LMFDB - Embedded newform 169.8.b.a.168.2

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.8.b.a.168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.0000i q^{2} -73.0000 q^{3} +28.0000 q^{4} -295.000i q^{5} -730.000i q^{6} -1373.00i q^{7} +1560.00i q^{8} +3142.00 q^{9} +O(q^{10})$	\(q+10.0000i q^{2} -73.0000 q^{3} +28.0000 q^{4} -295.000i q^{5} -730.000i q^{6} -1373.00i q^{7} +1560.00i q^{8} +3142.00 q^{9} +2950.00 q^{10} +7646.00i q^{11} -2044.00 q^{12} +13730.0 q^{14} +21535.0i q^{15} -12016.0 q^{16} +4147.00 q^{17} +31420.0i q^{18} -3186.00i q^{19} -8260.00i q^{20} +100229. i q^{21} -76460.0 q^{22} +17784.0 q^{23} -113880. i q^{24} -8900.00 q^{25} -69715.0 q^{27} -38444.0i q^{28} -93322.0 q^{29} -215350. q^{30} -124484. i q^{31} +79520.0i q^{32} -558158. i q^{33} +41470.0i q^{34} -405035. q^{35} +87976.0 q^{36} -273661. i q^{37} +31860.0 q^{38} +460200. q^{40} +585816. i q^{41} -1.00229e6 q^{42} +533559. q^{43} +214088. i q^{44} -926890. i q^{45} +177840. i q^{46} +530055. i q^{47} +877168. q^{48} -1.06159e6 q^{49} -89000.0i q^{50} -302731. q^{51} -615288. q^{53} -697150. i q^{54} +2.25557e6 q^{55} +2.14188e6 q^{56} +232578. i q^{57} -933220. i q^{58} +392514. i q^{59} +602980. i q^{60} +1.87806e6 q^{61} +1.24484e6 q^{62} -4.31397e6i q^{63} -2.33325e6 q^{64} +5.58158e6 q^{66} -3.97144e6i q^{67} +116116. q^{68} -1.29823e6 q^{69} -4.05035e6i q^{70} -3.74660e6i q^{71} +4.90152e6i q^{72} -2.48580e6i q^{73} +2.73661e6 q^{74} +649700. q^{75} -89208.0i q^{76} +1.04980e7 q^{77} -1.26446e6 q^{79} +3.54472e6i q^{80} -1.78236e6 q^{81} -5.85816e6 q^{82} +434308. i q^{83} +2.80641e6i q^{84} -1.22336e6i q^{85} +5.33559e6i q^{86} +6.81251e6 q^{87} -1.19278e7 q^{88} -5.83081e6i q^{89} +9.26890e6 q^{90} +497952. q^{92} +9.08733e6i q^{93} -5.30055e6 q^{94} -939870. q^{95} -5.80496e6i q^{96} -2.04533e6i q^{97} -1.06159e7i q^{98} +2.40237e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 146 q^{3} + 56 q^{4} + 6284 q^{9}+O(q^{10}) $ 2 * q - 146 * q^3 + 56 * q^4 + 6284 * q^9	$ 2 q - 146 q^{3} + 56 q^{4} + 6284 q^{9} + 5900 q^{10} - 4088 q^{12} + 27460 q^{14} - 24032 q^{16} + 8294 q^{17} - 152920 q^{22} + 35568 q^{23} - 17800 q^{25} - 139430 q^{27} - 186644 q^{29} - 430700 q^{30} - 810070 q^{35} + 175952 q^{36} + 63720 q^{38} + 920400 q^{40} - 2004580 q^{42} + 1067118 q^{43} + 1754336 q^{48} - 2123172 q^{49} - 605462 q^{51} - 1230576 q^{53} + 4511140 q^{55} + 4283760 q^{56} + 3756128 q^{61} + 2489680 q^{62} - 4666496 q^{64} + 11163160 q^{66} + 232232 q^{68} - 2596464 q^{69} + 5473220 q^{74} + 1299400 q^{75} + 20995916 q^{77} - 2528912 q^{79} - 3564718 q^{81} - 11716320 q^{82} + 13625012 q^{87} - 23855520 q^{88} + 18537800 q^{90} + 995904 q^{92} - 10601100 q^{94} - 1879740 q^{95}+O(q^{100}) $ 2 * q - 146 * q^3 + 56 * q^4 + 6284 * q^9 + 5900 * q^10 - 4088 * q^12 + 27460 * q^14 - 24032 * q^16 + 8294 * q^17 - 152920 * q^22 + 35568 * q^23 - 17800 * q^25 - 139430 * q^27 - 186644 * q^29 - 430700 * q^30 - 810070 * q^35 + 175952 * q^36 + 63720 * q^38 + 920400 * q^40 - 2004580 * q^42 + 1067118 * q^43 + 1754336 * q^48 - 2123172 * q^49 - 605462 * q^51 - 1230576 * q^53 + 4511140 * q^55 + 4283760 * q^56 + 3756128 * q^61 + 2489680 * q^62 - 4666496 * q^64 + 11163160 * q^66 + 232232 * q^68 - 2596464 * q^69 + 5473220 * q^74 + 1299400 * q^75 + 20995916 * q^77 - 2528912 * q^79 - 3564718 * q^81 - 11716320 * q^82 + 13625012 * q^87 - 23855520 * q^88 + 18537800 * q^90 + 995904 * q^92 - 10601100 * q^94 - 1879740 * q^95

Display 100 coefficients 10 coefficients

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 $

$2$	10.0000i	0.883883i	0.897044	+	0.441942i	$0.145710\pi$
$2$	10.0000i	0.883883i	−0.897044	+	0.441942i	$0.854290\pi$
$3$	−73.0000	−1.56098	−0.780492	−	0.625166i	$-0.785031\pi$
$3$	−73.0000	−1.56098	−0.780492	+	0.625166i	$0.785031\pi$
$4$	28.0000	0.218750
$5$	− 295.000i	− 1.05542i	−0.849423	−	0.527712i	$-0.823050\pi$
$5$	− 295.000i	− 1.05542i	0.849423	−	0.527712i	$-0.176950\pi$
$6$	− 730.000i	− 1.37973i
$7$	− 1373.00i	− 1.51296i	−0.654017	−	0.756480i	$-0.726918\pi$
$7$	− 1373.00i	− 1.51296i	0.654017	−	0.756480i	$-0.273082\pi$
$8$	1560.00i	1.07723i
$9$	3142.00	1.43667
$10$	2950.00	0.932872
$11$	7646.00i	1.73205i	0.500003	+	0.866024i	$0.333332\pi$
$11$	7646.00i	1.73205i	−0.500003	+	0.866024i	$0.666668\pi$
$12$	−2044.00	−0.341465
$13$	0	0
$13$	0	0
$14$	13730.0	1.33728
$15$	21535.0i	1.64750i
$16$	−12016.0	−0.733398
$17$	4147.00	0.204721	0.102361	−	0.994747i	$-0.467360\pi$
$17$	4147.00	0.204721	0.102361	+	0.994747i	$0.467360\pi$
$18$	31420.0i	1.26985i
$19$	− 3186.00i	− 0.106563i	−0.998580	−	0.0532817i	$-0.983032\pi$
$19$	− 3186.00i	− 0.106563i	0.998580	−	0.0532817i	$-0.0169681\pi$
$20$	− 8260.00i	− 0.230874i
$21$	100229.i	2.36171i
$22$	−76460.0	−1.53093
$23$	17784.0	0.304777	0.152388	−	0.988321i	$-0.451304\pi$
$23$	17784.0	0.304777	0.152388	+	0.988321i	$0.451304\pi$
$24$	− 113880.i	− 1.68154i
$25$	−8900.00	−0.113920
$26$	0	0
$27$	−69715.0	−0.681637
$28$	− 38444.0i	− 0.330960i
$29$	−93322.0	−0.710544	−0.355272	−	0.934763i	$-0.615612\pi$
$29$	−93322.0	−0.710544	−0.355272	+	0.934763i	$0.615612\pi$
$30$	−215350.	−1.45620
$31$	− 124484.i	− 0.750495i	−0.926925	−	0.375247i	$-0.877558\pi$
$31$	− 124484.i	− 0.750495i	0.926925	−	0.375247i	$-0.122442\pi$
$32$	79520.0i	0.428994i
$33$	− 558158.i	− 2.70370i
$34$	41470.0i	0.180950i
$35$	−405035.	−1.59681
$36$	87976.0	0.314272
$37$	− 273661.i	− 0.888192i	−0.895979	−	0.444096i	$-0.853525\pi$
$37$	− 273661.i	− 0.888192i	0.895979	−	0.444096i	$-0.146475\pi$
$38$	31860.0	0.0941897
$39$	0	0
$40$	460200.	1.13694
$41$	585816.i	1.32745i	0.747977	+	0.663724i	$0.231025\pi$
$41$	585816.i	1.32745i	−0.747977	+	0.663724i	$0.768975\pi$
$42$	−1.00229e6	−2.08747
$43$	533559.	1.02339	0.511697	−	0.859166i	$-0.329017\pi$
$43$	533559.	1.02339	0.511697	+	0.859166i	$0.329017\pi$
$44$	214088.i	0.378885i
$45$	− 926890.i	− 1.51630i
$46$	177840.i	0.269387i
$47$	530055.i	0.744695i	0.928093	+	0.372347i	$0.121447\pi$
$47$	530055.i	0.744695i	−0.928093	+	0.372347i	$0.878553\pi$
$48$	877168.	1.14482
$49$	−1.06159e6	−1.28905
$50$	− 89000.0i	− 0.100692i
$51$	−302731.	−0.319567
$52$	0	0
$53$	−615288.	−0.567692	−0.283846	−	0.958870i	$-0.591610\pi$
$53$	−615288.	−0.567692	−0.283846	+	0.958870i	$0.591610\pi$
$54$	− 697150.i	− 0.602488i
$55$	2.25557e6	1.82805
