LMFDB - Embedded newform 8001.2.a.r.1.7

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			1.7
Root			$1.09349$ 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)

$f(q)$	$=$	$q-1.09349 q^{2} -0.804288 q^{4} -1.42245 q^{5} -1.00000 q^{7} +3.06645 q^{8} +O(q^{10})$	\(q-1.09349 q^{2} -0.804288 q^{4} -1.42245 q^{5} -1.00000 q^{7} +3.06645 q^{8} +1.55543 q^{10} +4.17521 q^{11} +4.20386 q^{13} +1.09349 q^{14} -1.74454 q^{16} -5.66528 q^{17} +1.69710 q^{19} +1.14406 q^{20} -4.56553 q^{22} +0.943665 q^{23} -2.97663 q^{25} -4.59687 q^{26} +0.804288 q^{28} +7.23570 q^{29} -3.64045 q^{31} -4.22527 q^{32} +6.19491 q^{34} +1.42245 q^{35} -3.63729 q^{37} -1.85576 q^{38} -4.36188 q^{40} -11.0109 q^{41} +5.58412 q^{43} -3.35807 q^{44} -1.03188 q^{46} -8.66195 q^{47} +1.00000 q^{49} +3.25491 q^{50} -3.38112 q^{52} -6.51220 q^{53} -5.93903 q^{55} -3.06645 q^{56} -7.91213 q^{58} -3.57733 q^{59} +7.68429 q^{61} +3.98078 q^{62} +8.10936 q^{64} -5.97979 q^{65} +7.65604 q^{67} +4.55652 q^{68} -1.55543 q^{70} +6.76474 q^{71} -5.73850 q^{73} +3.97733 q^{74} -1.36496 q^{76} -4.17521 q^{77} +15.2530 q^{79} +2.48153 q^{80} +12.0402 q^{82} +0.799640 q^{83} +8.05859 q^{85} -6.10616 q^{86} +12.8031 q^{88} -16.6739 q^{89} -4.20386 q^{91} -0.758979 q^{92} +9.47172 q^{94} -2.41405 q^{95} -4.78664 q^{97} -1.09349 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.09349	−0.773211	−0.386606	−	0.922245i	$-0.626353\pi$
$2$	−1.09349	−0.773211	−0.386606	+	0.922245i	$0.626353\pi$
$3$	0	0
$3$	0	0
$4$	−0.804288	−0.402144
$5$	−1.42245	−0.636140	−0.318070	−	0.948067i	$-0.603035\pi$
$5$	−1.42245	−0.636140	−0.318070	+	0.948067i	$0.603035\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.06645	1.08415
$9$	0	0
$10$	1.55543	0.491870
$11$	4.17521	1.25887	0.629437	−	0.777052i	$-0.283286\pi$
$11$	4.17521	1.25887	0.629437	+	0.777052i	$0.283286\pi$
$12$	0	0
$13$	4.20386	1.16594	0.582971	−	0.812493i	$-0.301890\pi$
$13$	4.20386	1.16594	0.582971	+	0.812493i	$0.301890\pi$
$14$	1.09349	0.292246
$15$	0	0
$16$	−1.74454	−0.436136
$17$	−5.66528	−1.37403	−0.687017	−	0.726642i	$-0.741080\pi$
$17$	−5.66528	−1.37403	−0.687017	+	0.726642i	$0.741080\pi$
$18$	0	0
$19$	1.69710	0.389342	0.194671	−	0.980869i	$-0.437636\pi$
$19$	1.69710	0.389342	0.194671	+	0.980869i	$0.437636\pi$
$20$	1.14406	0.255820
$21$	0	0
$22$	−4.56553	−0.973375
$23$	0.943665	0.196768	0.0983839	−	0.995149i	$-0.468633\pi$
$23$	0.943665	0.196768	0.0983839	+	0.995149i	$0.468633\pi$
$24$	0	0
$25$	−2.97663	−0.595327
$26$	−4.59687	−0.901519
$27$	0	0
$28$	0.804288	0.151996
$29$	7.23570	1.34363	0.671817	−	0.740717i	$-0.265514\pi$
$29$	7.23570	1.34363	0.671817	+	0.740717i	$0.265514\pi$
$30$	0	0
$31$	−3.64045	−0.653844	−0.326922	−	0.945051i	$-0.606011\pi$
$31$	−3.64045	−0.653844	−0.326922	+	0.945051i	$0.606011\pi$
$32$	−4.22527	−0.746929
$33$	0	0
$34$	6.19491	1.06242
$35$	1.42245	0.240438
$36$	0	0
$37$	−3.63729	−0.597968	−0.298984	−	0.954258i	$-0.596648\pi$
$37$	−3.63729	−0.597968	−0.298984	+	0.954258i	$0.596648\pi$
$38$	−1.85576	−0.301044
$39$	0	0
$40$	−4.36188	−0.689673
$41$	−11.0109	−1.71961	−0.859804	−	0.510625i	$-0.829414\pi$
$41$	−11.0109	−1.71961	−0.859804	+	0.510625i	$0.829414\pi$
$42$	0	0
$43$	5.58412	0.851570	0.425785	−	0.904824i	$-0.359998\pi$
$43$	5.58412	0.851570	0.425785	+	0.904824i	$0.359998\pi$
$44$	−3.35807	−0.506249
$45$	0	0
$46$	−1.03188	−0.152143
$47$	−8.66195	−1.26348	−0.631738	−	0.775182i	$-0.717658\pi$
$47$	−8.66195	−1.26348	−0.631738	+	0.775182i	$0.717658\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.25491	0.460313
$51$	0	0
$52$	−3.38112	−0.468877
$53$	−6.51220	−0.894520	−0.447260	−	0.894404i	$-0.647600\pi$
$53$	−6.51220	−0.894520	−0.447260	+	0.894404i	$0.647600\pi$
$54$	0	0
$55$	−5.93903	−0.800819
$56$	−3.06645	−0.409772
$57$	0	0
$58$	−7.91213	−1.03891
$59$	−3.57733	−0.465728	−0.232864	−	0.972509i	$-0.574810\pi$
$59$	−3.57733	−0.465728	−0.232864	+	0.972509i	$0.574810\pi$
$60$	0	0
$61$	7.68429	0.983872	0.491936	−	0.870631i	$-0.336289\pi$
$61$	7.68429	0.983872	0.491936	+	0.870631i	$0.336289\pi$
$62$	3.98078	0.505560
$63$	0	0
$64$	8.10936	1.01367
$65$	−5.97979	−0.741702
$66$	0	0
$67$	7.65604	0.935334	0.467667	−	0.883905i	$-0.345095\pi$
$67$	7.65604	0.935334	0.467667	+	0.883905i	$0.345095\pi$
$68$	4.55652	0.552559
$69$	0	0
$70$	−1.55543	−0.185910
$71$	6.76474	0.802827	0.401414	−	0.915897i	$-0.368519\pi$
$71$	6.76474	0.802827	0.401414	+	0.915897i	$0.368519\pi$
$72$	0	0
$73$	−5.73850	−0.671640	−0.335820	−	0.941926i	$-0.609013\pi$
$73$	−5.73850	−0.671640	−0.335820	+	0.941926i	$0.609013\pi$
$74$	3.97733	0.462355
$75$	0	0
$76$	−1.36496	−0.156572
$77$	−4.17521	−0.475809
$78$	0	0
$79$	15.2530	1.71610	0.858051	−	0.513564i	$-0.171675\pi$
$79$	15.2530	1.71610	0.858051	+	0.513564i	$0.171675\pi$
$80$	2.48153	0.277443
$81$	0	0
$82$	12.0402	1.32962
