LMFDB - Embedded newform 177.3.c.a.58.18

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			58.18
Root			$3.07256i$ of defining polynomial
Character	$\chi$	$=$	177.58
Dual form			177.3.c.a.58.3

$q$-expansion

$f(q)$	$=$	$q+3.07256i q^{2} +1.73205 q^{3} -5.44061 q^{4} -6.03179 q^{5} +5.32183i q^{6} -9.30133 q^{7} -4.42637i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+3.07256i q^{2} +1.73205 q^{3} -5.44061 q^{4} -6.03179 q^{5} +5.32183i q^{6} -9.30133 q^{7} -4.42637i q^{8} +3.00000 q^{9} -18.5330i q^{10} -12.6629i q^{11} -9.42342 q^{12} +16.7819i q^{13} -28.5789i q^{14} -10.4474 q^{15} -8.16217 q^{16} +14.0828 q^{17} +9.21767i q^{18} -37.4075 q^{19} +32.8166 q^{20} -16.1104 q^{21} +38.9074 q^{22} +27.8025i q^{23} -7.66670i q^{24} +11.3824 q^{25} -51.5635 q^{26} +5.19615 q^{27} +50.6050 q^{28} +39.1159 q^{29} -32.1001i q^{30} +0.257147i q^{31} -42.7842i q^{32} -21.9327i q^{33} +43.2703i q^{34} +56.1036 q^{35} -16.3218 q^{36} +54.7014i q^{37} -114.937i q^{38} +29.0672i q^{39} +26.6989i q^{40} -11.3202 q^{41} -49.5001i q^{42} +7.36421i q^{43} +68.8937i q^{44} -18.0954 q^{45} -85.4249 q^{46} +28.7287i q^{47} -14.1373 q^{48} +37.5148 q^{49} +34.9732i q^{50} +24.3922 q^{51} -91.3041i q^{52} -47.3053 q^{53} +15.9655i q^{54} +76.3796i q^{55} +41.1711i q^{56} -64.7917 q^{57} +120.186i q^{58} +(-28.6844 + 51.5578i) q^{59} +56.8401 q^{60} -84.0505i q^{61} -0.790099 q^{62} -27.9040 q^{63} +98.8084 q^{64} -101.225i q^{65} +67.3895 q^{66} -81.5123i q^{67} -76.6193 q^{68} +48.1554i q^{69} +172.382i q^{70} -6.96917 q^{71} -13.2791i q^{72} +131.681i q^{73} -168.073 q^{74} +19.7150 q^{75} +203.520 q^{76} +117.781i q^{77} -89.3106 q^{78} +51.9880 q^{79} +49.2325 q^{80} +9.00000 q^{81} -34.7820i q^{82} +16.5553i q^{83} +87.6504 q^{84} -84.9447 q^{85} -22.6270 q^{86} +67.7508 q^{87} -56.0505 q^{88} +69.2544i q^{89} -55.5990i q^{90} -156.094i q^{91} -151.263i q^{92} +0.445392i q^{93} -88.2705 q^{94} +225.634 q^{95} -74.1045i q^{96} -110.495i q^{97} +115.266i q^{98} -37.9886i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			3.07256i			1.53628i	0.640282	+	0.768140i	$0.278817\pi$
$2$			3.07256i			1.53628i	−0.640282	+	0.768140i	$0.721183\pi$
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$	−5.44061			−1.36015
$5$	−6.03179			−1.20636			−0.603179	−	0.797606i	$-0.706099\pi$
$5$	−6.03179			−1.20636			−0.603179	+	0.797606i	$0.706099\pi$
$6$			5.32183i			0.886971i
$7$	−9.30133			−1.32876			−0.664381	−	0.747394i	$-0.731305\pi$
$7$	−9.30133			−1.32876			−0.664381	+	0.747394i	$0.731305\pi$
$8$		−	4.42637i		−	0.553296i
$9$	3.00000			0.333333
$10$		−	18.5330i		−	1.85330i
$11$		−	12.6629i		−	1.15117i	−0.817742	−	0.575584i	$-0.804775\pi$
$11$		−	12.6629i		−	1.15117i	0.817742	−	0.575584i	$-0.195225\pi$
$12$	−9.42342			−0.785285
$13$			16.7819i			1.29092i	0.763795	+	0.645459i	$0.223334\pi$
$13$			16.7819i			1.29092i	−0.763795	+	0.645459i	$0.776666\pi$
$14$		−	28.5789i		−	2.04135i
$15$	−10.4474			−0.696491
$16$	−8.16217			−0.510136
$17$	14.0828			0.828402			0.414201	−	0.910185i	$-0.364061\pi$
$17$	14.0828			0.828402			0.414201	+	0.910185i	$0.364061\pi$
$18$			9.21767i			0.512093i
$19$	−37.4075			−1.96882			−0.984408	−	0.175902i	$-0.943716\pi$
$19$	−37.4075			−1.96882			−0.984408	+	0.175902i	$0.943716\pi$
$20$	32.8166			1.64083
$21$	−16.1104			−0.767161
$22$	38.9074			1.76852
$23$			27.8025i			1.20881i	0.796679	+	0.604403i	$0.206588\pi$
$23$			27.8025i			1.20881i	−0.796679	+	0.604403i	$0.793412\pi$
$24$		−	7.66670i		−	0.319446i
$25$	11.3824			0.455298
$26$	−51.5635			−1.98321
$27$	5.19615			0.192450
$28$	50.6050			1.80732
$29$	39.1159			1.34882			0.674412	−	0.738355i	$-0.264397\pi$
$29$	39.1159			1.34882			0.674412	+	0.738355i	$0.264397\pi$
$30$		−	32.1001i		−	1.07000i
$31$			0.257147i			0.00829506i	0.999991	+	0.00414753i	$0.00132020\pi$
$31$			0.257147i			0.00829506i	−0.999991	+	0.00414753i	$0.998680\pi$
$32$		−	42.7842i		−	1.33701i
$33$		−	21.9327i		−	0.664627i
$34$			43.2703i			1.27266i
$35$	56.1036			1.60296
$36$	−16.3218			−0.453385
$37$			54.7014i			1.47842i	0.673477	+	0.739208i	$0.264800\pi$
$37$			54.7014i			1.47842i	−0.673477	+	0.739208i	$0.735200\pi$
$38$		−	114.937i		−	3.02465i
$39$			29.0672i			0.745312i
$40$			26.6989i			0.667473i
$41$	−11.3202			−0.276102			−0.138051	−	0.990425i	$-0.544084\pi$
$41$	−11.3202			−0.276102			−0.138051	+	0.990425i	$0.544084\pi$
$42$		−	49.5001i		−	1.17857i
$43$			7.36421i			0.171261i	0.996327	+	0.0856303i	$0.0272904\pi$
$43$			7.36421i			0.171261i	−0.996327	+	0.0856303i	$0.972710\pi$
$44$			68.8937i			1.56577i
$45$	−18.0954			−0.402119
$46$	−85.4249			−1.85706
$47$			28.7287i			0.611248i	0.952152	+	0.305624i	$0.0988651\pi$
$47$			28.7287i			0.611248i	−0.952152	+	0.305624i	$0.901135\pi$
$48$	−14.1373			−0.294527
$49$	37.5148			0.765608
$50$			34.9732i			0.699464i
$51$	24.3922			0.478278
$52$		−	91.3041i		−	1.75585i
$53$	−47.3053			−0.892552			−0.446276	−	0.894895i	$-0.647250\pi$
$53$	−47.3053			−0.892552			−0.446276	+	0.894895i	$0.647250\pi$
$54$			15.9655i			0.295657i
$55$			76.3796i			1.38872i
$56$			41.1711i			0.735199i
$57$	−64.7917			−1.13670
