LMFDB - Embedded newform 2646.2.m.a.1763.8

Newspace parameters

Level:	$ N $	$=$	$ 2646 = 2 \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2646.m (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$21.1284163748$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
Defining polynomial:	$ x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + 2785 x^{8} - 2640 x^{7} - 2601 x^{6} + 10260 x^{5} - 10611 x^{4} - 1944 x^{3} + \cdots + 6561 $ x^16 - 8x^15 + 23x^14 - 8x^13 - 131x^12 + 380x^11 - 289x^10 - 880x^9 + 2785x^8 - 2640x^7 - 2601x^6 + 10260x^5 - 10611x^4 - 1944x^3 + 16767x^2 - 17496*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1763.8
Root			$1.73109 + 0.0577511i$ of defining polynomial
Character	$\chi$	$=$	2646.1763
Dual form			2646.2.m.a.881.8

$q$-expansion

$f(q)$	$=$	$q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.14095 + 1.97618i) q^{5} +1.00000i q^{8} +O(q^{10})$	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.14095 + 1.97618i) q^{5} +1.00000i q^{8} +2.28190i q^{10} +(-0.946590 - 0.546514i) q^{11} +(-5.91448 + 3.41473i) q^{13} +(-0.500000 + 0.866025i) q^{16} +6.71727 q^{17} +2.86351i q^{19} +(-1.14095 + 1.97618i) q^{20} +(-0.546514 - 0.946590i) q^{22} +(-3.38264 + 1.95297i) q^{23} +(-0.103535 + 0.179327i) q^{25} -6.82946 q^{26} +(-1.59933 - 0.923371i) q^{29} +(1.75081 - 1.01083i) q^{31} +(-0.866025 + 0.500000i) q^{32} +(5.81732 + 3.35863i) q^{34} -7.15840 q^{37} +(-1.43175 + 2.47987i) q^{38} +(-1.97618 + 1.14095i) q^{40} +(2.45515 + 4.25245i) q^{41} +(-3.74246 + 6.48214i) q^{43} -1.09303i q^{44} -3.90593 q^{46} +(-3.40174 + 5.89199i) q^{47} +(-0.179327 + 0.103535i) q^{50} +(-5.91448 - 3.41473i) q^{52} +0.256424i q^{53} -2.49418i q^{55} +(-0.923371 - 1.59933i) q^{58} +(-0.971009 - 1.68184i) q^{59} +(-1.15315 - 0.665771i) q^{61} +2.02166 q^{62} -1.00000 q^{64} +(-13.4963 - 7.79207i) q^{65} +(-2.54959 - 4.41602i) q^{67} +(3.35863 + 5.81732i) q^{68} -0.233507i q^{71} +6.80432i q^{73} +(-6.19935 - 3.57920i) q^{74} +(-2.47987 + 1.43175i) q^{76} +(3.63624 - 6.29816i) q^{79} -2.28190 q^{80} +4.91031i q^{82} +(-2.91353 + 5.04638i) q^{83} +(7.66407 + 13.2746i) q^{85} +(-6.48214 + 3.74246i) q^{86} +(0.546514 - 0.946590i) q^{88} +17.9941 q^{89} +(-3.38264 - 1.95297i) q^{92} +(-5.89199 + 3.40174i) q^{94} +(-5.65882 + 3.26712i) q^{95} +(4.13903 + 2.38967i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{4}+O(q^{10}) $ 16 * q + 8 * q^4	$ 16 q + 8 q^{4} + 12 q^{11} - 6 q^{13} - 8 q^{16} + 36 q^{17} - 6 q^{23} - 8 q^{25} - 24 q^{26} - 6 q^{29} - 6 q^{31} + 4 q^{37} - 6 q^{41} - 2 q^{43} - 12 q^{46} + 18 q^{47} + 12 q^{50} - 6 q^{52} + 6 q^{58} - 30 q^{59} + 60 q^{61} + 36 q^{62} - 16 q^{64} + 42 q^{65} + 14 q^{67} + 18 q^{68} + 18 q^{74} - 16 q^{79} - 12 q^{85} + 24 q^{86} + 48 q^{89} - 6 q^{92} - 66 q^{95} - 6 q^{97}+O(q^{100}) $ 16 * q + 8 * q^4 + 12 * q^11 - 6 * q^13 - 8 * q^16 + 36 * q^17 - 6 * q^23 - 8 * q^25 - 24 * q^26 - 6 * q^29 - 6 * q^31 + 4 * q^37 - 6 * q^41 - 2 * q^43 - 12 * q^46 + 18 * q^47 + 12 * q^50 - 6 * q^52 + 6 * q^58 - 30 * q^59 + 60 * q^61 + 36 * q^62 - 16 * q^64 + 42 * q^65 + 14 * q^67 + 18 * q^68 + 18 * q^74 - 16 * q^79 - 12 * q^85 + 24 * q^86 + 48 * q^89 - 6 * q^92 - 66 * q^95 - 6 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$1081$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$

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 $

$2$	0.866025	+	0.500000i	0.612372	+	0.353553i
$2$	0.866025	+	0.500000i	0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$	1.14095	+	1.97618i	0.510248	+	0.883776i	0.999929	+	0.0118746i	$0.00377989\pi$
$5$	1.14095	+	1.97618i	0.510248	+	0.883776i	−0.489681	+	0.871902i	$0.662887\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$			1.00000i			0.353553i
$9$		0			0
$10$			2.28190i			0.721600i
$11$	−0.946590	−	0.546514i	−0.285408	−	0.164780i	0.350461	−	0.936577i	$-0.386025\pi$
$11$	−0.946590	−	0.546514i	−0.285408	−	0.164780i	−0.635869	+	0.771797i	$0.719358\pi$
$12$		0			0
$13$	−5.91448	+	3.41473i	−1.64038	+	0.947075i	−0.659685	+	0.751542i	$0.729310\pi$
$13$	−5.91448	+	3.41473i	−1.64038	+	0.947075i	−0.980697	+	0.195533i	$0.937356\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	6.71727			1.62918			0.814588	−	0.580040i	$-0.196963\pi$
$17$	6.71727			1.62918			0.814588	+	0.580040i	$0.196963\pi$
$18$		0			0
$19$			2.86351i			0.656934i	0.944515	+	0.328467i	$0.106532\pi$
$19$			2.86351i			0.656934i	−0.944515	+	0.328467i	$0.893468\pi$
$20$	−1.14095	+	1.97618i	−0.255124	+	0.441888i
$21$		0			0
$22$	−0.546514	−	0.946590i	−0.116517	−	0.201814i
$23$	−3.38264	+	1.95297i	−0.705328	+	0.407221i	−0.809329	−	0.587356i	$-0.800169\pi$
$23$	−3.38264	+	1.95297i	−0.705328	+	0.407221i	0.104001	+	0.994577i	$0.466836\pi$
$24$		0			0
$25$	−0.103535	+	0.179327i	−0.0207069	+	0.0358655i
$26$	−6.82946			−1.33937
$27$		0			0
$28$		0			0
$29$	−1.59933	−	0.923371i	−0.296987	−	0.171466i	0.344101	−	0.938933i	$-0.388184\pi$
$29$	−1.59933	−	0.923371i	−0.296987	−	0.171466i	−0.641089	+	0.767467i	$0.721517\pi$
$30$		0			0
$31$	1.75081	−	1.01083i	0.314455	−	0.181551i	−0.334463	−	0.942409i	$-0.608555\pi$
$31$	1.75081	−	1.01083i	0.314455	−	0.181551i	0.648918	+	0.760858i	$0.275222\pi$
$32$	−0.866025	+	0.500000i	−0.153093	+	0.0883883i
$33$		0			0
$34$	5.81732	+	3.35863i	0.997663	+	0.576001i
$35$		0			0
$36$		0			0
$37$	−7.15840			−1.17683			−0.588416	−	0.808558i	$-0.700248\pi$
$37$	−7.15840			−1.17683			−0.588416	+	0.808558i	$0.700248\pi$
$38$	−1.43175	+	2.47987i	−0.232261	+	0.402288i
$39$		0			0
$40$	−1.97618	+	1.14095i	−0.312462	+	0.180400i
