LMFDB - Embedded newform 8001.2.a.z.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.203292 q^{2} -1.95867 q^{4} +2.16833 q^{5} -1.00000 q^{7} -0.804765 q^{8} +O(q^{10})$	\(q+0.203292 q^{2} -1.95867 q^{4} +2.16833 q^{5} -1.00000 q^{7} -0.804765 q^{8} +0.440803 q^{10} -2.03638 q^{11} +1.83655 q^{13} -0.203292 q^{14} +3.75374 q^{16} +2.26480 q^{17} -5.79171 q^{19} -4.24705 q^{20} -0.413979 q^{22} -2.00013 q^{23} -0.298346 q^{25} +0.373355 q^{26} +1.95867 q^{28} +2.70238 q^{29} +0.925459 q^{31} +2.37263 q^{32} +0.460414 q^{34} -2.16833 q^{35} +7.11408 q^{37} -1.17741 q^{38} -1.74500 q^{40} +3.83498 q^{41} -0.434751 q^{43} +3.98860 q^{44} -0.406610 q^{46} -4.07454 q^{47} +1.00000 q^{49} -0.0606513 q^{50} -3.59720 q^{52} +6.46471 q^{53} -4.41554 q^{55} +0.804765 q^{56} +0.549370 q^{58} -13.7109 q^{59} -9.25010 q^{61} +0.188138 q^{62} -7.02515 q^{64} +3.98225 q^{65} +6.88314 q^{67} -4.43599 q^{68} -0.440803 q^{70} -14.1463 q^{71} +4.78576 q^{73} +1.44623 q^{74} +11.3441 q^{76} +2.03638 q^{77} +10.8460 q^{79} +8.13935 q^{80} +0.779619 q^{82} -4.97268 q^{83} +4.91082 q^{85} -0.0883813 q^{86} +1.63881 q^{88} +1.11999 q^{89} -1.83655 q^{91} +3.91761 q^{92} -0.828320 q^{94} -12.5583 q^{95} +7.04552 q^{97} +0.203292 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) $ 32 * q + 30 * q^4 - 32 * q^7	$ 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) $ 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * 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 $

$2$	0.203292	0.143749	0.0718744	−	0.997414i	$-0.477102\pi$
$2$	0.203292	0.143749	0.0718744	+	0.997414i	$0.477102\pi$
$3$	0	0
$3$	0	0
$4$	−1.95867	−0.979336
$5$	2.16833	0.969707	0.484853	−	0.874596i	$-0.338873\pi$
$5$	2.16833	0.969707	0.484853	+	0.874596i	$0.338873\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.804765	−0.284527
$9$	0	0
$10$	0.440803	0.139394
$11$	−2.03638	−0.613991	−0.306996	−	0.951711i	$-0.599324\pi$
$11$	−2.03638	−0.613991	−0.306996	+	0.951711i	$0.599324\pi$
$12$	0	0
$13$	1.83655	0.509367	0.254684	−	0.967024i	$-0.418029\pi$
$13$	1.83655	0.509367	0.254684	+	0.967024i	$0.418029\pi$
$14$	−0.203292	−0.0543320
$15$	0	0
$16$	3.75374	0.938436
$17$	2.26480	0.549294	0.274647	−	0.961545i	$-0.411439\pi$
$17$	2.26480	0.549294	0.274647	+	0.961545i	$0.411439\pi$
$18$	0	0
$19$	−5.79171	−1.32871	−0.664354	−	0.747418i	$-0.731293\pi$
$19$	−5.79171	−1.32871	−0.664354	+	0.747418i	$0.731293\pi$
$20$	−4.24705	−0.949669
$21$	0	0
$22$	−0.413979	−0.0882606
$23$	−2.00013	−0.417057	−0.208528	−	0.978016i	$-0.566867\pi$
$23$	−2.00013	−0.417057	−0.208528	+	0.978016i	$0.566867\pi$
$24$	0	0
$25$	−0.298346	−0.0596692
$26$	0.373355	0.0732210
$27$	0	0
$28$	1.95867	0.370154
$29$	2.70238	0.501819	0.250909	−	0.968011i	$-0.419270\pi$
$29$	2.70238	0.501819	0.250909	+	0.968011i	$0.419270\pi$
$30$	0	0
$31$	0.925459	0.166217	0.0831086	−	0.996540i	$-0.473515\pi$
$31$	0.925459	0.166217	0.0831086	+	0.996540i	$0.473515\pi$
$32$	2.37263	0.419427
$33$	0	0
$34$	0.460414	0.0789604
$35$	−2.16833	−0.366515
$36$	0	0
$37$	7.11408	1.16955	0.584774	−	0.811196i	$-0.301183\pi$
$37$	7.11408	1.16955	0.584774	+	0.811196i	$0.301183\pi$
$38$	−1.17741	−0.191000
$39$	0	0
$40$	−1.74500	−0.275908
$41$	3.83498	0.598923	0.299462	−	0.954108i	$-0.403193\pi$
$41$	3.83498	0.598923	0.299462	+	0.954108i	$0.403193\pi$
$42$	0	0
$43$	−0.434751	−0.0662989	−0.0331495	−	0.999450i	$-0.510554\pi$
$43$	−0.434751	−0.0662989	−0.0331495	+	0.999450i	$0.510554\pi$
$44$	3.98860	0.601304
$45$	0	0
$46$	−0.406610	−0.0599514
$47$	−4.07454	−0.594333	−0.297166	−	0.954826i	$-0.596042\pi$
$47$	−4.07454	−0.594333	−0.297166	+	0.954826i	$0.596042\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.0606513	−0.00857738
$51$	0	0
$52$	−3.59720	−0.498842
$53$	6.46471	0.887996	0.443998	−	0.896028i	$-0.353560\pi$
$53$	6.46471	0.887996	0.443998	+	0.896028i	$0.353560\pi$
$54$	0	0
$55$	−4.41554	−0.595391
$56$	0.804765	0.107541
$57$	0	0
$58$	0.549370	0.0721359
$59$	−13.7109	−1.78501	−0.892504	−	0.451040i	$-0.851053\pi$
$59$	−13.7109	−1.78501	−0.892504	+	0.451040i	$0.851053\pi$
$60$	0	0
$61$	−9.25010	−1.18435	−0.592177	−	0.805808i	$-0.701731\pi$
$61$	−9.25010	−1.18435	−0.592177	+	0.805808i	$0.701731\pi$
$62$	0.188138	0.0238935
$63$	0	0
$64$	−7.02515	−0.878144
$65$	3.98225	0.493937
$66$	0	0
$67$	6.88314	0.840909	0.420454	−	0.907314i	$-0.361871\pi$
$67$	6.88314	0.840909	0.420454	+	0.907314i	$0.361871\pi$
$68$	−4.43599	−0.537943
$69$	0	0
$70$	−0.440803	−0.0526861
$71$	−14.1463	−1.67886	−0.839430	−	0.543468i	$-0.817111\pi$
$71$	−14.1463	−1.67886	−0.839430	+	0.543468i	$0.817111\pi$
$72$	0	0
$73$	4.78576	0.560131	0.280065	−	0.959981i	$-0.409644\pi$
$73$	4.78576	0.560131	0.280065	+	0.959981i	$0.409644\pi$
$74$	1.44623	0.168121
$75$	0	0
$76$	11.3441	1.30125
$77$	2.03638	0.232067
$78$	0	0
$79$	10.8460	1.22027	0.610135	−	0.792297i	$-0.291115\pi$
