LMFDB - Embedded newform 8034.2.a.bb.1.2

Newspace parameters

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$64.1518129839$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
Defining polynomial:	$x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} - 7296 x - 2048$
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.81560$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.81560 q^{5} -1.00000 q^{6} +4.94203 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.81560 q^{5} -1.00000 q^{6} +4.94203 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.81560 q^{10} +2.65767 q^{11} +1.00000 q^{12} -1.00000 q^{13} -4.94203 q^{14} -3.81560 q^{15} +1.00000 q^{16} -4.45654 q^{17} -1.00000 q^{18} +3.06468 q^{19} -3.81560 q^{20} +4.94203 q^{21} -2.65767 q^{22} -3.71423 q^{23} -1.00000 q^{24} +9.55882 q^{25} +1.00000 q^{26} +1.00000 q^{27} +4.94203 q^{28} +5.31017 q^{29} +3.81560 q^{30} -4.66765 q^{31} -1.00000 q^{32} +2.65767 q^{33} +4.45654 q^{34} -18.8568 q^{35} +1.00000 q^{36} -5.87682 q^{37} -3.06468 q^{38} -1.00000 q^{39} +3.81560 q^{40} -6.56132 q^{41} -4.94203 q^{42} -5.51993 q^{43} +2.65767 q^{44} -3.81560 q^{45} +3.71423 q^{46} -10.0443 q^{47} +1.00000 q^{48} +17.4236 q^{49} -9.55882 q^{50} -4.45654 q^{51} -1.00000 q^{52} -10.4448 q^{53} -1.00000 q^{54} -10.1406 q^{55} -4.94203 q^{56} +3.06468 q^{57} -5.31017 q^{58} +10.2648 q^{59} -3.81560 q^{60} -4.64313 q^{61} +4.66765 q^{62} +4.94203 q^{63} +1.00000 q^{64} +3.81560 q^{65} -2.65767 q^{66} -1.96165 q^{67} -4.45654 q^{68} -3.71423 q^{69} +18.8568 q^{70} -13.9630 q^{71} -1.00000 q^{72} -1.93219 q^{73} +5.87682 q^{74} +9.55882 q^{75} +3.06468 q^{76} +13.1343 q^{77} +1.00000 q^{78} -5.22572 q^{79} -3.81560 q^{80} +1.00000 q^{81} +6.56132 q^{82} +3.46955 q^{83} +4.94203 q^{84} +17.0044 q^{85} +5.51993 q^{86} +5.31017 q^{87} -2.65767 q^{88} +8.23562 q^{89} +3.81560 q^{90} -4.94203 q^{91} -3.71423 q^{92} -4.66765 q^{93} +10.0443 q^{94} -11.6936 q^{95} -1.00000 q^{96} +14.4149 q^{97} -17.4236 q^{98} +2.65767 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) $	$ 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + 6q^{10} - 8q^{11} + 14q^{12} - 14q^{13} + 4q^{14} - 6q^{15} + 14q^{16} - 4q^{17} - 14q^{18} - q^{19} - 6q^{20} - 4q^{21} + 8q^{22} - 9q^{23} - 14q^{24} + 24q^{25} + 14q^{26} + 14q^{27} - 4q^{28} - 10q^{29} + 6q^{30} - 5q^{31} - 14q^{32} - 8q^{33} + 4q^{34} - 16q^{35} + 14q^{36} - 4q^{37} + q^{38} - 14q^{39} + 6q^{40} - 24q^{41} + 4q^{42} - 8q^{44} - 6q^{45} + 9q^{46} - 32q^{47} + 14q^{48} + 24q^{49} - 24q^{50} - 4q^{51} - 14q^{52} - 5q^{53} - 14q^{54} - 8q^{55} + 4q^{56} - q^{57} + 10q^{58} - 13q^{59} - 6q^{60} + 2q^{61} + 5q^{62} - 4q^{63} + 14q^{64} + 6q^{65} + 8q^{66} - 16q^{67} - 4q^{68} - 9q^{69} + 16q^{70} - 29q^{71} - 14q^{72} + 4q^{74} + 24q^{75} - q^{76} - 9q^{77} + 14q^{78} - 21q^{79} - 6q^{80} + 14q^{81} + 24q^{82} - 40q^{83} - 4q^{84} - 7q^{85} - 10q^{87} + 8q^{88} - 48q^{89} + 6q^{90} + 4q^{91} - 9q^{92} - 5q^{93} + 32q^{94} - 26q^{95} - 14q^{96} + 18q^{97} - 24q^{98} - 8q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−3.81560	−1.70639	−0.853195	−	0.521593i	$-0.825338\pi$
$5$	−3.81560	−1.70639	−0.853195	+	0.521593i	$0.825338\pi$
$6$	−1.00000	−0.408248
$7$	4.94203	1.86791	0.933956	−	0.357389i	$-0.116333\pi$
$7$	4.94203	1.86791	0.933956	+	0.357389i	$0.116333\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.81560	1.20660
$11$	2.65767	0.801318	0.400659	−	0.916227i	$-0.368781\pi$
$11$	2.65767	0.801318	0.400659	+	0.916227i	$0.368781\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−4.94203	−1.32081
$15$	−3.81560	−0.985184
$16$	1.00000	0.250000
$17$	−4.45654	−1.08087	−0.540435	−	0.841386i	$-0.681740\pi$
$17$	−4.45654	−1.08087	−0.540435	+	0.841386i	$0.681740\pi$
$18$	−1.00000	−0.235702
$19$	3.06468	0.703086	0.351543	−	0.936172i	$-0.385657\pi$
$19$	3.06468	0.703086	0.351543	+	0.936172i	$0.385657\pi$
$20$	−3.81560	−0.853195
$21$	4.94203	1.07844
$22$	−2.65767	−0.566617
$23$	−3.71423	−0.774470	−0.387235	−	0.921981i	$-0.626570\pi$
$23$	−3.71423	−0.774470	−0.387235	+	0.921981i	$0.626570\pi$
$24$	−1.00000	−0.204124
$25$	9.55882	1.91176
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	4.94203	0.933956
$29$	5.31017	0.986074	0.493037	−	0.870008i	$-0.335887\pi$
$29$	5.31017	0.986074	0.493037	+	0.870008i	$0.335887\pi$
$30$	3.81560	0.696631
$31$	−4.66765	−0.838335	−0.419167	−	0.907909i	$-0.637678\pi$
$31$	−4.66765	−0.838335	−0.419167	+	0.907909i	$0.637678\pi$
$32$	−1.00000	−0.176777
$33$	2.65767	0.462641
$34$	4.45654	0.764291
$35$	−18.8568	−3.18738
$36$	1.00000	0.166667
$37$	−5.87682	−0.966144	−0.483072	−	0.875581i	$-0.660479\pi$
$37$	−5.87682	−0.966144	−0.483072	+	0.875581i	$0.660479\pi$
$38$	−3.06468	−0.497157
$39$	−1.00000	−0.160128
$40$	3.81560	0.603300
$41$	−6.56132	−1.02471	−0.512353	−	0.858775i	$-0.671226\pi$
$41$	−6.56132	−1.02471	−0.512353	+	0.858775i	$0.671226\pi$
$42$	−4.94203	−0.762572
$43$	−5.51993	−0.841781	−0.420890	−	0.907111i	$-0.638282\pi$
$43$	−5.51993	−0.841781	−0.420890	+	0.907111i	$0.638282\pi$
$44$	2.65767	0.400659
$45$	−3.81560	−0.568796
$46$	3.71423	0.547633
$47$	−10.0443	−1.46511	−0.732554	−	0.680708i	$-0.761672\pi$
$47$	−10.0443	−1.46511	−0.732554	+	0.680708i	$0.761672\pi$
$48$	1.00000	0.144338
$49$	17.4236	2.48909
$50$	−9.55882	−1.35182
$51$	−4.45654	−0.624041
$52$	−1.00000	−0.138675
$53$	−10.4448	−1.43470	−0.717348	−	0.696715i	$-0.754644\pi$