$41$	2.45515	+	4.25245i	0.383431	+	0.664121i	0.991550	−	0.129724i	$-0.0414092\pi$
$41$	2.45515	+	4.25245i	0.383431	+	0.664121i	−0.608120	+	0.793845i	$0.708076\pi$
$42$		0			0
$43$	−3.74246	+	6.48214i	−0.570721	+	0.988517i	0.425772	+	0.904831i	$0.360003\pi$
$43$	−3.74246	+	6.48214i	−0.570721	+	0.988517i	−0.996492	+	0.0836863i	$0.973331\pi$
$44$		−	1.09303i		−	0.164780i
$45$		0			0
$46$	−3.90593			−0.575898
$47$	−3.40174	+	5.89199i	−0.496195	+	0.859435i	−0.999990	−	0.00438774i	$-0.998603\pi$
$47$	−3.40174	+	5.89199i	−0.496195	+	0.859435i	0.503795	+	0.863823i	$0.331937\pi$
$48$		0			0
$49$		0			0
$50$	−0.179327	+	0.103535i	−0.0253607	+	0.0146420i
$51$		0			0
$52$	−5.91448	−	3.41473i	−0.820191	−	0.473538i
$53$			0.256424i			0.0352225i	0.999845	+	0.0176112i	$0.00560612\pi$
$53$			0.256424i			0.0352225i	−0.999845	+	0.0176112i	$0.994394\pi$
$54$		0			0
$55$		−	2.49418i		−	0.336315i
$56$		0			0
$57$		0			0
$58$	−0.923371	−	1.59933i	−0.121245	−	0.210002i
$59$	−0.971009	−	1.68184i	−0.126415	−	0.218957i	0.795870	−	0.605467i	$-0.207014\pi$
$59$	−0.971009	−	1.68184i	−0.126415	−	0.218957i	−0.922285	+	0.386510i	$0.873680\pi$
$60$		0			0
$61$	−1.15315	−	0.665771i	−0.147646	−	0.0852432i	0.424357	−	0.905495i	$-0.360500\pi$
$61$	−1.15315	−	0.665771i	−0.147646	−	0.0852432i	−0.572003	+	0.820252i	$0.693833\pi$
$62$	2.02166			0.256752
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−13.4963	−	7.79207i	−1.67400	−	0.966487i
$66$		0			0
$67$	−2.54959	−	4.41602i	−0.311482	−	0.539503i	0.667201	−	0.744877i	$-0.267492\pi$
$67$	−2.54959	−	4.41602i	−0.311482	−	0.539503i	−0.978683	+	0.205375i	$0.934159\pi$
$68$	3.35863	+	5.81732i	0.407294	+	0.705454i
$69$		0			0
$70$		0			0
$71$		−	0.233507i		−	0.0277121i	−0.999904	−	0.0138561i	$-0.995589\pi$
$71$		−	0.233507i		−	0.0277121i	0.999904	−	0.0138561i	$-0.00441066\pi$
$72$		0			0
$73$			6.80432i			0.796386i	0.917302	+	0.398193i	$0.130362\pi$
$73$			6.80432i			0.796386i	−0.917302	+	0.398193i	$0.869638\pi$
$74$	−6.19935	−	3.57920i	−0.720660	−	0.416073i
$75$		0			0
$76$	−2.47987	+	1.43175i	−0.284461	+	0.164233i
$77$		0			0
$78$		0			0
$79$	3.63624	−	6.29816i	0.409109	−	0.708598i	−0.585681	−	0.810542i	$-0.699173\pi$
$79$	3.63624	−	6.29816i	0.409109	−	0.708598i	0.994790	+	0.101944i	$0.0325062\pi$
$80$	−2.28190			−0.255124
$81$		0			0
$82$			4.91031i			0.542253i
$83$	−2.91353	+	5.04638i	−0.319801	+	0.553912i	−0.980446	−	0.196786i	$-0.936950\pi$
$83$	−2.91353	+	5.04638i	−0.319801	+	0.553912i	0.660645	+	0.750698i	$0.270283\pi$
$84$		0			0
$85$	7.66407	+	13.2746i	0.831285	+	1.43983i
$86$	−6.48214	+	3.74246i	−0.698987	+	0.403560i
$87$		0			0
$88$	0.546514	−	0.946590i	0.0582586	−	0.100907i
$89$	17.9941			1.90737			0.953687	−	0.300800i	$-0.0972535\pi$
$89$	17.9941			1.90737			0.953687	+	0.300800i	$0.0972535\pi$
$90$		0			0
$91$		0			0
$92$	−3.38264	−	1.95297i	−0.352664	−	0.203611i
$93$		0			0
$94$	−5.89199	+	3.40174i	−0.607713	+	0.350863i
$95$	−5.65882	+	3.26712i	−0.580583	+	0.335200i
$96$		0			0
$97$	4.13903	+	2.38967i	0.420255	+	0.242634i	0.695186	−	0.718830i	$-0.255322\pi$
$97$	4.13903	+	2.38967i	0.420255	+	0.242634i	−0.274931	+	0.961464i	$0.588655\pi$
$98$		0			0
$99$		0			0
$100$	−0.207069			−0.0207069
$101$	−5.22981	+	9.05829i	−0.520385	+	0.901334i	0.479334	+	0.877633i	$0.340878\pi$
$101$	−5.22981	+	9.05829i	−0.520385	+	0.901334i	−0.999719	+	0.0237012i	$0.992455\pi$
$102$		0			0
$103$	−11.0398	+	6.37383i	−1.08778	+	0.628033i	−0.932986	−	0.359914i	$-0.882806\pi$
$103$	−11.0398	+	6.37383i	−1.08778	+	0.628033i	−0.154799	+	0.987946i	$0.549473\pi$
$104$	−3.41473	−	5.91448i	−0.334842	−	0.579963i
$105$		0			0
$106$	−0.128212	+	0.222069i	−0.0124530	+	0.0215693i
$107$		−	9.53627i		−	0.921906i	−0.887425	−	0.460953i	$-0.847508\pi$
$107$		−	9.53627i		−	0.921906i	0.887425	−	0.460953i	$-0.152492\pi$
$108$		0			0
$109$	5.76503			0.552189			0.276095	−	0.961130i	$-0.410960\pi$
$109$	5.76503			0.552189			0.276095	+	0.961130i	$0.410960\pi$
$110$	1.24709	−	2.16002i	0.118905	−	0.205950i
$111$		0			0
$112$		0			0
$113$	10.3333	−	5.96592i	0.972073	−	0.561227i	0.0722053	−	0.997390i	$-0.476996\pi$
$113$	10.3333	−	5.96592i	0.972073	−	0.561227i	0.899868	+	0.436163i	$0.143663\pi$
$114$		0			0
$115$	−7.71884	−	4.45647i	−0.719785	−	0.415568i
$116$		−	1.84674i		−	0.171466i
$117$		0			0
$118$		−	1.94202i		−	0.178777i
$119$		0			0
$120$		0			0
$121$	−4.90265	−	8.49163i	−0.445695	−	0.771966i
$122$	−0.665771	−	1.15315i	−0.0602760	−	0.104401i
$123$		0			0
$124$	1.75081	+	1.01083i	0.157228	+	0.0907754i
$125$	10.9370			0.978234
$126$		0			0
$127$	−10.9133			−0.968400			−0.484200	−	0.874957i	$-0.660889\pi$
$127$	−10.9133			−0.968400			−0.484200	+	0.874957i	$0.660889\pi$
$128$	−0.866025	−	0.500000i	−0.0765466	−	0.0441942i
$129$		0			0
$130$	−7.79207	−	13.4963i	−0.683410	−	1.18370i
$131$	0.989677	+	1.71417i	0.0864684	+	0.149768i	0.906016	−	0.423243i	$-0.139108\pi$
$131$	0.989677	+	1.71417i	0.0864684	+	0.149768i	−0.819548	+	0.573011i	$0.805775\pi$
$132$		0			0
$133$		0			0
$134$		−	5.09918i		−	0.440502i
$135$		0			0
$136$			6.71727i			0.576001i
$137$	2.86923	+	1.65655i	0.245135	+	0.141528i	0.617534	−	0.786544i	$-0.288132\pi$
$137$	2.86923	+	1.65655i	0.245135	+	0.141528i	−0.372400	+	0.928072i	$0.621465\pi$
$138$		0			0
$139$	3.00698	−	1.73608i	0.255048	−	0.147252i	−0.367025	−	0.930211i	$-0.619624\pi$
$139$	3.00698	−	1.73608i	0.255048	−	0.147252i	0.622074	+	0.782959i	$0.286290\pi$
$140$		0			0
$141$		0			0
$142$	0.116753	−	0.202223i	0.00979772	−	0.0169701i
$143$	7.46479			0.624237
$144$		0			0
$145$		−	4.21408i		−	0.349961i
$146$	−3.40216	+	5.89272i	−0.281565	+	0.487685i