$79$	10.8460	1.22027	0.610135	+	0.792297i	$0.291115\pi$
$80$	8.13935	0.910007
$81$	0	0
$82$	0.779619	0.0860945
$83$	−4.97268	−0.545822	−0.272911	−	0.962039i	$-0.587987\pi$
$83$	−4.97268	−0.545822	−0.272911	+	0.962039i	$0.587987\pi$
$84$	0	0
$85$	4.91082	0.532654
$86$	−0.0883813	−0.00953040
$87$	0	0
$88$	1.63881	0.174697
$89$	1.11999	0.118719	0.0593594	−	0.998237i	$-0.481094\pi$
$89$	1.11999	0.118719	0.0593594	+	0.998237i	$0.481094\pi$
$90$	0	0
$91$	−1.83655	−0.192523
$92$	3.91761	0.408439
$93$	0	0
$94$	−0.828320	−0.0854347
$95$	−12.5583	−1.28846
$96$	0	0
$97$	7.04552	0.715365	0.357682	−	0.933843i	$-0.383567\pi$
$97$	7.04552	0.715365	0.357682	+	0.933843i	$0.383567\pi$
$98$	0.203292	0.0205356
$99$	0	0
$100$	0.584362	0.0584362
$101$	10.6965	1.06434	0.532170	−	0.846638i	$-0.321377\pi$
$101$	10.6965	1.06434	0.532170	+	0.846638i	$0.321377\pi$
$102$	0	0
$103$	−14.9318	−1.47128	−0.735638	−	0.677375i	$-0.763118\pi$
$103$	−14.9318	−1.47128	−0.735638	+	0.677375i	$0.763118\pi$
$104$	−1.47799	−0.144929
$105$	0	0
$106$	1.31422	0.127648
$107$	−12.2693	−1.18612	−0.593059	−	0.805159i	$-0.702080\pi$
$107$	−12.2693	−1.18612	−0.593059	+	0.805159i	$0.702080\pi$
$108$	0	0
$109$	−1.53666	−0.147185	−0.0735926	−	0.997288i	$-0.523446\pi$
$109$	−1.53666	−0.147185	−0.0735926	+	0.997288i	$0.523446\pi$
$110$	−0.897643	−0.0855869
$111$	0	0
$112$	−3.75374	−0.354695
$113$	1.85798	0.174784	0.0873920	−	0.996174i	$-0.472147\pi$
$113$	1.85798	0.174784	0.0873920	+	0.996174i	$0.472147\pi$
$114$	0	0
$115$	−4.33695	−0.404422
$116$	−5.29307	−0.491449
$117$	0	0
$118$	−2.78731	−0.256593
$119$	−2.26480	−0.207614
$120$	0	0
$121$	−6.85316	−0.623015
$122$	−1.88047	−0.170249
$123$	0	0
$124$	−1.81267	−0.162783
$125$	−11.4886	−1.02757
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−6.17342	−0.545659
$129$	0	0
$130$	0.809558	0.0710029
$131$	−5.09715	−0.445340	−0.222670	−	0.974894i	$-0.571477\pi$
$131$	−5.09715	−0.445340	−0.222670	+	0.974894i	$0.571477\pi$
$132$	0	0
$133$	5.79171	0.502205
$134$	1.39928	0.120880
$135$	0	0
$136$	−1.82263	−0.156289
$137$	3.51026	0.299901	0.149951	−	0.988693i	$-0.452088\pi$
$137$	3.51026	0.299901	0.149951	+	0.988693i	$0.452088\pi$
$138$	0	0
$139$	−3.28843	−0.278921	−0.139460	−	0.990228i	$-0.544537\pi$
$139$	−3.28843	−0.278921	−0.139460	+	0.990228i	$0.544537\pi$
$140$	4.24705	0.358941
$141$	0	0
$142$	−2.87583	−0.241334
$143$	−3.73991	−0.312747
$144$	0	0
$145$	5.85964	0.486617
$146$	0.972905	0.0805182
$147$	0	0
$148$	−13.9342	−1.14538
$149$	−14.9074	−1.22126	−0.610630	−	0.791916i	$-0.709084\pi$
$149$	−14.9074	−1.22126	−0.610630	+	0.791916i	$0.709084\pi$
$150$	0	0
$151$	7.19878	0.585828	0.292914	−	0.956139i	$-0.405375\pi$
$151$	7.19878	0.585828	0.292914	+	0.956139i	$0.405375\pi$
$152$	4.66096	0.378054
$153$	0	0
$154$	0.413979	0.0333594
$155$	2.00670	0.161182
$156$	0	0
$157$	−9.94517	−0.793711	−0.396856	−	0.917881i	$-0.629899\pi$
$157$	−9.94517	−0.793711	−0.396856	+	0.917881i	$0.629899\pi$
$158$	2.20490	0.175412
$159$	0	0
$160$	5.14465	0.406721
$161$	2.00013	0.157633
$162$	0	0
$163$	−21.0465	−1.64849	−0.824244	−	0.566235i	$-0.808399\pi$
$163$	−21.0465	−1.64849	−0.824244	+	0.566235i	$0.808399\pi$
$164$	−7.51147	−0.586547
$165$	0	0
$166$	−1.01090	−0.0784614
$167$	−9.53253	−0.737649	−0.368825	−	0.929499i	$-0.620240\pi$
$167$	−9.53253	−0.737649	−0.368825	+	0.929499i	$0.620240\pi$
$168$	0	0
$169$	−9.62708	−0.740545
$170$	0.998330	0.0765684
$171$	0	0
$172$	0.851535	0.0649289
$173$	21.1959	1.61150	0.805748	−	0.592259i	$-0.201764\pi$
$173$	21.1959	1.61150	0.805748	+	0.592259i	$0.201764\pi$
$174$	0	0
$175$	0.298346	0.0225528
$176$	−7.64404	−0.576191
$177$	0	0
$178$	0.227685	0.0170657
$179$	23.4704	1.75426	0.877130	−	0.480253i	$-0.159455\pi$
$179$	23.4704	1.75426	0.877130	+	0.480253i	$0.159455\pi$
$180$	0	0
$181$	17.4217	1.29494	0.647471	−	0.762090i	$-0.275827\pi$
$181$	17.4217	1.29494	0.647471	+	0.762090i	$0.275827\pi$
$182$	−0.373355	−0.0276749
$183$	0	0
$184$	1.60964	0.118664
$185$	15.4257	1.13412
$186$	0	0
$187$	−4.61198	−0.337262
$188$	7.98069	0.582052
$189$	0	0
$190$	−2.55300	−0.185214
$191$	10.3025	0.745461	0.372730	−	0.927940i	$-0.378422\pi$
$191$	10.3025	0.745461	0.372730	+	0.927940i	$0.378422\pi$
$192$	0	0
$193$	−5.54183	−0.398910	−0.199455	−	0.979907i	$-0.563917\pi$
$193$	−5.54183	−0.398910	−0.199455	+	0.979907i	$0.563917\pi$
$194$	1.43230	0.102833
$195$	0	0
$196$	−1.95867	−0.139905
$197$	−11.0717	−0.788825	−0.394413	−	0.918933i	$-0.629052\pi$
$197$	−11.0717	−0.788825	−0.394413	+	0.918933i	$0.629052\pi$
$198$	0	0
$199$	13.3587	0.946970	0.473485	−	0.880802i	$-0.342996\pi$
$199$	13.3587	0.946970	0.473485	+	0.880802i	$0.342996\pi$
$200$	0.240098	0.0169775
$201$	0	0
$202$	2.17450	0.152998
$203$	−2.70238	−0.189670
$204$	0	0
$205$	8.31550	0.580780
$206$	−3.03552	−0.211494
$207$	0	0
$208$	6.89394	0.478009
$209$	11.7941	0.815816
$210$	0	0
