LMFDB - Embedded newform 8001.2.a.s.1.8

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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 $ x^16 - 4x^15 - 18x^14 + 83x^13 + 112x^12 - 668x^11 - 235x^10 + 2648x^9 - 298x^8 - 5422x^7 + 2075x^6 + 5385x^5 - 3163x^4 - 1882x^3 + 1614x^2 - 200*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.716430$ of defining polynomial
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)

$f(q)$	$=$	$q-0.716430 q^{2} -1.48673 q^{4} -0.617976 q^{5} -1.00000 q^{7} +2.49800 q^{8} +O(q^{10})$	\(q-0.716430 q^{2} -1.48673 q^{4} -0.617976 q^{5} -1.00000 q^{7} +2.49800 q^{8} +0.442737 q^{10} -2.03917 q^{11} -2.26367 q^{13} +0.716430 q^{14} +1.18382 q^{16} +1.11795 q^{17} +7.92719 q^{19} +0.918763 q^{20} +1.46092 q^{22} -6.62633 q^{23} -4.61811 q^{25} +1.62176 q^{26} +1.48673 q^{28} +3.70380 q^{29} -1.50424 q^{31} -5.84411 q^{32} -0.800934 q^{34} +0.617976 q^{35} +4.29904 q^{37} -5.67928 q^{38} -1.54370 q^{40} +2.12378 q^{41} -5.68717 q^{43} +3.03169 q^{44} +4.74730 q^{46} -7.25675 q^{47} +1.00000 q^{49} +3.30855 q^{50} +3.36546 q^{52} -7.02300 q^{53} +1.26016 q^{55} -2.49800 q^{56} -2.65351 q^{58} +1.93188 q^{59} +7.43893 q^{61} +1.07768 q^{62} +1.81927 q^{64} +1.39890 q^{65} +10.4720 q^{67} -1.66209 q^{68} -0.442737 q^{70} -9.88028 q^{71} +5.42411 q^{73} -3.07996 q^{74} -11.7856 q^{76} +2.03917 q^{77} +15.1144 q^{79} -0.731571 q^{80} -1.52154 q^{82} -8.98829 q^{83} -0.690868 q^{85} +4.07446 q^{86} -5.09384 q^{88} -10.5005 q^{89} +2.26367 q^{91} +9.85155 q^{92} +5.19896 q^{94} -4.89882 q^{95} -14.4370 q^{97} -0.716430 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * q^98

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 $

$2$	−0.716430	−0.506592	−0.253296	−	0.967389i	$-0.581515\pi$
$2$	−0.716430	−0.506592	−0.253296	+	0.967389i	$0.581515\pi$
$3$	0	0
$3$	0	0
$4$	−1.48673	−0.743364
$5$	−0.617976	−0.276367	−0.138184	−	0.990407i	$-0.544126\pi$
$5$	−0.617976	−0.276367	−0.138184	+	0.990407i	$0.544126\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.49800	0.883175
$9$	0	0
$10$	0.442737	0.140006
$11$	−2.03917	−0.614833	−0.307416	−	0.951575i	$-0.599464\pi$
$11$	−2.03917	−0.614833	−0.307416	+	0.951575i	$0.599464\pi$
$12$	0	0
$13$	−2.26367	−0.627829	−0.313915	−	0.949451i	$-0.601641\pi$
$13$	−2.26367	−0.627829	−0.313915	+	0.949451i	$0.601641\pi$
$14$	0.716430	0.191474
$15$	0	0
$16$	1.18382	0.295954
$17$	1.11795	0.271143	0.135572	−	0.990768i	$-0.456713\pi$
$17$	1.11795	0.271143	0.135572	+	0.990768i	$0.456713\pi$
$18$	0	0
$19$	7.92719	1.81862	0.909311	−	0.416117i	$-0.136609\pi$
$19$	7.92719	1.81862	0.909311	+	0.416117i	$0.136609\pi$
$20$	0.918763	0.205442
$21$	0	0
$22$	1.46092	0.311470
$23$	−6.62633	−1.38168	−0.690842	−	0.723005i	$-0.742760\pi$
$23$	−6.62633	−1.38168	−0.690842	+	0.723005i	$0.742760\pi$
$24$	0	0
$25$	−4.61811	−0.923621
$26$	1.62176	0.318054
$27$	0	0
$28$	1.48673	0.280965
$29$	3.70380	0.687778	0.343889	−	0.939010i	$-0.388256\pi$
$29$	3.70380	0.687778	0.343889	+	0.939010i	$0.388256\pi$
$30$	0	0
$31$	−1.50424	−0.270170	−0.135085	−	0.990834i	$-0.543131\pi$
$31$	−1.50424	−0.270170	−0.135085	+	0.990834i	$0.543131\pi$
$32$	−5.84411	−1.03310
$33$	0	0
$34$	−0.800934	−0.137359
$35$	0.617976	0.104457
$36$	0	0
$37$	4.29904	0.706757	0.353379	−	0.935480i	$-0.385033\pi$
$37$	4.29904	0.706757	0.353379	+	0.935480i	$0.385033\pi$
$38$	−5.67928	−0.921300
$39$	0	0
$40$	−1.54370	−0.244081
$41$	2.12378	0.331679	0.165839	−	0.986153i	$-0.446967\pi$
$41$	2.12378	0.331679	0.165839	+	0.986153i	$0.446967\pi$
$42$	0	0
$43$	−5.68717	−0.867285	−0.433642	−	0.901085i	$-0.642772\pi$
$43$	−5.68717	−0.867285	−0.433642	+	0.901085i	$0.642772\pi$
$44$	3.03169	0.457044
$45$	0	0
$46$	4.74730	0.699951
$47$	−7.25675	−1.05851	−0.529253	−	0.848464i	$-0.677528\pi$
$47$	−7.25675	−1.05851	−0.529253	+	0.848464i	$0.677528\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.30855	0.467899
$51$	0	0
$52$	3.36546	0.466706
$53$	−7.02300	−0.964683	−0.482341	−	0.875983i	$-0.660214\pi$
$53$	−7.02300	−0.964683	−0.482341	+	0.875983i	$0.660214\pi$
$54$	0	0
$55$	1.26016	0.169920
$56$	−2.49800	−0.333809
$57$	0	0
$58$	−2.65351	−0.348423
$59$	1.93188	0.251509	0.125754	−	0.992061i	$-0.459865\pi$
$59$	1.93188	0.251509	0.125754	+	0.992061i	$0.459865\pi$
$60$	0	0
$61$	7.43893	0.952457	0.476229	−	0.879322i	$-0.342003\pi$
$61$	7.43893	0.952457	0.476229	+	0.879322i	$0.342003\pi$
$62$	1.07768	0.136866
$63$	0	0
$64$	1.81927	0.227408
$65$	1.39890	0.173512
$66$	0	0
$67$	10.4720	1.27936	0.639682	−	0.768640i	$-0.279066\pi$
$67$	10.4720	1.27936	0.639682	+	0.768640i	$0.279066\pi$
$68$	−1.66209	−0.201558
$69$	0	0
$70$	−0.442737	−0.0529172
$71$	−9.88028	−1.17257	−0.586287	−	0.810104i	$-0.699411\pi$
$71$	−9.88028	−1.17257	−0.586287	+	0.810104i	$0.699411\pi$
$72$	0	0
$73$	5.42411	0.634845	0.317422	−	0.948284i	$-0.397183\pi$
