LMFDB - Embedded newform 8001.2.a.s.1.4

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			1.4
Root			$2.35655$ 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)

$f(q)$	$=$	$q-2.35655 q^{2} +3.55331 q^{4} -2.88159 q^{5} -1.00000 q^{7} -3.66045 q^{8} +O(q^{10})$	\(q-2.35655 q^{2} +3.55331 q^{4} -2.88159 q^{5} -1.00000 q^{7} -3.66045 q^{8} +6.79060 q^{10} +2.09099 q^{11} +0.909032 q^{13} +2.35655 q^{14} +1.51941 q^{16} -3.78525 q^{17} +7.58028 q^{19} -10.2392 q^{20} -4.92751 q^{22} +5.96978 q^{23} +3.30355 q^{25} -2.14218 q^{26} -3.55331 q^{28} +2.95240 q^{29} -0.925472 q^{31} +3.74036 q^{32} +8.92011 q^{34} +2.88159 q^{35} -0.864281 q^{37} -17.8633 q^{38} +10.5479 q^{40} +3.36343 q^{41} +7.26692 q^{43} +7.42993 q^{44} -14.0681 q^{46} -1.13179 q^{47} +1.00000 q^{49} -7.78497 q^{50} +3.23007 q^{52} +0.941428 q^{53} -6.02537 q^{55} +3.66045 q^{56} -6.95747 q^{58} +4.90991 q^{59} +3.87805 q^{61} +2.18092 q^{62} -11.8531 q^{64} -2.61946 q^{65} +2.67893 q^{67} -13.4502 q^{68} -6.79060 q^{70} -3.06017 q^{71} -13.6550 q^{73} +2.03672 q^{74} +26.9351 q^{76} -2.09099 q^{77} -2.32416 q^{79} -4.37830 q^{80} -7.92609 q^{82} +9.71119 q^{83} +10.9075 q^{85} -17.1248 q^{86} -7.65397 q^{88} -0.340872 q^{89} -0.909032 q^{91} +21.2125 q^{92} +2.66711 q^{94} -21.8433 q^{95} +8.16066 q^{97} -2.35655 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.35655	−1.66633	−0.833165	−	0.553024i	$-0.813474\pi$
$2$	−2.35655	−1.66633	−0.833165	+	0.553024i	$0.813474\pi$
$3$	0	0
$3$	0	0
$4$	3.55331	1.77666
$5$	−2.88159	−1.28869	−0.644343	−	0.764737i	$-0.722869\pi$
$5$	−2.88159	−1.28869	−0.644343	+	0.764737i	$0.722869\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.66045	−1.29417
$9$	0	0
$10$	6.79060	2.14738
$11$	2.09099	0.630457	0.315228	−	0.949016i	$-0.397919\pi$
$11$	2.09099	0.630457	0.315228	+	0.949016i	$0.397919\pi$
$12$	0	0
$13$	0.909032	0.252120	0.126060	−	0.992023i	$-0.459767\pi$
$13$	0.909032	0.252120	0.126060	+	0.992023i	$0.459767\pi$
$14$	2.35655	0.629814
$15$	0	0
$16$	1.51941	0.379851
$17$	−3.78525	−0.918057	−0.459028	−	0.888422i	$-0.651802\pi$
$17$	−3.78525	−0.918057	−0.459028	+	0.888422i	$0.651802\pi$
$18$	0	0
$19$	7.58028	1.73904	0.869518	−	0.493901i	$-0.164429\pi$
$19$	7.58028	1.73904	0.869518	+	0.493901i	$0.164429\pi$
$20$	−10.2392	−2.28955
$21$	0	0
$22$	−4.92751	−1.05055
$23$	5.96978	1.24478	0.622392	−	0.782705i	$-0.286161\pi$
$23$	5.96978	1.24478	0.622392	+	0.782705i	$0.286161\pi$
$24$	0	0
$25$	3.30355	0.660710
$26$	−2.14218	−0.420115
$27$	0	0
$28$	−3.55331	−0.671513
$29$	2.95240	0.548247	0.274124	−	0.961694i	$-0.411612\pi$
$29$	2.95240	0.548247	0.274124	+	0.961694i	$0.411612\pi$
$30$	0	0
$31$	−0.925472	−0.166220	−0.0831098	−	0.996540i	$-0.526485\pi$
$31$	−0.925472	−0.166220	−0.0831098	+	0.996540i	$0.526485\pi$
$32$	3.74036	0.661208
$33$	0	0
$34$	8.92011	1.52979
$35$	2.88159	0.487077
$36$	0	0
$37$	−0.864281	−0.142087	−0.0710435	−	0.997473i	$-0.522633\pi$
$37$	−0.864281	−0.142087	−0.0710435	+	0.997473i	$0.522633\pi$
$38$	−17.8633	−2.89781
$39$	0	0
$40$	10.5479	1.66777
$41$	3.36343	0.525280	0.262640	−	0.964894i	$-0.415407\pi$
$41$	3.36343	0.525280	0.262640	+	0.964894i	$0.415407\pi$
$42$	0	0
$43$	7.26692	1.10819	0.554097	−	0.832452i	$-0.313064\pi$
$43$	7.26692	1.10819	0.554097	+	0.832452i	$0.313064\pi$
$44$	7.42993	1.12010
$45$	0	0
$46$	−14.0681	−2.07422
$47$	−1.13179	−0.165088	−0.0825441	−	0.996587i	$-0.526305\pi$
$47$	−1.13179	−0.165088	−0.0825441	+	0.996587i	$0.526305\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−7.78497	−1.10096
$51$	0	0
$52$	3.23007	0.447931
$53$	0.941428	0.129315	0.0646575	−	0.997908i	$-0.479405\pi$
$53$	0.941428	0.129315	0.0646575	+	0.997908i	$0.479405\pi$
$54$	0	0
$55$	−6.02537	−0.812460
$56$	3.66045	0.489149
$57$	0	0
$58$	−6.95747	−0.913561
$59$	4.90991	0.639216	0.319608	−	0.947550i	$-0.396449\pi$
$59$	4.90991	0.639216	0.319608	+	0.947550i	$0.396449\pi$
$60$	0	0
$61$	3.87805	0.496533	0.248267	−	0.968692i	$-0.420139\pi$
$61$	3.87805	0.496533	0.248267	+	0.968692i	$0.420139\pi$
$62$	2.18092	0.276977
$63$	0	0
$64$	−11.8531	−1.48164
$65$	−2.61946	−0.324903
$66$	0	0
$67$	2.67893	0.327284	0.163642	−	0.986520i	$-0.447676\pi$
$67$	2.67893	0.327284	0.163642	+	0.986520i	$0.447676\pi$
$68$	−13.4502	−1.63107
$69$	0	0
$70$	−6.79060	−0.811632
$71$	−3.06017	−0.363175	−0.181587	−	0.983375i	$-0.558124\pi$
$71$	−3.06017	−0.363175	−0.181587	+	0.983375i	$0.558124\pi$
$72$	0	0
$73$	−13.6550	−1.59820	−0.799098	−	0.601201i	$-0.794689\pi$
$73$	−13.6550	−1.59820	−0.799098	+	0.601201i	$0.794689\pi$
$74$	2.03672	0.236764
$75$	0	0
$76$	26.9351	3.08967
$77$	−2.09099	−0.238290
$78$	0	0
$79$	−2.32416	−0.261489	−0.130744	−	0.991416i	$-0.541737\pi$
$79$	−2.32416	−0.261489	−0.130744	+	0.991416i	$0.541737\pi$
$80$	−4.37830	−0.489509
