LMFDB - Embedded newform 6021.2.a.t.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6021,2,Mod(1,6021)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6021, 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("6021.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6021 = 3^{3} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6021.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:	$48.0779270570$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	6021.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.37704 q^{2} -0.103765 q^{4} +2.60531 q^{5} -2.14080 q^{7} +2.89697 q^{8} +O(q^{10})$	\(q-1.37704 q^{2} -0.103765 q^{4} +2.60531 q^{5} -2.14080 q^{7} +2.89697 q^{8} -3.58761 q^{10} +5.33539 q^{11} -0.800507 q^{13} +2.94797 q^{14} -3.78170 q^{16} +4.24866 q^{17} -3.39603 q^{19} -0.270340 q^{20} -7.34703 q^{22} +0.350477 q^{23} +1.78762 q^{25} +1.10233 q^{26} +0.222141 q^{28} +6.14352 q^{29} +1.59280 q^{31} -0.586383 q^{32} -5.85057 q^{34} -5.57744 q^{35} +1.51297 q^{37} +4.67646 q^{38} +7.54748 q^{40} +4.20875 q^{41} +5.59560 q^{43} -0.553629 q^{44} -0.482620 q^{46} -0.326594 q^{47} -2.41696 q^{49} -2.46161 q^{50} +0.0830649 q^{52} -2.42798 q^{53} +13.9003 q^{55} -6.20183 q^{56} -8.45986 q^{58} -0.747504 q^{59} -2.16858 q^{61} -2.19334 q^{62} +8.37088 q^{64} -2.08556 q^{65} +2.07659 q^{67} -0.440864 q^{68} +7.68035 q^{70} +10.5876 q^{71} +0.550054 q^{73} -2.08342 q^{74} +0.352390 q^{76} -11.4220 q^{77} +6.13896 q^{79} -9.85249 q^{80} -5.79561 q^{82} +1.55908 q^{83} +11.0691 q^{85} -7.70536 q^{86} +15.4564 q^{88} +10.8215 q^{89} +1.71373 q^{91} -0.0363674 q^{92} +0.449732 q^{94} -8.84769 q^{95} +3.09238 q^{97} +3.32825 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) $ 40 * q + 46 * q^4 + 16 * q^7	$ 40 q + 46 q^{4} + 16 q^{7} + 22 q^{10} + 14 q^{13} + 50 q^{16} + 64 q^{19} + 12 q^{22} + 40 q^{25} + 48 q^{28} + 54 q^{31} + 32 q^{34} + 24 q^{37} + 40 q^{40} + 24 q^{43} + 52 q^{46} + 64 q^{49} + 18 q^{52} + 36 q^{55} + 8 q^{58} + 58 q^{61} + 120 q^{64} + 52 q^{67} - 30 q^{70} + 50 q^{73} + 112 q^{76} + 60 q^{79} + 50 q^{82} + 38 q^{85} + 16 q^{88} + 118 q^{91} + 44 q^{94} + 38 q^{97}+O(q^{100}) $ 40 * q + 46 * q^4 + 16 * q^7 + 22 * q^10 + 14 * q^13 + 50 * q^16 + 64 * q^19 + 12 * q^22 + 40 * q^25 + 48 * q^28 + 54 * q^31 + 32 * q^34 + 24 * q^37 + 40 * q^40 + 24 * q^43 + 52 * q^46 + 64 * q^49 + 18 * q^52 + 36 * q^55 + 8 * q^58 + 58 * q^61 + 120 * q^64 + 52 * q^67 - 30 * q^70 + 50 * q^73 + 112 * q^76 + 60 * q^79 + 50 * q^82 + 38 * q^85 + 16 * q^88 + 118 * q^91 + 44 * q^94 + 38 * q^97

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 $

$2$	−1.37704	−0.973713	−0.486857	−	0.873482i	$-0.661857\pi$
$2$	−1.37704	−0.973713	−0.486857	+	0.873482i	$0.661857\pi$
$3$	0	0
$3$	0	0
$4$	−0.103765	−0.0518827
$5$	2.60531	1.16513	0.582564	−	0.812785i	$-0.302050\pi$
$5$	2.60531	1.16513	0.582564	+	0.812785i	$0.302050\pi$
$6$	0	0
$7$	−2.14080	−0.809147	−0.404574	−	0.914505i	$-0.632580\pi$
$7$	−2.14080	−0.809147	−0.404574	+	0.914505i	$0.632580\pi$
$8$	2.89697	1.02423
$9$	0	0
$10$	−3.58761	−1.13450
$11$	5.33539	1.60868	0.804340	−	0.594169i	$-0.202519\pi$
$11$	5.33539	1.60868	0.804340	+	0.594169i	$0.202519\pi$
$12$	0	0
$13$	−0.800507	−0.222021	−0.111010	−	0.993819i	$-0.535409\pi$
$13$	−0.800507	−0.222021	−0.111010	+	0.993819i	$0.535409\pi$
$14$	2.94797	0.787877
$15$	0	0
$16$	−3.78170	−0.945425
$17$	4.24866	1.03045	0.515226	−	0.857054i	$-0.327708\pi$
$17$	4.24866	1.03045	0.515226	+	0.857054i	$0.327708\pi$
$18$	0	0
$19$	−3.39603	−0.779102	−0.389551	−	0.921005i	$-0.627370\pi$
$19$	−3.39603	−0.779102	−0.389551	+	0.921005i	$0.627370\pi$
$20$	−0.270340	−0.0604500
$21$	0	0
$22$	−7.34703	−1.56639
$23$	0.350477	0.0730795	0.0365397	−	0.999332i	$-0.488366\pi$
$23$	0.350477	0.0730795	0.0365397	+	0.999332i	$0.488366\pi$
$24$	0	0
$25$	1.78762	0.357523
$26$	1.10233	0.216184
$27$	0	0
$28$	0.222141	0.0419807
$29$	6.14352	1.14082	0.570411	−	0.821359i	$-0.306784\pi$
$29$	6.14352	1.14082	0.570411	+	0.821359i	$0.306784\pi$
$30$	0	0
$31$	1.59280	0.286075	0.143038	−	0.989717i	$-0.454313\pi$
$31$	1.59280	0.286075	0.143038	+	0.989717i	$0.454313\pi$
$32$	−0.586383	−0.103659
$33$	0	0
$34$	−5.85057	−1.00337
$35$	−5.57744	−0.942760
$36$	0	0
$37$	1.51297	0.248731	0.124365	−	0.992236i	$-0.460311\pi$
$37$	1.51297	0.248731	0.124365	+	0.992236i	$0.460311\pi$
$38$	4.67646	0.758622
$39$	0	0
$40$	7.54748	1.19336
$41$	4.20875	0.657297	0.328648	−	0.944452i	$-0.393407\pi$
$41$	4.20875	0.657297	0.328648	+	0.944452i	$0.393407\pi$
$42$	0	0
$43$	5.59560	0.853322	0.426661	−	0.904412i	$-0.359690\pi$
$43$	5.59560	0.853322	0.426661	+	0.904412i	$0.359690\pi$
$44$	−0.553629	−0.0834626
$45$	0	0
$46$	−0.482620	−0.0711584
$47$	−0.326594	−0.0476386	−0.0238193	−	0.999716i	$-0.507583\pi$
$47$	−0.326594	−0.0476386	−0.0238193	+	0.999716i	$0.507583\pi$
$48$	0	0
$49$	−2.41696	−0.345281
$50$	−2.46161	−0.348125
$51$	0	0
$52$	0.0830649	0.0115190
$53$	−2.42798	−0.333509	−0.166754	−	0.985998i	$-0.553329\pi$
$53$	−2.42798	−0.333509	−0.166754	+	0.985998i	$0.553329\pi$
$54$	0	0
$55$	13.9003	1.87432
