LMFDB - Embedded newform 539.2.a.l.1.1

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			1.1
Root			$-2.10267$ 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)

$f(q)$	$=$	$q-2.74816 q^{2} -2.10267 q^{3} +5.55241 q^{4} -2.44342 q^{5} +5.77848 q^{6} -9.76260 q^{8} +1.42122 q^{9} +O(q^{10})$	\(q-2.74816 q^{2} -2.10267 q^{3} +5.55241 q^{4} -2.44342 q^{5} +5.77848 q^{6} -9.76260 q^{8} +1.42122 q^{9} +6.71492 q^{10} +1.00000 q^{11} -11.6749 q^{12} -1.89199 q^{13} +5.13770 q^{15} +15.7244 q^{16} -3.95040 q^{17} -3.90573 q^{18} -2.88614 q^{19} -13.5669 q^{20} -2.74816 q^{22} -6.59322 q^{23} +20.5275 q^{24} +0.970299 q^{25} +5.19949 q^{26} +3.31966 q^{27} -0.675675 q^{29} -14.1192 q^{30} -7.81724 q^{31} -23.6881 q^{32} -2.10267 q^{33} +10.8563 q^{34} +7.89117 q^{36} +5.87673 q^{37} +7.93160 q^{38} +3.97822 q^{39} +23.8541 q^{40} +3.26890 q^{41} -1.06719 q^{43} +5.55241 q^{44} -3.47263 q^{45} +18.1192 q^{46} +9.70227 q^{47} -33.0632 q^{48} -2.66654 q^{50} +8.30638 q^{51} -10.5051 q^{52} +3.74286 q^{53} -9.12297 q^{54} -2.44342 q^{55} +6.06860 q^{57} +1.85686 q^{58} +6.48233 q^{59} +28.5266 q^{60} +6.09732 q^{61} +21.4831 q^{62} +33.6499 q^{64} +4.62292 q^{65} +5.77848 q^{66} +4.39944 q^{67} -21.9342 q^{68} +13.8634 q^{69} +8.39944 q^{71} -13.8748 q^{72} -2.66200 q^{73} -16.1502 q^{74} -2.04022 q^{75} -16.0250 q^{76} -10.9328 q^{78} +2.09038 q^{79} -38.4213 q^{80} -11.2438 q^{81} -8.98347 q^{82} -3.78398 q^{83} +9.65248 q^{85} +2.93281 q^{86} +1.42072 q^{87} -9.76260 q^{88} -2.44342 q^{89} +9.54335 q^{90} -36.6082 q^{92} +16.4371 q^{93} -26.6634 q^{94} +7.05206 q^{95} +49.8082 q^{96} +12.3633 q^{97} +1.42122 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.74816	−1.94325	−0.971623	−	0.236536i	$-0.923988\pi$
$2$	−2.74816	−1.94325	−0.971623	+	0.236536i	$0.923988\pi$
$3$	−2.10267	−1.21398	−0.606988	−	0.794711i	$-0.707622\pi$
$3$	−2.10267	−1.21398	−0.606988	+	0.794711i	$0.707622\pi$
$4$	5.55241	2.77620
$5$	−2.44342	−1.09273	−0.546365	−	0.837547i	$-0.683989\pi$
$5$	−2.44342	−1.09273	−0.546365	+	0.837547i	$0.683989\pi$
$6$	5.77848	2.35905
$7$	0	0
$7$	0	0
$8$	−9.76260	−3.45160
$9$	1.42122	0.473738
$10$	6.71492	2.12344
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−11.6749	−3.37025
$13$	−1.89199	−0.524743	−0.262371	−	0.964967i	$-0.584505\pi$
$13$	−1.89199	−0.524743	−0.262371	+	0.964967i	$0.584505\pi$
$14$	0	0
$15$	5.13770	1.32655
$16$	15.7244	3.93110
$17$	−3.95040	−0.958113	−0.479056	−	0.877784i	$-0.659021\pi$
$17$	−3.95040	−0.958113	−0.479056	+	0.877784i	$0.659021\pi$
$18$	−3.90573	−0.920590
$19$	−2.88614	−0.662127	−0.331063	−	0.943609i	$-0.607407\pi$
$19$	−2.88614	−0.662127	−0.331063	+	0.943609i	$0.607407\pi$
$20$	−13.5669	−3.03364
$21$	0	0
$22$	−2.74816	−0.585911
$23$	−6.59322	−1.37478	−0.687391	−	0.726288i	$-0.741244\pi$
$23$	−6.59322	−1.37478	−0.687391	+	0.726288i	$0.741244\pi$
$24$	20.5275	4.19016
$25$	0.970299	0.194060
$26$	5.19949	1.01970
$27$	3.31966	0.638869
$28$	0	0
$29$	−0.675675	−0.125470	−0.0627348	−	0.998030i	$-0.519982\pi$
$29$	−0.675675	−0.125470	−0.0627348	+	0.998030i	$0.519982\pi$
$30$	−14.1192	−2.57781
$31$	−7.81724	−1.40402	−0.702009	−	0.712168i	$-0.747713\pi$
$31$	−7.81724	−1.40402	−0.702009	+	0.712168i	$0.747713\pi$
$32$	−23.6881	−4.18750
$33$	−2.10267	−0.366028
$34$	10.8563	1.86185
$35$	0	0
$36$	7.89117	1.31519
$37$	5.87673	0.966129	0.483064	−	0.875585i	$-0.339524\pi$
$37$	5.87673	0.966129	0.483064	+	0.875585i	$0.339524\pi$
$38$	7.93160	1.28667
$39$	3.97822	0.637026
$40$	23.8541	3.77167
$41$	3.26890	0.510516	0.255258	−	0.966873i	$-0.417840\pi$
$41$	3.26890	0.510516	0.255258	+	0.966873i	$0.417840\pi$
$42$	0	0
$43$	−1.06719	−0.162745	−0.0813724	−	0.996684i	$-0.525930\pi$
$43$	−1.06719	−0.162745	−0.0813724	+	0.996684i	$0.525930\pi$
$44$	5.55241	0.837057
$45$	−3.47263	−0.517668
$46$	18.1192	2.67154
$47$	9.70227	1.41522	0.707611	−	0.706602i	$-0.249773\pi$
$47$	9.70227	1.41522	0.707611	+	0.706602i	$0.249773\pi$
$48$	−33.0632	−4.77227
$49$	0	0
$50$	−2.66654	−0.377106
$51$	8.30638	1.16313
$52$	−10.5051	−1.45679
$53$	3.74286	0.514122	0.257061	−	0.966395i	$-0.417246\pi$
$53$	3.74286	0.514122	0.257061	+	0.966395i	$0.417246\pi$
$54$	−9.12297	−1.24148
$55$	−2.44342	−0.329471
$56$	0	0
$57$	6.06860	0.803806
$58$	1.85686	0.243818
$59$	6.48233	0.843928	0.421964	−	0.906613i	$-0.361341\pi$
$59$	6.48233	0.843928	0.421964	+	0.906613i	$0.361341\pi$
$60$	28.5266	3.68277
$61$	6.09732	0.780682	0.390341	−	0.920670i	$-0.372357\pi$
$61$	6.09732	0.780682	0.390341	+	0.920670i	$0.372357\pi$
$62$	21.4831	2.72835
$63$	0	0
$64$	33.6499	4.20623
$65$	4.62292	0.573403
$66$	5.77848	0.711282
$67$	4.39944	0.537477	0.268738	−	0.963213i	$-0.413393\pi$
$67$	4.39944	0.537477	0.268738	+	0.963213i	$0.413393\pi$
$68$	−21.9342	−2.65992
$69$	13.8634	1.66895
$70$	0	0
$71$	8.39944	0.996830	0.498415	−	0.866939i	$-0.333916\pi$
$71$	8.39944	0.996830	0.498415	+	0.866939i	$0.333916\pi$
$72$	−13.8748	−1.63516
$73$	−2.66200	−0.311564	−0.155782	−	0.987791i	$-0.549790\pi$
$73$	−2.66200	−0.311564	−0.155782	+	0.987791i	$0.549790\pi$
$74$	−16.1502	−1.87743
$75$	−2.04022	−0.235584
$76$	−16.0250	−1.83820
$77$	0	0
$78$	−10.9328	−1.23790
$79$	2.09038	0.235186	0.117593	−	0.993062i	$-0.462482\pi$
$79$	2.09038	0.235186	0.117593	+	0.993062i	$0.462482\pi$
