LMFDB - Embedded newform 539.2.a.l.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,2,Mod(1,539)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("539.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	539.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$4.30393666895$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 26x^{8} + 245x^{6} - 1038x^{4} + 1884x^{2} - 968 $ x^10 - 26x^8 + 245x^6 - 1038x^4 + 1884x^2 - 968
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.903205$ of defining polynomial
Character	$\chi$	$=$	539.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.62810 q^{2} -0.903205 q^{3} +4.90690 q^{4} +0.337987 q^{5} -2.37371 q^{6} +7.63960 q^{8} -2.18422 q^{9} +O(q^{10})$	\(q+2.62810 q^{2} -0.903205 q^{3} +4.90690 q^{4} +0.337987 q^{5} -2.37371 q^{6} +7.63960 q^{8} -2.18422 q^{9} +0.888264 q^{10} +1.00000 q^{11} -4.43193 q^{12} +6.09040 q^{13} -0.305272 q^{15} +10.2638 q^{16} +3.93795 q^{17} -5.74034 q^{18} -8.10936 q^{19} +1.65847 q^{20} +2.62810 q^{22} +1.82729 q^{23} -6.90013 q^{24} -4.88576 q^{25} +16.0062 q^{26} +4.68241 q^{27} -6.61313 q^{29} -0.802284 q^{30} -6.18413 q^{31} +11.6951 q^{32} -0.903205 q^{33} +10.3493 q^{34} -10.7177 q^{36} -0.706234 q^{37} -21.3122 q^{38} -5.50088 q^{39} +2.58209 q^{40} -1.45556 q^{41} +2.45686 q^{43} +4.90690 q^{44} -0.738239 q^{45} +4.80228 q^{46} -4.70607 q^{47} -9.27034 q^{48} -12.8403 q^{50} -3.55678 q^{51} +29.8850 q^{52} +6.15627 q^{53} +12.3058 q^{54} +0.337987 q^{55} +7.32441 q^{57} -17.3799 q^{58} -6.77443 q^{59} -1.49794 q^{60} -4.28399 q^{61} -16.2525 q^{62} +10.2083 q^{64} +2.05848 q^{65} -2.37371 q^{66} -8.68510 q^{67} +19.3231 q^{68} -1.65041 q^{69} -4.68510 q^{71} -16.6866 q^{72} +11.0713 q^{73} -1.85605 q^{74} +4.41285 q^{75} -39.7918 q^{76} -14.4569 q^{78} +12.8253 q^{79} +3.46904 q^{80} +2.32348 q^{81} -3.82536 q^{82} +12.1808 q^{83} +1.33098 q^{85} +6.45686 q^{86} +5.97301 q^{87} +7.63960 q^{88} +0.337987 q^{89} -1.94016 q^{90} +8.96630 q^{92} +5.58554 q^{93} -12.3680 q^{94} -2.74086 q^{95} -10.5631 q^{96} +6.74935 q^{97} -2.18422 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9}+O(q^{10}) $ 10 * q + 2 * q^2 + 18 * q^4 - 6 * q^8 + 22 * q^9	$ 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9} + 10 q^{11} + 8 q^{15} + 42 q^{16} + 6 q^{18} + 2 q^{22} + 4 q^{23} + 18 q^{25} + 12 q^{29} - 4 q^{30} - 30 q^{32} - 2 q^{36} + 40 q^{37} - 16 q^{39} - 8 q^{43} + 18 q^{44} + 44 q^{46} - 62 q^{50} + 16 q^{53} - 8 q^{57} - 28 q^{58} + 36 q^{60} + 106 q^{64} - 32 q^{65} - 4 q^{67} + 36 q^{71} - 90 q^{72} - 28 q^{74} - 112 q^{78} + 8 q^{79} - 6 q^{81} + 88 q^{85} + 32 q^{86} - 6 q^{88} - 52 q^{92} + 44 q^{93} - 64 q^{95} + 22 q^{99}+O(q^{100}) $ 10 * q + 2 * q^2 + 18 * q^4 - 6 * q^8 + 22 * q^9 + 10 * q^11 + 8 * q^15 + 42 * q^16 + 6 * q^18 + 2 * q^22 + 4 * q^23 + 18 * q^25 + 12 * q^29 - 4 * q^30 - 30 * q^32 - 2 * q^36 + 40 * q^37 - 16 * q^39 - 8 * q^43 + 18 * q^44 + 44 * q^46 - 62 * q^50 + 16 * q^53 - 8 * q^57 - 28 * q^58 + 36 * q^60 + 106 * q^64 - 32 * q^65 - 4 * q^67 + 36 * q^71 - 90 * q^72 - 28 * q^74 - 112 * q^78 + 8 * q^79 - 6 * q^81 + 88 * q^85 + 32 * q^86 - 6 * q^88 - 52 * q^92 + 44 * q^93 - 64 * q^95 + 22 * q^99

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 $

$2$	2.62810	1.85835	0.929173	−	0.369646i	$-0.120521\pi$
$2$	2.62810	1.85835	0.929173	+	0.369646i	$0.120521\pi$
$3$	−0.903205	−0.521466	−0.260733	−	0.965411i	$-0.583964\pi$
$3$	−0.903205	−0.521466	−0.260733	+	0.965411i	$0.583964\pi$
$4$	4.90690	2.45345
$5$	0.337987	0.151153	0.0755763	−	0.997140i	$-0.475920\pi$
$5$	0.337987	0.151153	0.0755763	+	0.997140i	$0.475920\pi$
$6$	−2.37371	−0.969064
$7$	0	0
$7$	0	0
$8$	7.63960	2.70101
$9$	−2.18422	−0.728073
$10$	0.888264	0.280894
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−4.43193	−1.27939
$13$	6.09040	1.68917	0.844587	−	0.535419i	$-0.179846\pi$
$13$	6.09040	1.68917	0.844587	+	0.535419i	$0.179846\pi$
$14$	0	0
$15$	−0.305272	−0.0788209
$16$	10.2638	2.56596
$17$	3.93795	0.955093	0.477547	−	0.878606i	$-0.341526\pi$
$17$	3.93795	0.955093	0.477547	+	0.878606i	$0.341526\pi$
$18$	−5.74034	−1.35301
$19$	−8.10936	−1.86041	−0.930207	−	0.367035i	$-0.880373\pi$
$19$	−8.10936	−1.86041	−0.930207	+	0.367035i	$0.880373\pi$
$20$	1.65847	0.370845
$21$	0	0
$22$	2.62810	0.560312
$23$	1.82729	0.381015	0.190508	−	0.981686i	$-0.438987\pi$
$23$	1.82729	0.381015	0.190508	+	0.981686i	$0.438987\pi$
$24$	−6.90013	−1.40848
$25$	−4.88576	−0.977153
$26$	16.0062	3.13907
$27$	4.68241	0.901131
$28$	0	0
$29$	−6.61313	−1.22803	−0.614014	−	0.789295i	$-0.710446\pi$
$29$	−6.61313	−1.22803	−0.614014	+	0.789295i	$0.710446\pi$
$30$	−0.802284	−0.146476
$31$	−6.18413	−1.11070	−0.555352	−	0.831616i	$-0.687416\pi$
$31$	−6.18413	−1.11070	−0.555352	+	0.831616i	$0.687416\pi$
$32$	11.6951	2.06743
$33$	−0.903205	−0.157228
$34$	10.3493	1.77489
$35$	0	0
$36$	−10.7177	−1.78629
$37$	−0.706234	−0.116104	−0.0580521	−	0.998314i	$-0.518489\pi$
$37$	−0.706234	−0.116104	−0.0580521	+	0.998314i	$0.518489\pi$
$38$	−21.3122	−3.45729
$39$	−5.50088	−0.880846
$40$	2.58209	0.408264
$41$	−1.45556	−0.227321	−0.113660	−	0.993520i	$-0.536258\pi$
$41$	−1.45556	−0.227321	−0.113660	+	0.993520i	$0.536258\pi$
$42$	0	0
$43$	2.45686	0.374667	0.187333	−	0.982296i	$-0.440015\pi$
$43$	2.45686	0.374667	0.187333	+	0.982296i	$0.440015\pi$
$44$	4.90690	0.739742
$45$	−0.738239	−0.110050
$46$	4.80228	0.708058
$47$	−4.70607	−0.686451	−0.343226	−	0.939253i	$-0.611520\pi$
$47$	−4.70607	−0.686451	−0.343226	+	0.939253i	$0.611520\pi$
$48$	−9.27034	−1.33806
$49$	0	0
$50$	−12.8403	−1.81589
$51$	−3.55678	−0.498048
$52$	29.8850	4.14430
