LMFDB - Embedded newform 6069.2.a.be.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6069,2,Mod(1,6069)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6069.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6069, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$6069 = 3 \cdot 7 \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	6069.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,4,10,12,-6,4,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$48.4612089867$
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:	$x^{10} - 4x^{9} - 8x^{8} + 44x^{7} + 5x^{6} - 144x^{5} + 48x^{4} + 160x^{3} - 44x^{2} - 64x - 2$ x^10 - 4x^9 - 8x^8 + 44x^7 + 5x^6 - 144x^5 + 48x^4 + 160x^3 - 44x^2 - 64*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 357)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.23743$ of defining polynomial
Character	$\chi$	$=$	6069.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23743 q^{2} +1.00000 q^{3} -0.468767 q^{4} +1.81608 q^{5} +1.23743 q^{6} -1.00000 q^{7} -3.05493 q^{8} +1.00000 q^{9} +2.24727 q^{10} +2.35837 q^{11} -0.468767 q^{12} +1.26858 q^{13} -1.23743 q^{14} +1.81608 q^{15} -2.84272 q^{16} +1.23743 q^{18} -3.42380 q^{19} -0.851318 q^{20} -1.00000 q^{21} +2.91832 q^{22} +0.898969 q^{23} -3.05493 q^{24} -1.70185 q^{25} +1.56978 q^{26} +1.00000 q^{27} +0.468767 q^{28} +0.0242602 q^{29} +2.24727 q^{30} +6.75476 q^{31} +2.59218 q^{32} +2.35837 q^{33} -1.81608 q^{35} -0.468767 q^{36} +1.19320 q^{37} -4.23672 q^{38} +1.26858 q^{39} -5.54799 q^{40} +7.09225 q^{41} -1.23743 q^{42} +5.21815 q^{43} -1.10553 q^{44} +1.81608 q^{45} +1.11241 q^{46} +8.50780 q^{47} -2.84272 q^{48} +1.00000 q^{49} -2.10592 q^{50} -0.594667 q^{52} -1.07752 q^{53} +1.23743 q^{54} +4.28300 q^{55} +3.05493 q^{56} -3.42380 q^{57} +0.0300204 q^{58} +5.43516 q^{59} -0.851318 q^{60} +6.29919 q^{61} +8.35855 q^{62} -1.00000 q^{63} +8.89309 q^{64} +2.30384 q^{65} +2.91832 q^{66} -0.516299 q^{67} +0.898969 q^{69} -2.24727 q^{70} +7.25266 q^{71} -3.05493 q^{72} -13.3401 q^{73} +1.47650 q^{74} -1.70185 q^{75} +1.60496 q^{76} -2.35837 q^{77} +1.56978 q^{78} +10.7677 q^{79} -5.16262 q^{80} +1.00000 q^{81} +8.77616 q^{82} -7.22647 q^{83} +0.468767 q^{84} +6.45709 q^{86} +0.0242602 q^{87} -7.20466 q^{88} +10.2051 q^{89} +2.24727 q^{90} -1.26858 q^{91} -0.421407 q^{92} +6.75476 q^{93} +10.5278 q^{94} -6.21790 q^{95} +2.59218 q^{96} -0.205143 q^{97} +1.23743 q^{98} +2.35837 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$10 q + 4 q^{2} + 10 q^{3} + 12 q^{4} - 6 q^{5} + 4 q^{6} - 10 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{11} + 12 q^{12} + 6 q^{13} - 4 q^{14} - 6 q^{15} + 20 q^{16} + 4 q^{18} + 6 q^{19} - 16 q^{20} - 10 q^{21}+ \cdots - 2 q^{99}+O(q^{100})$ 10 * q + 4 * q^2 + 10 * q^3 + 12 * q^4 - 6 * q^5 + 4 * q^6 - 10 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^11 + 12 * q^12 + 6 * q^13 - 4 * q^14 - 6 * q^15 + 20 * q^16 + 4 * q^18 + 6 * q^19 - 16 * q^20 - 10 * q^21 + 24 * q^22 + 18 * q^23 + 12 * q^24 + 4 * q^25 + 12 * q^26 + 10 * q^27 - 12 * q^28 - 12 * q^29 - 8 * q^31 + 28 * q^32 - 2 * q^33 + 6 * q^35 + 12 * q^36 + 24 * q^37 + 32 * q^38 + 6 * q^39 - 36 * q^40 - 2 * q^41 - 4 * q^42 + 6 * q^43 + 28 * q^44 - 6 * q^45 - 28 * q^46 + 16 * q^47 + 20 * q^48 + 10 * q^49 + 52 * q^50 + 24 * q^52 - 12 * q^53 + 4 * q^54 + 18 * q^55 - 12 * q^56 + 6 * q^57 + 20 * q^58 + 20 * q^59 - 16 * q^60 - 4 * q^61 - 24 * q^62 - 10 * q^63 + 56 * q^64 - 38 * q^65 + 24 * q^66 + 20 * q^67 + 18 * q^69 - 20 * q^71 + 12 * q^72 + 20 * q^73 + 36 * q^74 + 4 * q^75 - 8 * q^76 + 2 * q^77 + 12 * q^78 + 20 * q^79 - 92 * q^80 + 10 * q^81 + 8 * q^82 + 36 * q^83 - 12 * q^84 - 12 * q^87 + 92 * q^88 + 32 * q^89 - 6 * q^91 + 40 * q^92 - 8 * q^93 + 60 * q^94 - 14 * q^95 + 28 * q^96 + 36 * q^97 + 4 * q^98 - 2 * q^99

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	1.23743	0.874995	0.437498	−	0.899220i	$-0.355865\pi$
$2$	1.23743	0.874995	0.437498	+	0.899220i	$0.355865\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.468767	−0.234383
$5$	1.81608	0.812176	0.406088	−	0.913834i	$-0.366893\pi$
$5$	1.81608	0.812176	0.406088	+	0.913834i	$0.366893\pi$
$6$	1.23743	0.505179
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.05493	−1.08008
$9$	1.00000	0.333333
$10$	2.24727	0.710650
$11$	2.35837	0.711076	0.355538	−	0.934662i	$-0.384298\pi$
$11$	2.35837	0.711076	0.355538	+	0.934662i	$0.384298\pi$
$12$	−0.468767	−0.135321
$13$	1.26858	0.351840	0.175920	−	0.984404i	$-0.443710\pi$
$13$	1.26858	0.351840	0.175920	+	0.984404i	$0.443710\pi$
$14$	−1.23743	−0.330717
$15$	1.81608	0.468910
$16$	−2.84272	−0.710681
$17$	0	0
$17$	0	0
$18$	1.23743	0.291665
$19$	−3.42380	−0.785474	−0.392737	−	0.919651i	$-0.628472\pi$
$19$	−3.42380	−0.785474	−0.392737	+	0.919651i	$0.628472\pi$
$20$	−0.851318	−0.190361
$21$	−1.00000	−0.218218
$22$	2.91832	0.622188
$23$	0.898969	0.187448	0.0937240	−	0.995598i	$-0.470123\pi$
$23$	0.898969	0.187448	0.0937240	+	0.995598i	$0.470123\pi$
$24$	−3.05493	−0.623584
$25$	−1.70185	−0.340370
$26$	1.56978	0.307858
$27$	1.00000	0.192450
$28$	0.468767	0.0885886
$29$	0.0242602	0.00450501	0.00225251	−	0.999997i	$-0.499283\pi$
$29$	0.0242602	0.00450501	0.00225251	+	0.999997i	$0.499283\pi$
$30$	2.24727	0.410294
$31$	6.75476	1.21319	0.606596	−	0.795011i	$-0.292535\pi$
$31$	6.75476	1.21319	0.606596	+	0.795011i	$0.292535\pi$
$32$	2.59218	0.458237
$33$	2.35837	0.410540
$34$	0	0
$35$	−1.81608	−0.306974
$36$	−0.468767	−0.0781278
$37$	1.19320	0.196161	0.0980806	−	0.995178i	$-0.468730\pi$