$81$	0	0
$82$	−7.92609	−0.875290
$83$	9.71119	1.06594	0.532971	−	0.846134i	$-0.321076\pi$
$83$	9.71119	1.06594	0.532971	+	0.846134i	$0.321076\pi$
$84$	0	0
$85$	10.9075	1.18309
$86$	−17.1248	−1.84662
$87$	0	0
$88$	−7.65397	−0.815915
$89$	−0.340872	−0.0361324	−0.0180662	−	0.999837i	$-0.505751\pi$
$89$	−0.340872	−0.0361324	−0.0180662	+	0.999837i	$0.505751\pi$
$90$	0	0
$91$	−0.909032	−0.0952924
$92$	21.2125	2.21155
$93$	0	0
$94$	2.66711	0.275091
$95$	−21.8433	−2.24107
$96$	0	0
$97$	8.16066	0.828590	0.414295	−	0.910143i	$-0.364028\pi$
$97$	8.16066	0.828590	0.414295	+	0.910143i	$0.364028\pi$
$98$	−2.35655	−0.238047
$99$	0	0
$100$	11.7386	1.17386
$101$	12.7682	1.27048	0.635241	−	0.772314i	$-0.280901\pi$
$101$	12.7682	1.27048	0.635241	+	0.772314i	$0.280901\pi$
$102$	0	0
$103$	−5.11308	−0.503806	−0.251903	−	0.967752i	$-0.581056\pi$
$103$	−5.11308	−0.503806	−0.251903	+	0.967752i	$0.581056\pi$
$104$	−3.32747	−0.326285
$105$	0	0
$106$	−2.21852	−0.215482
$107$	−3.96461	−0.383273	−0.191637	−	0.981466i	$-0.561380\pi$
$107$	−3.96461	−0.383273	−0.191637	+	0.981466i	$0.561380\pi$
$108$	0	0
$109$	12.4907	1.19639	0.598195	−	0.801350i	$-0.295885\pi$
$109$	12.4907	1.19639	0.598195	+	0.801350i	$0.295885\pi$
$110$	14.1991	1.35383
$111$	0	0
$112$	−1.51941	−0.143570
$113$	−0.691741	−0.0650736	−0.0325368	−	0.999471i	$-0.510359\pi$
$113$	−0.691741	−0.0650736	−0.0325368	+	0.999471i	$0.510359\pi$
$114$	0	0
$115$	−17.2024	−1.60414
$116$	10.4908	0.974047
$117$	0	0
$118$	−11.5704	−1.06514
$119$	3.78525	0.346993
$120$	0	0
$121$	−6.62777	−0.602524
$122$	−9.13881	−0.827389
$123$	0	0
$124$	−3.28849	−0.295315
$125$	4.88847	0.437238
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	20.4518	1.80770
$129$	0	0
$130$	6.17287	0.541396
$131$	−2.30452	−0.201347	−0.100673	−	0.994920i	$-0.532100\pi$
$131$	−2.30452	−0.201347	−0.100673	+	0.994920i	$0.532100\pi$
$132$	0	0
$133$	−7.58028	−0.657294
$134$	−6.31303	−0.545363
$135$	0	0
$136$	13.8557	1.18812
$137$	−6.70881	−0.573172	−0.286586	−	0.958054i	$-0.592520\pi$
$137$	−6.70881	−0.573172	−0.286586	+	0.958054i	$0.592520\pi$
$138$	0	0
$139$	−10.3703	−0.879600	−0.439800	−	0.898096i	$-0.644951\pi$
$139$	−10.3703	−0.879600	−0.439800	+	0.898096i	$0.644951\pi$
$140$	10.2392	0.865369
$141$	0	0
$142$	7.21142	0.605169
$143$	1.90077	0.158951
$144$	0	0
$145$	−8.50761	−0.706518
$146$	32.1786	2.66312
$147$	0	0
$148$	−3.07106	−0.252440
$149$	−11.5293	−0.944518	−0.472259	−	0.881460i	$-0.656561\pi$
$149$	−11.5293	−0.944518	−0.472259	+	0.881460i	$0.656561\pi$
$150$	0	0
$151$	−21.4583	−1.74625	−0.873126	−	0.487494i	$-0.837911\pi$
$151$	−21.4583	−1.74625	−0.873126	+	0.487494i	$0.837911\pi$
$152$	−27.7473	−2.25060
$153$	0	0
$154$	4.92751	0.397070
$155$	2.66683	0.214205
$156$	0	0
$157$	2.67835	0.213756	0.106878	−	0.994272i	$-0.465915\pi$
$157$	2.67835	0.213756	0.106878	+	0.994272i	$0.465915\pi$
$158$	5.47699	0.435726
$159$	0	0
$160$	−10.7782	−0.852089
$161$	−5.96978	−0.470484
$162$	0	0
$163$	−19.3157	−1.51292	−0.756460	−	0.654040i	$-0.773073\pi$
$163$	−19.3157	−1.51292	−0.756460	+	0.654040i	$0.773073\pi$
$164$	11.9513	0.933242
$165$	0	0
$166$	−22.8849	−1.77621
$167$	−0.608353	−0.0470758	−0.0235379	−	0.999723i	$-0.507493\pi$
$167$	−0.608353	−0.0470758	−0.0235379	+	0.999723i	$0.507493\pi$
$168$	0	0
$169$	−12.1737	−0.936435
$170$	−25.7041	−1.97141
$171$	0	0
$172$	25.8216	1.96888
$173$	8.03576	0.610948	0.305474	−	0.952200i	$-0.401185\pi$
$173$	8.03576	0.610948	0.305474	+	0.952200i	$0.401185\pi$
$174$	0	0
$175$	−3.30355	−0.249725
$176$	3.17706	0.239480
$177$	0	0
$178$	0.803281	0.0602085
$179$	17.4649	1.30539	0.652694	−	0.757622i	$-0.273639\pi$
$179$	17.4649	1.30539	0.652694	+	0.757622i	$0.273639\pi$
$180$	0	0
$181$	21.2958	1.58290	0.791452	−	0.611231i	$-0.209326\pi$
$181$	21.2958	1.58290	0.791452	+	0.611231i	$0.209326\pi$
$182$	2.14218	0.158789
$183$	0	0
$184$	−21.8521	−1.61096
$185$	2.49050	0.183105
$186$	0	0
$187$	−7.91490	−0.578795
$188$	−4.02160	−0.293305
$189$	0	0
$190$	51.4747	3.73436
$191$	−17.5241	−1.26800	−0.634001	−	0.773333i	$-0.718588\pi$
$191$	−17.5241	−1.26800	−0.634001	+	0.773333i	$0.718588\pi$
$192$	0	0
$193$	−7.80348	−0.561707	−0.280853	−	0.959751i	$-0.590617\pi$
$193$	−7.80348	−0.561707	−0.280853	+	0.959751i	$0.590617\pi$
$194$	−19.2310	−1.38070
$195$	0	0
$196$	3.55331	0.253808
$197$	−4.86752	−0.346796	−0.173398	−	0.984852i	$-0.555475\pi$
$197$	−4.86752	−0.346796	−0.173398	+	0.984852i	$0.555475\pi$
$198$	0	0
$199$	19.9314	1.41290	0.706449	−	0.707764i	$-0.250296\pi$
$199$	19.9314	1.41290	0.706449	+	0.707764i	$0.250296\pi$
$200$	−12.0925	−0.855069
$201$	0	0
$202$	−30.0888	−2.11704
$203$	−2.95240	−0.207218
$204$	0	0
$205$	−9.69203	−0.676921
$206$	12.0492	0.839508
$207$	0	0
$208$	1.38119	0.0957681
$209$	15.8503	1.09639
$210$	0	0
