LMFDB - Embedded newform 61.8.b.a.60.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [61,8,Mod(60,61)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(61, 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("61.60");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 61 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	61.b (of order $2$, degree $1$, 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:	$19.0554865545$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			60.17
Character	$\chi$	$=$	61.60
Dual form			61.8.b.a.60.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.698820i q^{2} +31.4227 q^{3} +127.512 q^{4} -277.058 q^{5} -21.9588i q^{6} -565.293i q^{7} -178.557i q^{8} -1199.61 q^{9} +O(q^{10})$	\(q-0.698820i q^{2} +31.4227 q^{3} +127.512 q^{4} -277.058 q^{5} -21.9588i q^{6} -565.293i q^{7} -178.557i q^{8} -1199.61 q^{9} +193.614i q^{10} -7720.67i q^{11} +4006.76 q^{12} -8475.06 q^{13} -395.038 q^{14} -8705.91 q^{15} +16196.7 q^{16} -2148.99i q^{17} +838.313i q^{18} +32360.9 q^{19} -35328.1 q^{20} -17763.0i q^{21} -5395.36 q^{22} +30452.1i q^{23} -5610.73i q^{24} -1363.92 q^{25} +5922.54i q^{26} -106417. q^{27} -72081.4i q^{28} -213628. i q^{29} +6083.86i q^{30} -265951. i q^{31} -34173.8i q^{32} -242605. i q^{33} -1501.75 q^{34} +156619. i q^{35} -152965. q^{36} -4954.13i q^{37} -22614.4i q^{38} -266309. q^{39} +49470.5i q^{40} -495.580 q^{41} -12413.2 q^{42} +825118. i q^{43} -984476. i q^{44} +332362. q^{45} +21280.5 q^{46} -199602. q^{47} +508945. q^{48} +503987. q^{49} +953.132i q^{50} -67527.0i q^{51} -1.08067e6 q^{52} -745306. i q^{53} +74366.0i q^{54} +2.13907e6i q^{55} -100937. q^{56} +1.01687e6 q^{57} -149288. q^{58} -1.80032e6i q^{59} -1.11011e6 q^{60} +(-622860. + 1.65976e6i) q^{61} -185852. q^{62} +678132. i q^{63} +2.04930e6 q^{64} +2.34808e6 q^{65} -169537. q^{66} +2.60727e6i q^{67} -274021. i q^{68} +956888. i q^{69} +109448. q^{70} +4.96656e6i q^{71} +214199. i q^{72} +643896. q^{73} -3462.04 q^{74} -42858.0 q^{75} +4.12639e6 q^{76} -4.36444e6 q^{77} +186102. i q^{78} -812946. i q^{79} -4.48743e6 q^{80} -720345. q^{81} +346.321i q^{82} +6.11167e6 q^{83} -2.26499e6i q^{84} +595394. i q^{85} +576609. q^{86} -6.71279e6i q^{87} -1.37858e6 q^{88} -1.20725e6i q^{89} -232261. i q^{90} +4.79089e6i q^{91} +3.88300e6i q^{92} -8.35690e6i q^{93} +139486. i q^{94} -8.96585e6 q^{95} -1.07383e6i q^{96} -3.70849e6 q^{97} -352196. i q^{98} +9.26182e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 52 q^{3} - 1974 q^{4} + 696 q^{5} + 18902 q^{9}+O(q^{10}) $ 34 * q + 52 * q^3 - 1974 * q^4 + 696 * q^5 + 18902 * q^9	$ 34 q + 52 q^{3} - 1974 q^{4} + 696 q^{5} + 18902 q^{9} - 8382 q^{12} + 19860 q^{13} + 3744 q^{14} - 24988 q^{15} + 110642 q^{16} + 48660 q^{19} - 148866 q^{20} - 147578 q^{22} + 220494 q^{25} + 371248 q^{27} + 382432 q^{34} + 241132 q^{36} - 908620 q^{39} + 569544 q^{41} + 421016 q^{42} - 306560 q^{45} - 342232 q^{46} - 1179240 q^{47} + 3464534 q^{48} - 1069538 q^{49} - 2494330 q^{52} + 3794484 q^{56} + 3106472 q^{57} - 6540834 q^{58} - 2370908 q^{60} + 2828770 q^{61} + 5869278 q^{62} + 3854362 q^{64} - 502500 q^{65} - 4330852 q^{66} - 16861390 q^{70} - 6245708 q^{73} + 7083030 q^{74} - 13741952 q^{75} - 2405382 q^{76} - 14437656 q^{77} - 5019522 q^{80} + 31255922 q^{81} + 1991136 q^{83} - 4399944 q^{86} + 25423434 q^{88} - 22196868 q^{95} - 6652408 q^{97}+O(q^{100}) $ 34 * q + 52 * q^3 - 1974 * q^4 + 696 * q^5 + 18902 * q^9 - 8382 * q^12 + 19860 * q^13 + 3744 * q^14 - 24988 * q^15 + 110642 * q^16 + 48660 * q^19 - 148866 * q^20 - 147578 * q^22 + 220494 * q^25 + 371248 * q^27 + 382432 * q^34 + 241132 * q^36 - 908620 * q^39 + 569544 * q^41 + 421016 * q^42 - 306560 * q^45 - 342232 * q^46 - 1179240 * q^47 + 3464534 * q^48 - 1069538 * q^49 - 2494330 * q^52 + 3794484 * q^56 + 3106472 * q^57 - 6540834 * q^58 - 2370908 * q^60 + 2828770 * q^61 + 5869278 * q^62 + 3854362 * q^64 - 502500 * q^65 - 4330852 * q^66 - 16861390 * q^70 - 6245708 * q^73 + 7083030 * q^74 - 13741952 * q^75 - 2405382 * q^76 - 14437656 * q^77 - 5019522 * q^80 + 31255922 * q^81 + 1991136 * q^83 - 4399944 * q^86 + 25423434 * q^88 - 22196868 * q^95 - 6652408 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/61\mathbb{Z}\right)^\times$.

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

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 $

$2$		−	0.698820i		−	0.0617675i	−0.999523	−	0.0308838i	$-0.990168\pi$
$2$		−	0.698820i		−	0.0617675i	0.999523	−	0.0308838i	$-0.00983217\pi$
$3$	31.4227			0.671923			0.335961	−	0.941876i	$-0.390939\pi$
$3$	31.4227			0.671923			0.335961	+	0.941876i	$0.390939\pi$
$4$	127.512			0.996185
$5$	−277.058			−0.991232			−0.495616	−	0.868542i	$-0.665058\pi$
$5$	−277.058			−0.991232			−0.495616	+	0.868542i	$0.665058\pi$
$6$		−	21.9588i		−	0.0415030i
$7$		−	565.293i		−	0.622917i	−0.950260	−	0.311459i	$-0.899183\pi$
$7$		−	565.293i		−	0.622917i	0.950260	−	0.311459i	$-0.100817\pi$
$8$		−	178.557i		−	0.123299i
$9$	−1199.61			−0.548520
$10$			193.614i			0.0612260i
$11$		−	7720.67i		−	1.74896i	−0.485058	−	0.874482i	$-0.661202\pi$
$11$		−	7720.67i		−	1.74896i	0.485058	−	0.874482i	$-0.338798\pi$
$12$	4006.76			0.669359
$13$	−8475.06			−1.06989			−0.534947	−	0.844885i	$-0.679669\pi$
$13$	−8475.06			−1.06989			−0.534947	+	0.844885i	$0.679669\pi$
$14$	−395.038			−0.0384760
$15$	−8705.91			−0.666032
$16$	16196.7			0.988569
$17$		−	2148.99i		−	0.106087i	−0.998592	−	0.0530436i	$-0.983108\pi$
$17$		−	2148.99i		−	0.106087i	0.998592	−	0.0530436i	$-0.0168922\pi$
$18$			838.313i			0.0338807i
$19$	32360.9			1.08239			0.541194	−	0.840898i	$-0.317972\pi$
$19$	32360.9			1.08239			0.541194	+	0.840898i	$0.317972\pi$
$20$	−35328.1			−0.987451
$21$		−	17763.0i		−	0.418552i
$22$	−5395.36			−0.108029
$23$			30452.1i			0.521879i	0.965355	+	0.260940i	$0.0840323\pi$
$23$			30452.1i			0.521879i	−0.965355	+	0.260940i	$0.915968\pi$
$24$		−	5610.73i		−	0.0828477i
$25$	−1363.92			−0.0174581
$26$			5922.54i			0.0660848i
$27$	−106417.			−1.04049
$28$		−	72081.4i		−	0.620540i
$29$		−	213628.i		−	1.62654i	−0.581883	−	0.813272i	$-0.697684\pi$
$29$		−	213628.i		−	1.62654i	0.581883	−	0.813272i	$-0.302316\pi$
$30$			6083.86i			0.0411391i
$31$		−	265951.i		−	1.60338i	−0.597742	−	0.801688i	$-0.703935\pi$
$31$		−	265951.i		−	1.60338i	0.597742	−	0.801688i	$-0.296065\pi$
$32$		−	34173.8i		−	0.184361i
$33$		−	242605.i		−	1.17517i
$34$	−1501.75			−0.00655274
$35$			156619.i			0.617456i
$36$	−152965.			−0.546427
$37$		−	4954.13i		−	0.0160791i	−0.999968	−	0.00803954i	$-0.997441\pi$
$37$		−	4954.13i		−	0.0160791i	0.999968	−	0.00803954i	$-0.00255909\pi$
$38$		−	22614.4i		−	0.0668565i
$39$	−266309.			−0.718887
$40$			49470.5i			0.122218i
$41$	−495.580			−0.00112298			−0.000561488	−	1.00000i	$-0.500179\pi$
$41$	−495.580			−0.00112298			−0.000561488		1.00000i	$0.500179\pi$
$42$	−12413.2			−0.0258529
$43$			825118.i			1.58262i	0.611416	+	0.791310i	$0.290600\pi$
$43$			825118.i			1.58262i	−0.611416	+	0.791310i	$0.709400\pi$
$44$		−	984476.i		−	1.74229i
$45$	332362.			0.543711
$46$	21280.5			0.0322352