$73$	5.42411	0.634845	0.317422	+	0.948284i	$0.397183\pi$
$74$	−3.07996	−0.358038
$75$	0	0
$76$	−11.7856	−1.35190
$77$	2.03917	0.232385
$78$	0	0
$79$	15.1144	1.70050	0.850250	−	0.526378i	$-0.176450\pi$
$79$	15.1144	1.70050	0.850250	+	0.526378i	$0.176450\pi$
$80$	−0.731571	−0.0817921
$81$	0	0
$82$	−1.52154	−0.168026
$83$	−8.98829	−0.986593	−0.493296	−	0.869861i	$-0.664208\pi$
$83$	−8.98829	−0.986593	−0.493296	+	0.869861i	$0.664208\pi$
$84$	0	0
$85$	−0.690868	−0.0749351
$86$	4.07446	0.439360
$87$	0	0
$88$	−5.09384	−0.543005
$89$	−10.5005	−1.11305	−0.556523	−	0.830832i	$-0.687865\pi$
$89$	−10.5005	−1.11305	−0.556523	+	0.830832i	$0.687865\pi$
$90$	0	0
$91$	2.26367	0.237297
$92$	9.85155	1.02709
$93$	0	0
$94$	5.19896	0.536231
$95$	−4.89882	−0.502608
$96$	0	0
$97$	−14.4370	−1.46585	−0.732925	−	0.680309i	$-0.761845\pi$
$97$	−14.4370	−1.46585	−0.732925	+	0.680309i	$0.761845\pi$
$98$	−0.716430	−0.0723704
$99$	0	0
$100$	6.86587	0.686587
$101$	−1.14263	−0.113696	−0.0568482	−	0.998383i	$-0.518105\pi$
$101$	−1.14263	−0.113696	−0.0568482	+	0.998383i	$0.518105\pi$
$102$	0	0
$103$	−10.3693	−1.02171	−0.510857	−	0.859666i	$-0.670672\pi$
$103$	−10.3693	−1.02171	−0.510857	+	0.859666i	$0.670672\pi$
$104$	−5.65464	−0.554483
$105$	0	0
$106$	5.03148	0.488701
$107$	9.21897	0.891231	0.445616	−	0.895224i	$-0.352985\pi$
$107$	9.21897	0.891231	0.445616	+	0.895224i	$0.352985\pi$
$108$	0	0
$109$	−0.865135	−0.0828649	−0.0414324	−	0.999141i	$-0.513192\pi$
$109$	−0.865135	−0.0828649	−0.0414324	+	0.999141i	$0.513192\pi$
$110$	−0.902815	−0.0860800
$111$	0	0
$112$	−1.18382	−0.111860
$113$	−16.4904	−1.55128	−0.775641	−	0.631174i	$-0.782573\pi$
$113$	−16.4904	−1.55128	−0.775641	+	0.631174i	$0.782573\pi$
$114$	0	0
$115$	4.09491	0.381853
$116$	−5.50654	−0.511269
$117$	0	0
$118$	−1.38405	−0.127413
$119$	−1.11795	−0.102482
$120$	0	0
$121$	−6.84179	−0.621981
$122$	−5.32947	−0.482508
$123$	0	0
$124$	2.23640	0.200834
$125$	5.94376	0.531626
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	10.3849	0.917900
$129$	0	0
$130$	−1.00221	−0.0878997
$131$	−17.2493	−1.50708	−0.753538	−	0.657404i	$-0.771654\pi$
$131$	−17.2493	−1.50708	−0.753538	+	0.657404i	$0.771654\pi$
$132$	0	0
$133$	−7.92719	−0.687374
$134$	−7.50249	−0.648116
$135$	0	0
$136$	2.79264	0.239467
$137$	7.14706	0.610614	0.305307	−	0.952254i	$-0.401241\pi$
$137$	7.14706	0.610614	0.305307	+	0.952254i	$0.401241\pi$
$138$	0	0
$139$	8.76101	0.743099	0.371550	−	0.928413i	$-0.378827\pi$
$139$	8.76101	0.743099	0.371550	+	0.928413i	$0.378827\pi$
$140$	−0.918763	−0.0776496
$141$	0	0
$142$	7.07853	0.594017
$143$	4.61601	0.386010
$144$	0	0
$145$	−2.28886	−0.190079
$146$	−3.88600	−0.321607
$147$	0	0
$148$	−6.39150	−0.525378
$149$	21.4514	1.75737	0.878685	−	0.477402i	$-0.158422\pi$
$149$	21.4514	1.75737	0.878685	+	0.477402i	$0.158422\pi$
$150$	0	0
$151$	−3.87577	−0.315406	−0.157703	−	0.987487i	$-0.550409\pi$
$151$	−3.87577	−0.315406	−0.157703	+	0.987487i	$0.550409\pi$
$152$	19.8021	1.60616
$153$	0	0
$154$	−1.46092	−0.117724
$155$	0.929586	0.0746661
$156$	0	0
$157$	−17.2049	−1.37310	−0.686549	−	0.727084i	$-0.740875\pi$
$157$	−17.2049	−1.37310	−0.686549	+	0.727084i	$0.740875\pi$
$158$	−10.8284	−0.861461
$159$	0	0
$160$	3.61152	0.285516
$161$	6.62633	0.522228
$162$	0	0
$163$	22.0674	1.72845	0.864225	−	0.503106i	$-0.167809\pi$
$163$	22.0674	1.72845	0.864225	+	0.503106i	$0.167809\pi$
$164$	−3.15748	−0.246558
$165$	0	0
$166$	6.43948	0.499800
$167$	−17.4323	−1.34895	−0.674475	−	0.738298i	$-0.735630\pi$
$167$	−17.4323	−1.34895	−0.674475	+	0.738298i	$0.735630\pi$
$168$	0	0
$169$	−7.87579	−0.605830
$170$	0.494958	0.0379616
$171$	0	0
$172$	8.45527	0.644708
$173$	2.35715	0.179210	0.0896052	−	0.995977i	$-0.471439\pi$
$173$	2.35715	0.179210	0.0896052	+	0.995977i	$0.471439\pi$
$174$	0	0
$175$	4.61811	0.349096
$176$	−2.41400	−0.181962
$177$	0	0
$178$	7.52284	0.563861
$179$	19.6844	1.47128	0.735639	−	0.677374i	$-0.236882\pi$
$179$	19.6844	1.47128	0.735639	+	0.677374i	$0.236882\pi$
$180$	0	0
$181$	5.42604	0.403314	0.201657	−	0.979456i	$-0.435367\pi$
$181$	5.42604	0.403314	0.201657	+	0.979456i	$0.435367\pi$
$182$	−1.62176	−0.120213
$183$	0	0
$184$	−16.5525	−1.22027
$185$	−2.65670	−0.195325
$186$	0	0
$187$	−2.27969	−0.166708
$188$	10.7888	0.786856
$189$	0	0
$190$	3.50966	0.254617
$191$	11.8304	0.856015	0.428007	−	0.903775i	$-0.359216\pi$
$191$	11.8304	0.856015	0.428007	+	0.903775i	$0.359216\pi$
$192$	0	0
$193$	−15.0353	−1.08227	−0.541133	−	0.840937i	$-0.682004\pi$
$193$	−15.0353	−1.08227	−0.541133	+	0.840937i	$0.682004\pi$
$194$	10.3431	0.742589
$195$	0	0
$196$	−1.48673	−0.106195
$197$	−1.48761	−0.105988	−0.0529941	−	0.998595i	$-0.516876\pi$
$197$	−1.48761	−0.105988	−0.0529941	+	0.998595i	$0.516876\pi$
$198$	0	0
$199$	9.11397	0.646072	0.323036	−	0.946387i	$-0.395297\pi$
$199$	9.11397	0.646072	0.323036	+	0.946387i	$0.395297\pi$