$37$	1.19320	0.196161	0.0980806	+	0.995178i	$0.468730\pi$
$38$	−4.23672	−0.687286
$39$	1.26858	0.203135
$40$	−5.54799	−0.877215
$41$	7.09225	1.10762	0.553811	−	0.832642i	$-0.313173\pi$
$41$	7.09225	1.10762	0.553811	+	0.832642i	$0.313173\pi$
$42$	−1.23743	−0.190940
$43$	5.21815	0.795760	0.397880	−	0.917437i	$-0.369746\pi$
$43$	5.21815	0.795760	0.397880	+	0.917437i	$0.369746\pi$
$44$	−1.10553	−0.166664
$45$	1.81608	0.270725
$46$	1.11241	0.164016
$47$	8.50780	1.24099	0.620495	−	0.784210i	$-0.286932\pi$
$47$	8.50780	1.24099	0.620495	+	0.784210i	$0.286932\pi$
$48$	−2.84272	−0.410312
$49$	1.00000	0.142857
$50$	−2.10592	−0.297822
$51$	0	0
$52$	−0.594667	−0.0824654
$53$	−1.07752	−0.148008	−0.0740042	−	0.997258i	$-0.523578\pi$
$53$	−1.07752	−0.148008	−0.0740042	+	0.997258i	$0.523578\pi$
$54$	1.23743	0.168393
$55$	4.28300	0.577519
$56$	3.05493	0.408232
$57$	−3.42380	−0.453494
$58$	0.0300204	0.00394187
$59$	5.43516	0.707597	0.353799	−	0.935322i	$-0.384890\pi$
$59$	5.43516	0.707597	0.353799	+	0.935322i	$0.384890\pi$
$60$	−0.851318	−0.109905
$61$	6.29919	0.806528	0.403264	−	0.915084i	$-0.367876\pi$
$61$	6.29919	0.806528	0.403264	+	0.915084i	$0.367876\pi$
$62$	8.35855	1.06154
$63$	−1.00000	−0.125988
$64$	8.89309	1.11164
$65$	2.30384	0.285756
$66$	2.91832	0.359221
$67$	−0.516299	−0.0630759	−0.0315380	−	0.999503i	$-0.510041\pi$
$67$	−0.516299	−0.0630759	−0.0315380	+	0.999503i	$0.510041\pi$
$68$	0	0
$69$	0.898969	0.108223
$70$	−2.24727	−0.268601
$71$	7.25266	0.860733	0.430366	−	0.902654i	$-0.358384\pi$
$71$	7.25266	0.860733	0.430366	+	0.902654i	$0.358384\pi$
$72$	−3.05493	−0.360027
$73$	−13.3401	−1.56134	−0.780668	−	0.624946i	$-0.785121\pi$
$73$	−13.3401	−1.56134	−0.780668	+	0.624946i	$0.785121\pi$
$74$	1.47650	0.171640
$75$	−1.70185	−0.196513
$76$	1.60496	0.184102
$77$	−2.35837	−0.268762
$78$	1.56978	0.177742
$79$	10.7677	1.21146	0.605728	−	0.795672i	$-0.292882\pi$
$79$	10.7677	1.21146	0.605728	+	0.795672i	$0.292882\pi$
$80$	−5.16262	−0.577198
$81$	1.00000	0.111111
$82$	8.77616	0.969165
$83$	−7.22647	−0.793208	−0.396604	−	0.917990i	$-0.629811\pi$
$83$	−7.22647	−0.793208	−0.396604	+	0.917990i	$0.629811\pi$
$84$	0.468767	0.0511466
$85$	0	0
$86$	6.45709	0.696286
$87$	0.0242602	0.00260097
$88$	−7.20466	−0.768019
$89$	10.2051	1.08174	0.540872	−	0.841105i	$-0.318094\pi$
$89$	10.2051	1.08174	0.540872	+	0.841105i	$0.318094\pi$
$90$	2.24727	0.236883
$91$	−1.26858	−0.132983
$92$	−0.421407	−0.0439347
$93$	6.75476	0.700436
$94$	10.5278	1.08586
$95$	−6.21790	−0.637943
$96$	2.59218	0.264563
$97$	−0.205143	−0.0208291	−0.0104145	−	0.999946i	$-0.503315\pi$
$97$	−0.205143	−0.0208291	−0.0104145	+	0.999946i	$0.503315\pi$
$98$	1.23743	0.124999
$99$	2.35837	0.237025
$100$	0.797770	0.0797770
$101$	10.0435	0.999361	0.499680	−	0.866210i	$-0.333451\pi$
$101$	10.0435	0.999361	0.499680	+	0.866210i	$0.333451\pi$
$102$	0	0
$103$	0.982712	0.0968295	0.0484147	−	0.998827i	$-0.484583\pi$
$103$	0.982712	0.0968295	0.0484147	+	0.998827i	$0.484583\pi$
$104$	−3.87541	−0.380015
$105$	−1.81608	−0.177231
$106$	−1.33335	−0.129507
$107$	−13.5270	−1.30771	−0.653854	−	0.756621i	$-0.726849\pi$
$107$	−13.5270	−1.30771	−0.653854	+	0.756621i	$0.726849\pi$
$108$	−0.468767	−0.0451071
$109$	9.56837	0.916483	0.458242	−	0.888828i	$-0.348479\pi$
$109$	9.56837	0.916483	0.458242	+	0.888828i	$0.348479\pi$
$110$	5.29991	0.505327
$111$	1.19320	0.113254
$112$	2.84272	0.268612
$113$	11.4380	1.07600	0.537999	−	0.842946i	$-0.319181\pi$
$113$	11.4380	1.07600	0.537999	+	0.842946i	$0.319181\pi$
$114$	−4.23672	−0.396805
$115$	1.63260	0.152241
$116$	−0.0113724	−0.00105590
$117$	1.26858	0.117280
$118$	6.72563	0.619144
$119$	0	0
$120$	−5.54799	−0.506460
$121$	−5.43807	−0.494370
$122$	7.79480	0.705708
$123$	7.09225	0.639486
$124$	−3.16641	−0.284352
$125$	−12.1711	−1.08862
$126$	−1.23743	−0.110239
$127$	−17.4380	−1.54737	−0.773685	−	0.633570i	$-0.781589\pi$
$127$	−17.4380	−1.54737	−0.773685	+	0.633570i	$0.781589\pi$
$128$	5.82022	0.514439
$129$	5.21815	0.459432
$130$	2.85084	0.250035
$131$	−2.30473	−0.201365	−0.100682	−	0.994919i	$-0.532103\pi$
$131$	−2.30473	−0.201365	−0.100682	+	0.994919i	$0.532103\pi$
$132$	−1.10553	−0.0962238
$133$	3.42380	0.296881
$134$	−0.638883	−0.0551911
$135$	1.81608	0.156303
$136$	0	0
$137$	−0.591167	−0.0505068	−0.0252534	−	0.999681i	$-0.508039\pi$
$137$	−0.591167	−0.0505068	−0.0252534	+	0.999681i	$0.508039\pi$
$138$	1.11241	0.0946948
$139$	−5.78876	−0.490996	−0.245498	−	0.969397i	$-0.578952\pi$
$139$	−5.78876	−0.490996	−0.245498	+	0.969397i	$0.578952\pi$
$140$	0.851318	0.0719495
$141$	8.50780	0.716486
$142$	8.97467	0.753137
$143$	2.99178	0.250185
$144$	−2.84272	−0.236894
$145$	0.0440586	0.00365886
$146$	−16.5074	−1.36616
$147$	1.00000	0.0824786
$148$	−0.559333	−0.0459769
$149$	20.6031	1.68787	0.843936	−	0.536444i	$-0.180233\pi$
$149$	20.6031	1.68787	0.843936	+	0.536444i	$0.180233\pi$
$150$	−2.10592	−0.171948
$151$	14.8292	1.20679	0.603393	−	0.797444i	$-0.293815\pi$
$151$	14.8292	1.20679	0.603393	+	0.797444i	$0.293815\pi$
$152$	10.4595	0.848374
$153$	0	0
$154$	−2.91832	−0.235165
$155$	12.2672	0.985325
$156$	−0.594667	−0.0476114
$157$	3.79172	0.302612	0.151306	−	0.988487i	$-0.451652\pi$
$157$	3.79172	0.302612	0.151306	+	0.988487i	$0.451652\pi$
$158$	13.3242	1.06002
$159$	−1.07752	−0.0854527
$160$	4.70761	0.372169
$161$	−0.898969	−0.0708487
$162$	1.23743	0.0972217
$163$	−21.6449	−1.69536	−0.847679	−	0.530509i	$-0.822001\pi$
$163$	−21.6449	−1.69536	−0.847679	+	0.530509i	$0.822001\pi$
$164$	−3.32461	−0.259608
$165$	4.28300	0.333431
$166$	−8.94225	−0.694053
