LMFDB - Embedded newform 171.3.p.f.46.3

Newspace parameters

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	171.p (of order $6$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$4.65941252056$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.19163381760000.1
Defining polynomial:	$ x^{8} - 14x^{6} + 177x^{4} - 266x^{2} + 361 $ x^8 - 14x^6 + 177x^4 - 266*x^2 + 361
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			46.3
Root			$1.06868 + 0.617004i$ of defining polynomial
Character	$\chi$	$=$	171.46
Dual form			171.3.p.f.145.3

$q$-expansion

$f(q)$	$=$	$q+(1.06868 + 0.617004i) q^{2} +(-1.23861 - 2.14534i) q^{4} +(-4.08850 + 7.08149i) q^{5} -10.4772 q^{7} -7.99294i q^{8} +O(q^{10})$	\(q+(1.06868 + 0.617004i) q^{2} +(-1.23861 - 2.14534i) q^{4} +(-4.08850 + 7.08149i) q^{5} -10.4772 q^{7} -7.99294i q^{8} +(-8.73861 + 5.04524i) q^{10} -3.90227 q^{11} +(-2.02277 + 1.16785i) q^{13} +(-11.1968 - 6.46449i) q^{14} +(-0.0227744 + 0.0394465i) q^{16} +(2.13736 - 3.70202i) q^{17} +(-17.4772 + 7.45296i) q^{19} +20.2563 q^{20} +(-4.17029 - 2.40772i) q^{22} +(20.0701 + 34.7623i) q^{23} +(-20.9317 - 36.2547i) q^{25} -2.88227 q^{26} +(12.9772 + 22.4772i) q^{28} +(10.6868 - 6.17004i) q^{29} +22.2148i q^{31} +(-27.7370 + 16.0140i) q^{32} +(4.56832 - 2.63752i) q^{34} +(42.8361 - 74.1944i) q^{35} -62.2749i q^{37} +(-23.2761 - 2.81867i) q^{38} +(56.6020 + 32.6792i) q^{40} +(-44.7873 - 25.8579i) q^{41} +(6.67029 - 11.5533i) q^{43} +(4.83341 + 8.37171i) q^{44} +49.5332i q^{46} +(24.1586 + 41.8438i) q^{47} +60.7723 q^{49} -51.6597i q^{50} +(5.01087 + 2.89303i) q^{52} +(-67.7396 + 39.1095i) q^{53} +(15.9545 - 27.6339i) q^{55} +83.7439i q^{56} +15.2277 q^{58} +(82.7011 + 47.7475i) q^{59} +(18.8861 + 32.7117i) q^{61} +(-13.7066 + 23.7406i) q^{62} -39.3406 q^{64} -19.0990i q^{65} +(-52.6247 + 30.3829i) q^{67} -10.5895 q^{68} +(91.5564 - 52.8601i) q^{70} +(-78.8877 - 45.5459i) q^{71} +(41.7950 - 72.3911i) q^{73} +(38.4239 - 66.5521i) q^{74} +(37.6366 + 28.2633i) q^{76} +40.8850 q^{77} +(0.466357 + 0.269251i) q^{79} +(-0.186227 - 0.322554i) q^{80} +(-31.9089 - 55.2678i) q^{82} -54.0816 q^{83} +(17.4772 + 30.2714i) q^{85} +(14.2568 - 8.23119i) q^{86} +31.1907i q^{88} +(-81.6811 + 47.1586i) q^{89} +(21.1931 - 12.2358i) q^{91} +(49.7180 - 86.1142i) q^{92} +59.6237i q^{94} +(18.6776 - 154.236i) q^{95} +(-126.909 - 73.2709i) q^{97} +(64.9462 + 37.4967i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{4} - 40 q^{7}+O(q^{10}) $ 8 * q + 12 * q^4 - 40 * q^7	$ 8 q + 12 q^{4} - 40 q^{7} - 48 q^{10} - 60 q^{13} - 44 q^{16} - 96 q^{19} + 120 q^{22} - 36 q^{25} + 60 q^{28} + 168 q^{34} + 168 q^{40} - 100 q^{43} + 48 q^{49} - 420 q^{52} + 40 q^{55} + 560 q^{58} - 68 q^{61} - 8 q^{64} - 180 q^{67} + 360 q^{70} - 60 q^{73} + 564 q^{76} + 420 q^{79} - 80 q^{82} + 96 q^{85} + 60 q^{91} - 840 q^{97}+O(q^{100}) $ 8 * q + 12 * q^4 - 40 * q^7 - 48 * q^10 - 60 * q^13 - 44 * q^16 - 96 * q^19 + 120 * q^22 - 36 * q^25 + 60 * q^28 + 168 * q^34 + 168 * q^40 - 100 * q^43 + 48 * q^49 - 420 * q^52 + 40 * q^55 + 560 * q^58 - 68 * q^61 - 8 * q^64 - 180 * q^67 + 360 * q^70 - 60 * q^73 + 564 * q^76 + 420 * q^79 - 80 * q^82 + 96 * q^85 + 60 * q^91 - 840 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/171\mathbb{Z}\right)^\times$.

$n$	$20$	$154$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	1.06868	+	0.617004i	0.534341	+	0.308502i	0.742782	−	0.669533i	$-0.233506\pi$
$2$	1.06868	+	0.617004i	0.534341	+	0.308502i	−0.208441	+	0.978035i	$0.566839\pi$
$3$		0			0
$3$		0			0
$4$	−1.23861	−	2.14534i	−0.309653	−	0.536335i
$5$	−4.08850	+	7.08149i	−0.817700	+	1.41630i	0.0896726	+	0.995971i	$0.471418\pi$
$5$	−4.08850	+	7.08149i	−0.817700	+	1.41630i	−0.907373	+	0.420327i	$0.861915\pi$
$6$		0			0
$7$	−10.4772			−1.49675			−0.748373	−	0.663278i	$-0.769165\pi$
$7$	−10.4772			−1.49675			−0.748373	+	0.663278i	$0.769165\pi$
$8$		−	7.99294i		−	0.999118i
$9$		0			0
$10$	−8.73861	+	5.04524i	−0.873861	+	0.504524i
$11$	−3.90227			−0.354752			−0.177376	−	0.984143i	$-0.556761\pi$
$11$	−3.90227			−0.354752			−0.177376	+	0.984143i	$0.556761\pi$
$12$		0			0
$13$	−2.02277	+	1.16785i	−0.155598	+	0.0898346i	−0.575777	−	0.817607i	$-0.695300\pi$
$13$	−2.02277	+	1.16785i	−0.155598	+	0.0898346i	0.420179	+	0.907441i	$0.361967\pi$
$14$	−11.1968	−	6.46449i	−0.799773	−	0.461749i
$15$		0			0
$16$	−0.0227744	+	0.0394465i	−0.00142340	+	0.00246540i
$17$	2.13736	−	3.70202i	0.125727	−	0.217766i	−0.796290	−	0.604915i	$-0.793207\pi$
$17$	2.13736	−	3.70202i	0.125727	−	0.217766i	0.922017	+	0.387149i	$0.126540\pi$
$18$		0			0
$19$	−17.4772	+	7.45296i	−0.919854	+	0.392261i
$19$	−17.4772	+	7.45296i	−0.919854	+	0.392261i
$20$	20.2563			1.01281
$21$		0			0
$22$	−4.17029	−	2.40772i	−0.189559	−	0.109442i
$23$	20.0701	+	34.7623i	0.872611	+	1.51141i	0.859286	+	0.511495i	$0.170908\pi$
$23$	20.0701	+	34.7623i	0.872611	+	1.51141i	0.0133248	+	0.999911i	$0.495758\pi$
$24$		0			0
$25$	−20.9317	−	36.2547i	−0.837267	−	1.45019i
$26$	−2.88227			−0.110857
$27$		0			0
$28$	12.9772	+	22.4772i	0.463472	+	0.802758i
$29$	10.6868	−	6.17004i	0.368511	−	0.212760i	−0.304297	−	0.952577i	$-0.598421\pi$
$29$	10.6868	−	6.17004i	0.368511	−	0.212760i	0.672808	+	0.739817i	$0.265088\pi$
$30$		0			0
$31$			22.2148i			0.716608i	0.933605	+	0.358304i	$0.116645\pi$
$31$			22.2148i			0.716608i	−0.933605	+	0.358304i	$0.883355\pi$
$32$	−27.7370	+	16.0140i	−0.866783	+	0.500437i
$33$		0			0
$34$	4.56832	−	2.63752i	0.134362	−	0.0775742i
$35$	42.8361	−	74.1944i	1.22389	−	2.11984i
$36$		0			0
$37$		−	62.2749i		−	1.68311i	−0.540174	−	0.841553i	$-0.681642\pi$
$37$		−	62.2749i		−	1.68311i	0.540174	−	0.841553i	$-0.318358\pi$
$38$	−23.2761	−	2.81867i	−0.612529	−	0.0741756i
$39$		0			0
$40$	56.6020	+	32.6792i	1.41505	+	0.816979i
$41$	−44.7873	−	25.8579i	−1.09237	−	0.630682i	−0.158166	−	0.987413i	$-0.550558\pi$
