LMFDB - Embedded newform 1078.2.c.c.1077.3

Newspace parameters

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1078.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$8.60787333789$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
Defining polynomial:	$ x^{16} - 32x^{14} + 512x^{12} - 2272x^{10} - 1087x^{8} + 72448x^{6} + 819200x^{4} + 1310720x^{2} + 1048576 $ x^16 - 32x^14 + 512x^12 - 2272x^10 - 1087x^8 + 72448x^6 + 819200x^4 + 1310720*x^2 + 1048576
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1077.3
Root			$-0.807586 - 1.94969i$ of defining polynomial
Character	$\chi$	$=$	1078.1077
Dual form			1078.2.c.c.1077.14

$q$-expansion

$f(q)$	$=$	$q-1.00000i q^{2} -0.765367i q^{3} -1.00000 q^{4} -2.05161i q^{5} -0.765367 q^{6} +1.00000i q^{8} +2.41421 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -0.765367i q^{3} -1.00000 q^{4} -2.05161i q^{5} -0.765367 q^{6} +1.00000i q^{8} +2.41421 q^{9} -2.05161 q^{10} +(2.49222 - 2.18834i) q^{11} +0.765367i q^{12} +6.59694 q^{13} -1.57024 q^{15} +1.00000 q^{16} -3.61108 q^{17} -2.41421i q^{18} -0.878539 q^{19} +2.05161i q^{20} +(-2.18834 - 2.49222i) q^{22} +6.61931 q^{23} +0.765367 q^{24} +0.790886 q^{25} -6.59694i q^{26} -4.14386i q^{27} +6.18955i q^{29} +1.57024i q^{30} -6.48376i q^{31} -1.00000i q^{32} +(-1.67488 - 1.90747i) q^{33} +3.61108i q^{34} -2.41421 q^{36} -11.0491 q^{37} +0.878539i q^{38} -5.04908i q^{39} +2.05161 q^{40} -2.36864 q^{41} +7.81288i q^{43} +(-2.49222 + 2.18834i) q^{44} -4.95303i q^{45} -6.61931i q^{46} -5.74713i q^{47} -0.765367i q^{48} -0.790886i q^{50} +2.76380i q^{51} -6.59694 q^{52} +0.429764 q^{53} -4.14386 q^{54} +(-4.48962 - 5.11308i) q^{55} +0.672404i q^{57} +6.18955 q^{58} -3.21844i q^{59} +1.57024 q^{60} -11.3342 q^{61} -6.48376 q^{62} -1.00000 q^{64} -13.5344i q^{65} +(-1.90747 + 1.67488i) q^{66} +2.96246 q^{67} +3.61108 q^{68} -5.06620i q^{69} -2.13403 q^{71} +2.41421i q^{72} -10.8252 q^{73} +11.0491i q^{74} -0.605318i q^{75} +0.878539 q^{76} -5.04908 q^{78} +8.48528i q^{79} -2.05161i q^{80} +4.07107 q^{81} +2.36864i q^{82} +12.3153 q^{83} +7.40854i q^{85} +7.81288 q^{86} +4.73728 q^{87} +(2.18834 + 2.49222i) q^{88} +6.72825i q^{89} -4.95303 q^{90} -6.61931 q^{92} -4.96246 q^{93} -5.74713 q^{94} +1.80242i q^{95} -0.765367 q^{96} +9.23880i q^{97} +(6.01676 - 5.28311i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} + 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^4 + 16 * q^9	$ 16 q - 16 q^{4} + 16 q^{9} + 16 q^{11} + 16 q^{16} - 8 q^{22} - 16 q^{23} - 64 q^{25} - 16 q^{36} - 80 q^{37} - 16 q^{44} + 32 q^{53} - 48 q^{58} - 16 q^{64} + 16 q^{67} - 48 q^{71} + 16 q^{78} - 48 q^{81} + 32 q^{86} + 8 q^{88} + 16 q^{92} - 48 q^{93} + 24 q^{99}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^9 + 16 * q^11 + 16 * q^16 - 8 * q^22 - 16 * q^23 - 64 * q^25 - 16 * q^36 - 80 * q^37 - 16 * q^44 + 32 * q^53 - 48 * q^58 - 16 * q^64 + 16 * q^67 - 48 * q^71 + 16 * q^78 - 48 * q^81 + 32 * q^86 + 8 * q^88 + 16 * q^92 - 48 * q^93 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times$.

$n$	$199$	$981$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		−	0.765367i		−	0.441885i	−0.975287	−	0.220942i	$-0.929087\pi$
$3$		−	0.765367i		−	0.441885i	0.975287	−	0.220942i	$-0.0709133\pi$
$4$	−1.00000			−0.500000
$5$		−	2.05161i		−	0.917509i	−0.888563	−	0.458755i	$-0.848296\pi$
$5$		−	2.05161i		−	0.917509i	0.888563	−	0.458755i	$-0.151704\pi$
$6$	−0.765367			−0.312460
$7$		0			0
$7$		0			0
$8$			1.00000i			0.353553i
$9$	2.41421			0.804738
$10$	−2.05161			−0.648777
$11$	2.49222	−	2.18834i	0.751434	−	0.659808i
$11$	2.49222	−	2.18834i	0.751434	−	0.659808i
$12$			0.765367i			0.220942i
$13$	6.59694			1.82966			0.914830	−	0.403838i	$-0.132324\pi$
$13$	6.59694			1.82966			0.914830	+	0.403838i	$0.132324\pi$
$14$		0			0
$15$	−1.57024			−0.405433
$16$	1.00000			0.250000
$17$	−3.61108			−0.875815			−0.437908	−	0.899020i	$-0.644280\pi$
$17$	−3.61108			−0.875815			−0.437908	+	0.899020i	$0.644280\pi$
$18$		−	2.41421i		−	0.569036i
$19$	−0.878539			−0.201551			−0.100775	−	0.994909i	$-0.532132\pi$
$19$	−0.878539			−0.201551			−0.100775	+	0.994909i	$0.532132\pi$
$20$			2.05161i			0.458755i
$21$		0			0
$22$	−2.18834	−	2.49222i	−0.466555	−	0.531344i
$23$	6.61931			1.38022			0.690111	−	0.723703i	$-0.257562\pi$
$23$	6.61931			1.38022			0.690111	+	0.723703i	$0.257562\pi$
$24$	0.765367			0.156230
$25$	0.790886			0.158177
$26$		−	6.59694i		−	1.29377i
$27$		−	4.14386i		−	0.797486i
$28$		0			0
$29$			6.18955i			1.14937i	0.818375	+	0.574685i	$0.194876\pi$
$29$			6.18955i			1.14937i	−0.818375	+	0.574685i	$0.805124\pi$
$30$			1.57024i			0.286685i
$31$		−	6.48376i		−	1.16452i	−0.813003	−	0.582259i	$-0.802169\pi$
$31$		−	6.48376i		−	1.16452i	0.813003	−	0.582259i	$-0.197831\pi$
$32$		−	1.00000i		−	0.176777i
$33$	−1.67488	−	1.90747i	−0.291559	−	0.332047i
$34$			3.61108i			0.619295i
$35$		0			0
$36$	−2.41421			−0.402369
$37$	−11.0491			−1.81646			−0.908229	−	0.418475i	$-0.862565\pi$
$37$	−11.0491			−1.81646			−0.908229	+	0.418475i	$0.862565\pi$
$38$			0.878539i			0.142518i
$39$		−	5.04908i		−	0.808499i
$40$	2.05161			0.324388
$41$	−2.36864			−0.369919			−0.184960	−	0.982746i	$-0.559215\pi$
