LMFDB - Embedded newform 177.3.b.a.119.13

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	177.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$4.82290067918$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			119.13
Character	$\chi$	$=$	177.119
Dual form			177.3.b.a.119.26

$q$-expansion

$f(q)$	$=$	$q-1.50450i q^{2} +(2.07476 - 2.16688i) q^{3} +1.73647 q^{4} +9.35091i q^{5} +(-3.26008 - 3.12148i) q^{6} +7.09053 q^{7} -8.63054i q^{8} +(-0.390753 - 8.99151i) q^{9} +O(q^{10})$	\(q-1.50450i q^{2} +(2.07476 - 2.16688i) q^{3} +1.73647 q^{4} +9.35091i q^{5} +(-3.26008 - 3.12148i) q^{6} +7.09053 q^{7} -8.63054i q^{8} +(-0.390753 - 8.99151i) q^{9} +14.0685 q^{10} +19.6440i q^{11} +(3.60276 - 3.76273i) q^{12} +5.28870 q^{13} -10.6677i q^{14} +(20.2623 + 19.4009i) q^{15} -6.03878 q^{16} -18.6801i q^{17} +(-13.5278 + 0.587888i) q^{18} -28.5841 q^{19} +16.2376i q^{20} +(14.7111 - 15.3643i) q^{21} +29.5544 q^{22} -22.2573i q^{23} +(-18.7014 - 17.9063i) q^{24} -62.4396 q^{25} -7.95686i q^{26} +(-20.2943 - 17.8085i) q^{27} +12.3125 q^{28} -17.6355i q^{29} +(29.1887 - 30.4847i) q^{30} -8.65134 q^{31} -25.4368i q^{32} +(42.5662 + 40.7565i) q^{33} -28.1042 q^{34} +66.3029i q^{35} +(-0.678531 - 15.6135i) q^{36} -2.64085 q^{37} +43.0049i q^{38} +(10.9728 - 11.4600i) q^{39} +80.7034 q^{40} +43.6618i q^{41} +(-23.1157 - 22.1329i) q^{42} +14.7841 q^{43} +34.1112i q^{44} +(84.0789 - 3.65389i) q^{45} -33.4861 q^{46} -43.3598i q^{47} +(-12.5290 + 13.0853i) q^{48} +1.27559 q^{49} +93.9405i q^{50} +(-40.4775 - 38.7567i) q^{51} +9.18367 q^{52} +30.3805i q^{53} +(-26.7929 + 30.5328i) q^{54} -183.689 q^{55} -61.1951i q^{56} +(-59.3051 + 61.9384i) q^{57} -26.5327 q^{58} -7.68115i q^{59} +(35.1849 + 33.6891i) q^{60} +67.1584 q^{61} +13.0160i q^{62} +(-2.77064 - 63.7546i) q^{63} -62.4248 q^{64} +49.4541i q^{65} +(61.3183 - 64.0409i) q^{66} +33.3864 q^{67} -32.4374i q^{68} +(-48.2289 - 46.1785i) q^{69} +99.7529 q^{70} +61.7198i q^{71} +(-77.6016 + 3.37240i) q^{72} +41.7753 q^{73} +3.97316i q^{74} +(-129.547 + 135.299i) q^{75} -49.6355 q^{76} +139.286i q^{77} +(-17.2416 - 16.5086i) q^{78} -16.0603 q^{79} -56.4681i q^{80} +(-80.6946 + 7.02691i) q^{81} +65.6893 q^{82} -50.6679i q^{83} +(25.5455 - 26.6797i) q^{84} +174.676 q^{85} -22.2427i q^{86} +(-38.2141 - 36.5894i) q^{87} +169.538 q^{88} +119.246i q^{89} +(-5.49729 - 126.497i) q^{90} +37.4996 q^{91} -38.6491i q^{92} +(-17.9494 + 18.7464i) q^{93} -65.2349 q^{94} -267.287i q^{95} +(-55.1185 - 52.7752i) q^{96} +58.4958 q^{97} -1.91912i q^{98} +(176.629 - 7.67594i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38q - 76q^{4} - 8q^{6} - 12q^{7} + 20q^{9} + O(q^{10}) $	$ 38q - 76q^{4} - 8q^{6} - 12q^{7} + 20q^{9} + 36q^{10} - 4q^{13} - 17q^{15} + 100q^{16} - 2q^{18} - 28q^{19} - 11q^{21} + 84q^{22} - 6q^{24} - 166q^{25} + 3q^{27} + 12q^{28} + 102q^{30} - 40q^{31} - 46q^{33} - 148q^{34} - 96q^{36} + 112q^{37} + 62q^{39} - 56q^{40} + 14q^{42} + 164q^{43} + 55q^{45} - 4q^{46} - 124q^{48} + 242q^{49} + 52q^{51} + 8q^{52} + 18q^{54} - 228q^{55} - 147q^{57} - 80q^{58} + 128q^{60} + 12q^{61} + 86q^{63} + 48q^{64} - 24q^{66} + 124q^{67} - 240q^{69} + 148q^{70} + 166q^{72} - 192q^{73} - 78q^{75} - 304q^{76} + 244q^{78} + 64q^{79} - 156q^{81} - 180q^{82} + 300q^{84} - 52q^{85} - 83q^{87} - 96q^{88} - 376q^{90} - 332q^{91} + 454q^{93} + 768q^{94} - 722q^{96} + 416q^{97} + 494q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	1.50450i		−	0.752251i	−0.926569	−	0.376126i	$-0.877256\pi$
$2$		−	1.50450i		−	0.752251i	0.926569	−	0.376126i	$-0.122744\pi$
$3$	2.07476	−	2.16688i	0.691586	−	0.722294i
$3$	2.07476	−	2.16688i	0.691586	−	0.722294i
$4$	1.73647			0.434118
$5$			9.35091i			1.87018i	0.354407	+	0.935091i	$0.384683\pi$
$5$			9.35091i			1.87018i	−0.354407	+	0.935091i	$0.615317\pi$
$6$	−3.26008	−	3.12148i	−0.543347	−	0.520247i
$7$	7.09053			1.01293			0.506466	−	0.862260i	$-0.330951\pi$
$7$	7.09053			1.01293			0.506466	+	0.862260i	$0.330951\pi$
$8$		−	8.63054i		−	1.07882i
$9$	−0.390753	−	8.99151i	−0.0434170	−	0.999057i
$10$	14.0685			1.40685
$11$			19.6440i			1.78582i	0.450239	+	0.892908i	$0.351339\pi$
$11$			19.6440i			1.78582i	−0.450239	+	0.892908i	$0.648661\pi$
$12$	3.60276	−	3.76273i	0.300230	−	0.313561i
$13$	5.28870			0.406823			0.203411	−	0.979093i	$-0.434797\pi$
$13$	5.28870			0.406823			0.203411	+	0.979093i	$0.434797\pi$
$14$		−	10.6677i		−	0.761980i
$15$	20.2623	+	19.4009i	1.35082	+	1.29339i
$16$	−6.03878			−0.377424
$17$		−	18.6801i		−	1.09883i	−0.835550	−	0.549414i	$-0.814851\pi$
$17$		−	18.6801i		−	1.09883i	0.835550	−	0.549414i	$-0.185149\pi$
$18$	−13.5278	+	0.587888i	−0.751542	+	0.0326605i
$19$	−28.5841			−1.50443			−0.752213	−	0.658920i	$-0.771014\pi$
$19$	−28.5841			−1.50443			−0.752213	+	0.658920i	$0.771014\pi$
$20$			16.2376i			0.811880i
$21$	14.7111	−	15.3643i	0.700530	−	0.731635i
$22$	29.5544			1.34338
$23$		−	22.2573i		−	0.967708i	−0.875149	−	0.483854i	$-0.839237\pi$
$23$		−	22.2573i		−	0.967708i	0.875149	−	0.483854i	$-0.160763\pi$
$24$	−18.7014	−	17.9063i	−0.779223	−	0.746095i
$25$	−62.4396			−2.49758
$26$		−	7.95686i		−	0.306033i
$27$	−20.2943	−	17.8085i	−0.751639	−	0.659574i
$28$	12.3125			0.439732
$29$		−	17.6355i		−	0.608121i	−0.952653	−	0.304061i	$-0.901657\pi$
$29$		−	17.6355i		−	0.608121i	0.952653	−	0.304061i	$-0.0983426\pi$
$30$	29.1887	−	30.4847i	0.972956	−	1.01616i
$31$	−8.65134			−0.279075			−0.139538	−	0.990217i	$-0.544562\pi$
$31$	−8.65134			−0.279075			−0.139538	+	0.990217i	$0.544562\pi$
$32$		−	25.4368i		−	0.794900i
$33$	42.5662	+	40.7565i	1.28988	+	1.23505i
$34$	−28.1042			−0.826595
$35$			66.3029i			1.89437i
$36$	−0.678531	−	15.6135i	−0.0188481	−	0.433708i
$37$	−2.64085			−0.0713742			−0.0356871	−	0.999363i	$-0.511362\pi$
$37$	−2.64085			−0.0713742			−0.0356871	+	0.999363i	$0.511362\pi$
$38$			43.0049i			1.13171i
$39$	10.9728	−	11.4600i	0.281353	−	0.293846i
$40$	80.7034			2.01758
$41$			43.6618i			1.06492i	0.846454	+	0.532461i	$0.178733\pi$