$47$	−199602.			−0.280429			−0.140214	−	0.990121i	$-0.544779\pi$
$47$	−199602.			−0.280429			−0.140214	+	0.990121i	$0.544779\pi$
$48$	508945.			0.664242
$49$	503987.			0.611974
$50$			953.132i			0.00107835i
$51$		−	67527.0i		−	0.0712823i
$52$	−1.08067e6			−1.06581
$53$		−	745306.i		−	0.687652i	−0.939033	−	0.343826i	$-0.888277\pi$
$53$		−	745306.i		−	0.687652i	0.939033	−	0.343826i	$-0.111723\pi$
$54$			74366.0i			0.0642682i
$55$			2.13907e6i			1.73363i
$56$	−100937.			−0.0768053
$57$	1.01687e6			0.727282
$58$	−149288.			−0.100468
$59$		−	1.80032e6i		−	1.14122i	−0.821223	−	0.570608i	$-0.806708\pi$
$59$		−	1.80032e6i		−	1.14122i	0.821223	−	0.570608i	$-0.193292\pi$
$60$	−1.11011e6			−0.663491
$61$	−622860.	+	1.65976e6i	−0.351347	+	0.936245i
$61$	−622860.	+	1.65976e6i	−0.351347	+	0.936245i
$62$	−185852.			−0.0990366
$63$			678132.i			0.341682i
$64$	2.04930e6			0.977181
$65$	2.34808e6			1.06051
$66$	−169537.			−0.0725873
$67$			2.60727e6i			1.05907i	0.848289	+	0.529534i	$0.177633\pi$
$67$			2.60727e6i			1.05907i	−0.848289	+	0.529534i	$0.822367\pi$
$68$		−	274021.i		−	0.105682i
$69$			956888.i			0.350662i
$70$	109448.			0.0381387
$71$			4.96656e6i			1.64684i	0.567431	+	0.823421i	$0.307937\pi$
$71$			4.96656e6i			1.64684i	−0.567431	+	0.823421i	$0.692063\pi$
$72$			214199.i			0.0676322i
$73$	643896.			0.193725			0.0968626	−	0.995298i	$-0.469119\pi$
$73$	643896.			0.193725			0.0968626	+	0.995298i	$0.469119\pi$
$74$	−3462.04			−0.000993165
$75$	−42858.0			−0.0117305
$76$	4.12639e6			1.07826
$77$	−4.36444e6			−1.08946
$78$			186102.i			0.0444039i
$79$		−	812946.i		−	0.185510i	−0.995689	−	0.0927549i	$-0.970433\pi$
$79$		−	812946.i		−	0.185510i	0.995689	−	0.0927549i	$-0.0295673\pi$
$80$	−4.48743e6			−0.979902
$81$	−720345.			−0.150606
$82$			346.321i			6.93634e-5i
$83$	6.11167e6			1.17324			0.586620	−	0.809863i	$-0.300458\pi$
$83$	6.11167e6			1.17324			0.586620	+	0.809863i	$0.300458\pi$
$84$		−	2.26499e6i		−	0.416955i
$85$			595394.i			0.105157i
$86$	576609.			0.0977545
$87$		−	6.71279e6i		−	1.09291i
$88$	−1.37858e6			−0.215646
$89$		−	1.20725e6i		−	0.181524i	−0.995873	−	0.0907619i	$-0.971070\pi$
$89$		−	1.20725e6i		−	0.181524i	0.995873	−	0.0907619i	$-0.0289302\pi$
$90$		−	232261.i		−	0.0335837i
$91$			4.79089e6i			0.666456i
$92$			3.88300e6i			0.519888i
$93$		−	8.35690e6i		−	1.07735i
$94$			139486.i			0.0173214i
$95$	−8.96585e6			−1.07290
$96$		−	1.07383e6i		−	0.123876i
$97$	−3.70849e6			−0.412568			−0.206284	−	0.978492i	$-0.566137\pi$
$97$	−3.70849e6			−0.412568			−0.206284	+	0.978492i	$0.566137\pi$
$98$		−	352196.i		−	0.0378001i
$99$			9.26182e6i			0.959341i
$100$	−173915.			−0.0173915
$101$		−	1.12675e7i		−	1.08819i	−0.839025	−	0.544093i	$-0.816874\pi$
$101$		−	1.12675e7i		−	1.08819i	0.839025	−	0.544093i	$-0.183126\pi$
$102$	−47189.2			−0.00440293
$103$	1.71279e7			1.54445			0.772226	−	0.635348i	$-0.219143\pi$
$103$	1.71279e7			1.54445			0.772226	+	0.635348i	$0.219143\pi$
$104$			1.51328e6i			0.131917i
$105$			4.92139e6i			0.414882i
$106$	−520834.			−0.0424746
$107$	6.25787e6			0.493837			0.246918	−	0.969036i	$-0.420582\pi$
$107$	6.25787e6			0.493837			0.246918	+	0.969036i	$0.420582\pi$
$108$	−1.35694e7			−1.03652
$109$	1.32666e7			0.981219			0.490609	−	0.871380i	$-0.336774\pi$
$109$	1.32666e7			0.981219			0.490609	+	0.871380i	$0.336774\pi$
$110$	1.49483e6			0.107082
$111$		−	155672.i		−	0.0108039i
$112$		−	9.15588e6i		−	0.615796i
$113$	−2.44346e7			−1.59306			−0.796529	−	0.604601i	$-0.793333\pi$
$113$	−2.44346e7			−1.59306			−0.796529	+	0.604601i	$0.793333\pi$
$114$		−	710607.i		−	0.0449224i
$115$		−	8.43700e6i		−	0.517304i
$116$		−	2.72401e7i		−	1.62034i
$117$	1.01668e7			0.586859
$118$	−1.25810e6			−0.0704900
$119$	−1.21481e6			−0.0660835
$120$			1.55450e6i			0.0821213i
$121$	−4.01216e7			−2.05887
$122$	1.15987e6	+	435267.i	0.0578296	+	0.0217018i
$123$	−15572.5			−0.000754553
$124$		−	3.39118e7i		−	1.59726i
$125$	2.20230e7			1.00854
$126$	473892.			0.0211049
$127$	−2.87775e7			−1.24664			−0.623320	−	0.781967i	$-0.714216\pi$
$127$	−2.87775e7			−1.24664			−0.623320	+	0.781967i	$0.714216\pi$
$128$		−	5.80634e6i		−	0.244719i
$129$			2.59275e7i			1.06340i
$130$		−	1.64089e6i		−	0.0655054i
$131$	−1.77481e6			−0.0689767			−0.0344883	−	0.999405i	$-0.510980\pi$
$131$	−1.77481e6			−0.0689767			−0.0344883	+	0.999405i	$0.510980\pi$
$132$		−	3.09349e7i		−	1.17069i
$133$		−	1.82934e7i		−	0.674238i
$134$	1.82201e6			0.0654160
$135$	2.94836e7			1.03136
$136$	−383716.			−0.0130805
$137$	4.30506e7			1.43040			0.715199	−	0.698921i	$-0.246336\pi$
$137$	4.30506e7			1.43040			0.715199	+	0.698921i	$0.246336\pi$
$138$	668692.			0.0216596
$139$			4.71104e7i			1.48787i	0.668251	+	0.743936i	$0.267043\pi$
$139$			4.71104e7i			1.48787i	−0.668251	+	0.743936i	$0.732957\pi$
$140$			1.99707e7i			0.615100i
$141$	−6.27204e6			−0.188426
$142$	3.47073e6			0.101721
$143$			6.54332e7i			1.87121i
$144$	−1.94298e7			−0.542250
$145$			5.91874e7i			1.61228i
$146$		−	449967.i		−	0.0119659i
$147$	1.58366e7			0.411200
$148$		−	631709.i		−	0.0160177i
$149$	−6.01322e6			−0.148921			−0.0744604	−	0.997224i	$-0.523723\pi$
$149$	−6.01322e6			−0.148921			−0.0744604	+	0.997224i	$0.523723\pi$
$150$			29950.0i			0.000724565i
$151$		−	1.43957e7i		−	0.340261i	−0.985422	−	0.170131i	$-0.945581\pi$
$151$		−	1.43957e7i		−	0.340261i	0.985422	−	0.170131i	$-0.0544189\pi$
$152$		−	5.77825e6i		−	0.133458i
$153$			2.57795e6i			0.0581909i
$154$			3.04996e6i			0.0672932i
$155$			7.36838e7i			1.58932i
$156$	−3.39576e7			−0.716144
$157$		−	4.04987e7i		−	0.835204i	−0.908630	−	0.417602i	$-0.862871\pi$
$157$		−	4.04987e7i		−	0.835204i	0.908630	−	0.417602i	$-0.137129\pi$
$158$	−568103.			−0.0114585
$159$		−	2.34195e7i		−	0.462049i
$160$			9.46813e6i			0.182744i
$161$	1.72144e7			0.325087
$162$			503391.i			0.00930258i
$163$	9.78447e7			1.76962			0.884812	−	0.465949i	$-0.154287\pi$
$163$	9.78447e7			1.76962			0.884812	+	0.465949i	$0.154287\pi$
$164$	−63192.2			−0.00111869
$165$			6.72155e7i			1.16487i
$166$		−	4.27096e6i		−	0.0724681i
$167$	2.36568e7			0.393051			0.196526	−	0.980499i	$-0.437034\pi$
$167$	2.36568e7			0.393051			0.196526	+	0.980499i	$0.437034\pi$
$168$	−3.17171e6			−0.0516072
$169$	9.07815e6			0.144675
$170$	416073.			0.00649529
$171$	−3.88206e7			−0.593712
$172$			1.05212e8i			1.57658i
$173$		−	6.03011e7i		−	0.885450i	−0.896657	−	0.442725i	$-0.854012\pi$
$173$		−	6.03011e7i		−	0.885450i	0.896657	−	0.442725i	$-0.145988\pi$
$174$	−4.69103e6			−0.0675065
$175$			771012.i			0.0108750i
$176$		−	1.25050e8i		−	1.72897i
$177$		−	5.65709e7i		−	0.766808i
$178$	−843653.			−0.0112123
$179$	−1.35205e8			−1.76201			−0.881004	−	0.473109i	$-0.843132\pi$
$179$	−1.35205e8			−1.76201			−0.881004	+	0.473109i	$0.843132\pi$
$180$	4.23801e7			0.541636
$181$		−	9.97065e7i		−	1.24982i	−0.780695	−	0.624912i	$-0.785135\pi$
$181$		−	9.97065e7i		−	1.24982i	0.780695	−	0.624912i	$-0.214865\pi$
$182$	3.34797e6			0.0411653
$183$	−1.95720e7	+	5.21540e7i	−0.236078	+	0.629085i
$184$	5.43742e6			0.0643474
$185$			1.37258e6i			0.0159381i
$186$	−5.83997e6			−0.0665449
$187$	−1.65916e7			−0.185543