$56$	2.14188e6	1.62981
$57$	232578.i	0.166344i
$58$	− 933220.i	− 0.628038i
$59$	392514.i	0.248813i	0.992231	+	0.124407i	$0.0397027\pi$
$59$	392514.i	0.248813i	−0.992231	+	0.124407i	$0.960297\pi$
$60$	602980.i	0.360391i
$61$	1.87806e6	1.05939	0.529695	−	0.848188i	$-0.322306\pi$
$61$	1.87806e6	1.05939	0.529695	+	0.848188i	$0.322306\pi$
$62$	1.24484e6	0.663350
$63$	− 4.31397e6i	− 2.17363i
$64$	−2.33325e6	−1.11258
$65$	0	0
$66$	5.58158e6	2.38976
$67$	− 3.97144e6i	− 1.61319i	−0.591103	−	0.806596i	$-0.701307\pi$
$67$	− 3.97144e6i	− 1.61319i	0.591103	−	0.806596i	$-0.298693\pi$
$68$	116116.	0.0447828
$69$	−1.29823e6	−0.475752
$70$	− 4.05035e6i	− 1.41140i
$71$	− 3.74660e6i	− 1.24232i	−0.783684	−	0.621160i	$-0.786662\pi$
$71$	− 3.74660e6i	− 1.24232i	0.783684	−	0.621160i	$-0.213338\pi$
$72$	4.90152e6i	1.54763i
$73$	− 2.48580e6i	− 0.747888i	−0.927451	−	0.373944i	$-0.878005\pi$
$73$	− 2.48580e6i	− 0.747888i	0.927451	−	0.373944i	$-0.121995\pi$
$74$	2.73661e6	0.785058
$75$	649700.	0.177827
$76$	− 89208.0i	− 0.0233107i
$77$	1.04980e7	2.62052
$78$	0	0
$79$	−1.26446e6	−0.288542	−0.144271	−	0.989538i	$-0.546084\pi$
$79$	−1.26446e6	−0.288542	−0.144271	+	0.989538i	$0.546084\pi$
$80$	3.54472e6i	0.774046i
$81$	−1.78236e6	−0.372647
$82$	−5.85816e6	−1.17331
$83$	434308.i	0.0833728i	0.999131	+	0.0416864i	$0.0132730\pi$
$83$	434308.i	0.0833728i	−0.999131	+	0.0416864i	$0.986727\pi$
$84$	2.80641e6i	0.516623i
$85$	− 1.22336e6i	− 0.216068i
$86$	5.33559e6i	0.904561i
$87$	6.81251e6	1.10915
$88$	−1.19278e7	−1.86582
$89$	− 5.83081e6i	− 0.876726i	−0.898798	−	0.438363i	$-0.855558\pi$
$89$	− 5.83081e6i	− 0.876726i	0.898798	−	0.438363i	$-0.144442\pi$
$90$	9.26890e6	1.34023
$91$	0	0
$92$	497952.	0.0666699
$93$	9.08733e6i	1.17151i
$94$	−5.30055e6	−0.658224
$95$	−939870.	−0.112470
$96$	− 5.80496e6i	− 0.669653i
$97$	− 2.04533e6i	− 0.227542i	−0.993507	−	0.113771i	$-0.963707\pi$
$97$	− 2.04533e6i	− 0.227542i	0.993507	−	0.113771i	$-0.0362931\pi$
$98$	− 1.06159e7i	− 1.13937i
$99$	2.40237e7i	2.48838i
$100$	−249200.	−0.0249200
$101$	1.55142e6	0.149832	0.0749160	−	0.997190i	$-0.476131\pi$
$101$	1.55142e6	0.149832	0.0749160	+	0.997190i	$0.476131\pi$
$102$	− 3.02731e6i	− 0.282460i
$103$	1.68251e7	1.51714	0.758572	−	0.651590i	$-0.225898\pi$
$103$	1.68251e7	1.51714	0.758572	+	0.651590i	$0.225898\pi$
$104$	0	0
$105$	2.95676e7	2.49260
$106$	− 6.15288e6i	− 0.501774i
$107$	2.19295e7	1.73055	0.865277	−	0.501294i	$-0.167142\pi$
$107$	2.19295e7	1.73055	0.865277	+	0.501294i	$0.167142\pi$
$108$	−1.95202e6	−0.149108
$109$	− 1.96595e7i	− 1.45405i	−0.686612	−	0.727024i	$-0.740903\pi$
$109$	− 1.96595e7i	− 1.45405i	0.686612	−	0.727024i	$-0.259097\pi$
$110$	2.25557e7i	1.61578i
$111$	1.99773e7i	1.38645i
$112$	1.64980e7i	1.10960i
$113$	−2.14963e7	−1.40149	−0.700744	−	0.713412i	$-0.747149\pi$
$113$	−2.14963e7	−1.40149	−0.700744	+	0.713412i	$0.747149\pi$
$114$	−2.32578e6	−0.147029
$115$	− 5.24628e6i	− 0.321669i
$116$	−2.61302e6	−0.155432
$117$	0	0
$118$	−3.92514e6	−0.219922
$119$	− 5.69383e6i	− 0.309735i
$120$	−3.35946e7	−1.77474
$121$	−3.89741e7	−1.99999
$122$	1.87806e7i	0.936378i
$123$	− 4.27646e7i	− 2.07213i
$124$	− 3.48555e6i	− 0.164171i
$125$	− 2.04214e7i	− 0.935190i
$126$	4.31397e7	1.92123
$127$	1.77419e7	0.768578	0.384289	−	0.923213i	$-0.374447\pi$
$127$	1.77419e7	0.768578	0.384289	+	0.923213i	$0.374447\pi$
$128$	− 1.31539e7i	− 0.554396i
$129$	−3.89498e7	−1.59750
$130$	0	0
$131$	−8.61184e6	−0.334693	−0.167346	−	0.985898i	$-0.553520\pi$
$131$	−8.61184e6	−0.334693	−0.167346	+	0.985898i	$0.553520\pi$
$132$	− 1.56284e7i	− 0.591434i
$133$	−4.37438e6	−0.161226
$134$	3.97144e7	1.42587
$135$	2.05659e7i	0.719416i
$136$	6.46932e6i	0.220532i
$137$	− 6.30262e6i	− 0.209411i	−0.994503	−	0.104705i	$-0.966610\pi$
$137$	− 6.30262e6i	− 0.209411i	0.994503	−	0.104705i	$-0.0333899\pi$
$138$	− 1.29823e7i	− 0.420509i
$139$	1.34997e7	0.426355	0.213177	−	0.977014i	$-0.431619\pi$
$139$	1.34997e7	0.426355	0.213177	+	0.977014i	$0.431619\pi$
$140$	−1.13410e7	−0.349303
$141$	− 3.86940e7i	− 1.16246i
$142$	3.74660e7	1.09807
$143$	0	0
$144$	−3.77543e7	−1.05365
$145$	2.75300e7i	0.749925i
$146$	2.48580e7	0.661046
$147$	7.74958e7	2.01218
$148$	− 7.66251e6i	− 0.194292i
$149$	− 1.43791e7i	− 0.356105i	−0.984021	−	0.178053i	$-0.943020\pi$
$149$	− 1.43791e7i	− 0.356105i	0.984021	−	0.178053i	$-0.0569797\pi$
$150$	6.49700e6i	0.157179i
$151$	− 8.24764e7i	− 1.94944i	−0.223424	−	0.974721i	$-0.571723\pi$
$151$	− 8.24764e7i	− 1.94944i	0.223424	−	0.974721i	$-0.428277\pi$
$152$	4.97016e6	0.114794
$153$	1.30299e7	0.294117
$154$	1.04980e8i	2.31623i
$155$	−3.67228e7	−0.792090
$156$	0	0
$157$	8.92107e6	0.183979	0.0919895	−	0.995760i	$-0.470677\pi$
$157$	8.92107e6	0.183979	0.0919895	+	0.995760i	$0.470677\pi$
$158$	− 1.26446e7i	− 0.255037i
$159$	4.49160e7	0.886158
$160$	2.34584e7	0.452771
$161$	− 2.44174e7i	− 0.461115i
$162$	− 1.78236e7i	− 0.329377i
$163$	− 2.09065e7i	− 0.378116i	−0.981966	−	0.189058i	$-0.939457\pi$
$163$	− 2.09065e7i	− 0.378116i	0.981966	−	0.189058i	$-0.0605434\pi$
$164$	1.64028e7i	0.290379i
$165$	−1.64657e8	−2.85355
$166$	−4.34308e6	−0.0736919
$167$	1.88221e7i	0.312724i	0.987700	+	0.156362i	$0.0499766\pi$
$167$	1.88221e7i	0.312724i	−0.987700	+	0.156362i	$0.950023\pi$
$168$	−1.56357e8	−2.54411
$169$	0	0
$170$	1.22336e7	0.190979
$171$	− 1.00104e7i	− 0.153097i
$172$	1.49397e7	0.223867
$173$	4.78358e6	0.0702412	0.0351206	−	0.999383i	$-0.488818\pi$
$173$	4.78358e6	0.0702412	0.0351206	+	0.999383i	$0.488818\pi$
$174$	6.81251e7i	0.980358i
$175$	1.22197e7i	0.172356i
$176$	− 9.18743e7i	− 1.27028i
$177$	− 2.86535e7i	− 0.388393i
$178$	5.83081e7	0.774924
$179$	−9.09914e7	−1.18581	−0.592904	−	0.805273i	$-0.702019\pi$
$179$	−9.09914e7	−1.18581	−0.592904	+	0.805273i	$0.702019\pi$
$180$	− 2.59529e7i	− 0.331690i
$181$	1.72015e7	0.215622	0.107811	−	0.994171i	$-0.465616\pi$
$181$	1.72015e7	0.215622	0.107811	+	0.994171i	$0.465616\pi$
$182$	0	0
$183$	−1.37099e8	−1.65369
$184$	2.77430e7i	0.328316i
$185$	−8.07300e7	−0.937419
$186$	−9.08733e7	−1.03548
$187$	3.17080e7i	0.354587i
$188$	1.48415e7i	0.162902i
$189$	9.57187e7i	1.03129i
$190$	− 9.39870e6i	− 0.0994100i
$191$	−6.68698e7	−0.694405	−0.347203	−	0.937790i	$-0.612868\pi$
$191$	−6.68698e7	−0.694405	−0.347203	+	0.937790i	$0.612868\pi$
$192$	1.70327e8	1.73672
$193$	4.86222e7i	0.486838i	0.969921	+	0.243419i	$0.0782689\pi$
