LMFDB - Embedded newform 108.3.d.a.55.1

Newspace parameters

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	108.d (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$2.94278685509$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			55.1
Root			$0.500000 + 0.866025i$ of $x^{2} - x + 1$
Character	$\chi$	$=$	108.55
Dual form			108.3.d.a.55.2

$q$-expansion

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +7.00000 q^{5} +8.66025i q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +7.00000 q^{5} +8.66025i q^{7} +8.00000 q^{8} +(-7.00000 - 12.1244i) q^{10} -8.66025i q^{11} +20.0000 q^{13} +(15.0000 - 8.66025i) q^{14} +(-8.00000 - 13.8564i) q^{16} -8.00000 q^{17} -10.3923i q^{19} +(-14.0000 + 24.2487i) q^{20} +(-15.0000 + 8.66025i) q^{22} +3.46410i q^{23} +24.0000 q^{25} +(-20.0000 - 34.6410i) q^{26} +(-30.0000 - 17.3205i) q^{28} +10.0000 q^{29} +53.6936i q^{31} +(-16.0000 + 27.7128i) q^{32} +(8.00000 + 13.8564i) q^{34} +60.6218i q^{35} -10.0000 q^{37} +(-18.0000 + 10.3923i) q^{38} +56.0000 q^{40} -50.0000 q^{41} -17.3205i q^{43} +(30.0000 + 17.3205i) q^{44} +(6.00000 - 3.46410i) q^{46} -86.6025i q^{47} -26.0000 q^{49} +(-24.0000 - 41.5692i) q^{50} +(-40.0000 + 69.2820i) q^{52} -47.0000 q^{53} -60.6218i q^{55} +69.2820i q^{56} +(-10.0000 - 17.3205i) q^{58} +34.6410i q^{59} -64.0000 q^{61} +(93.0000 - 53.6936i) q^{62} +64.0000 q^{64} +140.000 q^{65} -86.6025i q^{67} +(16.0000 - 27.7128i) q^{68} +(105.000 - 60.6218i) q^{70} -55.0000 q^{73} +(10.0000 + 17.3205i) q^{74} +(36.0000 + 20.7846i) q^{76} +75.0000 q^{77} -6.92820i q^{79} +(-56.0000 - 96.9948i) q^{80} +(50.0000 + 86.6025i) q^{82} -29.4449i q^{83} -56.0000 q^{85} +(-30.0000 + 17.3205i) q^{86} -69.2820i q^{88} +10.0000 q^{89} +173.205i q^{91} +(-12.0000 - 6.92820i) q^{92} +(-150.000 + 86.6025i) q^{94} -72.7461i q^{95} -25.0000 q^{97} +(26.0000 + 45.0333i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2q - 2q^{2} - 4q^{4} + 14q^{5} + 16q^{8} + O(q^{10}) $	$ 2q - 2q^{2} - 4q^{4} + 14q^{5} + 16q^{8} - 14q^{10} + 40q^{13} + 30q^{14} - 16q^{16} - 16q^{17} - 28q^{20} - 30q^{22} + 48q^{25} - 40q^{26} - 60q^{28} + 20q^{29} - 32q^{32} + 16q^{34} - 20q^{37} - 36q^{38} + 112q^{40} - 100q^{41} + 60q^{44} + 12q^{46} - 52q^{49} - 48q^{50} - 80q^{52} - 94q^{53} - 20q^{58} - 128q^{61} + 186q^{62} + 128q^{64} + 280q^{65} + 32q^{68} + 210q^{70} - 110q^{73} + 20q^{74} + 72q^{76} + 150q^{77} - 112q^{80} + 100q^{82} - 112q^{85} - 60q^{86} + 20q^{89} - 24q^{92} - 300q^{94} - 50q^{97} + 52q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$1$	$-1$

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 $

$2$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$2$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$5$	7.00000			1.40000			0.700000	−	0.714143i	$-0.253183\pi$
$5$	7.00000			1.40000			0.700000	+	0.714143i	$0.253183\pi$
$6$		0			0
$7$			8.66025i			1.23718i	0.785714	+	0.618590i	$0.212296\pi$
$7$			8.66025i			1.23718i	−0.785714	+	0.618590i	$0.787704\pi$
$8$	8.00000			1.00000
$9$		0			0
$10$	−7.00000	−	12.1244i	−0.700000	−	1.21244i
$11$		−	8.66025i		−	0.787296i	−0.919261	−	0.393648i	$-0.871213\pi$
$11$		−	8.66025i		−	0.787296i	0.919261	−	0.393648i	$-0.128787\pi$
$12$		0			0
$13$	20.0000			1.53846			0.769231	−	0.638971i	$-0.220640\pi$
$13$	20.0000			1.53846			0.769231	+	0.638971i	$0.220640\pi$
$14$	15.0000	−	8.66025i	1.07143	−	0.618590i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$17$	−8.00000			−0.470588			−0.235294	−	0.971924i	$-0.575605\pi$
$17$	−8.00000			−0.470588			−0.235294	+	0.971924i	$0.575605\pi$
$18$		0			0
$19$		−	10.3923i		−	0.546963i	−0.961877	−	0.273482i	$-0.911825\pi$
$19$		−	10.3923i		−	0.546963i	0.961877	−	0.273482i	$-0.0881753\pi$
$20$	−14.0000	+	24.2487i	−0.700000	+	1.21244i
$21$		0			0
$22$	−15.0000	+	8.66025i	−0.681818	+	0.393648i
$23$			3.46410i			0.150613i	0.997160	+	0.0753066i	$0.0239935\pi$
$23$			3.46410i			0.150613i	−0.997160	+	0.0753066i	$0.976006\pi$
$24$		0			0
$25$	24.0000			0.960000
$26$	−20.0000	−	34.6410i	−0.769231	−	1.33235i
$27$		0			0
$28$	−30.0000	−	17.3205i	−1.07143	−	0.618590i
$29$	10.0000			0.344828			0.172414	−	0.985025i	$-0.444843\pi$
$29$	10.0000			0.344828			0.172414	+	0.985025i	$0.444843\pi$
$30$		0			0
$31$			53.6936i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$31$			53.6936i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$32$	−16.0000	+	27.7128i	−0.500000	+	0.866025i
$33$		0			0
$34$	8.00000	+	13.8564i	0.235294	+	0.407541i
$35$			60.6218i			1.73205i
$36$		0			0
$37$	−10.0000			−0.270270			−0.135135	−	0.990827i	$-0.543147\pi$
$37$	−10.0000			−0.270270			−0.135135	+	0.990827i	$0.543147\pi$
$38$	−18.0000	+	10.3923i	−0.473684	+	0.273482i
$39$		0			0
$40$	56.0000			1.40000
$41$	−50.0000			−1.21951			−0.609756	−	0.792589i	$-0.708733\pi$
$41$	−50.0000			−1.21951			−0.609756	+	0.792589i	$0.708733\pi$
$42$		0			0
$43$		−	17.3205i		−	0.402803i	−0.979509	−	0.201401i	$-0.935450\pi$
$43$		−	17.3205i		−	0.402803i	0.979509	−	0.201401i	$-0.0645495\pi$
$44$	30.0000	+	17.3205i	0.681818	+	0.393648i
$45$		0			0
$46$	6.00000	−	3.46410i	0.130435	−	0.0753066i
$47$		−	86.6025i		−	1.84261i	−0.388844	−	0.921304i	$-0.627125\pi$
$47$		−	86.6025i		−	1.84261i	0.388844	−	0.921304i	$-0.372875\pi$