$167$	−25.4646	−1.97051	−0.985256	−	0.171089i	$-0.945272\pi$
$167$	−25.4646	−1.97051	−0.985256	+	0.171089i	$0.945272\pi$
$168$	3.05493	0.235693
$169$	−11.3907	−0.876209
$170$	0	0
$171$	−3.42380	−0.261825
$172$	−2.44609	−0.186513
$173$	−2.98246	−0.226752	−0.113376	−	0.993552i	$-0.536167\pi$
$173$	−2.98246	−0.226752	−0.113376	+	0.993552i	$0.536167\pi$
$174$	0.0300204	0.00227584
$175$	1.70185	0.128648
$176$	−6.70421	−0.505349
$177$	5.43516	0.408532
$178$	12.6282	0.946520
$179$	7.72272	0.577223	0.288612	−	0.957446i	$-0.406806\pi$
$179$	7.72272	0.577223	0.288612	+	0.957446i	$0.406806\pi$
$180$	−0.851318	−0.0634535
$181$	21.8503	1.62412	0.812058	−	0.583576i	$-0.198347\pi$
$181$	21.8503	1.62412	0.812058	+	0.583576i	$0.198347\pi$
$182$	−1.56978	−0.116360
$183$	6.29919	0.465649
$184$	−2.74629	−0.202459
$185$	2.16695	0.159317
$186$	8.35855	0.612878
$187$	0	0
$188$	−3.98817	−0.290867
$189$	−1.00000	−0.0727393
$190$	−7.69422	−0.558197
$191$	22.7283	1.64456	0.822282	−	0.569081i	$-0.192701\pi$
$191$	22.7283	1.64456	0.822282	+	0.569081i	$0.192701\pi$
$192$	8.89309	0.641803
$193$	−23.4939	−1.69113	−0.845565	−	0.533872i	$-0.820736\pi$
$193$	−23.4939	−1.69113	−0.845565	+	0.533872i	$0.820736\pi$
$194$	−0.253850	−0.0182253
$195$	2.30384	0.164981
$196$	−0.468767	−0.0334833
$197$	−6.79070	−0.483817	−0.241909	−	0.970299i	$-0.577773\pi$
$197$	−6.79070	−0.483817	−0.241909	+	0.970299i	$0.577773\pi$
$198$	2.91832	0.207396
$199$	−5.04115	−0.357357	−0.178679	−	0.983907i	$-0.557182\pi$
$199$	−5.04115	−0.357357	−0.178679	+	0.983907i	$0.557182\pi$
$200$	5.19902	0.367627
$201$	−0.516299	−0.0364169
$202$	12.4281	0.874436
$203$	−0.0242602	−0.00170273
$204$	0	0
$205$	12.8801	0.899585
$206$	1.21604	0.0847253
$207$	0.898969	0.0624827
$208$	−3.60622	−0.250046
$209$	−8.07460	−0.558532
$210$	−2.24727	−0.155077
$211$	−7.87431	−0.542090	−0.271045	−	0.962567i	$-0.587369\pi$
$211$	−7.87431	−0.542090	−0.271045	+	0.962567i	$0.587369\pi$
$212$	0.505104	0.0346907
$213$	7.25266	0.496944
$214$	−16.7388	−1.14424
$215$	9.47658	0.646297
$216$	−3.05493	−0.207861
$217$	−6.75476	−0.458543
$218$	11.8402	0.801918
$219$	−13.3401	−0.901438
$220$	−2.00773	−0.135361
$221$	0	0
$222$	1.47650	0.0990964
$223$	20.5049	1.37311	0.686556	−	0.727077i	$-0.259122\pi$
$223$	20.5049	1.37311	0.686556	+	0.727077i	$0.259122\pi$
$224$	−2.59218	−0.173197
$225$	−1.70185	−0.113457
$226$	14.1537	0.941492
$227$	8.83865	0.586642	0.293321	−	0.956014i	$-0.405240\pi$
$227$	8.83865	0.586642	0.293321	+	0.956014i	$0.405240\pi$
$228$	1.60496	0.106291
$229$	−6.26192	−0.413799	−0.206900	−	0.978362i	$-0.566337\pi$
$229$	−6.26192	−0.413799	−0.206900	+	0.978362i	$0.566337\pi$
$230$	2.02023	0.133210
$231$	−2.35837	−0.155170
$232$	−0.0741132	−0.00486577
$233$	−22.7169	−1.48823	−0.744116	−	0.668051i	$-0.767129\pi$
$233$	−22.7169	−1.48823	−0.744116	+	0.668051i	$0.767129\pi$
$234$	1.56978	0.102619
$235$	15.4509	1.00790
$236$	−2.54782	−0.165849
$237$	10.7677	0.699434
$238$	0	0
$239$	−5.42712	−0.351052	−0.175526	−	0.984475i	$-0.556163\pi$
$239$	−5.42712	−0.351052	−0.175526	+	0.984475i	$0.556163\pi$
$240$	−5.16262	−0.333246
$241$	26.9675	1.73713	0.868565	−	0.495575i	$-0.165043\pi$
$241$	26.9675	1.73713	0.868565	+	0.495575i	$0.165043\pi$
$242$	−6.72924	−0.432572
$243$	1.00000	0.0641500
$244$	−2.95285	−0.189037
$245$	1.81608	0.116025
$246$	8.77616	0.559547
$247$	−4.34336	−0.276361
$248$	−20.6353	−1.31034
$249$	−7.22647	−0.457959
$250$	−15.0609	−0.952534
$251$	11.3447	0.716074	0.358037	−	0.933707i	$-0.383446\pi$
$251$	11.3447	0.716074	0.358037	+	0.933707i	$0.383446\pi$
$252$	0.468767	0.0295295
$253$	2.12011	0.133290
$254$	−21.5783	−1.35394
$255$	0	0
$256$	−10.5841	−0.661504
$257$	26.4860	1.65215	0.826077	−	0.563557i	$-0.190568\pi$
$257$	26.4860	1.65215	0.826077	+	0.563557i	$0.190568\pi$
$258$	6.45709	0.402001
$259$	−1.19320	−0.0741419
$260$	−1.07996	−0.0669765
$261$	0.0242602	0.00150167
$262$	−2.85194	−0.176193
$263$	0.781732	0.0482036	0.0241018	−	0.999710i	$-0.492327\pi$
$263$	0.781732	0.0482036	0.0241018	+	0.999710i	$0.492327\pi$
$264$	−7.20466	−0.443416
$265$	−1.95686	−0.120209
$266$	4.23672	0.259770
$267$	10.2051	0.624545
$268$	0.242024	0.0147839
$269$	10.0993	0.615767	0.307884	−	0.951424i	$-0.400379\pi$
$269$	10.0993	0.615767	0.307884	+	0.951424i	$0.400379\pi$
$270$	2.24727	0.136765
$271$	16.8897	1.02597	0.512987	−	0.858397i	$-0.328539\pi$
$271$	16.8897	1.02597	0.512987	+	0.858397i	$0.328539\pi$
$272$	0	0
$273$	−1.26858	−0.0767778
$274$	−0.731527	−0.0441932
$275$	−4.01360	−0.242029
$276$	−0.421407	−0.0253657
$277$	−4.31657	−0.259358	−0.129679	−	0.991556i	$-0.541395\pi$
$277$	−4.31657	−0.259358	−0.129679	+	0.991556i	$0.541395\pi$
$278$	−7.16319	−0.429620
$279$	6.75476	0.404397
$280$	5.54799	0.331556
$281$	14.7023	0.877067	0.438534	−	0.898715i	$-0.355498\pi$
$281$	14.7023	0.877067	0.438534	+	0.898715i	$0.355498\pi$
$282$	10.5278	0.626922
$283$	−10.3543	−0.615499	−0.307750	−	0.951467i	$-0.599576\pi$
$283$	−10.3543	−0.615499	−0.307750	+	0.951467i	$0.599576\pi$
$284$	−3.39981	−0.201741
$285$	−6.21790	−0.368317
$286$	3.70212	0.218911
$287$	−7.09225	−0.418642
$288$	2.59218	0.152746
$289$	0	0
$290$	0.0545194	0.00320149
$291$	−0.205143	−0.0120257
$292$	6.25338	0.365951
$293$	10.9848	0.641741	0.320871	−	0.947123i	$-0.396025\pi$
$293$	10.9848	0.641741	0.320871	+	0.947123i	$0.396025\pi$
$294$	1.23743	0.0721684
$295$	9.87069	0.574694
$296$	−3.64514	−0.211870
$297$	2.35837	0.136847
$298$	25.4949	1.47688
$299$	1.14041	0.0659517
$300$	0.797770	0.0460593
$301$	−5.21815	−0.300769
$302$	18.3501	1.05593