$200$	−11.5360	−0.815719
$201$	0	0
$202$	0.818617	0.0575977
$203$	−3.70380	−0.259956
$204$	0	0
$205$	−1.31245	−0.0916652
$206$	7.42885	0.517593
$207$	0	0
$208$	−2.67977	−0.185809
$209$	−16.1649	−1.11815
$210$	0	0
$211$	14.9309	1.02789	0.513944	−	0.857824i	$-0.328184\pi$
$211$	14.9309	1.02789	0.513944	+	0.857824i	$0.328184\pi$
$212$	10.4413	0.717110
$213$	0	0
$214$	−6.60474	−0.451491
$215$	3.51454	0.239689
$216$	0	0
$217$	1.50424	0.102115
$218$	0.619808	0.0419787
$219$	0	0
$220$	−1.87351	−0.126312
$221$	−2.53068	−0.170232
$222$	0	0
$223$	−1.75342	−0.117418	−0.0587090	−	0.998275i	$-0.518698\pi$
$223$	−1.75342	−0.117418	−0.0587090	+	0.998275i	$0.518698\pi$
$224$	5.84411	0.390476
$225$	0	0
$226$	11.8142	0.785868
$227$	−0.504020	−0.0334530	−0.0167265	−	0.999860i	$-0.505324\pi$
$227$	−0.504020	−0.0334530	−0.0167265	+	0.999860i	$0.505324\pi$
$228$	0	0
$229$	15.5707	1.02894	0.514469	−	0.857509i	$-0.327989\pi$
$229$	15.5707	1.02894	0.514469	+	0.857509i	$0.327989\pi$
$230$	−2.93372	−0.193444
$231$	0	0
$232$	9.25207	0.607428
$233$	−3.69340	−0.241963	−0.120981	−	0.992655i	$-0.538604\pi$
$233$	−3.69340	−0.241963	−0.120981	+	0.992655i	$0.538604\pi$
$234$	0	0
$235$	4.48450	0.292537
$236$	−2.87218	−0.186963
$237$	0	0
$238$	0.800934	0.0519168
$239$	30.8421	1.99501	0.997506	−	0.0705784i	$-0.0224845\pi$
$239$	30.8421	1.99501	0.997506	+	0.0705784i	$0.0224845\pi$
$240$	0	0
$241$	−2.61954	−0.168739	−0.0843695	−	0.996435i	$-0.526888\pi$
$241$	−2.61954	−0.168739	−0.0843695	+	0.996435i	$0.526888\pi$
$242$	4.90166	0.315091
$243$	0	0
$244$	−11.0597	−0.708022
$245$	−0.617976	−0.0394811
$246$	0	0
$247$	−17.9445	−1.14178
$248$	−3.75759	−0.238607
$249$	0	0
$250$	−4.25829	−0.269318
$251$	25.5776	1.61445	0.807223	−	0.590246i	$-0.200969\pi$
$251$	25.5776	1.61445	0.807223	+	0.590246i	$0.200969\pi$
$252$	0	0
$253$	13.5122	0.849505
$254$	0.716430	0.0449528
$255$	0	0
$256$	−11.0786	−0.692409
$257$	−18.4468	−1.15068	−0.575339	−	0.817915i	$-0.695130\pi$
$257$	−18.4468	−1.15068	−0.575339	+	0.817915i	$0.695130\pi$
$258$	0	0
$259$	−4.29904	−0.267129
$260$	−2.07978	−0.128982
$261$	0	0
$262$	12.3579	0.763474
$263$	−15.1320	−0.933081	−0.466540	−	0.884500i	$-0.654500\pi$
$263$	−15.1320	−0.933081	−0.466540	+	0.884500i	$0.654500\pi$
$264$	0	0
$265$	4.34005	0.266607
$266$	5.67928	0.348219
$267$	0	0
$268$	−15.5691	−0.951033
$269$	28.6078	1.74425	0.872124	−	0.489285i	$-0.162742\pi$
$269$	28.6078	1.74425	0.872124	+	0.489285i	$0.162742\pi$
$270$	0	0
$271$	6.03510	0.366606	0.183303	−	0.983056i	$-0.441321\pi$
$271$	6.03510	0.366606	0.183303	+	0.983056i	$0.441321\pi$
$272$	1.32345	0.0802459
$273$	0	0
$274$	−5.12037	−0.309333
$275$	9.41710	0.567872
$276$	0	0
$277$	30.2445	1.81722	0.908609	−	0.417648i	$-0.137145\pi$
$277$	30.2445	1.81722	0.908609	+	0.417648i	$0.137145\pi$
$278$	−6.27665	−0.376448
$279$	0	0
$280$	1.54370	0.0922539
$281$	−0.763954	−0.0455737	−0.0227868	−	0.999740i	$-0.507254\pi$
$281$	−0.763954	−0.0455737	−0.0227868	+	0.999740i	$0.507254\pi$
$282$	0	0
$283$	−2.57222	−0.152903	−0.0764514	−	0.997073i	$-0.524359\pi$
$283$	−2.57222	−0.152903	−0.0764514	+	0.997073i	$0.524359\pi$
$284$	14.6893	0.871649
$285$	0	0
$286$	−3.30705	−0.195550
$287$	−2.12378	−0.125363
$288$	0	0
$289$	−15.7502	−0.926481
$290$	1.63981	0.0962928
$291$	0	0
$292$	−8.06418	−0.471921
$293$	−7.68306	−0.448849	−0.224424	−	0.974491i	$-0.572050\pi$
$293$	−7.68306	−0.448849	−0.224424	+	0.974491i	$0.572050\pi$
$294$	0	0
$295$	−1.19385	−0.0695089
$296$	10.7390	0.624191
$297$	0	0
$298$	−15.3684	−0.890270
$299$	14.9998	0.867462
$300$	0	0
$301$	5.68717	0.327803
$302$	2.77672	0.159782
$303$	0	0
$304$	9.38434	0.538229
$305$	−4.59708	−0.263228
$306$	0	0
$307$	−19.6054	−1.11894	−0.559469	−	0.828851i	$-0.688995\pi$
$307$	−19.6054	−1.11894	−0.559469	+	0.828851i	$0.688995\pi$
$308$	−3.03169	−0.172747
$309$	0	0
$310$	−0.665983	−0.0378253
$311$	15.4474	0.875942	0.437971	−	0.898989i	$-0.355697\pi$
$311$	15.4474	0.875942	0.437971	+	0.898989i	$0.355697\pi$
$312$	0	0
$313$	0.138060	0.00780361	0.00390181	−	0.999992i	$-0.498758\pi$
$313$	0.138060	0.00780361	0.00390181	+	0.999992i	$0.498758\pi$
$314$	12.3261	0.695601
$315$	0	0
$316$	−22.4710	−1.26409
$317$	16.5419	0.929086	0.464543	−	0.885551i	$-0.346219\pi$
$317$	16.5419	0.929086	0.464543	+	0.885551i	$0.346219\pi$
$318$	0	0
$319$	−7.55266	−0.422868
$320$	−1.12426	−0.0628482
$321$	0	0
$322$	−4.74730	−0.264557
$323$	8.86221	0.493107
$324$	0	0
$325$	10.4539	0.579876
$326$	−15.8097	−0.875620
$327$	0	0
$328$	5.30519	0.292930
$329$	7.25675	0.400078
$330$	0	0
$331$	−18.3281	−1.00740	−0.503702	−	0.863877i	$-0.668029\pi$
$331$	−18.3281	−1.00740	−0.503702	+	0.863877i	$0.668029\pi$
$332$	13.3631	0.733397
$333$	0	0
$334$	12.4890	0.683368
$335$	−6.47147	−0.353574
$336$	0	0
$337$	13.8550	0.754730	0.377365	−	0.926065i	$-0.376830\pi$
