LMFDB - Embedded newform 177.3.c.a.58.9

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	$4.82290067918$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
Defining polynomial:	$x^{20} + 60 x^{18} + 1522 x^{16} + 21286 x^{14} + 179593 x^{12} + 941588 x^{10} + 3057025 x^{8} + 5962230 x^{6} + 6517782 x^{4} + 3428244 x^{2} + 570861$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.9
Root			$-1.08628i$ of defining polynomial
Character	$\chi$	$=$	177.58
Dual form			177.3.c.a.58.12

$q$-expansion

$f(q)$	$=$	$q-1.08628i q^{2} +1.73205 q^{3} +2.82000 q^{4} +3.33671 q^{5} -1.88149i q^{6} -1.22637 q^{7} -7.40842i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.08628i q^{2} +1.73205 q^{3} +2.82000 q^{4} +3.33671 q^{5} -1.88149i q^{6} -1.22637 q^{7} -7.40842i q^{8} +3.00000 q^{9} -3.62460i q^{10} +1.21764i q^{11} +4.88438 q^{12} +12.6555i q^{13} +1.33218i q^{14} +5.77936 q^{15} +3.23240 q^{16} -0.815320 q^{17} -3.25883i q^{18} -7.74812 q^{19} +9.40954 q^{20} -2.12414 q^{21} +1.32270 q^{22} -33.8014i q^{23} -12.8318i q^{24} -13.8663 q^{25} +13.7474 q^{26} +5.19615 q^{27} -3.45837 q^{28} +7.53415 q^{29} -6.27799i q^{30} +7.71953i q^{31} -33.1450i q^{32} +2.10902i q^{33} +0.885664i q^{34} -4.09206 q^{35} +8.46000 q^{36} +16.7957i q^{37} +8.41661i q^{38} +21.9199i q^{39} -24.7198i q^{40} -19.0497 q^{41} +2.30741i q^{42} +48.9576i q^{43} +3.43375i q^{44} +10.0101 q^{45} -36.7177 q^{46} -4.79310i q^{47} +5.59868 q^{48} -47.4960 q^{49} +15.0627i q^{50} -1.41218 q^{51} +35.6884i q^{52} -48.7328 q^{53} -5.64447i q^{54} +4.06292i q^{55} +9.08548i q^{56} -13.4201 q^{57} -8.18418i q^{58} +(43.5656 + 39.7874i) q^{59} +16.2978 q^{60} +87.6284i q^{61} +8.38556 q^{62} -3.67912 q^{63} -23.0750 q^{64} +42.2277i q^{65} +2.29098 q^{66} -42.6674i q^{67} -2.29920 q^{68} -58.5457i q^{69} +4.44511i q^{70} -20.2245 q^{71} -22.2252i q^{72} +45.6107i q^{73} +18.2448 q^{74} -24.0172 q^{75} -21.8497 q^{76} -1.49328i q^{77} +23.8111 q^{78} -62.5528 q^{79} +10.7856 q^{80} +9.00000 q^{81} +20.6933i q^{82} +22.8390i q^{83} -5.99008 q^{84} -2.72049 q^{85} +53.1815 q^{86} +13.0495 q^{87} +9.02079 q^{88} +55.8982i q^{89} -10.8738i q^{90} -15.5203i q^{91} -95.3199i q^{92} +13.3706i q^{93} -5.20664 q^{94} -25.8533 q^{95} -57.4087i q^{96} -123.320i q^{97} +51.5939i q^{98} +3.65292i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20q - 40q^{4} - 8q^{7} + 60q^{9} + O(q^{10}) $	$ 20q - 40q^{4} - 8q^{7} + 60q^{9} - 24q^{12} + 24q^{15} - 8q^{16} + 16q^{17} - 60q^{19} - 164q^{20} + 40q^{22} + 100q^{25} - 156q^{26} + 200q^{28} - 60q^{29} - 32q^{35} - 120q^{36} + 28q^{41} + 180q^{46} - 96q^{48} + 284q^{49} - 24q^{51} - 8q^{53} + 24q^{57} - 152q^{59} - 72q^{60} - 8q^{62} - 24q^{63} + 204q^{64} - 120q^{66} + 384q^{68} + 92q^{71} + 104q^{74} + 240q^{75} + 120q^{76} - 468q^{78} - 420q^{79} + 376q^{80} + 180q^{81} + 168q^{84} - 348q^{85} + 232q^{86} + 144q^{87} - 212q^{88} + 152q^{94} + 788q^{95} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.08628i		−	0.543139i	−0.962419	−	0.271570i	$-0.912457\pi$
$2$		−	1.08628i		−	0.543139i	0.962419	−	0.271570i	$-0.0875427\pi$
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$	2.82000			0.705000
$5$	3.33671			0.667343			0.333671	−	0.942689i	$-0.391712\pi$
$5$	3.33671			0.667343			0.333671	+	0.942689i	$0.391712\pi$
$6$		−	1.88149i		−	0.313581i
$7$	−1.22637			−0.175196			−0.0875981	−	0.996156i	$-0.527919\pi$
$7$	−1.22637			−0.175196			−0.0875981	+	0.996156i	$0.527919\pi$
$8$		−	7.40842i		−	0.926052i
$9$	3.00000			0.333333
$10$		−	3.62460i		−	0.362460i
$11$			1.21764i			0.110695i	0.998467	+	0.0553473i	$0.0176266\pi$
$11$			1.21764i			0.110695i	−0.998467	+	0.0553473i	$0.982373\pi$
$12$	4.88438			0.407032
$13$			12.6555i			0.973497i	0.873542	+	0.486749i	$0.161817\pi$
$13$			12.6555i			0.973497i	−0.873542	+	0.486749i	$0.838183\pi$
$14$			1.33218i			0.0951559i
$15$	5.77936			0.385291
$16$	3.23240			0.202025
$17$	−0.815320			−0.0479600			−0.0239800	−	0.999712i	$-0.507634\pi$
$17$	−0.815320			−0.0479600			−0.0239800	+	0.999712i	$0.507634\pi$
$18$		−	3.25883i		−	0.181046i
$19$	−7.74812			−0.407796			−0.203898	−	0.978992i	$-0.565361\pi$
$19$	−7.74812			−0.407796			−0.203898	+	0.978992i	$0.565361\pi$
$20$	9.40954			0.470477
$21$	−2.12414			−0.101150
$22$	1.32270			0.0601226
$23$		−	33.8014i		−	1.46963i	−0.678270	−	0.734813i	$-0.737270\pi$
$23$		−	33.8014i		−	1.46963i	0.678270	−	0.734813i	$-0.262730\pi$
$24$		−	12.8318i		−	0.534656i
$25$	−13.8663			−0.554653
$26$	13.7474			0.528744
$27$	5.19615			0.192450
$28$	−3.45837			−0.123513
$29$	7.53415			0.259798			0.129899	−	0.991527i	$-0.458535\pi$
$29$	7.53415			0.259798			0.129899	+	0.991527i	$0.458535\pi$
$30$		−	6.27799i		−	0.209266i
$31$			7.71953i			0.249017i	0.992219	+	0.124509i	$0.0397354\pi$
$31$			7.71953i			0.249017i	−0.992219	+	0.124509i	$0.960265\pi$
$32$		−	33.1450i		−	1.03578i
$33$			2.10902i			0.0639096i
$34$			0.885664i			0.0260489i
$35$	−4.09206			−0.116916
$36$	8.46000			0.235000
$37$			16.7957i			0.453937i	0.973902	+	0.226969i	$0.0728815\pi$
$37$			16.7957i			0.453937i	−0.973902	+	0.226969i	$0.927119\pi$
$38$			8.41661i			0.221490i
$39$			21.9199i			0.562049i
$40$		−	24.7198i		−	0.617994i
$41$	−19.0497			−0.464627			−0.232314	−	0.972641i	$-0.574630\pi$
$41$	−19.0497			−0.464627			−0.232314	+	0.972641i	$0.574630\pi$
