LMFDB - Embedded newform 6069.2.a.bd.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	$-0.633107\pi$
$5$	−1.81608	−0.812176	−0.406088	+	0.913834i	$0.633107\pi$
$6$	−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	$-0.615702\pi$
$11$	−2.35837	−0.711076	−0.355538	+	0.934662i	$0.615702\pi$
$12$	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	$-0.529877\pi$
$23$	−0.898969	−0.187448	−0.0937240	+	0.995598i	$0.529877\pi$
$24$	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	$-0.500717\pi$
$29$	−0.0242602	−0.00450501	−0.00225251	+	0.999997i	$0.500717\pi$
$30$	2.24727	0.410294
$31$	−6.75476	−1.21319	−0.606596	−	0.795011i	$-0.707465\pi$
$31$	−6.75476	−1.21319	−0.606596	+	0.795011i	$0.707465\pi$
$32$	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	$-0.531270\pi$
$37$	−1.19320	−0.196161	−0.0980806	+	0.995178i	$0.531270\pi$
$38$	−4.23672	−0.687286
$39$	−1.26858	−0.203135
$40$	5.54799	0.877215
$41$	−7.09225	−1.10762	−0.553811	−	0.832642i	$-0.686827\pi$
$41$	−7.09225	−1.10762	−0.553811	+	0.832642i	$0.686827\pi$
$42$	−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	$-0.632124\pi$
$61$	−6.29919	−0.806528	−0.403264	+	0.915084i	$0.632124\pi$
$62$	−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	$-0.641616\pi$
$71$	−7.25266	−0.860733	−0.430366	+	0.902654i	$0.641616\pi$
$72$	−3.05493	−0.360027
$73$	13.3401	1.56134	0.780668	−	0.624946i	$-0.214879\pi$
$73$	13.3401	1.56134	0.780668	+	0.624946i	$0.214879\pi$
$74$	−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	$-0.707118\pi$
$79$	−10.7677	−1.21146	−0.605728	+	0.795672i	$0.707118\pi$
$80$	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	$-0.496685\pi$
$97$	0.205143	0.0208291	0.0104145	+	0.999946i	$0.496685\pi$
$98$	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	$-0.273151\pi$
$107$	13.5270	1.30771	0.653854	+	0.756621i	$0.273151\pi$
$108$	0.468767	0.0451071
$109$	−9.56837	−0.916483	−0.458242	−	0.888828i	$-0.651521\pi$
$109$	−9.56837	−0.916483	−0.458242	+	0.888828i	$0.651521\pi$
$110$	5.29991	0.505327
$111$	1.19320	0.113254
$112$	−2.84272	−0.268612
$113$	−11.4380	−1.07600	−0.537999	−	0.842946i	$-0.680819\pi$
$113$	−11.4380	−1.07600	−0.537999	+	0.842946i	$0.680819\pi$
$114$	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	$-0.467897\pi$
$131$	2.30473	0.201365	0.100682	+	0.994919i	$0.467897\pi$
$132$	−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	$-0.421048\pi$
$139$	5.78876	0.490996	0.245498	+	0.969397i	$0.421048\pi$
$140$	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	$-0.177999\pi$
$163$	21.6449	1.69536	0.847679	+	0.530509i	$0.177999\pi$
$164$	3.32461	0.259608
$165$	−4.28300	−0.333431
$166$	−8.94225	−0.694053
$167$	25.4646	1.97051	0.985256	−	0.171089i	$-0.0547284\pi$
$167$	25.4646	1.97051	0.985256	+	0.171089i	$0.0547284\pi$
$168$	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	$-0.463833\pi$
$173$	2.98246	0.226752	0.113376	+	0.993552i	$0.463833\pi$
$174$	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	$-0.801653\pi$
$181$	−21.8503	−1.62412	−0.812058	+	0.583576i	$0.801653\pi$
$182$	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	$-0.179264\pi$
$193$	23.4939	1.69113	0.845565	+	0.533872i	$0.179264\pi$
$194$	0.253850	0.0182253
$195$	2.30384	0.164981
$196$	−0.468767	−0.0334833