$211$	23.3900	1.61023	0.805117	−	0.593117i	$-0.202103\pi$
$211$	23.3900	1.61023	0.805117	+	0.593117i	$0.202103\pi$
$212$	−12.6622	−0.869646
$213$	0	0
$214$	−2.49425	−0.170503
$215$	−0.942684	−0.0642905
$216$	0	0
$217$	−0.925459	−0.0628242
$218$	−0.312390	−0.0211577
$219$	0	0
$220$	8.64860	0.583088
$221$	4.15941	0.279792
$222$	0	0
$223$	−24.2201	−1.62190	−0.810949	−	0.585117i	$-0.801048\pi$
$223$	−24.2201	−1.62190	−0.810949	+	0.585117i	$0.801048\pi$
$224$	−2.37263	−0.158528
$225$	0	0
$226$	0.377712	0.0251250
$227$	−29.0323	−1.92694	−0.963472	−	0.267808i	$-0.913701\pi$
$227$	−29.0323	−1.92694	−0.963472	+	0.267808i	$0.913701\pi$
$228$	0	0
$229$	−0.489031	−0.0323161	−0.0161580	−	0.999869i	$-0.505143\pi$
$229$	−0.489031	−0.0323161	−0.0161580	+	0.999869i	$0.505143\pi$
$230$	−0.881665	−0.0581353
$231$	0	0
$232$	−2.17478	−0.142781
$233$	−16.2677	−1.06573	−0.532866	−	0.846200i	$-0.678885\pi$
$233$	−16.2677	−1.06573	−0.532866	+	0.846200i	$0.678885\pi$
$234$	0	0
$235$	−8.83495	−0.576328
$236$	26.8552	1.74812
$237$	0	0
$238$	−0.460414	−0.0298442
$239$	−18.6270	−1.20488	−0.602442	−	0.798163i	$-0.705805\pi$
$239$	−18.6270	−1.20488	−0.602442	+	0.798163i	$0.705805\pi$
$240$	0	0
$241$	−18.1700	−1.17043	−0.585215	−	0.810878i	$-0.698990\pi$
$241$	−18.1700	−1.17043	−0.585215	+	0.810878i	$0.698990\pi$
$242$	−1.39319	−0.0895577
$243$	0	0
$244$	18.1179	1.15988
$245$	2.16833	0.138530
$246$	0	0
$247$	−10.6368	−0.676801
$248$	−0.744777	−0.0472934
$249$	0	0
$250$	−2.33553	−0.147712
$251$	28.9466	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$251$	28.9466	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$252$	0	0
$253$	4.07303	0.256069
$254$	−0.203292	−0.0127557
$255$	0	0
$256$	12.7953	0.799706
$257$	10.2025	0.636413	0.318207	−	0.948021i	$-0.396919\pi$
$257$	10.2025	0.636413	0.318207	+	0.948021i	$0.396919\pi$
$258$	0	0
$259$	−7.11408	−0.442048
$260$	−7.79992	−0.483730
$261$	0	0
$262$	−1.03621	−0.0640172
$263$	−0.138057	−0.00851293	−0.00425647	−	0.999991i	$-0.501355\pi$
$263$	−0.138057	−0.00851293	−0.00425647	+	0.999991i	$0.501355\pi$
$264$	0	0
$265$	14.0176	0.861095
$266$	1.17741	0.0721914
$267$	0	0
$268$	−13.4818	−0.823533
$269$	−0.767771	−0.0468118	−0.0234059	−	0.999726i	$-0.507451\pi$
$269$	−0.767771	−0.0468118	−0.0234059	+	0.999726i	$0.507451\pi$
$270$	0	0
$271$	−25.0186	−1.51977	−0.759886	−	0.650057i	$-0.774745\pi$
$271$	−25.0186	−1.51977	−0.759886	+	0.650057i	$0.774745\pi$
$272$	8.50146	0.515477
$273$	0	0
$274$	0.713606	0.0431105
$275$	0.607546	0.0366364
$276$	0	0
$277$	−24.0594	−1.44559	−0.722795	−	0.691063i	$-0.757143\pi$
$277$	−24.0594	−1.44559	−0.722795	+	0.691063i	$0.757143\pi$
$278$	−0.668510	−0.0400946
$279$	0	0
$280$	1.74500	0.104283
$281$	−10.1722	−0.606825	−0.303413	−	0.952859i	$-0.598126\pi$
$281$	−10.1722	−0.606825	−0.303413	+	0.952859i	$0.598126\pi$
$282$	0	0
$283$	−29.1780	−1.73445	−0.867227	−	0.497913i	$-0.834100\pi$
$283$	−29.1780	−1.73445	−0.867227	+	0.497913i	$0.834100\pi$
$284$	27.7080	1.64417
$285$	0	0
$286$	−0.760293	−0.0449571
$287$	−3.83498	−0.226372
$288$	0	0
$289$	−11.8707	−0.698276
$290$	1.19122	0.0699506
$291$	0	0
$292$	−9.37374	−0.548557
$293$	−12.9496	−0.756522	−0.378261	−	0.925699i	$-0.623478\pi$
$293$	−12.9496	−0.756522	−0.378261	+	0.925699i	$0.623478\pi$
$294$	0	0
$295$	−29.7298	−1.73093
$296$	−5.72516	−0.332768
$297$	0	0
$298$	−3.03054	−0.175555
$299$	−3.67335	−0.212435
$300$	0	0
$301$	0.434751	0.0250586
$302$	1.46345	0.0842122
$303$	0	0
$304$	−21.7406	−1.24691
$305$	−20.0573	−1.14848
$306$	0	0
$307$	−9.02772	−0.515239	−0.257620	−	0.966246i	$-0.582938\pi$
$307$	−9.02772	−0.515239	−0.257620	+	0.966246i	$0.582938\pi$
$308$	−3.98860	−0.227272
$309$	0	0
$310$	0.407945	0.0231697
$311$	24.7431	1.40305	0.701525	−	0.712645i	$-0.252503\pi$
$311$	24.7431	1.40305	0.701525	+	0.712645i	$0.252503\pi$
$312$	0	0
$313$	−16.8110	−0.950216	−0.475108	−	0.879928i	$-0.657591\pi$
$313$	−16.8110	−0.950216	−0.475108	+	0.879928i	$0.657591\pi$
$314$	−2.02177	−0.114095
$315$	0	0
$316$	−21.2438	−1.19505
$317$	3.14053	0.176389	0.0881947	−	0.996103i	$-0.471890\pi$
$317$	3.14053	0.176389	0.0881947	+	0.996103i	$0.471890\pi$
$318$	0	0
$319$	−5.50306	−0.308112
$320$	−15.2328	−0.851542
$321$	0	0
$322$	0.406610	0.0226595
$323$	−13.1170	−0.729851
$324$	0	0
$325$	−0.547928	−0.0303936
$326$	−4.27857	−0.236968
$327$	0	0
$328$	−3.08626	−0.170410
$329$	4.07454	0.224637
$330$	0	0
$331$	−20.2551	−1.11332	−0.556660	−	0.830740i	$-0.687917\pi$
$331$	−20.2551	−1.11332	−0.556660	+	0.830740i	$0.687917\pi$
$332$	9.73985	0.534544
$333$	0	0
$334$	−1.93788	−0.106036
$335$	14.9249	0.815435
$336$	0	0
$337$	11.4420	0.623283	0.311641	−	0.950200i	$-0.399121\pi$
$337$	11.4420	0.623283	0.311641	+	0.950200i	$0.399121\pi$
$338$	−1.95711	−0.106452
$339$	0	0
$340$	−9.61870	−0.521647
$341$	−1.88458	−0.102056
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0.349873	0.0188639
