LMFDB - Embedded newform 177.4.d.c.176.20

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.d (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$10.4433380710$
Analytic rank:	$0$
Dimension:	$52$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			176.20
Character	$\chi$	$=$	177.176
Dual form			177.4.d.c.176.19

$q$-expansion

$f(q)$	$=$	$q-2.52893 q^{2} +(0.100502 + 5.19518i) q^{3} -1.60453 q^{4} -17.9211i q^{5} +(-0.254162 - 13.1382i) q^{6} -4.91298 q^{7} +24.2892 q^{8} +(-26.9798 + 1.04425i) q^{9} +O(q^{10})$	\(q-2.52893 q^{2} +(0.100502 + 5.19518i) q^{3} -1.60453 q^{4} -17.9211i q^{5} +(-0.254162 - 13.1382i) q^{6} -4.91298 q^{7} +24.2892 q^{8} +(-26.9798 + 1.04425i) q^{9} +45.3212i q^{10} +38.0756 q^{11} +(-0.161258 - 8.33584i) q^{12} +51.2270i q^{13} +12.4246 q^{14} +(93.1035 - 1.80111i) q^{15} -48.5892 q^{16} +30.7171i q^{17} +(68.2299 - 2.64083i) q^{18} -45.5739 q^{19} +28.7550i q^{20} +(-0.493764 - 25.5238i) q^{21} -96.2904 q^{22} -137.131 q^{23} +(2.44110 + 126.187i) q^{24} -196.167 q^{25} -129.549i q^{26} +(-8.13658 - 140.060i) q^{27} +7.88305 q^{28} +12.5498i q^{29} +(-235.452 + 4.55486i) q^{30} +75.5324i q^{31} -71.4347 q^{32} +(3.82667 + 197.810i) q^{33} -77.6812i q^{34} +88.0462i q^{35} +(43.2900 - 1.67553i) q^{36} +152.739i q^{37} +115.253 q^{38} +(-266.133 + 5.14840i) q^{39} -435.289i q^{40} +272.870i q^{41} +(1.24869 + 64.5479i) q^{42} +364.987i q^{43} -61.0936 q^{44} +(18.7141 + 483.509i) q^{45} +346.793 q^{46} +430.845 q^{47} +(-4.88330 - 252.430i) q^{48} -318.863 q^{49} +496.092 q^{50} +(-159.581 + 3.08712i) q^{51} -82.1954i q^{52} +660.497i q^{53} +(20.5768 + 354.201i) q^{54} -682.358i q^{55} -119.332 q^{56} +(-4.58026 - 236.765i) q^{57} -31.7375i q^{58} +(-351.582 + 285.954i) q^{59} +(-149.388 + 2.88993i) q^{60} -251.217i q^{61} -191.016i q^{62} +(132.551 - 5.13038i) q^{63} +569.367 q^{64} +918.046 q^{65} +(-9.67736 - 500.246i) q^{66} +442.198i q^{67} -49.2865i q^{68} +(-13.7819 - 712.418i) q^{69} -222.662i q^{70} -639.306i q^{71} +(-655.316 + 25.3639i) q^{72} +36.1944i q^{73} -386.267i q^{74} +(-19.7151 - 1019.12i) q^{75} +73.1249 q^{76} -187.065 q^{77} +(673.032 - 13.0199i) q^{78} -485.902 q^{79} +870.774i q^{80} +(726.819 - 56.3473i) q^{81} -690.068i q^{82} -1376.64 q^{83} +(0.792260 + 40.9538i) q^{84} +550.485 q^{85} -923.025i q^{86} +(-65.1984 + 1.26128i) q^{87} +924.825 q^{88} +1250.69 q^{89} +(-47.3267 - 1222.76i) q^{90} -251.677i q^{91} +220.030 q^{92} +(-392.404 + 7.59114i) q^{93} -1089.58 q^{94} +816.736i q^{95} +(-7.17931 - 371.116i) q^{96} +1687.28i q^{97} +806.380 q^{98} +(-1027.27 + 39.7605i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52q - 8q^{3} + 268q^{4} - 16q^{7} - 4q^{9} + O(q^{10}) $	$ 52q - 8q^{3} + 268q^{4} - 16q^{7} - 4q^{9} + 28q^{12} + 114q^{15} + 484q^{16} - 184q^{19} - 758q^{21} - 60q^{22} + 36q^{25} + 742q^{27} - 4q^{28} - 888q^{36} + 1402q^{45} - 660q^{46} - 488q^{48} - 924q^{49} - 1772q^{51} - 630q^{57} - 1880q^{60} - 212q^{63} + 7648q^{64} + 1316q^{66} - 1556q^{75} - 5680q^{76} + 3224q^{78} - 1504q^{79} - 276q^{81} + 1228q^{84} - 848q^{85} + 3598q^{87} + 5760q^{88} + 888q^{94} + 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 $

$2$	−2.52893			−0.894110			−0.447055	−	0.894506i	$-0.647527\pi$
$2$	−2.52893			−0.894110			−0.447055	+	0.894506i	$0.647527\pi$
$3$	0.100502	+	5.19518i	0.0193416	+	0.999813i
$3$	0.100502	+	5.19518i	0.0193416	+	0.999813i
$4$	−1.60453			−0.200567
$5$		−	17.9211i		−	1.60291i	−0.598052	−	0.801457i	$-0.704058\pi$
$5$		−	17.9211i		−	1.60291i	0.598052	−	0.801457i	$-0.295942\pi$
$6$	−0.254162	−	13.1382i	−0.0172935	−	0.893943i
$7$	−4.91298			−0.265276			−0.132638	−	0.991165i	$-0.542345\pi$
$7$	−4.91298			−0.265276			−0.132638	+	0.991165i	$0.542345\pi$
$8$	24.2892			1.07344
$9$	−26.9798	+	1.04425i	−0.999252	+	0.0386759i
$10$			45.3212i			1.43318i
$11$	38.0756			1.04366			0.521829	−	0.853050i	$-0.325250\pi$
$11$	38.0756			1.04366			0.521829	+	0.853050i	$0.325250\pi$
$12$	−0.161258	−	8.33584i	−0.00387927	−	0.200529i
$13$			51.2270i			1.09291i	0.837489	+	0.546454i	$0.184023\pi$
$13$			51.2270i			1.09291i	−0.837489	+	0.546454i	$0.815977\pi$
$14$	12.4246			0.237186
$15$	93.1035	−	1.80111i	1.60261	−	0.0310029i
$16$	−48.5892			−0.759206
$17$			30.7171i			0.438234i	0.975699	+	0.219117i	$0.0703177\pi$
$17$			30.7171i			0.438234i	−0.975699	+	0.219117i	$0.929682\pi$
$18$	68.2299	−	2.64083i	0.893441	−	0.0345805i
$19$	−45.5739			−0.550283			−0.275141	−	0.961404i	$-0.588725\pi$
$19$	−45.5739			−0.550283			−0.275141	+	0.961404i	$0.588725\pi$
$20$			28.7550i			0.321491i
$21$	−0.493764	−	25.5238i	−0.00513086	−	0.265227i
$22$	−96.2904			−0.933145
$23$	−137.131			−1.24320			−0.621602	−	0.783333i	$-0.713518\pi$
$23$	−137.131			−1.24320			−0.621602	+	0.783333i	$0.713518\pi$
$24$	2.44110	+	126.187i	0.0207620	+	1.07324i
$25$	−196.167			−1.56934
$26$		−	129.549i		−	0.977181i
$27$	−8.13658	−	140.060i	−0.0579958	−	0.998317i
$28$	7.88305			0.0532056
$29$			12.5498i			0.0803598i	0.999192	+	0.0401799i	$0.0127931\pi$
$29$			12.5498i			0.0803598i	−0.999192	+	0.0401799i	$0.987207\pi$
$30$	−235.452	+	4.55486i	−1.43291	+	0.0277200i
$31$			75.5324i			0.437614i	0.975768	+	0.218807i	$0.0702164\pi$
$31$			75.5324i			0.437614i	−0.975768	+	0.218807i	$0.929784\pi$
$32$	−71.4347			−0.394625
$33$	3.82667	+	197.810i	0.0201860	+	1.04346i
$34$		−	77.6812i		−	0.391830i
$35$			88.0462i			0.425215i
$36$	43.2900	−	1.67553i	0.200417	−	0.00775710i
$37$			152.739i			0.678654i	0.940668	+	0.339327i	$0.110199\pi$
$37$			152.739i			0.678654i	−0.940668	+	0.339327i	$0.889801\pi$
$38$	115.253			0.492014
$39$	−266.133	+	5.14840i	−1.09270	+	0.0211386i
$40$		−	435.289i		−	1.72063i
$41$			272.870i			1.03939i	0.854351	+	0.519697i	$0.173955\pi$
$41$			272.870i			1.03939i	−0.854351	+	0.519697i	$0.826045\pi$
$42$	1.24869	+	64.5479i	0.00458756	+	0.237142i
$43$			364.987i			1.29442i	0.762313	+	0.647209i	$0.224064\pi$