$41$	−44.7873	−	25.8579i	−1.09237	−	0.630682i	−0.934207	+	0.356731i	$0.883891\pi$
$42$		0			0
$43$	6.67029	−	11.5533i	0.155123	−	0.268681i	−0.777981	−	0.628288i	$-0.783756\pi$
$43$	6.67029	−	11.5533i	0.155123	−	0.268681i	0.933104	+	0.359607i	$0.117089\pi$
$44$	4.83341	+	8.37171i	0.109850	+	0.190266i
$45$		0			0
$46$			49.5332i			1.07681i
$47$	24.1586	+	41.8438i	0.514012	+	0.890294i	0.999868	+	0.0162556i	$0.00517456\pi$
$47$	24.1586	+	41.8438i	0.514012	+	0.890294i	−0.485856	+	0.874039i	$0.661492\pi$
$48$		0			0
$49$	60.7723			1.24025
$50$		−	51.6597i		−	1.03319i
$51$		0			0
$52$	5.01087	+	2.89303i	0.0963629	+	0.0556351i
$53$	−67.7396	+	39.1095i	−1.27811	+	0.737915i	−0.976500	−	0.215518i	$-0.930856\pi$
$53$	−67.7396	+	39.1095i	−1.27811	+	0.737915i	−0.301606	+	0.953433i	$0.597523\pi$
$54$		0			0
$55$	15.9545	−	27.6339i	0.290081	−	0.502435i
$56$			83.7439i			1.49543i
$57$		0			0
$58$	15.2277			0.262547
$59$	82.7011	+	47.7475i	1.40171	+	0.809280i	0.994569	−	0.104083i	$-0.0331908\pi$
$59$	82.7011	+	47.7475i	1.40171	+	0.809280i	0.407146	+	0.913363i	$0.366524\pi$
$60$		0			0
$61$	18.8861	+	32.7117i	0.309609	+	0.536258i	0.978277	−	0.207303i	$-0.0664686\pi$
$61$	18.8861	+	32.7117i	0.309609	+	0.536258i	−0.668668	+	0.743561i	$0.733135\pi$
$62$	−13.7066	+	23.7406i	−0.221075	+	0.382913i
$63$		0			0
$64$	−39.3406			−0.614697
$65$		−	19.0990i		−	0.293831i
$66$		0			0
$67$	−52.6247	+	30.3829i	−0.785444	+	0.453476i	−0.838356	−	0.545123i	$-0.816483\pi$
$67$	−52.6247	+	30.3829i	−0.785444	+	0.453476i	0.0529122	+	0.998599i	$0.483150\pi$
$68$	−10.5895			−0.155727
$69$		0			0
$70$	91.5564	−	52.8601i	1.30795	−	0.755145i
$71$	−78.8877	−	45.5459i	−1.11110	−	0.641491i	−0.171982	−	0.985100i	$-0.555017\pi$
$71$	−78.8877	−	45.5459i	−1.11110	−	0.641491i	−0.939113	+	0.343609i	$0.888350\pi$
$72$		0			0
$73$	41.7950	−	72.3911i	0.572535	−	0.991659i	−0.423770	−	0.905770i	$-0.639294\pi$
$73$	41.7950	−	72.3911i	0.572535	−	0.991659i	0.996305	−	0.0858893i	$-0.0273731\pi$
$74$	38.4239	−	66.5521i	0.519241	−	0.899353i
$75$		0			0
$76$	37.6366	+	28.2633i	0.495219	+	0.371885i
$77$	40.8850			0.530974
$78$		0			0
$79$	0.466357	+	0.269251i	0.00590325	+	0.00340825i	0.502949	−	0.864316i	$-0.332248\pi$
$79$	0.466357	+	0.269251i	0.00590325	+	0.00340825i	−0.497045	+	0.867724i	$0.665582\pi$
$80$	−0.186227	−	0.322554i	−0.00232783	−	0.00403192i
$81$		0			0
$82$	−31.9089	−	55.2678i	−0.389133	−	0.673998i
$83$	−54.0816			−0.651586			−0.325793	−	0.945441i	$-0.605631\pi$
$83$	−54.0816			−0.651586			−0.325793	+	0.945441i	$0.605631\pi$
$84$		0			0
$85$	17.4772	+	30.2714i	0.205614	+	0.356135i
$86$	14.2568	−	8.23119i	0.165777	−	0.0957115i
$87$		0			0
$88$			31.1907i			0.354439i
$89$	−81.6811	+	47.1586i	−0.917766	+	0.529872i	−0.882922	−	0.469521i	$-0.844427\pi$
$89$	−81.6811	+	47.1586i	−0.917766	+	0.529872i	−0.0348441	+	0.999393i	$0.511093\pi$
$90$		0			0
$91$	21.1931	−	12.2358i	0.232891	−	0.134460i
$92$	49.7180	−	86.1142i	0.540414	−	0.936024i
$93$		0			0
$94$			59.6237i			0.634294i
$95$	18.6776	−	154.236i	0.196606	−	1.62354i
$96$		0			0
$97$	−126.909	−	73.2709i	−1.30834	−	0.755370i	−0.326520	−	0.945190i	$-0.605876\pi$
$97$	−126.909	−	73.2709i	−1.30834	−	0.755370i	−0.981819	+	0.189820i	$0.939209\pi$
$98$	64.9462	+	37.4967i	0.662716	+	0.382619i
$99$		0			0
$100$	−51.8525	+	89.8111i	−0.518525	+	0.898111i
$101$	59.3764	+	102.843i	0.587885	+	1.01825i	0.994509	+	0.104651i	$0.0333725\pi$
$101$	59.3764	+	102.843i	0.587885	+	1.01825i	−0.406624	+	0.913596i	$0.633294\pi$
$102$		0			0
$103$			26.1248i			0.253639i	0.991926	+	0.126819i	$0.0404769\pi$
$103$			26.1248i			0.253639i	−0.991926	+	0.126819i	$0.959523\pi$
$104$	9.33456	+	16.1679i	0.0897553	+	0.155461i
$105$		0			0
$106$	−96.5228			−0.910592
$107$			120.708i			1.12811i	0.825737	+	0.564056i	$0.190760\pi$
$107$			120.708i			1.12811i	−0.825737	+	0.564056i	$0.809240\pi$
$108$		0			0
$109$	−91.9545	−	53.0899i	−0.843619	−	0.487064i	0.0148739	−	0.999889i	$-0.495265\pi$
$109$	−91.9545	−	53.0899i	−0.843619	−	0.487064i	−0.858493	+	0.512826i	$0.828599\pi$
$110$	34.1005	−	19.6879i	0.310004	−	0.178981i
$111$		0			0
$112$	0.238613	−	0.413289i	0.00213047	−	0.00369008i
$113$		−	101.609i		−	0.899194i	−0.893232	−	0.449597i	$-0.851568\pi$
$113$		−	101.609i		−	0.899194i	0.893232	−	0.449597i	$-0.148432\pi$
$114$		0			0
$115$	−328.226			−2.85414
$116$	−26.4737	−	15.2846i	−0.228221	−	0.131764i
$117$		0			0
$118$	58.9208	+	102.054i	0.499329	+	0.864863i
$119$	−22.3936	+	38.7869i	−0.188182	+	0.325941i
$120$		0			0
$121$	−105.772			−0.874151
$122$			46.6112i			0.382059i
$123$		0			0
$124$	47.6584	−	27.5156i	0.384342	−	0.221900i
$125$	137.892			1.10313
$126$		0			0
$127$	95.7267	−	55.2678i	0.753754	−	0.435180i	−0.0732950	−	0.997310i	$-0.523351\pi$
$127$	95.7267	−	55.2678i	0.753754	−	0.435180i	0.827049	+	0.562130i	$0.190018\pi$
$128$	68.9056	+	39.7827i	0.538325	+	0.310802i
$129$		0			0
$130$	11.7842	−	20.4108i	0.0906474	−	0.157006i
$131$	−28.1582	+	48.7714i	−0.214948	+	0.372301i	−0.953256	−	0.302162i	$-0.902292\pi$
$131$	−28.1582	+	48.7714i	−0.214948	+	0.372301i	0.738309	+	0.674463i	$0.235625\pi$
$132$		0			0
$133$	183.113	−	78.0863i	1.37679	−	0.587115i
$134$	−74.9855			−0.559593
$135$		0			0
$136$	−29.5901	−	17.0838i	−0.217574	−	0.125616i
$137$	−69.9658	−	121.184i	−0.510700	−	0.884558i	−0.999923	−	0.0123992i	$-0.996053\pi$
$137$	−69.9658	−	121.184i	−0.510700	−	0.884558i	0.489224	−	0.872158i	$-0.337280\pi$
$138$		0			0
$139$	85.1475	+	147.480i	0.612572	+	1.06101i	0.990805	+	0.135296i	$0.0431984\pi$
$139$	85.1475	+	147.480i	0.612572	+	1.06101i	−0.378233	+	0.925710i	$0.623468\pi$
$140$	−212.230			−1.51593
$141$		0			0
$142$	−56.2039	−	97.3481i	−0.395802	−	0.685550i
$143$	7.89342	−	4.55727i	0.0551987	−	0.0318690i
$144$		0			0
$145$			100.905i			0.695895i
$146$	89.3312	−	51.5754i	0.611857	−	0.353256i
$147$		0			0
$148$	−133.601	+	77.1345i	−0.902709	+	0.521179i