$80$	−38.4213	−4.29564
$81$	−11.2438	−1.24931
$82$	−8.98347	−0.992058
$83$	−3.78398	−0.415345	−0.207673	−	0.978198i	$-0.566589\pi$
$83$	−3.78398	−0.415345	−0.207673	+	0.978198i	$0.566589\pi$
$84$	0	0
$85$	9.65248	1.04696
$86$	2.93281	0.316253
$87$	1.42072	0.152317
$88$	−9.76260	−1.04070
$89$	−2.44342	−0.259002	−0.129501	−	0.991579i	$-0.541338\pi$
$89$	−2.44342	−0.259002	−0.129501	+	0.991579i	$0.541338\pi$
$90$	9.54335	1.00596
$91$	0	0
$92$	−36.6082	−3.81667
$93$	16.4371	1.70444
$94$	−26.6634	−2.75012
$95$	7.05206	0.723526
$96$	49.8082	5.08352
$97$	12.3633	1.25531	0.627653	−	0.778493i	$-0.284016\pi$
$97$	12.3633	1.25531	0.627653	+	0.778493i	$0.284016\pi$
$98$	0	0
$99$	1.42122	0.142838
$100$	5.38750	0.538750
$101$	−10.2187	−1.01679	−0.508397	−	0.861123i	$-0.669762\pi$
$101$	−10.2187	−1.01679	−0.508397	+	0.861123i	$0.669762\pi$
$102$	−22.8273	−2.26024
$103$	5.36461	0.528591	0.264296	−	0.964442i	$-0.414861\pi$
$103$	5.36461	0.528591	0.264296	+	0.964442i	$0.414861\pi$
$104$	18.4707	1.81120
$105$	0	0
$106$	−10.2860	−0.999065
$107$	−16.0158	−1.54831	−0.774155	−	0.632996i	$-0.781825\pi$
$107$	−16.0158	−1.54831	−0.774155	+	0.632996i	$0.781825\pi$
$108$	18.4321	1.77363
$109$	3.57086	0.342026	0.171013	−	0.985269i	$-0.445296\pi$
$109$	3.57086	0.342026	0.171013	+	0.985269i	$0.445296\pi$
$110$	6.71492	0.640242
$111$	−12.3568	−1.17286
$112$	0	0
$113$	14.7244	1.38516	0.692578	−	0.721343i	$-0.256475\pi$
$113$	14.7244	1.38516	0.692578	+	0.721343i	$0.256475\pi$
$114$	−16.6775	−1.56199
$115$	16.1100	1.50227
$116$	−3.75162	−0.348329
$117$	−2.68892	−0.248591
$118$	−17.8145	−1.63996
$119$	0	0
$120$	−50.1573	−4.57872
$121$	1.00000	0.0909091
$122$	−16.7565	−1.51706
$123$	−6.87341	−0.619754
$124$	−43.4045	−3.89784
$125$	9.84625	0.880675
$126$	0	0
$127$	−17.1864	−1.52505	−0.762525	−	0.646959i	$-0.776041\pi$
$127$	−17.1864	−1.52505	−0.762525	+	0.646959i	$0.776041\pi$
$128$	−45.0993	−3.98625
$129$	2.24394	0.197568
$130$	−12.7045	−1.11426
$131$	13.6154	1.18958	0.594790	−	0.803881i	$-0.297235\pi$
$131$	13.6154	1.18958	0.594790	+	0.803881i	$0.297235\pi$
$132$	−11.6749	−1.01617
$133$	0	0
$134$	−12.0904	−1.04445
$135$	−8.11132	−0.698112
$136$	38.5662	3.30702
$137$	0.0111106	0.000949243 0	0.000474622	−	1.00000i	$-0.499849\pi$
$137$	0.0111106	0.000949243 0	0.000474622		1.00000i	$0.499849\pi$
$138$	−38.0988	−3.24318
$139$	−21.4191	−1.81675	−0.908374	−	0.418158i	$-0.862676\pi$
$139$	−21.4191	−1.81675	−0.908374	+	0.418158i	$0.862676\pi$
$140$	0	0
$141$	−20.4007	−1.71805
$142$	−23.0830	−1.93709
$143$	−1.89199	−0.158216
$144$	22.3478	1.86231
$145$	1.65096	0.137104
$146$	7.31562	0.605445
$147$	0	0
$148$	32.6300	2.68217
$149$	21.9761	1.80035	0.900177	−	0.435525i	$-0.143437\pi$
$149$	21.9761	1.80035	0.900177	+	0.435525i	$0.143437\pi$
$150$	5.60685	0.457798
$151$	−0.224758	−0.0182906	−0.00914528	−	0.999958i	$-0.502911\pi$
$151$	−0.224758	−0.0182906	−0.00914528	+	0.999958i	$0.502911\pi$
$152$	28.1763	2.28540
$153$	−5.61437	−0.453895
$154$	0	0
$155$	19.1008	1.53421
$156$	22.0887	1.76851
$157$	23.3823	1.86611	0.933056	−	0.359732i	$-0.117132\pi$
$157$	23.3823	1.86611	0.933056	+	0.359732i	$0.117132\pi$
$158$	−5.74471	−0.457024
$159$	−7.87000	−0.624132
$160$	57.8799	4.57581
$161$	0	0
$162$	30.8998	2.42772
$163$	−12.2642	−0.960608	−0.480304	−	0.877102i	$-0.659474\pi$
$163$	−12.2642	−0.960608	−0.480304	+	0.877102i	$0.659474\pi$
$164$	18.1503	1.41730
$165$	5.13770	0.399970
$166$	10.3990	0.807118
$167$	−17.6352	−1.36465	−0.682325	−	0.731049i	$-0.739031\pi$
$167$	−17.6352	−1.36465	−0.682325	+	0.731049i	$0.739031\pi$
$168$	0	0
$169$	−9.42038	−0.724645
$170$	−26.5266	−2.03450
$171$	−4.10183	−0.313675
$172$	−5.92547	−0.451813
$173$	20.5022	1.55875	0.779377	−	0.626555i	$-0.215536\pi$
$173$	20.5022	1.55875	0.779377	+	0.626555i	$0.215536\pi$
$174$	−3.90437	−0.295990
$175$	0	0
$176$	15.7244	1.18527
$177$	−13.6302	−1.02451
$178$	6.71492	0.503304
$179$	6.44901	0.482021	0.241011	−	0.970522i	$-0.422521\pi$
$179$	6.44901	0.482021	0.241011	+	0.970522i	$0.422521\pi$
$180$	−19.2814	−1.43715
$181$	−1.35627	−0.100811	−0.0504053	−	0.998729i	$-0.516051\pi$
$181$	−1.35627	−0.100811	−0.0504053	+	0.998729i	$0.516051\pi$
$182$	0	0
$183$	−12.8207	−0.947730
$184$	64.3669	4.74519
$185$	−14.3593	−1.05572
$186$	−45.1718	−3.31215
$187$	−3.95040	−0.288882
$188$	53.8710	3.92894
$189$	0	0
$190$	−19.3802	−1.40599
$191$	−8.47788	−0.613438	−0.306719	−	0.951800i	$-0.599231\pi$
$191$	−8.47788	−0.613438	−0.306719	+	0.951800i	$0.599231\pi$
$192$	−70.7545	−5.10627
$193$	18.2624	1.31456	0.657278	−	0.753649i	$-0.271708\pi$
$193$	18.2624	1.31456	0.657278	+	0.753649i	$0.271708\pi$
$194$	−33.9765	−2.43937
$195$	−9.72047	−0.696097
$196$	0	0
$197$	17.3802	1.23829	0.619145	−	0.785277i	$-0.287479\pi$
$197$	17.3802	1.23829	0.619145	+	0.785277i	$0.287479\pi$
$198$	−3.90573	−0.277568
$199$	−5.48005	−0.388470	−0.194235	−	0.980955i	$-0.562222\pi$
$199$	−5.48005	−0.388470	−0.194235	+	0.980955i	$0.562222\pi$
$200$	−9.47264	−0.669817
$201$	−9.25056	−0.652484
$202$	28.0825	1.97588
$203$	0	0
$204$	46.1204	3.22907
$205$	−7.98729	−0.557856
$206$	−14.7428	−1.02718
$207$	−9.37038	−0.651287
$208$	−29.7504	−2.06282
$209$	−2.88614	−0.199639
$210$	0	0
$211$	−7.03239	−0.484130	−0.242065	−	0.970260i	$-0.577825\pi$