$56$	−6.20183	−0.828755
$57$	0	0
$58$	−8.45986	−1.11083
$59$	−0.747504	−0.0973167	−0.0486584	−	0.998815i	$-0.515495\pi$
$59$	−0.747504	−0.0973167	−0.0486584	+	0.998815i	$0.515495\pi$
$60$	0	0
$61$	−2.16858	−0.277658	−0.138829	−	0.990316i	$-0.544334\pi$
$61$	−2.16858	−0.277658	−0.138829	+	0.990316i	$0.544334\pi$
$62$	−2.19334	−0.278555
$63$	0	0
$64$	8.37088	1.04636
$65$	−2.08556	−0.258682
$66$	0	0
$67$	2.07659	0.253696	0.126848	−	0.991922i	$-0.459514\pi$
$67$	2.07659	0.253696	0.126848	+	0.991922i	$0.459514\pi$
$68$	−0.440864	−0.0534626
$69$	0	0
$70$	7.68035	0.917978
$71$	10.5876	1.25651	0.628257	−	0.778006i	$-0.283769\pi$
$71$	10.5876	1.25651	0.628257	+	0.778006i	$0.283769\pi$
$72$	0	0
$73$	0.550054	0.0643790	0.0321895	−	0.999482i	$-0.489752\pi$
$73$	0.550054	0.0643790	0.0321895	+	0.999482i	$0.489752\pi$
$74$	−2.08342	−0.242192
$75$	0	0
$76$	0.352390	0.0404219
$77$	−11.4220	−1.30166
$78$	0	0
$79$	6.13896	0.690687	0.345343	−	0.938476i	$-0.387762\pi$
$79$	6.13896	0.690687	0.345343	+	0.938476i	$0.387762\pi$
$80$	−9.85249	−1.10154
$81$	0	0
$82$	−5.79561	−0.640018
$83$	1.55908	0.171132	0.0855658	−	0.996333i	$-0.472730\pi$
$83$	1.55908	0.171132	0.0855658	+	0.996333i	$0.472730\pi$
$84$	0	0
$85$	11.0691	1.20061
$86$	−7.70536	−0.830891
$87$	0	0
$88$	15.4564	1.64766
$89$	10.8215	1.14708	0.573540	−	0.819178i	$-0.305570\pi$
$89$	10.8215	1.14708	0.573540	+	0.819178i	$0.305570\pi$
$90$	0	0
$91$	1.71373	0.179647
$92$	−0.0363674	−0.00379156
$93$	0	0
$94$	0.449732	0.0463863
$95$	−8.84769	−0.907753
$96$	0	0
$97$	3.09238	0.313984	0.156992	−	0.987600i	$-0.449820\pi$
$97$	3.09238	0.313984	0.156992	+	0.987600i	$0.449820\pi$
$98$	3.32825	0.336204
$99$	0	0
$100$	−0.185493	−0.0185493
$101$	−12.4456	−1.23839	−0.619193	−	0.785239i	$-0.712540\pi$
$101$	−12.4456	−1.23839	−0.619193	+	0.785239i	$0.712540\pi$
$102$	0	0
$103$	−6.92245	−0.682089	−0.341045	−	0.940047i	$-0.610781\pi$
$103$	−6.92245	−0.682089	−0.341045	+	0.940047i	$0.610781\pi$
$104$	−2.31904	−0.227401
$105$	0	0
$106$	3.34342	0.324742
$107$	15.7301	1.52068	0.760342	−	0.649523i	$-0.225032\pi$
$107$	15.7301	1.52068	0.760342	+	0.649523i	$0.225032\pi$
$108$	0	0
$109$	−7.92697	−0.759266	−0.379633	−	0.925137i	$-0.623950\pi$
$109$	−7.92697	−0.759266	−0.379633	+	0.925137i	$0.623950\pi$
$110$	−19.1413	−1.82505
$111$	0	0
$112$	8.09588	0.764988
$113$	−12.5906	−1.18443	−0.592214	−	0.805781i	$-0.701746\pi$
$113$	−12.5906	−1.18443	−0.592214	+	0.805781i	$0.701746\pi$
$114$	0	0
$115$	0.913099	0.0851469
$116$	−0.637484	−0.0591889
$117$	0	0
$118$	1.02934	0.0947586
$119$	−9.09555	−0.833788
$120$	0	0
$121$	17.4664	1.58785
$122$	2.98621	0.270359
$123$	0	0
$124$	−0.165277	−0.0148423
$125$	−8.36924	−0.748568
$126$	0	0
$127$	−8.32971	−0.739142	−0.369571	−	0.929203i	$-0.620495\pi$
$127$	−8.32971	−0.739142	−0.369571	+	0.929203i	$0.620495\pi$
$128$	−10.3543	−0.915195
$129$	0	0
$130$	2.87190	0.251882
$131$	13.6906	1.19615	0.598076	−	0.801439i	$-0.295932\pi$
$131$	13.6906	1.19615	0.598076	+	0.801439i	$0.295932\pi$
$132$	0	0
$133$	7.27022	0.630408
$134$	−2.85954	−0.247027
$135$	0	0
$136$	12.3082	1.05542
$137$	10.8731	0.928951	0.464475	−	0.885586i	$-0.346243\pi$
$137$	10.8731	0.928951	0.464475	+	0.885586i	$0.346243\pi$
$138$	0	0
$139$	−2.67406	−0.226811	−0.113405	−	0.993549i	$-0.536176\pi$
$139$	−2.67406	−0.226811	−0.113405	+	0.993549i	$0.536176\pi$
$140$	0.578746	0.0489129
$141$	0	0
$142$	−14.5795	−1.22348
$143$	−4.27101	−0.357160
$144$	0	0
$145$	16.0057	1.32920
$146$	−0.757446	−0.0626867
$147$	0	0
$148$	−0.156994	−0.0129048
$149$	−16.0288	−1.31313	−0.656567	−	0.754268i	$-0.727992\pi$
$149$	−16.0288	−1.31313	−0.656567	+	0.754268i	$0.727992\pi$
$150$	0	0
$151$	−17.7707	−1.44616	−0.723081	−	0.690764i	$-0.757275\pi$
$151$	−17.7707	−1.44616	−0.723081	+	0.690764i	$0.757275\pi$
$152$	−9.83817	−0.797981
$153$	0	0
$154$	15.7285	1.26744
$155$	4.14973	0.333314
$156$	0	0
$157$	−4.58865	−0.366214	−0.183107	−	0.983093i	$-0.558616\pi$
$157$	−4.58865	−0.366214	−0.183107	+	0.983093i	$0.558616\pi$
$158$	−8.45358	−0.672531
$159$	0	0
$160$	−1.52771	−0.120776
$161$	−0.750301	−0.0591320
$162$	0	0
$163$	15.1282	1.18493	0.592465	−	0.805596i	$-0.298155\pi$
$163$	15.1282	1.18493	0.592465	+	0.805596i	$0.298155\pi$
$164$	−0.436723	−0.0341023
$165$	0	0
$166$	−2.14692	−0.166633
$167$	−17.0232	−1.31730	−0.658648	−	0.752451i	$-0.728871\pi$
$167$	−17.0232	−1.31730	−0.658648	+	0.752451i	$0.728871\pi$
$168$	0	0
$169$	−12.3592	−0.950707
$170$	−15.2425	−1.16905
$171$	0	0
$172$	−0.580630	−0.0442726
$173$	8.26333	0.628250	0.314125	−	0.949382i	$-0.398289\pi$
$173$	8.26333	0.628250	0.314125	+	0.949382i	$0.398289\pi$
$174$	0	0
$175$	−3.82693	−0.289289
$176$	−20.1768	−1.52089
$177$	0	0
$178$	−14.9017	−1.11693
$179$	8.35073	0.624163	0.312081	−	0.950055i	$-0.398974\pi$
$179$	8.35073	0.624163	0.312081	+	0.950055i	$0.398974\pi$
$180$	0	0
$181$	2.89671	0.215311	0.107655	−	0.994188i	$-0.465666\pi$
$181$	2.89671	0.215311	0.107655	+	0.994188i	$0.465666\pi$
$182$	−2.35987	−0.174925
$183$	0	0
$184$	1.01532	0.0748503
$185$	3.94175	0.289803
$186$	0	0
$187$	22.6683	1.65767
$188$	0.0338891	0.00247162