$337$	13.8550	0.754730	0.377365	+	0.926065i	$0.376830\pi$
$338$	5.64245	0.306909
$339$	0	0
$340$	1.02713	0.0557041
$341$	3.06740	0.166109
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−14.2065	−0.765964
$345$	0	0
$346$	−1.68873	−0.0907867
$347$	−27.5408	−1.47847	−0.739235	−	0.673447i	$-0.764813\pi$
$347$	−27.5408	−1.47847	−0.739235	+	0.673447i	$0.764813\pi$
$348$	0	0
$349$	−29.4626	−1.57710	−0.788550	−	0.614971i	$-0.789168\pi$
$349$	−29.4626	−1.57710	−0.788550	+	0.614971i	$0.789168\pi$
$350$	−3.30855	−0.176849
$351$	0	0
$352$	11.9171	0.635186
$353$	29.9244	1.59272	0.796358	−	0.604825i	$-0.206757\pi$
$353$	29.9244	1.59272	0.796358	+	0.604825i	$0.206757\pi$
$354$	0	0
$355$	6.10578	0.324061
$356$	15.6113	0.827399
$357$	0	0
$358$	−14.1025	−0.745339
$359$	−11.0464	−0.583007	−0.291504	−	0.956570i	$-0.594156\pi$
$359$	−11.0464	−0.583007	−0.291504	+	0.956570i	$0.594156\pi$
$360$	0	0
$361$	43.8403	2.30739
$362$	−3.88737	−0.204316
$363$	0	0
$364$	−3.36546	−0.176398
$365$	−3.35197	−0.175450
$366$	0	0
$367$	19.6359	1.02499	0.512494	−	0.858691i	$-0.328722\pi$
$367$	19.6359	1.02499	0.512494	+	0.858691i	$0.328722\pi$
$368$	−7.84436	−0.408915
$369$	0	0
$370$	1.90334	0.0989500
$371$	7.02300	0.364616
$372$	0	0
$373$	10.1118	0.523568	0.261784	−	0.965126i	$-0.415689\pi$
$373$	10.1118	0.523568	0.261784	+	0.965126i	$0.415689\pi$
$374$	1.63324	0.0844528
$375$	0	0
$376$	−18.1273	−0.934846
$377$	−8.38418	−0.431807
$378$	0	0
$379$	5.47081	0.281017	0.140508	−	0.990079i	$-0.455126\pi$
$379$	5.47081	0.281017	0.140508	+	0.990079i	$0.455126\pi$
$380$	7.28321	0.373621
$381$	0	0
$382$	−8.47562	−0.433651
$383$	9.87140	0.504405	0.252202	−	0.967674i	$-0.418845\pi$
$383$	9.87140	0.504405	0.252202	+	0.967674i	$0.418845\pi$
$384$	0	0
$385$	−1.26016	−0.0642236
$386$	10.7718	0.548268
$387$	0	0
$388$	21.4638	1.08966
$389$	−7.86204	−0.398621	−0.199311	−	0.979936i	$-0.563870\pi$
$389$	−7.86204	−0.398621	−0.199311	+	0.979936i	$0.563870\pi$
$390$	0	0
$391$	−7.40791	−0.374634
$392$	2.49800	0.126168
$393$	0	0
$394$	1.06577	0.0536928
$395$	−9.34033	−0.469963
$396$	0	0
$397$	−18.7639	−0.941732	−0.470866	−	0.882205i	$-0.656058\pi$
$397$	−18.7639	−0.941732	−0.470866	+	0.882205i	$0.656058\pi$
$398$	−6.52952	−0.327295
$399$	0	0
$400$	−5.46699	−0.273349
$401$	10.4959	0.524138	0.262069	−	0.965049i	$-0.415595\pi$
$401$	10.4959	0.524138	0.262069	+	0.965049i	$0.415595\pi$
$402$	0	0
$403$	3.40511	0.169620
$404$	1.69879	0.0845178
$405$	0	0
$406$	2.65351	0.131692
$407$	−8.76646	−0.434537
$408$	0	0
$409$	28.2977	1.39923	0.699616	−	0.714519i	$-0.253355\pi$
$409$	28.2977	1.39923	0.699616	+	0.714519i	$0.253355\pi$
$410$	0.940275	0.0464369
$411$	0	0
$412$	15.4163	0.759505
$413$	−1.93188	−0.0950614
$414$	0	0
$415$	5.55455	0.272662
$416$	13.2292	0.648613
$417$	0	0
$418$	11.5810	0.566445
$419$	27.6457	1.35058	0.675290	−	0.737552i	$-0.264018\pi$
$419$	27.6457	1.35058	0.675290	+	0.737552i	$0.264018\pi$
$420$	0	0
$421$	22.0977	1.07698	0.538488	−	0.842633i	$-0.318996\pi$
$421$	22.0977	1.07698	0.538488	+	0.842633i	$0.318996\pi$
$422$	−10.6970	−0.520720
$423$	0	0
$424$	−17.5434	−0.851984
$425$	−5.16282	−0.250434
$426$	0	0
$427$	−7.43893	−0.359995
$428$	−13.7061	−0.662509
$429$	0	0
$430$	−2.51792	−0.121425
$431$	−20.1964	−0.972827	−0.486414	−	0.873729i	$-0.661695\pi$
$431$	−20.1964	−0.972827	−0.486414	+	0.873729i	$0.661695\pi$
$432$	0	0
$433$	−14.7722	−0.709909	−0.354954	−	0.934884i	$-0.615504\pi$
$433$	−14.7722	−0.709909	−0.354954	+	0.934884i	$0.615504\pi$
$434$	−1.07768	−0.0517305
$435$	0	0
$436$	1.28622	0.0615988
$437$	−52.5281	−2.51276
$438$	0	0
$439$	−0.567166	−0.0270693	−0.0135347	−	0.999908i	$-0.504308\pi$
$439$	−0.567166	−0.0270693	−0.0135347	+	0.999908i	$0.504308\pi$
$440$	3.14787	0.150069
$441$	0	0
$442$	1.81305	0.0862381
$443$	7.32609	0.348073	0.174037	−	0.984739i	$-0.444319\pi$
$443$	7.32609	0.348073	0.174037	+	0.984739i	$0.444319\pi$
$444$	0	0
$445$	6.48904	0.307610
$446$	1.25621	0.0594830
$447$	0	0
$448$	−1.81927	−0.0859522
$449$	8.19856	0.386914	0.193457	−	0.981109i	$-0.438030\pi$
$449$	8.19856	0.386914	0.193457	+	0.981109i	$0.438030\pi$
$450$	0	0
$451$	−4.33074	−0.203927
$452$	24.5167	1.15317
$453$	0	0
$454$	0.361095	0.0169470
$455$	−1.39890	−0.0655812
$456$	0	0
$457$	−16.2094	−0.758243	−0.379122	−	0.925347i	$-0.623774\pi$
$457$	−16.2094	−0.758243	−0.379122	+	0.925347i	$0.623774\pi$
$458$	−11.1553	−0.521252
$459$	0	0
$460$	−6.08802	−0.283856
$461$	31.9638	1.48870	0.744351	−	0.667788i	$-0.232759\pi$
$461$	31.9638	1.48870	0.744351	+	0.667788i	$0.232759\pi$
$462$	0	0
$463$	40.0886	1.86308	0.931538	−	0.363645i	$-0.118468\pi$
$463$	40.0886	1.86308	0.931538	+	0.363645i	$0.118468\pi$
$464$	4.38462	0.203551
$465$	0	0
$466$	2.64606	0.122577
$467$	34.9976	1.61950	0.809749	−	0.586777i	$-0.199603\pi$
$467$	34.9976	1.61950	0.809749	+	0.586777i	$0.199603\pi$
$468$	0	0