$83$	0.799640	0.0877719	0.0438859	−	0.999037i	$-0.486026\pi$
$83$	0.799640	0.0877719	0.0438859	+	0.999037i	$0.486026\pi$
$84$	0	0
$85$	8.05859	0.874077
$86$	−6.10616	−0.658444
$87$	0	0
$88$	12.8031	1.36481
$89$	−16.6739	−1.76743	−0.883715	−	0.468026i	$-0.844965\pi$
$89$	−16.6739	−1.76743	−0.883715	+	0.468026i	$0.844965\pi$
$90$	0	0
$91$	−4.20386	−0.440685
$92$	−0.758979	−0.0791290
$93$	0	0
$94$	9.47172	0.976933
$95$	−2.41405	−0.247676
$96$	0	0
$97$	−4.78664	−0.486010	−0.243005	−	0.970025i	$-0.578133\pi$
$97$	−4.78664	−0.486010	−0.243005	+	0.970025i	$0.578133\pi$
$98$	−1.09349	−0.110459
$99$	0	0
$100$	2.39407	0.239407
$101$	−11.6560	−1.15981	−0.579905	−	0.814684i	$-0.696910\pi$
$101$	−11.6560	−1.15981	−0.579905	+	0.814684i	$0.696910\pi$
$102$	0	0
$103$	6.35802	0.626474	0.313237	−	0.949675i	$-0.398587\pi$
$103$	6.35802	0.626474	0.313237	+	0.949675i	$0.398587\pi$
$104$	12.8909	1.26406
$105$	0	0
$106$	7.12100	0.691653
$107$	14.0600	1.35923	0.679613	−	0.733570i	$-0.262148\pi$
$107$	14.0600	1.35923	0.679613	+	0.733570i	$0.262148\pi$
$108$	0	0
$109$	2.85604	0.273559	0.136780	−	0.990601i	$-0.456325\pi$
$109$	2.85604	0.273559	0.136780	+	0.990601i	$0.456325\pi$
$110$	6.49425	0.619202
$111$	0	0
$112$	1.74454	0.164844
$113$	−0.790426	−0.0743571	−0.0371785	−	0.999309i	$-0.511837\pi$
$113$	−0.790426	−0.0743571	−0.0371785	+	0.999309i	$0.511837\pi$
$114$	0	0
$115$	−1.34232	−0.125172
$116$	−5.81958	−0.540335
$117$	0	0
$118$	3.91176	0.360106
$119$	5.66528	0.519336
$120$	0	0
$121$	6.43238	0.584762
$122$	−8.40266	−0.760741
$123$	0	0
$124$	2.92797	0.262940
$125$	11.3464	1.01485
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−0.416937	−0.0368524
$129$	0	0
$130$	6.53882	0.573492
$131$	−13.3478	−1.16620	−0.583100	−	0.812400i	$-0.698160\pi$
$131$	−13.3478	−1.16620	−0.583100	+	0.812400i	$0.698160\pi$
$132$	0	0
$133$	−1.69710	−0.147158
$134$	−8.37177	−0.723211
$135$	0	0
$136$	−17.3723	−1.48966
$137$	−8.84264	−0.755478	−0.377739	−	0.925912i	$-0.623298\pi$
$137$	−8.84264	−0.755478	−0.377739	+	0.925912i	$0.623298\pi$
$138$	0	0
$139$	−5.60019	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$139$	−5.60019	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$140$	−1.14406	−0.0966908
$141$	0	0
$142$	−7.39715	−0.620755
$143$	17.5520	1.46777
$144$	0	0
$145$	−10.2924	−0.854739
$146$	6.27497	0.519320
$147$	0	0
$148$	2.92543	0.240469
$149$	−20.2950	−1.66263	−0.831317	−	0.555798i	$-0.812413\pi$
$149$	−20.2950	−1.66263	−0.831317	+	0.555798i	$0.812413\pi$
$150$	0	0
$151$	0.931160	0.0757768	0.0378884	−	0.999282i	$-0.487937\pi$
$151$	0.931160	0.0757768	0.0378884	+	0.999282i	$0.487937\pi$
$152$	5.20408	0.422107
$153$	0	0
$154$	4.56553	0.367901
$155$	5.17836	0.415936
$156$	0	0
$157$	19.1861	1.53122	0.765608	−	0.643307i	$-0.222438\pi$
$157$	19.1861	1.53122	0.765608	+	0.643307i	$0.222438\pi$
$158$	−16.6790	−1.32691
$159$	0	0
$160$	6.01023	0.475151
$161$	−0.943665	−0.0743712
$162$	0	0
$163$	−2.56236	−0.200700	−0.100350	−	0.994952i	$-0.531996\pi$
$163$	−2.56236	−0.200700	−0.100350	+	0.994952i	$0.531996\pi$
$164$	8.85590	0.691530
$165$	0	0
$166$	−0.874395	−0.0678662
$167$	23.1797	1.79370	0.896848	−	0.442339i	$-0.145851\pi$
$167$	23.1797	1.79370	0.896848	+	0.442339i	$0.145851\pi$
$168$	0	0
$169$	4.67246	0.359420
$170$	−8.81196	−0.675846
$171$	0	0
$172$	−4.49124	−0.342454
$173$	14.6190	1.11147	0.555733	−	0.831361i	$-0.312438\pi$
$173$	14.6190	1.11147	0.555733	+	0.831361i	$0.312438\pi$
$174$	0	0
$175$	2.97663	0.225012
$176$	−7.28384	−0.549040
$177$	0	0
$178$	18.2327	1.36660
$179$	−9.02380	−0.674471	−0.337235	−	0.941420i	$-0.609492\pi$
$179$	−9.02380	−0.674471	−0.337235	+	0.941420i	$0.609492\pi$
$180$	0	0
$181$	0.673016	0.0500249	0.0250124	−	0.999687i	$-0.492037\pi$
$181$	0.673016	0.0500249	0.0250124	+	0.999687i	$0.492037\pi$
$182$	4.59687	0.340742
$183$	0	0
$184$	2.89370	0.213327
$185$	5.17387	0.380391
$186$	0	0
$187$	−23.6538	−1.72973
$188$	6.96670	0.508099
$189$	0	0
$190$	2.63973	0.191506
$191$	12.3516	0.893731	0.446865	−	0.894601i	$-0.352540\pi$
$191$	12.3516	0.893731	0.446865	+	0.894601i	$0.352540\pi$
$192$	0	0
$193$	6.80443	0.489794	0.244897	−	0.969549i	$-0.421246\pi$
$193$	6.80443	0.489794	0.244897	+	0.969549i	$0.421246\pi$
$194$	5.23413	0.375788
$195$	0	0
$196$	−0.804288	−0.0574492
$197$	−3.78122	−0.269401	−0.134700	−	0.990886i	$-0.543007\pi$
$197$	−3.78122	−0.269401	−0.134700	+	0.990886i	$0.543007\pi$
$198$	0	0
$199$	0.390047	0.0276497	0.0138248	−	0.999904i	$-0.495599\pi$
$199$	0.390047	0.0276497	0.0138248	+	0.999904i	$0.495599\pi$
$200$	−9.12770	−0.645426
$201$	0	0
$202$	12.7456	0.896779
$203$	−7.23570	−0.507846
$204$	0	0
$205$	15.6624	1.09391
$206$	−6.95241	−0.484397
$207$	0	0
$208$	−7.33382	−0.508509
$209$	7.08577	0.490133
$210$	0	0
$211$	−4.79277	−0.329948	−0.164974	−	0.986298i	$-0.552754\pi$