$189$	0	0
$190$	12.1836	0.883891
$191$	14.5269	1.05113	0.525563	−	0.850754i	$-0.323855\pi$
$191$	14.5269	1.05113	0.525563	+	0.850754i	$0.323855\pi$
$192$	0	0
$193$	−7.11718	−0.512306	−0.256153	−	0.966636i	$-0.582455\pi$
$193$	−7.11718	−0.512306	−0.256153	+	0.966636i	$0.582455\pi$
$194$	−4.25833	−0.305730
$195$	0	0
$196$	0.250797	0.0179141
$197$	−6.08341	−0.433425	−0.216712	−	0.976235i	$-0.569533\pi$
$197$	−6.08341	−0.433425	−0.216712	+	0.976235i	$0.569533\pi$
$198$	0	0
$199$	10.7015	0.758607	0.379303	−	0.925272i	$-0.376164\pi$
$199$	10.7015	0.758607	0.379303	+	0.925272i	$0.376164\pi$
$200$	5.17866	0.366187
$201$	0	0
$202$	17.1381	1.20583
$203$	−13.1521	−0.923093
$204$	0	0
$205$	10.9651	0.765835
$206$	9.53248	0.664159
$207$	0	0
$208$	3.02728	0.209904
$209$	−18.1191	−1.25333
$210$	0	0
$211$	0.102720	0.00707154	0.00353577	−	0.999994i	$-0.498875\pi$
$211$	0.102720	0.00707154	0.00353577	+	0.999994i	$0.498875\pi$
$212$	0.251940	0.0173033
$213$	0	0
$214$	−21.6609	−1.48071
$215$	14.5783	0.994229
$216$	0	0
$217$	−3.40987	−0.231477
$218$	10.9157	0.739308
$219$	0	0
$220$	−1.44237	−0.0972447
$221$	−3.40108	−0.228782
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	1.25533	0.0838752
$225$	0	0
$226$	17.3378	1.15329
$227$	13.3106	0.883456	0.441728	−	0.897149i	$-0.354366\pi$
$227$	13.3106	0.883456	0.441728	+	0.897149i	$0.354366\pi$
$228$	0	0
$229$	−2.92332	−0.193178	−0.0965892	−	0.995324i	$-0.530793\pi$
$229$	−2.92332	−0.193178	−0.0965892	+	0.995324i	$0.530793\pi$
$230$	−1.25737	−0.0829087
$231$	0	0
$232$	17.7976	1.16847
$233$	10.8239	0.709098	0.354549	−	0.935037i	$-0.384634\pi$
$233$	10.8239	0.709098	0.354549	+	0.935037i	$0.384634\pi$
$234$	0	0
$235$	−0.850876	−0.0555050
$236$	0.0775650	0.00504905
$237$	0	0
$238$	12.5249	0.811870
$239$	−6.19186	−0.400518	−0.200259	−	0.979743i	$-0.564178\pi$
$239$	−6.19186	−0.400518	−0.200259	+	0.979743i	$0.564178\pi$
$240$	0	0
$241$	19.3002	1.24323	0.621616	−	0.783322i	$-0.286476\pi$
$241$	19.3002	1.24323	0.621616	+	0.783322i	$0.286476\pi$
$242$	−24.0519	−1.54611
$243$	0	0
$244$	0.225023	0.0144056
$245$	−6.29693	−0.402296
$246$	0	0
$247$	2.71854	0.172977
$248$	4.61428	0.293007
$249$	0	0
$250$	11.5248	0.728890
$251$	24.4159	1.54112	0.770560	−	0.637367i	$-0.219977\pi$
$251$	24.4159	1.54112	0.770560	+	0.637367i	$0.219977\pi$
$252$	0	0
$253$	1.86993	0.117561
$254$	11.4703	0.719712
$255$	0	0
$256$	−2.48355	−0.155222
$257$	−23.7990	−1.48454	−0.742270	−	0.670101i	$-0.766251\pi$
$257$	−23.7990	−1.48454	−0.742270	+	0.670101i	$0.766251\pi$
$258$	0	0
$259$	−3.23897	−0.201260
$260$	0.216409	0.0134211
$261$	0	0
$262$	−18.8525	−1.16471
$263$	−9.08171	−0.560002	−0.280001	−	0.960000i	$-0.590335\pi$
$263$	−9.08171	−0.560002	−0.280001	+	0.960000i	$0.590335\pi$
$264$	0	0
$265$	−6.32563	−0.388580
$266$	−10.0114	−0.613837
$267$	0	0
$268$	−0.215478	−0.0131624
$269$	11.4996	0.701141	0.350570	−	0.936536i	$-0.385988\pi$
$269$	11.4996	0.701141	0.350570	+	0.936536i	$0.385988\pi$
$270$	0	0
$271$	19.4140	1.17932	0.589659	−	0.807652i	$-0.299262\pi$
$271$	19.4140	1.17932	0.589659	+	0.807652i	$0.299262\pi$
$272$	−16.0672	−0.974216
$273$	0	0
$274$	−14.9727	−0.904532
$275$	9.53762	0.575140
$276$	0	0
$277$	−6.25680	−0.375935	−0.187967	−	0.982175i	$-0.560190\pi$
$277$	−6.25680	−0.375935	−0.187967	+	0.982175i	$0.560190\pi$
$278$	3.68228	0.220849
$279$	0	0
$280$	−16.1577	−0.965605
$281$	28.6580	1.70959	0.854796	−	0.518964i	$-0.173682\pi$
$281$	28.6580	1.70959	0.854796	+	0.518964i	$0.173682\pi$
$282$	0	0
$283$	19.7624	1.17475	0.587376	−	0.809314i	$-0.300161\pi$
$283$	19.7624	1.17475	0.587376	+	0.809314i	$0.300161\pi$
$284$	−1.09862	−0.0651913
$285$	0	0
$286$	5.88135	0.347771
$287$	−9.01011	−0.531850
$288$	0	0
$289$	1.05115	0.0618324
$290$	−22.0405	−1.29426
$291$	0	0
$292$	−0.0570766	−0.00334015
$293$	−8.38921	−0.490103	−0.245051	−	0.969510i	$-0.578805\pi$
$293$	−8.38921	−0.490103	−0.245051	+	0.969510i	$0.578805\pi$
$294$	0	0
$295$	−1.94748	−0.113386
$296$	4.38302	0.254758
$297$	0	0
$298$	22.0723	1.27862
$299$	−0.280559	−0.0162251
$300$	0	0
$301$	−11.9791	−0.690463
$302$	24.4710	1.40815
$303$	0	0
$304$	12.8428	0.736583
$305$	−5.64980	−0.323507
$306$	0	0
$307$	−14.5406	−0.829873	−0.414937	−	0.909850i	$-0.636196\pi$
$307$	−14.5406	−0.829873	−0.414937	+	0.909850i	$0.636196\pi$
$308$	1.18521	0.0675336
$309$	0	0
$310$	−5.71433	−0.324552
$311$	−0.955843	−0.0542009	−0.0271004	−	0.999633i	$-0.508627\pi$
$311$	−0.955843	−0.0542009	−0.0271004	+	0.999633i	$0.508627\pi$
$312$	0	0
$313$	−30.1554	−1.70449	−0.852243	−	0.523147i	$-0.824758\pi$
$313$	−30.1554	−1.70449	−0.852243	+	0.523147i	$0.824758\pi$
$314$	6.31875	0.356588
$315$	0	0
$316$	−0.637011	−0.0358347
$317$	6.17748	0.346962	0.173481	−	0.984837i	$-0.444498\pi$
$317$	6.17748	0.346962	0.173481	+	0.984837i	$0.444498\pi$
$318$	0	0
$319$	32.7780	1.83522
$320$	21.8087	1.21914
$321$	0	0
$322$	1.03319	0.0575776
$323$	−14.4286	−0.802827
$324$	0	0
$325$	−1.43100	−0.0793775
$326$	−20.8321	−1.15378
$327$	0	0
$328$	12.1926	0.673224
$329$	0.699172	0.0385466
$330$	0	0
$331$	−8.59557	−0.472456	−0.236228	−	0.971698i	$-0.575911\pi$