$211$	−7.58891	−0.522442	−0.261221	−	0.965279i	$-0.584125\pi$
$211$	−7.58891	−0.522442	−0.261221	+	0.965279i	$0.584125\pi$
$212$	3.34519	0.229748
$213$	0	0
$214$	9.34279	0.638660
$215$	−20.9403	−1.42811
$216$	0	0
$217$	0.925472	0.0628251
$218$	−29.4349	−1.99358
$219$	0	0
$220$	−21.4100	−1.44346
$221$	−3.44091	−0.231461
$222$	0	0
$223$	−0.0673464	−0.00450985	−0.00225493	−	0.999997i	$-0.500718\pi$
$223$	−0.0673464	−0.00450985	−0.00225493	+	0.999997i	$0.500718\pi$
$224$	−3.74036	−0.249913
$225$	0	0
$226$	1.63012	0.108434
$227$	16.9576	1.12552	0.562758	−	0.826622i	$-0.309740\pi$
$227$	16.9576	1.12552	0.562758	+	0.826622i	$0.309740\pi$
$228$	0	0
$229$	26.3712	1.74266	0.871329	−	0.490698i	$-0.163258\pi$
$229$	26.3712	1.74266	0.871329	+	0.490698i	$0.163258\pi$
$230$	40.5384	2.67302
$231$	0	0
$232$	−10.8071	−0.709523
$233$	−30.1882	−1.97769	−0.988846	−	0.148943i	$-0.952413\pi$
$233$	−30.1882	−1.97769	−0.988846	+	0.148943i	$0.952413\pi$
$234$	0	0
$235$	3.26135	0.212747
$236$	17.4465	1.13567
$237$	0	0
$238$	−8.92011	−0.578205
$239$	0.635680	0.0411188	0.0205594	−	0.999789i	$-0.493455\pi$
$239$	0.635680	0.0411188	0.0205594	+	0.999789i	$0.493455\pi$
$240$	0	0
$241$	14.1376	0.910683	0.455341	−	0.890317i	$-0.349517\pi$
$241$	14.1376	0.910683	0.455341	+	0.890317i	$0.349517\pi$
$242$	15.6186	1.00400
$243$	0	0
$244$	13.7799	0.882169
$245$	−2.88159	−0.184098
$246$	0	0
$247$	6.89072	0.438446
$248$	3.38765	0.215116
$249$	0	0
$250$	−11.5199	−0.728582
$251$	12.8099	0.808557	0.404278	−	0.914636i	$-0.367523\pi$
$251$	12.8099	0.808557	0.404278	+	0.914636i	$0.367523\pi$
$252$	0	0
$253$	12.4827	0.784783
$254$	2.35655	0.147863
$255$	0	0
$256$	−24.4893	−1.53058
$257$	−10.5321	−0.656976	−0.328488	−	0.944508i	$-0.606539\pi$
$257$	−10.5321	−0.656976	−0.328488	+	0.944508i	$0.606539\pi$
$258$	0	0
$259$	0.864281	0.0537038
$260$	−9.30774	−0.577242
$261$	0	0
$262$	5.43071	0.335510
$263$	−9.93958	−0.612901	−0.306450	−	0.951887i	$-0.599141\pi$
$263$	−9.93958	−0.612901	−0.306450	+	0.951887i	$0.599141\pi$
$264$	0	0
$265$	−2.71281	−0.166646
$266$	17.8633	1.09527
$267$	0	0
$268$	9.51909	0.581471
$269$	12.5316	0.764062	0.382031	−	0.924149i	$-0.375225\pi$
$269$	12.5316	0.764062	0.382031	+	0.924149i	$0.375225\pi$
$270$	0	0
$271$	−8.23955	−0.500517	−0.250259	−	0.968179i	$-0.580516\pi$
$271$	−8.23955	−0.500517	−0.250259	+	0.968179i	$0.580516\pi$
$272$	−5.75132	−0.348725
$273$	0	0
$274$	15.8096	0.955094
$275$	6.90769	0.416549
$276$	0	0
$277$	14.9032	0.895444	0.447722	−	0.894173i	$-0.352235\pi$
$277$	14.9032	0.895444	0.447722	+	0.894173i	$0.352235\pi$
$278$	24.4382	1.46570
$279$	0	0
$280$	−10.5479	−0.630359
$281$	28.8176	1.71911	0.859557	−	0.511039i	$-0.170739\pi$
$281$	28.8176	1.71911	0.859557	+	0.511039i	$0.170739\pi$
$282$	0	0
$283$	−28.9980	−1.72375	−0.861875	−	0.507121i	$-0.830710\pi$
$283$	−28.9980	−1.72375	−0.861875	+	0.507121i	$0.830710\pi$
$284$	−10.8737	−0.645237
$285$	0	0
$286$	−4.47926	−0.264864
$287$	−3.36343	−0.198537
$288$	0	0
$289$	−2.67192	−0.157172
$290$	20.0486	1.17729
$291$	0	0
$292$	−48.5204	−2.83944
$293$	−18.0306	−1.05336	−0.526678	−	0.850065i	$-0.676563\pi$
$293$	−18.0306	−1.05336	−0.526678	+	0.850065i	$0.676563\pi$
$294$	0	0
$295$	−14.1483	−0.823748
$296$	3.16366	0.183884
$297$	0	0
$298$	27.1694	1.57388
$299$	5.42672	0.313835
$300$	0	0
$301$	−7.26692	−0.418858
$302$	50.5675	2.90983
$303$	0	0
$304$	11.5175	0.660575
$305$	−11.1749	−0.639875
$306$	0	0
$307$	−0.156981	−0.00895936	−0.00447968	−	0.999990i	$-0.501426\pi$
$307$	−0.156981	−0.00895936	−0.00447968	+	0.999990i	$0.501426\pi$
$308$	−7.42993	−0.423360
$309$	0	0
$310$	−6.28451	−0.356936
$311$	3.98103	0.225743	0.112872	−	0.993610i	$-0.463995\pi$
$311$	3.98103	0.225743	0.112872	+	0.993610i	$0.463995\pi$
$312$	0	0
$313$	5.93875	0.335678	0.167839	−	0.985814i	$-0.446321\pi$
$313$	5.93875	0.335678	0.167839	+	0.985814i	$0.446321\pi$
$314$	−6.31166	−0.356188
$315$	0	0
$316$	−8.25847	−0.464575
$317$	−2.42916	−0.136435	−0.0682177	−	0.997670i	$-0.521731\pi$
$317$	−2.42916	−0.136435	−0.0682177	+	0.997670i	$0.521731\pi$
$318$	0	0
$319$	6.17344	0.345646
$320$	34.1559	1.90937
$321$	0	0
$322$	14.0681	0.783982
$323$	−28.6932	−1.59653
$324$	0	0
$325$	3.00303	0.166578
$326$	45.5183	2.52103
$327$	0	0
$328$	−12.3117	−0.679800
$329$	1.13179	0.0623975
$330$	0	0
$331$	23.0202	1.26531	0.632653	−	0.774436i	$-0.281966\pi$
$331$	23.0202	1.26531	0.632653	+	0.774436i	$0.281966\pi$
$332$	34.5069	1.89381
$333$	0	0
$334$	1.43361	0.0784438
$335$	−7.71959	−0.421766
$336$	0	0
$337$	13.2748	0.723122	0.361561	−	0.932348i	$-0.382244\pi$
$337$	13.2748	0.723122	0.361561	+	0.932348i	$0.382244\pi$
$338$	28.6878	1.56041
$339$	0	0
$340$	38.7578	2.10194
$341$	−1.93515	−0.104794
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−26.6002	−1.43419
$345$	0	0