$188$	−2.54516e7			−0.279359
$189$			6.01565e7i			0.648136i
$190$			6.26551e6i			0.0662703i
$191$		−	4.19902e6i		−	0.0436045i	−0.999762	−	0.0218023i	$-0.993060\pi$
$191$		−	4.19902e6i		−	0.0436045i	0.999762	−	0.0218023i	$-0.00694043\pi$
$192$	6.43945e7			0.656590
$193$			1.26788e8i			1.26948i	0.772725	+	0.634740i	$0.218893\pi$
$193$			1.26788e8i			1.26948i	−0.772725	+	0.634740i	$0.781107\pi$
$194$			2.59156e6i			0.0254833i
$195$	7.37831e7			0.712584
$196$	6.42642e7			0.609640
$197$	−3.21296e7			−0.299415			−0.149708	−	0.988730i	$-0.547833\pi$
$197$	−3.21296e7			−0.299415			−0.149708	+	0.988730i	$0.547833\pi$
$198$	6.47234e6			0.0592561
$199$	1.11464e8			1.00265			0.501327	−	0.865258i	$-0.332846\pi$
$199$	1.11464e8			1.00265			0.501327	+	0.865258i	$0.332846\pi$
$200$			243536.i			0.00215258i
$201$			8.19274e7i			0.711611i
$202$	−7.87396e6			−0.0672145
$203$	−1.20763e8			−1.01320
$204$		−	8.61048e6i		−	0.0710104i
$205$	137304.			0.00111313
$206$		−	1.19693e7i		−	0.0953970i
$207$		−	3.65307e7i		−	0.286261i
$208$	−1.37268e8			−1.05766
$209$		−	2.49848e8i		−	1.89306i
$210$	3.43916e6			0.0256263
$211$		−	9.78324e7i		−	0.716958i	−0.933538	−	0.358479i	$-0.883295\pi$
$211$		−	9.78324e7i		−	0.716958i	0.933538	−	0.358479i	$-0.116705\pi$
$212$		−	9.50352e7i		−	0.685029i
$213$			1.56063e8i			1.10655i
$214$		−	4.37312e6i		−	0.0305031i
$215$		−	2.28605e8i		−	1.56874i
$216$			1.90014e7i			0.128291i
$217$	−1.50340e8			−0.998770
$218$		−	9.27094e6i		−	0.0606074i
$219$	2.02330e7			0.130168
$220$			2.72757e8i			1.72702i
$221$			1.82128e7i			0.113502i
$222$	−108787.			−0.000667330
$223$		−	2.31324e8i		−	1.39686i	−0.715677	−	0.698431i	$-0.753882\pi$
$223$		−	2.31324e8i		−	1.39686i	0.715677	−	0.698431i	$-0.246118\pi$
$224$	−1.93182e7			−0.114841
$225$	1.63617e6			0.00957613
$226$			1.70754e7i			0.0983992i
$227$		−	1.27375e8i		−	0.722757i	−0.932419	−	0.361379i	$-0.882306\pi$
$227$		−	1.27375e8i		−	0.722757i	0.932419	−	0.361379i	$-0.117694\pi$
$228$	1.29663e8			0.724507
$229$	1.25311e8			0.689547			0.344774	−	0.938686i	$-0.387956\pi$
$229$	1.25311e8			0.689547			0.344774	+	0.938686i	$0.387956\pi$
$230$	−5.89594e6			−0.0319526
$231$	−1.37143e8			−0.732033
$232$	−3.81448e7			−0.200552
$233$		−	1.50292e8i		−	0.778376i	−0.921158	−	0.389188i	$-0.872756\pi$
$233$		−	1.50292e8i		−	0.778376i	0.921158	−	0.389188i	$-0.127244\pi$
$234$		−	7.10475e6i		−	0.0362488i
$235$	5.53013e7			0.277970
$236$		−	2.29562e8i		−	1.13686i
$237$		−	2.55450e7i		−	0.124648i
$238$			848931.i			0.00408181i
$239$	−1.08237e8			−0.512840			−0.256420	−	0.966565i	$-0.582543\pi$
$239$	−1.08237e8			−0.512840			−0.256420	+	0.966565i	$0.582543\pi$
$240$	−1.41007e8			−0.658418
$241$	−2.57751e7			−0.118615			−0.0593077	−	0.998240i	$-0.518889\pi$
$241$	−2.57751e7			−0.118615			−0.0593077	+	0.998240i	$0.518889\pi$
$242$			2.80378e7i			0.127172i
$243$	2.10098e8			0.939290
$244$	−7.94220e7	+	2.11638e8i	−0.350007	+	0.932673i
$245$	−1.39634e8			−0.606609
$246$			10882.4i			4.66069e-5i
$247$	−2.74261e8			−1.15804
$248$	−4.74873e7			−0.197695
$249$	1.92045e8			0.788326
$250$		−	1.53901e7i		−	0.0622949i
$251$			672492.i			0.00268429i	0.999999	+	0.00134214i	$0.000427218\pi$
$251$			672492.i			0.00268429i	−0.999999	+	0.00134214i	$0.999573\pi$
$252$			8.64698e7i			0.340379i
$253$	2.35111e8			0.912748
$254$			2.01103e7i			0.0770018i
$255$			1.87089e7i			0.0706574i
$256$	2.58253e8			0.962066
$257$	−3.90193e8			−1.43388			−0.716942	−	0.697133i	$-0.754459\pi$
$257$	−3.90193e8			−1.43388			−0.716942	+	0.697133i	$0.754459\pi$
$258$	1.81186e7			0.0656835
$259$	−2.80053e6			−0.0100159
$260$	2.99408e8			1.05647
$261$			2.56271e8i			0.892192i
$262$			1.24027e6i			0.00426052i
$263$	3.35310e8			1.13658			0.568292	−	0.822827i	$-0.307604\pi$
$263$	3.35310e8			1.13658			0.568292	+	0.822827i	$0.307604\pi$
$264$	−4.33186e7			−0.144898
$265$			2.06493e8i			0.681623i
$266$	−1.27838e7			−0.0416460
$267$		−	3.79352e7i		−	0.121970i
$268$			3.32457e8i			1.05503i
$269$	2.80063e8			0.877248			0.438624	−	0.898671i	$-0.355466\pi$
$269$	2.80063e8			0.877248			0.438624	+	0.898671i	$0.355466\pi$
$270$		−	2.06037e7i		−	0.0637048i
$271$	−7.37519e7			−0.225103			−0.112551	−	0.993646i	$-0.535902\pi$
$271$	−7.37519e7			−0.225103			−0.112551	+	0.993646i	$0.535902\pi$
$272$		−	3.48065e7i		−	0.104874i
$273$			1.50543e8i			0.447807i
$274$		−	3.00846e7i		−	0.0883522i
$275$			1.05304e7i			0.0305336i
$276$			1.22014e8i			0.349325i
$277$			1.16773e8i			0.330114i	0.986284	+	0.165057i	$0.0527807\pi$
$277$			1.16773e8i			0.330114i	−0.986284	+	0.165057i	$0.947219\pi$
$278$	3.29217e7			0.0919021
$279$			3.19038e8i			0.879484i
$280$	2.79653e7			0.0761319
$281$		−	5.27489e8i		−	1.41821i	−0.705101	−	0.709107i	$-0.749098\pi$
$281$		−	5.27489e8i		−	1.41821i	0.705101	−	0.709107i	$-0.250902\pi$
$282$			4.38303e6i			0.0116386i
$283$	−1.66903e8			−0.437735			−0.218868	−	0.975755i	$-0.570236\pi$
$283$	−1.66903e8			−0.437735			−0.218868	+	0.975755i	$0.570236\pi$
$284$			6.33295e8i			1.64056i
$285$	−2.81731e8			−0.720905
$286$	4.57260e7			0.115580
$287$			280148.i			0.000699521i
$288$			4.09954e7i			0.101126i
$289$	4.05721e8			0.988746
$290$	4.13614e7			0.0995868
$291$	−1.16531e8			−0.277214
$292$	8.21043e7			0.192986
$293$	−6.29710e8			−1.46253			−0.731264	−	0.682095i	$-0.761069\pi$
$293$	−6.29710e8			−1.46253			−0.731264	+	0.682095i	$0.761069\pi$
$294$		−	1.10670e7i		−	0.0253988i
$295$			4.98793e8i			1.13121i
$296$	−884592.			−0.00198254
$297$			8.21608e8i			1.81977i
$298$			4.20216e6i			0.00919847i
$299$		−	2.58083e8i		−	0.558356i
$300$	−5.46489e6			−0.0116858
$301$	4.66433e8			0.985841
$302$	−1.00600e7			−0.0210171
$303$		−	3.54056e8i		−	0.731177i
$304$	5.24140e8			1.07002
$305$	1.72568e8	−	4.59848e8i	0.348267	−	0.928037i
$306$	1.80152e6			0.00359431
$307$		−	3.65472e8i		−	0.720892i	−0.932780	−	0.360446i	$-0.882624\pi$
$307$		−	3.65472e8i		−	0.720892i	0.932780	−	0.360446i	$-0.117376\pi$
$308$	−5.56517e8			−1.08530
$309$	5.38206e8			1.03775
$310$	5.14917e7			0.0981683
$311$		−	3.65974e8i		−	0.689905i	−0.938620	−	0.344953i	$-0.887895\pi$
$311$		−	3.65974e8i		−	0.689905i	0.938620	−	0.344953i	$-0.112105\pi$
$312$			4.75513e7i			0.0886383i
$313$		−	6.02108e8i		−	1.10986i	−0.831896	−	0.554931i	$-0.812745\pi$
$313$		−	6.02108e8i		−	1.10986i	0.831896	−	0.554931i	$-0.187255\pi$
$314$	−2.83013e7			−0.0515885
$315$		−	1.87882e8i		−	0.338687i
$316$		−	1.03660e8i		−	0.184802i
$317$	4.10304e8			0.723433			0.361717	−	0.932288i	$-0.382191\pi$
$317$	4.10304e8			0.723433			0.361717	+	0.932288i	$0.382191\pi$
$318$	−1.63660e7			−0.0285396
$319$	−1.64936e9			−2.84477
$320$	−5.67774e8			−0.968614
$321$	1.96639e8			0.331820
$322$		−	1.20297e7i		−	0.0200798i
$323$		−	6.95432e7i		−	0.114827i
$324$	−9.18524e7			−0.150032
$325$	1.15593e7			0.0186784
$326$		−	6.83758e7i		−	0.109305i
$327$	4.16872e8			0.659303
$328$			88489.1i			0.000138462i
$329$			1.12834e8i			0.174684i
$330$	4.69715e7			0.0719508
$331$			9.81861e8i			1.48817i	0.668086	+	0.744084i	$0.267114\pi$
$331$			9.81861e8i			1.48817i	−0.668086	+	0.744084i	$0.732886\pi$
$332$	7.79309e8			1.16876