$42$			2.30741i			0.0549383i
$43$			48.9576i			1.13855i	0.822148	+	0.569274i	$0.192776\pi$
$43$			48.9576i			1.13855i	−0.822148	+	0.569274i	$0.807224\pi$
$44$			3.43375i			0.0780397i
$45$	10.0101			0.222448
$46$	−36.7177			−0.798211
$47$		−	4.79310i		−	0.101981i	−0.998699	−	0.0509904i	$-0.983762\pi$
$47$		−	4.79310i		−	0.101981i	0.998699	−	0.0509904i	$-0.0162378\pi$
$48$	5.59868			0.116639
$49$	−47.4960			−0.969306
$50$			15.0627i			0.301254i
$51$	−1.41218			−0.0276897
$52$			35.6884i			0.686316i
$53$	−48.7328			−0.919488			−0.459744	−	0.888052i	$-0.652059\pi$
$53$	−48.7328			−0.919488			−0.459744	+	0.888052i	$0.652059\pi$
$54$		−	5.64447i		−	0.104527i
$55$			4.06292i			0.0738713i
$56$			9.08548i			0.162241i
$57$	−13.4201			−0.235441
$58$		−	8.18418i		−	0.141107i
$59$	43.5656	+	39.7874i	0.738400	+	0.674363i
$59$	43.5656	+	39.7874i	0.738400	+	0.674363i
$60$	16.2978			0.271630
$61$			87.6284i			1.43653i	0.695769	+	0.718265i	$0.255064\pi$
$61$			87.6284i			1.43653i	−0.695769	+	0.718265i	$0.744936\pi$
$62$	8.38556			0.135251
$63$	−3.67912			−0.0583987
$64$	−23.0750			−0.360547
$65$			42.2277i			0.649657i
$66$	2.29098			0.0347118
$67$		−	42.6674i		−	0.636826i	−0.947952	−	0.318413i	$-0.896850\pi$
$67$		−	42.6674i		−	0.636826i	0.947952	−	0.318413i	$-0.103150\pi$
$68$	−2.29920			−0.0338118
$69$		−	58.5457i		−	0.848489i
$70$			4.44511i			0.0635016i
$71$	−20.2245			−0.284852			−0.142426	−	0.989805i	$-0.545490\pi$
$71$	−20.2245			−0.284852			−0.142426	+	0.989805i	$0.545490\pi$
$72$		−	22.2252i		−	0.308684i
$73$			45.6107i			0.624804i	0.949950	+	0.312402i	$0.101134\pi$
$73$			45.6107i			0.624804i	−0.949950	+	0.312402i	$0.898866\pi$
$74$	18.2448			0.246551
$75$	−24.0172			−0.320229
$76$	−21.8497			−0.287496
$77$		−	1.49328i		−	0.0193933i
$78$	23.8111			0.305271
$79$	−62.5528			−0.791808			−0.395904	−	0.918292i	$-0.629569\pi$
$79$	−62.5528			−0.791808			−0.395904	+	0.918292i	$0.629569\pi$
$80$	10.7856			0.134820
$81$	9.00000			0.111111
$82$			20.6933i			0.252357i
$83$			22.8390i			0.275169i	0.990490	+	0.137584i	$0.0439338\pi$
$83$			22.8390i			0.275169i	−0.990490	+	0.137584i	$0.956066\pi$
$84$	−5.99008			−0.0713105
$85$	−2.72049			−0.0320058
$86$	53.1815			0.618390
$87$	13.0495			0.149995
$88$	9.02079			0.102509
$89$			55.8982i			0.628070i	0.949411	+	0.314035i	$0.101681\pi$
$89$			55.8982i			0.628070i	−0.949411	+	0.314035i	$0.898319\pi$
$90$		−	10.8738i		−	0.120820i
$91$		−	15.5203i		−	0.170553i
$92$		−	95.3199i		−	1.03609i
$93$			13.3706i			0.143770i
$94$	−5.20664			−0.0553897
$95$	−25.8533			−0.272140
$96$		−	57.4087i		−	0.598008i
$97$		−	123.320i		−	1.27134i	−0.771962	−	0.635669i	$-0.780724\pi$
$97$		−	123.320i		−	1.27134i	0.771962	−	0.635669i	$-0.219276\pi$
$98$			51.5939i			0.526468i
$99$			3.65292i			0.0368982i
$100$	−39.1031			−0.391031
$101$		−	39.0946i		−	0.387075i	−0.981093	−	0.193538i	$-0.938004\pi$
$101$		−	39.0946i		−	0.387075i	0.981093	−	0.193538i	$-0.0619962\pi$
$102$			1.53401i			0.0150394i
$103$		−	56.8713i		−	0.552149i	−0.961136	−	0.276074i	$-0.910966\pi$
$103$		−	56.8713i		−	0.552149i	0.961136	−	0.276074i	$-0.0890336\pi$
$104$	93.7570			0.901509
$105$	−7.08765			−0.0675015
$106$			52.9374i			0.499410i
$107$	34.8029			0.325261			0.162630	−	0.986687i	$-0.448002\pi$
$107$	34.8029			0.325261			0.162630	+	0.986687i	$0.448002\pi$
$108$	14.6532			0.135677
$109$			9.83542i			0.0902332i	0.998982	+	0.0451166i	$0.0143659\pi$
$109$			9.83542i			0.0902332i	−0.998982	+	0.0451166i	$0.985634\pi$
$110$	4.41346			0.0401224
$111$			29.0910i			0.262081i
$112$	−3.96413			−0.0353940
$113$		−	145.812i		−	1.29037i	−0.764025	−	0.645186i	$-0.776780\pi$
$113$		−	145.812i		−	1.29037i	0.764025	−	0.645186i	$-0.223220\pi$
$114$			14.5780i			0.127877i
$115$		−	112.786i		−	0.980744i
$116$	21.2463			0.183158
$117$			37.9664i			0.324499i
$118$	43.2202	−	47.3244i	0.366273	−	0.401054i
$119$	0.999886			0.00840241
$120$		−	42.8159i		−	0.356799i
$121$	119.517			0.987747
$122$	95.1888			0.780236
$123$	−32.9951			−0.268253
$124$			21.7691i			0.175557i
$125$	−129.686			−1.03749
$126$			3.99655i			0.0317186i
$127$	145.827			1.14825			0.574123	−	0.818769i	$-0.305343\pi$
$127$	145.827			1.14825			0.574123	+	0.818769i	$0.305343\pi$
$128$		−	107.514i		−	0.839952i
$129$			84.7970i			0.657341i
$130$	45.8710			0.352854
$131$		−	30.5967i		−	0.233562i	−0.993158	−	0.116781i	$-0.962742\pi$
$131$		−	30.5967i		−	0.233562i	0.993158	−	0.116781i	$-0.0372576\pi$
$132$			5.94742i			0.0450562i
$133$	9.50208			0.0714442
$134$	−46.3486			−0.345885
$135$	17.3381			0.128430
$136$			6.04023i			0.0444134i
$137$	−39.4511			−0.287964			−0.143982	−	0.989580i	$-0.545991\pi$
$137$	−39.4511			−0.287964			−0.143982	+	0.989580i	$0.545991\pi$
$138$	−63.5969			−0.460847
$139$	93.5556			0.673062			0.336531	−	0.941672i	$-0.390746\pi$
$139$	93.5556			0.673062			0.336531	+	0.941672i	$0.390746\pi$
$140$	−11.5396			−0.0824257
$141$		−	8.30189i		−	0.0588786i
$142$			21.9694i			0.154714i
$143$	−15.4098			−0.107761
$144$	9.69720			0.0673417
$145$	25.1393			0.173375
$146$	49.5459			0.339356
$147$	−82.2655			−0.559629
$148$			47.3638i			0.320026i
$149$			133.435i			0.895536i	0.894150	+	0.447768i	$0.147781\pi$
$149$			133.435i			0.895536i	−0.894150	+	0.447768i	$0.852219\pi$
$150$			26.0894i			0.173929i