$53$	−10.4448	−1.43470	−0.717348	+	0.696715i	$0.754644\pi$
$54$	−1.00000	−0.136083
$55$	−10.1406	−1.36736
$56$	−4.94203	−0.660406
$57$	3.06468	0.405927
$58$	−5.31017	−0.697260
$59$	10.2648	1.33636	0.668179	−	0.744001i	$-0.267074\pi$
$59$	10.2648	1.33636	0.668179	+	0.744001i	$0.267074\pi$
$60$	−3.81560	−0.492592
$61$	−4.64313	−0.594492	−0.297246	−	0.954801i	$-0.596068\pi$
$61$	−4.64313	−0.594492	−0.297246	+	0.954801i	$0.596068\pi$
$62$	4.66765	0.592792
$63$	4.94203	0.622637
$64$	1.00000	0.125000
$65$	3.81560	0.473267
$66$	−2.65767	−0.327137
$67$	−1.96165	−0.239654	−0.119827	−	0.992795i	$-0.538234\pi$
$67$	−1.96165	−0.239654	−0.119827	+	0.992795i	$0.538234\pi$
$68$	−4.45654	−0.540435
$69$	−3.71423	−0.447140
$70$	18.8568	2.25382
$71$	−13.9630	−1.65710	−0.828550	−	0.559914i	$-0.810834\pi$
$71$	−13.9630	−1.65710	−0.828550	+	0.559914i	$0.810834\pi$
$72$	−1.00000	−0.117851
$73$	−1.93219	−0.226146	−0.113073	−	0.993587i	$-0.536069\pi$
$73$	−1.93219	−0.226146	−0.113073	+	0.993587i	$0.536069\pi$
$74$	5.87682	0.683167
$75$	9.55882	1.10376
$76$	3.06468	0.351543
$77$	13.1343	1.49679
$78$	1.00000	0.113228
$79$	−5.22572	−0.587940	−0.293970	−	0.955815i	$-0.594977\pi$
$79$	−5.22572	−0.587940	−0.293970	+	0.955815i	$0.594977\pi$
$80$	−3.81560	−0.426597
$81$	1.00000	0.111111
$82$	6.56132	0.724576
$83$	3.46955	0.380833	0.190417	−	0.981703i	$-0.439016\pi$
$83$	3.46955	0.380833	0.190417	+	0.981703i	$0.439016\pi$
$84$	4.94203	0.539220
$85$	17.0044	1.84439
$86$	5.51993	0.595229
$87$	5.31017	0.569310
$88$	−2.65767	−0.283309
$89$	8.23562	0.872974	0.436487	−	0.899710i	$-0.356222\pi$
$89$	8.23562	0.872974	0.436487	+	0.899710i	$0.356222\pi$
$90$	3.81560	0.402200
$91$	−4.94203	−0.518065
$92$	−3.71423	−0.387235
$93$	−4.66765	−0.484013
$94$	10.0443	1.03599
$95$	−11.6936	−1.19974
$96$	−1.00000	−0.102062
$97$	14.4149	1.46361	0.731806	−	0.681512i	$-0.238677\pi$
$97$	14.4149	1.46361	0.731806	+	0.681512i	$0.238677\pi$
$98$	−17.4236	−1.76005
$99$	2.65767	0.267106
$100$	9.55882	0.955882
$101$	1.10372	0.109824	0.0549121	−	0.998491i	$-0.482512\pi$
$101$	1.10372	0.109824	0.0549121	+	0.998491i	$0.482512\pi$
$102$	4.45654	0.441264
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	1.00000	0.0980581
$105$	−18.8568	−1.84024
$106$	10.4448	1.01448
$107$	4.79136	0.463198	0.231599	−	0.972811i	$-0.425604\pi$
$107$	4.79136	0.463198	0.231599	+	0.972811i	$0.425604\pi$
$108$	1.00000	0.0962250
$109$	−16.7042	−1.59997	−0.799986	−	0.600019i	$-0.795160\pi$
$109$	−16.7042	−1.59997	−0.799986	+	0.600019i	$0.795160\pi$
$110$	10.1406	0.966870
$111$	−5.87682	−0.557803
$112$	4.94203	0.466978
$113$	−2.10850	−0.198351	−0.0991754	−	0.995070i	$-0.531620\pi$
$113$	−2.10850	−0.198351	−0.0991754	+	0.995070i	$0.531620\pi$
$114$	−3.06468	−0.287034
$115$	14.1720	1.32155
$116$	5.31017	0.493037
$117$	−1.00000	−0.0924500
$118$	−10.2648	−0.944948
$119$	−22.0244	−2.01897
$120$	3.81560	0.348315
$121$	−3.93678	−0.357889
$122$	4.64313	0.420369
$123$	−6.56132	−0.591614
$124$	−4.66765	−0.419167
$125$	−17.3947	−1.55582
$126$	−4.94203	−0.440271
$127$	9.05858	0.803819	0.401910	−	0.915679i	$-0.368347\pi$
$127$	9.05858	0.803819	0.401910	+	0.915679i	$0.368347\pi$
$128$	−1.00000	−0.0883883
$129$	−5.51993	−0.486002
$130$	−3.81560	−0.334650
$131$	−13.6776	−1.19501	−0.597507	−	0.801863i	$-0.703842\pi$
$131$	−13.6776	−1.19501	−0.597507	+	0.801863i	$0.703842\pi$
$132$	2.65767	0.231321
$133$	15.1457	1.31330
$134$	1.96165	0.169461
$135$	−3.81560	−0.328395
$136$	4.45654	0.382145
$137$	14.1768	1.21121	0.605603	−	0.795767i	$-0.292932\pi$
$137$	14.1768	1.21121	0.605603	+	0.795767i	$0.292932\pi$
$138$	3.71423	0.316176
$139$	−9.95588	−0.844446	−0.422223	−	0.906492i	$-0.638750\pi$
$139$	−9.95588	−0.844446	−0.422223	+	0.906492i	$0.638750\pi$
$140$	−18.8568	−1.59369
$141$	−10.0443	−0.845881
$142$	13.9630	1.17175
$143$	−2.65767	−0.222246
$144$	1.00000	0.0833333
$145$	−20.2615	−1.68263
$146$	1.93219	0.159909
$147$	17.4236	1.43708
$148$	−5.87682	−0.483072
$149$	−5.41225	−0.443389	−0.221694	−	0.975116i	$-0.571159\pi$
$149$	−5.41225	−0.443389	−0.221694	+	0.975116i	$0.571159\pi$
$150$	−9.55882	−0.780475
$151$	−13.5624	−1.10369	−0.551846	−	0.833946i	$-0.686076\pi$
$151$	−13.5624	−1.10369	−0.551846	+	0.833946i	$0.686076\pi$
$152$	−3.06468	−0.248578
$153$	−4.45654	−0.360290
$154$	−13.1343	−1.05839
$155$	17.8099	1.43053
$156$	−1.00000	−0.0800641
$157$	8.54028	0.681588	0.340794	−	0.940138i	$-0.389304\pi$
$157$	8.54028	0.681588	0.340794	+	0.940138i	$0.389304\pi$
$158$	5.22572	0.415736
$159$	−10.4448	−0.828323
$160$	3.81560	0.301650
$161$	−18.3558	−1.44664
$162$	−1.00000	−0.0785674
$163$	−0.940059	−0.0736311	−0.0368155	−	0.999322i	$-0.511721\pi$
$163$	−0.940059	−0.0736311	−0.0368155	+	0.999322i	$0.511721\pi$
$164$	−6.56132	−0.512353
$165$	−10.1406	−0.789446
$166$	−3.46955	−0.269290
$167$	18.9730	1.46818	0.734088	−	0.679054i	$-0.237610\pi$
$167$	18.9730	1.46818	0.734088	+	0.679054i	$0.237610\pi$
$168$	−4.94203	−0.381286
$169$	1.00000	0.0769231
$170$	−17.0044	−1.30418
$171$	3.06468	0.234362
$172$	−5.51993	−0.420890
$173$	−3.59876	−0.273609	−0.136804	−	0.990598i	$-0.543683\pi$
$173$	−3.59876	−0.273609	−0.136804	+	0.990598i	$0.543683\pi$
$174$	−5.31017	−0.402563
$175$	47.2400	3.57101
$176$	2.65767	0.200330
$177$	10.2648	0.771547
$178$	−8.23562	−0.617286
$179$	−12.3216	−0.920958	−0.460479	−	0.887671i	$-0.652322\pi$
$179$	−12.3216	−0.920958	−0.460479	+	0.887671i	$0.652322\pi$
