LMFDB - Embedded newform 2169.2.a.h.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2169,2,Mod(1,2169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2169, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2169 = 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2169.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:	$17.3195521984$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} + \cdots - 1 $ x^12 - 3x^11 - 14x^10 + 44x^9 + 65x^8 - 219x^7 - 123x^6 + 444x^5 + 105x^4 - 328x^3 - 45x^2 + 18*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.54879$ of defining polynomial
Character	$\chi$	$=$	2169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.54879 q^{2} +0.398765 q^{4} -0.334961 q^{5} -4.24623 q^{7} +2.47998 q^{8} +O(q^{10})$	\(q-1.54879 q^{2} +0.398765 q^{4} -0.334961 q^{5} -4.24623 q^{7} +2.47998 q^{8} +0.518786 q^{10} -0.915418 q^{11} +4.81392 q^{13} +6.57654 q^{14} -4.63852 q^{16} +5.38915 q^{17} -4.34799 q^{19} -0.133571 q^{20} +1.41779 q^{22} -8.10534 q^{23} -4.88780 q^{25} -7.45578 q^{26} -1.69325 q^{28} +6.45221 q^{29} +10.7804 q^{31} +2.22414 q^{32} -8.34668 q^{34} +1.42232 q^{35} +5.16908 q^{37} +6.73415 q^{38} -0.830699 q^{40} +0.612344 q^{41} -1.85213 q^{43} -0.365036 q^{44} +12.5535 q^{46} +2.21116 q^{47} +11.0305 q^{49} +7.57020 q^{50} +1.91962 q^{52} -0.00846193 q^{53} +0.306629 q^{55} -10.5306 q^{56} -9.99315 q^{58} -8.85799 q^{59} +3.78644 q^{61} -16.6967 q^{62} +5.83230 q^{64} -1.61248 q^{65} +4.67684 q^{67} +2.14900 q^{68} -2.20289 q^{70} -1.48694 q^{71} -10.3630 q^{73} -8.00585 q^{74} -1.73383 q^{76} +3.88708 q^{77} -17.6786 q^{79} +1.55372 q^{80} -0.948396 q^{82} -7.45731 q^{83} -1.80516 q^{85} +2.86857 q^{86} -2.27022 q^{88} +0.520713 q^{89} -20.4410 q^{91} -3.23212 q^{92} -3.42463 q^{94} +1.45641 q^{95} -6.33494 q^{97} -17.0840 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8} - 7 q^{10} - 22 q^{11} - 5 q^{13} - 6 q^{14} + 15 q^{16} + 4 q^{17} - 6 q^{19} - 10 q^{20} - 12 q^{22} - 32 q^{23} + 4 q^{25} - 8 q^{26} - 11 q^{28} - 6 q^{29} + 8 q^{31} - q^{32} - 19 q^{34} - 15 q^{35} - 8 q^{37} + 10 q^{38} - 52 q^{40} + q^{41} - 2 q^{43} - 42 q^{44} - 25 q^{46} - 34 q^{47} - 9 q^{49} + 27 q^{50} - 41 q^{52} - 5 q^{53} - 3 q^{55} - q^{56} - 33 q^{58} - 26 q^{59} - 26 q^{61} + 17 q^{62} + 13 q^{64} + 25 q^{65} + 6 q^{67} + 35 q^{68} - 4 q^{70} - 94 q^{71} - 22 q^{73} - 26 q^{74} - 20 q^{76} + 7 q^{77} + 9 q^{79} - 19 q^{80} + 15 q^{82} + 8 q^{83} + 4 q^{85} - 9 q^{86} + 6 q^{88} + 3 q^{89} - 20 q^{91} - 36 q^{92} + 48 q^{94} - 33 q^{95} - 29 q^{97} - 28 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8 - 7 * q^10 - 22 * q^11 - 5 * q^13 - 6 * q^14 + 15 * q^16 + 4 * q^17 - 6 * q^19 - 10 * q^20 - 12 * q^22 - 32 * q^23 + 4 * q^25 - 8 * q^26 - 11 * q^28 - 6 * q^29 + 8 * q^31 - q^32 - 19 * q^34 - 15 * q^35 - 8 * q^37 + 10 * q^38 - 52 * q^40 + q^41 - 2 * q^43 - 42 * q^44 - 25 * q^46 - 34 * q^47 - 9 * q^49 + 27 * q^50 - 41 * q^52 - 5 * q^53 - 3 * q^55 - q^56 - 33 * q^58 - 26 * q^59 - 26 * q^61 + 17 * q^62 + 13 * q^64 + 25 * q^65 + 6 * q^67 + 35 * q^68 - 4 * q^70 - 94 * q^71 - 22 * q^73 - 26 * q^74 - 20 * q^76 + 7 * q^77 + 9 * q^79 - 19 * q^80 + 15 * q^82 + 8 * q^83 + 4 * q^85 - 9 * q^86 + 6 * q^88 + 3 * q^89 - 20 * q^91 - 36 * q^92 + 48 * q^94 - 33 * q^95 - 29 * q^97 - 28 * 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 $

$2$	−1.54879	−1.09516	−0.547582	−	0.836752i	$-0.684451\pi$
$2$	−1.54879	−1.09516	−0.547582	+	0.836752i	$0.684451\pi$
$3$	0	0
$3$	0	0
$4$	0.398765	0.199382
$5$	−0.334961	−0.149799	−0.0748996	−	0.997191i	$-0.523864\pi$
$5$	−0.334961	−0.149799	−0.0748996	+	0.997191i	$0.523864\pi$
$6$	0	0
$7$	−4.24623	−1.60493	−0.802463	−	0.596702i	$-0.796477\pi$
$7$	−4.24623	−1.60493	−0.802463	+	0.596702i	$0.796477\pi$
$8$	2.47998	0.876807
$9$	0	0
$10$	0.518786	0.164055
$11$	−0.915418	−0.276009	−0.138004	−	0.990432i	$-0.544069\pi$
$11$	−0.915418	−0.276009	−0.138004	+	0.990432i	$0.544069\pi$
$12$	0	0
$13$	4.81392	1.33514	0.667571	−	0.744546i	$-0.267334\pi$
$13$	4.81392	1.33514	0.667571	+	0.744546i	$0.267334\pi$
$14$	6.57654	1.75766
$15$	0	0
$16$	−4.63852	−1.15963
$17$	5.38915	1.30706	0.653530	−	0.756901i	$-0.273287\pi$
$17$	5.38915	1.30706	0.653530	+	0.756901i	$0.273287\pi$
$18$	0	0
$19$	−4.34799	−0.997498	−0.498749	−	0.866747i	$-0.666207\pi$
$19$	−4.34799	−0.997498	−0.498749	+	0.866747i	$0.666207\pi$
$20$	−0.133571	−0.0298673
$21$	0	0
$22$	1.41779	0.302275
$23$	−8.10534	−1.69008	−0.845040	−	0.534703i	$-0.820423\pi$
$23$	−8.10534	−1.69008	−0.845040	+	0.534703i	$0.820423\pi$
$24$	0	0
$25$	−4.88780	−0.977560
$26$	−7.45578	−1.46220
$27$	0	0
$28$	−1.69325	−0.319994
$29$	6.45221	1.19815	0.599073	−	0.800695i	$-0.295536\pi$
$29$	6.45221	1.19815	0.599073	+	0.800695i	$0.295536\pi$
$30$	0	0
$31$	10.7804	1.93623	0.968113	−	0.250515i	$-0.0806000\pi$
$31$	10.7804	1.93623	0.968113	+	0.250515i	$0.0806000\pi$
$32$	2.22414	0.393176
$33$	0	0
$34$	−8.34668	−1.43144
$35$	1.42232	0.240417