$469$	−10.4720	−0.483554
$470$	−3.21283	−0.148197
$471$	0	0
$472$	4.82582	0.222126
$473$	11.5971	0.533235
$474$	0	0
$475$	−36.6086	−1.67972
$476$	1.66209	0.0761818
$477$	0	0
$478$	−22.0962	−1.01066
$479$	31.4638	1.43762	0.718808	−	0.695209i	$-0.244688\pi$
$479$	31.4638	1.43762	0.718808	+	0.695209i	$0.244688\pi$
$480$	0	0
$481$	−9.73161	−0.443723
$482$	1.87671	0.0854820
$483$	0	0
$484$	10.1719	0.462358
$485$	8.92169	0.405113
$486$	0	0
$487$	−19.4829	−0.882857	−0.441428	−	0.897297i	$-0.645528\pi$
$487$	−19.4829	−0.882857	−0.441428	+	0.897297i	$0.645528\pi$
$488$	18.5824	0.841187
$489$	0	0
$490$	0.442737	0.0200008
$491$	−4.84336	−0.218578	−0.109289	−	0.994010i	$-0.534857\pi$
$491$	−4.84336	−0.218578	−0.109289	+	0.994010i	$0.534857\pi$
$492$	0	0
$493$	4.14067	0.186486
$494$	12.8560	0.578419
$495$	0	0
$496$	−1.78075	−0.0799579
$497$	9.88028	0.443191
$498$	0	0
$499$	−41.1439	−1.84185	−0.920927	−	0.389735i	$-0.872567\pi$
$499$	−41.1439	−1.84185	−0.920927	+	0.389735i	$0.872567\pi$
$500$	−8.83676	−0.395192
$501$	0	0
$502$	−18.3246	−0.817867
$503$	−7.38885	−0.329452	−0.164726	−	0.986339i	$-0.552674\pi$
$503$	−7.38885	−0.329452	−0.164726	+	0.986339i	$0.552674\pi$
$504$	0	0
$505$	0.706121	0.0314220
$506$	−9.68054	−0.430353
$507$	0	0
$508$	1.48673	0.0659629
$509$	8.13369	0.360520	0.180260	−	0.983619i	$-0.442306\pi$
$509$	8.13369	0.360520	0.180260	+	0.983619i	$0.442306\pi$
$510$	0	0
$511$	−5.42411	−0.239949
$512$	−12.8327	−0.567131
$513$	0	0
$514$	13.2158	0.582925
$515$	6.40796	0.282368
$516$	0	0
$517$	14.7977	0.650804
$518$	3.07996	0.135326
$519$	0	0
$520$	3.49444	0.153241
$521$	−28.9263	−1.26728	−0.633642	−	0.773626i	$-0.718441\pi$
$521$	−28.9263	−1.26728	−0.633642	+	0.773626i	$0.718441\pi$
$522$	0	0
$523$	17.3728	0.759660	0.379830	−	0.925056i	$-0.375982\pi$
$523$	17.3728	0.759660	0.379830	+	0.925056i	$0.375982\pi$
$524$	25.6450	1.12031
$525$	0	0
$526$	10.8410	0.472692
$527$	−1.68167	−0.0732547
$528$	0	0
$529$	20.9082	0.909052
$530$	−3.10934	−0.135061
$531$	0	0
$532$	11.7856	0.510969
$533$	−4.80754	−0.208238
$534$	0	0
$535$	−5.69710	−0.246307
$536$	26.1591	1.12990
$537$	0	0
$538$	−20.4955	−0.883623
$539$	−2.03917	−0.0878332
$540$	0	0
$541$	42.8394	1.84181	0.920905	−	0.389787i	$-0.127451\pi$
$541$	42.8394	1.84181	0.920905	+	0.389787i	$0.127451\pi$
$542$	−4.32373	−0.185720
$543$	0	0
$544$	−6.53344	−0.280119
$545$	0.534633	0.0229012
$546$	0	0
$547$	−6.81661	−0.291457	−0.145729	−	0.989325i	$-0.546553\pi$
$547$	−6.81661	−0.291457	−0.145729	+	0.989325i	$0.546553\pi$
$548$	−10.6257	−0.453909
$549$	0	0
$550$	−6.74669	−0.287680
$551$	29.3607	1.25081
$552$	0	0
$553$	−15.1144	−0.642729
$554$	−21.6681	−0.920589
$555$	0	0
$556$	−13.0252	−0.552393
$557$	1.16988	0.0495696	0.0247848	−	0.999693i	$-0.492110\pi$
$557$	1.16988	0.0495696	0.0247848	+	0.999693i	$0.492110\pi$
$558$	0	0
$559$	12.8739	0.544507
$560$	0.731571	0.0309145
$561$	0	0
$562$	0.547320	0.0230873
$563$	22.8170	0.961621	0.480810	−	0.876825i	$-0.340343\pi$
$563$	22.8170	0.961621	0.480810	+	0.876825i	$0.340343\pi$
$564$	0	0
$565$	10.1907	0.428724
$566$	1.84282	0.0774594
$567$	0	0
$568$	−24.6809	−1.03559
$569$	5.19397	0.217743	0.108871	−	0.994056i	$-0.465276\pi$
$569$	5.19397	0.217743	0.108871	+	0.994056i	$0.465276\pi$
$570$	0	0
$571$	6.91405	0.289344	0.144672	−	0.989480i	$-0.453787\pi$
$571$	6.91405	0.289344	0.144672	+	0.989480i	$0.453787\pi$
$572$	−6.86275	−0.286946
$573$	0	0
$574$	1.52154	0.0635078
$575$	30.6011	1.27615
$576$	0	0
$577$	16.8356	0.700873	0.350437	−	0.936586i	$-0.386033\pi$
$577$	16.8356	0.700873	0.350437	+	0.936586i	$0.386033\pi$
$578$	11.2839	0.469349
$579$	0	0
$580$	3.40291	0.141298
$581$	8.98829	0.372897
$582$	0	0
$583$	14.3211	0.593118
$584$	13.5494	0.560679
$585$	0	0
$586$	5.50437	0.227383
$587$	42.6694	1.76115	0.880576	−	0.473904i	$-0.157156\pi$
$587$	42.6694	1.76115	0.880576	+	0.473904i	$0.157156\pi$
$588$	0	0
$589$	−11.9244	−0.491337
$590$	0.855313	0.0352127
$591$	0	0
$592$	5.08927	0.209168
$593$	−23.0971	−0.948486	−0.474243	−	0.880394i	$-0.657278\pi$
$593$	−23.0971	−0.948486	−0.474243	+	0.880394i	$0.657278\pi$
$594$	0	0
$595$	0.690868	0.0283228
$596$	−31.8924	−1.30637
$597$	0	0
$598$	−10.7463	−0.439450
$599$	7.37647	0.301394	0.150697	−	0.988580i	$-0.451848\pi$
$599$	7.37647	0.301394	0.150697	+	0.988580i	$0.451848\pi$
$600$	0	0
$601$	−8.20517	−0.334696	−0.167348	−	0.985898i	$-0.553520\pi$
$601$	−8.20517	−0.334696	−0.167348	+	0.985898i	$0.553520\pi$
$602$	−4.07446	−0.166062
$603$	0	0
$604$	5.76222	0.234461
$605$	4.22806	0.171895
$606$	0	0
$607$	8.57849	0.348190	0.174095	−	0.984729i	$-0.444300\pi$
$607$	8.57849	0.348190	0.174095	+	0.984729i	$0.444300\pi$
$608$	−46.3274	−1.87882
$609$	0	0
$610$	3.29349	0.133349
$611$	16.4269	0.664561
$612$	0	0
$613$	42.3945	1.71230	0.856148	−	0.516730i	$-0.172851\pi$
$613$	42.3945	1.71230	0.856148	+	0.516730i	$0.172851\pi$