$41$			43.6618i			1.06492i	−0.846454	+	0.532461i	$0.821267\pi$
$42$	−23.1157	−	22.1329i	−0.550373	−	0.526975i
$43$	14.7841			0.343817			0.171908	−	0.985113i	$-0.445007\pi$
$43$	14.7841			0.343817			0.171908	+	0.985113i	$0.445007\pi$
$44$			34.1112i			0.775255i
$45$	84.0789	−	3.65389i	1.86842	−	0.0811976i
$46$	−33.4861			−0.727959
$47$		−	43.3598i		−	0.922549i	−0.887258	−	0.461274i	$-0.847392\pi$
$47$		−	43.3598i		−	0.922549i	0.887258	−	0.461274i	$-0.152608\pi$
$48$	−12.5290	+	13.0853i	−0.261021	+	0.272611i
$49$	1.27559			0.0260324
$50$			93.9405i			1.87881i
$51$	−40.4775	−	38.7567i	−0.793677	−	0.759935i
$52$	9.18367			0.176609
$53$			30.3805i			0.573217i	0.958048	+	0.286608i	$0.0925279\pi$
$53$			30.3805i			0.573217i	−0.958048	+	0.286608i	$0.907472\pi$
$54$	−26.7929	+	30.5328i	−0.496166	+	0.565422i
$55$	−183.689			−3.33980
$56$		−	61.1951i		−	1.09277i
$57$	−59.3051	+	61.9384i	−1.04044	+	1.08664i
$58$	−26.5327			−0.457460
$59$		−	7.68115i		−	0.130189i
$59$		−	7.68115i		−	0.130189i
$60$	35.1849	+	33.6891i	0.586416	+	0.561485i
$61$	67.1584			1.10096			0.550479	−	0.834849i	$-0.314445\pi$
$61$	67.1584			1.10096			0.550479	+	0.834849i	$0.314445\pi$
$62$			13.0160i			0.209935i
$63$	−2.77064	−	63.7546i	−0.0439784	−	1.01198i
$64$	−62.4248			−0.975388
$65$			49.4541i			0.760833i
$66$	61.3183	−	64.0409i	0.929065	−	0.970317i
$67$	33.3864			0.498304			0.249152	−	0.968464i	$-0.419848\pi$
$67$	33.3864			0.498304			0.249152	+	0.968464i	$0.419848\pi$
$68$		−	32.4374i		−	0.477021i
$69$	−48.2289	−	46.1785i	−0.698969	−	0.669253i
$70$	99.7529			1.42504
$71$			61.7198i			0.869293i	0.900601	+	0.434647i	$0.143127\pi$
$71$			61.7198i			0.869293i	−0.900601	+	0.434647i	$0.856873\pi$
$72$	−77.6016	+	3.37240i	−1.07780	+	0.0468390i
$73$	41.7753			0.572264			0.286132	−	0.958190i	$-0.407630\pi$
$73$	41.7753			0.572264			0.286132	+	0.958190i	$0.407630\pi$
$74$			3.97316i			0.0536914i
$75$	−129.547	+	135.299i	−1.72729	+	1.80399i
$76$	−49.6355			−0.653098
$77$			139.286i			1.80891i
$78$	−17.2416	−	16.5086i	−0.221046	−	0.211648i
$79$	−16.0603			−0.203294			−0.101647	−	0.994821i	$-0.532411\pi$
$79$	−16.0603			−0.203294			−0.101647	+	0.994821i	$0.532411\pi$
$80$		−	56.4681i		−	0.705852i
$81$	−80.6946	+	7.02691i	−0.996230	+	0.0867520i
$82$	65.6893			0.801089
$83$		−	50.6679i		−	0.610456i	−0.952279	−	0.305228i	$-0.901267\pi$
$83$		−	50.6679i		−	0.610456i	0.952279	−	0.305228i	$-0.0987327\pi$
$84$	25.5455	−	26.6797i	0.304113	−	0.317616i
$85$	174.676			2.05501
$86$		−	22.2427i		−	0.258637i
$87$	−38.2141	−	36.5894i	−0.439242	−	0.420568i
$88$	169.538			1.92657
$89$			119.246i			1.33984i	0.742434	+	0.669919i	$0.233671\pi$
$89$			119.246i			1.33984i	−0.742434	+	0.669919i	$0.766329\pi$
$90$	−5.49729	−	126.497i	−0.0610810	−	1.40552i
$91$	37.4996			0.412084
$92$		−	38.6491i		−	0.420099i
$93$	−17.9494	+	18.7464i	−0.193005	+	0.201574i
$94$	−65.2349			−0.693989
$95$		−	267.287i		−	2.81355i
$96$	−55.1185	−	52.7752i	−0.574151	−	0.549742i
$97$	58.4958			0.603050			0.301525	−	0.953458i	$-0.402504\pi$
$97$	58.4958			0.603050			0.301525	+	0.953458i	$0.402504\pi$
$98$		−	1.91912i		−	0.0195829i
$99$	176.629	−	7.67594i	1.78413	−	0.0775347i
$100$	−108.425			−1.08425
$101$			65.7293i			0.650785i	0.945579	+	0.325393i	$0.105496\pi$
$101$			65.7293i			0.650785i	−0.945579	+	0.325393i	$0.894504\pi$
$102$	−58.3095	+	60.8986i	−0.571662	+	0.597045i
$103$	−107.070			−1.03951			−0.519756	−	0.854315i	$-0.673977\pi$
$103$	−107.070			−1.03951			−0.519756	+	0.854315i	$0.673977\pi$
$104$		−	45.6443i		−	0.438887i
$105$	143.671	+	137.563i	1.36829	+	1.31012i
$106$	45.7075			0.431203
$107$			24.7229i			0.231055i	0.993304	+	0.115528i	$0.0368559\pi$
$107$			24.7229i			0.231055i	−0.993304	+	0.115528i	$0.963144\pi$
$108$	−35.2404	−	30.9240i	−0.326300	−	0.286333i
$109$	−54.3272			−0.498414			−0.249207	−	0.968450i	$-0.580170\pi$
$109$	−54.3272			−0.498414			−0.249207	+	0.968450i	$0.580170\pi$
$110$			276.361i			2.51237i
$111$	−5.47912	+	5.72240i	−0.0493614	+	0.0515532i
$112$	−42.8182			−0.382305
$113$		−	89.2190i		−	0.789549i	−0.918778	−	0.394774i	$-0.870823\pi$
$113$		−	89.2190i		−	0.789549i	0.918778	−	0.394774i	$-0.129177\pi$
$114$	93.1864	+	89.2247i	0.817425	+	0.782673i
$115$	208.126			1.80979
$116$		−	30.6236i		−	0.263996i
$117$	−2.06657	−	47.5534i	−0.0176630	−	0.406439i
$118$	−11.5563			−0.0979348
$119$		−	132.452i		−	1.11304i
$120$	167.440	−	174.875i	1.39533	−	1.45729i
$121$	−264.886			−2.18914
$122$		−	101.040i		−	0.828197i
$123$	94.6100	+	90.5877i	0.769187	+	0.736486i
$124$	−15.0228			−0.121152
$125$		−	350.094i		−	2.80075i
$126$	−95.9189	+	4.16844i	−0.761261	+	0.0330828i
$127$	−185.143			−1.45782			−0.728908	−	0.684612i	$-0.759972\pi$
$127$	−185.143			−1.45782			−0.728908	+	0.684612i	$0.759972\pi$
$128$		−	7.82879i		−	0.0611624i
$129$	30.6735	−	32.0354i	0.237779	−	0.248337i
$130$	74.4039			0.572338
$131$			23.1778i			0.176929i	0.996079	+	0.0884647i	$0.0281960\pi$
$131$			23.1778i			0.176929i	−0.996079	+	0.0884647i	$0.971804\pi$
$132$	73.9150	+	70.7725i	0.559962	+	0.536156i
$133$	−202.676			−1.52388
$134$		−	50.2299i		−	0.374850i
$135$	166.526	−	189.770i	1.23352	−	1.40570i
$136$	−161.219			−1.18543
$137$		−	152.484i		−	1.11302i	−0.830840	−	0.556512i	$-0.812139\pi$
$137$		−	152.484i		−	1.11302i	0.830840	−	0.556512i	$-0.187861\pi$
$138$	−69.4756	+	72.5605i	−0.503447	+	0.525801i
$139$	147.205			1.05903			0.529513	−	0.848302i	$-0.322375\pi$
$139$	147.205			1.05903			0.529513	+	0.848302i	$0.322375\pi$
$140$			115.133i			0.822379i
$141$	−93.9555	−	89.9611i	−0.666351	−	0.638022i
$142$	92.8576			0.653927
$143$			103.891i			0.726511i
$144$	2.35967	+	54.2978i	0.0163866	+	0.377068i
$145$	164.908			1.13730
$146$		−	62.8510i		−	0.430486i
$147$	2.64653	−	2.76405i	0.0180036	−	0.0188030i
$148$	−4.58576			−0.0309848
$149$			32.7895i			0.220064i	0.993928	+	0.110032i	$0.0350953\pi$
$149$			32.7895i			0.220064i	−0.993928	+	0.110032i	$0.964905\pi$
$150$	203.558	+	194.904i	1.35705	+	1.29936i