$333$			5.94304e6i			0.00881969i
$334$		−	1.65319e7i		−	0.0242778i
$335$		−	7.22364e8i		−	1.04978i
$336$		−	2.87703e8i		−	0.413768i
$337$			8.69464e8i			1.23750i	0.785586	+	0.618752i	$0.212362\pi$
$337$			8.69464e8i			1.23750i	−0.785586	+	0.618752i	$0.787638\pi$
$338$		−	6.34399e6i		−	0.00893623i
$339$	−7.67803e8			−1.07041
$340$			7.59197e7i			0.104756i
$341$	−2.05332e9			−2.80425
$342$			2.71286e7i			0.0366721i
$343$		−	7.50443e8i		−	1.00413i
$344$	1.47330e8			0.195136
$345$		−	2.65113e8i		−	0.347588i
$346$	−4.21396e7			−0.0546921
$347$	1.35319e9			1.73863			0.869313	−	0.494262i	$-0.164562\pi$
$347$	1.35319e9			1.73863			0.869313	+	0.494262i	$0.164562\pi$
$348$		−	8.55958e8i		−	1.08874i
$349$			5.19575e8i			0.654273i	0.944977	+	0.327137i	$0.106084\pi$
$349$			5.19575e8i			0.654273i	−0.944977	+	0.327137i	$0.893916\pi$
$350$	538798.			0.000671720
$351$	9.01887e8			1.11321
$352$	−2.63845e8			−0.322440
$353$	1.13600e9			1.37457			0.687284	−	0.726388i	$-0.258803\pi$
$353$	1.13600e9			1.37457			0.687284	+	0.726388i	$0.258803\pi$
$354$	−3.95329e7			−0.0473639
$355$		−	1.37603e9i		−	1.63240i
$356$		−	1.53939e8i		−	0.180831i
$357$	−3.81725e7			−0.0444030
$358$			9.44841e7i			0.108835i
$359$			2.59440e8i			0.295942i	0.988992	+	0.147971i	$0.0472742\pi$
$359$			2.59440e8i			0.295942i	−0.988992	+	0.147971i	$0.952726\pi$
$360$		−	5.93455e7i		−	0.0670392i
$361$	1.53357e8			0.171565
$362$	−6.96769e7			−0.0771985
$363$	−1.26073e9			−1.38340
$364$			6.10894e8i			0.663913i
$365$	−1.78397e8			−0.192027
$366$	3.64463e7	+	1.36773e7i	0.0388570	+	0.0145820i
$367$	1.27056e8			0.134173			0.0670865	−	0.997747i	$-0.478630\pi$
$367$	1.27056e8			0.134173			0.0670865	+	0.997747i	$0.478630\pi$
$368$			4.93224e8i			0.515913i
$369$	594504.			0.000615974
$370$	959186.			0.000984457
$371$	−4.21316e8			−0.428350
$372$		−	1.06560e9i		−	1.07323i
$373$		−	6.36415e8i		−	0.634979i	−0.948262	−	0.317489i	$-0.897160\pi$
$373$		−	6.36415e8i		−	0.634979i	0.948262	−	0.317489i	$-0.102840\pi$
$374$			1.15946e7i			0.0114605i
$375$	6.92024e8			0.677659
$376$			3.56403e7i			0.0345767i
$377$			1.81051e9i			1.74023i
$378$	4.20386e7			0.0400338
$379$	−2.30930e8			−0.217893			−0.108947	−	0.994048i	$-0.534748\pi$
$379$	−2.30930e8			−0.217893			−0.108947	+	0.994048i	$0.534748\pi$
$380$	−1.14325e9			−1.06881
$381$	−9.04269e8			−0.837645
$382$	−2.93436e6			−0.00269334
$383$		−	1.65959e9i		−	1.50940i	−0.656070	−	0.754700i	$-0.727782\pi$
$383$		−	1.65959e9i		−	1.50940i	0.656070	−	0.754700i	$-0.272218\pi$
$384$		−	1.82451e8i		−	0.164432i
$385$	1.20920e9			1.07991
$386$	8.86017e7			0.0784127
$387$		−	9.89822e8i		−	0.868098i
$388$	−4.72875e8			−0.410994
$389$		−	1.23808e9i		−	1.06641i	−0.845986	−	0.533205i	$-0.820987\pi$
$389$		−	1.23808e9i		−	1.06641i	0.845986	−	0.533205i	$-0.179013\pi$
$390$		−	5.15611e7i		−	0.0440145i
$391$	6.54412e7			0.0553647
$392$		−	8.99902e7i		−	0.0754561i
$393$	−5.57694e7			−0.0463470
$394$			2.24528e7i			0.0184941i
$395$			2.25233e8i			0.183883i
$396$			1.18099e9i			0.955681i
$397$		−	7.50590e7i		−	0.0602055i	−0.999547	−	0.0301027i	$-0.990417\pi$
$397$		−	7.50590e7i		−	0.0602055i	0.999547	−	0.0301027i	$-0.00958345\pi$
$398$		−	7.78936e7i		−	0.0619314i
$399$		−	5.74828e8i		−	0.453036i
$400$	−2.20910e7			−0.0172586
$401$			1.20818e9i			0.935674i	0.883815	+	0.467837i	$0.154967\pi$
$401$			1.20818e9i			0.935674i	−0.883815	+	0.467837i	$0.845033\pi$
$402$	5.72525e7			0.0439545
$403$			2.25395e9i			1.71544i
$404$		−	1.43674e9i		−	1.08403i
$405$	1.99577e8			0.149286
$406$			8.43913e7i			0.0625830i
$407$	−3.82492e7			−0.0281217
$408$	−1.20574e7			−0.00878907
$409$		−	3.15140e8i		−	0.227757i	−0.993495	−	0.113879i	$-0.963672\pi$
$409$		−	3.15140e8i		−	0.227757i	0.993495	−	0.113879i	$-0.0363275\pi$
$410$		−	95951.0i		−	6.87553e-5i
$411$	1.35277e9			0.961117
$412$	2.18401e9			1.53856
$413$	−1.01771e9			−0.710882
$414$	−2.55284e7			−0.0176816
$415$	−1.69329e9			−1.16295
$416$			2.89625e8i			0.197247i
$417$			1.48034e9i			0.999735i
$418$	−1.74599e8			−0.116930
$419$		−	8.34272e8i		−	0.554062i	−0.960861	−	0.277031i	$-0.910650\pi$
$419$		−	8.34272e8i		−	0.554062i	0.960861	−	0.277031i	$-0.0893505\pi$
$420$			6.27534e8i			0.413300i
$421$			2.16582e9i			1.41460i	0.706913	+	0.707301i	$0.250087\pi$
$421$			2.16582e9i			1.41460i	−0.706913	+	0.707301i	$0.749913\pi$
$422$	−6.83672e7			−0.0442847
$423$	2.39445e8			0.153821
$424$	−1.33079e8			−0.0847871
$425$			2.93104e6i			0.00185208i
$426$	1.09060e8			0.0683489
$427$	9.38247e8	+	3.52098e8i	0.583203	+	0.218860i
$428$	7.97952e8			0.491953
$429$			2.05609e9i			1.25731i
$430$	−1.59754e8			−0.0968974
$431$	4.99598e8			0.300573			0.150287	−	0.988642i	$-0.451980\pi$
$431$	4.99598e8			0.300573			0.150287	+	0.988642i	$0.451980\pi$
$432$	−1.72360e9			−1.02859
$433$			1.29590e9i			0.767122i	0.923516	+	0.383561i	$0.125302\pi$
$433$			1.29590e9i			0.767122i	−0.923516	+	0.383561i	$0.874698\pi$
$434$			1.05061e8i			0.0616916i
$435$			1.85983e9i			1.08333i
$436$	1.69164e9			0.977475
$437$			9.85458e8i			0.564876i
$438$		−	1.41392e7i		−	0.00804018i
$439$	2.64727e9			1.49339			0.746693	−	0.665169i	$-0.231640\pi$
$439$	2.64727e9			1.49339			0.746693	+	0.665169i	$0.231640\pi$
$440$	3.81946e8			0.213755
$441$	−6.04590e8			−0.335680
$442$	1.27275e7			0.00701074
$443$	1.07710e9			0.588633			0.294317	−	0.955708i	$-0.404908\pi$
$443$	1.07710e9			0.588633			0.294317	+	0.955708i	$0.404908\pi$
$444$		−	1.98500e7i		−	0.0107627i
$445$			3.34479e8i			0.179932i
$446$	−1.61654e8			−0.0862807
$447$	−1.88952e8			−0.100063
$448$		−	1.15845e9i		−	0.608703i
$449$	6.23744e8			0.325195			0.162598	−	0.986692i	$-0.448013\pi$
$449$	6.23744e8			0.325195			0.162598	+	0.986692i	$0.448013\pi$
$450$		−	1.14339e6i		−	0.000591494i
$451$			3.82621e6i			0.00196404i
$452$	−3.11570e9			−1.58698
$453$		−	4.52351e8i		−	0.228629i
$454$	−8.90119e7			−0.0446429
$455$		−	1.32735e9i		−	0.660613i
$456$		−	1.81568e8i		−	0.0896734i
$457$		−	1.98555e9i		−	0.973137i	−0.873642	−	0.486569i	$-0.838248\pi$
$457$		−	1.98555e9i		−	0.973137i	0.873642	−	0.486569i	$-0.161752\pi$
$458$		−	8.75696e7i		−	0.0425916i
$459$			2.28688e8i			0.110382i
$460$		−	1.07582e9i		−	0.515330i
$461$	−1.48639e9			−0.706610			−0.353305	−	0.935508i	$-0.614942\pi$
$461$	−1.48639e9			−0.706610			−0.353305	+	0.935508i	$0.614942\pi$
$462$			9.58379e7i			0.0452158i
$463$	−4.24664e9			−1.98844			−0.994219	−	0.107372i	$-0.965756\pi$
$463$	−4.24664e9			−1.98844			−0.994219	+	0.107372i	$0.965756\pi$
$464$		−	3.46008e9i		−	1.60795i
$465$			2.31535e9i			1.06790i
$466$	−1.05027e8			−0.0480784
$467$			3.69342e9i			1.67811i	0.544049	+	0.839054i	$0.316891\pi$
$467$			3.69342e9i			1.67811i	−0.544049	+	0.839054i	$0.683109\pi$
$468$	1.29638e9			0.584620
$469$	1.47387e9			0.659711
$470$		−	3.86457e7i		−	0.0171695i
$471$		−	1.27258e9i		−	0.561193i
$472$	−3.21459e8			−0.140711
$473$	6.37047e9			2.76794
$474$	−1.78513e7			−0.00769921
$475$	−4.41376e7			−0.0188965
$476$	−1.54902e8			−0.0658313
$477$			8.94078e8i			0.377191i
$478$			7.56380e7i			0.0316769i
$479$	−3.83923e9			−1.59613			−0.798067	−	0.602569i	$-0.794144\pi$