$331$	−8.59557	−0.472456	−0.236228	+	0.971698i	$0.575911\pi$
$332$	−0.161779	−0.00887877
$333$	0	0
$334$	23.4416	1.28267
$335$	5.41015	0.295588
$336$	0	0
$337$	9.39069	0.511543	0.255772	−	0.966737i	$-0.417670\pi$
$337$	9.39069	0.511543	0.255772	+	0.966737i	$0.417670\pi$
$338$	17.0191	0.925716
$339$	0	0
$340$	−1.14859	−0.0622908
$341$	8.49820	0.460203
$342$	0	0
$343$	20.1599	1.08853
$344$	16.2103	0.873999
$345$	0	0
$346$	−11.3789	−0.611735
$347$	−16.9533	−0.910102	−0.455051	−	0.890465i	$-0.650379\pi$
$347$	−16.9533	−0.910102	−0.455051	+	0.890465i	$0.650379\pi$
$348$	0	0
$349$	−15.1548	−0.811219	−0.405609	−	0.914046i	$-0.632941\pi$
$349$	−15.1548	−0.811219	−0.405609	+	0.914046i	$0.632941\pi$
$350$	5.26983	0.281684
$351$	0	0
$352$	−3.12858	−0.166754
$353$	−7.51732	−0.400107	−0.200053	−	0.979785i	$-0.564112\pi$
$353$	−7.51732	−0.400107	−0.200053	+	0.979785i	$0.564112\pi$
$354$	0	0
$355$	27.5839	1.46400
$356$	−1.12290	−0.0595136
$357$	0	0
$358$	−11.4993	−0.607756
$359$	−13.2650	−0.700102	−0.350051	−	0.936731i	$-0.613836\pi$
$359$	−13.2650	−0.700102	−0.350051	+	0.936731i	$0.613836\pi$
$360$	0	0
$361$	−7.46700	−0.393000
$362$	−3.98888	−0.209651
$363$	0	0
$364$	−0.177825	−0.00932059
$365$	1.43306	0.0750097
$366$	0	0
$367$	9.25727	0.483226	0.241613	−	0.970373i	$-0.422324\pi$
$367$	9.25727	0.483226	0.241613	+	0.970373i	$0.422324\pi$
$368$	−1.32540	−0.0690912
$369$	0	0
$370$	−5.42794	−0.282185
$371$	5.19783	0.269858
$372$	0	0
$373$	11.3006	0.585121	0.292560	−	0.956247i	$-0.405493\pi$
$373$	11.3006	0.585121	0.292560	+	0.956247i	$0.405493\pi$
$374$	−31.2151	−1.61409
$375$	0	0
$376$	−0.946130	−0.0487929
$377$	−4.91792	−0.253286
$378$	0	0
$379$	10.5003	0.539366	0.269683	−	0.962949i	$-0.413081\pi$
$379$	10.5003	0.539366	0.269683	+	0.962949i	$0.413081\pi$
$380$	0.918083	0.0470967
$381$	0	0
$382$	−20.0040	−1.02350
$383$	23.4094	1.19616	0.598081	−	0.801435i	$-0.295930\pi$
$383$	23.4094	1.19616	0.598081	+	0.801435i	$0.295930\pi$
$384$	0	0
$385$	−29.7578	−1.51660
$386$	9.80063	0.498839
$387$	0	0
$388$	−0.320882	−0.0162903
$389$	15.1258	0.766908	0.383454	−	0.923560i	$-0.374734\pi$
$389$	15.1258	0.766908	0.383454	+	0.923560i	$0.374734\pi$
$390$	0	0
$391$	1.48906	0.0753049
$392$	−7.00186	−0.353647
$393$	0	0
$394$	8.37708	0.422031
$395$	15.9939	0.804738
$396$	0	0
$397$	0.782515	0.0392733	0.0196366	−	0.999807i	$-0.493749\pi$
$397$	0.782515	0.0392733	0.0196366	+	0.999807i	$0.493749\pi$
$398$	−14.7363	−0.738665
$399$	0	0
$400$	−6.76023	−0.338011
$401$	−23.3580	−1.16644	−0.583221	−	0.812313i	$-0.698208\pi$
$401$	−23.3580	−1.16644	−0.583221	+	0.812313i	$0.698208\pi$
$402$	0	0
$403$	−1.27505	−0.0635145
$404$	1.29143	0.0642508
$405$	0	0
$406$	18.1109	0.898828
$407$	8.07228	0.400128
$408$	0	0
$409$	−5.86955	−0.290231	−0.145115	−	0.989415i	$-0.546355\pi$
$409$	−5.86955	−0.290231	−0.145115	+	0.989415i	$0.546355\pi$
$410$	−15.0993	−0.745703
$411$	0	0
$412$	0.718311	0.0353886
$413$	1.60026	0.0787436
$414$	0	0
$415$	4.06189	0.199390
$416$	0.469403	0.0230144
$417$	0	0
$418$	24.9507	1.22038
$419$	−7.54054	−0.368380	−0.184190	−	0.982891i	$-0.558966\pi$
$419$	−7.54054	−0.368380	−0.184190	+	0.982891i	$0.558966\pi$
$420$	0	0
$421$	−24.2216	−1.18049	−0.590244	−	0.807225i	$-0.700969\pi$
$421$	−24.2216	−1.18049	−0.590244	+	0.807225i	$0.700969\pi$
$422$	−0.141450	−0.00688566
$423$	0	0
$424$	−7.03378	−0.341590
$425$	7.59498	0.368410
$426$	0	0
$427$	4.64249	0.224666
$428$	−1.63224	−0.0788971
$429$	0	0
$430$	−20.0748	−0.968094
$431$	22.8420	1.10026	0.550130	−	0.835079i	$-0.314578\pi$
$431$	22.8420	1.10026	0.550130	+	0.835079i	$0.314578\pi$
$432$	0	0
$433$	0.919321	0.0441798	0.0220899	−	0.999756i	$-0.492968\pi$
$433$	0.919321	0.0441798	0.0220899	+	0.999756i	$0.492968\pi$
$434$	4.69552	0.225392
$435$	0	0
$436$	0.822545	0.0393928
$437$	−1.19023	−0.0569363
$438$	0	0
$439$	−8.37460	−0.399698	−0.199849	−	0.979827i	$-0.564045\pi$
$439$	−8.37460	−0.399698	−0.199849	+	0.979827i	$0.564045\pi$
$440$	40.2687	1.91974
$441$	0	0
$442$	4.68342	0.222768
$443$	−12.2264	−0.580893	−0.290447	−	0.956891i	$-0.593804\pi$
$443$	−12.2264	−0.580893	−0.290447	+	0.956891i	$0.593804\pi$
$444$	0	0
$445$	28.1934	1.33649
$446$	−1.37704	−0.0652047
$447$	0	0
$448$	−17.9204	−0.846659
$449$	21.2224	1.00154	0.500772	−	0.865579i	$-0.333049\pi$
$449$	21.2224	1.00154	0.500772	+	0.865579i	$0.333049\pi$
$450$	0	0
$451$	22.4553	1.05738
$452$	1.30647	0.0614513
$453$	0	0
$454$	−18.3292	−0.860232
$455$	4.46478	0.209312
$456$	0	0
$457$	9.91073	0.463604	0.231802	−	0.972763i	$-0.425538\pi$
$457$	9.91073	0.463604	0.231802	+	0.972763i	$0.425538\pi$
$458$	4.02552	0.188100
$459$	0	0
$460$	−0.0947480	−0.00441765
$461$	35.9182	1.67288	0.836439	−	0.548060i	$-0.184634\pi$
$461$	35.9182	1.67288	0.836439	+	0.548060i	$0.184634\pi$
$462$	0	0
$463$	14.7063	0.683459	0.341730	−	0.939798i	$-0.388987\pi$
$463$	14.7063	0.683459	0.341730	+	0.939798i	$0.388987\pi$
$464$	−23.2329	−1.07856
$465$	0	0
$466$	−14.9050	−0.690458
$467$	19.1601	0.886626	0.443313	−	0.896367i	$-0.353803\pi$
$467$	19.1601	0.886626	0.443313	+	0.896367i	$0.353803\pi$
$468$	0	0
$469$	−4.44557	−0.205277