$48$		0			0
$49$	−26.0000			−0.530612
$50$	−24.0000	−	41.5692i	−0.480000	−	0.831384i
$51$		0			0
$52$	−40.0000	+	69.2820i	−0.769231	+	1.33235i
$53$	−47.0000			−0.886792			−0.443396	−	0.896326i	$-0.646227\pi$
$53$	−47.0000			−0.886792			−0.443396	+	0.896326i	$0.646227\pi$
$54$		0			0
$55$		−	60.6218i		−	1.10221i
$56$			69.2820i			1.23718i
$57$		0			0
$58$	−10.0000	−	17.3205i	−0.172414	−	0.298629i
$59$			34.6410i			0.587136i	0.955938	+	0.293568i	$0.0948427\pi$
$59$			34.6410i			0.587136i	−0.955938	+	0.293568i	$0.905157\pi$
$60$		0			0
$61$	−64.0000			−1.04918			−0.524590	−	0.851355i	$-0.675781\pi$
$61$	−64.0000			−1.04918			−0.524590	+	0.851355i	$0.675781\pi$
$62$	93.0000	−	53.6936i	1.50000	−	0.866025i
$63$		0			0
$64$	64.0000			1.00000
$65$	140.000			2.15385
$66$		0			0
$67$		−	86.6025i		−	1.29258i	−0.763094	−	0.646288i	$-0.776321\pi$
$67$		−	86.6025i		−	1.29258i	0.763094	−	0.646288i	$-0.223679\pi$
$68$	16.0000	−	27.7128i	0.235294	−	0.407541i
$69$		0			0
$70$	105.000	−	60.6218i	1.50000	−	0.866025i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−55.0000			−0.753425			−0.376712	−	0.926330i	$-0.622945\pi$
$73$	−55.0000			−0.753425			−0.376712	+	0.926330i	$0.622945\pi$
$74$	10.0000	+	17.3205i	0.135135	+	0.234061i
$75$		0			0
$76$	36.0000	+	20.7846i	0.473684	+	0.273482i
$77$	75.0000			0.974026
$78$		0			0
$79$		−	6.92820i		−	0.0876988i	−0.999038	−	0.0438494i	$-0.986038\pi$
$79$		−	6.92820i		−	0.0876988i	0.999038	−	0.0438494i	$-0.0139622\pi$
$80$	−56.0000	−	96.9948i	−0.700000	−	1.21244i
$81$		0			0
$82$	50.0000	+	86.6025i	0.609756	+	1.05613i
$83$		−	29.4449i		−	0.354757i	−0.984143	−	0.177379i	$-0.943238\pi$
$83$		−	29.4449i		−	0.354757i	0.984143	−	0.177379i	$-0.0567617\pi$
$84$		0			0
$85$	−56.0000			−0.658824
$86$	−30.0000	+	17.3205i	−0.348837	+	0.201401i
$87$		0			0
$88$		−	69.2820i		−	0.787296i
$89$	10.0000			0.112360			0.0561798	−	0.998421i	$-0.482108\pi$
$89$	10.0000			0.112360			0.0561798	+	0.998421i	$0.482108\pi$
$90$		0			0
$91$			173.205i			1.90335i
$92$	−12.0000	−	6.92820i	−0.130435	−	0.0753066i
$93$		0			0
$94$	−150.000	+	86.6025i	−1.59574	+	0.921304i
$95$		−	72.7461i		−	0.765749i
$96$		0			0
$97$	−25.0000			−0.257732			−0.128866	−	0.991662i	$-0.541134\pi$
$97$	−25.0000			−0.257732			−0.128866	+	0.991662i	$0.541134\pi$
$98$	26.0000	+	45.0333i	0.265306	+	0.459524i
$99$		0			0
$100$	−48.0000	+	83.1384i	−0.480000	+	0.831384i
$101$	−155.000			−1.53465			−0.767327	−	0.641256i	$-0.778414\pi$
$101$	−155.000			−1.53465			−0.767327	+	0.641256i	$0.778414\pi$
$102$		0			0
$103$		−	138.564i		−	1.34528i	−0.739969	−	0.672641i	$-0.765160\pi$
$103$		−	138.564i		−	1.34528i	0.739969	−	0.672641i	$-0.234840\pi$
$104$	160.000			1.53846
$105$		0			0
$106$	47.0000	+	81.4064i	0.443396	+	0.767985i
$107$			129.904i			1.21405i	0.794681	+	0.607027i	$0.207638\pi$
$107$			129.904i			1.21405i	−0.794681	+	0.607027i	$0.792362\pi$
$108$		0			0
$109$	134.000			1.22936			0.614679	−	0.788777i	$-0.289286\pi$
$109$	134.000			1.22936			0.614679	+	0.788777i	$0.289286\pi$
$110$	−105.000	+	60.6218i	−0.954545	+	0.551107i
$111$		0			0
$112$	120.000	−	69.2820i	1.07143	−	0.618590i
$113$	−74.0000			−0.654867			−0.327434	−	0.944874i	$-0.606184\pi$
$113$	−74.0000			−0.654867			−0.327434	+	0.944874i	$0.606184\pi$
$114$		0			0
$115$			24.2487i			0.210858i
$116$	−20.0000	+	34.6410i	−0.172414	+	0.298629i
$117$		0			0
$118$	60.0000	−	34.6410i	0.508475	−	0.293568i
$119$		−	69.2820i		−	0.582202i
$120$		0			0
$121$	46.0000			0.380165
$122$	64.0000	+	110.851i	0.524590	+	0.908617i
$123$		0			0
$124$	−186.000	−	107.387i	−1.50000	−	0.866025i
$125$	−7.00000			−0.0560000
$126$		0			0
$127$			25.9808i			0.204573i	0.994755	+	0.102286i	$0.0326158\pi$
$127$			25.9808i			0.204573i	−0.994755	+	0.102286i	$0.967384\pi$
$128$	−64.0000	−	110.851i	−0.500000	−	0.866025i
$129$		0			0
$130$	−140.000	−	242.487i	−1.07692	−	1.86529i
$131$			164.545i			1.25607i	0.778186	+	0.628034i	$0.216140\pi$
$131$			164.545i			1.25607i	−0.778186	+	0.628034i	$0.783860\pi$
$132$		0			0
$133$	90.0000			0.676692
$134$	−150.000	+	86.6025i	−1.11940	+	0.646288i
$135$		0			0
$136$	−64.0000			−0.470588
$137$	−62.0000			−0.452555			−0.226277	−	0.974063i	$-0.572656\pi$
$137$	−62.0000			−0.452555			−0.226277	+	0.974063i	$0.572656\pi$
$138$		0			0
$139$			173.205i			1.24608i	0.782190	+	0.623040i	$0.214103\pi$
$139$			173.205i			1.24608i	−0.782190	+	0.623040i	$0.785897\pi$
$140$	−210.000	−	121.244i	−1.50000	−	0.866025i
$141$		0			0
$142$		0			0
$143$		−	173.205i		−	1.21122i
$144$		0			0
$145$	70.0000			0.482759
$146$	55.0000	+	95.2628i	0.376712	+	0.652485i
$147$		0			0
$148$	20.0000	−	34.6410i	0.135135	−	0.234061i
$149$	115.000			0.771812			0.385906	−	0.922538i	$-0.373889\pi$
$149$	115.000			0.771812			0.385906	+	0.922538i	$0.373889\pi$
$150$		0			0
$151$		−	43.3013i		−	0.286763i	−0.989667	−	0.143382i	$-0.954202\pi$
$151$		−	43.3013i		−	0.286763i	0.989667	−	0.143382i	$-0.0457977\pi$
$152$		−	83.1384i		−	0.546963i
$153$		0			0
$154$	−75.0000	−	129.904i	−0.487013	−	0.843531i
$155$			375.855i			2.42487i
$156$		0			0
$157$	20.0000			0.127389			0.0636943	−	0.997969i	$-0.479712\pi$
$157$	20.0000			0.127389			0.0636943	+	0.997969i	$0.479712\pi$