$151$	122.450			0.810928			0.405464	−	0.914111i	$-0.367110\pi$
$151$	122.450			0.810928			0.405464	+	0.914111i	$0.367110\pi$
$152$			246.696i			1.62300i
$153$	−167.962	+	7.29929i	−1.09779	+	0.0477078i
$154$	209.556			1.36076
$155$		−	80.8979i		−	0.521922i
$156$	19.0539	−	19.8999i	0.122140	−	0.127564i
$157$	154.489			0.984006			0.492003	−	0.870594i	$-0.336265\pi$
$157$	154.489			0.984006			0.492003	+	0.870594i	$0.336265\pi$
$158$			24.1627i			0.152928i
$159$	65.8309	+	63.0322i	0.414031	+	0.396429i
$160$	237.857			1.48661
$161$		−	157.816i		−	0.980223i
$162$	10.5720	+	121.405i	0.0652593	+	0.749415i
$163$	−90.2530			−0.553700			−0.276850	−	0.960913i	$-0.589290\pi$
$163$	−90.2530			−0.553700			−0.276850	+	0.960913i	$0.589290\pi$
$164$			75.8175i			0.462302i
$165$	−381.111	+	398.033i	−2.30976	+	2.41232i
$166$	−76.2300			−0.459217
$167$			256.147i			1.53381i	0.641758	+	0.766907i	$0.278205\pi$
$167$			256.147i			1.53381i	−0.641758	+	0.766907i	$0.721795\pi$
$168$	−132.602	−	126.965i	−0.789300	−	0.755744i
$169$	−141.030			−0.834495
$170$		−	262.800i		−	1.54588i
$171$	11.1693	+	257.014i	0.0653176	+	1.50301i
$172$	25.6722			0.149257
$173$			231.773i			1.33973i	0.742484	+	0.669864i	$0.233647\pi$
$173$			231.773i			1.33973i	−0.742484	+	0.669864i	$0.766353\pi$
$174$	−55.0489	+	57.4932i	−0.316373	+	0.330421i
$175$	−442.729			−2.52988
$176$		−	118.626i		−	0.674010i
$177$	−16.6441	−	15.9365i	−0.0940347	−	0.0900369i
$178$	179.405			1.00790
$179$		−	269.568i		−	1.50597i	−0.658039	−	0.752984i	$-0.728614\pi$
$179$		−	269.568i		−	1.50597i	0.658039	−	0.752984i	$-0.271386\pi$
$180$	146.001	−	6.34488i	0.811114	−	0.0352493i
$181$	−36.1729			−0.199850			−0.0999251	−	0.994995i	$-0.531860\pi$
$181$	−36.1729			−0.199850			−0.0999251	+	0.994995i	$0.531860\pi$
$182$		−	56.4183i		−	0.309991i
$183$	139.338	−	145.524i	0.761407	−	0.795215i
$184$	−192.092			−1.04398
$185$		−	24.6943i		−	0.133483i
$186$	28.2041	+	27.0050i	0.151635	+	0.145188i
$187$	366.951			1.96231
$188$		−	75.2930i		−	0.400495i
$189$	−143.897	−	126.272i	−0.761360	−	0.668104i
$190$	−402.135			−2.11650
$191$			29.5445i			0.154683i	0.997005	+	0.0773417i	$0.0246432\pi$
$191$			29.5445i			0.154683i	−0.997005	+	0.0773417i	$0.975357\pi$
$192$	−129.516	+	135.267i	−0.674565	+	0.704517i
$193$	142.657			0.739156			0.369578	−	0.929200i	$-0.379502\pi$
$193$	142.657			0.739156			0.369578	+	0.929200i	$0.379502\pi$
$194$		−	88.0072i		−	0.453645i
$195$	107.161	+	102.605i	0.549545	+	0.526182i
$196$	2.21502			0.0113011
$197$			8.10839i			0.0411593i	0.999788	+	0.0205797i	$0.00655118\pi$
$197$			8.10839i			0.0411593i	−0.999788	+	0.0205797i	$0.993449\pi$
$198$	−11.5485	−	265.739i	−0.0583256	−	1.34212i
$199$	192.191			0.965786			0.482893	−	0.875679i	$-0.339586\pi$
$199$	192.191			0.965786			0.482893	+	0.875679i	$0.339586\pi$
$200$			538.887i			2.69443i
$201$	69.2687	−	72.3443i	0.344620	−	0.359922i
$202$	98.8899			0.489554
$203$		−	125.045i		−	0.615986i
$204$	−70.2881	−	67.2998i	−0.344549	−	0.329901i
$205$	−408.278			−1.99160
$206$			161.087i			0.781975i
$207$	−200.127	+	8.69709i	−0.966795	+	0.0420149i
$208$	−31.9373			−0.153545
$209$		−	561.506i		−	2.68663i
$210$	206.963	−	216.153i	0.985539	−	1.02930i
$211$	124.794			0.591439			0.295720	−	0.955275i	$-0.404441\pi$
$211$	124.794			0.591439			0.295720	+	0.955275i	$0.404441\pi$
$212$			52.7548i			0.248844i
$213$	133.740	+	128.054i	0.627885	+	0.601191i
$214$	37.1957			0.173812
$215$			138.245i			0.643000i
$216$	−153.697	+	175.150i	−0.711560	+	0.810881i
$217$	−61.3426			−0.282685
$218$			81.7354i			0.374933i
$219$	86.6736	−	90.5220i	0.395770	−	0.413343i
$220$	−318.971			−1.44987
$221$		−	98.7933i		−	0.447028i
$222$	8.60937	+	8.24335i	0.0387810	+	0.0371322i
$223$	−265.562			−1.19086			−0.595430	−	0.803407i	$-0.703018\pi$
$223$	−265.562			−1.19086			−0.595430	+	0.803407i	$0.703018\pi$
$224$		−	180.360i		−	0.805180i
$225$	24.3984	+	561.426i	0.108437	+	2.49523i
$226$	−134.230			−0.593939
$227$			22.3352i			0.0983931i	0.998789	+	0.0491966i	$0.0156661\pi$
$227$			22.3352i			0.0983931i	−0.998789	+	0.0491966i	$0.984334\pi$
$228$	−102.982	+	107.554i	−0.451674	+	0.471729i
$229$	189.051			0.825551			0.412775	−	0.910833i	$-0.364559\pi$
$229$	189.051			0.825551			0.412775	+	0.910833i	$0.364559\pi$
$230$		−	313.126i		−	1.36142i
$231$	301.817	+	288.985i	1.30657	+	1.25102i
$232$	−152.204			−0.656052
$233$			222.374i			0.954396i	0.878796	+	0.477198i	$0.158348\pi$
$233$			222.374i			0.954396i	−0.878796	+	0.477198i	$0.841652\pi$
$234$	−71.5442	+	3.10916i	−0.305744	+	0.0132870i
$235$	405.454			1.72533
$236$		−	13.3381i		−	0.0565173i
$237$	−33.3212	+	34.8007i	−0.140596	+	0.146838i
$238$	−199.274			−0.837285
$239$		−	195.861i		−	0.819501i	−0.912198	−	0.409751i	$-0.865616\pi$
$239$		−	195.861i		−	0.819501i	0.912198	−	0.409751i	$-0.134384\pi$
$240$	−122.360	−	117.158i	−0.509832	−	0.488157i
$241$	26.5673			0.110238			0.0551189	−	0.998480i	$-0.482446\pi$
$241$	26.5673			0.110238			0.0551189	+	0.998480i	$0.482446\pi$
$242$			398.522i			1.64678i
$243$	−152.195	+	189.435i	−0.626318	+	0.779567i
$244$	116.619			0.477945
$245$			11.9279i			0.0486853i
$246$	136.290	−	142.341i	0.554022	−	0.578622i
$247$	−151.173			−0.612035
$248$			74.6657i			0.301071i
$249$	−109.791	−	105.124i	−0.440929	−	0.422183i
$250$	−526.717			−2.10687
$251$			358.916i			1.42994i	0.699154	+	0.714971i	$0.253560\pi$
$251$			358.916i			1.42994i	−0.699154	+	0.714971i	$0.746440\pi$
$252$	−4.81114	−	110.708i	−0.0190918	−	0.439317i
$253$	437.221			1.72815
$254$			278.548i			1.09664i
$255$	362.410	−	378.502i	1.42122	−	1.48432i
$256$	−261.478			−1.02140
$257$			18.4478i			0.0717814i	0.999356	+	0.0358907i	$0.0114268\pi$
$257$			18.4478i			0.0717814i	−0.999356	+	0.0358907i	$0.988573\pi$
$258$	−48.1974	−	46.1483i	−0.186812	−	0.178870i
$259$	−18.7250			−0.0722973
$260$			85.8757i			0.330291i
$261$	−158.570	+	6.89112i	−0.607548	+	0.0264028i
$262$	34.8710			0.133095
$263$		−	170.772i		−	0.649322i	−0.945831	−	0.324661i	$-0.894750\pi$