$470$	1.17169	0.0540460
$471$	0	0
$472$	−2.16549	−0.0996749
$473$	29.8547	1.37272
$474$	0	0
$475$	−6.07079	−0.278547
$476$	0.943803	0.0432592
$477$	0	0
$478$	8.52643	0.389990
$479$	−15.0738	−0.688741	−0.344371	−	0.938834i	$-0.611908\pi$
$479$	−15.0738	−0.688741	−0.344371	+	0.938834i	$0.611908\pi$
$480$	0	0
$481$	−1.21114	−0.0552233
$482$	−26.5771	−1.21055
$483$	0	0
$484$	−1.81240	−0.0823820
$485$	8.05660	0.365831
$486$	0	0
$487$	−19.2059	−0.870300	−0.435150	−	0.900358i	$-0.643305\pi$
$487$	−19.2059	−0.870300	−0.435150	+	0.900358i	$0.643305\pi$
$488$	−6.28229	−0.284386
$489$	0	0
$490$	8.67111	0.391721
$491$	−20.5092	−0.925569	−0.462785	−	0.886471i	$-0.653150\pi$
$491$	−20.5092	−0.925569	−0.462785	+	0.886471i	$0.653150\pi$
$492$	0	0
$493$	26.1017	1.17556
$494$	−3.74354	−0.168430
$495$	0	0
$496$	−6.02349	−0.270463
$497$	−22.6659	−1.01671
$498$	0	0
$499$	20.2242	0.905362	0.452681	−	0.891673i	$-0.350468\pi$
$499$	20.2242	0.905362	0.452681	+	0.891673i	$0.350468\pi$
$500$	0.868438	0.0388377
$501$	0	0
$502$	−33.6217	−1.50061
$503$	35.7867	1.59565	0.797824	−	0.602890i	$-0.205984\pi$
$503$	35.7867	1.59565	0.797824	+	0.602890i	$0.205984\pi$
$504$	0	0
$505$	−32.4247	−1.44288
$506$	−2.57496	−0.114471
$507$	0	0
$508$	0.864335	0.0383487
$509$	24.2364	1.07426	0.537130	−	0.843499i	$-0.319508\pi$
$509$	24.2364	1.07426	0.537130	+	0.843499i	$0.319508\pi$
$510$	0	0
$511$	−1.17756	−0.0520921
$512$	24.1284	1.06634
$513$	0	0
$514$	32.7721	1.44552
$515$	−18.0351	−0.794721
$516$	0	0
$517$	−1.74250	−0.0766352
$518$	4.46019	0.195969
$519$	0	0
$520$	−6.04181	−0.264951
$521$	−25.8128	−1.13088	−0.565439	−	0.824790i	$-0.691293\pi$
$521$	−25.8128	−1.13088	−0.565439	+	0.824790i	$0.691293\pi$
$522$	0	0
$523$	−27.4586	−1.20068	−0.600341	−	0.799744i	$-0.704968\pi$
$523$	−27.4586	−1.20068	−0.600341	+	0.799744i	$0.704968\pi$
$524$	−1.42061	−0.0620596
$525$	0	0
$526$	12.5059	0.545281
$527$	6.76727	0.294787
$528$	0	0
$529$	−22.8772	−0.994659
$530$	8.71064	0.378366
$531$	0	0
$532$	−0.754397	−0.0327073
$533$	−3.36913	−0.145933
$534$	0	0
$535$	40.9816	1.77179
$536$	6.01581	0.259843
$537$	0	0
$538$	−15.8353	−0.682710
$539$	−12.8954	−0.555446
$540$	0	0
$541$	44.2677	1.90322	0.951608	−	0.307314i	$-0.0994302\pi$
$541$	44.2677	1.90322	0.951608	+	0.307314i	$0.0994302\pi$
$542$	−26.7339	−1.14832
$543$	0	0
$544$	−2.49134	−0.106815
$545$	−20.6522	−0.884642
$546$	0	0
$547$	32.5336	1.39104	0.695518	−	0.718509i	$-0.255175\pi$
$547$	32.5336	1.39104	0.695518	+	0.718509i	$0.255175\pi$
$548$	−1.12825	−0.0481965
$549$	0	0
$550$	−13.1337	−0.560022
$551$	−20.8635	−0.888817
$552$	0	0
$553$	−13.1423	−0.558867
$554$	8.61586	0.366053
$555$	0	0
$556$	0.277475	0.0117675
$557$	−2.34483	−0.0993538	−0.0496769	−	0.998765i	$-0.515819\pi$
$557$	−2.34483	−0.0993538	−0.0496769	+	0.998765i	$0.515819\pi$
$558$	0	0
$559$	−4.47932	−0.189455
$560$	21.0922	0.891309
$561$	0	0
$562$	−39.4632	−1.66465
$563$	6.20837	0.261652	0.130826	−	0.991405i	$-0.458237\pi$
$563$	6.20837	0.261652	0.130826	+	0.991405i	$0.458237\pi$
$564$	0	0
$565$	−32.8024	−1.38001
$566$	−27.2136	−1.14387
$567$	0	0
$568$	30.6719	1.28696
$569$	25.7267	1.07852	0.539259	−	0.842140i	$-0.318704\pi$
$569$	25.7267	1.07852	0.539259	+	0.842140i	$0.318704\pi$
$570$	0	0
$571$	30.2742	1.26694	0.633469	−	0.773768i	$-0.281630\pi$
$571$	30.2742	1.26694	0.633469	+	0.773768i	$0.281630\pi$
$572$	0.443183	0.0185304
$573$	0	0
$574$	12.4073	0.517869
$575$	0.626517	0.0261276
$576$	0	0
$577$	37.0011	1.54038	0.770189	−	0.637816i	$-0.220162\pi$
$577$	37.0011	1.54038	0.770189	+	0.637816i	$0.220162\pi$
$578$	−1.44747	−0.0602070
$579$	0	0
$580$	−1.66084	−0.0689627
$581$	−3.33769	−0.138471
$582$	0	0
$583$	−12.9542	−0.536509
$584$	1.59349	0.0659390
$585$	0	0
$586$	11.5523	0.477220
$587$	−5.50336	−0.227148	−0.113574	−	0.993530i	$-0.536230\pi$
$587$	−5.50336	−0.227148	−0.113574	+	0.993530i	$0.536230\pi$
$588$	0	0
$589$	−5.40918	−0.222882
$590$	2.68175	0.110406
$591$	0	0
$592$	−5.72160	−0.235156
$593$	39.2959	1.61369	0.806845	−	0.590763i	$-0.201173\pi$
$593$	39.2959	1.61369	0.806845	+	0.590763i	$0.201173\pi$
$594$	0	0
$595$	−23.6967	−0.971469
$596$	1.66324	0.0681289
$597$	0	0
$598$	0.386340	0.0157986
$599$	34.5582	1.41201	0.706004	−	0.708208i	$-0.250496\pi$
$599$	34.5582	1.41201	0.706004	+	0.708208i	$0.250496\pi$
$600$	0	0
$601$	−5.42760	−0.221396	−0.110698	−	0.993854i	$-0.535309\pi$
$601$	−5.42760	−0.221396	−0.110698	+	0.993854i	$0.535309\pi$
$602$	16.4957	0.672313
$603$	0	0
$604$	1.84399	0.0750307
$605$	45.5052	1.85005
$606$	0	0
$607$	5.11847	0.207752	0.103876	−	0.994590i	$-0.466875\pi$
$607$	5.11847	0.207752	0.103876	+	0.994590i	$0.466875\pi$
$608$	1.99137	0.0807607
$609$	0	0
$610$	7.77999	0.315003
$611$	0.261440	0.0105767
$612$	0	0
$613$	−29.1051	−1.17555	−0.587773	−	0.809026i	$-0.699995\pi$
$613$	−29.1051	−1.17555	−0.587773	+	0.809026i	$0.699995\pi$
$614$	20.0229	0.808059
$615$	0	0
$616$	−33.0892	−1.33320
$617$	−9.29970	−0.374392	−0.187196	−	0.982323i	$-0.559940\pi$
$617$	−9.29970	−0.374392	−0.187196	+	0.982323i	$0.559940\pi$
$618$	0	0
$619$	−13.1636	−0.529088	−0.264544	−	0.964374i	$-0.585221\pi$
