LMFDB - Embedded newform 169.8.b.d.168.12

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			168.12
Root			$16.7213i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.8.b.d.168.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+16.7213i q^{2} +42.5564 q^{3} -151.602 q^{4} -94.5127i q^{5} +711.598i q^{6} +1415.38i q^{7} -394.655i q^{8} -375.955 q^{9} +1580.37 q^{10} -1433.00i q^{11} -6451.63 q^{12} -23667.0 q^{14} -4022.12i q^{15} -12805.9 q^{16} +2788.62 q^{17} -6286.46i q^{18} +28732.4i q^{19} +14328.3i q^{20} +60233.5i q^{21} +23961.7 q^{22} +50599.7 q^{23} -16795.1i q^{24} +69192.4 q^{25} -109070. q^{27} -214575. i q^{28} -228357. q^{29} +67255.0 q^{30} +154870. i q^{31} -264647. i q^{32} -60983.4i q^{33} +46629.4i q^{34} +133772. q^{35} +56995.5 q^{36} +73482.1i q^{37} -480443. q^{38} -37299.9 q^{40} -685869. i q^{41} -1.00718e6 q^{42} -787730. q^{43} +217246. i q^{44} +35532.5i q^{45} +846094. i q^{46} -690181. i q^{47} -544973. q^{48} -1.17976e6 q^{49} +1.15699e6i q^{50} +118674. q^{51} -1.69205e6 q^{53} -1.82379e6i q^{54} -135437. q^{55} +558588. q^{56} +1.22275e6i q^{57} -3.81843e6i q^{58} -493676. i q^{59} +609761. i q^{60} +2.20773e6 q^{61} -2.58963e6 q^{62} -532120. i q^{63} +2.78609e6 q^{64} +1.01972e6 q^{66} +2.25744e6i q^{67} -422760. q^{68} +2.15334e6 q^{69} +2.23683e6i q^{70} +403562. i q^{71} +148373. i q^{72} +509572. i q^{73} -1.22872e6 q^{74} +2.94458e6 q^{75} -4.35589e6i q^{76} +2.02825e6 q^{77} -1.91796e6 q^{79} +1.21032e6i q^{80} -3.81941e6 q^{81} +1.14686e7 q^{82} +4.57733e6i q^{83} -9.13152e6i q^{84} -263560. i q^{85} -1.31719e7i q^{86} -9.71806e6 q^{87} -565542. q^{88} -3.52649e6i q^{89} -594150. q^{90} -7.67102e6 q^{92} +6.59070e6i q^{93} +1.15407e7 q^{94} +2.71558e6 q^{95} -1.12624e7i q^{96} +4.86752e6i q^{97} -1.97272e7i q^{98} +538744. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	16.7213i	1.47797i	0.673723	+	0.738984i	$0.264694\pi$
$2$	16.7213i	1.47797i	−0.673723	+	0.738984i	$0.735306\pi$
$3$	42.5564	0.909998	0.454999	−	0.890492i	$-0.349640\pi$
$3$	42.5564	0.909998	0.454999	+	0.890492i	$0.349640\pi$
$4$	−151.602	−1.18439
$5$	− 94.5127i	− 0.338139i	−0.985604	−	0.169069i	$-0.945924\pi$
$5$	− 94.5127i	− 0.338139i	0.985604	−	0.169069i	$-0.0540762\pi$
$6$	711.598i	1.34495i
$7$	1415.38i	1.55966i	0.625990	+	0.779831i	$0.284695\pi$
$7$	1415.38i	1.55966i	−0.625990	+	0.779831i	$0.715305\pi$
$8$	− 394.655i	− 0.272523i
$9$	−375.955	−0.171904
$10$	1580.37	0.499758
$11$	− 1433.00i	− 0.324618i	−0.986740	−	0.162309i	$-0.948106\pi$
$11$	− 1433.00i	− 0.324618i	0.986740	−	0.162309i	$-0.0518941\pi$
$12$	−6451.63	−1.07779
$13$	0	0
$13$	0	0
$14$	−23667.0	−2.30513
$15$	− 4022.12i	− 0.307705i
$16$	−12805.9	−0.781610
$17$	2788.62	0.137663	0.0688316	−	0.997628i	$-0.478073\pi$
$17$	2788.62	0.137663	0.0688316	+	0.997628i	$0.478073\pi$
$18$	− 6286.46i	− 0.254069i
$19$	28732.4i	0.961025i	0.876988	+	0.480512i	$0.159549\pi$
$19$	28732.4i	0.961025i	−0.876988	+	0.480512i	$0.840451\pi$
$20$	14328.3i	0.400488i
$21$	60233.5i	1.41929i
$22$	23961.7	0.479775
$23$	50599.7	0.867163	0.433582	−	0.901114i	$-0.357250\pi$
$23$	50599.7	0.867163	0.433582	+	0.901114i	$0.357250\pi$
$24$	− 16795.1i	− 0.247995i
$25$	69192.4	0.885662
$26$	0	0
$27$	−109070.	−1.06643
$28$	− 214575.i	− 1.84725i
$29$	−228357.	−1.73869	−0.869344	−	0.494207i	$-0.835459\pi$
$29$	−228357.	−1.73869	−0.869344	+	0.494207i	$0.835459\pi$
$30$	67255.0	0.454779
$31$	154870.i	0.933686i	0.884340	+	0.466843i	$0.154609\pi$
$31$	154870.i	0.933686i	−0.884340	+	0.466843i	$0.845391\pi$
$32$	− 264647.i	− 1.42772i
$33$	− 60983.4i	− 0.295402i
$34$	46629.4i	0.203462i
$35$	133772.	0.527382
$36$	56995.5	0.203602
$37$	73482.1i	0.238493i	0.992865	+	0.119247i	$0.0380479\pi$
$37$	73482.1i	0.238493i	−0.992865	+	0.119247i	$0.961952\pi$
$38$	−480443.	−1.42036
$39$	0	0
$40$	−37299.9	−0.0921506
$41$	− 685869.i	− 1.55417i	−0.629398	−	0.777083i	$-0.716698\pi$
$41$	− 685869.i	− 1.55417i	0.629398	−	0.777083i	$-0.283302\pi$
$42$	−1.00718e6	−2.09766
$43$	−787730.	−1.51091	−0.755454	−	0.655202i	$-0.772584\pi$
$43$	−787730.	−1.51091	−0.755454	+	0.655202i	$0.772584\pi$
$44$	217246.i	0.384474i
$45$	35532.5i	0.0581275i
$46$	846094.i	1.28164i
$47$	− 690181.i	− 0.969663i	−0.874608	−	0.484831i	$-0.838881\pi$
$47$	− 690181.i	− 0.969663i	0.874608	−	0.484831i	$-0.161119\pi$
$48$	−544973.	−0.711263
$49$	−1.17976e6	−1.43255
$50$	1.15699e6i	1.30898i
$51$	118674.	0.125273
$52$	0	0
$53$	−1.69205e6	−1.56116	−0.780579	−	0.625057i	$-0.785076\pi$
$53$	−1.69205e6	−1.56116	−0.780579	+	0.625057i	$0.785076\pi$
$54$	− 1.82379e6i	− 1.57615i
$55$	−135437.	−0.109766
$56$	558588.	0.425044
$57$	1.22275e6i	0.874530i
$58$	− 3.81843e6i	− 2.56973i
$59$	− 493676.i	− 0.312939i	−0.987683	−	0.156470i	$-0.949989\pi$
$59$	− 493676.i	− 0.312939i	0.987683	−	0.156470i	$-0.0500113\pi$
$60$	609761.i	0.364443i
$61$	2.20773e6	1.24535	0.622674	−	0.782481i	$-0.286046\pi$
$61$	2.20773e6	1.24535	0.622674	+	0.782481i	$0.286046\pi$
$62$	−2.58963e6	−1.37996
$63$	− 532120.i	− 0.268113i
$64$	2.78609e6	1.32851
$65$	0	0
$66$	1.01972e6	0.436594
$67$	2.25744e6i	0.916968i	0.888703	+	0.458484i	$0.151607\pi$
$67$	2.25744e6i	0.916968i	−0.888703	+	0.458484i	$0.848393\pi$
$68$	−422760.	−0.163047
$69$	2.15334e6	0.789117
$70$	2.23683e6i	0.779455i
$71$	403562.i	0.133815i	0.997759	+	0.0669077i	$0.0213133\pi$
$71$	403562.i	0.133815i	−0.997759	+	0.0669077i	$0.978687\pi$
$72$	148373.i	0.0468479i
$73$	509572.i	0.153312i	0.997058	+	0.0766558i	$0.0244243\pi$
$73$	509572.i	0.153312i	−0.997058	+	0.0766558i	$0.975576\pi$
$74$	−1.22872e6	−0.352485
$75$	2.94458e6	0.805950
$76$	− 4.35589e6i	− 1.13823i
$77$	2.02825e6	0.506295
$78$	0	0
$79$	−1.91796e6	−0.437668	−0.218834	−	0.975762i	$-0.570225\pi$
$79$	−1.91796e6	−0.437668	−0.218834	+	0.975762i	$0.570225\pi$
$80$	1.21032e6i	0.264293i
$81$	−3.81941e6	−0.798544
$82$	1.14686e7	2.29701
$83$	4.57733e6i	0.878697i	0.898317	+	0.439348i	$0.144791\pi$
