LMFDB - Embedded newform 169.8.b.d.168.10

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			168.10
Root			$8.41902i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.8.b.d.168.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.41902i q^{2} -1.14749 q^{3} +57.1202 q^{4} +399.024i q^{5} -9.66073i q^{6} -944.231i q^{7} +1558.53i q^{8} -2185.68 q^{9} -3359.39 q^{10} +5673.16i q^{11} -65.5447 q^{12} +7949.50 q^{14} -457.876i q^{15} -5809.90 q^{16} -12214.2 q^{17} -18401.3i q^{18} -48622.1i q^{19} +22792.3i q^{20} +1083.49i q^{21} -47762.4 q^{22} -3416.52 q^{23} -1788.40i q^{24} -81095.5 q^{25} +5017.60 q^{27} -53934.6i q^{28} -126824. q^{29} +3854.87 q^{30} +170057. i q^{31} +150578. i q^{32} -6509.89i q^{33} -102832. i q^{34} +376771. q^{35} -124847. q^{36} +149165. i q^{37} +409350. q^{38} -621892. q^{40} -645325. i q^{41} -9121.96 q^{42} -129943. q^{43} +324052. i q^{44} -872141. i q^{45} -28763.7i q^{46} +338496. i q^{47} +6666.80 q^{48} -68029.2 q^{49} -682745. i q^{50} +14015.7 q^{51} -226916. q^{53} +42243.3i q^{54} -2.26373e6 q^{55} +1.47161e6 q^{56} +55793.3i q^{57} -1.06773e6i q^{58} +1.60203e6i q^{59} -26154.0i q^{60} -2.00107e6 q^{61} -1.43171e6 q^{62} +2.06379e6i q^{63} -2.01139e6 q^{64} +54806.9 q^{66} -1.38038e6i q^{67} -697677. q^{68} +3920.41 q^{69} +3.17204e6i q^{70} -5.13680e6i q^{71} -3.40645e6i q^{72} -5.76900e6i q^{73} -1.25582e6 q^{74} +93056.2 q^{75} -2.77730e6i q^{76} +5.35678e6 q^{77} -4.45623e6 q^{79} -2.31829e6i q^{80} +4.77433e6 q^{81} +5.43300e6 q^{82} -811479. i q^{83} +61889.4i q^{84} -4.87377e6i q^{85} -1.09399e6i q^{86} +145529. q^{87} -8.84179e6 q^{88} -5.15613e6i q^{89} +7.34257e6 q^{90} -195152. q^{92} -195138. i q^{93} -2.84980e6 q^{94} +1.94014e7 q^{95} -172787. i q^{96} -3.83876e6i q^{97} -572739. i q^{98} -1.23997e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	8.41902i	0.744143i	0.928204	+	0.372071i	$0.121352\pi$
$2$	8.41902i	0.744143i	−0.928204	+	0.372071i	$0.878648\pi$
$3$	−1.14749	−0.0245371	−0.0122686	−	0.999925i	$-0.503905\pi$
$3$	−1.14749	−0.0245371	−0.0122686	+	0.999925i	$0.503905\pi$
$4$	57.1202	0.446251
$5$	399.024i	1.42759i	0.700353	+	0.713797i	$0.253026\pi$
$5$	399.024i	1.42759i	−0.700353	+	0.713797i	$0.746974\pi$
$6$	− 9.66073i	− 0.0182591i
$7$	− 944.231i	− 1.04048i	−0.854019	−	0.520242i	$-0.825842\pi$
$7$	− 944.231i	− 1.04048i	0.854019	−	0.520242i	$-0.174158\pi$
$8$	1558.53i	1.07622i
$9$	−2185.68	−0.999398
$10$	−3359.39	−1.06233
$11$	5673.16i	1.28514i	0.766227	+	0.642571i	$0.222132\pi$
$11$	5673.16i	1.28514i	−0.766227	+	0.642571i	$0.777868\pi$
$12$	−65.5447	−0.0109497
$13$	0	0
$13$	0	0
$14$	7949.50	0.774268
$15$	− 457.876i	− 0.0350291i
$16$	−5809.90	−0.354608
$17$	−12214.2	−0.602968	−0.301484	−	0.953471i	$-0.597482\pi$
$17$	−12214.2	−0.602968	−0.301484	+	0.953471i	$0.597482\pi$
$18$	− 18401.3i	− 0.743695i
$19$	− 48622.1i	− 1.62628i	−0.582067	−	0.813141i	$-0.697756\pi$
$19$	− 48622.1i	− 1.62628i	0.582067	−	0.813141i	$-0.302244\pi$
$20$	22792.3i	0.637065i
$21$	1083.49i	0.0255305i
$22$	−47762.4	−0.956329
$23$	−3416.52	−0.0585512	−0.0292756	−	0.999571i	$-0.509320\pi$
$23$	−3416.52	−0.0585512	−0.0292756	+	0.999571i	$0.509320\pi$
$24$	− 1788.40i	− 0.0264073i
$25$	−81095.5	−1.03802
$26$	0	0
$27$	5017.60	0.0490595
$28$	− 53934.6i	− 0.464317i
$29$	−126824.	−0.965626	−0.482813	−	0.875724i	$-0.660385\pi$
$29$	−126824.	−0.965626	−0.482813	+	0.875724i	$0.660385\pi$
$30$	3854.87	0.0260666
$31$	170057.i	1.02525i	0.858614	+	0.512623i	$0.171326\pi$
$31$	170057.i	1.02525i	−0.858614	+	0.512623i	$0.828674\pi$
$32$	150578.i	0.812338i
$33$	− 6509.89i	− 0.0315337i
$34$	− 102832.i	− 0.448694i
$35$	376771.	1.48539
$36$	−124847.	−0.445983
$37$	149165.i	0.484129i	0.970260	+	0.242065i	$0.0778246\pi$
$37$	149165.i	0.484129i	−0.970260	+	0.242065i	$0.922175\pi$
$38$	409350.	1.21019
$39$	0	0
$40$	−621892.	−1.53640
$41$	− 645325.i	− 1.46230i	−0.682219	−	0.731148i	$-0.738985\pi$
$41$	− 645325.i	− 1.46230i	0.682219	−	0.731148i	$-0.261015\pi$
$42$	−9121.96	−0.0189983
$43$	−129943.	−0.249238	−0.124619	−	0.992205i	$-0.539771\pi$
$43$	−129943.	−0.249238	−0.124619	+	0.992205i	$0.539771\pi$
$44$	324052.i	0.573496i
$45$	− 872141.i	− 1.42673i
$46$	− 28763.7i	− 0.0435705i
$47$	338496.i	0.475566i	0.971318	+	0.237783i	$0.0764207\pi$
$47$	338496.i	0.475566i	−0.971318	+	0.237783i	$0.923579\pi$
$48$	6666.80	0.00870108
$49$	−68029.2	−0.0826056
$50$	− 682745.i	− 0.772437i
$51$	14015.7	0.0147951
$52$	0	0
$53$	−226916.	−0.209362	−0.104681	−	0.994506i	$-0.533382\pi$
$53$	−226916.	−0.209362	−0.104681	+	0.994506i	$0.533382\pi$
$54$	42243.3i	0.0365073i
$55$	−2.26373e6	−1.83466
$56$	1.47161e6	1.11979
$57$	55793.3i	0.0399043i
$58$	− 1.06773e6i	− 0.718564i
$59$	1.60203e6i	1.01552i	0.861498	+	0.507762i	$0.169527\pi$
$59$	1.60203e6i	1.01552i	−0.861498	+	0.507762i	$0.830473\pi$
$60$	− 26154.0i	− 0.0156318i
$61$	−2.00107e6	−1.12877	−0.564387	−	0.825510i	$-0.690887\pi$
$61$	−2.00107e6	−1.12877	−0.564387	+	0.825510i	$0.690887\pi$
$62$	−1.43171e6	−0.762930
$63$	2.06379e6i	1.03986i
$64$	−2.01139e6	−0.959104
$65$	0	0
$66$	54806.9	0.0234656
$67$	− 1.38038e6i	− 0.560706i	−0.959897	−	0.280353i	$-0.909548\pi$
$67$	− 1.38038e6i	− 0.560706i	0.959897	−	0.280353i	$-0.0904516\pi$
$68$	−697677.	−0.269075
$69$	3920.41	0.00143668
$70$	3.17204e6i	1.10534i
$71$	− 5.13680e6i	− 1.70329i	−0.524120	−	0.851644i	$-0.675606\pi$
$71$	− 5.13680e6i	− 1.70329i	0.524120	−	0.851644i	$-0.324394\pi$
$72$	− 3.40645e6i	− 1.07557i
$73$	− 5.76900e6i	− 1.73568i	−0.496839	−	0.867842i	$-0.665506\pi$
$73$	− 5.76900e6i	− 1.73568i	0.496839	−	0.867842i	$-0.334494\pi$
$74$	−1.25582e6	−0.360261
$75$	93056.2	0.0254701
$76$	− 2.77730e6i	− 0.725731i
$77$	5.35678e6	1.33717
$78$	0	0
$79$	−4.45623e6	−1.01689	−0.508444	−	0.861095i	$-0.669779\pi$
$79$	−4.45623e6	−1.01689	−0.508444	+	0.861095i	$0.669779\pi$
$80$	− 2.31829e6i	− 0.506237i
$81$	4.77433e6	0.998194
$82$	5.43300e6	1.08816
$83$	− 811479.i	− 0.155777i	−0.996962	−	0.0778886i	$-0.975182\pi$