$151$		−	95.5402i		−	0.632717i	−0.948640	−	0.316358i	$-0.897540\pi$
$151$		−	95.5402i		−	0.632717i	0.948640	−	0.316358i	$-0.102460\pi$
$152$			57.4013i			0.377640i
$153$	−2.44596			−0.0159867
$154$	−1.62212			−0.0105332
$155$			25.7579i			0.166180i
$156$			61.8142i			0.396245i
$157$		−	245.077i		−	1.56100i	−0.625157	−	0.780499i	$-0.714965\pi$
$157$		−	245.077i		−	1.56100i	0.625157	−	0.780499i	$-0.285035\pi$
$158$			67.9498i			0.430062i
$159$	−84.4078			−0.530866
$160$		−	110.595i		−	0.691220i
$161$			41.4531i			0.257473i
$162$		−	9.77650i		−	0.0603488i
$163$	204.601			1.25522			0.627611	−	0.778527i	$-0.284033\pi$
$163$	204.601			1.25522			0.627611	+	0.778527i	$0.284033\pi$
$164$	−53.7202			−0.327562
$165$			7.03718i			0.0426496i
$166$	24.8095			0.149455
$167$	−100.875			−0.604043			−0.302022	−	0.953301i	$-0.597661\pi$
$167$	−100.875			−0.604043			−0.302022	+	0.953301i	$0.597661\pi$
$168$			15.7365i			0.0936698i
$169$	8.83915			0.0523027
$170$			2.95521i			0.0173836i
$171$	−23.2443			−0.135932
$172$			138.060i			0.802677i
$173$			192.449i			1.11242i	0.831041	+	0.556212i	$0.187746\pi$
$173$			192.449i			1.11242i	−0.831041	+	0.556212i	$0.812254\pi$
$174$		−	14.1754i		−	0.0814679i
$175$	17.0053			0.0971732
$176$			3.93590i			0.0223631i
$177$	75.4579	+	68.9138i	0.426316	+	0.389343i
$178$	60.7210			0.341129
$179$			66.4049i			0.370977i	0.982646	+	0.185488i	$0.0593867\pi$
$179$			66.4049i			0.370977i	−0.982646	+	0.185488i	$0.940613\pi$
$180$	28.2286			0.156826
$181$	100.811			0.556968			0.278484	−	0.960441i	$-0.410168\pi$
$181$	100.811			0.556968			0.278484	+	0.960441i	$0.410168\pi$
$182$	−16.8594			−0.0926340
$183$			151.777i			0.829381i
$184$	−250.415			−1.36095
$185$			56.0424i			0.302932i
$186$	14.5242			0.0780872
$187$		−	0.992766i		−	0.00530891i
$188$		−	13.5165i		−	0.0718965i
$189$	−6.37242			−0.0337165
$190$			28.0838i			0.147810i
$191$		−	285.899i		−	1.49686i	−0.663216	−	0.748428i	$-0.730809\pi$
$191$		−	285.899i		−	1.49686i	0.663216	−	0.748428i	$-0.269191\pi$
$192$	−39.9671			−0.208162
$193$	11.7169			0.0607093			0.0303546	−	0.999539i	$-0.490336\pi$
$193$	11.7169			0.0607093			0.0303546	+	0.999539i	$0.490336\pi$
$194$	−133.960			−0.690513
$195$			73.1405i			0.375079i
$196$	−133.939			−0.683361
$197$	−269.535			−1.36820			−0.684099	−	0.729389i	$-0.739805\pi$
$197$	−269.535			−1.36820			−0.684099	+	0.729389i	$0.739805\pi$
$198$	3.96809			0.0200409
$199$	164.357			0.825912			0.412956	−	0.910751i	$-0.364496\pi$
$199$	164.357			0.825912			0.412956	+	0.910751i	$0.364496\pi$
$200$			102.728i			0.513638i
$201$		−	73.9021i		−	0.367672i
$202$	−42.4676			−0.210236
$203$	−9.23968			−0.0455157
$204$	−3.98233			−0.0195212
$205$	−63.5635			−0.310066
$206$	−61.7781			−0.299894
$207$		−	101.404i		−	0.489875i
$208$			40.9075i			0.196671i
$209$		−	9.43442i		−	0.0451408i
$210$			7.69916i			0.0366627i
$211$		−	155.167i		−	0.735391i	−0.929946	−	0.367695i	$-0.880147\pi$
$211$		−	155.167i		−	0.735391i	0.929946	−	0.367695i	$-0.119853\pi$
$212$	−137.427			−0.648239
$213$	−35.0298			−0.164459
$214$		−	37.8056i		−	0.176662i
$215$			163.357i			0.759802i
$216$		−	38.4953i		−	0.178219i
$217$		−	9.46703i		−	0.0436269i
$218$	10.6840			0.0490092
$219$			79.0001i			0.360731i
$220$			11.4574i			0.0520792i
$221$		−	10.3183i		−	0.0466889i
$222$	31.6009			0.142346
$223$	212.798			0.954251			0.477126	−	0.878835i	$-0.341679\pi$
$223$	212.798			0.954251			0.477126	+	0.878835i	$0.341679\pi$
$224$			40.6481i			0.181465i
$225$	−41.5990			−0.184884
$226$	−158.392			−0.700852
$227$		−	272.876i		−	1.20210i	−0.799213	−	0.601048i	$-0.794750\pi$
$227$		−	272.876i		−	1.20210i	0.799213	−	0.601048i	$-0.205250\pi$
$228$	−37.8448			−0.165986
$229$			350.552i			1.53080i	0.643557	+	0.765398i	$0.277458\pi$
$229$			350.552i			1.53080i	−0.643557	+	0.765398i	$0.722542\pi$
$230$	−122.517			−0.532681
$231$		−	2.58644i		−	0.0111967i
$232$		−	55.8161i		−	0.240587i
$233$		−	1.64412i		−	0.00705630i	−0.999994	−	0.00352815i	$-0.998877\pi$
$233$		−	1.64412i		−	0.00705630i	0.999994	−	0.00352815i	$-0.00112305\pi$
$234$	41.2421			0.176248
$235$		−	15.9932i		−	0.0680562i
$236$	122.855	+	112.200i	0.520572	+	0.475426i
$237$	−108.345			−0.457151
$238$		−	1.08615i		−	0.00456368i
$239$	−74.5410			−0.311887			−0.155943	−	0.987766i	$-0.549842\pi$
$239$	−74.5410			−0.311887			−0.155943	+	0.987766i	$0.549842\pi$
$240$	18.6812			0.0778384
$241$	−56.4597			−0.234272			−0.117136	−	0.993116i	$-0.537371\pi$
$241$	−56.4597			−0.234272			−0.117136	+	0.993116i	$0.537371\pi$
$242$		−	129.829i		−	0.536484i
$243$	15.5885			0.0641500
$244$			247.112i			1.01275i
$245$	−158.481			−0.646860
$246$			35.8418i			0.145698i
$247$		−	98.0560i		−	0.396988i
$248$	57.1895			0.230603
$249$			39.5583i			0.158869i
$250$			140.875i			0.563500i
$251$	−21.6224			−0.0861451			−0.0430726	−	0.999072i	$-0.513715\pi$
$251$	−21.6224			−0.0861451			−0.0430726	+	0.999072i	$0.513715\pi$
$252$	−10.3751			−0.0411711
$253$	41.1580			0.162680
$254$		−	158.409i		−	0.623657i
$255$	−4.71203			−0.0184785
$256$	−209.090			−0.816758
$257$	70.0007			0.272376			0.136188	−	0.990683i	$-0.456515\pi$
$257$	70.0007			0.272376			0.136188	+	0.990683i	$0.456515\pi$
$258$	92.1131			0.357028
$259$		−	20.5978i		−	0.0795281i
$260$			119.082i			0.458008i
$261$	22.6025			0.0865994
