LMFDB - Embedded newform 8001.2.a.r.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:	$1$
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} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} + \cdots + 4 $ x^16 - 5x^15 - 13x^14 + 98x^13 + 9x^12 - 712x^11 + 565x^10 + 2282x^9 - 3082x^8 - 2747x^7 + 5821x^6 - 158x^5 - 3341x^4 + 1002x^3 + 416x^2 - 148*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.639731$ 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)

$f(q)$	$=$	$q-0.639731 q^{2} -1.59074 q^{4} +2.86363 q^{5} -1.00000 q^{7} +2.29711 q^{8} +O(q^{10})$	\(q-0.639731 q^{2} -1.59074 q^{4} +2.86363 q^{5} -1.00000 q^{7} +2.29711 q^{8} -1.83195 q^{10} +2.14467 q^{11} -4.28315 q^{13} +0.639731 q^{14} +1.71195 q^{16} +2.78701 q^{17} -4.83017 q^{19} -4.55530 q^{20} -1.37201 q^{22} +0.469034 q^{23} +3.20037 q^{25} +2.74006 q^{26} +1.59074 q^{28} +5.86824 q^{29} -8.47495 q^{31} -5.68941 q^{32} -1.78294 q^{34} -2.86363 q^{35} -3.06505 q^{37} +3.09001 q^{38} +6.57807 q^{40} +11.3095 q^{41} -2.70985 q^{43} -3.41163 q^{44} -0.300056 q^{46} +4.49311 q^{47} +1.00000 q^{49} -2.04738 q^{50} +6.81339 q^{52} -6.46774 q^{53} +6.14155 q^{55} -2.29711 q^{56} -3.75410 q^{58} -12.4745 q^{59} +6.90425 q^{61} +5.42169 q^{62} +0.215788 q^{64} -12.2653 q^{65} +8.97692 q^{67} -4.43342 q^{68} +1.83195 q^{70} -7.06614 q^{71} -0.223717 q^{73} +1.96081 q^{74} +7.68357 q^{76} -2.14467 q^{77} -10.7386 q^{79} +4.90240 q^{80} -7.23505 q^{82} -10.6743 q^{83} +7.98097 q^{85} +1.73358 q^{86} +4.92655 q^{88} -12.7058 q^{89} +4.28315 q^{91} -0.746114 q^{92} -2.87438 q^{94} -13.8318 q^{95} -10.9050 q^{97} -0.639731 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * 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 $

$2$	−0.639731	−0.452358	−0.226179	−	0.974086i	$-0.572623\pi$
$2$	−0.639731	−0.452358	−0.226179	+	0.974086i	$0.572623\pi$
$3$	0	0
$3$	0	0
$4$	−1.59074	−0.795372
$5$	2.86363	1.28065	0.640327	−	0.768103i	$-0.278799\pi$
$5$	2.86363	1.28065	0.640327	+	0.768103i	$0.278799\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.29711	0.812151
$9$	0	0
$10$	−1.83195	−0.579314
$11$	2.14467	0.646643	0.323322	−	0.946289i	$-0.395200\pi$
$11$	2.14467	0.646643	0.323322	+	0.946289i	$0.395200\pi$
$12$	0	0
$13$	−4.28315	−1.18793	−0.593966	−	0.804490i	$-0.702439\pi$
$13$	−4.28315	−1.18793	−0.593966	+	0.804490i	$0.702439\pi$
$14$	0.639731	0.170975
$15$	0	0
$16$	1.71195	0.427988
$17$	2.78701	0.675950	0.337975	−	0.941155i	$-0.390258\pi$
$17$	2.78701	0.675950	0.337975	+	0.941155i	$0.390258\pi$
$18$	0	0
$19$	−4.83017	−1.10812	−0.554059	−	0.832478i	$-0.686922\pi$
$19$	−4.83017	−1.10812	−0.554059	+	0.832478i	$0.686922\pi$
$20$	−4.55530	−1.01860
$21$	0	0
$22$	−1.37201	−0.292515
$23$	0.469034	0.0978004	0.0489002	−	0.998804i	$-0.484428\pi$
$23$	0.469034	0.0978004	0.0489002	+	0.998804i	$0.484428\pi$
$24$	0	0
$25$	3.20037	0.640074
$26$	2.74006	0.537371
$27$	0	0
$28$	1.59074	0.300622
$29$	5.86824	1.08970	0.544852	−	0.838532i	$-0.316586\pi$
$29$	5.86824	1.08970	0.544852	+	0.838532i	$0.316586\pi$
$30$	0	0
$31$	−8.47495	−1.52215	−0.761073	−	0.648667i	$-0.775327\pi$
$31$	−8.47495	−1.52215	−0.761073	+	0.648667i	$0.775327\pi$
$32$	−5.68941	−1.00576
$33$	0	0
$34$	−1.78294	−0.305771
$35$	−2.86363	−0.484042
$36$	0	0
$37$	−3.06505	−0.503891	−0.251946	−	0.967741i	$-0.581070\pi$
$37$	−3.06505	−0.503891	−0.251946	+	0.967741i	$0.581070\pi$
$38$	3.09001	0.501266
$39$	0	0
$40$	6.57807	1.04008
$41$	11.3095	1.76625	0.883124	−	0.469139i	$-0.155436\pi$
$41$	11.3095	1.76625	0.883124	+	0.469139i	$0.155436\pi$
$42$	0	0
$43$	−2.70985	−0.413248	−0.206624	−	0.978420i	$-0.566248\pi$
$43$	−2.70985	−0.413248	−0.206624	+	0.978420i	$0.566248\pi$
$44$	−3.41163	−0.514322
$45$	0	0
$46$	−0.300056	−0.0442408
$47$	4.49311	0.655387	0.327693	−	0.944784i	$-0.393729\pi$
$47$	4.49311	0.655387	0.327693	+	0.944784i	$0.393729\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.04738	−0.289543
$51$	0	0
$52$	6.81339	0.944847
$53$	−6.46774	−0.888412	−0.444206	−	0.895925i	$-0.646514\pi$
$53$	−6.46774	−0.888412	−0.444206	+	0.895925i	$0.646514\pi$
$54$	0	0
$55$	6.14155	0.828126
$56$	−2.29711	−0.306964
$57$	0	0
$58$	−3.75410	−0.492937
$59$	−12.4745	−1.62404	−0.812019	−	0.583630i	$-0.801632\pi$
$59$	−12.4745	−1.62404	−0.812019	+	0.583630i	$0.801632\pi$
$60$	0	0
$61$	6.90425	0.883998	0.441999	−	0.897015i	$-0.354269\pi$
$61$	6.90425	0.883998	0.441999	+	0.897015i	$0.354269\pi$
$62$	5.42169	0.688555
$63$	0	0
$64$	0.215788	0.0269735
$65$	−12.2653	−1.52133
$66$	0	0
$67$	8.97692	1.09671	0.548353	−	0.836247i	$-0.315255\pi$
$67$	8.97692	1.09671	0.548353	+	0.836247i	$0.315255\pi$
$68$	−4.43342	−0.537631
$69$	0	0
$70$	1.83195	0.218960
$71$	−7.06614	−0.838597	−0.419298	−	0.907849i	$-0.637724\pi$
$71$	−7.06614	−0.838597	−0.419298	+	0.907849i	$0.637724\pi$
$72$	0	0
