LMFDB - Embedded newform 2646.2.m.a.881.8

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			881.8
Root			$1.73109 - 0.0577511i$ of defining polynomial
Character	$\chi$	$=$	2646.881
Dual form			2646.2.m.a.1763.8

$q$-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 $\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$

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	−0.489681	−	0.871902i	$-0.662887\pi$
$5$	1.14095	−	1.97618i	0.510248	−	0.883776i	0.999929	−	0.0118746i	$-0.00377989\pi$
$6$		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	−0.635869	−	0.771797i	$-0.719358\pi$
$11$	−0.946590	+	0.546514i	−0.285408	+	0.164780i	0.350461	+	0.936577i	$0.386025\pi$
$12$		0			0
$13$	−5.91448	−	3.41473i	−1.64038	−	0.947075i	−0.980697	−	0.195533i	$-0.937356\pi$
$13$	−5.91448	−	3.41473i	−1.64038	−	0.947075i	−0.659685	−	0.751542i	$-0.729310\pi$
$14$		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	$-0.893468\pi$
$19$		−	2.86351i		−	0.656934i	0.944515	−	0.328467i	$-0.106532\pi$
$20$	−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	0.104001	−	0.994577i	$-0.466836\pi$
$23$	−3.38264	−	1.95297i	−0.705328	−	0.407221i	−0.809329	+	0.587356i	$0.800169\pi$
$24$		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	−0.641089	−	0.767467i	$-0.721517\pi$
$29$	−1.59933	+	0.923371i	−0.296987	+	0.171466i	0.344101	+	0.938933i	$0.388184\pi$
$30$		0			0
$31$	1.75081	+	1.01083i	0.314455	+	0.181551i	0.648918	−	0.760858i	$-0.275222\pi$
$31$	1.75081	+	1.01083i	0.314455	+	0.181551i	−0.334463	+	0.942409i	$0.608555\pi$
$32$	−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	−0.608120	−	0.793845i	$-0.708076\pi$
$41$	2.45515	−	4.25245i	0.383431	−	0.664121i	0.991550	+	0.129724i	$0.0414092\pi$
$42$		0			0
$43$	−3.74246	−	6.48214i	−0.570721	−	0.988517i	−0.996492	−	0.0836863i	$-0.973331\pi$
$43$	−3.74246	−	6.48214i	−0.570721	−	0.988517i	0.425772	−	0.904831i	$-0.360003\pi$
$44$			1.09303i			0.164780i
$45$		0			0
$46$	−3.90593			−0.575898
$47$	−3.40174	−	5.89199i	−0.496195	−	0.859435i	0.503795	−	0.863823i	$-0.331937\pi$
$47$	−3.40174	−	5.89199i	−0.496195	−	0.859435i	−0.999990	+	0.00438774i	$0.998603\pi$
$48$		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	$-0.994394\pi$
$53$		−	0.256424i		−	0.0352225i	0.999845	−	0.0176112i	$-0.00560612\pi$
$54$		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	−0.922285	−	0.386510i	$-0.873680\pi$
$59$	−0.971009	+	1.68184i	−0.126415	+	0.218957i	0.795870	+	0.605467i	$0.207014\pi$
$60$		0			0
$61$	−1.15315	+	0.665771i	−0.147646	+	0.0852432i	−0.572003	−	0.820252i	$-0.693833\pi$
$61$	−1.15315	+	0.665771i	−0.147646	+	0.0852432i	0.424357	+	0.905495i	$0.360500\pi$
$62$	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	−0.978683	−	0.205375i	$-0.934159\pi$
$67$	−2.54959	+	4.41602i	−0.311482	+	0.539503i	0.667201	+	0.744877i	$0.267492\pi$
$68$	3.35863	−	5.81732i	0.407294	−	0.705454i
$69$		0			0
$70$		0			0
$71$			0.233507i			0.0277121i	0.999904	+	0.0138561i	$0.00441066\pi$