$193$	4.86222e7i	0.486838i	−0.969921	+	0.243419i	$0.921731\pi$
$194$	2.04533e7	0.201121
$195$	0	0
$196$	−2.97244e7	−0.281979
$197$	− 8.42682e7i	− 0.785293i	−0.919689	−	0.392646i	$-0.871560\pi$
$197$	− 8.42682e7i	− 0.785293i	0.919689	−	0.392646i	$-0.128440\pi$
$198$	−2.40237e8	−2.19944
$199$	−1.39905e8	−1.25849	−0.629243	−	0.777208i	$-0.716635\pi$
$199$	−1.39905e8	−1.25849	−0.629243	+	0.777208i	$0.716635\pi$
$200$	− 1.38840e7i	− 0.122718i
$201$	2.89915e8i	2.51817i
$202$	1.55142e7i	0.132434i
$203$	1.28131e8i	1.07502i
$204$	−8.47647e6	−0.0699052
$205$	1.72816e8	1.40102
$206$	1.68251e8i	1.34098i
$207$	5.58773e7	0.437864
$208$	0	0
$209$	2.43602e7	0.184573
$210$	2.95676e8i	2.20317i
$211$	2.26349e8	1.65878	0.829391	−	0.558669i	$-0.188688\pi$
$211$	2.26349e8	1.65878	0.829391	+	0.558669i	$0.188688\pi$
$212$	−1.72281e7	−0.124183
$213$	2.73502e8i	1.93924i
$214$	2.19295e8i	1.52961i
$215$	− 1.57400e8i	− 1.08011i
$216$	− 1.08755e8i	− 0.734282i
$217$	−1.70917e8	−1.13547
$218$	1.96595e8	1.28521
$219$	1.81464e8i	1.16744i
$220$	6.31560e7	0.399885
$221$	0	0
$222$	−1.99773e8	−1.22546
$223$	− 2.19897e8i	− 1.32786i	−0.747796	−	0.663929i	$-0.768888\pi$
$223$	− 2.19897e8i	− 1.32786i	0.747796	−	0.663929i	$-0.231112\pi$
$224$	1.09181e8	0.649051
$225$	−2.79638e7	−0.163666
$226$	− 2.14963e8i	− 1.23875i
$227$	− 2.30377e8i	− 1.30722i	−0.756831	−	0.653611i	$-0.773253\pi$
$227$	− 2.30377e8i	− 1.30722i	0.756831	−	0.653611i	$-0.226747\pi$
$228$	6.51218e6i	0.0363877i
$229$	5.41755e7i	0.298111i	0.988829	+	0.149056i	$0.0476234\pi$
$229$	5.41755e7i	0.298111i	−0.988829	+	0.149056i	$0.952377\pi$
$230$	5.24628e7	0.284318
$231$	−7.66351e8	−4.09059
$232$	− 1.45582e8i	− 0.765422i
$233$	−1.41580e8	−0.733259	−0.366629	−	0.930367i	$-0.619488\pi$
$233$	−1.41580e8	−0.733259	−0.366629	+	0.930367i	$0.619488\pi$
$234$	0	0
$235$	1.56366e8	0.785969
$236$	1.09904e7i	0.0544278i
$237$	9.23053e7	0.450409
$238$	5.69383e7	0.273770
$239$	− 2.57365e8i	− 1.21943i	−0.792621	−	0.609715i	$-0.791284\pi$
$239$	− 2.57365e8i	− 1.21943i	0.792621	−	0.609715i	$-0.208716\pi$
$240$	− 2.58765e8i	− 1.20827i
$241$	2.46818e8i	1.13584i	0.823083	+	0.567921i	$0.192252\pi$
$241$	2.46818e8i	1.13584i	−0.823083	+	0.567921i	$0.807748\pi$
$242$	− 3.89741e8i	− 1.76776i
$243$	2.82579e8	1.26333
$244$	5.25858e7	0.231742
$245$	3.13168e8i	1.36049i
$246$	4.27646e8	1.83152
$247$	0	0
$248$	1.94195e8	0.808458
$249$	− 3.17045e7i	− 0.130144i
$250$	2.04214e8	0.826599
$251$	−2.39628e7	−0.0956490	−0.0478245	−	0.998856i	$-0.515229\pi$
$251$	−2.39628e7	−0.0956490	−0.0478245	+	0.998856i	$0.515229\pi$
$252$	− 1.20791e8i	− 0.475481i
$253$	1.35976e8i	0.527888i
$254$	1.77419e8i	0.679333i
$255$	8.93056e7i	0.337278i
$256$	−1.67117e8	−0.622558
$257$	2.50050e8	0.918885	0.459443	−	0.888207i	$-0.348049\pi$
$257$	2.50050e8	0.918885	0.459443	+	0.888207i	$0.348049\pi$
$258$	− 3.89498e8i	− 1.41201i
$259$	−3.75737e8	−1.34380
$260$	0	0
$261$	−2.93218e8	−1.02082
$262$	− 8.61184e7i	− 0.295829i
$263$	−2.09182e8	−0.709055	−0.354527	−	0.935046i	$-0.615358\pi$
$263$	−2.09182e8	−0.709055	−0.354527	+	0.935046i	$0.615358\pi$
$264$	8.70726e8	2.91251
$265$	1.81510e8i	0.599156i
$266$	− 4.37438e7i	− 0.142505i
$267$	4.25649e8i	1.36856i
$268$	− 1.11200e8i	− 0.352886i
$269$	3.71414e8	1.16339	0.581695	−	0.813407i	$-0.302390\pi$
$269$	3.71414e8	1.16339	0.581695	+	0.813407i	$0.302390\pi$
$270$	−2.05659e8	−0.635880
$271$	− 3.30225e8i	− 1.00790i	−0.863733	−	0.503950i	$-0.831879\pi$
$271$	− 3.30225e8i	− 1.00790i	0.863733	−	0.503950i	$-0.168121\pi$
$272$	−4.98304e7	−0.150142
$273$	0	0
$274$	6.30262e7	0.185095
$275$	− 6.80494e7i	− 0.197315i
$276$	−3.63505e7	−0.104071
$277$	−5.06278e8	−1.43123	−0.715616	−	0.698494i	$-0.753854\pi$
$277$	−5.06278e8	−1.43123	−0.715616	+	0.698494i	$0.753854\pi$
$278$	1.34997e8i	0.376848i
$279$	− 3.91129e8i	− 1.07821i
$280$	− 6.31855e8i	− 1.72014i
$281$	− 1.06744e8i	− 0.286994i	−0.989651	−	0.143497i	$-0.954165\pi$
$281$	− 1.06744e8i	− 0.286994i	0.989651	−	0.143497i	$-0.0458347\pi$
$282$	3.86940e8	1.02748
$283$	5.56521e8	1.45958	0.729792	−	0.683669i	$-0.239617\pi$
$283$	5.56521e8	1.45958	0.729792	+	0.683669i	$0.239617\pi$
$284$	− 1.04905e8i	− 0.271757i
$285$	6.86105e7	0.175563
$286$	0	0
$287$	8.04325e8	2.00838
$288$	2.49852e8i	0.616324i
$289$	−3.93141e8	−0.958089
$290$	−2.75300e8	−0.662847
$291$	1.49309e8i	0.355190i
$292$	− 6.96025e7i	− 0.163600i
$293$	− 2.23708e8i	− 0.519571i	−0.965666	−	0.259786i	$-0.916348\pi$
$293$	− 2.23708e8i	− 0.519571i	0.965666	−	0.259786i	$-0.0836519\pi$
$294$	7.74958e8i	1.77853i
$295$	1.15792e8	0.262603
$296$	4.26911e8	0.956790
$297$	− 5.33041e8i	− 1.18063i
$298$	1.43791e8	0.314756
$299$	0	0
$300$	1.81916e7	0.0388997
$301$	− 7.32577e8i	− 1.54835i
$302$	8.24764e8	1.72308
$303$	−1.13254e8	−0.233885
$304$	3.82830e7i	0.0781534i
$305$	− 5.54029e8i	− 1.11811i
$306$	1.30299e8i	0.259965i
$307$	9.91919e8i	1.95655i	0.207300	+	0.978277i	$0.433532\pi$
$307$	9.91919e8i	1.95655i	−0.207300	+	0.978277i	$0.566468\pi$
$308$	2.93943e8	0.573239
$309$	−1.22823e9	−2.36824
$310$	− 3.67228e8i	− 0.700115i
$311$	2.48269e8	0.468016	0.234008	−	0.972235i	$-0.424816\pi$
$311$	2.48269e8	0.468016	0.234008	+	0.972235i	$0.424816\pi$
$312$	0	0
$313$	−2.00737e8	−0.370018	−0.185009	−	0.982737i	$-0.559231\pi$
$313$	−2.00737e8	−0.370018	−0.185009	+	0.982737i	$0.559231\pi$
$314$	8.92107e7i	0.162616i
$315$	−1.27262e9	−2.29410
$316$	−3.54048e7	−0.0631185
$317$	− 1.02635e8i	− 0.180962i	−0.995898	−	0.0904808i	$-0.971160\pi$
$317$	− 1.02635e8i	− 0.180962i	0.995898	−	0.0904808i	$-0.0288404\pi$
$318$	4.49160e8i	0.783261i
$319$	− 7.13540e8i	− 1.23070i
$320$	6.88308e8i	1.17424i
$321$	−1.60085e9	−2.70137
$322$	2.44174e8	0.407572
$323$	− 1.32123e7i	− 0.0218158i
$324$	−4.99061e7	−0.0815165
$325$	0	0
$326$	2.09065e8	0.334210
$327$	1.43514e9i	2.26975i
$328$	−9.13873e8	−1.42997
$329$	7.27766e8	1.12669
$330$	− 1.64657e9i	− 2.52220i
$331$	− 7.60053e8i	− 1.15198i	−0.817456	−	0.575991i	$-0.804616\pi$
$331$	− 7.60053e8i	− 1.15198i	0.817456	−	0.575991i	$-0.195384\pi$
$332$	1.21606e7i	0.0182378i
$333$	− 8.59843e8i	− 1.27604i
$334$	−1.88221e8	−0.276411
$335$	−1.17157e9	−1.70260
$336$	− 1.20435e9i	− 1.73207i
$337$	4.36659e7	0.0621495	0.0310748	−	0.999517i	$-0.490107\pi$
$337$	4.36659e7	0.0621495	0.0310748	+	0.999517i	$0.490107\pi$
$338$	0	0
$339$	1.56923e9	2.18770