$262$	−33.2365			−0.126857
$263$	450.103			1.71142			0.855709	−	0.517457i	$-0.173121\pi$
$263$	450.103			1.71142			0.855709	+	0.517457i	$0.173121\pi$
$264$	15.6245			0.0591836
$265$	−162.608			−0.613614
$266$		−	10.3219i		−	0.0388042i
$267$			96.8186i			0.362617i
$268$		−	120.322i		−	0.448963i
$269$			167.392i			0.622275i	0.950365	+	0.311137i	$0.100710\pi$
$269$			167.392i			0.622275i	−0.950365	+	0.311137i	$0.899290\pi$
$270$		−	18.8340i		−	0.0697555i
$271$	136.135			0.502342			0.251171	−	0.967943i	$-0.419184\pi$
$271$	136.135			0.502342			0.251171	+	0.967943i	$0.419184\pi$
$272$	−2.63544			−0.00968912
$273$		−	26.8820i		−	0.0984689i
$274$			42.8549i			0.156405i
$275$		−	16.8842i		−	0.0613971i
$276$		−	165.099i		−	0.598185i
$277$	−195.176			−0.704606			−0.352303	−	0.935886i	$-0.614601\pi$
$277$	−195.176			−0.704606			−0.352303	+	0.935886i	$0.614601\pi$
$278$		−	101.627i		−	0.365566i
$279$			23.1586i			0.0830057i
$280$			30.3157i			0.108270i
$281$	475.581			1.69246			0.846229	−	0.532820i	$-0.178868\pi$
$281$	475.581			1.69246			0.846229	+	0.532820i	$0.178868\pi$
$282$	−9.01816			−0.0319793
$283$		−	444.223i		−	1.56969i	−0.619691	−	0.784846i	$-0.712742\pi$
$283$		−	444.223i		−	1.56969i	0.619691	−	0.784846i	$-0.287258\pi$
$284$	−57.0330			−0.200821
$285$	−44.7791			−0.157120
$286$			16.7393i			0.0585292i
$287$	23.3621			0.0814009
$288$		−	99.4349i		−	0.345260i
$289$	−288.335			−0.997700
$290$		−	27.3083i		−	0.0941665i
$291$		−	213.596i		−	0.734007i
$292$			128.622i			0.440487i
$293$	−56.4950			−0.192816			−0.0964079	−	0.995342i	$-0.530735\pi$
$293$	−56.4950			−0.192816			−0.0964079	+	0.995342i	$0.530735\pi$
$294$			89.3632i			0.303956i
$295$	145.366	+	132.759i	0.492766	+	0.450031i
$296$	124.429			0.420369
$297$			6.32705i			0.0213032i
$298$	144.947			0.486401
$299$	427.772			1.43068
$300$	−67.7285			−0.225762
$301$		−	60.0403i		−	0.199469i
$302$	−103.783			−0.343653
$303$		−	67.7138i		−	0.223478i
$304$	−25.0450			−0.0823849
$305$			292.391i			0.958659i
$306$			2.65699i			0.00868298i
$307$	279.533			0.910530			0.455265	−	0.890356i	$-0.349545\pi$
$307$	279.533			0.910530			0.455265	+	0.890356i	$0.349545\pi$
$308$		−	4.21106i		−	0.0136723i
$309$		−	98.5040i		−	0.318783i
$310$	27.9802			0.0902588
$311$	210.784			0.677762			0.338881	−	0.940829i	$-0.389952\pi$
$311$	210.784			0.677762			0.338881	+	0.940829i	$0.389952\pi$
$312$	162.392			0.520487
$313$		−	372.662i		−	1.19061i	−0.803499	−	0.595306i	$-0.797031\pi$
$313$		−	372.662i		−	1.19061i	0.803499	−	0.595306i	$-0.202969\pi$
$314$	−266.221			−0.847839
$315$	−12.2762			−0.0389720
$316$	−176.399			−0.558225
$317$	299.104			0.943544			0.471772	−	0.881720i	$-0.343614\pi$
$317$	299.104			0.943544			0.471772	+	0.881720i	$0.343614\pi$
$318$			91.6903i			0.288334i
$319$			9.17389i			0.0287583i
$320$	−76.9948			−0.240609
$321$	60.2804			0.187789
$322$	45.0296			0.139844
$323$	6.31719			0.0195579
$324$	25.3800			0.0783333
$325$		−	175.485i		−	0.539954i
$326$		−	222.254i		−	0.681760i
$327$			17.0354i			0.0520962i
$328$			141.128i			0.430269i
$329$			5.87813i			0.0178666i
$330$	7.64434			0.0231647
$331$	268.182			0.810217			0.405109	−	0.914269i	$-0.367234\pi$
$331$	268.182			0.810217			0.405109	+	0.914269i	$0.367234\pi$
$332$			64.4060i			0.193994i
$333$			50.3870i			0.151312i
$334$			109.579i			0.328079i
$335$		−	142.369i		−	0.424982i
$336$	−6.86608			−0.0204347
$337$			372.241i			1.10457i	0.833655	+	0.552286i	$0.186244\pi$
$337$			372.241i			1.10457i	−0.833655	+	0.552286i	$0.813756\pi$
$338$		−	9.60178i		−	0.0284076i
$339$		−	252.554i		−	0.744997i
$340$	−7.67178			−0.0225641
$341$	−9.39962			−0.0275649
$342$			25.2498i			0.0738299i
$343$	118.340			0.345015
$344$	362.698			1.05436
$345$		−	195.350i		−	0.566233i
$346$	209.053			0.604201
$347$		−	162.680i		−	0.468820i	−0.972138	−	0.234410i	$-0.924684\pi$
$347$		−	162.680i		−	0.468820i	0.972138	−	0.234410i	$-0.0753158\pi$
$348$	36.7997			0.105746
$349$			474.478i			1.35954i	0.733427	+	0.679768i	$0.237920\pi$
$349$			474.478i			1.35954i	−0.733427	+	0.679768i	$0.762080\pi$
$350$		−	18.4725i		−	0.0527785i
$351$			65.7597i			0.187350i
$352$	40.3586			0.114655
$353$		−	553.231i		−	1.56723i	−0.621248	−	0.783614i	$-0.713374\pi$
$353$		−	553.231i		−	1.56723i	0.621248	−	0.783614i	$-0.286626\pi$
$354$	74.8595	−	81.9682i	0.211468	−	0.231549i
$355$	−67.4833			−0.190094
$356$			157.633i			0.442790i
$357$	1.73185			0.00485113
$358$	72.1342			0.201492
$359$	−347.072			−0.966775			−0.483387	−	0.875407i	$-0.660594\pi$
$359$	−347.072			−0.966775			−0.483387	+	0.875407i	$0.660594\pi$
$360$		−	74.1593i		−	0.205998i
$361$	−300.967			−0.833703
$362$		−	109.509i		−	0.302511i
$363$	207.010			0.570276
$364$		−	43.7673i		−	0.120240i
$365$			152.190i			0.416959i
$366$	164.872			0.450469
$367$		−	617.839i		−	1.68349i	−0.539879	−	0.841743i	$-0.681530\pi$
$367$		−	617.839i		−	1.68349i	0.539879	−	0.841743i	$-0.318470\pi$
$368$		−	109.260i		−	0.296901i
$369$	−57.1491			−0.154876
$370$	60.8776			0.164534
$371$	59.7647			0.161091
$372$			37.7052i			0.101358i
$373$	−427.265			−1.14548			−0.572741	−	0.819736i	$-0.694120\pi$
$373$	−427.265			−1.14548			−0.572741	+	0.819736i	$0.694120\pi$
$374$	−1.07842			−0.00288348
$375$	−224.623			−0.598993
$376$	−35.5093			−0.0944395
$377$			95.3482i			0.252913i