$71$			0.233507i			0.0277121i	−0.999904	+	0.0138561i	$0.995589\pi$
$72$		0			0
$73$		−	6.80432i		−	0.796386i	−0.917302	−	0.398193i	$-0.869638\pi$
$73$		−	6.80432i		−	0.796386i	0.917302	−	0.398193i	$-0.130362\pi$
$74$	−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	0.994790	−	0.101944i	$-0.0325062\pi$
$79$	3.63624	+	6.29816i	0.409109	+	0.708598i	−0.585681	+	0.810542i	$0.699173\pi$
$80$	−2.28190			−0.255124
$81$		0			0
$82$		−	4.91031i		−	0.542253i
$83$	−2.91353	−	5.04638i	−0.319801	−	0.553912i	0.660645	−	0.750698i	$-0.270283\pi$
$83$	−2.91353	−	5.04638i	−0.319801	−	0.553912i	−0.980446	+	0.196786i	$0.936950\pi$
$84$		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	−0.274931	−	0.961464i	$-0.588655\pi$
$97$	4.13903	−	2.38967i	0.420255	−	0.242634i	0.695186	+	0.718830i	$0.255322\pi$
$98$		0			0
$99$		0			0
$100$	−0.207069			−0.0207069
$101$	−5.22981	−	9.05829i	−0.520385	−	0.901334i	−0.999719	−	0.0237012i	$-0.992455\pi$
$101$	−5.22981	−	9.05829i	−0.520385	−	0.901334i	0.479334	−	0.877633i	$-0.340878\pi$
$102$		0			0
$103$	−11.0398	−	6.37383i	−1.08778	−	0.628033i	−0.154799	−	0.987946i	$-0.549473\pi$
$103$	−11.0398	−	6.37383i	−1.08778	−	0.628033i	−0.932986	+	0.359914i	$0.882806\pi$
$104$	−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	$0.152492\pi$
$107$			9.53627i			0.921906i	−0.887425	+	0.460953i	$0.847508\pi$
$108$		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	0.899868	−	0.436163i	$-0.143663\pi$
$113$	10.3333	+	5.96592i	0.972073	+	0.561227i	0.0722053	+	0.997390i	$0.476996\pi$
$114$		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	−0.819548	−	0.573011i	$-0.805775\pi$
$131$	0.989677	−	1.71417i	0.0864684	−	0.149768i	0.906016	+	0.423243i	$0.139108\pi$
$132$		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	−0.372400	−	0.928072i	$-0.621465\pi$
$137$	2.86923	−	1.65655i	0.245135	−	0.141528i	0.617534	+	0.786544i	$0.288132\pi$
$138$		0			0
$139$	3.00698	+	1.73608i	0.255048	+	0.147252i	0.622074	−	0.782959i	$-0.286290\pi$
$139$	3.00698	+	1.73608i	0.255048	+	0.147252i	−0.367025	+	0.930211i	$0.619624\pi$
$140$		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	−0.0545027	−	0.998514i	$-0.517357\pi$
$149$	−11.5534	−	6.67036i	−0.946492	−	0.546457i	−0.891989	+	0.452056i	$0.850691\pi$
$150$		0			0
$151$	2.66995	+	4.62450i	0.217278	+	0.376336i	0.953975	−	0.299887i	$-0.0969489\pi$
$151$	2.66995	+	4.62450i	0.217278	+	0.376336i	−0.736697	+	0.676223i	$0.763616\pi$
$152$	−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	0.965152	−	0.261689i	$-0.0842796\pi$
$157$	15.3003	+	8.83364i	1.22110	+	0.705002i	0.255946	+	0.966691i	$0.417613\pi$
$158$	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	−0.733959	−	0.679193i	$-0.762330\pi$
$167$	2.85878	−	4.95155i	0.221219	−	0.383163i	0.955178	+	0.296031i	$0.0956631\pi$
$168$		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	−0.995707	−	0.0925606i	$-0.970495\pi$
$173$	−7.60258	−	13.1681i	−0.578013	−	1.00115i	0.417694	−	0.908588i	$-0.362839\pi$