$340$	− 3.42542e7i	− 0.0472648i
$341$	9.51805e8	1.29989
$342$	1.00104e8	0.135320
$343$	3.26833e8i	0.437317i
$344$	8.32352e8i	1.10243i
$345$	3.82978e8i	0.502120i
$346$	4.78358e7i	0.0620851i
$347$	−1.82053e8	−0.233907	−0.116954	−	0.993137i	$-0.537313\pi$
$347$	−1.82053e8	−0.233907	−0.116954	+	0.993137i	$0.537313\pi$
$348$	1.90750e8	0.242626
$349$	5.80955e8i	0.731566i	0.930700	+	0.365783i	$0.119199\pi$
$349$	5.80955e8i	0.731566i	−0.930700	+	0.365783i	$0.880801\pi$
$350$	−1.22197e8	−0.152343
$351$	0	0
$352$	−6.08010e8	−0.743039
$353$	5.45624e8i	0.660210i	0.943944	+	0.330105i	$0.107084\pi$
$353$	5.45624e8i	0.660210i	−0.943944	+	0.330105i	$0.892916\pi$
$354$	2.86535e8	0.343294
$355$	−1.10525e9	−1.31117
$356$	− 1.63263e8i	− 0.191784i
$357$	4.15650e8i	0.483491i
$358$	− 9.09914e8i	− 1.04812i
$359$	1.05196e9i	1.19996i	0.800014	+	0.599981i	$0.204825\pi$
$359$	1.05196e9i	1.19996i	−0.800014	+	0.599981i	$0.795175\pi$
$360$	1.44595e9	1.63341
$361$	8.83721e8	0.988644
$362$	1.72015e8i	0.190584i
$363$	2.84511e9	3.12195
$364$	0	0
$365$	−7.33312e8	−0.789339
$366$	− 1.37099e9i	− 1.46167i
$367$	7.29203e8	0.770047	0.385024	−	0.922907i	$-0.374193\pi$
$367$	7.29203e8	0.770047	0.385024	+	0.922907i	$0.374193\pi$
$368$	−2.13693e8	−0.223523
$369$	1.84063e9i	1.90711i
$370$	− 8.07300e8i	− 0.828569i
$371$	8.44790e8i	0.858895i
$372$	2.54445e8i	0.256268i
$373$	1.32385e9	1.32087	0.660434	−	0.750884i	$-0.270372\pi$
$373$	1.32385e9	1.32087	0.660434	+	0.750884i	$0.270372\pi$
$374$	−3.17080e8	−0.313414
$375$	1.49076e9i	1.45982i
$376$	−8.26886e8	−0.802210
$377$	0	0
$378$	−9.57187e8	−0.911539
$379$	− 1.08474e8i	− 0.102350i	−0.998690	−	0.0511752i	$-0.983703\pi$
$379$	− 1.08474e8i	− 0.102350i	0.998690	−	0.0511752i	$-0.0162967\pi$
$380$	−2.63164e7	−0.0246027
$381$	−1.29516e9	−1.19974
$382$	− 6.68698e8i	− 0.613773i
$383$	2.84754e8i	0.258985i	0.991580	+	0.129492i	$0.0413348\pi$
$383$	2.84754e8i	0.258985i	−0.991580	+	0.129492i	$0.958665\pi$
$384$	9.60236e8i	0.865404i
$385$	− 3.09690e9i	− 2.76576i
$386$	−4.86222e8	−0.430308
$387$	1.67644e9	1.47028
$388$	− 5.72692e7i	− 0.0497749i
$389$	−3.22741e8	−0.277991	−0.138996	−	0.990293i	$-0.544387\pi$
$389$	−3.22741e8	−0.277991	−0.138996	+	0.990293i	$0.544387\pi$
$390$	0	0
$391$	7.37502e7	0.0623943
$392$	− 1.65607e9i	− 1.38860i
$393$	6.28664e8	0.522450
$394$	8.42682e8	0.694108
$395$	3.73015e8i	0.304534i
$396$	6.72664e8i	0.544334i
$397$	8.64634e8i	0.693531i	0.937952	+	0.346765i	$0.112720\pi$
$397$	8.64634e8i	0.693531i	−0.937952	+	0.346765i	$0.887280\pi$
$398$	− 1.39905e9i	− 1.11236i
$399$	3.19330e8	0.251671
$400$	1.06942e8	0.0835488
$401$	1.31166e9i	1.01582i	0.861411	+	0.507909i	$0.169581\pi$
$401$	1.31166e9i	1.01582i	−0.861411	+	0.507909i	$0.830419\pi$
$402$	−2.89915e9	−2.22577
$403$	0	0
$404$	4.34398e7	0.0327758
$405$	5.25796e8i	0.393301i
$406$	−1.28131e9	−0.950197
$407$	2.09241e9	1.53839
$408$	− 4.72260e8i	− 0.344248i
$409$	− 4.00024e7i	− 0.0289104i	−0.999896	−	0.0144552i	$-0.995399\pi$
$409$	− 4.00024e7i	− 0.0289104i	0.999896	−	0.0144552i	$-0.00460140\pi$
$410$	1.72816e9i	1.23834i
$411$	4.60091e8i	0.326887i
$412$	4.71102e8	0.331875
$413$	5.38922e8	0.376444
$414$	5.58773e8i	0.387021i
$415$	1.28121e8	0.0879937
$416$	0	0
$417$	−9.85475e8	−0.665533
$418$	2.43602e8i	0.163141i
$419$	2.64978e9	1.75979	0.879896	−	0.475166i	$-0.157612\pi$
$419$	2.64978e9	1.75979	0.879896	+	0.475166i	$0.157612\pi$
$420$	8.27892e8	0.545257
$421$	8.99741e8i	0.587665i	0.955857	+	0.293833i	$0.0949308\pi$
$421$	8.99741e8i	0.587665i	−0.955857	+	0.293833i	$0.905069\pi$
$422$	2.26349e9i	1.46617i
$423$	1.66543e9i	1.06988i
$424$	− 9.59849e8i	− 0.611537i
$425$	−3.69083e7	−0.0233218
$426$	−2.73502e9	−1.71406
$427$	− 2.57858e9i	− 1.60281i
$428$	6.14026e8	0.378559
$429$	0	0
$430$	1.57400e9	0.954695
$431$	3.69212e8i	0.222129i	0.993813	+	0.111065i	$0.0354261\pi$
$431$	3.69212e8i	0.222129i	−0.993813	+	0.111065i	$0.964574\pi$
$432$	8.37695e8	0.499911
$433$	−2.63280e9	−1.55851	−0.779255	−	0.626707i	$-0.784402\pi$
$433$	−2.63280e9	−1.55851	−0.779255	+	0.626707i	$0.784402\pi$
$434$	− 1.70917e9i	− 1.00362i
$435$	− 2.00969e9i	− 1.17062i
$436$	− 5.50465e8i	− 0.318073i
$437$	− 5.66598e7i	− 0.0324781i
$438$	−1.81464e9	−1.03188
$439$	−3.44814e8	−0.194518	−0.0972588	−	0.995259i	$-0.531007\pi$
$439$	−3.44814e8	−0.194518	−0.0972588	+	0.995259i	$0.531007\pi$
$440$	3.51869e9i	1.96923i
$441$	−3.33550e9	−1.85194
$442$	0	0
$443$	−1.68347e9	−0.920012	−0.460006	−	0.887916i	$-0.652153\pi$
$443$	−1.68347e9	−0.920012	−0.460006	+	0.887916i	$0.652153\pi$
$444$	5.59363e8i	0.303287i
$445$	−1.72009e9	−0.925318
$446$	2.19897e9	1.17367
$447$	1.04967e9i	0.555875i
$448$	3.20355e9i	1.68329i
$449$	− 3.20869e9i	− 1.67288i	−0.548058	−	0.836440i	$-0.684633\pi$
$449$	− 3.20869e9i	− 1.67288i	0.548058	−	0.836440i	$-0.315367\pi$
$450$	− 2.79638e8i	− 0.144661i
$451$	−4.47915e9	−2.29920
$452$	−6.01897e8	−0.306576
$453$	6.02078e9i	3.04305i
$454$	2.30377e9	1.15543
$455$	0	0
$456$	−3.62822e8	−0.179191
$457$	− 8.53834e8i	− 0.418473i	−0.977865	−	0.209236i	$-0.932902\pi$
$457$	− 8.53834e8i	− 0.418473i	0.977865	−	0.209236i	$-0.0670978\pi$
$458$	−5.41755e8	−0.263496
$459$	−2.89108e8	−0.139546
$460$	− 1.46896e8i	− 0.0703651i
$461$	− 3.91799e9i	− 1.86256i	−0.364308	−	0.931279i	$-0.618694\pi$
$461$	− 3.91799e9i	− 1.86256i	0.364308	−	0.931279i	$-0.381306\pi$
$462$	− 7.66351e9i	− 3.61560i
$463$	9.00831e8i	0.421803i	0.977507	+	0.210902i	$0.0676399\pi$
$463$	9.00831e8i	0.421803i	−0.977507	+	0.210902i	$0.932360\pi$
$464$	1.12136e9	0.521112
$465$	2.68076e9	1.23644
$466$	− 1.41580e9i	− 0.648115i
$467$	8.14889e7	0.0370245	0.0185123	−	0.999829i	$-0.494107\pi$
$467$	8.14889e7	0.0370245	0.0185123	+	0.999829i	$0.494107\pi$
$468$	0	0
$469$	−5.45278e9	−2.44069
$470$	1.56366e9i	0.694705i
$471$	−6.51238e8	−0.287188
$472$	−6.12322e8	−0.268030
$473$	4.07959e9i	1.77257i
$474$	9.23053e8i	0.398109i
$475$	2.83554e7i	0.0121397i
$476$	− 1.59427e8i	− 0.0677545i
$477$	−1.93323e9	−0.815587
$478$	2.57365e9	1.07783
$479$	− 4.28234e9i	− 1.78036i	−0.455612	−	0.890178i	$-0.650580\pi$
$479$	− 4.28234e9i	− 1.78036i	0.455612	−	0.890178i	$-0.349420\pi$
$480$	−1.71246e9	−0.706768
$481$	0	0
$482$	−2.46818e9	−1.00395
$483$	1.78247e9i	0.719793i
$484$	−1.09128e9	−0.437498
$485$	−6.03372e8	−0.240154
$486$	2.82579e9i	1.11664i