$43$			364.987i			1.29442i	−0.762313	+	0.647209i	$0.775936\pi$
$44$	−61.0936			−0.209323
$45$	18.7141	+	483.509i	0.0619942	+	1.60172i
$46$	346.793			1.11156
$47$	430.845			1.33713			0.668567	−	0.743652i	$-0.266908\pi$
$47$	430.845			1.33713			0.668567	+	0.743652i	$0.266908\pi$
$48$	−4.88330	−	252.430i	−0.0146842	−	0.759064i
$49$	−318.863			−0.929629
$50$	496.092			1.40316
$51$	−159.581	+	3.08712i	−0.438152	+	0.00847614i
$52$		−	82.1954i		−	0.219201i
$53$			660.497i			1.71181i	0.517129	+	0.855907i	$0.327001\pi$
$53$			660.497i			1.71181i	−0.517129	+	0.855907i	$0.672999\pi$
$54$	20.5768	+	354.201i	0.0518546	+	0.892605i
$55$		−	682.358i		−	1.67289i
$56$	−119.332			−0.284758
$57$	−4.58026	−	236.765i	−0.0106433	−	0.550180i
$58$		−	31.7375i		−	0.0718506i
$59$	−351.582	+	285.954i	−0.775797	+	0.630983i
$59$	−351.582	+	285.954i	−0.775797	+	0.630983i
$60$	−149.388	+	2.88993i	−0.321431	+	0.00621815i
$61$		−	251.217i		−	0.527295i	−0.964619	−	0.263648i	$-0.915074\pi$
$61$		−	251.217i		−	0.527295i	0.964619	−	0.263648i	$-0.0849256\pi$
$62$		−	191.016i		−	0.391275i
$63$	132.551	−	5.13038i	0.265078	−	0.0102598i
$64$	569.367			1.11204
$65$	918.046			1.75184
$66$	−9.67736	−	500.246i	−0.0180485	−	0.932971i
$67$			442.198i			0.806314i	0.915131	+	0.403157i	$0.132087\pi$
$67$			442.198i			0.806314i	−0.915131	+	0.403157i	$0.867913\pi$
$68$		−	49.2865i		−	0.0878952i
$69$	−13.7819	−	712.418i	−0.0240455	−	1.24297i
$70$		−	222.662i		−	0.380189i
$71$		−	639.306i		−	1.06862i	−0.845290	−	0.534308i	$-0.820572\pi$
$71$		−	639.306i		−	1.06862i	0.845290	−	0.534308i	$-0.179428\pi$
$72$	−655.316	+	25.3639i	−1.07264	+	0.0415162i
$73$			36.1944i			0.0580305i	0.999579	+	0.0290153i	$0.00923714\pi$
$73$			36.1944i			0.0580305i	−0.999579	+	0.0290153i	$0.990763\pi$
$74$		−	386.267i		−	0.606792i
$75$	−19.7151	−	1019.12i	−0.0303534	−	1.56904i
$76$	73.1249			0.110368
$77$	−187.065			−0.276858
$78$	673.032	−	13.0199i	0.976998	−	0.0189002i
$79$	−485.902			−0.692003			−0.346001	−	0.938234i	$-0.612461\pi$
$79$	−485.902			−0.692003			−0.346001	+	0.938234i	$0.612461\pi$
$80$			870.774i			1.21694i
$81$	726.819	−	56.3473i	0.997008	−	0.0772939i
$82$		−	690.068i		−	0.929332i
$83$	−1376.64			−1.82056			−0.910278	−	0.413998i	$-0.864132\pi$
$83$	−1376.64			−1.82056			−0.910278	+	0.413998i	$0.864132\pi$
$84$	0.792260	+	40.9538i	0.00102908	+	0.0531956i
$85$	550.485			0.702452
$86$		−	923.025i		−	1.15735i
$87$	−65.1984	+	1.26128i	−0.0803448	+	0.00155429i
$88$	924.825			1.12030
$89$	1250.69			1.48959			0.744793	−	0.667295i	$-0.232548\pi$
$89$	1250.69			1.48959			0.744793	+	0.667295i	$0.232548\pi$
$90$	−47.3267	−	1222.76i	−0.0554296	−	1.43211i
$91$		−	251.677i		−	0.289923i
$92$	220.030			0.249345
$93$	−392.404	+	7.59114i	−0.437532	+	0.00846414i
$94$	−1089.58			−1.19554
$95$			816.736i			0.882056i
$96$	−7.17931	−	371.116i	−0.00763266	−	0.394551i
$97$			1687.28i			1.76616i	0.469220	+	0.883081i	$0.344535\pi$
$97$			1687.28i			1.76616i	−0.469220	+	0.883081i	$0.655465\pi$
$98$	806.380			0.831191
$99$	−1027.27	+	39.7605i	−1.04288	+	0.0403644i
$100$	314.756			0.314756
$101$	916.279			0.902704			0.451352	−	0.892346i	$-0.350942\pi$
$101$	916.279			0.902704			0.451352	+	0.892346i	$0.350942\pi$
$102$	403.568	−	7.80710i	0.391757	−	0.00757860i
$103$			178.532i			0.170789i	0.996347	+	0.0853946i	$0.0272151\pi$
$103$			178.532i			0.170789i	−0.996347	+	0.0853946i	$0.972785\pi$
$104$			1244.26i			1.17317i
$105$	−457.416	+	8.84880i	−0.425136	+	0.00822433i
$106$		−	1670.35i		−	1.53055i
$107$		−	1657.49i		−	1.49753i	−0.662834	−	0.748766i	$-0.730646\pi$
$107$		−	1657.49i		−	1.49753i	0.662834	−	0.748766i	$-0.269354\pi$
$108$	13.0554	+	224.731i	0.0116320	+	0.200229i
$109$			116.077i			0.102002i	0.998699	+	0.0510009i	$0.0162411\pi$
$109$			116.077i			0.102002i	−0.998699	+	0.0510009i	$0.983759\pi$
$110$			1725.63i			1.49575i
$111$	−793.509	+	15.3506i	−0.678527	+	0.0131262i
$112$	238.718			0.201399
$113$	1038.47			0.864522			0.432261	−	0.901749i	$-0.357716\pi$
$113$	1038.47			0.864522			0.432261	+	0.901749i	$0.357716\pi$
$114$	11.5831	+	598.761i	0.00951632	+	0.491922i
$115$			2457.53i			1.99275i
$116$		−	20.1365i		−	0.0161175i
$117$	−53.4938	−	1382.09i	−0.0422692	−	1.09209i
$118$	889.124	−	723.155i	0.693648	−	0.564168i
$119$		−	150.912i		−	0.116253i
$120$	2261.41	−	43.7473i	1.72031	−	0.0332797i
$121$	118.753			0.0892212
$122$			635.309i			0.471460i
$123$	−1417.61	+	27.4239i	−1.03920	+	0.0201035i
$124$		−	121.194i		−	0.0877707i
$125$			1275.39i			0.912596i
$126$	−335.213	+	12.9744i	−0.237009	+	0.00917339i
$127$	1213.40			0.847813			0.423906	−	0.905706i	$-0.360659\pi$
$127$	1213.40			0.847813			0.423906	+	0.905706i	$0.360659\pi$
$128$	−868.409			−0.599666
$129$	−1896.17	+	36.6818i	−1.29418	+	0.0250361i
$130$	−2321.67			−1.56634
$131$	−2467.88			−1.64595			−0.822975	−	0.568077i	$-0.807687\pi$
$131$	−2467.88			−1.64595			−0.822975	+	0.568077i	$0.807687\pi$
$132$	−6.14001	−	317.392i	−0.00404863	−	0.209284i
$133$	223.904			0.145977
$134$		−	1118.29i		−	0.720934i
$135$	−2510.03	+	145.817i	−1.60022	+	0.0929623i
$136$			746.091i			0.470418i
$137$		−	653.589i		−	0.407590i	−0.979014	−	0.203795i	$-0.934672\pi$
$137$		−	653.589i		−	0.407590i	0.979014	−	0.203795i	$-0.0653276\pi$
$138$	34.8533	+	1801.65i	0.0214994	+	1.11135i
$139$	−706.567			−0.431153			−0.215576	−	0.976487i	$-0.569163\pi$
$139$	−706.567			−0.431153			−0.215576	+	0.976487i	$0.569163\pi$
$140$		−	141.273i		−	0.0852840i
$141$	43.3007	+	2238.32i	0.0258623	+	1.33688i
$142$			1616.76i			0.955460i
$143$			1950.50i			1.14062i
$144$	1310.93	−	50.7393i	0.758638	−	0.0293630i
$145$	224.906			0.128810
$146$		−	91.5329i		−	0.0518857i
$147$	−32.0463	−	1656.55i	−0.0179805	−	0.929455i
$148$		−	245.075i		−	0.136115i
$149$	−1113.52			−0.612235			−0.306118	−	0.951994i	$-0.599030\pi$
$149$	−1113.52			−0.612235			−0.306118	+	0.951994i	$0.599030\pi$
$150$	49.8581	+	2577.29i	0.0271393	+	1.40290i
$151$			1135.35i			0.611879i	0.952051	+	0.305939i	$0.0989705\pi$