$53$	6.15627	0.845629	0.422815	−	0.906216i	$-0.361042\pi$
$53$	6.15627	0.845629	0.422815	+	0.906216i	$0.361042\pi$
$54$	12.3058	1.67461
$55$	0.337987	0.0455742
$56$	0	0
$57$	7.32441	0.970142
$58$	−17.3799	−2.28210
$59$	−6.77443	−0.881956	−0.440978	−	0.897518i	$-0.645368\pi$
$59$	−6.77443	−0.881956	−0.440978	+	0.897518i	$0.645368\pi$
$60$	−1.49794	−0.193383
$61$	−4.28399	−0.548509	−0.274254	−	0.961657i	$-0.588431\pi$
$61$	−4.28399	−0.548509	−0.274254	+	0.961657i	$0.588431\pi$
$62$	−16.2525	−2.06407
$63$	0	0
$64$	10.2083	1.27604
$65$	2.05848	0.255323
$66$	−2.37371	−0.292184
$67$	−8.68510	−1.06105	−0.530527	−	0.847668i	$-0.678006\pi$
$67$	−8.68510	−1.06105	−0.530527	+	0.847668i	$0.678006\pi$
$68$	19.3231	2.34327
$69$	−1.65041	−0.198686
$70$	0	0
$71$	−4.68510	−0.556019	−0.278010	−	0.960578i	$-0.589675\pi$
$71$	−4.68510	−0.556019	−0.278010	+	0.960578i	$0.589675\pi$
$72$	−16.6866	−1.96653
$73$	11.0713	1.29580	0.647898	−	0.761727i	$-0.275648\pi$
$73$	11.0713	1.29580	0.647898	+	0.761727i	$0.275648\pi$
$74$	−1.85605	−0.215762
$75$	4.41285	0.509552
$76$	−39.7918	−4.56443
$77$	0	0
$78$	−14.4569	−1.63692
$79$	12.8253	1.44296	0.721479	−	0.692436i	$-0.243463\pi$
$79$	12.8253	1.44296	0.721479	+	0.692436i	$0.243463\pi$
$80$	3.46904	0.387851
$81$	2.32348	0.258164
$82$	−3.82536	−0.422441
$83$	12.1808	1.33702	0.668508	−	0.743705i	$-0.266933\pi$
$83$	12.1808	1.33702	0.668508	+	0.743705i	$0.266933\pi$
$84$	0	0
$85$	1.33098	0.144365
$86$	6.45686	0.696261
$87$	5.97301	0.640374
$88$	7.63960	0.814384
$89$	0.337987	0.0358266	0.0179133	−	0.999840i	$-0.494298\pi$
$89$	0.337987	0.0358266	0.0179133	+	0.999840i	$0.494298\pi$
$90$	−1.94016	−0.204511
$91$	0	0
$92$	8.96630	0.934801
$93$	5.58554	0.579194
$94$	−12.3680	−1.27566
$95$	−2.74086	−0.281206
$96$	−10.5631	−1.07809
$97$	6.74935	0.685293	0.342646	−	0.939464i	$-0.388677\pi$
$97$	6.74935	0.685293	0.342646	+	0.939464i	$0.388677\pi$
$98$	0	0
$99$	−2.18422	−0.219522
$100$	−23.9739	−2.39739
$101$	−14.1699	−1.40996	−0.704978	−	0.709230i	$-0.749043\pi$
$101$	−14.1699	−1.40996	−0.704978	+	0.709230i	$0.749043\pi$
$102$	−9.34756	−0.925546
$103$	−5.35216	−0.527364	−0.263682	−	0.964610i	$-0.584937\pi$
$103$	−5.35216	−0.527364	−0.263682	+	0.964610i	$0.584937\pi$
$104$	46.5283	4.56247
$105$	0	0
$106$	16.1793	1.57147
$107$	−8.76976	−0.847805	−0.423902	−	0.905708i	$-0.639340\pi$
$107$	−8.76976	−0.847805	−0.423902	+	0.905708i	$0.639340\pi$
$108$	22.9761	2.21088
$109$	10.7993	1.03439	0.517195	−	0.855868i	$-0.326976\pi$
$109$	10.7993	1.03439	0.517195	+	0.855868i	$0.326976\pi$
$110$	0.888264	0.0846926
$111$	0.637874	0.0605443
$112$	0	0
$113$	9.26383	0.871468	0.435734	−	0.900076i	$-0.356489\pi$
$113$	9.26383	0.871468	0.435734	+	0.900076i	$0.356489\pi$
$114$	19.2493	1.80286
$115$	0.617599	0.0575914
$116$	−32.4499	−3.01290
$117$	−13.3028	−1.22984
$118$	−17.8039	−1.63898
$119$	0	0
$120$	−2.33216	−0.212896
$121$	1.00000	0.0909091
$122$	−11.2587	−1.01932
$123$	1.31467	0.118540
$124$	−30.3449	−2.72505
$125$	−3.34126	−0.298852
$126$	0	0
$127$	−0.345429	−0.0306518	−0.0153259	−	0.999883i	$-0.504879\pi$
$127$	−0.345429	−0.0306518	−0.0153259	+	0.999883i	$0.504879\pi$
$128$	3.43812	0.303890
$129$	−2.21904	−0.195376
$130$	5.40988	0.474478
$131$	−2.59497	−0.226724	−0.113362	−	0.993554i	$-0.536162\pi$
$131$	−2.59497	−0.226724	−0.113362	+	0.993554i	$0.536162\pi$
$132$	−4.43193	−0.385750
$133$	0	0
$134$	−22.8253	−1.97181
$135$	1.58260	0.136208
$136$	30.0844	2.57971
$137$	−13.6199	−1.16363	−0.581815	−	0.813321i	$-0.697657\pi$
$137$	−13.6199	−1.16363	−0.581815	+	0.813321i	$0.697657\pi$
$138$	−4.33745	−0.369228
$139$	8.59787	0.729261	0.364631	−	0.931152i	$-0.381195\pi$
$139$	8.59787	0.729261	0.364631	+	0.931152i	$0.381195\pi$
$140$	0	0
$141$	4.25055	0.357961
$142$	−12.3129	−1.03328
$143$	6.09040	0.509305
$144$	−22.4185	−1.86821
$145$	−2.23515	−0.185619
$146$	29.0964	2.40804
$147$	0	0
$148$	−3.46542	−0.284855
$149$	−10.5777	−0.866556	−0.433278	−	0.901260i	$-0.642643\pi$
$149$	−10.5777	−0.866556	−0.433278	+	0.901260i	$0.642643\pi$
$150$	11.5974	0.946923
$151$	−3.91158	−0.318320	−0.159160	−	0.987253i	$-0.550879\pi$
$151$	−3.91158	−0.318320	−0.159160	+	0.987253i	$0.550879\pi$
$152$	−61.9523	−5.02499
$153$	−8.60135	−0.695378
$154$	0	0
$155$	−2.09016	−0.167886
$156$	−26.9923	−2.16111
$157$	8.43655	0.673310	0.336655	−	0.941628i	$-0.390704\pi$
$157$	8.43655	0.673310	0.336655	+	0.941628i	$0.390704\pi$
$158$	33.7061	2.68152
$159$	−5.56038	−0.440967
$160$	3.95281	0.312497
$161$	0	0
$162$	6.10633	0.479759
$163$	23.6802	1.85477	0.927387	−	0.374103i	$-0.122050\pi$
$163$	23.6802	1.85477	0.927387	+	0.374103i	$0.122050\pi$
$164$	−7.14230	−0.557720
$165$	−0.305272	−0.0237654
$166$	32.0123	2.48464
$167$	−3.58294	−0.277256	−0.138628	−	0.990345i	$-0.544269\pi$
$167$	−3.58294	−0.277256	−0.138628	+	0.990345i	$0.544269\pi$
$168$	0	0
$169$	24.0930	1.85331
$170$	3.49794	0.268280
$171$	17.7126	1.35452
$172$	12.0555	0.919226
$173$	24.3066	1.84800	0.924000	−	0.382392i	$-0.124900\pi$
$173$	24.3066	1.84800	0.924000	+	0.382392i	$0.124900\pi$
$174$	15.6977	1.19004
$175$	0	0
$176$	10.2638	0.773665
$177$	6.11870	0.459910
$178$	0.888264	0.0665782
$179$	11.8744	0.887532	0.443766	−	0.896143i	$-0.353642\pi$
$179$	11.8744	0.887532	0.443766	+	0.896143i	$0.353642\pi$
$180$	−3.62246	−0.270002
$181$	−7.41869	−0.551427	−0.275714	−	0.961240i	$-0.588914\pi$
$181$	−7.41869	−0.551427	−0.275714	+	0.961240i	$0.588914\pi$
$182$	0	0
$183$	3.86932	0.286029
$184$	13.9597	1.02913
$185$	−0.238698	−0.0175494
$186$	14.6793	1.07634
$187$	3.93795	0.287971
$188$	−23.0922	−1.68417
$189$	0	0
$190$	−7.20325	−0.522578
$191$	10.1486	0.734330	0.367165	−	0.930156i	$-0.380328\pi$