$487$	4.24063e9i	1.66371i	0.554990	+	0.831857i	$0.312722\pi$
$487$	4.24063e9i	1.66371i	−0.554990	+	0.831857i	$0.687278\pi$
$488$	2.92978e9i	1.14121i
$489$	1.52617e9i	0.590233i
$490$	−3.13168e9	−1.20252
$491$	−2.21387e9	−0.844046	−0.422023	−	0.906585i	$-0.638680\pi$
$491$	−2.21387e9	−0.844046	−0.422023	+	0.906585i	$0.638680\pi$
$492$	− 1.19741e9i	− 0.453278i
$493$	−3.87006e8	−0.145463
$494$	0	0
$495$	7.08700e9	2.62630
$496$	1.49580e9i	0.550412i
$497$	−5.14408e9	−1.87958
$498$	3.17045e8	0.115032
$499$	− 2.45975e9i	− 0.886215i	−0.896468	−	0.443108i	$-0.853876\pi$
$499$	− 2.45975e9i	− 0.886215i	0.896468	−	0.443108i	$-0.146124\pi$
$500$	− 5.71798e8i	− 0.204573i
$501$	− 1.37401e9i	− 0.488156i
$502$	− 2.39628e8i	− 0.0845426i
$503$	3.72798e9	1.30613	0.653063	−	0.757303i	$-0.273484\pi$
$503$	3.72798e9	1.30613	0.653063	+	0.757303i	$0.273484\pi$
$504$	6.72979e9	2.34150
$505$	− 4.57669e8i	− 0.158136i
$506$	−1.35976e9	−0.466592
$507$	0	0
$508$	4.96774e8	0.168126
$509$	1.48553e9i	0.499308i	0.968335	+	0.249654i	$0.0803168\pi$
$509$	1.48553e9i	0.499308i	−0.968335	+	0.249654i	$0.919683\pi$
$510$	−8.93056e8	−0.298115
$511$	−3.41301e9	−1.13152
$512$	− 3.35487e9i	− 1.10466i
$513$	2.22112e8i	0.0726376i
$514$	2.50050e9i	0.812187i
$515$	− 4.96340e9i	− 1.60123i
$516$	−1.09059e9	−0.349453
$517$	−4.05280e9	−1.28985
$518$	− 3.75737e9i	− 1.18776i
$519$	−3.49202e8	−0.109645
$520$	0	0
$521$	1.06857e9	0.331031	0.165516	−	0.986207i	$-0.447071\pi$
$521$	1.06857e9	0.331031	0.165516	+	0.986207i	$0.447071\pi$
$522$	− 2.93218e9i	− 0.902284i
$523$	−3.85266e8	−0.117762	−0.0588809	−	0.998265i	$-0.518753\pi$
$523$	−3.85266e8	−0.117762	−0.0588809	+	0.998265i	$0.518753\pi$
$524$	−2.41131e8	−0.0732140
$525$	− 8.92038e8i	− 0.269046i
$526$	− 2.09182e9i	− 0.626722i
$527$	− 5.16235e8i	− 0.153642i
$528$	6.70683e9i	1.98289i
$529$	−3.08855e9	−0.907111
$530$	−1.81510e9	−0.529584
$531$	1.23328e9i	0.357463i
$532$	−1.22483e8	−0.0352682
$533$	0	0
$534$	−4.25649e9	−1.20964
$535$	− 6.46920e9i	− 1.82647i
$536$	6.19544e9	1.73778
$537$	6.64237e9	1.85103
$538$	3.71414e9i	1.02830i
$539$	− 8.11689e9i	− 2.23269i
$540$	5.75846e8i	0.157372i
$541$	2.81334e9i	0.763891i	0.924185	+	0.381946i	$0.124746\pi$
$541$	2.81334e9i	0.763891i	−0.924185	+	0.381946i	$0.875254\pi$
$542$	3.30225e9	0.890867
$543$	−1.25571e9	−0.336582
$544$	3.29769e8i	0.0878242i
$545$	−5.79954e9	−1.53464
$546$	0	0
$547$	1.67344e9	0.437174	0.218587	−	0.975818i	$-0.429855\pi$
$547$	1.67344e9	0.437174	0.218587	+	0.975818i	$0.429855\pi$
$548$	− 1.76473e8i	− 0.0458086i
$549$	5.90088e9	1.52200
$550$	6.80494e8	0.174403
$551$	2.97324e8i	0.0757180i
$552$	− 2.02524e9i	− 0.512496i
$553$	1.73610e9i	0.436552i
$554$	− 5.06278e9i	− 1.26504i
$555$	5.89329e9	1.46330
$556$	3.77991e8	0.0932651
$557$	4.46631e9i	1.09511i	0.836771	+	0.547553i	$0.184441\pi$
$557$	4.46631e9i	1.09511i	−0.836771	+	0.547553i	$0.815559\pi$
$558$	3.91129e9	0.953016
$559$	0	0
$560$	4.86690e9	1.17110
$561$	− 2.31468e9i	− 0.553505i
$562$	1.06744e9	0.253669
$563$	5.06446e9	1.19606	0.598031	−	0.801473i	$-0.295950\pi$
$563$	5.06446e9	1.19606	0.598031	+	0.801473i	$0.295950\pi$
$564$	− 1.08343e9i	− 0.254287i
$565$	6.34142e9i	1.47917i
$566$	5.56521e9i	1.29010i
$567$	2.44718e9i	0.563800i
$568$	5.84470e9	1.33827
$569$	−4.07861e9	−0.928152	−0.464076	−	0.885795i	$-0.653614\pi$
$569$	−4.07861e9	−0.928152	−0.464076	+	0.885795i	$0.653614\pi$
$570$	6.86105e8i	0.155177i
$571$	7.82983e9	1.76005	0.880027	−	0.474923i	$-0.157524\pi$
$571$	7.82983e9	1.76005	0.880027	+	0.474923i	$0.157524\pi$
$572$	0	0
$573$	4.88149e9	1.08396
$574$	8.04325e9i	1.77517i
$575$	−1.58278e8	−0.0347202
$576$	−7.33107e9	−1.59841
$577$	− 7.94179e9i	− 1.72109i	−0.509376	−	0.860544i	$-0.670124\pi$
$577$	− 7.94179e9i	− 1.72109i	0.509376	−	0.860544i	$-0.329876\pi$
$578$	− 3.93141e9i	− 0.846839i
$579$	− 3.54942e9i	− 0.759946i
$580$	7.70840e8i	0.164046i
$581$	5.96305e8	0.126140
$582$	−1.49309e9	−0.313947
$583$	− 4.70449e9i	− 0.983270i
$584$	3.87785e9	0.805650
$585$	0	0
$586$	2.23708e9	0.459241
$587$	− 2.02009e9i	− 0.412227i	−0.978528	−	0.206114i	$-0.933918\pi$
$587$	− 2.02009e9i	− 0.412227i	0.978528	−	0.206114i	$-0.0660816\pi$
$588$	2.16988e9	0.440165
$589$	−3.96606e8	−0.0799753
$590$	1.15792e9i	0.232111i
$591$	6.15158e9i	1.22583i
$592$	3.28831e9i	0.651399i
$593$	− 5.19728e9i	− 1.02349i	−0.859137	−	0.511746i	$-0.828999\pi$
$593$	− 5.19728e9i	− 1.02349i	0.859137	−	0.511746i	$-0.171001\pi$
$594$	5.33041e9	1.04354
$595$	−1.67968e9	−0.326902
$596$	− 4.02614e8i	− 0.0778980i
$597$	1.02131e10	1.96448
$598$	0	0
$599$	−3.92347e9	−0.745893	−0.372946	−	0.927853i	$-0.621652\pi$
$599$	−3.92347e9	−0.745893	−0.372946	+	0.927853i	$0.621652\pi$
$600$	1.01353e9i	0.191561i
$601$	−9.51281e8	−0.178751	−0.0893754	−	0.995998i	$-0.528487\pi$
$601$	−9.51281e8	−0.178751	−0.0893754	+	0.995998i	$0.528487\pi$
$602$	7.32577e9	1.36856
$603$	− 1.24783e10i	− 2.31763i
$604$	− 2.30934e9i	− 0.426441i
$605$	1.14974e10i	2.11084i
$606$	− 1.13254e9i	− 0.206727i
$607$	8.27679e9	1.50211	0.751055	−	0.660240i	$-0.229545\pi$
$607$	8.27679e9	1.50211	0.751055	+	0.660240i	$0.229545\pi$
$608$	2.53351e8	0.0457151
$609$	− 9.35357e9i	− 1.67810i
$610$	5.54029e9	0.988275
$611$	0	0
$612$	3.64836e8	0.0643381
$613$	2.92674e9i	0.513183i	0.966520	+	0.256591i	$0.0825995\pi$
$613$	2.92674e9i	0.513183i	−0.966520	+	0.256591i	$0.917401\pi$
$614$	−9.91919e9	−1.72937
$615$	−1.26155e10	−2.18697
$616$	1.63768e10i	2.82291i
$617$	8.88587e9i	1.52301i	0.648161	+	0.761503i	$0.275538\pi$
$617$	8.88587e9i	1.52301i	−0.648161	+	0.761503i	$0.724462\pi$
$618$	− 1.22823e10i	− 2.09324i
$619$	4.16163e9i	0.705255i	0.935764	+	0.352627i	$0.114712\pi$
$619$	4.16163e9i	0.705255i	−0.935764	+	0.352627i	$0.885288\pi$
$620$	−1.02824e9	−0.173270
$621$	−1.23981e9	−0.207747
$622$	2.48269e9i	0.413671i
$623$	−8.00570e9	−1.32645
$624$	0	0
$625$	−6.71962e9	−1.10094
$626$	− 2.00737e9i	− 0.327052i
$627$	−1.77829e9	−0.288115
$628$	2.49790e8	0.0402454
$629$	− 1.13487e9i	− 0.181832i
$630$	− 1.27262e10i	− 2.02771i
$631$	7.30070e9i	1.15681i	0.815750	+	0.578405i	$0.196324\pi$
$631$	7.30070e9i	1.15681i	−0.815750	+	0.578405i	$0.803676\pi$
$632$	− 1.97255e9i	− 0.310827i
$633$	−1.65234e10	−2.58933
$634$	1.02635e9	0.159949
$635$	− 5.23387e9i	− 0.811176i
$636$	1.25765e9	0.193847
$637$	0	0
$638$	7.13540e9	1.08779
$639$	− 1.17718e10i	− 1.78480i
$640$	−3.88041e9	−0.585123