$151$			1135.35i			0.611879i	−0.952051	+	0.305939i	$0.901029\pi$
$152$	−1106.95			−0.590695
$153$	−32.0763	−	828.740i	−0.0169491	−	0.437906i
$154$	473.074			0.247541
$155$	1353.63			0.701457
$156$	427.020	−	8.26078i	0.219160	−	0.00423969i
$157$		−	1602.61i		−	0.814665i	−0.913280	−	0.407332i	$-0.866459\pi$
$157$		−	1602.61i		−	0.814665i	0.913280	−	0.407332i	$-0.133541\pi$
$158$	1228.81			0.618727
$159$	−3431.40	+	66.3811i	−1.71149	+	0.0331092i
$160$			1280.19i			0.632550i
$161$	673.720			0.329793
$162$	−1838.07	+	142.498i	−0.891436	+	0.0691093i
$163$	867.586			0.416899			0.208450	−	0.978033i	$-0.433158\pi$
$163$	867.586			0.416899			0.208450	+	0.978033i	$0.433158\pi$
$164$		−	437.829i		−	0.208468i
$165$	3544.97	−	68.5782i	1.67258	−	0.0323564i
$166$	3481.43			1.62778
$167$		−	3842.95i		−	1.78070i	−0.455277	−	0.890350i	$-0.650460\pi$
$167$		−	3842.95i		−	1.78070i	0.455277	−	0.890350i	$-0.349540\pi$
$168$	−11.9931	−	619.953i	−0.00550767	−	0.284705i
$169$	−427.204			−0.194449
$170$	−1392.13			−0.628070
$171$	1229.58	−	47.5905i	0.549871	−	0.0212827i
$172$		−	585.633i		−	0.259617i
$173$	2410.10			1.05917			0.529586	−	0.848256i	$-0.322347\pi$
$173$	2410.10			1.05917			0.529586	+	0.848256i	$0.322347\pi$
$174$	164.882	−	3.18967i	0.0718371	−	0.00138970i
$175$	963.765			0.416307
$176$	−1850.06			−0.792352
$177$	−1520.91	−	1797.79i	−0.645870	−	0.763448i
$178$	−3162.91			−1.33185
$179$	1229.32			0.513318			0.256659	−	0.966502i	$-0.417378\pi$
$179$	1229.32			0.513318			0.256659	+	0.966502i	$0.417378\pi$
$180$	−30.0274	−	775.805i	−0.0124340	−	0.321251i
$181$	−2553.60			−1.04866			−0.524329	−	0.851516i	$-0.675684\pi$
$181$	−2553.60			−1.04866			−0.524329	+	0.851516i	$0.675684\pi$
$182$			636.474i			0.259223i
$183$	1305.12	−	25.2477i	0.527197	−	0.0101987i
$184$	−3330.78			−1.33450
$185$	2737.26			1.08782
$186$	992.362	−	19.1974i	0.391202	−	0.00756787i
$187$			1169.57i			0.457367i
$188$	−691.306			−0.268184
$189$	39.9749	+	688.113i	0.0153849	+	0.264830i
$190$		−	2065.47i		−	0.788656i
$191$	−3086.92			−1.16943			−0.584716	−	0.811238i	$-0.698794\pi$
$191$	−3086.92			−1.16943			−0.584716	+	0.811238i	$0.698794\pi$
$192$	57.2224	+	2957.96i	0.0215087	+	1.11184i
$193$	−4387.11			−1.63622			−0.818111	−	0.575060i	$-0.804979\pi$
$193$	−4387.11			−1.63622			−0.818111	+	0.575060i	$0.804979\pi$
$194$		−	4267.02i		−	1.57914i
$195$	92.2652	+	4769.41i	0.0338833	+	1.75151i
$196$	511.626			0.186452
$197$			1321.27i			0.477852i	0.971038	+	0.238926i	$0.0767953\pi$
$197$			1321.27i			0.477852i	−0.971038	+	0.238926i	$0.923205\pi$
$198$	2597.90	−	100.551i	0.932447	−	0.0360902i
$199$	2247.81			0.800718			0.400359	−	0.916358i	$-0.368885\pi$
$199$	2247.81			0.800718			0.400359	+	0.916358i	$0.368885\pi$
$200$	−4764.73			−1.68459
$201$	−2297.30	+	44.4416i	−0.806163	+	0.0155954i
$202$	−2317.20			−0.807117
$203$		−	61.6569i		−	0.0213176i
$204$	256.052	−	4.95338i	0.0878787	−	0.00170003i
$205$	4890.14			1.66606
$206$		−	451.494i		−	0.152704i
$207$	3699.75	−	143.199i	1.24227	−	0.0480820i
$208$		−	2489.08i		−	0.829743i
$209$	−1735.26			−0.574307
$210$	1156.77	−	22.3780i	0.380118	−	0.00735346i
$211$			238.261i			0.0777372i	0.999244	+	0.0388686i	$0.0123754\pi$
$211$			238.261i			0.0777372i	−0.999244	+	0.0388686i	$0.987625\pi$
$212$		−	1059.79i		−	0.343333i
$213$	3321.31	−	64.2514i	1.06842	−	0.0206687i
$214$			4191.68i			1.33896i
$215$	6540.98			2.07484
$216$	−197.631	−	3401.94i	−0.0622549	−	1.07163i
$217$		−	371.090i		−	0.116088i
$218$		−	293.551i		−	0.0912008i
$219$	−188.036	+	3.63760i	−0.0580197	+	0.00112240i
$220$			1094.87i			0.335527i
$221$	−1573.54			−0.478950
$222$	2006.72	−	38.8205i	0.606678	−	0.0117363i
$223$	2110.45			0.633750			0.316875	−	0.948467i	$-0.397366\pi$
$223$	2110.45			0.633750			0.316875	+	0.948467i	$0.397366\pi$
$224$	350.958			0.104685
$225$	5292.54	−	204.847i	1.56816	−	0.0606955i
$226$	−2626.21			−0.772978
$227$	2633.15			0.769904			0.384952	−	0.922937i	$-0.374218\pi$
$227$	2633.15			0.769904			0.384952	+	0.922937i	$0.374218\pi$
$228$	7.34918	+	379.897i	0.00213470	+	0.110348i
$229$			811.669i			0.234221i	0.993119	+	0.117110i	$0.0373631\pi$
$229$			811.669i			0.234221i	−0.993119	+	0.117110i	$0.962637\pi$
$230$		−	6214.92i		−	1.78174i
$231$	−18.8004	−	971.836i	−0.00535486	−	0.276806i
$232$			304.824i			0.0862614i
$233$	−4623.35			−1.29994			−0.649970	−	0.759960i	$-0.725219\pi$
$233$	−4623.35			−1.29994			−0.649970	+	0.759960i	$0.725219\pi$
$234$	135.282	+	3495.21i	0.0377934	+	0.976450i
$235$		−	7721.24i		−	2.14331i
$236$	564.124	−	458.822i	0.155599	−	0.126554i
$237$	−48.8340	−	2524.35i	−0.0133844	−	0.691873i
$238$			381.646i			0.103943i
$239$			4107.24i			1.11161i	0.831312	+	0.555806i	$0.187590\pi$
$239$			4107.24i			1.11161i	−0.831312	+	0.555806i	$0.812410\pi$
$240$	−4523.83	+	87.5143i	−1.21672	+	0.0235376i
$241$	−1369.49			−0.366045			−0.183023	−	0.983109i	$-0.558588\pi$
$241$	−1369.49			−0.366045			−0.183023	+	0.983109i	$0.558588\pi$
$242$	−300.319			−0.0797736
$243$	365.781	+	3770.29i	0.0965632	+	0.995327i
$244$			403.086i			0.105758i
$245$			5714.38i			1.49012i
$246$	3585.03	−	69.3530i	0.929159	−	0.0179748i
$247$		−	2334.61i		−	0.601409i
$248$			1834.62i			0.469752i
$249$	−138.355	−	7151.91i	−0.0352124	−	1.82022i
$250$		−	3225.37i		−	0.815962i
$251$		−	581.170i		−	0.146148i	−0.997327	−	0.0730740i	$-0.976719\pi$
$251$		−	581.170i		−	0.146148i	0.997327	−	0.0730740i	$-0.0232809\pi$
$252$	−212.683	+	8.23187i	−0.0531658	+	0.00205777i
$253$	−5221.33			−1.29748
$254$	−3068.61			−0.758038
$255$	55.3247	+	2859.87i	0.0135865	+	0.702321i
$256$	−2358.79			−0.575877
$257$			1366.48i			0.331667i	0.986154	+	0.165834i	$0.0530315\pi$
$257$			1366.48i			0.331667i	−0.986154	+	0.165834i	$0.946969\pi$
$258$	4795.28	−	92.7656i	1.15714	−	0.0223850i
$259$		−	750.406i		−	0.180031i
$260$	−1473.03			−0.351360
$261$	−13.1051	−	338.591i	−0.00310799	−	0.0802997i
$262$	6241.08			1.47166