$41$	−2.36864			−0.369919			−0.184960	+	0.982746i	$0.559215\pi$
$42$		0			0
$43$			7.81288i			1.19145i	0.803188	+	0.595726i	$0.203136\pi$
$43$			7.81288i			1.19145i	−0.803188	+	0.595726i	$0.796864\pi$
$44$	−2.49222	+	2.18834i	−0.375717	+	0.329904i
$45$		−	4.95303i		−	0.738354i
$46$		−	6.61931i		−	0.975964i
$47$		−	5.74713i		−	0.838305i	−0.907916	−	0.419153i	$-0.862327\pi$
$47$		−	5.74713i		−	0.838305i	0.907916	−	0.419153i	$-0.137673\pi$
$48$		−	0.765367i		−	0.110471i
$49$		0			0
$50$		−	0.790886i		−	0.111848i
$51$			2.76380i			0.387009i
$52$	−6.59694			−0.914830
$53$	0.429764			0.0590326			0.0295163	−	0.999564i	$-0.490603\pi$
$53$	0.429764			0.0590326			0.0295163	+	0.999564i	$0.490603\pi$
$54$	−4.14386			−0.563908
$55$	−4.48962	−	5.11308i	−0.605380	−	0.689448i
$56$		0			0
$57$			0.672404i			0.0890621i
$58$	6.18955			0.812728
$59$		−	3.21844i		−	0.419006i	−0.977808	−	0.209503i	$-0.932815\pi$
$59$		−	3.21844i		−	0.419006i	0.977808	−	0.209503i	$-0.0671845\pi$
$60$	1.57024			0.202717
$61$	−11.3342			−1.45120			−0.725599	−	0.688118i	$-0.758437\pi$
$61$	−11.3342			−1.45120			−0.725599	+	0.688118i	$0.758437\pi$
$62$	−6.48376			−0.823439
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	13.5344i		−	1.67873i
$66$	−1.90747	+	1.67488i	−0.234793	+	0.206163i
$67$	2.96246			0.361922			0.180961	−	0.983490i	$-0.442079\pi$
$67$	2.96246			0.361922			0.180961	+	0.983490i	$0.442079\pi$
$68$	3.61108			0.437908
$69$		−	5.06620i		−	0.609899i
$70$		0			0
$71$	−2.13403			−0.253263			−0.126631	−	0.991950i	$-0.540417\pi$
$71$	−2.13403			−0.253263			−0.126631	+	0.991950i	$0.540417\pi$
$72$			2.41421i			0.284518i
$73$	−10.8252			−1.26700			−0.633499	−	0.773744i	$-0.718382\pi$
$73$	−10.8252			−1.26700			−0.633499	+	0.773744i	$0.718382\pi$
$74$			11.0491i			1.28443i
$75$		−	0.605318i		−	0.0698961i
$76$	0.878539			0.100775
$77$		0			0
$78$	−5.04908			−0.571695
$79$			8.48528i			0.954669i	0.878722	+	0.477334i	$0.158397\pi$
$79$			8.48528i			0.954669i	−0.878722	+	0.477334i	$0.841603\pi$
$80$		−	2.05161i		−	0.229377i
$81$	4.07107			0.452341
$82$			2.36864i			0.261572i
$83$	12.3153			1.35178			0.675892	−	0.737001i	$-0.263759\pi$
$83$	12.3153			1.35178			0.675892	+	0.737001i	$0.263759\pi$
$84$		0			0
$85$			7.40854i			0.803569i
$86$	7.81288			0.842484
$87$	4.73728			0.507889
$88$	2.18834	+	2.49222i	0.233277	+	0.265672i
$89$			6.72825i			0.713193i	0.934258	+	0.356597i	$0.116063\pi$
$89$			6.72825i			0.713193i	−0.934258	+	0.356597i	$0.883937\pi$
$90$	−4.95303			−0.522095
$91$		0			0
$92$	−6.61931			−0.690111
$93$	−4.96246			−0.514583
$94$	−5.74713			−0.592771
$95$			1.80242i			0.184924i
$96$	−0.765367			−0.0781149
$97$			9.23880i			0.938058i	0.883183	+	0.469029i	$0.155396\pi$
$97$			9.23880i			0.938058i	−0.883183	+	0.469029i	$0.844604\pi$
$98$		0			0
$99$	6.01676	−	5.28311i	0.604707	−	0.530973i
$100$	−0.790886			−0.0790886
$101$	1.85966			0.185043			0.0925216	−	0.995711i	$-0.470507\pi$
$101$	1.85966			0.185043			0.0925216	+	0.995711i	$0.470507\pi$
$102$	2.76380			0.273657
$103$			6.71011i			0.661167i	0.943777	+	0.330583i	$0.107245\pi$
$103$			6.71011i			0.661167i	−0.943777	+	0.330583i	$0.892755\pi$
$104$			6.59694i			0.646883i
$105$		0			0
$106$		−	0.429764i		−	0.0417423i
$107$		−	14.7533i		−	1.42626i	−0.701032	−	0.713130i	$-0.747277\pi$
$107$		−	14.7533i		−	1.42626i	0.701032	−	0.713130i	$-0.252723\pi$
$108$			4.14386i			0.398743i
$109$		−	6.00000i		−	0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$		−	6.00000i		−	0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$	−5.11308	+	4.48962i	−0.487513	+	0.428068i
$111$			8.45660i			0.802665i
$112$		0			0
$113$	−0.597322			−0.0561913			−0.0280956	−	0.999605i	$-0.508944\pi$
$113$	−0.597322			−0.0561913			−0.0280956	+	0.999605i	$0.508944\pi$
$114$	0.672404			0.0629764
$115$		−	13.5803i		−	1.26637i
$116$		−	6.18955i		−	0.574685i
$117$	15.9264			1.47240
$118$	−3.21844			−0.296282
$119$		0			0
$120$		−	1.57024i		−	0.143342i
$121$	1.42237	−	10.9077i	0.129306	−	0.991605i
$122$			11.3342i			1.02615i
$123$			1.81288i			0.163462i
$124$			6.48376i			0.582259i
$125$		−	11.8807i		−	1.06264i
$126$		0			0
$127$		−	7.33002i		−	0.650434i	−0.945639	−	0.325217i	$-0.894563\pi$
$127$		−	7.33002i		−	0.650434i	0.945639	−	0.325217i	$-0.105437\pi$
$128$			1.00000i			0.0883883i
$129$	5.97972			0.526485
$130$	−13.5344			−1.18704
$131$	−8.82050			−0.770651			−0.385325	−	0.922781i	$-0.625911\pi$
$131$	−8.82050			−0.770651			−0.385325	+	0.922781i	$0.625911\pi$
$132$	1.67488	+	1.90747i	0.145780	+	0.166024i
$133$		0			0
$134$		−	2.96246i		−	0.255917i
$135$	−8.50159			−0.731701
$136$		−	3.61108i		−	0.309647i
$137$	3.23620			0.276487			0.138244	−	0.990398i	$-0.455854\pi$
$137$	3.23620			0.276487			0.138244	+	0.990398i	$0.455854\pi$
$138$	−5.06620			−0.431264
$139$	−2.47122			−0.209606			−0.104803	−	0.994493i	$-0.533421\pi$
$139$	−2.47122			−0.209606			−0.104803	+	0.994493i	$0.533421\pi$
$140$		0			0
$141$	−4.39866			−0.370434
$142$			2.13403i			0.179084i
$143$	16.4410	−	14.4363i	1.37487	−	1.20723i
$144$	2.41421			0.201184
$145$	12.6986			1.05456
$146$			10.8252i			0.895903i
$147$		0			0
$148$	11.0491			0.908229