$263$		−	170.772i		−	0.649322i	0.945831	−	0.324661i	$-0.105250\pi$
$264$	351.751	−	367.369i	1.33239	−	1.39155i
$265$	−284.085			−1.07202
$266$			304.927i			1.14634i
$267$	258.391	+	247.406i	0.967757	+	0.926614i
$268$	57.9745			0.216323
$269$		−	113.815i		−	0.423105i	−0.977367	−	0.211552i	$-0.932148\pi$
$269$		−	113.815i		−	0.423105i	0.977367	−	0.211552i	$-0.0678519\pi$
$270$	−285.509	−	250.539i	−1.05744	−	0.927920i
$271$	285.172			1.05230			0.526148	−	0.850393i	$-0.323636\pi$
$271$	285.172			1.05230			0.526148	+	0.850393i	$0.323636\pi$
$272$			112.805i			0.414724i
$273$	77.8027	−	81.2573i	0.284992	−	0.297646i
$274$	−229.413			−0.837274
$275$		−	1226.56i		−	4.46022i
$276$	−83.7481	−	80.1876i	−0.303435	−	0.290535i
$277$	14.4234			0.0520702			0.0260351	−	0.999661i	$-0.491712\pi$
$277$	14.4234			0.0520702			0.0260351	+	0.999661i	$0.491712\pi$
$278$		−	221.470i		−	0.796654i
$279$	3.38053	+	77.7886i	0.0121166	+	0.278812i
$280$	572.230			2.04368
$281$			378.218i			1.34597i	0.739655	+	0.672987i	$0.234989\pi$
$281$			378.218i			1.34597i	−0.739655	+	0.672987i	$0.765011\pi$
$282$	−135.347	+	141.356i	−0.479953	+	0.501264i
$283$	190.998			0.674903			0.337452	−	0.941343i	$-0.390435\pi$
$283$	190.998			0.674903			0.337452	+	0.941343i	$0.390435\pi$
$284$			107.175i			0.377376i
$285$	−579.180	−	554.557i	−2.03221	−	1.94581i
$286$	156.304			0.546519
$287$			309.585i			1.07869i
$288$	−228.715	+	9.93949i	−0.794150	+	0.0345121i
$289$	−59.9455			−0.207424
$290$		−	248.105i		−	0.855534i
$291$	121.365	−	126.754i	0.417061	−	0.435579i
$292$	72.5415			0.248430
$293$			450.672i			1.53813i	0.639171	+	0.769065i	$0.279278\pi$
$293$			450.672i			1.53813i	−0.639171	+	0.769065i	$0.720722\pi$
$294$	−4.15851	−	3.98172i	−0.0141446	−	0.0135433i
$295$	71.8257			0.243477
$296$			22.7919i			0.0769998i
$297$	349.830	−	398.660i	1.17788	−	1.34229i
$298$	49.3320			0.165543
$299$		−	117.712i		−	0.393686i
$300$	−224.955	+	234.943i	−0.749849	+	0.783144i
$301$	104.827			0.348263
$302$		−	184.227i		−	0.610022i
$303$	142.428	+	136.372i	0.470058	+	0.450074i
$304$	172.613			0.567806
$305$			627.993i			2.05899i
$306$	10.9818	+	252.700i	0.0358882	+	0.825816i
$307$	−165.062			−0.537662			−0.268831	−	0.963187i	$-0.586637\pi$
$307$	−165.062			−0.537662			−0.268831	+	0.963187i	$0.586637\pi$
$308$			241.866i			0.785281i
$309$	−222.144	+	232.007i	−0.718912	+	0.750833i
$310$	−121.711			−0.392617
$311$		−	591.518i		−	1.90199i	−0.309212	−	0.950993i	$-0.600065\pi$
$311$		−	591.518i		−	1.90199i	0.309212	−	0.950993i	$-0.399935\pi$
$312$	−98.9058	−	94.7009i	−0.317006	−	0.303528i
$313$	350.121			1.11860			0.559298	−	0.828966i	$-0.311071\pi$
$313$	350.121			1.11860			0.559298	+	0.828966i	$0.311071\pi$
$314$		−	232.429i		−	0.740220i
$315$	596.163	−	25.9080i	1.89258	−	0.0822477i
$316$	−27.8882			−0.0882537
$317$		−	2.33568i		−	0.00736808i	−0.999993	−	0.00368404i	$-0.998827\pi$
$317$		−	2.33568i		−	0.00736808i	0.999993	−	0.00368404i	$-0.00117267\pi$
$318$	94.8321	−	99.0428i	0.298214	−	0.311455i
$319$	346.432			1.08599
$320$		−	583.729i		−	1.82415i
$321$	53.5717	+	51.2941i	0.166890	+	0.159795i
$322$	−237.434			−0.737374
$323$			533.953i			1.65311i
$324$	−140.124	+	12.2020i	−0.432481	+	0.0376606i
$325$	−330.224			−1.01607
$326$			135.786i			0.416521i
$327$	−112.716	+	117.721i	−0.344696	+	0.360002i
$328$	376.825			1.14886
$329$		−	307.444i		−	0.934480i
$330$	598.841	+	573.382i	1.81467	+	1.73752i
$331$	−37.9525			−0.114660			−0.0573301	−	0.998355i	$-0.518259\pi$
$331$	−37.9525			−0.114660			−0.0573301	+	0.998355i	$0.518259\pi$
$332$		−	87.9833i		−	0.265010i
$333$	1.03192	+	23.7452i	0.00309885	+	0.0713069i
$334$	385.374			1.15381
$335$			312.193i			0.931919i
$336$	−88.8373	+	92.7819i	−0.264397	+	0.276137i
$337$	340.605			1.01070			0.505349	−	0.862915i	$-0.331364\pi$
$337$	340.605			1.01070			0.505349	+	0.862915i	$0.331364\pi$
$338$			212.180i			0.627750i
$339$	−193.327	−	185.108i	−0.570286	−	0.546041i
$340$	303.320			0.892116
$341$		−	169.947i		−	0.498377i
$342$	386.679	−	16.8043i	1.13064	−	0.0491353i
$343$	−338.391			−0.986564
$344$		−	127.595i		−	0.370915i
$345$	431.811	−	450.984i	1.25163	−	1.30720i
$346$	348.703			1.00781
$347$		−	647.225i		−	1.86520i	−0.360911	−	0.932600i	$-0.617534\pi$
$347$		−	647.225i		−	1.86520i	0.360911	−	0.932600i	$-0.382466\pi$
$348$	−66.3577	−	63.5365i	−0.190683	−	0.182576i
$349$	384.765			1.10248			0.551239	−	0.834348i	$-0.314155\pi$
$349$	384.765			1.10248			0.551239	+	0.834348i	$0.314155\pi$
$350$			666.088i			1.90311i
$351$	−107.330	−	94.1838i	−0.305784	−	0.268330i
$352$	499.680			1.41954
$353$			537.674i			1.52316i	0.648074	+	0.761578i	$0.275575\pi$
$353$			537.674i			1.52316i	−0.648074	+	0.761578i	$0.724425\pi$
$354$	−23.9765	+	25.0411i	−0.0677304	+	0.0707377i
$355$	−577.137			−1.62574
$356$			207.067i			0.581648i
$357$	−287.007	−	274.805i	−0.803941	−	0.769763i
$358$	−405.566			−1.13287
$359$		−	61.1053i		−	0.170210i	−0.996372	−	0.0851048i	$-0.972877\pi$
$359$		−	61.1053i		−	0.170210i	0.996372	−	0.0851048i	$-0.0271225\pi$
$360$	−31.5351	−	725.646i	−0.0875974	−	2.01568i
$361$	456.051			1.26330
$362$			54.4222i			0.150338i
$363$	−549.575	+	573.977i	−1.51398	+	1.58120i
$364$	65.1171			0.178893
$365$			390.637i			1.07024i
$366$	−218.942	−	209.634i	−0.598202	−	0.572770i
$367$	444.355			1.21078			0.605388	−	0.795931i	$-0.293018\pi$
$367$	444.355			1.21078			0.605388	+	0.795931i	$0.293018\pi$
$368$			134.407i			0.365236i
$369$	392.586	−	17.0610i	1.06392	−	0.0462357i
$370$	−37.1527			−0.100413
$371$			215.414i			0.580630i
$372$	−31.1687	+	32.5526i	−0.0837868	+	0.0875071i
$373$	−8.79441			−0.0235775			−0.0117888	−	0.999931i	$-0.503753\pi$
$373$	−8.79441			−0.0235775			−0.0117888	+	0.999931i	$0.503753\pi$
$374$		−	552.079i		−	1.47615i
$375$	−758.612	−	726.361i	−2.02297	−	1.93696i
$376$	−374.218			−0.995261
$377$		−	93.2689i		−	0.247398i
$378$	−189.976	+	216.493i	−0.502582	+	0.572734i
$379$	−133.349			−0.351844			−0.175922	−	0.984404i	$-0.556291\pi$