$73$	−0.223717	−0.0261841	−0.0130920	−	0.999914i	$-0.504167\pi$
$73$	−0.223717	−0.0261841	−0.0130920	+	0.999914i	$0.504167\pi$
$74$	1.96081	0.227939
$75$	0	0
$76$	7.68357	0.881366
$77$	−2.14467	−0.244408
$78$	0	0
$79$	−10.7386	−1.20818	−0.604092	−	0.796915i	$-0.706464\pi$
$79$	−10.7386	−1.20818	−0.604092	+	0.796915i	$0.706464\pi$
$80$	4.90240	0.548105
$81$	0	0
$82$	−7.23505	−0.798977
$83$	−10.6743	−1.17166	−0.585828	−	0.810436i	$-0.699231\pi$
$83$	−10.6743	−1.17166	−0.585828	+	0.810436i	$0.699231\pi$
$84$	0	0
$85$	7.98097	0.865657
$86$	1.73358	0.186936
$87$	0	0
$88$	4.92655	0.525172
$89$	−12.7058	−1.34681	−0.673404	−	0.739274i	$-0.735169\pi$
$89$	−12.7058	−1.34681	−0.673404	+	0.739274i	$0.735169\pi$
$90$	0	0
$91$	4.28315	0.448996
$92$	−0.746114	−0.0777877
$93$	0	0
$94$	−2.87438	−0.296470
$95$	−13.8318	−1.41911
$96$	0	0
$97$	−10.9050	−1.10724	−0.553619	−	0.832770i	$-0.686754\pi$
$97$	−10.9050	−1.10724	−0.553619	+	0.832770i	$0.686754\pi$
$98$	−0.639731	−0.0646226
$99$	0	0
$100$	−5.09097	−0.509097
$101$	6.52396	0.649158	0.324579	−	0.945859i	$-0.394777\pi$
$101$	6.52396	0.649158	0.324579	+	0.945859i	$0.394777\pi$
$102$	0	0
$103$	−12.5772	−1.23927	−0.619633	−	0.784892i	$-0.712718\pi$
$103$	−12.5772	−1.23927	−0.619633	+	0.784892i	$0.712718\pi$
$104$	−9.83887	−0.964780
$105$	0	0
$106$	4.13762	0.401881
$107$	6.52171	0.630477	0.315239	−	0.949012i	$-0.397915\pi$
$107$	6.52171	0.630477	0.315239	+	0.949012i	$0.397915\pi$
$108$	0	0
$109$	11.2176	1.07446	0.537228	−	0.843437i	$-0.319472\pi$
$109$	11.2176	1.07446	0.537228	+	0.843437i	$0.319472\pi$
$110$	−3.92894	−0.374610
$111$	0	0
$112$	−1.71195	−0.161764
$113$	1.93198	0.181746	0.0908728	−	0.995863i	$-0.471034\pi$
$113$	1.93198	0.181746	0.0908728	+	0.995863i	$0.471034\pi$
$114$	0	0
$115$	1.34314	0.125248
$116$	−9.33487	−0.866721
$117$	0	0
$118$	7.98032	0.734648
$119$	−2.78701	−0.255485
$120$	0	0
$121$	−6.40038	−0.581852
$122$	−4.41686	−0.399884
$123$	0	0
$124$	13.4815	1.21067
$125$	−5.15348	−0.460941
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	11.2408	0.993554
$129$	0	0
$130$	7.84653	0.688186
$131$	7.36337	0.643341	0.321670	−	0.946852i	$-0.395756\pi$
$131$	7.36337	0.643341	0.321670	+	0.946852i	$0.395756\pi$
$132$	0	0
$133$	4.83017	0.418829
$134$	−5.74282	−0.496104
$135$	0	0
$136$	6.40208	0.548974
$137$	16.7976	1.43511	0.717556	−	0.696500i	$-0.245261\pi$
$137$	16.7976	1.43511	0.717556	+	0.696500i	$0.245261\pi$
$138$	0	0
$139$	−10.5934	−0.898524	−0.449262	−	0.893400i	$-0.648313\pi$
$139$	−10.5934	−0.898524	−0.449262	+	0.893400i	$0.648313\pi$
$140$	4.55530	0.384993
$141$	0	0
$142$	4.52043	0.379346
$143$	−9.18595	−0.768168
$144$	0	0
$145$	16.8045	1.39553
$146$	0.143119	0.0118446
$147$	0	0
$148$	4.87571	0.400781
$149$	−1.24912	−0.102332	−0.0511658	−	0.998690i	$-0.516294\pi$
$149$	−1.24912	−0.102332	−0.0511658	+	0.998690i	$0.516294\pi$
$150$	0	0
$151$	22.5520	1.83526	0.917629	−	0.397439i	$-0.130101\pi$
$151$	22.5520	1.83526	0.917629	+	0.397439i	$0.130101\pi$
$152$	−11.0954	−0.899959
$153$	0	0
$154$	1.37201	0.110560
$155$	−24.2691	−1.94934
$156$	0	0
$157$	−3.54482	−0.282908	−0.141454	−	0.989945i	$-0.545178\pi$
$157$	−3.54482	−0.282908	−0.141454	+	0.989945i	$0.545178\pi$
$158$	6.86980	0.546532
$159$	0	0
$160$	−16.2924	−1.28802
$161$	−0.469034	−0.0369651
$162$	0	0
$163$	−21.8432	−1.71090	−0.855448	−	0.517889i	$-0.826718\pi$
$163$	−21.8432	−1.71090	−0.855448	+	0.517889i	$0.826718\pi$
$164$	−17.9905	−1.40482
$165$	0	0
$166$	6.82868	0.530008
$167$	23.9080	1.85006	0.925029	−	0.379896i	$-0.124040\pi$
$167$	23.9080	1.85006	0.925029	+	0.379896i	$0.124040\pi$
$168$	0	0
$169$	5.34536	0.411181
$170$	−5.10567	−0.391587
$171$	0	0
$172$	4.31068	0.328686
$173$	−20.5551	−1.56277	−0.781386	−	0.624048i	$-0.785487\pi$
$173$	−20.5551	−1.56277	−0.781386	+	0.624048i	$0.785487\pi$
$174$	0	0
$175$	−3.20037	−0.241925
$176$	3.67158	0.276756
$177$	0	0
$178$	8.12828	0.609240
$179$	15.3641	1.14836	0.574182	−	0.818727i	$-0.305320\pi$
$179$	15.3641	1.14836	0.574182	+	0.818727i	$0.305320\pi$
$180$	0	0
$181$	−17.9925	−1.33737	−0.668687	−	0.743544i	$-0.733144\pi$
$181$	−17.9925	−1.33737	−0.668687	+	0.743544i	$0.733144\pi$
$182$	−2.74006	−0.203107
$183$	0	0
$184$	1.07742	0.0794288
$185$	−8.77717	−0.645310
$186$	0	0
$187$	5.97723	0.437098
$188$	−7.14738	−0.521276
$189$	0	0
$190$	8.84865	0.641948
$191$	−19.2035	−1.38952	−0.694758	−	0.719243i	$-0.744489\pi$
$191$	−19.2035	−1.38952	−0.694758	+	0.719243i	$0.744489\pi$
$192$	0	0
$193$	1.35160	0.0972900	0.0486450	−	0.998816i	$-0.484510\pi$
$193$	1.35160	0.0972900	0.0486450	+	0.998816i	$0.484510\pi$
$194$	6.97629	0.500869
$195$	0	0
$196$	−1.59074	−0.113625
$197$	0.603749	0.0430153	0.0215077	−	0.999769i	$-0.493153\pi$
$197$	0.603749	0.0430153	0.0215077	+	0.999769i	$0.493153\pi$
$198$	0	0
$199$	−3.01194	−0.213511	−0.106755	−	0.994285i	$-0.534046\pi$
