LMFDB - Embedded newform 2646.2.h.s.361.3

Newspace parameters

Level:	$ N $	$=$	$ 2646 = 2 \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2646.h (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$21.1284163748$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.3317760000.3
Defining polynomial:	$ x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 7x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 882)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.3
Root			$1.72286 + 0.178197i$ of defining polynomial
Character	$\chi$	$=$	2646.361
Dual form			2646.2.h.s.667.3

$q$-expansion

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +2.03151 q^{5} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +2.03151 q^{5} -1.00000 q^{8} +(1.01575 - 1.75934i) q^{10} -4.00000 q^{11} +(-2.12132 + 3.67423i) q^{13} +(-0.500000 + 0.866025i) q^{16} +(-0.707107 + 1.22474i) q^{17} +(-0.398461 - 0.690154i) q^{19} +(-1.01575 - 1.75934i) q^{20} +(-2.00000 + 3.46410i) q^{22} -6.74597 q^{23} -0.872983 q^{25} +(2.12132 + 3.67423i) q^{26} +(4.43649 + 7.68423i) q^{29} +(-2.73861 - 4.74342i) q^{31} +(0.500000 + 0.866025i) q^{32} +(0.707107 + 1.22474i) q^{34} +(4.87298 + 8.44025i) q^{37} -0.796921 q^{38} -2.03151 q^{40} +(-2.82843 + 4.89898i) q^{41} +(-4.43649 - 7.68423i) q^{43} +(2.00000 + 3.46410i) q^{44} +(-3.37298 + 5.84218i) q^{46} +(-3.44572 + 5.96816i) q^{47} +(-0.436492 + 0.756026i) q^{50} +4.24264 q^{52} +(-5.30948 + 9.19628i) q^{53} -8.12602 q^{55} +8.87298 q^{58} +(-1.32440 - 2.29393i) q^{59} +(0.398461 - 0.690154i) q^{61} -5.47723 q^{62} +1.00000 q^{64} +(-4.30948 + 7.46423i) q^{65} +(-0.436492 - 0.756026i) q^{67} +1.41421 q^{68} +2.12702 q^{71} +(7.68836 - 13.3166i) q^{73} +9.74597 q^{74} +(-0.398461 + 0.690154i) q^{76} +(-2.93649 + 5.08615i) q^{79} +(-1.01575 + 1.75934i) q^{80} +(2.82843 + 4.89898i) q^{82} +(-4.15283 - 7.19291i) q^{83} +(-1.43649 + 2.48808i) q^{85} -8.87298 q^{86} +4.00000 q^{88} +(-3.53553 - 6.12372i) q^{89} +(3.37298 + 5.84218i) q^{92} +(3.44572 + 5.96816i) q^{94} +(-0.809475 - 1.40205i) q^{95} +(-8.92295 - 15.4550i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8}+O(q^{10}) $ 8 * q + 4 * q^2 - 4 * q^4 - 8 * q^8	$ 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} - 32 q^{11} - 4 q^{16} - 16 q^{22} + 8 q^{23} + 24 q^{25} + 20 q^{29} + 4 q^{32} + 8 q^{37} - 20 q^{43} + 16 q^{44} + 4 q^{46} + 12 q^{50} + 4 q^{53} + 40 q^{58} + 8 q^{64} + 12 q^{65} + 12 q^{67} + 48 q^{71} + 16 q^{74} - 8 q^{79} + 4 q^{85} - 40 q^{86} + 32 q^{88} - 4 q^{92} + 40 q^{95}+O(q^{100}) $ 8 * q + 4 * q^2 - 4 * q^4 - 8 * q^8 - 32 * q^11 - 4 * q^16 - 16 * q^22 + 8 * q^23 + 24 * q^25 + 20 * q^29 + 4 * q^32 + 8 * q^37 - 20 * q^43 + 16 * q^44 + 4 * q^46 + 12 * q^50 + 4 * q^53 + 40 * q^58 + 8 * q^64 + 12 * q^65 + 12 * q^67 + 48 * q^71 + 16 * q^74 - 8 * q^79 + 4 * q^85 - 40 * q^86 + 32 * q^88 - 4 * q^92 + 40 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$

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 $

$2$	0.500000	−	0.866025i	0.353553	−	0.612372i
$2$	0.500000	−	0.866025i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$	2.03151			0.908517			0.454259	−	0.890870i	$-0.349904\pi$
$5$	2.03151			0.908517			0.454259	+	0.890870i	$0.349904\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−1.00000			−0.353553
$9$		0			0
$10$	1.01575	−	1.75934i	0.321209	−	0.556351i
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$	−2.12132	+	3.67423i	−0.588348	+	1.01905i	0.406100	+	0.913828i	$0.366888\pi$
$13$	−2.12132	+	3.67423i	−0.588348	+	1.01905i	−0.994449	+	0.105221i	$0.966445\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	−0.707107	+	1.22474i	−0.171499	+	0.297044i	−0.938944	−	0.344070i	$-0.888194\pi$
$17$	−0.707107	+	1.22474i	−0.171499	+	0.297044i	0.767445	+	0.641114i	$0.221528\pi$
$18$		0			0
$19$	−0.398461	−	0.690154i	−0.0914131	−	0.158332i	0.816693	−	0.577073i	$-0.195805\pi$
$19$	−0.398461	−	0.690154i	−0.0914131	−	0.158332i	−0.908106	+	0.418740i	$0.862472\pi$
$20$	−1.01575	−	1.75934i	−0.227129	−	0.393399i
$21$		0			0
$22$	−2.00000	+	3.46410i	−0.426401	+	0.738549i
$23$	−6.74597			−1.40663			−0.703316	−	0.710878i	$-0.748298\pi$
$23$	−6.74597			−1.40663			−0.703316	+	0.710878i	$0.748298\pi$
$24$		0			0
$25$	−0.872983			−0.174597
$26$	2.12132	+	3.67423i	0.416025	+	0.720577i
$27$		0			0
$28$		0			0
$29$	4.43649	+	7.68423i	0.823836	+	1.42693i	0.902806	+	0.430049i	$0.141504\pi$
$29$	4.43649	+	7.68423i	0.823836	+	1.42693i	−0.0789700	+	0.996877i	$0.525163\pi$
$30$		0			0
$31$	−2.73861	−	4.74342i	−0.491869	−	0.851943i	0.508087	−	0.861306i	$-0.330353\pi$
$31$	−2.73861	−	4.74342i	−0.491869	−	0.851943i	−0.999956	+	0.00936313i	$0.997020\pi$
$32$	0.500000	+	0.866025i	0.0883883	+	0.153093i
$33$		0			0
$34$	0.707107	+	1.22474i	0.121268	+	0.210042i
$35$		0			0
$36$		0			0
$37$	4.87298	+	8.44025i	0.801114	+	1.38757i	0.918884	+	0.394528i	$0.129092\pi$
$37$	4.87298	+	8.44025i	0.801114	+	1.38757i	−0.117770	+	0.993041i	$0.537575\pi$
$38$	−0.796921			−0.129278
$39$		0			0
$40$	−2.03151			−0.321209
$41$	−2.82843	+	4.89898i	−0.441726	+	0.765092i	−0.997818	−	0.0660290i	$-0.978967\pi$
$41$	−2.82843	+	4.89898i	−0.441726	+	0.765092i	0.556092	+	0.831121i	$0.312300\pi$
$42$		0			0
$43$	−4.43649	−	7.68423i	−0.676559	−	1.17183i	−0.976011	−	0.217723i	$-0.930137\pi$
$43$	−4.43649	−	7.68423i	−0.676559	−	1.17183i	0.299452	−	0.954111i	$-0.403196\pi$
$44$	2.00000	+	3.46410i	0.301511	+	0.522233i
$45$		0			0