$197$	6.79070	0.483817	0.241909	−	0.970299i	$-0.422227\pi$
$197$	6.79070	0.483817	0.241909	+	0.970299i	$0.422227\pi$
$198$	−2.91832	−0.207396
$199$	5.04115	0.357357	0.178679	−	0.983907i	$-0.442818\pi$
$199$	5.04115	0.357357	0.178679	+	0.983907i	$0.442818\pi$
$200$	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	$-0.412631\pi$
$211$	7.87431	0.542090	0.271045	+	0.962567i	$0.412631\pi$
$212$	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	$-0.594760\pi$
$227$	−8.83865	−0.586642	−0.293321	+	0.956014i	$0.594760\pi$
$228$	−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	$-0.232871\pi$
$233$	22.7169	1.48823	0.744116	+	0.668051i	$0.232871\pi$
$234$	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	$-0.834957\pi$
$241$	−26.9675	−1.73713	−0.868565	+	0.495575i	$0.834957\pi$
$242$	−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	$-0.599621\pi$
$269$	−10.0993	−0.615767	−0.307884	+	0.951424i	$0.599621\pi$
$270$	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	$-0.458605\pi$
$277$	4.31657	0.259358	0.129679	+	0.991556i	$0.458605\pi$
$278$	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	$-0.400424\pi$
$283$	10.3543	0.615499	0.307750	+	0.951467i	$0.400424\pi$
$284$	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	$-0.741498\pi$
$311$	−24.2650	−1.37594	−0.687970	+	0.725739i	$0.741498\pi$
$312$	3.87541	0.219402
$313$	16.0803	0.908914	0.454457	−	0.890769i	$-0.349833\pi$
$313$	16.0803	0.908914	0.454457	+	0.890769i	$0.349833\pi$
$314$	4.69199	0.264784
$315$	−1.81608	−0.102325
$316$	5.04752	0.283945
$317$	−4.29099	−0.241006	−0.120503	−	0.992713i	$-0.538451\pi$
$317$	−4.29099	−0.241006	−0.120503	+	0.992713i	$0.538451\pi$
$318$	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	$-0.679336\pi$
$337$	−19.6082	−1.06813	−0.534063	+	0.845444i	$0.679336\pi$
$338$	−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	$-0.519158\pi$
$347$	−2.24091	−0.120298	−0.0601491	+	0.998189i	$0.519158\pi$
$348$	−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	$-0.509183\pi$
$367$	−1.10513	−0.0576875	−0.0288437	+	0.999584i	$0.509183\pi$
$368$	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	$-0.752639\pi$
$379$	−27.7591	−1.42589	−0.712944	+	0.701221i	$0.752639\pi$
$380$	−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	$-0.139738\pi$
$397$	36.0710	1.81035	0.905177	+	0.425036i	$0.139738\pi$
$398$	6.23807	0.312686
$399$	3.42380	0.171404
$400$	4.83789	0.241894
$401$	−2.71025	−0.135343	−0.0676716	−	0.997708i	$-0.521557\pi$
$401$	−2.71025	−0.135343	−0.0676716	+	0.997708i	$0.521557\pi$
$402$	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	$-0.493376\pi$
$419$	0.851837	0.0416150	0.0208075	+	0.999784i	$0.493376\pi$
$420$	−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	$-0.677102\pi$
$431$	−21.9280	−1.05624	−0.528118	+	0.849171i	$0.677102\pi$
$432$	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	$-0.150843\pi$
$439$	37.2868	1.77960	0.889802	+	0.456348i	$0.150843\pi$
$440$	−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	$-0.427886\pi$
$449$	9.51925	0.449241	0.224621	+	0.974446i	$0.427886\pi$
$450$	−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	$-0.363726\pi$
$479$	18.1723	0.830315	0.415158	+	0.909750i	$0.363726\pi$