$36$	0	0
$37$	5.16908	0.849792	0.424896	−	0.905242i	$-0.360311\pi$
$37$	5.16908	0.849792	0.424896	+	0.905242i	$0.360311\pi$
$38$	6.73415	1.09242
$39$	0	0
$40$	−0.830699	−0.131345
$41$	0.612344	0.0956321	0.0478161	−	0.998856i	$-0.484774\pi$
$41$	0.612344	0.0956321	0.0478161	+	0.998856i	$0.484774\pi$
$42$	0	0
$43$	−1.85213	−0.282447	−0.141223	−	0.989978i	$-0.545104\pi$
$43$	−1.85213	−0.282447	−0.141223	+	0.989978i	$0.545104\pi$
$44$	−0.365036	−0.0550313
$45$	0	0
$46$	12.5535	1.85091
$47$	2.21116	0.322531	0.161265	−	0.986911i	$-0.448443\pi$
$47$	2.21116	0.322531	0.161265	+	0.986911i	$0.448443\pi$
$48$	0	0
$49$	11.0305	1.57579
$50$	7.57020	1.07059
$51$	0	0
$52$	1.91962	0.266204
$53$	−0.00846193	−0.00116234	−0.000581168	−	1.00000i	$-0.500185\pi$
$53$	−0.00846193	−0.00116234	−0.000581168		1.00000i	$0.500185\pi$
$54$	0	0
$55$	0.306629	0.0413459
$56$	−10.5306	−1.40721
$57$	0	0
$58$	−9.99315	−1.31216
$59$	−8.85799	−1.15321	−0.576606	−	0.817022i	$-0.695623\pi$
$59$	−8.85799	−1.15321	−0.576606	+	0.817022i	$0.695623\pi$
$60$	0	0
$61$	3.78644	0.484805	0.242402	−	0.970176i	$-0.422065\pi$
$61$	3.78644	0.484805	0.242402	+	0.970176i	$0.422065\pi$
$62$	−16.6967	−2.12048
$63$	0	0
$64$	5.83230	0.729037
$65$	−1.61248	−0.200003
$66$	0	0
$67$	4.67684	0.571367	0.285683	−	0.958324i	$-0.407779\pi$
$67$	4.67684	0.571367	0.285683	+	0.958324i	$0.407779\pi$
$68$	2.14900	0.260605
$69$	0	0
$70$	−2.20289	−0.263295
$71$	−1.48694	−0.176467	−0.0882334	−	0.996100i	$-0.528122\pi$
$71$	−1.48694	−0.176467	−0.0882334	+	0.996100i	$0.528122\pi$
$72$	0	0
$73$	−10.3630	−1.21290	−0.606449	−	0.795122i	$-0.707407\pi$
$73$	−10.3630	−1.21290	−0.606449	+	0.795122i	$0.707407\pi$
$74$	−8.00585	−0.930661
$75$	0	0
$76$	−1.73383	−0.198884
$77$	3.88708	0.442974
$78$	0	0
$79$	−17.6786	−1.98900	−0.994499	−	0.104743i	$-0.966598\pi$
$79$	−17.6786	−1.98900	−0.994499	+	0.104743i	$0.966598\pi$
$80$	1.55372	0.173712
$81$	0	0
$82$	−0.948396	−0.104733
$83$	−7.45731	−0.818546	−0.409273	−	0.912412i	$-0.634218\pi$
$83$	−7.45731	−0.818546	−0.409273	+	0.912412i	$0.634218\pi$
$84$	0	0
$85$	−1.80516	−0.195797
$86$	2.86857	0.309325
$87$	0	0
$88$	−2.27022	−0.242006
$89$	0.520713	0.0551955	0.0275978	−	0.999619i	$-0.491214\pi$
$89$	0.520713	0.0551955	0.0275978	+	0.999619i	$0.491214\pi$
$90$	0	0
$91$	−20.4410	−2.14280
$92$	−3.23212	−0.336972
$93$	0	0
$94$	−3.42463	−0.353224
$95$	1.45641	0.149424
$96$	0	0
$97$	−6.33494	−0.643216	−0.321608	−	0.946873i	$-0.604223\pi$
$97$	−6.33494	−0.643216	−0.321608	+	0.946873i	$0.604223\pi$
$98$	−17.0840	−1.72574
$99$	0	0
$100$	−1.94908	−0.194908
$101$	−7.20652	−0.717076	−0.358538	−	0.933515i	$-0.616725\pi$
$101$	−7.20652	−0.717076	−0.358538	+	0.933515i	$0.616725\pi$
$102$	0	0
$103$	7.82738	0.771255	0.385627	−	0.922655i	$-0.373985\pi$
$103$	7.82738	0.771255	0.385627	+	0.922655i	$0.373985\pi$
$104$	11.9385	1.17066
$105$	0	0
$106$	0.0131058	0.00127295
$107$	−3.28176	−0.317260	−0.158630	−	0.987338i	$-0.550708\pi$
$107$	−3.28176	−0.317260	−0.158630	+	0.987338i	$0.550708\pi$
$108$	0	0
$109$	6.89529	0.660449	0.330224	−	0.943902i	$-0.392876\pi$
$109$	6.89529	0.660449	0.330224	+	0.943902i	$0.392876\pi$
$110$	−0.474906	−0.0452805
$111$	0	0
$112$	19.6962	1.86112
$113$	14.4737	1.36157	0.680787	−	0.732481i	$-0.261638\pi$
$113$	14.4737	1.36157	0.680787	+	0.732481i	$0.261638\pi$
$114$	0	0
$115$	2.71497	0.253173
$116$	2.57292	0.238889
$117$	0	0
$118$	13.7192	1.26296
$119$	−22.8836	−2.09773
$120$	0	0
$121$	−10.1620	−0.923819
$122$	−5.86443	−0.530940
$123$	0	0
$124$	4.29886	0.386049
$125$	3.31203	0.296237
$126$	0	0
$127$	6.96745	0.618261	0.309130	−	0.951020i	$-0.399962\pi$
$127$	6.96745	0.618261	0.309130	+	0.951020i	$0.399962\pi$
$128$	−13.4813	−1.19159
$129$	0	0
$130$	2.49740	0.219036
$131$	−7.02108	−0.613434	−0.306717	−	0.951801i	$-0.599231\pi$
$131$	−7.02108	−0.613434	−0.306717	+	0.951801i	$0.599231\pi$
$132$	0	0
$133$	18.4626	1.60091
$134$	−7.24346	−0.625740
$135$	0	0
$136$	13.3650	1.14604
$137$	−5.78101	−0.493905	−0.246953	−	0.969028i	$-0.579429\pi$
$137$	−5.78101	−0.493905	−0.246953	+	0.969028i	$0.579429\pi$
$138$	0	0
$139$	2.34939	0.199272	0.0996362	−	0.995024i	$-0.468232\pi$
$139$	2.34939	0.199272	0.0996362	+	0.995024i	$0.468232\pi$
$140$	0.567173	0.0479349
$141$	0	0
$142$	2.30296	0.193260
$143$	−4.40675	−0.368511
$144$	0	0
$145$	−2.16124	−0.179481
$146$	16.0502	1.32832
$147$	0	0
$148$	2.06125	0.169434
$149$	−4.44128	−0.363844	−0.181922	−	0.983313i	$-0.558232\pi$
$149$	−4.44128	−0.363844	−0.181922	+	0.983313i	$0.558232\pi$
$150$	0	0
$151$	−5.33629	−0.434261	−0.217130	−	0.976143i	$-0.569670\pi$
$151$	−5.33629	−0.434261	−0.217130	+	0.976143i	$0.569670\pi$
$152$	−10.7830	−0.874613
$153$	0	0
$154$	−6.02028	−0.485128
$155$	−3.61103	−0.290045
$156$	0	0
$157$	−21.6127	−1.72488	−0.862442	−	0.506157i	$-0.831066\pi$
$157$	−21.6127	−1.72488	−0.862442	+	0.506157i	$0.831066\pi$
$158$	27.3805	2.17828
$159$	0	0
$160$	−0.745001	−0.0588975
$161$	34.4172	2.71245
$162$	0	0