$479$	−3.83923e9			−1.59613			−0.798067	+	0.602569i	$0.794144\pi$
$480$			2.97514e8i			0.122790i
$481$			4.19865e7i			0.0172029i
$482$			1.80122e7i			0.00732659i
$483$	5.40922e8			0.218434
$484$	−5.11598e9			−2.05102
$485$	1.02747e9			0.408951
$486$		−	1.46821e8i		−	0.0580176i
$487$	6.05435e8			0.237529			0.118764	−	0.992922i	$-0.462107\pi$
$487$	6.05435e8			0.237529			0.118764	+	0.992922i	$0.462107\pi$
$488$	2.96360e8	+	1.11216e8i	0.115438	+	0.0433209i
$489$	3.07455e9			1.18905
$490$			9.75787e7i			0.0374687i
$491$	−6.49454e8			−0.247607			−0.123804	−	0.992307i	$-0.539509\pi$
$491$	−6.49454e8			−0.247607			−0.123804	+	0.992307i	$0.539509\pi$
$492$	−1.98567e6			−0.000751674
$493$	−4.59085e8			−0.172555
$494$			1.91659e8i			0.0715294i
$495$		−	2.56606e9i		−	0.950930i
$496$		−	4.30753e9i		−	1.58505i
$497$	2.80756e9			1.02585
$498$		−	1.34205e8i		−	0.0486930i
$499$		−	4.87609e9i		−	1.75679i	−0.477937	−	0.878394i	$-0.658615\pi$
$499$		−	4.87609e9i		−	1.75679i	0.477937	−	0.878394i	$-0.341385\pi$
$500$	2.80819e9			1.00469
$501$	7.43362e8			0.264100
$502$	469951.			0.000165802
$503$	−9.27392e8			−0.324919			−0.162460	−	0.986715i	$-0.551943\pi$
$503$	−9.27392e8			−0.324919			−0.162460	+	0.986715i	$0.551943\pi$
$504$	1.21085e8			0.0421292
$505$			3.12175e9i			1.07865i
$506$		−	1.64300e8i		−	0.0563782i
$507$	2.85260e8			0.0972106
$508$	−3.66947e9			−1.24188
$509$			1.90704e9i			0.640985i	0.947251	+	0.320493i	$0.103848\pi$
$509$			1.90704e9i			0.640985i	−0.947251	+	0.320493i	$0.896152\pi$
$510$	1.30741e7			0.00436433
$511$		−	3.63990e8i		−	0.120675i
$512$		−	9.23683e8i		−	0.304143i
$513$	−3.44374e9			−1.12621
$514$			2.72675e8i			0.0885675i
$515$	−4.74543e9			−1.53091
$516$			3.30605e9i			1.05934i
$517$			1.54106e9i			0.490460i
$518$			1.95707e6i			0.000618659i
$519$		−	1.89483e9i		−	0.594954i
$520$		−	4.19266e8i		−	0.130761i
$521$			1.26121e9i			0.390710i	0.980733	+	0.195355i	$0.0625859\pi$
$521$			1.26121e9i			0.390710i	−0.980733	+	0.195355i	$0.937414\pi$
$522$	1.79087e8			0.0551085
$523$		−	3.66013e9i		−	1.11877i	−0.828909	−	0.559384i	$-0.811038\pi$
$523$		−	3.66013e9i		−	1.11877i	0.828909	−	0.559384i	$-0.188962\pi$
$524$	−2.26309e8			−0.0687135
$525$			2.42273e7i			0.00730714i
$526$		−	2.34321e8i		−	0.0702040i
$527$	−5.71525e8			−0.170098
$528$		−	3.92940e9i		−	1.16174i
$529$	2.47749e9			0.727642
$530$	1.44301e8			0.0421022
$531$			2.15969e9i			0.625979i
$532$		−	2.33262e9i		−	0.671666i
$533$	4.20007e6			0.00120147
$534$	−2.65099e7			−0.00753379
$535$	−1.73379e9			−0.489507
$536$	4.65544e8			0.130582
$537$	−4.24852e9			−1.18393
$538$		−	1.95713e8i		−	0.0541854i
$539$		−	3.89112e9i		−	1.07032i
$540$	3.75950e9			1.02743
$541$		−	1.61023e9i		−	0.437219i	−0.975812	−	0.218609i	$-0.929848\pi$
$541$		−	1.61023e9i		−	0.437219i	0.975812	−	0.218609i	$-0.0701520\pi$
$542$			5.15393e7i			0.0139040i
$543$		−	3.13305e9i		−	0.839785i
$544$	−7.34391e7			−0.0195583
$545$	−3.67561e9			−0.972616
$546$	1.05202e8			0.0276599
$547$			2.74896e9i			0.718145i	0.933310	+	0.359073i	$0.116907\pi$
$547$			2.74896e9i			0.718145i	−0.933310	+	0.359073i	$0.883093\pi$
$548$	5.48945e9			1.42494
$549$	7.47191e8	−	1.99106e9i	0.192721	−	0.513549i
$550$	7.35882e6			0.00188599
$551$		−	6.91321e9i		−	1.76055i
$552$	1.70859e8			0.0432365
$553$	−4.59552e8			−0.115557
$554$	8.16033e7			0.0203903
$555$			4.31302e7i			0.0107092i
$556$			6.00713e9i			1.48219i
$557$			5.75718e9i			1.41162i	0.708404	+	0.705808i	$0.249416\pi$
$557$			5.75718e9i			1.41162i	−0.708404	+	0.705808i	$0.750584\pi$
$558$	2.22950e8			0.0543235
$559$		−	6.99293e9i		−	1.69324i
$560$			2.53671e9i			0.610397i
$561$	−5.21354e8			−0.124670
$562$	−3.68620e8			−0.0875995
$563$	−1.75503e9			−0.414481			−0.207241	−	0.978290i	$-0.566448\pi$
$563$	−1.75503e9			−0.414481			−0.207241	+	0.978290i	$0.566448\pi$
$564$	−7.99758e8			−0.187708
$565$	6.76981e9			1.57909
$566$			1.16635e8i			0.0270378i
$567$			4.07206e8i			0.0938152i
$568$	8.86813e8			0.203055
$569$	−3.71086e9			−0.844464			−0.422232	−	0.906488i	$-0.638753\pi$
$569$	−3.71086e9			−0.844464			−0.422232	+	0.906488i	$0.638753\pi$
$570$			1.96879e8i			0.0445285i
$571$	4.77655e9			1.07371			0.536857	−	0.843673i	$-0.319612\pi$
$571$	4.77655e9			1.07371			0.536857	+	0.843673i	$0.319612\pi$
$572$			8.34349e9i			1.86407i
$573$		−	1.31945e8i		−	0.0292989i
$574$	195773.			4.32077e−5
$575$		−	4.15341e7i		−	0.00911104i
$576$	−2.45836e9			−0.536003
$577$			3.22357e7i			0.00698589i	0.999994	+	0.00349295i	$0.00111184\pi$
$577$			3.22357e7i			0.00698589i	−0.999994	+	0.00349295i	$0.998888\pi$
$578$		−	2.83526e8i		−	0.0610724i
$579$			3.98401e9i			0.852993i
$580$			7.54709e9i			1.60613i
$581$		−	3.45488e9i		−	0.730831i
$582$			8.14340e7i			0.0171228i
$583$	−5.75426e9			−1.20268
$584$		−	1.14972e8i		−	0.0238862i
$585$	−2.81679e9			−0.581713
$586$			4.40054e8i			0.0903367i
$587$			6.06320e9i			1.23728i	0.785674	+	0.618641i	$0.212317\pi$
$587$			6.06320e9i			1.23728i	−0.785674	+	0.618641i	$0.787683\pi$
$588$	2.01936e9			0.409631
$589$		−	8.60641e9i		−	1.73548i
$590$	3.48566e8			0.0698720
$591$	−1.00960e9			−0.201184
$592$		−	8.02406e7i		−	0.0158953i
$593$			9.46990e9i			1.86489i	0.361308	+	0.932447i	$0.382330\pi$
$593$			9.46990e9i			1.86489i	−0.361308	+	0.932447i	$0.617670\pi$
$594$	5.74156e8			0.112403
$595$	3.36572e8			0.0655041
$596$	−7.66756e8			−0.148353
$597$	3.50252e9			0.673705
$598$	−1.80354e8			−0.0344883
$599$			5.58050e9i			1.06091i	0.847713	+	0.530456i	$0.177979\pi$
$599$			5.58050e9i			1.06091i	−0.847713	+	0.530456i	$0.822021\pi$
$600$			7.65257e6i			0.00144637i
$601$	1.74714e9			0.328297			0.164149	−	0.986436i	$-0.447512\pi$
$601$	1.74714e9			0.328297			0.164149	+	0.986436i	$0.447512\pi$
$602$		−	3.25953e8i		−	0.0608929i
$603$		−	3.12771e9i		−	0.580919i
$604$		−	1.83561e9i		−	0.338963i
$605$	1.11160e10			2.04082
$606$	−2.47421e8			−0.0451630
$607$	8.93146e9			1.62092			0.810460	−	0.585793i	$-0.199217\pi$
$607$	8.93146e9			1.62092			0.810460	+	0.585793i	$0.199217\pi$
$608$		−	1.10590e9i		−	0.199550i
$609$	−3.79469e9			−0.680794
$610$	−3.21351e8	−	1.20594e8i	−0.0573225	−	0.0215116i
$611$	1.69164e9			0.300029
$612$			3.28719e8i			0.0579689i
$613$	−2.09431e9			−0.367223			−0.183611	−	0.982999i	$-0.558779\pi$
$613$	−2.09431e9			−0.367223			−0.183611	+	0.982999i	$0.558779\pi$
$614$	−2.55399e8			−0.0445277
$615$	4.31448e6			0.000747937
$616$			7.79300e8i			0.134330i
$617$			2.21215e9i			0.379155i	0.981866	+	0.189577i	$0.0607118\pi$
$617$			2.21215e9i			0.379155i	−0.981866	+	0.189577i	$0.939288\pi$
$618$		−	3.76109e8i		−	0.0640994i
$619$	−6.76535e8			−0.114650			−0.0573249	−	0.998356i	$-0.518257\pi$
$619$	−6.76535e8			−0.114650			−0.0573249	+	0.998356i	$0.518257\pi$
$620$			9.39554e9i			1.58326i
$621$		−	3.24061e9i		−	0.543008i
$622$	−2.55750e8			−0.0426137
$623$	−6.82452e8			−0.113074
$624$	−4.31334e9			−0.710669
$625$	−5.99510e9			−0.982237
$626$	−4.20765e8			−0.0685534
$627$		−	7.85091e9i		−	1.27199i
$628$		−	5.16406e9i		−	0.832018i
$629$	−1.06464e7			−0.00170578
$630$	−1.31296e8			−0.0209198
$631$			6.12618e9i			0.970704i	0.874319	+	0.485352i	$0.161308\pi$
