LMFDB - Embedded newform 169.8.b.d.168.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,8,Mod(168,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	169.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$52.7930693068$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 1279 x^{12} + 629380 x^{10} + 148562016 x^{8} + 16872573312 x^{6} + 790180980480 x^{4} + \cdots + 4669637050368 $ x^14 + 1279x^12 + 629380x^10 + 148562016x^8 + 16872573312x^6 + 790180980480x^4 + 10484997239808x^2 + 4669637050368
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{17}\cdot 3^{3}\cdot 13^{6} $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.9
Root			$4.51724i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.8.b.d.168.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.51724i q^{2} -43.8390 q^{3} +107.595 q^{4} +49.0275i q^{5} -198.031i q^{6} -850.411i q^{7} +1064.24i q^{8} -265.146 q^{9} -221.469 q^{10} -124.845i q^{11} -4716.83 q^{12} +3841.52 q^{14} -2149.31i q^{15} +8964.67 q^{16} -11326.0 q^{17} -1197.73i q^{18} +7864.97i q^{19} +5275.09i q^{20} +37281.1i q^{21} +563.954 q^{22} -86232.1 q^{23} -46655.1i q^{24} +75721.3 q^{25} +107500. q^{27} -91499.6i q^{28} +125681. q^{29} +9708.97 q^{30} -78827.3i q^{31} +176718. i q^{32} +5473.06i q^{33} -51162.4i q^{34} +41693.5 q^{35} -28528.3 q^{36} +168074. i q^{37} -35528.0 q^{38} -52176.9 q^{40} -346835. i q^{41} -168408. q^{42} +486181. q^{43} -13432.6i q^{44} -12999.4i q^{45} -389531. i q^{46} +443195. i q^{47} -393002. q^{48} +100343. q^{49} +342052. i q^{50} +496521. q^{51} -1.44734e6 q^{53} +485602. i q^{54} +6120.82 q^{55} +905040. q^{56} -344792. i q^{57} +567730. i q^{58} +2.66402e6i q^{59} -231254. i q^{60} -1.29692e6 q^{61} +356082. q^{62} +225483. i q^{63} +349200. q^{64} -24723.2 q^{66} +4.65420e6i q^{67} -1.21862e6 q^{68} +3.78033e6 q^{69} +188340. i q^{70} +4.09074e6i q^{71} -282178. i q^{72} +6.09222e6i q^{73} -759231. q^{74} -3.31954e6 q^{75} +846228. i q^{76} -106169. q^{77} -1131.00 q^{79} +439515. i q^{80} -4.13279e6 q^{81} +1.56674e6 q^{82} -5.26434e6i q^{83} +4.01125e6i q^{84} -555286. i q^{85} +2.19620e6i q^{86} -5.50971e6 q^{87} +132864. q^{88} -1.89261e6i q^{89} +58721.6 q^{90} -9.27810e6 q^{92} +3.45570e6i q^{93} -2.00202e6 q^{94} -385600. q^{95} -7.74714e6i q^{96} +1.68363e7i q^{97} +453276. i q^{98} +33102.1i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 52 q^{3} - 766 q^{4} + 6982 q^{9} + 1018 q^{10} + 38380 q^{12} - 47916 q^{14} + 1266 q^{16} + 76806 q^{17} + 251764 q^{22} + 137100 q^{23} + 39380 q^{25} - 432400 q^{27} - 443166 q^{29} + 315780 q^{30}+ \cdots + 86840772 q^{95}+O(q^{100}) $ 14 * q - 52 * q^3 - 766 * q^4 + 6982 * q^9 + 1018 * q^10 + 38380 * q^12 - 47916 * q^14 + 1266 * q^16 + 76806 * q^17 + 251764 * q^22 + 137100 * q^23 + 39380 * q^25 - 432400 * q^27 - 443166 * q^29 + 315780 * q^30 - 1319232 * q^35 - 1271350 * q^36 + 953640 * q^38 - 77182 * q^40 - 412812 * q^42 + 1765384 * q^43 - 2267260 * q^48 - 5307958 * q^49 - 38964 * q^51 - 1791210 * q^53 - 4907072 * q^55 + 14915904 * q^56 - 7717050 * q^61 - 3808284 * q^62 + 14570922 * q^64 - 31956396 * q^66 - 19139238 * q^68 + 27487680 * q^69 - 17280210 * q^74 - 20632304 * q^75 + 19599336 * q^77 - 30688440 * q^79 - 27772442 * q^81 + 74719438 * q^82 - 25539372 * q^87 - 54091072 * q^88 + 41193546 * q^90 - 47685588 * q^92 - 38277620 * q^94 + 86840772 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	4.51724i	0.399272i	0.979870	+	0.199636i	$0.0639759\pi$
$2$	4.51724i	0.399272i	−0.979870	+	0.199636i	$0.936024\pi$
$3$	−43.8390	−0.937423	−0.468712	−	0.883351i	$-0.655282\pi$
$3$	−43.8390	−0.937423	−0.468712	+	0.883351i	$0.655282\pi$
$4$	107.595	0.840582
$5$	49.0275i	0.175406i	0.996147	+	0.0877030i	$0.0279526\pi$
$5$	49.0275i	0.175406i	−0.996147	+	0.0877030i	$0.972047\pi$
$6$	− 198.031i	− 0.374287i
$7$	− 850.411i	− 0.937100i	−0.883437	−	0.468550i	$-0.844777\pi$
$7$	− 850.411i	− 0.937100i	0.883437	−	0.468550i	$-0.155223\pi$
$8$	1064.24i	0.734892i
$9$	−265.146	−0.121237
$10$	−221.469	−0.0700346
$11$	− 124.845i	− 0.0282811i	−0.999900	−	0.0141405i	$-0.995499\pi$
$11$	− 124.845i	− 0.0282811i	0.999900	−	0.0141405i	$-0.00450122\pi$
$12$	−4716.83	−0.787981
$13$	0	0
$13$	0	0
$14$	3841.52	0.374158
$15$	− 2149.31i	− 0.164430i
$16$	8964.67	0.547160
$17$	−11326.0	−0.559122	−0.279561	−	0.960128i	$-0.590189\pi$
$17$	−11326.0	−0.559122	−0.279561	+	0.960128i	$0.590189\pi$
$18$	− 1197.73i	− 0.0484066i
$19$	7864.97i	0.263063i	0.991312	+	0.131531i	$0.0419894\pi$
$19$	7864.97i	0.263063i	−0.991312	+	0.131531i	$0.958011\pi$
$20$	5275.09i	0.147443i
$21$	37281.1i	0.878459i
$22$	563.954	0.0112918
$23$	−86232.1	−1.47782	−0.738910	−	0.673804i	$-0.764659\pi$
$23$	−86232.1	−1.47782	−0.738910	+	0.673804i	$0.764659\pi$
$24$	− 46655.1i	− 0.688905i
$25$	75721.3	0.969233
$26$	0	0
$27$	107500.	1.05107
$28$	− 91499.6i	− 0.787709i
$29$	125681.	0.956920	0.478460	−	0.878109i	$-0.341195\pi$
$29$	125681.	0.956920	0.478460	+	0.878109i	$0.341195\pi$
$30$	9708.97	0.0656521
$31$	− 78827.3i	− 0.475237i	−0.971359	−	0.237619i	$-0.923633\pi$
$31$	− 78827.3i	− 0.475237i	0.971359	−	0.237619i	$-0.0763668\pi$
$32$	176718.i	0.953358i
$33$	5473.06i	0.0265113i
$34$	− 51162.4i	− 0.223241i
$35$	41693.5	0.164373
$36$	−28528.3	−0.101910
$37$	168074.i	0.545500i	0.962085	+	0.272750i	$0.0879331\pi$
$37$	168074.i	0.545500i	−0.962085	+	0.272750i	$0.912067\pi$
$38$	−35528.0	−0.105034
$39$	0	0
$40$	−52176.9	−0.128905
$41$	− 346835.i	− 0.785921i	−0.919555	−	0.392960i	$-0.871451\pi$
$41$	− 346835.i	− 0.785921i	0.919555	−	0.392960i	$-0.128549\pi$
$42$	−168408.	−0.350744
$43$	486181.	0.932520	0.466260	−	0.884648i	$-0.345601\pi$
$43$	486181.	0.932520	0.466260	+	0.884648i	$0.345601\pi$
$44$	− 13432.6i	− 0.0237726i
$45$	− 12999.4i	− 0.0212658i
$46$	− 389531.i	− 0.590052i
$47$	443195.i	0.622662i	0.950302	+	0.311331i	$0.100775\pi$
$47$	443195.i	0.622662i	−0.950302	+	0.311331i	$0.899225\pi$
$48$	−393002.	−0.512921
$49$	100343.	0.121844
$50$	342052.i	0.386987i
$51$	496521.	0.524134
$52$	0	0
$53$	−1.44734e6	−1.33538	−0.667692	−	0.744438i	$-0.732718\pi$
$53$	−1.44734e6	−1.33538	−0.667692	+	0.744438i	$0.732718\pi$
$54$	485602.i	0.419664i
$55$	6120.82	0.00496067
$56$	905040.	0.688668
$57$	− 344792.i	− 0.246601i
$58$	567730.i	0.382071i