$614$	14.0459	0.566846
$615$	0	0
$616$	5.09384	0.205237
$617$	22.9614	0.924390	0.462195	−	0.886778i	$-0.347062\pi$
$617$	22.9614	0.924390	0.462195	+	0.886778i	$0.347062\pi$
$618$	0	0
$619$	−16.0297	−0.644288	−0.322144	−	0.946691i	$-0.604403\pi$
$619$	−16.0297	−0.644288	−0.322144	+	0.946691i	$0.604403\pi$
$620$	−1.38204	−0.0555041
$621$	0	0
$622$	−11.0670	−0.443746
$623$	10.5005	0.420692
$624$	0	0
$625$	19.4174	0.776697
$626$	−0.0989103	−0.00395325
$627$	0	0
$628$	25.5790	1.02071
$629$	4.80612	0.191632
$630$	0	0
$631$	28.2130	1.12314	0.561571	−	0.827429i	$-0.310197\pi$
$631$	28.2130	1.12314	0.561571	+	0.827429i	$0.310197\pi$
$632$	37.7557	1.50184
$633$	0	0
$634$	−11.8511	−0.470668
$635$	0.617976	0.0245236
$636$	0	0
$637$	−2.26367	−0.0896899
$638$	5.41096	0.214222
$639$	0	0
$640$	−6.41759	−0.253678
$641$	27.3897	1.08183	0.540914	−	0.841078i	$-0.318078\pi$
$641$	27.3897	1.08183	0.540914	+	0.841078i	$0.318078\pi$
$642$	0	0
$643$	14.5051	0.572026	0.286013	−	0.958226i	$-0.407670\pi$
$643$	14.5051	0.572026	0.286013	+	0.958226i	$0.407670\pi$
$644$	−9.85155	−0.388205
$645$	0	0
$646$	−6.34916	−0.249804
$647$	13.0663	0.513690	0.256845	−	0.966453i	$-0.417317\pi$
$647$	13.0663	0.513690	0.256845	+	0.966453i	$0.417317\pi$
$648$	0	0
$649$	−3.93942	−0.154636
$650$	−7.48947	−0.293761
$651$	0	0
$652$	−32.8082	−1.28487
$653$	−28.8586	−1.12932	−0.564662	−	0.825323i	$-0.690993\pi$
$653$	−28.8586	−1.12932	−0.564662	+	0.825323i	$0.690993\pi$
$654$	0	0
$655$	10.6596	0.416507
$656$	2.51417	0.0981617
$657$	0	0
$658$	−5.19896	−0.202676
$659$	−41.5879	−1.62004	−0.810018	−	0.586405i	$-0.800543\pi$
$659$	−41.5879	−1.62004	−0.810018	+	0.586405i	$0.800543\pi$
$660$	0	0
$661$	45.4380	1.76733	0.883667	−	0.468115i	$-0.155067\pi$
$661$	45.4380	1.76733	0.883667	+	0.468115i	$0.155067\pi$
$662$	13.1308	0.510344
$663$	0	0
$664$	−22.4527	−0.871334
$665$	4.89882	0.189968
$666$	0	0
$667$	−24.5426	−0.950292
$668$	25.9170	1.00276
$669$	0	0
$670$	4.63636	0.179118
$671$	−15.1692	−0.585602
$672$	0	0
$673$	−14.1380	−0.544980	−0.272490	−	0.962159i	$-0.587847\pi$
$673$	−14.1380	−0.544980	−0.272490	+	0.962159i	$0.587847\pi$
$674$	−9.92614	−0.382341
$675$	0	0
$676$	11.7092	0.450352
$677$	36.8722	1.41711	0.708557	−	0.705654i	$-0.249347\pi$
$677$	36.8722	1.41711	0.708557	+	0.705654i	$0.249347\pi$
$678$	0	0
$679$	14.4370	0.554039
$680$	−1.72579	−0.0661808
$681$	0	0
$682$	−2.19758	−0.0841496
$683$	13.3883	0.512289	0.256145	−	0.966638i	$-0.417548\pi$
$683$	13.3883	0.512289	0.256145	+	0.966638i	$0.417548\pi$
$684$	0	0
$685$	−4.41671	−0.168754
$686$	0.716430	0.0273534
$687$	0	0
$688$	−6.73256	−0.256677
$689$	15.8978	0.605656
$690$	0	0
$691$	32.6469	1.24195	0.620974	−	0.783831i	$-0.286737\pi$
$691$	32.6469	1.24195	0.620974	+	0.783831i	$0.286737\pi$
$692$	−3.50443	−0.133219
$693$	0	0
$694$	19.7311	0.748982
$695$	−5.41410	−0.205368
$696$	0	0
$697$	2.37428	0.0899324
$698$	21.1079	0.798947
$699$	0	0
$700$	−6.86587	−0.259505
$701$	−36.7539	−1.38817	−0.694087	−	0.719891i	$-0.744192\pi$
$701$	−36.7539	−1.38817	−0.694087	+	0.719891i	$0.744192\pi$
$702$	0	0
$703$	34.0793	1.28532
$704$	−3.70979	−0.139818
$705$	0	0
$706$	−21.4388	−0.806858
$707$	1.14263	0.0429732
$708$	0	0
$709$	−21.1990	−0.796146	−0.398073	−	0.917354i	$-0.630321\pi$
$709$	−21.1990	−0.796146	−0.398073	+	0.917354i	$0.630321\pi$
$710$	−4.37436	−0.164167
$711$	0	0
$712$	−26.2301	−0.983015
$713$	9.96760	0.373289
$714$	0	0
$715$	−2.85258	−0.106681
$716$	−29.2653	−1.09370
$717$	0	0
$718$	7.91398	0.295347
$719$	17.8524	0.665782	0.332891	−	0.942965i	$-0.391976\pi$
$719$	17.8524	0.665782	0.332891	+	0.942965i	$0.391976\pi$
$720$	0	0
$721$	10.3693	0.386172
$722$	−31.4085	−1.16890
$723$	0	0
$724$	−8.06704	−0.299809
$725$	−17.1045	−0.635246
$726$	0	0
$727$	−26.0525	−0.966233	−0.483116	−	0.875556i	$-0.660495\pi$
$727$	−26.0525	−0.966233	−0.483116	+	0.875556i	$0.660495\pi$
$728$	5.65464	0.209575
$729$	0	0
$730$	2.40145	0.0888818
$731$	−6.35798	−0.235158
$732$	0	0
$733$	48.6871	1.79830	0.899150	−	0.437640i	$-0.144186\pi$
$733$	48.6871	1.79830	0.899150	+	0.437640i	$0.144186\pi$
$734$	−14.0678	−0.519251
$735$	0	0
$736$	38.7250	1.42742
$737$	−21.3543	−0.786594
$738$	0	0
$739$	−48.5162	−1.78470	−0.892349	−	0.451346i	$-0.850944\pi$
$739$	−48.5162	−1.78470	−0.892349	+	0.451346i	$0.850944\pi$
$740$	3.94980	0.145197
$741$	0	0
$742$	−5.03148	−0.184712
$743$	−6.51240	−0.238917	−0.119458	−	0.992839i	$-0.538116\pi$
$743$	−6.51240	−0.238917	−0.119458	+	0.992839i	$0.538116\pi$
$744$	0	0
$745$	−13.2565	−0.485680
$746$	−7.24438	−0.265236
$747$	0	0
$748$	3.38928	0.123924
$749$	−9.21897	−0.336854
$750$	0	0
$751$	45.2954	1.65285	0.826427	−	0.563044i	$-0.190370\pi$
$751$	45.2954	1.65285	0.826427	+	0.563044i	$0.190370\pi$
$752$	−8.59067	−0.313269
$753$	0	0
$754$	6.00667	0.218750
$755$	2.39514	0.0871679
$756$	0	0
