0.447096i	$0.147542\pi$
$774$	−9.00115e95	−0.151871
$775$	−8.05999e96	−1.30235
$776$	−3.14041e96	−0.485980
$777$	−4.68244e96	−0.694014
$778$	5.04025e96	0.715542
$779$	9.72899e96	1.32301
$780$	−1.22652e96	−0.159774
$781$	1.47448e97	1.84005
$782$	9.54564e95	0.114124
$783$	1.04405e97	1.19592
$784$	−3.16554e96	−0.347424
$785$	4.08634e96	0.429735
$786$	1.35684e96	0.136733
$787$	−1.79495e97	−1.73340	−0.866699	−	0.498832i	$-0.833762\pi$
$787$	−1.79495e97	−1.73340	−0.866699	+	0.498832i	$0.833762\pi$
$788$	7.76558e96	0.718697
$789$	−6.85829e96	−0.608326
$790$	−5.23399e95	−0.0444964
$791$	−6.86966e96	−0.559787
$792$	−4.69735e96	−0.366910
$793$	−1.06353e97	−0.796338
$794$	−2.72334e96	−0.195486
$795$	1.85816e96	0.127874
$796$	1.20973e97	0.798180
$797$	−2.24378e96	−0.141947	−0.0709737	−	0.997478i	$-0.522611\pi$
$797$	−2.24378e96	−0.141947	−0.0709737	+	0.997478i	$0.522611\pi$
$798$	−2.70701e96	−0.164207
$799$	1.98716e96	0.115589
$800$	−1.56659e97	−0.873859
$801$	−9.57931e96	−0.512442
$802$	−2.09361e96	−0.107413
$803$	1.21589e97	0.598308
$804$	2.46729e97	1.16451
$805$	2.19158e96	0.0992191
$806$	1.03607e97	0.449950
$807$	−3.72727e97	−1.55284
$808$	−1.20586e97	−0.481965
$809$	4.98406e97	1.91121	0.955603	−	0.294658i	$-0.0952057\pi$
$809$	4.98406e97	1.91121	0.955603	+	0.294658i	$0.0952057\pi$
$810$	1.97874e96	0.0728014
$811$	−6.56249e96	−0.231670	−0.115835	−	0.993268i	$-0.536954\pi$
$811$	−6.56249e96	−0.231670	−0.115835	+	0.993268i	$0.536954\pi$
$812$	1.50699e97	0.510486
$813$	2.36334e97	0.768232
$814$	2.87926e97	0.898181
$815$	−1.35953e97	−0.407013
$816$	−7.14101e96	−0.205182
$817$	−3.34742e97	−0.923148
$818$	−1.39420e97	−0.369054
$819$	5.58932e96	0.142020
$820$	−1.66609e97	−0.406381
$821$	−1.14583e96	−0.0268302	−0.0134151	−	0.999910i	$-0.504270\pi$
$821$	−1.14583e96	−0.0268302	−0.0134151	+	0.999910i	$0.504270\pi$
$822$	−1.71349e97	−0.385191
$823$	−4.59764e95	−0.00992295	−0.00496147	−	0.999988i	$-0.501579\pi$
$823$	−4.59764e95	−0.00992295	−0.00496147	+	0.999988i	$0.501579\pi$
$824$	1.48108e97	0.306915
$825$	−5.19908e97	−1.03447
$826$	1.92815e97	0.368391
$827$	−7.95450e97	−1.45941	−0.729707	−	0.683760i	$-0.760343\pi$
$827$	−7.95450e97	−1.45941	−0.729707	+	0.683760i	$0.760343\pi$
$828$	8.76934e96	0.154508
$829$	9.08914e97	1.53796	0.768982	−	0.639271i	$-0.220764\pi$
$829$	9.08914e97	1.53796	0.768982	+	0.639271i	$0.220764\pi$
$830$	−1.44074e96	−0.0234137
$831$	2.58584e97	0.403614
$832$	−5.06398e96	−0.0759208
$833$	−2.35768e97	−0.339529
$834$	−3.27186e97	−0.452619
$835$	−6.68839e96	−0.0888843
$836$	−7.90216e97	−1.00887
$837$	1.28793e98	1.57975
$838$	−5.57308e96	−0.0656781
$839$	8.73266e97	0.988828	0.494414	−	0.869227i	$-0.335383\pi$
$839$	8.73266e97	0.988828	0.494414	+	0.869227i	$0.335383\pi$
$840$	1.02480e97	0.111502
$841$	1.97831e97	0.206837
$842$	−6.32651e97	−0.635636
$843$	7.57171e97	0.731089
$844$	2.79910e97	0.259744
$845$	1.60650e97	0.143278
$846$	−3.84546e96	−0.0329641
$847$	−6.90890e97	−0.569265
$848$	3.15370e97	0.249781
$849$	9.28198e97	0.706697
$850$	−2.54067e97	−0.185958
$851$	−1.18827e98	−0.836132
$852$	−1.28717e98	−0.870780