$211$	−7.03239	−0.484130	−0.242065	+	0.970260i	$0.577825\pi$
$212$	20.7819	1.42731
$213$	−17.6612	−1.21013
$214$	44.0142	3.00875
$215$	2.60759	0.177836
$216$	−32.4085	−2.20512
$217$	0	0
$218$	−9.81331	−0.664641
$219$	5.59731	0.378231
$220$	−13.5669	−0.914678
$221$	7.47411	0.502763
$222$	33.9586	2.27915
$223$	−0.419109	−0.0280656	−0.0140328	−	0.999902i	$-0.504467\pi$
$223$	−0.419109	−0.0280656	−0.0140328	+	0.999902i	$0.504467\pi$
$224$	0	0
$225$	1.37900	0.0919336
$226$	−40.4651	−2.69170
$227$	25.5405	1.69518	0.847591	−	0.530651i	$-0.178052\pi$
$227$	25.5405	1.69518	0.847591	+	0.530651i	$0.178052\pi$
$228$	33.6954	2.23153
$229$	14.6118	0.965574	0.482787	−	0.875738i	$-0.339625\pi$
$229$	14.6118	0.965574	0.482787	+	0.875738i	$0.339625\pi$
$230$	−44.2729	−2.91927
$231$	0	0
$232$	6.59634	0.433071
$233$	−17.2716	−1.13150	−0.565749	−	0.824577i	$-0.691413\pi$
$233$	−17.2716	−1.13150	−0.565749	+	0.824577i	$0.691413\pi$
$234$	7.38960	0.483073
$235$	−23.7067	−1.54646
$236$	35.9925	2.34292
$237$	−4.39538	−0.285510
$238$	0	0
$239$	−15.8945	−1.02813	−0.514065	−	0.857752i	$-0.671861\pi$
$239$	−15.8945	−1.02813	−0.514065	+	0.857752i	$0.671861\pi$
$240$	80.7873	5.21480
$241$	8.85565	0.570443	0.285221	−	0.958462i	$-0.407933\pi$
$241$	8.85565	0.570443	0.285221	+	0.958462i	$0.407933\pi$
$242$	−2.74816	−0.176659
$243$	13.6830	0.877764
$244$	33.8548	2.16733
$245$	0	0
$246$	18.8893	1.20434
$247$	5.46055	0.347446
$248$	76.3166	4.84611
$249$	7.95645	0.504219
$250$	−27.0591	−1.71137
$251$	−11.1416	−0.703254	−0.351627	−	0.936140i	$-0.614371\pi$
$251$	−11.1416	−0.703254	−0.351627	+	0.936140i	$0.614371\pi$
$252$	0	0
$253$	−6.59322	−0.414512
$254$	47.2312	2.96355
$255$	−20.2960	−1.27098
$256$	56.6404	3.54003
$257$	13.7681	0.858831	0.429415	−	0.903107i	$-0.358720\pi$
$257$	13.7681	0.858831	0.429415	+	0.903107i	$0.358720\pi$
$258$	−6.16673	−0.383924
$259$	0	0
$260$	25.6683	1.59188
$261$	−0.960279	−0.0594398
$262$	−37.4173	−2.31165
$263$	−5.80622	−0.358027	−0.179013	−	0.983847i	$-0.557291\pi$
$263$	−5.80622	−0.358027	−0.179013	+	0.983847i	$0.557291\pi$
$264$	20.5275	1.26338
$265$	−9.14539	−0.561797
$266$	0	0
$267$	5.13770	0.314422
$268$	24.4275	1.49215
$269$	6.23095	0.379908	0.189954	−	0.981793i	$-0.439166\pi$
$269$	6.23095	0.379908	0.189954	+	0.981793i	$0.439166\pi$
$270$	22.2913	1.35660
$271$	−10.5576	−0.641329	−0.320664	−	0.947193i	$-0.603906\pi$
$271$	−10.5576	−0.641329	−0.320664	+	0.947193i	$0.603906\pi$
$272$	−62.1177	−3.76644
$273$	0	0
$274$	−0.0305338	−0.00184461
$275$	0.970299	0.0585112
$276$	76.9750	4.63335
$277$	−25.9920	−1.56171	−0.780853	−	0.624715i	$-0.785215\pi$
$277$	−25.9920	−1.56171	−0.780853	+	0.624715i	$0.785215\pi$
$278$	58.8633	3.53039
$279$	−11.1100	−0.665137
$280$	0	0
$281$	−12.8167	−0.764580	−0.382290	−	0.924042i	$-0.624864\pi$
$281$	−12.8167	−0.764580	−0.382290	+	0.924042i	$0.624864\pi$
$282$	56.0644	3.33859
$283$	31.2813	1.85948	0.929741	−	0.368213i	$-0.120030\pi$
$283$	31.2813	1.85948	0.929741	+	0.368213i	$0.120030\pi$
$284$	46.6371	2.76740
$285$	−14.8281	−0.878343
$286$	5.19949	0.307452
$287$	0	0
$288$	−33.6658	−1.98378
$289$	−1.39434	−0.0820202
$290$	−4.53710	−0.266428
$291$	−25.9960	−1.52391
$292$	−14.7805	−0.864965
$293$	−12.2025	−0.712875	−0.356438	−	0.934319i	$-0.616009\pi$
$293$	−12.2025	−0.712875	−0.356438	+	0.934319i	$0.616009\pi$
$294$	0	0
$295$	−15.8391	−0.922186
$296$	−57.3722	−3.33469
$297$	3.31966	0.192626
$298$	−60.3940	−3.49853
$299$	12.4743	0.721407
$300$	−11.3281	−0.654029
$301$	0	0
$302$	0.617673	0.0355431
$303$	21.4864	1.23436
$304$	−45.3829	−2.60289
$305$	−14.8983	−0.853075
$306$	15.4292	0.882029
$307$	21.5077	1.22751	0.613754	−	0.789498i	$-0.289659\pi$
$307$	21.5077	1.22751	0.613754	+	0.789498i	$0.289659\pi$
$308$	0	0
$309$	−11.2800	−0.641697
$310$	−52.4921	−2.98135
$311$	−0.407674	−0.0231171	−0.0115585	−	0.999933i	$-0.503679\pi$
$311$	−0.407674	−0.0231171	−0.0115585	+	0.999933i	$0.503679\pi$
$312$	−38.8378	−2.19876
$313$	−17.2967	−0.977666	−0.488833	−	0.872377i	$-0.662577\pi$
$313$	−17.2967	−0.977666	−0.488833	+	0.872377i	$0.662577\pi$
$314$	−64.2584	−3.62631
$315$	0	0
$316$	11.6066	0.652925
$317$	10.5834	0.594422	0.297211	−	0.954812i	$-0.403943\pi$
$317$	10.5834	0.594422	0.297211	+	0.954812i	$0.403943\pi$
$318$	21.6281	1.21284
$319$	−0.675675	−0.0378305
$320$	−82.2208	−4.59628
$321$	33.6760	1.87961
$322$	0	0
$323$	11.4014	0.634392
$324$	−62.4301	−3.46834
$325$	−1.83579	−0.101832
$326$	33.7041	1.86670
$327$	−7.50834	−0.415212
$328$	−31.9129	−1.76210
$329$	0	0
$330$	−14.1192	−0.777239
$331$	−5.19895	−0.285760	−0.142880	−	0.989740i	$-0.545636\pi$
$331$	−5.19895	−0.285760	−0.142880	+	0.989740i	$0.545636\pi$
$332$	−21.0102	−1.15308
$333$	8.35210	0.457692
$334$	48.4644	2.65185
$335$	−10.7497	−0.587317
$336$	0	0
$337$	11.4256	0.622393	0.311196	−	0.950346i	$-0.399270\pi$
$337$	11.4256	0.622393	0.311196	+	0.950346i	$0.399270\pi$
$338$	25.8888	1.40816
$339$	−30.9606	−1.68155
$340$	53.5945	2.90657
$341$	−7.81724	−0.423327
$342$	11.2725	0.609547
$343$	0	0
$344$	10.4185	0.561730
$345$	−33.8740	−1.82371
$346$	−56.3434	−3.02904
$347$	−25.9849	−1.39494	−0.697470	−	0.716614i	$-0.745691\pi$
$347$	−25.9849	−1.39494	−0.697470	+	0.716614i	$0.745691\pi$
$348$	7.88842	0.422863
$349$	26.4416	1.41539	0.707694	−	0.706519i	$-0.249736\pi$