$346$	−18.9366	−1.01804
$347$	−21.0772	−1.13148	−0.565742	−	0.824582i	$-0.691410\pi$
$347$	−21.0772	−1.13148	−0.565742	+	0.824582i	$0.691410\pi$
$348$	0	0
$349$	21.8288	1.16847	0.584234	−	0.811585i	$-0.301395\pi$
$349$	21.8288	1.16847	0.584234	+	0.811585i	$0.301395\pi$
$350$	7.78497	0.416124
$351$	0	0
$352$	7.82105	0.416863
$353$	−0.580410	−0.0308921	−0.0154461	−	0.999881i	$-0.504917\pi$
$353$	−0.580410	−0.0308921	−0.0154461	+	0.999881i	$0.504917\pi$
$354$	0	0
$355$	8.81814	0.468018
$356$	−1.21123	−0.0641948
$357$	0	0
$358$	−41.1568	−2.17521
$359$	−23.8413	−1.25830	−0.629148	−	0.777286i	$-0.716596\pi$
$359$	−23.8413	−1.25830	−0.629148	+	0.777286i	$0.716596\pi$
$360$	0	0
$361$	38.4607	2.02425
$362$	−50.1845	−2.63764
$363$	0	0
$364$	−3.23007	−0.169302
$365$	39.3480	2.05957
$366$	0	0
$367$	1.13290	0.0591367	0.0295683	−	0.999563i	$-0.490587\pi$
$367$	1.13290	0.0591367	0.0295683	+	0.999563i	$0.490587\pi$
$368$	9.07051	0.472833
$369$	0	0
$370$	−5.86899	−0.305114
$371$	−0.941428	−0.0488765
$372$	0	0
$373$	−0.250885	−0.0129903	−0.00649515	−	0.999979i	$-0.502067\pi$
$373$	−0.250885	−0.0129903	−0.00649515	+	0.999979i	$0.502067\pi$
$374$	18.6518	0.964464
$375$	0	0
$376$	4.14286	0.213652
$377$	2.68383	0.138224
$378$	0	0
$379$	−20.7844	−1.06762	−0.533812	−	0.845603i	$-0.679241\pi$
$379$	−20.7844	−1.06762	−0.533812	+	0.845603i	$0.679241\pi$
$380$	−77.6159	−3.98161
$381$	0	0
$382$	41.2964	2.11291
$383$	−13.1231	−0.670559	−0.335280	−	0.942119i	$-0.608831\pi$
$383$	−13.1231	−0.670559	−0.335280	+	0.942119i	$0.608831\pi$
$384$	0	0
$385$	6.02537	0.307081
$386$	18.3893	0.935989
$387$	0	0
$388$	28.9974	1.47212
$389$	1.05720	0.0536019	0.0268010	−	0.999641i	$-0.491468\pi$
$389$	1.05720	0.0536019	0.0268010	+	0.999641i	$0.491468\pi$
$390$	0	0
$391$	−22.5971	−1.14278
$392$	−3.66045	−0.184881
$393$	0	0
$394$	11.4705	0.577877
$395$	6.69728	0.336977
$396$	0	0
$397$	−0.516619	−0.0259284	−0.0129642	−	0.999916i	$-0.504127\pi$
$397$	−0.516619	−0.0259284	−0.0129642	+	0.999916i	$0.504127\pi$
$398$	−46.9692	−2.35435
$399$	0	0
$400$	5.01943	0.250972
$401$	−15.0413	−0.751127	−0.375564	−	0.926797i	$-0.622551\pi$
$401$	−15.0413	−0.751127	−0.375564	+	0.926797i	$0.622551\pi$
$402$	0	0
$403$	−0.841283	−0.0419073
$404$	45.3693	2.25721
$405$	0	0
$406$	6.95747	0.345294
$407$	−1.80720	−0.0895797
$408$	0	0
$409$	−7.70715	−0.381094	−0.190547	−	0.981678i	$-0.561026\pi$
$409$	−7.70715	−0.381094	−0.190547	+	0.981678i	$0.561026\pi$
$410$	22.8397	1.12797
$411$	0	0
$412$	−18.1684	−0.895091
$413$	−4.90991	−0.241601
$414$	0	0
$415$	−27.9836	−1.37366
$416$	3.40010	0.166704
$417$	0	0
$418$	−37.3519	−1.82694
$419$	28.0313	1.36942	0.684710	−	0.728815i	$-0.259929\pi$
$419$	28.0313	1.36942	0.684710	+	0.728815i	$0.259929\pi$
$420$	0	0
$421$	31.9491	1.55711	0.778553	−	0.627579i	$-0.215954\pi$
$421$	31.9491	1.55711	0.778553	+	0.627579i	$0.215954\pi$
$422$	17.8836	0.870561
$423$	0	0
$424$	−3.44605	−0.167355
$425$	−12.5048	−0.606570
$426$	0	0
$427$	−3.87805	−0.187672
$428$	−14.0875	−0.680945
$429$	0	0
$430$	49.3467	2.37971
$431$	29.7065	1.43091	0.715455	−	0.698659i	$-0.246219\pi$
$431$	29.7065	1.43091	0.715455	+	0.698659i	$0.246219\pi$
$432$	0	0
$433$	8.61337	0.413932	0.206966	−	0.978348i	$-0.433641\pi$
$433$	8.61337	0.413932	0.206966	+	0.978348i	$0.433641\pi$
$434$	−2.18092	−0.104687
$435$	0	0
$436$	44.3833	2.12557
$437$	45.2526	2.16473
$438$	0	0
$439$	−6.77586	−0.323394	−0.161697	−	0.986840i	$-0.551697\pi$
$439$	−6.77586	−0.323394	−0.161697	+	0.986840i	$0.551697\pi$
$440$	22.0556	1.05146
$441$	0	0
$442$	8.10866	0.385690
$443$	20.9325	0.994532	0.497266	−	0.867598i	$-0.334337\pi$
$443$	20.9325	0.994532	0.497266	+	0.867598i	$0.334337\pi$
$444$	0	0
$445$	0.982253	0.0465633
$446$	0.158705	0.00751490
$447$	0	0
$448$	11.8531	0.560008
$449$	−32.9230	−1.55373	−0.776865	−	0.629667i	$-0.783191\pi$
$449$	−32.9230	−1.55373	−0.776865	+	0.629667i	$0.783191\pi$
$450$	0	0
$451$	7.03290	0.331166
$452$	−2.45797	−0.115613
$453$	0	0
$454$	−39.9614	−1.87548
$455$	2.61946	0.122802
$456$	0	0
$457$	20.6965	0.968140	0.484070	−	0.875029i	$-0.339158\pi$
$457$	20.6965	0.968140	0.484070	+	0.875029i	$0.339158\pi$
$458$	−62.1450	−2.90385
$459$	0	0
$460$	−61.1257	−2.85000
$461$	−3.86099	−0.179824	−0.0899122	−	0.995950i	$-0.528659\pi$
$461$	−3.86099	−0.179824	−0.0899122	+	0.995950i	$0.528659\pi$
$462$	0	0
$463$	−6.29039	−0.292339	−0.146170	−	0.989260i	$-0.546695\pi$
$463$	−6.29039	−0.292339	−0.146170	+	0.989260i	$0.546695\pi$
$464$	4.48589	0.208252
$465$	0	0
$466$	71.1398	3.29549
$467$	−11.6118	−0.537331	−0.268665	−	0.963234i	$-0.586583\pi$
$467$	−11.6118	−0.537331	−0.268665	+	0.963234i	$0.586583\pi$
$468$	0	0
$469$	−2.67893	−0.123702
$470$	−7.68552	−0.354506
$471$	0	0
$472$	−17.9725	−0.827252
$473$	15.1950	0.698669
$474$	0	0