$158$	−12.0000	+	6.92820i	−0.0759494	+	0.0438494i
$159$		0			0
$160$	−112.000	+	193.990i	−0.700000	+	1.21244i
$161$	−30.0000			−0.186335
$162$		0			0
$163$		−	103.923i		−	0.637565i	−0.947828	−	0.318782i	$-0.896726\pi$
$163$		−	103.923i		−	0.637565i	0.947828	−	0.318782i	$-0.103274\pi$
$164$	100.000	−	173.205i	0.609756	−	1.05613i
$165$		0			0
$166$	−51.0000	+	29.4449i	−0.307229	+	0.177379i
$167$		−	245.951i		−	1.47276i	−0.676567	−	0.736381i	$-0.736533\pi$
$167$		−	245.951i		−	1.47276i	0.676567	−	0.736381i	$-0.263467\pi$
$168$		0			0
$169$	231.000			1.36686
$170$	56.0000	+	96.9948i	0.329412	+	0.570558i
$171$		0			0
$172$	60.0000	+	34.6410i	0.348837	+	0.201401i
$173$	127.000			0.734104			0.367052	−	0.930200i	$-0.380367\pi$
$173$	127.000			0.734104			0.367052	+	0.930200i	$0.380367\pi$
$174$		0			0
$175$			207.846i			1.18769i
$176$	−120.000	+	69.2820i	−0.681818	+	0.393648i
$177$		0			0
$178$	−10.0000	−	17.3205i	−0.0561798	−	0.0973062i
$179$		−	233.827i		−	1.30630i	−0.757231	−	0.653148i	$-0.773448\pi$
$179$		−	233.827i		−	1.30630i	0.757231	−	0.653148i	$-0.226552\pi$
$180$		0			0
$181$	56.0000			0.309392			0.154696	−	0.987962i	$-0.450560\pi$
$181$	56.0000			0.309392			0.154696	+	0.987962i	$0.450560\pi$
$182$	300.000	−	173.205i	1.64835	−	0.951676i
$183$		0			0
$184$			27.7128i			0.150613i
$185$	−70.0000			−0.378378
$186$		0			0
$187$			69.2820i			0.370492i
$188$	300.000	+	173.205i	1.59574	+	0.921304i
$189$		0			0
$190$	−126.000	+	72.7461i	−0.663158	+	0.382874i
$191$		−	34.6410i		−	0.181367i	−0.995880	−	0.0906833i	$-0.971095\pi$
$191$		−	34.6410i		−	0.181367i	0.995880	−	0.0906833i	$-0.0289051\pi$
$192$		0			0
$193$	65.0000			0.336788			0.168394	−	0.985720i	$-0.446142\pi$
$193$	65.0000			0.336788			0.168394	+	0.985720i	$0.446142\pi$
$194$	25.0000	+	43.3013i	0.128866	+	0.223202i
$195$		0			0
$196$	52.0000	−	90.0666i	0.265306	−	0.459524i
$197$	253.000			1.28426			0.642132	−	0.766594i	$-0.278050\pi$
$197$	253.000			1.28426			0.642132	+	0.766594i	$0.278050\pi$
$198$		0			0
$199$			129.904i			0.652783i	0.945235	+	0.326391i	$0.105833\pi$
$199$			129.904i			0.652783i	−0.945235	+	0.326391i	$0.894167\pi$
$200$	192.000			0.960000
$201$		0			0
$202$	155.000	+	268.468i	0.767327	+	1.32905i
$203$			86.6025i			0.426613i
$204$		0			0
$205$	−350.000			−1.70732
$206$	−240.000	+	138.564i	−1.16505	+	0.672641i
$207$		0			0
$208$	−160.000	−	277.128i	−0.769231	−	1.33235i
$209$	−90.0000			−0.430622
$210$		0			0
$211$		−	148.956i		−	0.705954i	−0.935632	−	0.352977i	$-0.885169\pi$
$211$		−	148.956i		−	0.705954i	0.935632	−	0.352977i	$-0.114831\pi$
$212$	94.0000	−	162.813i	0.443396	−	0.767985i
$213$		0			0
$214$	225.000	−	129.904i	1.05140	−	0.607027i
$215$		−	121.244i		−	0.563924i
$216$		0			0
$217$	−465.000			−2.14286
$218$	−134.000	−	232.095i	−0.614679	−	1.06466i
$219$		0			0
$220$	210.000	+	121.244i	0.954545	+	0.551107i
$221$	−160.000			−0.723982
$222$		0			0
$223$			34.6410i			0.155341i	0.996979	+	0.0776704i	$0.0247482\pi$
$223$			34.6410i			0.155341i	−0.996979	+	0.0776704i	$0.975252\pi$
$224$	−240.000	−	138.564i	−1.07143	−	0.618590i
$225$		0			0
$226$	74.0000	+	128.172i	0.327434	+	0.567132i
$227$			90.0666i			0.396769i	0.980124	+	0.198385i	$0.0635695\pi$
$227$			90.0666i			0.396769i	−0.980124	+	0.198385i	$0.936430\pi$
$228$		0			0
$229$	146.000			0.637555			0.318777	−	0.947830i	$-0.396728\pi$
$229$	146.000			0.637555			0.318777	+	0.947830i	$0.396728\pi$
$230$	42.0000	−	24.2487i	0.182609	−	0.105429i
$231$		0			0
$232$	80.0000			0.344828
$233$	334.000			1.43348			0.716738	−	0.697342i	$-0.245634\pi$
$233$	334.000			1.43348			0.716738	+	0.697342i	$0.245634\pi$
$234$		0			0
$235$		−	606.218i		−	2.57965i
$236$	−120.000	−	69.2820i	−0.508475	−	0.293568i
$237$		0			0
$238$	−120.000	+	69.2820i	−0.504202	+	0.291101i
$239$		−	17.3205i		−	0.0724707i	−0.999343	−	0.0362354i	$-0.988463\pi$
$239$		−	17.3205i		−	0.0724707i	0.999343	−	0.0362354i	$-0.0115366\pi$
$240$		0			0
$241$	134.000			0.556017			0.278008	−	0.960579i	$-0.410326\pi$
$241$	134.000			0.556017			0.278008	+	0.960579i	$0.410326\pi$
$242$	−46.0000	−	79.6743i	−0.190083	−	0.329233i
$243$		0			0
$244$	128.000	−	221.703i	0.524590	−	0.908617i
$245$	−182.000			−0.742857
$246$		0			0
$247$		−	207.846i		−	0.841482i
$248$			429.549i			1.73205i
$249$		0			0
$250$	7.00000	+	12.1244i	0.0280000	+	0.0484974i
$251$			207.846i			0.828072i	0.910260	+	0.414036i	$0.135881\pi$
$251$			207.846i			0.828072i	−0.910260	+	0.414036i	$0.864119\pi$
$252$		0			0
$253$	30.0000			0.118577
$254$	45.0000	−	25.9808i	0.177165	−	0.102286i
$255$		0			0
$256$	−128.000	+	221.703i	−0.500000	+	0.866025i
$257$	268.000			1.04280			0.521401	−	0.853312i	$-0.325410\pi$
$257$	268.000			1.04280			0.521401	+	0.853312i	$0.325410\pi$
$258$		0			0
$259$		−	86.6025i		−	0.334373i
$260$	−280.000	+	484.974i	−1.07692	+	1.86529i
$261$		0			0
$262$	285.000	−	164.545i	1.08779	−	0.628034i
$263$			433.013i			1.64644i	0.567725	+	0.823218i	$0.307824\pi$
$263$			433.013i			1.64644i	−0.567725	+	0.823218i	$0.692176\pi$
$264$		0			0
$265$	−329.000			−1.24151
$266$	−90.0000	−	155.885i	−0.338346	−	0.586032i
$267$		0			0
$268$	300.000	+	173.205i	1.11940	+	0.646288i
$269$	−350.000			−1.30112			−0.650558	−	0.759457i	$-0.725465\pi$