$83$	− 811479.i	− 0.155777i	0.996962	−	0.0778886i	$-0.0248179\pi$
$84$	61889.4i	0.0113930i
$85$	− 4.87377e6i	− 0.860793i
$86$	− 1.09399e6i	− 0.185468i
$87$	145529.	0.0236937
$88$	−8.84179e6	−1.38309
$89$	− 5.15613e6i	− 0.775281i	−0.921811	−	0.387640i	$-0.873290\pi$
$89$	− 5.15613e6i	− 0.775281i	0.921811	−	0.387640i	$-0.126710\pi$
$90$	7.34257e6	1.06169
$91$	0	0
$92$	−195152.	−0.0261286
$93$	− 195138.i	− 0.0251566i
$94$	−2.84980e6	−0.353889
$95$	1.94014e7	2.32167
$96$	− 172787.i	− 0.0199325i
$97$	− 3.83876e6i	− 0.427061i	−0.976936	−	0.213530i	$-0.931504\pi$
$97$	− 3.83876e6i	− 0.427061i	0.976936	−	0.213530i	$-0.0684963\pi$
$98$	− 572739.i	− 0.0614703i
$99$	− 1.23997e7i	− 1.28437i
$100$	−4.63219e6	−0.463219
$101$	9.61610e6	0.928697	0.464348	−	0.885653i	$-0.346289\pi$
$101$	9.61610e6	0.928697	0.464348	+	0.885653i	$0.346289\pi$
$102$	117998.i	0.0110097i
$103$	−2.86658e6	−0.258484	−0.129242	−	0.991613i	$-0.541254\pi$
$103$	−2.86658e6	−0.258484	−0.129242	+	0.991613i	$0.541254\pi$
$104$	0	0
$105$	−432341.	−0.0364472
$106$	− 1.91041e6i	− 0.155796i
$107$	−198466.	−0.0156618	−0.00783091	−	0.999969i	$-0.502493\pi$
$107$	−198466.	−0.0156618	−0.00783091	+	0.999969i	$0.502493\pi$
$108$	286606.	0.0218929
$109$	1.82103e7i	1.34687i	0.739249	+	0.673433i	$0.235181\pi$
$109$	1.82103e7i	1.34687i	−0.739249	+	0.673433i	$0.764819\pi$
$110$	− 1.90584e7i	− 1.36525i
$111$	− 171165.i	− 0.0118791i
$112$	5.48589e6i	0.368964i
$113$	−8.14016e6	−0.530711	−0.265356	−	0.964151i	$-0.585489\pi$
$113$	−8.14016e6	−0.530711	−0.265356	+	0.964151i	$0.585489\pi$
$114$	−469725.	−0.0296945
$115$	− 1.36327e6i	− 0.0835873i
$116$	−7.24421e6	−0.430912
$117$	0	0
$118$	−1.34876e7	−0.755694
$119$	1.15330e7i	0.627378i
$120$	713613.	0.0376989
$121$	−1.26976e7	−0.651588
$122$	− 1.68470e7i	− 0.839969i
$123$	740504.i	0.0358806i
$124$	9.71368e6i	0.457517i
$125$	− 1.18532e6i	− 0.0542813i
$126$	−1.73751e7	−0.773802
$127$	−3.41396e7	−1.47892	−0.739460	−	0.673200i	$-0.764919\pi$
$127$	−3.41396e7	−1.47892	−0.739460	+	0.673200i	$0.764919\pi$
$128$	2.34010e6i	0.0986277i
$129$	149108.	0.00611558
$130$	0	0
$131$	6.55064e6	0.254586	0.127293	−	0.991865i	$-0.459371\pi$
$131$	6.55064e6	0.254586	0.127293	+	0.991865i	$0.459371\pi$
$132$	− 371846.i	− 0.0140720i
$133$	−4.59105e7	−1.69212
$134$	1.16214e7	0.417246
$135$	2.00215e6i	0.0700370i
$136$	− 1.90362e7i	− 0.648924i
$137$	− 1.53188e7i	− 0.508984i	−0.967075	−	0.254492i	$-0.918092\pi$
$137$	− 1.53188e7i	− 0.508984i	0.967075	−	0.254492i	$-0.0819082\pi$
$138$	33006.0i	0.00106910i
$139$	1.12089e7	0.354007	0.177004	−	0.984210i	$-0.443360\pi$
$139$	1.12089e7	0.354007	0.177004	+	0.984210i	$0.443360\pi$
$140$	2.15212e7	0.662856
$141$	− 388420.i	− 0.0116690i
$142$	4.32468e7	1.26749
$143$	0	0
$144$	1.26986e7	0.354395
$145$	− 5.06059e7i	− 1.37852i
$146$	4.85693e7	1.29160
$147$	78062.8	0.00202690
$148$	8.52034e6i	0.216043i
$149$	3.85311e7i	0.954243i	0.878837	+	0.477121i	$0.158320\pi$
$149$	3.85311e7i	0.954243i	−0.878837	+	0.477121i	$0.841680\pi$
$150$	783442.i	0.0189534i
$151$	4.08938e7i	0.966581i	0.875460	+	0.483290i	$0.160558\pi$
$151$	4.08938e7i	0.966581i	−0.875460	+	0.483290i	$0.839442\pi$
$152$	7.57790e7	1.75023
$153$	2.66964e7	0.602605
$154$	4.50988e7i	0.995044i
$155$	−6.78568e7	−1.46363
$156$	0	0
$157$	−2.52905e7	−0.521565	−0.260783	−	0.965398i	$-0.583981\pi$
$157$	−2.52905e7	−0.521565	−0.260783	+	0.965398i	$0.583981\pi$
$158$	− 3.75171e7i	− 0.756710i
$159$	260383.	0.00513716
$160$	−6.00844e7	−1.15969
$161$	3.22598e6i	0.0609216i
$162$	4.01952e7i	0.742799i
$163$	6.83917e7i	1.23693i	0.785810	+	0.618467i	$0.212246\pi$
$163$	6.83917e7i	1.23693i	−0.785810	+	0.618467i	$0.787754\pi$
$164$	− 3.68611e7i	− 0.652551i
$165$	2.59761e6	0.0450173
$166$	6.83186e6	0.115921
$167$	− 2.92588e7i	− 0.486126i	−0.970010	−	0.243063i	$-0.921848\pi$
$167$	− 2.92588e7i	− 0.486126i	0.970010	−	0.243063i	$-0.0781522\pi$
$168$	−1.68866e6	−0.0274764
$169$	0	0
$170$	4.10323e7	0.640553
$171$	1.06272e8i	1.62530i
$172$	−7.42237e6	−0.111223
$173$	−9.01124e7	−1.32319	−0.661597	−	0.749860i	$-0.730121\pi$
$173$	−9.01124e7	−1.32319	−0.661597	+	0.749860i	$0.730121\pi$
$174$	1.22521e6i	0.0176315i
$175$	7.65729e7i	1.08005i
$176$	− 3.29605e7i	− 0.455722i
$177$	− 1.83832e6i	− 0.0249180i
$178$	4.34096e7	0.576920
$179$	−1.69729e7	−0.221193	−0.110596	−	0.993865i	$-0.535276\pi$
$179$	−1.69729e7	−0.221193	−0.110596	+	0.993865i	$0.535276\pi$
$180$	− 4.98168e7i	− 0.636682i
$181$	−4.68911e7	−0.587781	−0.293890	−	0.955839i	$-0.594950\pi$
$181$	−4.68911e7	−0.587781	−0.293890	+	0.955839i	$0.594950\pi$
$182$	0	0
$183$	2.29620e6	0.0276969
$184$	− 5.32474e6i	− 0.0630139i
$185$	−5.95206e7	−0.691140
$186$	1.64287e6	0.0187201
$187$	− 6.92932e7i	− 0.774898i
$188$	1.93349e7i	0.212222i
$189$	− 4.73778e6i	− 0.0510456i
$190$	1.63341e8i	1.72765i
$191$	−1.32211e8	−1.37294	−0.686470	−	0.727158i	$-0.740841\pi$
$191$	−1.32211e8	−1.37294	−0.686470	+	0.727158i	$0.740841\pi$
$192$	2.30804e6	0.0235337
$193$	1.37264e8i	1.37438i	0.726480	+	0.687188i	$0.241155\pi$
$193$	1.37264e8i	1.37438i	−0.726480	+	0.687188i	$0.758845\pi$
$194$	3.23186e7	0.317794
$195$	0	0
$196$	−3.88584e6	−0.0368628
$197$	1.66568e8i	1.55225i	0.630581	+	0.776123i	$0.282817\pi$
$197$	1.66568e8i	1.55225i	−0.630581	+	0.776123i	$0.717183\pi$
$198$	1.04394e8	0.955753
$199$	−5.79904e7	−0.521639	−0.260819	−	0.965388i	$-0.583993\pi$
$199$	−5.79904e7	−0.521639	−0.260819	+	0.965388i	$0.583993\pi$
$200$	− 1.26390e8i	− 1.11714i
$201$	1.58396e6i	0.0137581i
$202$	8.09581e7i	0.691083i
$203$	1.19751e8i	1.00472i
$204$	800577.	0.00660233
$205$	2.57501e8	2.08756
$206$	− 2.41338e7i	− 0.192349i
$207$	7.46742e6	0.0585160
$208$	0	0
$209$	2.75841e8	2.09000
$210$	− 3.63988e6i	− 0.0271219i
$211$	1.21082e8	0.887345	0.443673	−	0.896189i	$-0.353675\pi$
$211$	1.21082e8	0.887345	0.443673	+	0.896189i	$0.353675\pi$
$212$	−1.29615e7	−0.0934283
$213$	5.89442e6i	0.0417938i
$214$	− 1.67089e6i	− 0.0116546i
$215$	− 5.18505e7i	− 0.355810i
$216$	7.82009e6i	0.0527987i
$217$	1.60573e8	1.06675