$149$			3.90860i			0.320205i	0.987100	+	0.160103i	$0.0511825\pi$
$149$			3.90860i			0.320205i	−0.987100	+	0.160103i	$0.948817\pi$
$150$	−0.605318			−0.0494240
$151$			24.1895i			1.96852i	0.176733	+	0.984259i	$0.443447\pi$
$151$			24.1895i			1.96852i	−0.176733	+	0.984259i	$0.556553\pi$
$152$		−	0.878539i		−	0.0712589i
$153$	−8.71792			−0.704802
$154$		0			0
$155$	−13.3022			−1.06846
$156$			5.04908i			0.404250i
$157$			20.3029i			1.62034i	0.586192	+	0.810172i	$0.300626\pi$
$157$			20.3029i			1.62034i	−0.586192	+	0.810172i	$0.699374\pi$
$158$	8.48528			0.675053
$159$		−	0.328927i		−	0.0260856i
$160$	−2.05161			−0.162194
$161$		0			0
$162$		−	4.07107i		−	0.319853i
$163$	−13.2606			−1.03865			−0.519326	−	0.854576i	$-0.673817\pi$
$163$	−13.2606			−1.03865			−0.519326	+	0.854576i	$0.673817\pi$
$164$	2.36864			0.184960
$165$	−3.91338	+	3.43620i	−0.304656	+	0.267508i
$166$		−	12.3153i		−	0.955855i
$167$	20.4160			1.57984			0.789920	−	0.613210i	$-0.210122\pi$
$167$	20.4160			1.57984			0.789920	+	0.613210i	$0.210122\pi$
$168$		0			0
$169$	30.5196			2.34766
$170$	7.40854			0.568209
$171$	−2.12098			−0.162195
$172$		−	7.81288i		−	0.595726i
$173$	−15.5762			−1.18423			−0.592117	−	0.805852i	$-0.701708\pi$
$173$	−15.5762			−1.18423			−0.592117	+	0.805852i	$0.701708\pi$
$174$		−	4.73728i		−	0.359132i
$175$		0			0
$176$	2.49222	−	2.18834i	0.187859	−	0.164952i
$177$	−2.46329			−0.185152
$178$	6.72825			0.504304
$179$	−4.50159			−0.336465			−0.168232	−	0.985747i	$-0.553806\pi$
$179$	−4.50159			−0.336465			−0.168232	+	0.985747i	$0.553806\pi$
$180$			4.95303i			0.369177i
$181$		−	8.18338i		−	0.608266i	−0.952630	−	0.304133i	$-0.901633\pi$
$181$		−	8.18338i		−	0.608266i	0.952630	−	0.304133i	$-0.0983667\pi$
$182$		0			0
$183$			8.67483i			0.641262i
$184$			6.61931i			0.487982i
$185$			22.6684i			1.66662i
$186$			4.96246i			0.363865i
$187$	−8.99962	+	7.90226i	−0.658118	+	0.577870i
$188$			5.74713i			0.419153i
$189$		0			0
$190$	1.80242			0.130761
$191$	10.3662			0.750073			0.375037	−	0.927010i	$-0.377630\pi$
$191$	10.3662			0.750073			0.375037	+	0.927010i	$0.377630\pi$
$192$			0.765367i			0.0552356i
$193$			1.62333i			0.116850i	0.998292	+	0.0584248i	$0.0186078\pi$
$193$			1.62333i			0.116850i	−0.998292	+	0.0584248i	$0.981392\pi$
$194$	9.23880			0.663307
$195$	−10.3587			−0.741805
$196$		0			0
$197$		−	4.38713i		−	0.312570i	−0.987712	−	0.156285i	$-0.950048\pi$
$197$		−	4.38713i		−	0.312570i	0.987712	−	0.156285i	$-0.0499518\pi$
$198$	−5.28311	−	6.01676i	−0.375454	−	0.427593i
$199$			20.0640i			1.42230i	0.703040	+	0.711151i	$0.251826\pi$
$199$			20.0640i			1.42230i	−0.703040	+	0.711151i	$0.748174\pi$
$200$			0.790886i			0.0559241i
$201$		−	2.26737i		−	0.159928i
$202$		−	1.85966i		−	0.130845i
$203$		0			0
$204$		−	2.76380i		−	0.193505i
$205$			4.85953i			0.339404i
$206$	6.71011			0.467515
$207$	15.9804			1.11072
$208$	6.59694			0.457415
$209$	−2.18952	+	1.92254i	−0.151452	+	0.132985i
$210$		0			0
$211$		−	8.56380i		−	0.589556i	−0.955566	−	0.294778i	$-0.904754\pi$
$211$		−	8.56380i		−	0.589556i	0.955566	−	0.294778i	$-0.0952457\pi$
$212$	−0.429764			−0.0295163
$213$			1.63332i			0.111913i
$214$	−14.7533			−1.00852
$215$	16.0290			1.09317
$216$	4.14386			0.281954
$217$		0			0
$218$	−6.00000			−0.406371
$219$			8.28528i			0.559867i
$220$	4.48962	+	5.11308i	0.302690	+	0.344724i
$221$	−23.8221			−1.60245
$222$	8.45660			0.567570
$223$		−	8.24084i		−	0.551848i	−0.961180	−	0.275924i	$-0.911016\pi$
$223$		−	8.24084i		−	0.551848i	0.961180	−	0.275924i	$-0.0889837\pi$
$224$		0			0
$225$	1.90937			0.127291
$226$			0.597322i			0.0397332i
$227$	27.4795			1.82388			0.911938	−	0.410329i	$-0.134586\pi$
$227$	27.4795			1.82388			0.911938	+	0.410329i	$0.134586\pi$
$228$		−	0.672404i		−	0.0445311i
$229$		−	0.104343i		−	0.00689519i	−0.999994	−	0.00344759i	$-0.998903\pi$
$229$		−	0.104343i		−	0.00689519i	0.999994	−	0.00344759i	$-0.00109741\pi$
$230$	−13.5803			−0.895456
$231$		0			0
$232$	−6.18955			−0.406364
$233$			15.1577i			0.993013i	0.868033	+	0.496507i	$0.165384\pi$
$233$			15.1577i			0.993013i	−0.868033	+	0.496507i	$0.834616\pi$
$234$		−	15.9264i		−	1.04114i
$235$	−11.7909			−0.769153
$236$			3.21844i			0.209503i
$237$	6.49435			0.421854
$238$		0			0
$239$		−	25.9135i		−	1.67620i	−0.545515	−	0.838101i	$-0.683666\pi$
$239$		−	25.9135i		−	1.67620i	0.545515	−	0.838101i	$-0.316334\pi$
$240$	−1.57024			−0.101358
$241$	−7.31108			−0.470948			−0.235474	−	0.971881i	$-0.575664\pi$
$241$	−7.31108			−0.470948			−0.235474	+	0.971881i	$0.575664\pi$
$242$	−10.9077	−	1.42237i	−0.701170	−	0.0914334i
$243$		−	15.5474i		−	0.997369i
$244$	11.3342			0.725599
$245$		0			0
$246$	1.81288			0.115585
$247$	−5.79566			−0.368769
$248$	6.48376			0.411719
$249$		−	9.42575i		−	0.597333i
$250$	−11.8807			−0.751399
$251$			10.0161i			0.632208i	0.948724	+	0.316104i	$0.102375\pi$
$251$			10.0161i			0.632208i	−0.948724	+	0.316104i	$0.897625\pi$
$252$		0			0
$253$	16.4968	−	14.4853i	1.03715	−	0.910682i
$254$	−7.33002			−0.459926
$255$	5.67025			0.355085
$256$	1.00000			0.0625000
$257$			17.5115i			1.09234i	0.837674	+	0.546170i	$0.183915\pi$
$257$			17.5115i			1.09234i	−0.837674	+	0.546170i	$0.816085\pi$