$147$		0			0
$148$	−3.57920	−	6.19935i	−0.294208	−	0.509584i
$149$	−11.5534	+	6.67036i	−0.946492	+	0.546457i	−0.891989	−	0.452056i	$-0.850691\pi$
$149$	−11.5534	+	6.67036i	−0.946492	+	0.546457i	−0.0545027	+	0.998514i	$0.517357\pi$
$150$		0			0
$151$	2.66995	−	4.62450i	0.217278	−	0.376336i	−0.736697	−	0.676223i	$-0.763616\pi$
$151$	2.66995	−	4.62450i	0.217278	−	0.376336i	0.953975	+	0.299887i	$0.0969489\pi$
$152$	−2.86351			−0.232261
$153$		0			0
$154$		0			0
$155$	3.99518	+	2.30662i	0.320901	+	0.185272i
$156$		0			0
$157$	15.3003	−	8.83364i	1.22110	−	0.705002i	0.255946	−	0.966691i	$-0.417613\pi$
$157$	15.3003	−	8.83364i	1.22110	−	0.705002i	0.965152	+	0.261689i	$0.0842796\pi$
$158$	6.29816	−	3.63624i	0.501054	−	0.289284i
$159$		0			0
$160$	−1.97618	−	1.14095i	−0.156231	−	0.0902000i
$161$		0			0
$162$		0			0
$163$	−15.8983			−1.24525			−0.622625	−	0.782520i	$-0.713934\pi$
$163$	−15.8983			−1.24525			−0.622625	+	0.782520i	$0.713934\pi$
$164$	−2.45515	+	4.25245i	−0.191715	+	0.332061i
$165$		0			0
$166$	−5.04638	+	2.91353i	−0.391675	+	0.226134i
$167$	2.85878	+	4.95155i	0.221219	+	0.383163i	0.955178	−	0.296031i	$-0.0956631\pi$
$167$	2.85878	+	4.95155i	0.221219	+	0.383163i	−0.733959	+	0.679193i	$0.762330\pi$
$168$		0			0
$169$	16.8207	−	29.1344i	1.29390	−	2.24110i
$170$			15.3281i			1.17561i
$171$		0			0
$172$	−7.48493			−0.570721
$173$	−7.60258	+	13.1681i	−0.578013	+	1.00115i	0.417694	+	0.908588i	$0.362839\pi$
$173$	−7.60258	+	13.1681i	−0.578013	+	1.00115i	−0.995707	+	0.0925606i	$0.970495\pi$
$174$		0			0
$175$		0			0
$176$	0.946590	−	0.546514i	0.0713519	−	0.0411950i
$177$		0			0
$178$	15.5834	+	8.99707i	1.16802	+	0.674359i
$179$			4.05972i			0.303438i	0.988424	+	0.151719i	$0.0484809\pi$
$179$			4.05972i			0.303438i	−0.988424	+	0.151719i	$0.951519\pi$
$180$		0			0
$181$		−	3.68452i		−	0.273869i	−0.990580	−	0.136934i	$-0.956275\pi$
$181$		−	3.68452i		−	0.273869i	0.990580	−	0.136934i	$-0.0437249\pi$
$182$		0			0
$183$		0			0
$184$	−1.95297	−	3.38264i	−0.143975	−	0.249371i
$185$	−8.16737	−	14.1463i	−0.600477	−	1.04006i
$186$		0			0
$187$	−6.35850	−	3.67108i	−0.464979	−	0.268456i
$188$	−6.80349			−0.496195
$189$		0			0
$190$	−6.53424			−0.474044
$191$	22.3425	+	12.8994i	1.61664	+	0.933370i	0.987780	+	0.155854i	$0.0498130\pi$
$191$	22.3425	+	12.8994i	1.61664	+	0.933370i	0.628864	+	0.777516i	$0.283520\pi$
$192$		0			0
$193$	4.64331	+	8.04245i	0.334233	+	0.578908i	0.983337	−	0.181791i	$-0.0581894\pi$
$193$	4.64331	+	8.04245i	0.334233	+	0.578908i	−0.649104	+	0.760699i	$0.724856\pi$
$194$	2.38967	+	4.13903i	0.171568	+	0.297165i
$195$		0			0
$196$		0			0
$197$			5.86237i			0.417677i	0.977950	+	0.208838i	$0.0669683\pi$
$197$			5.86237i			0.417677i	−0.977950	+	0.208838i	$0.933032\pi$
$198$		0			0
$199$		−	16.0638i		−	1.13873i	−0.822083	−	0.569367i	$-0.807188\pi$
$199$		−	16.0638i		−	1.13873i	0.822083	−	0.569367i	$-0.192812\pi$
$200$	−0.179327	−	0.103535i	−0.0126804	−	0.00732101i
$201$		0			0
$202$	−9.05829	+	5.22981i	−0.637339	+	0.367968i
$203$		0			0
$204$		0			0
$205$	−5.60242	+	9.70367i	−0.391290	+	0.677734i
$206$	−12.7477			−0.888172
$207$		0			0
$208$		−	6.82946i		−	0.473538i
$209$	1.56495	−	2.71057i	0.108250	−	0.187494i
$210$		0			0
$211$	12.3741	+	21.4325i	0.851867	+	1.47548i	0.879522	+	0.475859i	$0.157863\pi$
$211$	12.3741	+	21.4325i	0.851867	+	1.47548i	−0.0276550	+	0.999618i	$0.508804\pi$
$212$	−0.222069	+	0.128212i	−0.0152518	+	0.00880562i
$213$		0			0
$214$	4.76813	−	8.25865i	0.325943	−	0.564550i
$215$	−17.0799			−1.16484
$216$		0			0
$217$		0			0
$218$	4.99266	+	2.88251i	0.338146	+	0.195228i
$219$		0			0
$220$	2.16002	−	1.24709i	0.145629	−	0.0840788i
$221$	−39.7291	+	22.9376i	−2.67247	+	1.54295i
$222$		0			0
$223$	−3.20041	−	1.84776i	−0.214315	−	0.123735i	0.389000	−	0.921238i	$-0.372821\pi$
$223$	−3.20041	−	1.84776i	−0.214315	−	0.123735i	−0.603315	+	0.797503i	$0.706154\pi$
$224$		0			0
$225$		0			0
$226$	11.9318			0.793694
$227$	2.30549	−	3.99322i	0.153020	−	0.265039i	−0.779316	−	0.626631i	$-0.784433\pi$
$227$	2.30549	−	3.99322i	0.153020	−	0.265039i	0.932336	+	0.361592i	$0.117767\pi$
$228$		0			0
$229$	−13.8220	+	7.98016i	−0.913386	+	0.527344i	−0.881519	−	0.472148i	$-0.843479\pi$
$229$	−13.8220	+	7.98016i	−0.913386	+	0.527344i	−0.0318672	+	0.999492i	$0.510145\pi$
$230$	−4.45647	−	7.71884i	−0.293851	−	0.508965i
$231$		0			0
$232$	0.923371	−	1.59933i	0.0606223	−	0.105001i
$233$			7.13153i			0.467202i	0.972333	+	0.233601i	$0.0750510\pi$
$233$			7.13153i			0.467202i	−0.972333	+	0.233601i	$0.924949\pi$
$234$		0			0
$235$	−15.5249			−1.01273
$236$	0.971009	−	1.68184i	0.0632073	−	0.109478i
$237$		0			0
$238$		0			0
$239$	−13.6219	+	7.86462i	−0.881129	+	0.508720i	−0.871031	−	0.491229i	$-0.836548\pi$
$239$	−13.6219	+	7.86462i	−0.881129	+	0.508720i	−0.0100987	+	0.999949i	$0.503215\pi$
$240$		0			0
$241$	1.39292	+	0.804201i	0.0897257	+	0.0518031i	0.544191	−	0.838961i	$-0.316837\pi$
$241$	1.39292	+	0.804201i	0.0897257	+	0.0518031i	−0.454466	+	0.890764i	$0.650170\pi$
$242$		−	9.80529i		−	0.630308i
$243$		0			0
$244$		−	1.33154i		−	0.0852432i
$245$		0			0
$246$		0			0
$247$	−9.77810	−	16.9362i	−0.622166	−	1.07762i
$248$	1.01083	+	1.75081i	0.0641879	+	0.111177i
$249$		0			0
$250$	9.47171	+	5.46850i	0.599044	+	0.345858i
$251$	−18.3728			−1.15968			−0.579841	−	0.814729i	$-0.696885\pi$
$251$	−18.3728			−1.15968			−0.579841	+	0.814729i	$0.696885\pi$
$252$		0			0
$253$	4.26929			0.268408
$254$	−9.45121	−	5.45666i	−0.593021	−	0.342381i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	8.73678	+	15.1326i	0.544986	+	0.943943i	0.998608	+	0.0527487i	$0.0167982\pi$