$619$	−13.1636	−0.529088	−0.264544	+	0.964374i	$0.585221\pi$
$620$	−0.430598	−0.0172932
$621$	0	0
$622$	1.31623	0.0527761
$623$	−23.1668	−0.928157
$624$	0	0
$625$	−30.7425	−1.22970
$626$	41.5252	1.65968
$627$	0	0
$628$	0.476143	0.0190002
$629$	6.42810	0.256305
$630$	0	0
$631$	24.9064	0.991508	0.495754	−	0.868463i	$-0.334892\pi$
$631$	24.9064	0.991508	0.495754	+	0.868463i	$0.334892\pi$
$632$	17.7844	0.707423
$633$	0	0
$634$	−8.50663	−0.337842
$635$	−21.7014	−0.861195
$636$	0	0
$637$	1.93480	0.0766594
$638$	−45.1366	−1.78698
$639$	0	0
$640$	−26.9760	−1.06632
$641$	24.6847	0.974986	0.487493	−	0.873127i	$-0.337911\pi$
$641$	24.6847	0.974986	0.487493	+	0.873127i	$0.337911\pi$
$642$	0	0
$643$	47.5821	1.87645	0.938227	−	0.346019i	$-0.112467\pi$
$643$	47.5821	1.87645	0.938227	+	0.346019i	$0.112467\pi$
$644$	0.0778553	0.00306793
$645$	0	0
$646$	19.8687	0.781724
$647$	22.0375	0.866385	0.433192	−	0.901301i	$-0.357387\pi$
$647$	22.0375	0.866385	0.433192	+	0.901301i	$0.357387\pi$
$648$	0	0
$649$	−3.98822	−0.156551
$650$	1.97054	0.0772909
$651$	0	0
$652$	−1.56978	−0.0614773
$653$	17.5794	0.687936	0.343968	−	0.938981i	$-0.388229\pi$
$653$	17.5794	0.687936	0.343968	+	0.938981i	$0.388229\pi$
$654$	0	0
$655$	35.6682	1.39367
$656$	−15.9162	−0.621425
$657$	0	0
$658$	−0.962787	−0.0375333
$659$	9.30700	0.362549	0.181275	−	0.983433i	$-0.441978\pi$
$659$	9.30700	0.362549	0.181275	+	0.983433i	$0.441978\pi$
$660$	0	0
$661$	30.6145	1.19076	0.595382	−	0.803442i	$-0.297001\pi$
$661$	30.6145	1.19076	0.595382	+	0.803442i	$0.297001\pi$
$662$	11.8364	0.460036
$663$	0	0
$664$	4.51661	0.175279
$665$	18.9411	0.734506
$666$	0	0
$667$	2.15316	0.0833707
$668$	1.76642	0.0683449
$669$	0	0
$670$	−7.44998	−0.287818
$671$	−11.5702	−0.446662
$672$	0	0
$673$	39.9395	1.53955	0.769777	−	0.638313i	$-0.220367\pi$
$673$	39.9395	1.53955	0.769777	+	0.638313i	$0.220367\pi$
$674$	−12.9313	−0.498097
$675$	0	0
$676$	1.28246	0.0493252
$677$	25.1474	0.966495	0.483247	−	0.875484i	$-0.339457\pi$
$677$	25.1474	0.966495	0.483247	+	0.875484i	$0.339457\pi$
$678$	0	0
$679$	−6.62018	−0.254059
$680$	32.0667	1.22970
$681$	0	0
$682$	−11.7023	−0.448106
$683$	16.7090	0.639354	0.319677	−	0.947527i	$-0.396426\pi$
$683$	16.7090	0.639354	0.319677	+	0.947527i	$0.396426\pi$
$684$	0	0
$685$	28.3277	1.08235
$686$	−27.7609	−1.05992
$687$	0	0
$688$	−21.1609	−0.806752
$689$	1.94361	0.0740458
$690$	0	0
$691$	26.8673	1.02208	0.511040	−	0.859557i	$-0.329260\pi$
$691$	26.8673	1.02208	0.511040	+	0.859557i	$0.329260\pi$
$692$	−0.857448	−0.0325953
$693$	0	0
$694$	23.3454	0.886179
$695$	−6.96674	−0.264263
$696$	0	0
$697$	17.8816	0.677313
$698$	20.8688	0.789895
$699$	0	0
$700$	0.397103	0.0150091
$701$	−5.33782	−0.201607	−0.100803	−	0.994906i	$-0.532141\pi$
$701$	−5.33782	−0.201607	−0.100803	+	0.994906i	$0.532141\pi$
$702$	0	0
$703$	−5.13809	−0.193787
$704$	44.6619	1.68326
$705$	0	0
$706$	10.3516	0.389589
$707$	26.6436	1.00204
$708$	0	0
$709$	41.2636	1.54969	0.774843	−	0.632153i	$-0.217829\pi$
$709$	41.2636	1.54969	0.774843	+	0.632153i	$0.217829\pi$
$710$	−37.9841	−1.42552
$711$	0	0
$712$	31.3496	1.17488
$713$	0.558239	0.0209062
$714$	0	0
$715$	−11.1273	−0.416137
$716$	−0.866517	−0.0323832
$717$	0	0
$718$	18.2665	0.681698
$719$	38.8073	1.44727	0.723634	−	0.690184i	$-0.242470\pi$
$719$	38.8073	1.44727	0.723634	+	0.690184i	$0.242470\pi$
$720$	0	0
$721$	14.8196	0.551911
$722$	10.2824	0.382670
$723$	0	0
$724$	−0.300578	−0.0111709
$725$	10.9822	0.407870
$726$	0	0
$727$	−28.0626	−1.04078	−0.520392	−	0.853927i	$-0.674214\pi$
$727$	−28.0626	−1.04078	−0.520392	+	0.853927i	$0.674214\pi$
$728$	4.96461	0.184001
$729$	0	0
$730$	−1.97338	−0.0730380
$731$	23.7738	0.879308
$732$	0	0
$733$	−7.49398	−0.276797	−0.138398	−	0.990377i	$-0.544195\pi$
$733$	−7.49398	−0.276797	−0.138398	+	0.990377i	$0.544195\pi$
$734$	−12.7476	−0.470523
$735$	0	0
$736$	−0.205513	−0.00757533
$737$	11.0794	0.408115
$738$	0	0
$739$	−41.7222	−1.53478	−0.767388	−	0.641183i	$-0.778444\pi$
$739$	−41.7222	−1.53478	−0.767388	+	0.641183i	$0.778444\pi$
$740$	−0.409017	−0.0150358
$741$	0	0
$742$	−7.15761	−0.262764
$743$	−47.1313	−1.72908	−0.864539	−	0.502565i	$-0.832390\pi$
$743$	−47.1313	−1.72908	−0.864539	+	0.502565i	$0.832390\pi$
$744$	0	0
$745$	−41.7600	−1.52997
$746$	−15.5613	−0.569740
$747$	0	0
$748$	−2.35218	−0.0860043
$749$	−33.6750	−1.23046
$750$	0	0
$751$	−29.1131	−1.06235	−0.531176	−	0.847261i	$-0.678250\pi$
$751$	−29.1131	−1.06235	−0.531176	+	0.847261i	$0.678250\pi$
$752$	1.23508	0.0450387
$753$	0	0
$754$	6.77217	0.246628
$755$	−46.2982	−1.68496
$756$	0	0
$757$	−7.90596	−0.287347	−0.143673	−	0.989625i	$-0.545891\pi$
$757$	−7.90596	−0.287347	−0.143673	+	0.989625i	$0.545891\pi$
$758$	−14.4594	−0.525187
$759$	0	0
$760$	−25.6314	−0.929750
$761$	42.8324	1.55267	0.776337	−	0.630318i	$-0.217076\pi$
$761$	42.8324	1.55267	0.776337	+	0.630318i	$0.217076\pi$
$762$	0	0
$763$	16.9701	0.614358
$764$	−1.50739	−0.0545353
$765$	0	0
$766$	−32.2356	−1.16472
$767$	0.598382	0.0216063
$768$	0	0
$769$	4.17005	0.150376	0.0751879	−	0.997169i	$-0.476044\pi$