$258$		−	5.97972i		−	0.372281i
$259$		0			0
$260$			13.5344i			0.839365i
$261$			14.9429i			0.924942i
$262$			8.82050i			0.544932i
$263$			8.48528i			0.523225i	0.965173	+	0.261612i	$0.0842542\pi$
$263$			8.48528i			0.523225i	−0.965173	+	0.261612i	$0.915746\pi$
$264$	1.90747	−	1.67488i	0.117396	−	0.103082i
$265$		−	0.881709i		−	0.0541629i
$266$		0			0
$267$	5.14958			0.315149
$268$	−2.96246			−0.180961
$269$			6.75523i			0.411873i	0.978565	+	0.205937i	$0.0660241\pi$
$269$			6.75523i			0.411873i	−0.978565	+	0.205937i	$0.933976\pi$
$270$			8.50159i			0.517391i
$271$	19.5432			1.18716			0.593581	−	0.804774i	$-0.297714\pi$
$271$	19.5432			1.18716			0.593581	+	0.804774i	$0.297714\pi$
$272$	−3.61108			−0.218954
$273$		0			0
$274$		−	3.23620i		−	0.195506i
$275$	1.97107	−	1.73072i	0.118860	−	0.104367i
$276$			5.06620i			0.304950i
$277$		−	26.6748i		−	1.60274i	−0.598172	−	0.801368i	$-0.704106\pi$
$277$		−	26.6748i		−	1.60274i	0.598172	−	0.801368i	$-0.295894\pi$
$278$			2.47122i			0.148214i
$279$		−	15.6532i		−	0.937132i
$280$		0			0
$281$			20.9533i			1.24997i	0.780636	+	0.624986i	$0.214895\pi$
$281$			20.9533i			1.24997i	−0.780636	+	0.624986i	$0.785105\pi$
$282$			4.39866i			0.261937i
$283$	−10.0629			−0.598180			−0.299090	−	0.954225i	$-0.596683\pi$
$283$	−10.0629			−0.598180			−0.299090	+	0.954225i	$0.596683\pi$
$284$	2.13403			0.126631
$285$	1.37951			0.0817153
$286$	−14.4363	−	16.4410i	−0.853637	−	0.972180i
$287$		0			0
$288$		−	2.41421i		−	0.142259i
$289$	−3.96011			−0.232947
$290$		−	12.6986i		−	0.745685i
$291$	7.07107			0.414513
$292$	10.8252			0.633499
$293$	29.6429			1.73176			0.865879	−	0.500253i	$-0.166760\pi$
$293$	29.6429			1.73176			0.865879	+	0.500253i	$0.166760\pi$
$294$		0			0
$295$	−6.60300			−0.384442
$296$		−	11.0491i		−	0.642215i
$297$	−9.06816	−	10.3274i	−0.526188	−	0.599258i
$298$	3.90860			0.226419
$299$	43.6672			2.52534
$300$			0.605318i			0.0349480i
$301$		0			0
$302$	24.1895			1.39195
$303$		−	1.42332i		−	0.0817677i
$304$	−0.878539			−0.0503876
$305$			23.2534i			1.33149i
$306$			8.71792i			0.498370i
$307$	−16.1877			−0.923883			−0.461941	−	0.886910i	$-0.652847\pi$
$307$	−16.1877			−0.923883			−0.461941	+	0.886910i	$0.652847\pi$
$308$		0			0
$309$	5.13569			0.292159
$310$			13.3022i			0.755513i
$311$			1.42639i			0.0808832i	0.999182	+	0.0404416i	$0.0128765\pi$
$311$			1.42639i			0.0808832i	−0.999182	+	0.0404416i	$0.987124\pi$
$312$	5.04908			0.285848
$313$			11.5468i			0.652664i	0.945255	+	0.326332i	$0.105813\pi$
$313$			11.5468i			0.652664i	−0.945255	+	0.326332i	$0.894187\pi$
$314$	20.3029			1.14576
$315$		0			0
$316$		−	8.48528i		−	0.477334i
$317$	31.7239			1.78179			0.890896	−	0.454207i	$-0.150077\pi$
$317$	31.7239			1.78179			0.890896	+	0.454207i	$0.150077\pi$
$318$	−0.328927			−0.0184453
$319$	13.5448	+	15.4257i	0.758364	+	0.863676i
$320$			2.05161i			0.114689i
$321$	−11.2917			−0.630242
$322$		0			0
$323$	3.17247			0.176521
$324$	−4.07107			−0.226170
$325$	5.21742			0.289411
$326$			13.2606i			0.734438i
$327$	−4.59220			−0.253949
$328$		−	2.36864i		−	0.130786i
$329$		0			0
$330$	3.43620	+	3.91338i	0.189157	+	0.215425i
$331$	31.9239			1.75470			0.877348	−	0.479854i	$-0.159310\pi$
$331$	31.9239			1.75470			0.877348	+	0.479854i	$0.159310\pi$
$332$	−12.3153			−0.675892
$333$	−26.6748			−1.46177
$334$		−	20.4160i		−	1.11712i
$335$		−	6.07782i		−	0.332067i
$336$		0			0
$337$		−	1.40854i		−	0.0767278i	−0.999264	−	0.0383639i	$-0.987785\pi$
$337$		−	1.40854i		−	0.0767278i	0.999264	−	0.0383639i	$-0.0122146\pi$
$338$		−	30.5196i		−	1.66005i
$339$			0.457170i			0.0248301i
$340$		−	7.40854i		−	0.401784i
$341$	−14.1887	−	16.1590i	−0.768359	−	0.875059i
$342$			2.12098i			0.114689i
$343$		0			0
$344$	−7.81288			−0.421242
$345$	−10.3939			−0.559588
$346$			15.5762i			0.837380i
$347$			11.0386i			0.592584i	0.955097	+	0.296292i	$0.0957502\pi$
$347$			11.0386i			0.592584i	−0.955097	+	0.296292i	$0.904250\pi$
$348$	−4.73728			−0.253945
$349$	−15.5762			−0.833773			−0.416887	−	0.908958i	$-0.636879\pi$
$349$	−15.5762			−0.833773			−0.416887	+	0.908958i	$0.636879\pi$
$350$		0			0
$351$		−	27.3368i		−	1.45913i
$352$	−2.18834	−	2.49222i	−0.116639	−	0.132836i
$353$			6.28791i			0.334672i	0.985900	+	0.167336i	$0.0535165\pi$
$353$			6.28791i			0.334672i	−0.985900	+	0.167336i	$0.946484\pi$
$354$			2.46329i			0.130922i
$355$			4.37821i			0.232371i
$356$		−	6.72825i		−	0.356597i
$357$		0			0
$358$			4.50159i			0.237917i
$359$			10.2877i			0.542964i	0.962443	+	0.271482i	$0.0875138\pi$
$359$			10.2877i			0.542964i	−0.962443	+	0.271482i	$0.912486\pi$
$360$	4.95303			0.261048
$361$	−18.2282			−0.959377
$362$	−8.18338			−0.430109
$363$	−8.34836	−	1.08864i	−0.438175	−	0.0571385i
$364$		0			0
$365$			22.2092i			1.16248i
$366$	8.67483			0.453441
$367$		−	4.52152i		−	0.236021i	−0.993012	−	0.118011i	$-0.962348\pi$
$367$		−	4.52152i		−	0.236021i	0.993012	−	0.118011i	$-0.0376517\pi$
$368$	6.61931			0.345056
$369$	−5.71840			−0.297688
$370$	22.6684			1.17848
$371$		0			0
$372$	4.96246			0.257291
$373$		−	19.7239i		−	1.02127i	−0.859799	−	0.510633i	$-0.829411\pi$
$373$		−	19.7239i		−	1.02127i	0.859799	−	0.510633i	$-0.170589\pi$