$211$	−4.79277	−0.329948	−0.164974	+	0.986298i	$0.552754\pi$
$212$	5.23769	0.359726
$213$	0	0
$214$	−15.3744	−1.05097
$215$	−7.94314	−0.541717
$216$	0	0
$217$	3.64045	0.247130
$218$	−3.12304	−0.211519
$219$	0	0
$220$	4.77669	0.322045
$221$	−23.8161	−1.60204
$222$	0	0
$223$	−9.45604	−0.633223	−0.316612	−	0.948555i	$-0.602545\pi$
$223$	−9.45604	−0.633223	−0.316612	+	0.948555i	$0.602545\pi$
$224$	4.22527	0.282312
$225$	0	0
$226$	0.864320	0.0574937
$227$	−22.1260	−1.46856	−0.734279	−	0.678848i	$-0.762479\pi$
$227$	−22.1260	−1.46856	−0.734279	+	0.678848i	$0.762479\pi$
$228$	0	0
$229$	15.9686	1.05523	0.527617	−	0.849482i	$-0.323086\pi$
$229$	15.9686	1.05523	0.527617	+	0.849482i	$0.323086\pi$
$230$	1.46781	0.0967842
$231$	0	0
$232$	22.1879	1.45671
$233$	19.8988	1.30361	0.651806	−	0.758385i	$-0.274011\pi$
$233$	19.8988	1.30361	0.651806	+	0.758385i	$0.274011\pi$
$234$	0	0
$235$	12.3212	0.803746
$236$	2.87720	0.187290
$237$	0	0
$238$	−6.19491	−0.401556
$239$	10.3167	0.667332	0.333666	−	0.942691i	$-0.391714\pi$
$239$	10.3167	0.667332	0.333666	+	0.942691i	$0.391714\pi$
$240$	0	0
$241$	−12.9848	−0.836426	−0.418213	−	0.908349i	$-0.637343\pi$
$241$	−12.9848	−0.836426	−0.418213	+	0.908349i	$0.637343\pi$
$242$	−7.03372	−0.452145
$243$	0	0
$244$	−6.18038	−0.395659
$245$	−1.42245	−0.0908771
$246$	0	0
$247$	7.13439	0.453950
$248$	−11.1633	−0.708867
$249$	0	0
$250$	−12.4071	−0.784694
$251$	−13.1533	−0.830232	−0.415116	−	0.909769i	$-0.636259\pi$
$251$	−13.1533	−0.830232	−0.415116	+	0.909769i	$0.636259\pi$
$252$	0	0
$253$	3.94000	0.247706
$254$	−1.09349	−0.0686114
$255$	0	0
$256$	−15.7628	−0.985175
$257$	19.5929	1.22217	0.611086	−	0.791564i	$-0.290733\pi$
$257$	19.5929	1.22217	0.611086	+	0.791564i	$0.290733\pi$
$258$	0	0
$259$	3.63729	0.226010
$260$	4.80947	0.298271
$261$	0	0
$262$	14.5956	0.901719
$263$	21.4530	1.32285	0.661425	−	0.750011i	$-0.269952\pi$
$263$	21.4530	1.32285	0.661425	+	0.750011i	$0.269952\pi$
$264$	0	0
$265$	9.26329	0.569039
$266$	1.85576	0.113784
$267$	0	0
$268$	−6.15766	−0.376139
$269$	−1.74919	−0.106650	−0.0533250	−	0.998577i	$-0.516982\pi$
$269$	−1.74919	−0.106650	−0.0533250	+	0.998577i	$0.516982\pi$
$270$	0	0
$271$	8.31387	0.505032	0.252516	−	0.967593i	$-0.418742\pi$
$271$	8.31387	0.505032	0.252516	+	0.967593i	$0.418742\pi$
$272$	9.88334	0.599265
$273$	0	0
$274$	9.66931	0.584144
$275$	−12.4281	−0.749441
$276$	0	0
$277$	−9.40653	−0.565184	−0.282592	−	0.959240i	$-0.591194\pi$
$277$	−9.40653	−0.565184	−0.282592	+	0.959240i	$0.591194\pi$
$278$	6.12373	0.367277
$279$	0	0
$280$	4.36188	0.260672
$281$	−17.3862	−1.03717	−0.518586	−	0.855025i	$-0.673541\pi$
$281$	−17.3862	−1.03717	−0.518586	+	0.855025i	$0.673541\pi$
$282$	0	0
$283$	−4.46118	−0.265190	−0.132595	−	0.991170i	$-0.542331\pi$
$283$	−4.46118	−0.265190	−0.132595	+	0.991170i	$0.542331\pi$
$284$	−5.44080	−0.322852
$285$	0	0
$286$	−19.1929	−1.13490
$287$	11.0109	0.649950
$288$	0	0
$289$	15.0954	0.887968
$290$	11.2546	0.660894
$291$	0	0
$292$	4.61541	0.270096
$293$	−27.1348	−1.58523	−0.792616	−	0.609721i	$-0.791281\pi$
$293$	−27.1348	−1.58523	−0.792616	+	0.609721i	$0.791281\pi$
$294$	0	0
$295$	5.08857	0.296268
$296$	−11.1536	−0.648289
$297$	0	0
$298$	22.1923	1.28557
$299$	3.96704	0.229420
$300$	0	0
$301$	−5.58412	−0.321863
$302$	−1.01821	−0.0585915
$303$	0	0
$304$	−2.96067	−0.169806
$305$	−10.9305	−0.625880
$306$	0	0
$307$	−26.7679	−1.52773	−0.763863	−	0.645379i	$-0.776700\pi$
$307$	−26.7679	−1.52773	−0.763863	+	0.645379i	$0.776700\pi$
$308$	3.35807	0.191344
$309$	0	0
$310$	−5.66247	−0.321606
$311$	−16.6974	−0.946824	−0.473412	−	0.880841i	$-0.656978\pi$
$311$	−16.6974	−0.946824	−0.473412	+	0.880841i	$0.656978\pi$
$312$	0	0
$313$	3.29040	0.185984	0.0929922	−	0.995667i	$-0.470357\pi$
$313$	3.29040	0.185984	0.0929922	+	0.995667i	$0.470357\pi$
$314$	−20.9797	−1.18395
$315$	0	0
$316$	−12.2678	−0.690120
$317$	−27.7367	−1.55785	−0.778924	−	0.627118i	$-0.784234\pi$
$317$	−27.7367	−1.55785	−0.778924	+	0.627118i	$0.784234\pi$
$318$	0	0
$319$	30.2105	1.69147
$320$	−11.5352	−0.644835
$321$	0	0
$322$	1.03188	0.0575047
$323$	−9.61458	−0.534969
$324$	0	0
$325$	−12.5134	−0.694116
$326$	2.80191	0.155183
$327$	0	0
$328$	−33.7642	−1.86432
$329$	8.66195	0.477549
$330$	0	0
$331$	−21.6399	−1.18943	−0.594717	−	0.803935i	$-0.702736\pi$
$331$	−21.6399	−1.18943	−0.594717	+	0.803935i	$0.702736\pi$
$332$	−0.643141	−0.0352970
$333$	0	0
$334$	−25.3466	−1.38691
$335$	−10.8903	−0.595003
$336$	0	0
$337$	1.41521	0.0770914	0.0385457	−	0.999257i	$-0.487727\pi$
$337$	1.41521	0.0770914	0.0385457	+	0.999257i	$0.487727\pi$
$338$	−5.10927	−0.277908
$339$	0	0
$340$	−6.48143	−0.351505
$341$	−15.1996	−0.823107
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	17.1234	0.923233
$345$	0	0
$346$	−15.9857	−0.859398
$347$	8.12219	0.436022	0.218011	−	0.975946i	$-0.430043\pi$