$59$	2.66402e6i	1.68871i	0.535784	+	0.844355i	$0.320016\pi$
$59$	2.66402e6i	1.68871i	−0.535784	+	0.844355i	$0.679984\pi$
$60$	− 231254.i	− 0.138217i
$61$	−1.29692e6	−0.731578	−0.365789	−	0.930698i	$-0.619201\pi$
$61$	−1.29692e6	−0.731578	−0.365789	+	0.930698i	$0.619201\pi$
$62$	356082.	0.189749
$63$	225483.i	0.113612i
$64$	349200.	0.166511
$65$	0	0
$66$	−24723.2	−0.0105852
$67$	4.65420e6i	1.89053i	0.326309	+	0.945263i	$0.394195\pi$
$67$	4.65420e6i	1.89053i	−0.326309	+	0.945263i	$0.605805\pi$
$68$	−1.21862e6	−0.469988
$69$	3.78033e6	1.38534
$70$	188340.i	0.0656295i
$71$	4.09074e6i	1.35643i	0.734864	+	0.678215i	$0.237246\pi$
$71$	4.09074e6i	1.35643i	−0.734864	+	0.678215i	$0.762754\pi$
$72$	− 282178.i	− 0.0890964i
$73$	6.09222e6i	1.83293i	0.400117	+	0.916464i	$0.368969\pi$
$73$	6.09222e6i	1.83293i	−0.400117	+	0.916464i	$0.631031\pi$
$74$	−759231.	−0.217803
$75$	−3.31954e6	−0.908581
$76$	846228.i	0.221126i
$77$	−106169.	−0.0265022
$78$	0	0
$79$	−1131.00	−0.000258087 0	−0.000129044	−	1.00000i	$-0.500041\pi$
$79$	−1131.00	−0.000258087 0	−0.000129044		1.00000i	$0.500041\pi$
$80$	439515.i	0.0959752i
$81$	−4.13279e6	−0.864064
$82$	1.56674e6	0.313796
$83$	− 5.26434e6i	− 1.01058i	−0.862950	−	0.505290i	$-0.831386\pi$
$83$	− 5.26434e6i	− 1.01058i	0.862950	−	0.505290i	$-0.168614\pi$
$84$	4.01125e6i	0.738417i
$85$	− 555286.i	− 0.0980733i
$86$	2.19620e6i	0.372329i
$87$	−5.50971e6	−0.897039
$88$	132864.	0.0207835
$89$	− 1.89261e6i	− 0.284574i	−0.989825	−	0.142287i	$-0.954554\pi$
$89$	− 1.89261e6i	− 0.284574i	0.989825	−	0.142287i	$-0.0454457\pi$
$90$	58721.6	0.00849081
$91$	0	0
$92$	−9.27810e6	−1.24223
$93$	3.45570e6i	0.445499i
$94$	−2.00202e6	−0.248611
$95$	−385600.	−0.0461428
$96$	− 7.74714e6i	− 0.893700i
$97$	1.68363e7i	1.87304i	0.350619	+	0.936518i	$0.385971\pi$
$97$	1.68363e7i	1.87304i	−0.350619	+	0.936518i	$0.614029\pi$
$98$	453276.i	0.0486487i
$99$	33102.1i	0.00342872i
$100$	8.14720e6	0.814720
$101$	−1.92121e7	−1.85546	−0.927728	−	0.373257i	$-0.878241\pi$
$101$	−1.92121e7	−1.85546	−0.927728	+	0.373257i	$0.878241\pi$
$102$	2.24291e6i	0.209272i
$103$	1.21968e7	1.09980	0.549902	−	0.835229i	$-0.314665\pi$
$103$	1.21968e7	1.09980	0.549902	+	0.835229i	$0.314665\pi$
$104$	0	0
$105$	−1.82780e6	−0.154087
$106$	− 6.53801e6i	− 0.533181i
$107$	−6.84075e6	−0.539834	−0.269917	−	0.962884i	$-0.586996\pi$
$107$	−6.84075e6	−0.539834	−0.269917	+	0.962884i	$0.586996\pi$
$108$	1.15664e7	0.883514
$109$	− 3.26600e6i	− 0.241559i	−0.992679	−	0.120779i	$-0.961461\pi$
$109$	− 3.26600e6i	− 0.241559i	0.992679	−	0.120779i	$-0.0385394\pi$
$110$	27649.2i	0.00198065i
$111$	− 7.36819e6i	− 0.511364i
$112$	− 7.62366e6i	− 0.512744i
$113$	173551.	0.0113150	0.00565748	−	0.999984i	$-0.498199\pi$
$113$	173551.	0.0113150	0.00565748	+	0.999984i	$0.498199\pi$
$114$	1.55751e6	0.0984610
$115$	− 4.22774e6i	− 0.259218i
$116$	1.35226e7	0.804370
$117$	0	0
$118$	−1.20340e7	−0.674254
$119$	9.63178e6i	0.523953i
$120$	2.28738e6	0.120838
$121$	1.94716e7	0.999200
$122$	− 5.85853e6i	− 0.292098i
$123$	1.52049e7i	0.736741i
$124$	− 8.48138e6i	− 0.399476i
$125$	7.54269e6i	0.345415i
$126$	−1.01856e6	−0.0453619
$127$	1.80804e7	0.783239	0.391620	−	0.920127i	$-0.371915\pi$
$127$	1.80804e7	0.783239	0.391620	+	0.920127i	$0.371915\pi$
$128$	2.41973e7i	1.01984i
$129$	−2.13137e7	−0.874166
$130$	0	0
$131$	1.46819e7	0.570601	0.285301	−	0.958438i	$-0.407907\pi$
$131$	1.46819e7	0.570601	0.285301	+	0.958438i	$0.407907\pi$
$132$	588871.i	0.0222849i
$133$	6.68846e6	0.246516
$134$	−2.10241e7	−0.754834
$135$	5.27043e6i	0.184365i
$136$	− 1.20536e7i	− 0.410894i
$137$	4.91779e7i	1.63399i	0.576648	+	0.816993i	$0.304360\pi$
$137$	4.91779e7i	1.63399i	−0.576648	+	0.816993i	$0.695640\pi$
$138$	1.70767e7i	0.553128i
$139$	−2.30684e7	−0.728561	−0.364280	−	0.931289i	$-0.618685\pi$
$139$	−2.30684e7	−0.728561	−0.364280	+	0.931289i	$0.618685\pi$
$140$	4.48599e6	0.138169
$141$	− 1.94292e7i	− 0.583698i
$142$	−1.84789e7	−0.541584
$143$	0	0
$144$	−2.37695e6	−0.0663363
$145$	6.16180e6i	0.167849i
$146$	−2.75200e7	−0.731836
$147$	−4.39895e6	−0.114219
$148$	1.80838e7i	0.458537i
$149$	− 5.02309e7i	− 1.24400i	−0.783018	−	0.621998i	$-0.786321\pi$
$149$	− 5.02309e7i	− 1.24400i	0.783018	−	0.621998i	$-0.213679\pi$
$150$	− 1.49952e7i	− 0.362771i
$151$	2.96842e7i	0.701626i	0.936446	+	0.350813i	$0.114095\pi$
$151$	2.96842e7i	0.701626i	−0.936446	+	0.350813i	$0.885905\pi$
$152$	−8.37020e6	−0.193323
$153$	3.00305e6	0.0677864
$154$	− 479593.i	− 0.0105816i
$155$	3.86470e6	0.0833595
$156$	0	0
$157$	3.56376e7	0.734952	0.367476	−	0.930033i	$-0.380222\pi$
$157$	3.56376e7	0.734952	0.367476	+	0.930033i	$0.380222\pi$
$158$	− 5108.99i	0 0.000103047i
$159$	6.34500e7	1.25182
$160$	−8.66404e6	−0.167225
$161$	7.33328e7i	1.38487i
$162$	− 1.86688e7i	− 0.344996i
$163$	1.04922e7i	0.189763i	0.995489	+	0.0948814i	$0.0302472\pi$
$163$	1.04922e7i	0.189763i	−0.995489	+	0.0948814i	$0.969753\pi$
$164$	− 3.73175e7i	− 0.660631i
$165$	−268330.	−0.00465025
$166$	2.37803e7	0.403496
$167$	− 1.94057e7i	− 0.322419i	−0.986920	−	0.161210i	$-0.948460\pi$
$167$	− 1.94057e7i	− 0.322419i	0.986920	−	0.161210i	$-0.0515395\pi$
$168$	−3.96760e7	−0.645573
$169$	0	0
$170$	2.50836e6	0.0391579
$171$	− 2.08537e6i	− 0.0318930i
$172$	5.23104e7	0.783860
$173$	−4.57500e7	−0.671785	−0.335892	−	0.941900i	$-0.609038\pi$
$173$	−4.57500e7	−0.671785	−0.335892	+	0.941900i	$0.609038\pi$
$174$	− 2.48887e7i	− 0.358162i
$175$	− 6.43943e7i	− 0.908268i
$176$	− 1.11919e6i	− 0.0154743i
$177$	− 1.16788e8i	− 1.58304i
$178$	8.54938e6	0.113623
$179$	3.92270e7	0.511211	0.255605	−	0.966781i	$-0.417725\pi$
$179$	3.92270e7	0.511211	0.255605	+	0.966781i	$0.417725\pi$
$180$	− 1.39867e6i	− 0.0178756i
$181$	6.74932e7	0.846029	0.423015	−	0.906123i	$-0.360972\pi$
$181$	6.74932e7	0.846029	0.423015	+	0.906123i	$0.360972\pi$
$182$	0	0
$183$	5.68558e7	0.685798
$184$	− 9.17715e7i	− 1.08604i
$185$	−8.24024e6	−0.0956839
$186$	−1.56103e7	−0.177875
$187$	1.41399e6i	0.0158126i
$188$	4.76853e7i	0.523398i
$189$	− 9.14188e7i	− 0.984962i
$190$	− 1.74185e6i	− 0.0184235i
$191$	1.71927e8	1.78536	0.892682	−	0.450688i	$-0.148821\pi$
$191$	1.71927e8	1.78536	0.892682	+	0.450688i	$0.148821\pi$
$192$	−1.53085e7	−0.156092
$193$	2.48491e7i	0.248806i	0.992232	+	0.124403i	$0.0397016\pi$