$180$	−3.81560	−0.284398
$181$	−14.3167	−1.06416	−0.532078	−	0.846695i	$-0.678589\pi$
$181$	−14.3167	−1.06416	−0.532078	+	0.846695i	$0.678589\pi$
$182$	4.94203	0.366328
$183$	−4.64313	−0.343230
$184$	3.71423	0.273816
$185$	22.4236	1.64862
$186$	4.66765	0.342249
$187$	−11.8440	−0.866121
$188$	−10.0443	−0.732554
$189$	4.94203	0.359480
$190$	11.6936	0.848343
$191$	−2.81037	−0.203351	−0.101676	−	0.994818i	$-0.532420\pi$
$191$	−2.81037	−0.203351	−0.101676	+	0.994818i	$0.532420\pi$
$192$	1.00000	0.0721688
$193$	−18.6496	−1.34243	−0.671214	−	0.741263i	$-0.734227\pi$
$193$	−18.6496	−1.34243	−0.671214	+	0.741263i	$0.734227\pi$
$194$	−14.4149	−1.03493
$195$	3.81560	0.273241
$196$	17.4236	1.24455
$197$	−2.28606	−0.162875	−0.0814375	−	0.996678i	$-0.525951\pi$
$197$	−2.28606	−0.162875	−0.0814375	+	0.996678i	$0.525951\pi$
$198$	−2.65767	−0.188872
$199$	5.66189	0.401361	0.200680	−	0.979657i	$-0.435685\pi$
$199$	5.66189	0.401361	0.200680	+	0.979657i	$0.435685\pi$
$200$	−9.55882	−0.675911
$201$	−1.96165	−0.138364
$202$	−1.10372	−0.0776575
$203$	26.2430	1.84190
$204$	−4.45654	−0.312020
$205$	25.0354	1.74855
$206$	−1.00000	−0.0696733
$207$	−3.71423	−0.258157
$208$	−1.00000	−0.0693375
$209$	8.14491	0.563395
$210$	18.8568	1.30124
$211$	13.0173	0.896148	0.448074	−	0.893996i	$-0.352110\pi$
$211$	13.0173	0.896148	0.448074	+	0.893996i	$0.352110\pi$
$212$	−10.4448	−0.717348
$213$	−13.9630	−0.956728
$214$	−4.79136	−0.327530
$215$	21.0618	1.43641
$216$	−1.00000	−0.0680414
$217$	−23.0677	−1.56594
$218$	16.7042	1.13135
$219$	−1.93219	−0.130565
$220$	−10.1406	−0.683680
$221$	4.45654	0.299780
$222$	5.87682	0.394426
$223$	3.43833	0.230247	0.115124	−	0.993351i	$-0.463274\pi$
$223$	3.43833	0.230247	0.115124	+	0.993351i	$0.463274\pi$
$224$	−4.94203	−0.330203
$225$	9.55882	0.637255
$226$	2.10850	0.140255
$227$	−14.8404	−0.984990	−0.492495	−	0.870315i	$-0.663915\pi$
$227$	−14.8404	−0.984990	−0.492495	+	0.870315i	$0.663915\pi$
$228$	3.06468	0.202963
$229$	4.64030	0.306640	0.153320	−	0.988177i	$-0.451004\pi$
$229$	4.64030	0.306640	0.153320	+	0.988177i	$0.451004\pi$
$230$	−14.1720	−0.934475
$231$	13.1343	0.864173
$232$	−5.31017	−0.348630
$233$	18.5533	1.21547	0.607734	−	0.794141i	$-0.292079\pi$
$233$	18.5533	1.21547	0.607734	+	0.794141i	$0.292079\pi$
$234$	1.00000	0.0653720
$235$	38.3250	2.50005
$236$	10.2648	0.668179
$237$	−5.22572	−0.339447
$238$	22.0244	1.42763
$239$	3.39069	0.219325	0.109663	−	0.993969i	$-0.465023\pi$
$239$	3.39069	0.219325	0.109663	+	0.993969i	$0.465023\pi$
$240$	−3.81560	−0.246296
$241$	−20.7142	−1.33432	−0.667158	−	0.744916i	$-0.732489\pi$
$241$	−20.7142	−1.33432	−0.667158	+	0.744916i	$0.732489\pi$
$242$	3.93678	0.253066
$243$	1.00000	0.0641500
$244$	−4.64313	−0.297246
$245$	−66.4817	−4.24736
$246$	6.56132	0.418334
$247$	−3.06468	−0.195001
$248$	4.66765	0.296396
$249$	3.46955	0.219874
$250$	17.3947	1.10013
$251$	14.5570	0.918830	0.459415	−	0.888222i	$-0.348059\pi$
$251$	14.5570	0.918830	0.459415	+	0.888222i	$0.348059\pi$
$252$	4.94203	0.311319
$253$	−9.87120	−0.620597
$254$	−9.05858	−0.568386
$255$	17.0044	1.06486
$256$	1.00000	0.0625000
$257$	24.7082	1.54125	0.770627	−	0.637286i	$-0.219943\pi$
$257$	24.7082	1.54125	0.770627	+	0.637286i	$0.219943\pi$
$258$	5.51993	0.343656
$259$	−29.0434	−1.80467
$260$	3.81560	0.236634
$261$	5.31017	0.328691
$262$	13.6776	0.845003
$263$	−14.4358	−0.890147	−0.445074	−	0.895494i	$-0.646822\pi$
$263$	−14.4358	−0.890147	−0.445074	+	0.895494i	$0.646822\pi$
$264$	−2.65767	−0.163568
$265$	39.8530	2.44815
$266$	−15.1457	−0.928645
$267$	8.23562	0.504012
$268$	−1.96165	−0.119827
$269$	−5.54942	−0.338354	−0.169177	−	0.985586i	$-0.554111\pi$
$269$	−5.54942	−0.338354	−0.169177	+	0.985586i	$0.554111\pi$
$270$	3.81560	0.232210
$271$	18.8197	1.14321	0.571606	−	0.820528i	$-0.306320\pi$
$271$	18.8197	1.14321	0.571606	+	0.820528i	$0.306320\pi$
$272$	−4.45654	−0.270218
$273$	−4.94203	−0.299105
$274$	−14.1768	−0.856452
$275$	25.4042	1.53193
$276$	−3.71423	−0.223570
$277$	13.4083	0.805627	0.402814	−	0.915282i	$-0.368032\pi$
$277$	13.4083	0.805627	0.402814	+	0.915282i	$0.368032\pi$
$278$	9.95588	0.597114
$279$	−4.66765	−0.279445
$280$	18.8568	1.12691
$281$	−31.6493	−1.88804	−0.944020	−	0.329889i	$-0.892989\pi$
$281$	−31.6493	−1.88804	−0.944020	+	0.329889i	$0.892989\pi$
$282$	10.0443	0.598128
$283$	0.0113313	0.000673573 0	0.000336787	−	1.00000i	$-0.499893\pi$
$283$	0.0113313	0.000673573 0	0.000336787		1.00000i	$0.499893\pi$
$284$	−13.9630	−0.828550
$285$	−11.6936	−0.692669
$286$	2.65767	0.157151
$287$	−32.4262	−1.91406
$288$	−1.00000	−0.0589256
$289$	2.86078	0.168281
$290$	20.2615	1.18980
$291$	14.4149	0.845017
$292$	−1.93219	−0.113073
$293$	−4.41047	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$293$	−4.41047	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$294$	−17.4236	−1.01617
$295$	−39.1662	−2.28035
$296$	5.87682	0.341583
$297$	2.65767	0.154214
$298$	5.41225	0.313523
$299$	3.71423	0.214799
$300$	9.55882	0.551879
$301$	−27.2796	−1.57237
$302$	13.5624	0.780428
$303$	1.10372	0.0634071
$304$	3.06468	0.175771
$305$	17.7163	1.01443
$306$	4.45654	0.254764
$307$	20.0406	1.14378	0.571888	−	0.820332i	$-0.306211\pi$
$307$	20.0406	1.14378	0.571888	+	0.820332i	$0.306211\pi$
$308$	13.1343	0.748396
$309$	1.00000	0.0568880
$310$	−17.8099	−1.01153
$311$	−4.04538	−0.229392	−0.114696	−	0.993401i	$-0.536589\pi$
$311$	−4.04538	−0.229392	−0.114696	+	0.993401i	$0.536589\pi$