$475$	25.0419	1.14900
$476$	13.4502	0.616487
$477$	0	0
$478$	−1.49801	−0.0685174
$479$	36.5063	1.66801	0.834007	−	0.551754i	$-0.186041\pi$
$479$	36.5063	1.66801	0.834007	+	0.551754i	$0.186041\pi$
$480$	0	0
$481$	−0.785659	−0.0358230
$482$	−33.3159	−1.51750
$483$	0	0
$484$	−23.5505	−1.07048
$485$	−23.5157	−1.06779
$486$	0	0
$487$	18.8621	0.854723	0.427361	−	0.904081i	$-0.359443\pi$
$487$	18.8621	0.854723	0.427361	+	0.904081i	$0.359443\pi$
$488$	−14.1954	−0.642597
$489$	0	0
$490$	6.79060	0.306768
$491$	−3.22865	−0.145707	−0.0728535	−	0.997343i	$-0.523211\pi$
$491$	−3.22865	−0.145707	−0.0728535	+	0.997343i	$0.523211\pi$
$492$	0	0
$493$	−11.1756	−0.503322
$494$	−16.2383	−0.730596
$495$	0	0
$496$	−1.40617	−0.0631387
$497$	3.06017	0.137267
$498$	0	0
$499$	−28.1242	−1.25901	−0.629506	−	0.776996i	$-0.716743\pi$
$499$	−28.1242	−1.25901	−0.629506	+	0.776996i	$0.716743\pi$
$500$	17.3702	0.776821
$501$	0	0
$502$	−30.1872	−1.34732
$503$	26.7342	1.19202	0.596010	−	0.802977i	$-0.296752\pi$
$503$	26.7342	1.19202	0.596010	+	0.802977i	$0.296752\pi$
$504$	0	0
$505$	−36.7926	−1.63725
$506$	−29.4162	−1.30771
$507$	0	0
$508$	−3.55331	−0.157653
$509$	−34.2422	−1.51776	−0.758879	−	0.651232i	$-0.774252\pi$
$509$	−34.2422	−1.51776	−0.758879	+	0.651232i	$0.774252\pi$
$510$	0	0
$511$	13.6550	0.604061
$512$	16.8065	0.742751
$513$	0	0
$514$	24.8194	1.09474
$515$	14.7338	0.649248
$516$	0	0
$517$	−2.36655	−0.104081
$518$	−2.03672	−0.0894883
$519$	0	0
$520$	9.58839	0.420479
$521$	7.34862	0.321949	0.160974	−	0.986959i	$-0.448536\pi$
$521$	7.34862	0.321949	0.160974	+	0.986959i	$0.448536\pi$
$522$	0	0
$523$	1.84572	0.0807077	0.0403538	−	0.999185i	$-0.487151\pi$
$523$	1.84572	0.0807077	0.0403538	+	0.999185i	$0.487151\pi$
$524$	−8.18868	−0.357724
$525$	0	0
$526$	23.4231	1.02129
$527$	3.50314	0.152599
$528$	0	0
$529$	12.6383	0.549490
$530$	6.39286	0.277688
$531$	0	0
$532$	−26.9351	−1.16779
$533$	3.05747	0.132434
$534$	0	0
$535$	11.4244	0.493919
$536$	−9.80611	−0.423560
$537$	0	0
$538$	−29.5312	−1.27318
$539$	2.09099	0.0900652
$540$	0	0
$541$	−10.2748	−0.441750	−0.220875	−	0.975302i	$-0.570891\pi$
$541$	−10.2748	−0.441750	−0.220875	+	0.975302i	$0.570891\pi$
$542$	19.4169	0.834027
$543$	0	0
$544$	−14.1582	−0.607027
$545$	−35.9930	−1.54177
$546$	0	0
$547$	−38.1107	−1.62950	−0.814748	−	0.579816i	$-0.803125\pi$
$547$	−38.1107	−1.62950	−0.814748	+	0.579816i	$0.803125\pi$
$548$	−23.8385	−1.01833
$549$	0	0
$550$	−16.2783	−0.694109
$551$	22.3800	0.953422
$552$	0	0
$553$	2.32416	0.0988334
$554$	−35.1200	−1.49211
$555$	0	0
$556$	−36.8490	−1.56275
$557$	2.06112	0.0873325	0.0436663	−	0.999046i	$-0.486096\pi$
$557$	2.06112	0.0873325	0.0436663	+	0.999046i	$0.486096\pi$
$558$	0	0
$559$	6.60586	0.279398
$560$	4.37830	0.185017
$561$	0	0
$562$	−67.9101	−2.86461
$563$	−3.75432	−0.158226	−0.0791129	−	0.996866i	$-0.525209\pi$
$563$	−3.75432	−0.158226	−0.0791129	+	0.996866i	$0.525209\pi$
$564$	0	0
$565$	1.99331	0.0838594
$566$	68.3351	2.87234
$567$	0	0
$568$	11.2016	0.470009
$569$	6.62415	0.277699	0.138849	−	0.990314i	$-0.455660\pi$
$569$	6.62415	0.277699	0.138849	+	0.990314i	$0.455660\pi$
$570$	0	0
$571$	13.6145	0.569748	0.284874	−	0.958565i	$-0.408048\pi$
$571$	13.6145	0.569748	0.284874	+	0.958565i	$0.408048\pi$
$572$	6.75405	0.282401
$573$	0	0
$574$	7.92609	0.330829
$575$	19.7215	0.822442
$576$	0	0
$577$	−14.7697	−0.614870	−0.307435	−	0.951569i	$-0.599471\pi$
$577$	−14.7697	−0.614870	−0.307435	+	0.951569i	$0.599471\pi$
$578$	6.29650	0.261900
$579$	0	0
$580$	−30.2302	−1.25524
$581$	−9.71119	−0.402888
$582$	0	0
$583$	1.96851	0.0815275
$584$	49.9834	2.06833
$585$	0	0
$586$	42.4899	1.75524
$587$	4.82816	0.199279	0.0996396	−	0.995024i	$-0.468231\pi$
$587$	4.82816	0.199279	0.0996396	+	0.995024i	$0.468231\pi$
$588$	0	0
$589$	−7.01534	−0.289062
$590$	33.3412	1.37264
$591$	0	0
$592$	−1.31319	−0.0539719
$593$	4.10629	0.168625	0.0843126	−	0.996439i	$-0.473131\pi$
$593$	4.10629	0.168625	0.0843126	+	0.996439i	$0.473131\pi$
$594$	0	0
$595$	−10.9075	−0.447165
$596$	−40.9673	−1.67808
$597$	0	0
$598$	−12.7883	−0.522953
$599$	−2.52365	−0.103114	−0.0515568	−	0.998670i	$-0.516418\pi$
$599$	−2.52365	−0.103114	−0.0515568	+	0.998670i	$0.516418\pi$
$600$	0	0
$601$	−29.8188	−1.21633	−0.608167	−	0.793809i	$-0.708095\pi$
$601$	−29.8188	−1.21633	−0.608167	+	0.793809i	$0.708095\pi$
$602$	17.1248	0.697956
$603$	0	0
$604$	−76.2481	−3.10249
$605$	19.0985	0.776464
$606$	0	0
$607$	30.7270	1.24717	0.623585	−	0.781756i	$-0.285676\pi$
$607$	30.7270	1.24717	0.623585	+	0.781756i	$0.285676\pi$
$608$	28.3530	1.14987
$609$	0	0
$610$	26.3343	1.06624
$611$	−1.02883	−0.0416220
$612$	0	0
$613$	−5.99195	−0.242013	−0.121006	−	0.992652i	$-0.538612\pi$
$613$	−5.99195	−0.242013	−0.121006	+	0.992652i	$0.538612\pi$
$614$	0.369932	0.0149292