$218$	−1.53313e8	−1.00226
$219$	6.61987e6i	0.0425887i
$220$	−1.29305e8	−0.818719
$221$	0	0
$222$	1.44104e6	0.00883979
$223$	− 4.62753e7i	− 0.279436i	−0.990191	−	0.139718i	$-0.955380\pi$
$223$	− 4.62753e7i	− 0.279436i	0.990191	−	0.139718i	$-0.0446196\pi$
$224$	1.42181e8	0.845224
$225$	1.77249e8	1.03740
$226$	− 6.85321e7i	− 0.394925i
$227$	3.36801e8i	1.91110i	0.294834	+	0.955548i	$0.404736\pi$
$227$	3.36801e8i	1.91110i	−0.294834	+	0.955548i	$0.595264\pi$
$228$	3.18692e6i	0.0178074i
$229$	− 2.88661e8i	− 1.58841i	−0.607648	−	0.794206i	$-0.707887\pi$
$229$	− 2.88661e8i	− 1.58841i	0.607648	−	0.794206i	$-0.292113\pi$
$230$	1.14774e7	0.0622009
$231$	−6.14684e6	−0.0328103
$232$	− 1.97659e8i	− 1.03922i
$233$	9.81949e7	0.508561	0.254281	−	0.967130i	$-0.418161\pi$
$233$	9.81949e7	0.508561	0.254281	+	0.967130i	$0.418161\pi$
$234$	0	0
$235$	−1.35068e8	−0.678914
$236$	9.15085e7i	0.453179i
$237$	5.11348e6	0.0249515
$238$	−9.70968e7	−0.466859
$239$	1.13856e8i	0.539465i	0.962935	+	0.269733i	$0.0869353\pi$
$239$	1.13856e8i	0.539465i	−0.962935	+	0.269733i	$0.913065\pi$
$240$	2.66022e6i	0.0124216i
$241$	− 7.42915e7i	− 0.341884i	−0.985281	−	0.170942i	$-0.945319\pi$
$241$	− 7.42915e7i	− 0.341884i	0.985281	−	0.170942i	$-0.0546811\pi$
$242$	− 1.06901e8i	− 0.484874i
$243$	−1.64520e7	−0.0735523
$244$	−1.14301e8	−0.503717
$245$	− 2.71453e7i	− 0.117927i
$246$	−6.23431e6	−0.0267003
$247$	0	0
$248$	−2.65039e8	−1.10339
$249$	931163.i	0.00382233i
$250$	9.97923e6	0.0403931
$251$	−1.42394e8	−0.568375	−0.284188	−	0.958769i	$-0.591724\pi$
$251$	−1.42394e8	−0.568375	−0.284188	+	0.958769i	$0.591724\pi$
$252$	1.17884e8i	0.464038i
$253$	− 1.93824e7i	− 0.0752466i
$254$	− 2.87421e8i	− 1.10053i
$255$	5.59259e6i	0.0211214i
$256$	−2.77159e8	−1.03250
$257$	2.92823e8	1.07607	0.538033	−	0.842924i	$-0.319168\pi$
$257$	2.92823e8	1.07607	0.538033	+	0.842924i	$0.319168\pi$
$258$	1.25534e6i	0.00455086i
$259$	1.40846e8	0.503728
$260$	0	0
$261$	2.77197e8	0.965044
$262$	5.51500e7i	0.189448i
$263$	−3.05065e8	−1.03406	−0.517032	−	0.855966i	$-0.672963\pi$
$263$	−3.05065e8	−1.03406	−0.517032	+	0.855966i	$0.672963\pi$
$264$	1.01459e7	0.0339371
$265$	− 9.05449e7i	− 0.298884i
$266$	− 3.86521e8i	− 1.25918i
$267$	5.91660e6i	0.0190232i
$268$	− 7.88473e7i	− 0.250216i
$269$	2.31855e8	0.726246	0.363123	−	0.931741i	$-0.381710\pi$
$269$	2.31855e8	0.726246	0.363123	+	0.931741i	$0.381710\pi$
$270$	−1.68561e7	−0.0521176
$271$	4.92648e8i	1.50364i	0.659368	+	0.751821i	$0.270824\pi$
$271$	4.92648e8i	1.50364i	−0.659368	+	0.751821i	$0.729176\pi$
$272$	7.09634e7	0.213817
$273$	0	0
$274$	1.28970e8	0.378757
$275$	− 4.60068e8i	− 1.33401i
$276$	223935.	0.000641120 0
$277$	4.33329e8	1.22501	0.612503	−	0.790468i	$-0.290163\pi$
$277$	4.33329e8	1.22501	0.612503	+	0.790468i	$0.290163\pi$
$278$	9.43681e7i	0.263432i
$279$	− 3.71690e8i	− 1.02463i
$280$	5.87209e8i	1.59860i
$281$	2.88578e8i	0.775874i	0.921686	+	0.387937i	$0.126812\pi$
$281$	2.88578e8i	0.775874i	−0.921686	+	0.387937i	$0.873188\pi$
$282$	3.27011e6	0.00868342
$283$	−5.98945e8	−1.57085	−0.785424	−	0.618958i	$-0.787555\pi$
$283$	−5.98945e8	−1.57085	−0.785424	+	0.618958i	$0.787555\pi$
$284$	− 2.93415e8i	− 0.760095i
$285$	−2.22629e7	−0.0569671
$286$	0	0
$287$	−6.09336e8	−1.52149
$288$	− 3.29116e8i	− 0.811849i
$289$	−2.61152e8	−0.636430
$290$	4.26052e8	1.02582
$291$	4.40493e6i	0.0104789i
$292$	− 3.29527e8i	− 0.774552i
$293$	− 3.12931e7i	− 0.0726794i	−0.999339	−	0.0363397i	$-0.988430\pi$
$293$	− 3.12931e7i	− 0.0726794i	0.999339	−	0.0363397i	$-0.0115698\pi$
$294$	657212.i	0.00150831i
$295$	−6.39251e8	−1.44975
$296$	−2.32478e8	−0.521029
$297$	2.84657e7i	0.0630484i
$298$	−3.24394e8	−0.710093
$299$	0	0
$300$	5.31539e6	0.0113661
$301$	1.22696e8i	0.259328i
$302$	−3.44286e8	−0.719274
$303$	−1.10344e7	−0.0227876
$304$	2.82490e8i	0.576693i
$305$	− 7.98474e8i	− 1.61143i
$306$	2.24757e8i	0.448424i
$307$	2.40062e8i	0.473520i	0.971568	+	0.236760i	$0.0760856\pi$
$307$	2.40062e8i	0.473520i	−0.971568	+	0.236760i	$0.923914\pi$
$308$	3.05980e8	0.596713
$309$	3.28937e6	0.00634247
$310$	− 5.71288e8i	− 1.08915i
$311$	1.13878e8	0.214674	0.107337	−	0.994223i	$-0.465768\pi$
$311$	1.13878e8	0.214674	0.107337	+	0.994223i	$0.465768\pi$
$312$	0	0
$313$	−2.86637e8	−0.528356	−0.264178	−	0.964474i	$-0.585101\pi$
$313$	−2.86637e8	−0.528356	−0.264178	+	0.964474i	$0.585101\pi$
$314$	− 2.12921e8i	− 0.388119i
$315$	−8.23503e8	−1.48449
$316$	−2.54541e8	−0.453788
$317$	9.47644e8i	1.67085i	0.549603	+	0.835426i	$0.314779\pi$
$317$	9.47644e8i	1.67085i	−0.549603	+	0.835426i	$0.685221\pi$
$318$	2.19217e6i	0.00382278i
$319$	− 7.19494e8i	− 1.24097i
$320$	− 8.02593e8i	− 1.36921i
$321$	227737.	0.000384296 0
$322$	−2.71596e7	−0.0453344
$323$	5.93880e8i	0.980596i
$324$	2.72711e8	0.445445
$325$	0	0
$326$	−5.75791e8	−0.920456
$327$	− 2.08961e7i	− 0.0330482i
$328$	1.00576e9	1.57375
$329$	3.19618e8	0.494818
$330$	2.18693e7i	0.0334993i
$331$	6.84847e8i	1.03800i	0.854776	+	0.518998i	$0.173695\pi$
$331$	6.84847e8i	1.03800i	−0.854776	+	0.518998i	$0.826305\pi$
$332$	− 4.63518e7i	− 0.0695158i
$333$	− 3.26028e8i	− 0.483838i
$334$	2.46330e8	0.361747
$335$	5.50804e8	0.800460
$336$	− 6.29500e6i	− 0.00905333i
$337$	1.12788e9	1.60531	0.802654	−	0.596445i	$-0.203421\pi$
$337$	1.12788e9	1.60531	0.802654	+	0.596445i	$0.203421\pi$
$338$	0	0
$339$	9.34074e6	0.0130221
$340$	− 2.78390e8i	− 0.384130i
$341$	−9.64760e8	−1.31759
$342$	−8.94710e8	−1.20946
$343$	− 7.13380e8i	− 0.954534i
$344$	− 2.02520e8i	− 0.268234i
$345$	1.56434e6i	0.00205099i
$346$	− 7.58658e8i	− 0.984645i
$347$	−3.84792e8	−0.494393	−0.247197	−	0.968965i	$-0.579509\pi$
$347$	−3.84792e8	−0.494393	−0.247197	+	0.968965i	$0.579509\pi$
$348$	8.31265e6	0.0105733
$349$	− 2.00731e8i	− 0.252770i	−0.991981	−	0.126385i	$-0.959663\pi$
$349$	− 2.00731e8i	− 0.252770i	0.991981	−	0.126385i	$-0.0403375\pi$
$350$	−6.44669e8	−0.803708
$351$	0	0
$352$	−8.54254e8	−1.04397
$353$	− 3.55684e8i	− 0.430381i	−0.976572	−	0.215191i	$-0.930963\pi$