$631$			6.12618e9i			0.970704i	−0.874319	+	0.485352i	$0.838692\pi$
$632$	−1.45157e8			−0.0228732
$633$		−	3.07416e9i		−	0.481741i
$634$		−	2.86729e8i		−	0.0446847i
$635$	7.97305e9			1.23571
$636$		−	2.98626e9i		−	0.460286i
$637$	−4.27132e9			−0.654748
$638$			1.15260e9i			0.175714i
$639$		−	5.95795e9i		−	0.903325i
$640$			1.60869e9i			0.242573i
$641$		−	1.12178e8i		−	0.0168230i	−0.999965	−	0.00841149i	$-0.997323\pi$
$641$		−	1.12178e8i		−	0.0168230i	0.999965	−	0.00841149i	$-0.00267749\pi$
$642$		−	1.37415e8i		−	0.0204957i
$643$			9.45067e9i			1.40192i	0.713199	+	0.700961i	$0.247245\pi$
$643$			9.45067e9i			1.40192i	−0.713199	+	0.700961i	$0.752755\pi$
$644$	2.19503e9			0.323847
$645$		−	7.18341e9i		−	1.05407i
$646$	−4.85982e7			−0.00709261
$647$			2.17313e9i			0.315443i	0.987484	+	0.157721i	$0.0504148\pi$
$647$			2.17313e9i			0.315443i	−0.987484	+	0.157721i	$0.949585\pi$
$648$			1.28622e8i			0.0185697i
$649$	−1.38997e10			−1.99594
$650$		−	8.07785e6i		−	0.00115372i
$651$	−4.72409e9			−0.671097
$652$	1.24763e10			1.76287
$653$			8.69040e9i			1.22136i	0.791878	+	0.610680i	$0.209104\pi$
$653$			8.69040e9i			1.22136i	−0.791878	+	0.610680i	$0.790896\pi$
$654$		−	2.91318e8i		−	0.0407235i
$655$	4.91725e8			0.0683719
$656$	−8.02677e6			−0.00111014
$657$	−7.72426e8			−0.106262
$658$	7.88503e7			0.0107898
$659$	−3.95611e9			−0.538480			−0.269240	−	0.963073i	$-0.586772\pi$
$659$	−3.95611e9			−0.538480			−0.269240	+	0.963073i	$0.586772\pi$
$660$			8.57076e9i			1.16042i
$661$		−	1.08178e10i		−	1.45692i	−0.685089	−	0.728459i	$-0.740237\pi$
$661$		−	1.08178e10i		−	1.45692i	0.685089	−	0.728459i	$-0.259763\pi$
$662$	6.86144e8			0.0919205
$663$			5.72296e8i			0.0762646i
$664$		−	1.09128e9i		−	0.144660i
$665$			5.06833e9i			0.668327i
$666$	4.15311e6			0.000544771
$667$	6.50544e9			0.848860
$668$	3.01652e9			0.391552
$669$		−	7.26883e9i		−	0.938583i
$670$	−5.04802e8			−0.0648424
$671$	1.28144e10	+	4.80890e9i	1.63746	+	0.614493i
$672$	−6.07031e8			−0.0771646
$673$			5.26495e9i			0.665796i	0.942963	+	0.332898i	$0.108026\pi$
$673$			5.26495e9i			0.665796i	−0.942963	+	0.332898i	$0.891974\pi$
$674$	6.07598e8			0.0764376
$675$	1.45143e8			0.0181649
$676$	1.15757e9			0.144123
$677$			8.19742e9i			1.01535i	0.861548	+	0.507676i	$0.169495\pi$
$677$			8.19742e9i			1.01535i	−0.861548	+	0.507676i	$0.830505\pi$
$678$			5.36556e8i			0.0661167i
$679$			2.09638e9i			0.256996i
$680$	1.06312e8			0.0129658
$681$		−	4.00246e9i		−	0.485637i
$682$			1.43490e9i			0.173211i
$683$	−2.95104e8			−0.0354408			−0.0177204	−	0.999843i	$-0.505641\pi$
$683$	−2.95104e8			−0.0354408			−0.0177204	+	0.999843i	$0.505641\pi$
$684$	−4.95007e9			−0.591446
$685$	−1.19275e10			−1.41786
$686$	−5.24424e8			−0.0620224
$687$	3.93760e9			0.463322
$688$			1.33642e10i			1.56453i
$689$			6.31651e9i			0.735716i
$690$	−1.85266e8			−0.0214697
$691$	−6.97755e9			−0.804507			−0.402254	−	0.915528i	$-0.631773\pi$
$691$	−6.97755e9			−0.804507			−0.402254	+	0.915528i	$0.631773\pi$
$692$		−	7.68910e9i		−	0.882072i
$693$	5.23564e9			0.597590
$694$		−	9.45637e8i		−	0.107391i
$695$		−	1.30523e10i		−	1.47483i
$696$	−1.19861e9			−0.134755
$697$			1.06500e6i			0.000119133i
$698$	3.63089e8			0.0404128
$699$		−	4.72258e9i		−	0.523009i
$700$			9.83130e7i			0.0108335i
$701$		−	5.73156e9i		−	0.628434i	−0.949351	−	0.314217i	$-0.898258\pi$
$701$		−	5.73156e9i		−	0.628434i	0.949351	−	0.314217i	$-0.101742\pi$
$702$		−	6.30256e8i		−	0.0687602i
$703$		−	1.60320e8i		−	0.0174038i
$704$		−	1.58220e10i		−	1.70905i
$705$	1.73772e9			0.186774
$706$		−	7.93859e8i		−	0.0849037i
$707$	−6.36944e9			−0.677849
$708$		−	7.21345e9i		−	0.763883i
$709$			2.21639e7i			0.00233553i	0.999999	+	0.00116776i	$0.000371711\pi$
$709$			2.21639e7i			0.00233553i	−0.999999	+	0.00116776i	$0.999628\pi$
$710$	−9.61594e8			−0.100829
$711$			9.75220e8i			0.101756i
$712$	−2.15563e8			−0.0223818
$713$	8.09876e9			0.836769
$714$			2.66757e7i			0.00274266i
$715$		−	1.81288e10i		−	1.85480i
$716$	−1.72402e10			−1.75529
$717$	−3.40109e9			−0.344589
$718$	1.81302e8			0.0182796
$719$	−8.46560e9			−0.849389			−0.424694	−	0.905337i	$-0.639618\pi$
$719$	−8.46560e9			−0.849389			−0.424694	+	0.905337i	$0.639618\pi$
$720$	5.38317e9			0.537495
$721$		−	9.68229e9i		−	0.962066i
$722$		−	1.07169e8i		−	0.0105971i
$723$	−8.09925e8			−0.0797005
$724$		−	1.27137e10i		−	1.24506i
$725$			2.91371e8i			0.0283964i
$726$			8.81024e8i			0.0854495i
$727$	−6.14232e9			−0.592873			−0.296437	−	0.955052i	$-0.595798\pi$
$727$	−6.14232e9			−0.592873			−0.296437	+	0.955052i	$0.595798\pi$
$728$	8.55445e8			0.0821736
$729$	8.17724e9			0.781737
$730$			1.24667e8i			0.0118610i
$731$	1.77317e9			0.167896
$732$	−2.49565e9	+	6.65025e9i	−0.235177	+	0.626684i
$733$	−1.62760e8			−0.0152645			−0.00763226	−	0.999971i	$-0.502429\pi$
$733$	−1.62760e8			−0.0152645			−0.00763226	+	0.999971i	$0.502429\pi$
$734$		−	8.87895e7i		−	0.00828754i
$735$	−4.38767e9			−0.407594
$736$	1.04067e9			0.0962141
$737$	2.01299e10			1.85227
$738$		−	415451.i		−	3.80472e-5i
$739$		−	1.06613e10i		−	0.971746i	−0.874029	−	0.485873i	$-0.838502\pi$
$739$		−	1.06613e10i		−	0.971746i	0.874029	−	0.485873i	$-0.161498\pi$
$740$			1.75020e8i			0.0158773i
$741$	−8.61802e9			−0.778115
$742$			2.94424e8i			0.0264581i
$743$		−	1.14686e10i		−	1.02577i	−0.858458	−	0.512884i	$-0.828577\pi$
$743$		−	1.14686e10i		−	1.02577i	0.858458	−	0.512884i	$-0.171423\pi$
$744$	−1.49218e9			−0.132836
$745$	1.66601e9			0.147615
$746$	−4.44739e8			−0.0392211
$747$	−7.33164e9			−0.643545
$748$	−2.11563e9			−0.184835
$749$		−	3.53753e9i		−	0.307619i
$750$		−	4.83600e8i		−	0.0418573i
$751$	−1.11474e10			−0.960358			−0.480179	−	0.877170i	$-0.659428\pi$
$751$	−1.11474e10			−0.960358			−0.480179	+	0.877170i	$0.659428\pi$
$752$	−3.23290e9			−0.277223
$753$			2.11315e7i			0.00180363i
$754$	1.26522e9			0.107490
$755$			3.98843e9i			0.337278i
$756$			7.67066e9i			0.645663i
$757$	9.81242e8			0.0822130			0.0411065	−	0.999155i	$-0.486912\pi$
$757$	9.81242e8			0.0822130			0.0411065	+	0.999155i	$0.486912\pi$
$758$			1.61379e8i			0.0134587i
$759$	7.38782e9			0.613296
$760$			1.60091e9i			0.132288i
$761$			6.09879e9i			0.501646i	0.968033	+	0.250823i	$0.0807013\pi$
$761$			6.09879e9i			0.501646i	−0.968033	+	0.250823i	$0.919299\pi$
$762$			6.31921e8i			0.0517393i
$763$		−	7.49949e9i		−	0.611218i
$764$		−	5.35424e8i		−	0.0434382i
$765$		−	7.14242e8i		−	0.0576807i
$766$	−1.15975e9			−0.0932319
$767$			1.52578e10i			1.22098i
$768$	8.11500e9			0.646434
$769$			4.30322e9i			0.341233i	0.985337	+	0.170617i	$0.0545760\pi$
$769$			4.30322e9i			0.341233i	−0.985337	+	0.170617i	$0.945424\pi$
$770$		−	8.45015e8i		−	0.0667032i
$771$	−1.22609e10			−0.963459
$772$			1.61669e10i			1.26464i
$773$	−9.40908e9			−0.732688			−0.366344	−	0.930480i	$-0.619391\pi$
$773$	−9.40908e9			−0.732688			−0.366344	+	0.930480i	$0.619391\pi$
$774$	−6.91707e8			−0.0536203
$775$			3.62735e8i			0.0279920i
$776$			6.62175e8i			0.0508694i
$777$	−8.80003e7			−0.00672993
$778$	−8.65193e8			−0.0658695
$779$	−1.60374e7			−0.00121550
$780$	9.40821e9			0.709865
$781$	3.83452e10			2.88027