$199$	−3.01194	−0.213511	−0.106755	+	0.994285i	$0.534046\pi$
$200$	7.35160	0.519837
$201$	0	0
$202$	−4.17358	−0.293652
$203$	−5.86824	−0.411870
$204$	0	0
$205$	32.3862	2.26195
$206$	8.04601	0.560592
$207$	0	0
$208$	−7.33255	−0.508421
$209$	−10.3591	−0.716557
$210$	0	0
$211$	−13.2682	−0.913419	−0.456709	−	0.889616i	$-0.650972\pi$
$211$	−13.2682	−0.913419	−0.456709	+	0.889616i	$0.650972\pi$
$212$	10.2885	0.706618
$213$	0	0
$214$	−4.17214	−0.285202
$215$	−7.76000	−0.529228
$216$	0	0
$217$	8.47495	0.575317
$218$	−7.17628	−0.486039
$219$	0	0
$220$	−9.76963	−0.658668
$221$	−11.9372	−0.802982
$222$	0	0
$223$	22.1491	1.48321	0.741606	−	0.670836i	$-0.234064\pi$
$223$	22.1491	1.48321	0.741606	+	0.670836i	$0.234064\pi$
$224$	5.68941	0.380140
$225$	0	0
$226$	−1.23595	−0.0822141
$227$	0.254160	0.0168692	0.00843460	−	0.999964i	$-0.497315\pi$
$227$	0.254160	0.0168692	0.00843460	+	0.999964i	$0.497315\pi$
$228$	0	0
$229$	2.55544	0.168869	0.0844343	−	0.996429i	$-0.473092\pi$
$229$	2.55544	0.168869	0.0844343	+	0.996429i	$0.473092\pi$
$230$	−0.859249	−0.0566572
$231$	0	0
$232$	13.4800	0.885005
$233$	13.3529	0.874775	0.437387	−	0.899273i	$-0.355904\pi$
$233$	13.3529	0.874775	0.437387	+	0.899273i	$0.355904\pi$
$234$	0	0
$235$	12.8666	0.839323
$236$	19.8437	1.29171
$237$	0	0
$238$	1.78294	0.115571
$239$	5.75955	0.372554	0.186277	−	0.982497i	$-0.440358\pi$
$239$	5.75955	0.372554	0.186277	+	0.982497i	$0.440358\pi$
$240$	0	0
$241$	−14.3999	−0.927582	−0.463791	−	0.885945i	$-0.653511\pi$
$241$	−14.3999	−0.927582	−0.463791	+	0.885945i	$0.653511\pi$
$242$	4.09452	0.263206
$243$	0	0
$244$	−10.9829	−0.703107
$245$	2.86363	0.182951
$246$	0	0
$247$	20.6883	1.31637
$248$	−19.4679	−1.23621
$249$	0	0
$250$	3.29684	0.208510
$251$	−16.6858	−1.05320	−0.526599	−	0.850114i	$-0.676533\pi$
$251$	−16.6858	−1.05320	−0.526599	+	0.850114i	$0.676533\pi$
$252$	0	0
$253$	1.00593	0.0632420
$254$	−0.639731	−0.0401403
$255$	0	0
$256$	−7.62265	−0.476416
$257$	10.1097	0.630629	0.315314	−	0.948987i	$-0.397890\pi$
$257$	10.1097	0.630629	0.315314	+	0.948987i	$0.397890\pi$
$258$	0	0
$259$	3.06505	0.190453
$260$	19.5110	1.21002
$261$	0	0
$262$	−4.71058	−0.291021
$263$	−27.7596	−1.71173	−0.855866	−	0.517198i	$-0.826975\pi$
$263$	−27.7596	−1.71173	−0.855866	+	0.517198i	$0.826975\pi$
$264$	0	0
$265$	−18.5212	−1.13775
$266$	−3.09001	−0.189461
$267$	0	0
$268$	−14.2800	−0.872289
$269$	9.31869	0.568170	0.284085	−	0.958799i	$-0.408310\pi$
$269$	9.31869	0.568170	0.284085	+	0.958799i	$0.408310\pi$
$270$	0	0
$271$	−25.8155	−1.56818	−0.784089	−	0.620649i	$-0.786869\pi$
$271$	−25.8155	−1.56818	−0.784089	+	0.620649i	$0.786869\pi$
$272$	4.77124	0.289299
$273$	0	0
$274$	−10.7459	−0.649185
$275$	6.86374	0.413899
$276$	0	0
$277$	30.1528	1.81171	0.905854	−	0.423590i	$-0.139230\pi$
$277$	30.1528	1.81171	0.905854	+	0.423590i	$0.139230\pi$
$278$	6.77696	0.406455
$279$	0	0
$280$	−6.57807	−0.393115
$281$	12.4617	0.743402	0.371701	−	0.928352i	$-0.378775\pi$
$281$	12.4617	0.743402	0.371701	+	0.928352i	$0.378775\pi$
$282$	0	0
$283$	−7.81660	−0.464649	−0.232324	−	0.972638i	$-0.574633\pi$
$283$	−7.81660	−0.464649	−0.232324	+	0.972638i	$0.574633\pi$
$284$	11.2404	0.666996
$285$	0	0
$286$	5.87654	0.347487
$287$	−11.3095	−0.667579
$288$	0	0
$289$	−9.23256	−0.543092
$290$	−10.7503	−0.631282
$291$	0	0
$292$	0.355876	0.0208261
$293$	9.34553	0.545972	0.272986	−	0.962018i	$-0.411989\pi$
$293$	9.34553	0.545972	0.272986	+	0.962018i	$0.411989\pi$
$294$	0	0
$295$	−35.7223	−2.07983
$296$	−7.04076	−0.409236
$297$	0	0
$298$	0.799099	0.0462906
$299$	−2.00894	−0.116180
$300$	0	0
$301$	2.70985	0.156193
$302$	−14.4272	−0.830194
$303$	0	0
$304$	−8.26903	−0.474262
$305$	19.7712	1.13210
$306$	0	0
$307$	−24.4605	−1.39603	−0.698017	−	0.716082i	$-0.745934\pi$
$307$	−24.4605	−1.39603	−0.698017	+	0.716082i	$0.745934\pi$
$308$	3.41163	0.194395
$309$	0	0
$310$	15.5257	0.881801
$311$	−4.64323	−0.263294	−0.131647	−	0.991297i	$-0.542026\pi$
$311$	−4.64323	−0.263294	−0.131647	+	0.991297i	$0.542026\pi$
$312$	0	0
$313$	9.52709	0.538503	0.269251	−	0.963070i	$-0.413224\pi$
$313$	9.52709	0.538503	0.269251	+	0.963070i	$0.413224\pi$
$314$	2.26773	0.127976
$315$	0	0
$316$	17.0823	0.960956
$317$	−19.7066	−1.10684	−0.553418	−	0.832904i	$-0.686677\pi$
$317$	−19.7066	−1.10684	−0.553418	+	0.832904i	$0.686677\pi$
$318$	0	0
$319$	12.5855	0.704650
$320$	0.617937	0.0345437
$321$	0	0
$322$	0.300056	0.0167215
$323$	−13.4617	−0.749032
$324$	0	0
$325$	−13.7077	−0.760364
$326$	13.9738	0.773938
$327$	0	0
$328$	25.9792	1.43446
$329$	−4.49311	−0.247713
$330$	0	0
$331$	−15.5325	−0.853741	−0.426870	−	0.904313i	$-0.640384\pi$
$331$	−15.5325	−0.853741	−0.426870	+	0.904313i	$0.640384\pi$
$332$	16.9801	0.931902
$333$	0	0
$334$	−15.2947	−0.836889
$335$	25.7066	1.40450
$336$	0	0
$337$	3.58228	0.195139	0.0975696	−	0.995229i	$-0.468893\pi$
