LMFDB - Embedded newform 177.3.b.a.119.16

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			119.16
Character	$\chi$	$=$	177.119
Dual form			177.3.b.a.119.23

$q$-expansion

$f(q)$	$=$	$q-1.19271i q^{2} +(0.625113 + 2.93415i) q^{3} +2.57744 q^{4} +0.192711i q^{5} +(3.49960 - 0.745581i) q^{6} +6.00162 q^{7} -7.84499i q^{8} +(-8.21847 + 3.66835i) q^{9} +O(q^{10})$	\(q-1.19271i q^{2} +(0.625113 + 2.93415i) q^{3} +2.57744 q^{4} +0.192711i q^{5} +(3.49960 - 0.745581i) q^{6} +6.00162 q^{7} -7.84499i q^{8} +(-8.21847 + 3.66835i) q^{9} +0.229849 q^{10} +7.32965i q^{11} +(1.61119 + 7.56258i) q^{12} +9.24069 q^{13} -7.15821i q^{14} +(-0.565443 + 0.120466i) q^{15} +0.952916 q^{16} +9.92749i q^{17} +(4.37529 + 9.80227i) q^{18} +0.266471 q^{19} +0.496700i q^{20} +(3.75169 + 17.6097i) q^{21} +8.74217 q^{22} +9.30188i q^{23} +(23.0184 - 4.90401i) q^{24} +24.9629 q^{25} -11.0215i q^{26} +(-15.9010 - 21.8211i) q^{27} +15.4688 q^{28} -28.6314i q^{29} +(0.143682 + 0.674411i) q^{30} -36.4525 q^{31} -32.5165i q^{32} +(-21.5063 + 4.58186i) q^{33} +11.8406 q^{34} +1.15658i q^{35} +(-21.1826 + 9.45494i) q^{36} +9.90726 q^{37} -0.317824i q^{38} +(5.77648 + 27.1136i) q^{39} +1.51182 q^{40} -38.7065i q^{41} +(21.0033 - 4.47469i) q^{42} -39.2418 q^{43} +18.8917i q^{44} +(-0.706931 - 1.58379i) q^{45} +11.0945 q^{46} +17.9574i q^{47} +(0.595680 + 2.79600i) q^{48} -12.9805 q^{49} -29.7735i q^{50} +(-29.1287 + 6.20581i) q^{51} +23.8173 q^{52} -23.3905i q^{53} +(-26.0263 + 18.9653i) q^{54} -1.41250 q^{55} -47.0827i q^{56} +(0.166575 + 0.781867i) q^{57} -34.1490 q^{58} +7.68115i q^{59} +(-1.45739 + 0.310494i) q^{60} -71.8344 q^{61} +43.4774i q^{62} +(-49.3241 + 22.0161i) q^{63} -34.9712 q^{64} +1.78078i q^{65} +(5.46485 + 25.6508i) q^{66} -58.1753 q^{67} +25.5875i q^{68} +(-27.2931 + 5.81473i) q^{69} +1.37947 q^{70} -25.1511i q^{71} +(28.7782 + 64.4738i) q^{72} -68.2547 q^{73} -11.8165i q^{74} +(15.6046 + 73.2448i) q^{75} +0.686812 q^{76} +43.9898i q^{77} +(32.3387 - 6.88968i) q^{78} +72.5230 q^{79} +0.183637i q^{80} +(54.0864 - 60.2964i) q^{81} -46.1657 q^{82} -85.5760i q^{83} +(9.66975 + 45.3878i) q^{84} -1.91314 q^{85} +46.8042i q^{86} +(84.0088 - 17.8979i) q^{87} +57.5011 q^{88} -94.8794i q^{89} +(-1.88900 + 0.843166i) q^{90} +55.4591 q^{91} +23.9750i q^{92} +(-22.7869 - 106.957i) q^{93} +21.4181 q^{94} +0.0513519i q^{95} +(95.4084 - 20.3265i) q^{96} -150.163 q^{97} +15.4821i q^{98} +(-26.8877 - 60.2385i) 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 $

$2$		−	1.19271i		−	0.596357i	−0.954510	−	0.298178i	$-0.903621\pi$
$2$		−	1.19271i		−	0.596357i	0.954510	−	0.298178i	$-0.0963790\pi$
$3$	0.625113	+	2.93415i	0.208371	+	0.978050i
$3$	0.625113	+	2.93415i	0.208371	+	0.978050i
$4$	2.57744			0.644359
$5$			0.192711i			0.0385422i	0.999814	+	0.0192711i	$0.00613456\pi$
$5$			0.192711i			0.0385422i	−0.999814	+	0.0192711i	$0.993865\pi$
$6$	3.49960	−	0.745581i	0.583266	−	0.124263i
$7$	6.00162			0.857375			0.428687	−	0.903453i	$-0.358976\pi$
$7$	6.00162			0.857375			0.428687	+	0.903453i	$0.358976\pi$
$8$		−	7.84499i		−	0.980624i
$9$	−8.21847	+	3.66835i	−0.913163	+	0.407595i
$10$	0.229849			0.0229849
$11$			7.32965i			0.666332i	0.942868	+	0.333166i	$0.108117\pi$
$11$			7.32965i			0.666332i	−0.942868	+	0.333166i	$0.891883\pi$
$12$	1.61119	+	7.56258i	0.134266	+	0.630215i
$13$	9.24069			0.710822			0.355411	−	0.934710i	$-0.384341\pi$
$13$	9.24069			0.710822			0.355411	+	0.934710i	$0.384341\pi$
$14$		−	7.15821i		−	0.511301i
$15$	−0.565443	+	0.120466i	−0.0376962	+	0.00803108i
$16$	0.952916			0.0595572
$17$			9.92749i			0.583970i	0.956423	+	0.291985i	$0.0943157\pi$
$17$			9.92749i			0.583970i	−0.956423	+	0.291985i	$0.905684\pi$
$18$	4.37529	+	9.80227i	0.243072	+	0.544571i
$19$	0.266471			0.0140248			0.00701240	−	0.999975i	$-0.497768\pi$
$19$	0.266471			0.0140248			0.00701240	+	0.999975i	$0.497768\pi$
$20$			0.496700i			0.0248350i
$21$	3.75169	+	17.6097i	0.178652	+	0.838555i
$22$	8.74217			0.397371
$23$			9.30188i			0.404430i	0.979341	+	0.202215i	$0.0648139\pi$
$23$			9.30188i			0.404430i	−0.979341	+	0.202215i	$0.935186\pi$
$24$	23.0184	−	4.90401i	0.959099	−	0.204334i
$25$	24.9629			0.998515
$26$		−	11.0215i		−	0.423903i
$27$	−15.9010	−	21.8211i	−0.588925	−	0.808188i
$28$	15.4688			0.552457
$29$		−	28.6314i		−	0.987290i	−0.869664	−	0.493645i	$-0.835664\pi$
$29$		−	28.6314i		−	0.987290i	0.869664	−	0.493645i	$-0.164336\pi$
$30$	0.143682	+	0.674411i	0.00478938	+	0.0224804i
$31$	−36.4525			−1.17589			−0.587944	−	0.808902i	$-0.700062\pi$
$31$	−36.4525			−1.17589			−0.587944	+	0.808902i	$0.700062\pi$
$32$		−	32.5165i		−	1.01614i
$33$	−21.5063	+	4.58186i	−0.651706	+	0.138844i
$34$	11.8406			0.348254
$35$			1.15658i			0.0330451i
$36$	−21.1826	+	9.45494i	−0.588405	+	0.262637i
$37$	9.90726			0.267764			0.133882	−	0.990997i	$-0.457256\pi$
$37$	9.90726			0.267764			0.133882	+	0.990997i	$0.457256\pi$
$38$		−	0.317824i		−	0.00836378i
$39$	5.77648	+	27.1136i	0.148115	+	0.695219i
$40$	1.51182			0.0377954
$41$		−	38.7065i		−	0.944061i	−0.881582	−	0.472030i	$-0.843521\pi$
$41$		−	38.7065i		−	0.944061i	0.881582	−	0.472030i	$-0.156479\pi$
$42$	21.0033	−	4.47469i	0.500078	−	0.106540i
$43$	−39.2418			−0.912599			−0.456300	−	0.889826i	$-0.650825\pi$
$43$	−39.2418			−0.912599			−0.456300	+	0.889826i	$0.650825\pi$
$44$			18.8917i			0.429357i
$45$	−0.706931	−	1.58379i	−0.0157096	−	0.0351953i
$46$	11.0945			0.241184
$47$			17.9574i			0.382073i	0.981583	+	0.191037i	$0.0611849\pi$
$47$			17.9574i			0.382073i	−0.981583	+	0.191037i	$0.938815\pi$
$48$	0.595680	+	2.79600i	0.0124100	+	0.0582499i
$49$	−12.9805			−0.264909
$50$		−	29.7735i		−	0.595471i
$51$	−29.1287	+	6.20581i	−0.571152	+	0.121682i
$52$	23.8173			0.458024
$53$		−	23.3905i		−	0.441331i	−0.975350	−	0.220665i	$-0.929177\pi$