$615$	0	0
$616$	7.65397	0.308387
$617$	36.8789	1.48469	0.742345	−	0.670018i	$-0.233713\pi$
$617$	36.8789	1.48469	0.742345	+	0.670018i	$0.233713\pi$
$618$	0	0
$619$	20.4588	0.822308	0.411154	−	0.911566i	$-0.365126\pi$
$619$	20.4588	0.822308	0.411154	+	0.911566i	$0.365126\pi$
$620$	9.47608	0.380568
$621$	0	0
$622$	−9.38148	−0.376163
$623$	0.340872	0.0136568
$624$	0	0
$625$	−30.6043	−1.22417
$626$	−13.9949	−0.559350
$627$	0	0
$628$	9.51702	0.379771
$629$	3.27152	0.130444
$630$	0	0
$631$	12.4014	0.493692	0.246846	−	0.969055i	$-0.420606\pi$
$631$	12.4014	0.493692	0.246846	+	0.969055i	$0.420606\pi$
$632$	8.50748	0.338410
$633$	0	0
$634$	5.72443	0.227346
$635$	2.88159	0.114352
$636$	0	0
$637$	0.909032	0.0360171
$638$	−14.5480	−0.575961
$639$	0	0
$640$	−58.9336	−2.32955
$641$	−5.76037	−0.227521	−0.113761	−	0.993508i	$-0.536290\pi$
$641$	−5.76037	−0.227521	−0.113761	+	0.993508i	$0.536290\pi$
$642$	0	0
$643$	−6.93658	−0.273552	−0.136776	−	0.990602i	$-0.543674\pi$
$643$	−6.93658	−0.273552	−0.136776	+	0.990602i	$0.543674\pi$
$644$	−21.2125	−0.835889
$645$	0	0
$646$	67.6170	2.66035
$647$	−33.1829	−1.30456	−0.652278	−	0.757980i	$-0.726186\pi$
$647$	−33.1829	−1.30456	−0.652278	+	0.757980i	$0.726186\pi$
$648$	0	0
$649$	10.2666	0.402998
$650$	−7.07679	−0.277574
$651$	0	0
$652$	−68.6346	−2.68794
$653$	46.9289	1.83647	0.918235	−	0.396035i	$-0.129614\pi$
$653$	46.9289	1.83647	0.918235	+	0.396035i	$0.129614\pi$
$654$	0	0
$655$	6.64068	0.259473
$656$	5.11042	0.199528
$657$	0	0
$658$	−2.66711	−0.103975
$659$	15.8666	0.618076	0.309038	−	0.951050i	$-0.399993\pi$
$659$	15.8666	0.618076	0.309038	+	0.951050i	$0.399993\pi$
$660$	0	0
$661$	7.41844	0.288544	0.144272	−	0.989538i	$-0.453916\pi$
$661$	7.41844	0.288544	0.144272	+	0.989538i	$0.453916\pi$
$662$	−54.2482	−2.10842
$663$	0	0
$664$	−35.5473	−1.37950
$665$	21.8433	0.847045
$666$	0	0
$667$	17.6252	0.682450
$668$	−2.16167	−0.0836375
$669$	0	0
$670$	18.1916	0.702801
$671$	8.10896	0.313043
$672$	0	0
$673$	21.1749	0.816232	0.408116	−	0.912930i	$-0.366186\pi$
$673$	21.1749	0.816232	0.408116	+	0.912930i	$0.366186\pi$
$674$	−31.2826	−1.20496
$675$	0	0
$676$	−43.2568	−1.66372
$677$	6.57777	0.252804	0.126402	−	0.991979i	$-0.459657\pi$
$677$	6.57777	0.252804	0.126402	+	0.991979i	$0.459657\pi$
$678$	0	0
$679$	−8.16066	−0.313177
$680$	−39.9265	−1.53111
$681$	0	0
$682$	4.56027	0.174622
$683$	−19.3432	−0.740145	−0.370073	−	0.929003i	$-0.620667\pi$
$683$	−19.3432	−0.740145	−0.370073	+	0.929003i	$0.620667\pi$
$684$	0	0
$685$	19.3320	0.738639
$686$	2.35655	0.0899734
$687$	0	0
$688$	11.0414	0.420949
$689$	0.855788	0.0326029
$690$	0	0
$691$	−26.3865	−1.00379	−0.501895	−	0.864929i	$-0.667364\pi$
$691$	−26.3865	−1.00379	−0.501895	+	0.864929i	$0.667364\pi$
$692$	28.5536	1.08544
$693$	0	0
$694$	49.6694	1.88543
$695$	29.8830	1.13353
$696$	0	0
$697$	−12.7314	−0.482237
$698$	−51.4405	−1.94705
$699$	0	0
$700$	−11.7386	−0.443676
$701$	−17.6124	−0.665210	−0.332605	−	0.943066i	$-0.607928\pi$
$701$	−17.6124	−0.665210	−0.332605	+	0.943066i	$0.607928\pi$
$702$	0	0
$703$	−6.55150	−0.247094
$704$	−24.7848	−0.934111
$705$	0	0
$706$	1.36776	0.0514765
$707$	−12.7682	−0.480197
$708$	0	0
$709$	48.0398	1.80417	0.902086	−	0.431557i	$-0.142036\pi$
$709$	48.0398	1.80417	0.902086	+	0.431557i	$0.142036\pi$
$710$	−20.7804	−0.779873
$711$	0	0
$712$	1.24775	0.0467613
$713$	−5.52486	−0.206908
$714$	0	0
$715$	−5.47725	−0.204838
$716$	62.0582	2.31922
$717$	0	0
$718$	56.1831	2.09674
$719$	−45.6237	−1.70148	−0.850739	−	0.525589i	$-0.823845\pi$
$719$	−45.6237	−1.70148	−0.850739	+	0.525589i	$0.823845\pi$
$720$	0	0
$721$	5.11308	0.190421
$722$	−90.6345	−3.37307
$723$	0	0
$724$	75.6706	2.81228
$725$	9.75341	0.362233
$726$	0	0
$727$	−5.30019	−0.196573	−0.0982867	−	0.995158i	$-0.531336\pi$
$727$	−5.30019	−0.196573	−0.0982867	+	0.995158i	$0.531336\pi$
$728$	3.32747	0.123324
$729$	0	0
$730$	−92.7255	−3.43193
$731$	−27.5071	−1.01739
$732$	0	0
$733$	−8.30042	−0.306583	−0.153292	−	0.988181i	$-0.548987\pi$
$733$	−8.30042	−0.306583	−0.153292	+	0.988181i	$0.548987\pi$
$734$	−2.66972	−0.0985412
$735$	0	0
$736$	22.3291	0.823062
$737$	5.60162	0.206338
$738$	0	0
$739$	28.5316	1.04955	0.524775	−	0.851241i	$-0.324149\pi$
$739$	28.5316	1.04955	0.524775	+	0.851241i	$0.324149\pi$
$740$	8.84954	0.325315
$741$	0	0
$742$	2.21852	0.0814444
$743$	7.19884	0.264100	0.132050	−	0.991243i	$-0.457844\pi$
$743$	7.19884	0.264100	0.132050	+	0.991243i	$0.457844\pi$
$744$	0	0
$745$	33.2227	1.21719
$746$	0.591221	0.0216461
$747$	0	0
$748$	−28.1241	−1.02832
$749$	3.96461	0.144864
$750$	0	0
$751$	25.8207	0.942211	0.471106	−	0.882077i	$-0.343855\pi$
$751$	25.8207	0.942211	0.471106	+	0.882077i	$0.343855\pi$
$752$	−1.71964	−0.0627090
$753$	0	0
$754$	−6.32456	−0.230327
$755$	61.8340	2.25037
$756$	0	0