$58$			120.186i			2.07217i
$59$	−28.6844	+	51.5578i	−0.486176	+	0.873861i
$59$	−28.6844	+	51.5578i	−0.486176	+	0.873861i
$60$	56.8401			0.947334
$61$		−	84.0505i		−	1.37788i	−0.724820	−	0.688938i	$-0.758077\pi$
$61$		−	84.0505i		−	1.37788i	0.724820	−	0.688938i	$-0.241923\pi$
$62$	−0.790099			−0.0127435
$63$	−27.9040			−0.442921
$64$	98.8084			1.54388
$65$		−	101.225i		−	1.55731i
$66$	67.3895			1.02105
$67$		−	81.5123i		−	1.21660i	−0.793707	−	0.608300i	$-0.791852\pi$
$67$		−	81.5123i		−	1.21660i	0.793707	−	0.608300i	$-0.208148\pi$
$68$	−76.6193			−1.12675
$69$			48.1554i			0.697905i
$70$			172.382i			2.46260i
$71$	−6.96917			−0.0981573			−0.0490787	−	0.998795i	$-0.515628\pi$
$71$	−6.96917			−0.0981573			−0.0490787	+	0.998795i	$0.515628\pi$
$72$		−	13.2791i		−	0.184432i
$73$			131.681i			1.80385i	0.431896	+	0.901923i	$0.357845\pi$
$73$			131.681i			1.80385i	−0.431896	+	0.901923i	$0.642155\pi$
$74$	−168.073			−2.27126
$75$	19.7150			0.262866
$76$	203.520			2.67789
$77$			117.781i			1.52963i
$78$	−89.3106			−1.14501
$79$	51.9880			0.658076			0.329038	−	0.944317i	$-0.393276\pi$
$79$	51.9880			0.658076			0.329038	+	0.944317i	$0.393276\pi$
$80$	49.2325			0.615406
$81$	9.00000			0.111111
$82$		−	34.7820i		−	0.424170i
$83$			16.5553i			0.199461i	0.995014	+	0.0997305i	$0.0317981\pi$
$83$			16.5553i			0.199461i	−0.995014	+	0.0997305i	$0.968202\pi$
$84$	87.6504			1.04346
$85$	−84.9447			−0.999349
$86$	−22.6270			−0.263104
$87$	67.7508			0.778744
$88$	−56.0505			−0.636937
$89$			69.2544i			0.778139i	0.921208	+	0.389070i	$0.127203\pi$
$89$			69.2544i			0.778139i	−0.921208	+	0.389070i	$0.872797\pi$
$90$		−	55.5990i		−	0.617767i
$91$		−	156.094i		−	1.71532i
$92$		−	151.263i		−	1.64416i
$93$			0.445392i			0.00478916i
$94$	−88.2705			−0.939048
$95$	225.634			2.37509
$96$		−	74.1045i		−	0.771922i
$97$		−	110.495i		−	1.13912i	−0.821949	−	0.569560i	$-0.807113\pi$
$97$		−	110.495i		−	1.13912i	0.821949	−	0.569560i	$-0.192887\pi$
$98$			115.266i			1.17619i
$99$		−	37.9886i		−	0.383723i
$100$	−61.9275			−0.619275
$101$		−	159.617i		−	1.58037i	−0.612868	−	0.790186i	$-0.709984\pi$
$101$		−	159.617i		−	1.58037i	0.612868	−	0.790186i	$-0.290016\pi$
$102$			74.9464i			0.734769i
$103$		−	26.7475i		−	0.259684i	−0.991535	−	0.129842i	$-0.958553\pi$
$103$		−	26.7475i		−	0.259684i	0.991535	−	0.129842i	$-0.0414470\pi$
$104$	74.2831			0.714260
$105$	97.1744			0.925470
$106$		−	145.348i		−	1.37121i
$107$	−35.3863			−0.330713			−0.165356	−	0.986234i	$-0.552877\pi$
$107$	−35.3863			−0.330713			−0.165356	+	0.986234i	$0.552877\pi$
$108$	−28.2703			−0.261762
$109$			68.2150i			0.625826i	0.949782	+	0.312913i	$0.101305\pi$
$109$			68.2150i			0.625826i	−0.949782	+	0.312913i	$0.898695\pi$
$110$	−234.681			−2.13346
$111$			94.7456i			0.853564i
$112$	75.9191			0.677849
$113$		−	120.908i		−	1.06998i	−0.844858	−	0.534990i	$-0.820315\pi$
$113$		−	120.908i		−	1.06998i	0.844858	−	0.534990i	$-0.179685\pi$
$114$		−	199.076i		−	1.74628i
$115$		−	167.699i		−	1.45825i
$116$	−212.815			−1.83461
$117$			50.3458i			0.430306i
$118$	−158.414	−	88.1345i	−1.34249	−	0.746902i
$119$	−130.989			−1.10075
$120$			46.2439i			0.385366i
$121$	−39.3479			−0.325189
$122$	258.250			2.11680
$123$	−19.6072			−0.159408
$124$		−	1.39904i		−	0.0112826i
$125$	82.1382			0.657105
$126$		−	85.7367i		−	0.680450i
$127$	−158.088			−1.24479			−0.622393	−	0.782705i	$-0.713839\pi$
$127$	−158.088			−1.24479			−0.622393	+	0.782705i	$0.713839\pi$
$128$			132.458i			1.03482i
$129$			12.7552i			0.0988774i
$130$	311.020			2.39246
$131$			130.513i			0.996282i	0.867096	+	0.498141i	$0.165984\pi$
$131$			130.513i			0.996282i	−0.867096	+	0.498141i	$0.834016\pi$
$132$			119.327i			0.903995i
$133$	347.939			2.61609
$134$	250.451			1.86904
$135$	−31.3421			−0.232164
$136$		−	62.3359i		−	0.458352i
$137$	208.921			1.52497			0.762487	−	0.647004i	$-0.223978\pi$
$137$	208.921			1.52497			0.762487	+	0.647004i	$0.223978\pi$
$138$	−147.960			−1.07218
$139$	−73.8315			−0.531162			−0.265581	−	0.964089i	$-0.585564\pi$
$139$	−73.8315			−0.531162			−0.265581	+	0.964089i	$0.585564\pi$
$140$	−305.238			−2.18027
$141$			49.7595i			0.352904i
$142$		−	21.4132i		−	0.150797i
$143$	212.507			1.48606
$144$	−24.4865			−0.170045
$145$	−235.939			−1.62716
$146$	−404.597			−2.77121
$147$	64.9775			0.442024
$148$		−	297.609i		−	2.01087i
$149$			66.3172i			0.445082i	0.974923	+	0.222541i	$0.0714351\pi$
$149$			66.3172i			0.445082i	−0.974923	+	0.222541i	$0.928565\pi$
$150$			60.5754i			0.403836i
$151$			103.185i			0.683347i	0.939819	+	0.341673i	$0.110994\pi$
$151$			103.185i			0.683347i	−0.939819	+	0.341673i	$0.889006\pi$
$152$			165.579i			1.08934i
$153$	42.2485			0.276134
$154$	−361.890			−2.34994
$155$		−	1.55106i		−	0.0100068i
$156$		−	158.143i		−	1.01374i
$157$			251.051i			1.59905i	0.600630	+	0.799527i	$0.294916\pi$
$157$			251.051i			1.59905i	−0.600630	+	0.799527i	$0.705084\pi$
$158$			159.736i			1.01099i
$159$	−81.9351			−0.515315
$160$			258.065i			1.61291i
$161$		−	258.601i		−	1.60622i
$162$			27.6530i			0.170698i
$163$	−96.4209			−0.591539			−0.295770	−	0.955259i	$-0.595576\pi$
$163$	−96.4209			−0.591539			−0.295770	+	0.955259i	$0.595576\pi$