$163$	−22.0999	−1.73100	−0.865500	−	0.500909i	$-0.832999\pi$
$163$	−22.0999	−1.73100	−0.865500	+	0.500909i	$0.832999\pi$
$164$	0.244181	0.0190674
$165$	0	0
$166$	11.5498	0.896442
$167$	2.58299	0.199877	0.0999387	−	0.994994i	$-0.468135\pi$
$167$	2.58299	0.199877	0.0999387	+	0.994994i	$0.468135\pi$
$168$	0	0
$169$	10.1739	0.782604
$170$	2.79582	0.214429
$171$	0	0
$172$	−0.738564	−0.0563149
$173$	−9.53817	−0.725174	−0.362587	−	0.931950i	$-0.618106\pi$
$173$	−9.53817	−0.725174	−0.362587	+	0.931950i	$0.618106\pi$
$174$	0	0
$175$	20.7547	1.56891
$176$	4.24618	0.320068
$177$	0	0
$178$	−0.806478	−0.0604481
$179$	−11.6661	−0.871963	−0.435981	−	0.899956i	$-0.643599\pi$
$179$	−11.6661	−0.871963	−0.435981	+	0.899956i	$0.643599\pi$
$180$	0	0
$181$	−6.78106	−0.504032	−0.252016	−	0.967723i	$-0.581094\pi$
$181$	−6.78106	−0.504032	−0.252016	+	0.967723i	$0.581094\pi$
$182$	31.6590	2.34672
$183$	0	0
$184$	−20.1011	−1.48187
$185$	−1.73144	−0.127298
$186$	0	0
$187$	−4.93332	−0.360760
$188$	0.881732	0.0643069
$189$	0	0
$190$	−2.25568	−0.163644
$191$	−9.30072	−0.672976	−0.336488	−	0.941688i	$-0.609239\pi$
$191$	−9.30072	−0.672976	−0.336488	+	0.941688i	$0.609239\pi$
$192$	0	0
$193$	−1.27177	−0.0915440	−0.0457720	−	0.998952i	$-0.514575\pi$
$193$	−1.27177	−0.0915440	−0.0457720	+	0.998952i	$0.514575\pi$
$194$	9.81153	0.704427
$195$	0	0
$196$	4.39858	0.314184
$197$	−9.51222	−0.677718	−0.338859	−	0.940837i	$-0.610041\pi$
$197$	−9.51222	−0.677718	−0.338859	+	0.940837i	$0.610041\pi$
$198$	0	0
$199$	−18.4776	−1.30984	−0.654922	−	0.755696i	$-0.727299\pi$
$199$	−18.4776	−1.30984	−0.654922	+	0.755696i	$0.727299\pi$
$200$	−12.1217	−0.857132
$201$	0	0
$202$	11.1614	0.785315
$203$	−27.3976	−1.92293
$204$	0	0
$205$	−0.205112	−0.0143256
$206$	−12.1230	−0.844650
$207$	0	0
$208$	−22.3295	−1.54827
$209$	3.98023	0.275318
$210$	0	0
$211$	5.54459	0.381705	0.190853	−	0.981619i	$-0.438875\pi$
$211$	5.54459	0.381705	0.190853	+	0.981619i	$0.438875\pi$
$212$	−0.00337432	−0.000231749 0
$213$	0	0
$214$	5.08278	0.347451
$215$	0.620391	0.0423103
$216$	0	0
$217$	−45.7763	−3.10750
$218$	−10.6794	−0.723299
$219$	0	0
$220$	0.122273	0.00824365
$221$	25.9429	1.74511
$222$	0	0
$223$	−12.8984	−0.863744	−0.431872	−	0.901935i	$-0.642147\pi$
$223$	−12.8984	−0.863744	−0.431872	+	0.901935i	$0.642147\pi$
$224$	−9.44422	−0.631019
$225$	0	0
$226$	−22.4169	−1.49115
$227$	−2.85469	−0.189473	−0.0947364	−	0.995502i	$-0.530201\pi$
$227$	−2.85469	−0.189473	−0.0947364	+	0.995502i	$0.530201\pi$
$228$	0	0
$229$	−16.1705	−1.06857	−0.534287	−	0.845303i	$-0.679420\pi$
$229$	−16.1705	−1.06857	−0.534287	+	0.845303i	$0.679420\pi$
$230$	−4.20494	−0.277265
$231$	0	0
$232$	16.0014	1.05054
$233$	−11.3792	−0.745476	−0.372738	−	0.927937i	$-0.621581\pi$
$233$	−11.3792	−0.745476	−0.372738	+	0.927937i	$0.621581\pi$
$234$	0	0
$235$	−0.740653	−0.0483148
$236$	−3.53226	−0.229930
$237$	0	0
$238$	35.4420	2.29736
$239$	−27.0358	−1.74880	−0.874400	−	0.485206i	$-0.838745\pi$
$239$	−27.0358	−1.74880	−0.874400	+	0.485206i	$0.838745\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	15.7389	1.01173
$243$	0	0
$244$	1.50990	0.0966615
$245$	−3.69479	−0.236052
$246$	0	0
$247$	−20.9309	−1.33180
$248$	26.7353	1.69770
$249$	0	0
$250$	−5.12966	−0.324428
$251$	16.5147	1.04240	0.521199	−	0.853435i	$-0.325485\pi$
$251$	16.5147	1.04240	0.521199	+	0.853435i	$0.325485\pi$
$252$	0	0
$253$	7.41977	0.466477
$254$	−10.7911	−0.677097
$255$	0	0
$256$	9.21519	0.575949
$257$	9.88138	0.616383	0.308192	−	0.951324i	$-0.400276\pi$
$257$	9.88138	0.616383	0.308192	+	0.951324i	$0.400276\pi$
$258$	0	0
$259$	−21.9491	−1.36385
$260$	−0.643000	−0.0398771
$261$	0	0
$262$	10.8742	0.671811
$263$	31.5776	1.94716	0.973579	−	0.228351i	$-0.0733335\pi$
$263$	31.5776	1.94716	0.973579	+	0.228351i	$0.0733335\pi$
$264$	0	0
$265$	0.00283442	0.000174117 0
$266$	−28.5948	−1.75326
$267$	0	0
$268$	1.86496	0.113921
$269$	−4.21527	−0.257010	−0.128505	−	0.991709i	$-0.541018\pi$
$269$	−4.21527	−0.257010	−0.128505	+	0.991709i	$0.541018\pi$
$270$	0	0
$271$	11.8060	0.717162	0.358581	−	0.933499i	$-0.383261\pi$
$271$	11.8060	0.717162	0.358581	+	0.933499i	$0.383261\pi$
$272$	−24.9976	−1.51570
$273$	0	0
$274$	8.95360	0.540907
$275$	4.47438	0.269815
$276$	0	0
$277$	8.01314	0.481463	0.240731	−	0.970592i	$-0.422613\pi$
$277$	8.01314	0.481463	0.240731	+	0.970592i	$0.422613\pi$
$278$	−3.63872	−0.218236
$279$	0	0
$280$	3.52734	0.210799
$281$	21.6719	1.29284	0.646419	−	0.762982i	$-0.276266\pi$
$281$	21.6719	1.29284	0.646419	+	0.762982i	$0.276266\pi$
$282$	0	0
$283$	2.15646	0.128188	0.0640941	−	0.997944i	$-0.479584\pi$
$283$	2.15646	0.128188	0.0640941	+	0.997944i	$0.479584\pi$
$284$	−0.592938	−0.0351844
$285$	0	0
$286$	6.82515	0.403580
$287$	−2.60016	−0.153482
$288$	0	0
$289$	12.0429	0.708406
$290$	3.34732	0.196561
$291$	0	0
$292$	−4.13240	−0.241831
$293$	−26.9520	−1.57455	−0.787276	−	0.616601i	$-0.788509\pi$
$293$	−26.9520	−1.57455	−0.787276	+	0.616601i	$0.788509\pi$
$294$	0	0
$295$	2.96708	0.172750