$193$	2.48491e7i	0.248806i	−0.992232	+	0.124403i	$0.960298\pi$
$194$	−7.60538e7	−0.747850
$195$	0	0
$196$	1.07964e7	0.102420
$197$	− 1.37653e8i	− 1.28279i	−0.767213	−	0.641393i	$-0.778357\pi$
$197$	− 1.37653e8i	− 1.28279i	0.767213	−	0.641393i	$-0.221643\pi$
$198$	−149530.	−0.00136899
$199$	−9.81839e7	−0.883190	−0.441595	−	0.897214i	$-0.645587\pi$
$199$	−9.81839e7	−0.883190	−0.441595	+	0.897214i	$0.645587\pi$
$200$	8.05855e7i	0.712282i
$201$	− 2.04035e8i	− 1.77222i
$202$	− 8.67859e7i	− 0.740831i
$203$	− 1.06880e8i	− 0.896730i
$204$	5.34229e7	0.440577
$205$	1.70044e7	0.137855
$206$	5.50959e7i	0.439121i
$207$	2.28641e7	0.179167
$208$	0	0
$209$	981900.	0.00743970
$210$	− 8.25662e6i	− 0.0615226i
$211$	−9.35140e7	−0.685312	−0.342656	−	0.939461i	$-0.611326\pi$
$211$	−9.35140e7	−0.685312	−0.342656	+	0.939461i	$0.611326\pi$
$212$	−1.55726e8	−1.12250
$213$	− 1.79334e8i	− 1.27155i
$214$	− 3.09013e7i	− 0.215541i
$215$	2.38362e7i	0.163570i
$216$	1.14405e8i	0.772426i
$217$	−6.70356e7	−0.445345
$218$	1.47533e7	0.0964477
$219$	− 2.67076e8i	− 1.71823i
$220$	658566.	0.00416985
$221$	0	0
$222$	3.32839e7	0.204173
$223$	− 6.52040e7i	− 0.393738i	−0.980430	−	0.196869i	$-0.936923\pi$
$223$	− 6.52040e7i	− 0.393738i	0.980430	−	0.196869i	$-0.0630774\pi$
$224$	1.50283e8	0.893392
$225$	−2.00772e7	−0.117507
$226$	783973.i	0.00451774i
$227$	1.31044e8i	0.743578i	0.928317	+	0.371789i	$0.121255\pi$
$227$	1.31044e8i	0.743578i	−0.928317	+	0.371789i	$0.878745\pi$
$228$	− 3.70977e7i	− 0.207289i
$229$	− 1.87132e8i	− 1.02973i	−0.857272	−	0.514864i	$-0.827842\pi$
$229$	− 1.87132e8i	− 1.02973i	0.857272	−	0.514864i	$-0.172158\pi$
$230$	1.90977e7	0.103499
$231$	4.65435e6	0.0248438
$232$	1.33754e8i	0.703233i
$233$	−8.77467e7	−0.454449	−0.227225	−	0.973842i	$-0.572965\pi$
$233$	−8.77467e7	−0.454449	−0.227225	+	0.973842i	$0.572965\pi$
$234$	0	0
$235$	−2.17287e7	−0.109219
$236$	2.86634e8i	1.41950i
$237$	49581.7	0.000241937 0
$238$	−4.35091e7	−0.209200
$239$	2.99804e8i	1.42051i	0.703943	+	0.710257i	$0.251421\pi$
$239$	2.99804e8i	1.42051i	−0.703943	+	0.710257i	$0.748579\pi$
$240$	− 1.92679e7i	− 0.0899694i
$241$	1.20929e8i	0.556509i	0.960507	+	0.278254i	$0.0897558\pi$
$241$	1.20929e8i	0.556509i	−0.960507	+	0.278254i	$0.910244\pi$
$242$	8.79579e7i	0.398952i
$243$	−5.39242e7	−0.241080
$244$	−1.39542e8	−0.614951
$245$	4.91958e6i	0.0213721i
$246$	−6.86841e7	−0.294160
$247$	0	0
$248$	8.38910e7	0.349248
$249$	2.30783e8i	0.947341i
$250$	−3.40722e7	−0.137915
$251$	3.42122e8	1.36560	0.682800	−	0.730606i	$-0.260762\pi$
$251$	3.42122e8	1.36560	0.682800	+	0.730606i	$0.260762\pi$
$252$	2.42608e7i	0.0954998i
$253$	1.07656e7i	0.0417943i
$254$	8.16735e7i	0.312725i
$255$	2.43432e7i	0.0919362i
$256$	−6.46077e7	−0.240682
$257$	1.33787e8	0.491641	0.245820	−	0.969315i	$-0.420943\pi$
$257$	1.33787e8	0.491641	0.245820	+	0.969315i	$0.420943\pi$
$258$	− 9.62790e7i	− 0.349030i
$259$	1.42932e8	0.511188
$260$	0	0
$261$	−3.33237e7	−0.116014
$262$	6.63217e7i	0.227825i
$263$	−4.16302e8	−1.41112	−0.705559	−	0.708651i	$-0.749304\pi$
$263$	−4.16302e8	−1.41112	−0.705559	+	0.708651i	$0.749304\pi$
$264$	−5.82464e6	−0.0194830
$265$	− 7.09596e7i	− 0.234234i
$266$	3.02134e7i	0.0984270i
$267$	8.29700e7i	0.266767i
$268$	5.00766e8i	1.58914i
$269$	−2.98622e8	−0.935381	−0.467690	−	0.883892i	$-0.654914\pi$
$269$	−2.98622e8	−0.935381	−0.467690	+	0.883892i	$0.654914\pi$
$270$	−2.38078e7	−0.0736116
$271$	5.32417e7i	0.162502i	0.996694	+	0.0812512i	$0.0258916\pi$
$271$	5.32417e7i	0.162502i	−0.996694	+	0.0812512i	$0.974108\pi$
$272$	−1.01534e8	−0.305929
$273$	0	0
$274$	−2.22149e8	−0.652404
$275$	− 9.45340e6i	− 0.0274109i
$276$	4.06742e8	1.16449
$277$	1.94332e8	0.549369	0.274685	−	0.961534i	$-0.411427\pi$
$277$	1.94332e8	0.549369	0.274685	+	0.961534i	$0.411427\pi$
$278$	− 1.04206e8i	− 0.290894i
$279$	2.09007e7i	0.0576165i
$280$	4.43718e7i	0.120796i
$281$	− 7.35301e7i	− 0.197694i	−0.995103	−	0.0988468i	$-0.968485\pi$
$281$	− 7.35301e7i	− 0.197694i	0.995103	−	0.0988468i	$-0.0315154\pi$
$282$	8.77665e7	0.233054
$283$	2.77363e8	0.727437	0.363719	−	0.931509i	$-0.381507\pi$
$283$	2.77363e8	0.727437	0.363719	+	0.931509i	$0.381507\pi$
$284$	4.40141e8i	1.14019i
$285$	1.69043e7	0.0432554
$286$	0	0
$287$	−2.94952e8	−0.736486
$288$	− 4.68561e7i	− 0.115583i
$289$	−2.82060e8	−0.687383
$290$	−2.78344e7	−0.0670175
$291$	− 7.38087e8i	− 1.75583i
$292$	6.55489e8i	1.54073i
$293$	8.92358e7i	0.207254i	0.994616	+	0.103627i	$0.0330448\pi$
$293$	8.92358e7i	0.207254i	−0.994616	+	0.103627i	$0.966955\pi$
$294$	− 1.98711e7i	− 0.0456044i
$295$	−1.30610e8	−0.296210
$296$	−1.78871e8	−0.400883
$297$	− 1.34207e7i	− 0.0297255i
$298$	2.26905e8	0.496693
$299$	0	0
$300$	−3.57165e8	−0.763737
$301$	− 4.13454e8i	− 0.873864i
$302$	−1.34091e8	−0.280140
$303$	8.42240e8	1.73935
$304$	7.05069e7i	0.143938i
$305$	− 6.35849e7i	− 0.128323i
$306$	1.35655e7i	0.0270652i
$307$	2.03142e8i	0.400696i	0.979725	+	0.200348i	$0.0642072\pi$
$307$	2.03142e8i	0.400696i	−0.979725	+	0.200348i	$0.935793\pi$
$308$	−1.14232e7	−0.0222773
$309$	−5.34694e8	−1.03098
$310$	1.74578e7i	0.0332831i
$311$	7.82046e8	1.47425	0.737125	−	0.675757i	$-0.236183\pi$
$311$	7.82046e8	1.47425	0.737125	+	0.675757i	$0.236183\pi$
$312$	0	0
$313$	3.84527e8	0.708797	0.354399	−	0.935094i	$-0.384686\pi$
$313$	3.84527e8	0.708797	0.354399	+	0.935094i	$0.384686\pi$
$314$	1.60984e8i	0.293446i
$315$	−1.10549e7	−0.0199281
$316$	−121689.	−0.000216944 0
$317$	4.19845e8i	0.740256i	0.928981	+	0.370128i	$0.120686\pi$
$317$	4.19845e8i	0.740256i	−0.928981	+	0.370128i	$0.879314\pi$
$318$	2.86619e8i	0.499817i
$319$	− 1.56906e7i	− 0.0270627i
$320$	1.71204e7i	0.0292071i
$321$	2.99891e8	0.506053
$322$	−3.31262e8	−0.552938
$323$	− 8.90789e7i	− 0.147084i
$324$	−4.44666e8	−0.726317
$325$	0	0
$326$	−4.73960e7	−0.0757669
$327$	1.43178e8i	0.226443i
$328$	3.69114e8	0.577567
$329$	3.76898e8	0.583497
$330$	− 1.21211e6i	− 0.00185671i
$331$	8.18268e8i	1.24022i	0.784516	+	0.620109i	$0.212911\pi$
$331$	8.18268e8i	1.24022i	−0.784516	+	0.620109i	$0.787089\pi$
$332$	− 5.66414e8i	− 0.849475i
$333$	− 4.45641e7i	− 0.0661349i
$334$	8.76601e7	0.128733
$335$	−2.28183e8	−0.331610
$336$	3.34213e8i	0.480658i
$337$	4.50190e8	0.640753	0.320377	−	0.947290i	$-0.396191\pi$
$337$	4.50190e8	0.640753	0.320377	+	0.947290i	$0.396191\pi$
$338$	0	0