$757$	18.3933	0.668516	0.334258	−	0.942482i	$-0.391514\pi$
$757$	18.3933	0.668516	0.334258	+	0.942482i	$0.391514\pi$
$758$	48.9795	1.77902
$759$	0	0
$760$	79.9562	2.90032
$761$	−54.4345	−1.97325	−0.986625	−	0.163009i	$-0.947880\pi$
$761$	−54.4345	−1.97325	−0.986625	+	0.163009i	$0.947880\pi$
$762$	0	0
$763$	−12.4907	−0.452193
$764$	−62.2687	−2.25280
$765$	0	0
$766$	30.9252	1.11737
$767$	4.46327	0.161159
$768$	0	0
$769$	38.3709	1.38369	0.691845	−	0.722046i	$-0.256798\pi$
$769$	38.3709	1.38369	0.691845	+	0.722046i	$0.256798\pi$
$770$	−14.1991	−0.511699
$771$	0	0
$772$	−27.7282	−0.997960
$773$	−11.0889	−0.398841	−0.199421	−	0.979914i	$-0.563906\pi$
$773$	−11.0889	−0.398841	−0.199421	+	0.979914i	$0.563906\pi$
$774$	0	0
$775$	−3.05734	−0.109823
$776$	−29.8717	−1.07233
$777$	0	0
$778$	−2.49133	−0.0893185
$779$	25.4958	0.913481
$780$	0	0
$781$	−6.39877	−0.228966
$782$	53.2511	1.90425
$783$	0	0
$784$	1.51941	0.0542645
$785$	−7.71791	−0.275464
$786$	0	0
$787$	−15.1820	−0.541179	−0.270589	−	0.962695i	$-0.587219\pi$
$787$	−15.1820	−0.541179	−0.270589	+	0.962695i	$0.587219\pi$
$788$	−17.2958	−0.616138
$789$	0	0
$790$	−15.7824	−0.561514
$791$	0.691741	0.0245955
$792$	0	0
$793$	3.52527	0.125186
$794$	1.21744	0.0432052
$795$	0	0
$796$	70.8224	2.51023
$797$	42.0898	1.49090	0.745449	−	0.666563i	$-0.232235\pi$
$797$	42.0898	1.49090	0.745449	+	0.666563i	$0.232235\pi$
$798$	0	0
$799$	4.28409	0.151560
$800$	12.3565	0.436867
$801$	0	0
$802$	35.4455	1.25163
$803$	−28.5524	−1.00759
$804$	0	0
$805$	17.2024	0.606307
$806$	1.98252	0.0698314
$807$	0	0
$808$	−46.7373	−1.64421
$809$	−35.4386	−1.24595	−0.622977	−	0.782240i	$-0.714077\pi$
$809$	−35.4386	−1.24595	−0.622977	+	0.782240i	$0.714077\pi$
$810$	0	0
$811$	18.0006	0.632087	0.316044	−	0.948745i	$-0.397645\pi$
$811$	18.0006	0.632087	0.316044	+	0.948745i	$0.397645\pi$
$812$	−10.4908	−0.368155
$813$	0	0
$814$	4.25876	0.149269
$815$	55.6598	1.94968
$816$	0	0
$817$	55.0853	1.92719
$818$	18.1623	0.635028
$819$	0	0
$820$	−34.4388	−1.20266
$821$	21.1160	0.736955	0.368477	−	0.929637i	$-0.379879\pi$
$821$	21.1160	0.736955	0.368477	+	0.929637i	$0.379879\pi$
$822$	0	0
$823$	34.8585	1.21509	0.607545	−	0.794285i	$-0.292154\pi$
$823$	34.8585	1.21509	0.607545	+	0.794285i	$0.292154\pi$
$824$	18.7162	0.652009
$825$	0	0
$826$	11.5704	0.402587
$827$	−15.8606	−0.551527	−0.275764	−	0.961225i	$-0.588931\pi$
$827$	−15.8606	−0.551527	−0.275764	+	0.961225i	$0.588931\pi$
$828$	0	0
$829$	16.1365	0.560445	0.280223	−	0.959935i	$-0.409592\pi$
$829$	16.1365	0.560445	0.280223	+	0.959935i	$0.409592\pi$
$830$	65.9448	2.28898
$831$	0	0
$832$	−10.7749	−0.373552
$833$	−3.78525	−0.131151
$834$	0	0
$835$	1.75302	0.0606659
$836$	56.3210	1.94790
$837$	0	0
$838$	−66.0572	−2.28191
$839$	39.0471	1.34806	0.674028	−	0.738705i	$-0.264563\pi$
$839$	39.0471	1.34806	0.674028	+	0.738705i	$0.264563\pi$
$840$	0	0
$841$	−20.2833	−0.699425
$842$	−75.2897	−2.59465
$843$	0	0
$844$	−26.9658	−0.928200
$845$	35.0795	1.20677
$846$	0	0
$847$	6.62777	0.227733
$848$	1.43041	0.0491205
$849$	0	0
$850$	29.4680	1.01075
$851$	−5.15957	−0.176868
$852$	0	0
$853$	35.6231	1.21971	0.609856	−	0.792512i	$-0.291227\pi$
$853$	35.6231	1.21971	0.609856	+	0.792512i	$0.291227\pi$
$854$	9.13881	0.312723
$855$	0	0
$856$	14.5123	0.496019
$857$	−6.14909	−0.210049	−0.105025	−	0.994470i	$-0.533492\pi$
$857$	−6.14909	−0.210049	−0.105025	+	0.994470i	$0.533492\pi$
$858$	0	0
$859$	−2.17991	−0.0743775	−0.0371888	−	0.999308i	$-0.511840\pi$
$859$	−2.17991	−0.0743775	−0.0371888	+	0.999308i	$0.511840\pi$
$860$	−74.4073	−2.53727
$861$	0	0
$862$	−70.0047	−2.38437
$863$	14.2536	0.485198	0.242599	−	0.970127i	$-0.422000\pi$
$863$	14.2536	0.485198	0.242599	+	0.970127i	$0.422000\pi$
$864$	0	0
$865$	−23.1558	−0.787319
$866$	−20.2978	−0.689747
$867$	0	0
$868$	3.28849	0.111619
$869$	−4.85979	−0.164857
$870$	0	0
$871$	2.43524	0.0825148
$872$	−45.7216	−1.54833
$873$	0	0
$874$	−106.640	−3.60715
$875$	−4.88847	−0.165260
$876$	0	0
$877$	45.8181	1.54717	0.773584	−	0.633694i	$-0.218462\pi$
$877$	45.8181	1.54717	0.773584	+	0.633694i	$0.218462\pi$
$878$	15.9676	0.538882
$879$	0	0
$880$	−9.15497	−0.308614
$881$	53.7021	1.80927	0.904634	−	0.426189i	$-0.140144\pi$
$881$	53.7021	1.80927	0.904634	+	0.426189i	$0.140144\pi$
$882$	0	0
$883$	23.3029	0.784205	0.392103	−	0.919922i	$-0.371748\pi$
$883$	23.3029	0.784205	0.392103	+	0.919922i	$0.371748\pi$
$884$	−12.2266	−0.411226
$885$	0	0
$886$	−49.3284	−1.65722
$887$	9.22062	0.309598	0.154799	−	0.987946i	$-0.450527\pi$
$887$	9.22062	0.309598	0.154799	+	0.987946i	$0.450527\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−2.31473	−0.0775898
$891$	0	0
$892$	−0.239303	−0.00801245
$893$	−8.57927	−0.287094
$894$	0	0
$895$	−50.3266	−1.68223
$896$	−20.4518	−0.683245
$897$	0	0
$898$	77.5845	2.58903