$345$	0	0
$346$	4.30895	0.231651
$347$	13.6007	0.730123	0.365061	−	0.930983i	$-0.381048\pi$
$347$	13.6007	0.730123	0.365061	+	0.930983i	$0.381048\pi$
$348$	0	0
$349$	6.18545	0.331100	0.165550	−	0.986201i	$-0.447060\pi$
$349$	6.18545	0.331100	0.165550	+	0.986201i	$0.447060\pi$
$350$	0.0606513	0.00324195
$351$	0	0
$352$	−4.83158	−0.257524
$353$	34.6672	1.84515	0.922574	−	0.385821i	$-0.126082\pi$
$353$	34.6672	1.84515	0.922574	+	0.385821i	$0.126082\pi$
$354$	0	0
$355$	−30.6739	−1.62800
$356$	−2.19370	−0.116266
$357$	0	0
$358$	4.77134	0.252173
$359$	−11.3002	−0.596400	−0.298200	−	0.954503i	$-0.596386\pi$
$359$	−11.3002	−0.596400	−0.298200	+	0.954503i	$0.596386\pi$
$360$	0	0
$361$	14.5439	0.765467
$362$	3.54168	0.186146
$363$	0	0
$364$	3.59720	0.188545
$365$	10.3771	0.543163
$366$	0	0
$367$	14.3988	0.751611	0.375806	−	0.926698i	$-0.377366\pi$
$367$	14.3988	0.751611	0.375806	+	0.926698i	$0.377366\pi$
$368$	−7.50799	−0.391381
$369$	0	0
$370$	3.13591	0.163028
$371$	−6.46471	−0.335631
$372$	0	0
$373$	−17.8622	−0.924870	−0.462435	−	0.886653i	$-0.653024\pi$
$373$	−17.8622	−0.924870	−0.462435	+	0.886653i	$0.653024\pi$
$374$	−0.937577	−0.0484810
$375$	0	0
$376$	3.27905	0.169104
$377$	4.96305	0.255610
$378$	0	0
$379$	8.53256	0.438288	0.219144	−	0.975693i	$-0.429674\pi$
$379$	8.53256	0.438288	0.219144	+	0.975693i	$0.429674\pi$
$380$	24.5977	1.26183
$381$	0	0
$382$	2.09441	0.107159
$383$	29.9038	1.52801	0.764006	−	0.645210i	$-0.223230\pi$
$383$	29.9038	1.52801	0.764006	+	0.645210i	$0.223230\pi$
$384$	0	0
$385$	4.41554	0.225037
$386$	−1.12661	−0.0573428
$387$	0	0
$388$	−13.7999	−0.700582
$389$	8.18046	0.414766	0.207383	−	0.978260i	$-0.433505\pi$
$389$	8.18046	0.414766	0.207383	+	0.978260i	$0.433505\pi$
$390$	0	0
$391$	−4.52989	−0.229087
$392$	−0.804765	−0.0406468
$393$	0	0
$394$	−2.25078	−0.113393
$395$	23.5177	1.18330
$396$	0	0
$397$	−1.31342	−0.0659188	−0.0329594	−	0.999457i	$-0.510493\pi$
$397$	−1.31342	−0.0659188	−0.0329594	+	0.999457i	$0.510493\pi$
$398$	2.71570	0.136126
$399$	0	0
$400$	−1.11991	−0.0559957
$401$	−33.1873	−1.65730	−0.828648	−	0.559770i	$-0.810889\pi$
$401$	−33.1873	−1.65730	−0.828648	+	0.559770i	$0.810889\pi$
$402$	0	0
$403$	1.69965	0.0846657
$404$	−20.9509	−1.04235
$405$	0	0
$406$	−0.549370	−0.0272648
$407$	−14.4870	−0.718092
$408$	0	0
$409$	−29.9044	−1.47868	−0.739340	−	0.673333i	$-0.764862\pi$
$409$	−29.9044	−1.47868	−0.739340	+	0.673333i	$0.764862\pi$
$410$	1.69047	0.0834864
$411$	0	0
$412$	29.2466	1.44087
$413$	13.7109	0.674669
$414$	0	0
$415$	−10.7824	−0.529288
$416$	4.35746	0.213642
$417$	0	0
$418$	2.39764	0.117273
$419$	−33.5181	−1.63747	−0.818734	−	0.574173i	$-0.805324\pi$
$419$	−33.5181	−1.63747	−0.818734	+	0.574173i	$0.805324\pi$
$420$	0	0
$421$	−33.0936	−1.61288	−0.806442	−	0.591314i	$-0.798610\pi$
$421$	−33.0936	−1.61288	−0.806442	+	0.591314i	$0.798610\pi$
$422$	4.75499	0.231469
$423$	0	0
$424$	−5.20257	−0.252659
$425$	−0.675693	−0.0327759
$426$	0	0
$427$	9.25010	0.447643
$428$	24.0315	1.16161
$429$	0	0
$430$	−0.191640	−0.00924169
$431$	1.90178	0.0916053	0.0458026	−	0.998951i	$-0.485415\pi$
$431$	1.90178	0.0916053	0.0458026	+	0.998951i	$0.485415\pi$
$432$	0	0
$433$	−22.4641	−1.07956	−0.539778	−	0.841807i	$-0.681492\pi$
$433$	−22.4641	−1.07956	−0.539778	+	0.841807i	$0.681492\pi$
$434$	−0.188138	−0.00903091
$435$	0	0
$436$	3.00981	0.144144
$437$	11.5842	0.554147
$438$	0	0
$439$	6.36163	0.303624	0.151812	−	0.988409i	$-0.451489\pi$
$439$	6.36163	0.303624	0.151812	+	0.988409i	$0.451489\pi$
$440$	3.55347	0.169405
$441$	0	0
$442$	0.845574	0.0402198
$443$	11.7960	0.560444	0.280222	−	0.959935i	$-0.409592\pi$
$443$	11.7960	0.560444	0.280222	+	0.959935i	$0.409592\pi$
$444$	0	0
$445$	2.42851	0.115122
$446$	−4.92374	−0.233146
$447$	0	0
$448$	7.02515	0.331907
$449$	−26.8230	−1.26586	−0.632928	−	0.774211i	$-0.718147\pi$
$449$	−26.8230	−1.26586	−0.632928	+	0.774211i	$0.718147\pi$
$450$	0	0
$451$	−7.80947	−0.367734
$452$	−3.63917	−0.171172
$453$	0	0
$454$	−5.90203	−0.276996
$455$	−3.98225	−0.186691
$456$	0	0
$457$	−10.2927	−0.481470	−0.240735	−	0.970591i	$-0.577388\pi$
$457$	−10.2927	−0.481470	−0.240735	+	0.970591i	$0.577388\pi$
$458$	−0.0994160	−0.00464540
$459$	0	0
$460$	8.49466	0.396066
$461$	−35.1566	−1.63741	−0.818703	−	0.574217i	$-0.805307\pi$
$461$	−35.1566	−1.63741	−0.818703	+	0.574217i	$0.805307\pi$
$462$	0	0
$463$	−35.3381	−1.64230	−0.821150	−	0.570712i	$-0.806667\pi$
$463$	−35.3381	−1.64230	−0.821150	+	0.570712i	$0.806667\pi$
$464$	10.1440	0.470924
$465$	0	0
$466$	−3.30709	−0.153198
$467$	−7.24322	−0.335176	−0.167588	−	0.985857i	$-0.553598\pi$
$467$	−7.24322	−0.335176	−0.167588	+	0.985857i	$0.553598\pi$
$468$	0	0
$469$	−6.88314	−0.317834
$470$	−1.79607	−0.0828466
$471$	0	0
$472$	11.0341	0.507884
$473$	0.885318	0.0407070
$474$	0	0
$475$	1.72793	0.0792830
$476$	4.43599	0.203323
$477$	0	0