$349$	26.4416	1.41539	0.707694	+	0.706519i	$0.249736\pi$
$350$	0	0
$351$	−6.28076	−0.335242
$352$	−23.6881	−1.26258
$353$	−37.3344	−1.98711	−0.993555	−	0.113351i	$-0.963842\pi$
$353$	−37.3344	−1.98711	−0.993555	+	0.113351i	$0.963842\pi$
$354$	37.4580	1.99087
$355$	−20.5234	−1.08927
$356$	−13.5669	−0.719042
$357$	0	0
$358$	−17.7229	−0.936686
$359$	25.3873	1.33989	0.669945	−	0.742410i	$-0.266317\pi$
$359$	25.3873	1.33989	0.669945	+	0.742410i	$0.266317\pi$
$360$	33.9018	1.78678
$361$	−10.6702	−0.561588
$362$	3.72725	0.195900
$363$	−2.10267	−0.110361
$364$	0	0
$365$	6.50439	0.340455
$366$	35.2333	1.84167
$367$	10.2663	0.535897	0.267948	−	0.963433i	$-0.413654\pi$
$367$	10.2663	0.535897	0.267948	+	0.963433i	$0.413654\pi$
$368$	−103.674	−5.40441
$369$	4.64581	0.241851
$370$	39.4618	2.05152
$371$	0	0
$372$	91.2653	4.73188
$373$	36.7010	1.90031	0.950154	−	0.311782i	$-0.100926\pi$
$373$	36.7010	1.90031	0.950154	+	0.311782i	$0.100926\pi$
$374$	10.8563	0.561368
$375$	−20.7034	−1.06912
$376$	−94.7194	−4.88478
$377$	1.27837	0.0658393
$378$	0	0
$379$	2.26506	0.116348	0.0581742	−	0.998306i	$-0.481472\pi$
$379$	2.26506	0.116348	0.0581742	+	0.998306i	$0.481472\pi$
$380$	39.1559	2.00866
$381$	36.1374	1.85137
$382$	23.2986	1.19206
$383$	−11.4550	−0.585322	−0.292661	−	0.956216i	$-0.594541\pi$
$383$	−11.4550	−0.585322	−0.292661	+	0.956216i	$0.594541\pi$
$384$	94.8288	4.83921
$385$	0	0
$386$	−50.1880	−2.55450
$387$	−1.51671	−0.0770985
$388$	68.6462	3.48498
$389$	14.1205	0.715939	0.357970	−	0.933733i	$-0.383469\pi$
$389$	14.1205	0.715939	0.357970	+	0.933733i	$0.383469\pi$
$390$	26.7134	1.35269
$391$	26.0458	1.31720
$392$	0	0
$393$	−28.6286	−1.44412
$394$	−47.7637	−2.40630
$395$	−5.10768	−0.256995
$396$	7.89117	0.396546
$397$	−19.4119	−0.974258	−0.487129	−	0.873330i	$-0.661956\pi$
$397$	−19.4119	−0.974258	−0.487129	+	0.873330i	$0.661956\pi$
$398$	15.0601	0.754893
$399$	0	0
$400$	15.2574	0.762869
$401$	3.95645	0.197576	0.0987878	−	0.995109i	$-0.468504\pi$
$401$	3.95645	0.197576	0.0987878	+	0.995109i	$0.468504\pi$
$402$	25.4221	1.26794
$403$	14.7901	0.736748
$404$	−56.7381	−2.82283
$405$	27.4733	1.36516
$406$	0	0
$407$	5.87673	0.291299
$408$	−81.0919	−4.01465
$409$	4.18623	0.206996	0.103498	−	0.994630i	$-0.466997\pi$
$409$	4.18623	0.206996	0.103498	+	0.994630i	$0.466997\pi$
$410$	21.9504	1.08405
$411$	−0.0233619	−0.00115236
$412$	29.7865	1.46748
$413$	0	0
$414$	25.7514	1.26561
$415$	9.24584	0.453860
$416$	44.8175	2.19736
$417$	45.0374	2.20549
$418$	7.93160	0.387947
$419$	21.3050	1.04082	0.520409	−	0.853917i	$-0.325779\pi$
$419$	21.3050	1.04082	0.520409	+	0.853917i	$0.325779\pi$
$420$	0	0
$421$	17.4277	0.849375	0.424688	−	0.905340i	$-0.360384\pi$
$421$	17.4277	0.849375	0.424688	+	0.905340i	$0.360384\pi$
$422$	19.3262	0.940783
$423$	13.7890	0.670445
$424$	−36.5401	−1.77454
$425$	−3.83307	−0.185931
$426$	48.5360	2.35158
$427$	0	0
$428$	−88.9265	−4.29843
$429$	3.97822	0.192070
$430$	−7.16609	−0.345579
$431$	−3.81498	−0.183761	−0.0918805	−	0.995770i	$-0.529288\pi$
$431$	−3.81498	−0.183761	−0.0918805	+	0.995770i	$0.529288\pi$
$432$	52.1997	2.51146
$433$	−22.6851	−1.09018	−0.545088	−	0.838379i	$-0.683504\pi$
$433$	−22.6851	−1.09018	−0.545088	+	0.838379i	$0.683504\pi$
$434$	0	0
$435$	−3.47141	−0.166442
$436$	19.8269	0.949535
$437$	19.0290	0.910279
$438$	−15.3823	−0.734996
$439$	−5.51513	−0.263223	−0.131611	−	0.991301i	$-0.542015\pi$
$439$	−5.51513	−0.263223	−0.131611	+	0.991301i	$0.542015\pi$
$440$	23.8541	1.13720
$441$	0	0
$442$	−20.5401	−0.976992
$443$	−18.7968	−0.893064	−0.446532	−	0.894768i	$-0.647341\pi$
$443$	−18.7968	−0.893064	−0.446532	+	0.894768i	$0.647341\pi$
$444$	−68.6101	−3.25609
$445$	5.97030	0.283019
$446$	1.15178	0.0545384
$447$	−46.2085	−2.18559
$448$	0	0
$449$	11.4269	0.539268	0.269634	−	0.962963i	$-0.413097\pi$
$449$	11.4269	0.539268	0.269634	+	0.962963i	$0.413097\pi$
$450$	−3.78973	−0.178650
$451$	3.26890	0.153926
$452$	81.7559	3.84548
$453$	0.472592	0.0222043
$454$	−70.1894	−3.29415
$455$	0	0
$456$	−59.2453	−2.77442
$457$	−12.3374	−0.577117	−0.288559	−	0.957462i	$-0.593176\pi$
$457$	−12.3374	−0.577117	−0.288559	+	0.957462i	$0.593176\pi$
$458$	−40.1556	−1.87635
$459$	−13.1140	−0.612108
$460$	89.4493	4.17059
$461$	10.4788	0.488046	0.244023	−	0.969769i	$-0.421533\pi$
$461$	10.4788	0.488046	0.244023	+	0.969769i	$0.421533\pi$
$462$	0	0
$463$	−4.05543	−0.188472	−0.0942360	−	0.995550i	$-0.530041\pi$
$463$	−4.05543	−0.188472	−0.0942360	+	0.995550i	$0.530041\pi$
$464$	−10.6246	−0.493234
$465$	−40.1626	−1.86250
$466$	47.4651	2.19878
$467$	13.5049	0.624932	0.312466	−	0.949929i	$-0.398845\pi$
$467$	13.5049	0.624932	0.312466	+	0.949929i	$0.398845\pi$
$468$	−14.9300	−0.690139
$469$	0	0
$470$	65.1500	3.00514
$471$	−49.1653	−2.26542
$472$	−63.2844	−2.91290
$473$	−1.06719	−0.0490694
$474$	12.0792	0.554817
$475$	−2.80042	−0.128492
$476$	0	0
$477$	5.31942	0.243559
$478$	43.6807	1.99791
$479$	6.40998	0.292879	0.146440	−	0.989220i	$-0.453219\pi$
$479$	6.40998	0.292879	0.146440	+	0.989220i	$0.453219\pi$
$480$	−121.702	−5.55492
$481$	−11.1187	−0.506969
$482$	−24.3368	−1.10851
$483$	0	0
$484$	5.55241	0.252382
$485$	−30.2088	−1.37171
$486$	−37.6031	−1.70571
$487$	7.67581	0.347824	0.173912	−	0.984761i	$-0.444359\pi$
$487$	7.67581	0.347824	0.173912	+	0.984761i	$0.444359\pi$