$337$	3.58228	0.195139	0.0975696	+	0.995229i	$0.468893\pi$
$338$	−3.41959	−0.186001
$339$	0	0
$340$	−12.6957	−0.688520
$341$	−18.1760	−0.984285
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−6.22483	−0.335620
$345$	0	0
$346$	13.1497	0.706933
$347$	5.91645	0.317612	0.158806	−	0.987310i	$-0.449236\pi$
$347$	5.91645	0.317612	0.158806	+	0.987310i	$0.449236\pi$
$348$	0	0
$349$	9.88737	0.529259	0.264629	−	0.964350i	$-0.414750\pi$
$349$	9.88737	0.529259	0.264629	+	0.964350i	$0.414750\pi$
$350$	2.04738	0.109437
$351$	0	0
$352$	−12.2019	−0.650365
$353$	−23.3599	−1.24332	−0.621660	−	0.783287i	$-0.713541\pi$
$353$	−23.3599	−1.24332	−0.621660	+	0.783287i	$0.713541\pi$
$354$	0	0
$355$	−20.2348	−1.07395
$356$	20.2116	1.07121
$357$	0	0
$358$	−9.82888	−0.519472
$359$	11.0409	0.582715	0.291358	−	0.956614i	$-0.405893\pi$
$359$	11.0409	0.582715	0.291358	+	0.956614i	$0.405893\pi$
$360$	0	0
$361$	4.33057	0.227925
$362$	11.5104	0.604973
$363$	0	0
$364$	−6.81339	−0.357119
$365$	−0.640642	−0.0335327
$366$	0	0
$367$	−28.4635	−1.48578	−0.742891	−	0.669413i	$-0.766546\pi$
$367$	−28.4635	−1.48578	−0.742891	+	0.669413i	$0.766546\pi$
$368$	0.802965	0.0418575
$369$	0	0
$370$	5.61503	0.291911
$371$	6.46774	0.335788
$372$	0	0
$373$	−9.77013	−0.505878	−0.252939	−	0.967482i	$-0.581397\pi$
$373$	−9.77013	−0.505878	−0.252939	+	0.967482i	$0.581397\pi$
$374$	−3.82382	−0.197725
$375$	0	0
$376$	10.3212	0.532273
$377$	−25.1345	−1.29449
$378$	0	0
$379$	21.5057	1.10467	0.552336	−	0.833622i	$-0.313737\pi$
$379$	21.5057	1.10467	0.552336	+	0.833622i	$0.313737\pi$
$380$	22.0029	1.12872
$381$	0	0
$382$	12.2851	0.628559
$383$	32.8746	1.67981	0.839906	−	0.542731i	$-0.182610\pi$
$383$	32.8746	1.67981	0.839906	+	0.542731i	$0.182610\pi$
$384$	0	0
$385$	−6.14155	−0.313002
$386$	−0.864658	−0.0440099
$387$	0	0
$388$	17.3471	0.880666
$389$	−8.50558	−0.431250	−0.215625	−	0.976476i	$-0.569179\pi$
$389$	−8.50558	−0.431250	−0.215625	+	0.976476i	$0.569179\pi$
$390$	0	0
$391$	1.30720	0.0661082
$392$	2.29711	0.116022
$393$	0	0
$394$	−0.386237	−0.0194583
$395$	−30.7513	−1.54727
$396$	0	0
$397$	−0.556513	−0.0279306	−0.0139653	−	0.999902i	$-0.504445\pi$
$397$	−0.556513	−0.0279306	−0.0139653	+	0.999902i	$0.504445\pi$
$398$	1.92683	0.0965834
$399$	0	0
$400$	5.47888	0.273944
$401$	−1.78222	−0.0889998	−0.0444999	−	0.999009i	$-0.514169\pi$
$401$	−1.78222	−0.0889998	−0.0444999	+	0.999009i	$0.514169\pi$
$402$	0	0
$403$	36.2994	1.80820
$404$	−10.3780	−0.516322
$405$	0	0
$406$	3.75410	0.186313
$407$	−6.57353	−0.325838
$408$	0	0
$409$	−4.43621	−0.219356	−0.109678	−	0.993967i	$-0.534982\pi$
$409$	−4.43621	−0.219356	−0.109678	+	0.993967i	$0.534982\pi$
$410$	−20.7185	−1.02321
$411$	0	0
$412$	20.0071	0.985677
$413$	12.4745	0.613829
$414$	0	0
$415$	−30.5672	−1.50049
$416$	24.3686	1.19477
$417$	0	0
$418$	6.62707	0.324140
$419$	16.1364	0.788316	0.394158	−	0.919043i	$-0.371036\pi$
$419$	16.1364	0.788316	0.394158	+	0.919043i	$0.371036\pi$
$420$	0	0
$421$	−4.52202	−0.220390	−0.110195	−	0.993910i	$-0.535147\pi$
$421$	−4.52202	−0.220390	−0.110195	+	0.993910i	$0.535147\pi$
$422$	8.48807	0.413193
$423$	0	0
$424$	−14.8571	−0.721525
$425$	8.91946	0.432658
$426$	0	0
$427$	−6.90425	−0.334120
$428$	−10.3744	−0.501464
$429$	0	0
$430$	4.96432	0.239401
$431$	−20.9007	−1.00675	−0.503376	−	0.864068i	$-0.667909\pi$
$431$	−20.9007	−1.00675	−0.503376	+	0.864068i	$0.667909\pi$
$432$	0	0
$433$	−21.2043	−1.01902	−0.509508	−	0.860466i	$-0.670172\pi$
$433$	−21.2043	−1.01902	−0.509508	+	0.860466i	$0.670172\pi$
$434$	−5.42169	−0.260249
$435$	0	0
$436$	−17.8444	−0.854591
$437$	−2.26552	−0.108374
$438$	0	0
$439$	−20.7907	−0.992287	−0.496143	−	0.868241i	$-0.665251\pi$
$439$	−20.7907	−0.992287	−0.496143	+	0.868241i	$0.665251\pi$
$440$	14.1078	0.672564
$441$	0	0
$442$	7.63659	0.363236
$443$	−19.0438	−0.904800	−0.452400	−	0.891815i	$-0.649432\pi$
$443$	−19.0438	−0.904800	−0.452400	+	0.891815i	$0.649432\pi$
$444$	0	0
$445$	−36.3846	−1.72480
$446$	−14.1695	−0.670943
$447$	0	0
$448$	−0.215788	−0.0101950
$449$	26.2728	1.23989	0.619945	−	0.784646i	$-0.287155\pi$
$449$	26.2728	1.23989	0.619945	+	0.784646i	$0.287155\pi$
$450$	0	0
$451$	24.2552	1.14213
$452$	−3.07329	−0.144555
$453$	0	0
$454$	−0.162594	−0.00763093
$455$	12.2653	0.575008
$456$	0	0
$457$	−36.8047	−1.72165	−0.860826	−	0.508899i	$-0.830053\pi$
$457$	−36.8047	−1.72165	−0.860826	+	0.508899i	$0.830053\pi$
$458$	−1.63480	−0.0763891
$459$	0	0
$460$	−2.13659	−0.0996191
$461$	−25.3045	−1.17855	−0.589273	−	0.807934i	$-0.700586\pi$
$461$	−25.3045	−1.17855	−0.589273	+	0.807934i	$0.700586\pi$
$462$	0	0
$463$	19.5841	0.910149	0.455075	−	0.890453i	$-0.349613\pi$
$463$	19.5841	0.910149	0.455075	+	0.890453i	$0.349613\pi$
$464$	10.0462	0.466381
$465$	0	0
$466$	−8.54224	−0.395712
$467$	−14.1679	−0.655614	−0.327807	−	0.944745i	$-0.606309\pi$