$269$	−350.000			−1.30112			−0.650558	+	0.759457i	$0.725465\pi$
$270$		0			0
$271$			36.3731i			0.134218i	0.997746	+	0.0671090i	$0.0213775\pi$
$271$			36.3731i			0.134218i	−0.997746	+	0.0671090i	$0.978622\pi$
$272$	64.0000	+	110.851i	0.235294	+	0.407541i
$273$		0			0
$274$	62.0000	+	107.387i	0.226277	+	0.391924i
$275$		−	207.846i		−	0.755804i
$276$		0			0
$277$	−520.000			−1.87726			−0.938628	−	0.344931i	$-0.887902\pi$
$277$	−520.000			−1.87726			−0.938628	+	0.344931i	$0.887902\pi$
$278$	300.000	−	173.205i	1.07914	−	0.623040i
$279$		0			0
$280$			484.974i			1.73205i
$281$	−440.000			−1.56584			−0.782918	−	0.622125i	$-0.786270\pi$
$281$	−440.000			−1.56584			−0.782918	+	0.622125i	$0.786270\pi$
$282$		0			0
$283$			329.090i			1.16286i	0.813596	+	0.581430i	$0.197507\pi$
$283$			329.090i			1.16286i	−0.813596	+	0.581430i	$0.802493\pi$
$284$		0			0
$285$		0			0
$286$	−300.000	+	173.205i	−1.04895	+	0.605612i
$287$		−	433.013i		−	1.50876i
$288$		0			0
$289$	−225.000			−0.778547
$290$	−70.0000	−	121.244i	−0.241379	−	0.418081i
$291$		0			0
$292$	110.000	−	190.526i	0.376712	−	0.652485i
$293$	−218.000			−0.744027			−0.372014	−	0.928227i	$-0.621333\pi$
$293$	−218.000			−0.744027			−0.372014	+	0.928227i	$0.621333\pi$
$294$		0			0
$295$			242.487i			0.821990i
$296$	−80.0000			−0.270270
$297$		0			0
$298$	−115.000	−	199.186i	−0.385906	−	0.668409i
$299$			69.2820i			0.231712i
$300$		0			0
$301$	150.000			0.498339
$302$	−75.0000	+	43.3013i	−0.248344	+	0.143382i
$303$		0			0
$304$	−144.000	+	83.1384i	−0.473684	+	0.273482i
$305$	−448.000			−1.46885
$306$		0			0
$307$			207.846i			0.677023i	0.940962	+	0.338512i	$0.109923\pi$
$307$			207.846i			0.677023i	−0.940962	+	0.338512i	$0.890077\pi$
$308$	−150.000	+	259.808i	−0.487013	+	0.843531i
$309$		0			0
$310$	651.000	−	375.855i	2.10000	−	1.21244i
$311$			294.449i			0.946780i	0.880853	+	0.473390i	$0.156970\pi$
$311$			294.449i			0.946780i	−0.880853	+	0.473390i	$0.843030\pi$
$312$		0			0
$313$	485.000			1.54952			0.774760	−	0.632255i	$-0.217870\pi$
$313$	485.000			1.54952			0.774760	+	0.632255i	$0.217870\pi$
$314$	−20.0000	−	34.6410i	−0.0636943	−	0.110322i
$315$		0			0
$316$	24.0000	+	13.8564i	0.0759494	+	0.0438494i
$317$	217.000			0.684543			0.342271	−	0.939601i	$-0.388804\pi$
$317$	217.000			0.684543			0.342271	+	0.939601i	$0.388804\pi$
$318$		0			0
$319$		−	86.6025i		−	0.271481i
$320$	448.000			1.40000
$321$		0			0
$322$	30.0000	+	51.9615i	0.0931677	+	0.161371i
$323$			83.1384i			0.257395i
$324$		0			0
$325$	480.000			1.47692
$326$	−180.000	+	103.923i	−0.552147	+	0.318782i
$327$		0			0
$328$	−400.000			−1.21951
$329$	750.000			2.27964
$330$		0			0
$331$		−	433.013i		−	1.30820i	−0.756410	−	0.654098i	$-0.773048\pi$
$331$		−	433.013i		−	1.30820i	0.756410	−	0.654098i	$-0.226952\pi$
$332$	102.000	+	58.8897i	0.307229	+	0.177379i
$333$		0			0
$334$	−426.000	+	245.951i	−1.27545	+	0.736381i
$335$		−	606.218i		−	1.80961i
$336$		0			0
$337$	−310.000			−0.919881			−0.459941	−	0.887950i	$-0.652129\pi$
$337$	−310.000			−0.919881			−0.459941	+	0.887950i	$0.652129\pi$
$338$	−231.000	−	400.104i	−0.683432	−	1.18374i
$339$		0			0
$340$	112.000	−	193.990i	0.329412	−	0.570558i
$341$	465.000			1.36364
$342$		0			0
$343$			199.186i			0.580717i
$344$		−	138.564i		−	0.402803i
$345$		0			0
$346$	−127.000	−	219.970i	−0.367052	−	0.635753i
$347$			216.506i			0.623938i	0.950092	+	0.311969i	$0.100988\pi$
$347$			216.506i			0.623938i	−0.950092	+	0.311969i	$0.899012\pi$
$348$		0			0
$349$	74.0000			0.212034			0.106017	−	0.994364i	$-0.466190\pi$
$349$	74.0000			0.212034			0.106017	+	0.994364i	$0.466190\pi$
$350$	360.000	−	207.846i	1.02857	−	0.593846i
$351$		0			0
$352$	240.000	+	138.564i	0.681818	+	0.393648i
$353$	394.000			1.11615			0.558074	−	0.829791i	$-0.311541\pi$
$353$	394.000			1.11615			0.558074	+	0.829791i	$0.311541\pi$
$354$		0			0
$355$		0			0
$356$	−20.0000	+	34.6410i	−0.0561798	+	0.0973062i
$357$		0			0
$358$	−405.000	+	233.827i	−1.13128	+	0.653148i
$359$			571.577i			1.59214i	0.605207	+	0.796068i	$0.293090\pi$
$359$			571.577i			1.59214i	−0.605207	+	0.796068i	$0.706910\pi$
$360$		0			0
$361$	253.000			0.700831
$362$	−56.0000	−	96.9948i	−0.154696	−	0.267942i
$363$		0			0
$364$	−600.000	−	346.410i	−1.64835	−	0.951676i
$365$	−385.000			−1.05479
$366$		0			0
$367$		−	562.917i		−	1.53383i	−0.641747	−	0.766916i	$-0.721790\pi$
$367$		−	562.917i		−	1.53383i	0.641747	−	0.766916i	$-0.278210\pi$
$368$	48.0000	−	27.7128i	0.130435	−	0.0753066i
$369$		0			0
$370$	70.0000	+	121.244i	0.189189	+	0.327685i
$371$		−	407.032i		−	1.09712i
$372$		0			0
$373$	−40.0000			−0.107239			−0.0536193	−	0.998561i	$-0.517076\pi$
$373$	−40.0000			−0.107239			−0.0536193	+	0.998561i	$0.517076\pi$
$374$	120.000	−	69.2820i	0.320856	−	0.185246i
$375$		0			0
$376$		−	692.820i		−	1.84261i
$377$	200.000			0.530504
$378$		0			0
$379$			685.892i			1.80974i	0.425686	+	0.904871i	$0.360033\pi$
$379$			685.892i			1.80974i	−0.425686	+	0.904871i	$0.639967\pi$
$380$	252.000	+	145.492i	0.663158	+	0.382874i
$381$		0			0
$382$	−60.0000	+	34.6410i	−0.157068	+	0.0906833i
$383$		−	360.267i		−	0.940644i	−0.882495	−	0.470322i	$-0.844138\pi$
$383$		−	360.267i		−	0.940644i	0.882495	−	0.470322i	$-0.155862\pi$
$384$		0			0