$757$	32.1126	1.16715	0.583577	−	0.812058i	$-0.301653\pi$
$757$	32.1126	1.16715	0.583577	+	0.812058i	$0.301653\pi$
$758$	−3.91946	−0.142361
$759$	0	0
$760$	−12.2372	−0.443891
$761$	17.0499	0.618059	0.309030	−	0.951052i	$-0.399996\pi$
$761$	17.0499	0.618059	0.309030	+	0.951052i	$0.399996\pi$
$762$	0	0
$763$	0.865135	0.0313200
$764$	−17.5885	−0.636330
$765$	0	0
$766$	−7.07217	−0.255528
$767$	−4.37313	−0.157905
$768$	0	0
$769$	23.0262	0.830345	0.415172	−	0.909743i	$-0.363721\pi$
$769$	23.0262	0.830345	0.415172	+	0.909743i	$0.363721\pi$
$770$	0.902815	0.0325352
$771$	0	0
$772$	22.3534	0.804518
$773$	52.0454	1.87194	0.935972	−	0.352074i	$-0.114524\pi$
$773$	52.0454	1.87194	0.935972	+	0.352074i	$0.114524\pi$
$774$	0	0
$775$	6.94675	0.249534
$776$	−36.0635	−1.29460
$777$	0	0
$778$	5.63260	0.201939
$779$	16.8356	0.603198
$780$	0	0
$781$	20.1476	0.720936
$782$	5.30725	0.189787
$783$	0	0
$784$	1.18382	0.0422792
$785$	10.6322	0.379479
$786$	0	0
$787$	−3.25919	−0.116178	−0.0580888	−	0.998311i	$-0.518501\pi$
$787$	−3.25919	−0.116178	−0.0580888	+	0.998311i	$0.518501\pi$
$788$	2.21168	0.0787878
$789$	0	0
$790$	6.69169	0.238080
$791$	16.4904	0.586330
$792$	0	0
$793$	−16.8393	−0.597981
$794$	13.4430	0.477074
$795$	0	0
$796$	−13.5500	−0.480267
$797$	14.7367	0.522000	0.261000	−	0.965339i	$-0.415948\pi$
$797$	14.7367	0.522000	0.261000	+	0.965339i	$0.415948\pi$
$798$	0	0
$799$	−8.11270	−0.287007
$800$	26.9887	0.954196
$801$	0	0
$802$	−7.51955	−0.265524
$803$	−11.0607	−0.390323
$804$	0	0
$805$	−4.09491	−0.144327
$806$	−2.43952	−0.0859285
$807$	0	0
$808$	−2.85430	−0.100414
$809$	−18.9304	−0.665557	−0.332779	−	0.943005i	$-0.607986\pi$
$809$	−18.9304	−0.665557	−0.332779	+	0.943005i	$0.607986\pi$
$810$	0	0
$811$	−48.2689	−1.69495	−0.847475	−	0.530835i	$-0.821878\pi$
$811$	−48.2689	−1.69495	−0.847475	+	0.530835i	$0.821878\pi$
$812$	5.50654	0.193242
$813$	0	0
$814$	6.28056	0.220133
$815$	−13.6371	−0.477687
$816$	0	0
$817$	−45.0833	−1.57726
$818$	−20.2733	−0.708840
$819$	0	0
$820$	1.95125	0.0681406
$821$	24.4326	0.852704	0.426352	−	0.904557i	$-0.359798\pi$
$821$	24.4326	0.852704	0.426352	+	0.904557i	$0.359798\pi$
$822$	0	0
$823$	30.8491	1.07533	0.537667	−	0.843157i	$-0.319306\pi$
$823$	30.8491	1.07533	0.537667	+	0.843157i	$0.319306\pi$
$824$	−25.9024	−0.902352
$825$	0	0
$826$	1.38405	0.0481574
$827$	−20.5018	−0.712917	−0.356458	−	0.934311i	$-0.616016\pi$
$827$	−20.5018	−0.712917	−0.356458	+	0.934311i	$0.616016\pi$
$828$	0	0
$829$	−24.0508	−0.835317	−0.417659	−	0.908604i	$-0.637149\pi$
$829$	−24.0508	−0.835317	−0.417659	+	0.908604i	$0.637149\pi$
$830$	−3.97944	−0.138129
$831$	0	0
$832$	−4.11822	−0.142774
$833$	1.11795	0.0387347
$834$	0	0
$835$	10.7727	0.372806
$836$	24.0328	0.831191
$837$	0	0
$838$	−19.8062	−0.684194
$839$	−44.0427	−1.52052	−0.760261	−	0.649618i	$-0.774929\pi$
$839$	−44.0427	−1.52052	−0.760261	+	0.649618i	$0.774929\pi$
$840$	0	0
$841$	−15.2819	−0.526962
$842$	−15.8314	−0.545587
$843$	0	0
$844$	−22.1982	−0.764095
$845$	4.86705	0.167432
$846$	0	0
$847$	6.84179	0.235087
$848$	−8.31394	−0.285502
$849$	0	0
$850$	3.69880	0.126868
$851$	−28.4868	−0.976516
$852$	0	0
$853$	26.4780	0.906591	0.453295	−	0.891360i	$-0.350248\pi$
$853$	26.4780	0.906591	0.453295	+	0.891360i	$0.350248\pi$
$854$	5.32947	0.182371
$855$	0	0
$856$	23.0289	0.787113
$857$	40.8062	1.39391	0.696956	−	0.717114i	$-0.254537\pi$
$857$	40.8062	1.39391	0.696956	+	0.717114i	$0.254537\pi$
$858$	0	0
$859$	−41.8868	−1.42916	−0.714580	−	0.699554i	$-0.753382\pi$
$859$	−41.8868	−1.42916	−0.714580	+	0.699554i	$0.753382\pi$
$860$	−5.22516	−0.178176
$861$	0	0
$862$	14.4693	0.492827
$863$	−4.40796	−0.150049	−0.0750244	−	0.997182i	$-0.523903\pi$
$863$	−4.40796	−0.150049	−0.0750244	+	0.997182i	$0.523903\pi$
$864$	0	0
$865$	−1.45666	−0.0495279
$866$	10.5833	0.359635
$867$	0	0
$868$	−2.23640	−0.0759083
$869$	−30.8208	−1.04552
$870$	0	0
$871$	−23.7053	−0.803222
$872$	−2.16110	−0.0731842
$873$	0	0
$874$	37.6327	1.27295
$875$	−5.94376	−0.200936
$876$	0	0
$877$	−20.5553	−0.694102	−0.347051	−	0.937846i	$-0.612817\pi$
$877$	−20.5553	−0.694102	−0.347051	+	0.937846i	$0.612817\pi$
$878$	0.406334	0.0137131
$879$	0	0
$880$	1.49180	0.0502884
$881$	−29.0290	−0.978012	−0.489006	−	0.872280i	$-0.662640\pi$
$881$	−29.0290	−0.978012	−0.489006	+	0.872280i	$0.662640\pi$
$882$	0	0
$883$	47.9692	1.61429	0.807145	−	0.590353i	$-0.201011\pi$
$883$	47.9692	1.61429	0.807145	+	0.590353i	$0.201011\pi$
$884$	3.76243	0.126544
$885$	0	0
$886$	−5.24863	−0.176331
$887$	5.23890	0.175905	0.0879526	−	0.996125i	$-0.471968\pi$
$887$	5.23890	0.175905	0.0879526	+	0.996125i	$0.471968\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−4.64894	−0.155833
$891$	0	0
$892$	2.60686	0.0872843
$893$	−57.5257	−1.92502
$894$	0	0
$895$	−12.1645	−0.406613
$896$	−10.3849	−0.346934
$897$	0	0
$898$	−5.87370	−0.196008