$83$	4.57733e6i	0.878697i	−0.898317	+	0.439348i	$0.855209\pi$
$84$	− 9.13152e6i	− 1.68099i
$85$	− 263560.i	− 0.0465493i
$86$	− 1.31719e7i	− 2.23307i
$87$	−9.71806e6	−1.58220
$88$	−565542.	−0.0884658
$89$	− 3.52649e6i	− 0.530246i	−0.964215	−	0.265123i	$-0.914587\pi$
$89$	− 3.52649e6i	− 0.530246i	0.964215	−	0.265123i	$-0.0854125\pi$
$90$	−594150.	−0.0859107
$91$	0	0
$92$	−7.67102e6	−1.02706
$93$	6.59070e6i	0.849652i
$94$	1.15407e7	1.43313
$95$	2.71558e6	0.324960
$96$	− 1.12624e7i	− 1.29922i
$97$	4.86752e6i	0.541511i	0.962648	+	0.270755i	$0.0872734\pi$
$97$	4.86752e6i	0.541511i	−0.962648	+	0.270755i	$0.912727\pi$
$98$	− 1.97272e7i	− 2.11726i
$99$	538744.i	0.0558033i
$100$	−1.04897e7	−1.04897
$101$	−1.39742e6	−0.134959	−0.0674795	−	0.997721i	$-0.521496\pi$
$101$	−1.39742e6	−0.134959	−0.0674795	+	0.997721i	$0.521496\pi$
$102$	1.98438e6i	0.185150i
$103$	−6.54055e6	−0.589772	−0.294886	−	0.955533i	$-0.595282\pi$
$103$	−6.54055e6	−0.589772	−0.294886	+	0.955533i	$0.595282\pi$
$104$	0	0
$105$	5.69283e6	0.479917
$106$	− 2.82932e7i	− 2.30734i
$107$	6.60631e6	0.521334	0.260667	−	0.965429i	$-0.416058\pi$
$107$	6.60631e6	0.521334	0.260667	+	0.965429i	$0.416058\pi$
$108$	1.65352e7	1.26307
$109$	− 1.24329e7i	− 0.919561i	−0.888033	−	0.459781i	$-0.847928\pi$
$109$	− 1.24329e7i	− 0.919561i	0.888033	−	0.459781i	$-0.152072\pi$
$110$	− 2.26468e6i	− 0.162231i
$111$	3.12713e6i	0.217028i
$112$	− 1.81252e7i	− 1.21905i
$113$	−1.18932e7	−0.775396	−0.387698	−	0.921786i	$-0.626730\pi$
$113$	−1.18932e7	−0.775396	−0.387698	+	0.921786i	$0.626730\pi$
$114$	−2.04459e7	−1.29253
$115$	− 4.78232e6i	− 0.293222i
$116$	3.46194e7	2.05929
$117$	0	0
$118$	8.25491e6	0.462514
$119$	3.94696e6i	0.214708i
$120$	−1.58735e6	−0.0838568
$121$	1.74337e7	0.894623
$122$	3.69161e7i	1.84059i
$123$	− 2.91881e7i	− 1.41429i
$124$	− 2.34786e7i	− 1.10585i
$125$	− 1.39234e7i	− 0.637616i
$126$	8.89774e6	0.396262
$127$	3.16001e7	1.36891	0.684457	−	0.729053i	$-0.260039\pi$
$127$	3.16001e7	1.36891	0.684457	+	0.729053i	$0.260039\pi$
$128$	1.27122e7i	0.535780i
$129$	−3.35229e7	−1.37492
$130$	0	0
$131$	−1.45168e7	−0.564184	−0.282092	−	0.959387i	$-0.591028\pi$
$131$	−1.45168e7	−0.564184	−0.282092	+	0.959387i	$0.591028\pi$
$132$	9.24520e6i	0.349871i
$133$	−4.06674e7	−1.49887
$134$	−3.77473e7	−1.35525
$135$	1.03085e7i	0.360601i
$136$	− 1.10054e6i	− 0.0375164i
$137$	5.29202e7i	1.75833i	0.476521	+	0.879163i	$0.341898\pi$
$137$	5.29202e7i	1.75833i	−0.476521	+	0.879163i	$0.658102\pi$
$138$	3.60067e7i	1.16629i
$139$	5.49458e7	1.73533	0.867666	−	0.497148i	$-0.165619\pi$
$139$	5.49458e7	1.73533	0.867666	+	0.497148i	$0.165619\pi$
$140$	−2.02800e7	−0.624627
$141$	− 2.93716e7i	− 0.882391i
$142$	−6.74808e6	−0.197775
$143$	0	0
$144$	4.81444e6	0.134362
$145$	2.15827e7i	0.587918i
$146$	−8.52070e6	−0.226590
$147$	−5.02065e7	−1.30361
$148$	− 1.11400e7i	− 0.282469i
$149$	4.53910e7i	1.12413i	0.827092	+	0.562066i	$0.189993\pi$
$149$	4.53910e7i	1.12413i	−0.827092	+	0.562066i	$0.810007\pi$
$150$	4.92371e7i	1.19117i
$151$	3.65804e7i	0.864627i	0.901723	+	0.432313i	$0.142303\pi$
$151$	3.65804e7i	0.864627i	−0.901723	+	0.432313i	$0.857697\pi$
$152$	1.13394e7	0.261901
$153$	−1.04840e6	−0.0236649
$154$	3.39149e7i	0.748287i
$155$	1.46372e7	0.315716
$156$	0	0
$157$	−6.28265e7	−1.29567	−0.647835	−	0.761781i	$-0.724325\pi$
$157$	−6.28265e7	−1.29567	−0.647835	+	0.761781i	$0.724325\pi$
$158$	− 3.20708e7i	− 0.646860i
$159$	−7.20074e7	−1.42065
$160$	−2.50125e7	−0.482767
$161$	7.16180e7i	1.35248i
$162$	− 6.38656e7i	− 1.18022i
$163$	2.82451e7i	0.510842i	0.966830	+	0.255421i	$0.0822140\pi$
$163$	2.82451e7i	0.510842i	−0.966830	+	0.255421i	$0.917786\pi$
$164$	1.03979e8i	1.84074i
$165$	−5.76370e6	−0.0998867
$166$	−7.65389e7	−1.29869
$167$	8.56901e7i	1.42371i	0.702324	+	0.711857i	$0.252146\pi$
$167$	8.56901e7i	1.42371i	−0.702324	+	0.711857i	$0.747854\pi$
$168$	2.37715e7	0.386789
$169$	0	0
$170$	4.40707e6	0.0687984
$171$	− 1.08021e7i	− 0.165204i
$172$	1.19421e8	1.78950
$173$	1.07204e7	0.157416	0.0787079	−	0.996898i	$-0.474921\pi$
$173$	1.07204e7	0.157416	0.0787079	+	0.996898i	$0.474921\pi$
$174$	− 1.62499e8i	− 2.33844i
$175$	9.79337e7i	1.38133i
$176$	1.83509e7i	0.253725i
$177$	− 2.10091e7i	− 0.284774i
$178$	5.89674e7	0.783686
$179$	1.14749e8	1.49543	0.747713	−	0.664022i	$-0.231152\pi$
$179$	1.14749e8	1.49543	0.747713	+	0.664022i	$0.231152\pi$
$180$	− 5.38680e6i	− 0.0688457i
$181$	−7.50660e7	−0.940953	−0.470477	−	0.882412i	$-0.655918\pi$
$181$	−7.50660e7	−0.940953	−0.470477	+	0.882412i	$0.655918\pi$
$182$	0	0
$183$	9.39528e7	1.13326
$184$	− 1.99695e7i	− 0.236322i
$185$	6.94499e6	0.0806438
$186$	−1.10205e8	−1.25576
$187$	− 3.99610e6i	− 0.0446880i
$188$	1.04633e8i	1.14846i
$189$	− 1.54376e8i	− 1.66327i
$190$	4.54080e7i	0.480280i
$191$	−1.40238e7	−0.145629	−0.0728147	−	0.997345i	$-0.523198\pi$
$191$	−1.40238e7	−0.145629	−0.0728147	+	0.997345i	$0.523198\pi$
$192$	1.18566e8	1.20894
$193$	− 6.00369e7i	− 0.601129i	−0.953761	−	0.300565i	$-0.902825\pi$
$193$	− 6.00369e7i	− 0.601129i	0.953761	−	0.300565i	$-0.0971751\pi$
$194$	−8.13913e7	−0.800336
$195$	0	0
$196$	1.78855e8	1.69670
$197$	4.54705e7i	0.423739i	0.977298	+	0.211869i	$0.0679551\pi$
$197$	4.54705e7i	0.423739i	−0.977298	+	0.211869i	$0.932045\pi$
$198$	−9.00851e6	−0.0824755
$199$	−1.27240e8	−1.14456	−0.572280	−	0.820058i	$-0.693941\pi$
$199$	−1.27240e8	−1.14456	−0.572280	+	0.820058i	$0.693941\pi$
$200$	− 2.73071e7i	− 0.241363i
$201$	9.60685e7i	0.834439i
$202$	− 2.33667e7i	− 0.199465i
$203$	− 3.23213e8i	− 2.71177i
$204$	−1.79911e7	−0.148372
$205$	−6.48233e7	−0.525524
$206$	− 1.09366e8i	− 0.871664i
$207$	−1.90232e7	−0.149069
$208$	0	0
$209$	4.11736e7	0.311966
$210$	9.51916e7i	0.709302i
$211$	6.62059e7	0.485186	0.242593	−	0.970128i	$-0.422002\pi$
$211$	6.62059e7	0.485186	0.242593	+	0.970128i	$0.422002\pi$
$212$	2.56518e8	1.84902
$213$	1.71741e7i	0.121772i
$214$	1.10466e8i	0.770515i
$215$	7.44505e7i	0.510896i
$216$	4.30451e7i	0.290627i
$217$	−2.19200e8	−1.45624