$385$	525.000			1.36364
$386$	−65.0000	−	112.583i	−0.168394	−	0.291667i
$387$		0			0
$388$	50.0000	−	86.6025i	0.128866	−	0.223202i
$389$	475.000			1.22108			0.610540	−	0.791986i	$-0.290953\pi$
$389$	475.000			1.22108			0.610540	+	0.791986i	$0.290953\pi$
$390$		0			0
$391$		−	27.7128i		−	0.0708768i
$392$	−208.000			−0.530612
$393$		0			0
$394$	−253.000	−	438.209i	−0.642132	−	1.11221i
$395$		−	48.4974i		−	0.122778i
$396$		0			0
$397$	260.000			0.654912			0.327456	−	0.944866i	$-0.393809\pi$
$397$	260.000			0.654912			0.327456	+	0.944866i	$0.393809\pi$
$398$	225.000	−	129.904i	0.565327	−	0.326391i
$399$		0			0
$400$	−192.000	−	332.554i	−0.480000	−	0.831384i
$401$	−740.000			−1.84539			−0.922693	−	0.385535i	$-0.874017\pi$
$401$	−740.000			−1.84539			−0.922693	+	0.385535i	$0.874017\pi$
$402$		0			0
$403$			1073.87i			2.66469i
$404$	310.000	−	536.936i	0.767327	−	1.32905i
$405$		0			0
$406$	150.000	−	86.6025i	0.369458	−	0.213307i
$407$			86.6025i			0.212783i
$408$		0			0
$409$	659.000			1.61125			0.805623	−	0.592428i	$-0.201830\pi$
$409$	659.000			1.61125			0.805623	+	0.592428i	$0.201830\pi$
$410$	350.000	+	606.218i	0.853659	+	1.47858i
$411$		0			0
$412$	480.000	+	277.128i	1.16505	+	0.672641i
$413$	−300.000			−0.726392
$414$		0			0
$415$		−	206.114i		−	0.496660i
$416$	−320.000	+	554.256i	−0.769231	+	1.33235i
$417$		0			0
$418$	90.0000	+	155.885i	0.215311	+	0.372930i
$419$		−	588.897i		−	1.40548i	−0.711445	−	0.702741i	$-0.751959\pi$
$419$		−	588.897i		−	1.40548i	0.711445	−	0.702741i	$-0.248041\pi$
$420$		0			0
$421$	−496.000			−1.17815			−0.589074	−	0.808079i	$-0.700507\pi$
$421$	−496.000			−1.17815			−0.589074	+	0.808079i	$0.700507\pi$
$422$	−258.000	+	148.956i	−0.611374	+	0.352977i
$423$		0			0
$424$	−376.000			−0.886792
$425$	−192.000			−0.451765
$426$		0			0
$427$		−	554.256i		−	1.29802i
$428$	−450.000	−	259.808i	−1.05140	−	0.607027i
$429$		0			0
$430$	−210.000	+	121.244i	−0.488372	+	0.281962i
$431$			571.577i			1.32616i	0.748547	+	0.663082i	$0.230752\pi$
$431$			571.577i			1.32616i	−0.748547	+	0.663082i	$0.769248\pi$
$432$		0			0
$433$	−235.000			−0.542725			−0.271363	−	0.962477i	$-0.587474\pi$
$433$	−235.000			−0.542725			−0.271363	+	0.962477i	$0.587474\pi$
$434$	465.000	+	805.404i	1.07143	+	1.85577i
$435$		0			0
$436$	−268.000	+	464.190i	−0.614679	+	1.06466i
$437$	36.0000			0.0823799
$438$		0			0
$439$			413.960i			0.942962i	0.881876	+	0.471481i	$0.156280\pi$
$439$			413.960i			0.942962i	−0.881876	+	0.471481i	$0.843720\pi$
$440$		−	484.974i		−	1.10221i
$441$		0			0
$442$	160.000	+	277.128i	0.361991	+	0.626987i
$443$		−	575.041i		−	1.29806i	−0.760763	−	0.649030i	$-0.775175\pi$
$443$		−	575.041i		−	1.29806i	0.760763	−	0.649030i	$-0.224825\pi$
$444$		0			0
$445$	70.0000			0.157303
$446$	60.0000	−	34.6410i	0.134529	−	0.0776704i
$447$		0			0
$448$			554.256i			1.23718i
$449$	−470.000			−1.04677			−0.523385	−	0.852096i	$-0.675331\pi$
$449$	−470.000			−1.04677			−0.523385	+	0.852096i	$0.675331\pi$
$450$		0			0
$451$			433.013i			0.960117i
$452$	148.000	−	256.344i	0.327434	−	0.567132i
$453$		0			0
$454$	156.000	−	90.0666i	0.343612	−	0.198385i
$455$			1212.44i			2.66469i
$456$		0			0
$457$	−325.000			−0.711160			−0.355580	−	0.934646i	$-0.615717\pi$
$457$	−325.000			−0.711160			−0.355580	+	0.934646i	$0.615717\pi$
$458$	−146.000	−	252.879i	−0.318777	−	0.552138i
$459$		0			0
$460$	−84.0000	−	48.4974i	−0.182609	−	0.105429i
$461$	655.000			1.42082			0.710412	−	0.703786i	$-0.248509\pi$
$461$	655.000			1.42082			0.710412	+	0.703786i	$0.248509\pi$
$462$		0			0
$463$		−	164.545i		−	0.355388i	−0.984086	−	0.177694i	$-0.943136\pi$
$463$		−	164.545i		−	0.355388i	0.984086	−	0.177694i	$-0.0568638\pi$
$464$	−80.0000	−	138.564i	−0.172414	−	0.298629i
$465$		0			0
$466$	−334.000	−	578.505i	−0.716738	−	1.24143i
$467$		−	57.1577i		−	0.122393i	−0.998126	−	0.0611967i	$-0.980508\pi$
$467$		−	57.1577i		−	0.122393i	0.998126	−	0.0611967i	$-0.0194917\pi$
$468$		0			0
$469$	750.000			1.59915
$470$	−1050.00	+	606.218i	−2.23404	+	1.28983i
$471$		0			0
$472$			277.128i			0.587136i
$473$	−150.000			−0.317125
$474$		0			0
$475$		−	249.415i		−	0.525085i
$476$	240.000	+	138.564i	0.504202	+	0.291101i
$477$		0			0
$478$	−30.0000	+	17.3205i	−0.0627615	+	0.0362354i
$479$			329.090i			0.687035i	0.939146	+	0.343517i	$0.111618\pi$
$479$			329.090i			0.687035i	−0.939146	+	0.343517i	$0.888382\pi$
$480$		0			0
$481$	−200.000			−0.415800
$482$	−134.000	−	232.095i	−0.278008	−	0.481524i
$483$		0			0
$484$	−92.0000	+	159.349i	−0.190083	+	0.329233i
$485$	−175.000			−0.360825
$486$		0			0
$487$		−	519.615i		−	1.06697i	−0.845809	−	0.533486i	$-0.820882\pi$
$487$		−	519.615i		−	1.06697i	0.845809	−	0.533486i	$-0.179118\pi$
$488$	−512.000			−1.04918
$489$		0			0
$490$	182.000	+	315.233i	0.371429	+	0.643333i
$491$			216.506i			0.440950i	0.975393	+	0.220475i	$0.0707607\pi$
$491$			216.506i			0.440950i	−0.975393	+	0.220475i	$0.929239\pi$
$492$		0			0
$493$	−80.0000			−0.162272
$494$	−360.000	+	207.846i	−0.728745	+	0.420741i
$495$		0			0
$496$	744.000	−	429.549i	1.50000	−	0.866025i
$497$		0			0
$498$		0			0
$499$		−	45.0333i		−	0.0902471i	−0.998981	−	0.0451236i	$-0.985632\pi$
$499$		−	45.0333i		−	0.0902471i	0.998981	−	0.0451236i	$-0.0143682\pi$