$257$	8.73678	+	15.1326i	0.544986	+	0.943943i	−0.453622	+	0.891194i	$0.649868\pi$
$258$		0			0
$259$		0			0
$260$		−	15.5841i		−	0.966487i
$261$		0			0
$262$			1.97935i			0.122285i
$263$	23.9148	+	13.8072i	1.47465	+	0.851389i	0.999592	−	0.0285666i	$-0.00909427\pi$
$263$	23.9148	+	13.8072i	1.47465	+	0.851389i	0.475057	+	0.879955i	$0.342428\pi$
$264$		0			0
$265$	−0.506740	+	0.292567i	−0.0311288	+	0.0179722i
$266$		0			0
$267$		0			0
$268$	2.54959	−	4.41602i	0.155741	−	0.269751i
$269$	6.77214			0.412904			0.206452	−	0.978457i	$-0.433808\pi$
$269$	6.77214			0.412904			0.206452	+	0.978457i	$0.433808\pi$
$270$		0			0
$271$		−	8.35798i		−	0.507711i	−0.967242	−	0.253856i	$-0.918301\pi$
$271$		−	8.35798i		−	0.507711i	0.967242	−	0.253856i	$-0.0816988\pi$
$272$	−3.35863	+	5.81732i	−0.203647	+	0.352727i
$273$		0			0
$274$	1.65655	+	2.86923i	0.100076	+	0.173336i
$275$	0.196010	−	0.113166i	0.0118198	−	0.00682419i
$276$		0			0
$277$	8.10617	−	14.0403i	0.487053	−	0.843600i	−0.512837	−	0.858486i	$-0.671405\pi$
$277$	8.10617	−	14.0403i	0.487053	−	0.843600i	0.999889	+	0.0148865i	$0.00473868\pi$
$278$	3.47216			0.208246
$279$		0			0
$280$		0			0
$281$	25.3352	+	14.6273i	1.51137	+	0.872590i	0.999912	+	0.0132818i	$0.00422786\pi$
$281$	25.3352	+	14.6273i	1.51137	+	0.872590i	0.511458	+	0.859308i	$0.329105\pi$
$282$		0			0
$283$	−1.24230	+	0.717242i	−0.0738470	+	0.0426356i	−0.536469	−	0.843920i	$-0.680242\pi$
$283$	−1.24230	+	0.717242i	−0.0738470	+	0.0426356i	0.462622	+	0.886556i	$0.346909\pi$
$284$	0.202223	−	0.116753i	0.0119997	−	0.00692803i
$285$		0			0
$286$	6.46469	+	3.73239i	0.382265	+	0.220701i
$287$		0			0
$288$		0			0
$289$	28.1217			1.65422
$290$	2.10704	−	3.64950i	0.123730	−	0.214306i
$291$		0			0
$292$	−5.89272	+	3.40216i	−0.344845	+	0.199096i
$293$	−10.8260	−	18.7511i	−0.632459	−	1.09545i	−0.987047	−	0.160429i	$-0.948712\pi$
$293$	−10.8260	−	18.7511i	−0.632459	−	1.09545i	0.354588	−	0.935023i	$-0.384621\pi$
$294$		0			0
$295$	2.21575	−	3.83779i	0.129006	−	0.223444i
$296$		−	7.15840i		−	0.416073i
$297$		0			0
$298$	−13.3407			−0.772808
$299$	13.3377	−	23.1016i	0.771338	−	1.33600i
$300$		0			0
$301$		0			0
$302$	4.62450	−	2.66995i	0.266110	−	0.153639i
$303$		0			0
$304$	−2.47987	−	1.43175i	−0.142230	−	0.0821167i
$305$		−	3.03844i		−	0.173981i
$306$		0			0
$307$		−	13.4732i		−	0.768957i	−0.923134	−	0.384479i	$-0.874381\pi$
$307$		−	13.4732i		−	0.768957i	0.923134	−	0.384479i	$-0.125619\pi$
$308$		0			0
$309$		0			0
$310$	2.30662	+	3.99518i	0.131007	+	0.226911i
$311$	14.3669	+	24.8842i	0.814672	+	1.41105i	0.909563	+	0.415565i	$0.136416\pi$
$311$	14.3669	+	24.8842i	0.814672	+	1.41105i	−0.0948916	+	0.995488i	$0.530250\pi$
$312$		0			0
$313$	20.4636	+	11.8147i	1.15667	+	0.667805i	0.950504	−	0.310712i	$-0.100567\pi$
$313$	20.4636	+	11.8147i	1.15667	+	0.667805i	0.206167	+	0.978517i	$0.433901\pi$
$314$	17.6673			0.997023
$315$		0			0
$316$	7.27248			0.409109
$317$	−0.760093	−	0.438840i	−0.0426911	−	0.0246477i	0.478503	−	0.878086i	$-0.341180\pi$
$317$	−0.760093	−	0.438840i	−0.0426911	−	0.0246477i	−0.521194	+	0.853438i	$0.674513\pi$
$318$		0			0
$319$	1.00927	+	1.74811i	0.0565083	+	0.0978753i
$320$	−1.14095	−	1.97618i	−0.0637811	−	0.110472i
$321$		0			0
$322$		0			0
$323$			19.2349i			1.07026i
$324$		0			0
$325$		−	1.41417i		−	0.0784441i
$326$	−13.7683	−	7.94915i	−0.762557	−	0.440262i
$327$		0			0
$328$	−4.25245	+	2.45515i	−0.234802	+	0.135563i
$329$		0			0
$330$		0			0
$331$	10.0915	−	17.4790i	0.554680	−	0.960733i	−0.443249	−	0.896399i	$-0.646174\pi$
$331$	10.0915	−	17.4790i	0.554680	−	0.960733i	0.997928	−	0.0643345i	$-0.0204925\pi$
$332$	−5.82706			−0.319801
$333$		0			0
$334$			5.71756i			0.312851i
$335$	5.81791	−	10.0769i	0.317866	−	0.550561i
$336$		0			0
$337$	0.757605	+	1.31221i	0.0412694	+	0.0714807i	0.885922	−	0.463834i	$-0.153526\pi$
$337$	0.757605	+	1.31221i	0.0412694	+	0.0714807i	−0.844653	+	0.535314i	$0.820193\pi$
$338$	29.1344	−	16.8207i	1.58470	−	0.914927i
$339$		0			0
$340$	−7.66407	+	13.2746i	−0.415642	+	0.719914i
$341$	−2.20974			−0.119664
$342$		0			0
$343$		0			0
$344$	−6.48214	−	3.74246i	−0.349494	−	0.201780i
$345$		0			0
$346$	−13.1681	+	7.60258i	−0.707919	+	0.408717i
$347$	31.2622	−	18.0492i	1.67824	−	0.968934i	0.715461	−	0.698652i	$-0.246217\pi$
$347$	31.2622	−	18.0492i	1.67824	−	0.968934i	0.962781	−	0.270281i	$-0.0871167\pi$
$348$		0			0
$349$	16.7962	+	9.69727i	0.899078	+	0.519083i	0.876901	−	0.480671i	$-0.159607\pi$
$349$	16.7962	+	9.69727i	0.899078	+	0.519083i	0.0221769	+	0.999754i	$0.492940\pi$
$350$		0			0
$351$		0			0
$352$	1.09303			0.0582586
$353$	2.01909	−	3.49717i	0.107465	−	0.186136i	−0.807277	−	0.590172i	$-0.799060\pi$
$353$	2.01909	−	3.49717i	0.107465	−	0.186136i	0.914743	+	0.404037i	$0.132393\pi$
$354$		0			0
$355$	0.461452	−	0.266419i	0.0244913	−	0.0141401i
$356$	8.99707	+	15.5834i	0.476844	+	0.825918i
$357$		0			0
$358$	−2.02986	+	3.51582i	−0.107281	+	0.185817i
$359$		−	24.5546i		−	1.29594i	−0.761666	−	0.647970i	$-0.775618\pi$
$359$		−	24.5546i		−	1.29594i	0.761666	−	0.647970i	$-0.224382\pi$
$360$		0			0
$361$	10.8003			0.568438
$362$	1.84226	−	3.19089i	0.0968272	−	0.167710i
$363$		0			0
$364$		0			0
$365$	−13.4466	+	7.76339i	−0.703827	+	0.406354i
$366$		0			0
$367$	−6.28109	−	3.62639i	−0.327870	−	0.189296i	0.327025	−	0.945016i	$-0.393954\pi$
$367$	−6.28109	−	3.62639i	−0.327870	−	0.189296i	−0.654895	+	0.755720i	$0.727287\pi$
$368$		−	3.90593i		−	0.203611i
$369$		0			0
$370$		−	16.3347i		−	0.849203i
$371$		0			0
$372$		0			0