$478$	−3.78672	−0.173201
$479$	9.85957	0.450495	0.225248	−	0.974302i	$-0.427681\pi$
$479$	9.85957	0.450495	0.225248	+	0.974302i	$0.427681\pi$
$480$	0	0
$481$	13.0654	0.595730
$482$	−3.69380	−0.168248
$483$	0	0
$484$	13.4231	0.610141
$485$	15.2770	0.693694
$486$	0	0
$487$	2.37567	0.107652	0.0538259	−	0.998550i	$-0.482858\pi$
$487$	2.37567	0.107652	0.0538259	+	0.998550i	$0.482858\pi$
$488$	7.44415	0.336981
$489$	0	0
$490$	0.440803	0.0199135
$491$	28.4513	1.28399	0.641995	−	0.766709i	$-0.278107\pi$
$491$	28.4513	1.28399	0.641995	+	0.766709i	$0.278107\pi$
$492$	0	0
$493$	6.12033	0.275646
$494$	−2.16237	−0.0972894
$495$	0	0
$496$	3.47393	0.155984
$497$	14.1463	0.634549
$498$	0	0
$499$	−12.8574	−0.575575	−0.287787	−	0.957694i	$-0.592920\pi$
$499$	−12.8574	−0.575575	−0.287787	+	0.957694i	$0.592920\pi$
$500$	22.5023	1.00633
$501$	0	0
$502$	5.88459	0.262642
$503$	5.24764	0.233981	0.116990	−	0.993133i	$-0.462675\pi$
$503$	5.24764	0.233981	0.116990	+	0.993133i	$0.462675\pi$
$504$	0	0
$505$	23.1935	1.03210
$506$	0.828013	0.0368096
$507$	0	0
$508$	1.95867	0.0869020
$509$	−23.4333	−1.03866	−0.519332	−	0.854572i	$-0.673819\pi$
$509$	−23.4333	−1.03866	−0.519332	+	0.854572i	$0.673819\pi$
$510$	0	0
$511$	−4.78576	−0.211710
$512$	14.9480	0.660616
$513$	0	0
$514$	2.07408	0.0914837
$515$	−32.3771	−1.42671
$516$	0	0
$517$	8.29731	0.364915
$518$	−1.44623	−0.0635438
$519$	0	0
$520$	−3.20477	−0.140539
$521$	8.39413	0.367754	0.183877	−	0.982949i	$-0.441135\pi$
$521$	8.39413	0.367754	0.183877	+	0.982949i	$0.441135\pi$
$522$	0	0
$523$	−17.3043	−0.756663	−0.378331	−	0.925670i	$-0.623502\pi$
$523$	−17.3043	−0.756663	−0.378331	+	0.925670i	$0.623502\pi$
$524$	9.98365	0.436138
$525$	0	0
$526$	−0.0280657	−0.00122372
$527$	2.09597	0.0913021
$528$	0	0
$529$	−18.9995	−0.826064
$530$	2.84966	0.123781
$531$	0	0
$532$	−11.3441	−0.491827
$533$	7.04313	0.305072
$534$	0	0
$535$	−26.6039	−1.15019
$536$	−5.53931	−0.239262
$537$	0	0
$538$	−0.156081	−0.00672915
$539$	−2.03638	−0.0877130
$540$	0	0
$541$	−5.60112	−0.240811	−0.120406	−	0.992725i	$-0.538419\pi$
$541$	−5.60112	−0.240811	−0.120406	+	0.992725i	$0.538419\pi$
$542$	−5.08607	−0.218465
$543$	0	0
$544$	5.37353	0.230388
$545$	−3.33198	−0.142726
$546$	0	0
$547$	2.16434	0.0925407	0.0462703	−	0.998929i	$-0.485266\pi$
$547$	2.16434	0.0925407	0.0462703	+	0.998929i	$0.485266\pi$
$548$	−6.87544	−0.293704
$549$	0	0
$550$	0.123509	0.00526644
$551$	−15.6514	−0.666771
$552$	0	0
$553$	−10.8460	−0.461219
$554$	−4.89107	−0.207802
$555$	0	0
$556$	6.44096	0.273157
$557$	34.6728	1.46914	0.734568	−	0.678535i	$-0.237385\pi$
$557$	34.6728	1.46914	0.734568	+	0.678535i	$0.237385\pi$
$558$	0	0
$559$	−0.798442	−0.0337705
$560$	−8.13935	−0.343950
$561$	0	0
$562$	−2.06793	−0.0872304
$563$	15.0614	0.634762	0.317381	−	0.948298i	$-0.397197\pi$
$563$	15.0614	0.634762	0.317381	+	0.948298i	$0.397197\pi$
$564$	0	0
$565$	4.02871	0.169489
$566$	−5.93165	−0.249326
$567$	0	0
$568$	11.3845	0.477682
$569$	17.9482	0.752426	0.376213	−	0.926533i	$-0.377226\pi$
$569$	17.9482	0.752426	0.376213	+	0.926533i	$0.377226\pi$
$570$	0	0
$571$	4.27764	0.179014	0.0895069	−	0.995986i	$-0.471471\pi$
$571$	4.27764	0.179014	0.0895069	+	0.995986i	$0.471471\pi$
$572$	7.32526	0.306285
$573$	0	0
$574$	−0.779619	−0.0325407
$575$	0.596732	0.0248854
$576$	0	0
$577$	38.9224	1.62036	0.810180	−	0.586182i	$-0.199370\pi$
$577$	38.9224	1.62036	0.810180	+	0.586182i	$0.199370\pi$
$578$	−2.41321	−0.100376
$579$	0	0
$580$	−11.4771	−0.476561
$581$	4.97268	0.206302
$582$	0	0
$583$	−13.1646	−0.545222
$584$	−3.85141	−0.159373
$585$	0	0
$586$	−2.63254	−0.108749
$587$	−24.3999	−1.00709	−0.503546	−	0.863968i	$-0.667972\pi$
$587$	−24.3999	−1.00709	−0.503546	+	0.863968i	$0.667972\pi$
$588$	0	0
$589$	−5.35999	−0.220854
$590$	−6.04381	−0.248820
$591$	0	0
$592$	26.7044	1.09755
$593$	−20.7529	−0.852219	−0.426110	−	0.904672i	$-0.640116\pi$
$593$	−20.7529	−0.852219	−0.426110	+	0.904672i	$0.640116\pi$
$594$	0	0
$595$	−4.91082	−0.201324
$596$	29.1986	1.19602
$597$	0	0
$598$	−0.746760	−0.0305373
$599$	35.5562	1.45279	0.726393	−	0.687280i	$-0.241195\pi$
$599$	35.5562	1.45279	0.726393	+	0.687280i	$0.241195\pi$
$600$	0	0
$601$	4.85431	0.198011	0.0990057	−	0.995087i	$-0.468434\pi$
$601$	4.85431	0.198011	0.0990057	+	0.995087i	$0.468434\pi$
$602$	0.0883813	0.00360215
$603$	0	0
$604$	−14.1000	−0.573723
$605$	−14.8599	−0.604141
$606$	0	0
$607$	2.55179	0.103574	0.0517869	−	0.998658i	$-0.483508\pi$
$607$	2.55179	0.103574	0.0517869	+	0.998658i	$0.483508\pi$
$608$	−13.7416	−0.557296
$609$	0	0
$610$	−4.07747	−0.165092
$611$	−7.48310	−0.302734
$612$	0	0
$613$	41.2815	1.66734	0.833672	−	0.552259i	$-0.186234\pi$
$613$	41.2815	1.66734	0.833672	+	0.552259i	$0.186234\pi$
$614$	−1.83526	−0.0740651
$615$	0	0
$616$	−1.63881	−0.0660294
$617$	−7.93093	−0.319287	−0.159644	−	0.987175i	$-0.551034\pi$
$617$	−7.93093	−0.319287	−0.159644	+	0.987175i	$0.551034\pi$