$149$	−59.4652	+	102.997i	−0.399096	+	0.691254i	−0.993615	−	0.112828i	$-0.964009\pi$
$149$	−59.4652	+	102.997i	−0.399096	+	0.691254i	0.594519	+	0.804082i	$0.297343\pi$
$150$		0			0
$151$		−	150.898i		−	0.999322i	−0.866221	−	0.499661i	$-0.833458\pi$
$151$		−	150.898i		−	0.999322i	0.866221	−	0.499661i	$-0.166542\pi$
$152$	59.5711	+	139.694i	0.391915	+	0.919043i
$153$		0			0
$154$	43.6931	+	25.2262i	0.283721	+	0.163807i
$155$	−157.314	−	90.8254i	−1.01493	−	0.585970i
$156$		0			0
$157$	5.11387	−	8.85749i	0.0325724	−	0.0564171i	−0.849280	−	0.527943i	$-0.822963\pi$
$157$	5.11387	−	8.85749i	0.0325724	−	0.0564171i	0.881852	+	0.471526i	$0.156297\pi$
$158$	0.332258	+	0.575488i	0.00210290	+	0.00364233i
$159$		0			0
$160$		−	261.893i		−	1.63683i
$161$	−210.278	−	364.213i	−1.30608	−	2.26219i
$162$		0			0
$163$	−42.8851			−0.263099			−0.131549	−	0.991310i	$-0.541995\pi$
$163$	−42.8851			−0.263099			−0.131549	+	0.991310i	$0.541995\pi$
$164$			128.112i			0.781170i
$165$		0			0
$166$	−57.7961	−	33.3686i	−0.348169	−	0.201016i
$167$	−1.48133	+	0.855246i	−0.00887024	+	0.00512123i	−0.504429	−	0.863453i	$-0.668297\pi$
$167$	−1.48133	+	0.855246i	−0.00887024	+	0.00512123i	0.495558	+	0.868575i	$0.334963\pi$
$168$		0			0
$169$	−81.7723	+	141.634i	−0.483860	+	0.838069i
$170$			43.1341i			0.253730i
$171$		0			0
$172$	−33.0476			−0.192137
$173$	176.089	+	101.665i	1.01786	+	0.587660i	0.913482	−	0.406879i	$-0.133383\pi$
$173$	176.089	+	101.665i	1.01786	+	0.587660i	0.104374	+	0.994538i	$0.466716\pi$
$174$		0			0
$175$	219.306	+	379.849i	1.25318	+	2.17057i
$176$	0.0888721	−	0.153931i	0.000504955	−	0.000874607i
$177$		0			0
$178$	−116.388			−0.653866
$179$			34.9725i			0.195377i	0.995217	+	0.0976885i	$0.0311449\pi$
$179$			34.9725i			0.195377i	−0.995217	+	0.0976885i	$0.968855\pi$
$180$		0			0
$181$	100.295	−	57.9054i	0.554116	−	0.319919i	−0.196664	−	0.980471i	$-0.563011\pi$
$181$	100.295	−	57.9054i	0.554116	−	0.319919i	0.750781	+	0.660552i	$0.229678\pi$
$182$	30.1982			0.165924
$183$		0			0
$184$	277.854	−	160.419i	1.51007	−	0.871841i
$185$	440.999	+	254.611i	2.38378	+	1.37628i
$186$		0			0
$187$	−8.34058	+	14.4463i	−0.0446020	+	0.0772530i
$188$	59.8462	−	103.657i	0.318331	−	0.551365i
$189$		0			0
$190$	115.125	−	153.305i	0.605920	−	0.806870i
$191$	−175.247			−0.917523			−0.458761	−	0.888559i	$-0.651707\pi$
$191$	−175.247			−0.917523			−0.458761	+	0.888559i	$0.651707\pi$
$192$		0			0
$193$	290.884	+	167.942i	1.50717	+	0.870166i	0.999965	+	0.00834091i	$0.00265502\pi$
$193$	290.884	+	167.942i	1.50717	+	0.870166i	0.507206	+	0.861825i	$0.330678\pi$
$194$	−90.4168	−	156.607i	−0.466066	−	0.807250i
$195$		0			0
$196$	−75.2733	−	130.377i	−0.384047	−	0.665190i
$197$	243.998			1.23857			0.619284	−	0.785167i	$-0.287423\pi$
$197$	243.998			1.23857			0.619284	+	0.785167i	$0.287423\pi$
$198$		0			0
$199$	169.738	+	293.994i	0.852953	+	1.47736i	0.878531	+	0.477685i	$0.158524\pi$
$199$	169.738	+	293.994i	0.852953	+	1.47736i	−0.0255788	+	0.999673i	$0.508143\pi$
$200$	−289.782	+	167.306i	−1.44891	+	0.836529i
$201$		0			0
$202$			146.542i			0.725454i
$203$	−111.968	+	64.6449i	−0.551568	+	0.318448i
$204$		0			0
$205$	366.226	−	211.440i	1.78647	−	1.03142i
$206$	−16.1191	+	27.9191i	−0.0782481	+	0.135530i
$207$		0			0
$208$		−	0.106388i		−	0.000511483i
$209$	68.2009	−	29.0835i	0.326320	−	0.139155i
$210$		0			0
$211$	−85.0782	−	49.1199i	−0.403214	−	0.232796i	0.284656	−	0.958630i	$-0.408121\pi$
$211$	−85.0782	−	49.1199i	−0.403214	−	0.232796i	−0.687870	+	0.725834i	$0.741454\pi$
$212$	167.806	+	96.8830i	0.791539	+	0.456995i
$213$		0			0
$214$	−74.4772	+	128.998i	−0.348024	+	0.602796i
$215$	54.5430	+	94.4712i	0.253688	+	0.439401i
$216$		0			0
$217$		−	232.750i		−	1.07258i
$218$	−65.5134	−	113.472i	−0.300520	−	0.520516i
$219$		0			0
$220$	−79.0455			−0.359298
$221$			9.98447i			0.0451786i
$222$		0			0
$223$	57.3535	+	33.1131i	0.257191	+	0.148489i	0.623052	−	0.782180i	$-0.285892\pi$
$223$	57.3535	+	33.1131i	0.257191	+	0.148489i	−0.365862	+	0.930669i	$0.619226\pi$
$224$	290.607	−	167.782i	1.29735	−	0.749028i
$225$		0			0
$226$	62.6931	−	108.588i	0.277403	−	0.480476i
$227$		−	375.221i		−	1.65296i	−0.562968	−	0.826479i	$-0.690341\pi$
$227$		−	375.221i		−	1.65296i	0.562968	−	0.826479i	$-0.309659\pi$
$228$		0			0
$229$	40.4534			0.176652			0.0883262	−	0.996092i	$-0.471848\pi$
$229$	40.4534			0.176652			0.0883262	+	0.996092i	$0.471848\pi$
$230$	−350.769	−	202.516i	−1.52508	−	0.880506i
$231$		0			0
$232$	−49.3168	−	85.4191i	−0.212572	−	0.368186i
$233$	134.459	−	232.890i	0.577078	−	0.999529i	−0.418734	−	0.908109i	$-0.637526\pi$
$233$	134.459	−	232.890i	0.577078	−	0.999529i	0.995812	−	0.0914198i	$-0.0291405\pi$
$234$		0			0
$235$	−395.089			−1.68123
$236$		−	236.563i		−	1.00238i
$237$		0			0
$238$	−47.8634	+	27.6339i	−0.201107	+	0.116109i
$239$	58.1786			0.243425			0.121713	−	0.992565i	$-0.461161\pi$
$239$	58.1786			0.243425			0.121713	+	0.992565i	$0.461161\pi$
$240$		0			0
$241$	125.159	−	72.2608i	0.519334	−	0.299837i	−0.217328	−	0.976099i	$-0.569734\pi$
$241$	125.159	−	72.2608i	0.519334	−	0.299837i	0.736662	+	0.676261i	$0.236401\pi$
$242$	−113.037	−	65.2619i	−0.467095	−	0.269677i
$243$		0			0
$244$	46.7852	−	81.0343i	0.191743	−	0.332108i
$245$	−248.467	+	430.358i	−1.01415	+	1.75656i
$246$		0			0
$247$	26.6486	−	35.4864i	0.107889	−	0.143670i
$248$	177.562			0.715976
$249$		0			0
$250$	147.362	+	85.0797i	0.589449	+	0.340319i
$251$	46.8273	+	81.1072i	0.186563	+	0.323136i	0.944102	−	0.329653i	$-0.106932\pi$
$251$	46.8273	+	81.1072i	0.186563	+	0.323136i	−0.757539	+	0.652790i	$0.773599\pi$
$252$		0			0
$253$	−78.3188	−	135.652i	−0.309561	−	0.536175i
$254$	136.402			0.537015
$255$		0			0
$256$	127.773	+	221.310i	0.499114	+	0.864492i
$257$	−358.565	+	207.018i	−1.39519	+	0.805516i	−0.993884	−	0.110427i	$-0.964778\pi$
$257$	−358.565	+	207.018i	−1.39519	+	0.805516i	−0.401310	+	0.915942i	$0.631445\pi$