$312$	1.00000	0.0566139
$313$	8.04036	0.454468	0.227234	−	0.973840i	$-0.427032\pi$
$313$	8.04036	0.454468	0.227234	+	0.973840i	$0.427032\pi$
$314$	−8.54028	−0.481956
$315$	−18.8568	−1.06246
$316$	−5.22572	−0.293970
$317$	−5.58000	−0.313404	−0.156702	−	0.987646i	$-0.550086\pi$
$317$	−5.58000	−0.313404	−0.156702	+	0.987646i	$0.550086\pi$
$318$	10.4448	0.585713
$319$	14.1127	0.790159
$320$	−3.81560	−0.213299
$321$	4.79136	0.267427
$322$	18.3558	1.02293
$323$	−13.6579	−0.759945
$324$	1.00000	0.0555556
$325$	−9.55882	−0.530228
$326$	0.940059	0.0520650
$327$	−16.7042	−0.923744
$328$	6.56132	0.362288
$329$	−49.6391	−2.73669
$330$	10.1406	0.558223
$331$	−10.5804	−0.581552	−0.290776	−	0.956791i	$-0.593913\pi$
$331$	−10.5804	−0.581552	−0.290776	+	0.956791i	$0.593913\pi$
$332$	3.46955	0.190417
$333$	−5.87682	−0.322048
$334$	−18.9730	−1.03816
$335$	7.48488	0.408942
$336$	4.94203	0.269610
$337$	−0.574016	−0.0312686	−0.0156343	−	0.999878i	$-0.504977\pi$
$337$	−0.574016	−0.0312686	−0.0156343	+	0.999878i	$0.504977\pi$
$338$	−1.00000	−0.0543928
$339$	−2.10850	−0.114518
$340$	17.0044	0.922193
$341$	−12.4051	−0.671773
$342$	−3.06468	−0.165719
$343$	51.5140	2.78149
$344$	5.51993	0.297614
$345$	14.1720	0.762996
$346$	3.59876	0.193471
$347$	9.56853	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$347$	9.56853	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$348$	5.31017	0.284655
$349$	−21.9927	−1.17724	−0.588622	−	0.808408i	$-0.700329\pi$
$349$	−21.9927	−1.17724	−0.588622	+	0.808408i	$0.700329\pi$
$350$	−47.2400	−2.52508
$351$	−1.00000	−0.0533761
$352$	−2.65767	−0.141654
$353$	−24.7904	−1.31946	−0.659731	−	0.751502i	$-0.729330\pi$
$353$	−24.7904	−1.31946	−0.659731	+	0.751502i	$0.729330\pi$
$354$	−10.2648	−0.545566
$355$	53.2772	2.82766
$356$	8.23562	0.436487
$357$	−22.0244	−1.16565
$358$	12.3216	0.651216
$359$	7.14448	0.377071	0.188536	−	0.982066i	$-0.439626\pi$
$359$	7.14448	0.377071	0.188536	+	0.982066i	$0.439626\pi$
$360$	3.81560	0.201100
$361$	−9.60774	−0.505670
$362$	14.3167	0.752472
$363$	−3.93678	−0.206627
$364$	−4.94203	−0.259033
$365$	7.37247	0.385892
$366$	4.64313	0.242700
$367$	10.7740	0.562398	0.281199	−	0.959649i	$-0.409268\pi$
$367$	10.7740	0.562398	0.281199	+	0.959649i	$0.409268\pi$
$368$	−3.71423	−0.193617
$369$	−6.56132	−0.341569
$370$	−22.4236	−1.16575
$371$	−51.6183	−2.67989
$372$	−4.66765	−0.242006
$373$	28.4760	1.47443	0.737215	−	0.675658i	$-0.236141\pi$
$373$	28.4760	1.47443	0.737215	+	0.675658i	$0.236141\pi$
$374$	11.8440	0.612440
$375$	−17.3947	−0.898256
$376$	10.0443	0.517994
$377$	−5.31017	−0.273488
$378$	−4.94203	−0.254191
$379$	−14.3097	−0.735038	−0.367519	−	0.930016i	$-0.619793\pi$
$379$	−14.3097	−0.735038	−0.367519	+	0.930016i	$0.619793\pi$
$380$	−11.6936	−0.599869
$381$	9.05858	0.464085
$382$	2.81037	0.143791
$383$	−28.4425	−1.45334	−0.726672	−	0.686984i	$-0.758934\pi$
$383$	−28.4425	−1.45334	−0.726672	+	0.686984i	$0.758934\pi$
$384$	−1.00000	−0.0510310
$385$	−50.1152	−2.55411
$386$	18.6496	0.949240
$387$	−5.51993	−0.280594
$388$	14.4149	0.731806
$389$	19.0716	0.966967	0.483484	−	0.875353i	$-0.339371\pi$
$389$	19.0716	0.966967	0.483484	+	0.875353i	$0.339371\pi$
$390$	−3.81560	−0.193211
$391$	16.5526	0.837102
$392$	−17.4236	−0.880027
$393$	−13.6776	−0.689942
$394$	2.28606	0.115170
$395$	19.9393	1.00325
$396$	2.65767	0.133553
$397$	−22.7585	−1.14221	−0.571107	−	0.820876i	$-0.693486\pi$
$397$	−22.7585	−1.14221	−0.571107	+	0.820876i	$0.693486\pi$
$398$	−5.66189	−0.283805
$399$	15.1457	0.758235
$400$	9.55882	0.477941
$401$	−11.5086	−0.574714	−0.287357	−	0.957824i	$-0.592777\pi$
$401$	−11.5086	−0.574714	−0.287357	+	0.957824i	$0.592777\pi$
$402$	1.96165	0.0978382
$403$	4.66765	0.232512
$404$	1.10372	0.0549121
$405$	−3.81560	−0.189599
$406$	−26.2430	−1.30242
$407$	−15.6187	−0.774188
$408$	4.45654	0.220632
$409$	−16.3617	−0.809034	−0.404517	−	0.914531i	$-0.632560\pi$
$409$	−16.3617	−0.809034	−0.404517	+	0.914531i	$0.632560\pi$
$410$	−25.0354	−1.23641
$411$	14.1768	0.699290
$412$	1.00000	0.0492665
$413$	50.7287	2.49620
$414$	3.71423	0.182544
$415$	−13.2384	−0.649849
$416$	1.00000	0.0490290
$417$	−9.95588	−0.487541
$418$	−8.14491	−0.398381
$419$	−31.7684	−1.55199	−0.775994	−	0.630740i	$-0.782751\pi$
$419$	−31.7684	−1.55199	−0.775994	+	0.630740i	$0.782751\pi$
$420$	−18.8568	−0.920118
$421$	−8.56383	−0.417376	−0.208688	−	0.977982i	$-0.566919\pi$
$421$	−8.56383	−0.417376	−0.208688	+	0.977982i	$0.566919\pi$
$422$	−13.0173	−0.633673
$423$	−10.0443	−0.488370
$424$	10.4448	0.507242
$425$	−42.5993	−2.06637
$426$	13.9630	0.676509
$427$	−22.9465	−1.11046
$428$	4.79136	0.231599
$429$	−2.65767	−0.128314
$430$	−21.0618	−1.01569
$431$	25.0472	1.20648	0.603240	−	0.797560i	$-0.293876\pi$
$431$	25.0472	1.20648	0.603240	+	0.797560i	$0.293876\pi$
$432$	1.00000	0.0481125
$433$	−26.4327	−1.27027	−0.635137	−	0.772399i	$-0.719056\pi$
$433$	−26.4327	−1.27027	−0.635137	+	0.772399i	$0.719056\pi$
$434$	23.0677	1.10728
$435$	−20.2615	−0.971464
$436$	−16.7042	−0.799986
$437$	−11.3829	−0.544519
$438$	1.93219	0.0923236
$439$	−26.3084	−1.25563	−0.627815	−	0.778363i	$-0.716050\pi$
$439$	−26.3084	−1.25563	−0.627815	+	0.778363i	$0.716050\pi$
$440$	10.1406	0.483435
$441$	17.4236	0.829697
$442$	−4.45654	−0.211976
$443$	32.4551	1.54199	0.770994	−	0.636842i	$-0.219760\pi$
$443$	32.4551	1.54199	0.770994	+	0.636842i	$0.219760\pi$
$444$	−5.87682	−0.278902