$373$	14.8921	+	25.7939i	0.771083	+	1.33556i	0.936970	+	0.349410i	$0.113618\pi$
$373$	14.8921	+	25.7939i	0.771083	+	1.33556i	−0.165887	+	0.986145i	$0.553049\pi$
$374$	−3.67108	−	6.35850i	−0.189827	−	0.328790i
$375$		0			0
$376$	−5.89199	−	3.40174i	−0.303856	−	0.175432i
$377$	12.6122			0.649564
$378$		0			0
$379$	6.11280			0.313993			0.156997	−	0.987599i	$-0.449819\pi$
$379$	6.11280			0.313993			0.156997	+	0.987599i	$0.449819\pi$
$380$	−5.65882	−	3.26712i	−0.290291	−	0.167600i
$381$		0			0
$382$	12.8994	+	22.3425i	0.659992	+	1.14314i
$383$	−16.2451	−	28.1374i	−0.830088	−	1.43775i	−0.897968	−	0.440061i	$-0.854957\pi$
$383$	−16.2451	−	28.1374i	−0.830088	−	1.43775i	0.0678797	−	0.997694i	$-0.478377\pi$
$384$		0			0
$385$		0			0
$386$			9.28662i			0.472677i
$387$		0			0
$388$			4.77934i			0.242634i
$389$	1.80316	+	1.04105i	0.0914236	+	0.0527834i	0.545015	−	0.838426i	$-0.316524\pi$
$389$	1.80316	+	1.04105i	0.0914236	+	0.0527834i	−0.453591	+	0.891210i	$0.649857\pi$
$390$		0			0
$391$	−22.7221	+	13.1186i	−1.14910	+	0.663435i
$392$		0			0
$393$		0			0
$394$	−2.93119	+	5.07696i	−0.147671	+	0.255774i
$395$	16.5951			0.834989
$396$		0			0
$397$		−	18.9364i		−	0.950393i	−0.879880	−	0.475196i	$-0.842377\pi$
$397$		−	18.9364i		−	0.950393i	0.879880	−	0.475196i	$-0.157623\pi$
$398$	8.03191	−	13.9117i	0.402603	−	0.697329i
$399$		0			0
$400$	−0.103535	−	0.179327i	−0.00517674	−	0.00896637i
$401$	3.35718	−	1.93827i	0.167650	−	0.0967926i	−0.413827	−	0.910355i	$-0.635808\pi$
$401$	3.35718	−	1.93827i	0.167650	−	0.0967926i	0.581477	+	0.813563i	$0.302475\pi$
$402$		0			0
$403$	−6.90343	+	11.9571i	−0.343884	+	0.595625i
$404$	−10.4596			−0.520385
$405$		0			0
$406$		0			0
$407$	6.77606	+	3.91216i	0.335877	+	0.193919i
$408$		0			0
$409$	−14.0286	+	8.09940i	−0.693669	+	0.400490i	−0.804985	−	0.593295i	$-0.797827\pi$
$409$	−14.0286	+	8.09940i	−0.693669	+	0.400490i	0.111316	+	0.993785i	$0.464493\pi$
$410$	−9.70367	+	5.60242i	−0.479230	+	0.276684i
$411$		0			0
$412$	−11.0398	−	6.37383i	−0.543892	−	0.314016i
$413$		0			0
$414$		0			0
$415$	−13.2968			−0.652713
$416$	3.41473	−	5.91448i	0.167421	−	0.289981i
$417$		0			0
$418$	2.71057	−	1.56495i	0.132578	−	0.0765441i
$419$	−1.63790	−	2.83692i	−0.0800165	−	0.138593i	0.823240	−	0.567693i	$-0.192164\pi$
$419$	−1.63790	−	2.83692i	−0.0800165	−	0.138593i	−0.903257	+	0.429100i	$0.858831\pi$
$420$		0			0
$421$	−0.844823	+	1.46328i	−0.0411741	+	0.0713157i	−0.885878	−	0.463918i	$-0.846443\pi$
$421$	−0.844823	+	1.46328i	−0.0411741	+	0.0713157i	0.844704	+	0.535234i	$0.179777\pi$
$422$			24.7482i			1.20472i
$423$		0			0
$424$	−0.256424			−0.0124530
$425$	−0.695470	+	1.20459i	−0.0337353	+	0.0584312i
$426$		0			0
$427$		0			0
$428$	8.25865	−	4.76813i	0.399197	−	0.230476i
$429$		0			0
$430$	−14.7916	−	8.53993i	−0.713314	−	0.411832i
$431$			0.0181384i			0.000873697i	1.00000		0.000436848i	$0.000139053\pi$
$431$			0.0181384i			0.000873697i	−1.00000		0.000436848i	$0.999861\pi$
$432$		0			0
$433$			5.36964i			0.258048i	0.991641	+	0.129024i	$0.0411845\pi$
$433$			5.36964i			0.258048i	−0.991641	+	0.129024i	$0.958815\pi$
$434$		0			0
$435$		0			0
$436$	2.88251	+	4.99266i	0.138047	+	0.239105i
$437$	−5.59233	−	9.68621i	−0.267518	−	0.463354i
$438$		0			0
$439$	18.9141	+	10.9201i	0.902720	+	0.521186i	0.878082	−	0.478511i	$-0.158823\pi$
$439$	18.9141	+	10.9201i	0.902720	+	0.521186i	0.0246384	+	0.999696i	$0.492157\pi$
$440$	2.49418			0.118905
$441$		0			0
$442$	−45.8753			−2.18206
$443$	1.81806	+	1.04966i	0.0863785	+	0.0498707i	0.542567	−	0.840012i	$-0.317452\pi$
$443$	1.81806	+	1.04966i	0.0863785	+	0.0498707i	−0.456189	+	0.889883i	$0.650786\pi$
$444$		0			0
$445$	20.5304	+	35.5597i	0.973235	+	1.68569i
$446$	−1.84776	−	3.20041i	−0.0874938	−	0.151544i
$447$		0			0
$448$		0			0
$449$		−	27.1356i		−	1.28061i	−0.768122	−	0.640303i	$-0.778809\pi$
$449$		−	27.1356i		−	1.28061i	0.768122	−	0.640303i	$-0.221191\pi$
$450$		0			0
$451$		−	5.36710i		−	0.252727i
$452$	10.3333	+	5.96592i	0.486036	+	0.280613i
$453$		0			0
$454$	3.99322	−	2.30549i	0.187411	−	0.108202i
$455$		0			0
$456$		0			0
$457$	−4.21598	+	7.30229i	−0.197215	+	0.341587i	−0.947624	−	0.319387i	$-0.896523\pi$
$457$	−4.21598	+	7.30229i	−0.197215	+	0.341587i	0.750409	+	0.660974i	$0.229856\pi$
$458$	−15.9603			−0.745777
$459$		0			0
$460$		−	8.91294i		−	0.415568i
$461$	4.67153	−	8.09133i	0.217575	−	0.376851i	−0.736491	−	0.676447i	$-0.763519\pi$
$461$	4.67153	−	8.09133i	0.217575	−	0.376851i	0.954066	+	0.299596i	$0.0968520\pi$
$462$		0			0
$463$	−12.7281	−	22.0458i	−0.591526	−	1.02455i	−0.994027	−	0.109134i	$-0.965192\pi$
$463$	−12.7281	−	22.0458i	−0.591526	−	1.02455i	0.402501	−	0.915420i	$-0.368141\pi$
$464$	1.59933	−	0.923371i	0.0742469	−	0.0428664i
$465$		0			0
$466$	−3.56577	+	6.17609i	−0.165181	+	0.286102i
$467$	20.6398			0.955094			0.477547	−	0.878606i	$-0.341526\pi$
$467$	20.6398			0.955094			0.477547	+	0.878606i	$0.341526\pi$
$468$		0			0
$469$		0			0
$470$	−13.4449	−	7.76244i	−0.620169	−	0.358055i
$471$		0			0
$472$	1.68184	−	0.971009i	0.0774128	−	0.0446943i
$473$	7.08516	−	4.09062i	0.325776	−	0.188087i
$474$		0			0
$475$	−0.513506	−	0.296473i	−0.0235613	−	0.0136031i
$476$		0			0
$477$		0			0
$478$	−15.7292			−0.719439
$479$	3.07442	−	5.32505i	0.140474	−	0.243308i	−0.787201	−	0.616696i	$-0.788471\pi$
$479$	3.07442	−	5.32505i	0.140474	−	0.243308i	0.927675	+	0.373388i	$0.121804\pi$
$480$		0			0
$481$	42.3382	−	24.4440i	1.93046	−	1.11455i
$482$	0.804201	+	1.39292i	0.0366303	+	0.0634456i
$483$		0			0
$484$	4.90265	−	8.49163i	0.222848	−	0.385983i
$485$			10.9060i			0.495215i
$486$		0			0