$191$	10.1486	0.734330	0.367165	+	0.930156i	$0.380328\pi$
$192$	−9.22019	−0.665410
$193$	24.1822	1.74068	0.870338	−	0.492456i	$-0.163901\pi$
$193$	24.1822	1.74068	0.870338	+	0.492456i	$0.163901\pi$
$194$	17.7380	1.27351
$195$	−1.85923	−0.133142
$196$	0	0
$197$	5.20325	0.370716	0.185358	−	0.982671i	$-0.440656\pi$
$197$	5.20325	0.370716	0.185358	+	0.982671i	$0.440656\pi$
$198$	−5.74034	−0.407948
$199$	−16.5234	−1.17131	−0.585657	−	0.810559i	$-0.699163\pi$
$199$	−16.5234	−1.17131	−0.585657	+	0.810559i	$0.699163\pi$
$200$	−37.3253	−2.63930
$201$	7.84443	0.553303
$202$	−37.2398	−2.62018
$203$	0	0
$204$	−17.4527	−1.22194
$205$	−0.491962	−0.0343601
$206$	−14.0660	−0.980025
$207$	−3.99119	−0.277407
$208$	62.5108	4.33435
$209$	−8.10936	−0.560936
$210$	0	0
$211$	−10.8350	−0.745915	−0.372957	−	0.927848i	$-0.621656\pi$
$211$	−10.8350	−0.745915	−0.372957	+	0.927848i	$0.621656\pi$
$212$	30.2082	2.07471
$213$	4.23161	0.289945
$214$	−23.0478	−1.57551
$215$	0.830386	0.0566319
$216$	35.7718	2.43396
$217$	0	0
$218$	28.3817	1.92225
$219$	−9.99964	−0.675713
$220$	1.65847	0.111814
$221$	23.9837	1.61332
$222$	1.67640	0.112512
$223$	−6.91273	−0.462911	−0.231455	−	0.972846i	$-0.574349\pi$
$223$	−6.91273	−0.462911	−0.231455	+	0.972846i	$0.574349\pi$
$224$	0	0
$225$	10.6716	0.711439
$226$	24.3462	1.61949
$227$	9.85600	0.654166	0.327083	−	0.944996i	$-0.393934\pi$
$227$	9.85600	0.654166	0.327083	+	0.944996i	$0.393934\pi$
$228$	35.9401	2.38019
$229$	18.6793	1.23436	0.617180	−	0.786822i	$-0.288275\pi$
$229$	18.6793	1.23436	0.617180	+	0.786822i	$0.288275\pi$
$230$	1.62311	0.107025
$231$	0	0
$232$	−50.5217	−3.31691
$233$	−2.83222	−0.185545	−0.0927725	−	0.995687i	$-0.529573\pi$
$233$	−2.83222	−0.185545	−0.0927725	+	0.995687i	$0.529573\pi$
$234$	−34.9610	−2.28547
$235$	−1.59059	−0.103759
$236$	−33.2414	−2.16383
$237$	−11.5839	−0.752454
$238$	0	0
$239$	1.10930	0.0717547	0.0358773	−	0.999356i	$-0.488577\pi$
$239$	1.10930	0.0717547	0.0358773	+	0.999356i	$0.488577\pi$
$240$	−3.13326	−0.202251
$241$	19.1346	1.23257	0.616285	−	0.787523i	$-0.288637\pi$
$241$	19.1346	1.23257	0.616285	+	0.787523i	$0.288637\pi$
$242$	2.62810	0.168940
$243$	−16.1458	−1.03576
$244$	−21.0211	−1.34574
$245$	0	0
$246$	3.45509	0.220288
$247$	−49.3892	−3.14256
$248$	−47.2443	−3.00002
$249$	−11.0018	−0.697209
$250$	−8.78117	−0.555370
$251$	−5.24482	−0.331050	−0.165525	−	0.986206i	$-0.552932\pi$
$251$	−5.24482	−0.331050	−0.165525	+	0.986206i	$0.552932\pi$
$252$	0	0
$253$	1.82729	0.114880
$254$	−0.907820	−0.0569617
$255$	−1.20215	−0.0752813
$256$	−11.3809	−0.711304
$257$	13.2315	0.825361	0.412681	−	0.910876i	$-0.364593\pi$
$257$	13.2315	0.825361	0.412681	+	0.910876i	$0.364593\pi$
$258$	−5.83187	−0.363076
$259$	0	0
$260$	10.1007	0.626421
$261$	14.4445	0.894094
$262$	−6.81983	−0.421331
$263$	−1.14218	−0.0704300	−0.0352150	−	0.999380i	$-0.511212\pi$
$263$	−1.14218	−0.0704300	−0.0352150	+	0.999380i	$0.511212\pi$
$264$	−6.90013	−0.424674
$265$	2.08074	0.127819
$266$	0	0
$267$	−0.305272	−0.0186823
$268$	−42.6169	−2.60324
$269$	24.1569	1.47287	0.736435	−	0.676508i	$-0.236508\pi$
$269$	24.1569	1.47287	0.736435	+	0.676508i	$0.236508\pi$
$270$	4.15922	0.253122
$271$	−3.26678	−0.198443	−0.0992213	−	0.995065i	$-0.531635\pi$
$271$	−3.26678	−0.198443	−0.0992213	+	0.995065i	$0.531635\pi$
$272$	40.4184	2.45073
$273$	0	0
$274$	−35.7945	−2.16243
$275$	−4.88576	−0.294623
$276$	−8.09841	−0.487467
$277$	13.8079	0.829636	0.414818	−	0.909904i	$-0.363845\pi$
$277$	13.8079	0.829636	0.414818	+	0.909904i	$0.363845\pi$
$278$	22.5960	1.35522
$279$	13.5075	0.808674
$280$	0	0
$281$	−14.9163	−0.889832	−0.444916	−	0.895572i	$-0.646766\pi$
$281$	−14.9163	−0.889832	−0.444916	+	0.895572i	$0.646766\pi$
$282$	11.1709	0.665215
$283$	−12.4483	−0.739975	−0.369988	−	0.929037i	$-0.620638\pi$
$283$	−12.4483	−0.739975	−0.369988	+	0.929037i	$0.620638\pi$
$284$	−22.9893	−1.36416
$285$	2.47556	0.146639
$286$	16.0062	0.946465
$287$	0	0
$288$	−25.5448	−1.50524
$289$	−1.49255	−0.0877972
$290$	−5.87420	−0.344945
$291$	−6.09605	−0.357357
$292$	54.3256	3.17917
$293$	−28.4204	−1.66034	−0.830170	−	0.557511i	$-0.811756\pi$
$293$	−28.4204	−1.66034	−0.830170	+	0.557511i	$0.811756\pi$
$294$	0	0
$295$	−2.28967	−0.133310
$296$	−5.39535	−0.313598
$297$	4.68241	0.271701
$298$	−27.7991	−1.61036
$299$	11.1289	0.643601
$300$	21.6534	1.25016
$301$	0	0
$302$	−10.2800	−0.591549
$303$	12.7983	0.735243
$304$	−83.2331	−4.77374
$305$	−1.44793	−0.0829085
$306$	−22.6052	−1.29225
$307$	−11.0964	−0.633303	−0.316651	−	0.948542i	$-0.602559\pi$
$307$	−11.0964	−0.633303	−0.316651	+	0.948542i	$0.602559\pi$
$308$	0	0
$309$	4.83410	0.275002
$310$	−5.49314	−0.311989
$311$	−21.5709	−1.22317	−0.611586	−	0.791178i	$-0.709468\pi$
$311$	−21.5709	−1.22317	−0.611586	+	0.791178i	$0.709468\pi$
$312$	−42.0246	−2.37917
$313$	−11.8986	−0.672549	−0.336275	−	0.941764i	$-0.609167\pi$
$313$	−11.8986	−0.672549	−0.336275	+	0.941764i	$0.609167\pi$
$314$	22.1721	1.25124
$315$	0	0
$316$	62.9324	3.54022
$317$	8.96066	0.503281	0.251640	−	0.967821i	$-0.419030\pi$
$317$	8.96066	0.503281	0.251640	+	0.967821i	$0.419030\pi$
$318$	−14.6132	−0.819468
$319$	−6.61313	−0.370264
$320$	3.45028	0.192876
$321$	7.92090	0.442101
$322$	0	0
$323$	−31.9342	−1.77687
$324$	11.4011	0.633393
$325$	−29.7563	−1.65058
$326$	62.2338	3.44681
$327$	−9.75402	−0.539399
$328$	−11.1199	−0.613996
$329$	0	0
$330$	−0.802284	−0.0441643
$331$	30.3961	1.67072	0.835362	−	0.549701i	$-0.185258\pi$
$331$	30.3961	1.67072	0.835362	+	0.549701i	$0.185258\pi$
$332$	59.7699	3.28030
$333$	1.54257	0.0845323
$334$	−9.41631	−0.515238
$335$	−2.93546	−0.160381
$336$	0	0
$337$	−23.6574	−1.28870	−0.644350	−	0.764731i	$-0.722872\pi$