$339$	−7.60830e6	−0.0106069
$340$	− 5.97458e7i	− 0.0824386i
$341$	−9.84117e6	−0.0134402
$342$	9.42011e6	0.0127340
$343$	− 7.85684e8i	− 1.05128i
$344$	5.17412e8i	0.685302i
$345$	1.85340e8i	0.242997i
$346$	− 2.06664e8i	− 0.268225i
$347$	−1.12644e9	−1.44729	−0.723647	−	0.690170i	$-0.757536\pi$
$347$	−1.12644e9	−1.44729	−0.723647	+	0.690170i	$0.757536\pi$
$348$	−5.92815e8	−0.754035
$349$	− 1.02048e9i	− 1.28504i	−0.766269	−	0.642520i	$-0.777889\pi$
$349$	− 1.02048e9i	− 1.28504i	0.766269	−	0.642520i	$-0.222111\pi$
$350$	2.90885e8	0.362646
$351$	0	0
$352$	2.20623e7	0.0269620
$353$	− 3.28291e6i	− 0.00397235i	−0.999998	−	0.00198618i	$-0.999368\pi$
$353$	− 3.28291e6i	− 0.00397235i	0.999998	−	0.00198618i	$-0.000632220\pi$
$354$	5.27559e8	0.632062
$355$	−2.00558e8	−0.237926
$356$	− 2.03634e8i	− 0.239208i
$357$	− 4.22247e8i	− 0.491166i
$358$	1.77198e8i	0.204112i
$359$	1.01744e9i	1.16059i	0.814406	+	0.580296i	$0.197063\pi$
$359$	1.01744e9i	1.16059i	−0.814406	+	0.580296i	$0.802937\pi$
$360$	1.38345e7	0.0156280
$361$	8.32014e8	0.930798
$362$	3.04883e8i	0.337795i
$363$	−8.53614e8	−0.936674
$364$	0	0
$365$	−2.98686e8	−0.321507
$366$	2.56832e8i	0.273820i
$367$	−1.36503e9	−1.44149	−0.720743	−	0.693202i	$-0.756199\pi$
$367$	−1.36503e9	−1.44149	−0.720743	+	0.693202i	$0.756199\pi$
$368$	−7.73043e8	−0.808604
$369$	9.19618e7i	0.0952829i
$370$	− 3.72232e7i	− 0.0382039i
$371$	1.23084e9i	1.25139i
$372$	3.71815e8i	0.374478i
$373$	9.65141e8	0.962963	0.481482	−	0.876456i	$-0.340099\pi$
$373$	9.65141e8	0.962963	0.481482	+	0.876456i	$0.340099\pi$
$374$	−6.38736e6	−0.00631351
$375$	− 3.30664e8i	− 0.323800i
$376$	−4.71665e8	−0.457590
$377$	0	0
$378$	4.12961e8	0.393267
$379$	8.15097e8i	0.769081i	0.923108	+	0.384540i	$0.125640\pi$
$379$	8.15097e8i	0.769081i	−0.923108	+	0.384540i	$0.874360\pi$
$380$	−4.14884e7	−0.0387868
$381$	−7.92625e8	−0.734227
$382$	7.76635e8i	0.712845i
$383$	− 1.28104e9i	− 1.16511i	−0.812790	−	0.582556i	$-0.802053\pi$
$383$	− 1.28104e9i	− 1.16511i	0.812790	−	0.582556i	$-0.197947\pi$
$384$	− 1.06079e9i	− 0.956023i
$385$	− 5.20521e6i	− 0.00464864i
$386$	−1.12250e8	−0.0993412
$387$	−1.28909e8	−0.113056
$388$	1.81150e9i	1.57444i
$389$	−9.51453e8	−0.819528	−0.409764	−	0.912192i	$-0.634389\pi$
$389$	−9.51453e8	−0.819528	−0.409764	+	0.912192i	$0.634389\pi$
$390$	0	0
$391$	9.76667e8	0.826281
$392$	1.06789e8i	0.0895419i
$393$	−6.43639e8	−0.534895
$394$	6.21812e8	0.512180
$395$	− 55449.9i	0 4.52701e-5i
$396$	3.56160e6i	0.00288212i
$397$	− 2.46693e8i	− 0.197875i	−0.995094	−	0.0989375i	$-0.968456\pi$
$397$	− 2.46693e8i	− 0.197875i	0.995094	−	0.0989375i	$-0.0315444\pi$
$398$	− 4.43520e8i	− 0.352633i
$399$	−2.93215e8	−0.231090
$400$	6.78817e8	0.530326
$401$	− 2.02504e9i	− 1.56830i	−0.620571	−	0.784150i	$-0.713099\pi$
$401$	− 2.02504e9i	− 1.56830i	0.620571	−	0.784150i	$-0.286901\pi$
$402$	9.21676e8	0.707599
$403$	0	0
$404$	−2.06712e9	−1.55966
$405$	− 2.02620e8i	− 0.151562i
$406$	4.82804e8	0.358039
$407$	2.09831e7	0.0154273
$408$	5.28417e8i	0.385182i
$409$	− 1.45005e9i	− 1.04798i	−0.851725	−	0.523989i	$-0.824443\pi$
$409$	− 1.45005e9i	− 1.04798i	0.851725	−	0.523989i	$-0.175557\pi$
$410$	7.68131e7i	0.0550417i
$411$	− 2.15591e9i	− 1.53174i
$412$	1.31231e9	0.924475
$413$	2.26551e9	1.58249
$414$	1.03283e8i	0.0715363i
$415$	2.58097e8	0.177262
$416$	0	0
$417$	1.01129e9	0.682970
$418$	4.43548e6i	0.00297046i
$419$	7.01637e8	0.465976	0.232988	−	0.972480i	$-0.425150\pi$
$419$	7.01637e8	0.465976	0.232988	+	0.972480i	$0.425150\pi$
$420$	−1.96661e8	−0.129523
$421$	6.04587e8i	0.394886i	0.980314	+	0.197443i	$0.0632637\pi$
$421$	6.04587e8i	0.394886i	−0.980314	+	0.197443i	$0.936736\pi$
$422$	− 4.22426e8i	− 0.273626i
$423$	− 1.17511e8i	− 0.0754899i
$424$	− 1.54032e9i	− 0.981364i
$425$	−8.57622e8	−0.541919
$426$	8.10094e8	0.507693
$427$	1.10292e9i	0.685561i
$428$	−7.36027e8	−0.453775
$429$	0	0
$430$	−1.07674e8	−0.0653087
$431$	4.89565e7i	0.0294537i	0.999892	+	0.0147269i	$0.00468788\pi$
$431$	4.89565e7i	0.0294537i	−0.999892	+	0.0147269i	$0.995312\pi$
$432$	9.63698e8	0.575106
$433$	−1.63499e9	−0.967851	−0.483925	−	0.875109i	$-0.660789\pi$
$433$	−1.63499e9	−0.967851	−0.483925	+	0.875109i	$0.660789\pi$
$434$	− 3.02816e8i	− 0.177814i
$435$	− 2.70127e8i	− 0.157346i
$436$	− 3.51403e8i	− 0.203050i
$437$	− 6.78213e8i	− 0.388760i
$438$	1.20645e9	0.686041
$439$	−1.47539e9	−0.832304	−0.416152	−	0.909295i	$-0.636622\pi$
$439$	−1.47539e9	−0.832304	−0.416152	+	0.909295i	$0.636622\pi$
$440$	6.51401e6i	0.00364556i
$441$	−2.66057e7	−0.0147720
$442$	0	0
$443$	2.19329e8	0.119863	0.0599313	−	0.998203i	$-0.480912\pi$
$443$	2.19329e8	0.119863	0.0599313	+	0.998203i	$0.480912\pi$
$444$	− 7.92776e8i	− 0.429843i
$445$	9.27898e7	0.0499161
$446$	2.94542e8	0.157208
$447$	2.20207e9i	1.16615i
$448$	− 2.96963e8i	− 0.156038i
$449$	− 1.00005e9i	− 0.521387i	−0.965422	−	0.260693i	$-0.916049\pi$
$449$	− 1.00005e9i	− 0.521387i	0.965422	−	0.260693i	$-0.0839512\pi$
$450$	− 9.06936e7i	− 0.0469173i
$451$	−4.33004e7	−0.0222267
$452$	1.86731e7	0.00951115
$453$	− 1.30132e9i	− 0.657721i
$454$	−5.91957e8	−0.296890
$455$	0	0
$456$	3.66941e8	0.181225
$457$	− 3.30790e9i	− 1.62123i	−0.585578	−	0.810616i	$-0.699132\pi$
$457$	− 3.30790e9i	− 1.62123i	0.585578	−	0.810616i	$-0.300868\pi$
$458$	8.45319e8	0.411142
$459$	−1.21754e9	−0.587678
$460$	− 4.54882e8i	− 0.217894i
$461$	− 3.94229e9i	− 1.87411i	−0.349180	−	0.937056i	$-0.613540\pi$
$461$	− 3.94229e9i	− 1.87411i	0.349180	−	0.937056i	$-0.386460\pi$
$462$	2.10249e7i	0.00991941i
$463$	5.62775e8i	0.263513i	0.991282	+	0.131756i	$0.0420616\pi$
$463$	5.62775e8i	0.263513i	−0.991282	+	0.131756i	$0.957938\pi$
$464$	1.12669e9	0.523589
$465$	−1.69424e8	−0.0781431
$466$	− 3.96373e8i	− 0.181449i
$467$	1.78505e7	0.00811038	0.00405519	−	0.999992i	$-0.498709\pi$
$467$	1.78505e7	0.00811038	0.00405519	+	0.999992i	$0.498709\pi$
$468$	0	0
$469$	3.95798e9	1.77161
$470$	− 9.81540e7i	− 0.0436079i
$471$	−1.56231e9	−0.688961
$472$	−2.83515e9	−1.24102
$473$	− 6.06971e7i	− 0.0263727i
$474$	223973.i	0 9.65987e-5i
$475$	5.95546e8i	0.254969i
$476$	1.03633e9i	0.440425i
$477$	3.83758e8	0.161898
$478$	−1.35429e9	−0.567171
$479$	− 6.41286e8i	− 0.266611i	−0.991075	−	0.133305i	$-0.957441\pi$
$479$	− 6.41286e8i	− 0.266611i	0.991075	−	0.133305i	$-0.0425591\pi$