$782$		−	4.57316e7i		−	0.00341974i
$783$			2.27336e10i			1.69240i
$784$	8.16294e9			0.604979
$785$			1.12205e10i			0.827882i
$786$			3.89727e7i			0.00286274i
$787$			2.81978e8i			0.0206207i	0.999947	+	0.0103104i	$0.00328195\pi$
$787$			2.81978e8i			0.0206207i	−0.999947	+	0.0103104i	$0.996718\pi$
$788$	−4.09690e9			−0.298273
$789$	1.05364e10			0.763697
$790$	1.57397e8			0.0113580
$791$			1.38127e10i			0.992343i
$792$	1.65376e9			0.118286
$793$	5.27878e9	−	1.40665e10i	0.375904	−	1.00168i
$794$	−5.24527e7			−0.00371874
$795$			6.48857e9i			0.457998i
$796$	1.42130e10			0.998828
$797$	−1.03990e10			−0.727591			−0.363795	−	0.931479i	$-0.618519\pi$
$797$	−1.03990e10			−0.727591			−0.363795	+	0.931479i	$0.618519\pi$
$798$	−4.01701e8			−0.0279829
$799$			4.28942e8i			0.0297499i
$800$			4.66102e7i			0.00321860i
$801$			1.44824e9i			0.0995694i
$802$	8.44297e8			0.0577943
$803$		−	4.97131e9i		−	0.338818i
$804$			1.04467e10i			0.708896i
$805$	−4.76937e9			−0.322237
$806$	1.57510e9			0.105959
$807$	8.80033e9			0.589443
$808$	−2.01189e9			−0.134173
$809$	−5.02948e9			−0.333967			−0.166983	−	0.985960i	$-0.553403\pi$
$809$	−5.02948e9			−0.333967			−0.166983	+	0.985960i	$0.553403\pi$
$810$		−	1.39469e8i		−	0.00922102i
$811$			1.71231e9i			0.112722i	0.998410	+	0.0563609i	$0.0179498\pi$
$811$			1.71231e9i			0.112722i	−0.998410	+	0.0563609i	$0.982050\pi$
$812$	−1.53986e10			−1.00934
$813$	−2.31749e9			−0.151252
$814$			2.67293e7i			0.00173701i
$815$	−2.71087e10			−1.75411
$816$		−	1.09372e9i		−	0.0704675i
$817$			2.67016e10i			1.71301i
$818$	−2.20226e8			−0.0140680
$819$		−	5.74721e9i		−	0.365564i
$820$	1.75079e7			0.00110888
$821$		−	9.87044e8i		−	0.0622494i	−0.999516	−	0.0311247i	$-0.990091\pi$
$821$		−	9.87044e8i		−	0.0622494i	0.999516	−	0.0311247i	$-0.00990891\pi$
$822$		−	9.45340e8i		−	0.0593658i
$823$		−	3.46022e9i		−	0.216373i	−0.994131	−	0.108187i	$-0.965496\pi$
$823$		−	3.46022e9i		−	0.216373i	0.994131	−	0.108187i	$-0.0345044\pi$
$824$		−	3.05830e9i		−	0.190430i
$825$			3.30892e8i			0.0205163i
$826$			7.11194e8i			0.0439094i
$827$	2.35734e10			1.44928			0.724642	−	0.689125i	$-0.242005\pi$
$827$	2.35734e10			1.44928			0.724642	+	0.689125i	$0.242005\pi$
$828$		−	4.65809e9i		−	0.285169i
$829$	1.10547e10			0.673916			0.336958	−	0.941520i	$-0.390602\pi$
$829$	1.10547e10			0.673916			0.336958	+	0.941520i	$0.390602\pi$
$830$			1.18330e9i			0.0718327i
$831$			3.66933e9i			0.221811i
$832$	−1.73679e10			−1.04548
$833$		−	1.08306e9i		−	0.0649226i
$834$	1.03449e9			0.0617511
$835$	−6.55431e9			−0.389605
$836$		−	3.18585e10i		−	1.88584i
$837$			2.83016e10i			1.66829i
$838$	−5.83005e8			−0.0342230
$839$	1.98223e10			1.15874			0.579371	−	0.815064i	$-0.303298\pi$
$839$	1.98223e10			1.15874			0.579371	+	0.815064i	$0.303298\pi$
$840$	8.78746e8			0.0511548
$841$	−2.83872e10			−1.64565
$842$	1.51351e9			0.0873765
$843$		−	1.65751e10i		−	0.952930i
$844$		−	1.24748e10i		−	0.714223i
$845$	−2.51517e9			−0.143407
$846$		−	1.67329e8i		−	0.00950112i
$847$			2.26805e10i			1.28251i
$848$		−	1.20715e10i		−	0.679792i
$849$	−5.24454e9			−0.294124
$850$	2.04827e6			0.000114399
$851$	1.50864e8			0.00839134
$852$			1.98998e10i			1.10233i
$853$	−2.44735e10			−1.35013			−0.675063	−	0.737760i	$-0.735884\pi$
$853$	−2.44735e10			−1.35013			−0.675063	+	0.737760i	$0.735884\pi$
$854$	2.46053e8	−	6.55666e8i	0.0135184	−	0.0360230i
$855$	1.07555e10			0.588506
$856$		−	1.11738e9i		−	0.0608898i
$857$	3.00387e10			1.63023			0.815114	−	0.579300i	$-0.196674\pi$
$857$	3.00387e10			1.63023			0.815114	+	0.579300i	$0.196674\pi$
$858$	1.43684e9			0.0776607
$859$	2.06518e10			1.11168			0.555842	−	0.831288i	$-0.312396\pi$
$859$	2.06518e10			1.11168			0.555842	+	0.831288i	$0.312396\pi$
$860$		−	2.91499e10i		−	1.56276i
$861$			8.80301e6i			0.000470024i
$862$		−	3.49129e8i		−	0.0185657i
$863$	1.64970e10			0.873711			0.436855	−	0.899532i	$-0.356092\pi$
$863$	1.64970e10			0.873711			0.436855	+	0.899532i	$0.356092\pi$
$864$			3.63666e9i			0.191825i
$865$			1.67069e10i			0.877687i
$866$	9.05602e8			0.0473832
$867$	1.27488e10			0.664361
$868$	−1.91701e10			−0.994960
$869$	−6.27649e9			−0.324450
$870$	1.29969e9			0.0669146
$871$		−	2.20967e10i		−	1.13309i
$872$		−	2.36883e9i		−	0.120984i
$873$	4.44875e9			0.226302
$874$	6.88658e8			0.0348910
$875$		−	1.24495e10i		−	0.628235i
$876$	2.57994e9			0.129672
$877$		−	1.90796e10i		−	0.955147i	−0.878592	−	0.477574i	$-0.841516\pi$
$877$		−	1.90796e10i		−	0.955147i	0.878592	−	0.477574i	$-0.158484\pi$
$878$		−	1.84996e9i		−	0.0922427i
$879$	−1.97872e10			−0.982706
$880$			3.46460e10i			1.71381i
$881$	4.39682e9			0.216632			0.108316	−	0.994117i	$-0.465454\pi$
$881$	4.39682e9			0.216632			0.108316	+	0.994117i	$0.465454\pi$
$882$			4.22499e8i			0.0207341i
$883$			3.10409e10i			1.51730i	0.651497	+	0.758651i	$0.274141\pi$
$883$			3.10409e10i			1.51730i	−0.651497	+	0.758651i	$0.725859\pi$
$884$			2.32234e9i			0.113069i
$885$			1.56734e10i			0.760085i
$886$		−	7.52702e8i		−	0.0363584i
$887$			2.02405e10i			0.973843i	0.873446	+	0.486922i	$0.161880\pi$
$887$			2.02405e10i			0.973843i	−0.873446	+	0.486922i	$0.838120\pi$
$888$	−2.77963e7			−0.00133211
$889$			1.62677e10i			0.776553i
$890$	2.33741e8			0.0111140
$891$			5.56155e9i			0.263405i
$892$		−	2.94965e10i		−	1.39153i
$893$	−6.45931e9			−0.303533
$894$			1.32043e8i			0.00618066i
$895$	3.74597e10			1.74656
$896$	−3.28228e9			−0.152440
$897$		−	8.10969e9i		−	0.375172i
$898$		−	4.35885e8i		−	0.0200865i
$899$	−5.68147e10			−2.60796
$900$	2.08631e8			0.00953960
$901$	−1.60165e9			−0.0729511
$902$	2.67383e6			0.000121314
$903$	1.46566e10			0.662409
$904$			4.36297e9i			0.196423i
$905$			2.76245e10i			1.23887i
$906$	−3.16112e8			−0.0141219
$907$			2.73683e10i			1.21793i	0.793197	+	0.608966i	$0.208415\pi$
$907$			2.73683e10i			1.21793i	−0.793197	+	0.608966i	$0.791585\pi$
$908$		−	1.62417e10i		−	0.720000i
$909$			1.35166e10i			0.596891i
$910$	−9.27581e8			−0.0408044
$911$	−2.37484e10			−1.04068			−0.520342	−	0.853958i	$-0.674196\pi$
$911$	−2.37484e10			−1.04068			−0.520342	+	0.853958i	$0.674196\pi$
$912$	1.64699e10			0.718968
$913$		−	4.71862e10i		−	2.05195i
$914$	−1.38754e9			−0.0601083
$915$	5.42257e9	−	1.44497e10i	0.234008	−	0.623569i
$916$	1.59786e10			0.686916
$917$			1.00329e9i			0.0429667i
$918$	1.59812e8			0.00681803
$919$	−1.96091e10			−0.833399			−0.416700	−	0.909044i	$-0.636813\pi$
$919$	−1.96091e10			−0.833399			−0.416700	+	0.909044i	$0.636813\pi$
$920$	−1.50648e9			−0.0637832
$921$		−	1.14841e10i		−	0.484384i
$922$			1.03872e9i			0.0436455i
$923$		−	4.20919e10i		−	1.76195i
$924$	−1.74873e10			−0.729240
$925$			6.75702e6i			0.000280711i
$926$			2.96764e9i			0.122821i
$927$	−2.05469e10			−0.847163
$928$	−7.30050e9			−0.299871
$929$	4.64392e10			1.90033			0.950167	−	0.311742i	$-0.100912\pi$
$929$	4.64392e10			1.90033			0.950167	+	0.311742i	$0.100912\pi$
$930$	1.61801e9			0.0659615
$931$	1.63095e10			0.662394
$932$		−	1.91640e10i		−	0.775407i
$933$		−	1.14999e10i		−	0.463563i
$934$	2.58104e9			0.103653
$935$	4.59684e9			0.183916
$936$		−	1.81535e9i		−	0.0723593i