$353$	− 3.55684e8i	− 0.430381i	0.976572	−	0.215191i	$-0.0690373\pi$
$354$	1.54768e7	0.0185426
$355$	2.04971e9	2.43160
$356$	− 2.94519e8i	− 0.345970i
$357$	− 1.32340e7i	− 0.0153941i
$358$	− 1.42895e8i	− 0.164599i
$359$	1.16954e8i	0.133409i	0.997773	+	0.0667044i	$0.0212485\pi$
$359$	1.16954e8i	0.133409i	−0.997773	+	0.0667044i	$0.978752\pi$
$360$	1.35926e9	1.53548
$361$	−1.47023e9	−1.64479
$362$	− 3.94777e8i	− 0.437393i
$363$	1.45703e7	0.0159881
$364$	0	0
$365$	2.30197e9	2.47785
$366$	1.93318e7i	0.0206104i
$367$	1.69538e9	1.79034	0.895169	−	0.445726i	$-0.147055\pi$
$367$	1.69538e9	1.79034	0.895169	+	0.445726i	$0.147055\pi$
$368$	1.98496e7	0.0207628
$369$	1.41048e9i	1.46142i
$370$	− 5.01104e8i	− 0.514307i
$371$	2.14261e8i	0.217838i
$372$	− 1.11463e7i	− 0.0112262i
$373$	−1.30728e9	−1.30433	−0.652164	−	0.758078i	$-0.726138\pi$
$373$	−1.30728e9	−1.30433	−0.652164	+	0.758078i	$0.726138\pi$
$374$	5.83380e8	0.576635
$375$	1.36014e6i	0.00133191i
$376$	−5.27555e8	−0.511812
$377$	0	0
$378$	3.98874e7	0.0379852
$379$	8.39591e8i	0.792192i	0.918209	+	0.396096i	$0.129635\pi$
$379$	8.39591e8i	0.792192i	−0.918209	+	0.396096i	$0.870365\pi$
$380$	1.10821e9	1.03605
$381$	3.91748e7	0.0362885
$382$	− 1.11309e9i	− 1.02166i
$383$	6.31720e8i	0.574551i	0.957848	+	0.287276i	$0.0927496\pi$
$383$	6.31720e8i	0.574551i	−0.957848	+	0.287276i	$0.907250\pi$
$384$	− 2.68523e6i	− 0.00242004i
$385$	2.13748e9i	1.90893i
$386$	−1.15563e9	−1.02273
$387$	2.84014e8	0.249088
$388$	− 2.19271e8i	− 0.190577i
$389$	−4.49938e8	−0.387551	−0.193776	−	0.981046i	$-0.562073\pi$
$389$	−4.49938e8	−0.387551	−0.193776	+	0.981046i	$0.562073\pi$
$390$	0	0
$391$	4.17300e7	0.0353045
$392$	− 1.06026e8i	− 0.0889016i
$393$	−7.51679e6	−0.00624681
$394$	−1.40234e9	−1.15509
$395$	− 1.77815e9i	− 1.45170i
$396$	− 7.08275e8i	− 0.573151i
$397$	− 1.42468e9i	− 1.14275i	−0.820690	−	0.571374i	$-0.806411\pi$
$397$	− 1.42468e9i	− 1.14275i	0.820690	−	0.571374i	$-0.193589\pi$
$398$	− 4.88222e8i	− 0.388174i
$399$	5.26817e7	0.0415198
$400$	4.71157e8	0.368092
$401$	8.70028e8i	0.673795i	0.941541	+	0.336898i	$0.109378\pi$
$401$	8.70028e8i	0.673795i	−0.941541	+	0.336898i	$0.890622\pi$
$402$	−1.33354e7	−0.0102380
$403$	0	0
$404$	5.49273e8	0.414432
$405$	1.90508e9i	1.42502i
$406$	−1.00819e9	−0.747653
$407$	−8.46238e8	−0.622174
$408$	2.18438e7i	0.0159228i
$409$	2.23157e9i	1.61280i	0.591373	+	0.806398i	$0.298586\pi$
$409$	2.23157e9i	1.61280i	−0.591373	+	0.806398i	$0.701414\pi$
$410$	2.16790e9i	1.55345i
$411$	1.75782e7i	0.0124890i
$412$	−1.63740e8	−0.115349
$413$	1.51269e9	1.05663
$414$	6.28683e7i	0.0435442i
$415$	3.23800e8	0.222387
$416$	0	0
$417$	−1.28621e7	−0.00868632
$418$	2.32231e9i	1.55526i
$419$	−8.31533e8	−0.552243	−0.276122	−	0.961123i	$-0.589049\pi$
$419$	−8.31533e8	−0.552243	−0.276122	+	0.961123i	$0.589049\pi$
$420$	−2.46954e7	−0.0162646
$421$	− 9.53842e8i	− 0.623002i	−0.950246	−	0.311501i	$-0.899168\pi$
$421$	− 9.53842e8i	− 0.623002i	0.950246	−	0.311501i	$-0.100832\pi$
$422$	1.01940e9i	0.660312i
$423$	− 7.39844e8i	− 0.475279i
$424$	− 3.53655e8i	− 0.225320i
$425$	9.90518e8	0.625894
$426$	−4.96252e7	−0.0311006
$427$	1.88947e9i	1.17447i
$428$	−1.13364e7	−0.00698911
$429$	0	0
$430$	4.36530e8	0.264773
$431$	− 1.01803e9i	− 0.612479i	−0.951954	−	0.306240i	$-0.900929\pi$
$431$	− 1.01803e9i	− 0.612479i	0.951954	−	0.306240i	$-0.0990709\pi$
$432$	−2.91518e7	−0.0173969
$433$	−1.81766e9	−1.07598	−0.537990	−	0.842951i	$-0.680816\pi$
$433$	−1.81766e9	−1.07598	−0.537990	+	0.842951i	$0.680816\pi$
$434$	1.35187e9i	0.793816i
$435$	5.80697e7i	0.0338250i
$436$	1.04018e9i	0.601040i
$437$	1.66118e8i	0.0952208i
$438$	−5.57328e7	−0.0316921
$439$	7.03870e8	0.397069	0.198535	−	0.980094i	$-0.436382\pi$
$439$	7.03870e8	0.397069	0.198535	+	0.980094i	$0.436382\pi$
$440$	− 3.52809e9i	− 1.97449i
$441$	1.48690e8	0.0825558
$442$	0	0
$443$	9.00901e7	0.0492339	0.0246169	−	0.999697i	$-0.492163\pi$
$443$	9.00901e7	0.0492339	0.0246169	+	0.999697i	$0.492163\pi$
$444$	− 9.77699e6i	− 0.00530109i
$445$	2.05742e9	1.10679
$446$	3.89592e8	0.207940
$447$	− 4.42139e7i	− 0.0234144i
$448$	1.89921e9i	0.997932i
$449$	− 1.03765e9i	− 0.540992i	−0.962721	−	0.270496i	$-0.912812\pi$
$449$	− 1.03765e9i	− 0.540992i	0.962721	−	0.270496i	$-0.0871876\pi$
$450$	1.49226e9i	0.771972i
$451$	3.66104e9	1.87926
$452$	−4.64967e8	−0.236831
$453$	− 4.69252e7i	− 0.0237171i
$454$	−2.83553e9	−1.42213
$455$	0	0
$456$	−8.69555e7	−0.0429457
$457$	− 1.66801e9i	− 0.817507i	−0.912645	−	0.408754i	$-0.865964\pi$
$457$	− 1.66801e9i	− 0.817507i	0.912645	−	0.408754i	$-0.134036\pi$
$458$	2.43024e9	1.18201
$459$	−6.12860e7	−0.0295813
$460$	− 7.78704e7i	− 0.0373010i
$461$	2.90964e9i	1.38320i	0.722280	+	0.691601i	$0.243094\pi$
$461$	2.90964e9i	1.38320i	−0.722280	+	0.691601i	$0.756906\pi$
$462$	− 5.17503e7i	− 0.0244155i
$463$	1.89500e9i	0.887310i	0.896198	+	0.443655i	$0.146318\pi$
$463$	1.89500e9i	0.887310i	−0.896198	+	0.443655i	$0.853682\pi$
$464$	7.36836e8	0.342419
$465$	7.78650e7	0.0359134
$466$	8.26704e8i	0.378442i
$467$	9.95155e8	0.452149	0.226075	−	0.974110i	$-0.427411\pi$
$467$	9.95155e8	0.452149	0.226075	+	0.974110i	$0.427411\pi$
$468$	0	0
$469$	−1.30339e9	−0.583405
$470$	− 1.13714e9i	− 0.505209i
$471$	2.90206e7	0.0127977
$472$	−2.49682e9	−1.09292
$473$	− 7.37188e8i	− 0.320305i
$474$	4.30504e7i	0.0185675i
$475$	3.94303e9i	1.68812i
$476$	6.58769e8i	0.279968i
$477$	4.95966e8	0.209236
$478$	−9.58556e8	−0.401439
$479$	1.39127e9i	0.578411i	0.957267	+	0.289206i	$0.0933912\pi$
$479$	1.39127e9i	0.578411i	−0.957267	+	0.289206i	$0.906609\pi$
$480$	6.89461e7	0.0284555
$481$	0	0
$482$	6.25461e8	0.254411
$483$	− 3.70178e6i	− 0.00149484i
$484$	−7.25289e8	−0.290772
$485$	1.53176e9	0.609669
$486$	− 1.38510e8i	− 0.0547335i
$487$	− 2.89212e9i	− 1.13466i	−0.823491	−	0.567329i	$-0.807977\pi$
$487$	− 2.89212e9i	− 1.13466i	0.823491	−	0.567329i	$-0.192023\pi$
$488$	− 3.11872e9i	− 1.21481i
$489$	− 7.84787e7i	− 0.0303508i
$490$	2.28537e8	0.0877547
$491$	3.80132e9	1.44927	0.724635	−	0.689133i	$-0.242009\pi$