$46$	−3.37298	+	5.84218i	−0.497319	+	0.861382i
$47$	−3.44572	+	5.96816i	−0.502610	+	0.870546i	0.497386	+	0.867530i	$0.334293\pi$
$47$	−3.44572	+	5.96816i	−0.502610	+	0.870546i	−0.999995	+	0.00301623i	$0.999040\pi$
$48$		0			0
$49$		0			0
$50$	−0.436492	+	0.756026i	−0.0617292	+	0.106918i
$51$		0			0
$52$	4.24264			0.588348
$53$	−5.30948	+	9.19628i	−0.729312	+	1.26321i	0.227862	+	0.973694i	$0.426827\pi$
$53$	−5.30948	+	9.19628i	−0.729312	+	1.26321i	−0.957174	+	0.289513i	$0.906507\pi$
$54$		0			0
$55$	−8.12602			−1.09571
$56$		0			0
$57$		0			0
$58$	8.87298			1.16508
$59$	−1.32440	−	2.29393i	−0.172422	−	0.298644i	0.766844	−	0.641833i	$-0.221826\pi$
$59$	−1.32440	−	2.29393i	−0.172422	−	0.298644i	−0.939266	+	0.343190i	$0.888493\pi$
$60$		0			0
$61$	0.398461	−	0.690154i	0.0510176	−	0.0883652i	−0.839389	−	0.543531i	$-0.817087\pi$
$61$	0.398461	−	0.690154i	0.0510176	−	0.0883652i	0.890406	+	0.455166i	$0.150420\pi$
$62$	−5.47723			−0.695608
$63$		0			0
$64$	1.00000			0.125000
$65$	−4.30948	+	7.46423i	−0.534525	+	0.925824i
$66$		0			0
$67$	−0.436492	−	0.756026i	−0.0533259	−	0.0923632i	0.838130	−	0.545470i	$-0.183649\pi$
$67$	−0.436492	−	0.756026i	−0.0533259	−	0.0923632i	−0.891456	+	0.453107i	$0.850316\pi$
$68$	1.41421			0.171499
$69$		0			0
$70$		0			0
$71$	2.12702			0.252430			0.126215	−	0.992003i	$-0.459717\pi$
$71$	2.12702			0.252430			0.126215	+	0.992003i	$0.459717\pi$
$72$		0			0
$73$	7.68836	−	13.3166i	0.899855	−	1.55859i	0.0721755	−	0.997392i	$-0.477006\pi$
$73$	7.68836	−	13.3166i	0.899855	−	1.55859i	0.827679	−	0.561202i	$-0.189661\pi$
$74$	9.74597			1.13295
$75$		0			0
$76$	−0.398461	+	0.690154i	−0.0457066	+	0.0791661i
$77$		0			0
$78$		0			0
$79$	−2.93649	+	5.08615i	−0.330381	+	0.572237i	−0.982587	−	0.185805i	$-0.940511\pi$
$79$	−2.93649	+	5.08615i	−0.330381	+	0.572237i	0.652205	+	0.758042i	$0.273844\pi$
$80$	−1.01575	+	1.75934i	−0.113565	+	0.196700i
$81$		0			0
$82$	2.82843	+	4.89898i	0.312348	+	0.541002i
$83$	−4.15283	−	7.19291i	−0.455832	−	0.789524i	0.542904	−	0.839795i	$-0.317325\pi$
$83$	−4.15283	−	7.19291i	−0.455832	−	0.789524i	−0.998736	+	0.0502709i	$0.983992\pi$
$84$		0			0
$85$	−1.43649	+	2.48808i	−0.155809	+	0.269870i
$86$	−8.87298			−0.956798
$87$		0			0
$88$	4.00000			0.426401
$89$	−3.53553	−	6.12372i	−0.374766	−	0.649113i	0.615526	−	0.788116i	$-0.288944\pi$
$89$	−3.53553	−	6.12372i	−0.374766	−	0.649113i	−0.990292	+	0.139003i	$0.955610\pi$
$90$		0			0
$91$		0			0
$92$	3.37298	+	5.84218i	0.351658	+	0.609089i
$93$		0			0
$94$	3.44572	+	5.96816i	0.355399	+	0.615569i
$95$	−0.809475	−	1.40205i	−0.0830504	−	0.143847i
$96$		0			0
$97$	−8.92295	−	15.4550i	−0.905988	−	1.56922i	−0.819586	−	0.572957i	$-0.805796\pi$
$97$	−8.92295	−	15.4550i	−0.905988	−	1.56922i	−0.0864021	−	0.996260i	$-0.527537\pi$
$98$		0			0
$99$		0			0
$100$	0.436492	+	0.756026i	0.0436492	+	0.0756026i
$101$	−7.86799			−0.782894			−0.391447	−	0.920201i	$-0.628025\pi$
$101$	−7.86799			−0.782894			−0.391447	+	0.920201i	$0.628025\pi$
$102$		0			0
$103$	18.0255			1.77611			0.888054	−	0.459740i	$-0.152057\pi$
$103$	18.0255			1.77611			0.888054	+	0.459740i	$0.152057\pi$
$104$	2.12132	−	3.67423i	0.208013	−	0.360288i
$105$		0			0
$106$	5.30948	+	9.19628i	0.515702	+	0.893222i
$107$	7.87298	+	13.6364i	0.761110	+	1.31828i	0.942279	+	0.334829i	$0.108679\pi$
$107$	7.87298	+	13.6364i	0.761110	+	1.31828i	−0.181169	+	0.983452i	$0.557988\pi$
$108$		0			0
$109$	−4.56351	+	7.90423i	−0.437105	+	0.757088i	−0.997465	−	0.0711614i	$-0.977329\pi$
$109$	−4.56351	+	7.90423i	−0.437105	+	0.757088i	0.560360	+	0.828249i	$0.310663\pi$
$110$	−4.06301	+	7.03734i	−0.387393	+	0.670984i
$111$		0			0
$112$		0			0
$113$	−1.93649	+	3.35410i	−0.182170	+	0.315527i	−0.942619	−	0.333870i	$-0.891645\pi$
$113$	−1.93649	+	3.35410i	−0.182170	+	0.315527i	0.760449	+	0.649397i	$0.224979\pi$
$114$		0			0
$115$	−13.7045			−1.27795
$116$	4.43649	−	7.68423i	0.411918	−	0.713463i
$117$		0			0
$118$	−2.64880			−0.243842
$119$		0			0
$120$		0			0
$121$	5.00000			0.454545
$122$	−0.398461	−	0.690154i	−0.0360749	−	0.0624836i
$123$		0			0
$124$	−2.73861	+	4.74342i	−0.245935	+	0.425971i
$125$	−11.9310			−1.06714
$126$		0			0
$127$	14.7460			1.30849			0.654246	−	0.756281i	$-0.272986\pi$
$127$	14.7460			1.30849			0.654246	+	0.756281i	$0.272986\pi$
$128$	0.500000	−	0.866025i	0.0441942	−	0.0765466i
$129$		0			0
$130$	4.30948	+	7.46423i	0.377966	+	0.654656i
$131$	7.86799			0.687429			0.343715	−	0.939074i	$-0.388315\pi$
$131$	7.86799			0.687429			0.343715	+	0.939074i	$0.388315\pi$
$132$		0			0
$133$		0			0
$134$	−0.872983			−0.0754143
$135$		0			0
$136$	0.707107	−	1.22474i	0.0606339	−	0.105021i
$137$	−15.4919			−1.32357			−0.661783	−	0.749696i	$-0.730200\pi$
$137$	−15.4919			−1.32357			−0.661783	+	0.749696i	$0.730200\pi$
$138$		0			0
$139$	−9.23159	+	15.9896i	−0.783013	+	1.35622i	0.147165	+	0.989112i	$0.452985\pi$
$139$	−9.23159	+	15.9896i	−0.783013	+	1.35622i	−0.930179	+	0.367107i	$0.880348\pi$
$140$		0			0
$141$		0			0
$142$	1.06351	−	1.84205i	0.0892476	−	0.154581i
$143$	8.48528	−	14.6969i	0.709575	−	1.22902i
$144$		0			0
$145$	9.01276	+	15.6106i	0.748469	+	1.29639i
$146$	−7.68836	−	13.3166i	−0.636293	−	1.10209i
$147$		0			0
$148$	4.87298	−	8.44025i	0.400557	−	0.693785i
$149$	0.254033			0.0208112			0.0104056	−	0.999946i	$-0.496688\pi$
$149$	0.254033			0.0208112			0.0104056	+	0.999946i	$0.496688\pi$
$150$		0			0
$151$	11.0000			0.895167			0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000			0.895167			0.447584	+	0.894242i	$0.352285\pi$
