LMFDB - Embedded newform 177.3.b.a.119.4

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			119.4
Character	$\chi$	$=$	177.119
Dual form			177.3.b.a.119.35

$q$-expansion

$f(q)$	$=$	$q-3.49731i q^{2} +(-2.91161 - 0.722867i) q^{3} -8.23119 q^{4} -8.21287i q^{5} +(-2.52809 + 10.1828i) q^{6} -3.56525 q^{7} +14.7978i q^{8} +(7.95493 + 4.20941i) q^{9} +O(q^{10})$	\(q-3.49731i q^{2} +(-2.91161 - 0.722867i) q^{3} -8.23119 q^{4} -8.21287i q^{5} +(-2.52809 + 10.1828i) q^{6} -3.56525 q^{7} +14.7978i q^{8} +(7.95493 + 4.20941i) q^{9} -28.7230 q^{10} -6.15586i q^{11} +(23.9660 + 5.95006i) q^{12} +2.36040 q^{13} +12.4688i q^{14} +(-5.93681 + 23.9127i) q^{15} +18.8277 q^{16} +0.880690i q^{17} +(14.7216 - 27.8209i) q^{18} +32.8857 q^{19} +67.6017i q^{20} +(10.3806 + 2.57720i) q^{21} -21.5290 q^{22} +20.2343i q^{23} +(10.6968 - 43.0854i) q^{24} -42.4512 q^{25} -8.25505i q^{26} +(-20.1188 - 18.0065i) q^{27} +29.3462 q^{28} -46.3821i q^{29} +(83.6300 + 20.7629i) q^{30} -55.1410 q^{31} -6.65529i q^{32} +(-4.44987 + 17.9235i) q^{33} +3.08005 q^{34} +29.2809i q^{35} +(-65.4785 - 34.6485i) q^{36} -10.4535 q^{37} -115.012i q^{38} +(-6.87256 - 1.70626i) q^{39} +121.532 q^{40} -5.03972i q^{41} +(9.01328 - 36.3042i) q^{42} +14.2432 q^{43} +50.6701i q^{44} +(34.5713 - 65.3327i) q^{45} +70.7656 q^{46} +75.4957i q^{47} +(-54.8190 - 13.6100i) q^{48} -36.2890 q^{49} +148.465i q^{50} +(0.636622 - 2.56423i) q^{51} -19.4289 q^{52} -72.2215i q^{53} +(-62.9744 + 70.3617i) q^{54} -50.5573 q^{55} -52.7578i q^{56} +(-95.7504 - 23.7720i) q^{57} -162.213 q^{58} -7.68115i q^{59} +(48.8670 - 196.830i) q^{60} -8.24650 q^{61} +192.845i q^{62} +(-28.3613 - 15.0076i) q^{63} +52.0353 q^{64} -19.3856i q^{65} +(62.6840 + 15.5626i) q^{66} -101.205 q^{67} -7.24913i q^{68} +(14.6267 - 58.9143i) q^{69} +102.404 q^{70} -97.6488i q^{71} +(-62.2900 + 117.715i) q^{72} +28.7308 q^{73} +36.5592i q^{74} +(123.601 + 30.6866i) q^{75} -270.689 q^{76} +21.9472i q^{77} +(-5.96731 + 24.0355i) q^{78} +149.129 q^{79} -154.630i q^{80} +(45.5617 + 66.9711i) q^{81} -17.6255 q^{82} +85.1008i q^{83} +(-85.4447 - 21.2134i) q^{84} +7.23299 q^{85} -49.8130i q^{86} +(-33.5281 + 135.047i) q^{87} +91.0932 q^{88} -98.6034i q^{89} +(-228.489 - 120.907i) q^{90} -8.41541 q^{91} -166.552i q^{92} +(160.549 + 39.8596i) q^{93} +264.032 q^{94} -270.086i q^{95} +(-4.81090 + 19.3776i) q^{96} +93.2794 q^{97} +126.914i q^{98} +(25.9126 - 48.9694i) 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 $

$2$		−	3.49731i		−	1.74866i	−0.485336	−	0.874328i	$-0.661303\pi$
$2$		−	3.49731i		−	1.74866i	0.485336	−	0.874328i	$-0.338697\pi$
$3$	−2.91161	−	0.722867i	−0.970536	−	0.240956i
$3$	−2.91161	−	0.722867i	−0.970536	−	0.240956i
$4$	−8.23119			−2.05780
$5$		−	8.21287i		−	1.64257i	−0.570516	−	0.821287i	$-0.693257\pi$
$5$		−	8.21287i		−	1.64257i	0.570516	−	0.821287i	$-0.306743\pi$
$6$	−2.52809	+	10.1828i	−0.421349	+	1.69713i
$7$	−3.56525			−0.509321			−0.254661	−	0.967031i	$-0.581964\pi$
$7$	−3.56525			−0.509321			−0.254661	+	0.967031i	$0.581964\pi$
$8$			14.7978i			1.84972i
$9$	7.95493	+	4.20941i	0.883881	+	0.467713i
$10$	−28.7230			−2.87230
$11$		−	6.15586i		−	0.559624i	−0.960055	−	0.279812i	$-0.909728\pi$
$11$		−	6.15586i		−	0.559624i	0.960055	−	0.279812i	$-0.0902721\pi$
$12$	23.9660	+	5.95006i	1.99717	+	0.495838i
$13$	2.36040			0.181569			0.0907846	−	0.995871i	$-0.471063\pi$
$13$	2.36040			0.181569			0.0907846	+	0.995871i	$0.471063\pi$
$14$			12.4688i			0.890627i
$15$	−5.93681	+	23.9127i	−0.395788	+	1.59418i
$16$	18.8277			1.17673
$17$			0.880690i			0.0518053i	0.999664	+	0.0259027i	$0.00824600\pi$
$17$			0.880690i			0.0518053i	−0.999664	+	0.0259027i	$0.991754\pi$
$18$	14.7216	−	27.8209i	0.817868	−	1.54560i
$19$	32.8857			1.73083			0.865414	−	0.501057i	$-0.167055\pi$
$19$	32.8857			1.73083			0.865414	+	0.501057i	$0.167055\pi$
$20$			67.6017i			3.38008i
$21$	10.3806	+	2.57720i	0.494314	+	0.122724i
$22$	−21.5290			−0.978590
$23$			20.2343i			0.879752i	0.898059	+	0.439876i	$0.144978\pi$
$23$			20.2343i			0.879752i	−0.898059	+	0.439876i	$0.855022\pi$
$24$	10.6968	−	43.0854i	0.445702	−	1.79522i
$25$	−42.4512			−1.69805
$26$		−	8.25505i		−	0.317502i
$27$	−20.1188	−	18.0065i	−0.745140	−	0.666908i
$28$	29.3462			1.04808
$29$		−	46.3821i		−	1.59938i	−0.600411	−	0.799691i	$-0.704996\pi$
$29$		−	46.3821i		−	1.59938i	0.600411	−	0.799691i	$-0.295004\pi$
$30$	83.6300	+	20.7629i	2.78767	+	0.692096i
$31$	−55.1410			−1.77874			−0.889371	−	0.457186i	$-0.848857\pi$
$31$	−55.1410			−1.77874			−0.889371	+	0.457186i	$0.848857\pi$
$32$		−	6.65529i		−	0.207978i
$33$	−4.44987	+	17.9235i	−0.134845	+	0.543135i
$34$	3.08005			0.0905897
$35$			29.2809i			0.836597i
$36$	−65.4785	−	34.6485i	−1.81885	−	0.962458i
$37$	−10.4535			−0.282527			−0.141264	−	0.989972i	$-0.545117\pi$
$37$	−10.4535			−0.282527			−0.141264	+	0.989972i	$0.545117\pi$
$38$		−	115.012i		−	3.02662i
$39$	−6.87256	−	1.70626i	−0.176219	−	0.0437501i
$40$	121.532			3.03831
$41$		−	5.03972i		−	0.122920i	−0.998110	−	0.0614599i	$-0.980424\pi$
$41$		−	5.03972i		−	0.122920i	0.998110	−	0.0614599i	$-0.0195756\pi$
$42$	9.01328	−	36.3042i	0.214602	−	0.864386i
$43$	14.2432			0.331238			0.165619	−	0.986190i	$-0.447038\pi$
$43$	14.2432			0.331238			0.165619	+	0.986190i	$0.447038\pi$
$44$			50.6701i			1.15159i
$45$	34.5713	−	65.3327i	0.768252	−	1.45184i
$46$	70.7656			1.53838
$47$			75.4957i			1.60629i	0.595783	+	0.803146i	$0.296842\pi$
$47$			75.4957i			1.60629i	−0.595783	+	0.803146i	$0.703158\pi$
$48$	−54.8190	−	13.6100i	−1.14206	−	0.283541i
$49$	−36.2890			−0.740592
$50$			148.465i			2.96930i
$51$	0.636622	−	2.56423i	0.0124828	−	0.0502789i
$52$	−19.4289			−0.373633
$53$		−	72.2215i		−	1.36267i	−0.731972	−	0.681335i	$-0.761400\pi$