$379$	−133.349			−0.351844			−0.175922	+	0.984404i	$0.556291\pi$
$380$		−	464.137i		−	1.22141i
$381$	−384.126	+	401.182i	−1.00821	+	1.05297i
$382$	44.4498			0.116361
$383$		−	612.768i		−	1.59992i	−0.600056	−	0.799958i	$-0.704855\pi$
$383$		−	612.768i		−	1.59992i	0.600056	−	0.799958i	$-0.295145\pi$
$384$	−16.9641	−	16.2428i	−0.0441772	−	0.0422991i
$385$	−1302.45			−3.38299
$386$		−	214.628i		−	0.556031i
$387$	−5.77693	−	132.932i	−0.0149275	−	0.343493i
$388$	101.576			0.261795
$389$		−	225.199i		−	0.578918i	−0.957190	−	0.289459i	$-0.906525\pi$
$389$		−	225.199i		−	0.578918i	0.957190	−	0.289459i	$-0.0934754\pi$
$390$	154.370	−	161.224i	0.395821	−	0.413396i
$391$	−415.768			−1.06334
$392$		−	11.0090i		−	0.0280842i
$393$	50.2234	+	48.0882i	0.127795	+	0.122362i
$394$	12.1991			0.0309622
$395$		−	150.178i		−	0.380198i
$396$	306.711	−	13.3290i	0.774524	−	0.0336592i
$397$	−455.986			−1.14858			−0.574289	−	0.818652i	$-0.694722\pi$
$397$	−455.986			−1.14858			−0.574289	+	0.818652i	$0.694722\pi$
$398$		−	289.153i		−	0.726514i
$399$	−420.505	+	439.176i	−1.05390	+	1.10069i
$400$	377.059			0.942647
$401$			398.057i			0.992661i	0.868134	+	0.496331i	$0.165320\pi$
$401$			398.057i			0.992661i	−0.868134	+	0.496331i	$0.834680\pi$
$402$	−108.842	−	104.215i	−0.270752	−	0.259241i
$403$	−45.7543			−0.113534
$404$			114.137i			0.282518i
$405$	−65.7081	−	754.568i	−0.162242	−	1.86313i
$406$	−188.131			−0.463376
$407$		−	51.8767i		−	0.127461i
$408$	−334.491	+	349.343i	−0.819831	+	0.856232i
$409$	786.441			1.92284			0.961419	−	0.275089i	$-0.0887073\pi$
$409$	786.441			1.92284			0.961419	+	0.275089i	$0.0887073\pi$
$410$			614.255i			1.49818i
$411$	−330.415	−	316.368i	−0.803930	−	0.769752i
$412$	−185.924			−0.451271
$413$		−	54.4634i		−	0.131873i
$414$	13.0848	+	301.091i	0.0316058	+	0.727273i
$415$	473.791			1.14166
$416$		−	134.527i		−	0.323383i
$417$	305.414	−	318.975i	0.732408	−	0.764929i
$418$	−844.787			−2.02102
$419$			152.439i			0.363817i	0.983315	+	0.181908i	$0.0582275\pi$
$419$			152.439i			0.363817i	−0.983315	+	0.181908i	$0.941773\pi$
$420$	249.480	+	238.873i	0.594000	+	0.568746i
$421$	−442.750			−1.05166			−0.525832	−	0.850589i	$-0.676246\pi$
$421$	−442.750			−1.05166			−0.525832	+	0.850589i	$0.676246\pi$
$422$		−	187.753i		−	0.444911i
$423$	−389.870	+	16.9429i	−0.921679	+	0.0400543i
$424$	262.200			0.618396
$425$			1166.38i			2.74441i
$426$	192.657	−	201.212i	0.452247	−	0.472327i
$427$	476.189			1.11520
$428$			42.9307i			0.100305i
$429$	225.120	+	215.549i	0.524754	+	0.502445i
$430$	207.990			0.483698
$431$			38.8281i			0.0900885i	0.998985	+	0.0450442i	$0.0143429\pi$
$431$			38.8281i			0.0900885i	−0.998985	+	0.0450442i	$0.985657\pi$
$432$	122.553	+	107.542i	0.283687	+	0.248939i
$433$	−491.902			−1.13603			−0.568016	−	0.823018i	$-0.692289\pi$
$433$	−491.902			−1.13603			−0.568016	+	0.823018i	$0.692289\pi$
$434$			92.2900i			0.212650i
$435$	342.145	−	357.337i	0.786540	−	0.821463i
$436$	−94.3375			−0.216371
$437$			636.204i			1.45584i
$438$	−136.191	−	130.401i	−0.310938	−	0.297718i
$439$	164.965			0.375775			0.187887	−	0.982191i	$-0.439836\pi$
$439$	164.965			0.375775			0.187887	+	0.982191i	$0.439836\pi$
$440$			1585.34i			3.60304i
$441$	−0.498439	−	11.4695i	−0.00113025	−	0.0260078i
$442$	−148.635			−0.336278
$443$			243.990i			0.550768i	0.961334	+	0.275384i	$0.0888050\pi$
$443$			243.990i			0.550768i	−0.961334	+	0.275384i	$0.911195\pi$
$444$	−9.51434	+	9.93679i	−0.0214287	+	0.0223802i
$445$	−1115.06			−2.50574
$446$			399.538i			0.895826i
$447$	71.0511	+	68.0304i	0.158951	+	0.152193i
$448$	−442.625			−0.988002
$449$			484.609i			1.07931i	0.841887	+	0.539654i	$0.181445\pi$
$449$			484.609i			1.07931i	−0.841887	+	0.539654i	$0.818555\pi$
$450$	844.667	−	36.7075i	1.87704	−	0.0815722i
$451$	−857.692			−1.90176
$452$		−	154.926i		−	0.342757i
$453$	254.054	−	265.335i	0.560827	−	0.585728i
$454$	33.6034			0.0740163
$455$			350.656i			0.770672i
$456$	534.561	+	511.835i	1.17228	+	1.12245i
$457$	−744.037			−1.62809			−0.814045	−	0.580802i	$-0.802739\pi$
$457$	−744.037			−1.62809			−0.814045	+	0.580802i	$0.802739\pi$
$458$		−	284.428i		−	0.621022i
$459$	−332.664	+	379.099i	−0.724759	+	0.825923i
$460$	361.405			0.785662
$461$		−	185.525i		−	0.402441i	−0.979546	−	0.201220i	$-0.935509\pi$
$461$		−	185.525i		−	0.402441i	0.979546	−	0.201220i	$-0.0644907\pi$
$462$	434.779	−	454.084i	0.941080	−	0.982866i
$463$	141.705			0.306058			0.153029	−	0.988222i	$-0.451097\pi$
$463$	141.705			0.306058			0.153029	+	0.988222i	$0.451097\pi$
$464$			106.497i			0.229520i
$465$	−175.296	−	167.844i	−0.376981	−	0.360954i
$466$	334.563			0.717946
$467$		−	205.961i		−	0.441031i	−0.975383	−	0.220515i	$-0.929226\pi$
$467$		−	205.961i		−	0.441031i	0.975383	−	0.220515i	$-0.0707739\pi$
$468$	−3.58854	−	82.5751i	−0.00766783	−	0.176442i
$469$	236.727			0.504748
$470$		−	610.006i		−	1.29789i
$471$	320.527	−	334.759i	0.680525	−	0.710741i
$472$	−66.2924			−0.140450
$473$			290.419i			0.613994i
$474$	52.3577	+	50.1318i	0.110459	+	0.105763i
$475$	1784.78			3.75743
$476$		−	229.998i		−	0.483190i
$477$	273.167	−	11.8713i	0.572676	−	0.0248873i
$478$	−294.673			−0.616471
$479$		−	285.339i		−	0.595697i	−0.954613	−	0.297849i	$-0.903731\pi$
$479$		−	285.339i		−	0.595697i	0.954613	−	0.297849i	$-0.0962691\pi$
$480$	493.496	−	515.408i	1.02812	−	1.07377i
$481$	−13.9666			−0.0290367
$482$		−	39.9706i		−	0.0829266i
$483$	−341.968	−	327.430i	−0.708009	−	0.677908i
$484$	−459.967			−0.950345
$485$			546.990i			1.12781i
$486$	285.005	+	228.978i	0.586431	+	0.471149i
$487$	−507.398			−1.04188			−0.520942	−	0.853592i	$-0.674419\pi$
$487$	−507.398			−1.04188			−0.520942	+	0.853592i	$0.674419\pi$
$488$		−	579.613i		−	1.18773i
$489$	−187.253	+	195.568i	−0.382931	+	0.399934i
$490$	17.9456			0.0366236
$491$		−	158.306i		−	0.322416i	−0.986920	−	0.161208i	$-0.948461\pi$
$491$		−	158.306i		−	0.322416i	0.986920	−	0.161208i	$-0.0515390\pi$
$492$	164.288	+	157.303i	0.333918	+	0.319722i
$493$	−329.433			−0.668221