$467$	−14.1679	−0.655614	−0.327807	+	0.944745i	$0.606309\pi$
$468$	0	0
$469$	−8.97692	−0.414516
$470$	−8.23116	−0.379675
$471$	0	0
$472$	−28.6553	−1.31897
$473$	−5.81174	−0.267224
$474$	0	0
$475$	−15.4583	−0.709277
$476$	4.43342	0.203206
$477$	0	0
$478$	−3.68456	−0.168528
$479$	13.0505	0.596292	0.298146	−	0.954520i	$-0.403632\pi$
$479$	13.0505	0.596292	0.298146	+	0.954520i	$0.403632\pi$
$480$	0	0
$481$	13.1281	0.598588
$482$	9.21210	0.419600
$483$	0	0
$484$	10.1814	0.462789
$485$	−31.2280	−1.41799
$486$	0	0
$487$	−35.8565	−1.62481	−0.812407	−	0.583091i	$-0.801843\pi$
$487$	−35.8565	−1.62481	−0.812407	+	0.583091i	$0.801843\pi$
$488$	15.8598	0.717940
$489$	0	0
$490$	−1.83195	−0.0827592
$491$	8.44943	0.381317	0.190659	−	0.981656i	$-0.438938\pi$
$491$	8.44943	0.381317	0.190659	+	0.981656i	$0.438938\pi$
$492$	0	0
$493$	16.3549	0.736586
$494$	−13.2350	−0.595470
$495$	0	0
$496$	−14.5087	−0.651461
$497$	7.06614	0.316960
$498$	0	0
$499$	17.8097	0.797272	0.398636	−	0.917109i	$-0.369484\pi$
$499$	17.8097	0.797272	0.398636	+	0.917109i	$0.369484\pi$
$500$	8.19786	0.366620
$501$	0	0
$502$	10.6744	0.476423
$503$	−26.4237	−1.17817	−0.589087	−	0.808070i	$-0.700512\pi$
$503$	−26.4237	−1.17817	−0.589087	+	0.808070i	$0.700512\pi$
$504$	0	0
$505$	18.6822	0.831347
$506$	−0.643522	−0.0286080
$507$	0	0
$508$	−1.59074	−0.0705778
$509$	22.4590	0.995477	0.497738	−	0.867327i	$-0.334164\pi$
$509$	22.4590	0.995477	0.497738	+	0.867327i	$0.334164\pi$
$510$	0	0
$511$	0.223717	0.00989665
$512$	−17.6051	−0.778043
$513$	0	0
$514$	−6.46752	−0.285270
$515$	−36.0163	−1.58707
$516$	0	0
$517$	9.63624	0.423801
$518$	−1.96081	−0.0861530
$519$	0	0
$520$	−28.1749	−1.23555
$521$	17.4966	0.766539	0.383269	−	0.923637i	$-0.374798\pi$
$521$	17.4966	0.766539	0.383269	+	0.923637i	$0.374798\pi$
$522$	0	0
$523$	23.9036	1.04523	0.522616	−	0.852568i	$-0.324956\pi$
$523$	23.9036	1.04523	0.522616	+	0.852568i	$0.324956\pi$
$524$	−11.7132	−0.511695
$525$	0	0
$526$	17.7587	0.774316
$527$	−23.6198	−1.02889
$528$	0	0
$529$	−22.7800	−0.990435
$530$	11.8486	0.514670
$531$	0	0
$532$	−7.68357	−0.333125
$533$	−48.4403	−2.09818
$534$	0	0
$535$	18.6757	0.807423
$536$	20.6210	0.890691
$537$	0	0
$538$	−5.96146	−0.257017
$539$	2.14467	0.0923776
$540$	0	0
$541$	−24.7352	−1.06345	−0.531724	−	0.846917i	$-0.678456\pi$
$541$	−24.7352	−1.06345	−0.531724	+	0.846917i	$0.678456\pi$
$542$	16.5150	0.709378
$543$	0	0
$544$	−15.8565	−0.679840
$545$	32.1232	1.37600
$546$	0	0
$547$	27.1964	1.16283	0.581416	−	0.813606i	$-0.302499\pi$
$547$	27.1964	1.16283	0.581416	+	0.813606i	$0.302499\pi$
$548$	−26.7206	−1.14145
$549$	0	0
$550$	−4.39095	−0.187231
$551$	−28.3446	−1.20752
$552$	0	0
$553$	10.7386	0.456651
$554$	−19.2897	−0.819541
$555$	0	0
$556$	16.8515	0.714661
$557$	−13.0921	−0.554731	−0.277366	−	0.960764i	$-0.589461\pi$
$557$	−13.0921	−0.554731	−0.277366	+	0.960764i	$0.589461\pi$
$558$	0	0
$559$	11.6067	0.490911
$560$	−4.90240	−0.207164
$561$	0	0
$562$	−7.97213	−0.336284
$563$	−44.9924	−1.89620	−0.948101	−	0.317969i	$-0.896999\pi$
$563$	−44.9924	−1.89620	−0.948101	+	0.317969i	$0.896999\pi$
$564$	0	0
$565$	5.53248	0.232753
$566$	5.00053	0.210188
$567$	0	0
$568$	−16.2317	−0.681067
$569$	−30.9356	−1.29689	−0.648445	−	0.761262i	$-0.724580\pi$
$569$	−30.9356	−1.29689	−0.648445	+	0.761262i	$0.724580\pi$
$570$	0	0
$571$	−24.1172	−1.00927	−0.504636	−	0.863332i	$-0.668373\pi$
$571$	−24.1172	−1.00927	−0.504636	+	0.863332i	$0.668373\pi$
$572$	14.6125	0.610979
$573$	0	0
$574$	7.23505	0.301985
$575$	1.50108	0.0625995
$576$	0	0
$577$	31.9611	1.33056	0.665279	−	0.746594i	$-0.268312\pi$
$577$	31.9611	1.33056	0.665279	+	0.746594i	$0.268312\pi$
$578$	5.90636	0.245672
$579$	0	0
$580$	−26.7316	−1.10997
$581$	10.6743	0.442844
$582$	0	0
$583$	−13.8712	−0.574486
$584$	−0.513903	−0.0212654
$585$	0	0
$586$	−5.97863	−0.246975
$587$	22.7895	0.940625	0.470312	−	0.882500i	$-0.344141\pi$
$587$	22.7895	0.940625	0.470312	+	0.882500i	$0.344141\pi$
$588$	0	0
$589$	40.9355	1.68672
$590$	22.8527	0.940829
$591$	0	0
$592$	−5.24722	−0.215660
$593$	35.5763	1.46094	0.730472	−	0.682943i	$-0.239300\pi$
$593$	35.5763	1.46094	0.730472	+	0.682943i	$0.239300\pi$
$594$	0	0
$595$	−7.98097	−0.327188
$596$	1.98702	0.0813917
$597$	0	0
$598$	1.28518	0.0525551
$599$	−15.9946	−0.653521	−0.326760	−	0.945107i	$-0.605957\pi$
$599$	−15.9946	−0.653521	−0.326760	+	0.945107i	$0.605957\pi$
$600$	0	0
$601$	−29.5924	−1.20710	−0.603550	−	0.797325i	$-0.706248\pi$
$601$	−29.5924	−1.20710	−0.603550	+	0.797325i	$0.706248\pi$
$602$	−1.73358	−0.0706553
$603$	0	0
$604$	−35.8745	−1.45971
$605$	−18.3283	−0.745151
$606$	0	0
$607$	−17.2875	−0.701680	−0.350840	−	0.936435i	$-0.614104\pi$
$607$	−17.2875	−0.701680	−0.350840	+	0.936435i	$0.614104\pi$
$608$	27.4808	1.11450
$609$	0	0
$610$	−12.6483	−0.512113
$611$	−19.2446	−0.778555
$612$	0	0