$641$	−3.46867e9	−0.520187	−0.260094	−	0.965583i	$-0.583753\pi$
$641$	−3.46867e9	−0.520187	−0.260094	+	0.965583i	$0.583753\pi$
$642$	− 1.60085e10i	− 2.38769i
$643$	− 3.72175e9i	− 0.552089i	−0.961145	−	0.276045i	$-0.910976\pi$
$643$	− 3.72175e9i	− 0.552089i	0.961145	−	0.276045i	$-0.0890237\pi$
$644$	− 6.83688e8i	− 0.100869i
$645$	1.14902e10i	1.68604i
$646$	1.32123e8	0.0192826
$647$	8.03095e9	1.16574	0.582870	−	0.812565i	$-0.301930\pi$
$647$	8.03095e9	1.16574	0.582870	+	0.812565i	$0.301930\pi$
$648$	− 2.78048e9i	− 0.401428i
$649$	−3.00116e9	−0.430956
$650$	0	0
$651$	1.24769e10	1.77245
$652$	− 5.85382e8i	− 0.0827128i
$653$	8.22151e9	1.15546	0.577731	−	0.816227i	$-0.303938\pi$
$653$	8.22151e9	1.15546	0.577731	+	0.816227i	$0.303938\pi$
$654$	−1.43514e10	−2.00619
$655$	2.54049e9i	0.353243i
$656$	− 7.03917e9i	− 0.973549i
$657$	− 7.81039e9i	− 1.07447i
$658$	7.27766e9i	0.995866i
$659$	6.47061e9	0.880737	0.440369	−	0.897817i	$-0.354848\pi$
$659$	6.47061e9	0.880737	0.440369	+	0.897817i	$0.354848\pi$
$660$	−4.61039e9	−0.624214
$661$	3.64380e9i	0.490738i	0.969430	+	0.245369i	$0.0789091\pi$
$661$	3.64380e9i	0.490738i	−0.969430	+	0.245369i	$0.921091\pi$
$662$	7.60053e9	1.01822
$663$	0	0
$664$	−6.77520e8	−0.0898120
$665$	1.29044e9i	0.170162i
$666$	8.59843e9	1.12787
$667$	−1.65964e9	−0.216557
$668$	5.27019e8i	0.0684083i
$669$	1.60524e10i	2.07276i
$670$	− 1.17157e10i	− 1.50490i
$671$	1.43597e10i	1.83491i
$672$	−7.97021e9	−1.01316
$673$	−2.32463e9	−0.293968	−0.146984	−	0.989139i	$-0.546957\pi$
$673$	−2.32463e9	−0.293968	−0.146984	+	0.989139i	$0.546957\pi$
$674$	4.36659e8i	0.0549330i
$675$	6.20464e8	0.0776521
$676$	0	0
$677$	2.19098e9	0.271380	0.135690	−	0.990751i	$-0.456675\pi$
$677$	2.19098e9	0.271380	0.135690	+	0.990751i	$0.456675\pi$
$678$	1.56923e10i	1.93367i
$679$	−2.80824e9	−0.344262
$680$	1.90845e9	0.232755
$681$	1.68175e10i	2.04055i
$682$	9.51805e9i	1.14895i
$683$	1.70757e9i	0.205072i	0.994729	+	0.102536i	$0.0326957\pi$
$683$	1.70757e9i	0.205072i	−0.994729	+	0.102536i	$0.967304\pi$
$684$	− 2.80292e8i	− 0.0334899i
$685$	−1.85927e9	−0.221017
$686$	−3.26833e9	−0.386537
$687$	− 3.95481e9i	− 0.465347i
$688$	−6.41124e9	−0.750556
$689$	0	0
$690$	−3.82978e9	−0.443816
$691$	1.48657e10i	1.71400i	0.515316	+	0.857000i	$0.327675\pi$
$691$	1.48657e10i	1.71400i	−0.515316	+	0.857000i	$0.672325\pi$
$692$	1.33940e8	0.0153653
$693$	3.29846e10	3.76482
$694$	− 1.82053e9i	− 0.206747i
$695$	− 3.98240e9i	− 0.449985i
$696$	1.06275e10i	1.19481i
$697$	2.42938e9i	0.271757i
$698$	−5.80955e9	−0.646619
$699$	1.03354e10	1.14460
$700$	3.42152e8i	0.0377030i
$701$	−8.96793e9	−0.983284	−0.491642	−	0.870797i	$-0.663603\pi$
$701$	−8.96793e9	−0.983284	−0.491642	+	0.870797i	$0.663603\pi$
$702$	0	0
$703$	−8.71884e8	−0.0946488
$704$	− 1.78400e10i	− 1.92704i
$705$	−1.14147e10	−1.22689
$706$	−5.45624e9	−0.583549
$707$	− 2.13010e9i	− 0.226690i
$708$	− 8.02299e8i	− 0.0849610i
$709$	− 1.31197e10i	− 1.38249i	−0.722619	−	0.691247i	$-0.757062\pi$
$709$	− 1.31197e10i	− 1.38249i	0.722619	−	0.691247i	$-0.242938\pi$
$710$	− 1.10525e10i	− 1.15892i
$711$	−3.97292e9	−0.414540
$712$	9.09606e9	0.944438
$713$	− 2.21382e9i	− 0.228733i
$714$	−4.15650e9	−0.427350
$715$	0	0
$716$	−2.54776e9	−0.259396
$717$	1.87876e10i	1.90351i
$718$	−1.05196e10	−1.06063
$719$	1.16702e10	1.17092	0.585462	−	0.810700i	$-0.300913\pi$
$719$	1.16702e10	1.17092	0.585462	+	0.810700i	$0.300913\pi$
$720$	1.11375e10i	1.11205i
$721$	− 2.31008e10i	− 2.29538i
$722$	8.83721e9i	0.873846i
$723$	− 1.80177e10i	− 1.77303i
$724$	4.81643e8	0.0471673
$725$	8.30566e8	0.0809452
$726$	2.84511e10i	2.75944i
$727$	−1.03092e10	−0.995068	−0.497534	−	0.867444i	$-0.665761\pi$
$727$	−1.03092e10	−0.995068	−0.497534	+	0.867444i	$0.665761\pi$
$728$	0	0
$729$	−1.67302e10	−1.59940
$730$	− 7.33312e9i	− 0.697684i
$731$	2.21267e9	0.209510
$732$	−3.83876e9	−0.361745
$733$	− 1.00112e10i	− 0.938910i	−0.882956	−	0.469455i	$-0.844450\pi$
$733$	− 1.00112e10i	− 0.938910i	0.882956	−	0.469455i	$-0.155550\pi$
$734$	7.29203e9i	0.680632i
$735$	− 2.28613e10i	− 2.12371i
$736$	1.41418e9i	0.130748i
$737$	3.03656e10	2.79413
$738$	−1.84063e10	−1.68566
$739$	− 3.30781e9i	− 0.301498i	−0.988572	−	0.150749i	$-0.951831\pi$
$739$	− 3.30781e9i	− 0.301498i	0.988572	−	0.150749i	$-0.0481685\pi$
$740$	−2.26044e9	−0.205060
$741$	0	0
$742$	−8.44790e9	−0.759163
$743$	− 2.17089e10i	− 1.94168i	−0.239735	−	0.970838i	$-0.577061\pi$
$743$	− 2.17089e10i	− 1.94168i	0.239735	−	0.970838i	$-0.422939\pi$
$744$	−1.41762e10	−1.26199
$745$	−4.24182e9	−0.375842
$746$	1.32385e10i	1.16749i
$747$	1.36460e9i	0.119779i
$748$	8.87823e8i	0.0775659i
$749$	− 3.01092e10i	− 2.61826i
$750$	−1.49076e10	−1.29031
$751$	−9.19095e9	−0.791809	−0.395904	−	0.918292i	$-0.629569\pi$
$751$	−9.19095e9	−0.791809	−0.395904	+	0.918292i	$0.629569\pi$
$752$	− 6.36914e9i	− 0.546158i
$753$	1.74929e9	0.149307
$754$	0	0
$755$	−2.43305e10	−2.05749
$756$	2.68012e9i	0.225595i
$757$	9.78965e9	0.820222	0.410111	−	0.912036i	$-0.365490\pi$
$757$	9.78965e9	0.820222	0.410111	+	0.912036i	$0.365490\pi$
$758$	1.08474e9	0.0904658
$759$	− 9.92628e9i	− 0.824025i
$760$	− 1.46620e9i	− 0.121156i
$761$	2.03733e10i	1.67577i	0.545845	+	0.837886i	$0.316209\pi$
$761$	2.03733e10i	1.67577i	−0.545845	+	0.837886i	$0.683791\pi$
$762$	− 1.29516e10i	− 1.06043i
$763$	−2.69924e10	−2.19992
$764$	−1.87235e9	−0.151901
$765$	− 3.84381e9i	− 0.310418i
$766$	−2.84754e9	−0.228913
$767$	0	0
$768$	1.21995e10	0.971803
$769$	− 8.96000e9i	− 0.710503i	−0.934771	−	0.355251i	$-0.884395\pi$
$769$	− 8.96000e9i	− 0.710503i	0.934771	−	0.355251i	$-0.115605\pi$
$770$	3.09690e10	2.44461
$771$	−1.82537e10	−1.43436
$772$	1.36142e9i	0.106496i
$773$	7.61579e9i	0.593044i	0.955026	+	0.296522i	$0.0958268\pi$
$773$	7.61579e9i	0.593044i	−0.955026	+	0.296522i	$0.904173\pi$
$774$	1.67644e10i	1.29956i
$775$	1.10791e9i	0.0854964i
$776$	3.19071e9	0.245116
$777$	2.74288e10	2.09765
$778$	− 3.22741e9i	− 0.245712i
$779$	1.86641e9	0.141457
$780$	0	0
$781$	2.86465e10	2.15176
$782$	7.37502e8i	0.0551493i
$783$	6.50594e9	0.484333
$784$	1.27560e10	0.945385
$785$	− 2.63172e9i	− 0.194176i
$786$	6.28664e9i	0.461785i
$787$	− 2.15840e10i	− 1.57841i	−0.614130	−	0.789205i	$-0.710493\pi$
$787$	− 2.15840e10i	− 1.57841i	0.614130	−	0.789205i	$-0.289507\pi$
$788$	− 2.35951e9i	− 0.171783i
$789$	1.52703e10	1.10682