$258$		0			0
$259$			652.469i			2.51918i
$260$	−40.9739	+	23.6563i	−0.157592	+	0.0909857i
$261$		0			0
$262$	−60.1843	+	34.7474i	−0.229711	+	0.132624i
$263$	−85.7696	+	148.557i	−0.326120	+	0.564857i	−0.981738	−	0.190236i	$-0.939075\pi$
$263$	−85.7696	+	148.557i	−0.326120	+	0.564857i	0.655618	+	0.755093i	$0.272408\pi$
$264$		0			0
$265$		−	639.597i		−	2.41357i
$266$	243.869	+	29.5319i	0.916800	+	0.111022i
$267$		0			0
$268$	130.363	+	75.2653i	0.486430	+	0.280841i
$269$	−5.85341	−	3.37947i	−0.0217599	−	0.0125631i	0.489081	−	0.872239i	$-0.337332\pi$
$269$	−5.85341	−	3.37947i	−0.0217599	−	0.0125631i	−0.510840	+	0.859676i	$0.670666\pi$
$270$		0			0
$271$	−123.499	+	213.906i	−0.455716	+	0.789323i	−0.998729	−	0.0504013i	$-0.983950\pi$
$271$	−123.499	+	213.906i	−0.455716	+	0.789323i	0.543013	+	0.839724i	$0.317283\pi$
$272$	0.0973545	+	0.168623i	0.000357921	+	0.000619937i
$273$		0			0
$274$		−	172.677i		−	0.630207i
$275$	81.6811	+	141.476i	0.297022	+	0.514458i
$276$		0			0
$277$	67.6356			0.244172			0.122086	−	0.992520i	$-0.461042\pi$
$277$	67.6356			0.244172			0.122086	+	0.992520i	$0.461042\pi$
$278$			210.145i			0.755918i
$279$		0			0
$280$	−593.032	−	342.387i	−2.11797	−	1.22281i
$281$	−22.1270	+	12.7750i	−0.0787439	+	0.0454628i	−0.538855	−	0.842399i	$-0.681143\pi$
$281$	−22.1270	+	12.7750i	−0.0787439	+	0.0454628i	0.460111	+	0.887861i	$0.347810\pi$
$282$		0			0
$283$	0.477226	−	0.826579i	0.00168631	−	0.00292077i	−0.865181	−	0.501460i	$-0.832797\pi$
$283$	0.477226	−	0.826579i	0.00168631	−	0.00292077i	0.866867	+	0.498539i	$0.166130\pi$
$284$			225.655i			0.794559i
$285$		0			0
$286$	11.2474			0.0393266
$287$	469.246	+	270.920i	1.63501	+	0.943971i
$288$		0			0
$289$	135.363	+	234.456i	0.468385	+	0.811267i
$290$	−62.2586	+	107.835i	−0.214685	+	0.371845i
$291$		0			0
$292$	−207.071			−0.709149
$293$		−	297.002i		−	1.01366i	−0.862046	−	0.506830i	$-0.830817\pi$
$293$		−	297.002i		−	1.01366i	0.862046	−	0.506830i	$-0.169183\pi$
$294$		0			0
$295$	−676.247	+	390.432i	−2.29236	+	1.32350i
$296$	−497.760			−1.68162
$297$		0			0
$298$	−127.099	+	73.3806i	−0.426506	+	0.246243i
$299$	−81.1944	−	46.8776i	−0.271553	−	0.156781i
$300$		0			0
$301$	−69.8861	+	121.046i	−0.232180	+	0.402147i
$302$	93.1044	−	161.262i	0.308293	−	0.533979i
$303$		0			0
$304$	0.104041	−	0.859152i	0.000342240	−	0.00282616i
$305$	−308.864			−1.01267
$306$		0			0
$307$	47.8634	+	27.6339i	0.155907	+	0.0900128i	0.575924	−	0.817503i	$-0.304643\pi$
$307$	47.8634	+	27.6339i	0.155907	+	0.0900128i	−0.420017	+	0.907516i	$0.637976\pi$
$308$	−50.6407	−	87.7123i	−0.164418	−	0.284780i
$309$		0			0
$310$	−112.079	−	194.127i	−0.361546	−	0.626216i
$311$	−110.770			−0.356175			−0.178088	−	0.984015i	$-0.556991\pi$
$311$	−110.770			−0.356175			−0.178088	+	0.984015i	$0.556991\pi$
$312$		0			0
$313$	−105.727	−	183.124i	−0.337785	−	0.585061i	0.646231	−	0.763142i	$-0.276344\pi$
$313$	−105.727	−	183.124i	−0.337785	−	0.585061i	−0.984016	+	0.178081i	$0.943011\pi$
$314$	10.9302	−	6.31056i	0.0348096	−	0.0200973i
$315$		0			0
$316$		−	1.33399i		−	0.00422150i
$317$	143.470	−	82.8324i	0.452587	−	0.261301i	−0.256335	−	0.966588i	$-0.582515\pi$
$317$	143.470	−	82.8324i	0.452587	−	0.261301i	0.708922	+	0.705287i	$0.249182\pi$
$318$		0			0
$319$	−41.7029	+	24.0772i	−0.130730	+	0.0754770i
$320$	160.844	−	278.590i	0.502637	−	0.870594i
$321$		0			0
$322$		−	518.970i		−	1.61171i
$323$	−9.76417	+	80.6308i	−0.0302296	+	0.249631i
$324$		0			0
$325$	84.6801	+	48.8901i	0.260554	+	0.150431i
$326$	−45.8305	−	26.4603i	−0.140584	−	0.0811664i
$327$		0			0
$328$	−206.681	+	357.982i	−0.630125	+	1.09141i
$329$	−253.115	−	438.407i	−0.769345	−	1.33255i
$330$		0			0
$331$			358.233i			1.08227i	0.840934	+	0.541137i	$0.182006\pi$
$331$			358.233i			1.08227i	−0.840934	+	0.541137i	$0.817994\pi$
$332$	66.9862	+	116.024i	0.201766	+	0.349468i
$333$		0			0
$334$	−2.11076			−0.00631964
$335$		−	496.882i		−	1.48323i
$336$		0			0
$337$	−550.179	−	317.646i	−1.63258	−	0.942570i	−0.983294	−	0.182025i	$-0.941735\pi$
$337$	−550.179	−	317.646i	−1.63258	−	0.942570i	−0.649285	−	0.760545i	$-0.724932\pi$
$338$	−174.777	+	100.908i	−0.517092	+	0.298543i
$339$		0			0
$340$	43.2950	−	74.9892i	0.127338	−	0.220556i
$341$		−	86.6884i		−	0.254218i
$342$		0			0
$343$	−123.341			−0.359594
$344$	−92.3447	−	53.3153i	−0.268444	−	0.154986i
$345$		0			0
$346$	125.455	+	217.295i	0.362588	+	0.628021i
$347$	−74.7992	+	129.556i	−0.215560	+	0.373360i	−0.953446	−	0.301565i	$-0.902491\pi$
$347$	−74.7992	+	129.556i	−0.215560	+	0.373360i	0.737886	+	0.674926i	$0.235824\pi$
$348$		0			0
$349$	211.863			0.607058			0.303529	−	0.952822i	$-0.401835\pi$
$349$	211.863			0.607058			0.303529	+	0.952822i	$0.401835\pi$
$350$			541.250i			1.54643i
$351$		0			0
$352$	108.238	−	62.4910i	0.307493	−	0.177531i
$353$	154.974			0.439019			0.219509	−	0.975610i	$-0.429554\pi$
$353$	154.974			0.439019			0.219509	+	0.975610i	$0.429554\pi$
$354$		0			0
$355$	645.065	−	372.429i	1.81709	−	1.04909i
$356$	202.343	+	116.823i	0.568378	+	0.328153i
$357$		0			0
$358$	−21.5782	+	37.3745i	−0.0602742	+	0.104398i
$359$	124.695	−	215.978i	0.347340	−	0.601610i	−0.638436	−	0.769675i	$-0.720418\pi$
$359$	124.695	−	215.978i	0.347340	−	0.601610i	0.985776	+	0.168064i	$0.0537517\pi$
$360$		0			0
$361$	249.907	−	260.514i	0.692263	−	0.721646i
$362$	142.911			0.394783
$363$		0			0
$364$	−52.5000	−	30.3109i	−0.144231	−	0.0832717i
$365$	341.758	+	591.942i	0.936323	+	1.62176i
$366$		0			0
$367$	−326.714	−	565.885i	−0.890228	−	1.54192i	−0.839601	−	0.543203i	$-0.817211\pi$
$367$	−326.714	−	565.885i	−0.890228	−	1.54192i	−0.0506268	−	0.998718i	$-0.516122\pi$
$368$	−1.82834			−0.00496830
$369$		0			0
$370$	314.192	+	544.197i	0.849168	+	1.47080i
$371$	709.723	−	409.759i	1.91300	−	1.10447i
$372$		0			0
$373$			324.494i			0.869956i	0.900441	+	0.434978i	$0.143244\pi$