$899$	−2.73236	−0.0911295
$900$	0	0
$901$	−3.56354	−0.118719
$902$	−16.5734	−0.551833
$903$	0	0
$904$	2.53209	0.0842160
$905$	−61.3657	−2.03987
$906$	0	0
$907$	45.5765	1.51334	0.756672	−	0.653795i	$-0.226824\pi$
$907$	45.5765	1.51334	0.756672	+	0.653795i	$0.226824\pi$
$908$	60.2557	1.99966
$909$	0	0
$910$	−6.17287	−0.204629
$911$	8.51625	0.282156	0.141078	−	0.989998i	$-0.454943\pi$
$911$	8.51625	0.282156	0.141078	+	0.989998i	$0.454943\pi$
$912$	0	0
$913$	20.3060	0.672030
$914$	−48.7722	−1.61324
$915$	0	0
$916$	93.7052	3.09611
$917$	2.30452	0.0761019
$918$	0	0
$919$	46.8342	1.54492	0.772458	−	0.635066i	$-0.219027\pi$
$919$	46.8342	1.54492	0.772458	+	0.635066i	$0.219027\pi$
$920$	62.9688	2.07602
$921$	0	0
$922$	9.09861	0.299647
$923$	−2.78179	−0.0915637
$924$	0	0
$925$	−2.85520	−0.0938783
$926$	14.8236	0.487133
$927$	0	0
$928$	11.0430	0.362506
$929$	35.8706	1.17688	0.588438	−	0.808542i	$-0.299743\pi$
$929$	35.8706	1.17688	0.588438	+	0.808542i	$0.299743\pi$
$930$	0	0
$931$	7.58028	0.248434
$932$	−107.268	−3.51368
$933$	0	0
$934$	27.3638	0.895370
$935$	22.8075	0.745885
$936$	0	0
$937$	37.9419	1.23951	0.619753	−	0.784797i	$-0.287233\pi$
$937$	37.9419	1.23951	0.619753	+	0.784797i	$0.287233\pi$
$938$	6.31303	0.206128
$939$	0	0
$940$	11.5886	0.377978
$941$	40.0327	1.30503	0.652515	−	0.757776i	$-0.273714\pi$
$941$	40.0327	1.30503	0.652515	+	0.757776i	$0.273714\pi$
$942$	0	0
$943$	20.0790	0.653861
$944$	7.46014	0.242807
$945$	0	0
$946$	−35.8078	−1.16421
$947$	42.7394	1.38884	0.694422	−	0.719568i	$-0.255660\pi$
$947$	42.7394	1.38884	0.694422	+	0.719568i	$0.255660\pi$
$948$	0	0
$949$	−12.4128	−0.402937
$950$	−59.0123	−1.91461
$951$	0	0
$952$	−13.8557	−0.449066
$953$	−43.9497	−1.42367	−0.711835	−	0.702347i	$-0.752135\pi$
$953$	−43.9497	−1.42367	−0.711835	+	0.702347i	$0.752135\pi$
$954$	0	0
$955$	50.4973	1.63405
$956$	2.25877	0.0730539
$957$	0	0
$958$	−86.0287	−2.77946
$959$	6.70881	0.216639
$960$	0	0
$961$	−30.1435	−0.972371
$962$	1.85144	0.0596929
$963$	0	0
$964$	50.2353	1.61797
$965$	22.4864	0.723863
$966$	0	0
$967$	32.3318	1.03972	0.519860	−	0.854251i	$-0.325984\pi$
$967$	32.3318	1.03972	0.519860	+	0.854251i	$0.325984\pi$
$968$	24.2606	0.779766
$969$	0	0
$970$	55.4158	1.77929
$971$	33.1010	1.06226	0.531131	−	0.847290i	$-0.321767\pi$
$971$	33.1010	1.06226	0.531131	+	0.847290i	$0.321767\pi$
$972$	0	0
$973$	10.3703	0.332458
$974$	−44.4494	−1.42425
$975$	0	0
$976$	5.89233	0.188609
$977$	26.3507	0.843033	0.421516	−	0.906821i	$-0.361498\pi$
$977$	26.3507	0.843033	0.421516	+	0.906821i	$0.361498\pi$
$978$	0	0
$979$	−0.712760	−0.0227799
$980$	−10.2392	−0.327079
$981$	0	0
$982$	7.60847	0.242796
$983$	−20.5951	−0.656883	−0.328442	−	0.944524i	$-0.606523\pi$
$983$	−20.5951	−0.656883	−0.328442	+	0.944524i	$0.606523\pi$
$984$	0	0
$985$	14.0262	0.446911
$986$	26.3357	0.838701
$987$	0	0
$988$	24.4849	0.778968
$989$	43.3819	1.37946
$990$	0	0
$991$	−15.2387	−0.484074	−0.242037	−	0.970267i	$-0.577815\pi$
$991$	−15.2387	−0.484074	−0.242037	+	0.970267i	$0.577815\pi$
$992$	−3.46160	−0.109906
$993$	0	0
$994$	−7.21142	−0.228733
$995$	−57.4340	−1.82078
$996$	0	0
$997$	30.6942	0.972096	0.486048	−	0.873932i	$-0.338438\pi$
$997$	30.6942	0.972096	0.486048	+	0.873932i	$0.338438\pi$
$998$	66.2759	2.09793
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.s.1.4		16
3.2	odd	2		2667.2.a.n.1.13	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.n.1.13	✓	16	3.2	odd	2
8001.2.a.s.1.4		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)

\(f(q)\)	\(=\)	\(q-2.35655 q^{2} +3.55331 q^{4} -2.88159 q^{5} -1.00000 q^{7} -3.66045 q^{8} +O(q^{10})\)	\(q-2.35655 q^{2} +3.55331 q^{4} -2.88159 q^{5} -1.00000 q^{7} -3.66045 q^{8} +6.79060 q^{10} +2.09099 q^{11} +0.909032 q^{13} +2.35655 q^{14} +1.51941 q^{16} -3.78525 q^{17} +7.58028 q^{19} -10.2392 q^{20} -4.92751 q^{22} +5.96978 q^{23} +3.30355 q^{25} -2.14218 q^{26} -3.55331 q^{28} +2.95240 q^{29} -0.925472 q^{31} +3.74036 q^{32} +8.92011 q^{34} +2.88159 q^{35} -0.864281 q^{37} -17.8633 q^{38} +10.5479 q^{40} +3.36343 q^{41} +7.26692 q^{43} +7.42993 q^{44} -14.0681 q^{46} -1.13179 q^{47} +1.00000 q^{49} -7.78497 q^{50} +3.23007 q^{52} +0.941428 q^{53} -6.02537 q^{55} +3.66045 q^{56} -6.95747 q^{58} +4.90991 q^{59} +3.87805 q^{61} +2.18092 q^{62} -11.8531 q^{64} -2.61946 q^{65} +2.67893 q^{67} -13.4502 q^{68} -6.79060 q^{70} -3.06017 q^{71} -13.6550 q^{73} +2.03672 q^{74} +26.9351 q^{76} -2.09099 q^{77} -2.32416 q^{79} -4.37830 q^{80} -7.92609 q^{82} +9.71119 q^{83} +10.9075 q^{85} -17.1248 q^{86} -7.65397 q^{88} -0.340872 q^{89} -0.909032 q^{91} +21.2125 q^{92} +2.66711 q^{94} -21.8433 q^{95} +8.16066 q^{97} -2.35655 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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