$853$	−5.65226e97	−0.367646	−0.183823	−	0.982959i	$-0.558847\pi$
$853$	−5.65226e97	−0.367646	−0.183823	+	0.982959i	$0.558847\pi$
$854$	4.01971e97	0.251394
$855$	−1.55717e97	−0.0936422
$856$	3.92010e97	0.226687
$857$	−3.44343e98	−1.91484	−0.957422	−	0.288692i	$-0.906780\pi$
$857$	−3.44343e98	−1.91484	−0.957422	+	0.288692i	$0.906780\pi$
$858$	6.68314e97	0.357402
$859$	1.63520e97	0.0841008	0.0420504	−	0.999115i	$-0.486611\pi$
$859$	1.63520e97	0.0841008	0.0420504	+	0.999115i	$0.486611\pi$
$860$	5.73244e97	0.283558
$861$	−1.47637e98	−0.702407
$862$	−1.36131e97	−0.0622967
$863$	1.28936e98	0.567559	0.283779	−	0.958890i	$-0.408412\pi$
$863$	1.28936e98	0.567559	0.283779	+	0.958890i	$0.408412\pi$
$864$	2.50330e98	1.05999
$865$	3.49472e97	0.142355
$866$	3.41612e97	0.133870
$867$	1.62360e98	0.612122
$868$	1.85900e98	0.674326
$869$	−1.35390e98	−0.472524
$870$	3.55077e97	0.119242
$871$	3.99076e98	1.28958
$872$	−7.54987e96	−0.0234767
$873$	−7.24122e97	−0.216688
$874$	−6.86962e97	−0.197833
$875$	−1.23327e98	−0.341812
$876$	−1.06143e98	−0.283141
$877$	−3.61956e98	−0.929329	−0.464665	−	0.885487i	$-0.653825\pi$
$877$	−3.61956e98	−0.929329	−0.464665	+	0.885487i	$0.653825\pi$
$878$	7.88834e97	0.194948
$879$	−5.87242e98	−1.39698
$880$	1.00814e98	0.230862
$881$	2.86429e98	0.631428	0.315714	−	0.948854i	$-0.397756\pi$
$881$	2.86429e98	0.631428	0.315714	+	0.948854i	$0.397756\pi$
$882$	4.56246e97	0.0968282
$883$	4.85729e98	0.992454	0.496227	−	0.868193i	$-0.334718\pi$
$883$	4.85729e98	0.992454	0.496227	+	0.868193i	$0.334718\pi$
$884$	−1.55042e98	−0.304998
$885$	−2.15675e98	−0.408508
$886$	−3.87385e98	−0.706501
$887$	5.57320e98	0.978730	0.489365	−	0.872079i	$-0.337229\pi$
$887$	5.57320e98	0.978730	0.489365	+	0.872079i	$0.337229\pi$
$888$	−5.55643e98	−0.939640
$889$	−6.36947e98	−1.03728
$890$	−1.28508e98	−0.201541
$891$	5.11848e98	0.773105
$892$	4.04778e97	0.0588835
$893$	−1.43008e98	−0.200372
$894$	2.65157e98	0.357846
$895$	−2.05285e98	−0.266861
$896$	4.56580e98	0.571740
$897$	−2.75813e98	−0.332711
$898$	1.54836e98	0.179935
$899$	1.42392e99	1.59417
$900$	−2.33405e98	−0.251759
$901$	2.34886e98	0.244105
$902$	9.07829e98	0.909044
$903$	5.07969e98	0.490114
$904$	−8.15190e98	−0.757908
$905$	4.18337e98	0.374800
$906$	−1.03397e98	−0.0892714
$907$	−1.07550e99	−0.894882	−0.447441	−	0.894313i	$-0.647665\pi$
$907$	−1.07550e99	−0.894882	−0.447441	+	0.894313i	$0.647665\pi$
$908$	−1.66438e99	−1.33467
$909$	−2.78049e98	−0.214897
$910$	7.49815e97	0.0558557
$911$	9.91254e98	0.711737	0.355868	−	0.934536i	$-0.384185\pi$
$911$	9.91254e98	0.711737	0.355868	+	0.934536i	$0.384185\pi$
$912$	5.13910e98	0.355680
$913$	−3.72684e98	−0.248639
$914$	−2.45613e98	−0.157962
$915$	−4.49628e98	−0.278771
$916$	−5.65957e98	−0.338287
$917$	3.93782e98	0.226926
$918$	4.05980e98	0.225567
$919$	−1.32361e99	−0.709073	−0.354537	−	0.935042i	$-0.615361\pi$
$919$	−1.32361e99	−0.709073	−0.354537	+	0.935042i	$0.615361\pi$
$920$	2.60065e98	0.134335
$921$	−1.08085e99	−0.538354
$922$	−1.10219e99	−0.529383
$923$	−2.08195e99	−0.964302
$924$	1.19915e99	0.535626
$925$	3.16270e99	1.36242
$926$	−3.84096e98	−0.159578