$373$			324.494i			0.869956i	−0.900441	+	0.434978i	$0.856756\pi$
$374$	−17.8269	+	10.2923i	−0.0476654	+	0.0275196i
$375$		0			0
$376$	334.455	−	193.098i	0.889509	−	0.513558i
$377$	−14.4113	+	24.9612i	−0.0382264	+	0.0662100i
$378$		0			0
$379$			21.4534i			0.0566053i	0.999599	+	0.0283026i	$0.00901021\pi$
$379$			21.4534i			0.0566053i	−0.999599	+	0.0283026i	$0.990990\pi$
$380$	−354.024	+	150.969i	−0.931641	+	0.397287i
$381$		0			0
$382$	−187.283	−	108.128i	−0.490270	−	0.283057i
$383$	−185.197	−	106.924i	−0.483544	−	0.279174i	0.238348	−	0.971180i	$-0.423394\pi$
$383$	−185.197	−	106.924i	−0.483544	−	0.279174i	−0.721892	+	0.692006i	$0.756727\pi$
$384$		0			0
$385$	−167.158	+	289.527i	−0.434178	+	0.752018i
$386$	207.242	+	358.953i	0.536896	+	0.929930i
$387$		0			0
$388$			363.017i			0.935611i
$389$	188.727	+	326.885i	0.485160	+	0.840321i	0.999855	−	0.0170523i	$-0.00542818\pi$
$389$	188.727	+	326.885i	0.485160	+	0.840321i	−0.514695	+	0.857373i	$0.672095\pi$
$390$		0			0
$391$	171.588			0.438844
$392$		−	485.749i		−	1.23916i
$393$		0			0
$394$	260.756	+	150.548i	0.661818	+	0.382101i
$395$	−3.81340	+	2.20167i	−0.00965418	+	0.00557385i
$396$		0			0
$397$	132.294	−	229.140i	0.333234	−	0.577179i	−0.649910	−	0.760011i	$-0.725193\pi$
$397$	132.294	−	229.140i	0.333234	−	0.577179i	0.983144	+	0.182833i	$0.0585267\pi$
$398$			418.915i			1.05255i
$399$		0			0
$400$	1.90683			0.00476707
$401$	−424.726	−	245.216i	−1.05917	−	0.611510i	−0.133965	−	0.990986i	$-0.542771\pi$
$401$	−424.726	−	245.216i	−1.05917	−	0.611510i	−0.925202	+	0.379476i	$0.876104\pi$
$402$		0			0
$403$	−25.9436	−	44.9356i	−0.0643761	−	0.111503i
$404$	147.089	−	254.765i	0.364081	−	0.630607i
$405$		0			0
$406$	−159.545			−0.392967
$407$			243.014i			0.597086i
$408$		0			0
$409$	−364.046	+	210.182i	−0.890087	+	0.513892i	−0.873971	−	0.485978i	$-0.838463\pi$
$409$	−364.046	+	210.182i	−0.890087	+	0.513892i	−0.0161160	+	0.999870i	$0.505130\pi$
$410$	521.838			1.27278
$411$		0			0
$412$	56.0466	−	32.3585i	0.136035	−	0.0785401i
$413$	−866.479	−	500.262i	−2.09801	−	1.21129i
$414$		0			0
$415$	221.113	−	382.979i	0.532802	−	0.922840i
$416$	37.4039	−	64.7854i	0.0899131	−	0.155734i
$417$		0			0
$418$	90.8297	+	10.9992i	0.217296	+	0.0263140i
$419$	646.178			1.54219			0.771095	−	0.636720i	$-0.219709\pi$
$419$	646.178			1.54219			0.771095	+	0.636720i	$0.219709\pi$
$420$		0			0
$421$	468.113	+	270.265i	1.11191	+	0.641960i	0.939323	−	0.343035i	$-0.111455\pi$
$421$	468.113	+	270.265i	1.11191	+	0.641960i	0.172584	+	0.984995i	$0.444788\pi$
$422$	−60.6143	−	104.987i	−0.143636	−	0.248785i
$423$		0			0
$424$	312.600	+	541.439i	0.737264	+	1.27698i
$425$	−178.954			−0.421069
$426$		0			0
$427$	−197.874	−	342.728i	−0.463406	−	0.802642i
$428$	258.960	−	149.510i	0.605046	−	0.349323i
$429$		0			0
$430$			134.613i			0.313053i
$431$	−198.483	+	114.594i	−0.460517	+	0.265880i	−0.712262	−	0.701914i	$-0.752329\pi$
$431$	−198.483	+	114.594i	−0.460517	+	0.265880i	0.251745	+	0.967794i	$0.418996\pi$
$432$		0			0
$433$	238.975	−	137.972i	0.551906	−	0.318643i	−0.197985	−	0.980205i	$-0.563440\pi$
$433$	238.975	−	137.972i	0.551906	−	0.318643i	0.749890	+	0.661562i	$0.230106\pi$
$434$	143.608	−	248.736i	0.330893	−	0.573123i
$435$		0			0
$436$			263.031i			0.603283i
$437$	−609.851	−	457.968i	−1.39554	−	1.04798i
$438$		0			0
$439$	10.6486	+	6.14794i	0.0242564	+	0.0140044i	0.512079	−	0.858938i	$-0.328875\pi$
$439$	10.6486	+	6.14794i	0.0242564	+	0.0140044i	−0.487823	+	0.872943i	$0.662209\pi$
$440$	−220.876	−	127.523i	−0.501992	−	0.289825i
$441$		0			0
$442$	−6.16046	+	10.6702i	−0.0139377	+	0.0241408i
$443$	31.4933	+	54.5480i	0.0710910	+	0.123133i	0.899380	−	0.437168i	$-0.144019\pi$
$443$	31.4933	+	54.5480i	0.0710910	+	0.123133i	−0.828289	+	0.560301i	$0.810685\pi$
$444$		0			0
$445$		−	771.232i		−	1.73311i
$446$	40.8618	+	70.7747i	0.0916183	+	0.158688i
$447$		0			0
$448$	412.180			0.920045
$449$			679.936i			1.51433i	0.653222	+	0.757167i	$0.273417\pi$
$449$			679.936i			1.51433i	−0.653222	+	0.757167i	$0.726583\pi$
$450$		0			0
$451$	174.772	+	100.905i	0.387522	+	0.223736i
$452$	−217.986	+	125.854i	−0.482269	+	0.278438i
$453$		0			0
$454$	231.513	−	400.992i	0.509940	−	0.883243i
$455$			200.105i			0.439790i
$456$		0			0
$457$	−434.089			−0.949867			−0.474933	−	0.880022i	$-0.657528\pi$
$457$	−434.089			−0.949867			−0.474933	+	0.880022i	$0.657528\pi$
$458$	43.2318	+	24.9599i	0.0943926	+	0.0544976i
$459$		0			0
$460$	406.545	+	704.156i	0.883792	+	1.53077i
$461$	−265.088	+	459.146i	−0.575028	+	0.995978i	0.421010	+	0.907056i	$0.361676\pi$
$461$	−265.088	+	459.146i	−0.575028	+	0.995978i	−0.996039	+	0.0889223i	$0.971658\pi$
$462$		0			0
$463$	−450.022			−0.971969			−0.485985	−	0.873967i	$-0.661539\pi$
$463$	−450.022			−0.971969			−0.485985	+	0.873967i	$0.661539\pi$
$464$			0.562076i			0.00121137i
$465$		0			0
$466$	287.388	−	165.924i	0.616713	−	0.356059i
$467$	−733.340			−1.57032			−0.785161	−	0.619292i	$-0.787420\pi$
$467$	−733.340			−1.57032			−0.785161	+	0.619292i	$0.787420\pi$
$468$		0			0
$469$	551.361	−	318.329i	1.17561	−	0.678739i
$470$	−422.224	−	243.771i	−0.898350	−	0.518663i
$471$		0			0
$472$	381.643	−	661.026i	0.808566	−	1.40048i
$473$	−26.0293	+	45.0841i	−0.0550302	+	0.0953152i
$474$		0			0
$475$	636.033	+	477.629i	1.33902	+	1.00554i
$476$	110.948			0.233084
$477$		0			0
$478$	62.1744	+	35.8964i	0.130072	+	0.0750971i
$479$	−131.657	−	228.037i	−0.274859	−	0.476069i	0.695241	−	0.718777i	$-0.255298\pi$
$479$	−131.657	−	228.037i	−0.274859	−	0.476069i	−0.970099	+	0.242708i	$0.921964\pi$
$480$		0			0
$481$	72.7277	+	125.968i	0.151201	+	0.261888i
$482$	178.341			0.370002
$483$		0			0
$484$	131.011	+	226.917i	0.270684	+	0.468838i
$485$	1037.73	−	599.136i	2.13966	−	1.23533i
$486$		0			0
$487$		−	307.749i		−	0.631929i	−0.948771	−	0.315964i	$-0.897672\pi$
$487$		−	307.749i		−	0.631929i	0.948771	−	0.315964i	$-0.102328\pi$