$263$			3753.52i			0.880046i	0.897986	+	0.440023i	$0.145030\pi$
$263$			3753.52i			0.880046i	−0.897986	+	0.440023i	$0.854970\pi$
$264$	92.9465	+	4804.63i	0.0216684	+	1.12009i
$265$	11836.8			2.74389
$266$	−566.237			−0.130520
$267$	125.697	+	6497.58i	0.0288110	+	1.48931i
$268$		−	709.521i		−	0.161720i
$269$	−2795.74			−0.633678			−0.316839	−	0.948479i	$-0.602621\pi$
$269$	−2795.74			−0.633678			−0.316839	+	0.948479i	$0.602621\pi$
$270$	6347.69	−	368.760i	1.43077	−	0.0831185i
$271$	−8440.91			−1.89206			−0.946031	−	0.324076i	$-0.894946\pi$
$271$	−8440.91			−1.89206			−0.946031	+	0.324076i	$0.894946\pi$
$272$		−	1492.52i		−	0.332710i
$273$	1307.51	−	25.2940i	0.289868	−	0.00560756i
$274$			1652.88i			0.364431i
$275$	−7469.18			−1.63785
$276$	22.1135	+	1143.10i	0.00482273	+	0.249299i
$277$	2260.98			0.490430			0.245215	−	0.969469i	$-0.421141\pi$
$277$	2260.98			0.490430			0.245215	+	0.969469i	$0.421141\pi$
$278$	1786.86			0.385498
$279$	−78.8747	−	2037.85i	−0.0169251	−	0.437286i
$280$			2138.57i			0.456443i
$281$		−	6580.58i		−	1.39703i	−0.715597	−	0.698513i	$-0.753845\pi$
$281$		−	6580.58i		−	1.39703i	0.715597	−	0.698513i	$-0.246155\pi$
$282$	−109.504	−	5660.54i	−0.0231237	−	1.19532i
$283$		−	5314.23i		−	1.11625i	−0.829758	−	0.558124i	$-0.811521\pi$
$283$		−	5314.23i		−	1.11625i	0.829758	−	0.558124i	$-0.188479\pi$
$284$			1025.79i			0.214329i
$285$	−4243.09	+	82.0834i	−0.881891	+	0.0170604i
$286$		−	4932.67i		−	1.01984i
$287$		−	1340.61i		−	0.275726i
$288$	1927.29	−	74.5957i	0.394330	−	0.0152625i
$289$	3969.46			0.807951
$290$	−568.771			−0.115170
$291$	−8765.74	+	169.575i	−1.76583	+	0.0341604i
$292$		−	58.0750i		−	0.0116390i
$293$		−	3526.60i		−	0.703161i	−0.936158	−	0.351580i	$-0.885644\pi$
$293$		−	3526.60i		−	0.703161i	0.936158	−	0.351580i	$-0.114356\pi$
$294$	81.0426	+	4189.29i	0.0160765	+	0.831035i
$295$	5124.61	+	6300.74i	1.01141	+	1.24354i
$296$			3709.91i			0.728494i
$297$	−309.805	−	5332.87i	−0.0605277	−	1.04190i
$298$	2816.01			0.547406
$299$		−	7024.78i		−	1.35871i
$300$	31.6336	+	1635.22i	0.00608788	+	0.314697i
$301$		−	1793.17i		−	0.343378i
$302$		−	2871.22i		−	0.547087i
$303$	92.0876	+	4760.23i	0.0174597	+	0.902535i
$304$	2214.40			0.417778
$305$	−4502.09			−0.845209
$306$	81.1185	+	2095.82i	0.0151544	+	0.391537i
$307$	−6528.33			−1.21365			−0.606827	−	0.794834i	$-0.707558\pi$
$307$	−6528.33			−1.21365			−0.606827	+	0.794834i	$0.707558\pi$
$308$	300.152			0.0555284
$309$	−927.506	+	17.9428i	−0.170757	+	0.00330333i
$310$	−3423.22			−0.627180
$311$			4105.29i			0.748519i	0.927324	+	0.374260i	$0.122103\pi$
$311$			4105.29i			0.748519i	−0.927324	+	0.374260i	$0.877897\pi$
$312$	−6464.16	+	125.050i	−1.17295	+	0.0226910i
$313$		−	5568.41i		−	1.00558i	−0.864410	−	0.502788i	$-0.832308\pi$
$313$		−	5568.41i		−	1.00558i	0.864410	−	0.502788i	$-0.167692\pi$
$314$			4052.89i			0.728400i
$315$	−91.9423	−	2375.47i	−0.0164456	−	0.424897i
$316$	779.645			0.138793
$317$		−	6372.94i		−	1.12915i	−0.825382	−	0.564574i	$-0.809040\pi$
$317$		−	6372.94i		−	1.12915i	0.825382	−	0.564574i	$-0.190960\pi$
$318$	8677.75	−	167.873i	1.53027	−	0.0296033i
$319$			477.841i			0.0838682i
$320$		−	10203.7i		−	1.78251i
$321$	8610.98	−	166.581i	1.49725	−	0.0289646i
$322$	−1703.79			−0.294871
$323$		−	1399.90i		−	0.241153i
$324$	−1166.21	+	90.4111i	−0.199967	+	0.0155026i
$325$		−	10049.0i		−	1.71514i
$326$	−2194.06			−0.372754
$327$	−603.042	+	11.6660i	−0.101983	+	0.00197287i
$328$			6627.78i			1.11573i
$329$	−2116.74			−0.354710
$330$	−8964.98	+	173.429i	−1.49547	+	0.0289302i
$331$	7783.40			1.29249			0.646246	−	0.763129i	$-0.276338\pi$
$331$	7783.40			1.29249			0.646246	+	0.763129i	$0.276338\pi$
$332$	2208.87			0.365143
$333$	−159.498	−	4120.88i	−0.0262476	−	0.678146i
$334$			9718.55i			1.59214i
$335$	7924.68			1.29245
$336$	23.9916	+	1240.18i	0.00389538	+	0.201362i
$337$			607.505i			0.0981986i	0.998794	+	0.0490993i	$0.0156351\pi$
$337$			607.505i			0.0981986i	−0.998794	+	0.0490993i	$0.984365\pi$
$338$	1080.37			0.173859
$339$	104.368	+	5395.04i	0.0167212	+	0.864360i
$340$	−883.271			−0.140888
$341$			2875.94i			0.456719i
$342$	−3109.50	+	120.353i	−0.491646	+	0.0190291i
$343$	3251.72			0.511885
$344$			8865.22i			1.38948i
$345$	−12767.3	+	246.987i	−1.99238	+	0.0385429i
$346$	−6094.97			−0.947017
$347$	8204.92			1.26935			0.634673	−	0.772780i	$-0.281135\pi$
$347$	8204.92			1.26935			0.634673	+	0.772780i	$0.281135\pi$
$348$	104.613	−	2.02376i	0.0161145	−	0.000311738i
$349$			12249.4i			1.87878i	0.342854	+	0.939389i	$0.388606\pi$
$349$			12249.4i			1.87878i	−0.342854	+	0.939389i	$0.611394\pi$
$350$	−2437.29			−0.372225
$351$	7174.85	−	416.813i	1.09107	−	0.0633841i
$352$	−2719.92			−0.411853
$353$	10546.0			1.59010			0.795051	−	0.606542i	$-0.207444\pi$
$353$	10546.0			1.59010			0.795051	+	0.606542i	$0.207444\pi$
$354$	3846.28	+	4546.48i	0.577479	+	0.682606i
$355$	−11457.1			−1.71290
$356$	−2006.78			−0.298761
$357$	784.017	−	15.1670i	0.116231	−	0.00224852i
$358$	−3108.87			−0.458963
$359$			4715.89i			0.693301i	0.937994	+	0.346651i	$0.112681\pi$
$359$			4715.89i			0.693301i	−0.937994	+	0.346651i	$0.887319\pi$
$360$	454.550	+	11744.0i	0.0665470	+	1.71934i
$361$	−4782.02			−0.697189
$362$	6457.85			0.937617
$363$	11.9349	+	616.946i	0.00172568	+	0.0892045i
$364$			403.825i			0.0581488i
$365$	648.644			0.0930180
$366$	−3300.54	+	63.8496i	−0.471372	+	0.00911878i
$367$			3339.88i			0.475042i	0.971382	+	0.237521i	$0.0763348\pi$
$367$			3339.88i			0.475042i	−0.971382	+	0.237521i	$0.923665\pi$
$368$	6663.06			0.943848
$369$	−284.944	−	7361.98i	−0.0401995	−	1.03862i
$370$	−6922.33			−0.972635
$371$		−	3245.01i		−	0.454104i
$372$	629.626	−	12.1802i	0.0877543	−	0.00169762i
$373$	8644.93			1.20005			0.600023	−	0.799982i	$-0.295158\pi$
$373$	8644.93			1.20005			0.600023	+	0.799982i	$0.295158\pi$
$374$		−	2957.76i		−	0.408936i
$375$	−6625.89	+	128.179i	−0.912425	+	0.0176510i
$376$	10464.9			1.43533
$377$	−642.887			−0.0878260