$296$	12.8192	0.745103
$297$	0	0
$298$	6.87863	0.398469
$299$	−39.0185	−2.25650
$300$	0	0
$301$	7.86457	0.453306
$302$	8.26481	0.475586
$303$	0	0
$304$	20.1682	1.15673
$305$	−1.26831	−0.0726234
$306$	0	0
$307$	16.6282	0.949020	0.474510	−	0.880250i	$-0.342625\pi$
$307$	16.6282	0.949020	0.474510	+	0.880250i	$0.342625\pi$
$308$	1.55003	0.0883211
$309$	0	0
$310$	5.59275	0.317647
$311$	10.1432	0.575167	0.287584	−	0.957756i	$-0.407148\pi$
$311$	10.1432	0.575167	0.287584	+	0.957756i	$0.407148\pi$
$312$	0	0
$313$	−26.7382	−1.51133	−0.755667	−	0.654956i	$-0.772687\pi$
$313$	−26.7382	−1.51133	−0.755667	+	0.654956i	$0.772687\pi$
$314$	33.4737	1.88903
$315$	0	0
$316$	−7.04961	−0.396571
$317$	25.6438	1.44030	0.720151	−	0.693817i	$-0.244072\pi$
$317$	25.6438	1.44030	0.720151	+	0.693817i	$0.244072\pi$
$318$	0	0
$319$	−5.90647	−0.330699
$320$	−1.95359	−0.109209
$321$	0	0
$322$	−53.3051	−2.97058
$323$	−23.4320	−1.30379
$324$	0	0
$325$	−23.5295	−1.30518
$326$	34.2282	1.89573
$327$	0	0
$328$	1.51860	0.0838509
$329$	−9.38910	−0.517638
$330$	0	0
$331$	21.9178	1.20471	0.602356	−	0.798228i	$-0.294229\pi$
$331$	21.9178	1.20471	0.602356	+	0.798228i	$0.294229\pi$
$332$	−2.97371	−0.163204
$333$	0	0
$334$	−4.00052	−0.218898
$335$	−1.56656	−0.0855903
$336$	0	0
$337$	5.64948	0.307747	0.153873	−	0.988091i	$-0.450825\pi$
$337$	5.64948	0.307747	0.153873	+	0.988091i	$0.450825\pi$
$338$	−15.7572	−0.857079
$339$	0	0
$340$	−0.719833	−0.0390384
$341$	−9.86861	−0.534415
$342$	0	0
$343$	−17.1144	−0.924093
$344$	−4.59325	−0.247651
$345$	0	0
$346$	14.7727	0.794184
$347$	−19.7057	−1.05786	−0.528928	−	0.848667i	$-0.677406\pi$
$347$	−19.7057	−1.05786	−0.528928	+	0.848667i	$0.677406\pi$
$348$	0	0
$349$	−31.5930	−1.69113	−0.845567	−	0.533869i	$-0.820738\pi$
$349$	−31.5930	−1.69113	−0.845567	+	0.533869i	$0.820738\pi$
$350$	−32.1448	−1.71821
$351$	0	0
$352$	−2.03602	−0.108520
$353$	−12.7948	−0.680999	−0.340499	−	0.940245i	$-0.610596\pi$
$353$	−12.7948	−0.680999	−0.340499	+	0.940245i	$0.610596\pi$
$354$	0	0
$355$	0.498066	0.0264346
$356$	0.207642	0.0110050
$357$	0	0
$358$	18.0683	0.954942
$359$	−27.2866	−1.44013	−0.720066	−	0.693906i	$-0.755888\pi$
$359$	−27.2866	−1.44013	−0.720066	+	0.693906i	$0.755888\pi$
$360$	0	0
$361$	−0.0949647	−0.00499814
$362$	10.5025	0.551998
$363$	0	0
$364$	−8.15117	−0.427237
$365$	3.47121	0.181691
$366$	0	0
$367$	25.9541	1.35479	0.677396	−	0.735619i	$-0.263109\pi$
$367$	25.9541	1.35479	0.677396	+	0.735619i	$0.263109\pi$
$368$	37.5967	1.95987
$369$	0	0
$370$	2.68165	0.139412
$371$	0.0359313	0.00186546
$372$	0	0
$373$	1.94963	0.100948	0.0504740	−	0.998725i	$-0.483927\pi$
$373$	1.94963	0.100948	0.0504740	+	0.998725i	$0.483927\pi$
$374$	7.64070	0.395091
$375$	0	0
$376$	5.48364	0.282797
$377$	31.0604	1.59969
$378$	0	0
$379$	22.0885	1.13461	0.567305	−	0.823508i	$-0.307986\pi$
$379$	22.0885	1.13461	0.567305	+	0.823508i	$0.307986\pi$
$380$	0.580765	0.0297926
$381$	0	0
$382$	14.4049	0.737019
$383$	32.6589	1.66879	0.834396	−	0.551166i	$-0.185817\pi$
$383$	32.6589	1.66879	0.834396	+	0.551166i	$0.185817\pi$
$384$	0	0
$385$	−1.30202	−0.0663571
$386$	1.96971	0.100256
$387$	0	0
$388$	−2.52615	−0.128246
$389$	1.59569	0.0809045	0.0404522	−	0.999181i	$-0.487120\pi$
$389$	1.59569	0.0809045	0.0404522	+	0.999181i	$0.487120\pi$
$390$	0	0
$391$	−43.6809	−2.20904
$392$	27.3555	1.38166
$393$	0	0
$394$	14.7325	0.742212
$395$	5.92165	0.297951
$396$	0	0
$397$	−20.5064	−1.02919	−0.514593	−	0.857434i	$-0.672057\pi$
$397$	−20.5064	−1.02919	−0.514593	+	0.857434i	$0.672057\pi$
$398$	28.6181	1.43449
$399$	0	0
$400$	22.6721	1.13361
$401$	−2.51432	−0.125559	−0.0627795	−	0.998027i	$-0.519996\pi$
$401$	−2.51432	−0.125559	−0.0627795	+	0.998027i	$0.519996\pi$
$402$	0	0
$403$	51.8962	2.58514
$404$	−2.87371	−0.142972
$405$	0	0
$406$	42.4333	2.10593
$407$	−4.73187	−0.234550
$408$	0	0
$409$	−3.99904	−0.197740	−0.0988698	−	0.995100i	$-0.531523\pi$
$409$	−3.99904	−0.197740	−0.0988698	+	0.995100i	$0.531523\pi$
$410$	0.317676	0.0156889
$411$	0	0
$412$	3.12128	0.153775
$413$	37.6131	1.85082
$414$	0	0
$415$	2.49791	0.122618
$416$	10.7068	0.524946
$417$	0	0
$418$	−6.16456	−0.301518
$419$	−19.3567	−0.945635	−0.472818	−	0.881160i	$-0.656763\pi$
$419$	−19.3567	−0.945635	−0.472818	+	0.881160i	$0.656763\pi$
$420$	0	0
$421$	−4.81504	−0.234671	−0.117335	−	0.993092i	$-0.537435\pi$
$421$	−4.81504	−0.234671	−0.117335	+	0.993092i	$0.537435\pi$
$422$	−8.58743	−0.418029
$423$	0	0
$424$	−0.0209855	−0.00101914
$425$	−26.3411	−1.27773
$426$	0	0
$427$	−16.0781	−0.778075
$428$	−1.30865	−0.0632561
$429$	0	0
$430$	−0.960859	−0.0463367
$431$	−26.8254	−1.29213	−0.646067	−	0.763281i	$-0.723587\pi$
$431$	−26.8254	−1.29213	−0.646067	+	0.763281i	$0.723587\pi$
$432$	0	0
$433$	1.59224	0.0765181	0.0382591	−	0.999268i	$-0.487819\pi$
$433$	1.59224	0.0765181	0.0382591	+	0.999268i	$0.487819\pi$
$434$	70.8981	3.40322
$435$	0	0
$436$	2.74960	0.131682
$437$	35.2419	1.68585
$438$	0	0
$439$	−36.5165	−1.74284	−0.871418	−	0.490541i	$-0.836799\pi$