$53$		−	72.2215i		−	1.36267i	0.731972	−	0.681335i	$-0.238600\pi$
$54$	−62.9744	+	70.3617i	−1.16619	+	1.30299i
$55$	−50.5573			−0.919224
$56$		−	52.7578i		−	0.942103i
$57$	−95.7504	−	23.7720i	−1.67983	−	0.417053i
$58$	−162.213			−2.79677
$59$		−	7.68115i		−	0.130189i
$59$		−	7.68115i		−	0.130189i
$60$	48.8670	−	196.830i	0.814451	−	3.28049i
$61$	−8.24650			−0.135188			−0.0675942	−	0.997713i	$-0.521532\pi$
$61$	−8.24650			−0.135188			−0.0675942	+	0.997713i	$0.521532\pi$
$62$			192.845i			3.11041i
$63$	−28.3613	−	15.0076i	−0.450179	−	0.238216i
$64$	52.0353			0.813052
$65$		−	19.3856i		−	0.298241i
$66$	62.6840	+	15.5626i	0.949757	+	0.235797i
$67$	−101.205			−1.51052			−0.755259	−	0.655427i	$-0.772489\pi$
$67$	−101.205			−1.51052			−0.755259	+	0.655427i	$0.772489\pi$
$68$		−	7.24913i		−	0.106605i
$69$	14.6267	−	58.9143i	0.211981	−	0.853831i
$70$	102.404			1.46292
$71$		−	97.6488i		−	1.37533i	−0.726026	−	0.687667i	$-0.758635\pi$
$71$		−	97.6488i		−	1.37533i	0.726026	−	0.687667i	$-0.241365\pi$
$72$	−62.2900	+	117.715i	−0.865139	+	1.63494i
$73$	28.7308			0.393573			0.196786	−	0.980446i	$-0.436949\pi$
$73$	28.7308			0.393573			0.196786	+	0.980446i	$0.436949\pi$
$74$			36.5592i			0.494043i
$75$	123.601	+	30.6866i	1.64802	+	0.409154i
$76$	−270.689			−3.56169
$77$			21.9472i			0.285028i
$78$	−5.96731	+	24.0355i	−0.0765040	+	0.308147i
$79$	149.129			1.88771			0.943855	−	0.330359i	$-0.107170\pi$
$79$	149.129			1.88771			0.943855	+	0.330359i	$0.107170\pi$
$80$		−	154.630i		−	1.93287i
$81$	45.5617	+	66.9711i	0.562490	+	0.826804i
$82$	−17.6255			−0.214945
$83$			85.1008i			1.02531i	0.858595	+	0.512655i	$0.171338\pi$
$83$			85.1008i			1.02531i	−0.858595	+	0.512655i	$0.828662\pi$
$84$	−85.4447	−	21.2134i	−1.01720	−	0.252541i
$85$	7.23299			0.0850940
$86$		−	49.8130i		−	0.579221i
$87$	−33.5281	+	135.047i	−0.385380	+	1.55226i
$88$	91.0932			1.03515
$89$		−	98.6034i		−	1.10790i	−0.832549	−	0.553952i	$-0.813119\pi$
$89$		−	98.6034i		−	1.10790i	0.832549	−	0.553952i	$-0.186881\pi$
$90$	−228.489	−	120.907i	−2.53877	−	1.34341i
$91$	−8.41541			−0.0924770
$92$		−	166.552i		−	1.81035i
$93$	160.549	+	39.8596i	1.72633	+	0.428598i
$94$	264.032			2.80885
$95$		−	270.086i		−	2.84301i
$96$	−4.81090	+	19.3776i	−0.0501135	+	0.201850i
$97$	93.2794			0.961644			0.480822	−	0.876818i	$-0.340338\pi$
$97$	93.2794			0.961644			0.480822	+	0.876818i	$0.340338\pi$
$98$			126.914i			1.29504i
$99$	25.9126	−	48.9694i	0.261743	−	0.494641i
$100$	349.424			3.49424
$101$		−	40.3505i		−	0.399510i	−0.979846	−	0.199755i	$-0.935985\pi$
$101$		−	40.3505i		−	0.399510i	0.979846	−	0.199755i	$-0.0640146\pi$
$102$	−8.96790	−	2.22647i	−0.0879206	−	0.0218281i
$103$	−119.073			−1.15605			−0.578023	−	0.816021i	$-0.696175\pi$
$103$	−119.073			−1.15605			−0.578023	+	0.816021i	$0.696175\pi$
$104$			34.9287i			0.335853i
$105$	21.1662	−	85.2545i	0.201583	−	0.811948i
$106$	−252.581			−2.38284
$107$			99.0486i			0.925688i	0.886440	+	0.462844i	$0.153171\pi$
$107$			99.0486i			0.925688i	−0.886440	+	0.462844i	$0.846829\pi$
$108$	165.602	+	148.215i	1.53335	+	1.37236i
$109$	20.2659			0.185926			0.0929630	−	0.995670i	$-0.470366\pi$
$109$	20.2659			0.185926			0.0929630	+	0.995670i	$0.470366\pi$
$110$			176.815i			1.60741i
$111$	30.4365	+	7.55650i	0.274203	+	0.0680766i
$112$	−67.1255			−0.599335
$113$		−	68.4754i		−	0.605977i	−0.952994	−	0.302988i	$-0.902016\pi$
$113$		−	68.4754i		−	0.605977i	0.952994	−	0.302988i	$-0.0979843\pi$
$114$	−83.1382	+	334.869i	−0.729282	+	2.93745i
$115$	166.182			1.44506
$116$			381.780i			3.29121i
$117$	18.7768	+	9.93590i	0.160485	+	0.0849222i
$118$	−26.8634			−0.227656
$119$		−	3.13988i		−	0.0263855i
$120$	−353.854	−	87.8517i	−2.94879	−	0.732098i
$121$	83.1053			0.686821
$122$			28.8406i			0.236398i
$123$	−3.64305	+	14.6737i	−0.0296183	+	0.119298i
$124$	453.876			3.66029
$125$			143.324i			1.14659i
$126$	−52.4863	+	99.1882i	−0.416558	+	0.787208i
$127$	76.7243			0.604128			0.302064	−	0.953288i	$-0.402324\pi$
$127$	76.7243			0.604128			0.302064	+	0.953288i	$0.402324\pi$
$128$		−	208.605i		−	1.62973i
$129$	−41.4707	−	10.2960i	−0.321478	−	0.0798136i
$130$	−67.7977			−0.521520
$131$		−	218.508i		−	1.66800i	−0.551766	−	0.833999i	$-0.686046\pi$
$131$		−	218.508i		−	1.66800i	0.551766	−	0.833999i	$-0.313954\pi$
$132$	36.6278	−	147.531i	0.277483	−	1.11766i
$133$	−117.246			−0.881547
$134$			353.944i			2.64138i
$135$	−147.885	+	165.233i	−1.09545	+	1.22395i
$136$	−13.0323			−0.0958256
$137$		−	56.4715i		−	0.412201i	−0.978531	−	0.206101i	$-0.933923\pi$
$137$		−	56.4715i		−	0.412201i	0.978531	−	0.206101i	$-0.0660774\pi$
$138$	−206.042	−	51.1542i	−1.49306	−	0.370682i
$139$	62.0097			0.446113			0.223056	−	0.974806i	$-0.428397\pi$
$139$	62.0097			0.446113			0.223056	+	0.974806i	$0.428397\pi$
$140$		−	241.017i		−	1.72155i
$141$	54.5734	−	219.814i	0.387045	−	1.55896i
$142$	−341.508			−2.40499
$143$		−	14.5303i		−	0.101610i
$144$	149.773	+	79.2537i	1.04009	+	0.550373i
$145$	−380.930			−2.62710
$146$		−	100.481i		−	0.688223i
$147$	105.659	+	26.2321i	0.718771	+	0.178450i
$148$	86.0448			0.581384
$149$		−	26.8893i		−	0.180465i	−0.995921	−	0.0902324i	$-0.971239\pi$
$149$		−	26.8893i		−	0.180465i	0.995921	−	0.0902324i	$-0.0287610\pi$
$150$	107.321	−	432.272i	0.715470	−	2.88181i
$151$	−124.391			−0.823785			−0.411892	−	0.911233i	$-0.635132\pi$
$151$	−124.391			−0.823785			−0.411892	+	0.911233i	$0.635132\pi$
$152$			486.636i			3.20155i
$153$	−3.70719	+	7.00583i	−0.0242300	+	0.0457897i
$154$	76.7561			0.498416
$155$			452.866i			2.92171i
$156$	56.5693	+	14.0445i	0.362624	+	0.0900289i
$157$	−223.209			−1.42172			−0.710858	−	0.703336i	$-0.751693\pi$
$157$	−223.209			−1.42172			−0.710858	+	0.703336i	$0.751693\pi$
$158$		−	521.551i		−	3.30096i