$927$	3.41511e98	0.136846
$928$	2.76761e99	1.06967
$929$	−1.24297e98	−0.0463377	−0.0231688	−	0.999732i	$-0.507376\pi$
$929$	−1.24297e98	−0.0463377	−0.0231688	+	0.999732i	$0.507376\pi$
$930$	4.38019e98	0.157512
$931$	1.69673e99	0.588569
$932$	3.33286e99	1.11528
$933$	−3.08339e99	−0.995389
$934$	−5.09760e98	−0.158761
$935$	7.50858e98	0.225615
$936$	6.63259e98	0.192284
$937$	4.03865e99	1.12969	0.564846	−	0.825197i	$-0.308936\pi$
$937$	4.03865e99	1.12969	0.564846	+	0.825197i	$0.308936\pi$
$938$	−1.50835e99	−0.407105
$939$	−2.10266e99	−0.547610
$940$	2.44901e98	0.0615470
$941$	−7.13430e99	−1.73021	−0.865105	−	0.501591i	$-0.832748\pi$
$941$	−7.13430e99	−1.73021	−0.865105	+	0.501591i	$0.832748\pi$
$942$	1.94373e99	0.454916
$943$	−3.74660e99	−0.846244
$944$	−3.66048e99	−0.797952
$945$	9.32089e98	0.196106
$946$	−3.12354e99	−0.634298
$947$	3.01348e99	0.590668	0.295334	−	0.955394i	$-0.404569\pi$
$947$	3.01348e99	0.590668	0.295334	+	0.955394i	$0.404569\pi$
$948$	1.18190e99	0.223616
$949$	−1.71682e99	−0.313551
$950$	1.82842e99	0.322355
$951$	−1.61823e99	−0.275418
$952$	1.29543e99	0.212851
$953$	4.15491e99	0.659097	0.329549	−	0.944139i	$-0.393103\pi$
$953$	4.15491e99	0.659097	0.329549	+	0.944139i	$0.393103\pi$
$954$	−4.54540e98	−0.0696148
$955$	1.07369e98	0.0158769
$956$	−6.70220e99	−0.956926
$957$	9.18494e99	1.26627
$958$	5.04988e99	0.672259
$959$	−4.97289e99	−0.639272
$960$	−2.14090e98	−0.0265773
$961$	9.22395e99	1.10582
$962$	−4.06548e99	−0.470703
$963$	9.03904e98	0.101075
$964$	4.12285e99	0.445263
$965$	−4.88413e99	−0.509475
$966$	1.04246e99	0.105033
$967$	4.64253e99	0.451823	0.225911	−	0.974148i	$-0.427464\pi$
$967$	4.64253e99	0.451823	0.225911	+	0.974148i	$0.427464\pi$
$968$	−8.19847e99	−0.770741
$969$	3.82757e99	0.347598
$970$	−9.71419e98	−0.0852224
$971$	1.49343e99	0.126573	0.0632863	−	0.997995i	$-0.479842\pi$
$971$	1.49343e99	0.126573	0.0632863	+	0.997995i	$0.479842\pi$
$972$	6.51376e99	0.533349
$973$	−9.49559e99	−0.751176
$974$	−8.03600e99	−0.614207
$975$	7.34103e99	0.542129
$976$	−7.63118e99	−0.544531
$977$	1.94135e100	1.33855	0.669275	−	0.743014i	$-0.266605\pi$
$977$	1.94135e100	1.33855	0.669275	+	0.743014i	$0.266605\pi$
$978$	−6.46681e99	−0.430862
$979$	−3.32417e100	−2.14024
$980$	−2.90564e99	−0.180787
$981$	−1.74086e98	−0.0104677
$982$	1.22653e100	0.712760
$983$	−7.70721e99	−0.432867	−0.216433	−	0.976297i	$-0.569442\pi$
$983$	−7.70721e99	−0.432867	−0.216433	+	0.976297i	$0.569442\pi$
$984$	−1.75194e100	−0.951005
$985$	5.31023e99	0.278612
$986$	4.48846e99	0.227625
$987$	2.17014e99	0.106381
$988$	1.11577e100	0.528711
$989$	1.28908e100	0.590479
$990$	−1.45302e99	−0.0643419
$991$	4.23472e99	0.181283	0.0906415	−	0.995884i	$-0.471108\pi$
$991$	4.23472e99	0.181283	0.0906415	+	0.995884i	$0.471108\pi$
$992$	3.41409e100	1.41297
$993$	−4.52217e99	−0.180945
$994$	7.86892e99	0.304419
$995$	8.27233e99	0.309425
$996$	3.25339e99	0.117665
$997$	1.09542e100	0.383082	0.191541	−	0.981485i	$-0.438652\pi$
$997$	1.09542e100	0.383082	0.191541	+	0.981485i	$0.438652\pi$
$998$	5.98149e99	0.202272