$152$	0.398461	+	0.690154i	0.0323194	+	0.0559789i
$153$		0			0
$154$		0			0
$155$	−5.56351	−	9.63628i	−0.446872	−	0.774005i
$156$		0			0
$157$	3.22689	+	5.58913i	0.257534	+	0.446061i	0.965581	−	0.260104i	$-0.0837568\pi$
$157$	3.22689	+	5.58913i	0.257534	+	0.446061i	−0.708047	+	0.706165i	$0.750424\pi$
$158$	2.93649	+	5.08615i	0.233615	+	0.404633i
$159$		0			0
$160$	1.01575	+	1.75934i	0.0803023	+	0.139088i
$161$		0			0
$162$		0			0
$163$	−5.00000	−	8.66025i	−0.391630	−	0.678323i	0.601035	−	0.799223i	$-0.294755\pi$
$163$	−5.00000	−	8.66025i	−0.391630	−	0.678323i	−0.992665	+	0.120900i	$0.961422\pi$
$164$	5.65685			0.441726
$165$		0			0
$166$	−8.30565			−0.644644
$167$	6.18433	−	10.7116i	0.478558	−	0.828887i	−0.521140	−	0.853471i	$-0.674493\pi$
$167$	6.18433	−	10.7116i	0.478558	−	0.828887i	0.999698	+	0.0245846i	$0.00782630\pi$
$168$		0			0
$169$	−2.50000	−	4.33013i	−0.192308	−	0.333087i
$170$	1.43649	+	2.48808i	0.110174	+	0.190827i
$171$		0			0
$172$	−4.43649	+	7.68423i	−0.338279	+	0.585917i
$173$	−6.36396	+	11.0227i	−0.483843	+	0.838041i	−0.999828	−	0.0185571i	$-0.994093\pi$
$173$	−6.36396	+	11.0227i	−0.483843	+	0.838041i	0.515985	+	0.856598i	$0.327426\pi$
$174$		0			0
$175$		0			0
$176$	2.00000	−	3.46410i	0.150756	−	0.261116i
$177$		0			0
$178$	−7.07107			−0.529999
$179$	0.436492	−	0.756026i	0.0326249	−	0.0565080i	−0.849252	−	0.527988i	$-0.822947\pi$
$179$	0.436492	−	0.756026i	0.0326249	−	0.0565080i	0.881877	+	0.471480i	$0.156280\pi$
$180$		0			0
$181$	7.86799			0.584823			0.292412	−	0.956293i	$-0.405542\pi$
$181$	7.86799			0.584823			0.292412	+	0.956293i	$0.405542\pi$
$182$		0			0
$183$		0			0
$184$	6.74597			0.497319
$185$	9.89949	+	17.1464i	0.727825	+	1.26063i
$186$		0			0
$187$	2.82843	−	4.89898i	0.206835	−	0.358249i
$188$	6.89144			0.502610
$189$		0			0
$190$	−1.61895			−0.117451
$191$	−6.93649	+	12.0144i	−0.501907	+	0.869328i	0.498091	+	0.867125i	$0.334035\pi$
$191$	−6.93649	+	12.0144i	−0.501907	+	0.869328i	−0.999998	+	0.00220333i	$0.999299\pi$
$192$		0			0
$193$	−8.06351	−	13.9664i	−0.580424	−	1.00532i	−0.995429	−	0.0955048i	$-0.969553\pi$
$193$	−8.06351	−	13.9664i	−0.580424	−	1.00532i	0.415005	−	0.909819i	$-0.363780\pi$
$194$	−17.8459			−1.28126
$195$		0			0
$196$		0			0
$197$	6.61895			0.471581			0.235790	−	0.971804i	$-0.424232\pi$
$197$	6.61895			0.471581			0.235790	+	0.971804i	$0.424232\pi$
$198$		0			0
$199$	−10.3372	+	17.9045i	−0.732782	+	1.26922i	0.222908	+	0.974839i	$0.428445\pi$
$199$	−10.3372	+	17.9045i	−0.732782	+	1.26922i	−0.955690	+	0.294376i	$0.904888\pi$
$200$	0.872983			0.0617292
$201$		0			0
$202$	−3.93399	+	6.81388i	−0.276795	+	0.479423i
$203$		0			0
$204$		0			0
$205$	−5.74597	+	9.95231i	−0.401316	+	0.695099i
$206$	9.01276	−	15.6106i	0.627949	−	1.08764i
$207$		0			0
$208$	−2.12132	−	3.67423i	−0.147087	−	0.254762i
$209$	1.59384	+	2.76062i	0.110248	+	0.190956i
$210$		0			0
$211$	1.30948	−	2.26808i	0.0901480	−	0.156141i	−0.817425	−	0.576035i	$-0.804599\pi$
$211$	1.30948	−	2.26808i	0.0901480	−	0.156141i	0.907573	+	0.419894i	$0.137933\pi$
$212$	10.6190			0.729312
$213$		0			0
$214$	15.7460			1.07637
$215$	−9.01276	−	15.6106i	−0.614665	−	1.06463i
$216$		0			0
$217$		0			0
$218$	4.56351	+	7.90423i	0.309080	+	0.535342i
$219$		0			0
$220$	4.06301	+	7.03734i	0.273928	+	0.474458i
$221$	−3.00000	−	5.19615i	−0.201802	−	0.349531i
$222$		0			0
$223$	−5.74667	−	9.95352i	−0.384825	−	0.666537i	0.606920	−	0.794763i	$-0.292405\pi$
$223$	−5.74667	−	9.95352i	−0.384825	−	0.666537i	−0.991745	+	0.128226i	$0.959072\pi$
$224$		0			0
$225$		0			0
$226$	1.93649	+	3.35410i	0.128814	+	0.223112i
$227$	−6.09452			−0.404507			−0.202254	−	0.979333i	$-0.564827\pi$
$227$	−6.09452			−0.404507			−0.202254	+	0.979333i	$0.564827\pi$
$228$		0			0
$229$	−7.86799			−0.519931			−0.259966	−	0.965618i	$-0.583711\pi$
$229$	−7.86799			−0.519931			−0.259966	+	0.965618i	$0.583711\pi$
$230$	−6.85224	+	11.8684i	−0.451823	+	0.782580i
$231$		0			0
$232$	−4.43649	−	7.68423i	−0.291270	−	0.504494i
$233$	11.3730	+	19.6986i	0.745069	+	1.29050i	0.950163	+	0.311755i	$0.100917\pi$
$233$	11.3730	+	19.6986i	0.745069	+	1.29050i	−0.205094	+	0.978742i	$0.565750\pi$
$234$		0			0
$235$	−7.00000	+	12.1244i	−0.456630	+	0.790906i
$236$	−1.32440	+	2.29393i	−0.0862110	+	0.149322i
$237$		0			0
$238$		0			0
$239$	−7.50000	+	12.9904i	−0.485135	+	0.840278i	−0.999854	−	0.0170808i	$-0.994563\pi$
$239$	−7.50000	+	12.9904i	−0.485135	+	0.840278i	0.514719	+	0.857359i	$0.327896\pi$
$240$		0			0
$241$	1.77347			0.114239			0.0571197	−	0.998367i	$-0.481808\pi$
$241$	1.77347			0.114239			0.0571197	+	0.998367i	$0.481808\pi$
$242$	2.50000	−	4.33013i	0.160706	−	0.278351i
$243$		0			0
$244$	−0.796921			−0.0510176
$245$		0			0
$246$		0			0
$247$	3.38105			0.215131
$248$	2.73861	+	4.74342i	0.173902	+	0.301207i
$249$		0			0
$250$	−5.96550	+	10.3325i	−0.377291	+	0.653488i
$251$	25.8935			1.63438			0.817192	−	0.576366i	$-0.195530\pi$
$251$	25.8935			1.63438			0.817192	+	0.576366i	$0.195530\pi$
$252$		0			0
$253$	26.9839			1.69646
$254$	7.37298	−	12.7704i	0.462622	−	0.801285i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	13.9625			0.870957			0.435479	−	0.900199i	$-0.356579\pi$
$257$	13.9625			0.870957			0.435479	+	0.900199i	$0.356579\pi$
$258$		0			0
$259$		0			0
$260$	8.61895			0.534525
$261$		0			0
$262$	3.93399	−	6.81388i	0.243043	−	0.420963i
$263$	−5.61895			−0.346479			−0.173240	−	0.984880i	$-0.555424\pi$