$445$	−31.4239	−1.48963
$446$	−3.43833	−0.162809
$447$	−5.41225	−0.255991
$448$	4.94203	0.233489
$449$	−26.6480	−1.25760	−0.628798	−	0.777569i	$-0.716453\pi$
$449$	−26.6480	−1.25760	−0.628798	+	0.777569i	$0.716453\pi$
$450$	−9.55882	−0.450607
$451$	−17.4378	−0.821115
$452$	−2.10850	−0.0991754
$453$	−13.5624	−0.637217
$454$	14.8404	0.696493
$455$	18.8568	0.884021
$456$	−3.06468	−0.143517
$457$	32.2187	1.50713	0.753565	−	0.657374i	$-0.228333\pi$
$457$	32.2187	1.50713	0.753565	+	0.657374i	$0.228333\pi$
$458$	−4.64030	−0.216827
$459$	−4.45654	−0.208014
$460$	14.1720	0.660774
$461$	−25.5852	−1.19162	−0.595811	−	0.803125i	$-0.703169\pi$
$461$	−25.5852	−1.19162	−0.595811	+	0.803125i	$0.703169\pi$
$462$	−13.1343	−0.611062
$463$	−26.6474	−1.23841	−0.619204	−	0.785230i	$-0.712545\pi$
$463$	−26.6474	−1.23841	−0.619204	+	0.785230i	$0.712545\pi$
$464$	5.31017	0.246518
$465$	17.8099	0.825914
$466$	−18.5533	−0.859466
$467$	38.6873	1.79023	0.895117	−	0.445830i	$-0.147092\pi$
$467$	38.6873	1.79023	0.895117	+	0.445830i	$0.147092\pi$
$468$	−1.00000	−0.0462250
$469$	−9.69453	−0.447652
$470$	−38.3250	−1.76780
$471$	8.54028	0.393515
$472$	−10.2648	−0.472474
$473$	−14.6702	−0.674534
$474$	5.22572	0.240025
$475$	29.2947	1.34413
$476$	−22.0244	−1.00949
$477$	−10.4448	−0.478232
$478$	−3.39069	−0.155086
$479$	−29.5370	−1.34958	−0.674790	−	0.738009i	$-0.735766\pi$
$479$	−29.5370	−1.34958	−0.674790	+	0.738009i	$0.735766\pi$
$480$	3.81560	0.174158
$481$	5.87682	0.267960
$482$	20.7142	0.943504
$483$	−18.3558	−0.835219
$484$	−3.93678	−0.178945
$485$	−55.0016	−2.49749
$486$	−1.00000	−0.0453609
$487$	3.40617	0.154348	0.0771741	−	0.997018i	$-0.475410\pi$
$487$	3.40617	0.154348	0.0771741	+	0.997018i	$0.475410\pi$
$488$	4.64313	0.210185
$489$	−0.940059	−0.0425109
$490$	66.4817	3.00334
$491$	30.1435	1.36036	0.680179	−	0.733047i	$-0.261902\pi$
$491$	30.1435	1.36036	0.680179	+	0.733047i	$0.261902\pi$
$492$	−6.56132	−0.295807
$493$	−23.6650	−1.06582
$494$	3.06468	0.137886
$495$	−10.1406	−0.455787
$496$	−4.66765	−0.209584
$497$	−69.0055	−3.09532
$498$	−3.46955	−0.155474
$499$	−0.262235	−0.0117392	−0.00586962	−	0.999983i	$-0.501868\pi$
$499$	−0.262235	−0.0117392	−0.00586962	+	0.999983i	$0.501868\pi$
$500$	−17.3947	−0.777912
$501$	18.9730	0.847652
$502$	−14.5570	−0.649711
$503$	−41.8959	−1.86805	−0.934023	−	0.357212i	$-0.883727\pi$
$503$	−41.8959	−1.86805	−0.934023	+	0.357212i	$0.883727\pi$
$504$	−4.94203	−0.220135
$505$	−4.21136	−0.187403
$506$	9.87120	0.438828
$507$	1.00000	0.0444116
$508$	9.05858	0.401910
$509$	12.8289	0.568631	0.284316	−	0.958731i	$-0.408234\pi$
$509$	12.8289	0.568631	0.284316	+	0.958731i	$0.408234\pi$
$510$	−17.0044	−0.752967
$511$	−9.54893	−0.422420
$512$	−1.00000	−0.0441942
$513$	3.06468	0.135309
$514$	−24.7082	−1.08983
$515$	−3.81560	−0.168136
$516$	−5.51993	−0.243001
$517$	−26.6944	−1.17402
$518$	29.0434	1.27609
$519$	−3.59876	−0.157968
$520$	−3.81560	−0.167325
$521$	−10.6316	−0.465780	−0.232890	−	0.972503i	$-0.574818\pi$
$521$	−10.6316	−0.465780	−0.232890	+	0.972503i	$0.574818\pi$
$522$	−5.31017	−0.232420
$523$	34.8435	1.52360	0.761800	−	0.647813i	$-0.224316\pi$
$523$	34.8435	1.52360	0.761800	+	0.647813i	$0.224316\pi$
$524$	−13.6776	−0.597507
$525$	47.2400	2.06172
$526$	14.4358	0.629429
$527$	20.8016	0.906132
$528$	2.65767	0.115660
$529$	−9.20451	−0.400196
$530$	−39.8530	−1.73110
$531$	10.2648	0.445453
$532$	15.1457	0.656651
$533$	6.56132	0.284202
$534$	−8.23562	−0.356390
$535$	−18.2819	−0.790396
$536$	1.96165	0.0847304
$537$	−12.3216	−0.531715
$538$	5.54942	0.239252
$539$	46.3063	1.99455
$540$	−3.81560	−0.164197
$541$	−32.8110	−1.41066	−0.705329	−	0.708881i	$-0.749200\pi$
$541$	−32.8110	−1.41066	−0.705329	+	0.708881i	$0.749200\pi$
$542$	−18.8197	−0.808373
$543$	−14.3167	−0.614390
$544$	4.45654	0.191073
$545$	63.7366	2.73017
$546$	4.94203	0.211499
$547$	−23.5696	−1.00776	−0.503881	−	0.863773i	$-0.668095\pi$
$547$	−23.5696	−1.00776	−0.503881	+	0.863773i	$0.668095\pi$
$548$	14.1768	0.605603
$549$	−4.64313	−0.198164
$550$	−25.4042	−1.08324
$551$	16.2740	0.693295
$552$	3.71423	0.158088
$553$	−25.8257	−1.09822
$554$	−13.4083	−0.569664
$555$	22.4236	0.951830
$556$	−9.95588	−0.422223
$557$	14.7282	0.624055	0.312027	−	0.950073i	$-0.398992\pi$
$557$	14.7282	0.624055	0.312027	+	0.950073i	$0.398992\pi$
$558$	4.66765	0.197597
$559$	5.51993	0.233468
$560$	−18.8568	−0.796846
$561$	−11.8440	−0.500055
$562$	31.6493	1.33505
$563$	22.8162	0.961586	0.480793	−	0.876834i	$-0.340349\pi$
$563$	22.8162	0.961586	0.480793	+	0.876834i	$0.340349\pi$
$564$	−10.0443	−0.422941
$565$	8.04519	0.338464
$566$	−0.0113313	−0.000476288 0
$567$	4.94203	0.207546
$568$	13.9630	0.585874
$569$	−35.6251	−1.49348	−0.746741	−	0.665115i	$-0.768383\pi$
$569$	−35.6251	−1.49348	−0.746741	+	0.665115i	$0.768383\pi$
$570$	11.6936	0.489791
$571$	−14.2946	−0.598210	−0.299105	−	0.954220i	$-0.596688\pi$
$571$	−14.2946	−0.598210	−0.299105	+	0.954220i	$0.596688\pi$
$572$	−2.65767	−0.111123
$573$	−2.81037	−0.117405
$574$	32.4262	1.35344
$575$	−35.5036	−1.48060
$576$	1.00000	0.0416667
$577$	−40.3880	−1.68137	−0.840686	−	0.541522i	$-0.817848\pi$
$577$	−40.3880	−1.68137	−0.840686	+	0.541522i	$0.817848\pi$
$578$	−2.86078	−0.118993
$579$	−18.6496	−0.775051
$580$	−20.2615	−0.841313
$581$	17.1466	0.711362
$582$	−14.4149	−0.597517
$583$	−27.7587	−1.14965
$584$	1.93219	0.0799546
$585$	3.81560	0.157756
$586$	4.41047	0.182195