$337$	−23.6574	−1.28870	−0.644350	+	0.764731i	$0.722872\pi$
$338$	63.3187	3.44409
$339$	−8.36714	−0.454441
$340$	6.53097	0.354191
$341$	−6.18413	−0.334890
$342$	46.5505	2.51716
$343$	0	0
$344$	18.7694	1.01198
$345$	−0.557819	−0.0300320
$346$	63.8802	3.43422
$347$	−19.7160	−1.05841	−0.529205	−	0.848494i	$-0.677510\pi$
$347$	−19.7160	−1.05841	−0.529205	+	0.848494i	$0.677510\pi$
$348$	29.3089	1.57112
$349$	12.4053	0.664040	0.332020	−	0.943272i	$-0.392270\pi$
$349$	12.4053	0.664040	0.332020	+	0.943272i	$0.392270\pi$
$350$	0	0
$351$	28.5178	1.52217
$352$	11.6951	0.623353
$353$	15.4831	0.824080	0.412040	−	0.911166i	$-0.364816\pi$
$353$	15.4831	0.824080	0.412040	+	0.911166i	$0.364816\pi$
$354$	16.0805	0.854671
$355$	−1.58351	−0.0840438
$356$	1.65847	0.0878987
$357$	0	0
$358$	31.2070	1.64934
$359$	−20.3206	−1.07248	−0.536241	−	0.844065i	$-0.680156\pi$
$359$	−20.3206	−1.07248	−0.536241	+	0.844065i	$0.680156\pi$
$360$	−5.63985	−0.297246
$361$	46.7617	2.46114
$362$	−19.4971	−1.02474
$363$	−0.903205	−0.0474060
$364$	0	0
$365$	3.74195	0.195863
$366$	10.1690	0.531540
$367$	−18.9552	−0.989455	−0.494728	−	0.869048i	$-0.664732\pi$
$367$	−18.9552	−0.989455	−0.494728	+	0.869048i	$0.664732\pi$
$368$	18.7549	0.977669
$369$	3.17927	0.165506
$370$	−0.627322	−0.0326129
$371$	0	0
$372$	27.4077	1.42102
$373$	0.634964	0.0328772	0.0164386	−	0.999865i	$-0.494767\pi$
$373$	0.634964	0.0328772	0.0164386	+	0.999865i	$0.494767\pi$
$374$	10.3493	0.535150
$375$	3.01785	0.155841
$376$	−35.9525	−1.85411
$377$	−40.2766	−2.07435
$378$	0	0
$379$	−3.77139	−0.193723	−0.0968617	−	0.995298i	$-0.530880\pi$
$379$	−3.77139	−0.193723	−0.0968617	+	0.995298i	$0.530880\pi$
$380$	−13.4491	−0.689925
$381$	0.311993	0.0159839
$382$	26.6716	1.36464
$383$	14.4360	0.737643	0.368822	−	0.929500i	$-0.379761\pi$
$383$	14.4360	0.737643	0.368822	+	0.929500i	$0.379761\pi$
$384$	−3.10533	−0.158468
$385$	0	0
$386$	63.5533	3.23478
$387$	−5.36631	−0.272785
$388$	33.1184	1.68133
$389$	−6.02971	−0.305719	−0.152859	−	0.988248i	$-0.548848\pi$
$389$	−6.02971	−0.305719	−0.152859	+	0.988248i	$0.548848\pi$
$390$	−4.88623	−0.247424
$391$	7.19576	0.363905
$392$	0	0
$393$	2.34379	0.118229
$394$	13.6746	0.688918
$395$	4.33479	0.218107
$396$	−10.7177	−0.538587
$397$	7.19976	0.361346	0.180673	−	0.983543i	$-0.442172\pi$
$397$	7.19976	0.361346	0.180673	+	0.983543i	$0.442172\pi$
$398$	−43.4251	−2.17670
$399$	0	0
$400$	−50.1467	−2.50733
$401$	−15.0018	−0.749152	−0.374576	−	0.927196i	$-0.622212\pi$
$401$	−15.0018	−0.749152	−0.374576	+	0.927196i	$0.622212\pi$
$402$	20.6159	1.02823
$403$	−37.6639	−1.87617
$404$	−69.5301	−3.45925
$405$	0.785307	0.0390222
$406$	0	0
$407$	−0.706234	−0.0350067
$408$	−27.1724	−1.34523
$409$	14.4208	0.713061	0.356530	−	0.934284i	$-0.383960\pi$
$409$	14.4208	0.713061	0.356530	+	0.934284i	$0.383960\pi$
$410$	−1.29292	−0.0638530
$411$	12.3016	0.606793
$412$	−26.2625	−1.29386
$413$	0	0
$414$	−10.4892	−0.515518
$415$	4.11696	0.202094
$416$	71.2281	3.49224
$417$	−7.76564	−0.380285
$418$	−21.3122	−1.04241
$419$	−0.872247	−0.0426120	−0.0213060	−	0.999773i	$-0.506782\pi$
$419$	−0.872247	−0.0426120	−0.0213060	+	0.999773i	$0.506782\pi$
$420$	0	0
$421$	5.41939	0.264125	0.132063	−	0.991241i	$-0.457840\pi$
$421$	5.41939	0.264125	0.132063	+	0.991241i	$0.457840\pi$
$422$	−28.4755	−1.38617
$423$	10.2791	0.499787
$424$	47.0315	2.28405
$425$	−19.2399	−0.933272
$426$	11.1211	0.538818
$427$	0	0
$428$	−43.0323	−2.08005
$429$	−5.50088	−0.265585
$430$	2.18234	0.105242
$431$	−25.4358	−1.22520	−0.612601	−	0.790393i	$-0.709877\pi$
$431$	−25.4358	−1.22520	−0.612601	+	0.790393i	$0.709877\pi$
$432$	48.0595	2.31226
$433$	8.34258	0.400919	0.200459	−	0.979702i	$-0.435757\pi$
$433$	8.34258	0.400919	0.200459	+	0.979702i	$0.435757\pi$
$434$	0	0
$435$	2.01880	0.0967942
$436$	52.9912	2.53782
$437$	−14.8181	−0.708846
$438$	−26.2800	−1.25571
$439$	−0.0528184	−0.00252089	−0.00126044	−	0.999999i	$-0.500401\pi$
$439$	−0.0528184	−0.00252089	−0.00126044	+	0.999999i	$0.500401\pi$
$440$	2.58209	0.123096
$441$	0	0
$442$	63.0315	2.99810
$443$	−8.24258	−0.391617	−0.195808	−	0.980642i	$-0.562733\pi$
$443$	−8.24258	−0.391617	−0.195808	+	0.980642i	$0.562733\pi$
$444$	3.12998	0.148542
$445$	0.114235	0.00541528
$446$	−18.1673	−0.860248
$447$	9.55380	0.451879
$448$	0	0
$449$	−30.4894	−1.43888	−0.719441	−	0.694553i	$-0.755602\pi$
$449$	−30.4894	−1.43888	−0.719441	+	0.694553i	$0.755602\pi$
$450$	28.0460	1.32210
$451$	−1.45556	−0.0685399
$452$	45.4566	2.13810
$453$	3.53296	0.165993
$454$	25.9025	1.21567
$455$	0	0
$456$	55.9556	2.62036
$457$	0.503010	0.0235298	0.0117649	−	0.999931i	$-0.496255\pi$
$457$	0.503010	0.0235298	0.0117649	+	0.999931i	$0.496255\pi$
$458$	49.0909	2.29387
$459$	18.4391	0.860664
$460$	3.03050	0.141298
$461$	−2.29972	−0.107109	−0.0535544	−	0.998565i	$-0.517055\pi$
$461$	−2.29972	−0.107109	−0.0535544	+	0.998565i	$0.517055\pi$
$462$	0	0
$463$	−0.601027	−0.0279321	−0.0139661	−	0.999902i	$-0.504446\pi$
$463$	−0.601027	−0.0279321	−0.0139661	+	0.999902i	$0.504446\pi$
$464$	−67.8760	−3.15107
$465$	1.88784	0.0875466
$466$	−7.44335	−0.344807
$467$	−12.9090	−0.597357	−0.298679	−	0.954354i	$-0.596546\pi$
$467$	−12.9090	−0.597357	−0.298679	+	0.954354i	$0.596546\pi$
$468$	−65.2753	−3.01735
$469$	0	0
$470$	−4.18023	−0.192820
$471$	−7.61993	−0.351108
$472$	−51.7540	−2.38217
$473$	2.45686	0.112966
$474$	−30.4435	−1.39832
$475$	39.6204	1.81791
$476$	0	0
$477$	−13.4467	−0.615680
$478$	2.91535	0.133345
$479$	12.3982	0.566486	0.283243	−	0.959048i	$-0.408590\pi$
$479$	12.3982	0.566486	0.283243	+	0.959048i	$0.408590\pi$
$480$	−3.57020	−0.162956
$481$	−4.30125	−0.196120