$480$	3.79822e8	0.156760
$481$	0	0
$482$	−5.46267e8	−0.222198
$483$	− 3.21483e9i	− 1.29820i
$484$	2.09504e9	0.839910
$485$	−8.25442e8	−0.328542
$486$	− 2.43589e8i	− 0.0962565i
$487$	2.14300e9i	0.840759i	0.907348	+	0.420380i	$0.138103\pi$
$487$	2.14300e9i	0.840759i	−0.907348	+	0.420380i	$0.861897\pi$
$488$	− 1.38024e9i	− 0.537631i
$489$	− 4.59968e8i	− 0.177888i
$490$	−2.22230e7	−0.00853327
$491$	1.50315e9	0.573084	0.286542	−	0.958068i	$-0.407494\pi$
$491$	1.50315e9	0.573084	0.286542	+	0.958068i	$0.407494\pi$
$492$	1.63596e9i	0.619291i
$493$	−1.42346e9	−0.535035
$494$	0	0
$495$	−1.62291e6	−0.000601418 0
$496$	− 7.06661e8i	− 0.260031i
$497$	3.47881e9	1.27111
$498$	−1.04250e9	−0.378247
$499$	− 3.29943e9i	− 1.18874i	−0.804192	−	0.594369i	$-0.797402\pi$
$499$	− 3.29943e9i	− 1.18874i	0.804192	−	0.594369i	$-0.202598\pi$
$500$	8.11552e8i	0.290350i
$501$	8.50724e8i	0.302243i
$502$	1.54545e9i	0.545245i
$503$	−1.85586e9	−0.650215	−0.325107	−	0.945677i	$-0.605400\pi$
$503$	−1.85586e9	−0.650215	−0.325107	+	0.945677i	$0.605400\pi$
$504$	−2.39968e8	−0.0834922
$505$	− 9.41922e8i	− 0.325458i
$506$	−4.86309e7	−0.0166873
$507$	0	0
$508$	1.94535e9	0.658377
$509$	4.17614e9i	1.40366i	0.712342	+	0.701832i	$0.247634\pi$
$509$	4.17614e9i	1.40366i	−0.712342	+	0.701832i	$0.752366\pi$
$510$	−1.09964e8	−0.0367075
$511$	5.18089e9	1.71764
$512$	2.80541e9i	0.923744i
$513$	8.45481e8i	0.276499i
$514$	6.04348e8i	0.196298i
$515$	5.97978e8i	0.192912i
$516$	−2.29323e9	−0.734808
$517$	5.53305e7	0.0176095
$518$	6.45659e8i	0.204103i
$519$	2.00563e9	0.629747
$520$	0	0
$521$	1.46095e9	0.452589	0.226295	−	0.974059i	$-0.427339\pi$
$521$	1.46095e9	0.452589	0.226295	+	0.974059i	$0.427339\pi$
$522$	− 1.50531e8i	− 0.0463213i
$523$	6.21674e8	0.190023	0.0950116	−	0.995476i	$-0.469711\pi$
$523$	6.21674e8	0.190023	0.0950116	+	0.995476i	$0.469711\pi$
$524$	1.57969e9	0.479637
$525$	2.82298e9i	0.851432i
$526$	− 1.88054e9i	− 0.563420i
$527$	8.92800e8i	0.265715i
$528$	4.90642e7i	0.0145059i
$529$	4.03115e9	1.18395
$530$	3.20542e8	0.0935232
$531$	− 7.06354e8i	− 0.204735i
$532$	7.19642e8	0.207217
$533$	0	0
$534$	−3.74796e8	−0.106512
$535$	− 3.35385e8i	− 0.0946902i
$536$	−4.95317e9	−1.38933
$537$	−1.71967e9	−0.479221
$538$	− 1.34895e9i	− 0.373471i
$539$	− 1.25273e7i	− 0.00344587i
$540$	5.67069e8i	0.154974i
$541$	− 3.06369e9i	− 0.831868i	−0.909395	−	0.415934i	$-0.863455\pi$
$541$	− 3.06369e9i	− 0.831868i	0.909395	−	0.415934i	$-0.136545\pi$
$542$	−2.40506e8	−0.0648826
$543$	−2.95883e9	−0.793087
$544$	− 2.00151e9i	− 0.533043i
$545$	1.60124e8	0.0423709
$546$	0	0
$547$	−7.27564e9	−1.90071	−0.950354	−	0.311171i	$-0.899279\pi$
$547$	−7.27564e9	−1.90071	−0.950354	+	0.311171i	$0.899279\pi$
$548$	5.29127e9i	1.37350i
$549$	3.43875e8	0.0886945
$550$	4.27033e7	0.0109444
$551$	9.88475e8i	0.251730i
$552$	4.02317e9i	1.01808i
$553$	961813.i	0 0.000241854i
$554$	8.77844e8i	0.219348i
$555$	3.61243e8	0.0896963
$556$	−2.48203e9	−0.612415
$557$	5.92128e9i	1.45185i	0.687773	+	0.725926i	$0.258589\pi$
$557$	5.92128e9i	1.45185i	−0.687773	+	0.725926i	$0.741411\pi$
$558$	−9.44137e7	−0.0230046
$559$	0	0
$560$	3.73769e8	0.0899383
$561$	− 6.19880e7i	− 0.0148231i
$562$	3.32153e8	0.0789335
$563$	6.42377e9	1.51709	0.758543	−	0.651623i	$-0.225911\pi$
$563$	6.42377e9	1.51709	0.758543	+	0.651623i	$0.225911\pi$
$564$	− 2.09048e9i	− 0.490646i
$565$	8.50877e6i	0.00198471i
$566$	1.25292e9i	0.290445i
$567$	3.51457e9i	0.809715i
$568$	−4.35352e9	−0.996830
$569$	5.59404e9	1.27301	0.636506	−	0.771272i	$-0.280379\pi$
$569$	5.59404e9	1.27301	0.636506	+	0.771272i	$0.280379\pi$
$570$	7.63608e7i	0.0172706i
$571$	2.51307e9	0.564908	0.282454	−	0.959281i	$-0.408852\pi$
$571$	2.51307e9	0.564908	0.282454	+	0.959281i	$0.408852\pi$
$572$	0	0
$573$	−7.53709e9	−1.67364
$574$	− 1.33237e9i	− 0.294058i
$575$	−6.52961e9	−1.43235
$576$	−9.25889e7	−0.0201874
$577$	− 5.64709e9i	− 1.22380i	−0.790936	−	0.611899i	$-0.790406\pi$
$577$	− 5.64709e9i	− 1.22380i	0.790936	−	0.611899i	$-0.209594\pi$
$578$	− 1.27413e9i	− 0.274453i
$579$	− 1.08936e9i	− 0.233237i
$580$	6.62976e8i	0.141091i
$581$	−4.47686e9	−0.947014
$582$	3.33412e9	0.701052
$583$	1.80693e8i	0.0377661i
$584$	−6.48357e9	−1.34700
$585$	0	0
$586$	−4.03100e8	−0.0827506
$587$	8.31882e9i	1.69757i	0.528736	+	0.848786i	$0.322666\pi$
$587$	8.31882e9i	1.69757i	−0.528736	+	0.848786i	$0.677334\pi$
$588$	−4.73303e8	−0.0960105
$589$	6.19974e8	0.125017
$590$	− 5.89998e8i	− 0.118268i
$591$	6.03456e9i	1.20251i
$592$	1.50673e9i	0.298476i
$593$	− 2.05874e9i	− 0.405425i	−0.979238	−	0.202713i	$-0.935024\pi$
$593$	− 2.05874e9i	− 0.405425i	0.979238	−	0.202713i	$-0.0649757\pi$
$594$	6.06248e7	0.0118685
$595$	−4.72222e8	−0.0919045
$596$	− 5.40457e9i	− 1.04568i
$597$	4.30428e9	0.827923
$598$	0	0
$599$	−3.14417e9	−0.597740	−0.298870	−	0.954294i	$-0.596610\pi$
$599$	−3.14417e9	−0.597740	−0.298870	+	0.954294i	$0.596610\pi$
$600$	− 3.53278e9i	− 0.667710i
$601$	−4.17491e9	−0.784489	−0.392244	−	0.919861i	$-0.628301\pi$
$601$	−4.17491e9	−0.784489	−0.392244	+	0.919861i	$0.628301\pi$
$602$	1.86767e9	0.348909
$603$	− 1.23404e9i	− 0.229202i
$604$	3.19385e9i	0.589774i
$605$	9.54642e8i	0.175266i
$606$	3.80460e9i	0.694473i
$607$	1.55868e9	0.282877	0.141438	−	0.989947i	$-0.454827\pi$
$607$	1.55868e9	0.282877	0.141438	+	0.989947i	$0.454827\pi$
$608$	−1.38988e9	−0.250793
$609$	4.68552e9i	0.840615i
$610$	2.87229e8	0.0512358
$611$	0	0
$612$	3.23112e8	0.0569801
$613$	2.10767e9i	0.369566i	0.982779	+	0.184783i	$0.0591582\pi$
$613$	2.10767e9i	0.369566i	−0.982779	+	0.184783i	$0.940842\pi$
$614$	−9.17641e8	−0.159987
$615$	−7.45456e8	−0.129229
$616$	− 1.12989e8i	− 0.0194763i
$617$	3.64167e9i	0.624169i	0.950054	+	0.312084i	$0.101027\pi$
$617$	3.64167e9i	0.624169i	−0.950054	+	0.312084i	$0.898973\pi$
$618$	− 2.41535e9i	− 0.411642i
$619$	3.99887e9i	0.677672i	0.940845	+	0.338836i	$0.110033\pi$
$619$	3.99887e9i	0.677672i	−0.940845	+	0.338836i	$0.889967\pi$
$620$	4.15821e8	0.0700705
$621$	−9.26991e9	−1.55330
$622$	3.53269e9i	0.588626i
$623$	−1.60950e9	−0.266675
$624$	0	0
$625$	5.54593e9	0.908645
$626$	1.73700e9i	0.283003i
$627$	−4.30455e7	−0.00697415
$628$	3.83440e9	0.617788
$629$	− 1.90361e9i	− 0.305001i
$630$	− 4.99375e7i	− 0.00795674i
$631$	− 5.76033e9i	− 0.912735i	−0.889791	−	0.456368i	$-0.849150\pi$