$587$	1.24044	0.0511984	0.0255992	−	0.999672i	$-0.491851\pi$
$587$	1.24044	0.0511984	0.0255992	+	0.999672i	$0.491851\pi$
$588$	17.4236	0.718539
$589$	−14.3049	−0.589421
$590$	39.1662	1.61245
$591$	−2.28606	−0.0940359
$592$	−5.87682	−0.241536
$593$	36.9172	1.51601	0.758005	−	0.652249i	$-0.226174\pi$
$593$	36.9172	1.51601	0.758005	+	0.652249i	$0.226174\pi$
$594$	−2.65767	−0.109046
$595$	84.0362	3.44515
$596$	−5.41225	−0.221694
$597$	5.66189	0.231726
$598$	−3.71423	−0.151886
$599$	30.8445	1.26027	0.630136	−	0.776485i	$-0.282999\pi$
$599$	30.8445	1.26027	0.630136	+	0.776485i	$0.282999\pi$
$600$	−9.55882	−0.390237
$601$	−8.74978	−0.356911	−0.178455	−	0.983948i	$-0.557110\pi$
$601$	−8.74978	−0.356911	−0.178455	+	0.983948i	$0.557110\pi$
$602$	27.2796	1.11183
$603$	−1.96165	−0.0798846
$604$	−13.5624	−0.551846
$605$	15.0212	0.610698
$606$	−1.10372	−0.0448356
$607$	−30.0722	−1.22059	−0.610297	−	0.792173i	$-0.708950\pi$
$607$	−30.0722	−1.22059	−0.610297	+	0.792173i	$0.708950\pi$
$608$	−3.06468	−0.124289
$609$	26.2430	1.06342
$610$	−17.7163	−0.717313
$611$	10.0443	0.406348
$612$	−4.45654	−0.180145
$613$	26.8435	1.08420	0.542100	−	0.840314i	$-0.317629\pi$
$613$	26.8435	1.08420	0.542100	+	0.840314i	$0.317629\pi$
$614$	−20.0406	−0.808771
$615$	25.0354	1.00952
$616$	−13.1343	−0.529196
$617$	−37.9223	−1.52669	−0.763346	−	0.645989i	$-0.776445\pi$
$617$	−37.9223	−1.52669	−0.763346	+	0.645989i	$0.776445\pi$
$618$	−1.00000	−0.0402259
$619$	16.3961	0.659017	0.329508	−	0.944153i	$-0.393117\pi$
$619$	16.3961	0.659017	0.329508	+	0.944153i	$0.393117\pi$
$620$	17.8099	0.715263
$621$	−3.71423	−0.149047
$622$	4.04538	0.162205
$623$	40.7007	1.63064
$624$	−1.00000	−0.0400320
$625$	18.5770	0.743079
$626$	−8.04036	−0.321358
$627$	8.14491	0.325276
$628$	8.54028	0.340794
$629$	26.1903	1.04428
$630$	18.8568	0.751274
$631$	−5.27790	−0.210110	−0.105055	−	0.994466i	$-0.533502\pi$
$631$	−5.27790	−0.210110	−0.105055	+	0.994466i	$0.533502\pi$
$632$	5.22572	0.207868
$633$	13.0173	0.517391
$634$	5.58000	0.221610
$635$	−34.5640	−1.37163
$636$	−10.4448	−0.414161
$637$	−17.4236	−0.690350
$638$	−14.1127	−0.558727
$639$	−13.9630	−0.552367
$640$	3.81560	0.150825
$641$	−1.36628	−0.0539647	−0.0269824	−	0.999636i	$-0.508590\pi$
$641$	−1.36628	−0.0539647	−0.0269824	+	0.999636i	$0.508590\pi$
$642$	−4.79136	−0.189100
$643$	−4.41093	−0.173950	−0.0869751	−	0.996210i	$-0.527720\pi$
$643$	−4.41093	−0.173950	−0.0869751	+	0.996210i	$0.527720\pi$
$644$	−18.3558	−0.723321
$645$	21.0618	0.829309
$646$	13.6579	0.537362
$647$	−0.0113644	−0.000446779 0	−0.000223389	−	1.00000i	$-0.500071\pi$
$647$	−0.0113644	−0.000446779 0	−0.000223389		1.00000i	$0.500071\pi$
$648$	−1.00000	−0.0392837
$649$	27.2804	1.07085
$650$	9.55882	0.374928
$651$	−23.0677	−0.904093
$652$	−0.940059	−0.0368155
$653$	40.4851	1.58430	0.792152	−	0.610323i	$-0.208960\pi$
$653$	40.4851	1.58430	0.792152	+	0.610323i	$0.208960\pi$
$654$	16.7042	0.653186
$655$	52.1882	2.03916
$656$	−6.56132	−0.256176
$657$	−1.93219	−0.0753819
$658$	49.6391	1.93513
$659$	11.5193	0.448727	0.224363	−	0.974506i	$-0.427970\pi$
$659$	11.5193	0.448727	0.224363	+	0.974506i	$0.427970\pi$
$660$	−10.1406	−0.394723
$661$	−3.18690	−0.123956	−0.0619780	−	0.998078i	$-0.519741\pi$
$661$	−3.18690	−0.123956	−0.0619780	+	0.998078i	$0.519741\pi$
$662$	10.5804	0.411219
$663$	4.45654	0.173078
$664$	−3.46955	−0.134645
$665$	−57.7901	−2.24100
$666$	5.87682	0.227722
$667$	−19.7232	−0.763685
$668$	18.9730	0.734088
$669$	3.43833	0.132933
$670$	−7.48488	−0.289166
$671$	−12.3399	−0.476377
$672$	−4.94203	−0.190643
$673$	11.8399	0.456393	0.228197	−	0.973615i	$-0.426717\pi$
$673$	11.8399	0.456393	0.228197	+	0.973615i	$0.426717\pi$
$674$	0.574016	0.0221103
$675$	9.55882	0.367919
$676$	1.00000	0.0384615
$677$	−14.0643	−0.540534	−0.270267	−	0.962785i	$-0.587112\pi$
$677$	−14.0643	−0.540534	−0.270267	+	0.962785i	$0.587112\pi$
$678$	2.10850	0.0809764
$679$	71.2389	2.73390
$680$	−17.0044	−0.652089
$681$	−14.8404	−0.568684
$682$	12.4051	0.475015
$683$	−39.0515	−1.49426	−0.747132	−	0.664675i	$-0.768570\pi$
$683$	−39.0515	−1.49426	−0.747132	+	0.664675i	$0.768570\pi$
$684$	3.06468	0.117181
$685$	−54.0930	−2.06679
$686$	−51.5140	−1.96681
$687$	4.64030	0.177039
$688$	−5.51993	−0.210445
$689$	10.4448	0.397913
$690$	−14.1720	−0.539519
$691$	36.2791	1.38012	0.690062	−	0.723751i	$-0.257583\pi$
$691$	36.2791	1.38012	0.690062	+	0.723751i	$0.257583\pi$
$692$	−3.59876	−0.136804
$693$	13.1343	0.498930
$694$	−9.56853	−0.363216
$695$	37.9877	1.44095
$696$	−5.31017	−0.201281
$697$	29.2408	1.10757
$698$	21.9927	0.832438
$699$	18.5533	0.701751
$700$	47.2400	1.78550
$701$	−32.6043	−1.23145	−0.615723	−	0.787963i	$-0.711136\pi$
$701$	−32.6043	−1.23145	−0.615723	+	0.787963i	$0.711136\pi$
$702$	1.00000	0.0377426
$703$	−18.0106	−0.679282
$704$	2.65767	0.100165
$705$	38.3250	1.44340
$706$	24.7904	0.933000
$707$	5.45462	0.205142
$708$	10.2648	0.385773
$709$	22.1362	0.831342	0.415671	−	0.909515i	$-0.363547\pi$
$709$	22.1362	0.831342	0.415671	+	0.909515i	$0.363547\pi$
$710$	−53.2772	−1.99946
$711$	−5.22572	−0.195980
$712$	−8.23562	−0.308643
$713$	17.3367	0.649265
$714$	22.0244	0.824241
$715$	10.1406	0.379238
$716$	−12.3216	−0.460479
$717$	3.39069	0.126628
$718$	−7.14448	−0.266630
$719$	52.8553	1.97117	0.985585	−	0.169183i	$-0.0541129\pi$
$719$	52.8553	1.97117	0.985585	+	0.169183i	$0.0541129\pi$
$720$	−3.81560	−0.142199
$721$	4.94203	0.184051