$899$	−5.57140	−0.185817
$900$	0	0
$901$	−7.85137	−0.261567
$902$	3.10268	0.103308
$903$	0	0
$904$	−41.1929	−1.37005
$905$	−3.35316	−0.111463
$906$	0	0
$907$	20.2955	0.673901	0.336950	−	0.941522i	$-0.390605\pi$
$907$	20.2955	0.673901	0.336950	+	0.941522i	$0.390605\pi$
$908$	0.749341	0.0248678
$909$	0	0
$910$	1.00221	0.0332230
$911$	−3.65224	−0.121004	−0.0605021	−	0.998168i	$-0.519270\pi$
$911$	−3.65224	−0.121004	−0.0605021	+	0.998168i	$0.519270\pi$
$912$	0	0
$913$	18.3286	0.606589
$914$	11.6129	0.384120
$915$	0	0
$916$	−23.1493	−0.764876
$917$	17.2493	0.569621
$918$	0	0
$919$	18.3743	0.606112	0.303056	−	0.952973i	$-0.401993\pi$
$919$	18.3743	0.606112	0.303056	+	0.952973i	$0.401993\pi$
$920$	10.2291	0.337243
$921$	0	0
$922$	−22.8998	−0.754166
$923$	22.3657	0.736176
$924$	0	0
$925$	−19.8534	−0.652776
$926$	−28.7207	−0.943820
$927$	0	0
$928$	−21.6454	−0.710545
$929$	34.1612	1.12079	0.560396	−	0.828225i	$-0.310649\pi$
$929$	34.1612	1.12079	0.560396	+	0.828225i	$0.310649\pi$
$930$	0	0
$931$	7.92719	0.259803
$932$	5.49108	0.179866
$933$	0	0
$934$	−25.0734	−0.820425
$935$	1.40880	0.0460726
$936$	0	0
$937$	12.3046	0.401973	0.200986	−	0.979594i	$-0.435585\pi$
$937$	12.3046	0.401973	0.200986	+	0.979594i	$0.435585\pi$
$938$	7.50249	0.244965
$939$	0	0
$940$	−6.66724	−0.217461
$941$	13.7984	0.449815	0.224908	−	0.974380i	$-0.427792\pi$
$941$	13.7984	0.449815	0.224908	+	0.974380i	$0.427792\pi$
$942$	0	0
$943$	−14.0729	−0.458275
$944$	2.28699	0.0744351
$945$	0	0
$946$	−8.30851	−0.270133
$947$	−15.8853	−0.516204	−0.258102	−	0.966118i	$-0.583097\pi$
$947$	−15.8853	−0.516204	−0.258102	+	0.966118i	$0.583097\pi$
$948$	0	0
$949$	−12.2784	−0.398574
$950$	26.2275	0.850932
$951$	0	0
$952$	−2.79264	−0.0905100
$953$	−32.1393	−1.04109	−0.520546	−	0.853833i	$-0.674272\pi$
$953$	−32.1393	−1.04109	−0.520546	+	0.853833i	$0.674272\pi$
$954$	0	0
$955$	−7.31088	−0.236575
$956$	−45.8539	−1.48302
$957$	0	0
$958$	−22.5416	−0.728285
$959$	−7.14706	−0.230790
$960$	0	0
$961$	−28.7373	−0.927008
$962$	6.97202	0.224787
$963$	0	0
$964$	3.89454	0.125435
$965$	9.29147	0.299103
$966$	0	0
$967$	−8.50545	−0.273517	−0.136758	−	0.990604i	$-0.543668\pi$
$967$	−8.50545	−0.273517	−0.136758	+	0.990604i	$0.543668\pi$
$968$	−17.0908	−0.549318
$969$	0	0
$970$	−6.39177	−0.205227
$971$	48.7435	1.56425	0.782127	−	0.623120i	$-0.214135\pi$
$971$	48.7435	1.56425	0.782127	+	0.623120i	$0.214135\pi$
$972$	0	0
$973$	−8.76101	−0.280865
$974$	13.9582	0.447249
$975$	0	0
$976$	8.80633	0.281884
$977$	−9.54315	−0.305312	−0.152656	−	0.988279i	$-0.548783\pi$
$977$	−9.54315	−0.305312	−0.152656	+	0.988279i	$0.548783\pi$
$978$	0	0
$979$	21.4122	0.684337
$980$	0.918763	0.0293488
$981$	0	0
$982$	3.46993	0.110730
$983$	−27.1267	−0.865206	−0.432603	−	0.901584i	$-0.642405\pi$
$983$	−27.1267	−0.865206	−0.432603	+	0.901584i	$0.642405\pi$
$984$	0	0
$985$	0.919311	0.0292917
$986$	−2.96650	−0.0944725
$987$	0	0
$988$	26.6787	0.848761
$989$	37.6850	1.19831
$990$	0	0
$991$	−8.07844	−0.256620	−0.128310	−	0.991734i	$-0.540955\pi$
$991$	−8.07844	−0.256620	−0.128310	+	0.991734i	$0.540955\pi$
$992$	8.79096	0.279113
$993$	0	0
$994$	−7.07853	−0.224517
$995$	−5.63222	−0.178553
$996$	0	0
$997$	42.4806	1.34538	0.672688	−	0.739927i	$-0.265140\pi$
$997$	42.4806	1.34538	0.672688	+	0.739927i	$0.265140\pi$
$998$	29.4767	0.933069
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.s.1.8		16
3.2	odd	2		2667.2.a.n.1.9	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.n.1.9	✓	16	3.2	odd	2
8001.2.a.s.1.8		16	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)

\(f(q)\)	\(=\)	\(q-0.716430 q^{2} -1.48673 q^{4} -0.617976 q^{5} -1.00000 q^{7} +2.49800 q^{8} +O(q^{10})\)	\(q-0.716430 q^{2} -1.48673 q^{4} -0.617976 q^{5} -1.00000 q^{7} +2.49800 q^{8} +0.442737 q^{10} -2.03917 q^{11} -2.26367 q^{13} +0.716430 q^{14} +1.18382 q^{16} +1.11795 q^{17} +7.92719 q^{19} +0.918763 q^{20} +1.46092 q^{22} -6.62633 q^{23} -4.61811 q^{25} +1.62176 q^{26} +1.48673 q^{28} +3.70380 q^{29} -1.50424 q^{31} -5.84411 q^{32} -0.800934 q^{34} +0.617976 q^{35} +4.29904 q^{37} -5.67928 q^{38} -1.54370 q^{40} +2.12378 q^{41} -5.68717 q^{43} +3.03169 q^{44} +4.74730 q^{46} -7.25675 q^{47} +1.00000 q^{49} +3.30855 q^{50} +3.36546 q^{52} -7.02300 q^{53} +1.26016 q^{55} -2.49800 q^{56} -2.65351 q^{58} +1.93188 q^{59} +7.43893 q^{61} +1.07768 q^{62} +1.81927 q^{64} +1.39890 q^{65} +10.4720 q^{67} -1.66209 q^{68} -0.442737 q^{70} -9.88028 q^{71} +5.42411 q^{73} -3.07996 q^{74} -11.7856 q^{76} +2.03917 q^{77} +15.1144 q^{79} -0.731571 q^{80} -1.52154 q^{82} -8.98829 q^{83} -0.690868 q^{85} +4.07446 q^{86} -5.09384 q^{88} -10.5005 q^{89} +2.26367 q^{91} +9.85155 q^{92} +5.19896 q^{94} -4.89882 q^{95} -14.4370 q^{97} -0.716430 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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