$374$	7.90226	+	8.99962i	0.408616	+	0.465359i
$375$	−9.09306			−0.469564
$376$	5.74713			0.296386
$377$			40.8321i			2.10296i
$378$		0			0
$379$	−35.6268			−1.83003			−0.915014	−	0.403423i	$-0.867820\pi$
$379$	−35.6268			−1.83003			−0.915014	+	0.403423i	$0.867820\pi$
$380$		−	1.80242i		−	0.0924622i
$381$	−5.61016			−0.287417
$382$		−	10.3662i		−	0.530382i
$383$		−	21.6373i		−	1.10561i	−0.833310	−	0.552806i	$-0.813557\pi$
$383$		−	21.6373i		−	1.10561i	0.833310	−	0.552806i	$-0.186443\pi$
$384$	0.765367			0.0390575
$385$		0			0
$386$	1.62333			0.0826252
$387$			18.8620i			0.958807i
$388$		−	9.23880i		−	0.469029i
$389$	−8.64698			−0.438419			−0.219210	−	0.975678i	$-0.570348\pi$
$389$	−8.64698			−0.438419			−0.219210	+	0.975678i	$0.570348\pi$
$390$			10.3587i			0.524536i
$391$	−23.9029			−1.20882
$392$		0			0
$393$			6.75092i			0.340539i
$394$	−4.38713			−0.221020
$395$	17.4085			0.875917
$396$	−6.01676	+	5.28311i	−0.302354	+	0.265486i
$397$			36.1718i			1.81541i	0.419608	+	0.907705i	$0.362168\pi$
$397$			36.1718i			1.81541i	−0.419608	+	0.907705i	$0.637832\pi$
$398$	20.0640			1.00572
$399$		0			0
$400$	0.790886			0.0395443
$401$	−8.99356			−0.449117			−0.224558	−	0.974461i	$-0.572094\pi$
$401$	−8.99356			−0.449117			−0.224558	+	0.974461i	$0.572094\pi$
$402$	−2.26737			−0.113086
$403$		−	42.7730i		−	2.13067i
$404$	−1.85966			−0.0925216
$405$		−	8.35225i		−	0.415027i
$406$		0			0
$407$	−27.5368	+	24.1791i	−1.36495	+	1.19851i
$408$	−2.76380			−0.136829
$409$	−9.58279			−0.473839			−0.236919	−	0.971529i	$-0.576138\pi$
$409$	−9.58279			−0.473839			−0.236919	+	0.971529i	$0.576138\pi$
$410$	4.85953			0.239995
$411$		−	2.47688i		−	0.122175i
$412$		−	6.71011i		−	0.330583i
$413$		0			0
$414$		−	15.9804i		−	0.785396i
$415$		−	25.2663i		−	1.24027i
$416$		−	6.59694i		−	0.323441i
$417$			1.89139i			0.0926218i
$418$	1.92254	+	2.18952i	0.0940344	+	0.107093i
$419$			37.9064i			1.85185i	0.377709	+	0.925924i	$0.376712\pi$
$419$			37.9064i			1.85185i	−0.377709	+	0.925924i	$0.623288\pi$
$420$		0			0
$421$	2.88394			0.140555			0.0702774	−	0.997527i	$-0.477612\pi$
$421$	2.88394			0.140555			0.0702774	+	0.997527i	$0.477612\pi$
$422$	−8.56380			−0.416879
$423$		−	13.8748i		−	0.674616i
$424$			0.429764i			0.0208712i
$425$	−2.85595			−0.138534
$426$	1.63332			0.0791345
$427$		0			0
$428$			14.7533i			0.713130i
$429$	−11.0491	−	12.5834i	−0.533454	−	0.607534i
$430$		−	16.0290i		−	0.772987i
$431$		−	32.4068i		−	1.56098i	−0.625169	−	0.780490i	$-0.714970\pi$
$431$		−	32.4068i		−	1.56098i	0.625169	−	0.780490i	$-0.285030\pi$
$432$		−	4.14386i		−	0.199372i
$433$			10.2319i			0.491714i	0.969306	+	0.245857i	$0.0790694\pi$
$433$			10.2319i			0.491714i	−0.969306	+	0.245857i	$0.920931\pi$
$434$		0			0
$435$		−	9.71905i		−	0.465993i
$436$			6.00000i			0.287348i
$437$	−5.81532			−0.278185
$438$	8.28528			0.395886
$439$	−28.8726			−1.37802			−0.689008	−	0.724754i	$-0.741953\pi$
$439$	−28.8726			−1.37802			−0.689008	+	0.724754i	$0.741953\pi$
$440$	5.11308	−	4.48962i	0.243757	−	0.214034i
$441$		0			0
$442$			23.8221i			1.13310i
$443$	−8.56036			−0.406715			−0.203358	−	0.979105i	$-0.565185\pi$
$443$	−8.56036			−0.406715			−0.203358	+	0.979105i	$0.565185\pi$
$444$		−	8.45660i		−	0.401332i
$445$	13.8038			0.654361
$446$	−8.24084			−0.390215
$447$	2.99152			0.141494
$448$		0			0
$449$	−10.8089			−0.510102			−0.255051	−	0.966928i	$-0.582092\pi$
$449$	−10.8089			−0.510102			−0.255051	+	0.966928i	$0.582092\pi$
$450$		−	1.90937i		−	0.0900084i
$451$	−5.90318	+	5.18338i	−0.277970	+	0.244076i
$452$	0.597322			0.0280956
$453$	18.5139			0.869858
$454$		−	27.4795i		−	1.28967i
$455$		0			0
$456$	−0.672404			−0.0314882
$457$			24.3896i			1.14090i	0.821334	+	0.570448i	$0.193230\pi$
$457$			24.3896i			1.14090i	−0.821334	+	0.570448i	$0.806770\pi$
$458$	−0.104343			−0.00487563
$459$			14.9638i			0.698451i
$460$			13.5803i			0.633183i
$461$	−1.13186			−0.0527158			−0.0263579	−	0.999653i	$-0.508391\pi$
$461$	−1.13186			−0.0527158			−0.0263579	+	0.999653i	$0.508391\pi$
$462$		0			0
$463$	6.38388			0.296684			0.148342	−	0.988936i	$-0.452606\pi$
$463$	6.38388			0.296684			0.148342	+	0.988936i	$0.452606\pi$
$464$			6.18955i			0.287343i
$465$			10.1810i			0.472135i
$466$	15.1577			0.702166
$467$		−	35.6840i		−	1.65126i	−0.564213	−	0.825629i	$-0.690820\pi$
$467$		−	35.6840i		−	1.65126i	0.564213	−	0.825629i	$-0.309180\pi$
$468$	−15.9264			−0.736199
$469$		0			0
$470$			11.7909i			0.543873i
$471$	15.5391			0.716006
$472$	3.21844			0.148141
$473$	17.0972	+	19.4714i	0.786130	+	0.895298i
$474$		−	6.49435i		−	0.298296i
$475$	−0.694824			−0.0318807
$476$		0			0
$477$	1.03754			0.0475058
$478$	−25.9135			−1.18525
$479$	−11.5819			−0.529189			−0.264595	−	0.964360i	$-0.585238\pi$
$479$	−11.5819			−0.529189			−0.264595	+	0.964360i	$0.585238\pi$
$480$			1.57024i			0.0716712i
$481$	−72.8901			−3.32350
$482$			7.31108i			0.333011i
$483$		0			0
$484$	−1.42237	+	10.9077i	−0.0646532	+	0.495802i
$485$	18.9544			0.860676
$486$	−15.5474			−0.705246
$487$	40.0638			1.81546			0.907732	−	0.419550i	$-0.137812\pi$
$487$	40.0638			1.81546			0.907732	+	0.419550i	$0.137812\pi$
$488$		−	11.3342i		−	0.513076i
$489$			10.1492i			0.458964i