$53$		−	23.3905i		−	0.441331i	0.975350	−	0.220665i	$-0.0708229\pi$
$54$	−26.0263	+	18.9653i	−0.481968	+	0.351209i
$55$	−1.41250			−0.0256819
$56$		−	47.0827i		−	0.840762i
$57$	0.166575	+	0.781867i	0.00292236	+	0.0137170i
$58$	−34.1490			−0.588777
$59$			7.68115i			0.130189i
$59$			7.68115i			0.130189i
$60$	−1.45739	+	0.310494i	−0.0242899	+	0.00517489i
$61$	−71.8344			−1.17761			−0.588807	−	0.808274i	$-0.700402\pi$
$61$	−71.8344			−1.17761			−0.588807	+	0.808274i	$0.700402\pi$
$62$			43.4774i			0.701248i
$63$	−49.3241	+	22.0161i	−0.782923	+	0.349461i
$64$	−34.9712			−0.546425
$65$			1.78078i			0.0273966i
$66$	5.46485	+	25.6508i	0.0828007	+	0.388649i
$67$	−58.1753			−0.868288			−0.434144	−	0.900843i	$-0.642949\pi$
$67$	−58.1753			−0.868288			−0.434144	+	0.900843i	$0.642949\pi$
$68$			25.5875i			0.376286i
$69$	−27.2931	+	5.81473i	−0.395552	+	0.0842714i
$70$	1.37947			0.0197067
$71$		−	25.1511i		−	0.354241i	−0.984189	−	0.177121i	$-0.943322\pi$
$71$		−	25.1511i		−	0.354241i	0.984189	−	0.177121i	$-0.0566783\pi$
$72$	28.7782	+	64.4738i	0.399697	+	0.895470i
$73$	−68.2547			−0.934996			−0.467498	−	0.883994i	$-0.654844\pi$
$73$	−68.2547			−0.934996			−0.467498	+	0.883994i	$0.654844\pi$
$74$		−	11.8165i		−	0.159683i
$75$	15.6046	+	73.2448i	0.208062	+	0.976597i
$76$	0.686812			0.00903701
$77$			43.9898i			0.571296i
$78$	32.3387	−	6.88968i	0.414599	−	0.0883292i
$79$	72.5230			0.918012			0.459006	−	0.888433i	$-0.348206\pi$
$79$	72.5230			0.918012			0.459006	+	0.888433i	$0.348206\pi$
$80$			0.183637i			0.00229547i
$81$	54.0864	−	60.2964i	0.667733	−	0.744401i
$82$	−46.1657			−0.562997
$83$		−	85.5760i		−	1.03104i	−0.856879	−	0.515518i	$-0.827599\pi$
$83$		−	85.5760i		−	1.03104i	0.856879	−	0.515518i	$-0.172401\pi$
$84$	9.66975	+	45.3878i	0.115116	+	0.540330i
$85$	−1.91314			−0.0225075
$86$			46.8042i			0.544235i
$87$	84.0088	−	17.8979i	0.965619	−	0.205723i
$88$	57.5011			0.653421
$89$		−	94.8794i		−	1.06606i	−0.846096	−	0.533030i	$-0.821053\pi$
$89$		−	94.8794i		−	1.06606i	0.846096	−	0.533030i	$-0.178947\pi$
$90$	−1.88900	+	0.843166i	−0.0209889	+	0.00936851i
$91$	55.4591			0.609441
$92$			23.9750i			0.260598i
$93$	−22.7869	−	106.957i	−0.245021	−	1.15008i
$94$	21.4181			0.227852
$95$			0.0513519i			0.000540546i
$96$	95.4084	−	20.3265i	0.993837	−	0.211735i
$97$	−150.163			−1.54807			−0.774034	−	0.633144i	$-0.781764\pi$
$97$	−150.163			−1.54807			−0.774034	+	0.633144i	$0.781764\pi$
$98$			15.4821i			0.157980i
$99$	−26.8877	−	60.2385i	−0.271593	−	0.608470i
$100$	64.3402			0.643402
$101$			106.598i			1.05543i	0.849422	+	0.527714i	$0.176951\pi$
$101$			106.598i			1.05543i	−0.849422	+	0.527714i	$0.823049\pi$
$102$	7.40175	+	34.7422i	0.0725661	+	0.340610i
$103$	45.3370			0.440165			0.220083	−	0.975481i	$-0.429367\pi$
$103$	45.3370			0.440165			0.220083	+	0.975481i	$0.429367\pi$
$104$		−	72.4931i		−	0.697049i
$105$	−3.39357	+	0.722992i	−0.0323197	+	0.00688564i
$106$	−27.8982			−0.263190
$107$		−	27.3786i		−	0.255875i	−0.991782	−	0.127938i	$-0.959164\pi$
$107$		−	27.3786i		−	0.255875i	0.991782	−	0.127938i	$-0.0408357\pi$
$108$	−40.9837	−	56.2424i	−0.379479	−	0.520763i
$109$	170.671			1.56579			0.782896	−	0.622152i	$-0.213741\pi$
$109$	170.671			1.56579			0.782896	+	0.622152i	$0.213741\pi$
$110$			1.68471i			0.0153156i
$111$	6.19316	+	29.0694i	0.0557943	+	0.261886i
$112$	5.71904			0.0510628
$113$		−	172.039i		−	1.52247i	−0.648477	−	0.761234i	$-0.724594\pi$
$113$		−	172.039i		−	1.52247i	0.648477	−	0.761234i	$-0.275406\pi$
$114$	0.932542	−	0.198676i	0.00818020	−	0.00174277i
$115$	−1.79257			−0.0155876
$116$		−	73.7956i		−	0.636169i
$117$	−75.9443	+	33.8981i	−0.649096	+	0.289727i
$118$	9.16140			0.0776390
$119$			59.5810i			0.500681i
$120$	0.945056	+	4.43589i	0.00787547	+	0.0369658i
$121$	67.2762			0.556002
$122$			85.6779i			0.702278i
$123$	113.571	−	24.1959i	0.923339	−	0.196715i
$124$	−93.9540			−0.757693
$125$			9.62839i			0.0770271i
$126$	26.2588	+	58.8295i	0.208404	+	0.466901i
$127$	70.8372			0.557773			0.278887	−	0.960324i	$-0.410035\pi$
$127$	70.8372			0.557773			0.278887	+	0.960324i	$0.410035\pi$
$128$		−	88.3555i		−	0.690277i
$129$	−24.5306	−	115.141i	−0.190159	−	0.892568i
$130$	2.12396			0.0163382
$131$			231.671i			1.76848i	0.467035	+	0.884239i	$0.345322\pi$
$131$			231.671i			1.76848i	−0.467035	+	0.884239i	$0.654678\pi$
$132$	−55.4311	+	11.8095i	−0.419932	+	0.0894655i
$133$	1.59926			0.0120245
$134$			69.3865i			0.517810i
$135$	4.20516	−	3.06429i	0.0311493	−	0.0226984i
$136$	77.8811			0.572655
$137$			57.1816i			0.417384i	0.977981	+	0.208692i	$0.0669206\pi$
$137$			57.1816i			0.417384i	−0.977981	+	0.208692i	$0.933079\pi$
$138$	6.93530	+	32.5528i	0.0502558	+	0.235890i
$139$	274.608			1.97560			0.987799	−	0.155735i	$-0.0497746\pi$
$139$	274.608			1.97560			0.987799	+	0.155735i	$0.0497746\pi$
$140$			2.98101i			0.0212929i
$141$	−52.6898	+	11.2254i	−0.373686	+	0.0796130i
$142$	−29.9981			−0.211254
$143$			67.7310i			0.473643i
$144$	−7.83150	+	3.49563i	−0.0543855	+	0.0242752i
$145$	5.51758			0.0380523
$146$			81.4083i			0.557591i
$147$	−8.11430	−	38.0868i	−0.0551993	−	0.259094i
$148$	25.5353			0.172536
$149$			195.063i			1.30915i	0.755997	+	0.654576i	$0.227153\pi$
$149$			195.063i			1.30915i	−0.755997	+	0.654576i	$0.772847\pi$
$150$	87.3600	−	18.6118i	0.582400	−	0.124079i
$151$	−23.0021			−0.152332			−0.0761659	−	0.997095i	$-0.524268\pi$
$151$	−23.0021			−0.152332			−0.0761659	+	0.997095i	$0.524268\pi$
$152$		−	2.09047i		−	0.0137531i
$153$	−36.4175	−	81.5888i	−0.238023	−	0.533260i
$154$	52.4672			0.340696
$155$		−	7.02479i		−	0.0453213i
$156$	14.8885	+	69.8834i	0.0954391	+	0.447971i
$157$	−102.811			−0.654848			−0.327424	−	0.944878i	$-0.606180\pi$
$157$	−102.811			−0.654848			−0.327424	+	0.944878i	$0.606180\pi$
$158$		−	86.4991i		−	0.547463i
$159$	68.6313	−	14.6217i	0.431643	−	0.0919606i