$618$	0	0
$619$	3.93425	0.158131	0.0790655	−	0.996869i	$-0.474806\pi$
$619$	3.93425	0.158131	0.0790655	+	0.996869i	$0.474806\pi$
$620$	−3.93047	−0.157851
$621$	0	0
$622$	5.03006	0.201687
$623$	−1.11999	−0.0448715
$624$	0	0
$625$	−23.4193	−0.936770
$626$	−3.41754	−0.136592
$627$	0	0
$628$	19.4793	0.777310
$629$	16.1119	0.642425
$630$	0	0
$631$	0.666745	0.0265427	0.0132713	−	0.999912i	$-0.495775\pi$
$631$	0.666745	0.0265427	0.0132713	+	0.999912i	$0.495775\pi$
$632$	−8.72848	−0.347200
$633$	0	0
$634$	0.638443	0.0253558
$635$	−2.16833	−0.0860475
$636$	0	0
$637$	1.83655	0.0727668
$638$	−1.11873	−0.0442908
$639$	0	0
$640$	−13.3860	−0.529129
$641$	−16.8540	−0.665692	−0.332846	−	0.942981i	$-0.608009\pi$
$641$	−16.8540	−0.665692	−0.332846	+	0.942981i	$0.608009\pi$
$642$	0	0
$643$	−25.0458	−0.987709	−0.493854	−	0.869545i	$-0.664412\pi$
$643$	−25.0458	−0.987709	−0.493854	+	0.869545i	$0.664412\pi$
$644$	−3.91761	−0.154375
$645$	0	0
$646$	−2.66658	−0.104915
$647$	6.04381	0.237607	0.118803	−	0.992918i	$-0.462094\pi$
$647$	6.04381	0.237607	0.118803	+	0.992918i	$0.462094\pi$
$648$	0	0
$649$	27.9206	1.09598
$650$	−0.111389	−0.00436904
$651$	0	0
$652$	41.2232	1.61442
$653$	−22.8710	−0.895010	−0.447505	−	0.894282i	$-0.647687\pi$
$653$	−22.8710	−0.895010	−0.447505	+	0.894282i	$0.647687\pi$
$654$	0	0
$655$	−11.0523	−0.431849
$656$	14.3955	0.562051
$657$	0	0
$658$	0.828320	0.0322913
$659$	41.9778	1.63522	0.817612	−	0.575770i	$-0.195298\pi$
$659$	41.9778	1.63522	0.817612	+	0.575770i	$0.195298\pi$
$660$	0	0
$661$	22.3913	0.870920	0.435460	−	0.900208i	$-0.356586\pi$
$661$	22.3913	0.870920	0.435460	+	0.900208i	$0.356586\pi$
$662$	−4.11769	−0.160039
$663$	0	0
$664$	4.00184	0.155301
$665$	12.5583	0.486991
$666$	0	0
$667$	−5.40511	−0.209287
$668$	18.6711	0.722407
$669$	0	0
$670$	3.03411	0.117218
$671$	18.8367	0.727183
$672$	0	0
$673$	−22.7915	−0.878550	−0.439275	−	0.898353i	$-0.644765\pi$
$673$	−22.7915	−0.878550	−0.439275	+	0.898353i	$0.644765\pi$
$674$	2.32605	0.0895962
$675$	0	0
$676$	18.8563	0.725242
$677$	−2.11742	−0.0813790	−0.0406895	−	0.999172i	$-0.512955\pi$
$677$	−2.11742	−0.0813790	−0.0406895	+	0.999172i	$0.512955\pi$
$678$	0	0
$679$	−7.04552	−0.270382
$680$	−3.95206	−0.151555
$681$	0	0
$682$	−0.383120	−0.0146704
$683$	−48.9058	−1.87133	−0.935665	−	0.352889i	$-0.885199\pi$
$683$	−48.9058	−1.87133	−0.935665	+	0.352889i	$0.885199\pi$
$684$	0	0
$685$	7.61139	0.290816
$686$	−0.203292	−0.00776171
$687$	0	0
$688$	−1.63194	−0.0622173
$689$	11.8728	0.452316
$690$	0	0
$691$	−7.85867	−0.298958	−0.149479	−	0.988765i	$-0.547760\pi$
$691$	−7.85867	−0.298958	−0.149479	+	0.988765i	$0.547760\pi$
$692$	−41.5159	−1.57820
$693$	0	0
$694$	2.76490	0.104954
$695$	−7.13040	−0.270471
$696$	0	0
$697$	8.68545	0.328985
$698$	1.25745	0.0475952
$699$	0	0
$700$	−0.584362	−0.0220868
$701$	37.9287	1.43255	0.716274	−	0.697819i	$-0.245846\pi$
$701$	37.9287	1.43255	0.716274	+	0.697819i	$0.245846\pi$
$702$	0	0
$703$	−41.2027	−1.55399
$704$	14.3059	0.539173
$705$	0	0
$706$	7.04755	0.265238
$707$	−10.6965	−0.402282
$708$	0	0
$709$	−15.3664	−0.577096	−0.288548	−	0.957465i	$-0.593172\pi$
$709$	−15.3664	−0.577096	−0.288548	+	0.957465i	$0.593172\pi$
$710$	−6.23575	−0.234023
$711$	0	0
$712$	−0.901330	−0.0337788
$713$	−1.85104	−0.0693220
$714$	0	0
$715$	−8.10936	−0.303273
$716$	−45.9708	−1.71801
$717$	0	0
$718$	−2.29723	−0.0857318
$719$	−20.3461	−0.758783	−0.379392	−	0.925236i	$-0.623867\pi$
$719$	−20.3461	−0.758783	−0.379392	+	0.925236i	$0.623867\pi$
$720$	0	0
$721$	14.9318	0.556090
$722$	2.95665	0.110035
$723$	0	0
$724$	−34.1233	−1.26818
$725$	−0.806243	−0.0299431
$726$	0	0
$727$	31.2751	1.15993	0.579965	−	0.814641i	$-0.303066\pi$
$727$	31.2751	1.15993	0.579965	+	0.814641i	$0.303066\pi$
$728$	1.47799	0.0547780
$729$	0	0
$730$	2.10958	0.0780790
$731$	−0.984623	−0.0364176
$732$	0	0
$733$	−29.8273	−1.10170	−0.550848	−	0.834606i	$-0.685696\pi$
$733$	−29.8273	−1.10170	−0.550848	+	0.834606i	$0.685696\pi$
$734$	2.92716	0.108043
$735$	0	0
$736$	−4.74558	−0.174925
$737$	−14.0167	−0.516311
$738$	0	0
$739$	49.0703	1.80508	0.902539	−	0.430607i	$-0.141701\pi$
$739$	49.0703	1.80508	0.902539	+	0.430607i	$0.141701\pi$
$740$	−30.2138	−1.11068
$741$	0	0
$742$	−1.31422	−0.0482466
$743$	−14.4447	−0.529925	−0.264962	−	0.964259i	$-0.585360\pi$
$743$	−14.4447	−0.529925	−0.264962	+	0.964259i	$0.585360\pi$
$744$	0	0
$745$	−32.3241	−1.18426
$746$	−3.63124	−0.132949
$747$	0	0
$748$	9.03336	0.330292
$749$	12.2693	0.448310
$750$	0	0
$751$	−39.1877	−1.42998	−0.714989	−	0.699136i	$-0.753568\pi$
$751$	−39.1877	−1.42998	−0.714989	+	0.699136i	$0.753568\pi$
$752$	−15.2948	−0.557743
$753$	0	0
$754$	1.00895	0.0367437
$755$	15.6093	0.568082
$756$	0	0
$757$	25.1509	0.914125	0.457063	−	0.889435i	$-0.348902\pi$
$757$	25.1509	0.914125	0.457063	+	0.889435i	$0.348902\pi$
$758$	1.73460	0.0630034