$303$	10.0435	0.576981
$304$	9.73293	0.558222
$305$	11.4398	0.655043
$306$	0	0
$307$	13.8104	0.788203	0.394102	−	0.919067i	$-0.371056\pi$
$307$	13.8104	0.788203	0.394102	+	0.919067i	$0.371056\pi$
$308$	1.10553	0.0629932
$309$	0.982712	0.0559045
$310$	15.1798	0.862155
$311$	24.2650	1.37594	0.687970	−	0.725739i	$-0.258502\pi$
$311$	24.2650	1.37594	0.687970	+	0.725739i	$0.258502\pi$
$312$	−3.87541	−0.219402
$313$	−16.0803	−0.908914	−0.454457	−	0.890769i	$-0.650167\pi$
$313$	−16.0803	−0.908914	−0.454457	+	0.890769i	$0.650167\pi$
$314$	4.69199	0.264784
$315$	−1.81608	−0.102325
$316$	−5.04752	−0.283945
$317$	4.29099	0.241006	0.120503	−	0.992713i	$-0.461549\pi$
$317$	4.29099	0.241006	0.120503	+	0.992713i	$0.461549\pi$
$318$	−1.33335	−0.0747707
$319$	0.0572147	0.00320341
$320$	16.1506	0.902844
$321$	−13.5270	−0.755006
$322$	−1.11241	−0.0619923
$323$	0	0
$324$	−0.468767	−0.0260426
$325$	−2.15893	−0.119756
$326$	−26.7840	−1.48343
$327$	9.56837	0.529132
$328$	−21.6663	−1.19632
$329$	−8.50780	−0.469050
$330$	5.29991	0.291750
$331$	−19.3135	−1.06156	−0.530782	−	0.847508i	$-0.678102\pi$
$331$	−19.3135	−1.06156	−0.530782	+	0.847508i	$0.678102\pi$
$332$	3.38753	0.185915
$333$	1.19320	0.0653870
$334$	−31.5107	−1.72419
$335$	−0.937640	−0.0512288
$336$	2.84272	0.155083
$337$	19.6082	1.06813	0.534063	−	0.845444i	$-0.320664\pi$
$337$	19.6082	1.06813	0.534063	+	0.845444i	$0.320664\pi$
$338$	−14.0952	−0.766678
$339$	11.4380	0.621227
$340$	0	0
$341$	15.9303	0.862672
$342$	−4.23672	−0.229095
$343$	−1.00000	−0.0539949
$344$	−15.9411	−0.859484
$345$	1.63260	0.0878963
$346$	−3.69059	−0.198407
$347$	2.24091	0.120298	0.0601491	−	0.998189i	$-0.480842\pi$
$347$	2.24091	0.120298	0.0601491	+	0.998189i	$0.480842\pi$
$348$	−0.0113724	−0.000609624 0
$349$	22.3354	1.19559	0.597794	−	0.801650i	$-0.296044\pi$
$349$	22.3354	1.19559	0.597794	+	0.801650i	$0.296044\pi$
$350$	2.10592	0.112566
$351$	1.26858	0.0677116
$352$	6.11333	0.325841
$353$	7.98083	0.424777	0.212388	−	0.977185i	$-0.431876\pi$
$353$	7.98083	0.424777	0.212388	+	0.977185i	$0.431876\pi$
$354$	6.72563	0.357463
$355$	13.1714	0.699067
$356$	−4.78383	−0.253543
$357$	0	0
$358$	9.55633	0.505068
$359$	−21.3945	−1.12916	−0.564579	−	0.825379i	$-0.690962\pi$
$359$	−21.3945	−1.12916	−0.564579	+	0.825379i	$0.690962\pi$
$360$	−5.54799	−0.292405
$361$	−7.27758	−0.383031
$362$	27.0382	1.42109
$363$	−5.43807	−0.285425
$364$	0.594667	0.0311690
$365$	−24.2266	−1.26808
$366$	7.79480	0.407441
$367$	1.10513	0.0576875	0.0288437	−	0.999584i	$-0.490817\pi$
$367$	1.10513	0.0576875	0.0288437	+	0.999584i	$0.490817\pi$
$368$	−2.55552	−0.133216
$369$	7.09225	0.369208
$370$	2.68145	0.139402
$371$	1.07752	0.0559419
$372$	−3.16641	−0.164171
$373$	−26.1722	−1.35514	−0.677572	−	0.735457i	$-0.736968\pi$
$373$	−26.1722	−1.35514	−0.677572	+	0.735457i	$0.736968\pi$
$374$	0	0
$375$	−12.1711	−0.628513
$376$	−25.9907	−1.34037
$377$	0.0307760	0.00158504
$378$	−1.23743	−0.0636465
$379$	27.7591	1.42589	0.712944	−	0.701221i	$-0.247361\pi$
$379$	27.7591	1.42589	0.712944	+	0.701221i	$0.247361\pi$
$380$	2.91475	0.149523
$381$	−17.4380	−0.893375
$382$	28.1247	1.43899
$383$	−17.8180	−0.910459	−0.455230	−	0.890374i	$-0.650443\pi$
$383$	−17.8180	−0.910459	−0.455230	+	0.890374i	$0.650443\pi$
$384$	5.82022	0.297012
$385$	−4.28300	−0.218282
$386$	−29.0721	−1.47973
$387$	5.21815	0.265253
$388$	0.0961640	0.00488199
$389$	−22.5784	−1.14477	−0.572386	−	0.819984i	$-0.693982\pi$
$389$	−22.5784	−1.14477	−0.572386	+	0.819984i	$0.693982\pi$
$390$	2.85084	0.144358
$391$	0	0
$392$	−3.05493	−0.154297
$393$	−2.30473	−0.116258
$394$	−8.40302	−0.423338
$395$	19.5549	0.983915
$396$	−1.10553	−0.0555548
$397$	−36.0710	−1.81035	−0.905177	−	0.425036i	$-0.860262\pi$
$397$	−36.0710	−1.81035	−0.905177	+	0.425036i	$0.860262\pi$
$398$	−6.23807	−0.312686
$399$	3.42380	0.171404
$400$	4.83789	0.241894
$401$	2.71025	0.135343	0.0676716	−	0.997708i	$-0.478443\pi$
$401$	2.71025	0.135343	0.0676716	+	0.997708i	$0.478443\pi$
$402$	−0.638883	−0.0318646
$403$	8.56894	0.426849
$404$	−4.70804	−0.234234
$405$	1.81608	0.0902418
$406$	−0.0300204	−0.00148988
$407$	2.81401	0.139486
$408$	0	0
$409$	−32.7262	−1.61821	−0.809103	−	0.587667i	$-0.800047\pi$
$409$	−32.7262	−1.61821	−0.809103	+	0.587667i	$0.800047\pi$
$410$	15.9382	0.787132
$411$	−0.591167	−0.0291601
$412$	−0.460662	−0.0226952
$413$	−5.43516	−0.267447
$414$	1.11241	0.0546721
$415$	−13.1239	−0.644225
$416$	3.28838	0.161226
$417$	−5.78876	−0.283477
$418$	−9.99176	−0.488713
$419$	−0.851837	−0.0416150	−0.0208075	−	0.999784i	$-0.506624\pi$
$419$	−0.851837	−0.0416150	−0.0208075	+	0.999784i	$0.506624\pi$
$420$	0.851318	0.0415401
$421$	−1.46190	−0.0712489	−0.0356244	−	0.999365i	$-0.511342\pi$
$421$	−1.46190	−0.0712489	−0.0356244	+	0.999365i	$0.511342\pi$
$422$	−9.74391	−0.474326
$423$	8.50780	0.413663
$424$	3.29174	0.159861
$425$	0	0
$426$	8.97467	0.434824
$427$	−6.29919	−0.304839
$428$	6.34102	0.306505
$429$	2.99178	0.144444
$430$	11.7266	0.565507
$431$	21.9280	1.05624	0.528118	−	0.849171i	$-0.322898\pi$
$431$	21.9280	1.05624	0.528118	+	0.849171i	$0.322898\pi$
$432$	−2.84272	−0.136771
$433$	−25.4725	−1.22413	−0.612065	−	0.790807i	$-0.709661\pi$
$433$	−25.4725	−1.22413	−0.612065	+	0.790807i	$0.709661\pi$
$434$	−8.35855	−0.401223
$435$	0.0440586	0.00211245
$436$	−4.48533	−0.214808
$437$	−3.07789	−0.147236
$438$	−16.5074	−0.788754
$439$	−37.2868	−1.77960	−0.889802	−	0.456348i	$-0.849157\pi$
$439$	−37.2868	−1.77960	−0.889802	+	0.456348i	$0.849157\pi$
$440$	−13.0842	−0.623767
$441$	1.00000	0.0476190
$442$	0	0