$482$	50.2877	2.29054
$483$	0	0
$484$	4.90690	0.223041
$485$	2.28120	0.103584
$486$	−42.4328	−1.92479
$487$	25.2264	1.14312	0.571559	−	0.820561i	$-0.306339\pi$
$487$	25.2264	1.14312	0.571559	+	0.820561i	$0.306339\pi$
$488$	−32.7280	−1.48153
$489$	−21.3881	−0.967201
$490$	0	0
$491$	−3.77515	−0.170370	−0.0851850	−	0.996365i	$-0.527148\pi$
$491$	−3.77515	−0.170370	−0.0851850	+	0.996365i	$0.527148\pi$
$492$	6.45096	0.290832
$493$	−26.0422	−1.17288
$494$	−129.800	−5.83997
$495$	−0.738239	−0.0331814
$496$	−63.4729	−2.85002
$497$	0	0
$498$	−28.9137	−1.29565
$499$	14.5058	0.649370	0.324685	−	0.945822i	$-0.394742\pi$
$499$	14.5058	0.649370	0.324685	+	0.945822i	$0.394742\pi$
$500$	−16.3952	−0.733217
$501$	3.23613	0.144580
$502$	−13.7839	−0.615205
$503$	39.9709	1.78222	0.891108	−	0.453791i	$-0.149929\pi$
$503$	39.9709	1.78222	0.891108	+	0.453791i	$0.149929\pi$
$504$	0	0
$505$	−4.78924	−0.213118
$506$	4.80228	0.213488
$507$	−21.7609	−0.966436
$508$	−1.69498	−0.0752027
$509$	34.7930	1.54217	0.771087	−	0.636730i	$-0.219713\pi$
$509$	34.7930	1.54217	0.771087	+	0.636730i	$0.219713\pi$
$510$	−3.15936	−0.139899
$511$	0	0
$512$	−36.7863	−1.62574
$513$	−37.9714	−1.67648
$514$	34.7738	1.53381
$515$	−1.80896	−0.0797125
$516$	−10.8886	−0.479345
$517$	−4.70607	−0.206973
$518$	0	0
$519$	−21.9539	−0.963669
$520$	15.7260	0.689629
$521$	−25.9277	−1.13591	−0.567956	−	0.823059i	$-0.692265\pi$
$521$	−25.9277	−1.13591	−0.567956	+	0.823059i	$0.692265\pi$
$522$	37.9616	1.66154
$523$	6.44125	0.281656	0.140828	−	0.990034i	$-0.455024\pi$
$523$	6.44125	0.281656	0.140828	+	0.990034i	$0.455024\pi$
$524$	−12.7332	−0.556254
$525$	0	0
$526$	−3.00177	−0.130883
$527$	−24.3528	−1.06083
$528$	−9.27034	−0.403440
$529$	−19.6610	−0.854827
$530$	5.46839	0.237532
$531$	14.7969	0.642129
$532$	0	0
$533$	−8.86497	−0.383985
$534$	−0.802284	−0.0347182
$535$	−2.96407	−0.128148
$536$	−66.3507	−2.86592
$537$	−10.7250	−0.462818
$538$	63.4866	2.73710
$539$	0	0
$540$	7.76564	0.334180
$541$	30.2550	1.30076	0.650382	−	0.759607i	$-0.274609\pi$
$541$	30.2550	1.30076	0.650382	+	0.759607i	$0.274609\pi$
$542$	−8.58541	−0.368775
$543$	6.70060	0.287550
$544$	46.0549	1.97459
$545$	3.65004	0.156351
$546$	0	0
$547$	−23.0018	−0.983485	−0.491742	−	0.870741i	$-0.663640\pi$
$547$	−23.0018	−0.983485	−0.491742	+	0.870741i	$0.663640\pi$
$548$	−66.8316	−2.85491
$549$	9.35718	0.399355
$550$	−12.8403	−0.547511
$551$	53.6282	2.28464
$552$	−12.6085	−0.536654
$553$	0	0
$554$	36.2885	1.54175
$555$	0.215593	0.00915143
$556$	42.1888	1.78920
$557$	−33.5794	−1.42281	−0.711403	−	0.702784i	$-0.751940\pi$
$557$	−33.5794	−1.42281	−0.711403	+	0.702784i	$0.751940\pi$
$558$	35.4991	1.50279
$559$	14.9632	0.632878
$560$	0	0
$561$	−3.55678	−0.150167
$562$	−39.2015	−1.65361
$563$	−42.4359	−1.78846	−0.894230	−	0.447608i	$-0.852276\pi$
$563$	−42.4359	−1.78846	−0.894230	+	0.447608i	$0.852276\pi$
$564$	20.8570	0.878238
$565$	3.13106	0.131725
$566$	−32.7154	−1.37513
$567$	0	0
$568$	−35.7923	−1.50181
$569$	−41.1943	−1.72696	−0.863478	−	0.504387i	$-0.831719\pi$
$569$	−41.1943	−1.72696	−0.863478	+	0.504387i	$0.831719\pi$
$570$	6.50601	0.272507
$571$	−14.9842	−0.627067	−0.313534	−	0.949577i	$-0.601513\pi$
$571$	−14.9842	−0.627067	−0.313534	+	0.949577i	$0.601513\pi$
$572$	29.8850	1.24955
$573$	−9.16630	−0.382928
$574$	0	0
$575$	−8.92769	−0.372310
$576$	−22.2972	−0.929049
$577$	8.33425	0.346959	0.173480	−	0.984837i	$-0.444499\pi$
$577$	8.33425	0.346959	0.173480	+	0.984837i	$0.444499\pi$
$578$	−3.92257	−0.163158
$579$	−21.8415	−0.907703
$580$	−10.9677	−0.455408
$581$	0	0
$582$	−16.0210	−0.664092
$583$	6.15627	0.254967
$584$	84.5802	3.49995
$585$	−4.49617	−0.185894
$586$	−74.6917	−3.08548
$587$	21.5300	0.888638	0.444319	−	0.895869i	$-0.353446\pi$
$587$	21.5300	0.888638	0.444319	+	0.895869i	$0.353446\pi$
$588$	0	0
$589$	50.1493	2.06637
$590$	−6.01748	−0.247736
$591$	−4.69960	−0.193316
$592$	−7.24867	−0.297918
$593$	1.16929	0.0480169	0.0240085	−	0.999712i	$-0.492357\pi$
$593$	1.16929	0.0480169	0.0240085	+	0.999712i	$0.492357\pi$
$594$	12.3058	0.504915
$595$	0	0
$596$	−51.9035	−2.12605
$597$	14.9240	0.610800
$598$	29.2478	1.19603
$599$	−30.6169	−1.25097	−0.625486	−	0.780235i	$-0.715099\pi$
$599$	−30.6169	−1.25097	−0.625486	+	0.780235i	$0.715099\pi$
$600$	33.7124	1.37630
$601$	−0.608559	−0.0248237	−0.0124118	−	0.999923i	$-0.503951\pi$
$601$	−0.608559	−0.0248237	−0.0124118	+	0.999923i	$0.503951\pi$
$602$	0	0
$603$	18.9702	0.772525
$604$	−19.1937	−0.780982
$605$	0.337987	0.0137411
$606$	33.6352	1.36634
$607$	−7.67100	−0.311356	−0.155678	−	0.987808i	$-0.549756\pi$
$607$	−7.67100	−0.311356	−0.155678	+	0.987808i	$0.549756\pi$
$608$	−94.8400	−3.84627
$609$	0	0
$610$	−3.80531	−0.154073
$611$	−28.6619	−1.15954
$612$	−42.2059	−1.70607
$613$	−23.4712	−0.947992	−0.473996	−	0.880527i	$-0.657189\pi$
$613$	−23.4712	−0.947992	−0.473996	+	0.880527i	$0.657189\pi$
$614$	−29.1623	−1.17690
$615$	0.444343	0.0179176
$616$	0	0
$617$	−5.37736	−0.216484	−0.108242	−	0.994125i	$-0.534522\pi$
$617$	−5.37736	−0.216484	−0.108242	+	0.994125i	$0.534522\pi$
$618$	12.7045	0.511050
$619$	41.7944	1.67986	0.839929	−	0.542697i	$-0.182597\pi$
$619$	41.7944	1.67986	0.839929	+	0.542697i	$0.182597\pi$
$620$	−10.2562	−0.411899
$621$	8.55611	0.343345
$622$	−56.6904	−2.27308
$623$	0	0
$624$	−56.4601	−2.26021
$625$	23.2995	0.931981
$626$	−31.2707	−1.24983
$627$	7.32441	0.292509
$628$	41.3972	1.65193
$629$	−2.78111	−0.110890
$630$	0	0
$631$	0.0411038	0.00163632	0.000818159	−	1.00000i	$-0.499740\pi$
$631$	0.0411038	0.00163632	0.000818159		1.00000i	$0.499740\pi$
$632$	97.9802	3.89744
$633$	9.78626	0.388969
$634$	23.5495	0.935270
$635$	−0.116751	−0.00463310