$174$		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	$-0.951519\pi$
$179$		−	4.05972i		−	0.303438i	0.988424	−	0.151719i	$-0.0484809\pi$
$180$		0			0
$181$			3.68452i			0.273869i	0.990580	+	0.136934i	$0.0437249\pi$
$181$			3.68452i			0.273869i	−0.990580	+	0.136934i	$0.956275\pi$
$182$		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	0.628864	−	0.777516i	$-0.283520\pi$
$191$	22.3425	−	12.8994i	1.61664	−	0.933370i	0.987780	−	0.155854i	$-0.0498130\pi$
$192$		0			0
$193$	4.64331	−	8.04245i	0.334233	−	0.578908i	−0.649104	−	0.760699i	$-0.724856\pi$
$193$	4.64331	−	8.04245i	0.334233	−	0.578908i	0.983337	+	0.181791i	$0.0581894\pi$
$194$	2.38967	−	4.13903i	0.171568	−	0.297165i
$195$		0			0
$196$		0			0
$197$		−	5.86237i		−	0.417677i	−0.977950	−	0.208838i	$-0.933032\pi$
$197$		−	5.86237i		−	0.417677i	0.977950	−	0.208838i	$-0.0669683\pi$
$198$		0			0
$199$			16.0638i			1.13873i	0.822083	+	0.569367i	$0.192812\pi$
$199$			16.0638i			1.13873i	−0.822083	+	0.569367i	$0.807188\pi$
$200$	−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	−0.0276550	−	0.999618i	$-0.508804\pi$
$211$	12.3741	−	21.4325i	0.851867	−	1.47548i	0.879522	−	0.475859i	$-0.157863\pi$
$212$	−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	−0.603315	−	0.797503i	$-0.706154\pi$
$223$	−3.20041	+	1.84776i	−0.214315	+	0.123735i	0.389000	+	0.921238i	$0.372821\pi$
$224$		0			0
$225$		0			0
$226$	11.9318			0.793694
$227$	2.30549	+	3.99322i	0.153020	+	0.265039i	0.932336	−	0.361592i	$-0.117767\pi$
$227$	2.30549	+	3.99322i	0.153020	+	0.265039i	−0.779316	+	0.626631i	$0.784433\pi$
$228$		0			0
$229$	−13.8220	−	7.98016i	−0.913386	−	0.527344i	−0.0318672	−	0.999492i	$-0.510145\pi$
$229$	−13.8220	−	7.98016i	−0.913386	−	0.527344i	−0.881519	+	0.472148i	$0.843479\pi$
$230$	−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	$-0.924949\pi$
$233$		−	7.13153i		−	0.467202i	0.972333	−	0.233601i	$-0.0750510\pi$
$234$		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	−0.0100987	−	0.999949i	$-0.503215\pi$
$239$	−13.6219	−	7.86462i	−0.881129	−	0.508720i	−0.871031	+	0.491229i	$0.836548\pi$
$240$		0			0
$241$	1.39292	−	0.804201i	0.0897257	−	0.0518031i	−0.454466	−	0.890764i	$-0.650170\pi$
$241$	1.39292	−	0.804201i	0.0897257	−	0.0518031i	0.544191	+	0.838961i	$0.316837\pi$
$242$			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	−0.453622	−	0.891194i	$-0.649868\pi$
$257$	8.73678	−	15.1326i	0.544986	−	0.943943i	0.998608	−	0.0527487i	$-0.0167982\pi$
$258$		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	0.475057	−	0.879955i	$-0.342428\pi$
$263$	23.9148	−	13.8072i	1.47465	−	0.851389i	0.999592	+	0.0285666i	$0.00909427\pi$
$264$		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	$0.0816988\pi$
$271$			8.35798i			0.507711i	−0.967242	+	0.253856i	$0.918301\pi$
$272$	−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	0.999889	−	0.0148865i	$-0.00473868\pi$
$277$	8.10617	+	14.0403i	0.487053	+	0.843600i	−0.512837	+	0.858486i	$0.671405\pi$