$490$		0			0
$491$		−	19.7876i		−	0.893003i	−0.894783	−	0.446502i	$-0.852670\pi$
$491$		−	19.7876i		−	0.893003i	0.894783	−	0.446502i	$-0.147330\pi$
$492$		−	1.81288i		−	0.0817308i
$493$		−	22.3510i		−	1.00664i
$494$			5.79566i			0.260759i
$495$	−10.8389	−	12.3441i	−0.487172	−	0.554825i
$496$		−	6.48376i		−	0.291130i
$497$		0			0
$498$	−9.42575			−0.422378
$499$	−26.6553			−1.19325			−0.596627	−	0.802519i	$-0.703493\pi$
$499$	−26.6553			−1.19325			−0.596627	+	0.802519i	$0.703493\pi$
$500$			11.8807i			0.531319i
$501$		−	15.6258i		−	0.698107i
$502$	10.0161			0.447039
$503$	10.9142			0.486638			0.243319	−	0.969946i	$-0.421764\pi$
$503$	10.9142			0.486638			0.243319	+	0.969946i	$0.421764\pi$
$504$		0			0
$505$		−	3.81530i		−	0.169779i
$506$	−14.4853	−	16.4968i	−0.643949	−	0.733373i
$507$		−	23.3587i		−	1.03739i
$508$			7.33002i			0.325217i
$509$			22.4340i			0.994369i	0.867645	+	0.497184i	$0.165633\pi$
$509$			22.4340i			0.994369i	−0.867645	+	0.497184i	$0.834367\pi$
$510$		−	5.67025i		−	0.251083i
$511$		0			0
$512$		−	1.00000i		−	0.0441942i
$513$			3.64054i			0.160734i
$514$	17.5115			0.772401
$515$	13.7665			0.606626
$516$	−5.97972			−0.263242
$517$	−12.5767	−	14.3231i	−0.553121	−	0.629931i
$518$		0			0
$519$			11.9215i			0.523295i
$520$	13.5344			0.593521
$521$			39.2106i			1.71785i	0.512102	+	0.858925i	$0.328867\pi$
$521$			39.2106i			1.71785i	−0.512102	+	0.858925i	$0.671133\pi$
$522$	14.9429			0.654033
$523$	18.0899			0.791015			0.395508	−	0.918463i	$-0.370569\pi$
$523$	18.0899			0.791015			0.395508	+	0.918463i	$0.370569\pi$
$524$	8.82050			0.385325
$525$		0			0
$526$	8.48528			0.369976
$527$			23.4134i			1.01990i
$528$	−1.67488	−	1.90747i	−0.0728898	−	0.0830118i
$529$	20.8153			0.905013
$530$	−0.881709			−0.0382990
$531$		−	7.77001i		−	0.337190i
$532$		0			0
$533$	−15.6258			−0.676827
$534$		−	5.14958i		−	0.222844i
$535$	−30.2681			−1.30861
$536$			2.96246i			0.127959i
$537$			3.44537i			0.148679i
$538$	6.75523			0.291238
$539$		0			0
$540$	8.50159			0.365850
$541$			9.62575i			0.413843i	0.978357	+	0.206922i	$0.0663446\pi$
$541$			9.62575i			0.413843i	−0.978357	+	0.206922i	$0.933655\pi$
$542$		−	19.5432i		−	0.839450i
$543$	−6.26328			−0.268783
$544$			3.61108i			0.154824i
$545$	−12.3097			−0.527289
$546$		0			0
$547$		−	10.2834i		−	0.439685i	−0.975535	−	0.219843i	$-0.929446\pi$
$547$		−	10.2834i		−	0.439685i	0.975535	−	0.219843i	$-0.0705544\pi$
$548$	−3.23620			−0.138244
$549$	−27.3632			−1.16783
$550$	−1.73072	−	1.97107i	−0.0737983	−	0.0840465i
$551$		−	5.43776i		−	0.231656i
$552$	5.06620			0.215632
$553$		0			0
$554$	−26.6748			−1.13330
$555$	17.3497			0.736452
$556$	2.47122			0.104803
$557$		−	37.2306i		−	1.57751i	−0.614707	−	0.788755i	$-0.710726\pi$
$557$		−	37.2306i		−	1.57751i	0.614707	−	0.788755i	$-0.289274\pi$
$558$	−15.6532			−0.662652
$559$			51.5411i			2.17995i
$560$		0			0
$561$	6.04812	+	6.88801i	0.255352	+	0.290812i
$562$	20.9533			0.883864
$563$	−16.1877			−0.682232			−0.341116	−	0.940021i	$-0.610805\pi$
$563$	−16.1877			−0.682232			−0.341116	+	0.940021i	$0.610805\pi$
$564$	4.39866			0.185217
$565$			1.22547i			0.0515560i
$566$			10.0629i			0.422977i
$567$		0			0
$568$		−	2.13403i		−	0.0895420i
$569$		−	30.4901i		−	1.27821i	−0.769118	−	0.639106i	$-0.779304\pi$
$569$		−	30.4901i		−	1.27821i	0.769118	−	0.639106i	$-0.220696\pi$
$570$		−	1.37951i		−	0.0577815i
$571$			40.4877i			1.69436i	0.531308	+	0.847179i	$0.321701\pi$
$571$			40.4877i			1.69436i	−0.531308	+	0.847179i	$0.678299\pi$
$572$	−16.4410	+	14.4363i	−0.687435	+	0.603613i
$573$		−	7.93396i		−	0.331446i
$574$		0			0
$575$	5.23512			0.218320
$576$	−2.41421			−0.100592
$577$		−	25.5604i		−	1.06409i	−0.846715	−	0.532047i	$-0.821423\pi$
$577$		−	25.5604i		−	1.06409i	0.846715	−	0.532047i	$-0.178577\pi$
$578$			3.96011i			0.164719i
$579$	1.24244			0.0516341
$580$	−12.6986			−0.527279
$581$		0			0
$582$		−	7.07107i		−	0.293105i
$583$	1.07107	−	0.940467i	0.0443591	−	0.0389502i
$584$		−	10.8252i		−	0.447951i
$585$		−	32.6748i		−	1.35094i
$586$		−	29.6429i		−	1.22454i
$587$		−	7.18094i		−	0.296389i	−0.988958	−	0.148195i	$-0.952654\pi$
$587$		−	7.18094i		−	0.296389i	0.988958	−	0.148195i	$-0.0473462\pi$
$588$		0			0
$589$			5.69624i			0.234709i
$590$			6.60300i			0.271841i
$591$	−3.35776			−0.138120
$592$	−11.0491			−0.454114
$593$	−19.7771			−0.812150			−0.406075	−	0.913840i	$-0.633103\pi$
$593$	−19.7771			−0.812150			−0.406075	+	0.913840i	$0.633103\pi$
$594$	−10.3274	+	9.06816i	−0.423740	+	0.372071i
$595$		0			0
$596$		−	3.90860i		−	0.160103i
$597$	15.3563			0.628493
$598$		−	43.6672i		−	1.78568i
$599$	−13.7598			−0.562210			−0.281105	−	0.959677i	$-0.590701\pi$
$599$	−13.7598			−0.562210			−0.281105	+	0.959677i	$0.590701\pi$
$600$	0.605318			0.0247120
$601$	−11.0384			−0.450266			−0.225133	−	0.974328i	$-0.572282\pi$
$601$	−11.0384			−0.450266			−0.225133	+	0.974328i	$0.572282\pi$
$602$		0			0
$603$	7.15201			0.291252
$604$		−	24.1895i		−	0.984259i
$605$	−22.3783	−	2.91815i	−0.909806	−	0.118640i
$606$	−1.42332			−0.0578185
$607$	11.8144			0.479530			0.239765	−	0.970831i	$-0.422930\pi$
$607$	11.8144			0.479530			0.239765	+	0.970831i	$0.422930\pi$