$759$	0	0
$760$	10.1065	0.366602
$761$	14.0558	0.509521	0.254760	−	0.967004i	$-0.418003\pi$
$761$	14.0558	0.509521	0.254760	+	0.967004i	$0.418003\pi$
$762$	0	0
$763$	1.53666	0.0556308
$764$	−20.1792	−0.730057
$765$	0	0
$766$	6.07919	0.219650
$767$	−25.1808	−0.909225
$768$	0	0
$769$	−3.79434	−0.136827	−0.0684137	−	0.997657i	$-0.521794\pi$
$769$	−3.79434	−0.136827	−0.0684137	+	0.997657i	$0.521794\pi$
$770$	0.897643	0.0323488
$771$	0	0
$772$	10.8546	0.390667
$773$	−3.70140	−0.133130	−0.0665651	−	0.997782i	$-0.521204\pi$
$773$	−3.70140	−0.133130	−0.0665651	+	0.997782i	$0.521204\pi$
$774$	0	0
$775$	−0.276107	−0.00991805
$776$	−5.66999	−0.203541
$777$	0	0
$778$	1.66302	0.0596221
$779$	−22.2111	−0.795795
$780$	0	0
$781$	28.8073	1.03081
$782$	−0.920889	−0.0329309
$783$	0	0
$784$	3.75374	0.134062
$785$	−21.5644	−0.769667
$786$	0	0
$787$	−31.4190	−1.11997	−0.559983	−	0.828504i	$-0.689192\pi$
$787$	−31.4190	−1.11997	−0.559983	+	0.828504i	$0.689192\pi$
$788$	21.6858	0.772525
$789$	0	0
$790$	4.78095	0.170099
$791$	−1.85798	−0.0660621
$792$	0	0
$793$	−16.9883	−0.603271
$794$	−0.267008	−0.00947576
$795$	0	0
$796$	−26.1652	−0.927402
$797$	1.53832	0.0544901	0.0272450	−	0.999629i	$-0.491327\pi$
$797$	1.53832	0.0544901	0.0272450	+	0.999629i	$0.491327\pi$
$798$	0	0
$799$	−9.22800	−0.326463
$800$	−0.707866	−0.0250268
$801$	0	0
$802$	−6.74671	−0.238234
$803$	−9.74562	−0.343916
$804$	0	0
$805$	4.33695	0.152857
$806$	0.345525	0.0121706
$807$	0	0
$808$	−8.60815	−0.302834
$809$	−8.97381	−0.315502	−0.157751	−	0.987479i	$-0.550424\pi$
$809$	−8.97381	−0.315502	−0.157751	+	0.987479i	$0.550424\pi$
$810$	0	0
$811$	43.0991	1.51341	0.756707	−	0.653754i	$-0.226807\pi$
$811$	43.0991	1.51341	0.756707	+	0.653754i	$0.226807\pi$
$812$	5.29307	0.185750
$813$	0	0
$814$	−2.94508	−0.103225
$815$	−45.6357	−1.59855
$816$	0	0
$817$	2.51795	0.0880920
$818$	−6.07932	−0.212559
$819$	0	0
$820$	−16.2873	−0.568779
$821$	24.2970	0.847971	0.423985	−	0.905669i	$-0.360631\pi$
$821$	24.2970	0.847971	0.423985	+	0.905669i	$0.360631\pi$
$822$	0	0
$823$	23.3470	0.813825	0.406913	−	0.913467i	$-0.366605\pi$
$823$	23.3470	0.813825	0.406913	+	0.913467i	$0.366605\pi$
$824$	12.0166	0.418618
$825$	0	0
$826$	2.78731	0.0969830
$827$	−53.3817	−1.85626	−0.928131	−	0.372253i	$-0.878585\pi$
$827$	−53.3817	−1.85626	−0.928131	+	0.372253i	$0.878585\pi$
$828$	0	0
$829$	−45.5788	−1.58302	−0.791509	−	0.611157i	$-0.790704\pi$
$829$	−45.5788	−1.58302	−0.791509	+	0.611157i	$0.790704\pi$
$830$	−2.19197	−0.0760845
$831$	0	0
$832$	−12.9020	−0.447298
$833$	2.26480	0.0784705
$834$	0	0
$835$	−20.6697	−0.715303
$836$	−23.1008	−0.798958
$837$	0	0
$838$	−6.81396	−0.235384
$839$	41.0272	1.41642	0.708208	−	0.706004i	$-0.249504\pi$
$839$	41.0272	1.41642	0.708208	+	0.706004i	$0.249504\pi$
$840$	0	0
$841$	−21.6972	−0.748178
$842$	−6.72765	−0.231850
$843$	0	0
$844$	−45.8133	−1.57696
$845$	−20.8747	−0.718111
$846$	0	0
$847$	6.85316	0.235477
$848$	24.2668	0.833327
$849$	0	0
$850$	−0.137363	−0.00471150
$851$	−14.2291	−0.487768
$852$	0	0
$853$	1.58986	0.0544359	0.0272179	−	0.999630i	$-0.491335\pi$
$853$	1.58986	0.0544359	0.0272179	+	0.999630i	$0.491335\pi$
$854$	1.88047	0.0643483
$855$	0	0
$856$	9.87390	0.337483
$857$	−15.7373	−0.537578	−0.268789	−	0.963199i	$-0.586623\pi$
$857$	−15.7373	−0.537578	−0.268789	+	0.963199i	$0.586623\pi$
$858$	0	0
$859$	3.73832	0.127550	0.0637749	−	0.997964i	$-0.479686\pi$
$859$	3.73832	0.127550	0.0637749	+	0.997964i	$0.479686\pi$
$860$	1.84641	0.0629620
$861$	0	0
$862$	0.386615	0.0131682
$863$	−22.7338	−0.773869	−0.386934	−	0.922107i	$-0.626466\pi$
$863$	−22.7338	−0.773869	−0.386934	+	0.922107i	$0.626466\pi$
$864$	0	0
$865$	45.9597	1.56268
$866$	−4.56677	−0.155185
$867$	0	0
$868$	1.81267	0.0615260
$869$	−22.0866	−0.749235
$870$	0	0
$871$	12.6412	0.428332
$872$	1.23665	0.0418782
$873$	0	0
$874$	2.35497	0.0796580
$875$	11.4886	0.388384
$876$	0	0
$877$	−39.9098	−1.34766	−0.673828	−	0.738888i	$-0.735351\pi$
$877$	−39.9098	−1.34766	−0.673828	+	0.738888i	$0.735351\pi$
$878$	1.29327	0.0436456
$879$	0	0
$880$	−16.5748	−0.558737
$881$	50.9994	1.71821	0.859106	−	0.511798i	$-0.171020\pi$
$881$	50.9994	1.71821	0.859106	+	0.511798i	$0.171020\pi$
$882$	0	0
$883$	−0.228422	−0.00768700	−0.00384350	−	0.999993i	$-0.501223\pi$
$883$	−0.228422	−0.00768700	−0.00384350	+	0.999993i	$0.501223\pi$
$884$	−8.14693	−0.274011
$885$	0	0
$886$	2.39802	0.0805632
$887$	27.4320	0.921078	0.460539	−	0.887639i	$-0.347656\pi$
$887$	27.4320	0.921078	0.460539	+	0.887639i	$0.347656\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	0.493696	0.0165487
$891$	0	0
$892$	47.4392	1.58838
$893$	23.5985	0.789695
$894$	0	0
$895$	50.8916	1.70112
$896$	6.17342	0.206240
$897$	0	0
$898$	−5.45289	−0.181965
$899$	2.50094	0.0834109
$900$	0	0
$901$	14.6412	0.487770
$902$	−1.58760	−0.0528613
$903$	0	0
$904$	−1.49524	−0.0497308