$631$	− 5.76033e9i	− 0.912735i	0.889791	−	0.456368i	$-0.150850\pi$
$632$	− 1.20365e6i	0 0.000189666i
$633$	4.09956e9	0.642427
$634$	−1.89654e9	−0.295563
$635$	8.86435e8i	0.137385i
$636$	6.82688e9	1.05226
$637$	0	0
$638$	7.08781e7	0.0108054
$639$	− 1.08464e9i	− 0.164450i
$640$	−1.18633e9	−0.178886
$641$	6.04766e9	0.906952	0.453476	−	0.891268i	$-0.350184\pi$
$641$	6.04766e9	0.906952	0.453476	+	0.891268i	$0.350184\pi$
$642$	1.35468e9i	0.202053i
$643$	9.02692e9i	1.33906i	0.742784	+	0.669532i	$0.233505\pi$
$643$	9.02692e9i	1.33906i	−0.742784	+	0.669532i	$0.766495\pi$
$644$	7.89020e9i	1.16409i
$645$	− 1.04495e9i	− 0.153334i
$646$	4.02391e8	0.0587266
$647$	−1.99531e9	−0.289631	−0.144816	−	0.989459i	$-0.546259\pi$
$647$	−1.99531e9	−0.289631	−0.144816	+	0.989459i	$0.546259\pi$
$648$	− 4.39827e9i	− 0.634994i
$649$	3.32589e8	0.0477585
$650$	0	0
$651$	2.93877e9	0.417477
$652$	1.12891e9i	0.159511i
$653$	5.54932e9	0.779909	0.389954	−	0.920834i	$-0.372491\pi$
$653$	5.54932e9	0.779909	0.389954	+	0.920834i	$0.372491\pi$
$654$	−6.46770e8	−0.0904123
$655$	7.19816e8i	0.100087i
$656$	− 3.10926e9i	− 0.430025i
$657$	− 1.61533e9i	− 0.222219i
$658$	1.70254e9i	0.232974i
$659$	−8.54955e7	−0.0116371	−0.00581854	−	0.999983i	$-0.501852\pi$
$659$	−8.54955e7	−0.0116371	−0.00581854	+	0.999983i	$0.501852\pi$
$660$	−2.88709e7	−0.00390891
$661$	− 1.03382e10i	− 1.39233i	−0.717882	−	0.696164i	$-0.754888\pi$
$661$	− 1.03382e10i	− 1.39233i	0.717882	−	0.696164i	$-0.245112\pi$
$662$	−3.69632e9	−0.495184
$663$	0	0
$664$	5.60251e9	0.742667
$665$	3.27918e8i	0.0432404i
$666$	2.01307e8	0.0264058
$667$	−1.08377e10	−1.41416
$668$	− 2.08794e9i	− 0.271020i
$669$	2.85848e9i	0.369099i
$670$	− 1.03076e9i	− 0.132402i
$671$	1.61914e8i	0.0206898i
$672$	−6.58825e9	−0.837486
$673$	−8.90146e9	−1.12566	−0.562832	−	0.826571i	$-0.690288\pi$
$673$	−8.90146e9	−1.12566	−0.562832	+	0.826571i	$0.690288\pi$
$674$	2.03362e9i	0.255835i
$675$	8.14000e9	1.01874
$676$	0	0
$677$	−8.37714e9	−1.03761	−0.518807	−	0.854892i	$-0.673624\pi$
$677$	−8.37714e9	−1.03761	−0.518807	+	0.854892i	$0.673624\pi$
$678$	− 3.43685e7i	− 0.00423504i
$679$	1.43178e10	1.75522
$680$	5.90957e8	0.0720733
$681$	− 5.74483e9i	− 0.697047i
$682$	− 4.44549e7i	− 0.00536630i
$683$	1.08780e10i	1.30640i	0.757185	+	0.653200i	$0.226574\pi$
$683$	1.08780e10i	1.30640i	−0.757185	+	0.653200i	$0.773426\pi$
$684$	− 2.24374e8i	− 0.0268087i
$685$	−2.41107e9	−0.286611
$686$	3.54912e9	0.419746
$687$	8.20365e9i	0.965292i
$688$	4.35845e9	0.510238
$689$	0	0
$690$	−8.37225e8	−0.0970220
$691$	− 1.00168e10i	− 1.15493i	−0.816414	−	0.577467i	$-0.804041\pi$
$691$	− 1.00168e10i	− 1.15493i	0.816414	−	0.577467i	$-0.195959\pi$
$692$	−4.92245e9	−0.564690
$693$	2.81504e7	0.00321305
$694$	− 5.08843e9i	− 0.577864i
$695$	− 1.13099e9i	− 0.127794i
$696$	− 5.86364e9i	− 0.659227i
$697$	3.92826e9i	0.439425i
$698$	4.60977e9	0.513080
$699$	3.84673e9	0.426011
$700$	− 6.92847e9i	− 0.763474i
$701$	−1.48395e10	−1.62707	−0.813537	−	0.581513i	$-0.802461\pi$
$701$	−1.48395e10	−1.62707	−0.813537	+	0.581513i	$0.802461\pi$
$702$	0	0
$703$	−1.32190e9	−0.143501
$704$	− 4.35957e7i	− 0.00470912i
$705$	9.52565e8	0.102384
$706$	1.48297e7	0.00158605
$707$	1.63382e10i	1.73875i
$708$	− 1.25657e10i	− 1.33067i
$709$	− 1.35650e10i	− 1.42941i	−0.699424	−	0.714707i	$-0.746560\pi$
$709$	− 1.35650e10i	− 1.42941i	0.699424	−	0.714707i	$-0.253440\pi$
$710$	− 9.05971e8i	− 0.0949971i
$711$	299879.	3.12898e−5 0
$712$	2.01419e9	0.209132
$713$	6.79744e9i	0.702315i
$714$	1.90739e9	0.196109
$715$	0	0
$716$	4.22061e9	0.429715
$717$	− 1.31431e10i	− 1.33162i
$718$	−4.59604e9	−0.463392
$719$	−8.17292e9	−0.820023	−0.410011	−	0.912080i	$-0.634475\pi$
$719$	−8.17292e9	−0.820023	−0.410011	+	0.912080i	$0.634475\pi$
$720$	− 1.16536e8i	− 0.0116358i
$721$	− 1.03723e10i	− 1.03063i
$722$	3.75841e9i	0.371641i
$723$	− 5.30141e9i	− 0.521684i
$724$	7.26190e9	0.711157
$725$	9.51671e9	0.927478
$726$	− 3.85598e9i	− 0.373987i
$727$	−7.62775e8	−0.0736251	−0.0368126	−	0.999322i	$-0.511720\pi$
$727$	−7.62775e8	−0.0736251	−0.0368126	+	0.999322i	$0.511720\pi$
$728$	0	0
$729$	1.14024e10	1.09006
$730$	− 1.34924e9i	− 0.128368i
$731$	−5.50650e9	−0.521392
$732$	6.11738e9	0.576470
$733$	8.21437e9i	0.770389i	0.922835	+	0.385195i	$0.125866\pi$
$733$	8.21437e9i	0.770389i	−0.922835	+	0.385195i	$0.874134\pi$
$734$	− 6.16617e9i	− 0.575545i
$735$	− 2.15669e8i	− 0.0200347i
$736$	− 1.52388e10i	− 1.40889i
$737$	5.81052e8	0.0534661
$738$	−4.15414e8	−0.0380438
$739$	8.22305e9i	0.749510i	0.927124	+	0.374755i	$0.122273\pi$
$739$	8.22305e9i	0.749510i	−0.927124	+	0.374755i	$0.877727\pi$
$740$	−8.86604e8	−0.0804301
$741$	0	0
$742$	−5.56000e9	−0.499644
$743$	1.37678e10i	1.23142i	0.787975	+	0.615708i	$0.211130\pi$
$743$	1.37678e10i	1.23142i	−0.787975	+	0.615708i	$0.788870\pi$
$744$	−3.67769e9	−0.327394
$745$	2.46269e9	0.218204
$746$	4.35978e9i	0.384484i
$747$	1.39582e9i	0.122520i
$748$	1.52138e8i	0.0132917i
$749$	5.81745e9i	0.505879i
$750$	1.49369e9	0.129284
$751$	1.23300e10	1.06224	0.531119	−	0.847297i	$-0.321772\pi$
$751$	1.23300e10	1.06224	0.531119	+	0.847297i	$0.321772\pi$
$752$	3.97310e9i	0.340696i
$753$	−1.49983e10	−1.28014
$754$	0	0
$755$	−1.45534e9	−0.123069
$756$	− 9.83616e9i	− 0.827941i
$757$	1.84529e10	1.54607	0.773036	−	0.634362i	$-0.218737\pi$
$757$	1.84529e10	1.54607	0.773036	+	0.634362i	$0.218737\pi$
$758$	−3.68199e9	−0.307072
$759$	− 4.71954e8i	− 0.0391790i
$760$	− 4.10370e8i	− 0.0339100i
$761$	1.45618e10i	1.19776i	0.800839	+	0.598880i	$0.204387\pi$
$761$	1.45618e10i	1.19776i	−0.800839	+	0.598880i	$0.795613\pi$
$762$	− 3.58048e9i	− 0.293156i
$763$	−2.77744e9	−0.226365
$764$	1.84984e10	1.50074
$765$	1.47232e8i	0.0118901i
$766$	5.78678e9	0.465196
$767$	0	0
$768$	2.83233e9	0.225621
$769$	4.69706e9i	0.372464i	0.982506	+	0.186232i	$0.0596276\pi$
$769$	4.69706e9i	0.372464i	−0.982506	+	0.186232i	$0.940372\pi$
$770$	2.35132e7	0.00185607
$771$	−5.86508e9	−0.460876
$772$	2.67363e9i	0.209142i
$773$	− 1.42447e10i	− 1.10924i	−0.832104	−	0.554619i	$-0.812864\pi$
$773$	− 1.42447e10i	− 1.10924i	0.832104	−	0.554619i	$-0.187136\pi$
$774$	− 5.82313e8i	− 0.0451402i
$775$	− 5.96890e9i	− 0.460616i
$776$	−1.79179e10	−1.37648
$777$	−6.26599e9	−0.479199
$778$	− 4.29795e9i	− 0.327214i
$779$	2.72784e9	0.206747
$780$	0	0
$781$	5.10707e8	0.0383613