$278$	3.47216			0.208246
$279$		0			0
$280$		0			0
$281$	25.3352	−	14.6273i	1.51137	−	0.872590i	0.511458	−	0.859308i	$-0.329105\pi$
$281$	25.3352	−	14.6273i	1.51137	−	0.872590i	0.999912	−	0.0132818i	$-0.00422786\pi$
$282$		0			0
$283$	−1.24230	−	0.717242i	−0.0738470	−	0.0426356i	0.462622	−	0.886556i	$-0.346909\pi$
$283$	−1.24230	−	0.717242i	−0.0738470	−	0.0426356i	−0.536469	+	0.843920i	$0.680242\pi$
$284$	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	0.354588	+	0.935023i	$0.384621\pi$
$293$	−10.8260	+	18.7511i	−0.632459	+	1.09545i	−0.987047	+	0.160429i	$0.948712\pi$
$294$		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	$0.125619\pi$
$307$			13.4732i			0.768957i	−0.923134	+	0.384479i	$0.874381\pi$
$308$		0			0
$309$		0			0
$310$	2.30662	−	3.99518i	0.131007	−	0.226911i
$311$	14.3669	−	24.8842i	0.814672	−	1.41105i	−0.0948916	−	0.995488i	$-0.530250\pi$
$311$	14.3669	−	24.8842i	0.814672	−	1.41105i	0.909563	−	0.415565i	$-0.136416\pi$
$312$		0			0
$313$	20.4636	−	11.8147i	1.15667	−	0.667805i	0.206167	−	0.978517i	$-0.433901\pi$
$313$	20.4636	−	11.8147i	1.15667	−	0.667805i	0.950504	+	0.310712i	$0.100567\pi$
$314$	17.6673			0.997023
$315$		0			0
$316$	7.27248			0.409109
$317$	−0.760093	+	0.438840i	−0.0426911	+	0.0246477i	−0.521194	−	0.853438i	$-0.674513\pi$
$317$	−0.760093	+	0.438840i	−0.0426911	+	0.0246477i	0.478503	+	0.878086i	$0.341180\pi$
$318$		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	0.997928	+	0.0643345i	$0.0204925\pi$
$331$	10.0915	+	17.4790i	0.554680	+	0.960733i	−0.443249	+	0.896399i	$0.646174\pi$
$332$	−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	−0.844653	−	0.535314i	$-0.820193\pi$
$337$	0.757605	−	1.31221i	0.0412694	−	0.0714807i	0.885922	+	0.463834i	$0.153526\pi$
$338$	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	0.962781	+	0.270281i	$0.0871167\pi$
$347$	31.2622	+	18.0492i	1.67824	+	0.968934i	0.715461	+	0.698652i	$0.246217\pi$
$348$		0			0
$349$	16.7962	−	9.69727i	0.899078	−	0.519083i	0.0221769	−	0.999754i	$-0.492940\pi$
$349$	16.7962	−	9.69727i	0.899078	−	0.519083i	0.876901	+	0.480671i	$0.159607\pi$
$350$		0			0
$351$		0			0
$352$	1.09303			0.0582586
$353$	2.01909	+	3.49717i	0.107465	+	0.186136i	0.914743	−	0.404037i	$-0.132393\pi$
$353$	2.01909	+	3.49717i	0.107465	+	0.186136i	−0.807277	+	0.590172i	$0.799060\pi$
$354$		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	$0.224382\pi$
$359$			24.5546i			1.29594i	−0.761666	+	0.647970i	$0.775618\pi$
$360$		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	−0.654895	−	0.755720i	$-0.727287\pi$
$367$	−6.28109	+	3.62639i	−0.327870	+	0.189296i	0.327025	+	0.945016i	$0.393954\pi$
$368$			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	−0.165887	−	0.986145i	$-0.553049\pi$
$373$	14.8921	−	25.7939i	0.771083	−	1.33556i	0.936970	−	0.349410i	$-0.113618\pi$
$374$	−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	0.0678797	+	0.997694i	$0.478377\pi$
$383$	−16.2451	+	28.1374i	−0.830088	+	1.43775i	−0.897968	+	0.440061i	$0.854957\pi$