$613$	37.0179	1.49514	0.747570	−	0.664183i	$-0.231220\pi$
$613$	37.0179	1.49514	0.747570	+	0.664183i	$0.231220\pi$
$614$	15.6481	0.631507
$615$	0	0
$616$	−4.92655	−0.198496
$617$	−23.5474	−0.947982	−0.473991	−	0.880530i	$-0.657187\pi$
$617$	−23.5474	−0.947982	−0.473991	+	0.880530i	$0.657187\pi$
$618$	0	0
$619$	−25.2575	−1.01519	−0.507593	−	0.861597i	$-0.669465\pi$
$619$	−25.2575	−1.01519	−0.507593	+	0.861597i	$0.669465\pi$
$620$	38.6059	1.55045
$621$	0	0
$622$	2.97042	0.119103
$623$	12.7058	0.509046
$624$	0	0
$625$	−30.7595	−1.23038
$626$	−6.09477	−0.243596
$627$	0	0
$628$	5.63891	0.225017
$629$	−8.54233	−0.340605
$630$	0	0
$631$	−32.2937	−1.28559	−0.642796	−	0.766037i	$-0.722226\pi$
$631$	−32.2937	−1.28559	−0.642796	+	0.766037i	$0.722226\pi$
$632$	−24.6677	−0.981228
$633$	0	0
$634$	12.6070	0.500686
$635$	2.86363	0.113640
$636$	0	0
$637$	−4.28315	−0.169705
$638$	−8.05131	−0.318754
$639$	0	0
$640$	32.1894	1.27240
$641$	−30.6157	−1.20925	−0.604624	−	0.796511i	$-0.706677\pi$
$641$	−30.6157	−1.20925	−0.604624	+	0.796511i	$0.706677\pi$
$642$	0	0
$643$	−21.0555	−0.830349	−0.415174	−	0.909742i	$-0.636279\pi$
$643$	−21.0555	−0.830349	−0.415174	+	0.909742i	$0.636279\pi$
$644$	0.746114	0.0294010
$645$	0	0
$646$	8.61190	0.338831
$647$	32.1967	1.26578	0.632891	−	0.774241i	$-0.281868\pi$
$647$	32.1967	1.26578	0.632891	+	0.774241i	$0.281868\pi$
$648$	0	0
$649$	−26.7537	−1.05017
$650$	8.76921	0.343957
$651$	0	0
$652$	34.7470	1.36080
$653$	−13.7608	−0.538501	−0.269250	−	0.963070i	$-0.586776\pi$
$653$	−13.7608	−0.538501	−0.269250	+	0.963070i	$0.586776\pi$
$654$	0	0
$655$	21.0860	0.823897
$656$	19.3614	0.755934
$657$	0	0
$658$	2.87438	0.112055
$659$	21.4170	0.834289	0.417144	−	0.908840i	$-0.363031\pi$
$659$	21.4170	0.834289	0.417144	+	0.908840i	$0.363031\pi$
$660$	0	0
$661$	31.9622	1.24319	0.621593	−	0.783340i	$-0.286486\pi$
$661$	31.9622	1.24319	0.621593	+	0.783340i	$0.286486\pi$
$662$	9.93660	0.386197
$663$	0	0
$664$	−24.5200	−0.951562
$665$	13.8318	0.536375
$666$	0	0
$667$	2.75241	0.106574
$668$	−38.0315	−1.47148
$669$	0	0
$670$	−16.4453	−0.635337
$671$	14.8074	0.571632
$672$	0	0
$673$	6.51078	0.250972	0.125486	−	0.992095i	$-0.459951\pi$
$673$	6.51078	0.250972	0.125486	+	0.992095i	$0.459951\pi$
$674$	−2.29170	−0.0882728
$675$	0	0
$676$	−8.50309	−0.327042
$677$	−24.9712	−0.959721	−0.479861	−	0.877345i	$-0.659313\pi$
$677$	−24.9712	−0.959721	−0.479861	+	0.877345i	$0.659313\pi$
$678$	0	0
$679$	10.9050	0.418497
$680$	18.3332	0.703045
$681$	0	0
$682$	11.6277	0.445250
$683$	10.6280	0.406670	0.203335	−	0.979109i	$-0.434822\pi$
$683$	10.6280	0.406670	0.203335	+	0.979109i	$0.434822\pi$
$684$	0	0
$685$	48.1020	1.83788
$686$	0.639731	0.0244251
$687$	0	0
$688$	−4.63914	−0.176865
$689$	27.7023	1.05537
$690$	0	0
$691$	−13.1760	−0.501240	−0.250620	−	0.968086i	$-0.580635\pi$
$691$	−13.1760	−0.501240	−0.250620	+	0.968086i	$0.580635\pi$
$692$	32.6978	1.24298
$693$	0	0
$694$	−3.78494	−0.143674
$695$	−30.3357	−1.15070
$696$	0	0
$697$	31.5197	1.19390
$698$	−6.32526	−0.239415
$699$	0	0
$700$	5.09097	0.192420
$701$	−11.7251	−0.442850	−0.221425	−	0.975177i	$-0.571071\pi$
$701$	−11.7251	−0.442850	−0.221425	+	0.975177i	$0.571071\pi$
$702$	0	0
$703$	14.8047	0.558371
$704$	0.462795	0.0174422
$705$	0	0
$706$	14.9440	0.562426
$707$	−6.52396	−0.245359
$708$	0	0
$709$	−14.2681	−0.535849	−0.267924	−	0.963440i	$-0.586338\pi$
$709$	−14.2681	−0.535849	−0.267924	+	0.963440i	$0.586338\pi$
$710$	12.9448	0.485811
$711$	0	0
$712$	−29.1866	−1.09381
$713$	−3.97504	−0.148866
$714$	0	0
$715$	−26.3052	−0.983757
$716$	−24.4403	−0.913377
$717$	0	0
$718$	−7.06319	−0.263596
$719$	−34.3652	−1.28160	−0.640802	−	0.767706i	$-0.721398\pi$
$719$	−34.3652	−1.28160	−0.640802	+	0.767706i	$0.721398\pi$
$720$	0	0
$721$	12.5772	0.468398
$722$	−2.77040	−0.103104
$723$	0	0
$724$	28.6215	1.06371
$725$	18.7805	0.697491
$726$	0	0
$727$	−43.7123	−1.62120	−0.810600	−	0.585600i	$-0.800859\pi$
$727$	−43.7123	−1.62120	−0.810600	+	0.585600i	$0.800859\pi$
$728$	9.83887	0.364653
$729$	0	0
$730$	0.409839	0.0151688
$731$	−7.55238	−0.279335
$732$	0	0
$733$	12.6969	0.468972	0.234486	−	0.972119i	$-0.424659\pi$
$733$	12.6969	0.468972	0.234486	+	0.972119i	$0.424659\pi$
$734$	18.2090	0.672106
$735$	0	0
$736$	−2.66853	−0.0983633
$737$	19.2526	0.709177
$738$	0	0
$739$	33.5612	1.23457	0.617284	−	0.786741i	$-0.288233\pi$
$739$	33.5612	1.23457	0.617284	+	0.786741i	$0.288233\pi$
$740$	13.9622	0.513262
$741$	0	0
$742$	−4.13762	−0.151897
$743$	−8.70060	−0.319194	−0.159597	−	0.987182i	$-0.551019\pi$
$743$	−8.70060	−0.319194	−0.159597	+	0.987182i	$0.551019\pi$
$744$	0	0
$745$	−3.57701	−0.131051
$746$	6.25026	0.228838
$747$	0	0
$748$	−9.50824	−0.347656
$749$	−6.52171	−0.238298
$750$	0	0
$751$	−31.8711	−1.16299	−0.581496	−	0.813549i	$-0.697533\pi$
$751$	−31.8711	−1.16299	−0.581496	+	0.813549i	$0.697533\pi$