$160$	6.26629			0.0391643
$161$			55.8264i			0.346748i
$162$	−71.9164	−	64.5096i	−0.443928	−	0.398207i
$163$	−205.066			−1.25807			−0.629036	−	0.777376i	$-0.716550\pi$
$163$	−205.066			−1.25807			−0.629036	+	0.777376i	$0.716550\pi$
$164$		−	99.7635i		−	0.608314i
$165$	−0.882975	−	4.14450i	−0.00535136	−	0.0251182i
$166$	−102.068			−0.614865
$167$			307.058i			1.83867i	0.393476	+	0.919335i	$0.371272\pi$
$167$			307.058i			1.83867i	−0.393476	+	0.919335i	$0.628728\pi$
$168$	138.148	−	29.4320i	0.822307	−	0.175191i
$169$	−83.6097			−0.494732
$170$			2.28182i			0.0134225i
$171$	−2.18999	+	0.977510i	−0.0128069	+	0.00571643i
$172$	−101.143			−0.588041
$173$			6.42487i			0.0371380i	0.999828	+	0.0185690i	$0.00591103\pi$
$173$			6.42487i			0.0371380i	−0.999828	+	0.0185690i	$0.994089\pi$
$174$	−21.3470	−	100.198i	−0.122684	−	0.575853i
$175$	149.818			0.856101
$176$			6.98454i			0.0396849i
$177$	−22.5376	+	4.80159i	−0.127331	+	0.0271276i
$178$	−113.164			−0.635752
$179$			219.882i			1.22839i	0.789153	+	0.614196i	$0.210520\pi$
$179$			219.882i			1.22839i	−0.789153	+	0.614196i	$0.789480\pi$
$180$	−1.82207	−	4.08211i	−0.0101226	−	0.0226784i
$181$	−61.5411			−0.340006			−0.170003	−	0.985444i	$-0.554378\pi$
$181$	−61.5411			−0.340006			−0.170003	+	0.985444i	$0.554378\pi$
$182$		−	66.1468i		−	0.363444i
$183$	−44.9046	−	210.773i	−0.245381	−	1.15176i
$184$	72.9732			0.396593
$185$			1.90924i			0.0103202i
$186$	−127.569	+	27.1783i	−0.685855	+	0.146120i
$187$	−72.7650			−0.389118
$188$			46.2841i			0.246192i
$189$	−95.4316	−	130.962i	−0.504929	−	0.692920i
$190$	0.0612481			0.000322358
$191$		−	32.8865i		−	0.172181i	−0.996287	−	0.0860904i	$-0.972563\pi$
$191$		−	32.8865i		−	0.172181i	0.996287	−	0.0860904i	$-0.0274374\pi$
$192$	−21.8610	−	102.611i	−0.113859	−	0.534431i
$193$	−117.139			−0.606937			−0.303468	−	0.952841i	$-0.598145\pi$
$193$	−117.139			−0.606937			−0.303468	+	0.952841i	$0.598145\pi$
$194$			179.101i			0.923200i
$195$	−5.22508	+	1.11319i	−0.0267953	+	0.00570867i
$196$	−33.4565			−0.170696
$197$		−	21.5523i		−	0.109402i	−0.998503	−	0.0547012i	$-0.982579\pi$
$197$		−	21.5523i		−	0.109402i	0.998503	−	0.0547012i	$-0.0174206\pi$
$198$	−71.8472	+	32.0693i	−0.362865	+	0.161966i
$199$	169.225			0.850378			0.425189	−	0.905105i	$-0.360208\pi$
$199$	169.225			0.850378			0.425189	+	0.905105i	$0.360208\pi$
$200$		−	195.833i		−	0.979167i
$201$	−36.3662	−	170.695i	−0.180926	−	0.849229i
$202$	127.141			0.629411
$203$		−	171.835i		−	0.846477i
$204$	−75.0775	+	15.9951i	−0.368027	+	0.0784072i
$205$	7.45916			0.0363862
$206$		−	54.0741i		−	0.262495i
$207$	−34.1226	−	76.4472i	−0.164843	−	0.369310i
$208$	8.80559			0.0423346
$209$			1.95314i			0.00934517i
$210$	0.862322	+	4.04756i	0.00410630	+	0.0192741i
$211$	201.272			0.953893			0.476947	−	0.878932i	$-0.341743\pi$
$211$	201.272			0.953893			0.476947	+	0.878932i	$0.341743\pi$
$212$		−	60.2876i		−	0.284375i
$213$	73.7972	−	15.7223i	0.346466	−	0.0738136i
$214$	−32.6549			−0.152593
$215$		−	7.56232i		−	0.0351736i
$216$	−171.186	+	124.743i	−0.792529	+	0.577514i
$217$	−218.774			−1.00818
$218$		−	203.562i		−	0.933771i
$219$	−42.6669	−	200.269i	−0.194826	−	0.914473i
$220$	−3.64064			−0.0165483
$221$			91.7368i			0.415099i
$222$	34.6715	−	7.38667i	0.156178	−	0.0332733i
$223$	25.6377			0.114967			0.0574837	−	0.998346i	$-0.481692\pi$
$223$	25.6377			0.114967			0.0574837	+	0.998346i	$0.481692\pi$
$224$		−	195.152i		−	0.871214i
$225$	−205.156	+	91.5726i	−0.911806	+	0.406989i
$226$	−205.193			−0.907934
$227$		−	87.6856i		−	0.386280i	−0.981171	−	0.193140i	$-0.938133\pi$
$227$		−	87.6856i		−	0.386280i	0.981171	−	0.193140i	$-0.0618672\pi$
$228$	0.429336	+	2.01521i	0.00188305	+	0.00883864i
$229$	149.568			0.653137			0.326568	−	0.945174i	$-0.394108\pi$
$229$	149.568			0.653137			0.326568	+	0.945174i	$0.394108\pi$
$230$			2.13803i			0.00929577i
$231$	−129.073	+	27.4986i	−0.558756	+	0.119042i
$232$	−224.613			−0.968160
$233$		−	102.416i		−	0.439552i	−0.975550	−	0.219776i	$-0.929467\pi$
$233$		−	102.416i		−	0.439552i	0.975550	−	0.219776i	$-0.0705326\pi$
$234$	40.4307	+	90.5797i	0.172781	+	0.387093i
$235$	−3.46059			−0.0147259
$236$			19.7977i			0.0838884i
$237$	45.3351	+	212.793i	0.191287	+	0.897862i
$238$	71.0631			0.298584
$239$			20.5515i			0.0859896i	0.999075	+	0.0429948i	$0.0136899\pi$
$239$			20.5515i			0.0859896i	−0.999075	+	0.0429948i	$0.986310\pi$
$240$	−0.538819	+	0.114794i	−0.00224508	+	0.000478309i
$241$	391.721			1.62540			0.812700	−	0.582682i	$-0.197997\pi$
$241$	391.721			1.62540			0.812700	+	0.582682i	$0.197997\pi$
$242$		−	80.2413i		−	0.331575i
$243$	210.729	+	121.005i	0.867197	+	0.497965i
$244$	−185.149			−0.758806
$245$		−	2.50149i		−	0.0102102i
$246$	−28.8588	−	135.457i	−0.117312	−	0.550639i
$247$	2.46238			0.00996914
$248$			285.970i			1.15310i
$249$	251.093	−	53.4947i	1.00840	−	0.214838i
$250$	11.4839			0.0459356
$251$			226.561i			0.902634i	0.892364	+	0.451317i	$0.149046\pi$
$251$			226.561i			0.902634i	−0.892364	+	0.451317i	$0.850954\pi$
$252$	−127.130	+	56.7450i	−0.504483	+	0.225178i
$253$	−68.1795			−0.269484
$254$		−	84.4884i		−	0.332632i
$255$	−1.19593	−	5.61343i	−0.00468991	−	0.0220134i
$256$	−245.268			−0.958077
$257$			430.021i			1.67323i	0.547788	+	0.836617i	$0.315470\pi$
$257$			430.021i			1.67323i	−0.547788	+	0.836617i	$0.684530\pi$
$258$	−137.330	+	29.2579i	−0.532289	+	0.113403i
$259$	59.4597			0.229574
$260$			4.58985i			0.0176533i
$261$	105.030	+	235.306i	0.402414	+	0.901556i
$262$	276.317			1.05464
$263$		−	90.6588i		−	0.344710i	−0.985035	−	0.172355i	$-0.944862\pi$
$263$		−	90.6588i		−	0.344710i	0.985035	−	0.172355i	$-0.0551377\pi$
$264$	35.9447	+	168.717i	0.136154	+	0.639078i
$265$	4.50761			0.0170098
$266$		−	1.90746i		−	0.00717089i
$267$	278.390	−	59.3104i	1.04266	−	0.222136i