$218$	2.07895e8	1.35908
$219$	2.16855e7i	0.139513i
$220$	2.05325e7	0.130006
$221$	0	0
$222$	−5.22897e7	−0.320761
$223$	1.91691e8i	1.15754i	0.815492	+	0.578769i	$0.196467\pi$
$223$	1.91691e8i	1.15754i	−0.815492	+	0.578769i	$0.803533\pi$
$224$	3.74577e8	2.22676
$225$	−2.60132e7	−0.152249
$226$	− 1.98870e8i	− 1.14601i
$227$	5.62115e7i	0.318959i	0.987201	+	0.159479i	$0.0509816\pi$
$227$	5.62115e7i	0.318959i	−0.987201	+	0.159479i	$0.949018\pi$
$228$	− 1.85371e8i	− 1.03579i
$229$	2.40345e8i	1.32255i	0.750145	+	0.661274i	$0.229984\pi$
$229$	2.40345e8i	1.32255i	−0.750145	+	0.661274i	$0.770016\pi$
$230$	7.99666e7	0.433372
$231$	8.63148e7	0.460727
$232$	9.01224e7i	0.473832i
$233$	−8.29632e7	−0.429675	−0.214837	−	0.976650i	$-0.568922\pi$
$233$	−8.29632e7	−0.429675	−0.214837	+	0.976650i	$0.568922\pi$
$234$	0	0
$235$	−6.52309e7	−0.327881
$236$	7.48423e7i	0.370642i
$237$	−8.16215e7	−0.398277
$238$	−6.59984e7	−0.317332
$239$	− 1.71336e8i	− 0.811812i	−0.913915	−	0.405906i	$-0.866956\pi$
$239$	− 1.71336e8i	− 0.811812i	0.913915	−	0.405906i	$-0.133044\pi$
$240$	5.15068e7i	0.240506i
$241$	− 4.48189e7i	− 0.206254i	−0.994668	−	0.103127i	$-0.967115\pi$
$241$	− 4.48189e7i	− 0.206254i	0.994668	−	0.103127i	$-0.0328847\pi$
$242$	2.91514e8i	1.32222i
$243$	7.59959e7	0.339757
$244$	−3.34696e8	−1.47498
$245$	1.11503e8i	0.484400i
$246$	4.88063e8	2.09027
$247$	0	0
$248$	6.11202e7	0.254451
$249$	1.94795e8i	0.799612i
$250$	2.32817e8	0.942376
$251$	2.46344e8	0.983294	0.491647	−	0.870795i	$-0.336395\pi$
$251$	2.46344e8	0.983294	0.491647	+	0.870795i	$0.336395\pi$
$252$	8.06704e7i	0.317550i
$253$	− 7.25096e7i	− 0.281497i
$254$	5.28396e8i	2.02321i
$255$	− 1.12162e7i	− 0.0423597i
$256$	1.44055e8	0.536646
$257$	−1.12599e8	−0.413777	−0.206889	−	0.978364i	$-0.566334\pi$
$257$	−1.12599e8	−0.413777	−0.206889	+	0.978364i	$0.566334\pi$
$258$	− 5.60547e8i	− 2.03209i
$259$	−1.04005e8	−0.371969
$260$	0	0
$261$	8.58520e7	0.298888
$262$	− 2.42739e8i	− 0.833846i
$263$	−3.49444e8	−1.18449	−0.592247	−	0.805757i	$-0.701759\pi$
$263$	−3.49444e8	−1.18449	−0.592247	+	0.805757i	$0.701759\pi$
$264$	−2.40674e7	−0.0805037
$265$	1.59920e8i	0.527888i
$266$	− 6.80011e8i	− 2.21529i
$267$	− 1.50074e8i	− 0.482522i
$268$	− 3.42232e8i	− 1.08605i
$269$	−3.40037e8	−1.06511	−0.532554	−	0.846396i	$-0.678768\pi$
$269$	−3.40037e8	−1.06511	−0.532554	+	0.846396i	$0.678768\pi$
$270$	−1.72372e8	−0.532957
$271$	4.42243e8i	1.34980i	0.737911	+	0.674898i	$0.235813\pi$
$271$	4.42243e8i	1.34980i	−0.737911	+	0.674898i	$0.764187\pi$
$272$	−3.57108e7	−0.107599
$273$	0	0
$274$	−8.84895e8	−2.59875
$275$	− 9.91528e7i	− 0.287502i
$276$	−3.26451e8	−0.934622
$277$	1.00629e8	0.284475	0.142237	−	0.989833i	$-0.454570\pi$
$277$	1.00629e8	0.284475	0.142237	+	0.989833i	$0.454570\pi$
$278$	9.18765e8i	2.56476i
$279$	− 5.82241e7i	− 0.160505i
$280$	− 5.27937e7i	− 0.143724i
$281$	− 2.13226e8i	− 0.573281i	−0.958038	−	0.286640i	$-0.907462\pi$
$281$	− 2.13226e8i	− 0.573281i	0.958038	−	0.286640i	$-0.0925385\pi$
$282$	4.91132e8	1.30415
$283$	1.73883e8	0.456042	0.228021	−	0.973656i	$-0.426774\pi$
$283$	1.73883e8	0.456042	0.228021	+	0.973656i	$0.426774\pi$
$284$	− 6.11808e7i	− 0.158490i
$285$	1.15565e8	0.295713
$286$	0	0
$287$	9.70767e8	2.42398
$288$	9.94954e7i	0.245431i
$289$	−4.02562e8	−0.981049
$290$	−3.60890e8	−0.868924
$291$	2.07144e8i	0.492773i
$292$	− 7.72520e7i	− 0.181581i
$293$	− 1.31462e8i	− 0.305327i	−0.988278	−	0.152663i	$-0.951215\pi$
$293$	− 1.31462e8i	− 0.305327i	0.988278	−	0.152663i	$-0.0487850\pi$
$294$	− 8.39518e8i	− 1.92670i
$295$	−4.66587e7	−0.105817
$296$	2.90001e7	0.0649948
$297$	1.56298e8i	0.346182i
$298$	−7.58996e8	−1.66143
$299$	0	0
$300$	−4.46403e8	−0.954560
$301$	− 1.11494e9i	− 2.35651i
$302$	−6.11671e8	−1.27789
$303$	−5.94691e7	−0.122812
$304$	− 3.67944e8i	− 0.751147i
$305$	− 2.08658e8i	− 0.421101i
$306$	− 1.75305e7i	− 0.0349760i
$307$	7.71793e8i	1.52236i	0.648542	+	0.761179i	$0.275379\pi$
$307$	7.71793e8i	1.52236i	−0.648542	+	0.761179i	$0.724621\pi$
$308$	−3.07486e8	−0.599650
$309$	−2.78342e8	−0.536691
$310$	2.44752e8i	0.466618i
$311$	3.88587e8	0.732533	0.366267	−	0.930510i	$-0.380636\pi$
$311$	3.88587e8	0.732533	0.366267	+	0.930510i	$0.380636\pi$
$312$	0	0
$313$	−5.42552e8	−1.00008	−0.500042	−	0.866001i	$-0.666682\pi$
$313$	−5.42552e8	−1.00008	−0.500042	+	0.866001i	$0.666682\pi$
$314$	− 1.05054e9i	− 1.91496i
$315$	−5.02921e7	−0.0906594
$316$	2.90767e8	0.518370
$317$	9.56858e7i	0.168710i	0.996436	+	0.0843549i	$0.0268829\pi$
$317$	9.56858e7i	0.168710i	−0.996436	+	0.0843549i	$0.973117\pi$
$318$	− 1.20406e9i	− 2.09968i
$319$	3.27237e8i	0.564410i
$320$	− 2.63321e8i	− 0.449221i
$321$	2.81141e8	0.474412
$322$	−1.19755e9	−1.99893
$323$	8.01238e7i	0.132298i
$324$	5.79031e8	0.945788
$325$	0	0
$326$	−4.72295e8	−0.755008
$327$	− 5.29100e8i	− 0.836798i
$328$	−2.70682e8	−0.423546
$329$	9.76871e8	1.51235
$330$	− 9.63766e7i	− 0.147629i
$331$	2.23414e8i	0.338620i	0.985563	+	0.169310i	$0.0541538\pi$
$331$	2.23414e8i	0.338620i	−0.985563	+	0.169310i	$0.945846\pi$
$332$	− 6.93932e8i	− 1.04072i
$333$	− 2.76260e7i	− 0.0409980i
$334$	−1.43285e9	−2.10420
$335$	2.13357e8	0.310063
$336$	− 7.71345e8i	− 1.10933i
$337$	5.78709e8	0.823674	0.411837	−	0.911257i	$-0.364887\pi$
$337$	5.78709e8	0.823674	0.411837	+	0.911257i	$0.364887\pi$
$338$	0	0
$339$	−5.06131e8	−0.705609
$340$	3.99562e7i	0.0551325i
$341$	2.21929e8	0.303091
$342$	1.80625e8	0.244167
$343$	− 5.04190e8i	− 0.674629i
$344$	3.10882e8i	0.411757i
$345$	− 2.03518e8i	− 0.266831i
$346$	1.79259e8i	0.232656i
$347$	5.47565e8	0.703530	0.351765	−	0.936088i	$-0.385582\pi$
$347$	5.47565e8	0.703530	0.351765	+	0.936088i	$0.385582\pi$
$348$	1.47328e9	1.87395
$349$	1.12482e9i	1.41643i	0.705997	+	0.708215i	$0.250499\pi$
$349$	1.12482e9i	1.41643i	−0.705997	+	0.708215i	$0.749501\pi$
$350$	−1.63758e9	−2.04157
$351$	0	0
$352$	−3.79240e8	−0.463463
$353$	9.30531e8i	1.12595i	0.826474	+	0.562976i	$0.190343\pi$
$353$	9.30531e8i	1.12595i	−0.826474	+	0.562976i	$0.809657\pi$