$443$	40.4670	1.92264	0.961321	−	0.275429i	$-0.0888199\pi$
$443$	40.4670	1.92264	0.961321	+	0.275429i	$0.0888199\pi$
$444$	−0.559333	−0.0265448
$445$	18.5334	0.878566
$446$	25.3734	1.20147
$447$	20.6031	0.974493
$448$	−8.89309	−0.420159
$449$	−9.51925	−0.449241	−0.224621	−	0.974446i	$-0.572114\pi$
$449$	−9.51925	−0.449241	−0.224621	+	0.974446i	$0.572114\pi$
$450$	−2.10592	−0.0992740
$451$	16.7262	0.787604
$452$	−5.36176	−0.252196
$453$	14.8292	0.696738
$454$	10.9372	0.513309
$455$	−2.30384	−0.108006
$456$	10.4595	0.489809
$457$	8.05510	0.376802	0.188401	−	0.982092i	$-0.439670\pi$
$457$	8.05510	0.376802	0.188401	+	0.982092i	$0.439670\pi$
$458$	−7.74869	−0.362073
$459$	0	0
$460$	−0.765309	−0.0356827
$461$	−33.2395	−1.54812	−0.774059	−	0.633113i	$-0.781777\pi$
$461$	−33.2395	−1.54812	−0.774059	+	0.633113i	$0.781777\pi$
$462$	−2.91832	−0.135773
$463$	−18.4862	−0.859125	−0.429563	−	0.903037i	$-0.641332\pi$
$463$	−18.4862	−0.859125	−0.429563	+	0.903037i	$0.641332\pi$
$464$	−0.0689652	−0.00320163
$465$	12.2672	0.568878
$466$	−28.1105	−1.30220
$467$	7.79833	0.360864	0.180432	−	0.983587i	$-0.442250\pi$
$467$	7.79833	0.360864	0.180432	+	0.983587i	$0.442250\pi$
$468$	−0.594667	−0.0274885
$469$	0.516299	0.0238405
$470$	19.1194	0.881910
$471$	3.79172	0.174713
$472$	−16.6040	−0.764262
$473$	12.3063	0.565846
$474$	13.3242	0.612001
$475$	5.82680	0.267352
$476$	0	0
$477$	−1.07752	−0.0493362
$478$	−6.71569	−0.307168
$479$	−18.1723	−0.830315	−0.415158	−	0.909750i	$-0.636274\pi$
$479$	−18.1723	−0.830315	−0.415158	+	0.909750i	$0.636274\pi$
$480$	4.70761	0.214872
$481$	1.51367	0.0690173
$482$	33.3704	1.51998
$483$	−0.898969	−0.0409045
$484$	2.54919	0.115872
$485$	−0.372555	−0.0169169
$486$	1.23743	0.0561310
$487$	−4.65708	−0.211033	−0.105516	−	0.994418i	$-0.533650\pi$
$487$	−4.65708	−0.211033	−0.105516	+	0.994418i	$0.533650\pi$
$488$	−19.2436	−0.871115
$489$	−21.6449	−0.978816
$490$	2.24727	0.101521
$491$	−41.8043	−1.88660	−0.943300	−	0.331942i	$-0.892296\pi$
$491$	−41.8043	−1.88660	−0.943300	+	0.331942i	$0.892296\pi$
$492$	−3.32461	−0.149885
$493$	0	0
$494$	−5.37460	−0.241815
$495$	4.28300	0.192506
$496$	−19.2019	−0.862192
$497$	−7.25266	−0.325326
$498$	−8.94225	−0.400712
$499$	−10.5845	−0.473828	−0.236914	−	0.971531i	$-0.576136\pi$
$499$	−10.5845	−0.473828	−0.236914	+	0.971531i	$0.576136\pi$
$500$	5.70541	0.255154
$501$	−25.4646	−1.13768
$502$	14.0383	0.626561
$503$	−27.3831	−1.22095	−0.610476	−	0.792035i	$-0.709022\pi$
$503$	−27.3831	−1.22095	−0.610476	+	0.792035i	$0.709022\pi$
$504$	3.05493	0.136077
$505$	18.2397	0.811657
$506$	2.62348	0.116628
$507$	−11.3907	−0.505879
$508$	8.17435	0.362678
$509$	−9.14311	−0.405261	−0.202631	−	0.979255i	$-0.564949\pi$
$509$	−9.14311	−0.405261	−0.202631	+	0.979255i	$0.564949\pi$
$510$	0	0
$511$	13.3401	0.590130
$512$	−24.7375	−1.09325
$513$	−3.42380	−0.151165
$514$	32.7746	1.44563
$515$	1.78468	0.0786426
$516$	−2.44609	−0.107683
$517$	20.0646	0.882439
$518$	−1.47650	−0.0648738
$519$	−2.98246	−0.130916
$520$	−7.03806	−0.308639
$521$	−17.6233	−0.772089	−0.386044	−	0.922480i	$-0.626159\pi$
$521$	−17.6233	−0.772089	−0.386044	+	0.922480i	$0.626159\pi$
$522$	0.0300204	0.00131396
$523$	−3.52171	−0.153994	−0.0769969	−	0.997031i	$-0.524533\pi$
$523$	−3.52171	−0.153994	−0.0769969	+	0.997031i	$0.524533\pi$
$524$	1.08038	0.0471966
$525$	1.70185	0.0742748
$526$	0.967338	0.0421779
$527$	0	0
$528$	−6.70421	−0.291763
$529$	−22.1919	−0.964863
$530$	−2.42148	−0.105182
$531$	5.43516	0.235866
$532$	−1.60496	−0.0695840
$533$	8.99706	0.389706
$534$	12.6282	0.546474
$535$	−24.5662	−1.06209
$536$	1.57725	0.0681270
$537$	7.72272	0.333260
$538$	12.4972	0.538793
$539$	2.35837	0.101582
$540$	−0.851318	−0.0366349
$541$	−34.7561	−1.49428	−0.747140	−	0.664667i	$-0.768574\pi$
$541$	−34.7561	−1.49428	−0.747140	+	0.664667i	$0.768574\pi$
$542$	20.8998	0.897722
$543$	21.8503	0.937684
$544$	0	0
$545$	17.3769	0.744346
$546$	−1.56978	−0.0671802
$547$	32.8195	1.40326	0.701630	−	0.712542i	$-0.252456\pi$
$547$	32.8195	1.40326	0.701630	+	0.712542i	$0.252456\pi$
$548$	0.277119	0.0118379
$549$	6.29919	0.268843
$550$	−4.96655	−0.211774
$551$	−0.0830623	−0.00353857
$552$	−2.74629	−0.116890
$553$	−10.7677	−0.457887
$554$	−5.34146	−0.226937
$555$	2.16695	0.0919819
$556$	2.71358	0.115081
$557$	−33.5274	−1.42060	−0.710300	−	0.703899i	$-0.751441\pi$
$557$	−33.5274	−1.42060	−0.710300	+	0.703899i	$0.751441\pi$
$558$	8.35855	0.353846
$559$	6.61962	0.279980
$560$	5.16262	0.218160
$561$	0	0
$562$	18.1931	0.767430
$563$	39.1139	1.64845	0.824226	−	0.566260i	$-0.191610\pi$
$563$	39.1139	1.64845	0.824226	+	0.566260i	$0.191610\pi$
$564$	−3.98817	−0.167932
$565$	20.7723	0.873899
$566$	−12.8127	−0.538559
$567$	−1.00000	−0.0419961
$568$	−22.1564	−0.929660
$569$	−18.6068	−0.780037	−0.390018	−	0.920807i	$-0.627531\pi$
$569$	−18.6068	−0.780037	−0.390018	+	0.920807i	$0.627531\pi$
$570$	−7.69422	−0.322275
$571$	−2.00086	−0.0837335	−0.0418667	−	0.999123i	$-0.513330\pi$
$571$	−2.00086	−0.0837335	−0.0418667	+	0.999123i	$0.513330\pi$
$572$	−1.40245	−0.0586392
$573$	22.7283	0.949489
$574$	−8.77616	−0.366310
$575$	−1.52991	−0.0638017
$576$	8.89309	0.370545
$577$	10.9555	0.456083	0.228042	−	0.973651i	$-0.426768\pi$
$577$	10.9555	0.456083	0.228042	+	0.973651i	$0.426768\pi$
$578$	0	0
$579$	−23.4939	−0.976375
$580$	−0.0206532	−0.000857577 0
$581$	7.22647	0.299805
$582$	−0.253850	−0.0105224
$583$	−2.54119	−0.105245
$584$	40.7529	1.68637
$585$	2.30384	0.0952520
$586$	13.5930	0.561520
$587$	4.27209	0.176328	0.0881640	−	0.996106i	$-0.471900\pi$