$268$	−149.943			−0.559489
$269$		−	177.836i		−	0.661099i	−0.943789	−	0.330549i	$-0.892766\pi$
$269$		−	177.836i		−	0.661099i	0.943789	−	0.330549i	$-0.107234\pi$
$270$	−3.65482	−	5.01555i	−0.0135364	−	0.0185761i
$271$	−430.864			−1.58990			−0.794952	−	0.606672i	$-0.792504\pi$
$271$	−430.864			−1.58990			−0.794952	+	0.606672i	$0.792504\pi$
$272$			9.46006i			0.0347796i
$273$	34.6682	+	162.725i	0.126990	+	0.596063i
$274$	68.2012			0.248910
$275$			182.969i			0.665342i
$276$	−70.3462	+	14.9871i	−0.254878	+	0.0543010i
$277$	−310.975			−1.12265			−0.561327	−	0.827594i	$-0.689709\pi$
$277$	−310.975			−1.12265			−0.561327	+	0.827594i	$0.689709\pi$
$278$		−	327.529i		−	1.17816i
$279$	299.584	−	133.721i	1.07378	−	0.479285i
$280$	9.07335			0.0324048
$281$		−	113.386i		−	0.403509i	−0.979436	−	0.201755i	$-0.935336\pi$
$281$		−	113.386i		−	0.403509i	0.979436	−	0.201755i	$-0.0646644\pi$
$282$	13.3887	+	62.8438i	0.0474777	+	0.222850i
$283$	331.753			1.17227			0.586135	−	0.810213i	$-0.300649\pi$
$283$	331.753			1.17227			0.586135	+	0.810213i	$0.300649\pi$
$284$		−	64.8254i		−	0.228258i
$285$	−0.150674	+	0.0321008i	−0.000528681	+	0.000112634i
$286$	80.7836			0.282460
$287$		−	232.302i		−	0.809414i
$288$	119.282	+	267.236i	0.414174	+	0.927903i
$289$	190.445			0.658979
$290$		−	6.58089i		−	0.0226927i
$291$	−93.8686	−	440.599i	−0.322573	−	1.51409i
$292$	−175.922			−0.602473
$293$			300.701i			1.02628i	0.858304	+	0.513142i	$0.171519\pi$
$293$			300.701i			1.02628i	−0.858304	+	0.513142i	$0.828481\pi$
$294$	−45.4267	+	9.67804i	−0.154512	+	0.0329185i
$295$	−1.48024			−0.00501776
$296$		−	77.7224i		−	0.262576i
$297$	159.941	−	116.548i	0.538521	−	0.392419i
$298$	232.655			0.780721
$299$			85.9558i			0.287477i
$300$	40.2199	+	188.784i	0.134066	+	0.629279i
$301$	−235.514			−0.782439
$302$			27.4349i			0.0908441i
$303$	−312.775	+	66.6359i	−1.03226	+	0.219921i
$304$	0.253925			0.000835278
$305$		−	13.8433i		−	0.0453878i
$306$	−97.3120	+	43.4357i	−0.318013	+	0.141947i
$307$	242.499			0.789899			0.394950	−	0.918703i	$-0.370762\pi$
$307$	242.499			0.789899			0.394950	+	0.918703i	$0.370762\pi$
$308$			113.381i			0.368120i
$309$	28.3408	+	133.026i	0.0917177	+	0.430504i
$310$	−8.37856			−0.0270276
$311$		−	356.024i		−	1.14477i	−0.819985	−	0.572386i	$-0.806018\pi$
$311$		−	356.024i		−	1.14477i	0.819985	−	0.572386i	$-0.193982\pi$
$312$	212.706	−	45.3164i	0.681749	−	0.145245i
$313$	−295.350			−0.943611			−0.471806	−	0.881703i	$-0.656398\pi$
$313$	−295.350			−0.943611			−0.471806	+	0.881703i	$0.656398\pi$
$314$			122.624i			0.390523i
$315$	−4.24273	−	9.50530i	−0.0134690	−	0.0301755i
$316$	186.923			0.591529
$317$			67.0964i			0.211660i	0.994384	+	0.105830i	$0.0337500\pi$
$317$			67.0964i			0.211660i	−0.994384	+	0.105830i	$0.966250\pi$
$318$	−17.4395	−	81.8575i	−0.0548413	−	0.257413i
$319$	209.858			0.657862
$320$		−	6.73934i		−	0.0210604i
$321$	80.3330	−	17.1147i	0.250259	−	0.0533170i
$322$	66.5848			0.206785
$323$			2.64539i			0.00819006i
$324$	139.404	−	155.410i	0.430260	−	0.479661i
$325$	230.674			0.709766
$326$			244.585i			0.750260i
$327$	106.689	+	500.775i	0.326266	+	1.53142i
$328$	−303.652			−0.925769
$329$			107.774i			0.327580i
$330$	−4.94319	+	1.05314i	−0.0149794	+	0.00319132i
$331$	608.621			1.83873			0.919367	−	0.393402i	$-0.128702\pi$
$331$	608.621			1.83873			0.919367	+	0.393402i	$0.128702\pi$
$332$		−	220.567i		−	0.664357i
$333$	−81.4225	+	36.3433i	−0.244512	+	0.109139i
$334$	366.232			1.09650
$335$		−	11.2110i		−	0.0334657i
$336$	3.57505	+	16.7805i	0.0106400	+	0.0499420i
$337$	39.1038			0.116035			0.0580175	−	0.998316i	$-0.481522\pi$
$337$	39.1038			0.116035			0.0580175	+	0.998316i	$0.481522\pi$
$338$			99.7224i			0.295037i
$339$	504.788	−	107.544i	1.48905	−	0.317238i
$340$	−4.93098			−0.0145029
$341$		−	267.184i		−	0.783531i
$342$	1.16589	+	2.61202i	0.00340903	+	0.00763750i
$343$	−371.984			−1.08450
$344$			307.851i			0.894917i
$345$	−1.12056	−	5.25968i	−0.00324800	−	0.0152454i
$346$	7.66303			0.0221475
$347$			294.930i			0.849941i	0.905207	+	0.424971i	$0.139716\pi$
$347$			294.930i			0.849941i	−0.905207	+	0.424971i	$0.860284\pi$
$348$	216.527	−	46.1306i	0.622205	−	0.132559i
$349$	−320.108			−0.917216			−0.458608	−	0.888639i	$-0.651652\pi$
$349$	−320.108			−0.917216			−0.458608	+	0.888639i	$0.651652\pi$
$350$		−	178.689i		−	0.510541i
$351$	−146.936	−	201.642i	−0.418621	−	0.574478i
$352$	238.335			0.677087
$353$		−	428.068i		−	1.21266i	−0.795214	−	0.606329i	$-0.792642\pi$
$353$		−	428.068i		−	1.21266i	0.795214	−	0.606329i	$-0.207358\pi$
$354$	5.72691	+	26.8809i	0.0161777	+	0.0759348i
$355$	4.84690			0.0136532
$356$		−	244.545i		−	0.686926i
$357$	−174.820	+	37.2449i	−0.489691	+	0.104327i
$358$	262.256			0.732560
$359$		−	390.412i		−	1.08750i	−0.839248	−	0.543749i	$-0.817004\pi$
$359$		−	390.412i		−	1.08750i	0.839248	−	0.543749i	$-0.182996\pi$
$360$	−12.4248	+	5.54587i	−0.0345134	+	0.0154052i
$361$	−360.929			−0.999803
$362$			73.4009i			0.202765i
$363$	42.0553	+	197.399i	0.115855	+	0.543798i
$364$	142.942			0.392699
$365$		−	13.1534i		−	0.0360368i
$366$	−251.392	+	53.5584i	−0.686862	+	0.146334i
$367$	287.119			0.782340			0.391170	−	0.920318i	$-0.372070\pi$
$367$	287.119			0.782340			0.391170	+	0.920318i	$0.372070\pi$
$368$			8.86391i			0.0240867i
$369$	141.989	+	318.108i	0.384794	+	0.862081i
$370$	2.27717			0.00615452
$371$		−	140.381i		−	0.378386i
$372$	−58.7319	−	275.675i	−0.157881	−	0.741062i
$373$	−488.823			−1.31052			−0.655259	−	0.755404i	$-0.727441\pi$
$373$	−488.823			−1.31052			−0.655259	+	0.755404i	$0.727441\pi$
$374$			86.7878i			0.232053i
$375$	−28.2511	+	6.01883i	−0.0753363	+	0.0160502i
$376$	140.876			0.374670
$377$		−	264.574i		−	0.701787i
$378$	−156.200	+	113.822i	−0.413227	+	0.301118i
$379$	371.086			0.979119			0.489559	−	0.871970i	$-0.337158\pi$