$263$	−5.61895			−0.346479			−0.173240	+	0.984880i	$0.555424\pi$
$264$		0			0
$265$	−10.7862	+	18.6823i	−0.662593	+	1.14764i
$266$		0			0
$267$		0			0
$268$	−0.436492	+	0.756026i	−0.0266630	+	0.0461816i
$269$	−1.72286	+	2.98408i	−0.105045	+	0.181943i	−0.913756	−	0.406262i	$-0.866832\pi$
$269$	−1.72286	+	2.98408i	−0.105045	+	0.181943i	0.808712	+	0.588205i	$0.200165\pi$
$270$		0			0
$271$	3.53553	+	6.12372i	0.214768	+	0.371990i	0.953201	−	0.302338i	$-0.0977670\pi$
$271$	3.53553	+	6.12372i	0.214768	+	0.371990i	−0.738433	+	0.674327i	$0.764434\pi$
$272$	−0.707107	−	1.22474i	−0.0428746	−	0.0742611i
$273$		0			0
$274$	−7.74597	+	13.4164i	−0.467951	+	0.810515i
$275$	3.49193			0.210572
$276$		0			0
$277$	−24.3649			−1.46395			−0.731973	−	0.681334i	$-0.761400\pi$
$277$	−24.3649			−1.46395			−0.731973	+	0.681334i	$0.761400\pi$
$278$	9.23159	+	15.9896i	0.553674	+	0.958992i
$279$		0			0
$280$		0			0
$281$	−3.37298	−	5.84218i	−0.201215	−	0.348515i	0.747705	−	0.664031i	$-0.231156\pi$
$281$	−3.37298	−	5.84218i	−0.201215	−	0.348515i	−0.948920	+	0.315516i	$0.897822\pi$
$282$		0			0
$283$	5.25839	+	9.10781i	0.312579	+	0.541403i	0.978920	−	0.204244i	$-0.0654737\pi$
$283$	5.25839	+	9.10781i	0.312579	+	0.541403i	−0.666341	+	0.745647i	$0.732140\pi$
$284$	−1.06351	−	1.84205i	−0.0631076	−	0.109306i
$285$		0			0
$286$	−8.48528	−	14.6969i	−0.501745	−	0.869048i
$287$		0			0
$288$		0			0
$289$	7.50000	+	12.9904i	0.441176	+	0.764140i
$290$	18.0255			1.05849
$291$		0			0
$292$	−15.3767			−0.899855
$293$	3.13707	−	5.43357i	0.183270	−	0.317433i	−0.759722	−	0.650248i	$-0.774665\pi$
$293$	3.13707	−	5.43357i	0.183270	−	0.317433i	0.942992	+	0.332815i	$0.107998\pi$
$294$		0			0
$295$	−2.69052	−	4.66013i	−0.156648	−	0.271323i
$296$	−4.87298	−	8.44025i	−0.283236	−	0.490580i
$297$		0			0
$298$	0.127017	−	0.219999i	0.00735788	−	0.0127442i
$299$	14.3104	−	24.7863i	0.827589	−	1.43343i
$300$		0			0
$301$		0			0
$302$	5.50000	−	9.52628i	0.316489	−	0.548176i
$303$		0			0
$304$	0.796921			0.0457066
$305$	0.809475	−	1.40205i	0.0463504	−	0.0802813i
$306$		0			0
$307$	24.1200			1.37660			0.688302	−	0.725425i	$-0.258357\pi$
$307$	24.1200			1.37660			0.688302	+	0.725425i	$0.258357\pi$
$308$		0			0
$309$		0			0
$310$	−11.1270			−0.631972
$311$	0.707107	+	1.22474i	0.0400963	+	0.0694489i	0.885377	−	0.464873i	$-0.153900\pi$
$311$	0.707107	+	1.22474i	0.0400963	+	0.0694489i	−0.845281	+	0.534322i	$0.820567\pi$
$312$		0			0
$313$	13.3452	−	23.1146i	0.754316	−	1.30651i	−0.191397	−	0.981513i	$-0.561302\pi$
$313$	13.3452	−	23.1146i	0.754316	−	1.30651i	0.945714	−	0.325001i	$-0.105365\pi$
$314$	6.45378			0.364208
$315$		0			0
$316$	5.87298			0.330381
$317$	0.690525	−	1.19602i	0.0387837	−	0.0671754i	−0.845982	−	0.533212i	$-0.820985\pi$
$317$	0.690525	−	1.19602i	0.0387837	−	0.0671754i	0.884766	+	0.466036i	$0.154318\pi$
$318$		0			0
$319$	−17.7460	−	30.7369i	−0.993583	−	1.72094i
$320$	2.03151			0.113565
$321$		0			0
$322$		0			0
$323$	1.12702			0.0627089
$324$		0			0
$325$	1.85188	−	3.20755i	0.102724	−	0.177923i
$326$	−10.0000			−0.553849
$327$		0			0
$328$	2.82843	−	4.89898i	0.156174	−	0.270501i
$329$		0			0
$330$		0			0
$331$	9.18246	−	15.9045i	0.504714	−	0.874190i	−0.495272	−	0.868738i	$-0.664931\pi$
$331$	9.18246	−	15.9045i	0.504714	−	0.874190i	0.999985	−	0.00545133i	$-0.00173522\pi$
$332$	−4.15283	+	7.19291i	−0.227916	+	0.394762i
$333$		0			0
$334$	−6.18433	−	10.7116i	−0.338392	−	0.586111i
$335$	−0.886735	−	1.53587i	−0.0484475	−	0.0839136i
$336$		0			0
$337$	−0.127017	+	0.219999i	−0.00691904	+	0.0119841i	−0.869464	−	0.493996i	$-0.835536\pi$
$337$	−0.127017	+	0.219999i	−0.00691904	+	0.0119841i	0.862545	+	0.505980i	$0.168869\pi$
$338$	−5.00000			−0.271964
$339$		0			0
$340$	2.87298			0.155809
$341$	10.9545	+	18.9737i	0.593217	+	1.02748i
$342$		0			0
$343$		0			0
$344$	4.43649	+	7.68423i	0.239200	+	0.414306i
$345$		0			0
$346$	6.36396	+	11.0227i	0.342129	+	0.592584i
$347$	−3.87298	−	6.70820i	−0.207913	−	0.360115i	0.743144	−	0.669131i	$-0.233334\pi$
$347$	−3.87298	−	6.70820i	−0.207913	−	0.360115i	−0.951057	+	0.309016i	$0.900000\pi$
$348$		0			0
$349$	8.21584	+	14.2302i	0.439784	+	0.761728i	0.997672	−	0.0681880i	$-0.0217218\pi$
$349$	8.21584	+	14.2302i	0.439784	+	0.761728i	−0.557889	+	0.829916i	$0.688388\pi$
$350$		0			0
$351$		0			0
$352$	−2.00000	−	3.46410i	−0.106600	−	0.184637i
$353$	−18.0255			−0.959402			−0.479701	−	0.877432i	$-0.659255\pi$
$353$	−18.0255			−0.959402			−0.479701	+	0.877432i	$0.659255\pi$
$354$		0			0
$355$	4.32105			0.229337
$356$	−3.53553	+	6.12372i	−0.187383	+	0.324557i
$357$		0			0
$358$	−0.436492	−	0.756026i	−0.0230693	−	0.0399572i
$359$	−14.1190	−	24.4547i	−0.745170	−	1.29067i	−0.950115	−	0.311898i	$-0.899035\pi$
$359$	−14.1190	−	24.4547i	−0.745170	−	1.29067i	0.204946	−	0.978773i	$-0.434298\pi$
$360$		0			0
$361$	9.18246	−	15.9045i	0.483287	−	0.837078i
$362$	3.93399	−	6.81388i	0.206766	−	0.358129i
$363$		0			0
$364$		0			0
$365$	15.6190	−	27.0528i	0.817533	−	1.41601i
$366$		0			0
$367$	4.06301			0.212088			0.106044	−	0.994361i	$-0.466182\pi$
$367$	4.06301			0.212088			0.106044	+	0.994361i	$0.466182\pi$
$368$	3.37298	−	5.84218i	0.175829	−	0.304545i
$369$		0			0
$370$	19.7990			1.02930
$371$		0			0
$372$		0			0
$373$	−8.87298			−0.459426			−0.229713	−	0.973258i	$-0.573779\pi$
$373$	−8.87298			−0.459426			−0.229713	+	0.973258i	$0.573779\pi$
$374$	−2.82843	−	4.89898i	−0.146254	−	0.253320i
$375$		0			0
$376$	3.44572	−	5.96816i	0.177699	−	0.307784i
$377$	−37.6449			−1.93881