$782$	4.41184e9i	0.329911i
$783$	1.35106e10	1.00579
$784$	8.99546e8	0.0666680
$785$	1.74722e9i	0.128915i
$786$	− 2.90747e9i	− 0.213568i
$787$	− 2.03693e10i	− 1.48958i	−0.667300	−	0.744789i	$-0.732550\pi$
$787$	− 2.03693e10i	− 1.48958i	0.667300	−	0.744789i	$-0.267450\pi$
$788$	− 1.48107e10i	− 1.07829i
$789$	1.82502e10	1.32282
$790$	250481.	1.80751e−5 0
$791$	− 1.47590e8i	− 0.0106032i
$792$	−3.52285e7	−0.00251974
$793$	0	0
$794$	1.11437e9	0.0790059
$795$	3.11079e9i	0.219577i
$796$	−1.05640e10	−0.742394
$797$	−1.18633e10	−0.830045	−0.415022	−	0.909811i	$-0.636226\pi$
$797$	−1.18633e10	−0.830045	−0.415022	+	0.909811i	$0.636226\pi$
$798$	− 1.32452e9i	− 0.0922678i
$799$	− 5.01964e9i	− 0.348144i
$800$	1.33813e10i	0.924026i
$801$	5.01818e8i	0.0345011i
$802$	9.14762e9	0.626178
$803$	7.60581e8	0.0518371
$804$	− 2.19531e10i	− 1.48970i
$805$	−3.59532e9	−0.242914
$806$	0	0
$807$	1.30913e10	0.876848
$808$	− 2.04463e10i	− 1.36356i
$809$	−8.79991e9	−0.584330	−0.292165	−	0.956368i	$-0.594376\pi$
$809$	−8.79991e9	−0.584330	−0.292165	+	0.956368i	$0.594376\pi$
$810$	9.15285e8	0.0605144
$811$	6.04080e9i	0.397669i	0.980033	+	0.198834i	$0.0637156\pi$
$811$	6.04080e9i	0.397669i	−0.980033	+	0.198834i	$0.936284\pi$
$812$	− 1.14997e10i	− 0.753775i
$813$	− 2.33406e9i	− 0.152333i
$814$	9.47860e7i	0.00615969i
$815$	−5.14407e8	−0.0332855
$816$	4.45115e9	0.286785
$817$	3.82380e9i	0.245311i
$818$	6.55024e9	0.418428
$819$	0	0
$820$	1.82958e9	0.115879
$821$	1.79851e10i	1.13426i	0.823629	+	0.567129i	$0.191946\pi$
$821$	1.79851e10i	1.13426i	−0.823629	+	0.567129i	$0.808054\pi$
$822$	9.73877e9	0.611579
$823$	−2.38986e10	−1.49442	−0.747210	−	0.664588i	$-0.768607\pi$
$823$	−2.38986e10	−1.49442	−0.747210	+	0.664588i	$0.768607\pi$
$824$	1.29803e10i	0.808237i
$825$	4.14427e8i	0.0256956i
$826$	1.02339e10i	0.631844i
$827$	2.40368e10i	1.47777i	0.673829	+	0.738887i	$0.264648\pi$
$827$	2.40368e10i	1.47777i	−0.673829	+	0.738887i	$0.735352\pi$
$828$	2.46005e9	0.150605
$829$	−2.77111e10	−1.68933	−0.844663	−	0.535298i	$-0.820199\pi$
$829$	−2.77111e10	−1.68933	−0.844663	+	0.535298i	$0.820199\pi$
$830$	1.16589e9i	0.0707756i
$831$	−8.51930e9	−0.514992
$832$	0	0
$833$	−1.13649e9	−0.0681254
$834$	4.56827e9i	0.272691i
$835$	9.51411e8	0.0565543
$836$	1.05647e8	0.00625368
$837$	− 8.47389e9i	− 0.499510i
$838$	3.16947e9i	0.186051i
$839$	− 5.42082e9i	− 0.316883i	−0.987368	−	0.158441i	$-0.949353\pi$
$839$	− 5.42082e9i	− 0.316883i	0.987368	−	0.158441i	$-0.0506469\pi$
$840$	− 1.94521e9i	− 0.113237i
$841$	−1.45424e9	−0.0843045
$842$	−2.73107e9	−0.157667
$843$	3.22348e9i	0.185323i
$844$	−1.00616e10	−0.576061
$845$	0	0
$846$	5.30828e8	0.0301410
$847$	− 1.65589e10i	− 0.936350i
$848$	−1.29750e10	−0.730669
$849$	−1.21593e10	−0.681917
$850$	− 3.87409e9i	− 0.216373i
$851$	− 1.44934e10i	− 0.806150i
$852$	− 1.92953e10i	− 1.06884i
$853$	9.15495e8i	0.0505050i	0.999681	+	0.0252525i	$0.00803898\pi$
$853$	9.15495e8i	0.0505050i	−0.999681	+	0.0252525i	$0.991961\pi$
$854$	−4.98216e9	−0.273725
$855$	1.02240e8	0.00559423
$856$	− 7.28019e9i	− 0.396720i
$857$	−5.57192e9	−0.302393	−0.151197	−	0.988504i	$-0.548313\pi$
$857$	−5.57192e9	−0.302393	−0.151197	+	0.988504i	$0.548313\pi$
$858$	0	0
$859$	−3.79832e9	−0.204463	−0.102232	−	0.994761i	$-0.532598\pi$
$859$	−3.79832e9	−0.204463	−0.102232	+	0.994761i	$0.532598\pi$
$860$	2.56464e9i	0.137494i
$861$	1.29304e10	0.690400
$862$	−2.21149e8	−0.0117600
$863$	8.87979e9i	0.470289i	0.971960	+	0.235145i	$0.0755563\pi$
$863$	8.87979e9i	0.470289i	−0.971960	+	0.235145i	$0.924444\pi$
$864$	1.89971e10i	1.00205i
$865$	− 2.24301e9i	− 0.117835i
$866$	− 7.38566e9i	− 0.386435i
$867$	1.23652e10	0.644369
$868$	−7.21266e9	−0.374349
$869$	141199.i	0 7.29899e-6i
$870$	1.22023e9	0.0628238
$871$	0	0
$872$	3.47580e9	0.177520
$873$	− 4.46408e9i	− 0.227082i
$874$	3.06365e9	0.155221
$875$	6.41439e9	0.323689
$876$	− 2.87360e10i	− 1.44431i
$877$	2.78832e10i	1.39587i	0.716162	+	0.697934i	$0.245897\pi$
$877$	2.78832e10i	1.39587i	−0.716162	+	0.697934i	$0.754103\pi$
$878$	− 6.66471e9i	− 0.332315i
$879$	− 3.91200e9i	− 0.194285i
$880$	5.48711e7	0.00271428
$881$	3.07787e10	1.51647	0.758237	−	0.651979i	$-0.226061\pi$
$881$	3.07787e10	1.51647	0.758237	+	0.651979i	$0.226061\pi$
$882$	− 1.20184e8i	− 0.00589804i
$883$	1.45010e9	0.0708820	0.0354410	−	0.999372i	$-0.488716\pi$
$883$	1.45010e9	0.0708820	0.0354410	+	0.999372i	$0.488716\pi$
$884$	0	0
$885$	5.72581e9	0.277674
$886$	9.90763e8i	0.0478577i
$887$	−9.25333e9	−0.445211	−0.222605	−	0.974909i	$-0.571456\pi$
$887$	−9.25333e9	−0.445211	−0.222605	+	0.974909i	$0.571456\pi$
$888$	7.84150e9	0.375798
$889$	− 1.53758e10i	− 0.733973i
$890$	4.19154e8i	0.0199301i
$891$	5.15957e8i	0.0244367i
$892$	− 7.01559e9i	− 0.330969i
$893$	−3.48572e9	−0.163799
$894$	−9.94729e9	−0.465611
$895$	1.92320e9i	0.0896694i
$896$	2.05777e10	0.955693
$897$	0	0
$898$	4.51748e9	0.208175
$899$	− 9.90706e9i	− 0.454764i
$900$	−2.16020e9	−0.0987744
$901$	1.63927e10	0.746642
$902$	− 1.95599e8i	− 0.00887448i
$903$	1.81254e10i	0.819181i
$904$	1.84700e8i	0.00831527i
$905$	3.30902e9i	0.148399i
$906$	5.87840e9	0.262609
$907$	−2.13718e10	−0.951078	−0.475539	−	0.879695i	$-0.657747\pi$
$907$	−2.13718e10	−0.951078	−0.475539	+	0.879695i	$0.657747\pi$
$908$	1.40996e10i	0.625038i
$909$	5.09402e9	0.224951
$910$	0	0
$911$	−2.45761e10	−1.07696	−0.538479	−	0.842639i	$-0.681001\pi$
$911$	−2.45761e10	−1.07696	−0.538479	+	0.842639i	$0.681001\pi$
$912$	− 3.09095e9i	− 0.134930i
$913$	−6.57225e8	−0.0285803
$914$	1.49426e10	0.647312
$915$	2.78750e9i	0.120293i
$916$	− 2.01343e10i	− 0.865572i
$917$	− 1.24857e10i	− 0.534710i
$918$	− 5.49994e9i	− 0.234643i
$919$	−1.04226e9	−0.0442966	−0.0221483	−	0.999755i	$-0.507051\pi$
$919$	−1.04226e9	−0.0442966	−0.0221483	+	0.999755i	$0.507051\pi$
$920$	4.49932e9	0.190498
$921$	− 8.90552e9i	− 0.375622i
$922$	1.78083e10	0.748280
$923$	0	0
$924$	5.00783e8	0.0208832
$925$	1.27268e10i	0.528716i
$926$	−2.54219e9	−0.105213
$927$	−3.23393e9	−0.133337
$928$	2.22100e10i	0.912287i
$929$	2.61948e10i	1.07191i	0.844245	+	0.535957i	$0.180049\pi$
$929$	2.61948e10i	1.07191i	−0.844245	+	0.535957i	$0.819951\pi$
$930$	− 7.65331e8i	− 0.0312003i
$931$	7.89198e8i	0.0320525i
$932$	−9.44107e9	−0.382002
$933$	−3.42841e10	−1.38200
$934$	8.06350e7i	0.00323824i