$354$	3.51299e8	0.420887
$355$	3.81417e7	0.0452482
$356$	5.34622e8i	0.628018i
$357$	1.67968e8i	0.195384i
$358$	1.91876e9i	2.21019i
$359$	8.37591e8i	0.955436i	0.878513	+	0.477718i	$0.158536\pi$
$359$	8.37591e8i	0.955436i	−0.878513	+	0.477718i	$0.841464\pi$
$360$	1.40231e7	0.0158411
$361$	6.83199e7	0.0764314
$362$	− 1.25520e9i	− 1.39070i
$363$	7.41914e8	0.814105
$364$	0	0
$365$	4.81610e7	0.0518406
$366$	1.57101e9i	1.67493i
$367$	5.81239e8	0.613795	0.306898	−	0.951743i	$-0.400709\pi$
$367$	5.81239e8	0.613795	0.306898	+	0.951743i	$0.400709\pi$
$368$	−6.47975e8	−0.677784
$369$	2.57856e8i	0.267168i
$370$	1.16129e8i	0.119189i
$371$	− 2.39489e9i	− 2.43488i
$372$	− 9.99163e8i	− 1.00632i
$373$	2.47661e8	0.247103	0.123551	−	0.992338i	$-0.460572\pi$
$373$	2.47661e8	0.247103	0.123551	+	0.992338i	$0.460572\pi$
$374$	6.68200e7	0.0660474
$375$	− 5.92528e8i	− 0.580229i
$376$	−2.72384e8	−0.264255
$377$	0	0
$378$	2.58137e9	2.45826
$379$	1.42176e9i	1.34149i	0.741687	+	0.670746i	$0.234026\pi$
$379$	1.42176e9i	1.34149i	−0.741687	+	0.670746i	$0.765974\pi$
$380$	−4.11687e8	−0.384879
$381$	1.34479e9	1.24571
$382$	− 2.34496e8i	− 0.215236i
$383$	1.11622e9i	1.01520i	0.861592	+	0.507601i	$0.169467\pi$
$383$	1.11622e9i	1.01520i	−0.861592	+	0.507601i	$0.830533\pi$
$384$	5.40986e8i	0.487559i
$385$	− 1.91695e8i	− 0.171198i
$386$	1.00390e9	0.888450
$387$	2.96151e8	0.259732
$388$	− 7.37926e8i	− 0.641360i
$389$	2.16836e9	1.86770	0.933851	−	0.357661i	$-0.116426\pi$
$389$	2.16836e9	1.86770	0.933851	+	0.357661i	$0.116426\pi$
$390$	0	0
$391$	1.41103e8	0.119377
$392$	4.65600e8i	0.390402i
$393$	−6.17781e8	−0.513406
$394$	−7.60326e8	−0.626272
$395$	1.81272e8i	0.147993i
$396$	− 8.16747e7i	− 0.0660928i
$397$	2.93811e8i	0.235669i	0.993033	+	0.117834i	$0.0375952\pi$
$397$	2.93811e8i	0.235669i	−0.993033	+	0.117834i	$0.962405\pi$
$398$	− 2.12762e9i	− 1.69162i
$399$	−1.73066e9	−1.36397
$400$	−8.86070e8	−0.692242
$401$	4.98847e8i	0.386333i	0.981166	+	0.193167i	$0.0618758\pi$
$401$	4.98847e8i	0.386333i	−0.981166	+	0.193167i	$0.938124\pi$
$402$	−1.60639e9	−1.23327
$403$	0	0
$404$	2.11851e8	0.159844
$405$	3.60983e8i	0.270019i
$406$	5.40454e9	4.00791
$407$	1.05300e8	0.0774191
$408$	− 4.68352e7i	− 0.0341398i
$409$	− 8.87949e7i	− 0.0641736i	−0.999485	−	0.0320868i	$-0.989785\pi$
$409$	− 8.87949e7i	− 0.0641736i	0.999485	−	0.0320868i	$-0.0102153\pi$
$410$	− 1.08393e9i	− 0.776708i
$411$	2.25209e9i	1.60007i
$412$	9.91560e8	0.698520
$413$	6.98741e8	0.488080
$414$	− 3.18093e8i	− 0.220320i
$415$	4.32616e8	0.297121
$416$	0	0
$417$	2.33829e9	1.57915
$418$	6.88477e8i	0.461076i
$419$	1.55617e8	0.103349	0.0516746	−	0.998664i	$-0.483544\pi$
$419$	1.55617e8	0.103349	0.0516746	+	0.998664i	$0.483544\pi$
$420$	−8.63045e8	−0.568409
$421$	− 2.18824e9i	− 1.42925i	−0.699509	−	0.714624i	$-0.746598\pi$
$421$	− 2.18824e9i	− 1.42925i	0.699509	−	0.714624i	$-0.253402\pi$
$422$	1.10705e9i	0.717089i
$423$	2.59477e8i	0.166689i
$424$	6.67775e8i	0.425451i
$425$	1.92951e8	0.121923
$426$	−2.87174e8	−0.179975
$427$	3.12478e9i	1.94232i
$428$	−1.00153e9	−0.617462
$429$	0	0
$430$	−1.24491e9	−0.755089
$431$	9.56674e7i	0.0575564i	0.999586	+	0.0287782i	$0.00916165\pi$
$431$	9.56674e7i	0.0575564i	−0.999586	+	0.0287782i	$0.990838\pi$
$432$	1.39674e9	0.833533
$433$	−9.63031e8	−0.570076	−0.285038	−	0.958516i	$-0.592006\pi$
$433$	−9.63031e8	−0.570076	−0.285038	+	0.958516i	$0.592006\pi$
$434$	− 3.66531e9i	− 2.15227i
$435$	9.18479e8i	0.535004i
$436$	1.88486e9i	1.08912i
$437$	1.45385e9i	0.833365i
$438$	−3.62610e8	−0.206196
$439$	−2.04315e9	−1.15259	−0.576296	−	0.817241i	$-0.695502\pi$
$439$	−2.04315e9	−1.15259	−0.576296	+	0.817241i	$0.695502\pi$
$440$	5.34509e7i	0.0299137i
$441$	4.43538e8	0.246261
$442$	0	0
$443$	9.64971e8	0.527353	0.263676	−	0.964611i	$-0.415065\pi$
$443$	9.64971e8	0.527353	0.263676	+	0.964611i	$0.415065\pi$
$444$	− 4.74080e8i	− 0.257046i
$445$	−3.33298e8	−0.179297
$446$	−3.20533e9	−1.71080
$447$	1.93168e9i	1.02296i
$448$	3.94338e9i	2.07203i
$449$	− 2.01105e9i	− 1.04848i	−0.851571	−	0.524240i	$-0.824350\pi$
$449$	− 2.01105e9i	− 1.04848i	0.851571	−	0.524240i	$-0.175650\pi$
$450$	− 4.34975e8i	− 0.225020i
$451$	−9.82852e8	−0.504511
$452$	1.80303e9	0.918372
$453$	1.55673e9i	0.786808i
$454$	−9.39930e8	−0.471411
$455$	0	0
$456$	4.82564e8	0.238329
$457$	− 3.09444e9i	− 1.51661i	−0.651897	−	0.758307i	$-0.726027\pi$
$457$	− 3.09444e9i	− 1.51661i	0.651897	−	0.758307i	$-0.273973\pi$
$458$	−4.01888e9	−1.95468
$459$	−3.04155e8	−0.146808
$460$	7.25009e8i	0.347289i
$461$	− 2.50825e9i	− 1.19239i	−0.802840	−	0.596194i	$-0.796679\pi$
$461$	− 2.50825e9i	− 1.19239i	0.802840	−	0.596194i	$-0.203321\pi$
$462$	1.44330e9i	0.680940i
$463$	8.14250e8i	0.381263i	0.981662	+	0.190631i	$0.0610535\pi$
$463$	8.14250e8i	0.381263i	−0.981662	+	0.190631i	$0.938947\pi$
$464$	2.92432e9	1.35898
$465$	6.22905e8	0.287300
$466$	− 1.38725e9i	− 0.635045i
$467$	−2.48449e9	−1.12883	−0.564413	−	0.825492i	$-0.690898\pi$
$467$	−2.48449e9	−1.12883	−0.564413	+	0.825492i	$0.690898\pi$
$468$	0	0
$469$	−3.19514e9	−1.43016
$470$	− 1.09075e9i	− 0.484597i
$471$	−2.67367e9	−1.17906
$472$	−1.94832e8	−0.0852831
$473$	1.12882e9i	0.490468i
$474$	− 1.36482e9i	− 0.588641i
$475$	1.98806e9i	0.851143i
$476$	− 5.98367e8i	− 0.254298i
$477$	6.36133e8	0.268370
$478$	2.86496e9	1.19983
$479$	− 4.44069e9i	− 1.84619i	−0.384575	−	0.923094i	$-0.625652\pi$
$479$	− 4.44069e9i	− 1.84619i	0.384575	−	0.923094i	$-0.374348\pi$
$480$	−1.06444e9	−0.439317
$481$	0	0
$482$	7.49430e8	0.304836
$483$	3.04780e9i	1.23076i
$484$	−2.64298e9	−1.05958
$485$	4.60043e8	0.183106
$486$	1.27075e9i	0.502150i
$487$	3.69425e9i	1.44935i	0.689089	+	0.724677i	$0.258011\pi$
$487$	3.69425e9i	1.44935i	−0.689089	+	0.724677i	$0.741989\pi$
$488$	− 8.71291e8i	− 0.339386i
$489$	1.20201e9i	0.464865i
$490$	−1.86447e9	−0.715928
$491$	3.09370e9	1.17949	0.589743	−	0.807591i	$-0.299229\pi$
$491$	3.09370e9	1.17949	0.589743	+	0.807591i	$0.299229\pi$
$492$	4.42497e9i	1.67507i