$379$	371.086			0.979119			0.489559	+	0.871970i	$0.337158\pi$
$380$			0.132356i			0.000348306i
$381$	44.2813	+	207.847i	0.116224	+	0.545530i
$382$	−39.2242			−0.102681
$383$			474.681i			1.23938i	0.784849	+	0.619688i	$0.212741\pi$
$383$			474.681i			1.23938i	−0.784849	+	0.619688i	$0.787259\pi$
$384$	259.248	−	55.2322i	0.675125	−	0.143834i
$385$	−8.47731			−0.0220190
$386$			139.713i			0.361951i
$387$	322.507	−	143.953i	0.833352	−	0.371971i
$388$	−387.034			−0.997511
$389$		−	605.349i		−	1.55617i	−0.628161	−	0.778083i	$-0.716192\pi$
$389$		−	605.349i		−	1.55617i	0.628161	−	0.778083i	$-0.283808\pi$
$390$	1.32772	+	6.23202i	0.00340440	+	0.0159795i
$391$	−92.3443			−0.236175
$392$			101.832i			0.259776i
$393$	−679.756	+	144.820i	−1.72966	+	0.368500i
$394$	−25.7057			−0.0652429
$395$			13.9760i			0.0353822i
$396$	−69.3014	−	155.261i	−0.175004	−	0.392073i
$397$	520.491			1.31106			0.655530	−	0.755169i	$-0.272445\pi$
$397$	520.491			1.31106			0.655530	+	0.755169i	$0.272445\pi$
$398$		−	201.837i		−	0.507128i
$399$	0.999718	+	4.69247i	0.00250556	+	0.0117606i
$400$	23.7875			0.0594687
$401$		−	550.265i		−	1.37223i	−0.727493	−	0.686115i	$-0.759314\pi$
$401$		−	550.265i		−	1.37223i	0.727493	−	0.686115i	$-0.240686\pi$
$402$	−203.590	+	43.3744i	−0.506444	+	0.107897i
$403$	−336.846			−0.835846
$404$			274.750i			0.680074i
$405$	11.6198	+	10.4230i	0.0286908	+	0.0257359i
$406$	−204.950			−0.504802
$407$			72.6168i			0.178420i
$408$	48.6845	+	228.515i	0.119325	+	0.560085i
$409$	−386.335			−0.944585			−0.472293	−	0.881442i	$-0.656573\pi$
$409$	−386.335			−0.944585			−0.472293	+	0.881442i	$0.656573\pi$
$410$		−	8.89664i		−	0.0216991i
$411$	−167.779	+	35.7450i	−0.408222	+	0.0869707i
$412$	116.853			0.283624
$413$			46.0993i			0.111621i
$414$	−91.1796	+	40.6984i	−0.220241	+	0.0983054i
$415$	16.4914			0.0397384
$416$		−	300.475i		−	0.722296i
$417$	171.661	+	805.741i	0.411657	+	1.93223i
$418$	2.32954			0.00557305
$419$			666.029i			1.58957i	0.606893	+	0.794784i	$0.292416\pi$
$419$			666.029i			1.58957i	−0.606893	+	0.794784i	$0.707584\pi$
$420$	−8.74671	+	1.86347i	−0.0208255	+	0.00443682i
$421$	−702.025			−1.66752			−0.833759	−	0.552129i	$-0.813816\pi$
$421$	−702.025			−1.66752			−0.833759	+	0.552129i	$0.813816\pi$
$422$		−	240.059i		−	0.568861i
$423$	−65.8742	−	147.583i	−0.155731	−	0.348895i
$424$	−183.499			−0.432780
$425$			247.819i			0.583103i
$426$	−18.7522	−	88.0188i	−0.0440192	−	0.206617i
$427$	−431.123			−1.00966
$428$		−	70.5667i		−	0.164875i
$429$	−198.733	+	42.3395i	−0.463247	+	0.0986936i
$430$	−9.01967			−0.0209760
$431$		−	709.664i		−	1.64655i	−0.567641	−	0.823276i	$-0.692144\pi$
$431$		−	709.664i		−	1.64655i	0.567641	−	0.823276i	$-0.307856\pi$
$432$	−15.1523	−	20.7936i	−0.0350747	−	0.0481334i
$433$	−52.7210			−0.121757			−0.0608787	−	0.998145i	$-0.519390\pi$
$433$	−52.7210			−0.121757			−0.0608787	+	0.998145i	$0.519390\pi$
$434$			260.935i			0.601232i
$435$	3.44911	+	16.1894i	0.00792900	+	0.0372170i
$436$	439.895			1.00893
$437$			2.47868i			0.00567205i
$438$	−238.864	+	50.8894i	−0.545352	+	0.116186i
$439$	−26.6734			−0.0607594			−0.0303797	−	0.999538i	$-0.509672\pi$
$439$	−26.6734			−0.0607594			−0.0303797	+	0.999538i	$0.509672\pi$
$440$			11.0811i			0.0251843i
$441$	106.680	−	47.6172i	0.241905	−	0.107975i
$442$	109.416			0.247547
$443$			281.292i			0.634972i	0.948263	+	0.317486i	$0.102839\pi$
$443$			281.292i			0.634972i	−0.948263	+	0.317486i	$0.897161\pi$
$444$	15.9625	+	74.9245i	0.0359515	+	0.168749i
$445$	18.2843			0.0410883
$446$		−	30.5785i		−	0.0685616i
$447$	−572.345	+	121.937i	−1.28041	+	0.272789i
$448$	−209.884			−0.468491
$449$		−	813.712i		−	1.81228i	−0.422983	−	0.906138i	$-0.639017\pi$
$449$		−	813.712i		−	1.81228i	0.422983	−	0.906138i	$-0.360983\pi$
$450$	109.220	+	244.693i	0.242711	+	0.543762i
$451$	283.705			0.629058
$452$		−	443.419i		−	0.981016i
$453$	−14.3789	−	67.4916i	−0.0317416	−	0.148988i
$454$	−104.584			−0.230361
$455$			10.6876i			0.0234892i
$456$	6.13374	−	1.30678i	0.0134512	−	0.00286574i
$457$	152.151			0.332935			0.166467	−	0.986047i	$-0.446764\pi$
$457$	152.151			0.332935			0.166467	+	0.986047i	$0.446764\pi$
$458$		−	178.392i		−	0.389502i
$459$	216.629	−	157.857i	0.471958	−	0.343914i
$460$	−4.62024			−0.0100440
$461$			151.258i			0.328108i	0.986451	+	0.164054i	$0.0524571\pi$
$461$			151.258i			0.328108i	−0.986451	+	0.164054i	$0.947543\pi$
$462$	32.7979	+	153.947i	0.0709912	+	0.333218i
$463$	−188.686			−0.407530			−0.203765	−	0.979020i	$-0.565318\pi$
$463$	−188.686			−0.407530			−0.203765	+	0.979020i	$0.565318\pi$
$464$		−	27.2833i		−	0.0588002i
$465$	20.6118	−	4.39129i	0.0443264	−	0.00944364i
$466$	−122.152			−0.262130
$467$		−	555.701i		−	1.18994i	−0.803749	−	0.594968i	$-0.797165\pi$
$467$		−	555.701i		−	1.18994i	0.803749	−	0.594968i	$-0.202835\pi$
$468$	−195.741	+	87.3701i	−0.418251	+	0.186688i
$469$	−349.146			−0.744448
$470$			4.12749i			0.00878190i
$471$	−64.2686	−	301.663i	−0.136451	−	0.640474i
$472$	60.2585			0.127666
$473$		−	287.628i		−	0.608094i
$474$	253.801	−	54.0717i	0.535446	−	0.114075i
$475$	6.65189			0.0140040
$476$			153.566i			0.322618i
$477$	85.8047	+	192.234i	0.179884	+	0.403007i
$478$	24.5121			0.0512805
$479$			264.217i			0.551601i	0.961215	+	0.275800i	$0.0889429\pi$
$479$			264.217i			0.551601i	−0.961215	+	0.275800i	$0.911057\pi$
$480$	3.91714	+	18.3862i	0.00816071	+	0.0383046i
$481$	91.5499			0.190332
$482$		−	467.211i		−	0.969318i
$483$	−163.803	+	34.8978i	−0.339136	+	0.0722522i
$484$	173.400			0.358265
$485$		−	28.9380i		−	0.0596659i
$486$	144.325	−	251.339i	0.296965	−	0.517159i
$487$	−163.268			−0.335252			−0.167626	−	0.985851i	$-0.553610\pi$
$487$	−163.268			−0.335252			−0.167626	+	0.985851i	$0.553610\pi$
$488$			563.541i			1.15480i
$489$	−128.189	−	601.694i	−0.262146	−	1.23046i