$378$			6.92222i			0.0183128i
$379$	−400.731			−1.05734			−0.528668	−	0.848828i	$-0.677308\pi$
$379$	−400.731			−1.05734			−0.528668	+	0.848828i	$0.677308\pi$
$380$	−72.9062			−0.191858
$381$	252.580			0.662940
$382$	−310.566			−0.813001
$383$	−18.1585			−0.0474111			−0.0237056	−	0.999719i	$-0.507546\pi$
$383$	−18.1585			−0.0474111			−0.0237056	+	0.999719i	$0.507546\pi$
$384$		−	186.220i		−	0.484947i
$385$		−	4.98266i		−	0.0129420i
$386$		−	12.7278i		−	0.0329736i
$387$			146.873i			0.379516i
$388$		−	347.762i		−	0.896293i
$389$	−308.582			−0.793269			−0.396635	−	0.917977i	$-0.629822\pi$
$389$	−308.582			−0.793269			−0.396635	+	0.917977i	$0.629822\pi$
$390$	79.4509			0.203720
$391$			27.5589i			0.0704832i
$392$			351.870i			0.897628i
$393$		−	52.9950i		−	0.134847i
$394$			292.790i			0.743122i
$395$	−208.721			−0.528407
$396$			10.3012i			0.0260132i
$397$			193.629i			0.487730i	0.969809	+	0.243865i	$0.0784154\pi$
$397$			193.629i			0.487730i	−0.969809	+	0.243865i	$0.921585\pi$
$398$		−	178.537i		−	0.448585i
$399$	16.4581			0.0412484
$400$	−44.8216			−0.112054
$401$		−	148.241i		−	0.369678i	−0.982769	−	0.184839i	$-0.940824\pi$
$401$		−	148.241i		−	0.369678i	0.982769	−	0.184839i	$-0.0591763\pi$
$402$	−80.2782			−0.199697
$403$	−97.6943			−0.242418
$404$		−	110.247i		−	0.272888i
$405$	30.0304			0.0741492
$406$			10.0369i			0.0247213i
$407$	−20.4511			−0.0502484
$408$			10.4620i			0.0256421i
$409$			486.132i			1.18859i	0.804249	+	0.594293i	$0.202568\pi$
$409$			486.132i			1.18859i	−0.804249	+	0.594293i	$0.797432\pi$
$410$			69.0476i			0.168409i
$411$	−68.3314			−0.166256
$412$		−	160.377i		−	0.389265i
$413$	−53.4277	−	48.7942i	−0.129365	−	0.118146i
$414$	−110.153			−0.266070
$415$			76.2073i			0.183632i
$416$	419.465			1.00833
$417$	162.043			0.388592
$418$	−10.2484			−0.0245177
$419$			393.978i			0.940281i	0.882592	+	0.470140i	$0.155797\pi$
$419$			393.978i			0.940281i	−0.882592	+	0.470140i	$0.844203\pi$
$420$	−19.9872			−0.0475885
$421$			545.228i			1.29508i	0.762032	+	0.647539i	$0.224202\pi$
$421$			545.228i			1.29508i	−0.762032	+	0.647539i	$0.775798\pi$
$422$	−168.555			−0.399420
$423$		−	14.3793i		−	0.0339936i
$424$			361.033i			0.851493i
$425$	11.3055			0.0266012
$426$			38.0521i			0.0893243i
$427$		−	107.465i		−	0.251675i
$428$	98.1441			0.229309
$429$	−26.6906			−0.0622158
$430$	177.452			0.412678
$431$			645.842i			1.49847i	0.662302	+	0.749237i	$0.269580\pi$
$431$			645.842i			1.49847i	−0.662302	+	0.749237i	$0.730420\pi$
$432$	16.7960			0.0388797
$433$	−440.441			−1.01719			−0.508593	−	0.861007i	$-0.669834\pi$
$433$	−440.441			−1.01719			−0.508593	+	0.861007i	$0.669834\pi$
$434$	−10.2838			−0.0236954
$435$	43.5426			0.100098
$436$			27.7359i			0.0636144i
$437$			261.897i			0.599307i
$438$	85.8160			0.195927
$439$	−817.423			−1.86201			−0.931006	−	0.365003i	$-0.881068\pi$
$439$	−817.423			−1.86201			−0.931006	+	0.365003i	$0.881068\pi$
$440$	30.0998			0.0684086
$441$	−142.488			−0.323102
$442$	−11.2085			−0.0253586
$443$		−	522.987i		−	1.18056i	−0.807200	−	0.590278i	$-0.799018\pi$
$443$		−	522.987i		−	1.18056i	0.807200	−	0.590278i	$-0.200982\pi$
$444$			82.0365i			0.184767i
$445$			186.517i			0.419138i
$446$		−	231.158i		−	0.518291i
$447$			231.116i			0.517038i
$448$	28.2986			0.0631665
$449$	−721.573			−1.60707			−0.803533	−	0.595260i	$-0.797049\pi$
$449$	−721.573			−1.60707			−0.803533	+	0.595260i	$0.797049\pi$
$450$			45.1881i			0.100418i
$451$		−	23.1957i		−	0.0514317i
$452$		−	411.190i		−	0.909713i
$453$		−	165.481i		−	0.365299i
$454$	−296.419			−0.652905
$455$		−	51.7869i		−	0.113817i
$456$			99.4219i			0.218031i
$457$			899.045i			1.96728i	0.180157	+	0.983638i	$0.442340\pi$
$457$			899.045i			1.96728i	−0.180157	+	0.983638i	$0.557660\pi$
$458$	380.797			0.831435
$459$	−4.23653			−0.00922990
$460$		−	318.055i		−	0.691425i
$461$	126.539			0.274489			0.137245	−	0.990537i	$-0.456175\pi$
$461$	126.539			0.274489			0.137245	+	0.990537i	$0.456175\pi$
$462$	−2.80959			−0.00608137
$463$			169.302i			0.365663i	0.983144	+	0.182831i	$0.0585263\pi$
$463$			169.302i			0.365663i	−0.983144	+	0.182831i	$0.941474\pi$
$464$	24.3534			0.0524858
$465$			44.6139i			0.0959440i
$466$	−1.78597			−0.00383255
$467$			98.8969i			0.211771i	0.994378	+	0.105885i	$0.0337676\pi$
$467$			98.8969i			0.211771i	−0.994378	+	0.105885i	$0.966232\pi$
$468$			107.065i			0.228772i
$469$			52.3261i			0.111570i
$470$	−17.3731			−0.0369640
$471$		−	424.485i		−	0.901243i
$472$	294.762	−	322.752i	0.624495	−	0.683797i
$473$	−59.6127			−0.126031
$474$			117.692i			0.248296i
$475$	107.438			0.226185
$476$	2.81968			0.00592370
$477$	−146.199			−0.306496
$478$			80.9722i			0.169398i
$479$	−91.0329			−0.190048			−0.0950240	−	0.995475i	$-0.530293\pi$
$479$	−91.0329			−0.190048			−0.0950240	+	0.995475i	$0.530293\pi$
$480$		−	191.557i		−	0.399076i
$481$	−212.557			−0.441907
$482$			61.3309i			0.127242i
$483$			71.7989i			0.148652i
$484$	337.039			0.696361
$485$		−	411.483i		−	0.848419i
$486$		−	16.9334i		−	0.0348424i
$487$	−72.6279			−0.149133			−0.0745667	−	0.997216i	$-0.523757\pi$
$487$	−72.6279			−0.149133			−0.0745667	+	0.997216i	$0.523757\pi$
$488$	649.187			1.33030
$489$	354.380			0.724703
$490$			172.154i			0.351335i
$491$	−269.921			−0.549736			−0.274868	−	0.961482i	$-0.588634\pi$