$384$		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	−0.453591	−	0.891210i	$-0.649857\pi$
$389$	1.80316	−	1.04105i	0.0914236	−	0.0527834i	0.545015	+	0.838426i	$0.316524\pi$
$390$		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	$0.157623\pi$
$397$			18.9364i			0.950393i	−0.879880	+	0.475196i	$0.842377\pi$
$398$	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	0.581477	−	0.813563i	$-0.302475\pi$
$401$	3.35718	+	1.93827i	0.167650	+	0.0967926i	−0.413827	+	0.910355i	$0.635808\pi$
$402$		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	0.111316	−	0.993785i	$-0.464493\pi$
$409$	−14.0286	−	8.09940i	−0.693669	−	0.400490i	−0.804985	+	0.593295i	$0.797827\pi$
$410$	−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	−0.903257	−	0.429100i	$-0.858831\pi$
$419$	−1.63790	+	2.83692i	−0.0800165	+	0.138593i	0.823240	+	0.567693i	$0.192164\pi$
$420$		0			0
$421$	−0.844823	−	1.46328i	−0.0411741	−	0.0713157i	0.844704	−	0.535234i	$-0.179777\pi$
$421$	−0.844823	−	1.46328i	−0.0411741	−	0.0713157i	−0.885878	+	0.463918i	$0.846443\pi$
$422$		−	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	$-0.999861\pi$
$431$		−	0.0181384i		−	0.000873697i	1.00000		0.000436848i	$-0.000139053\pi$
$432$		0			0
$433$		−	5.36964i		−	0.258048i	−0.991641	−	0.129024i	$-0.958815\pi$
$433$		−	5.36964i		−	0.258048i	0.991641	−	0.129024i	$-0.0411845\pi$
$434$		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	0.0246384	−	0.999696i	$-0.492157\pi$
$439$	18.9141	−	10.9201i	0.902720	−	0.521186i	0.878082	+	0.478511i	$0.158823\pi$
$440$	2.49418			0.118905
$441$		0			0
$442$	−45.8753			−2.18206
$443$	1.81806	−	1.04966i	0.0863785	−	0.0498707i	−0.456189	−	0.889883i	$-0.650786\pi$
$443$	1.81806	−	1.04966i	0.0863785	−	0.0498707i	0.542567	+	0.840012i	$0.317452\pi$
$444$		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	$0.221191\pi$
$449$			27.1356i			1.28061i	−0.768122	+	0.640303i	$0.778809\pi$
$450$		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	0.750409	−	0.660974i	$-0.229856\pi$
$457$	−4.21598	−	7.30229i	−0.197215	−	0.341587i	−0.947624	+	0.319387i	$0.896523\pi$
$458$	−15.9603			−0.745777
$459$		0			0
$460$			8.91294i			0.415568i
$461$	4.67153	+	8.09133i	0.217575	+	0.376851i	0.954066	−	0.299596i	$-0.0968520\pi$
$461$	4.67153	+	8.09133i	0.217575	+	0.376851i	−0.736491	+	0.676447i	$0.763519\pi$
$462$		0			0
$463$	−12.7281	+	22.0458i	−0.591526	+	1.02455i	0.402501	+	0.915420i	$0.368141\pi$
$463$	−12.7281	+	22.0458i	−0.591526	+	1.02455i	−0.994027	+	0.109134i	$0.965192\pi$
$464$	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	0.927675	−	0.373388i	$-0.121804\pi$
$479$	3.07442	+	5.32505i	0.140474	+	0.243308i	−0.787201	+	0.616696i	$0.788471\pi$
$480$		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	0.420618	−	0.907238i	$-0.361813\pi$
$491$	−3.42935	−	1.97994i	−0.154764	−	0.0893533i	−0.575382	+	0.817885i	$0.695147\pi$
$492$		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	−0.0784916	−	0.996915i	$-0.525010\pi$