$494$			227.440i			0.460404i
$495$	71.7770	+	1651.64i	0.145004	+	3.33665i
$496$	52.2435			0.105330
$497$			437.626i			0.880535i
$498$	−158.159	+	165.181i	−0.317588	+	0.331689i
$499$	703.877			1.41058			0.705288	−	0.708921i	$-0.250818\pi$
$499$	703.877			1.41058			0.705288	+	0.708921i	$0.250818\pi$
$500$		−	607.928i		−	1.21586i
$501$	555.040	+	531.443i	1.10786	+	1.06076i
$502$	539.989			1.07568
$503$		−	785.933i		−	1.56249i	−0.624224	−	0.781245i	$-0.714585\pi$
$503$		−	785.933i		−	1.56249i	0.624224	−	0.781245i	$-0.285415\pi$
$504$	−550.236	+	23.9121i	−1.09174	+	0.0474447i
$505$	−614.629			−1.21709
$506$		−	657.801i		−	1.30000i
$507$	−292.603	+	305.595i	−0.577125	+	0.602751i
$508$	−321.495			−0.632864
$509$		−	705.767i		−	1.38658i	−0.720661	−	0.693288i	$-0.756161\pi$
$509$		−	705.767i		−	1.38658i	0.720661	−	0.693288i	$-0.243839\pi$
$510$	−569.457	−	545.247i	−1.11658	−	1.06911i
$511$	296.209			0.579665
$512$			362.079i			0.707185i
$513$	580.093	+	509.040i	1.13079	+	0.992281i
$514$	27.7548			0.0539977
$515$		−	1001.20i		−	1.94408i
$516$	53.2636	−	55.6286i	0.103224	−	0.107807i
$517$	851.759			1.64750
$518$			28.1718i			0.0543857i
$519$	502.224	+	480.873i	0.967677	+	0.926537i
$520$	426.816			0.820800
$521$		−	8.30979i		−	0.0159497i	−0.999968	−	0.00797485i	$-0.997462\pi$
$521$		−	8.30979i		−	0.0159497i	0.999968	−	0.00797485i	$-0.00253850\pi$
$522$	10.3677	+	238.569i	0.0198615	+	0.457029i
$523$	535.808			1.02449			0.512245	−	0.858840i	$-0.328814\pi$
$523$	535.808			1.02449			0.512245	+	0.858840i	$0.328814\pi$
$524$			40.2475i			0.0768082i
$525$	−918.557	+	959.342i	−1.74963	+	1.82732i
$526$	−256.926			−0.488453
$527$			161.608i			0.306656i
$528$	−257.048	−	246.120i	−0.486833	−	0.466136i
$529$	33.6137			0.0635420
$530$			427.407i			0.806429i
$531$	−69.0651	+	3.00143i	−0.130066	+	0.00565241i
$532$	−351.942			−0.661545
$533$			230.914i			0.433235i
$534$	372.223	−	388.750i	0.697047	−	0.727997i
$535$	−231.182			−0.432116
$536$		−	288.142i		−	0.537579i
$537$	−584.123	−	559.289i	−1.08775	−	1.04151i
$538$	−171.235			−0.318281
$539$			25.0576i			0.0464891i
$540$	289.167	−	329.530i	0.535495	−	0.610241i
$541$	−957.545			−1.76995			−0.884977	−	0.465635i	$-0.845826\pi$
$541$	−957.545			−1.76995			−0.884977	+	0.465635i	$0.845826\pi$
$542$		−	429.042i		−	0.791591i
$543$	−75.0500	+	78.3823i	−0.138214	+	0.144351i
$544$	−475.161			−0.873458
$545$		−	508.008i		−	0.932126i
$546$	−122.252	−	117.054i	−0.223904	−	0.214385i
$547$	−106.032			−0.193843			−0.0969214	−	0.995292i	$-0.530900\pi$
$547$	−106.032			−0.193843			−0.0969214	+	0.995292i	$0.530900\pi$
$548$		−	264.785i		−	0.483184i
$549$	−26.2423	−	603.856i	−0.0478002	−	1.09992i
$550$	−1845.37			−3.35521
$551$			504.095i			0.914874i
$552$	−398.545	+	416.241i	−0.722002	+	0.754060i
$553$	−113.876			−0.205924
$554$		−	21.7001i		−	0.0391699i
$555$	−53.5097	−	51.2348i	−0.0964139	−	0.0923149i
$556$	255.617			0.459742
$557$			2.19030i			0.00393232i	0.999998	+	0.00196616i	$0.000625849\pi$
$557$			2.19030i			0.00393232i	−0.999998	+	0.00196616i	$0.999374\pi$
$558$	117.033	−	5.08602i	0.209737	−	0.00911473i
$559$	78.1887			0.139872
$560$		−	400.389i		−	0.714980i
$561$	761.335	−	795.140i	1.35710	−	1.41736i
$562$	569.031			1.01251
$563$		−	488.602i		−	0.867855i	−0.900948	−	0.433927i	$-0.857127\pi$
$563$		−	488.602i		−	0.867855i	0.900948	−	0.433927i	$-0.142873\pi$
$564$	−163.151	−	156.215i	−0.289275	−	0.276977i
$565$	834.279			1.47660
$566$		−	287.356i		−	0.507697i
$567$	−572.167	+	49.8245i	−1.00911	+	0.0878740i
$568$	532.675			0.937808
$569$			389.337i			0.684248i	0.939655	+	0.342124i	$0.111146\pi$
$569$			389.337i			0.684248i	−0.939655	+	0.342124i	$0.888854\pi$
$570$	−834.332	+	871.378i	−1.46374	+	1.52873i
$571$	−522.297			−0.914706			−0.457353	−	0.889285i	$-0.651202\pi$
$571$	−522.297			−0.914706			−0.457353	+	0.889285i	$0.651202\pi$
$572$			180.404i			0.315391i
$573$	64.0195	+	61.2978i	0.111727	+	0.106977i
$574$	465.772			0.811450
$575$			1389.73i			2.41693i
$576$	24.3927	+	561.294i	0.0423484	+	0.974468i
$577$	867.206			1.50296			0.751478	−	0.659758i	$-0.229341\pi$
$577$	867.206			1.50296			0.751478	+	0.659758i	$0.229341\pi$
$578$			90.1882i			0.156035i
$579$	295.979	−	309.121i	0.511190	−	0.533888i
$580$	286.358			0.493721
$581$		−	359.262i		−	0.618351i
$582$	−190.701	−	182.594i	−0.327665	−	0.313735i
$583$	−596.794			−1.02366
$584$		−	360.543i		−	0.617368i
$585$	444.668	−	19.3243i	0.760115	−	0.0330330i
$586$	678.037			1.15706
$587$			698.172i			1.18939i	0.803951	+	0.594695i	$0.202727\pi$
$587$			698.172i			1.18939i	−0.803951	+	0.594695i	$0.797273\pi$
$588$	4.59563	−	4.79969i	0.00781570	−	0.00816273i
$589$	247.291			0.419848
$590$		−	108.062i		−	0.183156i
$591$	17.5699	+	16.8230i	0.0297291	+	0.0284652i
$592$	15.9475			0.0269383
$593$		−	227.806i		−	0.384158i	−0.981379	−	0.192079i	$-0.938477\pi$
$593$		−	227.806i		−	0.384158i	0.981379	−	0.192079i	$-0.0615230\pi$
$594$	−599.785	−	526.320i	−1.00974	−	0.886061i
$595$	1238.54			2.08159
$596$			56.9381i			0.0955337i
$597$	398.751	−	416.456i	0.667924	−	0.697582i
$598$	−177.098			−0.296150
$599$			304.030i			0.507562i	0.967262	+	0.253781i	$0.0816742\pi$
$599$			304.030i			0.507562i	−0.967262	+	0.253781i	$0.918326\pi$
$600$	1167.70	+	1118.06i	1.94617	+	1.86343i
$601$	−939.805			−1.56374			−0.781868	−	0.623445i	$-0.785733\pi$
$601$	−939.805			−1.56374			−0.781868	+	0.623445i	$0.785733\pi$
$602$		−	157.713i		−	0.261981i
$603$	−13.0458	−	300.194i	−0.0216348	−	0.497834i
$604$	212.631			0.352038
$605$		−	2476.93i		−	4.09409i
$606$	205.173	−	214.283i	0.338569	−	0.353602i
$607$	−918.465			−1.51312			−0.756561	−	0.653923i	$-0.773122\pi$
$607$	−918.465			−1.51312			−0.756561	+	0.653923i	$0.773122\pi$
$608$			727.088i			1.19587i
$609$	−270.958	−	259.438i	−0.444923	−	0.426007i
$610$	944.817			1.54888
$611$		−	229.317i		−	0.375314i
$612$	−291.662	+	12.6750i	−0.476571	+	0.0207108i
$613$	−328.504			−0.535895			−0.267947	−	0.963434i	$-0.586345\pi$