$491$	−269.921			−0.549736			−0.274868	+	0.961482i	$0.588634\pi$
$492$	−93.0461			−0.189118
$493$	−6.14274			−0.0124599
$494$	−106.516			−0.215620
$495$			12.1888i			0.0246238i
$496$			24.9526i			0.0503077i
$497$	24.8028			0.0499050
$498$	42.9713			0.0862878
$499$	219.786			0.440452			0.220226	−	0.975449i	$-0.429320\pi$
$499$	219.786			0.440452			0.220226	+	0.975449i	$0.429320\pi$
$500$	−365.714			−0.731428
$501$	−174.721			−0.348745
$502$			23.4880i			0.0467888i
$503$			800.462i			1.59138i	0.605707	+	0.795688i	$0.292890\pi$
$503$			800.462i			1.59138i	−0.605707	+	0.795688i	$0.707110\pi$
$504$			27.2565i			0.0540803i
$505$		−	130.448i		−	0.258312i
$506$		−	44.7090i		−	0.0883577i
$507$	15.3099			0.0301970
$508$	411.233			0.809513
$509$			12.6193i			0.0247924i	0.999923	+	0.0123962i	$0.00394593\pi$
$509$			12.6193i			0.0247924i	−0.999923	+	0.0123962i	$0.996054\pi$
$510$			5.11857i			0.0100364i
$511$		−	55.9358i		−	0.109463i
$512$		−	202.926i		−	0.396339i
$513$	−40.2604			−0.0784803
$514$		−	76.0403i		−	0.147938i
$515$		−	189.763i		−	0.368473i
$516$			239.128i			0.463426i
$517$	5.83627			0.0112887
$518$	−22.3749			−0.0431948
$519$			333.332i			0.642258i
$520$	312.840			0.601616
$521$	629.736			1.20871			0.604353	−	0.796717i	$-0.293432\pi$
$521$	629.736			1.20871			0.604353	+	0.796717i	$0.293432\pi$
$522$		−	24.5525i		−	0.0470355i
$523$	852.513			1.63004			0.815022	−	0.579430i	$-0.196725\pi$
$523$	852.513			1.63004			0.815022	+	0.579430i	$0.196725\pi$
$524$		−	86.2826i		−	0.164661i
$525$	29.4541			0.0561030
$526$		−	488.937i		−	0.929538i
$527$		−	6.29389i		−	0.0119429i
$528$			6.81718i			0.0129113i
$529$	−613.534			−1.15980
$530$			176.637i			0.333277i
$531$	130.697	+	119.362i	0.246133	+	0.224788i
$532$	26.7959			0.0503682
$533$		−	241.083i		−	0.452313i
$534$	105.172			0.196951
$535$	116.127			0.217060
$536$	−316.098			−0.589734
$537$			115.017i			0.214184i
$538$	181.834			0.337982
$539$		−	57.8331i		−	0.107297i
$540$	48.8934			0.0905433
$541$		−	463.558i		−	0.856855i	−0.903576	−	0.428427i	$-0.859068\pi$
$541$		−	463.558i		−	0.856855i	0.903576	−	0.428427i	$-0.140932\pi$
$542$		−	147.880i		−	0.272842i
$543$	174.610			0.321566
$544$			27.0237i			0.0496760i
$545$			32.8180i			0.0602165i
$546$	−29.2013			−0.0534823
$547$	306.278			0.559922			0.279961	−	0.960011i	$-0.409678\pi$
$547$	306.278			0.559922			0.279961	+	0.960011i	$0.409678\pi$
$548$	−111.252			−0.203015
$549$			262.885i			0.478844i
$550$	−18.3409			−0.0333472
$551$	−58.3755			−0.105945
$552$	−433.731			−0.785745
$553$	76.7131			0.138722
$554$			212.015i			0.382699i
$555$			97.0682i			0.174898i
$556$	263.827			0.474508
$557$	−994.672			−1.78577			−0.892883	−	0.450288i	$-0.851321\pi$
$557$	−994.672			−1.78577			−0.892883	+	0.450288i	$0.851321\pi$
$558$	25.1567			0.0450836
$559$	−619.581			−1.10837
$560$	−13.2272			−0.0236199
$561$		−	1.71952i		−	0.00306510i
$562$		−	516.613i		−	0.919240i
$563$		−	206.256i		−	0.366352i	−0.983080	−	0.183176i	$-0.941362\pi$
$563$		−	206.256i		−	0.366352i	0.983080	−	0.183176i	$-0.0586378\pi$
$564$		−	23.4113i		−	0.0415094i
$565$		−	486.533i		−	0.861121i
$566$	−482.550			−0.852561
$567$	−11.0374			−0.0194662
$568$			149.831i			0.263788i
$569$		−	588.448i		−	1.03418i	−0.855931	−	0.517089i	$-0.827015\pi$
$569$		−	588.448i		−	1.03418i	0.855931	−	0.517089i	$-0.172985\pi$
$570$			48.6426i			0.0853379i
$571$		−	139.611i		−	0.244503i	−0.992499	−	0.122251i	$-0.960989\pi$
$571$		−	139.611i		−	0.244503i	0.992499	−	0.122251i	$-0.0390114\pi$
$572$	−43.4557			−0.0759714
$573$		−	495.192i		−	0.864210i
$574$		−	25.3777i		−	0.0442120i
$575$			468.701i			0.815133i
$576$	−69.2251			−0.120182
$577$	947.400			1.64194			0.820971	−	0.570970i	$-0.193433\pi$
$577$	947.400			1.64194			0.820971	+	0.570970i	$0.193433\pi$
$578$			313.212i			0.541890i
$579$	20.2942			0.0350505
$580$	70.8929			0.122229
$581$		−	28.0092i		−	0.0482085i
$582$	−232.025			−0.398668
$583$		−	59.3391i		−	0.101782i
$584$	337.903			0.578601
$585$			126.683i			0.216552i
$586$			61.3693i			0.104726i
$587$			672.802i			1.14617i	0.819496	+	0.573085i	$0.194254\pi$
$587$			672.802i			1.14617i	−0.819496	+	0.573085i	$0.805746\pi$
$588$	−231.989			−0.394539
$589$		−	59.8118i		−	0.101548i
$590$	144.213	−	157.908i	0.244429	−	0.267641i
$591$	−466.849			−0.789930
$592$			54.2903i			0.0917067i
$593$	235.268			0.396742			0.198371	−	0.980127i	$-0.436435\pi$
$593$	235.268			0.396742			0.198371	+	0.980127i	$0.436435\pi$
$594$	6.87293			0.0115706
$595$	3.33634			0.00560729
$596$			376.287i			0.631353i
$597$	284.674			0.476841
$598$		−	464.680i		−	0.777056i
$599$	−938.758			−1.56721			−0.783604	−	0.621260i	$-0.786621\pi$
$599$	−938.758			−1.56721			−0.783604	+	0.621260i	$0.786621\pi$
$600$			177.929i			0.296549i
$601$		−	396.943i		−	0.660470i	−0.943899	−	0.330235i	$-0.892872\pi$
$601$		−	396.943i		−	0.660470i	0.943899	−	0.330235i	$-0.107128\pi$
$602$	−65.2204			−0.108340
$603$		−	128.002i		−	0.212275i
$604$		−	269.423i		−	0.446065i
$605$	398.795			0.659166
$606$	−73.5561			−0.121380
$607$	794.727			1.30927			0.654635	−	0.755945i	$-0.272822\pi$
$607$	794.727			1.30927			0.654635	+	0.755945i	$0.272822\pi$
$608$			256.811i			0.422386i
$609$	−16.0036			−0.0262785
$610$	317.618			0.520685
$611$	60.6589			0.0992780
$612$	−6.89761			−0.0112706