$490$	−2.98356			−0.00608890
$491$			884.367i			1.80116i	0.434695	+	0.900578i	$0.356856\pi$
$491$			884.367i			1.80116i	−0.434695	+	0.900578i	$0.643144\pi$
$492$	292.721	−	62.3635i	0.594961	−	0.126755i
$493$	284.238			0.576548
$494$		−	2.93691i		−	0.00594516i
$495$	11.6086	−	5.18156i	0.0234517	−	0.0104678i
$496$	−34.7362			−0.0700326
$497$		−	150.948i		−	0.303717i
$498$	−63.8038	−	299.482i	−0.128120	−	0.601369i
$499$	543.803			1.08979			0.544893	−	0.838506i	$-0.316570\pi$
$499$	543.803			1.08979			0.544893	+	0.838506i	$0.316570\pi$
$500$			24.8165i			0.0496331i
$501$	−900.953	+	191.946i	−1.79831	+	0.383126i
$502$	270.222			0.538292
$503$			243.208i			0.483514i	0.970337	+	0.241757i	$0.0777238\pi$
$503$			243.208i			0.483514i	−0.970337	+	0.241757i	$0.922276\pi$
$504$	172.716	+	386.947i	0.342690	+	0.767753i
$505$	−20.5426			−0.0406785
$506$			81.3186i			0.160709i
$507$	−52.2655	−	245.323i	−0.103088	−	0.483873i
$508$	182.578			0.359406
$509$			807.215i			1.58588i	0.609297	+	0.792942i	$0.291452\pi$
$509$			807.215i			1.58588i	−0.609297	+	0.792942i	$0.708548\pi$
$510$	−6.69521	+	1.42640i	−0.0131279	+	0.00279686i
$511$	−409.639			−0.801642
$512$		−	60.8880i		−	0.118922i
$513$	−4.23715	−	5.81469i	−0.00825955	−	0.0113347i
$514$	512.892			0.997844
$515$			8.73694i			0.0169649i
$516$	−63.2259	−	296.769i	−0.122531	−	0.575134i
$517$	−131.622			−0.254587
$518$		−	70.9183i		−	0.136908i
$519$	−18.8515	+	4.01627i	−0.0363228	+	0.00773848i
$520$	13.9702			0.0268658
$521$			260.497i			0.499993i	0.968247	+	0.249997i	$0.0804295\pi$
$521$			260.497i			0.499993i	−0.968247	+	0.249997i	$0.919570\pi$
$522$	280.653	−	125.271i	0.537649	−	0.239982i
$523$	−619.355			−1.18424			−0.592118	−	0.805851i	$-0.701708\pi$
$523$	−619.355			−1.18424			−0.592118	+	0.805851i	$0.701708\pi$
$524$			597.116i			1.13953i
$525$	93.6530	+	439.587i	0.178387	+	0.837309i
$526$	−108.130			−0.205570
$527$		−	361.882i		−	0.686683i
$528$	−20.4937	+	4.36613i	−0.0388138	+	0.00826918i
$529$	442.475			0.836437
$530$		−	5.37629i		−	0.0101439i
$531$	−28.1771	−	63.1272i	−0.0530643	−	0.118884i
$532$	4.12199			0.00774810
$533$		−	357.675i		−	0.671059i
$534$	−70.7402	−	332.040i	−0.132472	−	0.621797i
$535$	5.27616			0.00986199
$536$			456.385i			0.851465i
$537$	−645.167	+	137.451i	−1.20143	+	0.255961i
$538$	−212.107			−0.394250
$539$		−	95.1428i		−	0.176517i
$540$	10.8385	−	7.89801i	0.0200713	−	0.0146259i
$541$	74.4432			0.137603			0.0688015	−	0.997630i	$-0.478082\pi$
$541$	74.4432			0.137603			0.0688015	+	0.997630i	$0.478082\pi$
$542$			513.897i			0.948150i
$543$	−38.4702	−	180.571i	−0.0708474	−	0.332543i
$544$	322.808			0.593396
$545$			32.8902i			0.0603491i
$546$	194.085	−	41.3492i	0.355466	−	0.0757312i
$547$	−451.226			−0.824911			−0.412455	−	0.910978i	$-0.635329\pi$
$547$	−451.226			−0.824911			−0.412455	+	0.910978i	$0.635329\pi$
$548$			147.382i			0.268945i
$549$	590.369	−	263.514i	1.07535	−	0.479989i
$550$	218.230			0.396781
$551$		−	7.62945i		−	0.0138465i
$552$	45.6165	+	214.114i	0.0826386	+	0.387888i
$553$	435.255			0.787080
$554$			370.904i			0.669502i
$555$	−5.60199	+	1.19349i	−0.0100937	+	0.00215043i
$556$	707.785			1.27299
$557$		−	731.546i		−	1.31337i	−0.754166	−	0.656684i	$-0.771958\pi$
$557$		−	731.546i		−	1.31337i	0.754166	−	0.656684i	$-0.228042\pi$
$558$	−159.490	−	357.317i	−0.285825	−	0.640354i
$559$	−362.621			−0.648696
$560$			1.10212i			0.00196807i
$561$	−45.4864	−	213.503i	−0.0810809	−	0.380577i
$562$	−135.237			−0.240635
$563$			938.349i			1.66669i	0.552750	+	0.833347i	$0.313579\pi$
$563$			938.349i			1.66669i	−0.552750	+	0.833347i	$0.686421\pi$
$564$	−135.805	+	28.9328i	−0.240788	+	0.0512993i
$565$	33.1538			0.0586792
$566$		−	395.686i		−	0.699091i
$567$	324.606	−	361.876i	0.572498	−	0.638230i
$568$	−197.310			−0.347377
$569$			262.597i			0.461505i	0.973012	+	0.230753i	$0.0741188\pi$
$569$			262.597i			0.461505i	−0.973012	+	0.230753i	$0.925881\pi$
$570$	0.0382870	+	0.179711i	6.71702e−5	+	0.000315283i
$571$	−305.783			−0.535521			−0.267761	−	0.963485i	$-0.586284\pi$
$571$	−305.783			−0.535521			−0.267761	+	0.963485i	$0.586284\pi$
$572$			174.572i			0.305196i
$573$	96.4940	−	20.5578i	0.168401	−	0.0358775i
$574$	−277.069			−0.482699
$575$			232.202i			0.403829i
$576$	287.410	−	128.287i	0.498975	−	0.222720i
$577$	631.668			1.09475			0.547373	−	0.836889i	$-0.315628\pi$
$577$	631.668			1.09475			0.547373	+	0.836889i	$0.315628\pi$
$578$		−	227.146i		−	0.392986i
$579$	−73.2250	−	343.703i	−0.126468	−	0.593615i
$580$	14.2212			0.0245193
$581$		−	513.595i		−	0.883984i
$582$	−525.509	+	111.958i	−0.902936	+	0.192368i
$583$	171.444			0.294073
$584$			535.458i			0.916880i
$585$	−6.53253	−	14.6353i	−0.0111667	−	0.0250176i
$586$	358.650			0.612031
$587$		−	588.946i		−	1.00331i	−0.865066	−	0.501657i	$-0.832724\pi$
$587$		−	588.946i		−	1.00331i	0.865066	−	0.501657i	$-0.167276\pi$
$588$	−20.9141	−	98.1663i	−0.0355682	−	0.166950i
$589$	−9.71354			−0.0164916
$590$			1.76550i			0.00299238i
$591$	63.2376	−	13.4726i	0.107001	−	0.0227963i
$592$	9.44079			0.0159473
$593$		−	123.195i		−	0.207749i	−0.994590	−	0.103875i	$-0.966876\pi$
$593$		−	123.195i		−	0.207749i	0.994590	−	0.103875i	$-0.0331241\pi$
$594$	−139.009	−	190.764i	−0.234022	−	0.321151i
$595$	−11.4819			−0.0192973
$596$			502.764i			0.843563i
$597$	105.785	+	496.532i	0.177194	+	0.831712i
$598$	102.521			0.171439
$599$			1039.18i			1.73485i	0.497564	+	0.867427i	$0.334228\pi$
$599$			1039.18i			1.73485i	−0.497564	+	0.867427i	$0.665772\pi$
$600$	574.605	−	122.418i	0.957675	−	0.204030i
$601$	703.372			1.17034			0.585168	−	0.810912i	$-0.301029\pi$
$601$	703.372			1.17034			0.585168	+	0.810912i	$0.301029\pi$
$602$			280.901i			0.466613i
$603$	478.112	−	213.408i	0.792889	−	0.353910i
$604$	−59.2864			−0.0981564
$605$			12.9649i			0.0214295i