$636$	−27.2842	−1.08189
$637$	0	0
$638$	−17.3799	−0.688079
$639$	10.2333	0.404823
$640$	1.16204	0.0459337
$641$	−3.02724	−0.119569	−0.0597845	−	0.998211i	$-0.519041\pi$
$641$	−3.02724	−0.119569	−0.0597845	+	0.998211i	$0.519041\pi$
$642$	20.8169	0.821577
$643$	−20.8037	−0.820417	−0.410208	−	0.911992i	$-0.634544\pi$
$643$	−20.8037	−0.820417	−0.410208	+	0.911992i	$0.634544\pi$
$644$	0	0
$645$	−0.750009	−0.0295316
$646$	−83.9263	−3.30204
$647$	−21.1241	−0.830475	−0.415237	−	0.909713i	$-0.636302\pi$
$647$	−21.1241	−0.830475	−0.415237	+	0.909713i	$0.636302\pi$
$648$	17.7505	0.697304
$649$	−6.77443	−0.265920
$650$	−78.2024	−3.06735
$651$	0	0
$652$	116.196	4.55059
$653$	36.6457	1.43406	0.717028	−	0.697045i	$-0.245502\pi$
$653$	36.6457	1.43406	0.717028	+	0.697045i	$0.245502\pi$
$654$	−25.6345	−1.00239
$655$	−0.877067	−0.0342698
$656$	−14.9397	−0.583296
$657$	−24.1821	−0.943434
$658$	0	0
$659$	−47.5488	−1.85224	−0.926120	−	0.377230i	$-0.876877\pi$
$659$	−47.5488	−1.85224	−0.926120	+	0.377230i	$0.876877\pi$
$660$	−1.49794	−0.0583071
$661$	−28.8388	−1.12170	−0.560849	−	0.827918i	$-0.689525\pi$
$661$	−28.8388	−1.12170	−0.560849	+	0.827918i	$0.689525\pi$
$662$	79.8840	3.10478
$663$	−21.6622	−0.841290
$664$	93.0565	3.61129
$665$	0	0
$666$	4.05403	0.157090
$667$	−12.0841	−0.467897
$668$	−17.5811	−0.680233
$669$	6.24362	0.241392
$670$	−7.71466	−0.298043
$671$	−4.28399	−0.165382
$672$	0	0
$673$	−0.747654	−0.0288199	−0.0144100	−	0.999896i	$-0.504587\pi$
$673$	−0.747654	−0.0288199	−0.0144100	+	0.999896i	$0.504587\pi$
$674$	−62.1739	−2.39485
$675$	−22.8772	−0.880543
$676$	118.222	4.54699
$677$	10.0797	0.387393	0.193696	−	0.981062i	$-0.437952\pi$
$677$	10.0797	0.387393	0.193696	+	0.981062i	$0.437952\pi$
$678$	−21.9897	−0.844508
$679$	0	0
$680$	10.1681	0.389930
$681$	−8.90199	−0.341125
$682$	−16.2525	−0.622341
$683$	37.9385	1.45168	0.725838	−	0.687866i	$-0.241452\pi$
$683$	37.9385	1.45168	0.725838	+	0.687866i	$0.241452\pi$
$684$	86.9140	3.32324
$685$	−4.60337	−0.175886
$686$	0	0
$687$	−16.8712	−0.643677
$688$	25.2167	0.961380
$689$	37.4942	1.42841
$690$	−1.46600	−0.0558098
$691$	32.8262	1.24877	0.624384	−	0.781118i	$-0.285350\pi$
$691$	32.8262	1.24877	0.624384	+	0.781118i	$0.285350\pi$
$692$	119.270	4.53397
$693$	0	0
$694$	−51.8156	−1.96689
$695$	2.90597	0.110230
$696$	45.6314	1.72966
$697$	−5.73194	−0.217113
$698$	32.6023	1.23402
$699$	2.55808	0.0967553
$700$	0	0
$701$	−23.9841	−0.905865	−0.452933	−	0.891545i	$-0.649622\pi$
$701$	−23.9841	−0.905865	−0.452933	+	0.891545i	$0.649622\pi$
$702$	74.9475	2.82871
$703$	5.72710	0.216002
$704$	10.2083	0.384740
$705$	1.43663	0.0541067
$706$	40.6910	1.53142
$707$	0	0
$708$	30.0238	1.12836
$709$	26.5116	0.995663	0.497832	−	0.867274i	$-0.334130\pi$
$709$	26.5116	0.995663	0.497832	+	0.867274i	$0.334130\pi$
$710$	−4.16161	−0.156182
$711$	−28.0133	−1.05058
$712$	2.58209	0.0967679
$713$	−11.3002	−0.423195
$714$	0	0
$715$	2.05848	0.0769827
$716$	58.2663	2.17751
$717$	−1.00193	−0.0374176
$718$	−53.4046	−1.99304
$719$	−38.5807	−1.43882	−0.719409	−	0.694587i	$-0.755587\pi$
$719$	−38.5807	−1.43882	−0.719409	+	0.694587i	$0.755587\pi$
$720$	−7.57716	−0.282384
$721$	0	0
$722$	122.894	4.57365
$723$	−17.2825	−0.642744
$724$	−36.4028	−1.35290
$725$	32.3102	1.19997
$726$	−2.37371	−0.0880967
$727$	26.7675	0.992752	0.496376	−	0.868108i	$-0.334664\pi$
$727$	26.7675	0.992752	0.496376	+	0.868108i	$0.334664\pi$
$728$	0	0
$729$	7.61256	0.281946
$730$	9.83422	0.363981
$731$	9.67497	0.357842
$732$	18.9864	0.701756
$733$	−17.4234	−0.643548	−0.321774	−	0.946817i	$-0.604279\pi$
$733$	−17.4234	−0.643548	−0.321774	+	0.946817i	$0.604279\pi$
$734$	−49.8162	−1.83875
$735$	0	0
$736$	21.3704	0.787722
$737$	−8.68510	−0.319920
$738$	8.35544	0.307568
$739$	10.5453	0.387914	0.193957	−	0.981010i	$-0.437868\pi$
$739$	10.5453	0.387914	0.193957	+	0.981010i	$0.437868\pi$
$740$	−1.17127	−0.0430566
$741$	44.6086	1.63874
$742$	0	0
$743$	−8.55301	−0.313780	−0.156890	−	0.987616i	$-0.550147\pi$
$743$	−8.55301	−0.313780	−0.156890	+	0.987616i	$0.550147\pi$
$744$	42.6713	1.56441
$745$	−3.57512	−0.130982
$746$	1.66875	0.0610971
$747$	−26.6056	−0.973447
$748$	19.3231	0.706523
$749$	0	0
$750$	7.93119	0.289606
$751$	30.8901	1.12720	0.563598	−	0.826049i	$-0.309417\pi$
$751$	30.8901	1.12720	0.563598	+	0.826049i	$0.309417\pi$
$752$	−48.3023	−1.76140
$753$	4.73715	0.172631
$754$	−105.851	−3.85486
$755$	−1.32207	−0.0481149
$756$	0	0
$757$	45.3951	1.64991	0.824957	−	0.565195i	$-0.191199\pi$
$757$	45.3951	1.64991	0.824957	+	0.565195i	$0.191199\pi$
$758$	−9.91158	−0.360005
$759$	−1.65041	−0.0599062
$760$	−20.9391	−0.759540
$761$	7.93220	0.287542	0.143771	−	0.989611i	$-0.454077\pi$
$761$	7.93220	0.287542	0.143771	+	0.989611i	$0.454077\pi$
$762$	0.819948	0.0297036
$763$	0	0
$764$	49.7983	1.80164
$765$	−2.90715	−0.105108
$766$	37.9391	1.37080
$767$	−41.2590	−1.48978
$768$	10.2793	0.370921
$769$	−22.8803	−0.825086	−0.412543	−	0.910938i	$-0.635359\pi$
$769$	−22.8803	−0.825086	−0.412543	+	0.910938i	$0.635359\pi$
$770$	0	0
$771$	−11.9508	−0.430398
$772$	118.660	4.27066
$773$	47.5131	1.70893	0.854463	−	0.519511i	$-0.173886\pi$
$773$	47.5131	1.70893	0.854463	+	0.519511i	$0.173886\pi$
$774$	−14.1032	−0.506929
$775$	30.2142	1.08533
$776$	51.5624	1.85098
$777$	0	0
$778$	−15.8467	−0.568131
$779$	11.8037	0.422911
$780$	−9.12304	−0.326657
$781$	−4.68510	−0.167646
$782$	18.9112	0.676261
$783$	−30.9654	−1.10661
$784$	0	0
$785$	2.85145	0.101772
$786$	6.15971	0.219710
$787$	24.8735	0.886645	0.443323	−	0.896362i	$-0.353800\pi$
$787$	24.8735	0.886645	0.443323	+	0.896362i	$0.353800\pi$
$788$	25.5318	0.909532
$789$	1.03163	0.0367268