$164$	61.5888			0.375542
$165$			132.293i			0.801778i
$166$	−50.8670			−0.306428
$167$	78.3907			0.469405			0.234703	−	0.972067i	$-0.424588\pi$
$167$	78.3907			0.469405			0.234703	+	0.972067i	$0.424588\pi$
$168$			71.3105i			0.424467i
$169$	−112.633			−0.666470
$170$		−	260.997i		−	1.53528i
$171$	−112.222			−0.656272
$172$		−	40.0658i		−	0.232941i
$173$			52.2561i			0.302058i	0.988529	+	0.151029i	$0.0482587\pi$
$173$			52.2561i			0.302058i	−0.988529	+	0.151029i	$0.951741\pi$
$174$			208.168i			1.19637i
$175$	−105.872			−0.604982
$176$			103.356i			0.587252i
$177$	−49.6828	+	89.3007i	−0.280694	+	0.504524i
$178$	−212.788			−1.19544
$179$		−	117.505i		−	0.656453i	−0.944599	−	0.328226i	$-0.893549\pi$
$179$		−	117.505i		−	0.656453i	0.944599	−	0.328226i	$-0.106451\pi$
$180$	98.4499			0.546944
$181$	160.180			0.884972			0.442486	−	0.896776i	$-0.354097\pi$
$181$	160.180			0.884972			0.442486	+	0.896776i	$0.354097\pi$
$182$	479.609			2.63521
$183$		−	145.580i		−	0.795517i
$184$	123.064			0.668828
$185$		−	329.947i		−	1.78350i
$186$	−1.36849			−0.00735748
$187$		−	178.329i		−	0.953630i
$188$		−	156.302i		−	0.831391i
$189$	−48.3311			−0.255720
$190$			693.274i			3.64881i
$191$		−	14.2938i		−	0.0748369i	−0.999300	−	0.0374184i	$-0.988087\pi$
$191$		−	14.2938i		−	0.0748369i	0.999300	−	0.0374184i	$-0.0119134\pi$
$192$	171.141			0.891360
$193$	71.9463			0.372779			0.186389	−	0.982476i	$-0.440321\pi$
$193$	71.9463			0.372779			0.186389	+	0.982476i	$0.440321\pi$
$194$	339.501			1.75001
$195$		−	175.327i		−	0.899113i
$196$	−204.103			−1.04134
$197$	−44.5489			−0.226137			−0.113068	−	0.993587i	$-0.536068\pi$
$197$	−44.5489			−0.226137			−0.113068	+	0.993587i	$0.536068\pi$
$198$	116.722			0.589505
$199$	−125.217			−0.629233			−0.314617	−	0.949219i	$-0.601876\pi$
$199$	−125.217			−0.629233			−0.314617	+	0.949219i	$0.601876\pi$
$200$		−	50.3829i		−	0.251915i
$201$		−	141.183i		−	0.702405i
$202$	490.434			2.42789
$203$	−363.830			−1.79227
$204$	−132.708			−0.650532
$205$	68.2810			0.333078
$206$	82.1831			0.398947
$207$			83.4076i			0.402935i
$208$		−	136.977i		−	0.658544i
$209$			473.686i			2.26644i
$210$			298.574i			1.42178i
$211$		−	106.781i		−	0.506071i	−0.967457	−	0.253035i	$-0.918571\pi$
$211$		−	106.781i		−	0.506071i	0.967457	−	0.253035i	$-0.0814289\pi$
$212$	257.370			1.21401
$213$	−12.0710			−0.0566711
$214$		−	108.726i		−	0.508067i
$215$		−	44.4193i		−	0.206602i
$216$		−	23.0001i		−	0.106482i
$217$		−	2.39181i		−	0.0110222i
$218$	−209.595			−0.961443
$219$			228.078i			1.04145i
$220$		−	415.552i		−	1.88887i
$221$			236.337i			1.06940i
$222$	−291.111			−1.31131
$223$	−37.6647			−0.168900			−0.0844501	−	0.996428i	$-0.526913\pi$
$223$	−37.6647			−0.168900			−0.0844501	+	0.996428i	$0.526913\pi$
$224$			397.950i			1.77656i
$225$	34.1473			0.151766
$226$	371.496			1.64379
$227$		−	189.636i		−	0.835399i	−0.908585	−	0.417699i	$-0.862837\pi$
$227$		−	189.636i		−	0.835399i	0.908585	−	0.417699i	$-0.137163\pi$
$228$	352.507			1.54608
$229$			206.938i			0.903658i	0.892104	+	0.451829i	$0.149228\pi$
$229$			206.938i			0.903658i	−0.892104	+	0.451829i	$0.850772\pi$
$230$	515.265			2.24028
$231$			204.003i			0.883131i
$232$		−	173.142i		−	0.746300i
$233$			213.272i			0.915330i	0.889125	+	0.457665i	$0.151314\pi$
$233$			213.272i			0.915330i	−0.889125	+	0.457665i	$0.848686\pi$
$234$	−154.690			−0.661070
$235$		−	173.285i		−	0.737384i
$236$	156.061	−	280.506i	0.661274	−	1.18858i
$237$	90.0458			0.379940
$238$		−	402.472i		−	1.69106i
$239$	−142.148			−0.594760			−0.297380	−	0.954759i	$-0.596113\pi$
$239$	−142.148			−0.594760			−0.297380	+	0.954759i	$0.596113\pi$
$240$	85.2732			0.355305
$241$	−391.267			−1.62351			−0.811756	−	0.583996i	$-0.801488\pi$
$241$	−391.267			−1.62351			−0.811756	+	0.583996i	$0.801488\pi$
$242$		−	120.899i		−	0.499581i
$243$	15.5885			0.0641500
$244$			457.286i			1.87412i
$245$	−226.281			−0.923596
$246$		−	60.2441i		−	0.244895i
$247$		−	627.770i		−	2.54158i
$248$	1.13823			0.00458963
$249$			28.6745i			0.115159i
$250$			252.374i			1.00950i
$251$	135.007			0.537877			0.268938	−	0.963157i	$-0.413327\pi$
$251$	135.007			0.537877			0.268938	+	0.963157i	$0.413327\pi$
$252$	151.815			0.602440
$253$	352.060			1.39154
$254$		−	485.734i		−	1.91234i
$255$	−147.128			−0.576974
$256$	−11.7500			−0.0458983
$257$	−133.333			−0.518807			−0.259403	−	0.965769i	$-0.583526\pi$
$257$	−133.333			−0.518807			−0.259403	+	0.965769i	$0.583526\pi$
$258$	−39.1910			−0.151903
$259$		−	508.796i		−	1.96446i
$260$			550.727i			2.11818i
$261$	117.348			0.449608
$262$	−401.009			−1.53057
$263$	272.345			1.03553			0.517765	−	0.855523i	$-0.326764\pi$
$263$	272.345			1.03553			0.517765	+	0.855523i	$0.326764\pi$
$264$	−97.0823			−0.367736
$265$	285.335			1.07674
$266$			1069.06i			4.01904i
$267$			119.952i			0.449259i
$268$			443.477i			1.65476i
$269$			275.151i			1.02287i	0.859323	+	0.511434i	$0.170886\pi$
$269$			275.151i			1.02287i	−0.859323	+	0.511434i	$0.829114\pi$
$270$		−	96.3004i		−	0.356668i
$271$	−142.359			−0.525308			−0.262654	−	0.964890i	$-0.584598\pi$
$271$	−142.359			−0.525308			−0.262654	+	0.964890i	$0.584598\pi$