$378$		0			0
$379$	−12.3649			−0.635143			−0.317572	−	0.948234i	$-0.602867\pi$
$379$	−12.3649			−0.635143			−0.317572	+	0.948234i	$0.602867\pi$
$380$	−0.809475	+	1.40205i	−0.0415252	+	0.0719237i
$381$		0			0
$382$	6.93649	+	12.0144i	0.354902	+	0.614708i
$383$	−23.8620			−1.21929			−0.609646	−	0.792674i	$-0.708688\pi$
$383$	−23.8620			−1.21929			−0.609646	+	0.792674i	$0.708688\pi$
$384$		0			0
$385$		0			0
$386$	−16.1270			−0.820844
$387$		0			0
$388$	−8.92295	+	15.4550i	−0.452994	+	0.784608i
$389$	−20.8730			−1.05830			−0.529151	−	0.848528i	$-0.677490\pi$
$389$	−20.8730			−1.05830			−0.529151	+	0.848528i	$0.677490\pi$
$390$		0			0
$391$	4.77012	−	8.26209i	0.241235	−	0.417832i
$392$		0			0
$393$		0			0
$394$	3.30948	−	5.73218i	0.166729	−	0.288783i
$395$	−5.96550	+	10.3325i	−0.300157	+	0.519887i
$396$		0			0
$397$	3.53553	+	6.12372i	0.177443	+	0.307341i	0.941004	−	0.338395i	$-0.109884\pi$
$397$	3.53553	+	6.12372i	0.177443	+	0.307341i	−0.763561	+	0.645736i	$0.776551\pi$
$398$	10.3372	+	17.9045i	0.518155	+	0.897471i
$399$		0			0
$400$	0.436492	−	0.756026i	0.0218246	−	0.0378013i
$401$	−3.87298			−0.193408			−0.0967038	−	0.995313i	$-0.530830\pi$
$401$	−3.87298			−0.193408			−0.0967038	+	0.995313i	$0.530830\pi$
$402$		0			0
$403$	23.2379			1.15756
$404$	3.93399	+	6.81388i	0.195724	+	0.339003i
$405$		0			0
$406$		0			0
$407$	−19.4919	−	33.7610i	−0.966179	−	1.67347i
$408$		0			0
$409$	−15.8144	−	27.3913i	−0.781971	−	1.35441i	−0.930792	−	0.365549i	$-0.880881\pi$
$409$	−15.8144	−	27.3913i	−0.781971	−	1.35441i	0.148821	−	0.988864i	$-0.452452\pi$
$410$	5.74597	+	9.95231i	0.283773	+	0.491509i
$411$		0			0
$412$	−9.01276	−	15.6106i	−0.444027	−	0.769077i
$413$		0			0
$414$		0			0
$415$	−8.43649	−	14.6124i	−0.414131	−	0.717296i
$416$	−4.24264			−0.208013
$417$		0			0
$418$	3.18768			0.155915
$419$	1.99230	−	3.45077i	0.0973304	−	0.168581i	−0.813248	−	0.581917i	$-0.802303\pi$
$419$	1.99230	−	3.45077i	0.0973304	−	0.168581i	0.910579	+	0.413335i	$0.135636\pi$
$420$		0			0
$421$	2.56351	+	4.44013i	0.124938	+	0.216399i	0.921709	−	0.387883i	$-0.126794\pi$
$421$	2.56351	+	4.44013i	0.124938	+	0.216399i	−0.796771	+	0.604282i	$0.793460\pi$
$422$	−1.30948	−	2.26808i	−0.0637442	−	0.110408i
$423$		0			0
$424$	5.30948	−	9.19628i	0.257851	−	0.446611i
$425$	0.617292	−	1.06918i	0.0299431	−	0.0518629i
$426$		0			0
$427$		0			0
$428$	7.87298	−	13.6364i	0.380555	−	0.659141i
$429$		0			0
$430$	−18.0255			−0.869268
$431$	4.74597	−	8.22026i	0.228605	−	0.395956i	−0.728790	−	0.684737i	$-0.759917\pi$
$431$	4.74597	−	8.22026i	0.228605	−	0.395956i	0.957395	+	0.288782i	$0.0932502\pi$
$432$		0			0
$433$	29.6985			1.42722			0.713609	−	0.700544i	$-0.247059\pi$
$433$	29.6985			1.42722			0.713609	+	0.700544i	$0.247059\pi$
$434$		0			0
$435$		0			0
$436$	9.12702			0.437105
$437$	2.68800	+	4.65576i	0.128585	+	0.222715i
$438$		0			0
$439$	5.47723	−	9.48683i	0.261414	−	0.452782i	−0.705204	−	0.709004i	$-0.749145\pi$
$439$	5.47723	−	9.48683i	0.261414	−	0.452782i	0.966618	+	0.256223i	$0.0824780\pi$
$440$	8.12602			0.387393
$441$		0			0
$442$	−6.00000			−0.285391
$443$	12.1825	−	21.1006i	0.578806	−	1.00252i	−0.416811	−	0.908993i	$-0.636852\pi$
$443$	12.1825	−	21.1006i	0.578806	−	1.00252i	0.995617	−	0.0935281i	$-0.0298145\pi$
$444$		0			0
$445$	−7.18246	−	12.4404i	−0.340481	−	0.589731i
$446$	−11.4933			−0.544225
$447$		0			0
$448$		0			0
$449$	−9.00000			−0.424736			−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000			−0.424736			−0.212368	+	0.977190i	$0.568118\pi$
$450$		0			0
$451$	11.3137	−	19.5959i	0.532742	−	0.922736i
$452$	3.87298			0.182170
$453$		0			0
$454$	−3.04726	+	5.27801i	−0.143015	+	0.247709i
$455$		0			0
$456$		0			0
$457$	−16.9365	+	29.3349i	−0.792256	+	1.37223i	0.132312	+	0.991208i	$0.457760\pi$
$457$	−16.9365	+	29.3349i	−0.792256	+	1.37223i	−0.924567	+	0.381019i	$0.875573\pi$
$458$	−3.93399	+	6.81388i	−0.183823	+	0.318392i
$459$		0			0
$460$	6.85224	+	11.8684i	0.319487	+	0.553368i
$461$	4.98895	+	8.64112i	0.232359	+	0.402457i	0.958502	−	0.285087i	$-0.0920224\pi$
$461$	4.98895	+	8.64112i	0.232359	+	0.402457i	−0.726143	+	0.687544i	$0.758689\pi$
$462$		0			0
$463$	−10.8095	+	18.7226i	−0.502359	+	0.870111i	0.497637	+	0.867385i	$0.334201\pi$
$463$	−10.8095	+	18.7226i	−0.502359	+	0.870111i	−0.999996	+	0.00272598i	$0.999132\pi$
$464$	−8.87298			−0.411918
$465$		0			0
$466$	22.7460			1.05369
$467$	−19.5295	−	33.8262i	−0.903720	−	1.56529i	−0.822627	−	0.568582i	$-0.807492\pi$
$467$	−19.5295	−	33.8262i	−0.903720	−	1.56529i	−0.0810929	−	0.996707i	$-0.525841\pi$
$468$		0			0
$469$		0			0
$470$	7.00000	+	12.1244i	0.322886	+	0.559255i
$471$		0			0
$472$	1.32440	+	2.29393i	0.0609604	+	0.105587i
$473$	17.7460	+	30.7369i	0.815960	+	1.41328i
$474$		0			0
$475$	0.347849	+	0.602493i	0.0159604	+	0.0276443i
$476$		0			0
$477$		0			0
$478$	7.50000	+	12.9904i	0.343042	+	0.594166i
$479$	−18.0255			−0.823607			−0.411803	−	0.911273i	$-0.635101\pi$
$479$	−18.0255			−0.823607			−0.411803	+	0.911273i	$0.635101\pi$
$480$		0			0
$481$	−41.3486			−1.88534
$482$	0.886735	−	1.53587i	0.0403897	−	0.0699570i
$483$		0			0
$484$	−2.50000	−	4.33013i	−0.113636	−	0.196824i
$485$	−18.1270	−	31.3969i	−0.823105	−	1.42566i
$486$		0			0
$487$	−15.2460	+	26.4068i	−0.690861	+	1.19661i	0.280696	+	0.959797i	$0.409435\pi$
$487$	−15.2460	+	26.4068i	−0.690861	+	1.19661i	−0.971556	+	0.236809i	$0.923898\pi$
$488$	−0.398461	+	0.690154i	−0.0180375	+	0.0312418i
$489$		0			0
$490$		0			0