$790$	11.3922	0.405318
$791$	0	0
$792$	−16.6866	−0.592932
$793$	−26.0912	−0.926527
$794$	18.9217	0.671505
$795$	−1.87934	−0.0666532
$796$	−81.0786	−2.87376
$797$	51.0891	1.80967	0.904835	−	0.425763i	$-0.139994\pi$
$797$	51.0891	1.80967	0.904835	+	0.425763i	$0.139994\pi$
$798$	0	0
$799$	−18.5323	−0.655625
$800$	−57.1397	−2.02019
$801$	−0.738239	−0.0260844
$802$	−39.4261	−1.39218
$803$	11.0713	0.390697
$804$	38.4918	1.35750
$805$	0	0
$806$	−98.9843	−3.48657
$807$	−21.8186	−0.768052
$808$	−108.252	−3.80830
$809$	−7.27441	−0.255755	−0.127877	−	0.991790i	$-0.540816\pi$
$809$	−7.27441	−0.255755	−0.127877	+	0.991790i	$0.540816\pi$
$810$	2.06386	0.0725167
$811$	16.0694	0.564274	0.282137	−	0.959374i	$-0.408957\pi$
$811$	16.0694	0.564274	0.282137	+	0.959374i	$0.408957\pi$
$812$	0	0
$813$	2.95057	0.103481
$814$	−1.85605	−0.0650546
$815$	8.00360	0.280354
$816$	−36.5062	−1.27797
$817$	−19.9235	−0.697036
$818$	37.8992	1.32511
$819$	0	0
$820$	−2.41401	−0.0843008
$821$	−23.6506	−0.825411	−0.412706	−	0.910864i	$-0.635416\pi$
$821$	−23.6506	−0.825411	−0.412706	+	0.910864i	$0.635416\pi$
$822$	32.3298	1.12763
$823$	−9.58412	−0.334081	−0.167041	−	0.985950i	$-0.553421\pi$
$823$	−9.58412	−0.334081	−0.167041	+	0.985950i	$0.553421\pi$
$824$	−40.8884	−1.42442
$825$	4.41285	0.153636
$826$	0	0
$827$	55.7714	1.93936	0.969681	−	0.244375i	$-0.0785828\pi$
$827$	55.7714	1.93936	0.969681	+	0.244375i	$0.0785828\pi$
$828$	−19.5844	−0.680604
$829$	−43.1792	−1.49967	−0.749837	−	0.661622i	$-0.769868\pi$
$829$	−43.1792	−1.49967	−0.749837	+	0.661622i	$0.769868\pi$
$830$	10.8198	0.375560
$831$	−12.4714	−0.432627
$832$	62.1726	2.15545
$833$	0	0
$834$	−20.4089	−0.706701
$835$	−1.21099	−0.0419080
$836$	−39.7918	−1.37623
$837$	−28.9567	−1.00089
$838$	−2.29235	−0.0791879
$839$	−47.6252	−1.64421	−0.822103	−	0.569339i	$-0.807199\pi$
$839$	−47.6252	−1.64421	−0.822103	+	0.569339i	$0.807199\pi$
$840$	0	0
$841$	14.7335	0.508051
$842$	14.2427	0.490836
$843$	13.4725	0.464017
$844$	−53.1664	−1.83006
$845$	8.14313	0.280132
$846$	27.0145	0.928777
$847$	0	0
$848$	63.1869	2.16985
$849$	11.2434	0.385872
$850$	−50.5643	−1.73434
$851$	−1.29049	−0.0442375
$852$	20.7641	0.711365
$853$	−14.5131	−0.496918	−0.248459	−	0.968642i	$-0.579924\pi$
$853$	−14.5131	−0.496918	−0.248459	+	0.968642i	$0.579924\pi$
$854$	0	0
$855$	5.98664	0.204739
$856$	−66.9975	−2.28993
$857$	7.82656	0.267350	0.133675	−	0.991025i	$-0.457322\pi$
$857$	7.82656	0.267350	0.133675	+	0.991025i	$0.457322\pi$
$858$	−14.4569	−0.493549
$859$	−20.7714	−0.708713	−0.354356	−	0.935110i	$-0.615300\pi$
$859$	−20.7714	−0.708713	−0.354356	+	0.935110i	$0.615300\pi$
$860$	4.07462	0.138943
$861$	0	0
$862$	−66.8479	−2.27685
$863$	−11.9401	−0.406447	−0.203223	−	0.979132i	$-0.565142\pi$
$863$	−11.9401	−0.406447	−0.203223	+	0.979132i	$0.565142\pi$
$864$	54.7615	1.86302
$865$	8.21534	0.279330
$866$	21.9251	0.745046
$867$	1.34808	0.0457833
$868$	0	0
$869$	12.8253	0.435068
$870$	5.30561	0.179877
$871$	−52.8958	−1.79230
$872$	82.5027	2.79389
$873$	−14.7421	−0.498944
$874$	−38.9434	−1.31728
$875$	0	0
$876$	−49.0672	−1.65783
$877$	54.3713	1.83599	0.917993	−	0.396596i	$-0.129808\pi$
$877$	54.3713	1.83599	0.917993	+	0.396596i	$0.129808\pi$
$878$	−0.138812	−0.00468468
$879$	25.6695	0.865810
$880$	3.46904	0.116941
$881$	−47.0818	−1.58622	−0.793112	−	0.609075i	$-0.791541\pi$
$881$	−47.0818	−1.58622	−0.793112	+	0.609075i	$0.791541\pi$
$882$	0	0
$883$	40.3391	1.35752	0.678759	−	0.734361i	$-0.262518\pi$
$883$	40.3391	1.35752	0.678759	+	0.734361i	$0.262518\pi$
$884$	117.685	3.95819
$885$	2.06804	0.0695165
$886$	−21.6623	−0.727760
$887$	−31.6268	−1.06192	−0.530962	−	0.847396i	$-0.678169\pi$
$887$	−31.6268	−1.06192	−0.530962	+	0.847396i	$0.678169\pi$
$888$	4.87311	0.163531
$889$	0	0
$890$	0.300222	0.0100635
$891$	2.32348	0.0778395
$892$	−33.9201	−1.13573
$893$	38.1632	1.27708
$894$	25.1083	0.839748
$895$	4.01339	0.134153
$896$	0	0
$897$	−10.0517	−0.335616
$898$	−80.1291	−2.67394
$899$	40.8965	1.36397
$900$	52.3644	1.74548
$901$	24.2431	0.807655
$902$	−3.82536	−0.127371
$903$	0	0
$904$	70.7720	2.35384
$905$	−2.50743	−0.0833496
$906$	9.28497	0.308473
$907$	16.6244	0.552005	0.276002	−	0.961157i	$-0.410990\pi$
$907$	16.6244	0.552005	0.276002	+	0.961157i	$0.410990\pi$
$908$	48.3624	1.60496
$909$	30.9501	1.02655
$910$	0	0
$911$	35.8816	1.18881	0.594406	−	0.804165i	$-0.297387\pi$
$911$	35.8816	1.18881	0.594406	+	0.804165i	$0.297387\pi$
$912$	75.1765	2.48934
$913$	12.1808	0.403126
$914$	1.32196	0.0437265
$915$	1.30778	0.0432340
$916$	91.6572	3.02844
$917$	0	0
$918$	48.4598	1.59941
$919$	−2.75613	−0.0909164	−0.0454582	−	0.998966i	$-0.514475\pi$
$919$	−2.75613	−0.0909164	−0.0454582	+	0.998966i	$0.514475\pi$
$920$	4.71821	0.155555
$921$	10.0223	0.330246
$922$	−6.04390	−0.199045
$923$	−28.5342	−0.939213
$924$	0	0
$925$	3.45049	0.113452
$926$	−1.57956	−0.0519075
$927$	11.6903	0.383960
$928$	−77.3414	−2.53886
$929$	−2.97025	−0.0974506	−0.0487253	−	0.998812i	$-0.515516\pi$
$929$	−2.97025	−0.0974506	−0.0487253	+	0.998812i	$0.515516\pi$
$930$	4.96143	0.162692
$931$	0	0
$932$	−13.8974	−0.455225
$933$	19.4829	0.637842
$934$	−33.9261	−1.11010
$935$	1.33098	0.0435276
$936$	−101.628	−3.32181
$937$	32.2479	1.05349	0.526747	−	0.850022i	$-0.323411\pi$
$937$	32.2479	1.05349	0.526747	+	0.850022i	$0.323411\pi$
$938$	0	0
$939$	10.7469	0.350711
$940$	−7.80487	−0.254567
$941$	−48.7786	−1.59014	−0.795069	−	0.606519i	$-0.792565\pi$
$941$	−48.7786	−1.59014	−0.795069	+	0.606519i	$0.792565\pi$
$942$	−20.0259	−0.652480
$943$	−2.65973	−0.0866128
$944$	−69.5316	−2.26306
$945$	0	0
$946$	6.45686	0.209930
$947$	23.5517	0.765328	0.382664	−	0.923888i	$-0.375007\pi$