$487$	19.7273			0.893929			0.446965	−	0.894552i	$-0.352505\pi$
$487$	19.7273			0.893929			0.446965	+	0.894552i	$0.352505\pi$
$488$	0.665771	−	1.15315i	0.0301380	−	0.0522006i
$489$		0			0
$490$		0			0
$491$	−3.42935	+	1.97994i	−0.154764	+	0.0893533i	−0.575382	−	0.817885i	$-0.695147\pi$
$491$	−3.42935	+	1.97994i	−0.154764	+	0.0893533i	0.420618	+	0.907238i	$0.361813\pi$
$492$		0			0
$493$	−10.7431	−	6.20253i	−0.483845	−	0.279348i
$494$		−	19.5562i		−	0.879875i
$495$		0			0
$496$			2.02166i			0.0907754i
$497$		0			0
$498$		0			0
$499$	18.4092	+	31.8856i	0.824108	+	1.42740i	0.902599	+	0.430482i	$0.141656\pi$
$499$	18.4092	+	31.8856i	0.824108	+	1.42740i	−0.0784916	+	0.996915i	$0.525010\pi$
$500$	5.46850	+	9.47171i	0.244559	+	0.423588i
$501$		0			0
$502$	−15.9113	−	9.18641i	−0.710158	−	0.410010i
$503$	12.3802			0.552004			0.276002	−	0.961157i	$-0.410990\pi$
$503$	12.3802			0.552004			0.276002	+	0.961157i	$0.410990\pi$
$504$		0			0
$505$	−23.8678			−1.06210
$506$	3.69731	+	2.13465i	0.164366	+	0.0948966i
$507$		0			0
$508$	−5.45666	−	9.45121i	−0.242100	−	0.419329i
$509$	−7.54528	−	13.0688i	−0.334438	−	0.579264i	0.648938	−	0.760841i	$-0.275213\pi$
$509$	−7.54528	−	13.0688i	−0.334438	−	0.579264i	−0.983377	+	0.181577i	$0.941880\pi$
$510$		0			0
$511$		0			0
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			17.4736i			0.770726i
$515$	−25.1917	−	14.5445i	−1.11008	−	0.640905i
$516$		0			0
$517$	6.44011	−	3.71820i	0.283236	−	0.163526i
$518$		0			0
$519$		0			0
$520$	7.79207	−	13.4963i	0.341705	−	0.591850i
$521$	24.9816			1.09446			0.547231	−	0.836981i	$-0.315682\pi$
$521$	24.9816			1.09446			0.547231	+	0.836981i	$0.315682\pi$
$522$		0			0
$523$		−	25.0359i		−	1.09474i	−0.836889	−	0.547372i	$-0.815628\pi$
$523$		−	25.0359i		−	1.09474i	0.836889	−	0.547372i	$-0.184372\pi$
$524$	−0.989677	+	1.71417i	−0.0432342	+	0.0748839i
$525$		0			0
$526$	13.8072	+	23.9148i	0.602023	+	1.04273i
$527$	11.7607	−	6.79003i	0.512303	−	0.295778i
$528$		0			0
$529$	−3.87185	+	6.70625i	−0.168341	+	0.291576i
$530$	−0.585133			−0.0254166
$531$		0			0
$532$		0			0
$533$	−29.0419	−	16.7674i	−1.25795	−	0.726275i
$534$		0			0
$535$	18.8454	−	10.8804i	0.814759	−	0.470401i
$536$	4.41602	−	2.54959i	0.190743	−	0.110126i
$537$		0			0
$538$	5.86484	+	3.38607i	0.252851	+	0.145984i
$539$		0			0
$540$		0			0
$541$	−14.4608			−0.621720			−0.310860	−	0.950456i	$-0.600617\pi$
$541$	−14.4608			−0.621720			−0.310860	+	0.950456i	$0.600617\pi$
$542$	4.17899	−	7.23822i	0.179503	−	0.310908i
$543$		0			0
$544$	−5.81732	+	3.35863i	−0.249416	+	0.144000i
$545$	6.57761	+	11.3928i	0.281754	+	0.488012i
$546$		0			0
$547$	16.9160	−	29.2994i	0.723277	−	1.25275i	−0.236402	−	0.971655i	$-0.575968\pi$
$547$	16.9160	−	29.2994i	0.723277	−	1.25275i	0.959679	−	0.281098i	$-0.0906985\pi$
$548$			3.31310i			0.141528i
$549$		0			0
$550$	0.226333			0.00965086
$551$	2.64408	−	4.57968i	0.112642	−	0.195101i
$552$		0			0
$553$		0			0
$554$	14.0403	−	8.10617i	0.596515	−	0.344398i
$555$		0			0
$556$	3.00698	+	1.73608i	0.127524	+	0.0736261i
$557$		−	5.88269i		−	0.249257i	−0.992203	−	0.124629i	$-0.960226\pi$
$557$		−	5.88269i		−	0.249257i	0.992203	−	0.124629i	$-0.0397740\pi$
$558$		0			0
$559$		−	51.1180i		−	2.16206i
$560$		0			0
$561$		0			0
$562$	14.6273	+	25.3352i	0.617014	+	1.06870i
$563$	−7.78184	−	13.4785i	−0.327966	−	0.568053i	0.654142	−	0.756371i	$-0.273030\pi$
$563$	−7.78184	−	13.4785i	−0.327966	−	0.568053i	−0.982108	+	0.188318i	$0.939696\pi$
$564$		0			0
$565$	23.5795	+	13.6136i	0.991997	+	0.572730i
$566$	−1.43448			−0.0602958
$567$		0			0
$568$	0.233507			0.00979772
$569$	22.6993	+	13.1055i	0.951606	+	0.549410i	0.893580	−	0.448905i	$-0.148186\pi$
$569$	22.6993	+	13.1055i	0.951606	+	0.549410i	0.0580267	+	0.998315i	$0.481519\pi$
$570$		0			0
$571$	−12.8090	−	22.1859i	−0.536041	−	0.928451i	−0.999112	−	0.0421295i	$-0.986586\pi$
$571$	−12.8090	−	22.1859i	−0.536041	−	0.928451i	0.463071	−	0.886321i	$-0.346748\pi$
$572$	3.73239	+	6.46469i	0.156059	+	0.270302i
$573$		0			0
$574$		0			0
$575$		−	0.808799i		−	0.0337292i
$576$		0			0
$577$			32.3939i			1.34858i	0.738468	+	0.674288i	$0.235549\pi$
$577$			32.3939i			1.34858i	−0.738468	+	0.674288i	$0.764451\pi$
$578$	24.3541	+	14.0608i	1.01300	+	0.584853i
$579$		0			0
$580$	3.64950	−	2.10704i	0.151537	−	0.0874901i
$581$		0			0
$582$		0			0
$583$	0.140139	−	0.242728i	0.00580397	−	0.0100528i
$584$	−6.80432			−0.281565
$585$		0			0
$586$		−	21.6519i		−	0.894432i
$587$	−4.76851	+	8.25931i	−0.196818	+	0.340898i	−0.947495	−	0.319771i	$-0.896394\pi$
$587$	−4.76851	+	8.25931i	−0.196818	+	0.340898i	0.750677	+	0.660669i	$0.229727\pi$
$588$		0			0
$589$	2.89453	+	5.01347i	0.119267	+	0.206576i
$590$	3.83779	−	2.21575i	0.157999	−	0.0912208i
$591$		0			0
$592$	3.57920	−	6.19935i	0.147104	−	0.254792i
$593$	3.78817			0.155562			0.0777808	−	0.996970i	$-0.475217\pi$
$593$	3.78817			0.155562			0.0777808	+	0.996970i	$0.475217\pi$
$594$		0			0
$595$		0			0
$596$	−11.5534	−	6.67036i	−0.473246	−	0.273229i
$597$		0			0
$598$	23.1016	−	13.3377i	0.944693	−	0.545419i
$599$	31.2971	−	18.0694i	1.27876	−	0.738295i	0.302143	−	0.953263i	$-0.402298\pi$
$599$	31.2971	−	18.0694i	1.27876	−	0.738295i	0.976621	+	0.214968i	$0.0689647\pi$
$600$		0			0
$601$	13.8275	+	7.98332i	0.564036	+	0.325646i	0.754764	−	0.655997i	$-0.227751\pi$
$601$	13.8275	+	7.98332i	0.564036	+	0.325646i	−0.190728	+	0.981643i	$0.561085\pi$
$602$		0			0
$603$		0			0
$604$	5.33991			0.217278
$605$	11.1873	−	19.3771i	0.454830	−	0.787789i
$606$		0			0
$607$	−19.6190	+	11.3270i	−0.796309	+	0.459749i	−0.842179	−	0.539198i	$-0.818727\pi$