$752$	7.69199	0.280498
$753$	0	0
$754$	16.0794	0.585575
$755$	64.5806	2.35033
$756$	0	0
$757$	−37.7868	−1.37338	−0.686692	−	0.726948i	$-0.740938\pi$
$757$	−37.7868	−1.37338	−0.686692	+	0.726948i	$0.740938\pi$
$758$	−13.7578	−0.499707
$759$	0	0
$760$	−31.7732	−1.15254
$761$	0.671830	0.0243538	0.0121769	−	0.999926i	$-0.496124\pi$
$761$	0.671830	0.0243538	0.0121769	+	0.999926i	$0.496124\pi$
$762$	0	0
$763$	−11.2176	−0.406106
$764$	30.5479	1.10518
$765$	0	0
$766$	−21.0309	−0.759877
$767$	53.4300	1.92925
$768$	0	0
$769$	32.5165	1.17257	0.586287	−	0.810103i	$-0.300589\pi$
$769$	32.5165	1.17257	0.586287	+	0.810103i	$0.300589\pi$
$770$	3.92894	0.141589
$771$	0	0
$772$	−2.15004	−0.0773817
$773$	−46.1274	−1.65909	−0.829544	−	0.558441i	$-0.811400\pi$
$773$	−46.1274	−1.65909	−0.829544	+	0.558441i	$0.811400\pi$
$774$	0	0
$775$	−27.1229	−0.974285
$776$	−25.0501	−0.899245
$777$	0	0
$778$	5.44128	0.195080
$779$	−54.6269	−1.95721
$780$	0	0
$781$	−15.1546	−0.542273
$782$	−0.836260	−0.0299046
$783$	0	0
$784$	1.71195	0.0611412
$785$	−10.1511	−0.362307
$786$	0	0
$787$	15.8116	0.563624	0.281812	−	0.959470i	$-0.409065\pi$
$787$	15.8116	0.563624	0.281812	+	0.959470i	$0.409065\pi$
$788$	−0.960409	−0.0342132
$789$	0	0
$790$	19.6726	0.699918
$791$	−1.93198	−0.0686934
$792$	0	0
$793$	−29.5719	−1.05013
$794$	0.356019	0.0126346
$795$	0	0
$796$	4.79122	0.169820
$797$	31.9076	1.13023	0.565113	−	0.825013i	$-0.308832\pi$
$797$	31.9076	1.13023	0.565113	+	0.825013i	$0.308832\pi$
$798$	0	0
$799$	12.5223	0.443008
$800$	−18.2082	−0.643758
$801$	0	0
$802$	1.14014	0.0402598
$803$	−0.479800	−0.0169318
$804$	0	0
$805$	−1.34314	−0.0473395
$806$	−23.2219	−0.817956
$807$	0	0
$808$	14.9863	0.527215
$809$	−41.5302	−1.46013	−0.730063	−	0.683380i	$-0.760509\pi$
$809$	−41.5302	−1.46013	−0.730063	+	0.683380i	$0.760509\pi$
$810$	0	0
$811$	−41.3815	−1.45310	−0.726550	−	0.687113i	$-0.758878\pi$
$811$	−41.3815	−1.45310	−0.726550	+	0.687113i	$0.758878\pi$
$812$	9.33487	0.327590
$813$	0	0
$814$	4.20529	0.147395
$815$	−62.5510	−2.19106
$816$	0	0
$817$	13.0890	0.457928
$818$	2.83798	0.0992276
$819$	0	0
$820$	−51.5182	−1.79909
$821$	−20.8050	−0.726099	−0.363050	−	0.931770i	$-0.618264\pi$
$821$	−20.8050	−0.726099	−0.363050	+	0.931770i	$0.618264\pi$
$822$	0	0
$823$	−20.2012	−0.704168	−0.352084	−	0.935968i	$-0.614527\pi$
$823$	−20.2012	−0.704168	−0.352084	+	0.935968i	$0.614527\pi$
$824$	−28.8912	−1.00647
$825$	0	0
$826$	−7.98032	−0.277671
$827$	23.5294	0.818197	0.409098	−	0.912490i	$-0.365843\pi$
$827$	23.5294	0.818197	0.409098	+	0.912490i	$0.365843\pi$
$828$	0	0
$829$	26.4591	0.918962	0.459481	−	0.888188i	$-0.348035\pi$
$829$	26.4591	0.918962	0.459481	+	0.888188i	$0.348035\pi$
$830$	19.5548	0.678757
$831$	0	0
$832$	−0.924252	−0.0320427
$833$	2.78701	0.0965642
$834$	0	0
$835$	68.4637	2.36928
$836$	16.4787	0.569929
$837$	0	0
$838$	−10.3230	−0.356601
$839$	35.7496	1.23421	0.617107	−	0.786879i	$-0.288304\pi$
$839$	35.7496	1.23421	0.617107	+	0.786879i	$0.288304\pi$
$840$	0	0
$841$	5.43623	0.187456
$842$	2.89288	0.0996951
$843$	0	0
$844$	21.1063	0.726508
$845$	15.3071	0.526581
$846$	0	0
$847$	6.40038	0.219920
$848$	−11.0725	−0.380230
$849$	0	0
$850$	−5.70606	−0.195716
$851$	−1.43761	−0.0492808
$852$	0	0
$853$	0.126874	0.00434409	0.00217205	−	0.999998i	$-0.499309\pi$
$853$	0.126874	0.00434409	0.00217205	+	0.999998i	$0.499309\pi$
$854$	4.41686	0.151142
$855$	0	0
$856$	14.9811	0.512043
$857$	−16.1476	−0.551591	−0.275796	−	0.961216i	$-0.588941\pi$
$857$	−16.1476	−0.551591	−0.275796	+	0.961216i	$0.588941\pi$
$858$	0	0
$859$	−24.3559	−0.831011	−0.415506	−	0.909591i	$-0.636395\pi$
$859$	−24.3559	−0.831011	−0.415506	+	0.909591i	$0.636395\pi$
$860$	12.3442	0.420933
$861$	0	0
$862$	13.3708	0.455413
$863$	14.7584	0.502381	0.251191	−	0.967938i	$-0.419178\pi$
$863$	14.7584	0.502381	0.251191	+	0.967938i	$0.419178\pi$
$864$	0	0
$865$	−58.8620	−2.00137
$866$	13.5651	0.460960
$867$	0	0
$868$	−13.4815	−0.457591
$869$	−23.0307	−0.781264
$870$	0	0
$871$	−38.4495	−1.30281
$872$	25.7682	0.872620
$873$	0	0
$874$	1.44932	0.0490241
$875$	5.15348	0.174219
$876$	0	0
$877$	28.3397	0.956964	0.478482	−	0.878097i	$-0.341187\pi$
$877$	28.3397	0.956964	0.478482	+	0.878097i	$0.341187\pi$
$878$	13.3005	0.448869
$879$	0	0
$880$	10.5140	0.354428
$881$	8.26928	0.278599	0.139300	−	0.990250i	$-0.455515\pi$
$881$	8.26928	0.278599	0.139300	+	0.990250i	$0.455515\pi$
$882$	0	0
$883$	16.7295	0.562994	0.281497	−	0.959562i	$-0.409169\pi$
$883$	16.7295	0.562994	0.281497	+	0.959562i	$0.409169\pi$
$884$	18.9890	0.638669
$885$	0	0
$886$	12.1829	0.409294
$887$	−37.7803	−1.26854	−0.634269	−	0.773113i	$-0.718699\pi$
$887$	−37.7803	−1.26854	−0.634269	+	0.773113i	$0.718699\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	23.2764	0.780226
$891$	0	0
$892$	−35.2335	−1.17971
$893$	−21.7025	−0.726246