$493$	−6.36802e8	−0.239354
$494$	0	0
$495$	5.09182e7	0.0188692
$496$	− 1.98325e9i	− 0.729779i
$497$	−5.71195e8	−0.208707
$498$	−3.25722e9	−1.18180
$499$	− 2.44557e9i	− 0.881107i	−0.897726	−	0.440554i	$-0.854782\pi$
$499$	− 2.44557e9i	− 0.881107i	0.897726	−	0.440554i	$-0.145218\pi$
$500$	2.11081e9i	0.755186i
$501$	3.64666e9i	1.29558i
$502$	4.11919e9i	1.45328i
$503$	3.44504e9	1.20700	0.603498	−	0.797365i	$-0.293773\pi$
$503$	3.44504e9	1.20700	0.603498	+	0.797365i	$0.293773\pi$
$504$	−2.10004e8	−0.0730669
$505$	1.32074e8i	0.0456349i
$506$	1.21245e9	0.416043
$507$	0	0
$508$	−4.79064e9	−1.62133
$509$	− 1.88074e9i	− 0.632146i	−0.948735	−	0.316073i	$-0.897636\pi$
$509$	− 1.88074e9i	− 0.632146i	0.948735	−	0.316073i	$-0.102364\pi$
$510$	1.87549e8	0.0626064
$511$	−7.21239e8	−0.239114
$512$	4.03595e9i	1.32893i
$513$	− 3.13385e9i	− 1.02487i
$514$	− 1.88279e9i	− 0.611550i
$515$	6.18165e8i	0.199425i
$516$	5.08214e9	1.62844
$517$	−9.89032e8	−0.314770
$518$	− 1.73910e9i	− 0.549758i
$519$	4.56220e8	0.143248
$520$	0	0
$521$	7.91865e8	0.245312	0.122656	−	0.992449i	$-0.460859\pi$
$521$	7.91865e8	0.245312	0.122656	+	0.992449i	$0.460859\pi$
$522$	1.43556e9i	0.441747i
$523$	2.06100e8	0.0629972	0.0314986	−	0.999504i	$-0.489972\pi$
$523$	2.06100e8	0.0629972	0.0314986	+	0.999504i	$0.489972\pi$
$524$	2.20077e9	0.668214
$525$	4.16770e9i	1.25701i
$526$	− 5.84316e9i	− 1.75064i
$527$	4.31873e8i	0.128534i
$528$	7.80947e8i	0.230889i
$529$	−8.44491e8	−0.248028
$530$	−2.67407e9	−0.780202
$531$	1.85600e8i	0.0537956i
$532$	6.16525e9	1.77525
$533$	0	0
$534$	2.50944e9	0.713153
$535$	− 6.24380e8i	− 0.176283i
$536$	8.90911e8	0.249895
$537$	4.88332e9	1.36083
$538$	− 5.68587e9i	− 1.57420i
$539$	1.69061e9i	0.465031i
$540$	− 1.56279e9i	− 0.427093i
$541$	− 5.13810e9i	− 1.39512i	−0.716525	−	0.697561i	$-0.754269\pi$
$541$	− 5.13810e9i	− 1.39512i	0.716525	−	0.697561i	$-0.245731\pi$
$542$	−7.39488e9	−1.99496
$543$	−3.19454e9	−0.856265
$544$	− 7.38000e8i	− 0.196544i
$545$	−1.17507e9	−0.310939
$546$	0	0
$547$	3.45399e8	0.0902331	0.0451165	−	0.998982i	$-0.485634\pi$
$547$	3.45399e8	0.0902331	0.0451165	+	0.998982i	$0.485634\pi$
$548$	− 8.02280e9i	− 2.08254i
$549$	−8.30006e8	−0.214081
$550$	1.65796e9	0.424919
$551$	− 6.56126e9i	− 1.67092i
$552$	− 8.49828e8i	− 0.215052i
$553$	− 2.71465e9i	− 0.682615i
$554$	1.68265e9i	0.420444i
$555$	2.95554e8	0.0733856
$556$	−8.32989e9	−2.05531
$557$	3.59385e9i	0.881185i	0.897707	+	0.440592i	$0.145232\pi$
$557$	3.59385e9i	0.881185i	−0.897707	+	0.440592i	$0.854768\pi$
$558$	9.73583e8	0.237221
$559$	0	0
$560$	−1.71307e9	−0.412207
$561$	− 1.70060e8i	− 0.0406660i
$562$	3.56541e9	0.847291
$563$	−1.27744e9	−0.301689	−0.150845	−	0.988557i	$-0.548199\pi$
$563$	−1.27744e9	−0.301689	−0.150845	+	0.988557i	$0.548199\pi$
$564$	4.45279e9i	1.04510i
$565$	1.12406e9i	0.262192i
$566$	2.90755e9i	0.674016i
$567$	− 5.40593e9i	− 1.24546i
$568$	1.59268e8	0.0364678
$569$	4.78849e9	1.08970	0.544849	−	0.838534i	$-0.316587\pi$
$569$	4.78849e9	1.08970	0.544849	+	0.838534i	$0.316587\pi$
$570$	1.93240e9i	0.437054i
$571$	−1.93902e9	−0.435869	−0.217935	−	0.975963i	$-0.569932\pi$
$571$	−1.93902e9	−0.435869	−0.217935	+	0.975963i	$0.569932\pi$
$572$	0	0
$573$	−5.96802e8	−0.132522
$574$	1.62325e10i	3.58256i
$575$	3.50112e9	0.768014
$576$	−1.04744e9	−0.228377
$577$	6.58473e9i	1.42699i	0.700658	+	0.713497i	$0.252890\pi$
$577$	6.58473e9i	1.42699i	−0.700658	+	0.713497i	$0.747110\pi$
$578$	− 6.73137e9i	− 1.44996i
$579$	− 2.55495e9i	− 0.547026i
$580$	− 3.27197e9i	− 0.696324i
$581$	−6.47868e9	−1.37047
$582$	−3.46372e9	−0.728303
$583$	2.42471e9i	0.506780i
$584$	2.01105e8	0.0417809
$585$	0	0
$586$	2.19822e9	0.451263
$587$	1.22975e9i	0.250948i	0.992097	+	0.125474i	$0.0400451\pi$
$587$	1.22975e9i	0.250948i	−0.992097	+	0.125474i	$0.959955\pi$
$588$	7.61140e9	1.54399
$589$	−4.44979e9	−0.897296
$590$	− 7.80193e8i	− 0.156394i
$591$	1.93506e9i	0.385601i
$592$	− 9.41005e8i	− 0.186409i
$593$	− 8.85789e9i	− 1.74437i	−0.489174	−	0.872186i	$-0.662702\pi$
$593$	− 8.85789e9i	− 1.74437i	0.489174	−	0.872186i	$-0.337298\pi$
$594$	−2.61350e9	−0.511647
$595$	3.73038e8	0.0726012
$596$	− 6.88136e9i	− 1.33141i
$597$	−5.41488e9	−1.04155
$598$	0	0
$599$	3.97030e9	0.754795	0.377398	−	0.926051i	$-0.376819\pi$
$599$	3.97030e9	0.754795	0.377398	+	0.926051i	$0.376819\pi$
$600$	− 1.16209e9i	− 0.219640i
$601$	5.95118e8	0.111826	0.0559130	−	0.998436i	$-0.482193\pi$
$601$	5.95118e8	0.111826	0.0559130	+	0.998436i	$0.482193\pi$
$602$	1.86432e10	3.48284
$603$	− 8.48696e8i	− 0.157631i
$604$	− 5.54565e9i	− 1.02406i
$605$	− 1.64770e9i	− 0.302507i
$606$	− 9.94400e8i	− 0.181513i
$607$	−1.95656e8	−0.0355085	−0.0177542	−	0.999842i	$-0.505652\pi$
$607$	−1.95656e8	−0.0355085	−0.0177542	+	0.999842i	$0.505652\pi$
$608$	7.60395e9	1.37207
$609$	− 1.37548e10i	− 2.46770i
$610$	3.48904e9	0.622373
$611$	0	0
$612$	1.58939e8	0.0280285
$613$	3.70839e9i	0.650240i	0.945673	+	0.325120i	$0.105405\pi$
$613$	3.70839e9i	0.650240i	−0.945673	+	0.325120i	$0.894595\pi$
$614$	−1.29054e10	−2.25000
$615$	−2.75864e9	−0.478226
$616$	− 8.00458e8i	− 0.137977i
$617$	− 1.71550e8i	− 0.0294030i	−0.999892	−	0.0147015i	$-0.995320\pi$
$617$	− 1.71550e8i	− 0.0294030i	0.999892	−	0.0147015i	$-0.00467980\pi$
$618$	− 4.65424e9i	− 0.793212i
$619$	− 3.49467e9i	− 0.592228i	−0.955153	−	0.296114i	$-0.904309\pi$
$619$	− 3.49467e9i	− 0.592228i	0.955153	−	0.296114i	$-0.0956908\pi$
$620$	−2.21902e9	−0.373930
$621$	−5.51892e9	−0.924769
$622$	6.49769e9i	1.08266i
$623$	4.99133e9	0.827004
$624$	0	0
$625$	4.08972e9	0.670060
$626$	− 9.07218e9i	− 1.47809i
$627$	1.75220e9	0.283888
$628$	9.52462e9	1.53458
$629$	2.04914e8i	0.0328317i
$630$	− 8.40949e8i	− 0.133992i
$631$	3.07515e9i	0.487263i	0.969868	+	0.243631i	$0.0783387\pi$
$631$	3.07515e9i	0.487263i	−0.969868	+	0.243631i	$0.921661\pi$
$632$	7.56933e8i	0.119275i
$633$	2.81748e9	0.441518
$634$	−1.59999e9	−0.249348
$635$	− 2.98661e9i	− 0.462883i
$636$	1.09165e10	1.68260
$637$	0	0