$607$	−19.6190	+	11.3270i	−0.796309	+	0.459749i	0.0458701	+	0.998947i	$0.485394\pi$
$608$	−1.43175	−	2.47987i	−0.0580653	−	0.100572i
$609$		0			0
$610$	1.51922	−	2.63137i	0.0615115	−	0.106541i
$611$		−	46.4641i		−	1.87974i
$612$		0			0
$613$	3.69517			0.149246			0.0746232	−	0.997212i	$-0.476225\pi$
$613$	3.69517			0.149246			0.0746232	+	0.997212i	$0.476225\pi$
$614$	6.73661	−	11.6682i	0.271868	−	0.470888i
$615$		0			0
$616$		0			0
$617$	−22.5187	+	13.0011i	−0.906567	+	0.523407i	−0.879325	−	0.476222i	$-0.842006\pi$
$617$	−22.5187	+	13.0011i	−0.906567	+	0.523407i	−0.0272418	+	0.999629i	$0.508672\pi$
$618$		0			0
$619$	20.5526	+	11.8660i	0.826079	+	0.476937i	0.852508	−	0.522714i	$-0.175080\pi$
$619$	20.5526	+	11.8660i	0.826079	+	0.476937i	−0.0264296	+	0.999651i	$0.508414\pi$
$620$			4.61324i			0.185272i
$621$		0			0
$622$			28.7338i			1.15212i
$623$		0			0
$624$		0			0
$625$	12.9962	+	22.5101i	0.519849	+	0.900406i
$626$	11.8147	+	20.4636i	0.472209	+	0.817890i
$627$		0			0
$628$	15.3003	+	8.83364i	0.610549	+	0.352501i
$629$	−48.0848			−1.91727
$630$		0			0
$631$	−0.664631			−0.0264586			−0.0132293	−	0.999912i	$-0.504211\pi$
$631$	−0.664631			−0.0264586			−0.0132293	+	0.999912i	$0.504211\pi$
$632$	6.29816	+	3.63624i	0.250527	+	0.144642i
$633$		0			0
$634$	−0.438840	−	0.760093i	−0.0174286	−	0.0301872i
$635$	−12.4515	−	21.5667i	−0.494125	−	0.855849i
$636$		0			0
$637$		0			0
$638$			2.01854i			0.0799148i
$639$		0			0
$640$		−	2.28190i		−	0.0902000i
$641$	7.27466	+	4.20003i	0.287332	+	0.165891i	0.636738	−	0.771080i	$-0.280283\pi$
$641$	7.27466	+	4.20003i	0.287332	+	0.165891i	−0.349406	+	0.936971i	$0.613617\pi$
$642$		0			0
$643$	0.237974	−	0.137394i	0.00938478	−	0.00541831i	−0.495300	−	0.868722i	$-0.664942\pi$
$643$	0.237974	−	0.137394i	0.00938478	−	0.00541831i	0.504685	+	0.863304i	$0.331609\pi$
$644$		0			0
$645$		0			0
$646$	−9.61747	+	16.6580i	−0.378394	+	0.655398i
$647$	−34.1015			−1.34067			−0.670335	−	0.742059i	$-0.733850\pi$
$647$	−34.1015			−1.34067			−0.670335	+	0.742059i	$0.733850\pi$
$648$		0			0
$649$			2.12268i			0.0833225i
$650$	0.707086	−	1.22471i	0.0277342	−	0.0480370i
$651$		0			0
$652$	−7.94915	−	13.7683i	−0.311313	−	0.539209i
$653$	−1.48356	+	0.856531i	−0.0580560	+	0.0335187i	−0.528747	−	0.848779i	$-0.677338\pi$
$653$	−1.48356	+	0.856531i	−0.0580560	+	0.0335187i	0.470691	+	0.882298i	$0.344005\pi$
$654$		0			0
$655$	−2.25834	+	3.91157i	−0.0882408	+	0.152838i
$656$	−4.91031			−0.191715
$657$		0			0
$658$		0			0
$659$	−30.0556	−	17.3526i	−1.17080	−	0.675961i	−0.216930	−	0.976187i	$-0.569604\pi$
$659$	−30.0556	−	17.3526i	−1.17080	−	0.675961i	−0.953868	+	0.300226i	$0.902938\pi$
$660$		0			0
$661$	33.2075	−	19.1724i	1.29162	−	0.745718i	0.312681	−	0.949858i	$-0.398773\pi$
$661$	33.2075	−	19.1724i	1.29162	−	0.745718i	0.978942	+	0.204140i	$0.0654397\pi$
$662$	17.4790	−	10.0915i	0.679341	−	0.392218i
$663$		0			0
$664$	−5.04638	−	2.91353i	−0.195838	−	0.113067i
$665$		0			0
$666$		0			0
$667$	7.21325			0.279298
$668$	−2.85878	+	4.95155i	−0.110610	+	0.191581i
$669$		0			0
$670$	10.0769	−	5.81791i	0.389305	−	0.224765i
$671$	0.727706	+	1.26042i	0.0280928	+	0.0486581i
$672$		0			0
$673$	−8.33538	+	14.4373i	−0.321305	+	0.556517i	−0.980758	−	0.195229i	$-0.937455\pi$
$673$	−8.33538	+	14.4373i	−0.321305	+	0.556517i	0.659452	+	0.751746i	$0.270788\pi$
$674$			1.51521i			0.0583637i
$675$		0			0
$676$	33.6415			1.29390
$677$	−10.4682	+	18.1315i	−0.402327	+	0.696850i	−0.994006	−	0.109323i	$-0.965132\pi$
$677$	−10.4682	+	18.1315i	−0.402327	+	0.696850i	0.591680	+	0.806173i	$0.298465\pi$
$678$		0			0
$679$		0			0
$680$	−13.2746	+	7.66407i	−0.509056	+	0.293903i
$681$		0			0
$682$	−1.91369	−	1.10487i	−0.0732789	−	0.0423076i
$683$			7.64415i			0.292495i	0.989248	+	0.146248i	$0.0467197\pi$
$683$			7.64415i			0.292495i	−0.989248	+	0.146248i	$0.953280\pi$
$684$		0			0
$685$			7.56016i			0.288859i
$686$		0			0
$687$		0			0
$688$	−3.74246	−	6.48214i	−0.142680	−	0.247129i
$689$	−0.875617	−	1.51661i	−0.0333583	−	0.0577783i
$690$		0			0
$691$	−9.10461	−	5.25655i	−0.346356	−	0.199969i	0.316723	−	0.948518i	$-0.397417\pi$
$691$	−9.10461	−	5.25655i	−0.346356	−	0.199969i	−0.663079	+	0.748549i	$0.730751\pi$
$692$	−15.2052			−0.578013
$693$		0			0
$694$	36.0985			1.37028
$695$	6.86162	+	3.96156i	0.260276	+	0.150270i
$696$		0			0
$697$	16.4919	+	28.5648i	0.624676	+	1.08197i
$698$	9.69727	+	16.7962i	0.367047	+	0.635744i
$699$		0			0
$700$		0			0
$701$		−	15.7336i		−	0.594250i	−0.954839	−	0.297125i	$-0.903972\pi$
$701$		−	15.7336i		−	0.594250i	0.954839	−	0.297125i	$-0.0960277\pi$
$702$		0			0
$703$		−	20.4981i		−	0.773102i
$704$	0.946590	+	0.546514i	0.0356759	+	0.0205975i
$705$		0			0
$706$	3.49717	−	2.01909i	0.131618	−	0.0759895i
$707$		0			0
$708$		0			0
$709$	1.44973	−	2.51100i	0.0544456	−	0.0943025i	−0.837518	−	0.546410i	$-0.815994\pi$
$709$	1.44973	−	2.51100i	0.0544456	−	0.0943025i	0.891964	+	0.452107i	$0.149328\pi$
$710$	0.532839			0.0199971
$711$		0			0
$712$			17.9941i			0.674359i
$713$	−3.94824	+	6.83855i	−0.147863	+	0.256106i
$714$		0			0
$715$	8.51695	+	14.7518i	0.318516	+	0.551686i
$716$	−3.51582	+	2.02986i	−0.131392	+	0.0758595i
$717$		0			0
$718$	12.2773	−	21.2649i	0.458184	−	0.793598i
$719$	−41.1289			−1.53385			−0.766924	−	0.641738i	$-0.778214\pi$
$719$	−41.1289			−1.53385			−0.766924	+	0.641738i	$0.778214\pi$
$720$		0			0
$721$		0			0
$722$	9.35335	+	5.40016i	0.348096	+	0.200973i
$723$		0			0
$724$	3.19089	−	1.84226i	0.118589	−	0.0684671i
$725$	0.331172	−	0.191202i	0.0122994	−	0.00710106i
$726$		0			0