$499$	18.4092	−	31.8856i	0.824108	−	1.42740i	0.902599	−	0.430482i	$-0.141656\pi$
$500$	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	−0.983377	−	0.181577i	$-0.941880\pi$
$509$	−7.54528	+	13.0688i	−0.334438	+	0.579264i	0.648938	+	0.760841i	$0.275213\pi$
$510$		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	$0.184372\pi$
$523$			25.0359i			1.09474i	−0.836889	+	0.547372i	$0.815628\pi$
$524$	−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	0.959679	+	0.281098i	$0.0906985\pi$
$547$	16.9160	+	29.2994i	0.723277	+	1.25275i	−0.236402	+	0.971655i	$0.575968\pi$
$548$		−	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	$0.0397740\pi$
$557$			5.88269i			0.249257i	−0.992203	+	0.124629i	$0.960226\pi$
$558$		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	−0.982108	−	0.188318i	$-0.939696\pi$
$563$	−7.78184	+	13.4785i	−0.327966	+	0.568053i	0.654142	+	0.756371i	$0.273030\pi$
$564$		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	0.0580267	−	0.998315i	$-0.481519\pi$
$569$	22.6993	−	13.1055i	0.951606	−	0.549410i	0.893580	+	0.448905i	$0.148186\pi$
$570$		0			0
$571$	−12.8090	+	22.1859i	−0.536041	+	0.928451i	0.463071	+	0.886321i	$0.346748\pi$
$571$	−12.8090	+	22.1859i	−0.536041	+	0.928451i	−0.999112	+	0.0421295i	$0.986586\pi$
$572$	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	$-0.764451\pi$
$577$		−	32.3939i		−	1.34858i	0.738468	−	0.674288i	$-0.235549\pi$
$578$	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	0.750677	−	0.660669i	$-0.229727\pi$
$587$	−4.76851	−	8.25931i	−0.196818	−	0.340898i	−0.947495	+	0.319771i	$0.896394\pi$
$588$		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	0.976621	−	0.214968i	$-0.0689647\pi$
$599$	31.2971	+	18.0694i	1.27876	+	0.738295i	0.302143	+	0.953263i	$0.402298\pi$
$600$		0			0
$601$	13.8275	−	7.98332i	0.564036	−	0.325646i	−0.190728	−	0.981643i	$-0.561085\pi$
$601$	13.8275	−	7.98332i	0.564036	−	0.325646i	0.754764	+	0.655997i	$0.227751\pi$
$602$		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	0.0458701	−	0.998947i	$-0.485394\pi$
$607$	−19.6190	−	11.3270i	−0.796309	−	0.459749i	−0.842179	+	0.539198i	$0.818727\pi$
$608$	−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	−0.0272418	−	0.999629i	$-0.508672\pi$
$617$	−22.5187	−	13.0011i	−0.906567	−	0.523407i	−0.879325	+	0.476222i	$0.842006\pi$
$618$		0			0
$619$	20.5526	−	11.8660i	0.826079	−	0.476937i	−0.0264296	−	0.999651i	$-0.508414\pi$
$619$	20.5526	−	11.8660i	0.826079	−	0.476937i	0.852508	+	0.522714i	$0.175080\pi$
$620$		−	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	−0.349406	−	0.936971i	$-0.613617\pi$
$641$	7.27466	−	4.20003i	0.287332	−	0.165891i	0.636738	+	0.771080i	$0.280283\pi$
$642$		0			0
$643$	0.237974	+	0.137394i	0.00938478	+	0.00541831i	0.504685	−	0.863304i	$-0.331609\pi$
$643$	0.237974	+	0.137394i	0.00938478	+	0.00541831i	−0.495300	+	0.868722i	$0.664942\pi$
$644$		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	0.470691	−	0.882298i	$-0.344005\pi$