$613$	−328.504			−0.535895			−0.267947	+	0.963434i	$0.586345\pi$
$614$			248.337i			0.404457i
$615$	−847.078	+	884.690i	−1.37736	+	1.43852i
$616$	1202.11			1.95148
$617$		−	968.760i		−	1.57011i	−0.619424	−	0.785057i	$-0.712634\pi$
$617$		−	968.760i		−	1.57011i	0.619424	−	0.785057i	$-0.287366\pi$
$618$	349.056	+	334.216i	0.564815	+	0.540803i
$619$	812.768			1.31303			0.656517	−	0.754311i	$-0.272029\pi$
$619$	812.768			1.31303			0.656517	+	0.754311i	$0.272029\pi$
$620$		−	140.477i		−	0.226576i
$621$	−396.369	+	451.695i	−0.638275	+	0.727367i
$622$	−889.940			−1.43077
$623$			845.514i			1.35717i
$624$	−66.2622	+	69.2043i	−0.106189	+	0.110904i
$625$	1712.71			2.74034
$626$		−	526.758i		−	0.841466i
$627$	−1216.72	−	1164.99i	−1.94054	−	1.85804i
$628$	268.265			0.427174
$629$			49.3312i			0.0784281i
$630$	−38.9787	−	896.930i	−0.0618710	−	1.42370i
$631$	−22.1005			−0.0350246			−0.0175123	−	0.999847i	$-0.505575\pi$
$631$	−22.1005			−0.0350246			−0.0175123	+	0.999847i	$0.505575\pi$
$632$			138.609i			0.219317i
$633$	258.917	−	270.413i	0.409031	−	0.427193i
$634$	−3.51404			−0.00554265
$635$		−	1731.25i		−	2.72638i
$636$	114.314	+	109.454i	0.179738	+	0.172097i
$637$	6.74619			0.0105906
$638$		−	521.208i		−	0.816940i
$639$	554.955	−	24.1172i	0.868473	−	0.0377421i
$640$	73.2063			0.114385
$641$		−	482.527i		−	0.752772i	−0.926463	−	0.376386i	$-0.877167\pi$
$641$		−	482.527i		−	0.752772i	0.926463	−	0.376386i	$-0.122833\pi$
$642$	77.1722	−	80.5987i	0.120206	−	0.125543i
$643$	−1172.14			−1.82292			−0.911462	−	0.411384i	$-0.865045\pi$
$643$	−1172.14			−1.82292			−0.911462	+	0.411384i	$0.865045\pi$
$644$		−	274.043i		−	0.425532i
$645$	299.561	+	286.825i	0.464435	+	0.444690i
$646$	803.334			1.24355
$647$			212.268i			0.328080i	0.986454	+	0.164040i	$0.0524526\pi$
$647$			212.268i			0.328080i	−0.986454	+	0.164040i	$0.947547\pi$
$648$	60.6460	+	696.438i	0.0935896	+	1.07475i
$649$	150.888			0.232493
$650$			496.823i			0.764343i
$651$	−127.271	+	132.922i	−0.195501	+	0.204181i
$652$	−156.722			−0.240371
$653$			370.397i			0.567224i	0.958939	+	0.283612i	$0.0915327\pi$
$653$			370.397i			0.567224i	−0.958939	+	0.283612i	$0.908467\pi$
$654$	177.111	+	169.581i	0.270812	+	0.259298i
$655$	−216.733			−0.330890
$656$		−	263.664i		−	0.401927i
$657$	−16.3238	−	375.623i	−0.0248460	−	0.571724i
$658$	−462.550			−0.702964
$659$			1206.27i			1.83046i	0.402931	+	0.915230i	$0.367991\pi$
$659$			1206.27i			1.83046i	−0.402931	+	0.915230i	$0.632009\pi$
$660$	−661.788	+	691.172i	−1.00271	+	1.04723i
$661$	279.734			0.423198			0.211599	−	0.977357i	$-0.432133\pi$
$661$	279.734			0.423198			0.211599	+	0.977357i	$0.432133\pi$
$662$			57.0996i			0.0862532i
$663$	−214.073	−	204.972i	−0.322886	−	0.309159i
$664$	−437.291			−0.658571
$665$		−	1895.21i		−	2.84994i
$666$	35.7247	−	1.55252i	0.0536407	−	0.00233112i
$667$	−392.519			−0.588484
$668$			444.792i			0.665856i
$669$	−550.976	+	575.441i	−0.823582	+	0.860151i
$670$	469.695			0.701038
$671$			1319.26i			1.96611i
$672$	−390.819	−	374.204i	−0.581576	−	0.556851i
$673$	−72.6008			−0.107876			−0.0539382	−	0.998544i	$-0.517177\pi$
$673$	−72.6008			−0.107876			−0.0539382	+	0.998544i	$0.517177\pi$
$674$		−	512.441i		−	0.760299i
$675$	1267.16	+	1111.96i	1.87728	+	1.64734i
$676$	−244.894			−0.362269
$677$			1191.38i			1.75979i	0.475172	+	0.879893i	$0.342386\pi$
$677$			1191.38i			1.75979i	−0.475172	+	0.879893i	$0.657614\pi$
$678$	−278.495	+	290.861i	−0.410760	+	0.428999i
$679$	414.766			0.610849
$680$		−	1507.55i		−	2.21698i
$681$	48.3978	+	46.3402i	0.0710687	+	0.0680473i
$682$	−255.685			−0.374905
$683$		−	1224.58i		−	1.79295i	−0.443094	−	0.896475i	$-0.646119\pi$
$683$		−	1224.58i		−	1.79295i	0.443094	−	0.896475i	$-0.353881\pi$
$684$	19.3952	+	446.298i	0.0283555	+	0.652482i
$685$	1425.87			2.08156
$686$			509.111i			0.742144i
$687$	392.235	−	409.651i	0.570939	−	0.596290i
$688$	−89.2781			−0.129765
$689$			160.673i			0.233198i
$690$	−678.507	−	649.661i	−0.983343	−	0.941537i
$691$	−1352.87			−1.95785			−0.978923	−	0.204229i	$-0.934531\pi$
$691$	−1352.87			−1.95785			−0.978923	+	0.204229i	$0.934531\pi$
$692$			402.467i			0.581600i
$693$	1252.39	−	54.4264i	1.80721	−	0.0785374i
$694$	−973.751			−1.40310
$695$			1376.50i			1.98057i
$696$	−315.787	+	329.808i	−0.453716	+	0.473862i
$697$	815.606			1.17017
$698$		−	578.879i		−	0.829340i
$699$	481.859	+	461.373i	0.689355	+	0.660047i
$700$	−768.787			−1.09827
$701$			799.448i			1.14044i	0.821492	+	0.570220i	$0.193142\pi$
$701$			799.448i			1.14044i	−0.821492	+	0.570220i	$0.806858\pi$
$702$	−141.700	+	161.479i	−0.201852	+	0.230026i
$703$	75.4862			0.107377
$704$		−	1226.27i		−	1.74186i
$705$	841.218	−	878.570i	1.19322	−	1.24620i
$706$	808.932			1.14580
$707$			466.056i			0.659202i
$708$	−28.9021	−	27.6733i	−0.0408221	−	0.0390866i
$709$	437.816			0.617512			0.308756	−	0.951141i	$-0.400087\pi$
$709$	437.816			0.617512			0.308756	+	0.951141i	$0.400087\pi$
$710$			868.304i			1.22296i
$711$	6.27559	+	144.406i	0.00882642	+	0.203103i
$712$	1029.15			1.44544
$713$			192.555i			0.270063i
$714$	−413.445	+	431.803i	−0.579055	+	0.604766i
$715$	−971.476			−1.35871
$716$		−	468.098i		−	0.653768i
$717$	−424.407	−	406.364i	−0.591921	−	0.566756i
$718$	−91.9330			−0.128040
$719$		−	995.948i		−	1.38519i	−0.721329	−	0.692593i	$-0.756468\pi$
$719$		−	995.948i		−	1.38519i	0.721329	−	0.692593i	$-0.243532\pi$
$720$	−507.734	+	22.0651i	−0.705186	+	0.0306459i
$721$	−759.181			−1.05296
$722$		−	686.130i		−	0.950318i
$723$	55.1208	−	57.5682i	0.0762390	−	0.0796241i
$724$	−62.8132			−0.0867585
$725$			1101.15i			1.51883i
$726$	863.549	+	826.836i	1.18946	+	1.13889i
$727$	810.654			1.11507			0.557533	−	0.830155i	$-0.311748\pi$
$727$	810.654			1.11507			0.557533	+	0.830155i	$0.311748\pi$
$728$		−	323.642i		−	0.444563i
$729$	94.7142	+	722.821i	0.129923	+	0.991524i
$730$	587.714			0.805088
$731$		−	276.169i		−	0.377796i
$732$	241.956	−	252.699i	0.330541	−	0.345217i
$733$	−753.454			−1.02790			−0.513952	−	0.857819i	$-0.671819\pi$