$347$	8.12219	0.436022	0.218011	+	0.975946i	$0.430043\pi$
$348$	0	0
$349$	−20.4468	−1.09449	−0.547245	−	0.836973i	$-0.684323\pi$
$349$	−20.4468	−1.09449	−0.547245	+	0.836973i	$0.684323\pi$
$350$	−3.25491	−0.173982
$351$	0	0
$352$	−17.6414	−0.940288
$353$	−16.4225	−0.874083	−0.437041	−	0.899441i	$-0.643974\pi$
$353$	−16.4225	−0.874083	−0.437041	+	0.899441i	$0.643974\pi$
$354$	0	0
$355$	−9.62252	−0.510710
$356$	13.4106	0.710761
$357$	0	0
$358$	9.86740	0.521509
$359$	−27.7302	−1.46354	−0.731771	−	0.681551i	$-0.761306\pi$
$359$	−27.7302	−1.46354	−0.731771	+	0.681551i	$0.761306\pi$
$360$	0	0
$361$	−16.1198	−0.848413
$362$	−0.735934	−0.0386798
$363$	0	0
$364$	3.38112	0.177219
$365$	8.16273	0.427257
$366$	0	0
$367$	−29.1317	−1.52066	−0.760330	−	0.649537i	$-0.774963\pi$
$367$	−29.1317	−1.52066	−0.760330	+	0.649537i	$0.774963\pi$
$368$	−1.64627	−0.0858175
$369$	0	0
$370$	−5.65756	−0.294122
$371$	6.51220	0.338097
$372$	0	0
$373$	25.0145	1.29520	0.647600	−	0.761981i	$-0.275773\pi$
$373$	25.0145	1.29520	0.647600	+	0.761981i	$0.275773\pi$
$374$	25.8650	1.33745
$375$	0	0
$376$	−26.5614	−1.36980
$377$	30.4179	1.56660
$378$	0	0
$379$	−34.2723	−1.76045	−0.880225	−	0.474556i	$-0.842609\pi$
$379$	−34.2723	−1.76045	−0.880225	+	0.474556i	$0.842609\pi$
$380$	1.94159	0.0996015
$381$	0	0
$382$	−13.5063	−0.691043
$383$	−23.1775	−1.18432	−0.592158	−	0.805822i	$-0.701724\pi$
$383$	−23.1775	−1.18432	−0.592158	+	0.805822i	$0.701724\pi$
$384$	0	0
$385$	5.93903	0.302681
$386$	−7.44055	−0.378714
$387$	0	0
$388$	3.84984	0.195446
$389$	−24.4655	−1.24045	−0.620226	−	0.784423i	$-0.712959\pi$
$389$	−24.4655	−1.24045	−0.620226	+	0.784423i	$0.712959\pi$
$390$	0	0
$391$	−5.34613	−0.270366
$392$	3.06645	0.154879
$393$	0	0
$394$	4.13471	0.208304
$395$	−21.6967	−1.09168
$396$	0	0
$397$	−19.4737	−0.977355	−0.488677	−	0.872465i	$-0.662521\pi$
$397$	−19.4737	−0.977355	−0.488677	+	0.872465i	$0.662521\pi$
$398$	−0.426511	−0.0213791
$399$	0	0
$400$	5.19287	0.259643
$401$	−6.75197	−0.337177	−0.168589	−	0.985687i	$-0.553921\pi$
$401$	−6.75197	−0.337177	−0.168589	+	0.985687i	$0.553921\pi$
$402$	0	0
$403$	−15.3039	−0.762344
$404$	9.37475	0.466411
$405$	0	0
$406$	7.91213	0.392672
$407$	−15.1865	−0.752765
$408$	0	0
$409$	19.7435	0.976252	0.488126	−	0.872773i	$-0.337681\pi$
$409$	19.7435	0.976252	0.488126	+	0.872773i	$0.337681\pi$
$410$	−17.1266	−0.845824
$411$	0	0
$412$	−5.11368	−0.251933
$413$	3.57733	0.176029
$414$	0	0
$415$	−1.13745	−0.0558352
$416$	−17.7624	−0.870875
$417$	0	0
$418$	−7.74819	−0.378976
$419$	35.0809	1.71381	0.856907	−	0.515471i	$-0.172383\pi$
$419$	35.0809	1.71381	0.856907	+	0.515471i	$0.172383\pi$
$420$	0	0
$421$	−11.7230	−0.571344	−0.285672	−	0.958327i	$-0.592217\pi$
$421$	−11.7230	−0.571344	−0.285672	+	0.958327i	$0.592217\pi$
$422$	5.24082	0.255119
$423$	0	0
$424$	−19.9693	−0.969797
$425$	16.8635	0.817998
$426$	0	0
$427$	−7.68429	−0.371869
$428$	−11.3083	−0.546605
$429$	0	0
$430$	8.68571	0.418862
$431$	32.9486	1.58708	0.793539	−	0.608519i	$-0.208236\pi$
$431$	32.9486	1.58708	0.793539	+	0.608519i	$0.208236\pi$
$432$	0	0
$433$	3.71785	0.178668	0.0893341	−	0.996002i	$-0.471526\pi$
$433$	3.71785	0.178668	0.0893341	+	0.996002i	$0.471526\pi$
$434$	−3.98078	−0.191084
$435$	0	0
$436$	−2.29708	−0.110010
$437$	1.60150	0.0766100
$438$	0	0
$439$	37.1121	1.77126	0.885632	−	0.464387i	$-0.153725\pi$
$439$	37.1121	1.77126	0.885632	+	0.464387i	$0.153725\pi$
$440$	−18.2117	−0.868211
$441$	0	0
$442$	26.0425	1.23872
$443$	9.57128	0.454745	0.227373	−	0.973808i	$-0.426986\pi$
$443$	9.57128	0.454745	0.227373	+	0.973808i	$0.426986\pi$
$444$	0	0
$445$	23.7178	1.12433
$446$	10.3400	0.489616
$447$	0	0
$448$	−8.10936	−0.383131
$449$	−16.6303	−0.784834	−0.392417	−	0.919787i	$-0.628361\pi$
$449$	−16.6303	−0.784834	−0.392417	+	0.919787i	$0.628361\pi$
$450$	0	0
$451$	−45.9726	−2.16477
$452$	0.635731	0.0299023
$453$	0	0
$454$	24.1945	1.13551
$455$	5.97979	0.280337
$456$	0	0
$457$	4.32034	0.202097	0.101048	−	0.994882i	$-0.467780\pi$
$457$	4.32034	0.202097	0.101048	+	0.994882i	$0.467780\pi$
$458$	−17.4614	−0.815919
$459$	0	0
$460$	1.07961	0.0503371
$461$	13.5843	0.632685	0.316342	−	0.948645i	$-0.397545\pi$
$461$	13.5843	0.632685	0.316342	+	0.948645i	$0.397545\pi$
$462$	0	0
$463$	13.3015	0.618171	0.309086	−	0.951034i	$-0.399977\pi$
$463$	13.3015	0.618171	0.309086	+	0.951034i	$0.399977\pi$
$464$	−12.6230	−0.586007
$465$	0	0
$466$	−21.7590	−1.00797
$467$	0.341269	0.0157920	0.00789601	−	0.999969i	$-0.497487\pi$
$467$	0.341269	0.0157920	0.00789601	+	0.999969i	$0.497487\pi$
$468$	0	0
$469$	−7.65604	−0.353523
$470$	−13.4731	−0.621466
$471$	0	0
$472$	−10.9697	−0.504921
$473$	23.3149	1.07202
$474$	0	0
$475$	−5.05165	−0.231786
$476$	−4.55652	−0.208848
$477$	0	0
$478$	−11.2812	−0.515989
$479$	−28.6366	−1.30844	−0.654221	−	0.756304i	$-0.727003\pi$