$638$	−5.47182e9	−0.834180
$639$	− 1.51721e8i	− 0.0230035i
$640$	1.20147e9	0.181168
$641$	1.25219e9	0.187788	0.0938942	−	0.995582i	$-0.470068\pi$
$641$	1.25219e9	0.187788	0.0938942	+	0.995582i	$0.470068\pi$
$642$	4.70104e9i	0.701166i
$643$	− 1.04790e10i	− 1.55447i	−0.629210	−	0.777235i	$-0.716622\pi$
$643$	− 1.04790e10i	− 1.55447i	0.629210	−	0.777235i	$-0.283378\pi$
$644$	− 1.08574e10i	− 1.60187i
$645$	3.16834e9i	0.464915i
$646$	−1.33977e9	−0.195532
$647$	8.34931e9	1.21195	0.605976	−	0.795483i	$-0.292783\pi$
$647$	8.34931e9	1.21195	0.605976	+	0.795483i	$0.292783\pi$
$648$	1.50735e9i	0.217622i
$649$	−7.07439e8	−0.101586
$650$	0	0
$651$	−9.32836e9	−1.32517
$652$	− 4.28201e9i	− 0.605036i
$653$	2.47822e9	0.348292	0.174146	−	0.984720i	$-0.444284\pi$
$653$	2.47822e9	0.348292	0.174146	+	0.984720i	$0.444284\pi$
$654$	8.84724e9	1.23676
$655$	1.37202e9i	0.190772i
$656$	8.78317e9i	1.21475i
$657$	− 1.91576e8i	− 0.0263549i
$658$	1.63346e10i	2.23520i
$659$	−6.06308e9	−0.825266	−0.412633	−	0.910897i	$-0.635391\pi$
$659$	−6.06308e9	−0.825266	−0.412633	+	0.910897i	$0.635391\pi$
$660$	8.73789e8	0.118305
$661$	− 5.29524e9i	− 0.713150i	−0.934267	−	0.356575i	$-0.883944\pi$
$661$	− 5.29524e9i	− 0.713150i	0.934267	−	0.356575i	$-0.116056\pi$
$662$	−3.73577e9	−0.500469
$663$	0	0
$664$	1.80647e9	0.239465
$665$	3.84358e9i	0.506828i
$666$	4.61942e8	0.0605938
$667$	−1.15548e10	−1.50773
$668$	− 1.29908e10i	− 1.68623i
$669$	8.15768e9i	1.05336i
$670$	3.56760e9i	0.458263i
$671$	− 3.16368e9i	− 0.404263i
$672$	1.59406e10	2.02634
$673$	−4.62818e9	−0.585272	−0.292636	−	0.956224i	$-0.594532\pi$
$673$	−4.62818e9	−0.585272	−0.292636	+	0.956224i	$0.594532\pi$
$674$	9.67677e9i	1.21736i
$675$	−7.54682e9	−0.944497
$676$	0	0
$677$	−1.40888e10	−1.74507	−0.872536	−	0.488550i	$-0.837526\pi$
$677$	−1.40888e10	−1.74507	−0.872536	+	0.488550i	$0.837526\pi$
$678$	− 8.46317e9i	− 1.04287i
$679$	−6.88941e9	−0.844574
$680$	−1.04015e8	−0.0126857
$681$	2.39216e9i	0.290252i
$682$	3.71094e9i	0.447959i
$683$	1.20616e10i	1.44855i	0.689511	+	0.724276i	$0.257826\pi$
$683$	1.20616e10i	1.44855i	−0.689511	+	0.724276i	$0.742174\pi$
$684$	1.63762e9i	0.195666i
$685$	5.00163e9	0.594558
$686$	8.43071e9	0.997080
$687$	1.02282e10i	1.20351i
$688$	1.00876e10	1.18094
$689$	0	0
$690$	3.40309e9	0.394368
$691$	− 2.06932e9i	− 0.238591i	−0.992859	−	0.119296i	$-0.961936\pi$
$691$	− 2.06932e9i	− 0.238591i	0.992859	−	0.119296i	$-0.0380636\pi$
$692$	−1.62523e9	−0.186442
$693$	−7.62529e8	−0.0870343
$694$	9.15601e9i	1.03980i
$695$	− 5.19307e9i	− 0.586783i
$696$	3.83528e9i	0.431186i
$697$	− 1.91263e9i	− 0.213952i
$698$	−1.88085e10	−2.09344
$699$	−3.53061e9	−0.391003
$700$	− 1.48469e10i	− 1.63604i
$701$	−1.66276e10	−1.82312	−0.911561	−	0.411166i	$-0.865122\pi$
$701$	−1.66276e10	−1.82312	−0.911561	+	0.411166i	$0.865122\pi$
$702$	0	0
$703$	−2.11132e9	−0.229198
$704$	− 3.99248e9i	− 0.431259i
$705$	−2.77599e9	−0.298371
$706$	−1.55597e10	−1.66412
$707$	− 1.97788e9i	− 0.210490i
$708$	3.18502e9i	0.337284i
$709$	− 6.48948e8i	− 0.0683830i	−0.999415	−	0.0341915i	$-0.989114\pi$
$709$	− 6.48948e8i	− 0.0683830i	0.999415	−	0.0341915i	$-0.0108856\pi$
$710$	6.37779e8i	0.0668754i
$711$	7.21067e8	0.0752371
$712$	−1.39175e9	−0.144504
$713$	7.83638e9i	0.809659i
$714$	−2.80865e9	−0.288771
$715$	0	0
$716$	−1.73962e10	−1.77117
$717$	− 7.29143e9i	− 0.738747i
$718$	−1.40056e10	−1.41210
$719$	−6.31401e9	−0.633511	−0.316755	−	0.948507i	$-0.602593\pi$
$719$	−6.31401e9	−0.633511	−0.316755	+	0.948507i	$0.602593\pi$
$720$	− 4.55026e8i	− 0.0454331i
$721$	− 9.25738e9i	− 0.919845i
$722$	1.14240e9i	0.112963i
$723$	− 1.90733e9i	− 0.187690i
$724$	1.13801e10	1.11446
$725$	−1.58006e10	−1.53989
$726$	1.24058e10i	1.20322i
$727$	−6.79560e9	−0.655930	−0.327965	−	0.944690i	$-0.606363\pi$
$727$	−6.79560e9	−0.655930	−0.327965	+	0.944690i	$0.606363\pi$
$728$	0	0
$729$	1.15872e10	1.10772
$730$	8.05314e8i	0.0766188i
$731$	−2.19668e9	−0.207997
$732$	−1.42434e10	−1.34223
$733$	− 2.19137e9i	− 0.205519i	−0.994706	−	0.102760i	$-0.967233\pi$
$733$	− 2.19137e9i	− 0.205519i	0.994706	−	0.102760i	$-0.0327672\pi$
$734$	9.71907e9i	0.907170i
$735$	4.74515e9i	0.440803i
$736$	− 1.33911e10i	− 1.23806i
$737$	3.23492e9	0.297664
$738$	−4.31168e9	−0.394866
$739$	− 1.32973e10i	− 1.21202i	−0.795459	−	0.606008i	$-0.792770\pi$
$739$	− 1.32973e10i	− 1.21202i	0.795459	−	0.606008i	$-0.207230\pi$
$740$	−1.05287e9	−0.0955137
$741$	0	0
$742$	4.00457e10	3.59867
$743$	− 6.93757e9i	− 0.620506i	−0.950654	−	0.310253i	$-0.899586\pi$
$743$	− 6.93757e9i	− 0.620506i	0.950654	−	0.310253i	$-0.100414\pi$
$744$	2.60105e9	0.231550
$745$	4.29002e9	0.380113
$746$	4.14122e9i	0.365210i
$747$	− 1.72087e9i	− 0.151052i
$748$	6.05817e8i	0.0529280i
$749$	9.35046e9i	0.813105i
$750$	9.90783e9	0.857559
$751$	2.15610e10	1.85751	0.928753	−	0.370700i	$-0.120882\pi$
$751$	2.15610e10	1.85751	0.928753	+	0.370700i	$0.120882\pi$
$752$	8.83839e9i	0.757898i
$753$	1.04835e10	0.894795
$754$	0	0
$755$	3.45731e9	0.292364
$756$	2.34037e10i	1.96996i
$757$	−3.63572e9	−0.304617	−0.152309	−	0.988333i	$-0.548671\pi$
$757$	−3.63572e9	−0.304617	−0.152309	+	0.988333i	$0.548671\pi$
$758$	−2.37736e10	−1.98268
$759$	− 3.08574e9i	− 0.256161i
$760$	− 1.07172e9i	− 0.0885590i
$761$	8.44474e9i	0.694608i	0.937753	+	0.347304i	$0.112903\pi$
$761$	8.44474e9i	0.694608i	−0.937753	+	0.347304i	$0.887097\pi$
$762$	2.24866e10i	1.84112i
$763$	1.75973e10	1.43421
$764$	2.12604e9	0.172482
$765$	9.90867e7i	0.00800203i
$766$	−1.86646e10	−1.50044
$767$	0	0
$768$	6.13045e9	0.488346
$769$	9.24151e9i	0.732825i	0.930452	+	0.366413i	$0.119414\pi$
$769$	9.24151e9i	0.732825i	−0.930452	+	0.366413i	$0.880586\pi$
$770$	3.20539e9	0.253025
$771$	−4.79178e9	−0.376536
$772$	9.10172e9i	0.711972i
$773$	1.17156e10i	0.912297i	0.889904	+	0.456149i	$0.150771\pi$
$773$	1.17156e10i	0.912297i	−0.889904	+	0.456149i	$0.849229\pi$
$774$	4.95203e9i	0.383875i
$775$	1.07158e10i	0.826931i
$776$	1.92099e9	0.147574
$777$	−4.42609e9	−0.338491