$733$	−753.454			−1.02790			−0.513952	+	0.857819i	$0.671819\pi$
$734$		−	668.533i		−	0.910807i
$735$	25.8463	+	24.7475i	0.0351651	+	0.0336701i
$736$	−566.153			−0.769230
$737$			655.841i			0.889879i
$738$	−25.6683	−	590.646i	−0.0347809	−	0.800334i
$739$	536.212			0.725592			0.362796	−	0.931869i	$-0.381822\pi$
$739$	536.212			0.725592			0.362796	+	0.931869i	$0.381822\pi$
$740$		−	42.8810i		−	0.0579473i
$741$	−313.647	+	327.573i	−0.423275	+	0.442069i
$742$	324.091			0.436780
$743$			329.135i			0.442981i	0.975163	+	0.221490i	$0.0710922\pi$
$743$			329.135i			0.442981i	−0.975163	+	0.221490i	$0.928908\pi$
$744$	161.792	+	154.913i	0.217462	+	0.208217i
$745$	−306.612			−0.411560
$746$			13.2312i			0.0177362i
$747$	−455.581	+	19.7986i	−0.609881	+	0.0265042i
$748$	637.200			0.851872
$749$			175.299i			0.234044i
$750$	−1092.81	+	1141.33i	−1.45708	+	1.52178i
$751$	152.366			0.202884			0.101442	−	0.994841i	$-0.467654\pi$
$751$	152.366			0.202884			0.101442	+	0.994841i	$0.467654\pi$
$752$			261.840i			0.348192i
$753$	777.728	+	744.663i	1.03284	+	0.988929i
$754$	−140.323			−0.186105
$755$			1145.02i			1.51658i
$756$	−249.873	−	219.267i	−0.330520	−	0.290036i
$757$	−1062.25			−1.40324			−0.701618	−	0.712554i	$-0.747539\pi$
$757$	−1062.25			−1.40324			−0.701618	+	0.712554i	$0.747539\pi$
$758$			200.624i			0.264675i
$759$	907.129	−	947.407i	1.19516	−	1.24823i
$760$	−2306.83			−3.03531
$761$		−	347.069i		−	0.456069i	−0.973653	−	0.228035i	$-0.926770\pi$
$761$		−	347.069i		−	0.456069i	0.973653	−	0.228035i	$-0.0732299\pi$
$762$	603.580	+	577.919i	0.792099	+	0.758424i
$763$	−385.208			−0.504860
$764$			51.3032i			0.0671508i
$765$	−68.2550	−	1570.60i	−0.0892223	−	2.05307i
$766$	−921.911			−1.20354
$767$		−	40.6232i		−	0.0529638i
$768$	−542.503	+	566.591i	−0.706385	+	0.737749i
$769$	−599.875			−0.780071			−0.390036	−	0.920800i	$-0.627537\pi$
$769$	−599.875			−0.780071			−0.390036	+	0.920800i	$0.627537\pi$
$770$			1959.54i			2.54486i
$771$	39.9742	+	38.2748i	0.0518473	+	0.0496430i
$772$	247.720			0.320881
$773$			915.795i			1.18473i	0.805670	+	0.592364i	$0.201805\pi$
$773$			915.795i			1.18473i	−0.805670	+	0.592364i	$0.798195\pi$
$774$	−199.996	+	8.69141i	−0.258393	+	0.0112292i
$775$	540.186			0.697014
$776$		−	504.851i		−	0.650581i
$777$	−38.8499	+	40.5749i	−0.0499998	+	0.0522199i
$778$	−338.813			−0.435492
$779$		−	1248.03i		−	1.60210i
$780$	186.082	+	178.171i	0.238567	+	0.228425i
$781$	−1212.42			−1.55240
$782$			625.524i			0.799902i
$783$	−314.062	+	357.900i	−0.401101	+	0.457088i
$784$	−7.70299			−0.00982524
$785$			1444.61i			1.84027i
$786$	72.3489	−	75.5613i	0.0920469	−

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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	177.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.50450i q^{2} +(2.07476 - 2.16688i) q^{3} +1.73647 q^{4} +9.35091i q^{5} +(-3.26008 - 3.12148i) q^{6} +7.09053 q^{7} -8.63054i q^{8} +(-0.390753 - 8.99151i) q^{9} +O(q^{10})\)	\(q-1.50450i q^{2} +(2.07476 - 2.16688i) q^{3} +1.73647 q^{4} +9.35091i q^{5} +(-3.26008 - 3.12148i) q^{6} +7.09053 q^{7} -8.63054i q^{8} +(-0.390753 - 8.99151i) q^{9} +14.0685 q^{10} +19.6440i q^{11} +(3.60276 - 3.76273i) q^{12} +5.28870 q^{13} -10.6677i q^{14} +(20.2623 + 19.4009i) q^{15} -6.03878 q^{16} -18.6801i q^{17} +(-13.5278 + 0.587888i) q^{18} -28.5841 q^{19} +16.2376i q^{20} +(14.7111 - 15.3643i) q^{21} +29.5544 q^{22} -22.2573i q^{23} +(-18.7014 - 17.9063i) q^{24} -62.4396 q^{25} -7.95686i q^{26} +(-20.2943 - 17.8085i) q^{27} +12.3125 q^{28} -17.6355i q^{29} +(29.1887 - 30.4847i) q^{30} -8.65134 q^{31} -25.4368i q^{32} +(42.5662 + 40.7565i) q^{33} -28.1042 q^{34} +66.3029i q^{35} +(-0.678531 - 15.6135i) q^{36} -2.64085 q^{37} +43.0049i q^{38} +(10.9728 - 11.4600i) q^{39} +80.7034 q^{40} +43.6618i q^{41} +(-23.1157 - 22.1329i) q^{42} +14.7841 q^{43} +34.1112i q^{44} +(84.0789 - 3.65389i) q^{45} -33.4861 q^{46} -43.3598i q^{47} +(-12.5290 + 13.0853i) q^{48} +1.27559 q^{49} +93.9405i q^{50} +(-40.4775 - 38.7567i) q^{51} +9.18367 q^{52} +30.3805i q^{53} +(-26.7929 + 30.5328i) q^{54} -183.689 q^{55} -61.1951i q^{56} +(-59.3051 + 61.9384i) q^{57} -26.5327 q^{58} -7.68115i q^{59} +(35.1849 + 33.6891i) q^{60} +67.1584 q^{61} +13.0160i q^{62} +(-2.77064 - 63.7546i) q^{63} -62.4248 q^{64} +49.4541i q^{65} +(61.3183 - 64.0409i) q^{66} +33.3864 q^{67} -32.4374i q^{68} +(-48.2289 - 46.1785i) q^{69} +99.7529 q^{70} +61.7198i q^{71} +(-77.6016 + 3.37240i) q^{72} +41.7753 q^{73} +3.97316i q^{74} +(-129.547 + 135.299i) q^{75} -49.6355 q^{76} +139.286i q^{77} +(-17.2416 - 16.5086i) q^{78} -16.0603 q^{79} -56.4681i q^{80} +(-80.6946 + 7.02691i) q^{81} +65.6893 q^{82} -50.6679i q^{83} +(25.5455 - 26.6797i) q^{84} +174.676 q^{85} -22.2427i q^{86} +(-38.2141 - 36.5894i) q^{87} +169.538 q^{88} +119.246i q^{89} +(-5.49729 - 126.497i) q^{90} +37.4996 q^{91} -38.6491i q^{92} +(-17.9494 + 18.7464i) q^{93} -65.2349 q^{94} -267.287i q^{95} +(-55.1185 - 52.7752i) q^{96} +58.4958 q^{97} -1.91912i q^{98} +(176.629 - 7.67594i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38q - 76q^{4} - 8q^{6} - 12q^{7} + 20q^{9} + O(q^{10}) \)	\( 38q - 76q^{4} - 8q^{6} - 12q^{7} + 20q^{9} + 36q^{10} - 4q^{13} - 17q^{15} + 100q^{16} - 2q^{18} - 28q^{19} - 11q^{21} + 84q^{22} - 6q^{24} - 166q^{25} + 3q^{27} + 12q^{28} + 102q^{30} - 40q^{31} - 46q^{33} - 148q^{34} - 96q^{36} + 112q^{37} + 62q^{39} - 56q^{40} + 14q^{42} + 164q^{43} + 55q^{45} - 4q^{46} - 124q^{48} + 242q^{49} + 52q^{51} + 8q^{52} + 18q^{54} - 228q^{55} - 147q^{57} - 80q^{58} + 128q^{60} + 12q^{61} + 86q^{63} + 48q^{64} - 24q^{66} + 124q^{67} - 240q^{69} + 148q^{70} + 166q^{72} - 192q^{73} - 78q^{75} - 304q^{76} + 244q^{78} + 64q^{79} - 156q^{81} - 180q^{82} + 300q^{84} - 52q^{85} - 83q^{87} - 96q^{88} - 376q^{90} - 332q^{91} + 454q^{93} + 768q^{94} - 722q^{96} + 416q^{97} + 494q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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