$439$	−36.5165	−1.74284	−0.871418	+	0.490541i	$0.836799\pi$
$440$	0.760436	0.0362524
$441$	0	0
$442$	−40.1803	−1.91118
$443$	1.27451	0.0605540	0.0302770	−	0.999542i	$-0.490361\pi$
$443$	1.27451	0.0605540	0.0302770	+	0.999542i	$0.490361\pi$
$444$	0	0
$445$	−0.174419	−0.00826825
$446$	19.9770	0.945940
$447$	0	0
$448$	−24.7653	−1.17005
$449$	−31.9473	−1.50768	−0.753842	−	0.657055i	$-0.771802\pi$
$449$	−31.9473	−1.50768	−0.753842	+	0.657055i	$0.771802\pi$
$450$	0	0
$451$	−0.560551	−0.0263953
$452$	5.77162	0.271474
$453$	0	0
$454$	4.42133	0.207504
$455$	6.84696	0.320990
$456$	0	0
$457$	12.8561	0.601384	0.300692	−	0.953721i	$-0.402782\pi$
$457$	12.8561	0.601384	0.300692	+	0.953721i	$0.402782\pi$
$458$	25.0447	1.17026
$459$	0	0
$460$	1.08264	0.0504782
$461$	−35.6691	−1.66127	−0.830637	−	0.556814i	$-0.812024\pi$
$461$	−35.6691	−1.66127	−0.830637	+	0.556814i	$0.812024\pi$
$462$	0	0
$463$	2.74466	0.127555	0.0637776	−	0.997964i	$-0.479685\pi$
$463$	2.74466	0.127555	0.0637776	+	0.997964i	$0.479685\pi$
$464$	−29.9287	−1.38940
$465$	0	0
$466$	17.6240	0.816418
$467$	25.5934	1.18432	0.592160	−	0.805821i	$-0.298275\pi$
$467$	25.5934	1.18432	0.592160	+	0.805821i	$0.298275\pi$
$468$	0	0
$469$	−19.8590	−0.917001
$470$	1.14712	0.0529126
$471$	0	0
$472$	−21.9677	−1.01114
$473$	1.69547	0.0779578
$474$	0	0
$475$	21.2521	0.975114
$476$	−9.12517	−0.418251
$477$	0	0
$478$	41.8729	1.91522
$479$	0.610882	0.0279119	0.0139560	−	0.999903i	$-0.495558\pi$
$479$	0.610882	0.0279119	0.0139560	+	0.999903i	$0.495558\pi$
$480$	0	0
$481$	24.8836	1.13459
$482$	−1.54879	−0.0705457
$483$	0	0
$484$	−4.05225	−0.184193
$485$	2.12196	0.0963533
$486$	0	0
$487$	26.6270	1.20658	0.603291	−	0.797521i	$-0.293856\pi$
$487$	26.6270	1.20658	0.603291	+	0.797521i	$0.293856\pi$
$488$	9.39032	0.425080
$489$	0	0
$490$	5.72247	0.258515
$491$	31.7856	1.43447	0.717233	−	0.696833i	$-0.245408\pi$
$491$	31.7856	1.43447	0.717233	+	0.696833i	$0.245408\pi$
$492$	0	0
$493$	34.7719	1.56605
$494$	32.4177	1.45854
$495$	0	0
$496$	−50.0053	−2.24530
$497$	6.31387	0.283216
$498$	0	0
$499$	−17.2468	−0.772074	−0.386037	−	0.922483i	$-0.626156\pi$
$499$	−17.2468	−0.772074	−0.386037	+	0.922483i	$0.626156\pi$
$500$	1.32072	0.0590645
$501$	0	0
$502$	−25.5779	−1.14160
$503$	23.6924	1.05639	0.528196	−	0.849122i	$-0.322868\pi$
$503$	23.6924	1.05639	0.528196	+	0.849122i	$0.322868\pi$
$504$	0	0
$505$	2.41391	0.107417
$506$	−11.4917	−0.510868
$507$	0	0
$508$	2.77837	0.123270
$509$	−20.3227	−0.900787	−0.450394	−	0.892830i	$-0.648716\pi$
$509$	−20.3227	−0.900787	−0.450394	+	0.892830i	$0.648716\pi$
$510$	0	0
$511$	44.0037	1.94661
$512$	12.6902	0.560832
$513$	0	0
$514$	−15.3042	−0.675040
$515$	−2.62187	−0.115533
$516$	0	0
$517$	−2.02413	−0.0890213
$518$	33.9947	1.49364
$519$	0	0
$520$	−3.99892	−0.175364
$521$	−16.3912	−0.718112	−0.359056	−	0.933316i	$-0.616901\pi$
$521$	−16.3912	−0.718112	−0.359056	+	0.933316i	$0.616901\pi$
$522$	0	0
$523$	0.258754	0.0113145	0.00565727	−	0.999984i	$-0.498199\pi$
$523$	0.258754	0.0113145	0.00565727	+	0.999984i	$0.498199\pi$
$524$	−2.79976	−0.122308
$525$	0	0
$526$	−48.9072	−2.13246
$527$	58.0974	2.53076
$528$	0	0
$529$	42.6965	1.85637
$530$	−0.00438994	−0.000190687 0
$531$	0	0
$532$	7.36223	0.319193
$533$	2.94778	0.127682
$534$	0	0
$535$	1.09926	0.0475253
$536$	11.5985	0.500978
$537$	0	0
$538$	6.52859	0.281468
$539$	−10.0975	−0.434931
$540$	0	0
$541$	−17.9084	−0.769941	−0.384970	−	0.922929i	$-0.625788\pi$
$541$	−17.9084	−0.769941	−0.384970	+	0.922929i	$0.625788\pi$
$542$	−18.2850	−0.785410
$543$	0	0
$544$	11.9862	0.513905
$545$	−2.30965	−0.0989347
$546$	0	0
$547$	12.2295	0.522897	0.261448	−	0.965217i	$-0.415800\pi$
$547$	12.2295	0.522897	0.261448	+	0.965217i	$0.415800\pi$
$548$	−2.30526	−0.0984760
$549$	0	0
$550$	−6.92989	−0.295492
$551$	−28.0542	−1.19515
$552$	0	0
$553$	75.0675	3.19219
$554$	−12.4107	−0.527280
$555$	0	0
$556$	0.936853	0.0397314
$557$	−25.2541	−1.07005	−0.535024	−	0.844837i	$-0.679698\pi$
$557$	−25.2541	−1.07005	−0.535024	+	0.844837i	$0.679698\pi$
$558$	0	0
$559$	−8.91600	−0.377107
$560$	−6.59747	−0.278794
$561$	0	0
$562$	−33.5654	−1.41587
$563$	10.4115	0.438794	0.219397	−	0.975636i	$-0.429591\pi$
$563$	10.4115	0.438794	0.219397	+	0.975636i	$0.429591\pi$
$564$	0	0
$565$	−4.84814	−0.203963
$566$	−3.33991	−0.140387
$567$	0	0
$568$	−3.68758	−0.154727
$569$	28.5127	1.19532	0.597658	−	0.801751i	$-0.296098\pi$
$569$	28.5127	1.19532	0.597658	+	0.801751i	$0.296098\pi$
$570$	0	0
$571$	35.8278	1.49935	0.749674	−	0.661807i	$-0.230210\pi$
$571$	35.8278	1.49935	0.749674	+	0.661807i	$0.230210\pi$
$572$	−1.75726	−0.0734746
$573$	0	0
$574$	4.02711	0.168088
$575$	39.6173	1.65215
$576$	0	0
$577$	1.00615	0.0418864	0.0209432	−	0.999781i	$-0.493333\pi$
$577$	1.00615	0.0418864	0.0209432	+	0.999781i	$0.493333\pi$
$578$	−18.6520	−0.775820
$579$	0	0
$580$	−0.861827	−0.0357854
$581$	31.6655	1.31371
$582$	0	0
$583$	0.00774620	0.000320815 0
$584$	−25.7001	−1.06348
$585$	0	0
$586$	41.7431	1.72439