$613$			14.2456i			0.0232392i	0.999932	+	0.0116196i	$0.00369872\pi$
$613$			14.2456i			0.0232392i	−0.999932	+	0.0116196i	$0.996301\pi$
$614$		−	303.650i		−	0.494544i
$615$	−110.095			−0.179017
$616$	−11.0629			−0.0179592
$617$	739.753			1.19895			0.599476	−	0.800393i	$-0.295376\pi$
$617$	739.753			1.19895			0.599476	+	0.800393i	$0.295376\pi$
$618$	−107.003			−0.173144
$619$	202.230			0.326705			0.163352	−	0.986568i	$-0.447769\pi$
$619$	202.230			0.326705			0.163352	+	0.986568i	$0.447769\pi$
$620$			72.6372i			0.117157i
$621$		−	175.637i		−	0.282830i
$622$		−	228.970i		−	0.368119i
$623$		−	68.5521i		−	0.110036i
$624$			70.8539i			0.113548i
$625$	−86.0664			−0.137706
$626$	−404.814			−0.646668
$627$		−	16.3409i		−	0.0260620i
$628$		−	691.116i		−	1.10050i
$629$		−	13.6938i		−	0.0217708i
$630$			13.3353i			0.0211672i
$631$	−504.848			−0.800076			−0.400038	−	0.916499i	$-0.631003\pi$
$631$	−504.848			−0.800076			−0.400038	+	0.916499i	$0.631003\pi$
$632$			463.417i			0.733255i
$633$		−	268.758i		−	0.424578i
$634$		−	324.910i		−	0.512476i
$635$	486.584			0.766274
$636$	−238.030			−0.374261
$637$		−	601.084i		−	0.943617i
$638$	9.96539			0.0156197
$639$	−60.6734			−0.0949506
$640$		−	358.743i		−	0.560536i
$641$	757.354			1.18152			0.590760	−	0.806847i	$-0.298828\pi$
$641$	757.354			1.18152			0.590760	+	0.806847i	$0.298828\pi$
$642$		−	65.4812i		−	0.101996i
$643$	390.062			0.606628			0.303314	−	0.952891i	$-0.401907\pi$
$643$	390.062			0.606628			0.303314	+	0.952891i	$0.401907\pi$
$644$			116.898i			0.181518i
$645$			282.943i			0.438672i
$646$		−	6.86223i		−	0.0106226i
$647$	−992.824			−1.53450			−0.767252	−	0.641346i	$-0.778376\pi$
$647$	−992.824			−1.53450			−0.767252	+	0.641346i	$0.778376\pi$
$648$		−	66.6757i		−	0.102895i
$649$	−48.4467	+	53.0473i	−0.0746483	+	0.0817369i
$650$	−190.625			−0.293270
$651$		−	16.3974i		−	0.0251880i
$652$	576.976			0.884932
$653$	1166.56			1.78646			0.893228	−	0.449604i	$-0.148435\pi$
$653$	1166.56			1.78646			0.893228	+	0.449604i	$0.148435\pi$
$654$	18.5052			0.0282955
$655$		−	102.092i		−	0.155866i
$656$	−61.5763			−0.0938663
$657$			136.832i			0.208268i
$658$	6.38528			0.00970407
$659$		−	689.431i		−	1.04618i	−0.852278	−	0.523089i	$-0.824780\pi$
$659$		−	689.431i		−	1.04618i	0.852278	−	0.523089i	$-0.175220\pi$
$660$			19.8449i			0.0300680i
$661$	1035.80			1.56703			0.783513	−	0.621376i	$-0.213426\pi$
$661$	1035.80			1.56703			0.783513	+	0.621376i	$0.213426\pi$
$662$		−	291.320i		−	0.440061i
$663$		−	17.8717i		−	0.0269559i
$664$	169.201			0.254821
$665$	31.7057			0.0476778
$666$	54.7343			0.0821836
$667$		−	254.665i		−	0.381806i
$668$	−284.468			−0.425851
$669$	368.577			0.550937
$670$	−154.652			−0.230824
$671$	−106.700			−0.159016
$672$			70.4046i			0.104769i
$673$			504.708i			0.749937i	0.927037	+	0.374969i	$0.122347\pi$
$673$			504.708i			0.749937i	−0.927037	+	0.374969i	$0.877653\pi$
$674$	404.357			0.599936
$675$	−72.0516			−0.106743
$676$	24.9264			0.0368734
$677$	184.647			0.272743			0.136372	−	0.990658i	$-0.456456\pi$
$677$	184.647			0.272743			0.136372	+	0.990658i	$0.456456\pi$
$678$	−274.344			−0.404637
$679$			151.236i			0.222734i
$680$			20.1545i			0.0296390i
$681$		−	472.635i		−	0.694030i
$682$			10.2106i			0.0149715i
$683$		−	150.545i		−	0.220417i	−0.993908	−	0.110209i	$-0.964848\pi$
$683$		−	150.545i		−	0.220417i	0.993908	−	0.110209i	$-0.0351519\pi$
$684$	−65.5491			−0.0958320
$685$	−131.637			−0.192171
$686$		−	128.550i		−	0.187391i
$687$			607.175i			0.883806i
$688$			158.251i			0.230015i
$689$		−	616.737i		−	0.895119i
$690$	−212.205			−0.307543
$691$		−	8.87982i		−	0.0128507i	−0.999979	−	0.00642534i	$-0.997955\pi$
$691$		−	8.87982i		−	0.0128507i	0.999979	−	0.00642534i	$-0.00204526\pi$
$692$			542.707i			0.784259i
$693$		−	4.47985i		−	0.00646442i
$694$	−176.716			−0.254634
$695$	312.168			0.449163
$696$		−	96.6764i		−	0.138903i
$697$	15.5316			0.0222835
$698$	515.415			0.738417
$699$		−	2.84770i		−	0.00407396i
$700$	47.9550			0.0685071
$701$			672.703i			0.959633i	0.877369	+	0.479817i	$0.159297\pi$
$701$			672.703i			0.959633i	−0.877369	+	0.479817i	$0.840703\pi$
$702$	71.4334			0.101757
$703$		−	130.135i		−	0.185114i
$704$		−	28.0971i		−	0.0399106i
$705$		−	27.7010i		−	0.0392922i
$706$	−600.963			−0.851223
$707$			47.9446i			0.0678141i
$708$	212.791	+	194.337i	0.300553	+	0.274487i
$709$	−885.467			−1.24890			−0.624448	−	0.781067i	$-0.714676\pi$
$709$	−885.467			−1.24890			−0.624448	+	0.781067i	$0.714676\pi$
$710$			73.3057i			0.103247i
$711$	−187.658			−0.263936
$712$	414.117			0.581626
$713$	260.931			0.365962
$714$		−	1.88128i		−	0.00263484i
$715$	−51.4181			−0.0719135
$716$			187.262i			0.261539i
$717$	−129.109			−0.180068
$718$			377.017i			0.525093i
$719$			900.012i			1.25175i	0.779922	+	0.625877i	$0.215259\pi$
$719$			900.012i			1.25175i	−0.779922	+	0.625877i	$0.784741\pi$
$720$	32.3568			0.0449400
$721$			69.7455i			0.0967344i
$722$			326.934i			0.452816i
$723$	−97.7910			−0.135257
$724$	284.288			0.392662
$725$	−104.471			−0.144098
$726$		−	224.871i		−	0.309739i
$727$	−702.800			−0.966713			−0.483357	−	0.875424i	$-0.660583\pi$
$727$	−702.800			−0.966713			−0.483357	+	0.875424i	$0.660583\pi$
$728$	−114.981			−0.157941
$729$	27.0000			0.0370370
$730$	165.321			0.226467
$731$		−	39.9161i		−	0.0546048i