$500$	14.0000	−	24.2487i	0.0280000	−	0.0484974i
$501$		0			0
$502$	360.000	−	207.846i	0.717131	−	0.414036i
$503$			384.515i			0.764444i	0.924071	+	0.382222i	$0.124841\pi$
$503$			384.515i			0.764444i	−0.924071	+	0.382222i	$0.875159\pi$
$504$		0			0
$505$	−1085.00			−2.14851
$506$	−30.0000	−	51.9615i	−0.0592885	−	0.102691i
$507$		0			0
$508$	−90.0000	−	51.9615i	−0.177165	−	0.102286i
$509$	265.000			0.520629			0.260314	−	0.965524i	$-0.416174\pi$
$509$	265.000			0.520629			0.260314	+	0.965524i	$0.416174\pi$
$510$		0			0
$511$		−	476.314i		−	0.932121i
$512$	512.000			1.00000
$513$		0			0
$514$	−268.000	−	464.190i	−0.521401	−	0.903093i
$515$		−	969.948i		−	1.88340i
$516$		0			0
$517$	−750.000			−1.45068
$518$	−150.000	+	86.6025i	−0.289575	+	0.167186i
$519$		0			0
$520$	1120.00			2.15385
$521$	−380.000			−0.729367			−0.364683	−	0.931132i	$-0.618823\pi$
$521$	−380.000			−0.729367			−0.364683	+	0.931132i	$0.618823\pi$
$522$		0			0
$523$		−	623.538i		−	1.19223i	−0.802898	−	0.596117i	$-0.796709\pi$
$523$		−	623.538i		−	1.19223i	0.802898	−	0.596117i	$-0.203291\pi$
$524$	−570.000	−	329.090i	−1.08779	−	0.628034i
$525$		0			0
$526$	750.000	−	433.013i	1.42586	−	0.823218i
$527$		−	429.549i		−	0.815083i
$528$		0			0
$529$	517.000			0.977316
$530$	329.000	+	569.845i	0.620755	+	1.07518i
$531$		0			0
$532$	−180.000	+	311.769i	−0.338346	+	0.586032i
$533$	−1000.00			−1.87617
$534$		0			0
$535$			909.327i			1.69968i
$536$		−	692.820i		−	1.29258i
$537$		0			0
$538$	350.000	+	606.218i	0.650558	+	1.12680i
$539$			225.167i			0.417749i
$540$		0			0
$541$	−532.000			−0.983364			−0.491682	−	0.870775i	$-0.663618\pi$
$541$	−532.000			−0.983364			−0.491682	+	0.870775i	$0.663618\pi$
$542$	63.0000	−	36.3731i	0.116236	−	0.0671090i
$543$		0			0
$544$	128.000	−	221.703i	0.235294	−	0.407541i
$545$	938.000			1.72110
$546$		0			0
$547$		−	900.666i		−	1.64656i	−0.567638	−	0.823278i	$-0.692143\pi$
$547$		−	900.666i		−	1.64656i	0.567638	−	0.823278i	$-0.307857\pi$
$548$	124.000	−	214.774i	0.226277	−	0.391924i
$549$		0			0
$550$	−360.000	+	207.846i	−0.654545	+	0.377902i
$551$		−	103.923i		−	0.188608i
$552$		0			0
$553$	60.0000			0.108499
$554$	520.000	+	900.666i	0.938628	+	1.62575i
$555$		0			0
$556$	−600.000	−	346.410i	−1.07914	−	0.623040i
$557$	−89.0000			−0.159785			−0.0798923	−	0.996804i	$-0.525458\pi$
$557$	−89.0000			−0.159785			−0.0798923	+	0.996804i	$0.525458\pi$
$558$		0			0
$559$		−	346.410i		−	0.619696i
$560$	840.000	−	484.974i	1.50000	−	0.866025i
$561$		0			0
$562$	440.000	+	762.102i	0.782918	+	1.35605i
$563$		−	303.109i		−	0.538382i	−0.963087	−	0.269191i	$-0.913244\pi$
$563$		−	303.109i		−	0.538382i	0.963087	−	0.269191i	$-0.0867562\pi$
$564$		0			0
$565$	−518.000			−0.916814
$566$	570.000	−	329.090i	1.00707	−	0.581430i
$567$		0			0
$568$		0			0
$569$	100.000			0.175747			0.0878735	−	0.996132i	$-0.471993\pi$
$569$	100.000			0.175747			0.0878735	+	0.996132i	$0.471993\pi$
$570$		0			0
$571$			339.482i			0.594539i	0.954794	+	0.297270i	$0.0960760\pi$
$571$			339.482i			0.594539i	−0.954794	+	0.297270i	$0.903924\pi$
$572$	600.000	+	346.410i	1.04895	+	0.605612i
$573$		0			0
$574$	−750.000	+	433.013i	−1.30662	+	0.754378i
$575$			83.1384i			0.144589i
$576$		0			0
$577$	−730.000			−1.26516			−0.632582	−	0.774493i	$-0.718005\pi$
$577$	−730.000			−1.26516			−0.632582	+	0.774493i	$0.718005\pi$
$578$	225.000	+	389.711i	0.389273	+	0.674241i
$579$		0			0
$580$	−140.000	+	242.487i	−0.241379	+	0.418081i
$581$	255.000			0.438898
$582$		0			0
$583$			407.032i			0.698168i
$584$	−440.000			−0.753425
$585$		0			0
$586$	218.000	+	377.587i	0.372014	+	0.644347i
$587$			801.940i			1.36617i	0.730341	+	0.683083i	$0.239361\pi$
$587$			801.940i			1.36617i	−0.730341	+	0.683083i	$0.760639\pi$
$588$		0			0
$589$	558.000			0.947368
$590$	420.000	−	242.487i	0.711864	−	0.410995i
$591$		0			0
$592$	80.0000	+	138.564i	0.135135	+	0.234061i
$593$	982.000			1.65599			0.827993	−	0.560738i	$-0.189483\pi$
$593$	982.000			1.65599			0.827993	+	0.560738i	$0.189483\pi$
$594$		0			0
$595$		−	484.974i		−	0.815083i
$596$	−230.000	+	398.372i	−0.385906	+	0.668409i
$597$		0			0
$598$	120.000	−	69.2820i	0.200669	−	0.115856i
$599$		−	225.167i		−	0.375904i	−0.982178	−	0.187952i	$-0.939815\pi$
$599$		−	225.167i		−	0.375904i	0.982178	−	0.187952i	$-0.0601850\pi$
$600$		0			0
$601$	251.000			0.417637			0.208819	−	0.977954i	$-0.433038\pi$
$601$	251.000			0.417637			0.208819	+	0.977954i	$0.433038\pi$
$602$	−150.000	−	259.808i	−0.249169	−	0.431574i
$603$		0			0
$604$	150.000	+	86.6025i	0.248344	+	0.143382i
$605$	322.000			0.532231
$606$		0			0
$607$			381.051i			0.627761i	0.949462	+	0.313881i	$0.101629\pi$
$607$			381.051i			0.627761i	−0.949462	+	0.313881i	$0.898371\pi$
$608$	288.000	+	166.277i	0.473684	+	0.273482i
$609$		0			0
$610$	448.000	+	775.959i	0.734426	+	1.27206i
$611$		−	1732.05i		−	2.83478i
$612$		0			0
$613$	650.000			1.06036			0.530179	−	0.847885i	$-0.322125\pi$
$613$	650.000			1.06036			0.530179	+	0.847885i	$0.322125\pi$
$614$	360.000	−	207.846i	0.586319	−	0.338512i
$615$		0			0
$616$	600.000			0.974026
$617$	−758.000			−1.22853			−0.614263	−	0.789102i	$-0.710546\pi$
$617$	−758.000			−1.22853			−0.614263	+	0.789102i	$0.710546\pi$
$618$		0			0