$947$	23.5517	0.765328	0.382664	+	0.923888i	$0.375007\pi$
$948$	−56.8409	−1.84611
$949$	67.4286	2.18882
$950$	104.126	3.37830
$951$	−8.09332	−0.262444
$952$	0	0
$953$	−25.5414	−0.827366	−0.413683	−	0.910421i	$-0.635758\pi$
$953$	−25.5414	−0.827366	−0.413683	+	0.910421i	$0.635758\pi$
$954$	−35.3391	−1.14415
$955$	3.43011	0.110996
$956$	5.44322	0.176046
$957$	5.97301	0.193080
$958$	32.5835	1.05273
$959$	0	0
$960$	−3.11631	−0.100578
$961$	7.24352	0.233662
$962$	−11.3041	−0.364459
$963$	19.1551	0.617264
$964$	93.8917	3.02405
$965$	8.17329	0.263107
$966$	0	0
$967$	13.4875	0.433728	0.216864	−	0.976202i	$-0.430417\pi$
$967$	13.4875	0.433728	0.216864	+	0.976202i	$0.430417\pi$
$968$	7.63960	0.245546
$969$	28.8432	0.926576
$970$	5.99520	0.192494
$971$	35.2268	1.13048	0.565240	−	0.824926i	$-0.308783\pi$
$971$	35.2268	1.13048	0.565240	+	0.824926i	$0.308783\pi$
$972$	−79.2259	−2.54117
$973$	0	0
$974$	66.2974	2.12431
$975$	26.8760	0.860721
$976$	−43.9702	−1.40745
$977$	−16.6842	−0.533773	−0.266887	−	0.963728i	$-0.585995\pi$
$977$	−16.6842	−0.533773	−0.266887	+	0.963728i	$0.585995\pi$
$978$	−56.2099	−1.79739
$979$	0.337987	0.0108021
$980$	0	0
$981$	−23.5881	−0.753111
$982$	−9.92145	−0.316606
$983$	30.9021	0.985623	0.492812	−	0.870136i	$-0.335969\pi$
$983$	30.9021	0.985623	0.492812	+	0.870136i	$0.335969\pi$
$984$	10.0436	0.320178
$985$	1.75863	0.0560347
$986$	−68.4414	−2.17962
$987$	0	0
$988$	−242.348	−7.71011
$989$	4.48938	0.142754
$990$	−1.94016	−0.0616624
$991$	42.7014	1.35646	0.678228	−	0.734852i	$-0.262748\pi$
$991$	42.7014	1.35646	0.678228	+	0.734852i	$0.262748\pi$
$992$	−72.3243	−2.29630
$993$	−27.4540	−0.871225
$994$	0	0
$995$	−5.58470	−0.177047
$996$	−53.9845	−1.71056
$997$	30.1349	0.954380	0.477190	−	0.878800i	$-0.341655\pi$
$997$	30.1349	0.954380	0.477190	+	0.878800i	$0.341655\pi$
$998$	38.1227	1.20675
$999$	−3.30688	−0.104625

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.a.l.1.9	✓	10
3.2	odd	2		4851.2.a.cg.1.1		10
4.3	odd	2		8624.2.a.df.1.6		10
7.2	even	3		539.2.e.o.67.2		20
7.3	odd	6		539.2.e.o.177.1		20
7.4	even	3		539.2.e.o.177.2		20
7.5	odd	6		539.2.e.o.67.1		20
7.6	odd	2	inner	539.2.a.l.1.10	yes	10
11.10	odd	2		5929.2.a.bv.1.1		10
21.20	even	2		4851.2.a.cg.1.2		10
28.27	even	2		8624.2.a.df.1.5		10
77.76	even	2		5929.2.a.bv.1.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.a.l.1.9	✓	10	1.1	even	1	trivial
539.2.a.l.1.10	yes	10	7.6	odd	2	inner
539.2.e.o.67.1		20	7.5	odd	6
539.2.e.o.67.2		20	7.2	even	3
539.2.e.o.177.1		20	7.3	odd	6
539.2.e.o.177.2		20	7.4	even	3
4851.2.a.cg.1.1		10	3.2	odd	2
4851.2.a.cg.1.2		10	21.20	even	2
5929.2.a.bv.1.1		10	11.10	odd	2
5929.2.a.bv.1.2		10	77.76	even	2
8624.2.a.df.1.5		10	28.27	even	2
8624.2.a.df.1.6		10	4.3	odd	2

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:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

\(f(q)\)	\(=\)	\(q+2.62810 q^{2} -0.903205 q^{3} +4.90690 q^{4} +0.337987 q^{5} -2.37371 q^{6} +7.63960 q^{8} -2.18422 q^{9} +O(q^{10})\)	\(q+2.62810 q^{2} -0.903205 q^{3} +4.90690 q^{4} +0.337987 q^{5} -2.37371 q^{6} +7.63960 q^{8} -2.18422 q^{9} +0.888264 q^{10} +1.00000 q^{11} -4.43193 q^{12} +6.09040 q^{13} -0.305272 q^{15} +10.2638 q^{16} +3.93795 q^{17} -5.74034 q^{18} -8.10936 q^{19} +1.65847 q^{20} +2.62810 q^{22} +1.82729 q^{23} -6.90013 q^{24} -4.88576 q^{25} +16.0062 q^{26} +4.68241 q^{27} -6.61313 q^{29} -0.802284 q^{30} -6.18413 q^{31} +11.6951 q^{32} -0.903205 q^{33} +10.3493 q^{34} -10.7177 q^{36} -0.706234 q^{37} -21.3122 q^{38} -5.50088 q^{39} +2.58209 q^{40} -1.45556 q^{41} +2.45686 q^{43} +4.90690 q^{44} -0.738239 q^{45} +4.80228 q^{46} -4.70607 q^{47} -9.27034 q^{48} -12.8403 q^{50} -3.55678 q^{51} +29.8850 q^{52} +6.15627 q^{53} +12.3058 q^{54} +0.337987 q^{55} +7.32441 q^{57} -17.3799 q^{58} -6.77443 q^{59} -1.49794 q^{60} -4.28399 q^{61} -16.2525 q^{62} +10.2083 q^{64} +2.05848 q^{65} -2.37371 q^{66} -8.68510 q^{67} +19.3231 q^{68} -1.65041 q^{69} -4.68510 q^{71} -16.6866 q^{72} +11.0713 q^{73} -1.85605 q^{74} +4.41285 q^{75} -39.7918 q^{76} -14.4569 q^{78} +12.8253 q^{79} +3.46904 q^{80} +2.32348 q^{81} -3.82536 q^{82} +12.1808 q^{83} +1.33098 q^{85} +6.45686 q^{86} +5.97301 q^{87} +7.63960 q^{88} +0.337987 q^{89} -1.94016 q^{90} +8.96630 q^{92} +5.58554 q^{93} -12.3680 q^{94} -2.74086 q^{95} -10.5631 q^{96} +6.74935 q^{97} -2.18422 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9}+O(q^{10}) \) 10 * q + 2 * q^2 + 18 * q^4 - 6 * q^8 + 22 * q^9	\( 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9} + 10 q^{11} + 8 q^{15} + 42 q^{16} + 6 q^{18} + 2 q^{22} + 4 q^{23} + 18 q^{25} + 12 q^{29} - 4 q^{30} - 30 q^{32} - 2 q^{36} + 40 q^{37} - 16 q^{39} - 8 q^{43} + 18 q^{44} + 44 q^{46} - 62 q^{50} + 16 q^{53} - 8 q^{57} - 28 q^{58} + 36 q^{60} + 106 q^{64} - 32 q^{65} - 4 q^{67} + 36 q^{71} - 90 q^{72} - 28 q^{74} - 112 q^{78} + 8 q^{79} - 6 q^{81} + 88 q^{85} + 32 q^{86} - 6 q^{88} - 52 q^{92} + 44 q^{93} - 64 q^{95} + 22 q^{99}+O(q^{100}) \) 10 * q + 2 * q^2 + 18 * q^4 - 6 * q^8 + 22 * q^9 + 10 * q^11 + 8 * q^15 + 42 * q^16 + 6 * q^18 + 2 * q^22 + 4 * q^23 + 18 * q^25 + 12 * q^29 - 4 * q^30 - 30 * q^32 - 2 * q^36 + 40 * q^37 - 16 * q^39 - 8 * q^43 + 18 * q^44 + 44 * q^46 - 62 * q^50 + 16 * q^53 - 8 * q^57 - 28 * q^58 + 36 * q^60 + 106 * q^64 - 32 * q^65 - 4 * q^67 + 36 * q^71 - 90 * q^72 - 28 * q^74 - 112 * q^78 + 8 * q^79 - 6 * q^81 + 88 * q^85 + 32 * q^86 - 6 * q^88 - 52 * q^92 + 44 * q^93 - 64 * q^95 + 22 * q^99

Introduction

L-functions

Modular forms

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Database

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