$479$	−28.6366	−1.30844	−0.654221	+	0.756304i	$0.727003\pi$
$480$	0	0
$481$	−15.2907	−0.697195
$482$	14.1987	0.646734
$483$	0	0
$484$	−5.17349	−0.235159
$485$	6.80877	0.309170
$486$	0	0
$487$	−19.6800	−0.891784	−0.445892	−	0.895087i	$-0.647113\pi$
$487$	−19.6800	−0.891784	−0.445892	+	0.895087i	$0.647113\pi$
$488$	23.5635	1.06667
$489$	0	0
$490$	1.55543	0.0702672
$491$	8.04981	0.363283	0.181641	−	0.983365i	$-0.441859\pi$
$491$	8.04981	0.363283	0.181641	+	0.983365i	$0.441859\pi$
$492$	0	0
$493$	−40.9923	−1.84620
$494$	−7.80136	−0.351000
$495$	0	0
$496$	6.35092	0.285165
$497$	−6.76474	−0.303440
$498$	0	0
$499$	−29.0690	−1.30131	−0.650654	−	0.759375i	$-0.725505\pi$
$499$	−29.0690	−1.30131	−0.650654	+	0.759375i	$0.725505\pi$
$500$	−9.12575	−0.408116
$501$	0	0
$502$	14.3830	0.641944
$503$	−0.159747	−0.00712275	−0.00356137	−	0.999994i	$-0.501134\pi$
$503$	−0.159747	−0.00712275	−0.00356137	+	0.999994i	$0.501134\pi$
$504$	0	0
$505$	16.5800	0.737801
$506$	−4.30834	−0.191529
$507$	0	0
$508$	−0.804288	−0.0356845
$509$	35.8621	1.58956	0.794780	−	0.606898i	$-0.207586\pi$
$509$	35.8621	1.58956	0.794780	+	0.606898i	$0.207586\pi$
$510$	0	0
$511$	5.73850	0.253856
$512$	18.0703	0.798601
$513$	0	0
$514$	−21.4246	−0.944997
$515$	−9.04397	−0.398525
$516$	0	0
$517$	−36.1655	−1.59055
$518$	−3.97733	−0.174754
$519$	0	0
$520$	−18.3367	−0.804119
$521$	6.99437	0.306429	0.153214	−	0.988193i	$-0.451037\pi$
$521$	6.99437	0.306429	0.153214	+	0.988193i	$0.451037\pi$
$522$	0	0
$523$	16.4954	0.721293	0.360647	−	0.932703i	$-0.382556\pi$
$523$	16.4954	0.721293	0.360647	+	0.932703i	$0.382556\pi$
$524$	10.7355	0.468980
$525$	0	0
$526$	−23.4586	−1.02284
$527$	20.6242	0.898403
$528$	0	0
$529$	−22.1095	−0.961282
$530$	−10.1293	−0.439988
$531$	0	0
$532$	1.36496	0.0591786
$533$	−46.2881	−2.00496
$534$	0	0
$535$	−19.9996	−0.864658
$536$	23.4769	1.01405
$537$	0	0
$538$	1.91271	0.0824629
$539$	4.17521	0.179839
$540$	0	0
$541$	−29.9488	−1.28760	−0.643799	−	0.765195i	$-0.722643\pi$
$541$	−29.9488	−1.28760	−0.643799	+	0.765195i	$0.722643\pi$
$542$	−9.09111	−0.390496
$543$	0	0
$544$	23.9373	1.02630
$545$	−4.06258	−0.174022
$546$	0	0
$547$	29.2283	1.24971	0.624856	−	0.780740i	$-0.285158\pi$
$547$	29.2283	1.24971	0.624856	+	0.780740i	$0.285158\pi$
$548$	7.11203	0.303811
$549$	0	0
$550$	13.5899	0.579476
$551$	12.2797	0.523134
$552$	0	0
$553$	−15.2530	−0.648626
$554$	10.2859	0.437007
$555$	0	0
$556$	4.50417	0.191019
$557$	7.22344	0.306067	0.153033	−	0.988221i	$-0.451096\pi$
$557$	7.22344	0.306067	0.153033	+	0.988221i	$0.451096\pi$
$558$	0	0
$559$	23.4749	0.992881
$560$	−2.48153	−0.104864
$561$	0	0
$562$	19.0115	0.801953
$563$	−8.83211	−0.372229	−0.186115	−	0.982528i	$-0.559590\pi$
$563$	−8.83211	−0.372229	−0.186115	+	0.982528i	$0.559590\pi$
$564$	0	0
$565$	1.12434	0.0473015
$566$	4.87824	0.205048
$567$	0	0
$568$	20.7437	0.870388
$569$	18.4768	0.774587	0.387294	−	0.921956i	$-0.373410\pi$
$569$	18.4768	0.774587	0.387294	+	0.921956i	$0.373410\pi$
$570$	0	0
$571$	8.58231	0.359159	0.179579	−	0.983744i	$-0.442526\pi$
$571$	8.58231	0.359159	0.179579	+	0.983744i	$0.442526\pi$
$572$	−14.1169	−0.590256
$573$	0	0
$574$	−12.0402	−0.502549
$575$	−2.80894	−0.117141
$576$	0	0
$577$	−11.0503	−0.460028	−0.230014	−	0.973187i	$-0.573877\pi$
$577$	−11.0503	−0.460028	−0.230014	+	0.973187i	$0.573877\pi$
$578$	−16.5067	−0.686587
$579$	0	0
$580$	8.27808	0.343728
$581$	−0.799640	−0.0331747
$582$	0	0
$583$	−27.1898	−1.12609
$584$	−17.5968	−0.728161
$585$	0	0
$586$	29.6715	1.22572
$587$	−5.85881	−0.241819	−0.120910	−	0.992664i	$-0.538581\pi$
$587$	−5.85881	−0.241819	−0.120910	+	0.992664i	$0.538581\pi$
$588$	0	0
$589$	−6.17822	−0.254569
$590$	−5.56428	−0.229078
$591$	0	0
$592$	6.34542	0.260795
$593$	11.8786	0.487795	0.243898	−	0.969801i	$-0.421574\pi$
$593$	11.8786	0.487795	0.243898	+	0.969801i	$0.421574\pi$
$594$	0	0
$595$	−8.05859	−0.330370
$596$	16.3231	0.668619
$597$	0	0
$598$	−4.33790	−0.177390
$599$	19.8426	0.810748	0.405374	−	0.914151i	$-0.367141\pi$
$599$	19.8426	0.810748	0.405374	+	0.914151i	$0.367141\pi$
$600$	0	0
$601$	−11.0541	−0.450905	−0.225453	−	0.974254i	$-0.572386\pi$
$601$	−11.0541	−0.450905	−0.225453	+	0.974254i	$0.572386\pi$
$602$	6.10616	0.248868
$603$	0	0
$604$	−0.748921	−0.0304732
$605$	−9.14975	−0.371990
$606$	0	0
$607$	−0.738923	−0.0299920	−0.0149960	−	0.999888i	$-0.504774\pi$
$607$	−0.738923	−0.0299920	−0.0149960	+	0.999888i	$0.504774\pi$
$608$	−7.17071	−0.290811
$609$	0	0
$610$	11.9524	0.483938
$611$	−36.4136	−1.47314
$612$	0	0
$613$	−6.05094	−0.244395	−0.122198	−	0.992506i	$-0.538994\pi$
$613$	−6.05094	−0.244395	−0.122198	+	0.992506i	$0.538994\pi$
$614$	29.2703	1.18126
$615$	0	0
$616$	−12.8031	−0.515851
$617$	23.4533	0.944193	0.472096	−	0.881547i	$-0.343497\pi$
$617$	23.4533	0.944193	0.472096	+	0.881547i	$0.343497\pi$
$618$	0	0