$159$	−52.2066	+	210.281i	−0.328343	+	1.32252i
$160$	−54.6591			−0.341619
$161$		−	72.1403i		−	0.448076i
$162$	234.219	−	159.343i	1.44580	−	0.983601i
$163$	61.7561			0.378872			0.189436	−	0.981893i	$-0.439334\pi$
$163$	61.7561			0.378872			0.189436	+	0.981893i	$0.439334\pi$
$164$			41.4829i			0.252944i
$165$	147.203	+	36.5462i	0.892140	+	0.221492i
$166$	297.624			1.79291
$167$		−	60.8323i		−	0.364265i	−0.983274	−	0.182133i	$-0.941700\pi$
$167$		−	60.8323i		−	0.364265i	0.983274	−	0.182133i	$-0.0583000\pi$
$168$	−38.1369	+	153.610i	−0.227005	+	0.914345i
$169$	−163.429			−0.967033
$170$		−	25.2960i		−	0.148800i
$171$	261.604	+	138.430i	1.52985	+	0.809530i
$172$	−117.239			−0.681620
$173$			110.975i			0.641477i	0.947168	+	0.320738i	$0.103931\pi$
$173$			110.975i			0.641477i	−0.947168	+	0.320738i	$0.896069\pi$
$174$	472.300	+	117.258i	2.71437	+	0.673898i
$175$	151.349			0.864851
$176$		−	115.901i		−	0.658528i
$177$	−5.55245	+	22.3645i	−0.0313698	+	0.126353i
$178$	−344.847			−1.93734
$179$			54.6001i			0.305029i	0.988301	+	0.152514i	$0.0487370\pi$
$179$			54.6001i			0.305029i	−0.988301	+	0.152514i	$0.951263\pi$
$180$	−284.563	+	537.766i	−1.58091	+	2.98759i
$181$	−124.891			−0.690008			−0.345004	−	0.938601i	$-0.612122\pi$
$181$	−124.891			−0.690008			−0.345004	+	0.938601i	$0.612122\pi$
$182$			29.4313i			0.161710i
$183$	24.0106	+	5.96112i	0.131205	+	0.0325744i
$184$	−299.423			−1.62730
$185$			85.8533i			0.464072i
$186$	139.402	−	561.490i	0.749471	−	3.01876i
$187$	5.42141			0.0289915
$188$		−	621.419i		−	3.30542i
$189$	71.7284	+	64.1977i	0.379516	+	0.339670i
$190$	−944.576			−4.97145
$191$		−	96.6411i		−	0.505974i	−0.967470	−	0.252987i	$-0.918587\pi$
$191$		−	96.6411i		−	0.505974i	0.967470	−	0.252987i	$-0.0814130\pi$
$192$	−151.506	−	37.6146i	−0.789096	−	0.195909i
$193$	−6.90714			−0.0357883			−0.0178941	−	0.999840i	$-0.505696\pi$
$193$	−6.90714			−0.0357883			−0.0178941	+	0.999840i	$0.505696\pi$
$194$		−	326.227i		−	1.68158i
$195$	−14.0133	+	56.4434i	−0.0718628	+	0.289453i
$196$	298.702			1.52399
$197$			24.7912i			0.125844i	0.998018	+	0.0629218i	$0.0200419\pi$
$197$			24.7912i			0.125844i	−0.998018	+	0.0629218i	$0.979958\pi$
$198$	−171.261	−	90.6244i	−0.864957	−	0.457699i
$199$	−20.1503			−0.101258			−0.0506289	−	0.998718i	$-0.516123\pi$
$199$	−20.1503			−0.101258			−0.0506289	+	0.998718i	$0.516123\pi$
$200$		−	628.184i		−	3.14092i
$201$	294.668	+	73.1576i	1.46601	+	0.363968i
$202$	−141.118			−0.698605
$203$			165.364i			0.814599i
$204$	−5.24016	+	21.1066i	−0.0256871	+	0.103464i
$205$	−41.3905			−0.201905
$206$			416.434i			2.02153i
$207$	−85.1745	+	160.962i	−0.411471	+	0.777596i
$208$	44.4410			0.213659
$209$		−	202.440i		−	0.968613i
$210$	−298.162	−	74.0248i	−1.41982	−	0.352499i
$211$	325.183			1.54115			0.770575	−	0.637349i	$-0.219969\pi$
$211$	325.183			1.54115			0.770575	+	0.637349i	$0.219969\pi$
$212$			594.469i			2.80410i
$213$	−70.5871	+	284.315i	−0.331395	+	1.33481i
$214$	346.404			1.61871
$215$		−	116.978i		−	0.544082i
$216$	266.457	−	297.714i	1.23360	−	1.37830i
$217$	196.591			0.905951
$218$		−	70.8763i		−	0.325121i
$219$	−83.6528	−	20.7686i	−0.381976	−	0.0948336i
$220$	416.147			1.89158
$221$			2.07878i			0.00940625i
$222$	26.4274	−	106.446i	0.119042	−	0.479486i
$223$	−146.044			−0.654908			−0.327454	−	0.944867i	$-0.606191\pi$
$223$	−146.044			−0.654908			−0.327454	+	0.944867i	$0.606191\pi$
$224$			23.7278i			0.105928i
$225$	−337.696	−	178.695i	−1.50087	−	0.794198i
$226$	−239.480			−1.05964
$227$		−	115.837i		−	0.510296i	−0.966902	−	0.255148i	$-0.917876\pi$
$227$		−	115.837i		−	0.510296i	0.966902	−	0.255148i	$-0.0821242\pi$
$228$	788.140	+	195.672i	3.45675	+	0.858211i
$229$	−425.811			−1.85944			−0.929719	−	0.368271i	$-0.879950\pi$
$229$	−425.811			−1.85944			−0.929719	+	0.368271i	$0.879950\pi$
$230$		−	581.189i		−	2.52691i
$231$	15.8649	−	63.9016i	0.0686792	−	0.276630i
$232$	686.353			2.95842
$233$		−	160.262i		−	0.687820i	−0.939003	−	0.343910i	$-0.888248\pi$
$233$		−	160.262i		−	0.687820i	0.939003	−	0.343910i	$-0.111752\pi$
$234$	34.7489	−	65.6683i	0.148500	−	0.280634i
$235$	620.036			2.63845
$236$			63.2250i			0.267902i
$237$	−434.206	−	107.801i	−1.83209	−	0.454855i
$238$	−10.9811			−0.0461392
$239$			215.732i			0.902644i	0.892361	+	0.451322i	$0.149047\pi$
$239$			215.732i			0.902644i	−0.892361	+	0.451322i	$0.850953\pi$
$240$	−111.777	+	450.221i	−0.465736	+	1.87592i
$241$	308.468			1.27995			0.639975	−	0.768396i	$-0.278945\pi$
$241$	308.468			1.27995			0.639975	+	0.768396i	$0.278945\pi$
$242$		−	290.645i		−	1.20101i
$243$	−84.2465	−	227.929i	−0.346694	−	0.937978i
$244$	67.8785			0.278191
$245$			298.037i			1.21648i
$246$	51.3184	+	12.7409i	0.208612	+	0.0517921i
$247$	77.6235			0.314265
$248$		−	815.965i		−	3.29018i
$249$	61.5166	−	247.780i	0.247054	−	0.995101i
$250$	501.250			2.00500
$251$		−	136.796i		−	0.545004i	−0.962155	−	0.272502i	$-0.912149\pi$
$251$		−	136.796i		−	0.545004i	0.962155	−	0.272502i	$-0.0878512\pi$
$252$	233.447	+	123.530i	0.926377	+	0.490200i
$253$	124.560			0.492330
$254$		−	268.329i		−	1.05641i
$255$	−21.0596	−	5.22849i	−0.0825868	−	0.0205039i
$256$	−521.415			−2.03678
$257$			14.0185i			0.0545468i	0.999628	+	0.0272734i	$0.00868247\pi$
$257$			14.0185i			0.0545468i	−0.999628	+	0.0272734i	$0.991318\pi$
$258$	−36.0082	+	145.036i	−0.139567	+	0.562155i
$259$	37.2693			0.143897
$260$			159.567i			0.613719i
$261$	195.241	−	368.966i	0.748051	−	1.41366i
$262$	−764.190			−2.91675
$263$		−	157.760i		−	0.599847i	−0.953963	−	0.299924i	$-0.903039\pi$
$263$		−	157.760i		−	0.599847i	0.953963	−	0.299924i	$-0.0969612\pi$
$264$	−265.228	−	65.8483i	−1.00465	−	0.249425i
$265$	−593.146			−2.23828
$266$			410.045i			1.54152i