$935$	−6.93245e7	−0.00277362
$936$	0	0
$937$	−1.84464e10	−0.732525	−0.366263	−	0.930512i	$-0.619363\pi$
$937$	−1.84464e10	−0.732525	−0.366263	+	0.930512i	$0.619363\pi$
$938$	1.78792e10i	0.707355i
$939$	−1.68573e10	−0.664443
$940$	−2.33789e9	−0.0918072
$941$	− 2.74685e10i	− 1.07466i	−0.843373	−	0.537329i	$-0.819433\pi$
$941$	− 2.74685e10i	− 1.07466i	0.843373	−	0.537329i	$-0.180567\pi$
$942$	− 7.05735e9i	− 0.275083i
$943$	2.99083e10i	1.16145i
$944$	2.38821e10i	0.923995i
$945$	4.48203e9	0.172768
$946$	2.74184e8	0.0105299
$947$	1.25081e10i	0.478593i	0.970947	+	0.239297i	$0.0769168\pi$
$947$	1.25081e10i	0.478593i	−0.970947	+	0.239297i	$0.923083\pi$
$948$	5.33472e6	0.000203368 0
$949$	0	0
$950$	−2.69023e9	−0.101802
$951$	− 1.84056e10i	− 0.693933i
$952$	−1.02505e10	−0.385049
$953$	4.44972e10	1.66536	0.832679	−	0.553755i	$-0.186806\pi$
$953$	4.44972e10	1.66536	0.832679	+	0.553755i	$0.186806\pi$
$954$	1.73353e9i	0.0646415i
$955$	8.42913e9i	0.313163i
$956$	3.22573e10i	1.19406i
$957$	6.87858e8i	0.0253692i
$958$	2.89685e9	0.106450
$959$	4.18215e10	1.53121
$960$	− 7.50539e8i	− 0.0273794i
$961$	2.12989e10	0.774150
$962$	0	0
$963$	1.81380e9	0.0654481
$964$	1.30113e10i	0.467791i
$965$	−1.21829e9	−0.0436421
$966$	1.45222e10	0.518337
$967$	1.03373e10i	0.367632i	0.982961	+	0.183816i	$0.0588450\pi$
$967$	1.03373e10i	0.367632i	−0.982961	+	0.183816i	$0.941155\pi$
$968$	2.07224e10i	0.734305i
$969$	3.90513e9i	0.137880i
$970$	− 3.72872e9i	− 0.131177i
$971$	−3.56148e10	−1.24843	−0.624214	−	0.781254i	$-0.714581\pi$
$971$	−3.56148e10	−1.24843	−0.624214	+	0.781254i	$0.714581\pi$
$972$	−5.80194e9	−0.202648
$973$	1.96176e10i	0.682734i
$974$	−9.68047e9	−0.335691
$975$	0	0
$976$	−1.16265e10	−0.400290
$977$	4.78316e10i	1.64091i	0.571713	+	0.820454i	$0.306279\pi$
$977$	4.78316e10i	1.64091i	−0.571713	+	0.820454i	$0.693721\pi$
$978$	2.07779e9	0.0710257
$979$	−2.36282e8	−0.00804807
$980$	5.29320e8i	0.0179650i
$981$	8.65966e8i	0.0292860i
$982$	6.79011e9i	0.228816i
$983$	1.01346e10i	0.340306i	0.985418	+	0.170153i	$0.0544262\pi$
$983$	1.01346e10i	0.340306i	−0.985418	+	0.170153i	$0.945574\pi$
$984$	−1.61816e10	−0.541425
$985$	6.74878e9	0.225008
$986$	− 6.43013e9i	− 0.213624i
$987$	−1.65228e10	−0.546983
$988$	0	0
$989$	−4.19244e10	−1.37810
$990$	− 7.33108e6i	0 0.000240129i
$991$	−1.22781e10	−0.400749	−0.200375	−	0.979719i	$-0.564216\pi$
$991$	−1.22781e10	−0.400749	−0.200375	+	0.979719i	$0.564216\pi$
$992$	1.39302e10	0.453071
$993$	− 3.58720e10i	− 1.16261i
$994$	1.57146e10i	0.507518i
$995$	− 4.81371e9i	− 0.154917i
$996$	2.48310e10i	0.796318i
$997$	−3.28608e10	−1.05013	−0.525067	−	0.851061i	$-0.675960\pi$
$997$	−3.28608e10	−1.05013	−0.525067	+	0.851061i	$0.675960\pi$
$998$	1.49043e10	0.474630
$999$	1.80679e10i	0.573360i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.b.d.168.9		14
13.5	odd	4		169.8.a.g.1.9		14
13.8	odd	4		169.8.a.g.1.6		14
13.9	even	3		13.8.e.a.10.5	yes	14
13.10	even	6		13.8.e.a.4.5	✓	14
13.12	even	2	inner	169.8.b.d.168.6		14
39.23	odd	6		117.8.q.b.82.3		14
39.35	odd	6		117.8.q.b.10.3		14
52.23	odd	6		208.8.w.a.17.3		14
52.35	odd	6		208.8.w.a.49.3		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.e.a.4.5	✓	14	13.10	even	6
13.8.e.a.10.5	yes	14	13.9	even	3
117.8.q.b.10.3		14	39.35	odd	6
117.8.q.b.82.3		14	39.23	odd	6
169.8.a.g.1.6		14	13.8	odd	4
169.8.a.g.1.9		14	13.5	odd	4
169.8.b.d.168.6		14	13.12	even	2	inner
169.8.b.d.168.9		14	1.1	even	1	trivial
208.8.w.a.17.3		14	52.23	odd	6
208.8.w.a.49.3		14	52.35	odd	6

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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.51724i q^{2} -43.8390 q^{3} +107.595 q^{4} +49.0275i q^{5} -198.031i q^{6} -850.411i q^{7} +1064.24i q^{8} -265.146 q^{9} -221.469 q^{10} -124.845i q^{11} -4716.83 q^{12} +3841.52 q^{14} -2149.31i q^{15} +8964.67 q^{16} -11326.0 q^{17} -1197.73i q^{18} +7864.97i q^{19} +5275.09i q^{20} +37281.1i q^{21} +563.954 q^{22} -86232.1 q^{23} -46655.1i q^{24} +75721.3 q^{25} +107500. q^{27} -91499.6i q^{28} +125681. q^{29} +9708.97 q^{30} -78827.3i q^{31} +176718. i q^{32} +5473.06i q^{33} -51162.4i q^{34} +41693.5 q^{35} -28528.3 q^{36} +168074. i q^{37} -35528.0 q^{38} -52176.9 q^{40} -346835. i q^{41} -168408. q^{42} +486181. q^{43} -13432.6i q^{44} -12999.4i q^{45} -389531. i q^{46} +443195. i q^{47} -393002. q^{48} +100343. q^{49} +342052. i q^{50} +496521. q^{51} -1.44734e6 q^{53} +485602. i q^{54} +6120.82 q^{55} +905040. q^{56} -344792. i q^{57} +567730. i q^{58} +2.66402e6i q^{59} -231254. i q^{60} -1.29692e6 q^{61} +356082. q^{62} +225483. i q^{63} +349200. q^{64} -24723.2 q^{66} +4.65420e6i q^{67} -1.21862e6 q^{68} +3.78033e6 q^{69} +188340. i q^{70} +4.09074e6i q^{71} -282178. i q^{72} +6.09222e6i q^{73} -759231. q^{74} -3.31954e6 q^{75} +846228. i q^{76} -106169. q^{77} -1131.00 q^{79} +439515. i q^{80} -4.13279e6 q^{81} +1.56674e6 q^{82} -5.26434e6i q^{83} +4.01125e6i q^{84} -555286. i q^{85} +2.19620e6i q^{86} -5.50971e6 q^{87} +132864. q^{88} -1.89261e6i q^{89} +58721.6 q^{90} -9.27810e6 q^{92} +3.45570e6i q^{93} -2.00202e6 q^{94} -385600. q^{95} -7.74714e6i q^{96} +1.68363e7i q^{97} +453276. i q^{98} +33102.1i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 52 q^{3} - 766 q^{4} + 6982 q^{9} + 1018 q^{10} + 38380 q^{12} - 47916 q^{14} + 1266 q^{16} + 76806 q^{17} + 251764 q^{22} + 137100 q^{23} + 39380 q^{25} - 432400 q^{27} - 443166 q^{29} + 315780 q^{30}+ \cdots + 86840772 q^{95}+O(q^{100}) \) 14 * q - 52 * q^3 - 766 * q^4 + 6982 * q^9 + 1018 * q^10 + 38380 * q^12 - 47916 * q^14 + 1266 * q^16 + 76806 * q^17 + 251764 * q^22 + 137100 * q^23 + 39380 * q^25 - 432400 * q^27 - 443166 * q^29 + 315780 * q^30 - 1319232 * q^35 - 1271350 * q^36 + 953640 * q^38 - 77182 * q^40 - 412812 * q^42 + 1765384 * q^43 - 2267260 * q^48 - 5307958 * q^49 - 38964 * q^51 - 1791210 * q^53 - 4907072 * q^55 + 14915904 * q^56 - 7717050 * q^61 - 3808284 * q^62 + 14570922 * q^64 - 31956396 * q^66 - 19139238 * q^68 + 27487680 * q^69 - 17280210 * q^74 - 20632304 * q^75 + 19599336 * q^77 - 30688440 * q^79 - 27772442 * q^81 + 74719438 * q^82 - 25539372 * q^87 - 54091072 * q^88 + 41193546 * q^90 - 47685588 * q^92 - 38277620 * q^94 + 86840772 * q^95

Introduction

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