$491$	3.80132e9	1.44927	0.724635	+	0.689133i	$0.242009\pi$
$492$	4.22977e7i	0.0160117i
$493$	1.54906e9	0.582241
$494$	0	0
$495$	4.94780e9	1.83355
$496$	− 9.88014e8i	− 0.363561i
$497$	−4.85032e9	−1.77224
$498$	−7.83948e6	−0.00284436
$499$	4.75421e8i	0.171288i	0.996326	+	0.0856439i	$0.0272947\pi$
$499$	4.75421e8i	0.171288i	−0.996326	+	0.0856439i	$0.972705\pi$
$500$	− 6.77057e7i	− 0.0242231i
$501$	3.35742e7i	0.0119281i
$502$	− 1.19882e9i	− 0.422952i
$503$	−1.70972e9	−0.599013	−0.299506	−	0.954094i	$-0.596822\pi$
$503$	−1.70972e9	−0.599013	−0.299506	+	0.954094i	$0.596822\pi$
$504$	−3.21648e9	−1.11911
$505$	3.83706e9i	1.32580i
$506$	1.63181e8	0.0559942
$507$	0	0
$508$	−1.95006e9	−0.659970
$509$	7.59754e8i	0.255365i	0.991815	+	0.127682i	$0.0407538\pi$
$509$	7.59754e8i	0.255365i	−0.991815	+	0.127682i	$0.959246\pi$
$510$	−4.70841e7	−0.0157173
$511$	−5.44727e9	−1.80595
$512$	− 2.03387e9i	− 0.669698i
$513$	− 2.43966e8i	− 0.0797846i
$514$	2.46528e9i	0.800747i
$515$	− 1.14384e9i	− 0.369011i
$516$	8.51709e6	0.00272908
$517$	−1.92034e9	−0.611169
$518$	1.18579e9i	0.374846i
$519$	1.03403e8	0.0324674
$520$	0	0
$521$	−4.09809e9	−1.26955	−0.634774	−	0.772698i	$-0.718907\pi$
$521$	−4.09809e9	−1.26955	−0.634774	+	0.772698i	$0.718907\pi$
$522$	2.33373e9i	0.718131i
$523$	−3.85404e9	−1.17804	−0.589021	−	0.808118i	$-0.700486\pi$
$523$	−3.85404e9	−1.17804	−0.589021	+	0.808118i	$0.700486\pi$
$524$	3.74174e8	0.113609
$525$	− 8.78666e7i	− 0.0265012i
$526$	− 2.56835e9i	− 0.769491i
$527$	− 2.07711e9i	− 0.618190i
$528$	3.78218e7i	0.0111821i
$529$	−3.39315e9	−0.996572
$530$	7.62299e8	0.222413
$531$	− 3.50154e9i	− 1.01491i
$532$	−2.62241e9	−0.755111
$533$	0	0
$534$	−4.98120e7	−0.0141560
$535$	− 7.91927e7i	− 0.0223587i
$536$	2.15136e9	0.603442
$537$	1.94762e7	0.00542743
$538$	1.95199e9i	0.540431i
$539$	− 3.85941e8i	− 0.106160i
$540$	1.14363e8i	0.0312541i
$541$	1.31488e8i	0.0357022i	0.999841	+	0.0178511i	$0.00568248\pi$
$541$	1.31488e8i	0.0357022i	−0.999841	+	0.0178511i	$0.994318\pi$
$542$	−4.14761e9	−1.11892
$543$	5.38070e7	0.0144225
$544$	− 1.83919e9i	− 0.489814i
$545$	−7.26635e9	−1.92278
$546$	0	0
$547$	2.11326e9	0.552073	0.276037	−	0.961147i	$-0.410979\pi$
$547$	2.11326e9	0.552073	0.276037	+	0.961147i	$0.410979\pi$
$548$	− 8.75015e8i	− 0.227135i
$549$	4.37370e9	1.12809
$550$	3.87332e9	0.992691
$551$	6.16645e9i	1.57038i
$552$	6.11008e6i	0.00154618i
$553$	4.20771e9i	1.05805i
$554$	3.64820e9i	0.911579i
$555$	6.82992e7	0.0169586
$556$	6.40255e8	0.157976
$557$	− 2.39033e9i	− 0.586091i	−0.956098	−	0.293046i	$-0.905331\pi$
$557$	− 2.39033e9i	− 0.586091i	0.956098	−	0.293046i	$-0.0946688\pi$
$558$	3.12927e9	0.762470
$559$	0	0
$560$	−2.18901e9	−0.526731
$561$	7.95131e7i	0.0190138i
$562$	−2.42954e9	−0.577361
$563$	5.42284e9	1.28070	0.640350	−	0.768083i	$-0.278789\pi$
$563$	5.42284e9	1.28070	0.640350	+	0.768083i	$0.278789\pi$
$564$	− 2.21866e7i	− 0.00520732i
$565$	− 3.24812e9i	− 0.757640i
$566$	− 5.04253e9i	− 1.16894i
$567$	− 4.50807e9i	− 1.03860i
$568$	8.00585e9	1.83311
$569$	−1.61095e9	−0.366598	−0.183299	−	0.983057i	$-0.558678\pi$
$569$	−1.61095e9	−0.366598	−0.183299	+	0.983057i	$0.558678\pi$
$570$	− 1.87432e8i	− 0.0423917i
$571$	5.46762e9	1.22906	0.614528	−	0.788895i	$-0.289347\pi$
$571$	5.46762e9	1.22906	0.614528	+	0.788895i	$0.289347\pi$
$572$	0	0
$573$	1.51711e8	0.0336880
$574$	− 5.13001e9i	− 1.13221i
$575$	2.77064e8	0.0607775
$576$	4.39626e9	0.958527
$577$	− 3.08434e9i	− 0.668417i	−0.942499	−	0.334209i	$-0.891531\pi$
$577$	− 3.08434e9i	− 0.668417i	0.942499	−	0.334209i	$-0.108469\pi$
$578$	− 2.19864e9i	− 0.473595i
$579$	− 1.57509e8i	− 0.0337233i
$580$	− 2.89062e9i	− 0.615167i
$581$	−7.66224e8	−0.162084
$582$	−3.70852e7	−0.00779777
$583$	− 1.28733e9i	− 0.269060i
$584$	8.99117e9	1.86797
$585$	0	0
$586$	2.63457e8	0.0540839
$587$	1.47215e9i	0.300413i	0.988655	+	0.150206i	$0.0479938\pi$
$587$	1.47215e9i	0.300413i	−0.988655	+	0.150206i	$0.952006\pi$
$588$	4.45896e6	0.000904509 0
$589$	8.26852e9	1.66734
$590$	− 5.38186e9i	− 1.07882i
$591$	− 1.91135e8i	− 0.0380877i
$592$	− 8.66635e8i	− 0.171676i
$593$	− 5.98992e9i	− 1.17959i	−0.807554	−	0.589793i	$-0.799209\pi$
$593$	− 5.98992e9i	− 1.17959i	0.807554	−	0.589793i	$-0.200791\pi$
$594$	−2.39653e8	−0.0469170
$595$	−4.60196e9	−0.895640
$596$	2.20090e9i	0.425832i
$597$	6.65433e7	0.0127995
$598$	0	0
$599$	−6.44870e9	−1.22597	−0.612983	−	0.790096i	$-0.710031\pi$
$599$	−6.44870e9	−1.22597	−0.612983	+	0.790096i	$0.710031\pi$
$600$	1.45031e8i	0.0274114i
$601$	5.19885e9	0.976892	0.488446	−	0.872594i	$-0.337564\pi$
$601$	5.19885e9	0.976892	0.488446	+	0.872594i	$0.337564\pi$
$602$	−1.03298e9	−0.192977
$603$	3.01706e9i	0.560369i
$604$	2.33586e9i	0.431338i
$605$	− 5.06665e9i	− 0.930202i
$606$	− 9.28985e7i	− 0.0169572i
$607$	−3.63606e9	−0.659890	−0.329945	−	0.944000i	$-0.607030\pi$
$607$	−3.63606e9	−0.659890	−0.329945	+	0.944000i	$0.607030\pi$
$608$	7.32142e9	1.32109
$609$	− 1.37413e8i	− 0.0246529i
$610$	6.72237e9	1.19913
$611$	0	0
$612$	1.52490e9	0.268913
$613$	− 1.64970e9i	− 0.289263i	−0.989486	−	0.144632i	$-0.953800\pi$
$613$	− 1.64970e9i	− 0.289263i	0.989486	−	0.144632i	$-0.0461997\pi$
$614$	−2.02108e9	−0.352367
$615$	−2.95479e8	−0.0512229
$616$	8.34869e9i	1.43908i
$617$	6.85487e8i	0.117490i	0.998273	+	0.0587450i	$0.0187099\pi$
$617$	6.85487e8i	0.117490i	−0.998273	+	0.0587450i	$0.981290\pi$
$618$	2.76933e7i	0.00471970i
$619$	3.07866e9i	0.521728i	0.965376	+	0.260864i	$0.0840074\pi$
$619$	3.07866e9i	0.521728i	−0.965376	+	0.260864i	$0.915993\pi$
$620$	−3.87599e9	−0.653149
$621$	−1.71427e7	−0.00287249
$622$	9.58744e8i	0.159748i
$623$	−4.86858e9	−0.806667
$624$	0	0
$625$	−5.86262e9	−0.960531
$626$	− 2.41320e9i	− 0.393172i
$627$	−3.16524e8	−0.0512827
$628$	−1.44460e9	−0.232749
$629$	− 1.82193e9i	− 0.291914i
$630$	− 6.93308e9i	− 1.10467i
$631$	− 8.91118e9i	− 1.41199i	−0.708215	−	0.705996i	$-0.750499\pi$
$631$	− 8.91118e9i	− 1.41199i	0.708215	−	0.705996i	$-0.249501\pi$
$632$	− 6.94517e9i	− 1.09439i
$633$	−1.38941e8	−0.0217729
$634$	−7.97823e9	−1.24335