$778$	3.62578e10i	2.76041i
$779$	1.97067e10	1.49359
$780$	0	0
$781$	5.78305e8	0.0434389
$782$	2.35943e9i	0.176435i
$783$	2.49069e10	1.85419
$784$	1.51079e10	1.11969
$785$	5.93790e9i	0.438116i
$786$	− 1.03301e10i	− 0.758798i
$787$	− 1.29788e10i	− 0.949121i	−0.880223	−	0.474560i	$-0.842607\pi$
$787$	− 1.29788e10i	− 0.949121i	0.880223	−	0.474560i	$-0.157393\pi$
$788$	− 6.89342e9i	− 0.501872i
$789$	−1.48711e10	−1.07789
$790$	−3.03110e9	−0.218728
$791$	− 1.68334e10i	− 1.20936i
$792$	2.12618e8	0.0152077
$793$	0	0
$794$	−4.91291e9	−0.348311
$795$	6.80561e9i	0.480377i
$796$	1.92899e10	1.35560
$797$	6.23728e9	0.436406	0.218203	−	0.975903i	$-0.429980\pi$
$797$	6.23728e9	0.436406	0.218203	+	0.975903i	$0.429980\pi$
$798$	− 2.89388e10i	− 2.01591i
$799$	− 1.92465e9i	− 0.133487i
$800$	− 1.83116e10i	− 1.26448i
$801$	1.32580e9i	0.0911516i
$802$	−8.34137e9	−0.570988
$803$	7.30217e8	0.0497677
$804$	− 1.45642e10i	− 0.988301i
$805$	6.76881e9	0.457327
$806$	0	0
$807$	−1.44708e10	−0.969246
$808$	5.51499e8i	0.0367794i
$809$	1.36198e10	0.904379	0.452189	−	0.891922i	$-0.350643\pi$
$809$	1.36198e10	0.904379	0.452189	+	0.891922i	$0.350643\pi$
$810$	−6.03611e9	−0.399079
$811$	− 3.36478e9i	− 0.221505i	−0.993848	−	0.110752i	$-0.964674\pi$
$811$	− 3.36478e9i	− 0.221505i	0.993848	−	0.110752i	$-0.0353261\pi$
$812$	4.89997e10i	3.21179i
$813$	1.88203e10i	1.22831i
$814$	1.76075e9i	0.114423i
$815$	2.66952e9	0.172735
$816$	−1.51972e9	−0.0979148
$817$	− 2.26334e10i	− 1.45202i
$818$	1.48477e9	0.0948465
$819$	0	0
$820$	9.82734e9	0.622426
$821$	− 2.07010e10i	− 1.30554i	−0.757555	−	0.652771i	$-0.773606\pi$
$821$	− 2.07010e10i	− 1.30554i	0.757555	−	0.652771i	$-0.226394\pi$
$822$	−3.76579e10	−2.36486
$823$	7.89394e9	0.493622	0.246811	−	0.969064i	$-0.420617\pi$
$823$	7.89394e9	0.493622	0.246811	+	0.969064i	$0.420617\pi$
$824$	2.58126e9i	0.160726i
$825$	− 4.21959e9i	− 0.261626i
$826$	1.16839e10i	0.721366i
$827$	2.15815e9i	0.132682i	0.997797	+	0.0663411i	$0.0211325\pi$
$827$	2.15815e9i	0.132682i	−0.997797	+	0.0663411i	$0.978867\pi$
$828$	2.88396e9	0.176556
$829$	−6.94515e9	−0.423390	−0.211695	−	0.977336i	$-0.567898\pi$
$829$	−6.94515e9	−0.423390	−0.211695	+	0.977336i	$0.567898\pi$
$830$	7.23390e9i	0.439136i
$831$	4.28240e9	0.258871
$832$	0	0
$833$	−3.28992e9	−0.197209
$834$	3.90993e10i	2.33393i
$835$	8.09880e9	0.481413
$836$	−6.24200e9	−0.369489
$837$	− 1.68917e10i	− 0.995711i
$838$	2.60211e9i	0.152747i
$839$	− 1.03645e10i	− 0.605875i	−0.953010	−	0.302938i	$-0.902033\pi$
$839$	− 1.03645e10i	− 0.605875i	0.953010	−	0.302938i	$-0.0979674\pi$
$840$	− 2.24671e9i	− 0.130788i
$841$	3.48972e10	2.02304
$842$	3.65902e10	2.11238
$843$	− 9.07411e9i	− 0.521684i
$844$	−1.00369e10	−0.574649
$845$	0	0
$846$	−4.33879e9	−0.246362
$847$	2.46753e10i	1.39531i
$848$	2.16682e10	1.22022
$849$	7.39984e9	0.414997
$850$	3.22640e9i	0.180199i
$851$	3.71818e9i	0.206812i
$852$	− 2.60363e9i	− 0.144225i
$853$	1.25745e10i	0.693696i	0.937921	+	0.346848i	$0.112748\pi$
$853$	1.25745e10i	0.693696i	−0.937921	+	0.346848i	$0.887252\pi$
$854$	−5.22503e10	−2.87069
$855$	−1.02093e9	−0.0558620
$856$	− 2.60722e9i	− 0.142075i
$857$	−2.63377e10	−1.42937	−0.714687	−	0.699445i	$-0.753431\pi$
$857$	−2.63377e10	−1.42937	−0.714687	+	0.699445i	$0.753431\pi$
$858$	0	0
$859$	−6.43919e9	−0.346621	−0.173311	−	0.984867i	$-0.555446\pi$
$859$	−6.43919e9	−0.346621	−0.173311	+	0.984867i	$0.555446\pi$
$860$	− 1.12868e10i	− 0.605101i
$861$	4.13123e10	2.20581
$862$	−1.59968e9	−0.0850665
$863$	− 3.08603e10i	− 1.63442i	−0.576343	−	0.817208i	$-0.695521\pi$
$863$	− 3.08603e10i	− 1.63442i	0.576343	−	0.817208i	$-0.304479\pi$
$864$	2.88651e10i	1.52256i
$865$	− 1.01321e9i	− 0.0532284i
$866$	− 1.61031e10i	− 0.842554i
$867$	−1.71316e10	−0.892752
$868$	3.32312e10	1.72475
$869$	2.74844e9i	0.142075i
$870$	−1.53582e10	−0.790719
$871$	0	0
$872$	−4.90672e9	−0.250601
$873$	− 1.82997e9i	− 0.0930881i
$874$	−2.43103e10	−1.23169
$875$	1.97069e10	0.994465
$876$	− 3.28757e9i	− 0.165238i
$877$	9.81457e9i	0.491330i	0.969355	+	0.245665i	$0.0790062\pi$
$877$	9.81457e9i	0.491330i	−0.969355	+	0.245665i	$0.920994\pi$
$878$	− 3.41642e10i	− 1.70349i
$879$	− 5.59456e9i	− 0.277847i
$880$	1.73439e9	0.0857942
$881$	1.77084e10	0.872498	0.436249	−	0.899826i	$-0.356307\pi$
$881$	1.77084e10	0.872498	0.436249	+	0.899826i	$0.356307\pi$
$882$	7.41654e9i	0.363966i
$883$	3.61270e10	1.76591	0.882956	−	0.469456i	$-0.155550\pi$
$883$	3.61270e10	1.76591	0.882956	+	0.469456i	$0.155550\pi$
$884$	0	0
$885$	−1.98562e9	−0.0962932
$886$	1.61356e10i	0.779411i
$887$	−1.76108e10	−0.847320	−0.423660	−	0.905821i	$-0.639255\pi$
$887$	−1.76108e10	−0.847320	−0.423660	+	0.905821i	$0.639255\pi$
$888$	1.23414e9	0.0591451
$889$	4.47263e10i	2.13504i
$890$	− 5.57317e9i	− 0.264995i
$891$	5.47323e9i	0.259222i
$892$	− 2.90607e10i	− 1.37098i
$893$	1.98306e10	0.931870
$894$	−3.23001e10	−1.51190
$895$	− 1.08453e10i	− 0.505662i
$896$	−1.79927e10	−0.835636
$897$	0	0
$898$	3.36273e10	1.54962
$899$	− 3.53657e10i	− 1.62339i
$900$	3.94365e9	0.180322
$901$	−4.71848e9	−0.214914
$902$	− 1.64346e10i	− 0.745651i
$903$	− 4.74478e10i	− 2.14442i
$904$	4.69371e9i	0.211313i
$905$	7.09468e9i	0.318173i
$906$	−2.60305e10	−1.16288
$907$	−3.08827e10	−1.37432	−0.687162	−	0.726504i	$-0.741144\pi$
$907$	−3.08827e10	−1.37432	−0.687162	+	0.726504i	$0.741144\pi$
$908$	− 8.52177e9i	− 0.377772i
$909$	5.25366e8	0.0232000
$910$	0	0
$911$	5.82328e9	0.255184	0.127592	−	0.991827i	$-0.459275\pi$
$911$	5.82328e9	0.255184	0.127592	+	0.991827i	$0.459275\pi$
$912$	− 1.56584e10i	− 0.683542i
$913$	6.55933e9	0.285241
$914$	5.17430e10	2.24151
$915$	− 8.87973e9i	− 0.383201i
$916$	− 3.64368e10i	− 1.56641i
$917$	− 2.05468e10i	− 0.879937i
$918$	− 5.08587e9i	− 0.216978i
$919$	−2.43323e10	−1.03414	−0.517068	−	0.855944i	$-0.672977\pi$
$919$	−2.43323e10	−1.03414	−0.517068	+	0.855944i	$0.672977\pi$
$920$	−1.88737e9	−0.0799096
$921$	3.28447e10i	1.38534i
$922$	4.19412e10	1.76231
$923$	0	0
$924$	−1.30855e10	−0.545680