$488$	261.463	−	150.956i	0.535785	−	0.309336i
$489$		0			0
$490$	−531.065	+	306.611i	−1.08381	+	0.625736i
$491$	−296.882	+	514.214i	−0.604647	+	1.04728i	0.387460	+	0.921887i	$0.373353\pi$
$491$	−296.882	+	514.214i	−0.604647	+	1.04728i	−0.992107	+	0.125393i	$0.959981\pi$
$492$		0			0
$493$		−	52.7505i		−	0.106999i
$494$	50.3741	−	21.4814i	0.101972	−	0.0434847i
$495$		0			0
$496$	−0.876297	−	0.505930i	−0.00176673	−	0.00102002i
$497$	826.525	+	477.194i	1.66303	+	0.960149i
$498$		0			0
$499$	327.577	−	567.380i	0.656467	−	1.13703i	−0.325057	−	0.945695i	$-0.605383\pi$
$499$	327.577	−	567.380i	0.656467	−	1.13703i	0.981524	−	0.191340i	$-0.0612833\pi$
$500$	−170.794	−	295.825i	−0.341589	−	0.591649i
$501$		0			0
$502$			115.570i			0.230220i
$503$	383.121	+	663.586i	0.761673	+	1.31926i	0.941988	+	0.335647i	$0.108955\pi$
$503$	383.121	+	663.586i	0.761673	+	1.31926i	−0.180315	+	0.983609i	$0.557712\pi$
$504$		0			0
$505$	−971.041			−1.92285
$506$		−	193.292i		−	0.382000i
$507$		0			0
$508$	−237.137	−	136.911i	−0.466804	−	0.269510i
$509$	−757.837	+	437.537i	−1.48887	+	0.859602i	−0.999919	−	0.0127070i	$-0.995955\pi$
$509$	−757.837	+	437.537i	−1.48887	+	0.859602i	−0.488955	+	0.872309i	$0.662622\pi$
$510$		0			0
$511$	−437.896	+	758.458i	−0.856939	+	1.48426i
$512$		−	2.91512i		−	0.00569359i
$513$		0			0
$514$	−510.922			−0.994012
$515$	−185.003	−	106.811i	−0.359228	−	0.207401i
$516$		0			0
$517$	−94.2733	−	163.286i	−0.182347	−	0.315834i
$518$	−402.576	+	697.281i	−0.777173	+	1.34610i
$519$		0			0
$520$	−152.657			−0.293572
$521$		−	445.728i		−	0.855525i	−0.903891	−	0.427762i	$-0.859302\pi$
$521$		−	445.728i		−	0.855525i	0.903891	−	0.427762i	$-0.140698\pi$
$522$		0			0
$523$	764.415	−	441.335i	1.46160	−	0.843853i	0.462510	−	0.886614i	$-0.346949\pi$
$523$	764.415	−	441.335i	1.46160	−	0.843853i	0.999085	+	0.0427611i	$0.0136154\pi$
$524$	139.508			0.266237
$525$		0			0
$526$	−183.321	+	105.840i	−0.348519	+	0.201217i
$527$	82.2398	+	47.4812i	0.156053	+	0.0900971i
$528$		0			0
$529$	−541.114	+	937.237i	−1.02290	+	1.77171i
$530$	394.633	−	683.525i	0.744591	−	1.28967i
$531$		0			0
$532$	−394.328	−	296.121i	−0.741217	−	0.556618i
$533$	120.793			0.226628
$534$		0			0
$535$	−854.792	−	493.514i	−1.59774	−	0.922457i
$536$	242.849	+	420.627i	0.453076	+	0.784751i
$537$		0			0
$538$	−4.17029	−	7.22315i	−0.00775147	−	0.0134259i
$539$	−237.150			−0.439981
$540$		0			0
$541$	−72.3654	−	125.341i	−0.133762	−	0.231683i	0.791362	−	0.611348i	$-0.209373\pi$
$541$	−72.3654	−	125.341i	−0.133762	−	0.231683i	−0.925124	+	0.379665i	$0.876039\pi$
$542$	−263.962	+	152.399i	−0.487015	+	0.281178i
$543$		0			0
$544$			136.911i			0.251674i
$545$	751.912	−	434.116i	1.37965	−	0.796544i
$546$		0			0
$547$	−91.2148	+	52.6629i	−0.166755	+	0.0962758i	−0.581055	−	0.813865i	$-0.697360\pi$
$547$	−91.2148	+	52.6629i	−0.166755	+	0.0962758i	0.414300	+	0.910140i	$0.364026\pi$
$548$	−173.321	+	300.201i	−0.316279	+	0.547812i
$549$		0			0
$550$			201.590i			0.366528i
$551$	−140.791	+	187.484i	−0.255519	+	0.340261i
$552$		0			0
$553$	−4.88613	−	2.82101i	−0.00883567	−	0.00510128i
$554$	72.2809	+	41.7314i	0.130471	+	0.0753275i
$555$		0			0
$556$	210.930	−	365.341i	0.379370	−	0.657088i
$557$	−298.372	−	516.795i	−0.535676	−	0.927819i	−0.999130	−	0.0416975i	$-0.986723\pi$
$557$	−298.372	−	516.795i	−0.535676	−	0.927819i	0.463454	−	0.886121i	$-0.346610\pi$
$558$		0			0
$559$			31.1596i			0.0557416i
$560$	1.95114	+	3.37947i	0.00348417	+	0.00603477i
$561$		0			0
$562$	−31.5290			−0.0561014
$563$		−	709.552i		−	1.26031i	−0.776471	−	0.630153i	$-0.782992\pi$
$563$		−	709.552i		−	1.26031i	0.776471	−	0.630153i	$-0.217008\pi$
$564$		0			0
$565$	719.542	+	415.428i	1.27353	+	0.735271i
$566$	1.02000	−	0.588900i	0.00180213	−	0.00104046i
$567$		0			0
$568$	−364.046	+	630.545i	−0.640925	+	1.11012i
$569$			275.069i			0.483425i	0.970348	+	0.241712i	$0.0777090\pi$
$569$			275.069i			0.483425i	−0.970348	+	0.241712i	$0.922291\pi$
$570$		0			0
$571$	−924.109			−1.61840			−0.809202	−	0.587531i	$-0.800100\pi$
$571$	−924.109			−1.61840			−0.809202	+	0.587531i	$0.800100\pi$
$572$	−19.5538	−	11.2894i	−0.0341849	−	0.0197367i
$573$		0			0
$574$	334.317	+	579.054i	0.582433	+	1.00880i
$575$	840.200	−	1455.27i	1.46122	−	2.53090i
$576$		0			0
$577$	−443.905			−0.769332			−0.384666	−	0.923056i	$-0.625683\pi$
$577$	−443.905			−0.769332			−0.384666	+	0.923056i	$0.625683\pi$
$578$			334.079i			0.577991i
$579$		0			0
$580$	216.475	−	124.982i	0.373233	−	0.215486i
$581$	566.626			0.975259
$582$		0			0
$583$	264.339	−	152.616i	0.453411	−	0.261777i
$584$	−578.618	−	334.065i	−0.990785	−	0.572030i
$585$		0			0
$586$	183.252	−	317.401i	0.312716	−	0.541640i
$587$	−255.163	+	441.955i	−0.434690	+	0.752905i	−0.997270	−	0.0738374i	$-0.976475\pi$
$587$	−255.163	+	441.955i	−0.434690	+	0.752905i	0.562580	+	0.826743i	$0.309809\pi$
$588$		0			0
$589$	−165.566	−	388.254i	−0.281097	−	0.659174i
$590$	−963.591			−1.63321
$591$		0			0
$592$	2.45653	+	1.41828i	0.00414954	+	0.00239574i
$593$	57.1501	+	98.9869i	0.0963746	+	0.166926i	0.910181	−	0.414210i	$-0.135942\pi$
$593$	57.1501	+	98.9869i	0.0963746	+	0.166926i	−0.813807	+	0.581135i	$0.802609\pi$
$594$		0			0
$595$	−183.113	−	317.161i	−0.307753	−	0.533043i
$596$	294.618			0.494325
$597$		0			0
$598$	−57.8473	−	100.194i	−0.0967346	−	0.167549i
$599$	−530.087	+	306.046i	−0.884954	+	0.510928i	−0.872288	−	0.488992i	$-0.837365\pi$
$599$	−530.087	+	306.046i	−0.884954	+	0.510928i	−0.0126651	+	0.999920i	$0.504032\pi$
$600$		0			0
$601$			46.8204i			0.0779041i	0.999241	+	0.0389520i	$0.0124020\pi$
$601$			46.8204i			0.0779041i	−0.999241	+	0.0389520i	$0.987598\pi$
$602$	−149.372	+	86.2400i	−0.248126	+	0.143256i
$603$		0			0
$604$	−323.727	+	186.904i	−0.535971	+	0.309443i
$605$	432.450	−	749.025i	0.714793	−	1.23806i
$606$		0			0
$607$			177.626i			0.292629i	0.989238	+	0.146315i	$0.0467412\pi$