$267$	−71.2772	+	287.094i	−0.266956	+	1.07526i
$268$	833.035			3.10834
$269$			4.90919i			0.0182498i	0.999958	+	0.00912488i	$0.00290458\pi$
$269$			4.90919i			0.0182498i	−0.999958	+	0.00912488i	$0.997095\pi$
$270$	577.871	+	517.200i	2.14026	+	1.91556i
$271$	−169.663			−0.626061			−0.313031	−	0.949743i	$-0.601344\pi$
$271$	−169.663			−0.626061			−0.313031	+	0.949743i	$0.601344\pi$
$272$			16.5814i			0.0609611i
$273$	24.5024	+	6.08322i	0.0897523	+	0.0222829i
$274$	−197.499			−0.720798
$275$			261.324i			0.950268i
$276$	−120.395	+	484.935i	−0.436215	+	1.75701i
$277$	247.335			0.892907			0.446453	−	0.894807i	$-0.352687\pi$
$277$	247.335			0.892907			0.446453	+	0.894807i	$0.352687\pi$
$278$		−	216.867i		−	0.780098i
$279$	−438.643	−	232.111i	−1.57220	−	0.831940i
$280$	−433.293			−1.54747
$281$			36.5748i			0.130159i	0.997880	+	0.0650797i	$0.0207302\pi$
$281$			36.5748i			0.130159i	−0.997880	+	0.0650797i	$0.979270\pi$
$282$	−768.758	−	190.860i	−2.72609	−	0.676809i
$283$	−298.925			−1.05627			−0.528135	−	0.849160i	$-0.677109\pi$
$283$	−298.925			−1.05627			−0.528135	+	0.849160i	$0.677109\pi$
$284$			803.766i			2.83016i
$285$	−195.236	+	786.385i	−0.685040	+	2.75925i
$286$	−50.8170			−0.177682
$287$			17.9678i			0.0626057i
$288$	28.0149	−	52.9424i	0.0972739	−	0.183828i
$289$	288.224			0.997316
$290$			1332.23i			4.59390i
$291$	−271.593	−	67.4287i	−0.933310	−	0.231714i
$292$	−236.489			−0.809893
$293$		−	168.158i		−	0.573918i	−0.957943	−	0.286959i	$-0.907356\pi$
$293$		−	168.158i		−	0.573918i	0.957943	−	0.286959i	$-0.0926444\pi$
$294$	91.7420	−	369.524i	0.312048	−	1.25688i
$295$	−63.0842			−0.213845
$296$		−	154.689i		−	0.522597i
$297$	−110.846	+	123.848i	−0.373218	+	0.416998i
$298$	−94.0401			−0.315571
$299$			47.7610i			0.159736i
$300$	−1017.39	−	252.587i	−3.39128	−	0.841957i
$301$	−50.7806			−0.168706
$302$			435.036i			1.44052i
$303$	−29.1681	+	117.485i	−0.0962642	+	0.387739i
$304$	619.164			2.03672
$305$			67.7274i			0.222057i
$306$	24.5016	+	12.9652i	0.0800705	+	0.0423699i
$307$	159.132			0.518345			0.259172	−	0.965831i	$-0.416550\pi$
$307$	159.132			0.518345			0.259172	+	0.965831i	$0.416550\pi$
$308$		−	180.651i		−	0.586531i
$309$	346.693	+	86.0737i	1.12198	+	0.278556i
$310$	1583.81			5.10907
$311$			371.820i			1.19556i	0.801659	+	0.597782i	$0.203951\pi$
$311$			371.820i			1.19556i	−0.801659	+	0.597782i	$0.796049\pi$
$312$	25.2488	−	101.699i	0.0809257	−	0.325957i
$313$	−1.08898			−0.00347918			−0.00173959	−	0.999998i	$-0.500554\pi$
$313$	−1.08898			−0.00347918			−0.00173959	+	0.999998i	$0.500554\pi$
$314$			780.633i			2.48609i
$315$	−123.255	+	232.927i	−0.391287	+	0.739452i
$316$	−1227.51			−3.88453
$317$		−	333.579i		−	1.05230i	−0.850392	−	0.526150i	$-0.823635\pi$
$317$		−	333.579i		−	1.05230i	0.850392	−	0.526150i	$-0.176365\pi$
$318$	735.417	+	182.583i	2.31263	+	0.574159i
$319$	−285.522			−0.895053
$320$		−	427.359i		−	1.33550i
$321$	71.5990	−	288.391i	0.223050	−	0.898413i
$322$	−252.297			−0.783531
$323$			28.9622i			0.0896661i
$324$	−375.027	−	551.252i	−1.15749	−	1.70140i
$325$	−100.202			−0.308313
$326$		−	215.980i		−	0.662516i
$327$	−59.0065	−	14.6496i	−0.180448	−	0.0448000i
$328$	74.5767			0.227368
$329$		−	269.161i		−	0.818118i
$330$	127.814	−	514.815i	0.387314	−	1.56005i
$331$	333.908			1.00878			0.504392	−	0.863475i	$-0.331717\pi$
$331$	333.908			1.00878			0.504392	+	0.863475i	$0.331717\pi$
$332$		−	700.481i		−	2.10988i
$333$	−83.1569	−	44.0031i	−0.249720	−	0.132142i
$334$	−212.749			−0.636974
$335$			831.181i			2.48114i
$336$	195.443	+	48.5229i	0.581676	+	0.144413i
$337$	368.536			1.09358			0.546789	−	0.837271i	$-0.315850\pi$
$337$	368.536			1.09358			0.546789	+	0.837271i	$0.315850\pi$
$338$			571.560i			1.69101i
$339$	−49.4986	+	199.373i	−0.146014	+	0.588122i
$340$	−59.5362			−0.175106
$341$			339.441i			0.995427i
$342$	484.132	−	914.909i	1.41559	−	2.67517i
$343$	304.076			0.886520
$344$			210.768i			0.612698i
$345$	−483.856	−	120.127i	−1.40248	−	0.348195i
$346$	388.116			1.12172
$347$		−	454.904i		−	1.31096i	−0.755211	−	0.655482i	$-0.772466\pi$
$347$		−	454.904i		−	1.31096i	0.755211	−	0.655482i	$-0.227534\pi$
$348$	275.976	−	1111.59i	0.793035	−	3.19423i
$349$	483.918			1.38658			0.693292	−	0.720657i	$-0.256160\pi$
$349$	483.918			1.38658			0.693292	+	0.720657i	$0.256160\pi$
$350$		−	529.315i		−	1.51233i
$351$	−47.4884	−	42.5026i	−0.135294	−	0.121090i
$352$	−40.9691			−0.116389
$353$		−	294.298i		−	0.833705i	−0.908974	−	0.416853i	$-0.863133\pi$
$353$		−	294.298i		−	0.833705i	0.908974	−	0.416853i	$-0.136867\pi$
$354$	78.2156	+	19.4186i	0.220948	+	0.0548549i
$355$	−801.976			−2.25909
$356$			811.623i			2.27984i
$357$	−2.26972	+	9.14210i	−0.00635775	+	0.0256081i
$358$	190.954			0.533390
$359$			152.605i			0.425083i	0.977152	+	0.212541i	$0.0681740\pi$
$359$			152.605i			0.425083i	−0.977152	+	0.212541i	$0.931826\pi$
$360$	966.781	+	511.580i	2.68550	+	1.42105i
$361$	720.472			1.99577
$362$			436.784i			1.20659i
$363$	−241.970	−	60.0741i	−0.666584	−	0.165493i
$364$	69.2688			0.190299
$365$		−	235.962i		−	0.646472i
$366$	20.8479	−	83.9725i	0.0569615	−	0.229433i
$367$	−252.819			−0.688881			−0.344441	−	0.938808i	$-0.611931\pi$
$367$	−252.819			−0.688881			−0.344441	+	0.938808i	$0.611931\pi$
$368$			380.966i			1.03523i
$369$	21.2142	−	40.0906i	0.0574912	−	0.108647i
$370$	300.256			0.811502
$371$			257.487i			0.694036i
$372$	−1321.51	−	328.092i	−3.55245	−	0.881968i
$373$	630.560			1.69051			0.845255	−	0.534363i	$-0.179449\pi$
$373$	630.560			1.69051			0.845255	+	0.534363i	$0.179449\pi$
$374$		−	18.9604i		−	0.0506962i
$375$	103.604	−	417.304i	0.276279	−	1.11281i
$376$	−1117.17			−2.97120
$377$		−	109.480i		−	0.290399i
$378$	224.519	−	250.857i	0.593967	−	0.663642i