$732$			428.011i			0.584714i
$733$	−389.940			−0.531978			−0.265989	−	0.963976i	$-0.585698\pi$
$733$	−389.940			−0.531978			−0.265989	+	0.963976i	$0.585698\pi$
$734$	−671.145			−0.914366
$735$	−274.497			−0.373465
$736$	−1120.35			−1.52221
$737$	51.9535			0.0704933
$738$			62.0799i			0.0841191i
$739$			1203.79i			1.62894i	0.580206	+	0.814470i	$0.302972\pi$
$739$			1203.79i			1.62894i	−0.580206	+	0.814470i	$0.697028\pi$
$740$			158.039i			0.213567i
$741$		−	169.838i		−	0.229201i
$742$		−	64.9210i		−	0.0874947i
$743$	−1379.37			−1.85649			−0.928247	−	0.371965i	$-0.878684\pi$
$743$	−1379.37			−1.85649			−0.928247	+	0.371965i	$0.878684\pi$
$744$	99.0551			0.133139
$745$			445.234i			0.597630i
$746$			464.128i			0.622156i
$747$			68.5170i			0.0917229i
$748$		−	2.79960i		−	0.00374278i
$749$	−42.6813			−0.0569844
$750$			244.003i			0.325337i
$751$			855.872i			1.13964i	0.821769	+	0.569821i	$0.192988\pi$
$751$			855.872i			1.13964i	−0.821769	+	0.569821i	$0.807012\pi$
$752$		−	15.4932i		−	0.0206027i
$753$	−37.4511			−0.0497359
$754$	103.575			0.137367
$755$		−	318.791i		−	0.422239i
$756$	−17.9702			−0.0237702
$757$	−126.963			−0.167719			−0.0838593	−	0.996478i	$-0.526725\pi$
$757$	−126.963			−0.167719			−0.0838593	+	0.996478i	$0.526725\pi$
$758$			435.305i			0.574281i
$759$	71.2877			0.0939231
$760$			191.532i			0.252015i
$761$	615.423			0.808703			0.404351	−	0.914604i	$-0.367497\pi$
$761$	615.423			0.808703			0.404351	+	0.914604i	$0.367497\pi$
$762$		−	274.372i		−	0.360069i
$763$		−	12.0619i		−	0.0158085i
$764$		−	806.236i		−	1.05528i
$765$	−8.16147			−0.0106686
$766$			19.7251i			0.0257508i
$767$	−503.528	+	551.343i	−0.656490	+	0.718831i
$768$	−362.155			−0.471556
$769$			986.655i			1.28304i	0.767108	+	0.641518i	$0.221695\pi$
$769$			986.655i			1.28304i	−0.767108	+	0.641518i	$0.778305\pi$
$770$	−5.41255			−0.00702929
$771$	121.245			0.157257
$772$	33.0416			0.0428000
$773$		−	6.18406i		−	0.00800008i	−0.999992	−	0.00400004i	$-0.998727\pi$
$773$		−	6.18406i		−	0.00800008i	0.999992	−	0.00400004i	$-0.00127326\pi$
$774$	159.545			0.206130
$775$		−	107.042i		−	0.138118i
$776$	−913.604			−1.17733
$777$		−	35.6764i		−	0.0459155i
$778$			335.206i			0.430856i
$779$	147.599			0.189473
$780$			206.256i			0.264431i
$781$		−	24.6262i		−	0.0315316i
$782$	29.9367			0.0382822
$783$	39.1486			0.0499982
$784$	−153.526			−0.195824
$785$		−	817.751i		−	1.04172i
$786$	−57.5673			−0.0732408
$787$	545.010			0.692516			0.346258	−	0.938139i	$-0.387452\pi$
$787$	545.010			0.692516			0.346258	+	0.938139i	$0.387452\pi$
$788$	−760.089			−0.964580
$789$	779.601			0.988088
$790$			226.729i			0.286999i
$791$			178.820i			0.226068i

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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.08628i q^{2} +1.73205 q^{3} +2.82000 q^{4} +3.33671 q^{5} -1.88149i q^{6} -1.22637 q^{7} -7.40842i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.08628i q^{2} +1.73205 q^{3} +2.82000 q^{4} +3.33671 q^{5} -1.88149i q^{6} -1.22637 q^{7} -7.40842i q^{8} +3.00000 q^{9} -3.62460i q^{10} +1.21764i q^{11} +4.88438 q^{12} +12.6555i q^{13} +1.33218i q^{14} +5.77936 q^{15} +3.23240 q^{16} -0.815320 q^{17} -3.25883i q^{18} -7.74812 q^{19} +9.40954 q^{20} -2.12414 q^{21} +1.32270 q^{22} -33.8014i q^{23} -12.8318i q^{24} -13.8663 q^{25} +13.7474 q^{26} +5.19615 q^{27} -3.45837 q^{28} +7.53415 q^{29} -6.27799i q^{30} +7.71953i q^{31} -33.1450i q^{32} +2.10902i q^{33} +0.885664i q^{34} -4.09206 q^{35} +8.46000 q^{36} +16.7957i q^{37} +8.41661i q^{38} +21.9199i q^{39} -24.7198i q^{40} -19.0497 q^{41} +2.30741i q^{42} +48.9576i q^{43} +3.43375i q^{44} +10.0101 q^{45} -36.7177 q^{46} -4.79310i q^{47} +5.59868 q^{48} -47.4960 q^{49} +15.0627i q^{50} -1.41218 q^{51} +35.6884i q^{52} -48.7328 q^{53} -5.64447i q^{54} +4.06292i q^{55} +9.08548i q^{56} -13.4201 q^{57} -8.18418i q^{58} +(43.5656 + 39.7874i) q^{59} +16.2978 q^{60} +87.6284i q^{61} +8.38556 q^{62} -3.67912 q^{63} -23.0750 q^{64} +42.2277i q^{65} +2.29098 q^{66} -42.6674i q^{67} -2.29920 q^{68} -58.5457i q^{69} +4.44511i q^{70} -20.2245 q^{71} -22.2252i q^{72} +45.6107i q^{73} +18.2448 q^{74} -24.0172 q^{75} -21.8497 q^{76} -1.49328i q^{77} +23.8111 q^{78} -62.5528 q^{79} +10.7856 q^{80} +9.00000 q^{81} +20.6933i q^{82} +22.8390i q^{83} -5.99008 q^{84} -2.72049 q^{85} +53.1815 q^{86} +13.0495 q^{87} +9.02079 q^{88} +55.8982i q^{89} -10.8738i q^{90} -15.5203i q^{91} -95.3199i q^{92} +13.3706i q^{93} -5.20664 q^{94} -25.8533 q^{95} -57.4087i q^{96} -123.320i q^{97} +51.5939i q^{98} +3.65292i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20q - 40q^{4} - 8q^{7} + 60q^{9} + O(q^{10}) \)	\( 20q - 40q^{4} - 8q^{7} + 60q^{9} - 24q^{12} + 24q^{15} - 8q^{16} + 16q^{17} - 60q^{19} - 164q^{20} + 40q^{22} + 100q^{25} - 156q^{26} + 200q^{28} - 60q^{29} - 32q^{35} - 120q^{36} + 28q^{41} + 180q^{46} - 96q^{48} + 284q^{49} - 24q^{51} - 8q^{53} + 24q^{57} - 152q^{59} - 72q^{60} - 8q^{62} - 24q^{63} + 204q^{64} - 120q^{66} + 384q^{68} + 92q^{71} + 104q^{74} + 240q^{75} + 120q^{76} - 468q^{78} - 420q^{79} + 376q^{80} + 180q^{81} + 168q^{84} - 348q^{85} + 232q^{86} + 144q^{87} - 212q^{88} + 152q^{94} + 788q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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