$722$	9.60774	0.357563
$723$	−20.7142	−0.770368
$724$	−14.3167	−0.532078
$725$	50.7590	1.88514
$726$	3.93678	0.146108
$727$	10.6192	0.393843	0.196922	−	0.980419i	$-0.436906\pi$
$727$	10.6192	0.393843	0.196922	+	0.980419i	$0.436906\pi$
$728$	4.94203	0.183164
$729$	1.00000	0.0370370
$730$	−7.37247	−0.272867
$731$	24.5998	0.909856
$732$	−4.64313	−0.171615
$733$	−7.45444	−0.275336	−0.137668	−	0.990478i	$-0.543961\pi$
$733$	−7.45444	−0.275336	−0.137668	+	0.990478i	$0.543961\pi$
$734$	−10.7740	−0.397676
$735$	−66.4817	−2.45221
$736$	3.71423	0.136908
$737$	−5.21342	−0.192039
$738$	6.56132	0.241525
$739$	7.53788	0.277286	0.138643	−	0.990342i	$-0.455726\pi$
$739$	7.53788	0.277286	0.138643	+	0.990342i	$0.455726\pi$
$740$	22.4236	0.824309
$741$	−3.06468	−0.112584
$742$	51.6183	1.89497
$743$	−10.8326	−0.397409	−0.198704	−	0.980059i	$-0.563673\pi$
$743$	−10.8326	−0.397409	−0.198704	+	0.980059i	$0.563673\pi$
$744$	4.66765	0.171124
$745$	20.6510	0.756594
$746$	−28.4760	−1.04258
$747$	3.46955	0.126944
$748$	−11.8440	−0.433061
$749$	23.6790	0.865212
$750$	17.3947	0.635163
$751$	33.7846	1.23282	0.616408	−	0.787427i	$-0.288587\pi$
$751$	33.7846	1.23282	0.616408	+	0.787427i	$0.288587\pi$
$752$	−10.0443	−0.366277
$753$	14.5570	0.530487
$754$	5.31017	0.193385
$755$	51.7487	1.88333
$756$	4.94203	0.179740
$757$	−37.6181	−1.36725	−0.683626	−	0.729833i	$-0.739598\pi$
$757$	−37.6181	−1.36725	−0.683626	+	0.729833i	$0.739598\pi$
$758$	14.3097	0.519751
$759$	−9.87120	−0.358302
$760$	11.6936	0.424171
$761$	−19.3563	−0.701665	−0.350832	−	0.936438i	$-0.614101\pi$
$761$	−19.3563	−0.701665	−0.350832	+	0.936438i	$0.614101\pi$
$762$	−9.05858	−0.328158
$763$	−82.5526	−2.98861
$764$	−2.81037	−0.101676
$765$	17.0044	0.614795
$766$	28.4425	1.02767
$767$	−10.2648	−0.370639
$768$	1.00000	0.0360844
$769$	18.5088	0.667445	0.333722	−	0.942671i	$-0.391695\pi$
$769$	18.5088	0.667445	0.333722	+	0.942671i	$0.391695\pi$
$770$	50.1152	1.80603
$771$	24.7082	0.889844
$772$	−18.6496	−0.671214
$773$	−43.5265	−1.56554	−0.782769	−	0.622312i	$-0.786193\pi$
$773$	−43.5265	−1.56554	−0.782769	+	0.622312i	$0.786193\pi$
$774$	5.51993	0.198410
$775$	−44.6172	−1.60270
$776$	−14.4149	−0.517465
$777$	−29.0434	−1.04193
$778$	−19.0716	−0.683749
$779$	−20.1083	−0.720456
$780$	3.81560	0.136620
$781$	−37.1090	−1.32787
$782$	−16.5526	−0.591920
$783$	5.31017	0.189770
$784$	17.4236	0.622273
$785$	−32.5863	−1.16306
$786$	13.6776	0.487863
$787$	−15.5787	−0.555321	−0.277661	−	0.960679i	$-0.589559\pi$
$787$	−15.5787	−0.555321	−0.277661	+	0.960679i	$0.589559\pi$
$788$	−2.28606	−0.0814375
$789$	−14.4358	−0.513927
$790$	−19.9393	−0.709408
$791$	−10.4203	−0.370502
$792$	−2.65767	−0.0944362
$793$	4.64313	0.164882
$794$	22.7585	0.807667
$795$	39.8530	1.41344
$796$	5.66189	0.200680
$797$	50.3766	1.78443	0.892215	−	0.451611i	$-0.149151\pi$
$797$	50.3766	1.78443	0.892215	+	0.451611i	$0.149151\pi$
$798$	−15.1457	−0.536153
$799$	44.7628	1.58359
$800$	−9.55882	−0.337955
$801$	8.23562	0.290991
$802$	11.5086	0.406384
$803$	−5.13512	−0.181215
$804$	−1.96165	−0.0691821
$805$	70.0385	2.46853
$806$	−4.66765	−0.164411
$807$	−5.54942	−0.195349
$808$	−1.10372	−0.0388288
$809$	9.00214	0.316499	0.158249	−	0.987399i	$-0.449415\pi$
$809$	9.00214	0.316499	0.158249	+	0.987399i	$0.449415\pi$
$810$	3.81560	0.134067
$811$	−42.0442	−1.47637	−0.738186	−	0.674597i	$-0.764317\pi$
$811$	−42.0442	−1.47637	−0.738186	+	0.674597i	$0.764317\pi$
$812$	26.2430	0.920949
$813$	18.8197	0.660034
$814$	15.6187	0.547434
$815$	3.58689	0.125643
$816$	−4.45654	−0.156010
$817$	−16.9168	−0.591844
$818$	16.3617	0.572073
$819$	−4.94203	−0.172688
$820$	25.0354	0.874273
$821$	1.80159	0.0628760	0.0314380	−	0.999506i	$-0.489991\pi$
$821$	1.80159	0.0628760	0.0314380	+	0.999506i	$0.489991\pi$
$822$	−14.1768	−0.494473
$823$	−19.9754	−0.696299	−0.348150	−	0.937439i	$-0.613190\pi$
$823$	−19.9754	−0.696299	−0.348150	+	0.937439i	$0.613190\pi$
$824$	−1.00000	−0.0348367
$825$	25.4042	0.884461
$826$	−50.7287	−1.76508
$827$	−26.8586	−0.933965	−0.466982	−	0.884267i	$-0.654659\pi$
$827$	−26.8586	−0.933965	−0.466982	+	0.884267i	$0.654659\pi$
$828$	−3.71423	−0.129078
$829$	−11.1674	−0.387861	−0.193930	−	0.981015i	$-0.562124\pi$
$829$	−11.1674	−0.387861	−0.193930	+	0.981015i	$0.562124\pi$
$830$	13.2384	0.459513
$831$	13.4083	0.465129
$832$	−1.00000	−0.0346688
$833$	−77.6492	−2.69039
$834$	9.95588	0.344744
$835$	−72.3935	−2.50528
$836$	8.14491	0.281698
$837$	−4.66765	−0.161338
$838$	31.7684	1.09742
$839$	−7.51830	−0.259560	−0.129780	−	0.991543i	$-0.541427\pi$
$839$	−7.51830	−0.259560	−0.129780	+	0.991543i	$0.541427\pi$
$840$	18.8568	0.650622
$841$	−0.802092	−0.0276584
$842$	8.56383	0.295129
$843$	−31.6493	−1.09006
$844$	13.0173	0.448074
$845$	−3.81560	−0.131261
$846$	10.0443	0.345329
$847$	−19.4557	−0.668505
$848$	−10.4448	−0.358674
$849$	0.0113313	0.000388888 0
$850$	42.5993	1.46114
$851$	21.8279	0.748249
$852$	−13.9630	−0.478364
$853$	−42.4262	−1.45264	−0.726322	−	0.687354i	$-0.758772\pi$
$853$	−42.4262	−1.45264	−0.726322	+	0.687354i	$0.758772\pi$
$854$	22.9465	0.785212
$855$	−11.6936	−0.399913
$856$	−4.79136	−0.163765
$857$	25.5464	0.872647	0.436324	−	0.899790i	$-0.356280\pi$
$857$	25.5464	0.872647	0.436324	+	0.899790i	$0.356280\pi$
$858$	2.65767	0.0907314
$859$	55.0691	1.87894	0.939468	−	0.342638i	$-0.111320\pi$
$859$	55.0691	1.87894	0.939468	+	0.342638i	$0.111320\pi$
$860$	21.0618	0.718203
$861$	−32.4262	−1.10508