$379$	14.5692			0.0384412			0.0192206	−	0.999815i	$-0.493882\pi$
$379$	14.5692			0.0384412			0.0192206	+	0.999815i	$0.493882\pi$
$380$			2223.13i			5.85034i
$381$	−223.391	−	55.4615i	−0.586328	−	0.145568i
$382$	−337.984			−0.884775
$383$		−	594.960i		−	1.55342i	−0.629858	−	0.776710i	$-0.716887\pi$
$383$		−	594.960i		−	1.55342i	0.629858	−	0.776710i	$-0.283113\pi$
$384$	−150.794	+	607.376i	−0.392692	+	1.58171i
$385$	180.249			0.468180
$386$			24.1564i			0.0625814i
$387$	113.304	+	59.9556i	0.292775	+	0.154924i
$388$	−767.801			−1.97887
$389$		−	405.842i		−	1.04330i	−0.853161	−	0.521648i	$-0.825318\pi$
$389$		−	405.842i		−	1.04330i	0.853161	−	0.521648i	$-0.174682\pi$
$390$	197.400	+	49.0087i	0.506154	+	0.125663i
$391$	−17.8202			−0.0455758
$392$		−	536.997i		−	1.36989i
$393$	−157.952	+	636.209i	−0.401914	+	1.61885i
$394$	86.7025			0.220057
$395$		−	1224.78i		−	3.10070i
$396$	−213.291	+	403.077i	−0.538615	+	1.01787i
$397$	203.579			0.512793			0.256396	−	0.966572i	$-0.417465\pi$
$397$	203.579			0.512793			0.256396	+	0.966572i	$0.417465\pi$
$398$			70.4720i			0.177065i
$399$	341.374	+	84.7531i	0.855573	+	0.212414i
$400$	−799.260			−1.99815
$401$			266.091i			0.663568i	0.943355	+	0.331784i	$0.107651\pi$
$401$			266.091i			0.663568i	−0.943355	+	0.331784i	$0.892349\pi$
$402$	255.855	−	1030.55i	0.636455	−	2.56355i
$403$	−130.155			−0.322965
$404$			332.133i			0.822111i
$405$	550.025	−	374.192i	1.35809	−	0.923931i
$406$	578.328			1.42445
$407$			64.3504i			0.158109i
$408$	37.9449	+	9.42061i	0.0930022	+	0.0230897i
$409$	−50.9470			−0.124565			−0.0622824	−	0.998059i	$-0.519838\pi$
$409$	−50.9470			−0.124565			−0.0622824	+	0.998059i	$0.519838\pi$
$410$			144.756i			0.353062i
$411$	−40.8214	+	164.423i	−0.0993222	+	0.400056i
$412$	980.110			2.37891
$413$			27.3852i			0.0663080i
$414$	562.935	+	297.882i	1.35975	+	0.719521i
$415$	698.921			1.68415
$416$		−	15.7092i		−	0.0377624i
$417$	−180.548	−	44.8248i	−0.432969	−	0.107493i
$418$	−707.996			−1.69377
$419$		−	159.456i		−	0.380564i	−0.981730	−	0.190282i	$-0.939060\pi$
$419$		−	159.456i		−	0.380564i	0.981730	−	0.190282i	$-0.0609402\pi$
$420$	−174.223	+	701.746i	−0.414817	+	1.67082i
$421$	413.018			0.981041			0.490520	−	0.871430i	$-0.336807\pi$
$421$	413.018			0.981041			0.490520	+	0.871430i	$0.336807\pi$
$422$		−	1137.27i		−	2.69494i
$423$	−317.793	+	600.563i	−0.751283	+	1.41977i
$424$	1068.72			2.52056
$425$		−	37.3864i		−	0.0879679i
$426$	994.338	+	246.865i	2.33413	+	0.579496i
$427$	29.4008			0.0688543
$428$		−	815.288i		−	1.90488i
$429$	−10.5035	+	42.3065i	−0.0244836	+	0.0986166i
$430$	−409.107			−0.951413
$431$		−	797.605i		−	1.85059i	−0.379246	−	0.925296i	$-0.623816\pi$
$431$		−	797.605i		−	1.85059i	0.379246	−	0.925296i	$-0.376184\pi$
$432$	−378.791	−	339.022i	−0.876831	−	0.784773i
$433$	139.027			0.321079			0.160539	−	0.987029i	$-0.448677\pi$
$433$	139.027			0.321079			0.160539	+	0.987029i	$0.448677\pi$
$434$		−	687.541i		−	1.58420i
$435$	1109.12	+	275.362i	2.54970	+	0.633016i
$436$	−166.813			−0.382598
$437$			665.420i			1.52270i
$438$	−72.6341	+	292.560i	−0.165831	+	0.667945i
$439$	−179.734			−0.409417			−0.204708	−	0.978823i	$-0.565625\pi$
$439$	−179.734			−0.409417			−0.204708	+	0.978823i	$0.565625\pi$
$440$		−	748.136i		−	1.70031i
$441$	−288.676	−	152.755i	−0.654595	−	0.346384i
$442$	7.27015			0.0164483
$443$		−	1.01657i		−	0.00229473i	−0.999999	−	0.00114737i	$-0.999635\pi$
$443$		−	1.01657i		−	0.00229473i	0.999999	−	0.00114737i	$-0.000365218\pi$
$444$	−250.529	−	62.1990i	−0.564254	−	0.140088i
$445$	−809.816			−1.81981
$446$			510.763i			1.14521i
$447$	−19.4374	+	78.2910i	−0.0434840	+	0.175148i
$448$	−185.519			−0.414104
$449$			486.766i			1.08411i	0.840343	+	0.542055i	$0.182354\pi$
$449$			486.766i			1.08411i	−0.840343	+	0.542055i	$0.817646\pi$
$450$	−624.951	+	1181.03i	−1.38878	+	2.62451i
$451$	−31.0238			−0.0687889
$452$			563.634i			1.24698i
$453$	362.179	+	89.9185i	0.799513	+	0.198496i
$454$	−405.119			−0.892332
$455$			69.1146i			0.151900i
$456$	351.773	−	1416.89i	0.771433	−	3.10722i
$457$	−333.251			−0.729214			−0.364607	−	0.931161i	$-0.618797\pi$
$457$	−333.251			−0.729214			−0.364607	+	0.931161i	$0.618797\pi$
$458$			1489.19i			3.25152i
$459$	15.8582	−	17.7184i	0.0345494	−	0.0386022i
$460$	−1367.87			−2.97364
$461$			459.339i			0.996398i	0.867063	+	0.498199i	$0.166005\pi$
$461$			459.339i			0.996398i	−0.867063	+	0.498199i	$0.833995\pi$
$462$	−223.484	−	55.4845i	−0.483731	−	0.120096i
$463$	493.142			1.06510			0.532551	−	0.846398i	$-0.321233\pi$
$463$	493.142			1.06510			0.532551	+	0.846398i	$0.321233\pi$
$464$		−	873.270i		−	1.88205i
$465$	327.362	−	1318.57i	0.704004	−	2.83563i
$466$	−560.487			−1.20276
$467$			531.264i			1.13761i	0.822473	+	0.568805i	$0.192594\pi$
$467$			531.264i			1.13761i	−0.822473	+	0.568805i	$0.807406\pi$
$468$	−154.555	−	81.7843i	−0.330247	−	0.174753i
$469$	360.820			0.769338
$470$		−	2168.46i		−	4.61374i
$471$	649.898	+	161.351i	1.37983	+	0.342571i
$472$	113.664			0.240814
$473$		−	87.6794i		−	0.185369i
$474$	−377.012	+	1518.55i	−0.795385	+	3.20370i
$475$	−1396.04			−2.93903
$476$			25.8449i			0.0542961i
$477$	304.010	−	574.517i	0.637338	−	1.20444i
$478$	754.482			1.57841
$479$		−	601.359i		−	1.25545i	−0.778437	−	0.627723i	$-0.783987\pi$
$479$		−	601.359i		−	1.25545i	0.778437	−	0.627723i	$-0.216013\pi$
$480$	159.146	+	39.5112i	0.331554	+	0.0823151i
$481$	−24.6744			−0.0512982
$482$		−	1078.81i		−	2.23819i
$483$	−52.1478	+	210.044i	−0.107967	+	0.434874i
$484$	−684.056			−1.41334
$485$		−	766.092i		−	1.57957i
$486$	−797.138	+	294.636i	−1.64020	+	0.606248i
$487$	−555.132			−1.13990			−0.569951	−	0.821679i	$-0.693038\pi$
$487$	−555.132			−1.13990			−0.569951	+	0.821679i	$0.693038\pi$