$272$	−114.947			−0.422598
$273$		−	270.363i		−	0.990342i
$274$			641.923i			2.34278i
$275$		−	144.134i		−	0.524124i
$276$		−	261.995i		−	0.949257i
$277$	−282.049			−1.01823			−0.509114	−	0.860699i	$-0.670027\pi$
$277$	−282.049			−1.01823			−0.509114	+	0.860699i	$0.670027\pi$
$278$		−	226.852i		−	0.816013i
$279$			0.771441i			0.00276502i
$280$		−	248.336i		−	0.886913i
$281$	−273.179			−0.972168			−0.486084	−	0.873912i	$-0.661575\pi$
$281$	−273.179			−0.972168			−0.486084	+	0.873912i	$0.661575\pi$
$282$	−152.889			−0.542159
$283$			240.930i			0.851343i	0.904878	+	0.425671i	$0.139962\pi$
$283$			240.930i			0.851343i	−0.904878	+	0.425671i	$0.860038\pi$
$284$	37.9166			0.133509
$285$	390.810			1.37126
$286$			652.941i			2.28301i
$287$	105.293			0.366874
$288$		−	128.353i		−	0.445669i
$289$	−90.6737			−0.313750
$290$		−	724.936i		−	2.49978i
$291$		−	191.382i		−	0.657672i
$292$		−	716.424i		−	2.45351i
$293$	−321.271			−1.09649			−0.548245	−	0.836318i	$-0.684704\pi$
$293$	−321.271			−1.09649			−0.548245	+	0.836318i	$0.684704\pi$
$294$			199.647i			0.679072i
$295$	173.018	−	310.986i	0.586502	−	1.05419i
$296$	242.129			0.818002
$297$		−	65.7981i		−	0.221542i
$298$	−203.763			−0.683770
$299$	−466.581			−1.56047
$300$	−107.262			−0.357539
$301$		−	68.4969i		−	0.227565i
$302$	−317.043			−1.04981
$303$		−	276.466i		−	0.912428i
$304$	305.326			1.00436
$305$			506.974i			1.66221i
$306$			129.811i			0.424219i
$307$	−569.638			−1.85550			−0.927750	−	0.373203i	$-0.878259\pi$
$307$	−569.638			−1.85550			−0.927750	+	0.373203i	$0.878259\pi$
$308$		−	640.803i		−	2.08053i
$309$		−	46.3280i		−	0.149929i
$310$	4.76571			0.0153733
$311$	−357.447			−1.14935			−0.574673	−	0.818383i	$-0.694871\pi$
$311$	−357.447			−1.14935			−0.574673	+	0.818383i	$0.694871\pi$
$312$	128.662			0.412378
$313$			540.979i			1.72837i	0.503176	+	0.864184i	$0.332165\pi$
$313$			540.979i			1.72837i	−0.503176	+	0.864184i	$0.667835\pi$
$314$	−771.370			−2.45659
$315$	168.311			0.534320
$316$	−282.847			−0.895084
$317$	431.328			1.36066			0.680328	−	0.732908i	$-0.261837\pi$
$317$	431.328			1.36066			0.680328	+	0.732908i	$0.261837\pi$
$318$		−	251.750i		−	0.791668i
$319$		−	495.319i		−	1.55272i
$320$	−595.991			−1.86247
$321$	−61.2908			−0.190937
$322$	794.566			2.46760
$323$	−526.804			−1.63097
$324$	−48.9655			−0.151128
$325$			191.019i			0.587752i
$326$		−	296.259i		−	0.908770i
$327$			118.152i			0.361321i
$328$			50.1074i			0.152766i
$329$		−	267.215i		−	0.812203i
$330$	−406.479			−1.23176
$331$	241.399			0.729301			0.364650	−	0.931145i	$-0.381189\pi$
$331$	241.399			0.729301			0.364650	+	0.931145i	$0.381189\pi$
$332$		−	90.0708i		−	0.271297i
$333$			164.104i			0.492805i
$334$			240.860i			0.721138i
$335$			491.664i			1.46766i
$336$	131.496			0.391356
$337$		−	122.088i		−	0.362278i	−0.983457	−	0.181139i	$-0.942022\pi$
$337$		−	122.088i		−	0.362278i	0.983457	−	0.181139i	$-0.0579784\pi$
$338$		−	346.073i		−	1.02388i
$339$		−	209.419i		−	0.617754i
$340$	462.151			1.35927
$341$	3.25621			0.00954901
$342$		−	344.810i		−	1.00822i
$343$	106.828			0.311452
$344$	32.5967			0.0947579
$345$		−	290.463i		−	0.841922i
$346$	−160.560			−0.464046
$347$		−	403.930i		−	1.16406i	−0.813166	−	0.582032i	$-0.802258\pi$
$347$		−	403.930i		−	1.16406i	0.813166	−	0.582032i	$-0.197742\pi$
$348$	−368.606			−1.05921
$349$			449.445i			1.28781i	0.765106	+	0.643904i	$0.222686\pi$
$349$			449.445i			1.28781i	−0.765106	+	0.643904i	$0.777314\pi$
$350$		−	325.298i		−	0.929422i
$351$			87.2015i			0.248437i
$352$	−541.771			−1.53912
$353$			213.176i			0.603899i	0.953324	+	0.301950i	$0.0976374\pi$
$353$			213.176i			0.603899i	−0.953324	+	0.301950i	$0.902363\pi$
$354$	−274.382	−	152.653i	−0.775089	−	0.431224i
$355$	42.0365			0.118413
$356$		−	376.786i		−	1.05839i
$357$	−226.880			−0.635518
$358$	361.041			1.00849
$359$	405.372			1.12917			0.564585	−	0.825375i	$-0.309036\pi$
$359$	405.372			1.12917			0.564585	+	0.825375i	$0.309036\pi$
$360$			80.0968i			0.222491i
$361$	1038.32			2.87623
$362$			492.162i			1.35956i
$363$	−68.1525			−0.187748
$364$			849.249i			2.33310i
$365$		−	794.270i		−	2.17608i
$366$	447.302			1.22214
$367$			541.192i			1.47464i	0.675546	+	0.737318i	$0.263908\pi$
$367$			541.192i			1.47464i	−0.675546	+	0.737318i	$0.736092\pi$
$368$		−	226.929i		−	0.616655i
$369$	−33.9606			−0.0920341
$370$	1013.78			2.73995
$371$	440.002			1.18599
$372$		−	2.42320i		−	0.00651399i
$373$	−107.068			−0.287046			−0.143523	−	0.989647i	$-0.545843\pi$
$373$	−107.068			−0.287046			−0.143523	+	0.989647i	$0.545843\pi$
$374$	547.926			1.46504
$375$	142.268			0.379380
$376$	127.164			0.338201
$377$			656.441i			1.74122i
$378$		−	148.500i		−	0.392858i
$379$	−204.702			−0.540112			−0.270056	−	0.962845i	$-0.587042\pi$
$379$	−204.702			−0.540112			−0.270056	+	0.962845i	$0.587042\pi$
$380$	−1227.59			−3.23049
$381$	−273.816			−0.718678
$382$	43.9187			0.114970
$383$	−4.98048			−0.0130039			−0.00650193	−	0.999979i	$-0.502070\pi$
$383$	−4.98048			−0.0130039			−0.00650193	+	0.999979i	$0.502070\pi$
$384$			229.423i			0.597456i
$385$		−	710.432i		−	1.84528i