$378$	−101.094	−	1740.19i	−0.0137558	−	0.236787i
$379$	−10504.2			−1.42365			−0.711825	−	0.702356i	$-0.752131\pi$
$379$	−10504.2			−1.42365			−0.711825	+	0.702356i	$0.752131\pi$
$380$		−	1310.48i		−	0.176911i
$381$	121.949	+	6303.85i	0.0163980	+	0.847654i
$382$	7806.59			1.04560
$383$			3695.29i			0.493004i	0.969142	+	0.246502i	$0.0792812\pi$
$383$			3695.29i			0.493004i	−0.969142	+	0.246502i	$0.920719\pi$
$384$	−87.2766	−	4511.54i	−0.0115985	−	0.599554i
$385$			3352.42i			0.443779i
$386$	11094.7			1.46296
$387$	−381.137	−	9847.27i	−0.0500628	−	1.29345i
$388$		−	2707.30i		−	0.354233i
$389$			12641.4i			1.64767i	0.566832	+	0.823833i	$0.308169\pi$
$389$			12641.4i			1.64767i	−0.566832	+	0.823833i	$0.691831\pi$
$390$	−233.332	−	12061.5i	−0.0302954	−	1.56604i
$391$		−	4212.25i		−	0.544815i
$392$	−7744.90			−0.997900
$393$	−248.026	−	12821.1i	−0.0318353	−	1.64564i
$394$		−	3341.40i		−	0.427252i
$395$			8707.91i			1.10922i
$396$	1648.29	−	63.7970i	0.209166	−	0.00809575i
$397$			13140.9i			1.66127i	0.556821	+	0.830633i	$0.312021\pi$
$397$			13140.9i			1.66127i	−0.556821	+	0.830633i	$0.687979\pi$
$398$	−5684.54			−0.715931
$399$	22.5027	+	1163.22i	0.00282342	+	0.145950i
$400$	9531.60			1.19145
$401$	−4936.62			−0.614770			−0.307385	−	0.951585i	$-0.599454\pi$
$401$	−4936.62			−0.614770			−0.307385	+	0.951585i	$0.599454\pi$
$402$	5809.69	−	112.390i	0.720799	−	0.0139440i
$403$	−3869.30			−0.478272
$404$	−1470.20			−0.181052
$405$	−1009.81	−	13025.4i	−0.123896	−	1.59812i
$406$			155.926i			0.0190603i
$407$			5815.65i			0.708282i
$408$	−3876.08	+	74.9835i	−0.470330	+	0.00909862i
$409$			5911.21i			0.714647i	0.933981	+	0.357323i	$0.116311\pi$
$409$			5911.21i			0.714647i	−0.933981	+	0.357323i	$0.883689\pi$
$410$	−12366.8			−1.48964
$411$	3395.51	−	65.6868i	0.407514	−	0.00788344i
$412$		−	286.460i		−	0.0342546i
$413$	1727.31	−	1404.89i	0.205800	−	0.167385i
$414$	−9356.40	+	362.138i	−1.11073	+	0.0429907i
$415$			24671.0i			2.91820i
$416$		−	3659.39i		−	0.431289i
$417$	−71.0113	−	3670.74i	−0.00833918	−	0.431072i
$418$	4388.33			0.513494
$419$	−14149.6			−1.64976			−0.824882	−	0.565304i	$-0.808759\pi$
$419$	−14149.6			−1.64976			−0.824882	+	0.565304i	$0.808759\pi$
$420$	733.939	−	14.1982i	0.0852680	−	0.00164953i
$421$		−	4735.46i		−	0.548200i	−0.961701	−	0.274100i	$-0.911620\pi$
$421$		−	4735.46i		−	0.548200i	0.961701	−	0.274100i	$-0.0883799\pi$
$422$		−	602.544i		−	0.0695057i
$423$	−11624.1	+	449.910i	−1.33613	+	0.0517148i
$424$			16042.9i			1.83753i
$425$		−	6025.67i		−	0.687737i
$426$	−8399.35	+	162.487i	−0.955281	+	0.0184801i
$427$			1234.22i			0.139879i
$428$			2659.50i			0.300355i
$429$	−10133.2	+	196.029i	−1.14041	+	0.0220614i
$430$	−16541.6			−1.85514
$431$	10921.4			1.22057			0.610287	−	0.792180i	$-0.291054\pi$
$431$	10921.4			1.22057			0.610287	+	0.792180i	$0.291054\pi$
$432$	395.350	+	6805.40i	0.0440308	+	0.757929i
$433$	1855.12			0.205893			0.102946	−	0.994687i	$-0.467173\pi$
$433$	1855.12			0.205893			0.102946	+	0.994687i	$0.467173\pi$
$434$			938.458i			0.103796i
$435$	22.6035	+	1168.43i	0.00249139	+	0.128786i
$436$		−	186.250i		−	0.0204581i
$437$	6249.58			0.684114
$438$	475.530	−	9.19921i	0.0518760	−	0.00100355i
$439$	9335.14			1.01490			0.507451	−	0.861681i	$-0.330588\pi$
$439$	9335.14			1.01490			0.507451	+	0.861681i	$0.330588\pi$
$440$		−	16573.9i		−	1.79575i
$441$	8602.85	−	332.972i	0.928933	−	0.0359542i
$442$	3979.37			0.428234
$443$	−2399.08			−0.257300			−0.128650	−	0.991690i	$-0.541064\pi$
$443$	−2399.08			−0.257300			−0.128650	+	0.991690i	$0.541064\pi$
$444$	1273.21	−	24.6305i	0.136090	−	0.00263268i
$445$		−	22413.8i		−	2.38768i
$446$	−5337.17			−0.566642
$447$	−111.911	−	5784.94i	−0.0118416	−	0.612121i
$448$	−2797.29			−0.294999
$449$			8641.91i			0.908323i	0.890919	+	0.454162i	$0.150061\pi$
$449$			8641.91i			0.908323i	−0.890919	+	0.454162i	$0.849939\pi$
$450$	−13384.5	+	518.044i	−1.40211	+	0.0542685i
$451$			10389.7i			1.08477i
$452$	−1666.26			−0.173394
$453$	−5898.36	+	114.105i	−0.611764	+	0.0118347i
$454$	−6659.04			−0.688379
$455$	−4510.34			−0.464721
$456$	−111.251	−	5750.82i	−0.0114250	−	0.590585i
$457$		−	14348.7i		−	1.46872i	−0.678762	−	0.734359i	$-0.737483\pi$
$457$		−	14348.7i		−	1.46872i	0.678762	−	0.734359i	$-0.262517\pi$
$458$		−	2052.65i		−	0.209419i
$459$	4302.23	−	249.932i	0.437497	−	0.0254157i
$460$		−	3943.19i		−	0.399679i
$461$			8909.98i			0.900172i	0.892985	+	0.450086i	$0.148607\pi$
$461$			8909.98i			0.900172i	−0.892985	+	0.450086i	$0.851393\pi$
$462$	47.5447	+	2457.70i	0.00478784	+	0.247495i
$463$		−	13537.1i		−	1.35880i	−0.733769	−	0.679400i	$-0.762240\pi$
$463$		−	13537.1i		−	1.35880i	0.733769	−	0.679400i	$-0.237760\pi$
$464$		−	609.784i		−	0.0610097i
$465$	136.042	+	7032.33i	0.0135673	+	0.701326i
$466$	11692.1			1.16229
$467$	10252.9			1.01595			0.507976	−	0.861371i	$-0.330394\pi$
$467$	10252.9			1.01595			0.507976	+	0.861371i	$0.330394\pi$
$468$	85.8325	+	2217.62i	0.00847780	+	0.219037i
$469$		−	2172.51i		−	0.213896i
$470$			19526.4i			1.91636i
$471$	8325.86	−	161.065i	0.814512	−	0.0157569i
$472$	−8539.62	+	6945.57i	−0.832771	+	0.677322i
$473$			13897.1i			1.35093i
$474$	123.498	+	6383.89i	0.0119671	+	0.618611i
$475$	8940.10			0.863578
$476$			242.144i			0.0233165i
$477$	−689.723	−	17820.1i	−0.0662060	−	1.71053i
$478$		−	10386.9i		−	0.993905i
$479$		−	9800.56i		−	0.934862i	−0.884029	−	0.467431i	$-0.845180\pi$
$479$		−	9800.56i		−	0.934862i	0.884029	−	0.467431i	$-0.154820\pi$
$480$	−6650.82	+	128.661i	−0.632431	+	0.0122345i
$481$	−7824.38			−0.741707
$482$	3463.35			0.327285
$483$	67.7101	+	3500.10i	0.00637871	+	0.329731i
$484$	−190.544			−0.0178948
$485$	30238.0			2.83101
$486$	−925.033	−	9534.79i	−0.0863382	−	0.889932i
$487$	−1932.83			−0.179846			−0.0899229	−	0.995949i	$-0.528662\pi$
$487$	−1932.83			−0.179846			−0.0899229	+	0.995949i	$0.528662\pi$
$488$		−	6101.84i		−	0.566019i
$489$	87.1939	+	4507.26i	0.00806349	+	0.416821i