$480$	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	$-0.466350\pi$
$487$	4.65708	0.211033	0.105516	+	0.994418i	$0.466350\pi$
$488$	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	$-0.423864\pi$
$499$	10.5845	0.473828	0.236914	+	0.971531i	$0.423864\pi$
$500$	−5.70541	−0.255154
$501$	−25.4646	−1.13768
$502$	14.0383	0.626561
$503$	27.3831	1.22095	0.610476	−	0.792035i	$-0.290978\pi$
$503$	27.3831	1.22095	0.610476	+	0.792035i	$0.290978\pi$
$504$	−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	$-0.373841\pi$
$521$	17.6233	0.772089	0.386044	+	0.922480i	$0.373841\pi$
$522$	−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	$-0.231426\pi$
$541$	34.7561	1.49428	0.747140	+	0.664667i	$0.231426\pi$
$542$	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	$-0.747544\pi$
$547$	−32.8195	−1.40326	−0.701630	+	0.712542i	$0.747544\pi$
$548$	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	$-0.486670\pi$
$571$	2.00086	0.0837335	0.0418667	+	0.999123i	$0.486670\pi$
$572$	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	$-0.250696\pi$
$601$	34.5940	1.41112	0.705559	+	0.708651i	$0.250696\pi$
$602$	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	$-0.482810\pi$
$607$	2.65970	0.107954	0.0539770	+	0.998542i	$0.482810\pi$
$608$	−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	$-0.666571\pi$
$617$	−24.8265	−0.999478	−0.499739	+	0.866176i	$0.666571\pi$
$618$	−1.21604	−0.0489162
$619$	−18.8851	−0.759055	−0.379527	−	0.925181i	$-0.623913\pi$
$619$	−18.8851	−0.759055	−0.379527	+	0.925181i	$0.623913\pi$
$620$	−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	$-0.383937\pi$
$641$	18.0566	0.713191	0.356596	+	0.934259i	$0.383937\pi$
$642$	−16.7388	−0.660626
$643$	26.5565	1.04729	0.523643	−	0.851938i	$-0.324572\pi$
$643$	26.5565	1.04729	0.523643	+	0.851938i	$0.324572\pi$
$644$	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	$-0.501757\pi$
$653$	−0.282151	−0.0110414	−0.00552070	+	0.999985i	$0.501757\pi$
$654$	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	$-0.803693\pi$
$673$	−42.3264	−1.63156	−0.815781	+	0.578360i	$0.803693\pi$
$674$	−24.2638	−0.934606
$675$	1.70185	0.0655042
$676$	5.33959	0.205369
$677$	41.9506	1.61229	0.806146	−	0.591717i	$-0.201550\pi$
$677$	41.9506	1.61229	0.806146	+	0.591717i	$0.201550\pi$
$678$	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	$-0.616155\pi$
$683$	−18.6529	−0.713734	−0.356867	+	0.934155i	$0.616155\pi$
$684$	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	$-0.307624\pi$
$691$	29.8746	1.13648	0.568242	+	0.822862i	$0.307624\pi$
$692$	−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	$-0.869492\pi$
$709$	−48.8404	−1.83424	−0.917120	+	0.398612i	$0.869492\pi$
$710$	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	$-0.282939\pi$
$719$	33.8010	1.26056	0.630282	+	0.776366i	$0.282939\pi$
$720$	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	$-0.819004\pi$
$743$	−45.9378	−1.68529	−0.842647	+	0.538466i	$0.819004\pi$
$744$	−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	$-0.857410\pi$
$751$	−49.4009	−1.80266	−0.901332	+	0.433128i	$0.857410\pi$
$752$	−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	$-0.399194\pi$
$787$	17.4731	0.622850	0.311425	+	0.950271i	$0.399194\pi$
$788$	−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	$-0.313706\pi$