$587$	4.27209	0.176328	0.0881640	+	0.996106i	$0.471900\pi$
$588$	−0.468767	−0.0193316
$589$	−23.1270	−0.952930
$590$	12.2143	0.502854
$591$	−6.79070	−0.279332
$592$	−3.39194	−0.139408
$593$	−19.2572	−0.790799	−0.395399	−	0.918509i	$-0.629394\pi$
$593$	−19.2572	−0.790799	−0.395399	+	0.918509i	$0.629394\pi$
$594$	2.91832	0.119740
$595$	0	0
$596$	−9.65805	−0.395609
$597$	−5.04115	−0.206320
$598$	1.41118	0.0577075
$599$	−4.47849	−0.182986	−0.0914931	−	0.995806i	$-0.529164\pi$
$599$	−4.47849	−0.182986	−0.0914931	+	0.995806i	$0.529164\pi$
$600$	5.19902	0.212249
$601$	−34.5940	−1.41112	−0.705559	−	0.708651i	$-0.749304\pi$
$601$	−34.5940	−1.41112	−0.705559	+	0.708651i	$0.749304\pi$
$602$	−6.45709	−0.263171
$603$	−0.516299	−0.0210253
$604$	−6.95145	−0.282851
$605$	−9.87598	−0.401516
$606$	12.4281	0.504856
$607$	−2.65970	−0.107954	−0.0539770	−	0.998542i	$-0.517190\pi$
$607$	−2.65970	−0.107954	−0.0539770	+	0.998542i	$0.517190\pi$
$608$	−8.87511	−0.359933
$609$	−0.0242602	−0.000983074 0
$610$	14.1560	0.573160
$611$	10.7928	0.436630
$612$	0	0
$613$	−8.81178	−0.355905	−0.177952	−	0.984039i	$-0.556947\pi$
$613$	−8.81178	−0.355905	−0.177952	+	0.984039i	$0.556947\pi$
$614$	17.0895	0.689674
$615$	12.8801	0.519375
$616$	7.20466	0.290284
$617$	24.8265	0.999478	0.499739	−	0.866176i	$-0.333429\pi$
$617$	24.8265	0.999478	0.499739	+	0.866176i	$0.333429\pi$
$618$	1.21604	0.0489162
$619$	18.8851	0.759055	0.379527	−	0.925181i	$-0.376087\pi$
$619$	18.8851	0.759055	0.379527	+	0.925181i	$0.376087\pi$
$620$	−5.75045	−0.230944
$621$	0.898969	0.0360744
$622$	30.0262	1.20394
$623$	−10.2051	−0.408861
$624$	−3.60622	−0.144364
$625$	−13.5945	−0.543778
$626$	−19.8983	−0.795295
$627$	−8.07460	−0.322469
$628$	−1.77743	−0.0709273
$629$	0	0
$630$	−2.24727	−0.0895335
$631$	−33.5614	−1.33606	−0.668030	−	0.744135i	$-0.732862\pi$
$631$	−33.5614	−1.33606	−0.668030	+	0.744135i	$0.732862\pi$
$632$	−32.8944	−1.30847
$633$	−7.87431	−0.312976
$634$	5.30980	0.210879
$635$	−31.6688	−1.25674
$636$	0.505104	0.0200287
$637$	1.26858	0.0502629
$638$	0.0707992	0.00280297
$639$	7.25266	0.286911
$640$	10.5700	0.417815
$641$	−18.0566	−0.713191	−0.356596	−	0.934259i	$-0.616063\pi$
$641$	−18.0566	−0.713191	−0.356596	+	0.934259i	$0.616063\pi$
$642$	−16.7388	−0.660626
$643$	−26.5565	−1.04729	−0.523643	−	0.851938i	$-0.675428\pi$
$643$	−26.5565	−1.04729	−0.523643	+	0.851938i	$0.675428\pi$
$644$	0.421407	0.0166058
$645$	9.47658	0.373140
$646$	0	0
$647$	−12.7877	−0.502735	−0.251367	−	0.967892i	$-0.580880\pi$
$647$	−12.7877	−0.502735	−0.251367	+	0.967892i	$0.580880\pi$
$648$	−3.05493	−0.120009
$649$	12.8181	0.503156
$650$	−2.67152	−0.104786
$651$	−6.75476	−0.264740
$652$	10.1464	0.397364
$653$	0.282151	0.0110414	0.00552070	−	0.999985i	$-0.498243\pi$
$653$	0.282151	0.0110414	0.00552070	+	0.999985i	$0.498243\pi$
$654$	11.8402	0.462988
$655$	−4.18557	−0.163544
$656$	−20.1613	−0.787167
$657$	−13.3401	−0.520445
$658$	−10.5278	−0.410417
$659$	38.5587	1.50203	0.751017	−	0.660283i	$-0.229564\pi$
$659$	38.5587	1.50203	0.751017	+	0.660283i	$0.229564\pi$
$660$	−2.00773	−0.0781507
$661$	19.9752	0.776944	0.388472	−	0.921461i	$-0.373003\pi$
$661$	19.9752	0.776944	0.388472	+	0.921461i	$0.373003\pi$
$662$	−23.8991	−0.928863
$663$	0	0
$664$	22.0763	0.856728
$665$	6.21790	0.241120
$666$	1.47650	0.0572133
$667$	0.0218092	0.000844456 0
$668$	11.9370	0.461855
$669$	20.5049	0.792766
$670$	−1.16026	−0.0448249
$671$	14.8558	0.573503
$672$	−2.59218	−0.0999955
$673$	42.3264	1.63156	0.815781	−	0.578360i	$-0.196307\pi$
$673$	42.3264	1.63156	0.815781	+	0.578360i	$0.196307\pi$
$674$	24.2638	0.934606
$675$	−1.70185	−0.0655042
$676$	5.33959	0.205369
$677$	−41.9506	−1.61229	−0.806146	−	0.591717i	$-0.798450\pi$
$677$	−41.9506	−1.61229	−0.806146	+	0.591717i	$0.798450\pi$
$678$	14.1537	0.543571
$679$	0.205143	0.00787265
$680$	0	0
$681$	8.83865	0.338698
$682$	19.7126	0.754834
$683$	18.6529	0.713734	0.356867	−	0.934155i	$-0.383845\pi$
$683$	18.6529	0.713734	0.356867	+	0.934155i	$0.383845\pi$
$684$	1.60496	0.0613673
$685$	−1.07361	−0.0410204
$686$	−1.23743	−0.0472453
$687$	−6.26192	−0.238907
$688$	−14.8338	−0.565532
$689$	−1.36691	−0.0520753
$690$	2.02023	0.0769088
$691$	−29.8746	−1.13648	−0.568242	−	0.822862i	$-0.692376\pi$
$691$	−29.8746	−1.13648	−0.568242	+	0.822862i	$0.692376\pi$
$692$	1.39808	0.0531470
$693$	−2.35837	−0.0895872
$694$	2.77296	0.105260
$695$	−10.5129	−0.398776
$696$	−0.0741132	−0.00280926
$697$	0	0
$698$	27.6385	1.04613
$699$	−22.7169	−0.859231
$700$	−0.797770	−0.0301529
$701$	29.6430	1.11960	0.559800	−	0.828628i	$-0.310878\pi$
$701$	29.6430	1.11960	0.559800	+	0.828628i	$0.310878\pi$
$702$	1.56978	0.0592474
$703$	−4.08529	−0.154079
$704$	20.9732	0.790458
$705$	15.4509	0.581913
$706$	9.87572	0.371677
$707$	−10.0435	−0.377723
$708$	−2.54782	−0.0957530
$709$	48.8404	1.83424	0.917120	−	0.398612i	$-0.130508\pi$
$709$	48.8404	1.83424	0.917120	+	0.398612i	$0.130508\pi$
$710$	16.2987	0.611680
$711$	10.7677	0.403818
$712$	−31.1760	−1.16837
$713$	6.07233	0.227410
$714$	0	0
$715$	5.43331	0.203194
$716$	−3.62015	−0.135292
$717$	−5.42712	−0.202680
$718$	−26.4742	−0.988008
$719$	−33.8010	−1.26056	−0.630282	−	0.776366i	$-0.717061\pi$
$719$	−33.8010	−1.26056	−0.630282	+	0.776366i	$0.717061\pi$
$720$	−5.16262	−0.192399
$721$	−0.982712	−0.0365981
$722$	−9.00550	−0.335150
$723$	26.9675	1.00293
$724$	−10.2427	−0.380666
$725$	−0.0412873	−0.00153337
$726$	−6.72924	−0.249745
$727$	12.4398	0.461367	0.230684	−	0.973029i	$-0.425904\pi$
$727$	12.4398	0.461367	0.230684	+	0.973029i	$0.425904\pi$
$728$	3.87541	0.143632