$490$		−	14451.2i		−	1.33233i
$491$		−	10380.0i		−	0.954061i	−0.878887	−	0.477031i	$-0.841713\pi$
$491$		−	10380.0i		−	0.954061i	0.878887	−	0.477031i	$-0.158287\pi$
$492$	2274.60	−	44.0026i	0.208429	−	0.00403209i
$493$	−385.492			−0.0352164
$494$			5904.07i			0.537726i
$495$	712.552	+	18409.9i	0.0647007	+	1.67164i
$496$		−	3670.06i		−	0.332239i
$497$			3140.90i			0.283478i
$498$	349.890	+	18086.6i	0.0314838	+	1.62747i
$499$	−8109.16			−0.727487			−0.363743	−	0.931499i	$-0.618501\pi$
$499$	−8109.16			−0.727487			−0.363743	+	0.931499i	$0.618501\pi$
$500$		−	2046.41i		−	0.183036i
$501$	19964.8	−	386.224i	1.78037	−	0.0344415i
$502$			1469.74i			0.130672i
$503$	−3825.79			−0.339132			−0.169566	−	0.985519i	$-0.554237\pi$
$503$	−3825.79			−0.339132			−0.169566	+	0.985519i	$0.554237\pi$
$504$	3219.56	−	124.613i	0.284545	−	0.0110133i
$505$		−	16420.8i		−	1.44696i
$506$	13204.4			1.16009
$507$	−42.9348	−	2219.40i	−0.00376095	−	0.194413i
$508$	−1946.95			−0.170043
$509$	19008.0			1.65524			0.827618	−	0.561291i	$-0.189695\pi$
$509$	19008.0			1.65524			0.827618	+	0.561291i	$0.189695\pi$
$510$	−139.912	−	7232.39i	−0.0121479	−	0.627952i
$511$		−	177.822i		−	0.0153941i
$512$	12912.5			1.11456
$513$	370.816	+	6383.08i	0.0319141	+	0.549357i
$514$		−	3455.72i		−	0.296547i
$515$	3199.50			0.273760
$516$	3042.47	−	58.8572i	0.259568	−	0.00502140i
$517$	16404.7			1.39551
$518$			1897.72i			0.160967i
$519$	242.220	+	12520.9i	0.0204861	+	1.05897i
$520$	22298.5			1.88049
$521$		−	18365.1i		−	1.54432i	−0.635428	−	0.772160i	$-0.719176\pi$
$521$		−	18365.1i		−	1.54432i	0.635428	−	0.772160i	$-0.280824\pi$
$522$	33.1418	+	856.270i	0.00277889	+	0.0717968i
$523$	13117.1			1.09669			0.548346	−	0.836252i	$-0.315258\pi$
$523$	13117.1			1.09669			0.548346	+	0.836252i	$0.315258\pi$
$524$	3959.79			0.330123
$525$	96.8601	+	5006.93i	0.00805204	+	0.416230i
$526$		−	9492.38i		−	0.786858i
$527$	−2320.13			−0.191777
$528$	−185.935	−	9611.42i	−0.0153253	−	0.792203i
$529$	6637.78			0.545556
$530$	−29934.5			−2.45334
$531$	9186.99	−	8082.11i	0.750813	−	0.660515i
$532$	−359.261			−0.0292781
$533$	−13978.3			−1.13596
$534$	−317.878	−	16431.9i	−0.0257602	−	1.33161i
$535$	−29704.2			−2.40042
$536$			10740.6i			0.865529i
$537$	123.549	+	6386.55i	0.00992838	+	0.513222i
$538$	7070.22			0.566578
$539$	−12140.9			−0.970214
$540$	4027.43	−	233.968i	0.320950	−	0.0186451i
$541$			1040.15i			0.0826605i	0.999146	+	0.0413303i	$0.0131596\pi$
$541$			1040.15i			0.0826605i	−0.999146	+	0.0413303i	$0.986840\pi$
$542$	21346.4			1.69171
$543$	−256.641	−	13266.4i	−0.0202827	−	1.04846i
$544$		−	2194.26i		−	0.172938i
$545$	2080.24			0.163500
$546$	−3306.60	+	63.9667i	−0.259174	+	0.00501378i
$547$	270.011			0.0211057			0.0105529	−	0.999944i	$-0.496641\pi$
$547$	270.011			0.0211057			0.0105529	+	0.999944i	$0.496641\pi$
$548$			1048.70i			0.0817490i
$549$	262.333	+	6777.78i	0.0203936	+	0.526901i
$550$	18889.0			1.46442
$551$		−	571.943i		−	0.0442206i
$552$	−334.750	−	17304.0i	−0.0258114	−	1.33425i
$553$	2387.23			0.183572
$554$	−5717.86			−0.438499
$555$	275.100	+	14220.6i	0.0210402	+	1.08762i
$556$	1133.71			0.0864749
$557$			19341.3i			1.47130i	0.677359	+	0.735652i	$0.263124\pi$
$557$			19341.3i			1.47130i	−0.677359	+	0.735652i	$0.736876\pi$
$558$	199.468	+	5153.57i	0.0151329	+	0.390982i
$559$	−18697.2			−1.41468
$560$		−	4278.10i		−	0.322826i
$561$	−6076.14	+	117.544i	−0.457281	+	0.00884619i
$562$			16641.8i			1.24910i
$563$	−24978.5			−1.86983			−0.934917	−	0.354866i	$-0.884527\pi$
$563$	−24978.5			−1.86983			−0.934917	+	0.354866i	$0.884527\pi$
$564$	−69.4774	−	3591.46i	−0.00518711	−	0.268134i
$565$		−	18610.5i		−	1.38576i
$566$			13439.3i			0.998048i
$567$	−3570.85	+	276.833i	−0.264483	+	0.0205042i
$568$		−	15528.2i		−	1.14709i
$569$	−3828.07			−0.282041			−0.141020	−	0.990007i	$-0.545038\pi$
$569$	−3828.07			−0.282041			−0.141020	+	0.990007i	$0.545038\pi$
$570$	10730.5	−	207.583i	0.788508	−	0.0152538i
$571$		−	19204.9i		−	1.40753i	−0.710432	−	0.703765i	$-0.751501\pi$
$571$		−	19204.9i		−	1.40753i	0.710432	−	0.703765i	$-0.248499\pi$
$572$		−	3129.64i		−	0.228771i
$573$	−310.241	−	16037.1i	−0.0226187	−	1.16921i
$574$			3390.29i			0.246530i
$575$	26900.5			1.95100
$576$	−15361.4	+	594.561i	−1.11121	+	0.0430093i
$577$	−12909.8			−0.931445			−0.465723	−	0.884931i	$-0.654206\pi$
$577$	−12909.8			−0.931445			−0.465723	+	0.884931i	$0.654206\pi$
$578$	−10038.5			−0.722397
$579$	−440.912	−	22791.8i	−0.0316471	−	1.63592i
$580$	−360.869			−0.0258350
$581$	6763.42			0.482950
$582$	22167.9	−	428.843i	1.57885	−	0.0305431i
$583$			25148.8i			1.78655i
$584$			879.130i			0.0622923i
$585$	−24768.7	+	958.669i	−1.75053	+	0.0677540i
$586$			8918.51i			0.628703i
$587$	4715.16			0.331543			0.165771	−	0.986164i	$-0.446989\pi$
$587$	4715.16			0.331543			0.165771	+	0.986164i	$0.446989\pi$
$588$	51.4193	+	2657.99i	0.00360628	+	0.186418i
$589$		−	3442.31i		−	0.240811i
$590$	−12959.8	−	15934.1i	−0.904313	−	1.11186i
$591$	−6864.25	+	132.790i	−0.477763	+	0.00924241i
$592$		−	7421.48i		−	0.515238i
$593$		−	15022.0i		−	1.04027i	−0.854084	−	0.520135i	$-0.825882\pi$
$593$		−	15022.0i		−	1.04027i	0.854084	−	0.520135i	$-0.174118\pi$
$594$	783.475	+	13486.4i	0.0541185	+	0.931574i
$595$	−2704.52			−0.186344
$596$	1786.68			0.122794
$597$	225.909	+	11677.8i	0.0154872	+	0.800569i
$598$			17765.2i			1.21484i
$599$		−	20568.7i		−	1.40303i	−0.712655	−	0.701515i	$-0.752507\pi$
$599$		−	20568.7i		−	1.40303i	0.712655	−	0.701515i	$-0.247493\pi$
$600$	−478.864	−	24753.6i	−0.0325825	−	1.68427i
$601$			12269.7i			0.832764i	0.909190	+	0.416382i	$0.136702\pi$
$601$			12269.7i			0.832764i	−0.909190	+	0.416382i	$0.863298\pi$
$602$			4534.81i			0.307018i
$603$	−461.765	−	11930.4i	−0.0311849	−	0.805711i
$604$		−	1821.71i		−	0.122722i
$605$		−	2128.20i		−	0.143014i
$606$	−232.883	−	12038.3i	−0.0156109	−	0.806966i
$607$	−11719.7			−0.783673			−0.391837	−	0.920035i	$-0.628160\pi$