$635$	− 1.36225e10i	− 2.11130i
$636$	1.48731e7	0.00229246
$637$	0	0
$638$	6.05743e9	0.923455
$639$	1.12274e10i	1.70226i
$640$	−9.33756e8	−0.140800
$641$	−1.19624e9	−0.179397	−0.0896984	−	0.995969i	$-0.528590\pi$
$641$	−1.19624e9	−0.179397	−0.0896984	+	0.995969i	$0.528590\pi$
$642$	1.91732e6i	0 0.000285971i
$643$	− 7.69066e9i	− 1.14084i	−0.821353	−	0.570421i	$-0.806780\pi$
$643$	− 7.69066e9i	− 1.14084i	0.821353	−	0.570421i	$-0.193220\pi$
$644$	1.84269e8i	0.0271863i
$645$	5.94978e7i	0.00873056i
$646$	−4.99989e9	−0.729703
$647$	−2.75099e9	−0.399322	−0.199661	−	0.979865i	$-0.563984\pi$
$647$	−2.75099e9	−0.399322	−0.199661	+	0.979865i	$0.563984\pi$
$648$	7.44094e9i	1.07427i
$649$	−9.08860e9	−1.30509
$650$	0	0
$651$	−1.84256e8	−0.0261750
$652$	3.90655e9i	0.551984i
$653$	−4.79202e9	−0.673477	−0.336738	−	0.941598i	$-0.609324\pi$
$653$	−4.79202e9	−0.673477	−0.336738	+	0.941598i	$0.609324\pi$
$654$	1.75925e8	0.0245926
$655$	2.61387e9i	0.363445i
$656$	3.74928e9i	0.518542i
$657$	1.26092e10i	1.73464i
$658$	2.69087e9i	0.368215i
$659$	−4.70316e9	−0.640164	−0.320082	−	0.947390i	$-0.603710\pi$
$659$	−4.70316e9	−0.640164	−0.320082	+	0.947390i	$0.603710\pi$
$660$	1.48376e8	0.0200890
$661$	− 4.42747e8i	− 0.0596280i	−0.999555	−	0.0298140i	$-0.990508\pi$
$661$	− 4.42747e8i	− 0.0596280i	0.999555	−	0.0298140i	$-0.00949150\pi$
$662$	−5.76574e9	−0.772417
$663$	0	0
$664$	1.26471e9	0.167650
$665$	− 1.83194e10i	− 2.41566i
$666$	2.74483e9	0.360044
$667$	4.33297e8	0.0565386
$668$	− 1.67127e9i	− 0.216934i
$669$	5.31003e7i	0.00685656i
$670$	4.63722e9i	0.595657i
$671$	− 1.13524e10i	− 1.45063i
$672$	−1.63151e8	−0.0207394
$673$	−4.22977e9	−0.534890	−0.267445	−	0.963573i	$-0.586179\pi$
$673$	−4.22977e9	−0.534890	−0.267445	+	0.963573i	$0.586179\pi$
$674$	9.49564e9i	1.19458i
$675$	−4.06905e8	−0.0509249
$676$	0	0
$677$	2.02881e9	0.251294	0.125647	−	0.992075i	$-0.459899\pi$
$677$	2.02881e9	0.251294	0.125647	+	0.992075i	$0.459899\pi$
$678$	7.86398e7i	0.00969033i
$679$	−3.62468e9	−0.444350
$680$	7.59591e9	0.926400
$681$	− 3.86475e8i	− 0.0468929i
$682$	− 8.12233e9i	− 0.980472i
$683$	1.47606e9i	0.177268i	0.996064	+	0.0886340i	$0.0282502\pi$
$683$	1.47606e9i	0.177268i	−0.996064	+	0.0886340i	$0.971750\pi$
$684$	6.07030e9i	0.725294i
$685$	6.11259e9	0.726622
$686$	6.00595e9	0.710309
$687$	3.31235e8i	0.0389751i
$688$	7.54957e8	0.0883818
$689$	0	0
$690$	−1.31702e7	−0.00152623
$691$	8.97569e9i	1.03489i	0.855716	+	0.517445i	$0.173117\pi$
$691$	8.97569e9i	1.03489i	−0.855716	+	0.517445i	$0.826883\pi$
$692$	−5.14724e9	−0.590477
$693$	−1.17082e10	−1.33636
$694$	− 3.23957e9i	− 0.367899i
$695$	4.47263e9i	0.505378i
$696$	2.26812e8i	0.0254996i
$697$	7.88214e9i	0.881717i
$698$	1.68996e9	0.188097
$699$	−1.12678e8	−0.0124786
$700$	4.37386e9i	0.481972i
$701$	5.08142e9	0.557149	0.278575	−	0.960415i	$-0.410138\pi$
$701$	5.08142e9	0.557149	0.278575	+	0.960415i	$0.410138\pi$
$702$	0	0
$703$	7.25272e9	0.787331
$704$	− 1.14109e10i	− 1.23258i
$705$	1.54989e8	0.0166586
$706$	2.99451e9	0.320265
$707$	− 9.07982e9i	− 0.966294i
$708$	− 1.05005e8i	− 0.0111197i
$709$	3.63793e9i	0.383348i	0.981459	+	0.191674i	$0.0613916\pi$
$709$	3.63793e9i	0.383348i	−0.981459	+	0.191674i	$0.938608\pi$
$710$	1.72565e10i	1.80946i
$711$	9.73991e9	1.01628
$712$	8.03598e9	0.834371
$713$	− 5.81002e8i	− 0.0600294i
$714$	1.11417e8	0.0114554
$715$	0	0
$716$	−9.69495e8	−0.0987075
$717$	− 1.30649e8i	− 0.0132369i
$718$	−9.84638e8	−0.0992753
$719$	−8.62579e9	−0.865462	−0.432731	−	0.901523i	$-0.642450\pi$
$719$	−8.62579e9	−0.865462	−0.432731	+	0.901523i	$0.642450\pi$
$720$	5.06706e9i	0.505932i
$721$	2.70672e9i	0.268949i
$722$	− 1.23779e10i	− 1.22396i
$723$	8.52486e7i	0.00838887i
$724$	−2.67843e9	−0.262298
$725$	1.02849e10	1.00234
$726$	1.22668e8i	0.0118974i
$727$	1.23787e10	1.19482	0.597412	−	0.801934i	$-0.296196\pi$
$727$	1.23787e10	1.19482	0.597412	+	0.801934i	$0.296196\pi$
$728$	0	0
$729$	−1.04226e10	−0.996389
$730$	1.93804e10i	1.84388i
$731$	1.58715e9	0.150282
$732$	1.31159e8	0.0123598
$733$	1.32984e10i	1.24720i	0.781745	+	0.623598i	$0.214330\pi$
$733$	1.32984e10i	1.24720i	−0.781745	+	0.623598i	$0.785670\pi$
$734$	1.42734e10i	1.33227i
$735$	3.11490e7i	0.00289360i
$736$	− 5.14452e8i	− 0.0475634i
$737$	7.83109e9	0.720587
$738$	−1.18748e10	−1.08750
$739$	− 1.28451e10i	− 1.17079i	−0.810747	−	0.585397i	$-0.800939\pi$
$739$	− 1.28451e10i	− 1.17079i	0.810747	−	0.585397i	$-0.199061\pi$
$740$	−3.39982e9	−0.308422
$741$	0	0
$742$	−1.80386e9	−0.162103
$743$	− 2.01042e10i	− 1.79815i	−0.437793	−	0.899076i	$-0.644240\pi$
$743$	− 2.01042e10i	− 1.79815i	0.437793	−	0.899076i	$-0.355760\pi$
$744$	3.04129e8	0.0270740
$745$	−1.53748e10	−1.36227
$746$	− 1.10060e10i	− 0.970606i
$747$	1.77364e9i	0.155683i
$748$	− 3.95804e9i	− 0.345799i
$749$	1.87397e8i	0.0162959i
$750$	−1.14511e7	−0.000991131 0
$751$	−1.99539e8	−0.0171905	−0.00859524	−	0.999963i	$-0.502736\pi$
$751$	−1.99539e8	−0.0171905	−0.00859524	+	0.999963i	$0.502736\pi$
$752$	− 1.96663e9i	− 0.168640i
$753$	1.63396e8	0.0139463
$754$	0	0
$755$	−1.63176e10	−1.37988
$756$	− 2.70623e8i	− 0.0227792i
$757$	2.34770e10	1.96701	0.983507	−	0.180868i	$-0.0578908\pi$
$757$	2.34770e10	1.96701	0.983507	+	0.180868i	$0.0578908\pi$
$758$	−7.06853e9	−0.589504
$759$	2.22411e7i	0.00184634i
$760$	3.02377e10i	2.49862i
$761$	1.60366e10i	1.31907i	0.751676	+	0.659533i	$0.229246\pi$
$761$	1.60366e10i	1.31907i	−0.751676	+	0.659533i	$0.770754\pi$
$762$	3.29813e8i	0.0270038i
$763$	1.71947e10	1.40139
$764$	−7.55193e9	−0.612676
$765$	1.06525e10i	0.860274i
$766$	−5.31846e9	−0.427548
$767$	0	0
$768$	3.18037e8	0.0253345
$769$	1.49614e10i	1.18639i	0.805057	+	0.593197i	$0.202135\pi$
$769$	1.49614e10i	1.18639i	−0.805057	+	0.593197i	$0.797865\pi$
$770$	−1.79955e10	−1.42052
$771$	−3.36011e8	−0.0264036
$772$	7.84053e9i	0.613317i
$773$	− 3.25072e9i	− 0.253135i	−0.991958	−	0.126567i	$-0.959604\pi$
$773$	− 3.25072e9i	− 0.253135i	0.991958	−	0.126567i	$-0.0403960\pi$
$774$	2.39112e9i	0.185357i
$775$	− 1.37909e10i	− 1.06423i