$999$	−5.05377e100	−1.65261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1.68.a.a.1.3	✓	5
3.2	odd	2		9.68.a.a.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.68.a.a.1.3	✓	5	1.1	even	1	trivial
9.68.a.a.1.3		5	3.2	odd	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1 \)
Weight:	\( k \)	\(=\)	\( 68 \)
Character orbit:	\([\chi]\)	\(=\)	1.a (trivial)

\(f(q)\)	\(=\)	\(q+5.06726e9 q^{2} -7.82458e15 q^{3} -1.21897e20 q^{4} -8.33551e22 q^{5} -3.96491e25 q^{6} -1.15070e28 q^{7} -1.36548e30 q^{8} -3.14854e31 q^{9} +O(q^{10})\)	\(q+5.06726e9 q^{2} -7.82458e15 q^{3} -1.21897e20 q^{4} -8.33551e22 q^{5} -3.96491e25 q^{6} -1.15070e28 q^{7} -1.36548e30 q^{8} -3.14854e31 q^{9} -4.22381e32 q^{10} -1.09259e35 q^{11} +9.53792e35 q^{12} +1.54273e37 q^{13} -5.83087e37 q^{14} +6.52218e38 q^{15} +1.10696e40 q^{16} +8.24456e40 q^{17} -1.59545e41 q^{18} -5.93328e42 q^{19} +1.01607e43 q^{20} +9.00371e43 q^{21} -5.53645e44 q^{22} +2.28489e45 q^{23} +1.06843e46 q^{24} -6.08146e46 q^{25} +7.81738e46 q^{26} +9.71773e47 q^{27} +1.40266e48 q^{28} +1.07438e49 q^{29} +3.30496e48 q^{30} +1.32534e50 q^{31} +2.57601e50 q^{32} +8.54908e50 q^{33} +4.17773e50 q^{34} +9.59164e50 q^{35} +3.83798e51 q^{36} -5.20056e52 q^{37} -3.00655e52 q^{38} -1.20712e53 q^{39} +1.13819e53 q^{40} -1.63973e54 q^{41} +4.56241e53 q^{42} +5.64177e54 q^{43} +1.33184e55 q^{44} +2.62447e54 q^{45} +1.15781e55 q^{46} +2.41027e55 q^{47} -8.66148e55 q^{48} -2.85968e56 q^{49} -3.08163e56 q^{50} -6.45102e56 q^{51} -1.88053e57 q^{52} +2.84898e57 q^{53} +4.92422e57 q^{54} +9.10731e57 q^{55} +1.57125e58 q^{56} +4.64254e58 q^{57} +5.44415e58 q^{58} -3.30680e59 q^{59} -7.95034e58 q^{60} -6.89383e59 q^{61} +6.71583e59 q^{62} +3.62302e59 q^{63} -3.28249e59 q^{64} -1.28594e60 q^{65} +4.33204e60 q^{66} +2.58683e61 q^{67} -1.00499e61 q^{68} -1.78783e61 q^{69} +4.86033e60 q^{70} -1.34953e62 q^{71} +4.29927e61 q^{72} -1.11285e62 q^{73} -2.63526e62 q^{74} +4.75848e62 q^{75} +7.23248e62 q^{76} +1.25724e63 q^{77} -6.11677e62 q^{78} +1.23916e63 q^{79} -9.22705e62 q^{80} -4.68471e63 q^{81} -8.30894e63 q^{82} +3.41100e63 q^{83} -1.09752e64 q^{84} -6.87226e63 q^{85} +2.85883e64 q^{86} -8.40656e64 q^{87} +1.49191e65 q^{88} +3.04246e65 q^{89} +1.32989e64 q^{90} -1.77521e65 q^{91} -2.78521e65 q^{92} -1.03702e66 q^{93} +1.22134e65 q^{94} +4.94569e65 q^{95} -2.01562e66 q^{96} +2.29986e66 q^{97} -1.44907e66 q^{98} +3.44008e66 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5554901256 q^{2} + 34\!\cdots\!72 q^{3}+ \cdots + 27\!\cdots\!85 q^{9}+O(q^{10}) \) 5 * q + 5554901256 * q^2 + 3443360269119372 * q^3 + 350472004382296600640 * q^4 + 330744836907160597528950 * q^5 + 146648871573834733762548960 * q^6 + 33632726520317603069152899256 * q^7 + 3261706912683890613885787783680 * q^8 + 27806018581845734541016911293385 * q^9	\( 5 q + 5554901256 q^{2} + 34\!\cdots\!72 q^{3}+ \cdots + 16\!