$607$	−11719.7			−0.783673			−0.391837	+	0.920035i	$0.628160\pi$
$608$	3255.56			0.217155
$609$	320.319	−	6.19662i	0.0213136	−	0.000412315i
$610$	11385.4			0.755710
$611$			22070.9i			1.46136i
$612$	51.4675	+	1329.74i	0.00339943	+	0.0878294i
$613$		−	331.561i		−	0.0218460i	−0.999940	−	0.0109230i	$-0.996523\pi$
$613$		−	331.561i		−	0.0218460i	0.999940	−	0.0109230i	$-0.00347697\pi$
$614$	16509.7			1.08514
$615$	491.467	+	25405.1i	0.0322242	+	1.66575i
$616$	−4543.65			−0.297190
$617$		−	22795.3i		−	1.48736i	−0.668533	−	0.743682i	$-0.733077\pi$
$617$		−	22795.3i		−	1.48736i	0.668533	−	0.743682i	$-0.266923\pi$
$618$	2345.59	−	45.3760i	0.152676	−	0.00295354i
$619$	−14271.8			−0.926708			−0.463354	−	0.886173i	$-0.653354\pi$
$619$	−14271.8			−0.926708			−0.463354	+	0.886173i	$0.653354\pi$
$620$	−2171.94			−0.140689
$621$	1115.77	+	19206.5i	0.0721006	+	1.24111i
$622$		−	10382.0i		−	0.669259i
$623$	−6144.64			−0.395152
$624$	12931.2	−	250.157i	0.829588	−	0.0160485i
$625$	−1664.40			−0.106522
$626$			14082.1i			0.899096i
$627$	−174.396	−	9014.97i	−0.0111080	−	0.574199i
$628$			2571.44i			0.163395i
$629$	−4691.70			−0.297409
$630$	232.515	+	6007.39i	0.0147042	+	0.379905i
$631$	−5033.55			−0.317563			−0.158782	−	0.987314i	$-0.550757\pi$
$631$	−5033.55			−0.317563			−0.158782	+	0.987314i	$0.550757\pi$
$632$	−11802.1			−0.742823
$633$	−1237.81	+	23.9456i	−0.0777227	+	0.00150356i
$634$			16116.7i			1.00958i
$635$		−	21745.6i		−	1.35897i
$636$	5505.79	−	106.511i	0.343269	−	0.00664060i
$637$		−	16334.4i		−	1.01600i
$638$		−	1208.42i		−	0.0749874i
$639$	667.595	+	17248.4i	0.0413297	+	1.06782i
$640$			15562.9i			0.961213i
$641$			15611.3i			0.961948i	0.876735	+	0.480974i	$0.159717\pi$
$641$			15611.3i			0.961948i	−0.876735	+	0.480974i	$0.840283\pi$
$642$	−21776.5	+	421.271i	−1.33871	+	0.0258976i
$643$	23418.0			1.43626			0.718130	−	0.695909i	$-0.244998\pi$
$643$	23418.0			1.43626			0.718130	+	0.695909i	$0.244998\pi$
$644$	−1081.01			−0.0661454
$645$	657.380	+	33981.6i	0.0401307	+	2.07445i
$646$			3540.24i			0.215617i
$647$			17281.7i			1.05010i	0.851072	+	0.525048i	$0.175953\pi$
$647$			17281.7i			1.05010i	−0.851072	+	0.525048i	$0.824047\pi$
$648$	17653.8	−	1368.63i	1.07023	−	0.0829703i
$649$	−13386.7	+	10887.9i	−0.809666	+	0.658530i
$650$			25413.3i			1.53352i
$651$	1927.88	−	37.2952i	0.116067	−	0.00224533i
$652$	−1392.07			−0.0836161
$653$		−	2766.26i		−	0.165777i	−0.996559	−	0.0828884i	$-0.973585\pi$
$653$		−	2766.26i		−	0.165777i	0.996559	−	0.0828884i	$-0.0264145\pi$
$654$	1525.05	−	29.5024i	0.0911837	−	0.00176397i
$655$			44227.2i			2.63832i
$656$		−	13258.5i		−	0.789114i
$657$	−37.7959	−	976.517i	−0.00224438	−	0.0579871i
$658$	5353.07			0.317150
$659$	14505.0			0.857412			0.428706	−	0.903444i	$-0.358970\pi$
$659$	14505.0			0.857412			0.428706	+	0.903444i	$0.358970\pi$
$660$	−5688.03	+	110.036i	−0.335464	+	0.00648961i
$661$	2069.11			0.121754			0.0608769	−	0.998145i	$-0.480610\pi$
$661$	2069.11			0.121754			0.0608769	+	0.998145i	$0.480610\pi$
$662$	−19683.7			−1.15563
$663$	−158.144	−	8174.84i	−0.00926365	−	0.478860i
$664$	−33437.5			−1.95426
$665$		−	4012.61i		−	0.233989i
$666$	403.359	+	10421.4i	0.0234682	+	0.606338i
$667$		−	1720.96i		−	0.0999037i
$668$			6166.15i			0.357149i
$669$	212.104	+	10964.2i	0.0122577	+	0.633631i
$670$	−20040.9			−1.15560
$671$		−	9565.24i		−	0.550316i
$672$	35.2719	+	1823.29i	0.00202476	+	0.104665i
$673$			19658.6i			1.12598i	0.826465	+	0.562988i	$0.190348\pi$
$673$			19658.6i			1.12598i	−0.826465	+	0.562988i	$0.809652\pi$
$674$		−	1536.34i		−	0.0878004i
$675$	1596.13	+	27475.1i	0.0910148	+	1.56669i
$676$	685.464			0.0390000
$677$			6768.57i			0.384250i	0.981370	+	0.192125i	$0.0615379\pi$
$677$			6768.57i			0.384250i	−0.981370	+	0.192125i	$0.938462\pi$
$678$	−263.939	−	13643.6i	−0.0149506	−	0.772834i
$679$		−	8289.60i		−	0.468521i
$680$	13370.8			0.754040
$681$	264.636	+	13679.7i	0.0148912	+	0.769760i
$682$		−	7273.05i		−	0.408357i
$683$	−27264.9			−1.52747			−0.763734	−	0.645531i	$-0.776636\pi$
$683$	−27264.9			−1.52747			−0.763734	+	0.645531i	$0.776636\pi$
$684$	−1972.89	+	76.3606i	−0.110286	+	0.00426860i
$685$	−11713.1			−0.653332
$686$	−8223.36			−0.457681
$687$	−4216.76	+	81.5741i	−0.234177	+	0.00453020i
$688$		−	17734.4i		−	0.982730i
$689$	−33835.3			−1.87086
$690$	32287.6	−	624.611i	1.78141	−	0.0344616i
$691$		−	5793.00i		−	0.318923i	−0.987204	−	0.159462i	$-0.949024\pi$
$691$		−	5793.00i		−	0.318923i	0.987204	−	0.159462i	$-0.0509758\pi$
$692$	−3867.09			−0.212435
$693$	5046.98	−	195.343i	0.276650	−	0.0107077i
$694$	−20749.6			−1.13494
$695$			12662.5i			0.691101i
$696$	−1583.61	+	30.6353i	−0.0862453	+	0.00166843i
$697$	−8381.76			−0.455498
$698$		−	30977.7i		−	1.67983i
$699$	−464.655	−	24019.2i	−0.0251429	−	1.29970i
$700$	−1546.39			−0.0834974
$701$	1161.05			0.0625565			0.0312783	−	0.999511i	$-0.490042\pi$
$701$	1161.05			0.0625565			0.0312783	+	0.999511i	$0.490042\pi$
$702$	−18144.7	+	1054.09i	−0.975536	+	0.0566724i
$703$		−	6960.93i		−	0.373452i
$704$	21679.0			1.16059
$705$	40113.2	−	775.998i	2.14291	−	0.0414550i
$706$	−26670.0			−1.42173
$707$	−4501.66			−0.239466
$708$	2440.36	+	2884.61i	0.129540	+	0.153122i
$709$	2589.53			0.137167			0.0685837	−	0.997645i	$-0.478152\pi$
$709$	2589.53			0.137167			0.0685837	+	0.997645i	$0.478152\pi$
$710$	28974.1			1.53152
$711$	13109.5	−	507.403i	0.691485	−	0.0267638i
$712$	30378.3			1.59898
$713$		−	10357.8i		−	0.544043i
$714$	−1982.72	+	38.3561i	−0.103924	+	0.00201042i
$715$	34955.2			1.82832
$716$	−1972.49			−0.102954
$717$	−21337.9	+	412.785i	−1.11141	+	0.0215003i
$718$		−	11926.1i		−	0.619888i
$719$	−25261.8			−1.31030			−0.655152	−	0.755497i	$-0.727395\pi$
$719$	−25261.8			−1.31030			−0.655152	+	0.755497i	$0.727395\pi$
$720$	−909.305	−	23493.3i	−0.0470664	−	1.21603i
$721$		−	877.125i		−	0.0453063i
$722$	12093.4			0.623364
$723$	−137.637	−	7114.77i	−0.00707989	−	0.365977i
$724$	4097.33			0.210326