$776$	5.98282e9	0.459611
$777$	−1.61620e8	−0.0123601
$778$	− 3.78804e9i	− 0.288394i
$779$	−3.13771e10	−2.37811
$780$	0	0
$781$	2.91419e10	2.18897
$782$	3.51326e8i	0.0262716i
$783$	−6.36353e8	−0.0473731
$784$	3.95243e8	0.0292926
$785$	− 1.00915e10i	− 0.744583i
$786$	− 6.32840e7i	− 0.00464852i
$787$	1.13640e10i	0.831033i	0.909586	+	0.415517i	$0.136399\pi$
$787$	1.13640e10i	0.831033i	−0.909586	+	0.415517i	$0.863601\pi$
$788$	9.51441e9i	0.692692i
$789$	3.50059e8	0.0253730
$790$	1.49702e10	1.08027
$791$	7.68619e9i	0.552196i
$792$	1.93254e10	1.38226
$793$	0	0
$794$	1.19944e10	0.850367
$795$	1.03899e8i	0.00733377i
$796$	−3.31242e9	−0.232782
$797$	5.72603e9	0.400636	0.200318	−	0.979731i	$-0.435803\pi$
$797$	5.72603e9	0.400636	0.200318	+	0.979731i	$0.435803\pi$
$798$	4.43528e8i	0.0308967i
$799$	− 4.13445e9i	− 0.286751i
$800$	− 1.22112e10i	− 0.843226i
$801$	1.12697e10i	0.774814i
$802$	−7.32478e9	−0.501400
$803$	3.27285e10	2.23060
$804$	9.04763e7i	0.00613958i
$805$	−1.28725e9	−0.0869712
$806$	0	0
$807$	−2.66051e8	−0.0178200
$808$	1.49870e10i	0.999480i
$809$	1.88681e10	1.25288	0.626440	−	0.779470i	$-0.284512\pi$
$809$	1.88681e10	1.25288	0.626440	+	0.779470i	$0.284512\pi$
$810$	−1.60389e10	−1.06042
$811$	− 2.28943e10i	− 1.50714i	−0.657367	−	0.753570i	$-0.728330\pi$
$811$	− 2.28943e10i	− 1.50714i	0.657367	−	0.753570i	$-0.271670\pi$
$812$	6.84021e9i	0.448356i
$813$	− 5.65308e8i	− 0.0368951i
$814$	− 7.12449e9i	− 0.462987i
$815$	−2.72900e10	−1.76584
$816$	−8.14297e7	−0.00524647
$817$	6.31810e9i	0.405331i
$818$	−1.87876e10	−1.20015
$819$	0	0
$820$	1.47085e10	0.931578
$821$	3.01467e9i	0.190125i	0.995471	+	0.0950625i	$0.0303051\pi$
$821$	3.01467e9i	0.190125i	−0.995471	+	0.0950625i	$0.969695\pi$
$822$	−1.47991e8	−0.00929361
$823$	−9.51460e9	−0.594964	−0.297482	−	0.954727i	$-0.596147\pi$
$823$	−9.51460e9	−0.594964	−0.297482	+	0.954727i	$0.596147\pi$
$824$	− 4.46766e9i	− 0.278185i
$825$	5.27923e8i	0.0327327i
$826$	1.27354e10i	0.786287i
$827$	3.09369e10i	1.90199i	0.309210	+	0.950994i	$0.399936\pi$
$827$	3.09369e10i	1.90199i	−0.309210	+	0.950994i	$0.600064\pi$
$828$	4.26540e8	0.0261128
$829$	−1.75499e10	−1.06988	−0.534938	−	0.844891i	$-0.679665\pi$
$829$	−1.75499e10	−1.06988	−0.534938	+	0.844891i	$0.679665\pi$
$830$	2.72608e9i	0.165487i
$831$	−4.97240e8	−0.0300581
$832$	0	0
$833$	8.30923e8	0.0498085
$834$	− 1.08286e8i	− 0.00646386i
$835$	1.16750e10	0.693991
$836$	1.57561e10	0.932666
$837$	8.53278e8i	0.0502981i
$838$	− 7.00069e9i	− 0.410948i
$839$	− 1.26056e10i	− 0.736878i	−0.929652	−	0.368439i	$-0.879892\pi$
$839$	− 1.26056e10i	− 0.736878i	0.929652	−	0.368439i	$-0.120108\pi$
$840$	− 6.73816e8i	− 0.0392251i
$841$	−1.16552e9	−0.0675671
$842$	8.03041e9	0.463602
$843$	− 3.31140e8i	− 0.0190377i
$844$	6.91625e9	0.395979
$845$	0	0
$846$	6.22876e9	0.353676
$847$	1.19895e10i	0.677966i
$848$	1.31836e9	0.0742417
$849$	6.87282e8	0.0385441
$850$	8.33918e9i	0.465755i
$851$	− 5.09625e8i	− 0.0283464i
$852$	3.36690e8i	0.0186506i
$853$	− 1.77001e10i	− 0.976457i	−0.872716	−	0.488229i	$-0.837643\pi$
$853$	− 1.77001e10i	− 0.976457i	0.872716	−	0.488229i	$-0.162357\pi$
$854$	−1.59075e10	−0.873974
$855$	−4.24053e10	−2.32027
$856$	− 3.09315e8i	− 0.0168555i
$857$	6.38662e9	0.346608	0.173304	−	0.984868i	$-0.444556\pi$
$857$	6.38662e9	0.346608	0.173304	+	0.984868i	$0.444556\pi$
$858$	0	0
$859$	9.57498e9	0.515421	0.257710	−	0.966222i	$-0.417032\pi$
$859$	9.57498e9	0.515421	0.257710	+	0.966222i	$0.417032\pi$
$860$	− 2.96171e9i	− 0.158781i
$861$	6.99206e8	0.0373331
$862$	8.57084e9	0.455772
$863$	− 3.52438e10i	− 1.86658i	−0.359130	−	0.933288i	$-0.616926\pi$
$863$	− 3.52438e10i	− 1.86658i	0.359130	−	0.933288i	$-0.383074\pi$
$864$	7.55541e8i	0.0398529i
$865$	− 3.59571e10i	− 1.88898i
$866$	− 1.53029e10i	− 0.800683i
$867$	2.99669e8	0.0156162
$868$	9.17195e9	0.476039
$869$	− 2.52809e10i	− 1.30684i
$870$	−4.88890e8	−0.0251706
$871$	0	0
$872$	−2.83813e10	−1.44952
$873$	8.39031e9i	0.426804i
$874$	−1.39855e9	−0.0708579
$875$	−1.11922e9	−0.0564788
$876$	3.78128e8i	0.0190053i
$877$	− 6.34498e9i	− 0.317638i	−0.987308	−	0.158819i	$-0.949231\pi$
$877$	− 6.34498e9i	− 0.317638i	0.987308	−	0.158819i	$-0.0507686\pi$
$878$	5.92589e9i	0.295476i
$879$	3.59084e7i	0.00178335i
$880$	1.31521e10	0.650586
$881$	2.92173e8	0.0143954	0.00719771	−	0.999974i	$-0.497709\pi$
$881$	2.92173e8	0.0143954	0.00719771	+	0.999974i	$0.497709\pi$
$882$	1.25183e9i	0.0614333i
$883$	3.68103e10	1.79931	0.899656	−	0.436600i	$-0.143817\pi$
$883$	3.68103e10	1.79931	0.899656	+	0.436600i	$0.143817\pi$
$884$	0	0
$885$	7.33533e8	0.0355728
$886$	7.58470e8i	0.0366370i
$887$	4.03158e9	0.193973	0.0969867	−	0.995286i	$-0.469080\pi$
$887$	4.03158e9	0.193973	0.0969867	+	0.995286i	$0.469080\pi$
$888$	2.66766e8	0.0127846
$889$	3.22356e10i	1.53879i
$890$	1.73215e10i	0.823607i
$891$	2.70856e10i	1.28282i
$892$	− 2.64325e9i	− 0.124699i
$893$	1.64584e10	0.773404
$894$	3.72238e8	0.0174237
$895$	− 6.77261e9i	− 0.315773i
$896$	2.20959e9	0.102620
$897$	0	0
$898$	8.73603e9	0.402575
$899$	− 2.15673e10i	− 0.990004i
$900$	1.01245e10	0.462940
$901$	2.77159e9	0.126239
$902$	3.08223e10i	1.39844i
$903$	− 1.40793e8i	− 0.00636316i
$904$	− 1.26867e10i	− 0.571161i
$905$	− 1.87107e10i	− 0.839112i
$906$	3.95064e8	0.0176489
$907$	2.17702e10	0.968807	0.484403	−	0.874845i	$-0.339037\pi$
$907$	2.17702e10	0.968807	0.484403	+	0.874845i	$0.339037\pi$
$908$	1.92381e10i	0.852830i
$909$	−2.10177e10	−0.928138
$910$	0	0
$911$	−3.27391e10	−1.43467	−0.717335	−	0.696728i	$-0.754638\pi$
$911$	−3.27391e10	−1.43467	−0.717335	+	0.696728i	$0.754638\pi$
$912$	− 3.24154e8i	− 0.0141504i
$913$	4.60365e9	0.200196
$914$	1.40430e10	0.608342
$915$	9.16240e8i	0.0395399i
$916$	− 1.64883e10i	− 0.708831i
$917$	− 6.18532e9i	− 0.264892i
$918$	− 5.15968e8i	− 0.0220127i
$919$	1.03618e10	0.440385	0.220193	−	0.975456i	$-0.429331\pi$
$919$	1.03618e10	0.440385	0.220193	+	0.975456i	$0.429331\pi$
$920$	2.12470e9	0.0899582
$921$	− 2.75468e8i	− 0.0116188i
$922$	−2.44963e10	−1.02930