$488$	−59.5257	−2.69460
$489$	25.7876	1.16616
$490$	0	0
$491$	18.9051	0.853175	0.426588	−	0.904446i	$-0.359716\pi$
$491$	18.9051	0.853175	0.426588	+	0.904446i	$0.359716\pi$
$492$	−38.1640	−1.72056
$493$	2.66918	0.120214
$494$	−15.0065	−0.675174
$495$	−3.47263	−0.156083
$496$	−122.921	−5.51934
$497$	0	0
$498$	−21.8656	−0.979822
$499$	−16.5765	−0.742068	−0.371034	−	0.928619i	$-0.620997\pi$
$499$	−16.5765	−0.742068	−0.371034	+	0.928619i	$0.620997\pi$
$500$	54.6704	2.44493
$501$	37.0809	1.65665
$502$	30.6190	1.36659
$503$	−1.21118	−0.0540039	−0.0270020	−	0.999635i	$-0.508596\pi$
$503$	−1.21118	−0.0540039	−0.0270020	+	0.999635i	$0.508596\pi$
$504$	0	0
$505$	24.9685	1.11108
$506$	18.1192	0.805499
$507$	19.8079	0.879702
$508$	−95.4261	−4.23385
$509$	−10.7702	−0.477380	−0.238690	−	0.971096i	$-0.576718\pi$
$509$	−10.7702	−0.477380	−0.238690	+	0.971096i	$0.576718\pi$
$510$	55.7767	2.46983
$511$	0	0
$512$	−65.4587	−2.89289
$513$	−9.58102	−0.423012
$514$	−37.8370	−1.66892
$515$	−13.1080	−0.577608
$516$	12.4593	0.548490
$517$	9.70227	0.426706
$518$	0	0
$519$	−43.1094	−1.89229
$520$	−45.1317	−1.97916
$521$	−38.1463	−1.67122	−0.835611	−	0.549322i	$-0.814886\pi$
$521$	−38.1463	−1.67122	−0.835611	+	0.549322i	$0.814886\pi$
$522$	2.63901	0.115506
$523$	5.58225	0.244095	0.122047	−	0.992524i	$-0.461054\pi$
$523$	5.58225	0.244095	0.122047	+	0.992524i	$0.461054\pi$
$524$	75.5981	3.30252
$525$	0	0
$526$	15.9564	0.695734
$527$	30.8812	1.34521
$528$	−33.0632	−1.43889
$529$	20.4705	0.890023
$530$	25.1330	1.09171
$531$	9.21279	0.399801
$532$	0	0
$533$	−6.18471	−0.267890
$534$	−14.1192	−0.611000
$535$	39.1334	1.69189
$536$	−42.9500	−1.85516
$537$	−13.5601	−0.585163
$538$	−17.1237	−0.738254
$539$	0	0
$540$	−45.0374	−1.93810
$541$	−20.0012	−0.859917	−0.429959	−	0.902849i	$-0.641472\pi$
$541$	−20.0012	−0.859917	−0.429959	+	0.902849i	$0.641472\pi$
$542$	29.0140	1.24626
$543$	2.85178	0.122382
$544$	93.5773	4.01209
$545$	−8.72511	−0.373743
$546$	0	0
$547$	−4.04355	−0.172890	−0.0864449	−	0.996257i	$-0.527551\pi$
$547$	−4.04355	−0.172890	−0.0864449	+	0.996257i	$0.527551\pi$
$548$	0.0616906	0.00263529
$549$	8.66561	0.369839
$550$	−2.66654	−0.113702
$551$	1.95009	0.0830768
$552$	−135.342	−5.76055
$553$	0	0
$554$	71.4302	3.03478
$555$	30.1929	1.28162
$556$	−118.928	−5.04366
$557$	17.9326	0.759827	0.379913	−	0.925022i	$-0.375954\pi$
$557$	17.9326	0.759827	0.379913	+	0.925022i	$0.375954\pi$
$558$	30.5321	1.29252
$559$	2.01911	0.0853992
$560$	0	0
$561$	8.30638	0.350696
$562$	35.2224	1.48577
$563$	−14.9267	−0.629085	−0.314542	−	0.949243i	$-0.601851\pi$
$563$	−14.9267	−0.629085	−0.314542	+	0.949243i	$0.601851\pi$
$564$	−113.273	−4.76965
$565$	−35.9779	−1.51360
$566$	−85.9663	−3.61343
$567$	0	0
$568$	−82.0003	−3.44066
$569$	35.2133	1.47622	0.738110	−	0.674681i	$-0.235719\pi$
$569$	35.2133	1.47622	0.738110	+	0.674681i	$0.235719\pi$
$570$	40.7502	1.70684
$571$	−6.78432	−0.283915	−0.141958	−	0.989873i	$-0.545340\pi$
$571$	−6.78432	−0.283915	−0.141958	+	0.989873i	$0.545340\pi$
$572$	−10.5051	−0.439240
$573$	17.8262	0.744699
$574$	0	0
$575$	−6.39739	−0.266790
$576$	47.8237	1.99266
$577$	35.4345	1.47516	0.737579	−	0.675261i	$-0.235969\pi$
$577$	35.4345	1.47516	0.737579	+	0.675261i	$0.235969\pi$
$578$	3.83188	0.159385
$579$	−38.3997	−1.59584
$580$	9.16678	0.380630
$581$	0	0
$582$	71.4412	2.96133
$583$	3.74286	0.155014
$584$	25.9881	1.07539
$585$	6.57016	0.271643
$586$	33.5344	1.38529
$587$	24.5453	1.01309	0.506547	−	0.862212i	$-0.330922\pi$
$587$	24.5453	1.01309	0.506547	+	0.862212i	$0.330922\pi$
$588$	0	0
$589$	22.5617	0.929638
$590$	43.5283	1.79203
$591$	−36.5448	−1.50325
$592$	92.4082	3.79795
$593$	−19.5710	−0.803683	−0.401842	−	0.915709i	$-0.631630\pi$
$593$	−19.5710	−0.803683	−0.401842	+	0.915709i	$0.631630\pi$
$594$	−9.12297	−0.374320
$595$	0	0
$596$	122.020	4.99815
$597$	11.5227	0.471594
$598$	−34.2814	−1.40187
$599$	36.4275	1.48839	0.744193	−	0.667965i	$-0.232834\pi$
$599$	36.4275	1.48839	0.744193	+	0.667965i	$0.232834\pi$
$600$	19.9178	0.813142
$601$	−15.5885	−0.635866	−0.317933	−	0.948113i	$-0.602989\pi$
$601$	−15.5885	−0.635866	−0.317933	+	0.948113i	$0.602989\pi$
$602$	0	0
$603$	6.25255	0.254624
$604$	−1.24795	−0.0507783
$605$	−2.44342	−0.0993391
$606$	−59.0483	−2.39867
$607$	10.4999	0.426177	0.213089	−	0.977033i	$-0.431648\pi$
$607$	10.4999	0.426177	0.213089	+	0.977033i	$0.431648\pi$
$608$	68.3672	2.77265
$609$	0	0
$610$	40.9430	1.65774
$611$	−18.3566	−0.742628
$612$	−31.1733	−1.26010
$613$	10.3208	0.416853	0.208427	−	0.978038i	$-0.433166\pi$
$613$	10.3208	0.416853	0.208427	+	0.978038i	$0.433166\pi$
$614$	−59.1066	−2.38535
$615$	16.7946	0.677225
$616$	0	0
$617$	18.8079	0.757179	0.378590	−	0.925565i	$-0.376409\pi$
$617$	18.8079	0.757179	0.378590	+	0.925565i	$0.376409\pi$
$618$	30.9993	1.24697
$619$	−16.1480	−0.649043	−0.324521	−	0.945878i	$-0.605203\pi$
$619$	−16.1480	−0.649043	−0.324521	+	0.945878i	$0.605203\pi$
$620$	106.055	4.25929
$621$	−21.8872	−0.878305
$622$	1.12035	0.0449221
$623$	0	0
$624$	62.5552	2.50421
$625$	−28.9100	−1.15640
$626$	47.5341	1.89985
$627$	6.06860	0.242357
$628$	129.828	5.18071
$629$	−23.2154	−0.925660
$630$	0	0
$631$	−20.5398	−0.817678	−0.408839	−	0.912607i	$-0.634066\pi$
$631$	−20.5398	−0.817678	−0.408839	+	0.912607i	$0.634066\pi$