$386$			221.059i			0.572692i
$387$			22.0926i			0.0570869i
$388$			601.159i			1.54938i
$389$	138.904			0.357079			0.178539	−	0.983933i	$-0.442863\pi$
$389$	138.904			0.357079			0.178539	+	0.983933i	$0.442863\pi$
$390$	538.702			1.38129
$391$			391.539i			1.00138i
$392$		−	166.054i		−	0.423608i
$393$			226.055i			0.575204i
$394$		−	136.879i		−	0.347409i
$395$	−313.580			−0.793874
$396$			206.681i			0.521922i
$397$		−	266.530i		−	0.671359i	−0.941976	−	0.335679i	$-0.891034\pi$
$397$		−	266.530i		−	0.671359i	0.941976	−	0.335679i	$-0.108966\pi$
$398$		−	384.738i		−	0.966678i
$399$	602.649			1.51040
$400$	−92.9055			−0.232264
$401$			515.725i			1.28610i	0.765825	+	0.643049i	$0.222331\pi$
$401$			515.725i			1.28610i	−0.765825	+	0.643049i	$0.777669\pi$
$402$	433.794			1.07909
$403$	−4.31542			−0.0107082
$404$			868.417i			2.14955i
$405$	−54.2861			−0.134040
$406$		−	1117.89i		−	2.75342i
$407$	692.676			1.70191
$408$		−	107.969i		−	0.264630i
$409$			53.2453i			0.130184i	0.997879	+	0.0650920i	$0.0207341\pi$
$409$			53.2453i			0.130184i	−0.997879	+	0.0650920i	$0.979266\pi$
$410$			209.797i			0.511701i
$411$	361.862			0.880444
$412$			145.523i			0.353210i
$413$	266.803	−	479.556i	0.646012	−	1.16115i
$414$	−256.275			−0.619021
$415$		−	99.8578i		−	0.240621i
$416$	718.002			1.72597
$417$	−127.880			−0.306667
$418$	−1455.43			−3.48188
$419$		−	275.166i		−	0.656721i	−0.944553	−	0.328360i	$-0.893504\pi$
$419$		−	275.166i		−	0.656721i	0.944553	−	0.328360i	$-0.106496\pi$
$420$	−528.688			−1.25878
$421$		−	508.314i		−	1.20740i	−0.797213	−	0.603699i	$-0.793693\pi$
$421$		−	508.314i		−	1.20740i	0.797213	−	0.603699i	$-0.206307\pi$
$422$	328.091			0.777466
$423$			86.1860i			0.203749i
$424$			209.391i			0.493846i
$425$	160.297			0.377170
$426$		−	37.0887i		−	0.0870627i
$427$			781.781i			1.83087i
$428$	192.523			0.449820
$429$	368.073			0.857980
$430$	136.481			0.317398
$431$			331.069i			0.768141i	0.923304	+	0.384070i	$0.125478\pi$
$431$			331.069i			0.768141i	−0.923304	+	0.384070i	$0.874522\pi$
$432$	−42.4119			−0.0981757
$433$	216.406			0.499783			0.249891	−	0.968274i	$-0.419605\pi$
$433$	216.406			0.499783			0.249891	+	0.968274i	$0.419605\pi$
$434$	7.34897			0.0169331
$435$	−408.658			−0.939444
$436$		−	371.132i		−	0.851219i
$437$		−	1040.02i		−	2.37992i
$438$	−700.782			−1.59996
$439$	594.843			1.35500			0.677498	−	0.735525i	$-0.263064\pi$
$439$	594.843			1.35500			0.677498	+	0.735525i	$0.263064\pi$
$440$	338.085			0.768374
$441$	112.544			0.255203
$442$	−726.160			−1.64290
$443$		−	101.223i		−	0.228495i	−0.993452	−	0.114247i	$-0.963554\pi$
$443$		−	101.223i		−	0.228495i	0.993452	−	0.114247i	$-0.0364456\pi$
$444$		−	515.474i		−	1.16098i
$445$		−	417.728i		−	0.938714i
$446$		−	115.727i		−	0.259478i
$447$			114.865i			0.256968i
$448$	−919.049			−2.05145
$449$	612.041			1.36312			0.681561	−	0.731762i	$-0.261302\pi$
$449$	612.041			1.36312			0.681561	+	0.731762i	$0.261302\pi$
$450$			104.920i			0.233155i
$451$			143.346i			0.317840i
$452$			657.813i			1.45534i
$453$			178.722i			0.394530i
$454$	582.666			1.28341
$455$			941.528i			2.06929i
$456$			286.792i			0.628930i
$457$		−	406.851i		−	0.890264i	−0.895465	−	0.445132i	$-0.853157\pi$
$457$		−	406.851i		−	0.890264i	0.895465	−	0.445132i	$-0.146843\pi$
$458$	−635.828			−1.38827
$459$	73.1766			0.159426
$460$			912.385i			1.98345i
$461$	−852.748			−1.84978			−0.924889	−	0.380236i	$-0.875843\pi$
$461$	−852.748			−1.84978			−0.924889	+	0.380236i	$0.875843\pi$
$462$	−626.812			−1.35674
$463$			727.640i			1.57158i	0.618495	+	0.785788i	$0.287743\pi$
$463$			727.640i			1.57158i	−0.618495	+	0.785788i	$0.712257\pi$
$464$	−319.271			−0.688084
$465$		−	2.68651i		−	0.00577743i
$466$	−655.290			−1.40620
$467$		−	24.1720i		−	0.0517602i	−0.999665	−	0.0258801i	$-0.991761\pi$
$467$		−	24.1720i		−	0.0517602i	0.999665	−	0.0258801i	$-0.00823880\pi$
$468$		−	273.912i		−	0.585282i
$469$			758.173i			1.61657i
$470$	532.429			1.13283
$471$			434.834i			0.923214i
$472$	228.214	+	126.968i	0.483504	+	0.268999i
$473$	93.2519			0.197150
$474$			276.671i			0.583694i
$475$	−425.789			−0.896397
$476$	712.661			1.49719
$477$	−141.916			−0.297517
$478$		−	436.757i		−	0.913717i
$479$	−540.528			−1.12845			−0.564225	−	0.825621i	$-0.690825\pi$
$479$	−540.528			−1.12845			−0.564225	+	0.825621i	$0.690825\pi$
$480$			446.982i			0.931213i
$481$	−917.996			−1.90851
$482$		−	1202.19i		−	2.49417i
$483$		−	447.909i		−	0.927349i
$484$	214.077			0.442307
$485$			666.481i			1.37419i
$486$			47.8964i			0.0985524i
$487$	−564.800			−1.15975			−0.579877	−	0.814704i	$-0.696899\pi$
$487$	−564.800			−1.15975			−0.579877	+	0.814704i	$0.696899\pi$
$488$	−372.038			−0.762374
$489$	−167.006			−0.341525
$490$		−	695.262i		−	1.41890i
$491$	8.09712			0.0164911			0.00824554	−	0.999966i	$-0.497375\pi$
$491$	8.09712			0.0164911			0.00824554	+	0.999966i	$0.497375\pi$
$492$	106.675			0.216819
$493$	550.863			1.11737
$494$	1928.86			3.90458
$495$			229.139i			0.462907i
$496$		−	2.09888i		−	0.00423161i
$497$	64.8226			0.130428
$498$	−88.1042			−0.176916
$499$	719.073			1.44103			0.720514	−	0.693440i	$-0.243906\pi$