$894$	0	0
$895$	43.9970	1.47066
$896$	−11.2408	−0.375528
$897$	0	0
$898$	−16.8075	−0.560874
$899$	−49.7330	−1.65869
$900$	0	0
$901$	−18.0257	−0.600522
$902$	−15.5168	−0.516653
$903$	0	0
$904$	4.43798	0.147605
$905$	−51.5239	−1.71271
$906$	0	0
$907$	30.6436	1.01750	0.508751	−	0.860913i	$-0.330107\pi$
$907$	30.6436	1.01750	0.508751	+	0.860913i	$0.330107\pi$
$908$	−0.404304	−0.0134173
$909$	0	0
$910$	−7.84653	−0.260110
$911$	51.2704	1.69867	0.849333	−	0.527858i	$-0.177005\pi$
$911$	51.2704	1.69867	0.849333	+	0.527858i	$0.177005\pi$
$912$	0	0
$913$	−22.8929	−0.757643
$914$	23.5451	0.778804
$915$	0	0
$916$	−4.06506	−0.134313
$917$	−7.36337	−0.243160
$918$	0	0
$919$	−20.9534	−0.691189	−0.345594	−	0.938384i	$-0.612323\pi$
$919$	−20.9534	−0.691189	−0.345594	+	0.938384i	$0.612323\pi$
$920$	3.08534	0.101721
$921$	0	0
$922$	16.1881	0.533125
$923$	30.2653	0.996195
$924$	0	0
$925$	−9.80929	−0.322527
$926$	−12.5285	−0.411714
$927$	0	0
$928$	−33.3868	−1.09598
$929$	−48.2526	−1.58312	−0.791558	−	0.611094i	$-0.790730\pi$
$929$	−48.2526	−1.58312	−0.791558	+	0.611094i	$0.790730\pi$
$930$	0	0
$931$	−4.83017	−0.158303
$932$	−21.2410	−0.695771
$933$	0	0
$934$	9.06367	0.296572
$935$	17.1166	0.559772
$936$	0	0
$937$	20.3601	0.665135	0.332568	−	0.943079i	$-0.392085\pi$
$937$	20.3601	0.665135	0.332568	+	0.943079i	$0.392085\pi$
$938$	5.74282	0.187510
$939$	0	0
$940$	−20.4674	−0.667574
$941$	−51.4709	−1.67790	−0.838951	−	0.544207i	$-0.816831\pi$
$941$	−51.4709	−1.67790	−0.838951	+	0.544207i	$0.816831\pi$
$942$	0	0
$943$	5.30455	0.172740
$944$	−21.3557	−0.695070
$945$	0	0
$946$	3.71795	0.120881
$947$	18.0087	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$947$	18.0087	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$948$	0	0
$949$	0.958213	0.0311049
$950$	9.88918	0.320847
$951$	0	0
$952$	−6.40208	−0.207492
$953$	−8.95630	−0.290123	−0.145061	−	0.989423i	$-0.546338\pi$
$953$	−8.95630	−0.290123	−0.145061	+	0.989423i	$0.546338\pi$
$954$	0	0
$955$	−54.9917	−1.77949
$956$	−9.16196	−0.296319
$957$	0	0
$958$	−8.34881	−0.269738
$959$	−16.7976	−0.542422
$960$	0	0
$961$	40.8247	1.31693
$962$	−8.39843	−0.270776
$963$	0	0
$964$	22.9066	0.737773
$965$	3.87047	0.124595
$966$	0	0
$967$	41.1995	1.32489	0.662443	−	0.749112i	$-0.269520\pi$
$967$	41.1995	1.32489	0.662443	+	0.749112i	$0.269520\pi$
$968$	−14.7024	−0.472552
$969$	0	0
$970$	19.9775	0.641439
$971$	33.4073	1.07209	0.536046	−	0.844189i	$-0.319917\pi$
$971$	33.4073	1.07209	0.536046	+	0.844189i	$0.319917\pi$
$972$	0	0
$973$	10.5934	0.339610
$974$	22.9385	0.734998
$975$	0	0
$976$	11.8198	0.378341
$977$	−1.91159	−0.0611572	−0.0305786	−	0.999532i	$-0.509735\pi$
$977$	−1.91159	−0.0611572	−0.0305786	+	0.999532i	$0.509735\pi$
$978$	0	0
$979$	−27.2497	−0.870905
$980$	−4.55530	−0.145514
$981$	0	0
$982$	−5.40536	−0.172492
$983$	−0.369783	−0.0117943	−0.00589713	−	0.999983i	$-0.501877\pi$
$983$	−0.369783	−0.0117943	−0.00589713	+	0.999983i	$0.501877\pi$
$984$	0	0
$985$	1.72891	0.0550877
$986$	−10.4627	−0.333201
$987$	0	0
$988$	−32.9099	−1.04700
$989$	−1.27101	−0.0404159
$990$	0	0
$991$	−39.1171	−1.24260	−0.621298	−	0.783574i	$-0.713394\pi$
$991$	−39.1171	−1.24260	−0.621298	+	0.783574i	$0.713394\pi$
$992$	48.2175	1.53091
$993$	0	0
$994$	−4.52043	−0.143379
$995$	−8.62508	−0.273433
$996$	0	0
$997$	−20.0511	−0.635026	−0.317513	−	0.948254i	$-0.602848\pi$
$997$	−20.0511	−0.635026	−0.317513	+	0.948254i	$0.602848\pi$
$998$	−11.3934	−0.360652
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-0.639731 q^{2} -1.59074 q^{4} +2.86363 q^{5} -1.00000 q^{7} +2.29711 q^{8} +O(q^{10})\)	\(q-0.639731 q^{2} -1.59074 q^{4} +2.86363 q^{5} -1.00000 q^{7} +2.29711 q^{8} -1.83195 q^{10} +2.14467 q^{11} -4.28315 q^{13} +0.639731 q^{14} +1.71195 q^{16} +2.78701 q^{17} -4.83017 q^{19} -4.55530 q^{20} -1.37201 q^{22} +0.469034 q^{23} +3.20037 q^{25} +2.74006 q^{26} +1.59074 q^{28} +5.86824 q^{29} -8.47495 q^{31} -5.68941 q^{32} -1.78294 q^{34} -2.86363 q^{35} -3.06505 q^{37} +3.09001 q^{38} +6.57807 q^{40} +11.3095 q^{41} -2.70985 q^{43} -3.41163 q^{44} -0.300056 q^{46} +4.49311 q^{47} +1.00000 q^{49} -2.04738 q^{50} +6.81339 q^{52} -6.46774 q^{53} +6.14155 q^{55} -2.29711 q^{56} -3.75410 q^{58} -12.4745 q^{59} +6.90425 q^{61} +5.42169 q^{62} +0.215788 q^{64} -12.2653 q^{65} +8.97692 q^{67} -4.43342 q^{68} +1.83195 q^{70} -7.06614 q^{71} -0.223717 q^{73} +1.96081 q^{74} +7.68357 q^{76} -2.14467 q^{77} -10.7386 q^{79} +4.90240 q^{80} -7.23505 q^{82} -10.6743 q^{83} +7.98097 q^{85} +1.73358 q^{86} +4.92655 q^{88} -12.7058 q^{89} +4.28315 q^{91} -0.746114 q^{92} -2.87438 q^{94} -13.8318 q^{95} -10.9050 q^{97} -0.639731 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

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Database

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