$632$	−20.4075	−0.811768
$633$	14.7868	0.587722
$634$	−29.0849	−1.15511
$635$	41.9937	1.66647
$636$	−43.6975	−1.73272
$637$	0	0
$638$	1.85686	0.0735140
$639$	11.9374	0.472237
$640$	110.196	4.35590
$641$	33.6667	1.32976	0.664878	−	0.746952i	$-0.268484\pi$
$641$	33.6667	1.32976	0.664878	+	0.746952i	$0.268484\pi$
$642$	−92.5472	−3.65255
$643$	−22.3937	−0.883123	−0.441562	−	0.897231i	$-0.645575\pi$
$643$	−22.3937	−0.883123	−0.441562	+	0.897231i	$0.645575\pi$
$644$	0	0
$645$	−5.48290	−0.215889
$646$	−31.3330	−1.23278
$647$	−25.1811	−0.989970	−0.494985	−	0.868902i	$-0.664826\pi$
$647$	−25.1811	−0.989970	−0.494985	+	0.868902i	$0.664826\pi$
$648$	109.769	4.31212
$649$	6.48233	0.254454
$650$	5.04506	0.197884
$651$	0	0
$652$	−68.0960	−2.66684
$653$	−7.68403	−0.300699	−0.150350	−	0.988633i	$-0.548040\pi$
$653$	−7.68403	−0.300699	−0.150350	+	0.988633i	$0.548040\pi$
$654$	20.6341	0.806859
$655$	−33.2681	−1.29989
$656$	51.4015	2.00689
$657$	−3.78328	−0.147600
$658$	0	0
$659$	−26.4280	−1.02949	−0.514745	−	0.857344i	$-0.672113\pi$
$659$	−26.4280	−1.02949	−0.514745	+	0.857344i	$0.672113\pi$
$660$	28.5266	1.11040
$661$	−31.6085	−1.22943	−0.614715	−	0.788749i	$-0.710729\pi$
$661$	−31.6085	−1.22943	−0.614715	+	0.788749i	$0.710729\pi$
$662$	14.2876	0.555302
$663$	−15.7156	−0.610342
$664$	36.9414	1.43361
$665$	0	0
$666$	−22.9530	−0.889409
$667$	4.45487	0.172493
$668$	−97.9177	−3.78855
$669$	0.881247	0.0340710
$670$	29.5419	1.14130
$671$	6.09732	0.235385
$672$	0	0
$673$	22.6584	0.873418	0.436709	−	0.899603i	$-0.356144\pi$
$673$	22.6584	0.873418	0.436709	+	0.899603i	$0.356144\pi$
$674$	−31.3995	−1.20946
$675$	3.22106	0.123979
$676$	−52.3058	−2.01176
$677$	−26.9107	−1.03426	−0.517130	−	0.855907i	$-0.673000\pi$
$677$	−26.9107	−1.03426	−0.517130	+	0.855907i	$0.673000\pi$
$678$	85.0847	3.26766
$679$	0	0
$680$	−94.2333	−3.61368
$681$	−53.7032	−2.05791
$682$	21.4831	0.822629
$683$	−12.1197	−0.463747	−0.231874	−	0.972746i	$-0.574486\pi$
$683$	−12.1197	−0.463747	−0.231874	+	0.972746i	$0.574486\pi$
$684$	−22.7750	−0.870825
$685$	−0.0271479	−0.00103727
$686$	0	0
$687$	−30.7237	−1.17218
$688$	−16.7809	−0.639767
$689$	−7.08145	−0.269782
$690$	93.0913	3.54392
$691$	10.0908	0.383872	0.191936	−	0.981407i	$-0.438523\pi$
$691$	10.0908	0.383872	0.191936	+	0.981407i	$0.438523\pi$
$692$	113.837	4.32742
$693$	0	0
$694$	71.4107	2.71071
$695$	52.3360	1.98522
$696$	−13.8699	−0.525738
$697$	−12.9135	−0.489132
$698$	−72.6659	−2.75045
$699$	36.3164	1.37361
$700$	0	0
$701$	−23.0714	−0.871395	−0.435698	−	0.900093i	$-0.643498\pi$
$701$	−23.0714	−0.871395	−0.435698	+	0.900093i	$0.643498\pi$
$702$	17.2606	0.651458
$703$	−16.9611	−0.639700
$704$	33.6499	1.26823
$705$	49.8474	1.87736
$706$	102.601	3.86144
$707$	0	0
$708$	−75.6804	−2.84424
$709$	42.1958	1.58470	0.792348	−	0.610070i	$-0.208859\pi$
$709$	42.1958	1.58470	0.792348	+	0.610070i	$0.208859\pi$
$710$	56.4015	2.11671
$711$	2.97088	0.111417
$712$	23.8541	0.893971
$713$	51.5408	1.93022
$714$	0	0
$715$	4.62292	0.172887
$716$	35.8075	1.33819
$717$	33.4208	1.24812
$718$	−69.7685	−2.60374
$719$	51.2748	1.91223	0.956114	−	0.292995i	$-0.0946519\pi$
$719$	51.2748	1.91223	0.956114	+	0.292995i	$0.0946519\pi$
$720$	−54.6050	−2.03501
$721$	0	0
$722$	29.3234	1.09130
$723$	−18.6205	−0.692504
$724$	−7.53056	−0.279871
$725$	−0.655606	−0.0243486
$726$	5.77848	0.214459
$727$	−0.742649	−0.0275433	−0.0137717	−	0.999905i	$-0.504384\pi$
$727$	−0.742649	−0.0275433	−0.0137717	+	0.999905i	$0.504384\pi$
$728$	0	0
$729$	4.96059	0.183725
$730$	−17.8751	−0.661588
$731$	4.21582	0.155928
$732$	−71.1855	−2.63109
$733$	−8.53582	−0.315278	−0.157639	−	0.987497i	$-0.550388\pi$
$733$	−8.53582	−0.315278	−0.157639	+	0.987497i	$0.550388\pi$
$734$	−28.2135	−1.04138
$735$	0	0
$736$	156.181	5.75689
$737$	4.39944	0.162055
$738$	−12.7674	−0.469976
$739$	10.7081	0.393902	0.196951	−	0.980413i	$-0.436896\pi$
$739$	10.7081	0.393902	0.196951	+	0.980413i	$0.436896\pi$
$740$	−79.7288	−2.93089
$741$	−11.4817	−0.421792
$742$	0	0
$743$	47.7623	1.75223	0.876114	−	0.482103i	$-0.160127\pi$
$743$	47.7623	1.75223	0.876114	+	0.482103i	$0.160127\pi$
$744$	−160.468	−5.88306
$745$	−53.6969	−1.96730
$746$	−100.860	−3.69276
$747$	−5.37784	−0.196765
$748$	−21.9342	−0.801995
$749$	0	0
$750$	56.8963	2.07756
$751$	11.0243	0.402284	0.201142	−	0.979562i	$-0.435535\pi$
$751$	11.0243	0.402284	0.201142	+	0.979562i	$0.435535\pi$
$752$	152.563	5.56338
$753$	23.4272	0.853733
$754$	−3.51317	−0.127942
$755$	0.549179	0.0199867
$756$	0	0
$757$	31.6362	1.14984	0.574919	−	0.818210i	$-0.305034\pi$
$757$	31.6362	1.14984	0.574919	+	0.818210i	$0.305034\pi$
$758$	−6.22476	−0.226093
$759$	13.8634	0.503208
$760$	−68.8464	−2.49732
$761$	24.6478	0.893484	0.446742	−	0.894663i	$-0.352584\pi$
$761$	24.6478	0.893484	0.446742	+	0.894663i	$0.352584\pi$
$762$	−99.3115	−3.59767
$763$	0	0
$764$	−47.0726	−1.70303
$765$	13.7183	0.495985
$766$	31.4802	1.13743
$767$	−12.2645	−0.442845
$768$	−119.096	−4.29751
$769$	24.1344	0.870307	0.435154	−	0.900356i	$-0.356694\pi$
$769$	24.1344	0.870307	0.435154	+	0.900356i	$0.356694\pi$
$770$	0	0
$771$	−28.9498	−1.04260
$772$	101.400	3.64947
$773$	−15.7682	−0.567142	−0.283571	−	0.958951i	$-0.591519\pi$
$773$	−15.7682	−0.567142	−0.283571	+	0.958951i	$0.591519\pi$
$774$	4.16816	0.149821