$499$	719.073			1.44103			0.720514	+	0.693440i	$0.243906\pi$
$500$	−446.882			−0.893764
$501$	135.777			0.271011
$502$			414.817i			0.826329i
$503$		−	524.427i		−	1.04260i	−0.853374	−	0.521299i	$-0.825448\pi$
$503$		−	524.427i		−	1.04260i	0.853374	−	0.521299i	$-0.174552\pi$
$504$			123.513i			0.245066i
$505$			962.778i			1.90649i
$506$			1081.72i			2.13779i
$507$	−195.087			−0.384787
$508$	860.095			1.69310
$509$			749.520i			1.47253i	0.676691	+	0.736267i	$0.263413\pi$
$509$			749.520i			1.47253i	−0.676691	+	0.736267i	$0.736587\pi$
$510$		−	452.061i		−	0.886394i
$511$		−	1224.81i		−	2.39688i
$512$			493.728i			0.964312i
$513$	−194.375			−0.378899
$514$		−	409.674i		−	0.797032i
$515$			161.335i			0.313272i
$516$		−	69.3960i		−	0.134488i
$517$	363.787			0.703650
$518$	1563.31			3.01796
$519$			90.5102i			0.174393i
$520$	−448.060			−0.861653
$521$	−191.543			−0.367646			−0.183823	−	0.982959i	$-0.558847\pi$
$521$	−191.543			−0.367646			−0.183823	+	0.982959i	$0.558847\pi$
$522$			360.558i			0.690724i
$523$	985.876			1.88504			0.942520	−	0.334149i	$-0.108449\pi$
$523$	985.876			1.88504			0.942520	+	0.334149i	$0.108449\pi$
$524$		−	710.071i		−	1.35510i
$525$	−183.375			−0.349287
$526$			836.795i			1.59086i
$527$			3.62136i			0.00687165i
$528$			179.019i			0.339050i
$529$	−243.981			−0.461212
$530$			876.709i			1.65417i
$531$	−86.0532	+	154.673i	−0.162059	+	0.291287i
$532$	−1893.00			−3.55828
$533$		−	189.975i		−	0.356426i
$534$	−368.560			−0.690187
$535$	213.443			0.398958
$536$	−360.804			−0.673141
$537$		−	203.525i		−	0.379003i
$538$	−845.419			−1.57141
$539$		−	475.044i		−	0.881343i
$540$	170.520			0.315778
$541$		−	239.083i		−	0.441928i	−0.975282	−	0.220964i	$-0.929080\pi$
$541$		−	239.083i		−	0.441928i	0.975282	−	0.220964i	$-0.0709204\pi$
$542$		−	437.405i		−	0.807020i
$543$	277.440			0.510939
$544$		−	602.523i		−	1.10758i
$545$		−	411.458i		−	0.754969i
$546$	830.707			1.52144
$547$	106.125			0.194013			0.0970065	−	0.995284i	$-0.469073\pi$
$547$	106.125			0.194013			0.0970065	+	0.995284i	$0.469073\pi$
$548$	−1136.66			−2.07420
$549$		−	252.151i		−	0.459292i
$550$	442.861			0.805201
$551$	−1463.23			−2.65559
$552$	213.154			0.386148
$553$	−483.557			−0.874426
$554$		−	866.613i		−	1.56428i
$555$		−	571.485i		−	1.02970i
$556$	401.689			0.722462
$557$	8.64276			0.0155166			0.00775831	−	0.999970i	$-0.497530\pi$
$557$	8.64276			0.0155166			0.00775831	+	0.999970i	$0.497530\pi$
$558$	−2.37030			−0.00424784
$559$	−123.586			−0.221084
$560$	−457.928			−0.817728
$561$		−	308.875i		−	0.550579i
$562$		−	839.359i		−	1.49352i
$563$		−	327.402i		−	0.581530i	−0.956794	−	0.290765i	$-0.906090\pi$
$563$		−	327.402i		−	0.581530i	0.956794	−	0.290765i	$-0.0939099\pi$
$564$		−	270.722i		−	0.480004i
$565$			729.290i			1.29078i
$566$	−740.272			−1.30790
$567$	−83.7120			−0.147640
$568$			30.8481i			0.0543101i
$569$		−	616.454i		−	1.08340i	−0.840572	−	0.541700i	$-0.817781\pi$
$569$		−	616.454i		−	1.08340i	0.840572	−	0.541700i	$-0.182219\pi$
$570$			1200.79i			2.10664i
$571$			518.165i			0.907470i	0.891137	+	0.453735i	$0.149909\pi$
$571$			518.165i			0.907470i	−0.891137	+	0.453735i	$0.850091\pi$
$572$	−1156.17			−2.02128
$573$		−	24.7577i		−	0.0432071i
$574$			323.519i			0.563621i
$575$			316.461i			0.550367i
$576$	296.425			0.514627
$577$	787.940			1.36558			0.682790	−	0.730614i	$-0.260766\pi$
$577$	787.940			1.36558			0.682790	+	0.730614i	$0.260766\pi$
$578$		−	278.600i		−	0.482007i
$579$	124.615			0.215224
$580$	1283.65			2.21319
$581$		−	153.986i		−	0.265036i
$582$	588.034			1.01037
$583$			599.020i			1.02748i
$584$	582.868			0.998062
$585$		−	303.675i		−	0.519103i
$586$		−	987.125i		−	1.68451i
$587$		−	782.879i		−	1.33370i	−0.745194	−	0.666848i	$-0.767643\pi$
$587$		−	782.879i		−	1.33370i	0.745194	−	0.666848i	$-0.232357\pi$
$588$	−353.517			−0.601220
$589$		−	9.61922i		−	0.0163314i
$590$	955.521	+	531.608i	1.61953	+	0.901031i
$591$	−77.1610			−0.130560
$592$		−	446.482i		−	0.754193i
$593$	1037.25			1.74916			0.874580	−	0.484881i	$-0.161137\pi$
$593$	1037.25			1.74916			0.874580	+	0.484881i	$0.161137\pi$
$594$	202.169			0.340351
$595$	790.098			1.32790
$596$		−	360.806i		−	0.605380i
$597$	−216.883			−0.363288
$598$		−	1433.60i		−	2.39732i
$599$	673.671			1.12466			0.562330	−	0.826913i	$-0.309905\pi$
$599$	673.671			1.12466			0.562330	+	0.826913i	$0.309905\pi$
$600$		−	87.2658i		−	0.145443i
$601$			962.944i			1.60224i	0.598506	+	0.801118i	$0.295761\pi$
$601$			962.944i			1.60224i	−0.598506	+	0.801118i	$0.704239\pi$
$602$	210.461			0.349603
$603$		−	244.537i		−	0.405534i
$604$		−	561.392i		−	0.929457i
$605$	237.338			0.392294
$606$	849.457			1.40174
$607$	−790.355			−1.30207			−0.651034	−	0.759049i	$-0.725664\pi$
$607$	−790.355			−1.30207			−0.651034	+	0.759049i	$0.725664\pi$
$608$			1600.45i			2.63232i
$609$	−630.172			−1.03477
$610$	−1557.71			−2.55362
$611$	−482.123			−0.789071
$612$	−229.858			−0.375585
$613$		−	953.011i		−	1.55467i	−0.629088	−	0.777334i	$-0.716572\pi$
$613$		−	953.011i		−	1.55467i	0.629088	−	0.777334i	$-0.283428\pi$
$614$		−	1750.25i		−	2.85056i
$615$	118.266			0.192303
$616$	521.344			0.846338