$769$	4.17005	0.150376	0.0751879	+	0.997169i	$0.476044\pi$
$770$	40.9777	1.47673
$771$	0	0
$772$	0.738517	0.0265798
$773$	5.35228	0.192508	0.0962540	−	0.995357i	$-0.469314\pi$
$773$	5.35228	0.192508	0.0962540	+	0.995357i	$0.469314\pi$
$774$	0	0
$775$	2.84731	0.102278
$776$	8.95852	0.321592
$777$	0	0
$778$	−20.8288	−0.746748
$779$	−14.2930	−0.512101
$780$	0	0
$781$	56.4888	2.02133
$782$	−2.05049	−0.0733254
$783$	0	0
$784$	9.14024	0.326437
$785$	−11.9548	−0.426687
$786$	0	0
$787$	−24.7881	−0.883599	−0.441799	−	0.897114i	$-0.645660\pi$
$787$	−24.7881	−0.883599	−0.441799	+	0.897114i	$0.645660\pi$
$788$	0.631247	0.0224872
$789$	0	0
$790$	−22.0242	−0.783584
$791$	26.9541	0.958376
$792$	0	0
$793$	1.73596	0.0616457
$794$	−1.07755	−0.0382409
$795$	0	0
$796$	−1.11044	−0.0393586
$797$	−46.9787	−1.66407	−0.832035	−	0.554724i	$-0.812824\pi$
$797$	−46.9787	−1.66407	−0.832035	+	0.554724i	$0.812824\pi$
$798$	0	0
$799$	−1.38759	−0.0490893
$800$	−1.04823	−0.0370604
$801$	0	0
$802$	32.1649	1.13578
$803$	2.93475	0.103565
$804$	0	0
$805$	−1.95476	−0.0688964
$806$	1.75579	0.0618449
$807$	0	0
$808$	−36.0546	−1.26840
$809$	−21.3518	−0.750690	−0.375345	−	0.926885i	$-0.622476\pi$
$809$	−21.3518	−0.750690	−0.375345	+	0.926885i	$0.622476\pi$
$810$	0	0
$811$	−13.9881	−0.491190	−0.245595	−	0.969373i	$-0.578983\pi$
$811$	−13.9881	−0.491190	−0.245595	+	0.969373i	$0.578983\pi$
$812$	1.36473	0.0478926
$813$	0	0
$814$	−11.1158	−0.389610
$815$	39.4135	1.38059
$816$	0	0
$817$	−19.0028	−0.664825
$818$	8.08260	0.282601
$819$	0	0
$820$	−1.13780	−0.0397336
$821$	6.25434	0.218278	0.109139	−	0.994027i	$-0.465191\pi$
$821$	6.25434	0.218278	0.109139	+	0.994027i	$0.465191\pi$
$822$	0	0
$823$	20.6634	0.720280	0.360140	−	0.932898i	$-0.382729\pi$
$823$	20.6634	0.720280	0.360140	+	0.932898i	$0.382729\pi$
$824$	−20.0541	−0.698618
$825$	0	0
$826$	−2.20362	−0.0766736
$827$	−16.9118	−0.588081	−0.294041	−	0.955793i	$-0.595000\pi$
$827$	−16.9118	−0.588081	−0.294041	+	0.955793i	$0.595000\pi$
$828$	0	0
$829$	29.9761	1.04111	0.520556	−	0.853827i	$-0.325725\pi$
$829$	29.9761	1.04111	0.520556	+	0.853827i	$0.325725\pi$
$830$	−5.59338	−0.194149
$831$	0	0
$832$	−6.70094	−0.232313
$833$	−10.2689	−0.355795
$834$	0	0
$835$	−44.3507	−1.53482
$836$	1.88014	0.0650259
$837$	0	0
$838$	10.3836	0.358696
$839$	−43.7524	−1.51050	−0.755251	−	0.655436i	$-0.772485\pi$
$839$	−43.7524	−1.51050	−0.755251	+	0.655436i	$0.772485\pi$
$840$	0	0
$841$	8.74278	0.301475
$842$	33.3541	1.14946
$843$	0	0
$844$	−0.0106588	−0.000366891 0
$845$	−32.1995	−1.10770
$846$	0	0
$847$	−37.3920	−1.28481
$848$	9.18190	0.315308
$849$	0	0
$850$	−10.4586	−0.358726
$851$	0.530261	0.0181771
$852$	0	0
$853$	−29.5000	−1.01006	−0.505030	−	0.863102i	$-0.668518\pi$
$853$	−29.5000	−1.01006	−0.505030	+	0.863102i	$0.668518\pi$
$854$	−6.39289	−0.218760
$855$	0	0
$856$	45.5695	1.55753
$857$	−35.0296	−1.19659	−0.598293	−	0.801277i	$-0.704154\pi$
$857$	−35.0296	−1.19659	−0.598293	+	0.801277i	$0.704154\pi$
$858$	0	0
$859$	5.88772	0.200887	0.100443	−	0.994943i	$-0.467974\pi$
$859$	5.88772	0.200887	0.100443	+	0.994943i	$0.467974\pi$
$860$	−1.51272	−0.0515833
$861$	0	0
$862$	−31.4543	−1.07134
$863$	27.4134	0.933163	0.466582	−	0.884478i	$-0.345485\pi$
$863$	27.4134	0.933163	0.466582	+	0.884478i	$0.345485\pi$
$864$	0	0
$865$	21.5285	0.731991
$866$	−1.26594	−0.0430184
$867$	0	0
$868$	0.353826	0.0120096
$869$	32.7537	1.11109
$870$	0	0
$871$	−1.66232	−0.0563257
$872$	−22.9642	−0.777665
$873$	0	0
$874$	1.63899	0.0554397
$875$	17.9169	0.605702
$876$	0	0
$877$	−22.7408	−0.767901	−0.383950	−	0.923354i	$-0.625437\pi$
$877$	−22.7408	−0.767901	−0.383950	+	0.923354i	$0.625437\pi$
$878$	11.5321	0.389191
$879$	0	0
$880$	−52.5668	−1.77203
$881$	39.6786	1.33681	0.668403	−	0.743799i	$-0.266978\pi$
$881$	39.6786	1.33681	0.668403	+	0.743799i	$0.266978\pi$
$882$	0	0
$883$	−7.10176	−0.238993	−0.119497	−	0.992835i	$-0.538128\pi$
$883$	−7.10176	−0.238993	−0.119497	+	0.992835i	$0.538128\pi$
$884$	0.352915	0.0118698
$885$	0	0
$886$	16.8362	0.565623
$887$	6.07103	0.203845	0.101923	−	0.994792i	$-0.467501\pi$
$887$	6.07103	0.203845	0.101923	+	0.994792i	$0.467501\pi$
$888$	0	0
$889$	17.8323	0.598075
$890$	−38.8234	−1.30136
$891$	0	0
$892$	−0.103765	−0.00347432
$893$	1.10912	0.0371153
$894$	0	0
$895$	21.7562	0.727229
$896$	22.1664	0.740528
$897$	0	0
$898$	−29.2240	−0.975217
$899$	9.78538	0.326361
$900$	0	0
$901$	−10.3157	−0.343665
$902$	−30.9218	−1.02958
$903$	0	0
$904$	−36.4746	−1.21313
$905$	7.54681	0.250864
$906$	0	0
$907$	−31.7562	−1.05445	−0.527224	−	0.849726i	$-0.676767\pi$
$907$	−31.7562	−1.05445	−0.527224	+	0.849726i	$0.676767\pi$
$908$	−1.38118	−0.0458361
$909$	0	0
$910$	−6.14817	−0.203810
$911$	19.3621	0.641494	0.320747	−	0.947165i	$-0.396066\pi$
$911$	19.3621	0.641494	0.320747	+	0.947165i	$0.396066\pi$
$912$	0	0
$913$	8.31832	0.275296
$914$	−13.6475	−0.451418
$915$	0	0
$916$	0.303339	0.0100226
$917$	−29.3088	−0.967863
$918$	0	0
$919$	13.9505	0.460184	0.230092	−	0.973169i	$-0.426097\pi$
$919$	13.9505	0.460184	0.230092	+	0.973169i	$0.426097\pi$
$920$	2.64522	0.0872102
$921$	0	0
$922$	−49.4607	−1.62890