$905$	37.7759	1.25571
$906$	0	0
$907$	−15.4064	−0.511562	−0.255781	−	0.966735i	$-0.582333\pi$
$907$	−15.4064	−0.511562	−0.255781	+	0.966735i	$0.582333\pi$
$908$	56.8649	1.88713
$909$	0	0
$910$	−0.809558	−0.0268366
$911$	−52.7023	−1.74610	−0.873052	−	0.487627i	$-0.837863\pi$
$911$	−52.7023	−1.74610	−0.873052	+	0.487627i	$0.837863\pi$
$912$	0	0
$913$	10.1263	0.335130
$914$	−2.09241	−0.0692108
$915$	0	0
$916$	0.957852	0.0316483
$917$	5.09715	0.168323
$918$	0	0
$919$	49.5827	1.63558	0.817792	−	0.575514i	$-0.195198\pi$
$919$	49.5827	1.63558	0.817792	+	0.575514i	$0.195198\pi$
$920$	3.49022	0.115069
$921$	0	0
$922$	−7.14704	−0.235375
$923$	−25.9804	−0.855156
$924$	0	0
$925$	−2.12246	−0.0697860
$926$	−7.18394	−0.236079
$927$	0	0
$928$	6.41175	0.210476
$929$	−43.3260	−1.42148	−0.710740	−	0.703455i	$-0.751640\pi$
$929$	−43.3260	−1.42148	−0.710740	+	0.703455i	$0.751640\pi$
$930$	0	0
$931$	−5.79171	−0.189816
$932$	31.8631	1.04371
$933$	0	0
$934$	−1.47249	−0.0481812
$935$	−10.0003	−0.327045
$936$	0	0
$937$	5.45956	0.178356	0.0891780	−	0.996016i	$-0.471576\pi$
$937$	5.45956	0.178356	0.0891780	+	0.996016i	$0.471576\pi$
$938$	−1.39928	−0.0456882
$939$	0	0
$940$	17.3048	0.564419
$941$	13.6724	0.445708	0.222854	−	0.974852i	$-0.428463\pi$
$941$	13.6724	0.445708	0.222854	+	0.974852i	$0.428463\pi$
$942$	0	0
$943$	−7.67047	−0.249785
$944$	−51.4672	−1.67511
$945$	0	0
$946$	0.179978	0.00585158
$947$	−2.37187	−0.0770756	−0.0385378	−	0.999257i	$-0.512270\pi$
$947$	−2.37187	−0.0770756	−0.0385378	+	0.999257i	$0.512270\pi$
$948$	0	0
$949$	8.78929	0.285313
$950$	0.351274	0.0113968
$951$	0	0
$952$	1.82263	0.0590717
$953$	−23.0787	−0.747591	−0.373796	−	0.927511i	$-0.621944\pi$
$953$	−23.0787	−0.747591	−0.373796	+	0.927511i	$0.621944\pi$
$954$	0	0
$955$	22.3392	0.722878
$956$	36.4843	1.17999
$957$	0	0
$958$	2.00437	0.0647582
$959$	−3.51026	−0.113352
$960$	0	0
$961$	−30.1435	−0.972372
$962$	2.65608	0.0856355
$963$	0	0
$964$	35.5890	1.14624
$965$	−12.0165	−0.386825
$966$	0	0
$967$	9.81160	0.315520	0.157760	−	0.987477i	$-0.449573\pi$
$967$	9.81160	0.315520	0.157760	+	0.987477i	$0.449573\pi$
$968$	5.51518	0.177265
$969$	0	0
$970$	3.10569	0.0997177
$971$	−15.2562	−0.489594	−0.244797	−	0.969574i	$-0.578721\pi$
$971$	−15.2562	−0.489594	−0.244797	+	0.969574i	$0.578721\pi$
$972$	0	0
$973$	3.28843	0.105422
$974$	0.482953	0.0154748
$975$	0	0
$976$	−34.7225	−1.11144
$977$	3.90224	0.124844	0.0624219	−	0.998050i	$-0.480118\pi$
$977$	3.90224	0.124844	0.0624219	+	0.998050i	$0.480118\pi$
$978$	0	0
$979$	−2.28073	−0.0728924
$980$	−4.24705	−0.135667
$981$	0	0
$982$	5.78392	0.184572
$983$	53.3262	1.70084	0.850420	−	0.526104i	$-0.176348\pi$
$983$	53.3262	1.70084	0.850420	+	0.526104i	$0.176348\pi$
$984$	0	0
$985$	−24.0071	−0.764929
$986$	1.24421	0.0396238
$987$	0	0
$988$	20.8339	0.662816
$989$	0.869560	0.0276504
$990$	0	0
$991$	19.5111	0.619790	0.309895	−	0.950771i	$-0.399706\pi$
$991$	19.5111	0.619790	0.309895	+	0.950771i	$0.399706\pi$
$992$	2.19578	0.0697159
$993$	0	0
$994$	2.87583	0.0912158
$995$	28.9660	0.918283
$996$	0	0
$997$	45.8824	1.45311	0.726556	−	0.687108i	$-0.241120\pi$
$997$	45.8824	1.45311	0.726556	+	0.687108i	$0.241120\pi$
$998$	−2.61379	−0.0827382
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.17	yes	32
3.2	odd	2	inner	8001.2.a.z.1.16	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.16	✓	32	3.2	odd	2	inner
8001.2.a.z.1.17	yes	32	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+0.203292 q^{2} -1.95867 q^{4} +2.16833 q^{5} -1.00000 q^{7} -0.804765 q^{8} +O(q^{10})\)	\(q+0.203292 q^{2} -1.95867 q^{4} +2.16833 q^{5} -1.00000 q^{7} -0.804765 q^{8} +0.440803 q^{10} -2.03638 q^{11} +1.83655 q^{13} -0.203292 q^{14} +3.75374 q^{16} +2.26480 q^{17} -5.79171 q^{19} -4.24705 q^{20} -0.413979 q^{22} -2.00013 q^{23} -0.298346 q^{25} +0.373355 q^{26} +1.95867 q^{28} +2.70238 q^{29} +0.925459 q^{31} +2.37263 q^{32} +0.460414 q^{34} -2.16833 q^{35} +7.11408 q^{37} -1.17741 q^{38} -1.74500 q^{40} +3.83498 q^{41} -0.434751 q^{43} +3.98860 q^{44} -0.406610 q^{46} -4.07454 q^{47} +1.00000 q^{49} -0.0606513 q^{50} -3.59720 q^{52} +6.46471 q^{53} -4.41554 q^{55} +0.804765 q^{56} +0.549370 q^{58} -13.7109 q^{59} -9.25010 q^{61} +0.188138 q^{62} -7.02515 q^{64} +3.98225 q^{65} +6.88314 q^{67} -4.43599 q^{68} -0.440803 q^{70} -14.1463 q^{71} +4.78576 q^{73} +1.44623 q^{74} +11.3441 q^{76} +2.03638 q^{77} +10.8460 q^{79} +8.13935 q^{80} +0.779619 q^{82} -4.97268 q^{83} +4.91082 q^{85} -0.0883813 q^{86} +1.63881 q^{88} +1.11999 q^{89} -1.83655 q^{91} +3.91761 q^{92} -0.828320 q^{94} -12.5583 q^{95} +7.04552 q^{97} +0.203292 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) 32 * q + 30 * q^4 - 32 * q^7	\( 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) \) 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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