$491$	−6.87298	+	11.9044i	−0.310173	+	0.537236i	−0.978400	−	0.206722i	$-0.933720\pi$
$491$	−6.87298	+	11.9044i	−0.310173	+	0.537236i	0.668226	+	0.743958i	$0.267054\pi$
$492$		0			0
$493$	−12.5483			−0.565147
$494$	1.69052	−	2.92808i	0.0760603	−	0.131740i
$495$		0			0
$496$	5.47723			0.245935
$497$		0			0
$498$		0			0
$499$	−14.2540			−0.638098			−0.319049	−	0.947738i	$-0.603363\pi$
$499$	−14.2540			−0.638098			−0.319049	+	0.947738i	$0.603363\pi$
$500$	5.96550	+	10.3325i	0.266785	+	0.462086i
$501$		0			0
$502$	12.9468	−	22.4244i	0.577842	−	1.00085i
$503$	18.0255			0.803718			0.401859	−	0.915702i	$-0.368364\pi$
$503$	18.0255			0.803718			0.401859	+	0.915702i	$0.368364\pi$
$504$		0			0
$505$	−15.9839			−0.711273
$506$	13.4919	−	23.3687i	0.599790	−	1.03887i
$507$		0			0
$508$	−7.37298	−	12.7704i	−0.327123	−	0.566594i
$509$	33.7615			1.49645			0.748226	−	0.663444i	$-0.230906\pi$
$509$	33.7615			1.49645			0.748226	+	0.663444i	$0.230906\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	6.98125	−	12.0919i	0.307930	−	0.533350i
$515$	36.6190			1.61362
$516$		0			0
$517$	13.7829	−	23.8726i	0.606170	−	1.04992i
$518$		0			0
$519$		0			0
$520$	4.30948	−	7.46423i	0.188983	−	0.327328i
$521$	−0.707107	+	1.22474i	−0.0309789	+	0.0536570i	−0.881099	−	0.472931i	$-0.843196\pi$
$521$	−0.707107	+	1.22474i	−0.0309789	+	0.0536570i	0.850120	+	0.526589i	$0.176529\pi$
$522$		0			0
$523$	−15.5057	−	26.8567i	−0.678019	−	1.17436i	−0.975577	−	0.219659i	$-0.929506\pi$
$523$	−15.5057	−	26.8567i	−0.678019	−	1.17436i	0.297558	−	0.954704i	$-0.403828\pi$
$524$	−3.93399	−	6.81388i	−0.171857	−	0.297666i
$525$		0			0
$526$	−2.80948	+	4.86615i	−0.122499	+	0.212174i
$527$	7.74597			0.337420
$528$		0			0
$529$	22.5081			0.978612
$530$	10.7862	+	18.6823i	0.468524	+	0.811507i
$531$		0			0
$532$		0			0
$533$	−12.0000	−	20.7846i	−0.519778	−	0.900281i
$534$		0			0
$535$	15.9940	+	27.7024i	0.691481	+	1.19768i
$536$	0.436492	+	0.756026i	0.0188536	+	0.0326553i
$537$		0			0
$538$	1.72286	+	2.98408i	0.0742778	+	0.128653i
$539$		0			0
$540$		0			0
$541$	−8.05544	−	13.9524i	−0.346330	−	0.599862i	0.639264	−	0.768987i	$-0.279239\pi$
$541$	−8.05544	−	13.9524i	−0.346330	−	0.599862i	−0.985595	+	0.169125i	$0.945906\pi$
$542$	7.07107			0.303728
$543$		0			0
$544$	−1.41421			−0.0606339
$545$	−9.27079	+	16.0575i	−0.397117	+	0.687827i
$546$		0			0
$547$	−14.4919	−	25.1008i	−0.619630	−	1.07323i	−0.989553	−	0.144169i	$-0.953949\pi$
$547$	−14.4919	−	25.1008i	−0.619630	−	1.07323i	0.369923	−	0.929062i	$-0.379384\pi$
$548$	7.74597	+	13.4164i	0.330891	+	0.573121i
$549$		0			0
$550$	1.74597	−	3.02410i	0.0744483	−	0.128948i
$551$	3.53553	−	6.12372i	0.150619	−	0.260879i
$552$		0			0
$553$		0			0
$554$	−12.1825	+	21.1006i	−0.517583	+	0.896480i
$555$		0			0
$556$	18.4632			0.783013
$557$	1.69052	−	2.92808i	0.0716298	−	0.124067i	−0.827986	−	0.560749i	$-0.810513\pi$
$557$	1.69052	−	2.92808i	0.0716298	−	0.124067i	0.899616	+	0.436682i	$0.143847\pi$
$558$		0			0
$559$	37.6449			1.59221
$560$		0			0
$561$		0			0
$562$	−6.74597			−0.284561
$563$	2.60960	+	4.51995i	0.109981	+	0.190493i	0.915762	−	0.401720i	$-0.131588\pi$
$563$	2.60960	+	4.51995i	0.109981	+	0.190493i	−0.805781	+	0.592213i	$0.798254\pi$
$564$		0			0
$565$	−3.93399	+	6.81388i	−0.165504	+	0.286662i
$566$	10.5168			0.442054
$567$		0			0
$568$	−2.12702			−0.0892476
$569$	3.74597	−	6.48820i	0.157039	−	0.272000i	−0.776761	−	0.629796i	$-0.783138\pi$
$569$	3.74597	−	6.48820i	0.157039	−	0.272000i	0.933800	+	0.357796i	$0.116472\pi$
$570$		0			0
$571$	−0.254033	−	0.439999i	−0.0106310	−	0.0184134i	0.860661	−	0.509178i	$-0.170051\pi$
$571$	−0.254033	−	0.439999i	−0.0106310	−	0.0184134i	−0.871292	+	0.490765i	$0.836717\pi$
$572$	−16.9706			−0.709575
$573$		0			0
$574$		0			0
$575$	5.88912			0.245593
$576$		0			0
$577$	17.5879	−	30.4631i	0.732192	−	1.26819i	−0.223752	−	0.974646i	$-0.571831\pi$
$577$	17.5879	−	30.4631i	0.732192	−	1.26819i	0.955944	−	0.293548i	$-0.0948360\pi$
$578$	15.0000			0.623918
$579$		0			0
$580$	9.01276	−	15.6106i	0.374234	−	0.648193i
$581$		0			0
$582$		0			0
$583$	21.2379	−	36.7851i	0.879584	−	1.52348i
$584$	−7.68836	+	13.3166i	−0.318147	+	0.551046i
$585$		0			0
$586$	−3.13707	−	5.43357i	−0.129591	−	0.224459i
$587$	14.8884	+	25.7875i	0.614512	+	1.06437i	0.990470	+	0.137729i	$0.0439804\pi$
$587$	14.8884	+	25.7875i	0.614512	+	1.06437i	−0.375958	+	0.926637i	$0.622686\pi$
$588$		0			0
$589$	−2.18246	+	3.78013i	−0.0899266	+	0.155757i
$590$	−5.38105			−0.221534
$591$		0			0
$592$	−9.74597			−0.400557
$593$	−11.4035	−	19.7515i	−0.468287	−	0.811096i	0.531057	−	0.847336i	$-0.321795\pi$
$593$	−11.4035	−	19.7515i	−0.468287	−	0.811096i	−0.999343	+	0.0362403i	$0.988462\pi$
$594$		0			0
$595$		0			0
$596$	−0.127017	−	0.219999i	−0.00520280	−	0.00901152i
$597$		0			0
$598$	−14.3104	−	24.7863i	−0.585194	−	1.01359i
$599$	0.618950	+	1.07205i	0.0252896	+	0.0438029i	0.878393	−	0.477939i	$-0.158616\pi$
$599$	0.618950	+	1.07205i	0.0252896	+	0.0438029i	−0.853104	+	0.521742i	$0.825283\pi$
$600$		0			0
$601$	−18.8224	−	32.6014i	−0.767783	−	1.32984i	−0.938762	−	0.344565i	$-0.888026\pi$
$601$	−18.8224	−	32.6014i	−0.767783	−	1.32984i	0.170979	−	0.985275i	$-0.445307\pi$
$602$		0			0
$603$		0			0
$604$	−5.50000	−	9.52628i	−0.223792	−	0.387619i
$605$	10.1575			0.412962
$606$		0			0
$607$	−15.7360			−0.638704			−0.319352	−	0.947636i	$-0.603465\pi$
$607$	−15.7360			−0.638704			−0.319352	+	0.947636i	$0.603465\pi$
$608$	0.398461	−	0.690154i	0.0161597	−	0.0279894i