$923$	−8.47543	−0.278972
$924$	0	0
$925$	2.70461	0.0889270
$926$	−20.2511	−0.665493
$927$	0	0
$928$	−3.60245	−0.118256
$929$	−0.426220	−0.0139838	−0.00699190	−	0.999976i	$-0.502226\pi$
$929$	−0.426220	−0.0139838	−0.00699190	+	0.999976i	$0.502226\pi$
$930$	0	0
$931$	8.20808	0.269009
$932$	−1.12315	−0.0367899
$933$	0	0
$934$	−26.3843	−0.863319
$935$	59.0578	1.93140
$936$	0	0
$937$	14.8674	0.485696	0.242848	−	0.970064i	$-0.421918\pi$
$937$	14.8674	0.485696	0.242848	+	0.970064i	$0.421918\pi$
$938$	6.12172	0.199881
$939$	0	0
$940$	0.0882914	0.00287975
$941$	19.5985	0.638892	0.319446	−	0.947605i	$-0.396503\pi$
$941$	19.5985	0.638892	0.319446	+	0.947605i	$0.396503\pi$
$942$	0	0
$943$	1.47507	0.0480349
$944$	2.82684	0.0920057
$945$	0	0
$946$	−41.1111	−1.33664
$947$	−30.4348	−0.988997	−0.494498	−	0.869179i	$-0.664648\pi$
$947$	−30.4348	−0.988997	−0.494498	+	0.869179i	$0.664648\pi$
$948$	0	0
$949$	−0.440322	−0.0142935
$950$	8.35971	0.271225
$951$	0	0
$952$	−26.3495	−0.853992
$953$	−20.6788	−0.669853	−0.334927	−	0.942244i	$-0.608712\pi$
$953$	−20.6788	−0.669853	−0.334927	+	0.942244i	$0.608712\pi$
$954$	0	0
$955$	37.8469	1.22470
$956$	0.642501	0.0207800
$957$	0	0
$958$	20.7573	0.670637
$959$	−23.2771	−0.751658
$960$	0	0
$961$	−28.4630	−0.918161
$962$	1.66779	0.0537717
$963$	0	0
$964$	−2.00269	−0.0645022
$965$	−18.5424	−0.596902
$966$	0	0
$967$	18.8082	0.604832	0.302416	−	0.953176i	$-0.402207\pi$
$967$	18.8082	0.604832	0.302416	+	0.953176i	$0.402207\pi$
$968$	50.5995	1.62633
$969$	0	0
$970$	−11.0942	−0.356215
$971$	−26.4333	−0.848286	−0.424143	−	0.905595i	$-0.639425\pi$
$971$	−26.4333	−0.848286	−0.424143	+	0.905595i	$0.639425\pi$
$972$	0	0
$973$	5.72463	0.183523
$974$	26.4472	0.847423
$975$	0	0
$976$	8.20091	0.262505
$977$	0.425347	0.0136080	0.00680402	−	0.999977i	$-0.497834\pi$
$977$	0.425347	0.0136080	0.00680402	+	0.999977i	$0.497834\pi$
$978$	0	0
$979$	57.7371	1.84528
$980$	0.653403	0.0208722
$981$	0	0
$982$	28.2420	0.901239
$983$	−19.1510	−0.610822	−0.305411	−	0.952221i	$-0.598794\pi$
$983$	−19.1510	−0.610822	−0.305411	+	0.952221i	$0.598794\pi$
$984$	0	0
$985$	−15.8491	−0.504995
$986$	−35.9431	−1.14466
$987$	0	0
$988$	−0.282091	−0.00897449
$989$	1.96113	0.0623603
$990$	0	0
$991$	−14.4730	−0.459750	−0.229875	−	0.973220i	$-0.573832\pi$
$991$	−14.4730	−0.459750	−0.229875	+	0.973220i	$0.573832\pi$
$992$	−0.933989	−0.0296542
$993$	0	0
$994$	31.2118	0.989979
$995$	27.8806	0.883874
$996$	0	0
$997$	29.6318	0.938448	0.469224	−	0.883079i	$-0.344534\pi$
$997$	29.6318	0.938448	0.469224	+	0.883079i	$0.344534\pi$
$998$	−27.8496	−0.881562
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.t.1.12	✓	40
3.2	odd	2	inner	6021.2.a.t.1.29	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.t.1.12	✓	40	1.1	even	1	trivial
6021.2.a.t.1.29	yes	40	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 6021 = 3^{3} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6021.a (trivial)

\(f(q)\)	\(=\)	\(q-1.37704 q^{2} -0.103765 q^{4} +2.60531 q^{5} -2.14080 q^{7} +2.89697 q^{8} +O(q^{10})\)	\(q-1.37704 q^{2} -0.103765 q^{4} +2.60531 q^{5} -2.14080 q^{7} +2.89697 q^{8} -3.58761 q^{10} +5.33539 q^{11} -0.800507 q^{13} +2.94797 q^{14} -3.78170 q^{16} +4.24866 q^{17} -3.39603 q^{19} -0.270340 q^{20} -7.34703 q^{22} +0.350477 q^{23} +1.78762 q^{25} +1.10233 q^{26} +0.222141 q^{28} +6.14352 q^{29} +1.59280 q^{31} -0.586383 q^{32} -5.85057 q^{34} -5.57744 q^{35} +1.51297 q^{37} +4.67646 q^{38} +7.54748 q^{40} +4.20875 q^{41} +5.59560 q^{43} -0.553629 q^{44} -0.482620 q^{46} -0.326594 q^{47} -2.41696 q^{49} -2.46161 q^{50} +0.0830649 q^{52} -2.42798 q^{53} +13.9003 q^{55} -6.20183 q^{56} -8.45986 q^{58} -0.747504 q^{59} -2.16858 q^{61} -2.19334 q^{62} +8.37088 q^{64} -2.08556 q^{65} +2.07659 q^{67} -0.440864 q^{68} +7.68035 q^{70} +10.5876 q^{71} +0.550054 q^{73} -2.08342 q^{74} +0.352390 q^{76} -11.4220 q^{77} +6.13896 q^{79} -9.85249 q^{80} -5.79561 q^{82} +1.55908 q^{83} +11.0691 q^{85} -7.70536 q^{86} +15.4564 q^{88} +10.8215 q^{89} +1.71373 q^{91} -0.0363674 q^{92} +0.449732 q^{94} -8.84769 q^{95} +3.09238 q^{97} +3.32825 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) \) 40 * q + 46 * q^4 + 16 * q^7	\( 40 q + 46 q^{4} + 16 q^{7} + 22 q^{10} + 14 q^{13} + 50 q^{16} + 64 q^{19} + 12 q^{22} + 40 q^{25} + 48 q^{28} + 54 q^{31} + 32 q^{34} + 24 q^{37} + 40 q^{40} + 24 q^{43} + 52 q^{46} + 64 q^{49} + 18 q^{52} + 36 q^{55} + 8 q^{58} + 58 q^{61} + 120 q^{64} + 52 q^{67} - 30 q^{70} + 50 q^{73} + 112 q^{76} + 60 q^{79} + 50 q^{82} + 38 q^{85} + 16 q^{88} + 118 q^{91} + 44 q^{94} + 38 q^{97}+O(q^{100}) \) 40 * q + 46 * q^4 + 16 * q^7 + 22 * q^10 + 14 * q^13 + 50 * q^16 + 64 * q^19 + 12 * q^22 + 40 * q^25 + 48 * q^28 + 54 * q^31 + 32 * q^34 + 24 * q^37 + 40 * q^40 + 24 * q^43 + 52 * q^46 + 64 * q^49 + 18 * q^52 + 36 * q^55 + 8 * q^58 + 58 * q^61 + 120 * q^64 + 52 * q^67 - 30 * q^70 + 50 * q^73 + 112 * q^76 + 60 * q^79 + 50 * q^82 + 38 * q^85 + 16 * q^88 + 118 * q^91 + 44 * q^94 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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