$862$	−25.0472	−0.853110
$863$	49.7915	1.69492	0.847460	−	0.530859i	$-0.178130\pi$
$863$	49.7915	1.69492	0.847460	+	0.530859i	$0.178130\pi$
$864$	−1.00000	−0.0340207
$865$	13.7315	0.466883
$866$	26.4327	0.898220
$867$	2.86078	0.0971573
$868$	−23.0677	−0.782968
$869$	−13.8883	−0.471127
$870$	20.2615	0.686929
$871$	1.96165	0.0664680
$872$	16.7042	0.565675
$873$	14.4149	0.487871
$874$	11.3829	0.385033
$875$	−85.9649	−2.90614
$876$	−1.93219	−0.0652826
$877$	40.4708	1.36660	0.683301	−	0.730137i	$-0.260544\pi$
$877$	40.4708	1.36660	0.683301	+	0.730137i	$0.260544\pi$
$878$	26.3084	0.887864
$879$	−4.41047	−0.148762
$880$	−10.1406	−0.341840
$881$	−43.7069	−1.47252	−0.736262	−	0.676697i	$-0.763411\pi$
$881$	−43.7069	−1.47252	−0.736262	+	0.676697i	$0.763411\pi$
$882$	−17.4236	−0.586685
$883$	−33.3296	−1.12163	−0.560815	−	0.827941i	$-0.689512\pi$
$883$	−33.3296	−1.12163	−0.560815	+	0.827941i	$0.689512\pi$
$884$	4.45654	0.149890
$885$	−39.1662	−1.31656
$886$	−32.4551	−1.09035
$887$	−12.7487	−0.428061	−0.214031	−	0.976827i	$-0.568659\pi$
$887$	−12.7487	−0.428061	−0.214031	+	0.976827i	$0.568659\pi$
$888$	5.87682	0.197213
$889$	44.7678	1.50146
$890$	31.4239	1.05333
$891$	2.65767	0.0890353
$892$	3.43833	0.115124
$893$	−30.7825	−1.03010
$894$	5.41225	0.181013
$895$	47.0142	1.57151
$896$	−4.94203	−0.165102
$897$	3.71423	0.124014
$898$	26.6480	0.889254
$899$	−24.7860	−0.826660
$900$	9.55882	0.318627
$901$	46.5475	1.55072
$902$	17.4378	0.580616
$903$	−27.2796	−0.907809
$904$	2.10850	0.0701276
$905$	54.6270	1.81586
$906$	13.5624	0.450580
$907$	−26.3431	−0.874710	−0.437355	−	0.899289i	$-0.644085\pi$
$907$	−26.3431	−0.874710	−0.437355	+	0.899289i	$0.644085\pi$
$908$	−14.8404	−0.492495
$909$	1.10372	0.0366081
$910$	−18.8568	−0.625097
$911$	−33.1966	−1.09985	−0.549926	−	0.835213i	$-0.685344\pi$
$911$	−33.1966	−1.09985	−0.549926	+	0.835213i	$0.685344\pi$
$912$	3.06468	0.101482
$913$	9.22094	0.305168
$914$	−32.2187	−1.06570
$915$	17.7163	0.585684
$916$	4.64030	0.153320
$917$	−67.5949	−2.23218
$918$	4.45654	0.147088
$919$	−12.7869	−0.421801	−0.210900	−	0.977508i	$-0.567640\pi$
$919$	−12.7869	−0.421801	−0.210900	+	0.977508i	$0.567640\pi$
$920$	−14.1720	−0.467238
$921$	20.0406	0.660359
$922$	25.5852	0.842604
$923$	13.9630	0.459597
$924$	13.1343	0.432086
$925$	−56.1755	−1.84704
$926$	26.6474	0.875687
$927$	1.00000	0.0328443
$928$	−5.31017	−0.174315
$929$	−43.9133	−1.44075	−0.720375	−	0.693585i	$-0.756030\pi$
$929$	−43.9133	−1.44075	−0.720375	+	0.693585i	$0.756030\pi$
$930$	−17.8099	−0.584010
$931$	53.3979	1.75005
$932$	18.5533	0.607734
$933$	−4.04538	−0.132440
$934$	−38.6873	−1.26589
$935$	45.1921	1.47794
$936$	1.00000	0.0326860
$937$	−39.0533	−1.27582	−0.637908	−	0.770112i	$-0.720200\pi$
$937$	−39.0533	−1.27582	−0.637908	+	0.770112i	$0.720200\pi$
$938$	9.69453	0.316538
$939$	8.04036	0.262387
$940$	38.3250	1.25002
$941$	41.1817	1.34249	0.671243	−	0.741237i	$-0.265761\pi$
$941$	41.1817	1.34249	0.671243	+	0.741237i	$0.265761\pi$
$942$	−8.54028	−0.278257
$943$	24.3702	0.793604
$944$	10.2648	0.334089
$945$	−18.8568	−0.613412
$946$	14.6702	0.476968
$947$	−47.1613	−1.53254	−0.766268	−	0.642521i	$-0.777888\pi$
$947$	−47.1613	−1.53254	−0.766268	+	0.642521i	$0.777888\pi$
$948$	−5.22572	−0.169724
$949$	1.93219	0.0627215
$950$	−29.2947	−0.950447
$951$	−5.58000	−0.180944
$952$	22.0244	0.713814
$953$	−24.2561	−0.785732	−0.392866	−	0.919596i	$-0.628516\pi$
$953$	−24.2561	−0.785732	−0.392866	+	0.919596i	$0.628516\pi$
$954$	10.4448	0.338161
$955$	10.7233	0.346996
$956$	3.39069	0.109663
$957$	14.1127	0.456198
$958$	29.5370	0.954298
$959$	70.0621	2.26242
$960$	−3.81560	−0.123148
$961$	−9.21303	−0.297195
$962$	−5.87682	−0.189476
$963$	4.79136	0.154399
$964$	−20.7142	−0.667158
$965$	71.1595	2.29071
$966$	18.3558	0.590589
$967$	−53.9653	−1.73541	−0.867704	−	0.497082i	$-0.834405\pi$
$967$	−53.9653	−1.73541	−0.867704	+	0.497082i	$0.834405\pi$
$968$	3.93678	0.126533
$969$	−13.6579	−0.438754
$970$	55.0016	1.76599
$971$	11.2320	0.360453	0.180226	−	0.983625i	$-0.442317\pi$
$971$	11.2320	0.360453	0.180226	+	0.983625i	$0.442317\pi$
$972$	1.00000	0.0320750
$973$	−49.2022	−1.57735
$974$	−3.40617	−0.109141
$975$	−9.55882	−0.306127
$976$	−4.64313	−0.148623
$977$	−51.8538	−1.65895	−0.829475	−	0.558544i	$-0.811360\pi$
$977$	−51.8538	−1.65895	−0.829475	+	0.558544i	$0.811360\pi$
$978$	0.940059	0.0300598
$979$	21.8876	0.699530
$980$	−66.4817	−2.12368
$981$	−16.7042	−0.533324
$982$	−30.1435	−0.961918
$983$	30.3957	0.969474	0.484737	−	0.874660i	$-0.338915\pi$
$983$	30.3957	0.969474	0.484737	+	0.874660i	$0.338915\pi$
$984$	6.56132	0.209167
$985$	8.72269	0.277928
$986$	23.6650	0.753647
$987$	−49.6391	−1.58003
$988$	−3.06468	−0.0975005
$989$	20.5023	0.651934
$990$	10.1406	0.322290
$991$	−58.4368	−1.85631	−0.928153	−	0.372198i	$-0.878604\pi$
$991$	−58.4368	−1.85631	−0.928153	+	0.372198i	$0.878604\pi$
$992$	4.66765	0.148198
$993$	−10.5804	−0.335759
$994$	69.0055	2.18872
$995$	−21.6035	−0.684878
$996$	3.46955	0.109937
$997$	24.6228	0.779811	0.389906	−	0.920855i	$-0.372508\pi$
$997$	24.6228	0.779811	0.389906	+	0.920855i	$0.372508\pi$
$998$	0.262235	0.00830090
$999$	−5.87682	−0.185934

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.bb.1.2	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.bb.1.2	✓	14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

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