$729$	1.00000	0.0370370
$730$	−29.9788	−1.10956
$731$	0	0
$732$	−2.95285	−0.109140
$733$	13.2154	0.488123	0.244061	−	0.969760i	$-0.421520\pi$
$733$	13.2154	0.488123	0.244061	+	0.969760i	$0.421520\pi$
$734$	1.36752	0.0504763
$735$	1.81608	0.0669872
$736$	2.33029	0.0858956
$737$	−1.21762	−0.0448518
$738$	8.77616	0.323055
$739$	32.3074	1.18845	0.594223	−	0.804300i	$-0.297460\pi$
$739$	32.3074	1.18845	0.594223	+	0.804300i	$0.297460\pi$
$740$	−1.01579	−0.0373413
$741$	−4.34336	−0.159557
$742$	1.33335	0.0489489
$743$	45.9378	1.68529	0.842647	−	0.538466i	$-0.180996\pi$
$743$	45.9378	1.68529	0.842647	+	0.538466i	$0.180996\pi$
$744$	−20.6353	−0.756527
$745$	37.4169	1.37085
$746$	−32.3862	−1.18574
$747$	−7.22647	−0.264403
$748$	0	0
$749$	13.5270	0.494267
$750$	−15.0609	−0.549946
$751$	49.4009	1.80266	0.901332	−	0.433128i	$-0.142590\pi$
$751$	49.4009	1.80266	0.901332	+	0.433128i	$0.142590\pi$
$752$	−24.1853	−0.881948
$753$	11.3447	0.413425
$754$	0.0380831	0.00138691
$755$	26.9311	0.980123
$756$	0.468767	0.0170489
$757$	44.2730	1.60913	0.804564	−	0.593866i	$-0.202399\pi$
$757$	44.2730	1.60913	0.804564	+	0.593866i	$0.202399\pi$
$758$	34.3499	1.24765
$759$	2.12011	0.0769550
$760$	18.9952	0.689029
$761$	−5.71316	−0.207102	−0.103551	−	0.994624i	$-0.533021\pi$
$761$	−5.71316	−0.207102	−0.103551	+	0.994624i	$0.533021\pi$
$762$	−21.5783	−0.781699
$763$	−9.56837	−0.346398
$764$	−10.6543	−0.385458
$765$	0	0
$766$	−22.0486	−0.796648
$767$	6.89492	0.248961
$768$	−10.5841	−0.381920
$769$	−27.7810	−1.00181	−0.500904	−	0.865503i	$-0.666999\pi$
$769$	−27.7810	−1.00181	−0.500904	+	0.865503i	$0.666999\pi$
$770$	−5.29991	−0.190996
$771$	26.4860	0.953872
$772$	11.0132	0.396373
$773$	−14.0849	−0.506600	−0.253300	−	0.967388i	$-0.581516\pi$
$773$	−14.0849	−0.506600	−0.253300	+	0.967388i	$0.581516\pi$
$774$	6.45709	0.232095
$775$	−11.4956	−0.412934
$776$	0.626695	0.0224971
$777$	−1.19320	−0.0428059
$778$	−27.9392	−1.00167
$779$	−24.2824	−0.870009
$780$	−1.07996	−0.0386689
$781$	17.1045	0.612047
$782$	0	0
$783$	0.0242602	0.000866990 0
$784$	−2.84272	−0.101526
$785$	6.88607	0.245774
$786$	−2.85194	−0.101725
$787$	−17.4731	−0.622850	−0.311425	−	0.950271i	$-0.600806\pi$
$787$	−17.4731	−0.622850	−0.311425	+	0.950271i	$0.600806\pi$
$788$	3.18325	0.113399
$789$	0.781732	0.0278304
$790$	24.1979	0.860921
$791$	−11.4380	−0.406689
$792$	−7.20466	−0.256006
$793$	7.99101	0.283769
$794$	−44.6354	−1.58405
$795$	−1.95686	−0.0694027
$796$	2.36312	0.0837586
$797$	−16.4792	−0.583724	−0.291862	−	0.956460i	$-0.594275\pi$
$797$	−16.4792	−0.583724	−0.291862	+	0.956460i	$0.594275\pi$
$798$	4.23672	0.149978
$799$	0	0
$800$	−4.41150	−0.155970
$801$	10.2051	0.360581
$802$	3.35374	0.118425
$803$	−31.4608	−1.11023
$804$	0.242024	0.00853551
$805$	−1.63260	−0.0575416
$806$	10.6035	0.373491
$807$	10.0993	0.355513
$808$	−30.6820	−1.07939
$809$	−31.4247	−1.10483	−0.552417	−	0.833568i	$-0.686294\pi$
$809$	−31.4247	−1.10483	−0.552417	+	0.833568i	$0.686294\pi$
$810$	2.24727	0.0789611
$811$	21.9386	0.770367	0.385183	−	0.922840i	$-0.374138\pi$
$811$	21.9386	0.770367	0.385183	+	0.922840i	$0.374138\pi$
$812$	0.0113724	0.000399093 0
$813$	16.8897	0.592346
$814$	3.48215	0.122049
$815$	−39.3089	−1.37693
$816$	0	0
$817$	−17.8659	−0.625049
$818$	−40.4964	−1.41592
$819$	−1.26858	−0.0443277
$820$	−6.03776	−0.210848
$821$	−21.5124	−0.750789	−0.375395	−	0.926865i	$-0.622493\pi$
$821$	−21.5124	−0.750789	−0.375395	+	0.926865i	$0.622493\pi$
$822$	−0.731527	−0.0255149
$823$	−43.1249	−1.50324	−0.751620	−	0.659597i	$-0.770727\pi$
$823$	−43.1249	−1.50324	−0.751620	+	0.659597i	$0.770727\pi$
$824$	−3.00211	−0.104584
$825$	−4.01360	−0.139735
$826$	−6.72563	−0.234015
$827$	−26.4330	−0.919164	−0.459582	−	0.888135i	$-0.652001\pi$
$827$	−26.4330	−0.919164	−0.459582	+	0.888135i	$0.652001\pi$
$828$	−0.421407	−0.0146449
$829$	4.62524	0.160641	0.0803205	−	0.996769i	$-0.474406\pi$
$829$	4.62524	0.160641	0.0803205	+	0.996769i	$0.474406\pi$
$830$	−16.2399	−0.563694
$831$	−4.31657	−0.149740
$832$	11.2816	0.391118
$833$	0	0
$834$	−7.16319	−0.248041
$835$	−46.2458	−1.60040
$836$	3.78511	0.130911
$837$	6.75476	0.233479
$838$	−1.05409	−0.0364129
$839$	31.6127	1.09139	0.545695	−	0.837984i	$-0.316266\pi$
$839$	31.6127	1.09139	0.545695	+	0.837984i	$0.316266\pi$
$840$	5.54799	0.191424
$841$	−28.9994	−0.999980
$842$	−1.80900	−0.0623424
$843$	14.7023	0.506375
$844$	3.69121	0.127057
$845$	−20.6865	−0.711636
$846$	10.5278	0.361953
$847$	5.43807	0.186854
$848$	3.06309	0.105187
$849$	−10.3543	−0.355359
$850$	0	0
$851$	1.07265	0.0367700
$852$	−3.39981	−0.116475
$853$	5.05011	0.172912	0.0864562	−	0.996256i	$-0.472446\pi$
$853$	5.05011	0.172912	0.0864562	+	0.996256i	$0.472446\pi$
$854$	−7.79480	−0.266733
$855$	−6.21790	−0.212648
$856$	41.3241	1.41243
$857$	11.7212	0.400388	0.200194	−	0.979756i	$-0.435843\pi$
$857$	11.7212	0.400388	0.200194	+	0.979756i	$0.435843\pi$
$858$	3.70212	0.126388
$859$	−48.5065	−1.65502	−0.827510	−	0.561450i	$-0.810244\pi$
$859$	−48.5065	−1.65502	−0.827510	+	0.561450i	$0.810244\pi$
$860$	−4.44230	−0.151481
$861$	−7.09225	−0.241703
$862$	27.1344	0.924201
$863$	13.5157	0.460081	0.230041	−	0.973181i	$-0.426114\pi$
$863$	13.5157	0.460081	0.230041	+	0.973181i	$0.426114\pi$
$864$	2.59218	0.0881877
$865$	−5.41639	−0.184163
$866$	−31.5204	−1.07111
$867$	0	0
$868$	3.16641	0.107475
$869$	25.3941	0.861437
$870$	0.0545194	0.00184838
$871$	−0.654965	−0.0221926
$872$	−29.2307	−0.989875
$873$	−0.205143	−0.00694302
$874$	−3.80868	−0.128830
$875$	12.1711	0.411458
$876$	6.25338	0.211282