$790$	−3.73015e9	−0.269173
$791$	2.95145e10i	2.12040i
$792$	−3.74770e10	−2.68057
$793$	0	0
$794$	−8.64634e9	−0.613000
$795$	− 1.32502e10i	− 0.935273i
$796$	−3.91735e9	−0.275294
$797$	1.58880e10	1.11164	0.555822	−	0.831301i	$-0.312403\pi$
$797$	1.58880e10	1.11164	0.555822	+	0.831301i	$0.312403\pi$
$798$	3.19330e9i	0.222448i
$799$	2.19814e9i	0.152455i
$800$	− 7.07728e8i	− 0.0488710i
$801$	− 1.83204e10i	− 1.25957i
$802$	−1.31166e10	−0.897865
$803$	1.90064e10	1.29538
$804$	8.11762e9i	0.550849i
$805$	−7.20314e9	−0.486672
$806$	0	0
$807$	−2.71132e10	−1.81603
$808$	2.42022e9i	0.161404i
$809$	8.28899e9	0.550404	0.275202	−	0.961386i	$-0.411255\pi$
$809$	8.28899e9	0.550404	0.275202	+	0.961386i	$0.411255\pi$
$810$	−5.25796e9	−0.347632
$811$	− 6.46851e9i	− 0.425825i	−0.977071	−	0.212913i	$-0.931705\pi$
$811$	− 6.46851e9i	− 0.425825i	0.977071	−	0.212913i	$-0.0682949\pi$
$812$	3.58767e9i	0.235162i
$813$	2.41064e10i	1.57332i
$814$	2.09241e10i	1.35976i
$815$	−6.16742e9	−0.399072
$816$	3.63762e9	0.234370
$817$	− 1.69992e9i	− 0.109056i
$818$	4.00024e8	0.0255534
$819$	0	0
$820$	4.83884e9	0.306473
$821$	1.65268e10i	1.04228i	0.853470	+	0.521142i	$0.174494\pi$
$821$	1.65268e10i	1.04228i	−0.853470	+	0.521142i	$0.825506\pi$
$822$	−4.60091e9	−0.288930
$823$	2.05119e10	1.28264	0.641322	−	0.767272i	$-0.278386\pi$
$823$	2.05119e10	1.28264	0.641322	+	0.767272i	$0.278386\pi$
$824$	2.62471e10i	1.63432i
$825$	4.96761e9i	0.308005i
$826$	5.38922e9i	0.332733i
$827$	− 8.63679e9i	− 0.530986i	−0.964113	−	0.265493i	$-0.914465\pi$
$827$	− 8.63679e9i	− 0.530986i	0.964113	−	0.265493i	$-0.0855347\pi$
$828$	1.56457e9	0.0957828
$829$	−8.81187e9	−0.537189	−0.268595	−	0.963253i	$-0.586559\pi$
$829$	−8.81187e9	−0.537189	−0.268595	+	0.963253i	$0.586559\pi$
$830$	1.28121e9i	0.0777762i
$831$	3.69583e10	2.23413
$832$	0	0
$833$	−4.40240e9	−0.263895
$834$	− 9.85475e9i	− 0.588254i
$835$	5.55252e9	0.330056
$836$	6.82084e8	0.0403753
$837$	8.67840e9i	0.511565i
$838$	2.64978e10i	1.55545i
$839$	− 1.73321e10i	− 1.01317i	−0.862189	−	0.506586i	$-0.830907\pi$
$839$	− 1.73321e10i	− 1.01317i	0.862189	−	0.506586i	$-0.169093\pi$
$840$	4.61254e10i	2.68511i
$841$	−8.54088e9	−0.495127
$842$	−8.99741e9	−0.519428
$843$	7.79234e9i	0.447993i
$844$	6.33776e9	0.362859
$845$	0	0
$846$	−1.66543e10	−0.945651
$847$	5.35115e10i	3.02590i
$848$	7.39330e9	0.416345
$849$	−4.06260e10	−2.27839
$850$	− 3.69083e8i	− 0.0206138i
$851$	− 4.86679e9i	− 0.270700i
$852$	7.65805e9i	0.424209i
$853$	1.62475e10i	0.896322i	0.893953	+	0.448161i	$0.147921\pi$
$853$	1.62475e10i	0.896322i	−0.893953	+	0.448161i	$0.852079\pi$
$854$	2.57858e10	1.41670
$855$	−2.95307e9	−0.161582
$856$	3.42100e10i	1.86421i
$857$	2.45158e10	1.33049	0.665247	−	0.746623i	$-0.268326\pi$
$857$	2.45158e10	1.33049	0.665247	+	0.746623i	$0.268326\pi$
$858$	0	0
$859$	9.28369e9	0.499741	0.249870	−	0.968279i	$-0.419612\pi$
$859$	9.28369e9	0.499741	0.249870	+	0.968279i	$0.419612\pi$
$860$	− 4.40720e9i	− 0.236275i
$861$	−5.87158e10	−3.13504
$862$	−3.69212e9	−0.196336
$863$	− 6.17565e9i	− 0.327073i	−0.986537	−	0.163536i	$-0.947710\pi$
$863$	− 6.17565e9i	− 0.327073i	0.986537	−	0.163536i	$-0.0522901\pi$
$864$	− 5.54374e9i	− 0.292418i
$865$	− 1.41116e9i	− 0.0741343i
$866$	− 2.63280e10i	− 1.37754i
$867$	2.86993e10	1.49556
$868$	−4.78566e9	−0.248384
$869$	− 9.66803e9i	− 0.499768i
$870$	2.00969e10	1.03469
$871$	0	0
$872$	3.06688e10	1.56635
$873$	− 6.42643e9i	− 0.326904i
$874$	5.66598e8	0.0287068
$875$	−2.80385e10	−1.41491
$876$	5.08098e9i	0.255378i
$877$	− 1.34392e10i	− 0.672782i	−0.941722	−	0.336391i	$-0.890794\pi$
$877$	− 1.34392e10i	− 0.672782i	0.941722	−	0.336391i	$-0.109206\pi$
$878$	− 3.44814e9i	− 0.171931i
$879$	1.63307e10i	0.811043i
$880$	−2.71029e10	−1.34069
$881$	−3.85616e9	−0.189994	−0.0949968	−	0.995478i	$-0.530284\pi$
$881$	−3.85616e9	−0.189994	−0.0949968	+	0.995478i	$0.530284\pi$
$882$	− 3.33550e10i	− 1.63690i
$883$	−2.27185e10	−1.11050	−0.555248	−	0.831685i	$-0.687377\pi$
$883$	−2.27185e10	−1.11050	−0.555248	+	0.831685i	$0.687377\pi$
$884$	0	0
$885$	−8.45279e9	−0.409920
$886$	− 1.68347e10i	− 0.813183i
$887$	2.21671e10	1.06654	0.533270	−	0.845945i	$-0.320963\pi$
$887$	2.21671e10	1.06654	0.533270	+	0.845945i	$0.320963\pi$
$888$	−3.11645e10	−1.49353
$889$	− 2.43597e10i	− 1.16283i
$890$	− 1.72009e10i	− 0.817873i
$891$	− 1.36279e10i	− 0.645442i
$892$	− 6.15710e9i	− 0.290469i
$893$	1.68876e9	0.0793572
$894$	−1.04967e10	−0.491328
$895$	2.68425e10i	1.25153i
$896$	−1.80603e10	−0.838779
$897$	0	0
$898$	3.20869e10	1.47863
$899$	1.16171e10i	0.533260i
$900$	−7.82986e8	−0.0358018
$901$	−2.55160e9	−0.116219
$902$	− 4.47915e10i	− 2.03223i
$903$	5.34781e10i	2.41696i
$904$	− 3.35343e10i	− 1.50973i
$905$	− 5.07446e9i	− 0.227572i
$906$	−6.02078e10	−2.68970
$907$	3.55279e10	1.58104	0.790522	−	0.612434i	$-0.209809\pi$
$907$	3.55279e10	1.58104	0.790522	+	0.612434i	$0.209809\pi$
$908$	− 6.45056e9i	− 0.285955i
$909$	4.87456e9	0.215259
$910$	0	0
$911$	−4.10088e9	−0.179706	−0.0898530	−	0.995955i	$-0.528640\pi$
$911$	−4.10088e9	−0.179706	−0.0898530	+	0.995955i	$0.528640\pi$
$912$	− 2.79466e9i	− 0.121996i
$913$	−3.32072e9	−0.144406
$914$	8.53834e9	0.369881
$915$	4.04441e10i	1.74535i
$916$	1.51691e9i	0.0652119i
$917$	1.18240e10i	0.506377i
$918$	− 2.89108e9i	− 0.123342i
$919$	9.41768e8	0.0400258	0.0200129	−	0.999800i	$-0.493629\pi$
$919$	9.41768e8	0.0400258	0.0200129	+	0.999800i	$0.493629\pi$
$920$	8.18420e9	0.346512
$921$	− 7.24101e10i	− 3.05415i
$922$	3.91799e10	1.64628
$923$	0	0
$924$	−2.14578e10	−0.894816
$925$	2.43558e9i	0.101183i
$926$	−9.00831e9	−0.372825
$927$	5.28644e10	2.17964
$928$	− 7.42097e9i	− 0.304819i
$929$	− 1.39001e10i	− 0.568803i	−0.958705	−	0.284402i	$-0.908205\pi$
$929$	− 1.39001e10i	− 0.568803i	0.958705	−	0.284402i	$-0.0917950\pi$
$930$	2.68076e10i	1.09287i
$931$	3.38221e9i	0.137365i
$932$	−3.96425e9	−0.160400
$933$	−1.81236e10	−0.730565
$934$	8.14889e8i	0.0327254i
$935$	9.35385e9	0.374240
$936$	0	0
$937$	−1.57057e10	−0.623689	−0.311844	−	0.950133i	$-0.600947\pi$
$937$	−1.57057e10	−0.623689	−0.311844	+	0.950133i	$0.600947\pi$
$938$	− 5.45278e10i	− 2.15729i
$939$	1.46538e10	0.577592
$940$	4.37825e9	0.171931
$941$	− 3.84758e10i	− 1.50530i	−0.658419	−	0.752651i	$-0.728775\pi$
$941$	− 3.84758e10i	− 1.50530i	0.658419	−	0.752651i	$-0.271225\pi$
$942$	− 6.51238e9i	− 0.253841i