$617$	419.167			0.679364			0.339682	−	0.940540i	$-0.389681\pi$
$617$	419.167			0.679364			0.339682	+	0.940540i	$0.389681\pi$
$618$	142.345			0.230332
$619$	−209.634			−0.338666			−0.169333	−	0.985559i	$-0.554161\pi$
$619$	−209.634			−0.338666			−0.169333	+	0.985559i	$0.554161\pi$
$620$			8.43869i			0.0136108i
$621$			144.466i			0.232635i
$622$		−	1098.28i		−	1.76572i
$623$		−	644.158i		−	1.03396i
$624$		−	237.251i		−	0.380210i
$625$	−780.001			−1.24800
$626$	−1662.19			−2.65526
$627$			820.448i			1.30853i
$628$		−	1365.87i		−	2.17496i
$629$			770.351i			1.22472i
$630$			517.145i			0.820865i
$631$	−612.575			−0.970800			−0.485400	−	0.874292i	$-0.661326\pi$
$631$	−612.575			−0.970800			−0.485400	+	0.874292i	$0.661326\pi$
$632$		−	230.118i		−	0.364111i
$633$		−	184.950i		−	0.292180i
$634$			1325.28i			2.09035i
$635$	953.552			1.50166
$636$	445.777			0.700908
$637$			629.571i			0.988337i
$638$	1521.90			2.38542
$639$	−20.9075			−0.0327191
$640$		−	798.955i		−	1.24837i
$641$	198.748			0.310060			0.155030	−	0.987910i	$-0.450453\pi$
$641$	198.748			0.310060			0.155030	+	0.987910i	$0.450453\pi$
$642$		−	188.320i		−	0.293333i
$643$	−157.384			−0.244766			−0.122383	−	0.992483i	$-0.539054\pi$
$643$	−157.384			−0.244766			−0.122383	+	0.992483i	$0.539054\pi$
$644$			1406.95i			2.18470i
$645$		−	76.9365i		−	0.119281i
$646$		−	1618.63i		−	2.50563i
$647$	1214.33			1.87686			0.938430	−	0.345470i	$-0.112280\pi$
$647$	1214.33			1.87686			0.938430	+	0.345470i	$0.112280\pi$
$648$		−	39.8373i		−	0.0614774i
$649$	652.869	+	363.226i	1.00596	+	0.559671i
$650$	−586.918			−0.902951
$651$		−	4.14273i		−	0.00636365i
$652$	524.589			0.804584
$653$	−404.646			−0.619673			−0.309836	−	0.950790i	$-0.600274\pi$
$653$	−404.646			−0.619673			−0.309836	+	0.950790i	$0.600274\pi$
$654$	−363.028			−0.555089
$655$		−	787.226i		−	1.20187i
$656$	92.3974			0.140850
$657$			395.042i			0.601282i
$658$	821.033			1.24777
$659$			1245.68i			1.89025i	0.326703	+	0.945127i	$0.394062\pi$
$659$			1245.68i			1.89025i	−0.326703	+	0.945127i	$0.605938\pi$
$660$		−	719.757i		−	1.09054i
$661$	96.3638			0.145785			0.0728924	−	0.997340i	$-0.476777\pi$
$661$	96.3638			0.145785			0.0728924	+	0.997340i	$0.476777\pi$
$662$			741.711i			1.12041i
$663$			409.348i			0.617418i
$664$	73.2797			0.110361
$665$	−2098.70			−3.15593
$666$	−504.220			−0.757087
$667$			1087.52i			1.63047i
$668$	−426.493			−0.638463
$669$	−65.2372			−0.0975146
$670$	−1510.67			−2.25473
$671$	−1064.32			−1.58617
$672$			689.270i			1.02570i
$673$			786.566i			1.16875i	0.811485	+	0.584373i	$0.198660\pi$
$673$			786.566i			1.16875i	−0.811485	+	0.584373i	$0.801340\pi$
$674$	375.122			0.556561
$675$	59.1449			0.0876221
$676$	612.795			0.906502
$677$	−797.643			−1.17820			−0.589101	−	0.808059i	$-0.700518\pi$
$677$	−797.643			−1.17820			−0.589101	+	0.808059i	$0.700518\pi$
$678$	643.451			0.949042
$679$			1027.75i			1.51362i
$680$			375.997i			0.552936i
$681$		−	328.458i		−	0.482318i
$682$			10.0049i			0.0146700i
$683$		−	743.492i		−	1.08857i	−0.838901	−	0.544284i	$-0.816801\pi$
$683$		−	743.492i		−	1.08857i	0.838901	−	0.544284i	$-0.183199\pi$
$684$	610.559			0.892630
$685$	−1260.17			−1.83966
$686$			328.235i			0.478477i
$687$			358.427i			0.521727i
$688$		−	60.1079i		−	0.0873662i
$689$		−	793.874i		−	1.15221i
$690$	892.465			1.29343
$691$		−	652.050i		−	0.943632i	−0.881697	−	0.471816i	$-0.843599\pi$
$691$		−	652.050i		−	0.943632i	0.881697	−	0.471816i	$-0.156401\pi$
$692$		−	284.305i		−	0.410845i
$693$			353.344i			0.509876i
$694$	1241.10			1.78833
$695$	445.336			0.640771
$696$		−	299.890i		−	0.430877i
$697$	−159.420			−0.228724
$698$	−1380.95			−1.97843
$699$			369.398i			0.528466i
$700$	576.008			0.822869
$701$			24.8059i			0.0353864i	0.999843	+	0.0176932i	$0.00563221\pi$
$701$			24.8059i			0.0353864i	−0.999843	+	0.0176932i	$0.994368\pi$
$702$	−267.932			−0.381669
$703$		−	2046.24i		−	2.91073i
$704$		−	1251.20i		−	1.77727i
$705$		−	300.139i		−	0.425729i
$706$	−654.997			−0.927758
$707$			1484.66i			2.09994i
$708$	270.305	−	485.851i	0.381787	−	0.686230i
$709$	521.844			0.736028			0.368014	−	0.929820i	$-0.380038\pi$
$709$	521.844			0.736028			0.368014	+	0.929820i	$0.380038\pi$
$710$			129.160i			0.181915i
$711$	155.964			0.219359
$712$	306.546			0.430542
$713$	−7.14934			−0.0100271
$714$		−	697.102i		−	0.976333i
$715$	−1281.80			−1.79272
$716$			639.299i			0.892876i
$717$	−246.207			−0.343385
$718$			1245.53i			1.73472i
$719$			997.450i			1.38727i	0.720325	+	0.693637i	$0.243993\pi$
$719$			997.450i			1.38727i	−0.720325	+	0.693637i	$0.756007\pi$
$720$	147.697			0.205135
$721$			248.787i			0.345058i
$722$			3190.30i			4.41870i
$723$	−677.693			−0.937335
$724$	−871.477			−1.20370
$725$	445.235			0.614117
$726$		−	209.403i		−	0.288433i
$727$	−570.949			−0.785349			−0.392675	−	0.919677i	$-0.628450\pi$
$727$	−570.949			−0.785349			−0.392675	+	0.919677i	$0.628450\pi$
$728$	−690.932			−0.949082
$729$	27.0000			0.0370370
$730$	2440.44			3.34307
$731$			103.709i			0.141873i
$732$			792.043i			1.08203i
$733$	−337.572			−0.460535			−0.230268	−	0.973127i	$-0.573960\pi$
$733$	−337.572			−0.460535			−0.230268	+	0.973127i	$0.573960\pi$