$606$	79.4775	+	373.051i	0.131151	+	0.615595i
$607$	−290.004			−0.477766			−0.238883	−	0.971048i	$-0.576781\pi$
$607$	−290.004			−0.477766			−0.238883	+	0.971048i	$0.576781\pi$
$608$		−	8.66472i		−	0.0142512i
$609$	504.189	−	107.416i	0.827897	−	0.176381i
$610$	−16.5111			−0.0270673
$611$			165.939i			0.271586i
$612$	−93.8638	−	210.290i	−0.153372	−	0.343611i
$613$	−637.904			−1.04063			−0.520313	−	0.853976i	$-0.674185\pi$
$613$	−637.904			−1.04063			−0.520313	+	0.853976i	$0.674185\pi$
$614$		−	289.232i		−	0.471062i
$615$	4.66282	+	21.8863i	0.00758182	+	0.0355875i
$616$	345.100			0.560227
$617$		−	871.415i		−	1.41234i	−0.708041	−	0.706171i	$-0.750421\pi$
$617$		−	871.415i		−	1.41234i	0.708041	−	0.706171i	$-0.249579\pi$
$618$	158.661	−	33.8024i	0.256734	−	0.0546965i
$619$	529.878			0.856023			0.428011	−	0.903773i	$-0.359214\pi$
$619$	529.878			0.856023			0.428011	+	0.903773i	$0.359214\pi$
$620$		−	18.1060i		−	0.0292032i
$621$	202.977	−	147.909i	0.326855	−	0.238179i
$622$	−424.634			−0.682692
$623$		−	569.430i		−	0.914013i
$624$	5.50449	+	25.8369i	0.00882130	+	0.0414053i
$625$	622.216			0.995546
$626$			352.268i			0.562729i
$627$	−5.73081	+	1.22093i	−0.00914004	+	0.00194726i
$628$	−264.989			−0.421957
$629$			98.3543i			0.156366i
$630$	−11.3371	+	5.06037i	−0.0179954	+	0.00803233i
$631$	79.4381			0.125892			0.0629462	−	0.998017i	$-0.479950\pi$
$631$	79.4381			0.125892			0.0629462	+	0.998017i	$0.479950\pi$
$632$		−	568.942i		−	0.900225i
$633$	125.817	+	590.561i	0.198764	+	0.932955i
$634$	80.0267			0.126225
$635$			13.6511i			0.0214978i
$636$	176.893	−	37.6866i	0.278133	−	0.0592556i
$637$	−119.949			−0.188303
$638$		−	250.301i		−	0.392321i
$639$	92.2632	+	206.704i	0.144387	+	0.323480i
$640$	17.0271			0.0266048
$641$			617.174i			0.962830i	0.876493	+	0.481415i	$0.159877\pi$
$641$			617.174i			0.962830i	−0.876493	+	0.481415i	$0.840123\pi$
$642$	−20.4130	−	95.8142i	−0.0317959	−	0.149243i
$643$	478.843			0.744701			0.372351	−	0.928092i	$-0.378552\pi$
$643$	478.843			0.744701			0.372351	+	0.928092i	$0.378552\pi$
$644$			143.889i			0.223430i
$645$	22.1890	−	4.72730i	0.0344015	−	0.00732915i
$646$	3.15519			0.00488420
$647$			952.499i			1.47218i	0.676885	+	0.736088i	$0.263329\pi$
$647$			952.499i			1.47218i	−0.676885	+	0.736088i	$0.736671\pi$
$648$	−473.025	−	424.307i	−0.729977	−	0.654795i
$649$	−56.3001			−0.0867490
$650$		−	275.128i		−	0.423274i
$651$	−136.759	−	641.916i	−0.210075	−	0.986046i
$652$	−528.544			−0.810650
$653$			838.343i			1.28383i	0.766775	+	0.641916i	$0.221860\pi$
$653$			838.343i			1.28383i	−0.766775	+	0.641916i	$0.778140\pi$
$654$	597.281	−	127.249i	0.913274	−	0.194571i
$655$	−44.6454			−0.0681610
$656$		−	36.8840i		−	0.0562256i
$657$	560.949	−	250.382i	0.853804	−	0.381099i
$658$	128.543			0.195354
$659$		−	697.326i		−	1.05816i	−0.848572	−	0.529079i	$-0.822537\pi$
$659$		−	697.326i		−	1.05816i	0.848572	−	0.529079i	$-0.177463\pi$
$660$	−2.27581	−	10.6822i	−0.00344820	−	0.0161851i
$661$	122.681			0.185599			0.0927997	−	0.995685i	$-0.470418\pi$
$661$	122.681			0.185599			0.0927997	+	0.995685i	$0.470418\pi$
$662$		−	725.910i		−	1.09654i
$663$	−269.170	+	57.3459i	−0.405987	+	0.0864946i
$664$	−671.343			−1.01106
$665$			0.308195i			0.000463451i
$666$	43.3472	+	97.1137i	0.0650858	+	0.145816i
$667$	266.326			0.399289
$668$			791.422i			1.18476i
$669$	16.0265	+	75.2250i	0.0239559	+	0.112444i
$670$	−13.3715			−0.0199575
$671$		−	526.521i		−	0.784681i
$672$	572.605	−	121.992i	0.852091	−	0.181536i
$673$	415.437			0.617291			0.308645	−	0.951177i	$-0.400124\pi$
$673$	415.437			0.617291			0.308645	+	0.951177i	$0.400124\pi$
$674$		−	46.6396i		−	0.0691982i
$675$	−396.934	−	544.717i	−0.588050	−	0.806987i
$676$	−215.499			−0.318785
$677$		−	605.231i		−	0.893990i	−0.894536	−	0.446995i	$-0.852494\pi$
$677$		−	605.231i		−	0.893990i	0.894536	−	0.446995i	$-0.147506\pi$
$678$	−128.269	−	602.067i	−0.189187	−	0.888004i
$679$	−901.219			−1.32727
$680$			15.0085i			0.0220714i
$681$	257.283	−	54.8134i	0.377801	−	0.0804896i
$682$	−318.674			−0.467264
$683$			155.804i			0.228118i	0.993474	+	0.114059i	$0.0363853\pi$
$683$			155.804i			0.228118i	−0.993474	+	0.114059i	$0.963615\pi$
$684$	−5.64455	+	2.51947i	−0.00825226	+	0.00368343i
$685$	−11.0195			−0.0160869
$686$			443.670i			0.646749i
$687$	93.4971	+	438.856i	0.136095	+	0.638800i
$688$	−37.3941			−0.0543519
$689$		−	216.145i		−	0.313708i
$690$	−6.27329	+	1.33651i	−0.00909172	+	0.00193697i
$691$	−58.8402			−0.0851523			−0.0425761	−	0.999093i	$-0.513557\pi$
$691$	−58.8402			−0.0851523			−0.0425761	+	0.999093i	$0.513557\pi$
$692$			16.5597i			0.0239302i
$693$	−161.370	−	361.529i	−0.232857	−	0.521686i
$694$	351.766			0.506868
$695$			52.9200i			0.0761438i
$696$	−140.409	−	659.049i	−0.201737	−	0.946909i
$697$	384.258			0.551303
$698$			381.798i			0.546988i
$699$	300.503	−	64.0213i	0.429903	−	0.0915899i
$700$	386.145			0.551636
$701$			1175.46i			1.67683i	0.545032	+	0.838416i	$0.316518\pi$
$701$			1175.46i			1.67683i	−0.545032	+	0.838416i	$0.683482\pi$
$702$	−240.501	+	175.252i	−0.342594	+	0.249647i
$703$	2.64000			0.00375534
$704$		−	256.327i		−	0.364101i
$705$	−2.16326	−	10.1539i	−0.00306846	−	0.0144027i
$706$	−510.562			−0.723176
$707$			639.762i			0.904897i
$708$	−58.0893	+	12.3758i	−0.0820470	+	0.0174799i
$709$	1266.22			1.78592			0.892961	−	0.450135i	$-0.148624\pi$
$709$	1266.22			1.78592			0.892961	+	0.450135i	$0.148624\pi$
$710$		−	5.78096i		−	0.00814219i
$711$	−596.028	+	266.040i	−0.838295	+	0.374177i
$712$	−744.328			−1.04540
$713$		−	339.077i		−	0.475564i
$714$	44.4225	+	208.510i	0.0622164	+	0.292030i
$715$	−13.0525			−0.0182552
$716$			566.732i			0.791525i
$717$	−60.3012	+	12.8470i	−0.0841021	+	0.0179177i
$718$	−465.649			−0.648537
$719$			419.586i			0.583570i	0.956484	+	0.291785i	$0.0942491\pi$