$587$	7.69459	0.317590	0.158795	−	0.987312i	$-0.449239\pi$
$587$	7.69459	0.317590	0.158795	+	0.987312i	$0.449239\pi$
$588$	0	0
$589$	−46.8733	−1.93138
$590$	−4.59540	−0.189190
$591$	0	0
$592$	−23.9769	−0.985443
$593$	27.9165	1.14639	0.573197	−	0.819418i	$-0.305703\pi$
$593$	27.9165	1.14639	0.573197	+	0.819418i	$0.305703\pi$
$594$	0	0
$595$	7.66511	0.314239
$596$	−1.77103	−0.0725441
$597$	0	0
$598$	60.4316	2.47123
$599$	−43.2997	−1.76918	−0.884588	−	0.466372i	$-0.845561\pi$
$599$	−43.2997	−1.76918	−0.884588	+	0.466372i	$0.845561\pi$
$600$	0	0
$601$	−25.4104	−1.03651	−0.518255	−	0.855226i	$-0.673418\pi$
$601$	−25.4104	−1.03651	−0.518255	+	0.855226i	$0.673418\pi$
$602$	−12.1806	−0.496444
$603$	0	0
$604$	−2.12792	−0.0865840
$605$	3.40388	0.138387
$606$	0	0
$607$	−2.58042	−0.104736	−0.0523679	−	0.998628i	$-0.516677\pi$
$607$	−2.58042	−0.104736	−0.0523679	+	0.998628i	$0.516677\pi$
$608$	−9.67055	−0.392192
$609$	0	0
$610$	1.96436	0.0795344
$611$	10.6443	0.430624
$612$	0	0
$613$	24.6593	0.995980	0.497990	−	0.867183i	$-0.334072\pi$
$613$	24.6593	0.995980	0.497990	+	0.867183i	$0.334072\pi$
$614$	−25.7536	−1.03933
$615$	0	0
$616$	9.63989	0.388402
$617$	17.2509	0.694497	0.347248	−	0.937773i	$-0.387116\pi$
$617$	17.2509	0.694497	0.347248	+	0.937773i	$0.387116\pi$
$618$	0	0
$619$	−0.0379403	−0.00152495	−0.000762475	−	1.00000i	$-0.500243\pi$
$619$	−0.0379403	−0.00152495	−0.000762475		1.00000i	$0.500243\pi$
$620$	−1.43995	−0.0578299
$621$	0	0
$622$	−15.7097	−0.629902
$623$	−2.21107	−0.0885847
$624$	0	0
$625$	23.3296	0.933184
$626$	41.4120	1.65516
$627$	0	0
$628$	−8.61840	−0.343911
$629$	27.8569	1.11073
$630$	0	0
$631$	39.8274	1.58551	0.792753	−	0.609544i	$-0.208647\pi$
$631$	39.8274	1.58551	0.792753	+	0.609544i	$0.208647\pi$
$632$	−43.8427	−1.74397
$633$	0	0
$634$	−39.7171	−1.57737
$635$	−2.33383	−0.0926150
$636$	0	0
$637$	53.1000	2.10390
$638$	9.14790	0.362169
$639$	0	0
$640$	4.51572	0.178499
$641$	−17.4965	−0.691071	−0.345536	−	0.938406i	$-0.612303\pi$
$641$	−17.4965	−0.691071	−0.345536	+	0.938406i	$0.612303\pi$
$642$	0	0
$643$	19.8884	0.784323	0.392162	−	0.919896i	$-0.371727\pi$
$643$	19.8884	0.784323	0.392162	+	0.919896i	$0.371727\pi$
$644$	13.7244	0.540815
$645$	0	0
$646$	36.2913	1.42786
$647$	39.9620	1.57107	0.785535	−	0.618817i	$-0.212388\pi$
$647$	39.9620	1.57107	0.785535	+	0.618817i	$0.212388\pi$
$648$	0	0
$649$	8.10876	0.318297
$650$	36.4424	1.42939
$651$	0	0
$652$	−8.81267	−0.345131
$653$	45.0393	1.76252	0.881262	−	0.472629i	$-0.156695\pi$
$653$	45.0393	1.76252	0.881262	+	0.472629i	$0.156695\pi$
$654$	0	0
$655$	2.35179	0.0918920
$656$	−2.84037	−0.110898
$657$	0	0
$658$	14.5418	0.566898
$659$	28.6163	1.11473	0.557367	−	0.830266i	$-0.311812\pi$
$659$	28.6163	1.11473	0.557367	+	0.830266i	$0.311812\pi$
$660$	0	0
$661$	−21.0849	−0.820106	−0.410053	−	0.912062i	$-0.634490\pi$
$661$	−21.0849	−0.820106	−0.410053	+	0.912062i	$0.634490\pi$
$662$	−33.9462	−1.31936
$663$	0	0
$664$	−18.4940	−0.717707
$665$	−6.18425	−0.239815
$666$	0	0
$667$	−52.2974	−2.02496
$668$	1.03000	0.0398521
$669$	0	0
$670$	2.42628	0.0937354
$671$	−3.46618	−0.133810
$672$	0	0
$673$	11.9180	0.459406	0.229703	−	0.973261i	$-0.426224\pi$
$673$	11.9180	0.459406	0.229703	+	0.973261i	$0.426224\pi$
$674$	−8.74988	−0.337033
$675$	0	0
$676$	4.05698	0.156038
$677$	−46.0077	−1.76822	−0.884111	−	0.467278i	$-0.845235\pi$
$677$	−46.0077	−1.76822	−0.884111	+	0.467278i	$0.845235\pi$
$678$	0	0
$679$	26.8997	1.03231
$680$	−4.47676	−0.171676
$681$	0	0
$682$	15.2844	0.585272
$683$	−41.5189	−1.58868	−0.794338	−	0.607477i	$-0.792182\pi$
$683$	−41.5189	−1.58868	−0.794338	+	0.607477i	$0.792182\pi$
$684$	0	0
$685$	1.93642	0.0739866
$686$	26.5068	1.01203
$687$	0	0
$688$	8.59113	0.327534
$689$	−0.0407351	−0.00155188
$690$	0	0
$691$	28.5903	1.08763	0.543814	−	0.839206i	$-0.316980\pi$
$691$	28.5903	1.08763	0.543814	+	0.839206i	$0.316980\pi$
$692$	−3.80349	−0.144587
$693$	0	0
$694$	30.5200	1.15852
$695$	−0.786954	−0.0298509
$696$	0	0
$697$	3.30001	0.124997
$698$	48.9311	1.85207
$699$	0	0
$700$	8.27626	0.312813
$701$	−19.2033	−0.725298	−0.362649	−	0.931926i	$-0.618128\pi$
$701$	−19.2033	−0.725298	−0.362649	+	0.931926i	$0.618128\pi$
$702$	0	0
$703$	−22.4751	−0.847665
$704$	−5.33899	−0.201221
$705$	0	0
$706$	19.8165	0.745805
$707$	30.6006	1.15085
$708$	0	0
$709$	4.05386	0.152246	0.0761229	−	0.997098i	$-0.475746\pi$
$709$	4.05386	0.152246	0.0761229	+	0.997098i	$0.475746\pi$
$710$	−0.771402	−0.0289502
$711$	0	0
$712$	1.29136	0.0483958
$713$	−87.3792	−3.27238
$714$	0	0
$715$	1.47609	0.0552027
$716$	−4.65202	−0.173854
$717$	0	0
$718$	42.2613	1.57718
$719$	−13.0438	−0.486452	−0.243226	−	0.969970i	$-0.578206\pi$
$719$	−13.0438	−0.486452	−0.243226	+	0.969970i	$0.578206\pi$
$720$	0	0
$721$	−33.2369	−1.23781
$722$	0.147081	0.00547378
$723$	0	0
$724$	−2.70405	−0.100495
$725$	−31.5371	−1.17126
$726$	0	0
$727$	−43.3367	−1.60727	−0.803635	−	0.595123i	$-0.797103\pi$
$727$	−43.3367	−1.60727	−0.803635	+	0.595123i	$0.797103\pi$