$608$			0.878539i			0.0356294i
$609$		0			0
$610$	23.2534			0.941503
$611$		−	37.9135i		−	1.53381i
$612$	8.71792			0.352401
$613$		−	21.1472i		−	0.854129i	−0.904221	−	0.427064i	$-0.859548\pi$
$613$		−	21.1472i		−	0.854129i	0.904221	−	0.427064i	$-0.140452\pi$
$614$			16.1877i			0.653284i
$615$	3.71932			0.149978
$616$		0			0
$617$	−43.8743			−1.76631			−0.883157	−	0.469078i	$-0.844586\pi$
$617$	−43.8743			−1.76631			−0.883157	+	0.469078i	$0.844586\pi$
$618$		−	5.13569i		−	0.206588i
$619$			4.76850i			0.191662i	0.995398	+	0.0958310i	$0.0305509\pi$
$619$			4.76850i			0.191662i	−0.995398	+	0.0958310i	$0.969449\pi$
$620$	13.3022			0.534228
$621$		−	27.4295i		−	1.10071i
$622$	1.42639			0.0571931
$623$		0			0
$624$		−	5.04908i		−	0.202125i
$625$	−20.4201			−0.816803
$626$	11.5468			0.461503
$627$	1.47145	+	1.67578i	0.0587639	+	0.0669243i
$628$		−	20.3029i		−	0.810172i
$629$	39.8991			1.59088
$630$		0			0
$631$	−6.66195			−0.265208			−0.132604	−	0.991169i	$-0.542334\pi$
$631$	−6.66195			−0.265208			−0.132604	+	0.991169i	$0.542334\pi$
$632$	−8.48528			−0.337526
$633$	−6.55445			−0.260516
$634$		−	31.7239i		−	1.25992i
$635$	−15.0384			−0.596779
$636$			0.328927i			0.0130428i
$637$		0			0
$638$	15.4257	−	13.5448i	0.610711	−	0.536244i
$639$	−5.15201			−0.203810
$640$	2.05161			0.0810971
$641$	−11.5215			−0.455071			−0.227535	−	0.973770i	$-0.573067\pi$
$641$	−11.5215			−0.455071			−0.227535	+	0.973770i	$0.573067\pi$
$642$			11.2917i			0.445649i
$643$		−	21.7793i		−	0.858892i	−0.903093	−	0.429446i	$-0.858709\pi$
$643$		−	21.7793i		−	0.858892i	0.903093	−	0.429446i	$-0.141291\pi$
$644$		0			0
$645$		−	12.2681i		−	0.483055i
$646$		−	3.17247i		−	0.124819i
$647$		−	17.5615i		−	0.690413i	−0.938527	−	0.345207i	$-0.887809\pi$
$647$		−	17.5615i		−	0.690413i	0.938527	−	0.345207i	$-0.112191\pi$
$648$			4.07107i			0.159927i
$649$	−7.04304	−	8.02109i	−0.276463	−	0.314855i
$650$		−	5.21742i		−	0.204644i
$651$		0			0
$652$	13.2606			0.519326
$653$	26.4426			1.03478			0.517390	−	0.855750i	$-0.326904\pi$
$653$	26.4426			1.03478			0.517390	+	0.855750i	$0.326904\pi$
$654$			4.59220i			0.179569i
$655$			18.0962i			0.707079i
$656$	−2.36864			−0.0924798
$657$	−26.1344			−1.01960
$658$		0			0
$659$			38.9324i			1.51659i	0.651910	+	0.758296i	$0.273968\pi$
$659$			38.9324i			1.51659i	−0.651910	+	0.758296i	$0.726032\pi$
$660$	3.91338	−	3.43620i	0.152328	−	0.133754i
$661$			11.8914i			0.462521i	0.972892	+	0.231261i	$0.0742850\pi$
$661$			11.8914i			0.462521i	−0.972892	+	0.231261i	$0.925715\pi$
$662$		−	31.9239i		−	1.24076i
$663$			18.2326i			0.708096i
$664$			12.3153i			0.477928i
$665$		0			0
$666$			26.6748i			1.03363i
$667$			40.9706i			1.58639i
$668$	−20.4160			−0.789920
$669$	−6.30727			−0.243853
$670$	−6.07782			−0.234807
$671$	−28.2474	+	24.8031i	−1.09048	+	0.957512i
$672$		0			0
$673$			17.4386i			0.672210i	0.941825	+	0.336105i	$0.109110\pi$
$673$			17.4386i			0.672210i	−0.941825	+	0.336105i	$0.890890\pi$
$674$	−1.40854			−0.0542548
$675$		−	3.27732i		−	0.126144i
$676$	−30.5196			−1.17383
$677$	41.4300			1.59228			0.796141	−	0.605111i	$-0.206871\pi$
$677$	41.4300			1.59228			0.796141	+	0.605111i	$0.206871\pi$
$678$	0.457170			0.0175575
$679$		0			0
$680$	−7.40854			−0.284104
$681$		−	21.0319i		−	0.805943i
$682$	−16.1590	+	14.1887i	−0.618760	+	0.543312i
$683$	21.4153			0.819433			0.409717	−	0.912213i	$-0.365628\pi$
$683$	21.4153			0.819433			0.409717	+	0.912213i	$0.365628\pi$
$684$	2.12098			0.0810977
$685$		−	6.63943i		−	0.253679i
$686$		0			0
$687$	−0.0798608			−0.00304688
$688$			7.81288i			0.297863i
$689$	2.83512			0.108010
$690$			10.3939i			0.395688i
$691$		−	13.4791i		−	0.512769i	−0.966575	−	0.256384i	$-0.917469\pi$
$691$		−	13.4791i		−	0.512769i	0.966575	−	0.256384i	$-0.0825313\pi$
$692$	15.5762			0.592117
$693$		0			0
$694$	11.0386			0.419020
$695$			5.06999i			0.192316i
$696$			4.73728i			0.179566i
$697$	8.55334			0.323981
$698$			15.5762i			0.589567i
$699$	11.6012			0.438797
$700$		0			0
$701$		−	26.4644i		−	0.999545i	−0.866157	−	0.499773i	$-0.833417\pi$
$701$		−	26.4644i		−	0.999545i	0.866157	−	0.499773i	$-0.166583\pi$
$702$	−27.3368			−1.03176
$703$	9.70704			0.366108
$704$	−2.49222	+	2.18834i	−0.0939293	+	0.0824760i
$705$			9.02435i			0.339877i
$706$	6.28791			0.236649
$707$		0			0
$708$	2.46329			0.0925761
$709$	18.7778			0.705214			0.352607	−	0.935772i	$-0.385295\pi$
$709$	18.7778			0.705214			0.352607	+	0.935772i	$0.385295\pi$
$710$	4.37821			0.164311
$711$			20.4853i			0.768258i
$712$	−6.72825			−0.252152
$713$		−	42.9181i		−	1.60729i
$714$		0			0
$715$	−29.6177	−	33.7307i	−1.10764	−	1.26146i
$716$	4.50159			0.168232
$717$	−19.8333			−0.740688
$718$	10.2877			0.383934
$719$			3.46741i			0.129313i	0.997908	+	0.0646564i	$0.0205951\pi$
$719$			3.46741i			0.129313i	−0.997908	+	0.0646564i	$0.979405\pi$
$720$		−	4.95303i		−	0.184589i
$721$		0			0
$722$			18.2282i			0.678382i
$723$			5.59566i			0.208105i
$724$			8.18338i			0.304133i
$725$			4.89523i			0.181804i
$726$	−1.08864	+	8.34836i	−0.0404030	+	0.309837i
$727$		−	21.6647i		−	0.803500i	−0.915749	−	0.401750i	$-0.868402\pi$
$727$		−	21.6647i		−	0.803500i	0.915749	−	0.401750i	$-0.131598\pi$