$719$			419.586i			0.583570i	−0.956484	+	0.291785i	$0.905751\pi$
$720$	−0.673646	−	1.50922i	−0.000935619	−	0.00209613i
$721$	272.096			0.377386
$722$			430.485i			0.596239i
$723$	244.870	+	1149.37i	0.338686	+	1.58972i
$724$	−158.618			−0.219086
$725$		−	714.722i		−	0.985823i
$726$	235.440	−	50.1599i	0.324297	−	0.0690907i
$727$	−862.945			−1.18699			−0.593497	−	0.804836i	$-0.702253\pi$
$727$	−862.945			−1.18699			−0.593497	+	0.804836i	$0.702253\pi$
$728$		−	435.076i		−	0.597632i
$729$	−223.319	+	693.952i	−0.306336	+	0.951924i
$730$	−15.6883			−0.0214908
$731$		−	389.572i		−	0.532931i
$732$	−115.739	−	543.254i	−0.158113	−	0.742150i
$733$	−161.777			−0.220705			−0.110352	−	0.993893i	$-0.535198\pi$
$733$	−161.777			−0.220705			−0.110352	+	0.993893i	$0.535198\pi$
$734$		−	342.451i		−	0.466554i
$735$	7.33975	−	1.56371i	0.00998605	−	0.00212750i
$736$	302.465			0.410958
$737$		−	426.405i		−	0.578568i
$738$	379.412	−	169.352i	0.514108	−	0.229474i
$739$	−346.715			−0.469168			−0.234584	−	0.972096i	$-0.575373\pi$
$739$	−346.715			−0.469168			−0.234584	+	0.972096i	$0.575373\pi$
$740$			4.92094i			0.00664992i
$741$	1.53926	+	7.22498i	0.00207728	+	0.00975031i
$742$	−167.434			−0.225653
$743$			658.712i			0.886557i	0.896384	+	0.443279i	$0.146185\pi$
$743$			658.712i			0.886557i	−0.896384	+	0.443279i	$0.853815\pi$
$744$	−839.078	+	178.763i	−1.12779	+	0.240273i
$745$	−37.5909			−0.0504575
$746$			583.026i			0.781536i
$747$	313.923	+	703.304i	0.420245	+	0.941504i
$748$	−187.547			−0.250732
$749$		−	164.316i		−	0.219381i
$750$	7.17874	+	33.6955i	0.00957165	+	0.0449273i
$751$	−518.163			−0.689964			−0.344982	−	0.938609i	$-0.612115\pi$
$751$	−518.163			−0.689964			−0.344982	+	0.938609i	$0.612115\pi$
$752$			17.1119i			0.0227552i
$753$	−664.764	+	141.626i	−0.882821	+	0.188083i
$754$	−315.561			−0.418515
$755$		−	4.43276i		−	0.00587120i
$756$	−245.969	−	337.546i	−0.325355	−	0.446489i
$757$	−396.939			−0.524358			−0.262179	−	0.965019i	$-0.584441\pi$
$757$	−396.939			−0.524358			−0.262179	+	0.965019i	$0.584441\pi$
$758$		−	442.599i		−	0.583904i
$759$	−42.6199	−	200.049i	−0.0561527	−	0.263569i
$760$	0.402855			0.000530073
$761$		−	143.135i		−	0.188088i	−0.995568	−	0.0940442i	$-0.970020\pi$
$761$		−	143.135i		−	0.188088i	0.995568	−	0.0940442i	$-0.0299795\pi$
$762$	247.902	−	52.8148i	0.325330	−	0.0693108i
$763$	1024.31			1.34247
$764$		−	84.7629i		−	0.110946i
$765$	15.7230	−	7.01805i	0.0205530	−	0.00917393i
$766$	566.158			0.739110
$767$			70.9791i			0.0925411i
$768$	−153.320	−	719.652i	−0.199635	−	0.937047i
$769$	−1406.57			−1.82909			−0.914546	−	0.404482i	$-0.867452\pi$
$769$	−1406.57			−1.82909			−0.914546	+	0.404482i	$0.867452\pi$
$770$			10.1110i			0.0131312i
$771$	−1261.75	+	268.812i	−1.63651	+	0.348654i
$772$	−301.918			−0.391085
$773$			1460.02i			1.88877i	0.328840	+	0.944386i	$0.393342\pi$
$773$			1460.02i			1.88877i	−0.328840	+	0.944386i	$0.606658\pi$
$774$	−171.694	−	384.659i	−0.221827	−	0.496975i
$775$	−909.959			−1.17414
$776$			1178.02i			1.51807i
$777$	37.1690	+	174.464i	0.0478366	+	0.224535i
$778$	−722.007			−0.928030
$779$		−	10.3142i		−	0.0132403i
$780$	−13.4673	+	2.86917i	−0.0172658	+	0.00367843i
$781$	184.349			0.236042
$782$			110.140i			0.140844i
$783$	−624.768	+	455.267i	−0.797916	+	0.581439i
$784$	−12.3694			−0.0157772
$785$		−	19.8128i		−	0.0252393i
$786$	172.729	+	810.754i

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)

\(f(q)\)	\(=\)	\(q-1.19271i q^{2} +(0.625113 + 2.93415i) q^{3} +2.57744 q^{4} +0.192711i q^{5} +(3.49960 - 0.745581i) q^{6} +6.00162 q^{7} -7.84499i q^{8} +(-8.21847 + 3.66835i) q^{9} +O(q^{10})\)	\(q-1.19271i q^{2} +(0.625113 + 2.93415i) q^{3} +2.57744 q^{4} +0.192711i q^{5} +(3.49960 - 0.745581i) q^{6} +6.00162 q^{7} -7.84499i q^{8} +(-8.21847 + 3.66835i) q^{9} +0.229849 q^{10} +7.32965i q^{11} +(1.61119 + 7.56258i) q^{12} +9.24069 q^{13} -7.15821i q^{14} +(-0.565443 + 0.120466i) q^{15} +0.952916 q^{16} +9.92749i q^{17} +(4.37529 + 9.80227i) q^{18} +0.266471 q^{19} +0.496700i q^{20} +(3.75169 + 17.6097i) q^{21} +8.74217 q^{22} +9.30188i q^{23} +(23.0184 - 4.90401i) q^{24} +24.9629 q^{25} -11.0215i q^{26} +(-15.9010 - 21.8211i) q^{27} +15.4688 q^{28} -28.6314i q^{29} +(0.143682 + 0.674411i) q^{30} -36.4525 q^{31} -32.5165i q^{32} +(-21.5063 + 4.58186i) q^{33} +11.8406 q^{34} +1.15658i q^{35} +(-21.1826 + 9.45494i) q^{36} +9.90726 q^{37} -0.317824i q^{38} +(5.77648 + 27.1136i) q^{39} +1.51182 q^{40} -38.7065i q^{41} +(21.0033 - 4.47469i) q^{42} -39.2418 q^{43} +18.8917i q^{44} +(-0.706931 - 1.58379i) q^{45} +11.0945 q^{46} +17.9574i q^{47} +(0.595680 + 2.79600i) q^{48} -12.9805 q^{49} -29.7735i q^{50} +(-29.1287 + 6.20581i) q^{51} +23.8173 q^{52} -23.3905i q^{53} +(-26.0263 + 18.9653i) q^{54} -1.41250 q^{55} -47.0827i q^{56} +(0.166575 + 0.781867i) q^{57} -34.1490 q^{58} +7.68115i q^{59} +(-1.45739 + 0.310494i) q^{60} -71.8344 q^{61} +43.4774i q^{62} +(-49.3241 + 22.0161i) q^{63} -34.9712 q^{64} +1.78078i q^{65} +(5.46485 + 25.6508i) q^{66} -58.1753 q^{67} +25.5875i q^{68} +(-27.2931 + 5.81473i) q^{69} +1.37947 q^{70} -25.1511i q^{71} +(28.7782 + 64.4738i) q^{72} -68.2547 q^{73} -11.8165i q^{74} +(15.6046 + 73.2448i) q^{75} +0.686812 q^{76} +43.9898i q^{77} +(32.3387 - 6.88968i) q^{78} +72.5230 q^{79} +0.183637i q^{80} +(54.0864 - 60.2964i) q^{81} -46.1657 q^{82} -85.5760i q^{83} +(9.66975 + 45.3878i) q^{84} -1.91314 q^{85} +46.8042i q^{86} +(84.0088 - 17.8979i) q^{87} +57.5011 q^{88} -94.8794i q^{89} +(-1.88900 + 0.843166i) q^{90} +55.4591 q^{91} +23.9750i q^{92} +(-22.7869 - 106.957i) q^{93} +21.4181 q^{94} +0.0513519i q^{95} +(95.4084 - 20.3265i) q^{96} -150.163 q^{97} +15.4821i q^{98} +(-26.8877 - 60.2385i) 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

Related objects

Downloads

Learn more about