$728$	−50.6935	−1.87882
$729$	0	0
$730$	−5.37619	−0.198982
$731$	−9.98139	−0.369175
$732$	0	0
$733$	−6.99674	−0.258431	−0.129215	−	0.991617i	$-0.541246\pi$
$733$	−6.99674	−0.258431	−0.129215	+	0.991617i	$0.541246\pi$
$734$	−40.1975	−1.48372
$735$	0	0
$736$	−18.0274	−0.664499
$737$	−4.28126	−0.157702
$738$	0	0
$739$	−1.44307	−0.0530842	−0.0265421	−	0.999648i	$-0.508450\pi$
$739$	−1.44307	−0.0530842	−0.0265421	+	0.999648i	$0.508450\pi$
$740$	−0.690438	−0.0253810
$741$	0	0
$742$	−0.0556503	−0.00204299
$743$	4.24266	0.155648	0.0778240	−	0.996967i	$-0.475203\pi$
$743$	4.24266	0.155648	0.0778240	+	0.996967i	$0.475203\pi$
$744$	0	0
$745$	1.48766	0.0545036
$746$	−3.01958	−0.110555
$747$	0	0
$748$	−1.96723	−0.0719292
$749$	13.9351	0.509179
$750$	0	0
$751$	19.0193	0.694024	0.347012	−	0.937861i	$-0.387196\pi$
$751$	19.0193	0.694024	0.347012	+	0.937861i	$0.387196\pi$
$752$	−10.2565	−0.374016
$753$	0	0
$754$	−48.1063	−1.75193
$755$	1.78745	0.0650519
$756$	0	0
$757$	0.151878	0.00552010	0.00276005	−	0.999996i	$-0.499121\pi$
$757$	0.151878	0.00552010	0.00276005	+	0.999996i	$0.499121\pi$
$758$	−34.2105	−1.24258
$759$	0	0
$760$	3.61187	0.131016
$761$	16.2184	0.587916	0.293958	−	0.955818i	$-0.405027\pi$
$761$	16.2184	0.587916	0.293958	+	0.955818i	$0.405027\pi$
$762$	0	0
$763$	−29.2790	−1.05997
$764$	−3.70880	−0.134180
$765$	0	0
$766$	−50.5819	−1.82760
$767$	−42.6417	−1.53970
$768$	0	0
$769$	5.32816	0.192138	0.0960691	−	0.995375i	$-0.469373\pi$
$769$	5.32816	0.192138	0.0960691	+	0.995375i	$0.469373\pi$
$770$	2.01656	0.0726719
$771$	0	0
$772$	−0.507137	−0.0182523
$773$	6.19612	0.222859	0.111430	−	0.993772i	$-0.464457\pi$
$773$	6.19612	0.222859	0.111430	+	0.993772i	$0.464457\pi$
$774$	0	0
$775$	−52.6927	−1.89278
$776$	−15.7106	−0.563976
$777$	0	0
$778$	−2.47139	−0.0886036
$779$	−2.66247	−0.0953929
$780$	0	0
$781$	1.36117	0.0487064
$782$	67.6527	2.41925
$783$	0	0
$784$	−51.1652	−1.82733
$785$	7.23943	0.258386
$786$	0	0
$787$	26.4924	0.944351	0.472176	−	0.881504i	$-0.343469\pi$
$787$	26.4924	0.944351	0.472176	+	0.881504i	$0.343469\pi$
$788$	−3.79314	−0.135125
$789$	0	0
$790$	−9.17142	−0.326304
$791$	−61.4589	−2.18523
$792$	0	0
$793$	18.2277	0.647283
$794$	31.7602	1.12713
$795$	0	0
$796$	−7.36823	−0.261160
$797$	−41.8090	−1.48095	−0.740476	−	0.672083i	$-0.765400\pi$
$797$	−41.8090	−1.48095	−0.740476	+	0.672083i	$0.765400\pi$
$798$	0	0
$799$	11.9163	0.421567
$800$	−10.8712	−0.384353
$801$	0	0
$802$	3.89416	0.137508
$803$	9.48648	0.334770
$804$	0	0
$805$	−11.5284	−0.406323
$806$	−80.3766	−2.83115
$807$	0	0
$808$	−17.8721	−0.628737
$809$	−1.66235	−0.0584451	−0.0292226	−	0.999573i	$-0.509303\pi$
$809$	−1.66235	−0.0584451	−0.0292226	+	0.999573i	$0.509303\pi$
$810$	0	0
$811$	−13.4156	−0.471085	−0.235542	−	0.971864i	$-0.575687\pi$
$811$	−13.4156	−0.471085	−0.235542	+	0.971864i	$0.575687\pi$
$812$	−10.9252	−0.383399
$813$	0	0
$814$	7.32869	0.256870
$815$	7.40262	0.259302
$816$	0	0
$817$	8.05304	0.281740
$818$	6.19369	0.216557
$819$	0	0
$820$	−0.0817914	−0.00285628
$821$	−23.1210	−0.806930	−0.403465	−	0.914995i	$-0.632194\pi$
$821$	−23.1210	−0.806930	−0.403465	+	0.914995i	$0.632194\pi$
$822$	0	0
$823$	20.7084	0.721849	0.360924	−	0.932595i	$-0.382461\pi$
$823$	20.7084	0.721849	0.360924	+	0.932595i	$0.382461\pi$
$824$	19.4118	0.676241
$825$	0	0
$826$	−58.2550	−2.02695
$827$	1.22288	0.0425235	0.0212618	−	0.999774i	$-0.493232\pi$
$827$	1.22288	0.0425235	0.0212618	+	0.999774i	$0.493232\pi$
$828$	0	0
$829$	26.5008	0.920411	0.460205	−	0.887813i	$-0.347776\pi$
$829$	26.5008	0.920411	0.460205	+	0.887813i	$0.347776\pi$
$830$	−3.86875	−0.134286
$831$	0	0
$832$	28.0762	0.973368
$833$	59.4450	2.05965
$834$	0	0
$835$	−0.865201	−0.0299415
$836$	1.58718	0.0548936
$837$	0	0
$838$	29.9795	1.03562
$839$	−27.5572	−0.951380	−0.475690	−	0.879613i	$-0.657802\pi$
$839$	−27.5572	−0.951380	−0.475690	+	0.879613i	$0.657802\pi$
$840$	0	0
$841$	12.6310	0.435553
$842$	7.45750	0.257003
$843$	0	0
$844$	2.21099	0.0761053
$845$	−3.40785	−0.117234
$846$	0	0
$847$	43.1503	1.48266
$848$	0.0392508	0.00134788
$849$	0	0
$850$	40.7969	1.39932
$851$	−41.8972	−1.43622
$852$	0	0
$853$	14.3366	0.490875	0.245437	−	0.969412i	$-0.421068\pi$
$853$	14.3366	0.490875	0.245437	+	0.969412i	$0.421068\pi$
$854$	24.9017	0.852119
$855$	0	0
$856$	−8.13872	−0.278176
$857$	−33.5953	−1.14759	−0.573796	−	0.818998i	$-0.694530\pi$
$857$	−33.5953	−1.14759	−0.573796	+	0.818998i	$0.694530\pi$
$858$	0	0
$859$	−31.1946	−1.06435	−0.532173	−	0.846635i	$-0.678625\pi$
$859$	−31.1946	−1.06435	−0.532173	+	0.846635i	$0.678625\pi$
$860$	0.247390	0.00843594
$861$	0	0
$862$	41.5470	1.41510
$863$	−23.3830	−0.795965	−0.397983	−	0.917393i	$-0.630290\pi$
$863$	−23.3830	−0.795965	−0.397983	+	0.917393i	$0.630290\pi$
$864$	0	0
$865$	3.19492	0.108631
$866$	−2.46605	−0.0837999
$867$	0	0
$868$	−18.2540	−0.619580
$869$	16.1833	0.548981
$870$	0	0
$871$	22.5139	0.762856
$872$	17.1002	0.579086
$873$	0	0
$874$	−54.5825	−1.84628