$734$	−1662.84			−2.26545
$735$	−391.930			−0.533239
$736$	1189.51			1.61618
$737$	−1032.18			−1.40051
$738$		−	104.346i		−	0.141390i
$739$		−	1041.04i		−	1.40872i	−0.709843	−	0.704360i	$-0.751234\pi$
$739$		−	1041.04i		−	1.40872i	0.709843	−	0.704360i	$-0.248766\pi$
$740$			1795.12i			2.42583i
$741$		−	1087.33i		−	1.46738i
$742$			1351.93i			1.82201i
$743$	955.497			1.28600			0.643000	−	0.765867i	$-0.277690\pi$
$743$	955.497			1.28600			0.643000	+	0.765867i	$0.277690\pi$
$744$	1.97147			0.00264982
$745$		−	400.011i		−	0.536928i
$746$		−	328.974i		−	0.440983i
$747$			49.6658i			0.0664870i
$748$			970.219i			1.29708i
$749$	329.140			0.439439
$750$			437.125i			0.582834i
$751$		−	450.913i		−	0.600416i	−0.953874	−	0.300208i	$-0.902944\pi$
$751$		−	450.913i		−	0.600416i	0.953874	−	0.300208i	$-0.0970561\pi$
$752$		−	234.488i		−	0.311820i
$753$	233.839			0.310543
$754$	−2016.95			−2.67500
$755$		−	622.392i		−	0.824360i
$756$	262.951			0.347819
$757$	410.015			0.541632			0.270816	−	0.962631i	$-0.412707\pi$
$757$	410.015			0.541632			0.270816	+	0.962631i	$0.412707\pi$
$758$		−	628.960i		−	0.829763i
$759$	609.785			0.803406
$760$		−	998.740i		−	1.31413i
$761$	1469.25			1.93068			0.965339	−	0.260998i	$-0.0840517\pi$
$761$	1469.25			1.93068			0.965339	+	0.260998i	$0.0840517\pi$
$762$		−	841.316i		−	1.10409i
$763$		−	634.490i		−	0.831573i
$764$			77.7673i			0.101790i
$765$	−254.834			−0.333116
$766$		−	15.3028i		−	0.0199776i
$767$	−865.240	−	481.380i	−1.12808	−	0.627614i
$768$	−20.3515			−0.0264994
$769$		−	469.742i		−	0.610848i	−0.952216	−	0.305424i	$-0.901202\pi$
$769$		−	469.742i		−	0.610848i	0.952216	−	0.305424i	$-0.0987982\pi$
$770$	2182.84			2.83486
$771$	−230.940			−0.299533
$772$	−391.432			−0.507037
$773$		−	373.454i		−	0.483122i	−0.970386	−	0.241561i	$-0.922341\pi$
$773$		−	373.454i		−	0.483122i	0.970386	−	0.241561i	$-0.0776595\pi$
$774$	−67.8809			−0.0877014
$775$			2.92696i			0.00377672i
$776$	−489.091			−0.630271
$777$		−	881.260i		−	1.13418i
$778$			426.789i			0.548572i
$779$	423.460			0.543594
$780$			953.886i			1.22293i
$781$			88.2496i			0.112996i
$782$	−1203.03			−1.53840
$783$	203.252			0.259581
$784$	−306.202			−0.390564
$785$		−	1514.29i		−	1.92903i
$786$	−694.567			−0.883674
$787$	−97.7362			−0.124188			−0.0620942	−	0.998070i	$-0.519778\pi$
$787$	−97.7362			−0.124188			−0.0620942	+	0.998070i	$0.519778\pi$
$788$	242.374			0.307581
$789$	471.715			0.597864
$790$		−	963.494i		−	1.21961i
$791$			1124.60i			1.42175i
$792$	−168.151

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)

\(f(q)\)	\(=\)	\(q+3.07256i q^{2} +1.73205 q^{3} -5.44061 q^{4} -6.03179 q^{5} +5.32183i q^{6} -9.30133 q^{7} -4.42637i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+3.07256i q^{2} +1.73205 q^{3} -5.44061 q^{4} -6.03179 q^{5} +5.32183i q^{6} -9.30133 q^{7} -4.42637i q^{8} +3.00000 q^{9} -18.5330i q^{10} -12.6629i q^{11} -9.42342 q^{12} +16.7819i q^{13} -28.5789i q^{14} -10.4474 q^{15} -8.16217 q^{16} +14.0828 q^{17} +9.21767i q^{18} -37.4075 q^{19} +32.8166 q^{20} -16.1104 q^{21} +38.9074 q^{22} +27.8025i q^{23} -7.66670i q^{24} +11.3824 q^{25} -51.5635 q^{26} +5.19615 q^{27} +50.6050 q^{28} +39.1159 q^{29} -32.1001i q^{30} +0.257147i q^{31} -42.7842i q^{32} -21.9327i q^{33} +43.2703i q^{34} +56.1036 q^{35} -16.3218 q^{36} +54.7014i q^{37} -114.937i q^{38} +29.0672i q^{39} +26.6989i q^{40} -11.3202 q^{41} -49.5001i q^{42} +7.36421i q^{43} +68.8937i q^{44} -18.0954 q^{45} -85.4249 q^{46} +28.7287i q^{47} -14.1373 q^{48} +37.5148 q^{49} +34.9732i q^{50} +24.3922 q^{51} -91.3041i q^{52} -47.3053 q^{53} +15.9655i q^{54} +76.3796i q^{55} +41.1711i q^{56} -64.7917 q^{57} +120.186i q^{58} +(-28.6844 + 51.5578i) q^{59} +56.8401 q^{60} -84.0505i q^{61} -0.790099 q^{62} -27.9040 q^{63} +98.8084 q^{64} -101.225i q^{65} +67.3895 q^{66} -81.5123i q^{67} -76.6193 q^{68} +48.1554i q^{69} +172.382i q^{70} -6.96917 q^{71} -13.2791i q^{72} +131.681i q^{73} -168.073 q^{74} +19.7150 q^{75} +203.520 q^{76} +117.781i q^{77} -89.3106 q^{78} +51.9880 q^{79} +49.2325 q^{80} +9.00000 q^{81} -34.7820i q^{82} +16.5553i q^{83} +87.6504 q^{84} -84.9447 q^{85} -22.6270 q^{86} +67.7508 q^{87} -56.0505 q^{88} +69.2544i q^{89} -55.5990i q^{90} -156.094i q^{91} -151.263i q^{92} +0.445392i q^{93} -88.2705 q^{94} +225.634 q^{95} -74.1045i q^{96} -110.495i q^{97} +115.266i q^{98} -37.9886i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20q - 40q^{4} - 8q^{7} + 60q^{9} + O(q^{10}) \)	\( 20q - 40q^{4} - 8q^{7} + 60q^{9} - 24q^{12} + 24q^{15} - 8q^{16} + 16q^{17} - 60q^{19} - 164q^{20} + 40q^{22} + 100q^{25} - 156q^{26} + 200q^{28} - 60q^{29} - 32q^{35} - 120q^{36} + 28q^{41} + 180q^{46} - 96q^{48} + 284q^{49} - 24q^{51} - 8q^{53} + 24q^{57} - 152q^{59} - 72q^{60} - 8q^{62} - 24q^{63} + 204q^{64} - 120q^{66} + 384q^{68} + 92q^{71} + 104q^{74} + 240q^{75} + 120q^{76} - 468q^{78} - 420q^{79} + 376q^{80} + 180q^{81} + 168q^{84} - 348q^{85} + 232q^{86} + 144q^{87} - 212q^{88} + 152q^{94} + 788q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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