$725$		−	2461.85i		−	0.126112i
$726$	−30.1826	−	1560.21i	−0.00154295	−	0.0797587i
$727$	−6751.37			−0.344422			−0.172211	−	0.985060i	$-0.555091\pi$
$727$	−6751.37			−0.344422			−0.172211	+	0.985060i	$0.555091\pi$
$728$		−	6113.03i		−	0.311214i
$729$	−19550.6	+	2279.22i	−0.993273	+	0.115796i
$730$	−1640.37			−0.0831684
$731$	−11211.3			−0.567258
$732$	−2094.10	+	40.5108i	−0.105738	+	0.00204552i
$733$	5684.58			0.286446			0.143223	−	0.989690i	$-0.454253\pi$
$733$	5684.58			0.286446			0.143223	+	0.989690i	$0.454253\pi$
$734$		−	8446.31i		−	0.424740i
$735$	−29687.2	+	574.305i	−1.48984	+	0.0288212i
$736$	9795.88			0.490599
$737$			16837.0i			0.841516i
$738$	720.603	+	18617.9i	0.0359428	+	0.928637i
$739$			22799.0i			1.13488i	0.823416	+	0.567438i	$0.192065\pi$
$739$			22799.0i			1.13488i	−0.823416	+	0.567438i	$0.807935\pi$
$740$	−4392.03			−0.218181
$741$	12128.7	−	234.633i	0.601296	−	0.0116322i
$742$			8206.39i			0.406019i
$743$			4680.60i			0.231110i	0.993301	+	0.115555i	$0.0368646\pi$
$743$			4680.60i			0.231110i	−0.993301	+	0.115555i	$0.963135\pi$
$744$	−9531.17	+	184.382i	−0.469664	+	0.00908573i
$745$			19955.5i			0.981361i
$746$	−21862.4			−1.07297
$747$	37141.5	−	1437.56i	1.81919	−	0.0704117i
$748$		−	1876.62i		−	0.0917325i
$749$			8143.24i			0.397260i
$750$	16756.4	−	324.156i	0.815809	−	0.0157820i
$751$			24749.5i			1.20256i	0.799039	+	0.601279i	$0.205342\pi$
$751$			24749.5i			1.20256i	−0.799039	+	0.601279i	$0.794658\pi$
$752$	−20934.4			−1.01516
$753$	3019.28	−	58.4086i	0.146121	−	0.00282673i
$754$	1625.81			0.0785261
$755$	20346.8			0.980790
$756$	−64.1411	−	1104.10i	−0.00308570	−	0.0531160i
$757$	24788.0			1.19014			0.595069	−	0.803675i	$-0.297125\pi$
$757$	24788.0			1.19014			0.595069	+	0.803675i	$0.297125\pi$
$758$	26564.3			1.27290
$759$	−524.753	−	27125.8i	−0.0250953	−	1.29724i
$760$			19837.8i			0.946834i
$761$		−	21981.9i		−	1.04710i	−0.851995	−	0.523549i	$-0.824608\pi$
$761$		−	21981.9i		−	1.04710i	0.851995	−	0.523549i	$-0.175392\pi$
$762$	−308.401	−	15942.0i	−0.0146616	−	0.757896i
$763$		−	570.286i		−	0.0270586i
$764$	4953.06			0.234549
$765$	−14852.0	+	574.843i	−0.701927	+	0.0271680i
$766$		−	9345.12i		−	0.440800i
$767$	−14648.5	−	18010.5i	−0.689606	−	0.847875i
$768$	−237.063	−	12254.4i	−0.0111384	−	0.575769i
$769$			33444.5i			1.56832i	0.620556	+	0.784162i	$0.286907\pi$
$769$			33444.5i			1.56832i	−0.620556	+	0.784162i	$0.713093\pi$
$770$		−	8478.01i		−	0.396787i
$771$	−7099.09	+	137.333i	−0.331605	+	0.00641497i
$772$	7039.26			0.328172
$773$	25499.6			1.18649			0.593246	−	0.805021i	$-0.297846\pi$
$773$	25499.6			1.18649			0.593246	+	0.805021i	$0.297846\pi$
$774$	963.868	+	24903.0i	0.0447617	+	1.15649i
$775$		−	14817.0i		−	0.686762i
$776$			40982.7i			1.89587i
$777$	3898.50	−	75.4171i	0.179997	−	0.00348208i
$778$		−	31969.1i		−	1.47320i
$779$		−	12435.8i		−	0.571960i
$780$	−148.043	−	7652.68i	−0.00679586	−	0.351295i
$781$		−	24342.0i		−	1.11527i
$782$			10652.5i			0.487124i
$783$	1757.72	−	102.112i

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.52893 q^{2} +(0.100502 + 5.19518i) q^{3} -1.60453 q^{4} -17.9211i q^{5} +(-0.254162 - 13.1382i) q^{6} -4.91298 q^{7} +24.2892 q^{8} +(-26.9798 + 1.04425i) q^{9} +O(q^{10})\)	\(q-2.52893 q^{2} +(0.100502 + 5.19518i) q^{3} -1.60453 q^{4} -17.9211i q^{5} +(-0.254162 - 13.1382i) q^{6} -4.91298 q^{7} +24.2892 q^{8} +(-26.9798 + 1.04425i) q^{9} +45.3212i q^{10} +38.0756 q^{11} +(-0.161258 - 8.33584i) q^{12} +51.2270i q^{13} +12.4246 q^{14} +(93.1035 - 1.80111i) q^{15} -48.5892 q^{16} +30.7171i q^{17} +(68.2299 - 2.64083i) q^{18} -45.5739 q^{19} +28.7550i q^{20} +(-0.493764 - 25.5238i) q^{21} -96.2904 q^{22} -137.131 q^{23} +(2.44110 + 126.187i) q^{24} -196.167 q^{25} -129.549i q^{26} +(-8.13658 - 140.060i) q^{27} +7.88305 q^{28} +12.5498i q^{29} +(-235.452 + 4.55486i) q^{30} +75.5324i q^{31} -71.4347 q^{32} +(3.82667 + 197.810i) q^{33} -77.6812i q^{34} +88.0462i q^{35} +(43.2900 - 1.67553i) q^{36} +152.739i q^{37} +115.253 q^{38} +(-266.133 + 5.14840i) q^{39} -435.289i q^{40} +272.870i q^{41} +(1.24869 + 64.5479i) q^{42} +364.987i q^{43} -61.0936 q^{44} +(18.7141 + 483.509i) q^{45} +346.793 q^{46} +430.845 q^{47} +(-4.88330 - 252.430i) q^{48} -318.863 q^{49} +496.092 q^{50} +(-159.581 + 3.08712i) q^{51} -82.1954i q^{52} +660.497i q^{53} +(20.5768 + 354.201i) q^{54} -682.358i q^{55} -119.332 q^{56} +(-4.58026 - 236.765i) q^{57} -31.7375i q^{58} +(-351.582 + 285.954i) q^{59} +(-149.388 + 2.88993i) q^{60} -251.217i q^{61} -191.016i q^{62} +(132.551 - 5.13038i) q^{63} +569.367 q^{64} +918.046 q^{65} +(-9.67736 - 500.246i) q^{66} +442.198i q^{67} -49.2865i q^{68} +(-13.7819 - 712.418i) q^{69} -222.662i q^{70} -639.306i q^{71} +(-655.316 + 25.3639i) q^{72} +36.1944i q^{73} -386.267i q^{74} +(-19.7151 - 1019.12i) q^{75} +73.1249 q^{76} -187.065 q^{77} +(673.032 - 13.0199i) q^{78} -485.902 q^{79} +870.774i q^{80} +(726.819 - 56.3473i) q^{81} -690.068i q^{82} -1376.64 q^{83} +(0.792260 + 40.9538i) q^{84} +550.485 q^{85} -923.025i q^{86} +(-65.1984 + 1.26128i) q^{87} +924.825 q^{88} +1250.69 q^{89} +(-47.3267 - 1222.76i) q^{90} -251.677i q^{91} +220.030 q^{92} +(-392.404 + 7.59114i) q^{93} -1089.58 q^{94} +816.736i q^{95} +(-7.17931 - 371.116i) q^{96} +1687.28i q^{97} +806.380 q^{98} +(-1027.27 + 39.7605i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52q - 8q^{3} + 268q^{4} - 16q^{7} - 4q^{9} + O(q^{10}) \)	\( 52q - 8q^{3} + 268q^{4} - 16q^{7} - 4q^{9} + 28q^{12} + 114q^{15} + 484q^{16} - 184q^{19} - 758q^{21} - 60q^{22} + 36q^{25} + 742q^{27} - 4q^{28} - 888q^{36} + 1402q^{45} - 660q^{46} - 488q^{48} - 924q^{49} - 1772q^{51} - 630q^{57} - 1880q^{60} - 212q^{63} + 7648q^{64} + 1316q^{66} - 1556q^{75} - 5680q^{76} + 3224q^{78} - 1504q^{79} - 276q^{81} + 1228q^{84} - 848q^{85} + 3598q^{87} + 5760q^{88} + 888q^{94} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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