$923$	0	0
$924$	−3.51108e8	−0.0146416
$925$	− 1.20966e10i	− 0.502537i
$926$	−1.59540e10	−0.660286
$927$	6.26544e9	0.258329
$928$	− 1.90969e10i	− 0.784415i
$929$	2.15053e10i	0.880018i	0.897993	+	0.440009i	$0.145025\pi$
$929$	2.15053e10i	0.880018i	−0.897993	+	0.440009i	$0.854975\pi$
$930$	6.55546e8i	0.0267247i
$931$	3.30772e9i	0.134340i
$932$	5.60891e9	0.226946
$933$	−1.30674e8	−0.00526750
$934$	8.37823e9i	0.336464i
$935$	2.76497e10	1.10624
$936$	0	0
$937$	2.31411e8	0.00918957	0.00459479	−	0.999989i	$-0.498537\pi$
$937$	2.31411e8	0.00918957	0.00459479	+	0.999989i	$0.498537\pi$
$938$	− 1.09733e10i	− 0.434137i
$939$	3.28912e8	0.0129643
$940$	−7.71511e9	−0.302966
$941$	2.99031e10i	1.16991i	0.811065	+	0.584956i	$0.198888\pi$
$941$	2.99031e10i	1.16991i	−0.811065	+	0.584956i	$0.801112\pi$
$942$	2.44325e8i	0.00952334i
$943$	2.20476e9i	0.0856192i
$944$	− 9.30767e9i	− 0.360113i
$945$	1.89049e9	0.0728724
$946$	6.20640e9	0.238353
$947$	− 3.37889e10i	− 1.29285i	−0.762976	−	0.646427i	$-0.776263\pi$
$947$	− 3.37889e10i	− 1.29285i	0.762976	−	0.646427i	$-0.223737\pi$
$948$	2.92083e8	0.0111347
$949$	0	0
$950$	−3.31965e10	−1.25620
$951$	− 1.08741e9i	− 0.0409979i
$952$	−1.79746e10	−0.675195
$953$	4.50079e10	1.68447	0.842236	−	0.539110i	$-0.181239\pi$
$953$	4.50079e10	1.68447	0.842236	+	0.539110i	$0.181239\pi$
$954$	4.17554e9i	0.155702i
$955$	− 5.27555e10i	− 1.96000i
$956$	6.50348e9i	0.240737i
$957$	8.25611e8i	0.0304497i
$958$	−1.17131e10	−0.430421
$959$	−1.44645e10	−0.529589
$960$	9.20966e8i	0.0335965i
$961$	−1.40672e9	−0.0511298
$962$	0	0
$963$	4.33783e8	0.0156524
$964$	− 4.24354e9i	− 0.152566i
$965$	−5.47716e10	−1.96205
$966$	3.11653e7	0.00111238
$967$	5.77459e9i	0.205366i	0.994714	+	0.102683i	$0.0327427\pi$
$967$	5.77459e9i	0.205366i	−0.994714	+	0.102683i	$0.967257\pi$
$968$	− 1.97896e10i	− 0.701250i
$969$	− 6.81471e8i	− 0.0240610i
$970$	1.28959e10i	0.453681i
$971$	−5.40196e10	−1.89358	−0.946790	−	0.321851i	$-0.895695\pi$
$971$	−5.40196e10	−1.89358	−0.946790	+	0.321851i	$0.895695\pi$
$972$	−9.39741e8	−0.0328228
$973$	− 1.05838e10i	− 0.368338i
$974$	2.43488e10	0.844347
$975$	0	0
$976$	1.16260e10	0.400273
$977$	− 3.21144e10i	− 1.10171i	−0.834600	−	0.550857i	$-0.814301\pi$
$977$	− 3.21144e10i	− 1.10171i	0.834600	−	0.550857i	$-0.185699\pi$
$978$	6.60713e8	0.0225854
$979$	2.92516e10	0.996345
$980$	− 1.55055e9i	− 0.0526252i
$981$	− 3.98019e10i	− 1.34605i
$982$	3.20034e10i	1.07846i
$983$	− 1.36597e10i	− 0.458673i	−0.973347	−	0.229337i	$-0.926344\pi$
$983$	− 1.36597e10i	− 0.458673i	0.973347	−	0.229337i	$-0.0736556\pi$
$984$	−1.15410e9	−0.0386153
$985$	−6.64648e10	−2.21598
$986$	1.30415e10i	0.433271i
$987$	−3.66758e8	−0.0121414
$988$	0	0
$989$	4.43953e8	0.0145932
$990$	4.16556e10i	1.36443i
$991$	−3.04984e10	−0.995450	−0.497725	−	0.867335i	$-0.665831\pi$
$991$	−3.04984e10	−0.995450	−0.497725	+	0.867335i	$0.665831\pi$
$992$	−2.56068e10	−0.832847
$993$	− 7.85854e8i	− 0.0254694i
$994$	− 4.08349e10i	− 1.31880i
$995$	− 2.31396e10i	− 0.744688i
$996$	5.31882e7i	0.00170572i
$997$	3.41604e10	1.09167	0.545833	−	0.837894i	$-0.316213\pi$
$997$	3.41604e10	1.09167	0.545833	+	0.837894i	$0.316213\pi$
$998$	−4.00258e9	−0.127463
$999$	7.48452e8i	0.0237511i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.b.d.168.10		14
13.3	even	3		13.8.e.a.4.3	✓	14
13.4	even	6		13.8.e.a.10.3	yes	14
13.5	odd	4		169.8.a.g.1.10		14
13.8	odd	4		169.8.a.g.1.5		14
13.12	even	2	inner	169.8.b.d.168.5		14
39.17	odd	6		117.8.q.b.10.5		14
39.29	odd	6		117.8.q.b.82.5		14
52.3	odd	6		208.8.w.a.17.4		14
52.43	odd	6		208.8.w.a.49.4		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.e.a.4.3	✓	14	13.3	even	3
13.8.e.a.10.3	yes	14	13.4	even	6
117.8.q.b.10.5		14	39.17	odd	6
117.8.q.b.82.5		14	39.29	odd	6
169.8.a.g.1.5		14	13.8	odd	4
169.8.a.g.1.10		14	13.5	odd	4
169.8.b.d.168.5		14	13.12	even	2	inner
169.8.b.d.168.10		14	1.1	even	1	trivial
208.8.w.a.17.4		14	52.3	odd	6
208.8.w.a.49.4		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)

\(f(q)\)	\(=\)	\(q+8.41902i q^{2} -1.14749 q^{3} +57.1202 q^{4} +399.024i q^{5} -9.66073i q^{6} -944.231i q^{7} +1558.53i q^{8} -2185.68 q^{9} -3359.39 q^{10} +5673.16i q^{11} -65.5447 q^{12} +7949.50 q^{14} -457.876i q^{15} -5809.90 q^{16} -12214.2 q^{17} -18401.3i q^{18} -48622.1i q^{19} +22792.3i q^{20} +1083.49i q^{21} -47762.4 q^{22} -3416.52 q^{23} -1788.40i q^{24} -81095.5 q^{25} +5017.60 q^{27} -53934.6i q^{28} -126824. q^{29} +3854.87 q^{30} +170057. i q^{31} +150578. i q^{32} -6509.89i q^{33} -102832. i q^{34} +376771. q^{35} -124847. q^{36} +149165. i q^{37} +409350. q^{38} -621892. q^{40} -645325. i q^{41} -9121.96 q^{42} -129943. q^{43} +324052. i q^{44} -872141. i q^{45} -28763.7i q^{46} +338496. i q^{47} +6666.80 q^{48} -68029.2 q^{49} -682745. i q^{50} +14015.7 q^{51} -226916. q^{53} +42243.3i q^{54} -2.26373e6 q^{55} +1.47161e6 q^{56} +55793.3i q^{57} -1.06773e6i q^{58} +1.60203e6i q^{59} -26154.0i q^{60} -2.00107e6 q^{61} -1.43171e6 q^{62} +2.06379e6i q^{63} -2.01139e6 q^{64} +54806.9 q^{66} -1.38038e6i q^{67} -697677. q^{68} +3920.41 q^{69} +3.17204e6i q^{70} -5.13680e6i q^{71} -3.40645e6i q^{72} -5.76900e6i q^{73} -1.25582e6 q^{74} +93056.2 q^{75} -2.77730e6i q^{76} +5.35678e6 q^{77} -4.45623e6 q^{79} -2.31829e6i q^{80} +4.77433e6 q^{81} +5.43300e6 q^{82} -811479. i q^{83} +61889.4i q^{84} -4.87377e6i q^{85} -1.09399e6i q^{86} +145529. q^{87} -8.84179e6 q^{88} -5.15613e6i q^{89} +7.34257e6 q^{90} -195152. q^{92} -195138. i q^{93} -2.84980e6 q^{94} +1.94014e7 q^{95} -172787. i q^{96} -3.83876e6i q^{97} -572739. i q^{98} -1.23997e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 52 q^{3} - 766 q^{4} + 6982 q^{9} + 1018 q^{10} + 38380 q^{12} - 47916 q^{14} + 1266 q^{16} + 76806 q^{17} + 251764 q^{22} + 137100 q^{23} + 39380 q^{25} - 432400 q^{27} - 443166 q^{29} + 315780 q^{30}+ \cdots + 86840772 q^{95}+O(q^{100}) \) 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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