$937$	−7.19894e9			−0.285877			−0.142939	−	0.989732i	$-0.545655\pi$
$937$	−7.19894e9			−0.285877			−0.142939	+	0.989732i	$0.545655\pi$
$938$		−	1.02997e9i		−	0.0407487i
$939$		−	1.89199e10i		−	0.745742i
$940$	7.05156e9			0.276910
$941$			5.54012e9i			0.216748i	0.994110	+	0.108374i	$0.0345644\pi$
$941$			5.54012e9i			0.216748i	−0.994110	+	0.108374i	$0.965436\pi$
$942$	−8.89305e8			−0.0346635
$943$		−	1.50915e7i		−	0.000586058i
$944$		−	2.91593e10i		−	1.12817i
$945$		−	1.66668e10i		−	0.642454i
$946$		−	4.45181e9i		−	0.170969i
$947$			4.07617e10i			1.55965i	0.625996	+	0.779826i	$0.284693\pi$
$947$			4.07617e10i			1.55965i	−0.625996	+	0.779826i	$0.715307\pi$
$948$		−	3.25728e9i		−	0.124173i
$949$	−5.45706e9			−0.207266
$950$			3.08442e7i			0.00116719i
$951$	1.28929e10			0.486091
$952$			2.16912e8i			0.00814805i
$953$		−	1.65701e10i		−	0.620154i	−0.950711	−	0.310077i	$-0.899645\pi$
$953$		−	1.65701e10i		−	0.620154i	0.950711	−	0.310077i	$-0.100355\pi$
$954$	6.24800e8			0.0232981
$955$			1.16337e9i			0.0432222i
$956$	−1.38014e10			−0.510883
$957$	−5.18272e10			−1.91146
$958$			2.68293e9i			0.0985893i
$959$		−	2.43362e10i		−	0.891020i
$960$	−1.78410e10			−0.650834
$961$	−4.32172e10			−1.57082
$962$	2.93410e7			0.00106258
$963$	−7.50702e9			−0.270879
$964$	−3.28663e9			−0.118163
$965$		−	3.51275e10i		−	1.25835i
$966$		−	3.78007e8i		−	0.0134921i
$967$	3.37519e10			1.20034			0.600172	−	0.799871i	$-0.295099\pi$
$967$	3.37519e10			1.20034			0.600172	+	0.799871i	$0.295099\pi$
$968$			7.16398e9i			0.253858i
$969$		−	2.18524e9i		−	0.0771552i
$970$		−	7.18013e8i		−	0.0252599i
$971$	1.91827e10			0.672421			0.336211	−	0.941787i	$-0.390855\pi$
$971$	1.91827e10			0.672421			0.336211	+	0.941787i	$0.390855\pi$
$972$	2.67899e10			0.935706
$973$	2.66312e10			0.926820
$974$		−	4.23090e8i		−	0.0146716i
$975$	3.63224e8			0.0125504
$976$	−1.00883e10	+	2.68826e10i	−0.347331	+	0.925543i
$977$	3.61034e9			0.123856			0.0619279	−	0.998081i	$-0.480275\pi$
$977$	3.61034e9			0.123856			0.0619279	+	0.998081i	$0.480275\pi$
$978$		−	2.14855e9i		−	0.0734447i
$979$	−9.32082e9			−0.317479
$980$	−1.78049e10			−0.604295
$981$	−1.59147e10			−0.538218
$982$			4.53851e8i			0.0152941i
$983$		−	2.30404e10i		−	0.773664i	−0.922150	−	0.386832i	$-0.873569\pi$
$983$		−	2.30404e10i		−	0.773664i	0.922150	−	0.386832i	$-0.126431\pi$
$984$			2.78057e6i			9.30359e-5i
$985$	8.90177e9			0.296790
$986$			3.20818e8i			0.0106583i
$987$			3.54554e9i			0.117374i
$988$	−3.49714e10			−1.15362
$989$	−2.51266e10			−0.825936
$990$	−1.79321e9			−0.0587366
$991$	4.15405e10			1.35586			0.677928	−	0.735128i	$-0.262878\pi$
$991$	4.15405e10			1.35586			0.677928	+	0.735128i	$0.262878\pi$
$992$	−9.08856e9			−0.295600
$993$			3.08527e10i			0.999934i
$994$		−	1.96198e9i		−	0.0633639i
$995$	−3.08821e10			−0.993862
$996$	2.44880e10			0.785319
$997$			1.18580e10i			0.378948i	0.981886	+	0.189474i	$0.0606783\pi$
$997$			1.18580e10i			0.378948i	−0.981886	+	0.189474i	$0.939322\pi$
$998$	−3.40751e9			−0.108512
$999$			5.27201e8i			0.0167301i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.8.b.a.60.17	✓	34
61.60	even	2	inner	61.8.b.a.60.18	yes	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.8.b.a.60.17	✓	34	1.1	even	1	trivial
61.8.b.a.60.18	yes	34	61.60	even	2	inner

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:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	61.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.698820i q^{2} +31.4227 q^{3} +127.512 q^{4} -277.058 q^{5} -21.9588i q^{6} -565.293i q^{7} -178.557i q^{8} -1199.61 q^{9} +O(q^{10})\)	\(q-0.698820i q^{2} +31.4227 q^{3} +127.512 q^{4} -277.058 q^{5} -21.9588i q^{6} -565.293i q^{7} -178.557i q^{8} -1199.61 q^{9} +193.614i q^{10} -7720.67i q^{11} +4006.76 q^{12} -8475.06 q^{13} -395.038 q^{14} -8705.91 q^{15} +16196.7 q^{16} -2148.99i q^{17} +838.313i q^{18} +32360.9 q^{19} -35328.1 q^{20} -17763.0i q^{21} -5395.36 q^{22} +30452.1i q^{23} -5610.73i q^{24} -1363.92 q^{25} +5922.54i q^{26} -106417. q^{27} -72081.4i q^{28} -213628. i q^{29} +6083.86i q^{30} -265951. i q^{31} -34173.8i q^{32} -242605. i q^{33} -1501.75 q^{34} +156619. i q^{35} -152965. q^{36} -4954.13i q^{37} -22614.4i q^{38} -266309. q^{39} +49470.5i q^{40} -495.580 q^{41} -12413.2 q^{42} +825118. i q^{43} -984476. i q^{44} +332362. q^{45} +21280.5 q^{46} -199602. q^{47} +508945. q^{48} +503987. q^{49} +953.132i q^{50} -67527.0i q^{51} -1.08067e6 q^{52} -745306. i q^{53} +74366.0i q^{54} +2.13907e6i q^{55} -100937. q^{56} +1.01687e6 q^{57} -149288. q^{58} -1.80032e6i q^{59} -1.11011e6 q^{60} +(-622860. + 1.65976e6i) q^{61} -185852. q^{62} +678132. i q^{63} +2.04930e6 q^{64} +2.34808e6 q^{65} -169537. q^{66} +2.60727e6i q^{67} -274021. i q^{68} +956888. i q^{69} +109448. q^{70} +4.96656e6i q^{71} +214199. i q^{72} +643896. q^{73} -3462.04 q^{74} -42858.0 q^{75} +4.12639e6 q^{76} -4.36444e6 q^{77} +186102. i q^{78} -812946. i q^{79} -4.48743e6 q^{80} -720345. q^{81} +346.321i q^{82} +6.11167e6 q^{83} -2.26499e6i q^{84} +595394. i q^{85} +576609. q^{86} -6.71279e6i q^{87} -1.37858e6 q^{88} -1.20725e6i q^{89} -232261. i q^{90} +4.79089e6i q^{91} +3.88300e6i q^{92} -8.35690e6i q^{93} +139486. i q^{94} -8.96585e6 q^{95} -1.07383e6i q^{96} -3.70849e6 q^{97} -352196. i q^{98} +9.26182e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 52 q^{3} - 1974 q^{4} + 696 q^{5} + 18902 q^{9}+O(q^{10}) \) 34 * q + 52 * q^3 - 1974 * q^4 + 696 * q^5 + 18902 * q^9	\( 34 q + 52 q^{3} - 1974 q^{4} + 696 q^{5} + 18902 q^{9} - 8382 q^{12} + 19860 q^{13} + 3744 q^{14} - 24988 q^{15} + 110642 q^{16} + 48660 q^{19} - 148866 q^{20} - 147578 q^{22} + 220494 q^{25} + 371248 q^{27} + 382432 q^{34} + 241132 q^{36} - 908620 q^{39} + 569544 q^{41} + 421016 q^{42} - 306560 q^{45} - 342232 q^{46} - 1179240 q^{47} + 3464534 q^{48} - 1069538 q^{49} - 2494330 q^{52} + 3794484 q^{56} + 3106472 q^{57} - 6540834 q^{58} - 2370908 q^{60} + 2828770 q^{61} + 5869278 q^{62} + 3854362 q^{64} - 502500 q^{65} - 4330852 q^{66} - 16861390 q^{70} - 6245708 q^{73} + 7083030 q^{74} - 13741952 q^{75} - 2405382 q^{76} - 14437656 q^{77} - 5019522 q^{80} + 31255922 q^{81} + 1991136 q^{83} - 4399944 q^{86} + 25423434 q^{88} - 22196868 q^{95} - 6652408 q^{97}+O(q^{100}) \) 34 * q + 52 * q^3 - 1974 * q^4 + 696 * q^5 + 18902 * q^9 - 8382 * q^12 + 19860 * q^13 + 3744 * q^14 - 24988 * q^15 + 110642 * q^16 + 48660 * q^19 - 148866 * q^20 - 147578 * q^22 + 220494 * q^25 + 371248 * q^27 + 382432 * q^34 + 241132 * q^36 - 908620 * q^39 + 569544 * q^41 + 421016 * q^42 - 306560 * q^45 - 342232 * q^46 - 1179240 * q^47 + 3464534 * q^48 - 1069538 * q^49 - 2494330 * q^52 + 3794484 * q^56 + 3106472 * q^57 - 6540834 * q^58 - 2370908 * q^60 + 2828770 * q^61 + 5869278 * q^62 + 3854362 * q^64 - 502500 * q^65 - 4330852 * q^66 - 16861390 * q^70 - 6245708 * q^73 + 7083030 * q^74 - 13741952 * q^75 - 2405382 * q^76 - 14437656 * q^77 - 5019522 * q^80 + 31255922 * q^81 + 1991136 * q^83 - 4399944 * q^86 + 25423434 * q^88 - 22196868 * q^95 - 6652408 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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