$727$	33.6212	+	19.4112i	1.24694	+	0.719921i	0.970498	−	0.241109i	$-0.0775112\pi$
$727$	33.6212	+	19.4112i	1.24694	+	0.719921i	0.276442	+	0.961031i	$0.410845\pi$
$728$		0			0
$729$		0			0
$730$	−15.5268			−0.574672
$731$	−25.1391	+	43.5423i	−0.929804	+	1.61047i
$732$		0			0
$733$	22.5362	−	13.0113i	0.832394	−	0.480583i	−0.0222778	−	0.999752i	$-0.507092\pi$
$733$	22.5362	−	13.0113i	0.832394	−	0.480583i	0.854672	+	0.519169i	$0.173758\pi$
$734$	−3.62639	−	6.28109i	−0.133852	−	0.231839i
$735$		0			0
$736$	1.95297	−	3.38264i	0.0719873	−	0.124686i
$737$			5.57355i			0.205304i
$738$		0			0
$739$	7.40008			0.272216			0.136108	−	0.990694i	$-0.456541\pi$
$739$	7.40008			0.272216			0.136108	+	0.990694i	$0.456541\pi$
$740$	8.16737	−	14.1463i	0.300239	−	0.520028i
$741$		0			0
$742$		0			0
$743$	24.3241	−	14.0435i	0.892366	−	0.515208i	0.0176504	−	0.999844i	$-0.494381\pi$
$743$	24.3241	−	14.0435i	0.892366	−	0.515208i	0.874716	+	0.484636i	$0.161048\pi$
$744$		0			0
$745$	−26.3637	−	15.2211i	−0.965892	−	0.557658i
$746$			29.7842i			1.09048i
$747$		0			0
$748$		−	7.34216i		−	0.268456i
$749$		0			0
$750$		0			0
$751$	−21.1897	−	36.7016i	−0.773221	−	1.33926i	−0.935789	−	0.352561i	$-0.885311\pi$
$751$	−21.1897	−	36.7016i	−0.773221	−	1.33926i	0.162567	−	0.986697i	$-0.448023\pi$
$752$	−3.40174	−	5.89199i	−0.124049	−	0.214859i
$753$		0			0
$754$	10.9225	+	6.30612i	0.397775	+	0.229655i
$755$	12.1851			0.443463
$756$		0			0
$757$	−41.6462			−1.51366			−0.756828	−	0.653614i	$-0.773252\pi$
$757$	−41.6462			−1.51366			−0.756828	+	0.653614i	$0.773252\pi$
$758$	5.29384	+	3.05640i	0.192281	+	0.111013i
$759$		0			0
$760$	−3.26712	−	5.65882i	−0.118511	−	0.205267i
$761$	−17.4823	−	30.2802i	−0.633732	−	1.09766i	−0.986782	−	0.162051i	$-0.948189\pi$
$761$	−17.4823	−	30.2802i	−0.633732	−	1.09766i	0.353051	−	0.935604i	$-0.385144\pi$
$762$		0			0
$763$		0			0
$764$			25.7989i			0.933370i
$765$		0			0
$766$		−	32.4903i		−	1.17392i
$767$	11.4860	+	6.63146i	0.414737	+	0.239448i
$768$		0			0
$769$	−18.9307	+	10.9296i	−0.682658	+	0.394133i	−0.800856	−	0.598857i	$-0.795622\pi$
$769$	−18.9307	+	10.9296i	−0.682658	+	0.394133i	0.118198	+	0.992990i	$0.462288\pi$
$770$		0			0
$771$		0			0
$772$	−4.64331	+	8.04245i	−0.167116	+	0.289454i
$773$	−21.2736			−0.765158			−0.382579	−	0.923923i	$-0.624964\pi$
$773$	−21.2736			−0.765158			−0.382579	+	0.923923i	$0.624964\pi$
$774$		0			0
$775$			0.418625i			0.0150374i
$776$	−2.38967	+	4.13903i	−0.0857842	+	0.148583i
$777$		0			0
$778$	1.04105	+	1.80316i	0.0373235	+	0.0646463i
$779$	−12.1769	+	7.03035i	−0.436284	+	0.251889i
$780$		0			0
$781$	−0.127615	+	0.221035i	−0.00456641	+	0.00790925i
$782$	−26.2372			−0.938239
$783$		0			0
$784$		0			0
$785$	34.9138	+	20.1575i	1.24613	+	0.719452i
$786$		0			0
$787$	−40.7238	+	23.5119i	−1.45165	+	0.838108i	−0.998575	−	0.0533671i	$-0.983005\pi$
$787$	−40.7238	+	23.5119i	−1.45165	+	0.838108i	−0.453070	+	0.891475i	$0.649671\pi$
$788$	−5.07696	+	2.93119i	−0.180859	+	0.104419i
$789$		0			0
$790$	14.3718	+	8.29754i	0.511325	+	0.295213i
$791$		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.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.14095 + 1.97618i) q^{5} +1.00000i q^{8} +O(q^{10})\)	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.14095 + 1.97618i) q^{5} +1.00000i q^{8} +2.28190i q^{10} +(-0.946590 - 0.546514i) q^{11} +(-5.91448 + 3.41473i) q^{13} +(-0.500000 + 0.866025i) q^{16} +6.71727 q^{17} +2.86351i q^{19} +(-1.14095 + 1.97618i) q^{20} +(-0.546514 - 0.946590i) q^{22} +(-3.38264 + 1.95297i) q^{23} +(-0.103535 + 0.179327i) q^{25} -6.82946 q^{26} +(-1.59933 - 0.923371i) q^{29} +(1.75081 - 1.01083i) q^{31} +(-0.866025 + 0.500000i) q^{32} +(5.81732 + 3.35863i) q^{34} -7.15840 q^{37} +(-1.43175 + 2.47987i) q^{38} +(-1.97618 + 1.14095i) q^{40} +(2.45515 + 4.25245i) q^{41} +(-3.74246 + 6.48214i) q^{43} -1.09303i q^{44} -3.90593 q^{46} +(-3.40174 + 5.89199i) q^{47} +(-0.179327 + 0.103535i) q^{50} +(-5.91448 - 3.41473i) q^{52} +0.256424i q^{53} -2.49418i q^{55} +(-0.923371 - 1.59933i) q^{58} +(-0.971009 - 1.68184i) q^{59} +(-1.15315 - 0.665771i) q^{61} +2.02166 q^{62} -1.00000 q^{64} +(-13.4963 - 7.79207i) q^{65} +(-2.54959 - 4.41602i) q^{67} +(3.35863 + 5.81732i) q^{68} -0.233507i q^{71} +6.80432i q^{73} +(-6.19935 - 3.57920i) q^{74} +(-2.47987 + 1.43175i) q^{76} +(3.63624 - 6.29816i) q^{79} -2.28190 q^{80} +4.91031i q^{82} +(-2.91353 + 5.04638i) q^{83} +(7.66407 + 13.2746i) q^{85} +(-6.48214 + 3.74246i) q^{86} +(0.546514 - 0.946590i) q^{88} +17.9941 q^{89} +(-3.38264 - 1.95297i) q^{92} +(-5.89199 + 3.40174i) q^{94} +(-5.65882 + 3.26712i) q^{95} +(4.13903 + 2.38967i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4}+O(q^{10}) \) 16 * q + 8 * q^4	\( 16 q + 8 q^{4} + 12 q^{11} - 6 q^{13} - 8 q^{16} + 36 q^{17} - 6 q^{23} - 8 q^{25} - 24 q^{26} - 6 q^{29} - 6 q^{31} + 4 q^{37} - 6 q^{41} - 2 q^{43} - 12 q^{46} + 18 q^{47} + 12 q^{50} - 6 q^{52} + 6 q^{58} - 30 q^{59} + 60 q^{61} + 36 q^{62} - 16 q^{64} + 42 q^{65} + 14 q^{67} + 18 q^{68} + 18 q^{74} - 16 q^{79} - 12 q^{85} + 24 q^{86} + 48 q^{89} - 6 q^{92} - 66 q^{95} - 6 q^{97}+O(q^{100}) \) 16 * q + 8 * q^4 + 12 * q^11 - 6 * q^13 - 8 * q^16 + 36 * q^17 - 6 * q^23 - 8 * q^25 - 24 * q^26 - 6 * q^29 - 6 * q^31 + 4 * q^37 - 6 * q^41 - 2 * q^43 - 12 * q^46 + 18 * q^47 + 12 * q^50 - 6 * q^52 + 6 * q^58 - 30 * q^59 + 60 * q^61 + 36 * q^62 - 16 * q^64 + 42 * q^65 + 14 * q^67 + 18 * q^68 + 18 * q^74 - 16 * q^79 - 12 * q^85 + 24 * q^86 + 48 * q^89 - 6 * q^92 - 66 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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