$607$			177.626i			0.292629i	−0.989238	+	0.146315i	$0.953259\pi$
$608$	365.415	−	486.603i	0.601012	−	0.800334i
$609$		0			0
$610$	−330.077	−	190.570i	−0.541110	−	0.312410i
$611$	−97.7346	−	56.4271i	−0.159958	−	0.0923520i
$612$		0			0
$613$	−194.067	+	336.134i	−0.316586	+	0.548343i	−0.979773	−	0.200110i	$-0.935870\pi$
$613$	−194.067	+	336.134i	−0.316586	+	0.548343i	0.663187	+	0.748454i	$0.269203\pi$
$614$	34.1005	+	59.0637i	0.0555382	+	0.0961950i
$615$		0			0
$616$		−	326.792i		−	0.530506i
$617$	113.174	+	196.024i	0.183427	+	0.317705i	0.943045	−	0.332664i	$-0.107948\pi$
$617$	113.174	+	196.024i	0.183427	+	0.317705i	−0.759618	+	0.650369i	$0.774614\pi$
$618$		0			0
$619$	−168.929			−0.272906			−0.136453	−	0.990647i	$-0.543570\pi$
$619$	−168.929			−0.272906			−0.136453	+	0.990647i	$0.543570\pi$
$620$			449.990i			0.725790i
$621$		0			0
$622$	−118.378	−	68.3458i	−0.190319	−	0.109881i
$623$	855.792	−	494.092i	1.37366	−	0.793084i
$624$		0			0
$625$	−40.4783	+	70.1104i	−0.0647652	+	0.112177i
$626$		−	260.935i		−	0.416829i
$627$		0			0
$628$	−25.3364			−0.0403446
$629$	−230.543	−	133.104i	−0.366523	−	0.211612i
$630$		0			0
$631$	−355.351	−	615.487i	−0.563156	−	0.975415i	−0.997219	−	0.0745321i	$-0.976254\pi$
$631$	−355.351	−	615.487i	−0.563156	−	0.975415i	0.434063	−	0.900883i	$-0.357080\pi$
$632$	2.15211	−	3.72757i	0.00340524	−	0.00589805i
$633$		0			0
$634$	204.432			0.322447
$635$			903.850i			1.42339i
$636$		0			0
$637$	−122.929	+	70.9728i	−0.192980	+	0.111417i
$638$	−59.4228			−0.0931392
$639$		0			0
$640$	−563.442	+	325.303i	−0.880377	+	0.508286i
$641$	901.332	+	520.385i	1.40613	+	0.811832i	0.995013	−	0.0997480i	$-0.0318037\pi$
$641$	901.332	+	520.385i	1.40613	+	0.811832i	0.411122	+	0.911580i	$0.365137\pi$
$642$		0			0
$643$	7.89596	−	13.6762i	0.0122799	−	0.0212694i	−0.859820	−	0.510597i	$-0.829424\pi$
$643$	7.89596	−	13.6762i	0.0122799	−	0.0212694i	0.872100	+	0.489328i	$0.162758\pi$
$644$	−520.907	+	902.238i	−0.808862	+	1.40099i
$645$		0			0
$646$	−60.1843	+	80.1441i	−0.0931645	+	0.124062i
$647$	−214.998			−0.332299			−0.166150	−	0.986101i	$-0.553133\pi$
$647$	−214.998			−0.332299			−0.166150	+	0.986101i	$0.553133\pi$
$648$		0			0
$649$	−322.723	−	186.324i	−0.497261	−	0.287094i
$650$	60.3307	+	104.496i	0.0928165	+	0.160763i
$651$		0			0
$652$	53.1180	+	92.0031i	0.0814694	+	0.141109i
$653$	371.833			0.569423			0.284712	−	0.958613i	$-0.408102\pi$
$653$	371.833			0.569423			0.284712	+	0.958613i	$0.408102\pi$
$654$		0			0
$655$	−230.249	−	398.804i	−0.351526	−	0.608861i
$656$	2.04001	−	1.17780i	0.00310977	−	0.00179543i
$657$		0			0
$658$		−	624.691i		−	0.949378i
$659$	−113.255	+	65.3877i	−0.171859	+	0.0992226i	−0.583462	−	0.812140i	$-0.698302\pi$
$659$	−113.255	+	65.3877i	−0.171859	+	0.0992226i	0.411603	+	0.911363i	$0.364969\pi$
$660$		0			0
$661$	−194.222	+	112.134i	−0.293830	+	0.169643i	−0.639668	−	0.768652i	$-0.720928\pi$
$661$	−194.222	+	112.134i	−0.293830	+	0.169643i	0.345838	+	0.938294i	$0.387595\pi$
$662$	−221.031	+	382.837i	−0.333884	+	0.578303i
$663$		0			0
$664$			432.272i			0.651011i
$665$	−195.689	+	1615.97i	−0.294270	+	2.43003i
$666$		0			0
$667$	428.970	+	247.666i	0.643133	+	0.371313i
$668$	3.66959	+	2.11864i	0.00549339	+	0.00317161i
$669$		0			0
$670$	306.578	−	531.009i	0.457579	−	0.792551i
$671$	−73.6989	−	127.650i	−0.109834	−	0.190239i
$672$		0			0
$673$		−	101.375i		−	0.150631i	−0.997160	−	0.0753154i	$-0.976004\pi$
$673$		−	101.375i		−	0.150631i	0.997160	−	0.0753154i	$-0.0239964\pi$
$674$	−391.978	−	678.925i	−0.581569	−	1.00731i
$675$		0			0
$676$	405.137			0.599315
$677$		−	60.9135i		−	0.0899756i	−0.998988	−	0.0449878i	$-0.985675\pi$
$677$		−	60.9135i		−	0.0899756i	0.998988	−	0.0449878i	$-0.0143249\pi$
$678$		0			0
$679$	1329.65	+	767.676i	1.95825	+	1.13060i
$680$	241.958	−	139.694i	0.355821	−	0.205433i
$681$		0			0
$682$	53.4871	−	92.6423i	0.0784268	−	0.135839i
$683$		−	118.773i		−	0.173898i	−0.996213	−	0.0869492i	$-0.972288\pi$
$683$		−	118.773i		−	0.173898i	0.996213	−	0.0869492i	$-0.0277118\pi$
$684$		0			0
$685$	1144.22			1.67040
$686$	−131.812	−	76.1016i	−0.192146	−	0.110935i
$687$		0			0
$688$	0.303824	+	0.526239i	0.000441605	+	0.000764882i
$689$	91.3480	−	158.219i	0.132580	−	0.229636i
$690$		0			0
$691$	−55.9089			−0.0809101			−0.0404551	−	0.999181i	$-0.512881\pi$
$691$	−55.9089			−0.0809101			−0.0404551	+	0.999181i	$0.512881\pi$
$692$		−	503.695i		−	0.727883i
$693$		0			0
$694$	−159.873	+	92.3028i	−0.230365	+	0.133001i
$695$	−1392.50			−2.00360
$696$		0			0
$697$	−191.453	+	110.536i	−0.274682	+	0.158588i
$698$	226.415	+	130.720i	0.324376	+	0.187279i
$699$		0			0
$700$	543.270	−	940.972i	0.776100	−	1.34425i
$701$	−16.5402	+	28.6485i	−0.0235952	+	0.0408681i	−0.877582	−	0.479427i	$-0.840845\pi$
$701$	−16.5402	+	28.6485i	−0.0235952	+	0.0408681i	0.853987	+	0.520295i	$0.174178\pi$
$702$		0			0
$703$	464.132	+	1088.39i	0.660217	+	1.54821i
$704$	153.518			0.218065
$705$		0			0
$706$	165.617	+	95.6193i	0.234586	+	0.135438i
$707$	−622.100	−	1077.51i	−0.879915	−	1.52406i
$708$		0			0
$709$	371.042	+	642.664i	0.523332	+	0.906438i	0.999631	+	0.0271545i	$0.00864459\pi$
$709$	371.042	+	642.664i	0.523332	+	0.906438i	−0.476299	+	0.879283i	$0.658022\pi$
$710$	919.159			1.29459
$711$		0			0
$712$	376.936	+	652.873i	0.529405	+	0.916956i
$713$	−772.240	+	445.853i	−1.08309	+	0.625320i
$714$		0			0
$715$			74.5296i			0.104237i
$716$	75.0279	−	43.3174i	0.104788	−	0.0604991i
$717$		0			0
$718$	266.519	−	153.875i	0.371196	−	0.214310i
$719$	−672.563	+	1164.91i	−0.935414	+	1.62018i	−0.161520	+	0.986869i	$0.551640\pi$
$719$	−672.563	+	1164.91i	−0.935414	+	1.62018i	−0.773894	+	0.633315i	$0.781694\pi$
$720$		0			0
$721$		−	273.715i		−	0.379633i
$722$	427.809	−	124.213i	0.592533	−	0.172040i
$723$		0			0
$724$	−248.453	−	143.445i	−0.343168	−	0.198128i
$725$	−447.386	−	258.298i	−0.617084	−	0.356274i