$728$		0			0
$729$	0.313708			0.0116188
$730$	22.2092			0.821999
$731$		−	28.2129i		−	1.04349i
$732$		−	8.67483i		−	0.320631i
$733$	14.9085			0.550656			0.275328	−	0.961350i	$-0.411213\pi$
$733$	14.9085			0.550656			0.275328	+	0.961350i	$0.411213\pi$
$734$	−4.52152			−0.166892
$735$		0			0
$736$		−	6.61931i		−	0.243991i
$737$	7.38311	−	6.48286i	0.271960	−	0.238799i
$738$			5.71840i			0.210497i
$739$			4.75577i			0.174944i	0.996167	+	0.0874719i	$0.0278788\pi$
$739$			4.75577i			0.174944i	−0.996167	+	0.0874719i	$0.972121\pi$
$740$		−	22.6684i		−	0.833308i
$741$			4.43581i			0.162954i
$742$		0			0
$743$			14.8792i			0.545864i	0.962033	+	0.272932i	$0.0879934\pi$
$743$			14.8792i			0.545864i	−0.962033	+	0.272932i	$0.912007\pi$
$744$		−	4.96246i		−	0.181933i
$745$	8.01894			0.293791
$746$	−19.7239			−0.722144
$747$	29.7318			1.08783
$748$	8.99962	−	7.90226i	0.329059	−	0.288935i
$749$		0			0
$750$			9.09306i			0.332032i
$751$	16.4820			0.601438			0.300719	−	0.953713i	$-0.402773\pi$
$751$	16.4820			0.601438			0.300719	+	0.953713i	$0.402773\pi$
$752$		−	5.74713i		−	0.209576i
$753$	7.66596			0.279363
$754$	40.8321			1.48702
$755$	49.6276			1.80613
$756$		0			0
$757$	−3.82008			−0.138843			−0.0694216	−	0.997587i	$-0.522115\pi$
$757$	−3.82008			−0.138843			−0.0694216	+	0.997587i	$0.522115\pi$
$758$			35.6268i			1.29402i
$759$	−11.0866	−	12.6261i	−0.402416	−	0.458299i
$760$	−1.80242			−0.0653807
$761$	−1.85400			−0.0672075			−0.0336038	−	0.999435i	$-0.510698\pi$
$761$	−1.85400			−0.0672075			−0.0336038	+	0.999435i	$0.510698\pi$
$762$			5.61016i			0.203235i
$763$		0			0
$764$	−10.3662			−0.375037
$765$			17.8858i			0.646662i
$766$	−21.6373			−0.781786
$767$		−	21.2319i		−	0.766638i
$768$		−	0.765367i		−	0.0276178i
$769$	8.92308			0.321775			0.160887	−	0.986973i	$-0.448564\pi$
$769$	8.92308			0.321775			0.160887	+	0.986973i	$0.448564\pi$
$770$		0			0
$771$	13.4028			0.482688
$772$		−	1.62333i		−	0.0584248i
$773$			20.3267i			0.731099i	0.930792	+	0.365550i	$0.119119\pi$
$773$			20.3267i			0.731099i	−0.930792	+	0.365550i	$0.880881\pi$
$774$	18.8620			0.677979
$775$		−	5.12792i		−	0.184200i
$776$	−9.23880			−0.331653
$777$		0			0
$778$			8.64698i			0.310009i
$779$	2.08094			0.0745574
$780$	10.3587			0.370903
$781$	−5.31849	+	4.66998i	−0.190310	+	0.167105i
$782$			23.9029i			0.854765i
$783$	25.6486			0.916607
$784$		0			0
$785$	41.6536			1.48668
$786$	6.75092			0.240797
$787$	38.4016			1.36887			0.684435	−	0.729074i	$-0.260049\pi$
$787$	38.4016			1.36887			0.684435	+	0.729074i	$0.260049\pi$
$788$			4.38713i			0.156285i
$789$	6.49435			0.231205
$790$		−	17.4085i		−	0.619367i
$791$		0			0
$792$	5.28311	+	6.01676i	0.187727	+	0.213796i
$793$

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:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -0.765367i q^{3} -1.00000 q^{4} -2.05161i q^{5} -0.765367 q^{6} +1.00000i q^{8} +2.41421 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -0.765367i q^{3} -1.00000 q^{4} -2.05161i q^{5} -0.765367 q^{6} +1.00000i q^{8} +2.41421 q^{9} -2.05161 q^{10} +(2.49222 - 2.18834i) q^{11} +0.765367i q^{12} +6.59694 q^{13} -1.57024 q^{15} +1.00000 q^{16} -3.61108 q^{17} -2.41421i q^{18} -0.878539 q^{19} +2.05161i q^{20} +(-2.18834 - 2.49222i) q^{22} +6.61931 q^{23} +0.765367 q^{24} +0.790886 q^{25} -6.59694i q^{26} -4.14386i q^{27} +6.18955i q^{29} +1.57024i q^{30} -6.48376i q^{31} -1.00000i q^{32} +(-1.67488 - 1.90747i) q^{33} +3.61108i q^{34} -2.41421 q^{36} -11.0491 q^{37} +0.878539i q^{38} -5.04908i q^{39} +2.05161 q^{40} -2.36864 q^{41} +7.81288i q^{43} +(-2.49222 + 2.18834i) q^{44} -4.95303i q^{45} -6.61931i q^{46} -5.74713i q^{47} -0.765367i q^{48} -0.790886i q^{50} +2.76380i q^{51} -6.59694 q^{52} +0.429764 q^{53} -4.14386 q^{54} +(-4.48962 - 5.11308i) q^{55} +0.672404i q^{57} +6.18955 q^{58} -3.21844i q^{59} +1.57024 q^{60} -11.3342 q^{61} -6.48376 q^{62} -1.00000 q^{64} -13.5344i q^{65} +(-1.90747 + 1.67488i) q^{66} +2.96246 q^{67} +3.61108 q^{68} -5.06620i q^{69} -2.13403 q^{71} +2.41421i q^{72} -10.8252 q^{73} +11.0491i q^{74} -0.605318i q^{75} +0.878539 q^{76} -5.04908 q^{78} +8.48528i q^{79} -2.05161i q^{80} +4.07107 q^{81} +2.36864i q^{82} +12.3153 q^{83} +7.40854i q^{85} +7.81288 q^{86} +4.73728 q^{87} +(2.18834 + 2.49222i) q^{88} +6.72825i q^{89} -4.95303 q^{90} -6.61931 q^{92} -4.96246 q^{93} -5.74713 q^{94} +1.80242i q^{95} -0.765367 q^{96} +9.23880i q^{97} +(6.01676 - 5.28311i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^4 + 16 * q^9	\( 16 q - 16 q^{4} + 16 q^{9} + 16 q^{11} + 16 q^{16} - 8 q^{22} - 16 q^{23} - 64 q^{25} - 16 q^{36} - 80 q^{37} - 16 q^{44} + 32 q^{53} - 48 q^{58} - 16 q^{64} + 16 q^{67} - 48 q^{71} + 16 q^{78} - 48 q^{81} + 32 q^{86} + 8 q^{88} + 16 q^{92} - 48 q^{93} + 24 q^{99}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^9 + 16 * q^11 + 16 * q^16 - 8 * q^22 - 16 * q^23 - 64 * q^25 - 16 * q^36 - 80 * q^37 - 16 * q^44 + 32 * q^53 - 48 * q^58 - 16 * q^64 + 16 * q^67 - 48 * q^71 + 16 * q^78 - 48 * q^81 + 32 * q^86 + 8 * q^88 + 16 * q^92 - 48 * q^93 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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