$609$		0			0
$610$	−0.809475	−	1.40205i	−0.0327747	−	0.0567674i
$611$	−14.6190	−	25.3208i	−0.591419	−	1.02437i
$612$		0			0
$613$	−1.69052	+	2.92808i	−0.0682797	+	0.118264i	−0.898144	−	0.439701i	$-0.855084\pi$
$613$	−1.69052	+	2.92808i	−0.0682797	+	0.118264i	0.829864	+	0.557965i	$0.188418\pi$
$614$	12.0600	−	20.8886i	0.486703	−	0.842994i
$615$		0			0
$616$		0			0
$617$	−13.8730	+	24.0287i	−0.558505	+	0.967360i	0.439116	+	0.898430i	$0.355292\pi$
$617$	−13.8730	+	24.0287i	−0.558505	+	0.967360i	−0.997622	+	0.0689293i	$0.978042\pi$
$618$		0			0
$619$	−22.3466			−0.898184			−0.449092	−	0.893485i	$-0.648253\pi$
$619$	−22.3466			−0.898184			−0.449092	+	0.893485i	$0.648253\pi$
$620$	−5.56351	+	9.63628i	−0.223436	+	0.387002i
$621$		0			0
$622$	1.41421			0.0567048
$623$		0			0
$624$		0			0
$625$	−19.8730			−0.794919
$626$	−13.3452	−	23.1146i	−0.533382	−	0.923845i
$627$		0			0
$628$	3.22689	−	5.58913i	0.128767	−	0.223031i
$629$	−13.7829			−0.549559
$630$		0			0
$631$	−27.6190			−1.09949			−0.549747	−	0.835332i	$-0.685276\pi$
$631$	−27.6190			−1.09949			−0.549747	+	0.835332i	$0.685276\pi$
$632$	2.93649	−	5.08615i	0.116807	−	0.202316i
$633$		0			0
$634$	−0.690525	−	1.19602i	−0.0274243	−	0.0475002i
$635$	29.9565			1.18879
$636$		0			0
$637$		0			0
$638$	−35.4919			−1.40514
$639$		0			0
$640$	1.01575	−	1.75934i	0.0401512	−	0.0695439i
$641$	37.1109			1.46579			0.732896	−	0.680341i	$-0.238168\pi$
$641$	37.1109			1.46579			0.732896	+	0.680341i	$0.238168\pi$
$642$		0			0
$643$	−20.1468	+	34.8953i	−0.794514	+	1.37614i	0.128634	+	0.991692i	$0.458941\pi$
$643$	−20.1468	+	34.8953i	−0.794514	+	1.37614i	−0.923147	+	0.384446i	$0.874392\pi$
$644$		0			0
$645$		0			0
$646$	0.563508	−	0.976025i	0.0221709	−	0.0384012i
$647$	−2.73861	+	4.74342i	−0.107666	+	0.186483i	−0.914824	−	0.403852i	$-0.867671\pi$
$647$	−2.73861	+	4.74342i	−0.107666	+	0.186483i	0.807158	+	0.590335i	$0.201004\pi$
$648$		0			0
$649$	5.29760	+	9.17571i	0.207949	+	0.360178i
$650$	−1.85188	−	3.20755i	−0.0726366	−	0.125810i
$651$		0			0
$652$	−5.00000	+	8.66025i	−0.195815	+	0.339162i
$653$	45.4919			1.78024			0.890118	−	0.455729i	$-0.150621\pi$
$653$	45.4919			1.78024			0.890118	+	0.455729i	$0.150621\pi$
$654$		0			0
$655$	15.9839			0.624541
$656$	−2.82843	−	4.89898i	−0.110432	−	0.191273i
$657$		0			0
$658$		0			0
$659$	18.4365	+	31.9329i	0.718184	+	1.24393i	0.961719	+	0.274039i	$0.0883596\pi$
$659$	18.4365	+	31.9329i	0.718184	+	1.24393i	−0.243535	+	0.969892i	$0.578307\pi$
$660$		0			0
$661$	12.4977	+	21.6466i	0.486104	+	0.841956i	0.999872	−	0.0159726i	$-0.00508447\pi$
$661$	12.4977	+	21.6466i	0.486104	+	0.841956i	−0.513769	+	0.857929i	$0.671751\pi$
$662$	−9.18246	−	15.9045i	−0.356886	−	0.618145i
$663$		0			0
$664$	4.15283	+	7.19291i	0.161161	+	0.279139i
$665$		0			0
$666$		0			0
$667$	−29.9284	−	51.8376i	−1.15883	−	2.00716i
$668$	−12.3687			−0.478558
$669$		0			0
$670$	−1.77347			−0.0685152
$671$	−1.59384	+	2.76062i	−0.0615296	+	0.106572i
$672$		0			0
$673$	16.1190	+	27.9188i	0.621340	+	1.07619i	0.989236	+	0.146326i	$0.0467447\pi$
$673$	16.1190	+	27.9188i	0.621340	+	1.07619i	−0.367897	+	0.929867i	$0.619922\pi$
$674$	0.127017	+	0.219999i	0.00489250	+	0.00847406i
$675$		0			0
$676$	−2.50000	+	4.33013i	−0.0961538	+	0.166543i
$677$	−6.36396	+	11.0227i	−0.244587	+	0.423637i	−0.962015	−	0.272995i	$-0.911986\pi$
$677$	−6.36396	+	11.0227i	−0.244587	+	0.423637i	0.717428	+	0.696632i	$0.245319\pi$
$678$		0			0
$679$		0			0
$680$	1.43649	−	2.48808i	0.0550869	−	0.0954134i
$681$		0			0
$682$	21.9089			0.838935
$683$	−3.87298	+	6.70820i	−0.148196	+	0.256682i	−0.930561	−	0.366138i	$-0.880680\pi$
$683$	−3.87298	+	6.70820i	−0.148196	+	0.256682i	0.782365	+	0.622820i	$0.214013\pi$
$684$		0			0
$685$	−31.4720			−1.20248
$686$		0			0
$687$		0			0
$688$	8.87298			0.338279
$689$	−22.5262	−	39.0165i	−0.858180	−	1.48641i
$690$		0			0
$691$	−10.6458	+	18.4391i	−0.404986	+	0.701455i	−0.994320	−	0.106434i	$-0.966057\pi$
$691$	−10.6458	+	18.4391i	−0.404986	+	0.701455i	0.589334	+	0.807889i	$0.299390\pi$
$692$	12.7279			0.483843
$693$		0			0
$694$	−7.74597			−0.294033
$695$	−18.7540	+	32.4829i	−0.711381	+	1.23215i
$696$		0			0
$697$	−4.00000	−	6.92820i	−0.151511	−	0.262424i
$698$	16.4317			0.621948
$699$		0			0
$700$		0			0
$701$	31.7460			1.19903			0.599514	−	0.800364i	$-0.295360\pi$
$701$	31.7460			1.19903			0.599514	+	0.800364i	$0.295360\pi$
$702$		0			0
$703$	3.88338	−	6.72622i	0.146465	−	0.253684i
$704$	−4.00000			−0.150756
$705$		0			0
$706$	−9.01276	+	15.6106i	−0.339200	+	0.587511i
$707$		0			0
$708$		0			0
$709$	−12.3649	+	21.4167i	−0.464374	+	0.804320i	−0.999173	−	0.0406597i	$-0.987054\pi$
$709$	−12.3649	+	21.4167i	−0.464374	+	0.804320i	0.534799	+	0.844979i	$0.320387\pi$
$710$	2.16052	−	3.74214i	0.0810830	−	0.140440i
$711$		0			0
$712$	3.53553	+	6.12372i	0.132500	+	0.229496i
$713$	18.4746	+	31.9989i	0.691879	+	1.19837i
$714$		0			0
$715$	17.2379	−	29.8569i	0.644661	−	1.11659i
$716$	−0.872983			−0.0326249
$717$		0			0
$718$	−28.2379			−1.05383
$719$	−4.68030	−	8.10653i	−0.174546	−	0.302322i	0.765458	−	0.643486i	$-0.222512\pi$
$719$	−4.68030	−	8.10653i	−0.174546	−	0.302322i	−0.940004	+	0.341163i	$0.889179\pi$
$720$		0			0
$721$		0			0
$722$	−9.18246	−	15.9045i	−0.341736	−	0.591904i
$723$		0			0
$724$	−3.93399	−	6.81388i	−0.146206	−	0.253236i
$725$	−3.87298	−	6.70820i	−0.143839	−	0.249136i
$726$		0			0
$727$	18.1153	+	31.3767i	0.671861	+	1.16370i	0.977376	+	0.211509i	$0.0678379\pi$