$726$		0			0
$727$	107.104	−	185.510i	0.147323	−	0.255171i	−0.782914	−	0.622130i	$-0.786268\pi$
$727$	107.104	−	185.510i	0.147323	−	0.255171i	0.930237	+	0.366958i	$0.119601\pi$
$728$	−97.8002	−	169.395i	−0.134341	−	0.232685i
$729$		0			0
$730$			843.464i			1.15543i
$731$	−28.5137	−	49.3871i	−0.0390064	−	0.0675610i
$732$		0			0
$733$	503.818			0.687337			0.343668	−	0.939091i	$-0.388330\pi$
$733$	503.818			0.687337			0.343668	+	0.939091i	$0.388330\pi$
$734$		−	806.334i		−	1.09855i
$735$		0			0
$736$	−1113.37	−	642.803i	−1.51273	−	0.873374i
$737$	205.356	−	118.562i	0.278638	−	0.160872i
$738$		0			0
$739$	708.464	−	1227.10i	0.958680	−	1.66048i	0.232966	−	0.972485i	$-0.425157\pi$
$739$	708.464	−	1227.10i	0.958680	−	1.66048i	0.725713	−	0.687997i	$-0.241510\pi$
$740$		−	1261.46i		−	1.70467i
$741$		0			0
$742$	1011.29			1.36293
$743$	−539.026	−	311.207i	−0.725472	−	0.418852i	0.0912911	−	0.995824i	$-0.470901\pi$
$743$	−539.026	−	311.207i	−0.725472	−	0.418852i	−0.816764	+	0.576973i	$0.804234\pi$
$744$		0			0
$745$	−486.247	−	842.205i	−0.652681	−	1.13048i
$746$	−200.214	+	346.780i	−0.268383	+	0.464853i
$747$		0			0
$748$	41.3230			0.0552446
$749$		−	1264.68i		−	1.68850i
$750$		0			0
$751$	−484.987	+	280.007i	−0.645788	+	0.372846i	−0.786841	−	0.617156i	$-0.788285\pi$
$751$	−484.987	+	280.007i	−0.645788	+	0.372846i	0.141052	+	0.990002i	$0.454951\pi$
$752$	−2.20079			−0.00292658
$753$		0			0
$754$	−30.8023	+	17.7837i	−0.0408518	+	0.0235858i
$755$	1068.58	+	616.945i	1.41534	+	0.817146i
$756$		0			0
$757$	−686.860	+	1189.68i	−0.907345	+	1.57157i	−0.0896071	+	0.995977i	$0.528561\pi$
$757$	−686.860	+	1189.68i	−0.907345	+	1.57157i	−0.817738	+	0.575591i	$0.804772\pi$
$758$	−13.2368	+	22.9269i	−0.0174628	+	0.0302465i
$759$		0			0
$760$	−1232.80	−	149.289i	−1.62211	−	0.196433i
$761$	1196.22			1.57191			0.785953	−	0.618286i	$-0.212173\pi$
$761$	1196.22			1.57191			0.785953	+	0.618286i	$0.212173\pi$
$762$		0			0
$763$	963.428	+	556.235i	1.26268	+	0.729011i
$764$	217.063	+	375.964i	0.284114	+	0.492100i
$765$		0			0
$766$	−131.945	−	228.535i	−0.172251	−	0.298348i
$767$	−223.048			−0.290805
$768$		0			0
$769$	73.0921	+	126.599i	0.0950483	+	0.164628i	0.909629	−	0.415422i	$-0.136366\pi$
$769$	73.0921	+	126.599i	0.0950483	+	0.164628i	−0.814580	+	0.580051i	$0.803033\pi$
$770$	−357.278	+	206.275i	−0.463998	+	0.267889i
$771$		0			0
$772$		−	832.060i		−	1.07780i
$773$	−486.540	+	280.904i	−0.629418	+	0.363395i	−0.780527	−	0.625123i	$-0.785049\pi$
$773$	−486.540	+	280.904i	−0.629418	+	0.363395i	0.151109	+	0.988517i	$0.451716\pi$
$774$		0			0
$775$	805.393	−	464.994i	1.03922	−	0.599992i
$776$	−585.650	+	1014.38i	−0.754704	+	1.30719i
$777$		0			0
$778$			465.781i			0.598691i
$779$	975.476	+	118.127i	1.25222	+	0.151640i
$780$		0			0
$781$	307.842	+	177.732i	0.394163	+	0.227570i
$782$	183.373	+	105.870i	0.234492	+	0.135384i
$783$		0			0
$784$	−1.38405	+	2.39725i	−0.00176537	+	0.00305772i
$785$	41.8161	+	72.4277i	0.0532690	+	0.0922646i

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:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	171.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.06868 + 0.617004i) q^{2} +(-1.23861 - 2.14534i) q^{4} +(-4.08850 + 7.08149i) q^{5} -10.4772 q^{7} -7.99294i q^{8} +O(q^{10})\)	\(q+(1.06868 + 0.617004i) q^{2} +(-1.23861 - 2.14534i) q^{4} +(-4.08850 + 7.08149i) q^{5} -10.4772 q^{7} -7.99294i q^{8} +(-8.73861 + 5.04524i) q^{10} -3.90227 q^{11} +(-2.02277 + 1.16785i) q^{13} +(-11.1968 - 6.46449i) q^{14} +(-0.0227744 + 0.0394465i) q^{16} +(2.13736 - 3.70202i) q^{17} +(-17.4772 + 7.45296i) q^{19} +20.2563 q^{20} +(-4.17029 - 2.40772i) q^{22} +(20.0701 + 34.7623i) q^{23} +(-20.9317 - 36.2547i) q^{25} -2.88227 q^{26} +(12.9772 + 22.4772i) q^{28} +(10.6868 - 6.17004i) q^{29} +22.2148i q^{31} +(-27.7370 + 16.0140i) q^{32} +(4.56832 - 2.63752i) q^{34} +(42.8361 - 74.1944i) q^{35} -62.2749i q^{37} +(-23.2761 - 2.81867i) q^{38} +(56.6020 + 32.6792i) q^{40} +(-44.7873 - 25.8579i) q^{41} +(6.67029 - 11.5533i) q^{43} +(4.83341 + 8.37171i) q^{44} +49.5332i q^{46} +(24.1586 + 41.8438i) q^{47} +60.7723 q^{49} -51.6597i q^{50} +(5.01087 + 2.89303i) q^{52} +(-67.7396 + 39.1095i) q^{53} +(15.9545 - 27.6339i) q^{55} +83.7439i q^{56} +15.2277 q^{58} +(82.7011 + 47.7475i) q^{59} +(18.8861 + 32.7117i) q^{61} +(-13.7066 + 23.7406i) q^{62} -39.3406 q^{64} -19.0990i q^{65} +(-52.6247 + 30.3829i) q^{67} -10.5895 q^{68} +(91.5564 - 52.8601i) q^{70} +(-78.8877 - 45.5459i) q^{71} +(41.7950 - 72.3911i) q^{73} +(38.4239 - 66.5521i) q^{74} +(37.6366 + 28.2633i) q^{76} +40.8850 q^{77} +(0.466357 + 0.269251i) q^{79} +(-0.186227 - 0.322554i) q^{80} +(-31.9089 - 55.2678i) q^{82} -54.0816 q^{83} +(17.4772 + 30.2714i) q^{85} +(14.2568 - 8.23119i) q^{86} +31.1907i q^{88} +(-81.6811 + 47.1586i) q^{89} +(21.1931 - 12.2358i) q^{91} +(49.7180 - 86.1142i) q^{92} +59.6237i q^{94} +(18.6776 - 154.236i) q^{95} +(-126.909 - 73.2709i) q^{97} +(64.9462 + 37.4967i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{4} - 40 q^{7}+O(q^{10}) \) 8 * q + 12 * q^4 - 40 * q^7	\( 8 q + 12 q^{4} - 40 q^{7} - 48 q^{10} - 60 q^{13} - 44 q^{16} - 96 q^{19} + 120 q^{22} - 36 q^{25} + 60 q^{28} + 168 q^{34} + 168 q^{40} - 100 q^{43} + 48 q^{49} - 420 q^{52} + 40 q^{55} + 560 q^{58} - 68 q^{61} - 8 q^{64} - 180 q^{67} + 360 q^{70} - 60 q^{73} + 564 q^{76} + 420 q^{79} - 80 q^{82} + 96 q^{85} + 60 q^{91} - 840 q^{97}+O(q^{100}) \) 8 * q + 12 * q^4 - 40 * q^7 - 48 * q^10 - 60 * q^13 - 44 * q^16 - 96 * q^19 + 120 * q^22 - 36 * q^25 + 60 * q^28 + 168 * q^34 + 168 * q^40 - 100 * q^43 + 48 * q^49 - 420 * q^52 + 40 * q^55 + 560 * q^58 - 68 * q^61 - 8 * q^64 - 180 * q^67 + 360 * q^70 - 60 * q^73 + 564 * q^76 + 420 * q^79 - 80 * q^82 + 96 * q^85 + 60 * q^91 - 840 * q^97

Introduction

L-functions

Modular forms

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