$488$		−	122.030i		−	0.250061i
$489$	−179.809	−	44.6414i	−0.367709	−	0.0912913i
$490$	1042.33			2.12720
$491$		−	10.0690i		−	0.0205071i	−0.999947	−	0.0102535i	$-0.996736\pi$
$491$		−	10.0690i		−	0.0205071i	0.999947	−	0.0102535i	$-0.00326386\pi$
$492$	29.9866	−	120.782i	0.0609484	−	0.245492i
$493$	40.8483			0.0828565
$494$		−	271.473i		−	0.549541i
$495$	−402.180	−	212.817i	−0.812484	−	0.429932i
$496$	−1038.18			−2.09311
$497$			348.142i			0.700487i
$498$	−866.564	−	215.143i	−1.74009	−	0.432013i
$499$	409.942			0.821526			0.410763	−	0.911742i	$-0.365262\pi$
$499$	409.942			0.821526			0.410763	+	0.911742i	$0.365262\pi$
$500$		−	1179.73i		−	2.35946i
$501$	−43.9737	+	177.120i	−0.0877718	+	0.353532i
$502$	−478.419			−0.953025
$503$		−	50.1524i		−	0.0997065i	−0.998757	−	0.0498532i	$-0.984125\pi$
$503$		−	50.1524i		−	0.0997065i	0.998757	−	0.0498532i	$-0.0158754\pi$
$504$	222.079	−	419.684i	0.440634	−	0.832707i
$505$	−331.393			−0.656224
$506$		−	435.624i		−	0.860916i
$507$	475.840	+	118.137i	0.938540	+	0.233012i
$508$	−631.532			−1.24317
$509$			615.820i			1.20986i	0.796278	+	0.604931i	$0.206799\pi$
$509$			615.820i			1.20986i	−0.796278	+	0.604931i	$0.793201\pi$
$510$	−18.2857	+	73.6521i	−0.0358543	+	0.144416i
$511$	−102.432			−0.200455
$512$			989.132i			1.93190i
$513$	−661.621	−	592.158i	−1.28971	−	1.15430i
$514$	49.0272			0.0953836
$515$			977.928i			1.89889i
$516$	341.353	+	84.7480i	0.661537	+	0.164240i
$517$	464.741			0.898919
$518$		−	130.342i		−	0.251626i
$519$	80.2206	−	323.117i	0.154568	−	0.622576i
$520$	286.865			0.551663
$521$			428.898i			0.823220i	0.911360	+	0.411610i	$0.135033\pi$
$521$			428.898i			0.823220i	−0.911360	+	0.411610i	$0.864967\pi$
$522$	−1290.39	−	682.820i	−2.47201	−	1.30808i
$523$	−636.438			−1.21690			−0.608450	−	0.793592i	$-0.708208\pi$
$523$	−636.438			−1.21690			−0.608450	+	0.793592i	$0.708208\pi$
$524$			1798.58i			3.43240i
$525$	−440.669	−	109.405i	−0.839369	−	0.208391i
$526$	−551.735			−1.04893
$527$		−	48.5622i		−	0.0921483i
$528$	−83.7810	+	337.458i	−0.158676	+	0.639126i
$529$	119.573			0.226036
$530$			2074.41i			3.91399i
$531$	32.3331	−	61.1029i	0.0608910	−	0.115071i
$532$	965.072			1.81405
$533$		−	11.8957i		−	0.0223185i
$534$	1004.06	+	249.278i	1.88026	+	0.466814i
$535$	813.473			1.52051
$536$		−	1497.61i		−	2.79404i
$537$	39.4687	−	158.974i	0.0734984	−	0.296041i
$538$	17.1690			0.0319126
$539$			223.390i			0.414453i
$540$	1217.27	−	1360.06i	2.25421	−	2.51864i
$541$	−724.827			−1.33979			−0.669896	−	0.742455i	$-0.733661\pi$
$541$	−724.827			−1.33979			−0.669896	+	0.742455i	$0.733661\pi$
$542$			593.363i			1.09477i
$543$	363.635	+	90.2799i	0.669677	+	0.166261i
$544$	5.86125			0.0107744
$545$		−	166.442i		−	0.305397i
$546$	21.2749	−	85.6924i	0.0389651	−	0.156946i
$547$	869.429			1.58945			0.794725	−	0.606970i	$-0.207615\pi$
$547$	869.429			1.58945			0.794725	+	0.606970i	$0.207615\pi$
$548$			464.828i			0.848226i
$549$	−65.6003	−	34.7129i	−0.119490	−	0.0632294i
$550$	913.931			1.66169
$551$		−	1525.31i		−	2.76826i
$552$	871.802	+	216.443i	1.57935	+	0.392107i
$553$	−531.682			−0.961451
$554$		−	865.008i		−	1.56139i
$555$	62.0605	−	249.971i	0.111821	−	0.450398i
$556$	−510.414			−0.918010
$557$			382.734i			0.687135i	0.939128	+	0.343568i	$0.111635\pi$
$557$			382.734i			0.687135i	−0.939128	+	0.343568i	$0.888365\pi$
$558$	−811.766	+	1534.07i	−1.45478	+	2.74923i
$559$	33.6197			0.0601426
$560$			551.293i			0.984452i
$561$	−15.7850	−	3.91896i	−0.0281373	−	0.00698567i
$562$	127.913			0.227604
$563$			870.684i			1.54651i	0.634096	+	0.773254i	$0.281372\pi$
$563$			870.684i			1.54651i	−0.634096	+	0.773254i	$0.718628\pi$
$564$	−449.204	+	1809.33i	−0.796461	+	3.20803i
$565$	−562.379			−0.995361
$566$			1045.43i			1.84705i
$567$	−162.439	−	238.769i	−0.286488	−	0.421109i
$568$	1444.99			2.54399
$569$			678.235i			1.19198i	0.802993	+	0.595989i	$0.203240\pi$
$569$			678.235i			1.19198i	−0.802993	+	0.595989i	$0.796760\pi$
$570$	2750.23	+	682.803i	4.82497	+	1.19790i
$571$	−615.610			−1.07813			−0.539063	−	0.842266i	$-0.681221\pi$
$571$	−615.610			−1.07813			−0.539063	+	0.842266i	$0.681221\pi$
$572$			119.602i			0.209094i
$573$	−69.8587	+	281.381i	−0.121917	+	0.491066i
$574$	62.8391			0.109476
$575$		−	858.970i		−	1.49386i
$576$	413.937	+	219.038i	0.718641	+	0.380274i
$577$	755.566			1.30947			0.654737	−	0.755857i	$-0.272780\pi$
$577$	755.566			1.30947			0.654737	+	0.755857i	$0.272780\pi$
$578$		−	1008.01i		−	1.74396i
$579$	20.1109	+	4.99294i	0.0347338	+	0.00862339i
$580$	3135.51			5.40605
$581$		−	303.405i		−	0.522212i
$582$	−235.819	+	949.846i	−0.405187	+	1.63204i
$583$	−444.586			−0.762583
$584$			425.152i			0.728001i
$585$	81.6022	−	154.211i	0.139491	−	0.263609i
$586$	−588.101			−1.00359
$587$			464.614i			0.791505i	0.918357	+	0.395753i	$0.129516\pi$
$587$			464.614i			0.791505i	−0.918357	+	0.395753i	$0.870484\pi$
$588$	−869.703	−	215.922i	−1.47909	−	0.367214i
$589$	−1813.35			−3.07870
$590$			220.625i			0.373941i
$591$	17.9207	−	72.1822i	0.0303227	−	0.122136i
$592$	−196.816			−0.332459
$593$		−	401.514i		−	0.677090i	−0.940950	−	0.338545i	$-0.890065\pi$
$593$		−	401.514i		−	0.677090i	0.940950	−	0.338545i	$-0.109935\pi$
$594$	433.137	+	387.662i	0.729187	+	0.652630i
$595$	−25.7874			−0.0433402
$596$			221.331i			0.371360i
$597$	58.6698	+	14.5660i	0.0982744	+	0.0243987i
$598$	167.035			0.279323
$599$			488.252i			0.815111i	0.913180	+	0.407556i	$0.133619\pi$
$599$			488.252i			0.815111i	−0.913180	+	0.407556i	$0.866381\pi$
$600$	−454.094	+	1829.03i	−0.756823	+	3.04838i
$601$	353.891			0.588836			0.294418	−	0.955677i	$-0.404874\pi$
$601$	353.891			0.588836			0.294418	+	0.955677i	$0.404874\pi$
$602$			177.596i			0.295009i
$603$	−805.076	−	426.012i	−1.33512	−	0.706488i