$619$	37.3804	1.50244	0.751222	−	0.660050i	$-0.229465\pi$
$619$	37.3804	1.50244	0.751222	+	0.660050i	$0.229465\pi$
$620$	−4.16490	−0.167266
$621$	0	0
$622$	18.2584	0.732095
$623$	16.6739	0.668026
$624$	0	0
$625$	−1.25650	−0.0502599
$626$	−3.59801	−0.143805
$627$	0	0
$628$	−15.4311	−0.615770
$629$	20.6063	0.821627
$630$	0	0
$631$	−14.0387	−0.558872	−0.279436	−	0.960164i	$-0.590147\pi$
$631$	−14.0387	−0.558872	−0.279436	+	0.960164i	$0.590147\pi$
$632$	46.7727	1.86052
$633$	0	0
$634$	30.3297	1.20455
$635$	−1.42245	−0.0564483
$636$	0	0
$637$	4.20386	0.166563
$638$	−33.0348	−1.30786
$639$	0	0
$640$	0.593072	0.0234432
$641$	−31.1199	−1.22916	−0.614582	−	0.788853i	$-0.710675\pi$
$641$	−31.1199	−1.22916	−0.614582	+	0.788853i	$0.710675\pi$
$642$	0	0
$643$	11.0173	0.434481	0.217240	−	0.976118i	$-0.430294\pi$
$643$	11.0173	0.434481	0.217240	+	0.976118i	$0.430294\pi$
$644$	0.758979	0.0299080
$645$	0	0
$646$	10.5134	0.413644
$647$	20.5596	0.808281	0.404141	−	0.914697i	$-0.367571\pi$
$647$	20.5596	0.808281	0.404141	+	0.914697i	$0.367571\pi$
$648$	0	0
$649$	−14.9361	−0.586293
$650$	13.6832	0.536698
$651$	0	0
$652$	2.06088	0.0807102
$653$	−33.7921	−1.32239	−0.661195	−	0.750215i	$-0.729950\pi$
$653$	−33.7921	−1.32239	−0.661195	+	0.750215i	$0.729950\pi$
$654$	0	0
$655$	18.9865	0.741866
$656$	19.2089	0.749982
$657$	0	0
$658$	−9.47172	−0.369246
$659$	13.5115	0.526333	0.263166	−	0.964750i	$-0.415233\pi$
$659$	13.5115	0.526333	0.263166	+	0.964750i	$0.415233\pi$
$660$	0	0
$661$	−35.6025	−1.38478	−0.692389	−	0.721524i	$-0.743442\pi$
$661$	−35.6025	−1.38478	−0.692389	+	0.721524i	$0.743442\pi$
$662$	23.6629	0.919684
$663$	0	0
$664$	2.45206	0.0951582
$665$	2.41405	0.0936127
$666$	0	0
$667$	6.82807	0.264384
$668$	−18.6431	−0.721324
$669$	0	0
$670$	11.9084	0.460063
$671$	32.0835	1.23857
$672$	0	0
$673$	34.8182	1.34214	0.671071	−	0.741393i	$-0.265835\pi$
$673$	34.8182	1.34214	0.671071	+	0.741393i	$0.265835\pi$
$674$	−1.54751	−0.0596079
$675$	0	0
$676$	−3.75801	−0.144539
$677$	−46.2794	−1.77866	−0.889332	−	0.457262i	$-0.848830\pi$
$677$	−46.2794	−1.77866	−0.889332	+	0.457262i	$0.848830\pi$
$678$	0	0
$679$	4.78664	0.183694
$680$	24.7113	0.947634
$681$	0	0
$682$	16.6206	0.636435
$683$	−40.8139	−1.56170	−0.780851	−	0.624717i	$-0.785214\pi$
$683$	−40.8139	−1.56170	−0.780851	+	0.624717i	$0.785214\pi$
$684$	0	0
$685$	12.5782	0.480589
$686$	1.09349	0.0417495
$687$	0	0
$688$	−9.74174	−0.371400
$689$	−27.3764	−1.04296
$690$	0	0
$691$	−5.01351	−0.190723	−0.0953614	−	0.995443i	$-0.530401\pi$
$691$	−5.01351	−0.190723	−0.0953614	+	0.995443i	$0.530401\pi$
$692$	−11.7579	−0.446969
$693$	0	0
$694$	−8.88150	−0.337137
$695$	7.96600	0.302168
$696$	0	0
$697$	62.3796	2.36280
$698$	22.3582	0.846272
$699$	0	0
$700$	−2.39407	−0.0904874
$701$	−37.6558	−1.42224	−0.711121	−	0.703070i	$-0.751812\pi$
$701$	−37.6558	−1.42224	−0.711121	+	0.703070i	$0.751812\pi$
$702$	0	0
$703$	−6.17287	−0.232814
$704$	33.8583	1.27608
$705$	0	0
$706$	17.9578	0.675851
$707$	11.6560	0.438367
$708$	0	0
$709$	−35.1779	−1.32113	−0.660567	−	0.750767i	$-0.729684\pi$
$709$	−35.1779	−1.32113	−0.660567	+	0.750767i	$0.729684\pi$
$710$	10.5221	0.394887
$711$	0	0
$712$	−51.1297	−1.91617
$713$	−3.43537	−0.128655
$714$	0	0
$715$	−24.9669	−0.933708
$716$	7.25774	0.271234
$717$	0	0
$718$	30.3225	1.13163
$719$	−42.7083	−1.59275	−0.796375	−	0.604803i	$-0.793252\pi$
$719$	−42.7083	−1.59275	−0.796375	+	0.604803i	$0.793252\pi$
$720$	0	0
$721$	−6.35802	−0.236785
$722$	17.6268	0.656002
$723$	0	0
$724$	−0.541299	−0.0201172
$725$	−21.5380	−0.799901
$726$	0	0
$727$	13.3544	0.495286	0.247643	−	0.968851i	$-0.420344\pi$
$727$	13.3544	0.495286	0.247643	+	0.968851i	$0.420344\pi$
$728$	−12.8909	−0.477770
$729$	0	0
$730$	−8.92583	−0.330360
$731$	−31.6356	−1.17009
$732$	0	0
$733$	−35.1811	−1.29944	−0.649722	−	0.760172i	$-0.725115\pi$
$733$	−35.1811	−1.29944	−0.649722	+	0.760172i	$0.725115\pi$
$734$	31.8551	1.17579
$735$	0	0
$736$	−3.98724	−0.146971
$737$	31.9656	1.17747
$738$	0	0
$739$	34.7496	1.27828	0.639142	−	0.769089i	$-0.279290\pi$
$739$	34.7496	1.27828	0.639142	+	0.769089i	$0.279290\pi$
$740$	−4.16129	−0.152972
$741$	0	0
$742$	−7.12100	−0.261420
$743$	12.0839	0.443315	0.221658	−	0.975125i	$-0.428853\pi$
$743$	12.0839	0.443315	0.221658	+	0.975125i	$0.428853\pi$
$744$	0	0
$745$	28.8687	1.05767
$746$	−27.3530	−1.00146
$747$	0	0
$748$	19.0244	0.695602
$749$	−14.0600	−0.513739
$750$	0	0
$751$	−8.82113	−0.321888	−0.160944	−	0.986964i	$-0.551454\pi$
$751$	−8.82113	−0.321888	−0.160944	+	0.986964i	$0.551454\pi$
$752$	15.1111	0.551047
$753$	0	0
$754$	−33.2615	−1.21131
$755$	−1.32453	−0.0482046
$756$	0	0
$757$	−1.73023	−0.0628864	−0.0314432	−	0.999506i	$-0.510010\pi$
$757$	−1.73023	−0.0628864	−0.0314432	+	0.999506i	$0.510010\pi$
$758$	37.4763	1.36120
$759$	0	0
$760$	−7.40256	−0.268519