$877$	40.1461	1.35564	0.677820	−	0.735228i	$-0.262925\pi$
$877$	40.1461	1.35564	0.677820	+	0.735228i	$0.262925\pi$
$878$	−46.1398	−1.55714
$879$	10.9848	0.370509
$880$	−12.1754	−0.410432
$881$	−26.3597	−0.888082	−0.444041	−	0.896006i	$-0.646456\pi$
$881$	−26.3597	−0.888082	−0.444041	+	0.896006i	$0.646456\pi$
$882$	1.23743	0.0416664
$883$	50.3856	1.69561	0.847805	−	0.530308i	$-0.177924\pi$
$883$	50.3856	1.69561	0.847805	+	0.530308i	$0.177924\pi$
$884$	0	0
$885$	9.87069	0.331800
$886$	50.0750	1.68230
$887$	20.6679	0.693959	0.346980	−	0.937873i	$-0.387207\pi$
$887$	20.6679	0.693959	0.346980	+	0.937873i	$0.387207\pi$
$888$	−3.64514	−0.122323
$889$	17.4380	0.584851
$890$	22.9338	0.768741
$891$	2.35837	0.0790085
$892$	−9.61203	−0.321834
$893$	−29.1290	−0.974765
$894$	25.4949	0.852677
$895$	14.0251	0.468807
$896$	−5.82022	−0.194440
$897$	1.14041	0.0380773
$898$	−11.7794	−0.393084
$899$	0.163872	0.00546544
$900$	0.797770	0.0265923
$901$	0	0
$902$	20.6975	0.689150
$903$	−5.21815	−0.173649
$904$	−34.9423	−1.16216
$905$	39.6818	1.31907
$906$	18.3501	0.609643
$907$	−3.68850	−0.122475	−0.0612373	−	0.998123i	$-0.519505\pi$
$907$	−3.68850	−0.122475	−0.0612373	+	0.998123i	$0.519505\pi$
$908$	−4.14327	−0.137499
$909$	10.0435	0.333120
$910$	−2.85084	−0.0945044
$911$	−5.69091	−0.188548	−0.0942741	−	0.995546i	$-0.530053\pi$
$911$	−5.69091	−0.188548	−0.0942741	+	0.995546i	$0.530053\pi$
$912$	9.73293	0.322289
$913$	−17.0427	−0.564032
$914$	9.96762	0.329700
$915$	11.4398	0.378189
$916$	2.93538	0.0969877
$917$	2.30473	0.0761088
$918$	0	0
$919$	14.4789	0.477614	0.238807	−	0.971067i	$-0.423244\pi$
$919$	14.4789	0.477614	0.238807	+	0.971067i	$0.423244\pi$
$920$	−4.98748	−0.164432
$921$	13.8104	0.455069
$922$	−41.1316	−1.35460
$923$	9.20056	0.302840
$924$	1.10553	0.0363692
$925$	−2.03065	−0.0667673
$926$	−22.8754	−0.751731
$927$	0.982712	0.0322765
$928$	0.0628869	0.00206436
$929$	−10.2976	−0.337853	−0.168926	−	0.985629i	$-0.554030\pi$
$929$	−10.2976	−0.337853	−0.168926	+	0.985629i	$0.554030\pi$
$930$	15.1798	0.497765
$931$	−3.42380	−0.112211
$932$	10.6489	0.348817
$933$	24.2650	0.794399
$934$	9.64989	0.315754
$935$	0	0
$936$	−3.87541	−0.126672
$937$	32.4551	1.06026	0.530132	−	0.847915i	$-0.322142\pi$
$937$	32.4551	1.06026	0.530132	+	0.847915i	$0.322142\pi$
$938$	0.638883	0.0208603
$939$	−16.0803	−0.524762
$940$	−7.24285	−0.236236
$941$	−14.6166	−0.476489	−0.238244	−	0.971205i	$-0.576572\pi$
$941$	−14.6166	−0.476489	−0.238244	+	0.971205i	$0.576572\pi$
$942$	4.69199	0.152873
$943$	6.37571	0.207622
$944$	−15.4507	−0.502876
$945$	−1.81608	−0.0590771
$946$	15.2282	0.495113
$947$	−0.695796	−0.0226103	−0.0113052	−	0.999936i	$-0.503599\pi$
$947$	−0.695796	−0.0226103	−0.0113052	+	0.999936i	$0.503599\pi$
$948$	−5.04752	−0.163936
$949$	−16.9229	−0.549341
$950$	7.21025	0.233931
$951$	4.29099	0.139145
$952$	0	0
$953$	−36.8109	−1.19242	−0.596210	−	0.802828i	$-0.703328\pi$
$953$	−36.8109	−1.19242	−0.596210	+	0.802828i	$0.703328\pi$
$954$	−1.33335	−0.0431689
$955$	41.2765	1.33568
$956$	2.54406	0.0822806
$957$	0.0572147	0.00184949
$958$	−22.4870	−0.726522
$959$	0.591167	0.0190898
$960$	16.1506	0.521258
$961$	14.6268	0.471833
$962$	1.87306	0.0603898
$963$	−13.5270	−0.435903
$964$	−12.6415	−0.407154
$965$	−42.6669	−1.37350
$966$	−1.11241	−0.0357913
$967$	−3.34730	−0.107642	−0.0538210	−	0.998551i	$-0.517140\pi$
$967$	−3.34730	−0.107642	−0.0538210	+	0.998551i	$0.517140\pi$
$968$	16.6129	0.533959
$969$	0	0
$970$	−0.461011	−0.0148022
$971$	5.99767	0.192474	0.0962372	−	0.995358i	$-0.469319\pi$
$971$	5.99767	0.192474	0.0962372	+	0.995358i	$0.469319\pi$
$972$	−0.468767	−0.0150357
$973$	5.78876	0.185579
$974$	−5.76281	−0.184652
$975$	−2.15893	−0.0691410
$976$	−17.9069	−0.573184
$977$	−30.5729	−0.978115	−0.489058	−	0.872252i	$-0.662659\pi$
$977$	−30.5729	−0.978115	−0.489058	+	0.872252i	$0.662659\pi$
$978$	−26.7840	−0.856459
$979$	24.0675	0.769202
$980$	−0.851318	−0.0271944
$981$	9.56837	0.305494
$982$	−51.7298	−1.65077
$983$	1.11337	0.0355111	0.0177555	−	0.999842i	$-0.494348\pi$
$983$	1.11337	0.0355111	0.0177555	+	0.999842i	$0.494348\pi$
$984$	−21.6663	−0.690696
$985$	−12.3325	−0.392945
$986$	0	0
$987$	−8.50780	−0.270806
$988$	2.03602	0.0647745
$989$	4.69096	0.149164
$990$	5.29991	0.168442
$991$	40.7818	1.29548	0.647738	−	0.761863i	$-0.275715\pi$
$991$	40.7818	1.29548	0.647738	+	0.761863i	$0.275715\pi$
$992$	17.5096	0.555929
$993$	−19.3135	−0.612894
$994$	−8.97467	−0.284659
$995$	−9.15513	−0.290237
$996$	3.38753	0.107338
$997$	−25.1182	−0.795502	−0.397751	−	0.917493i	$-0.630209\pi$
$997$	−25.1182	−0.795502	−0.397751	+	0.917493i	$0.630209\pi$
$998$	−13.0976	−0.414597
$999$	1.19320	0.0377512

Display

a_p

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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

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a_p

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6069.2.a.be.1.6		10
17.2	even	8		357.2.k.b.106.6	yes	20
17.9	even	8		357.2.k.b.64.5	✓	20
17.16	even	2		6069.2.a.bd.1.6		10
51.2	odd	8		1071.2.n.b.820.5		20
51.26	odd	8		1071.2.n.b.64.6		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.k.b.64.5	✓	20	17.9	even	8
357.2.k.b.106.6	yes	20	17.2	even	8
1071.2.n.b.64.6		20	51.26	odd	8
1071.2.n.b.820.5		20	51.2	odd	8
6069.2.a.bd.1.6		10	17.16	even	2
6069.2.a.be.1.6		10	1.1	even	1	trivial

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	6069.2.a.be.1.6

Level	$6069$
Weight	$2$
Character	6069.1
Self dual	yes
Analytic conductor	$48.461$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$1$