$809$	31.4247	1.10483	0.552417	+	0.833568i	$0.313706\pi$
$810$	−2.24727	−0.0789611
$811$	−21.9386	−0.770367	−0.385183	−	0.922840i	$-0.625862\pi$
$811$	−21.9386	−0.770367	−0.385183	+	0.922840i	$0.625862\pi$
$812$	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	$-0.377507\pi$
$821$	21.5124	0.750789	0.375395	+	0.926865i	$0.377507\pi$
$822$	0.731527	0.0255149
$823$	43.1249	1.50324	0.751620	−	0.659597i	$-0.229273\pi$
$823$	43.1249	1.50324	0.751620	+	0.659597i	$0.229273\pi$
$824$	−3.00211	−0.104584
$825$	−4.01360	−0.139735
$826$	6.72563	0.234015
$827$	26.4330	0.919164	0.459582	−	0.888135i	$-0.347999\pi$
$827$	26.4330	0.919164	0.459582	+	0.888135i	$0.347999\pi$
$828$	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	$-0.683734\pi$
$839$	−31.6127	−1.09139	−0.545695	+	0.837984i	$0.683734\pi$
$840$	−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	$-0.527554\pi$
$853$	−5.05011	−0.172912	−0.0864562	+	0.996256i	$0.527554\pi$
$854$	−7.79480	−0.266733
$855$	6.21790	0.212648
$856$	−41.3241	−1.41243
$857$	−11.7212	−0.400388	−0.200194	−	0.979756i	$-0.564157\pi$
$857$	−11.7212	−0.400388	−0.200194	+	0.979756i	$0.564157\pi$
$858$	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	$-0.737075\pi$
$877$	−40.1461	−1.35564	−0.677820	+	0.735228i	$0.737075\pi$
$878$	46.1398	1.55714
$879$	−10.9848	−0.370509
$880$	−12.1754	−0.410432
$881$	26.3597	0.888082	0.444041	−	0.896006i	$-0.353544\pi$
$881$	26.3597	0.888082	0.444041	+	0.896006i	$0.353544\pi$
$882$	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	$-0.612793\pi$
$887$	−20.6679	−0.693959	−0.346980	+	0.937873i	$0.612793\pi$
$888$	−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	$-0.480495\pi$
$907$	3.68850	0.122475	0.0612373	+	0.998123i	$0.480495\pi$
$908$	4.14327	0.137499
$909$	10.0435	0.333120
$910$	−2.85084	−0.0945044
$911$	5.69091	0.188548	0.0942741	−	0.995546i	$-0.469947\pi$
$911$	5.69091	0.188548	0.0942741	+	0.995546i	$0.469947\pi$
$912$	−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	$-0.445970\pi$
$929$	10.2976	0.337853	0.168926	+	0.985629i	$0.445970\pi$
$930$	−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	$-0.423428\pi$
$941$	14.6166	0.476489	0.238244	+	0.971205i	$0.423428\pi$
$942$	−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	$-0.496401\pi$
$947$	0.695796	0.0226103	0.0113052	+	0.999936i	$0.496401\pi$
$948$	−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	$-0.505652\pi$
$983$	−1.11337	−0.0355111	−0.0177555	+	0.999842i	$0.505652\pi$
$984$	−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	$-0.724285\pi$
$991$	−40.7818	−1.29548	−0.647738	+	0.761863i	$0.724285\pi$
$992$	−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	$-0.369791\pi$
$997$	25.1182	0.795502	0.397751	+	0.917493i	$0.369791\pi$
$998$	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

) (See only

a_p

) (See only

a_p

)

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.k.b.64.5	✓	20	17.8	even	8
357.2.k.b.106.6	yes	20	17.15	even	8
1071.2.n.b.64.6		20	51.8	odd	8
1071.2.n.b.820.5		20	51.32	odd	8
6069.2.a.bd.1.6		10	1.1	even	1	trivial
6069.2.a.be.1.6		10	17.16	even	2

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.bd.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	$1$