$619$		−	173.205i		−	0.279814i	−0.990165	−	0.139907i	$-0.955320\pi$
$619$		−	173.205i		−	0.279814i	0.990165	−	0.139907i	$-0.0446804\pi$
$620$	−1302.00	−	751.710i	−2.10000	−	1.21244i
$621$		0			0
$622$	510.000	−	294.449i	0.819936	−	0.473390i
$623$			86.6025i			0.139009i
$624$		0			0
$625$	−649.000			−1.03840
$626$	−485.000	−	840.045i	−0.774760	−	1.34192i
$627$		0			0
$628$	−40.0000	+	69.2820i	−0.0636943	+	0.110322i
$629$	80.0000			0.127186
$630$		0			0
$631$			119.512i			0.189400i	0.995506	+	0.0947001i	$0.0301892\pi$
$631$			119.512i			0.189400i	−0.995506	+	0.0947001i	$0.969811\pi$
$632$		−	55.4256i		−	0.0876988i
$633$		0			0
$634$	−217.000	−	375.855i	−0.342271	−	0.592831i
$635$			181.865i			0.286402i
$636$		0			0
$637$	−520.000			−0.816327
$638$	−150.000	+	86.6025i	−0.235110	+	0.135741i
$639$		0			0
$640$	−448.000	−	775.959i	−0.700000	−	1.21244i
$641$	910.000			1.41966			0.709828	−	0.704375i	$-0.248772\pi$
$641$	910.000			1.41966			0.709828	+	0.704375i	$0.248772\pi$
$642$		0			0
$643$		−	34.6410i		−	0.0538741i	−0.999637	−	0.0269370i	$-0.991425\pi$
$643$		−	34.6410i		−	0.0538741i	0.999637	−	0.0269370i	$-0.00857536\pi$
$644$	60.0000	−	103.923i	0.0931677	−	0.161371i
$645$		0			0
$646$	144.000	−	83.1384i	0.222910	−	0.128697i
$647$			914.523i			1.41348i	0.707472	+	0.706741i	$0.249835\pi$
$647$			914.523i			1.41348i	−0.707472	+	0.706741i	$0.750165\pi$
$648$		0			0
$649$	300.000			0.462250
$650$	−480.000	−	831.384i	−0.738462	−	1.27905i
$651$		0			0
$652$	360.000	+	207.846i	0.552147	+	0.318782i
$653$	103.000			0.157734			0.0788668	−	0.996885i	$-0.474870\pi$
$653$	103.000			0.157734			0.0788668	+	0.996885i	$0.474870\pi$
$654$		0			0
$655$			1151.81i			1.75849i
$656$	400.000	+	692.820i	0.609756	+	1.05613i
$657$		0			0
$658$	−750.000	−	1299.04i	−1.13982	−	1.97422i
$659$		−	60.6218i		−	0.0919906i	−0.998942	−	0.0459953i	$-0.985354\pi$
$659$		−	60.6218i		−	0.0919906i	0.998942	−	0.0459953i	$-0.0146459\pi$
$660$		0			0
$661$	578.000			0.874433			0.437216	−	0.899356i	$-0.355964\pi$
$661$	578.000			0.874433			0.437216	+	0.899356i	$0.355964\pi$
$662$	−750.000	+	433.013i	−1.13293	+	0.654098i
$663$		0			0
$664$		−	235.559i		−	0.354757i
$665$	630.000			0.947368
$666$		0			0
$667$			34.6410i			0.0519356i
$668$	852.000	+	491.902i	1.27545	+	0.736381i
$669$		0			0
$670$	−1050.00	+	606.218i	−1.56716	+	0.904803i
$671$			554.256i			0.826015i
$672$		0			0
$673$	845.000			1.25557			0.627786	−	0.778386i	$-0.283961\pi$
$673$	845.000			1.25557			0.627786	+	0.778386i	$0.283961\pi$
$674$	310.000	+	536.936i	0.459941	+	0.796641i
$675$		0			0
$676$	−462.000	+	800.207i	−0.683432	+	1.18374i
$677$	−1154.00			−1.70458			−0.852290	−	0.523070i	$-0.824786\pi$
$677$	−1154.00			−1.70458			−0.852290	+	0.523070i	$0.824786\pi$
$678$		0			0
$679$		−	216.506i		−	0.318861i
$680$	−448.000			−0.658824
$681$		0			0
$682$	−465.000	−	805.404i	−0.681818	−	1.18094i
$683$			187.061i			0.273882i	0.990579	+	0.136941i	$0.0437271\pi$
$683$			187.061i			0.273882i	−0.990579	+	0.136941i	$0.956273\pi$
$684$		0			0
$685$	−434.000			−0.633577
$686$	345.000	−	199.186i	0.502915	−	0.290358i
$687$		0			0
$688$	−240.000	+	138.564i	−0.348837	+	0.201401i
$689$	−940.000			−1.36430
$690$		0			0
$691$			491.902i			0.711870i	0.934511	+	0.355935i	$0.115838\pi$
$691$			491.902i			0.711870i	−0.934511	+	0.355935i	$0.884162\pi$
$692$	−254.000	+	439.941i	−0.367052	+	0.635753i
$693$		0			0
$694$	375.000	−	216.506i	0.540346	−	0.311969i
$695$			1212.44i			1.74451i
$696$		0			0
$697$	400.000			0.573888
$698$	−74.0000	−	128.172i	−0.106017	−	0.183627i
$699$		0			0
$700$	−720.000	−	415.692i	−1.02857	−	0.593846i
$701$	−215.000			−0.306705			−0.153352	−	0.988172i	$-0.549007\pi$
$701$	−215.000			−0.306705			−0.153352	+	0.988172i	$0.549007\pi$
$702$		0			0
$703$			103.923i			0.147828i
$704$		−	554.256i		−	0.787296i
$705$		0			0
$706$	−394.000	−	682.428i	−0.558074	−	0.966612i
$707$		−	1342.34i		−	1.89864i
$708$		0			0
$709$	−532.000			−0.750353			−0.375176	−	0.926953i	$-0.622418\pi$
$709$	−532.000			−0.750353			−0.375176	+	0.926953i	$0.622418\pi$
$710$		0			0
$711$		0			0
$712$	80.0000			0.112360
$713$	−186.000			−0.260870
$714$		0			0
$715$		−	1212.44i		−	1.69571i
$716$	810.000	+	467.654i	1.13128	+	0.653148i
$717$		0			0
$718$	990.000	−	571.577i	1.37883	−	0.796068i
$719$		−	1143.15i		−	1.58992i	−0.606661	−	0.794961i	$-0.707491\pi$
$719$		−	1143.15i		−	1.58992i	0.606661	−	0.794961i	$-0.292509\pi$
$720$		0			0
$721$	1200.00			1.66436
$722$	−253.000	−	438.209i	−0.350416	−	0.606937i
$723$		0			0
$724$	−112.000	+	193.990i	−0.154696	+	0.267942i
$725$	240.000			0.331034
$726$		0			0
$727$		−	1238.42i		−	1.70346i	−0.523980	−	0.851731i	$-0.675553\pi$
$727$		−	1238.42i		−	1.70346i	0.523980	−	0.851731i	$-0.324447\pi$
$728$			1385.64i			1.90335i
$729$		0			0
$730$	385.000	+	666.840i	0.527397	+	0.913479i
$731$			138.564i			0.189554i
$732$		0			0
$733$	950.000			1.29604			0.648022	−	0.761622i	$-0.275597\pi$
$733$	950.000			1.29604			0.648022	+	0.761622i	$0.275597\pi$
$734$	−975.000	+	562.917i	−1.32834	+	0.766916i
$735$		0			0
$736$	−96.0000	−	55.4256i	−0.130435	−	0.0753066i
$737$	−750.000			−1.01764
$738$		0			0
$739$			581.969i			0.787509i	0.919216	+	0.393754i	$0.128824\pi$