$604$	1023.89			1.69518
$605$		−	682.533i		−	1.12815i
$606$	410.881	+	102.010i	0.678022	+	0.168333i
$607$	571.185			0.940996			0.470498	−	0.882401i	$-0.344074\pi$
$607$	571.185			0.940996			0.470498	+	0.882401i	$0.344074\pi$
$608$		−	218.864i		−	0.359974i
$609$	119.536	−	481.474i	0.196282	−	0.790598i
$610$	236.864			0.388301
$611$			178.200i			0.291653i
$612$	30.5146	−	57.6663i	0.0498604	−	0.0942260i
$613$	48.0285			0.0783499			0.0391750	−	0.999232i	$-0.487527\pi$
$613$	48.0285			0.0783499			0.0391750	+	0.999232i	$0.487527\pi$
$614$		−	556.534i		−	0.906407i
$615$	120.513	+	29.9199i	0.195956	+	0.0486502i
$616$	−324.770			−0.527224
$617$		−	529.530i		−	0.858233i	−0.903249	−	0.429116i	$-0.858825\pi$
$617$		−	529.530i		−	0.858233i	0.903249	−	0.429116i	$-0.141175\pi$
$618$	301.027	−	1212.49i	0.487098	−	1.96196i
$619$	−35.3405			−0.0570929			−0.0285465	−	0.999592i	$-0.509088\pi$
$619$	−35.3405			−0.0570929			−0.0285465	+	0.999592i	$0.509088\pi$
$620$		−	3727.62i		−	6.01230i
$621$	364.349	−	407.089i	0.586714	−	0.655538i
$622$	1300.37			2.09063
$623$			351.545i			0.564278i
$624$	−129.395	−	32.1249i	−0.207363	−	0.0514823i
$625$	115.824			0.185318
$626$			3.80852i			0.00608390i
$627$	−146.337	+	589.426i	−0.233393	+	0.940074i
$628$	1837.28			2.92560
$629$		−	9.20630i		−	0.0146364i
$630$	814.620	+	431.063i	1.29305	+	0.684226i
$631$	−33.5343			−0.0531447			−0.0265724	−	0.999647i	$-0.508459\pi$
$631$	−33.5343			−0.0531447			−0.0265724	+	0.999647i	$0.508459\pi$
$632$			2206.78i			3.49174i
$633$	−946.805	−	235.064i	−1.49574	−	0.371349i
$634$	−1166.63			−1.84011
$635$		−	630.127i		−	0.992325i
$636$	429.722	−	1730.86i	0.675664	−	2.72148i
$637$	−85.6566			−0.134469
$638$			998.559i			1.56514i
$639$	411.044	−	776.789i	0.643261	−	1.21563i
$640$	−1713.24			−2.67694
$641$			177.729i			0.277268i	0.990344	+	0.138634i	$0.0442712\pi$
$641$			177.729i			0.277268i	−0.990344	+	0.138634i	$0.955729\pi$
$642$	−1008.59	−	250.404i	−1.57102	−	0.390037i
$643$	−182.397			−0.283666			−0.141833	−	0.989891i	$-0.545300\pi$
$643$	−182.397			−0.283666			−0.141833	+	0.989891i	$0.545300\pi$
$644$			593.800i			0.922050i
$645$	−84.5594	+	340.593i	−0.131100	+	0.528052i
$646$	101.290			0.156795
$647$			713.043i			1.10208i	0.834480	+	0.551038i	$0.185768\pi$
$647$			713.043i			1.10208i	−0.834480	+	0.551038i	$0.814232\pi$
$648$	−991.025	+	674.212i	−1.52936	+	1.04045i
$649$	−47.2841			−0.0728568
$650$			350.437i			0.539134i
$651$	−572.397	−	142.109i	−0.879258	−	0.218294i
$652$	−508.326			−0.779641
$653$		−	173.772i		−	0.266114i	−0.991108	−	0.133057i	$-0.957521\pi$
$653$		−	173.772i		−	0.266114i	0.991108	−	0.133057i	$-0.0424793\pi$
$654$	−51.2342	+	206.364i	−0.0783397	+	0.315541i
$655$	−1794.57			−2.73981
$656$		−	94.8864i		−	0.144644i
$657$	228.551	+	120.940i	0.347871	+	0.184079i
$658$	−941.339			−1.43061
$659$		−	741.153i		−	1.12466i	−0.826912	−	0.562332i	$-0.809904\pi$
$659$		−	741.153i		−	1.12466i	0.826912	−	0.562332i	$-0.190096\pi$
$660$	−1211.66	−	300.819i	−1.83584	−	0.455786i
$661$	583.315			0.882474			0.441237	−	0.897391i	$-0.354540\pi$
$661$	583.315			0.882474			0.441237	+	0.897391i	$0.354540\pi$
$662$		−	1167.78i		−	1.76402i
$663$	1.50268	−	6.05260i	0.00226649	−	0.00912911i
$664$	−1259.30			−1.89654
$665$			962.924i			1.44801i
$666$	−153.893	+	290.825i	−0.231070	+	0.436675i
$667$	938.509			1.40706
$668$			500.722i			0.749584i
$669$	425.224	+	105.571i	0.635612	+	0.157804i
$670$	2906.90			4.33865
$671$			50.7643i			0.0756547i
$672$	17.1520	−	69.0860i	0.0255239	−	0.102807i
$673$	−879.243			−1.30645			−0.653226	−	0.757163i	$-0.726585\pi$
$673$	−879.243			−1.30645			−0.653226	+	0.757163i	$0.726585\pi$
$674$		−	1288.88i		−	1.91229i
$675$	854.066	+	764.398i	1.26528	+	1.13244i
$676$	1345.21			1.98996
$677$			916.427i			1.35366i	0.736140	+	0.676830i	$0.236647\pi$
$677$			916.427i			1.35366i	−0.736140	+	0.676830i	$0.763353\pi$
$678$	697.271	+	173.112i	1.02842	+	0.255328i
$679$	−332.564			−0.489785
$680$			107.032i			0.157401i
$681$	−83.7349	+	337.272i	−0.122959	+	0.495261i
$682$	1187.13			1.74066
$683$			744.048i			1.08938i	0.838637	+	0.544691i	$0.183353\pi$
$683$			744.048i			1.08938i	−0.838637	+	0.544691i	$0.816647\pi$
$684$	−2153.31	−	1139.44i	−3.14811	−	1.66585i
$685$	−463.793			−0.677070
$686$		−	1063.45i		−	1.55022i
$687$	1239.80	+	307.805i	1.80465	+	0.448042i
$688$	268.168			0.389779
$689$		−	170.472i		−	0.247419i
$690$	−420.122	+	1692.19i	−0.608873	+	2.45246i
$691$	252.740			0.365760			0.182880	−	0.983135i	$-0.441458\pi$
$691$	252.740			0.365760			0.182880	+	0.983135i	$0.441458\pi$
$692$		−	913.460i		−	1.32003i
$693$	−92.3847	+	174.588i	−0.133311	+	0.251931i
$694$	−1590.94			−2.29242
$695$		−	509.277i		−	0.732773i
$696$	−1998.39	−	496.142i	−2.87125	−	0.712848i
$697$	4.43843			0.00636790
$698$		−	1692.41i		−	2.42466i
$699$	−115.848	+	466.620i	−0.165734	+	0.667554i
$700$	−1245.78			−1.77969
$701$		−	590.422i		−	0.842256i	−0.907001	−	0.421128i	$-0.861634\pi$
$701$		−	590.422i		−	0.842256i	0.907001	−	0.421128i	$-0.138366\pi$
$702$	−148.645	+	166.082i	−0.211745	+	0.236583i
$703$	−343.771			−0.489006
$704$		−	320.322i		−	0.455003i
$705$	−1805.30	−	448.204i	−2.56071	−	0.635750i
$706$	−1029.25			−1.45786
$707$			143.860i			0.203479i
$708$	45.7033	−	184.086i	0.0645526	−	0.260009i
$709$	648.001			0.913964			0.456982	−	0.889476i	$-0.348930\pi$
$709$	648.001			0.913964			0.456982	+	0.889476i	$0.348930\pi$
$710$			2804.76i			3.95037i
$711$	1186.31	+	627.746i	1.66851	+	0.882906i
$712$	1459.11			2.04932
$713$		−	1115.74i		−	1.56485i
$714$	31.9728	+	7.93791i	0.0447798	+	0.0111175i
$715$	−119.335			−0.166903
$716$		−	449.424i		−	0.627687i
$717$	155.946	−	628.127i	0.217497	−	0.876049i
$718$	533.706			0.743323