\cdots\!20 q^{99}+O(q^{100}) \) 5 * q + 5554901256 * q^2 + 3443360269119372 * q^3 + 350472004382296600640 * q^4 + 330744836907160597528950 * q^5 + 146648871573834733762548960 * q^6 + 33632726520317603069152899256 * q^7 + 3261706912683890613885787783680 * q^8 + 27806018581845734541016911293385 * q^9 - 5287067006725590176865792208438800 * q^10 + 203006981551281631527218766884109060 * q^11 - 2089518810401181456110201520986913024 * q^12 + 177822884499813229570219072356766702 * q^13 + 290753425875732621001638220329334092480 * q^14 - 933024301033068664231979455289562936600 * q^15 + 18266362102387359184851236859301091348480 * q^16 + 75241497376497768433807587544150344102906 * q^17 - 1011568566626978308983328666525487111002968 * q^18 + 3998884102997801557241177083441346393295100 * q^19 - 15074831378020267075078660994426474705846400 * q^20 - 175719863949295325296418063144372762630915040 * q^21 + 1841247880573095918698534631505063585473750432 * q^22 - 419660874260151197180020017656265233426170968 * q^23 - 13019747846573945420584224979196286639928166400 * q^24 + 175279270003539476977813620578789764300129011875 * q^25 + 21416337629300654169124097666599354769371062960 * q^26 + 1886613800166950113417786970194782722207999659320 * q^27 + 14982728306233364463143494158089725946900023694848 * q^28 + 18733547497571939101380180097730717676662894233950 * q^29 + 183881202860964913168534714290880945912096094414400 * q^30 + 365849822471060800690376948081970588059264217288160 * q^31 + 1591165751988992347021834813546084975107380979204096 * q^32 + 2437038903031213839498705428654461423996034595356784 * q^33 + 7817592154883765712044327776251337623941567500782480 * q^34 + 4524513410554564733118320802153390591800073874285200 * q^35 - 38131548457864843751915538710226876689225094472585920 * q^36 - 56389392588768330454829645685931746748224606527380394 * q^37 - 311451303114725973135630000941921262878276604287141280 * q^38 - 717177850582388225418943791249053833224228521090828280 * q^39 - 1886768345048751997029226502271137564956948018338432000 * q^40 + 113555410342193387626817165052765483601006465327103010 * q^41 + 2805540126450529847390961864461873572829871059166350592 * q^42 + 6508304671085340439704487085091235272293711787089861092 * q^43 + 49369449025916528958230735645862721892607029455332199680 * q^44 + 99011086960300194550392336618799141939148872728728967150 * q^45 + 79552980744915855625545425031873854934595203714399466560 * q^46 - 128023594814179612806658270243785955024442659789917751344 * q^47 - 198512903890152994486195174416803543869354744423782432768 * q^48 - 771329931628098140389185999842408514636676231229111640435 * q^49 - 3919481096957516748709549462396523762064894577356951885000 * q^50 - 3397415009313164473667296477133432796692029796730856284840 * q^51 + 672997930141532412373441872733517069231734104886640838016 * q^52 + 1046765637663599376128164919371991945530174947374966490822 * q^53 + 19066352042639687223500180230679796188217876586398799185600 * q^54 + 66942753107037791618627925698973654692865254681839842449400 * q^55 + 164746697572073652525116376573879338548073684495562965299200 * q^56 - 9288557942524759644038647059781434529266663439727382990960 * q^57 - 61137956327686914952144218440883774026433838412764071303120 * q^58 - 305790738450157663862297559072875409310668010528073986627500 * q^59 - 1346157500908637758724100110370785768539984220742218248588800 * q^60 - 1137073938560830615075915586814815057732339483683661907261890 * q^61 - 2197846158587850633685459639420082896328263471188214086352128 * q^62 - 2062780018748959038039139080074136999529046449885010133153768 * q^63 + 6319522016046144185339524760669817745425888834305772325437440 * q^64 + 17054811715744261908524663200762497384422620874171150779332900 * q^65 + 27998862803339055552343423092993933514291039899414979026779520 * q^66 - 11903044989154510038048691316368615691989662189985228212145844 * q^67 + 81678078346863753465296539273887923556029130514802676363013248 * q^68 + 3593002077831059525793506029116145162632074080747869594484320 * q^69 - 262230520857255225386866306600227822969034558447869634844476800 * q^70 - 112169920751076079402369696382938772997794660792753351047772040 * q^71 - 747723581155501189674233444839980291686130221322311676193789440 * q^72 - 304061800126793465330511223999919410949872771539142074085150718 * q^73 + 921008183492866283853128598587548033201480125306857491530532080 * q^74 + 21623812027928499471467875193108522496460255319651250940492500 * q^75 + 2076567511595431334156222539483464646991367936386801535927366400 * q^76 + 5018131462902473316385236580394702266689060843204393245422638432 * q^77 + 6187393864780941106550100388692932558738785969110711253222726464 * q^78 + 2994504004186984867153587379277397704940492543518650040519815600 * q^79 - 15198313539268031787508043655897930060910106707913153957798092800 * q^80 - 11195187555119704986426983039082721262159900559267345597680888595 * q^81 - 17699192859543411700126655967680257958811789647196784631772687408 * q^82 - 47352447490969920981134910324833485175129837165394254819464203588 * q^83 - 52124143836539534782259645807914007868879894236203487647926097920 * q^84 - 39334683419450918052255554891934302881190270223971152032299949300 * q^85 + 49699285824187070765304255878405188987409992188766856866277888160 * q^86 + 357574798440063707158439877351651513807909832776407769090038489160 * q^87 + 430257938373980762031490456670953174964896424231159472833711400960 * q^88 - 11988583288945008097106276879882407358382120120051430459779195150 * q^89 + 477054381540878916877701150770415125576892082956059127548575220400 * q^90 + 232376792470844928685229521265495056030669651812230900159714321360 * q^91 - 986292139147147043544160128863480288571347084605459783959415954944 * q^92 - 1849280210034321990512094428557344731461785100166176155774311127936 * q^93 - 4927754861315037106505826725122767884474273389521981763161929704320 * q^94 - 2022377488008256909110920831467942087293430657450319294506545963000 * q^95 - 2242556165580376632567217570505477297218451346654758522676359331840 * q^96 + 2897556354128426205551279700626881901316694084191208635967319112106 * q^97 + 12426885478574475523720488645206508880013087100212747539199689691208 * q^98 + 16175146031966622867876651602334534091105243319832315048256094545620 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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