$727$	18.1153	+	31.3767i	0.671861	+	1.16370i	−0.305515	+	0.952187i	$0.598829\pi$
$728$		0			0
$729$		0			0
$730$	−15.6190	−	27.0528i	−0.578083	−	1.00127i
$731$	12.5483			0.464115
$732$		0			0
$733$	9.64146			0.356115			0.178058	−	0.984020i	$-0.443019\pi$
$733$	9.64146			0.356115			0.178058	+	0.984020i	$0.443019\pi$
$734$	2.03151	−	3.51867i	0.0749843	−	0.129877i
$735$		0			0
$736$	−3.37298	−	5.84218i	−0.124330	−	0.215346i
$737$	1.74597	+	3.02410i	0.0643135	+	0.111394i
$738$		0			0
$739$	15.0000	−	25.9808i	0.551784	−	0.955718i	−0.446362	−	0.894852i	$-0.647281\pi$
$739$	15.0000	−	25.9808i	0.551784	−	0.955718i	0.998146	−	0.0608653i	$-0.0193860\pi$
$740$	9.89949	−	17.1464i	0.363913	−	0.630315i
$741$		0			0
$742$		0			0
$743$	−13.8730	+	24.0287i	−0.508950	+	0.881528i	0.490996	+	0.871162i	$0.336633\pi$
$743$	−13.8730	+	24.0287i	−0.508950	+	0.881528i	−0.999946	+	0.0103660i	$0.996700\pi$
$744$		0			0
$745$	0.516070			0.0189073
$746$	−4.43649	+	7.68423i	−0.162432	+	0.281340i
$747$		0			0
$748$	−5.65685			−0.206835
$749$		0			0
$750$		0			0
$751$	−13.0000			−0.474377			−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000			−0.474377			−0.237188	+	0.971464i	$0.576226\pi$
$752$	−3.44572	−	5.96816i	−0.125652	−	0.217636i
$753$		0			0
$754$	−18.8224	+	32.6014i	−0.685473	+	1.18727i
$755$	22.3466			0.813275
$756$		0			0
$757$	−15.2379			−0.553831			−0.276915	−	0.960894i	$-0.589312\pi$
$757$	−15.2379			−0.553831			−0.276915	+	0.960894i	$0.589312\pi$
$758$	−6.18246	+	10.7083i	−0.224557	+	0.388944i
$759$		0			0
$760$	0.809475	+	1.40205i	0.0293627	+	0.0508578i
$761$	2.28954			0.0829958			0.0414979	−	0.999139i	$-0.486787\pi$
$761$	2.28954			0.0829958			0.0414979	+	0.999139i	$0.486787\pi$
$762$		0			0
$763$		0			0
$764$	13.8730			0.501907
$765$		0			0
$766$	−11.9310	+	20.6651i	−0.431085	+	0.746660i
$767$	11.2379			0.405777
$768$		0			0
$769$	4.85993	−	8.41765i	0.175254	−	0.303548i	−0.764995	−	0.644036i	$-0.777259\pi$
$769$	4.85993	−	8.41765i	0.175254	−	0.303548i	0.940249	+	0.340488i	$0.110592\pi$
$770$		0			0
$771$		0			0
$772$	−8.06351	+	13.9664i	−0.290212	+	0.502662i
$773$	−1.72286	+	2.98408i	−0.0619670	+	0.107330i	−0.895345	−	0.445374i	$-0.853071\pi$
$773$	−1.72286	+	2.98408i	−0.0619670	+	0.107330i	0.833378	+	0.552704i	$0.186404\pi$
$774$		0			0
$775$	2.39076	+	4.14092i	0.0858788	+	0.148746i
$776$	8.92295	+	15.4550i	0.320315	+	0.554802i
$777$		0			0
$778$	−10.4365	+	18.0765i	−0.374166	+	0.648075i
$779$	4.50807			0.161518
$780$		0			0
$781$	−8.50807			−0.304443
$782$	−4.77012	−	8.26209i	−0.170579	−	0.295452i
$783$		0			0
$784$		0			0
$785$	6.55544	+	11.3544i	0.233974	+	0.405254i
$786$		0			0
$787$	−4.77012	−	8.26209i	−0.170036	−	0.294512i	0.768396	−	0.639975i	$-0.221055\pi$
$787$	−4.77012	−	8.26209i	−0.170036	−	0.294512i	−0.938432	+	0.345463i	$0.887722\pi$
$788$	−3.30948	−	5.73218i	−0.117895	−	0.204200i
$789$		0			0
$790$	5.96550	+	10.3325i	0.212243	+	0.367616i
$791$		0			0
$792$		0			0

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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.h (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +2.03151 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +2.03151 q^{5} -1.00000 q^{8} +(1.01575 - 1.75934i) q^{10} -4.00000 q^{11} +(-2.12132 + 3.67423i) q^{13} +(-0.500000 + 0.866025i) q^{16} +(-0.707107 + 1.22474i) q^{17} +(-0.398461 - 0.690154i) q^{19} +(-1.01575 - 1.75934i) q^{20} +(-2.00000 + 3.46410i) q^{22} -6.74597 q^{23} -0.872983 q^{25} +(2.12132 + 3.67423i) q^{26} +(4.43649 + 7.68423i) q^{29} +(-2.73861 - 4.74342i) q^{31} +(0.500000 + 0.866025i) q^{32} +(0.707107 + 1.22474i) q^{34} +(4.87298 + 8.44025i) q^{37} -0.796921 q^{38} -2.03151 q^{40} +(-2.82843 + 4.89898i) q^{41} +(-4.43649 - 7.68423i) q^{43} +(2.00000 + 3.46410i) q^{44} +(-3.37298 + 5.84218i) q^{46} +(-3.44572 + 5.96816i) q^{47} +(-0.436492 + 0.756026i) q^{50} +4.24264 q^{52} +(-5.30948 + 9.19628i) q^{53} -8.12602 q^{55} +8.87298 q^{58} +(-1.32440 - 2.29393i) q^{59} +(0.398461 - 0.690154i) q^{61} -5.47723 q^{62} +1.00000 q^{64} +(-4.30948 + 7.46423i) q^{65} +(-0.436492 - 0.756026i) q^{67} +1.41421 q^{68} +2.12702 q^{71} +(7.68836 - 13.3166i) q^{73} +9.74597 q^{74} +(-0.398461 + 0.690154i) q^{76} +(-2.93649 + 5.08615i) q^{79} +(-1.01575 + 1.75934i) q^{80} +(2.82843 + 4.89898i) q^{82} +(-4.15283 - 7.19291i) q^{83} +(-1.43649 + 2.48808i) q^{85} -8.87298 q^{86} +4.00000 q^{88} +(-3.53553 - 6.12372i) q^{89} +(3.37298 + 5.84218i) q^{92} +(3.44572 + 5.96816i) q^{94} +(-0.809475 - 1.40205i) q^{95} +(-8.92295 - 15.4550i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8}+O(q^{10}) \) 8 * q + 4 * q^2 - 4 * q^4 - 8 * q^8	\( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} - 32 q^{11} - 4 q^{16} - 16 q^{22} + 8 q^{23} + 24 q^{25} + 20 q^{29} + 4 q^{32} + 8 q^{37} - 20 q^{43} + 16 q^{44} + 4 q^{46} + 12 q^{50} + 4 q^{53} + 40 q^{58} + 8 q^{64} + 12 q^{65} + 12 q^{67} + 48 q^{71} + 16 q^{74} - 8 q^{79} + 4 q^{85} - 40 q^{86} + 32 q^{88} - 4 q^{92} + 40 q^{95}+O(q^{100}) \) 8 * q + 4 * q^2 - 4 * q^4 - 8 * q^8 - 32 * q^11 - 4 * q^16 - 16 * q^22 + 8 * q^23 + 24 * q^25 + 20 * q^29 + 4 * q^32 + 8 * q^37 - 20 * q^43 + 16 * q^44 + 4 * q^46 + 12 * q^50 + 4 * q^53 + 40 * q^58 + 8 * q^64 + 12 * q^65 + 12 * q^67 + 48 * q^71 + 16 * q^74 - 8 * q^79 + 4 * q^85 - 40 * q^86 + 32 * q^88 - 4 * q^92 + 40 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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