$719$		−	532.079i		−	0.740027i	−0.929026	−	0.370014i	$-0.879353\pi$
$719$		−	532.079i		−	0.740027i	0.929026	−	0.370014i	$-0.120647\pi$
$720$	650.900	−	1230.07i	0.904028	−	1.70843i
$721$	424.524			0.588798
$722$		−	2519.71i		−	3.48991i
$723$	−898.138	−	222.981i	−1.24224	−	0.308411i
$724$	1028.00			1.41990
$725$			1968.98i			2.71583i
$726$	−210.098	+	846.245i	−0.289391	+	1.16563i
$727$	−1288.32			−1.77210			−0.886049	−	0.463592i	$-0.846560\pi$
$727$	−1288.32			−1.77210			−0.886049	+	0.463592i	$0.846560\pi$
$728$		−	124.529i		−	0.171057i
$729$	80.5307	+	724.538i	0.110467	+	0.993880i
$730$	−825.234			−1.13046
$731$			12.5439i			0.0171599i
$732$	−197.636	−	49.0671i	−0.269994	−	0.0670316i
$733$	394.335			0.537974			0.268987	−	0.963144i	$-0.413311\pi$
$733$	394.335			0.537974			0.268987	+	0.963144i	$0.413311\pi$
$734$			884.188i			1.20462i
$735$	215.441	−	867.766i	0.293117	−	1.18063i
$736$	134.665			0.182969
$737$			623.002i			0.845322i
$738$	−140.209	−	74.1928i	−0.189985	−	0.100532i
$739$	−199.786			−0.270347			−0.135173	−	0.990822i	$-0.543159\pi$
$739$	−199.786			−0.270347			−0.135173	+	0.990822i	$0.543159\pi$
$740$		−	706.675i		−	0.954966i
$741$	−226.009	−	56.1115i	−0.305006	−	0.0757240i
$742$	900.514			1.21363
$743$			112.421i			0.151306i	0.997134	+	0.0756531i	$0.0241042\pi$
$743$			112.421i			0.151306i	−0.997134	+	0.0756531i	$0.975896\pi$
$744$	−589.835	+	2375.77i	−0.792789	+	3.19324i
$745$	−220.838			−0.296427
$746$		−	2205.27i		−	2.95612i
$747$	−358.224	+	676.970i	−0.479551	+	0.906252i
$748$	−44.6247			−0.0596587
$749$		−	353.133i		−	0.471472i
$750$	−1459.44	−	362.337i	−1.94592	−	0.483116i
$751$	785.625			1.04611			0.523053	−	0.852300i	$-0.324793\pi$
$751$	785.625			1.04611			0.523053	+	0.852300i	$0.324793\pi$
$752$			1421.41i			1.89018i
$753$	−98.8854	+	398.297i	−0.131322	+	0.528947i
$754$	−382.887			−0.507807
$755$			1021.61i			1.35313i
$756$	−590.410	−	528.423i	−0.780966	−	0.698973i
$757$	1361.82			1.79897			0.899486	−	0.436950i	$-0.143941\pi$
$757$	1361.82			1.79897			0.899486	+	0.436950i	$0.143941\pi$
$758$		−	50.9531i		−	0.0672205i
$759$	−362.669	−	90.0401i	−0.477824	−	0.118630i
$760$	3996.68			5.25879
$761$		−	1092.13i		−	1.43512i	−0.696494	−	0.717562i	$-0.745258\pi$
$761$		−	1092.13i		−	1.43512i	0.696494	−	0.717562i	$-0.254742\pi$
$762$	−193.966	+	781.269i	−0.254549	+	1.02529i
$763$	−72.2531			−0.0946961
$764$			795.471i			1.04119i
$765$	57.5379	+	30.4467i	0.0752130	+	0.0397996i
$766$	−2080.76			−2.71640
$767$		−	18.1306i		−	0.0236383i
$768$	1518.16	+	376.914i	1.97677	+	0.490773i
$769$	428.468			0.557176			0.278588	−	0.960411i	$-0.410134\pi$
$769$	428.468			0.557176			0.278588	+	0.960411i	$0.410134\pi$
$770$		−	630.388i		−	0.818686i
$771$	10.1335	−	40.8165i	0.0131434	−	0.0529396i
$772$	56.8540			0.0736450
$773$		−	493.717i		−	0.638702i	−0.947637	−	0.319351i	$-0.896535\pi$
$773$		−	493.717i		−	0.638702i	0.947637	−	0.319351i	$-0.103465\pi$
$774$	209.683	−	396.259i	0.270909	−	0.511962i
$775$	2340.80			3.02039
$776$			1380.33i			1.77878i
$777$	−108.514	−	26.9408i	−0.139657	−	0.0346728i
$778$	−1419.36			−1.82436
$779$		−	165.735i		−	0.212753i
$780$	115.346	−	464.596i	0.147879	−	0.595637i
$781$	−601.113			−0.769670
$782$			62.3226i			0.0796965i
$783$	−835.180	+	933.151i	−1.06664	+	1.19176i
$784$	−683.240			−0.871480
$785$			1833.19i			2.33527i
$786$

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)

\(f(q)\)	\(=\)	\(q-3.49731i q^{2} +(-2.91161 - 0.722867i) q^{3} -8.23119 q^{4} -8.21287i q^{5} +(-2.52809 + 10.1828i) q^{6} -3.56525 q^{7} +14.7978i q^{8} +(7.95493 + 4.20941i) q^{9} +O(q^{10})\)	\(q-3.49731i q^{2} +(-2.91161 - 0.722867i) q^{3} -8.23119 q^{4} -8.21287i q^{5} +(-2.52809 + 10.1828i) q^{6} -3.56525 q^{7} +14.7978i q^{8} +(7.95493 + 4.20941i) q^{9} -28.7230 q^{10} -6.15586i q^{11} +(23.9660 + 5.95006i) q^{12} +2.36040 q^{13} +12.4688i q^{14} +(-5.93681 + 23.9127i) q^{15} +18.8277 q^{16} +0.880690i q^{17} +(14.7216 - 27.8209i) q^{18} +32.8857 q^{19} +67.6017i q^{20} +(10.3806 + 2.57720i) q^{21} -21.5290 q^{22} +20.2343i q^{23} +(10.6968 - 43.0854i) q^{24} -42.4512 q^{25} -8.25505i q^{26} +(-20.1188 - 18.0065i) q^{27} +29.3462 q^{28} -46.3821i q^{29} +(83.6300 + 20.7629i) q^{30} -55.1410 q^{31} -6.65529i q^{32} +(-4.44987 + 17.9235i) q^{33} +3.08005 q^{34} +29.2809i q^{35} +(-65.4785 - 34.6485i) q^{36} -10.4535 q^{37} -115.012i q^{38} +(-6.87256 - 1.70626i) q^{39} +121.532 q^{40} -5.03972i q^{41} +(9.01328 - 36.3042i) q^{42} +14.2432 q^{43} +50.6701i q^{44} +(34.5713 - 65.3327i) q^{45} +70.7656 q^{46} +75.4957i q^{47} +(-54.8190 - 13.6100i) q^{48} -36.2890 q^{49} +148.465i q^{50} +(0.636622 - 2.56423i) q^{51} -19.4289 q^{52} -72.2215i q^{53} +(-62.9744 + 70.3617i) q^{54} -50.5573 q^{55} -52.7578i q^{56} +(-95.7504 - 23.7720i) q^{57} -162.213 q^{58} -7.68115i q^{59} +(48.8670 - 196.830i) q^{60} -8.24650 q^{61} +192.845i q^{62} +(-28.3613 - 15.0076i) q^{63} +52.0353 q^{64} -19.3856i q^{65} +(62.6840 + 15.5626i) q^{66} -101.205 q^{67} -7.24913i q^{68} +(14.6267 - 58.9143i) q^{69} +102.404 q^{70} -97.6488i q^{71} +(-62.2900 + 117.715i) q^{72} +28.7308 q^{73} +36.5592i q^{74} +(123.601 + 30.6866i) q^{75} -270.689 q^{76} +21.9472i q^{77} +(-5.96731 + 24.0355i) q^{78} +149.129 q^{79} -154.630i q^{80} +(45.5617 + 66.9711i) q^{81} -17.6255 q^{82} +85.1008i q^{83} +(-85.4447 - 21.2134i) q^{84} +7.23299 q^{85} -49.8130i q^{86} +(-33.5281 + 135.047i) q^{87} +91.0932 q^{88} -98.6034i q^{89} +(-228.489 - 120.907i) q^{90} -8.41541 q^{91} -166.552i q^{92} +(160.549 + 39.8596i) q^{93} +264.032 q^{94} -270.086i q^{95} +(-4.81090 + 19.3776i) q^{96} +93.2794 q^{97} +126.914i q^{98} +(25.9126 - 48.9694i) 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