$775$	−7.58506	−0.272463
$776$	−120.698	−4.33281
$777$	0	0
$778$	−38.8055	−1.39125
$779$	−9.43451	−0.338026
$780$	−53.9720	−1.93251
$781$	8.39944	0.300556
$782$	−71.5783	−2.55963
$783$	−2.24301	−0.0801587
$784$	0	0
$785$	−57.1328	−2.03916
$786$	78.6761	2.80628
$787$	55.5828	1.98131	0.990656	−	0.136387i	$-0.0435491\pi$
$787$	55.5828	1.98131	0.990656	+	0.136387i	$0.0435491\pi$
$788$	96.5020	3.43774
$789$	12.2086	0.434636
$790$	14.0367	0.499405
$791$	0	0
$792$	−13.8748	−0.493018
$793$	−11.5361	−0.409658
$794$	53.3472	1.89322
$795$	19.2297	0.682008
$796$	−30.4274	−1.07847
$797$	−32.2398	−1.14199	−0.570997	−	0.820952i	$-0.693443\pi$
$797$	−32.2398	−1.14199	−0.570997	+	0.820952i	$0.693443\pi$
$798$	0	0
$799$	−38.3279	−1.35594
$800$	−22.9845	−0.812625
$801$	−3.47263	−0.122699
$802$	−10.8730	−0.383938
$803$	−2.66200	−0.0939400
$804$	−51.3629	−1.81143
$805$	0	0
$806$	−40.6457	−1.43168
$807$	−13.1016	−0.461199
$808$	99.7606	3.50957
$809$	−49.4935	−1.74010	−0.870050	−	0.492963i	$-0.835914\pi$
$809$	−49.4935	−1.74010	−0.870050	+	0.492963i	$0.835914\pi$
$810$	−75.5012	−2.65284
$811$	13.7840	0.484022	0.242011	−	0.970273i	$-0.422193\pi$
$811$	13.7840	0.484022	0.242011	+	0.970273i	$0.422193\pi$
$812$	0	0
$813$	22.1991	0.778558
$814$	−16.1502	−0.566065
$815$	29.9667	1.04969
$816$	130.613	4.57237
$817$	3.08006	0.107758
$818$	−11.5044	−0.402243
$819$	0	0
$820$	−44.3487	−1.54872
$821$	−2.18076	−0.0761091	−0.0380545	−	0.999276i	$-0.512116\pi$
$821$	−2.18076	−0.0761091	−0.0380545	+	0.999276i	$0.512116\pi$
$822$	0.0642024	0.00223932
$823$	46.2602	1.61253	0.806265	−	0.591555i	$-0.201486\pi$
$823$	46.2602	1.61253	0.806265	+	0.591555i	$0.201486\pi$
$824$	−52.3726	−1.82448
$825$	−2.04022	−0.0710313
$826$	0	0
$827$	5.27184	0.183320	0.0916599	−	0.995790i	$-0.470783\pi$
$827$	5.27184	0.183320	0.0916599	+	0.995790i	$0.470783\pi$
$828$	−52.0282	−1.80810
$829$	−13.9453	−0.484339	−0.242170	−	0.970234i	$-0.577859\pi$
$829$	−13.9453	−0.484339	−0.242170	+	0.970234i	$0.577859\pi$
$830$	−25.4091	−0.881962
$831$	54.6525	1.89587
$832$	−63.6652	−2.20719
$833$	0	0
$834$	−123.770	−4.28581
$835$	43.0901	1.49120
$836$	−16.0250	−0.554238
$837$	−25.9506	−0.896983
$838$	−58.5497	−2.02257
$839$	22.6533	0.782077	0.391039	−	0.920374i	$-0.372116\pi$
$839$	22.6533	0.782077	0.391039	+	0.920374i	$0.372116\pi$
$840$	0	0
$841$	−28.5435	−0.984257
$842$	−47.8943	−1.65054
$843$	26.9493	0.928182
$844$	−39.0467	−1.34404
$845$	23.0179	0.791841
$846$	−37.8945	−1.30284
$847$	0	0
$848$	58.8543	2.02107
$849$	−65.7743	−2.25737
$850$	10.5339	0.361310
$851$	−38.7466	−1.32822
$852$	−98.0624	−3.35956
$853$	−19.7192	−0.675172	−0.337586	−	0.941295i	$-0.609610\pi$
$853$	−19.7192	−0.675172	−0.337586	+	0.941295i	$0.609610\pi$
$854$	0	0
$855$	10.0225	0.342762
$856$	156.356	5.34415
$857$	13.6176	0.465168	0.232584	−	0.972576i	$-0.425282\pi$
$857$	13.6176	0.465168	0.232584	+	0.972576i	$0.425282\pi$
$858$	−10.9328	−0.373240
$859$	−21.2078	−0.723600	−0.361800	−	0.932256i	$-0.617838\pi$
$859$	−21.2078	−0.723600	−0.361800	+	0.932256i	$0.617838\pi$
$860$	14.4784	0.493710
$861$	0	0
$862$	10.4842	0.357093
$863$	−9.69536	−0.330034	−0.165017	−	0.986291i	$-0.552768\pi$
$863$	−9.69536	−0.330034	−0.165017	+	0.986291i	$0.552768\pi$
$864$	−78.6363	−2.67526
$865$	−50.0955	−1.70330
$866$	62.3424	2.11848
$867$	2.93184	0.0995705
$868$	0	0
$869$	2.09038	0.0709113
$870$	9.54002	0.323437
$871$	−8.32368	−0.282037
$872$	−34.8609	−1.18054
$873$	17.5710	0.594687
$874$	−52.2947	−1.76890
$875$	0	0
$876$	31.0785	1.05005
$877$	28.9184	0.976505	0.488252	−	0.872702i	$-0.337635\pi$
$877$	28.9184	0.976505	0.488252	+	0.872702i	$0.337635\pi$
$878$	15.1565	0.511507
$879$	25.6577	0.865414
$880$	−38.4213	−1.29518
$881$	−29.7222	−1.00137	−0.500683	−	0.865631i	$-0.666918\pi$
$881$	−29.7222	−1.00137	−0.500683	+	0.865631i	$0.666918\pi$
$882$	0	0
$883$	−26.4560	−0.890316	−0.445158	−	0.895452i	$-0.646853\pi$
$883$	−26.4560	−0.890316	−0.445158	+	0.895452i	$0.646853\pi$
$884$	41.4993	1.39577
$885$	33.3043	1.11951
$886$	51.6568	1.73544
$887$	−2.52809	−0.0848850	−0.0424425	−	0.999099i	$-0.513514\pi$
$887$	−2.52809	−0.0848850	−0.0424425	+	0.999099i	$0.513514\pi$
$888$	120.635	4.04823
$889$	0	0
$890$	−16.4074	−0.549976
$891$	−11.2438	−0.376681
$892$	−2.32706	−0.0779158
$893$	−28.0022	−0.937056
$894$	126.989	4.24713
$895$	−15.7576	−0.526719
$896$	0	0
$897$	−26.2293	−0.875771
$898$	−31.4030	−1.04793
$899$	5.28191	0.176162
$900$	7.65679	0.255226
$901$	−14.7858	−0.492587
$902$	−8.98347	−0.299117
$903$	0	0
$904$	−143.749	−4.78100
$905$	3.31393	0.110159
$906$	−1.29876	−0.0431484
$907$	38.4844	1.27785	0.638926	−	0.769268i	$-0.279379\pi$
$907$	38.4844	1.27785	0.638926	+	0.769268i	$0.279379\pi$
$908$	141.811	4.70617
$909$	−14.5229	−0.481695
$910$	0	0
$911$	31.0724	1.02948	0.514738	−	0.857348i	$-0.327889\pi$
$911$	31.0724	1.02948	0.514738	+	0.857348i	$0.327889\pi$
$912$	95.4252	3.15984
$913$	−3.78398	−0.125231
$914$	33.9051	1.12148
$915$	31.3262	1.03561
$916$	81.1305	2.68063
$917$	0	0
$918$	36.0394	1.18948
$919$	−25.3894	−0.837520	−0.418760	−	0.908097i	$-0.637535\pi$
$919$	−25.3894	−0.837520	−0.418760	+	0.908097i	$0.637535\pi$
$920$	−157.275	−5.18522
$921$	−45.2235	−1.49016
$922$	−28.7975	−0.948394
$923$	−15.8916	−0.523080
$924$	0	0
$925$	5.70219	0.187487