$943$	1.04182e10i	0.404576i
$944$	− 4.71645e9i	− 0.182479i
$945$	2.82370e10	1.08845
$946$	−4.07959e10	−1.56674
$947$	1.18417e10i	0.453095i	0.974000	+	0.226548i	$0.0727439\pi$
$947$	1.18417e10i	0.453095i	−0.974000	+	0.226548i	$0.927256\pi$
$948$	2.58455e9	0.0985270
$949$	0	0
$950$	−2.83554e8	−0.0107301
$951$	7.49233e9i	0.282478i
$952$	8.88238e9	0.333657
$953$	−8.06299e9	−0.301767	−0.150883	−	0.988552i	$-0.548212\pi$
$953$	−8.06299e9	−0.301767	−0.150883	+	0.988552i	$0.548212\pi$
$954$	− 1.93323e10i	− 0.720884i
$955$	1.97266e10i	0.732892i
$956$	− 7.20622e9i	− 0.266750i
$957$	5.20884e10i	1.92110i
$958$	4.28234e10	1.57363
$959$	−8.65349e9	−0.316830
$960$	− 5.02465e10i	− 1.83297i
$961$	1.20163e10	0.436758
$962$	0	0
$963$	6.89024e10	2.48624
$964$	6.91091e9i	0.248465i
$965$	1.43436e10	0.513820
$966$	−1.78247e10	−0.636214
$967$	2.18672e10i	0.777679i	0.921305	+	0.388840i	$0.127124\pi$
$967$	2.18672e10i	0.777679i	−0.921305	+	0.388840i	$0.872876\pi$
$968$	− 6.07997e10i	− 2.15446i
$969$	9.64501e8i	0.0340541i
$970$	− 6.03372e9i	− 0.212268i
$971$	1.48206e10	0.519516	0.259758	−	0.965674i	$-0.416357\pi$
$971$	1.48206e10	0.519516	0.259758	+	0.965674i	$0.416357\pi$
$972$	7.91221e9	0.276354
$973$	− 1.85350e10i	− 0.645058i
$974$	−4.24063e10	−1.47053
$975$	0	0
$976$	−2.25668e10	−0.776955
$977$	− 4.40385e10i	− 1.51078i	−0.655274	−	0.755391i	$-0.727447\pi$
$977$	− 4.40385e10i	− 1.51078i	0.655274	−	0.755391i	$-0.272553\pi$
$978$	−1.52617e10	−0.521697
$979$	4.45824e10	1.51853
$980$	8.76870e9i	0.297608i
$981$	− 6.17700e10i	− 2.08899i
$982$	− 2.21387e10i	− 0.746038i
$983$	− 2.32688e10i	− 0.781333i	−0.920532	−	0.390666i	$-0.872245\pi$
$983$	− 2.32688e10i	− 0.781333i	0.920532	−	0.390666i	$-0.127755\pi$
$984$	6.67127e10	2.23216
$985$	−2.48591e10	−0.828817
$986$	− 3.87006e9i	− 0.128573i
$987$	−5.31269e10	−1.75875
$988$	0	0
$989$	9.48881e9	0.311907
$990$	7.08700e10i	2.32134i
$991$	1.25560e10	0.409821	0.204911	−	0.978781i	$-0.434310\pi$
$991$	1.25560e10	0.409821	0.204911	+	0.978781i	$0.434310\pi$
$992$	9.89897e9	0.321958
$993$	5.54839e10i	1.79823i
$994$	− 5.14408e10i	− 1.66133i
$995$	4.12721e10i	1.32824i
$996$	− 8.87726e8i	− 0.0284689i
$997$	3.50176e10	1.11906	0.559530	−	0.828810i	$-0.310982\pi$
$997$	3.50176e10	1.11906	0.559530	+	0.828810i	$0.310982\pi$
$998$	2.45975e10	0.783311
$999$	1.90783e10i	0.605424i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.b.a.168.2		2
13.5	odd	4		13.8.a.a.1.1	✓	1
13.8	odd	4		169.8.a.a.1.1		1
13.12	even	2	inner	169.8.b.a.168.1		2
39.5	even	4		117.8.a.a.1.1		1
52.31	even	4		208.8.a.d.1.1		1
65.44	odd	4		325.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.a.a.1.1	✓	1	13.5	odd	4
117.8.a.a.1.1		1	39.5	even	4
169.8.a.a.1.1		1	13.8	odd	4
169.8.b.a.168.1		2	13.12	even	2	inner
169.8.b.a.168.2		2	1.1	even	1	trivial
208.8.a.d.1.1		1	52.31	even	4
325.8.a.a.1.1		1	65.44	odd	4

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)

\(f(q)\)	\(=\)	\(q+10.0000i q^{2} -73.0000 q^{3} +28.0000 q^{4} -295.000i q^{5} -730.000i q^{6} -1373.00i q^{7} +1560.00i q^{8} +3142.00 q^{9} +O(q^{10})\)	\(q+10.0000i q^{2} -73.0000 q^{3} +28.0000 q^{4} -295.000i q^{5} -730.000i q^{6} -1373.00i q^{7} +1560.00i q^{8} +3142.00 q^{9} +2950.00 q^{10} +7646.00i q^{11} -2044.00 q^{12} +13730.0 q^{14} +21535.0i q^{15} -12016.0 q^{16} +4147.00 q^{17} +31420.0i q^{18} -3186.00i q^{19} -8260.00i q^{20} +100229. i q^{21} -76460.0 q^{22} +17784.0 q^{23} -113880. i q^{24} -8900.00 q^{25} -69715.0 q^{27} -38444.0i q^{28} -93322.0 q^{29} -215350. q^{30} -124484. i q^{31} +79520.0i q^{32} -558158. i q^{33} +41470.0i q^{34} -405035. q^{35} +87976.0 q^{36} -273661. i q^{37} +31860.0 q^{38} +460200. q^{40} +585816. i q^{41} -1.00229e6 q^{42} +533559. q^{43} +214088. i q^{44} -926890. i q^{45} +177840. i q^{46} +530055. i q^{47} +877168. q^{48} -1.06159e6 q^{49} -89000.0i q^{50} -302731. q^{51} -615288. q^{53} -697150. i q^{54} +2.25557e6 q^{55} +2.14188e6 q^{56} +232578. i q^{57} -933220. i q^{58} +392514. i q^{59} +602980. i q^{60} +1.87806e6 q^{61} +1.24484e6 q^{62} -4.31397e6i q^{63} -2.33325e6 q^{64} +5.58158e6 q^{66} -3.97144e6i q^{67} +116116. q^{68} -1.29823e6 q^{69} -4.05035e6i q^{70} -3.74660e6i q^{71} +4.90152e6i q^{72} -2.48580e6i q^{73} +2.73661e6 q^{74} +649700. q^{75} -89208.0i q^{76} +1.04980e7 q^{77} -1.26446e6 q^{79} +3.54472e6i q^{80} -1.78236e6 q^{81} -5.85816e6 q^{82} +434308. i q^{83} +2.80641e6i q^{84} -1.22336e6i q^{85} +5.33559e6i q^{86} +6.81251e6 q^{87} -1.19278e7 q^{88} -5.83081e6i q^{89} +9.26890e6 q^{90} +497952. q^{92} +9.08733e6i q^{93} -5.30055e6 q^{94} -939870. q^{95} -5.80496e6i q^{96} -2.04533e6i q^{97} -1.06159e7i q^{98} +2.40237e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 146 q^{3} + 56 q^{4} + 6284 q^{9}+O(q^{10}) \) 2 * q - 146 * q^3 + 56 * q^4 + 6284 * q^9	\( 2 q - 146 q^{3} + 56 q^{4} + 6284 q^{9} + 5900 q^{10} - 4088 q^{12} + 27460 q^{14} - 24032 q^{16} + 8294 q^{17} - 152920 q^{22} + 35568 q^{23} - 17800 q^{25} - 139430 q^{27} - 186644 q^{29} - 430700 q^{30} - 810070 q^{35} + 175952 q^{36} + 63720 q^{38} + 920400 q^{40} - 2004580 q^{42} + 1067118 q^{43} + 1754336 q^{48} - 2123172 q^{49} - 605462 q^{51} - 1230576 q^{53} + 4511140 q^{55} + 4283760 q^{56} + 3756128 q^{61} + 2489680 q^{62} - 4666496 q^{64} + 11163160 q^{66} + 232232 q^{68} - 2596464 q^{69} + 5473220 q^{74} + 1299400 q^{75} + 20995916 q^{77} - 2528912 q^{79} - 3564718 q^{81} - 11716320 q^{82} + 13625012 q^{87} - 23855520 q^{88} + 18537800 q^{90} + 995904 q^{92} - 10601100 q^{94} - 1879740 q^{95}+O(q^{100}) \) 2 * q - 146 * q^3 + 56 * q^4 + 6284 * q^9 + 5900 * q^10 - 4088 * q^12 + 27460 * q^14 - 24032 * q^16 + 8294 * q^17 - 152920 * q^22 + 35568 * q^23 - 17800 * q^25 - 139430 * q^27 - 186644 * q^29 - 430700 * q^30 - 810070 * q^35 + 175952 * q^36 + 63720 * q^38 + 920400 * q^40 - 2004580 * q^42 + 1067118 * q^43 + 1754336 * q^48 - 2123172 * q^49 - 605462 * q^51 - 1230576 * q^53 + 4511140 * q^55 + 4283760 * q^56 + 3756128 * q^61 + 2489680 * q^62 - 4666496 * q^64 + 11163160 * q^66 + 232232 * q^68 - 2596464 * q^69 + 5473220 * q^74 + 1299400 * q^75 + 20995916 * q^77 - 2528912 * q^79 - 3564718 * q^81 - 11716320 * q^82 + 13625012 * q^87 - 23855520 * q^88 + 18537800 * q^90 + 995904 * q^92 - 10601100 * q^94 - 1879740 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more