$925$	5.08440e9i	0.211224i
$926$	−1.36153e10	−0.563494
$927$	2.45895e9	0.101384
$928$	6.04341e10i	2.48236i
$929$	4.65831e9i	0.190622i	0.995448	+	0.0953111i	$0.0303846\pi$
$929$	4.65831e9i	0.190622i	−0.995448	+	0.0953111i	$0.969615\pi$
$930$	1.04158e10i	0.424621i
$931$	− 3.38975e10i	− 1.37671i
$932$	1.25774e10	0.508902
$933$	1.65369e10	0.666604
$934$	− 4.15438e10i	− 1.66837i
$935$	−3.77682e8	−0.0151107
$936$	0	0
$937$	4.24433e10	1.68547	0.842734	−	0.538331i	$-0.180945\pi$
$937$	4.24433e10	1.68547	0.842734	+	0.538331i	$0.180945\pi$
$938$	− 5.34269e10i	− 2.11373i
$939$	−2.30890e10	−0.910073
$940$	9.88913e9	0.388339
$941$	− 2.82397e10i	− 1.10483i	−0.833569	−	0.552416i	$-0.813706\pi$
$941$	− 2.82397e10i	− 1.10483i	0.833569	−	0.552416i	$-0.186294\pi$
$942$	− 4.47072e10i	− 1.74261i
$943$	− 3.47048e10i	− 1.34772i
$944$	6.32197e9i	0.244597i
$945$	−1.45905e10	−0.562417
$946$	−1.88753e10	−0.724896
$947$	1.66024e10i	0.635251i	0.948216	+	0.317625i	$0.102885\pi$
$947$	1.66024e10i	0.635251i	−0.948216	+	0.317625i	$0.897115\pi$
$948$	1.23740e10	0.471715
$949$	0	0
$950$	−3.32430e10	−1.25796
$951$	4.07204e9i	0.153525i
$952$	1.55769e9	0.0585129
$953$	−4.25790e10	−1.59357	−0.796783	−	0.604265i	$-0.793467\pi$
$953$	−4.25790e10	−1.59357	−0.796783	+	0.604265i	$0.793467\pi$
$954$	1.06370e10i	0.396642i
$955$	1.32543e9i	0.0492430i
$956$	2.59748e10i	0.961502i
$957$	1.39260e10i	0.513611i
$958$	7.42540e10	2.72861
$959$	−7.49023e10	−2.74240
$960$	− 1.12060e10i	− 0.408790i
$961$	3.52794e9	0.128230
$962$	0	0
$963$	−2.48367e9	−0.0896195
$964$	6.79463e9i	0.244285i
$965$	−5.67425e9	−0.203265
$966$	−5.09632e10	−1.81902
$967$	2.16365e10i	0.769474i	0.923026	+	0.384737i	$0.125708\pi$
$967$	2.16365e10i	0.769474i	−0.923026	+	0.384737i	$0.874292\pi$
$968$	− 6.88029e9i	− 0.243805i
$969$	3.40978e9i	0.120391i
$970$	7.69251e9i	0.270624i
$971$	5.78647e9	0.202837	0.101418	−	0.994844i	$-0.467662\pi$
$971$	5.78647e9	0.202837	0.101418	+	0.994844i	$0.467662\pi$
$972$	−1.15211e10	−0.402404
$973$	7.77693e10i	2.70653i
$974$	−6.17726e10	−2.14210
$975$	0	0
$976$	−2.82719e10	−0.973377
$977$	− 1.10283e10i	− 0.378335i	−0.981945	−	0.189167i	$-0.939421\pi$
$977$	− 1.10283e10i	− 0.378335i	0.981945	−	0.189167i	$-0.0605789\pi$
$978$	−2.00992e10	−0.687055
$979$	−5.05346e9	−0.172127
$980$	− 1.69040e10i	− 0.573719i
$981$	4.67422e9i	0.158077i
$982$	5.17307e10i	1.74324i
$983$	1.85197e10i	0.621864i	0.950432	+	0.310932i	$0.100641\pi$
$983$	1.85197e10i	0.621864i	−0.950432	+	0.310932i	$0.899359\pi$
$984$	−1.15192e10	−0.385426
$985$	4.29754e9	0.143282
$986$	− 1.06482e10i	− 0.353757i
$987$	4.15721e10	1.37623
$988$	0	0
$989$	−3.98590e10	−1.31020
$990$	8.51418e8i	0.0278882i
$991$	−1.54024e10	−0.502725	−0.251363	−	0.967893i	$-0.580879\pi$
$991$	−1.54024e10	−0.502725	−0.251363	+	0.967893i	$0.580879\pi$
$992$	4.09859e10	1.33304
$993$	9.50768e9i	0.308143i
$994$	− 9.55112e9i	− 0.308462i
$995$	1.20258e10i	0.387020i
$996$	− 2.95312e10i	− 0.947053i
$997$	−3.62981e10	−1.15998	−0.579990	−	0.814624i	$-0.696943\pi$
$997$	−3.62981e10	−1.15998	−0.579990	+	0.814624i	$0.696943\pi$
$998$	4.08932e10	1.30225
$999$	− 8.01470e9i	− 0.254336i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.b.d.168.12		14
13.3	even	3		13.8.e.a.4.2	✓	14
13.4	even	6		13.8.e.a.10.2	yes	14
13.5	odd	4		169.8.a.g.1.12		14
13.8	odd	4		169.8.a.g.1.3		14
13.12	even	2	inner	169.8.b.d.168.3		14
39.17	odd	6		117.8.q.b.10.6		14
39.29	odd	6		117.8.q.b.82.6		14
52.3	odd	6		208.8.w.a.17.5		14
52.43	odd	6		208.8.w.a.49.5		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.e.a.4.2	✓	14	13.3	even	3
13.8.e.a.10.2	yes	14	13.4	even	6
117.8.q.b.10.6		14	39.17	odd	6
117.8.q.b.82.6		14	39.29	odd	6
169.8.a.g.1.3		14	13.8	odd	4
169.8.a.g.1.12		14	13.5	odd	4
169.8.b.d.168.3		14	13.12	even	2	inner
169.8.b.d.168.12		14	1.1	even	1	trivial
208.8.w.a.17.5		14	52.3	odd	6
208.8.w.a.49.5		14	52.43	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)

\(f(q)\)	\(=\)	\(q+16.7213i q^{2} +42.5564 q^{3} -151.602 q^{4} -94.5127i q^{5} +711.598i q^{6} +1415.38i q^{7} -394.655i q^{8} -375.955 q^{9} +1580.37 q^{10} -1433.00i q^{11} -6451.63 q^{12} -23667.0 q^{14} -4022.12i q^{15} -12805.9 q^{16} +2788.62 q^{17} -6286.46i q^{18} +28732.4i q^{19} +14328.3i q^{20} +60233.5i q^{21} +23961.7 q^{22} +50599.7 q^{23} -16795.1i q^{24} +69192.4 q^{25} -109070. q^{27} -214575. i q^{28} -228357. q^{29} +67255.0 q^{30} +154870. i q^{31} -264647. i q^{32} -60983.4i q^{33} +46629.4i q^{34} +133772. q^{35} +56995.5 q^{36} +73482.1i q^{37} -480443. q^{38} -37299.9 q^{40} -685869. i q^{41} -1.00718e6 q^{42} -787730. q^{43} +217246. i q^{44} +35532.5i q^{45} +846094. i q^{46} -690181. i q^{47} -544973. q^{48} -1.17976e6 q^{49} +1.15699e6i q^{50} +118674. q^{51} -1.69205e6 q^{53} -1.82379e6i q^{54} -135437. q^{55} +558588. q^{56} +1.22275e6i q^{57} -3.81843e6i q^{58} -493676. i q^{59} +609761. i q^{60} +2.20773e6 q^{61} -2.58963e6 q^{62} -532120. i q^{63} +2.78609e6 q^{64} +1.01972e6 q^{66} +2.25744e6i q^{67} -422760. q^{68} +2.15334e6 q^{69} +2.23683e6i q^{70} +403562. i q^{71} +148373. i q^{72} +509572. i q^{73} -1.22872e6 q^{74} +2.94458e6 q^{75} -4.35589e6i q^{76} +2.02825e6 q^{77} -1.91796e6 q^{79} +1.21032e6i q^{80} -3.81941e6 q^{81} +1.14686e7 q^{82} +4.57733e6i q^{83} -9.13152e6i q^{84} -263560. i q^{85} -1.31719e7i q^{86} -9.71806e6 q^{87} -565542. q^{88} -3.52649e6i q^{89} -594150. q^{90} -7.67102e6 q^{92} +6.59070e6i q^{93} +1.15407e7 q^{94} +2.71558e6 q^{95} -1.12624e7i q^{96} +4.86752e6i q^{97} -1.97272e7i q^{98} +538744. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 52 q^{3} - 766 q^{4} + 6982 q^{9} + 1018 q^{10} + 38380 q^{12} - 47916 q^{14} + 1266 q^{16} + 76806 q^{17} + 251764 q^{22} + 137100 q^{23} + 39380 q^{25} - 432400 q^{27} - 443166 q^{29} + 315780 q^{30}+ \cdots + 86840772 q^{95}+O(q^{100}) \) 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

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