$926$	11.1450	0.366247
$927$	7.62427	0.250414
$928$	16.0054	0.525404
$929$	13.2344	0.434206	0.217103	−	0.976149i	$-0.430339\pi$
$929$	13.2344	0.434206	0.217103	+	0.976149i	$0.430339\pi$
$930$	110.374	3.61929
$931$	0	0
$932$	−95.8988	−3.14127
$933$	0.857203	0.0280636
$934$	−37.1137	−1.21440
$935$	9.65248	0.315670
$936$	26.2509	0.858036
$937$	−12.3994	−0.405069	−0.202535	−	0.979275i	$-0.564918\pi$
$937$	−12.3994	−0.405069	−0.202535	+	0.979275i	$0.564918\pi$
$938$	0	0
$939$	36.3692	1.18686
$940$	−131.629	−4.29328
$941$	−29.5103	−0.962009	−0.481004	−	0.876718i	$-0.659728\pi$
$941$	−29.5103	−0.962009	−0.481004	+	0.876718i	$0.659728\pi$
$942$	135.114	4.40226
$943$	−21.5526	−0.701848
$944$	101.931	3.31757
$945$	0	0
$946$	2.93281	0.0953539
$947$	0.423958	0.0137768	0.00688838	−	0.999976i	$-0.497807\pi$
$947$	0.423958	0.0137768	0.00688838	+	0.999976i	$0.497807\pi$
$948$	−24.4049	−0.792635
$949$	5.03648	0.163491
$950$	7.69602	0.249692
$951$	−22.2534	−0.721614
$952$	0	0
$953$	57.5354	1.86376	0.931878	−	0.362773i	$-0.118170\pi$
$953$	57.5354	1.86376	0.931878	+	0.362773i	$0.118170\pi$
$954$	−14.6186	−0.473296
$955$	20.7150	0.670322
$956$	−88.2527	−2.85430
$957$	1.42072	0.0459254
$958$	−17.6157	−0.569137
$959$	0	0
$960$	172.883	5.57978
$961$	30.1092	0.971265
$962$	30.5560	0.985166
$963$	−22.7620	−0.733494
$964$	49.1702	1.58366
$965$	−44.6227	−1.43645
$966$	0	0
$967$	−3.79491	−0.122036	−0.0610180	−	0.998137i	$-0.519435\pi$
$967$	−3.79491	−0.122036	−0.0610180	+	0.998137i	$0.519435\pi$
$968$	−9.76260	−0.313782
$969$	−23.9734	−0.770137
$970$	83.0187	2.66557
$971$	1.02111	0.0327690	0.0163845	−	0.999866i	$-0.494784\pi$
$971$	1.02111	0.0327690	0.0163845	+	0.999866i	$0.494784\pi$
$972$	75.9735	2.43685
$973$	0	0
$974$	−21.0944	−0.675908
$975$	3.86007	0.123621
$976$	95.8768	3.06894
$977$	−46.4645	−1.48653	−0.743266	−	0.668996i	$-0.766724\pi$
$977$	−46.4645	−1.48653	−0.743266	+	0.668996i	$0.766724\pi$
$978$	−70.8686	−2.26613
$979$	−2.44342	−0.0780920
$980$	0	0
$981$	5.07496	0.162031
$982$	−51.9543	−1.65793
$983$	−2.77930	−0.0886459	−0.0443229	−	0.999017i	$-0.514113\pi$
$983$	−2.77930	−0.0886459	−0.0443229	+	0.999017i	$0.514113\pi$
$984$	67.1023	2.13914
$985$	−42.4672	−1.35312
$986$	−7.33536	−0.233605
$987$	0	0
$988$	30.3192	0.964582
$989$	7.03621	0.223738
$990$	9.54335	0.303307
$991$	1.66335	0.0528381	0.0264190	−	0.999651i	$-0.491590\pi$
$991$	1.66335	0.0528381	0.0264190	+	0.999651i	$0.491590\pi$
$992$	185.175	5.87932
$993$	10.9317	0.346906
$994$	0	0
$995$	13.3901	0.424493
$996$	44.1774	1.39982
$997$	33.6708	1.06637	0.533183	−	0.846000i	$-0.320996\pi$
$997$	33.6708	1.06637	0.533183	+	0.846000i	$0.320996\pi$
$998$	45.5550	1.44202
$999$	19.5088	0.617230

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.a.l.1.1	✓	10
3.2	odd	2		4851.2.a.cg.1.10		10
4.3	odd	2		8624.2.a.df.1.7		10
7.2	even	3		539.2.e.o.67.10		20
7.3	odd	6		539.2.e.o.177.9		20
7.4	even	3		539.2.e.o.177.10		20
7.5	odd	6		539.2.e.o.67.9		20
7.6	odd	2	inner	539.2.a.l.1.2	yes	10
11.10	odd	2		5929.2.a.bv.1.9		10
21.20	even	2		4851.2.a.cg.1.9		10
28.27	even	2		8624.2.a.df.1.4		10
77.76	even	2		5929.2.a.bv.1.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.a.l.1.1	✓	10	1.1	even	1	trivial
539.2.a.l.1.2	yes	10	7.6	odd	2	inner
539.2.e.o.67.9		20	7.5	odd	6
539.2.e.o.67.10		20	7.2	even	3
539.2.e.o.177.9		20	7.3	odd	6
539.2.e.o.177.10		20	7.4	even	3
4851.2.a.cg.1.9		10	21.20	even	2
4851.2.a.cg.1.10		10	3.2	odd	2
5929.2.a.bv.1.9		10	11.10	odd	2
5929.2.a.bv.1.10		10	77.76	even	2
8624.2.a.df.1.4		10	28.27	even	2
8624.2.a.df.1.7		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)

\(f(q)\)	\(=\)	\(q-2.74816 q^{2} -2.10267 q^{3} +5.55241 q^{4} -2.44342 q^{5} +5.77848 q^{6} -9.76260 q^{8} +1.42122 q^{9} +O(q^{10})\)	\(q-2.74816 q^{2} -2.10267 q^{3} +5.55241 q^{4} -2.44342 q^{5} +5.77848 q^{6} -9.76260 q^{8} +1.42122 q^{9} +6.71492 q^{10} +1.00000 q^{11} -11.6749 q^{12} -1.89199 q^{13} +5.13770 q^{15} +15.7244 q^{16} -3.95040 q^{17} -3.90573 q^{18} -2.88614 q^{19} -13.5669 q^{20} -2.74816 q^{22} -6.59322 q^{23} +20.5275 q^{24} +0.970299 q^{25} +5.19949 q^{26} +3.31966 q^{27} -0.675675 q^{29} -14.1192 q^{30} -7.81724 q^{31} -23.6881 q^{32} -2.10267 q^{33} +10.8563 q^{34} +7.89117 q^{36} +5.87673 q^{37} +7.93160 q^{38} +3.97822 q^{39} +23.8541 q^{40} +3.26890 q^{41} -1.06719 q^{43} +5.55241 q^{44} -3.47263 q^{45} +18.1192 q^{46} +9.70227 q^{47} -33.0632 q^{48} -2.66654 q^{50} +8.30638 q^{51} -10.5051 q^{52} +3.74286 q^{53} -9.12297 q^{54} -2.44342 q^{55} +6.06860 q^{57} +1.85686 q^{58} +6.48233 q^{59} +28.5266 q^{60} +6.09732 q^{61} +21.4831 q^{62} +33.6499 q^{64} +4.62292 q^{65} +5.77848 q^{66} +4.39944 q^{67} -21.9342 q^{68} +13.8634 q^{69} +8.39944 q^{71} -13.8748 q^{72} -2.66200 q^{73} -16.1502 q^{74} -2.04022 q^{75} -16.0250 q^{76} -10.9328 q^{78} +2.09038 q^{79} -38.4213 q^{80} -11.2438 q^{81} -8.98347 q^{82} -3.78398 q^{83} +9.65248 q^{85} +2.93281 q^{86} +1.42072 q^{87} -9.76260 q^{88} -2.44342 q^{89} +9.54335 q^{90} -36.6082 q^{92} +16.4371 q^{93} -26.6634 q^{94} +7.05206 q^{95} +49.8082 q^{96} +12.3633 q^{97} +1.42122 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9}+O(q^{10}) \) 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

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