$761$	−5.09144	−0.184565	−0.0922824	−	0.995733i	$-0.529416\pi$
$761$	−5.09144	−0.184565	−0.0922824	+	0.995733i	$0.529416\pi$
$762$	0	0
$763$	−2.85604	−0.103396
$764$	−9.93425	−0.359409
$765$	0	0
$766$	25.3443	0.915727
$767$	−15.0386	−0.543012
$768$	0	0
$769$	−2.44337	−0.0881101	−0.0440550	−	0.999029i	$-0.514028\pi$
$769$	−2.44337	−0.0881101	−0.0440550	+	0.999029i	$0.514028\pi$
$770$	−6.49425	−0.234037
$771$	0	0
$772$	−5.47272	−0.196968
$773$	−48.5123	−1.74486	−0.872432	−	0.488735i	$-0.837459\pi$
$773$	−48.5123	−1.74486	−0.872432	+	0.488735i	$0.837459\pi$
$774$	0	0
$775$	10.8363	0.389251
$776$	−14.6780	−0.526910
$777$	0	0
$778$	26.7527	0.959131
$779$	−18.6866	−0.669516
$780$	0	0
$781$	28.2442	1.01066
$782$	5.84592	0.209050
$783$	0	0
$784$	−1.74454	−0.0623051
$785$	−27.2913	−0.974067
$786$	0	0
$787$	−3.21954	−0.114764	−0.0573821	−	0.998352i	$-0.518275\pi$
$787$	−3.21954	−0.114764	−0.0573821	+	0.998352i	$0.518275\pi$
$788$	3.04119	0.108338
$789$	0	0
$790$	23.7251	0.844100
$791$	0.790426	0.0281043
$792$	0	0
$793$	32.3037	1.14714
$794$	21.2942	0.755702
$795$	0	0
$796$	−0.313710	−0.0111192
$797$	−14.0731	−0.498494	−0.249247	−	0.968440i	$-0.580183\pi$
$797$	−14.0731	−0.498494	−0.249247	+	0.968440i	$0.580183\pi$
$798$	0	0
$799$	49.0724	1.73606
$800$	12.5771	0.444666
$801$	0	0
$802$	7.38318	0.260709
$803$	−23.9594	−0.845510
$804$	0	0
$805$	1.34232	0.0473105
$806$	16.7347	0.589453
$807$	0	0
$808$	−35.7424	−1.25741
$809$	35.3954	1.24443	0.622217	−	0.782845i	$-0.286232\pi$
$809$	35.3954	1.24443	0.622217	+	0.782845i	$0.286232\pi$
$810$	0	0
$811$	−0.107616	−0.00377891	−0.00188946	−	0.999998i	$-0.500601\pi$
$811$	−0.107616	−0.00377891	−0.00188946	+	0.999998i	$0.500601\pi$
$812$	5.81958	0.204227
$813$	0	0
$814$	16.6062	0.582047
$815$	3.64484	0.127673
$816$	0	0
$817$	9.47683	0.331552
$818$	−21.5892	−0.754849
$819$	0	0
$820$	−12.5971	−0.439910
$821$	28.3172	0.988279	0.494139	−	0.869383i	$-0.335483\pi$
$821$	28.3172	0.988279	0.494139	+	0.869383i	$0.335483\pi$
$822$	0	0
$823$	−15.4891	−0.539917	−0.269958	−	0.962872i	$-0.587010\pi$
$823$	−15.4891	−0.539917	−0.269958	+	0.962872i	$0.587010\pi$
$824$	19.4966	0.679195
$825$	0	0
$826$	−3.91176	−0.136107
$827$	−44.3930	−1.54370	−0.771848	−	0.635808i	$-0.780667\pi$
$827$	−44.3930	−1.54370	−0.771848	+	0.635808i	$0.780667\pi$
$828$	0	0
$829$	−51.3977	−1.78512	−0.892559	−	0.450931i	$-0.851092\pi$
$829$	−51.3977	−1.78512	−0.892559	+	0.450931i	$0.851092\pi$
$830$	1.24378	0.0431724
$831$	0	0
$832$	34.0906	1.18188
$833$	−5.66528	−0.196290
$834$	0	0
$835$	−32.9719	−1.14104
$836$	−5.69900	−0.197104
$837$	0	0
$838$	−38.3605	−1.32514
$839$	−2.45582	−0.0847842	−0.0423921	−	0.999101i	$-0.513498\pi$
$839$	−2.45582	−0.0847842	−0.0423921	+	0.999101i	$0.513498\pi$
$840$	0	0
$841$	23.3553	0.805355
$842$	12.8189	0.441770
$843$	0	0
$844$	3.85477	0.132687
$845$	−6.64635	−0.228641
$846$	0	0
$847$	−6.43238	−0.221019
$848$	11.3608	0.390132
$849$	0	0
$850$	−18.4400	−0.632486
$851$	−3.43239	−0.117661
$852$	0	0
$853$	3.93264	0.134651	0.0673256	−	0.997731i	$-0.478553\pi$
$853$	3.93264	0.134651	0.0673256	+	0.997731i	$0.478553\pi$
$854$	8.40266	0.287533
$855$	0	0
$856$	43.1141	1.47361
$857$	26.4657	0.904050	0.452025	−	0.892005i	$-0.350702\pi$
$857$	26.4657	0.904050	0.452025	+	0.892005i	$0.350702\pi$
$858$	0	0
$859$	41.0636	1.40107	0.700535	−	0.713618i	$-0.252945\pi$
$859$	41.0636	1.40107	0.700535	+	0.713618i	$0.252945\pi$
$860$	6.38857	0.217848
$861$	0	0
$862$	−36.0288	−1.22715
$863$	−48.4599	−1.64960	−0.824798	−	0.565428i	$-0.808711\pi$
$863$	−48.4599	−1.64960	−0.824798	+	0.565428i	$0.808711\pi$
$864$	0	0
$865$	−20.7949	−0.707047
$866$	−4.06541	−0.138148
$867$	0	0
$868$	−2.92797	−0.0993818
$869$	63.6847	2.16036
$870$	0	0
$871$	32.1849	1.09054
$872$	8.75792	0.296581
$873$	0	0
$874$	−1.75122	−0.0592357
$875$	−11.3464	−0.383577
$876$	0	0
$877$	−24.0677	−0.812710	−0.406355	−	0.913715i	$-0.633200\pi$
$877$	−24.0677	−0.812710	−0.406355	+	0.913715i	$0.633200\pi$
$878$	−40.5816	−1.36956
$879$	0	0
$880$	10.3609	0.349266
$881$	−47.7061	−1.60726	−0.803629	−	0.595131i	$-0.797100\pi$
$881$	−47.7061	−1.60726	−0.803629	+	0.595131i	$0.797100\pi$
$882$	0	0
$883$	41.0881	1.38273	0.691363	−	0.722508i	$-0.257011\pi$
$883$	41.0881	1.38273	0.691363	+	0.722508i	$0.257011\pi$
$884$	19.1550	0.644252
$885$	0	0
$886$	−10.4661	−0.351614
$887$	−3.77105	−0.126619	−0.0633097	−	0.997994i	$-0.520166\pi$
$887$	−3.77105	−0.126619	−0.0633097	+	0.997994i	$0.520166\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−25.9351	−0.869346
$891$	0	0
$892$	7.60538	0.254647
$893$	−14.7002	−0.491924
$894$	0	0
$895$	12.8359	0.429058
$896$	0.416937	0.0139289
$897$	0	0
$898$	18.1850	0.606843
$899$	−26.3412	−0.878528
$900$	0	0
$901$	36.8935	1.22910
$902$	50.2704	1.67382
$903$	0	0
$904$	−2.42380	−0.0806145
$905$	−0.957333	−0.0318228
$906$	0	0