$739$			581.969i			0.787509i	−0.919216	+	0.393754i	$0.871176\pi$
$740$	140.000	−	242.487i	0.189189	−	0.327685i
$741$		0			0
$742$	−705.000	+	407.032i	−0.950135	+	0.548561i
$743$			866.025i			1.16558i	0.812623	+	0.582790i	$0.198039\pi$
$743$			866.025i			1.16558i	−0.812623	+	0.582790i	$0.801961\pi$
$744$		0			0
$745$	805.000			1.08054
$746$	40.0000	+	69.2820i	0.0536193	+	0.0928714i
$747$		0			0
$748$	−240.000	−	138.564i	−0.320856	−	0.185246i
$749$	−1125.00			−1.50200
$750$		0			0
$751$		−	174.937i		−	0.232939i	−0.993194	−	0.116469i	$-0.962842\pi$
$751$		−	174.937i		−	0.232939i	0.993194	−	0.116469i	$-0.0371577\pi$
$752$	−1200.00	+	692.820i	−1.59574	+	0.921304i
$753$		0			0
$754$	−200.000	−	346.410i	−0.265252	−	0.459430i
$755$		−	303.109i		−	0.401469i
$756$		0			0
$757$	830.000			1.09643			0.548217	−	0.836336i	$-0.315307\pi$
$757$	830.000			1.09643			0.548217	+	0.836336i	$0.315307\pi$
$758$	1188.00	−	685.892i	1.56728	−	0.904871i
$759$		0			0
$760$		−	581.969i		−	0.765749i
$761$	−560.000			−0.735874			−0.367937	−	0.929851i	$-0.619936\pi$
$761$	−560.000			−0.735874			−0.367937	+	0.929851i	$0.619936\pi$
$762$		0			0
$763$			1160.47i			1.52094i
$764$	120.000	+	69.2820i	0.157068	+	0.0906833i
$765$		0			0
$766$	−624.000	+	360.267i	−0.814621	+	0.470322i
$767$			692.820i			0.903286i
$768$		0			0
$769$	−331.000			−0.430429			−0.215215	−	0.976567i	$-0.569045\pi$
$769$	−331.000			−0.430429			−0.215215	+	0.976567i	$0.569045\pi$
$770$	−525.000	−	909.327i	−0.681818	−	1.18094i
$771$		0			0
$772$	−130.000	+	225.167i	−0.168394	+	0.291667i
$773$	298.000			0.385511			0.192755	−	0.981247i	$-0.438258\pi$
$773$	298.000			0.385511			0.192755	+	0.981247i	$0.438258\pi$
$774$		0			0
$775$			1288.65i			1.66277i
$776$	−200.000			−0.257732
$777$		0			0
$778$	−475.000	−	822.724i	−0.610540	−	1.05749i
$779$			519.615i			0.667029i
$780$		0			0
$781$		0			0
$782$	−48.0000	+	27.7128i	−0.0613811	+	0.0354384i
$783$		0			0
$784$	208.000	+	360.267i	0.265306	+	0.459524i
$785$	140.000			0.178344
$786$		0			0
$787$			1108.51i			1.40853i	0.709938	+	0.704265i	$0.248723\pi$
$787$			1108.51i			1.40853i	−0.709938	+	0.704265i	$0.751277\pi$
$788$	−506.000	+	876.418i	−0.642132	+	1.11221i
$789$		0			0
$790$	−84.0000	+	48.4974i	−0.106329	+	0.0613891i
$791$		−	640.859i		−	0.810188i
$792$		0			0
$793$	−1280.00			−1.61412
$794$	−260.000	−	450.333i	−0.327456	−	0.567170i
$795$		0			0
$796$	−450.000	−	259.808i	−0.565327	−	0.326391i
$797$	1303.00			1.63488			0.817440	−	0.576013i	$-0.195392\pi$
$797$	1303.00			1.63488			0.817440	+	0.576013i	$0.195392\pi$
$798$		0			0
$799$			692.820i			0.867109i
$800$	−384.000	+	665.108i	−0.480000	+	0.831384i
$801$		0			0
$802$	740.000	+	1281.72i	0.922693	+	1.59815i
$803$

Introduction	Features
Universe	News

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	108.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +7.00000 q^{5} +8.66025i q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +7.00000 q^{5} +8.66025i q^{7} +8.00000 q^{8} +(-7.00000 - 12.1244i) q^{10} -8.66025i q^{11} +20.0000 q^{13} +(15.0000 - 8.66025i) q^{14} +(-8.00000 - 13.8564i) q^{16} -8.00000 q^{17} -10.3923i q^{19} +(-14.0000 + 24.2487i) q^{20} +(-15.0000 + 8.66025i) q^{22} +3.46410i q^{23} +24.0000 q^{25} +(-20.0000 - 34.6410i) q^{26} +(-30.0000 - 17.3205i) q^{28} +10.0000 q^{29} +53.6936i q^{31} +(-16.0000 + 27.7128i) q^{32} +(8.00000 + 13.8564i) q^{34} +60.6218i q^{35} -10.0000 q^{37} +(-18.0000 + 10.3923i) q^{38} +56.0000 q^{40} -50.0000 q^{41} -17.3205i q^{43} +(30.0000 + 17.3205i) q^{44} +(6.00000 - 3.46410i) q^{46} -86.6025i q^{47} -26.0000 q^{49} +(-24.0000 - 41.5692i) q^{50} +(-40.0000 + 69.2820i) q^{52} -47.0000 q^{53} -60.6218i q^{55} +69.2820i q^{56} +(-10.0000 - 17.3205i) q^{58} +34.6410i q^{59} -64.0000 q^{61} +(93.0000 - 53.6936i) q^{62} +64.0000 q^{64} +140.000 q^{65} -86.6025i q^{67} +(16.0000 - 27.7128i) q^{68} +(105.000 - 60.6218i) q^{70} -55.0000 q^{73} +(10.0000 + 17.3205i) q^{74} +(36.0000 + 20.7846i) q^{76} +75.0000 q^{77} -6.92820i q^{79} +(-56.0000 - 96.9948i) q^{80} +(50.0000 + 86.6025i) q^{82} -29.4449i q^{83} -56.0000 q^{85} +(-30.0000 + 17.3205i) q^{86} -69.2820i q^{88} +10.0000 q^{89} +173.205i q^{91} +(-12.0000 - 6.92820i) q^{92} +(-150.000 + 86.6025i) q^{94} -72.7461i q^{95} -25.0000 q^{97} +(26.0000 + 45.0333i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q - 2q^{2} - 4q^{4} + 14q^{5} + 16q^{8} + O(q^{10}) \)	\( 2q - 2q^{2} - 4q^{4} + 14q^{5} + 16q^{8} - 14q^{10} + 40q^{13} + 30q^{14} - 16q^{16} - 16q^{17} - 28q^{20} - 30q^{22} + 48q^{25} - 40q^{26} - 60q^{28} + 20q^{29} - 32q^{32} + 16q^{34} - 20q^{37} - 36q^{38} + 112q^{40} - 100q^{41} + 60q^{44} + 12q^{46} - 52q^{49} - 48q^{50} - 80q^{52} - 94q^{53} - 20q^{58} - 128q^{61} + 186q^{62} + 128q^{64} + 280q^{65} + 32q^{68} + 210q^{70} - 110q^{73} + 20q^{74} + 72q^{76} + 150q^{77} - 112q^{80} + 100q^{82} - 112q^{85} - 60q^{86} + 20q^{89} - 24q^{92} - 300q^{94} - 50q^{97} + 52q^{98} + O(q^{100}) \)

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(1\)	\(-1\)

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L-functions

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Knowledge

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