$653$	−1.48356	−	0.856531i	−0.0580560	−	0.0335187i	−0.528747	+	0.848779i	$0.677338\pi$
$654$		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	−0.953868	−	0.300226i	$-0.902938\pi$
$659$	−30.0556	+	17.3526i	−1.17080	+	0.675961i	−0.216930	+	0.976187i	$0.569604\pi$
$660$		0			0
$661$	33.2075	+	19.1724i	1.29162	+	0.745718i	0.978942	−	0.204140i	$-0.0654397\pi$
$661$	33.2075	+	19.1724i	1.29162	+	0.745718i	0.312681	+	0.949858i	$0.398773\pi$
$662$	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	0.659452	−	0.751746i	$-0.270788\pi$
$673$	−8.33538	−	14.4373i	−0.321305	−	0.556517i	−0.980758	+	0.195229i	$0.937455\pi$
$674$		−	1.51521i		−	0.0583637i
$675$		0			0
$676$	33.6415			1.29390
$677$	−10.4682	−	18.1315i	−0.402327	−	0.696850i	0.591680	−	0.806173i	$-0.298465\pi$
$677$	−10.4682	−	18.1315i	−0.402327	−	0.696850i	−0.994006	+	0.109323i	$0.965132\pi$
$678$		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	$-0.953280\pi$
$683$		−	7.64415i		−	0.292495i	0.989248	−	0.146248i	$-0.0467197\pi$
$684$		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	−0.663079	−	0.748549i	$-0.730751\pi$
$691$	−9.10461	+	5.25655i	−0.346356	+	0.199969i	0.316723	+	0.948518i	$0.397417\pi$
$692$	−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	$0.0960277\pi$
$701$			15.7336i			0.594250i	−0.954839	+	0.297125i	$0.903972\pi$
$702$		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	0.891964	−	0.452107i	$-0.149328\pi$
$709$	1.44973	+	2.51100i	0.0544456	+	0.0943025i	−0.837518	+	0.546410i	$0.815994\pi$
$710$	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	0.276442	−	0.961031i	$-0.410845\pi$
$727$	33.6212	−	19.4112i	1.24694	−	0.719921i	0.970498	+	0.241109i	$0.0775112\pi$
$728$		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	0.854672	−	0.519169i	$-0.173758\pi$
$733$	22.5362	+	13.0113i	0.832394	+	0.480583i	−0.0222778	+	0.999752i	$0.507092\pi$
$734$	−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	0.874716	−	0.484636i	$-0.161048\pi$
$743$	24.3241	+	14.0435i	0.892366	+	0.515208i	0.0176504	+	0.999844i	$0.494381\pi$
$744$		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	0.162567	+	0.986697i	$0.448023\pi$
$751$	−21.1897	+	36.7016i	−0.773221	+	1.33926i	−0.935789	+	0.352561i	$0.885311\pi$
$752$	−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	0.353051	+	0.935604i	$0.385144\pi$
$761$	−17.4823	+	30.2802i	−0.633732	+	1.09766i	−0.986782	+	0.162051i	$0.948189\pi$
$762$		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	0.118198	−	0.992990i	$-0.462288\pi$
$769$	−18.9307	−	10.9296i	−0.682658	−	0.394133i	−0.800856	+	0.598857i	$0.795622\pi$
$770$		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	−0.453070	−	0.891475i	$-0.649671\pi$
$787$	−40.7238	−	23.5119i	−1.45165	−	0.838108i	−0.998575	+	0.0533671i	$0.983005\pi$
$788$	−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

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Database

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