$907$	56.4235	1.87351	0.936755	−	0.349985i	$-0.113813\pi$
$907$	56.4235	1.87351	0.936755	+	0.349985i	$0.113813\pi$
$908$	17.7957	0.590572
$909$	0	0
$910$	−6.53882	−0.216760
$911$	−50.5017	−1.67320	−0.836598	−	0.547818i	$-0.815459\pi$
$911$	−50.5017	−1.67320	−0.836598	+	0.547818i	$0.815459\pi$
$912$	0	0
$913$	3.33867	0.110494
$914$	−4.72423	−0.156264
$915$	0	0
$916$	−12.8433	−0.424356
$917$	13.3478	0.440782
$918$	0	0
$919$	−33.2800	−1.09780	−0.548902	−	0.835887i	$-0.684954\pi$
$919$	−33.2800	−1.09780	−0.548902	+	0.835887i	$0.684954\pi$
$920$	−4.11615	−0.135705
$921$	0	0
$922$	−14.8543	−0.489199
$923$	28.4380	0.936050
$924$	0	0
$925$	10.8269	0.355986
$926$	−14.5450	−0.477977
$927$	0	0
$928$	−30.5727	−1.00360
$929$	−0.685368	−0.0224862	−0.0112431	−	0.999937i	$-0.503579\pi$
$929$	−0.685368	−0.0224862	−0.0112431	+	0.999937i	$0.503579\pi$
$930$	0	0
$931$	1.69710	0.0556203
$932$	−16.0044	−0.524240
$933$	0	0
$934$	−0.373172	−0.0122106
$935$	33.6463	1.10035
$936$	0	0
$937$	−27.9004	−0.911466	−0.455733	−	0.890116i	$-0.650623\pi$
$937$	−27.9004	−0.911466	−0.455733	+	0.890116i	$0.650623\pi$
$938$	8.37177	0.273348
$939$	0	0
$940$	−9.90980	−0.323222
$941$	28.6611	0.934324	0.467162	−	0.884172i	$-0.345276\pi$
$941$	28.6611	0.934324	0.467162	+	0.884172i	$0.345276\pi$
$942$	0	0
$943$	−10.3906	−0.338363
$944$	6.24080	0.203121
$945$	0	0
$946$	−25.4945	−0.828897
$947$	−1.99501	−0.0648292	−0.0324146	−	0.999475i	$-0.510320\pi$
$947$	−1.99501	−0.0648292	−0.0324146	+	0.999475i	$0.510320\pi$
$948$	0	0
$949$	−24.1238	−0.783093
$950$	5.52391	0.179219
$951$	0	0
$952$	17.3723	0.563040
$953$	−12.9815	−0.420512	−0.210256	−	0.977646i	$-0.567430\pi$
$953$	−12.9815	−0.420512	−0.210256	+	0.977646i	$0.567430\pi$
$954$	0	0
$955$	−17.5696	−0.568538
$956$	−8.29760	−0.268364
$957$	0	0
$958$	31.3138	1.01170
$959$	8.84264	0.285544
$960$	0	0
$961$	−17.7471	−0.572488
$962$	16.7202	0.539079
$963$	0	0
$964$	10.4435	0.336364
$965$	−9.67897	−0.311577
$966$	0	0
$967$	−55.2360	−1.77627	−0.888135	−	0.459583i	$-0.847999\pi$
$967$	−55.2360	−1.77627	−0.888135	+	0.459583i	$0.847999\pi$
$968$	19.7246	0.633972
$969$	0	0
$970$	−7.44529	−0.239054
$971$	−49.2465	−1.58039	−0.790197	−	0.612852i	$-0.790022\pi$
$971$	−49.2465	−1.58039	−0.790197	+	0.612852i	$0.790022\pi$
$972$	0	0
$973$	5.60019	0.179534
$974$	21.5198	0.689537
$975$	0	0
$976$	−13.4056	−0.429102
$977$	15.3223	0.490202	0.245101	−	0.969497i	$-0.421179\pi$
$977$	15.3223	0.490202	0.245101	+	0.969497i	$0.421179\pi$
$978$	0	0
$979$	−69.6170	−2.22497
$980$	1.14406	0.0365457
$981$	0	0
$982$	−8.80236	−0.280894
$983$	−32.0313	−1.02164	−0.510821	−	0.859687i	$-0.670658\pi$
$983$	−32.0313	−1.02164	−0.510821	+	0.859687i	$0.670658\pi$
$984$	0	0
$985$	5.37860	0.171377
$986$	44.8245	1.42750
$987$	0	0
$988$	−5.73811	−0.182554
$989$	5.26954	0.167562
$990$	0	0
$991$	−18.7472	−0.595524	−0.297762	−	0.954640i	$-0.596240\pi$
$991$	−18.7472	−0.595524	−0.297762	+	0.954640i	$0.596240\pi$
$992$	15.3819	0.488375
$993$	0	0
$994$	7.39715	0.234623
$995$	−0.554823	−0.0175891
$996$	0	0
$997$	30.1185	0.953864	0.476932	−	0.878940i	$-0.341749\pi$
$997$	30.1185	0.953864	0.476932	+	0.878940i	$0.341749\pi$
$998$	31.7865	1.00619
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.r.1.7		16
3.2	odd	2		2667.2.a.o.1.10	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.o.1.10	✓	16	3.2	odd	2
8001.2.a.r.1.7		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)

\(f(q)\)	\(=\)	\(q-1.09349 q^{2} -0.804288 q^{4} -1.42245 q^{5} -1.00000 q^{7} +3.06645 q^{8} +O(q^{10})\)	\(q-1.09349 q^{2} -0.804288 q^{4} -1.42245 q^{5} -1.00000 q^{7} +3.06645 q^{8} +1.55543 q^{10} +4.17521 q^{11} +4.20386 q^{13} +1.09349 q^{14} -1.74454 q^{16} -5.66528 q^{17} +1.69710 q^{19} +1.14406 q^{20} -4.56553 q^{22} +0.943665 q^{23} -2.97663 q^{25} -4.59687 q^{26} +0.804288 q^{28} +7.23570 q^{29} -3.64045 q^{31} -4.22527 q^{32} +6.19491 q^{34} +1.42245 q^{35} -3.63729 q^{37} -1.85576 q^{38} -4.36188 q^{40} -11.0109 q^{41} +5.58412 q^{43} -3.35807 q^{44} -1.03188 q^{46} -8.66195 q^{47} +1.00000 q^{49} +3.25491 q^{50} -3.38112 q^{52} -6.51220 q^{53} -5.93903 q^{55} -3.06645 q^{56} -7.91213 q^{58} -3.57733 q^{59} +7.68429 q^{61} +3.98078 q^{62} +8.10936 q^{64} -5.97979 q^{65} +7.65604 q^{67} +4.55652 q^{68} -1.55543 q^{70} +6.76474 q^{71} -5.73850 q^{73} +3.97733 q^{74} -1.36496 q^{76} -4.17521 q^{77} +15.2530 q^{79} +2.48153 q^{80} +12.0402 q^{82} +0.799640 q^{83} +8.05859 q^{85} -6.10616 q^{86} +12.8031 q^{88} -16.6739 q^{89} -4.20386 q^{91} -0.758979 q^{92} +9.47172 q^{94} -2.41405 q^{95} -4.78664 q^{97} -1.09349 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) \) 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

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Database

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