$875$	−14.0637	−0.475438
$876$	0	0
$877$	−35.6526	−1.20390	−0.601951	−	0.798533i	$-0.705610\pi$
$877$	−35.6526	−1.20390	−0.601951	+	0.798533i	$0.705610\pi$
$878$	56.5565	1.90869
$879$	0	0
$880$	−1.42231	−0.0479459
$881$	−29.7222	−1.00137	−0.500683	−	0.865631i	$-0.666918\pi$
$881$	−29.7222	−1.00137	−0.500683	+	0.865631i	$0.666918\pi$
$882$	0	0
$883$	−41.7428	−1.40476	−0.702379	−	0.711803i	$-0.747879\pi$
$883$	−41.7428	−1.40476	−0.702379	+	0.711803i	$0.747879\pi$
$884$	10.3451	0.347944
$885$	0	0
$886$	−1.97396	−0.0663165
$887$	2.72462	0.0914838	0.0457419	−	0.998953i	$-0.485435\pi$
$887$	2.72462	0.0914838	0.0457419	+	0.998953i	$0.485435\pi$
$888$	0	0
$889$	−29.5854	−0.992263
$890$	0.270139	0.00905508
$891$	0	0
$892$	−5.14345	−0.172215
$893$	−9.61410	−0.321724
$894$	0	0
$895$	3.90768	0.130619
$896$	57.2448	1.91241
$897$	0	0
$898$	49.4797	1.65116
$899$	69.5577	2.31988
$900$	0	0
$901$	−0.0456026	−0.00151924
$902$	0.868178	0.0289072
$903$	0	0
$904$	35.8946	1.19384
$905$	2.27139	0.0755037
$906$	0	0
$907$	−19.4428	−0.645587	−0.322794	−	0.946469i	$-0.604622\pi$
$907$	−19.4428	−0.645587	−0.322794	+	0.946469i	$0.604622\pi$
$908$	−1.13835	−0.0377775
$909$	0	0
$910$	−10.6045	−0.351537
$911$	−35.4707	−1.17520	−0.587598	−	0.809153i	$-0.699926\pi$
$911$	−35.4707	−1.17520	−0.587598	+	0.809153i	$0.699926\pi$
$912$	0	0
$913$	6.82655	0.225926
$914$	−19.9115	−0.658614
$915$	0	0
$916$	−6.44821	−0.213055
$917$	29.8131	0.984516
$918$	0	0
$919$	30.5669	1.00831	0.504154	−	0.863614i	$-0.331804\pi$
$919$	30.5669	1.00831	0.504154	+	0.863614i	$0.331804\pi$
$920$	6.73310	0.221984
$921$	0	0
$922$	55.2441	1.81937
$923$	−7.15799	−0.235608
$924$	0	0
$925$	−25.2654	−0.830723
$926$	−4.25092	−0.139694
$927$	0	0
$928$	14.3506	0.471082
$929$	59.7630	1.96076	0.980380	−	0.197116i	$-0.0631577\pi$
$929$	59.7630	1.96076	0.980380	+	0.197116i	$0.0631577\pi$
$930$	0	0
$931$	−47.9605	−1.57184
$932$	−4.53762	−0.148635
$933$	0	0
$934$	−39.6389	−1.29702
$935$	1.65247	0.0540416
$936$	0	0
$937$	29.4112	0.960824	0.480412	−	0.877043i	$-0.340487\pi$
$937$	29.4112	0.960824	0.480412	+	0.877043i	$0.340487\pi$
$938$	30.7574	1.00427
$939$	0	0
$940$	−0.295346	−0.00963313
$941$	−45.0066	−1.46717	−0.733587	−	0.679595i	$-0.762155\pi$
$941$	−45.0066	−1.46717	−0.733587	+	0.679595i	$0.762155\pi$
$942$	0	0
$943$	−4.96326	−0.161626
$944$	41.0879	1.33730
$945$	0	0
$946$	−2.62594	−0.0853765
$947$	10.4456	0.339437	0.169719	−	0.985493i	$-0.445714\pi$
$947$	10.4456	0.339437	0.169719	+	0.985493i	$0.445714\pi$
$948$	0	0
$949$	−49.8867	−1.61939
$950$	−32.9152	−1.06791
$951$	0	0
$952$	−56.7509	−1.83931
$953$	48.6114	1.57468	0.787339	−	0.616521i	$-0.211458\pi$
$953$	48.6114	1.57468	0.787339	+	0.616521i	$0.211458\pi$
$954$	0	0
$955$	3.11538	0.100811
$956$	−10.7809	−0.348680
$957$	0	0
$958$	−0.946131	−0.0305681
$959$	24.5475	0.792681
$960$	0	0
$961$	85.2180	2.74897
$962$	−38.5395	−1.24256
$963$	0	0
$964$	0.398765	0.0128434
$965$	0.425994	0.0137132
$966$	0	0
$967$	−28.8087	−0.926426	−0.463213	−	0.886247i	$-0.653303\pi$
$967$	−28.8087	−0.926426	−0.463213	+	0.886247i	$0.653303\pi$
$968$	−25.2016	−0.810011
$969$	0	0
$970$	−3.28648	−0.105523
$971$	4.01498	0.128847	0.0644234	−	0.997923i	$-0.479479\pi$
$971$	4.01498	0.128847	0.0644234	+	0.997923i	$0.479479\pi$
$972$	0	0
$973$	−9.97605	−0.319817
$974$	−41.2397	−1.32140
$975$	0	0
$976$	−17.5635	−0.562193
$977$	−4.44163	−0.142100	−0.0710501	−	0.997473i	$-0.522635\pi$
$977$	−4.44163	−0.142100	−0.0710501	+	0.997473i	$0.522635\pi$
$978$	0	0
$979$	−0.476670	−0.0152344
$980$	−1.47335	−0.0470645
$981$	0	0
$982$	−49.2294	−1.57097
$983$	16.5283	0.527172	0.263586	−	0.964636i	$-0.415095\pi$
$983$	16.5283	0.527172	0.263586	+	0.964636i	$0.415095\pi$
$984$	0	0
$985$	3.18623	0.101522
$986$	−53.8545	−1.71508
$987$	0	0
$988$	−8.34651	−0.265538
$989$	15.0121	0.477358
$990$	0	0
$991$	34.8094	1.10576	0.552878	−	0.833262i	$-0.313530\pi$
$991$	34.8094	1.10576	0.552878	+	0.833262i	$0.313530\pi$
$992$	23.9772	0.761278
$993$	0	0
$994$	−9.77890	−0.310168
$995$	6.18929	0.196214
$996$	0	0
$997$	−14.7239	−0.466312	−0.233156	−	0.972439i	$-0.574905\pi$
$997$	−14.7239	−0.466312	−0.233156	+	0.972439i	$0.574905\pi$
$998$	26.7118	0.845547
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.2.a.h.1.5		12
3.2	odd	2		241.2.a.b.1.8	✓	12
12.11	even	2		3856.2.a.n.1.2		12
15.14	odd	2		6025.2.a.h.1.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.8	✓	12	3.2	odd	2
2169.2.a.h.1.5		12	1.1	even	1	trivial
3856.2.a.n.1.2		12	12.11	even	2
6025.2.a.h.1.5		12	15.14	odd	2

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:	\( N \)	\(=\)	\( 2169 = 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2169.a (trivial)

\(f(q)\)	\(=\)	\(q-1.54879 q^{2} +0.398765 q^{4} -0.334961 q^{5} -4.24623 q^{7} +2.47998 q^{8} +O(q^{10})\)	\(q-1.54879 q^{2} +0.398765 q^{4} -0.334961 q^{5} -4.24623 q^{7} +2.47998 q^{8} +0.518786 q^{10} -0.915418 q^{11} +4.81392 q^{13} +6.57654 q^{14} -4.63852 q^{16} +5.38915 q^{17} -4.34799 q^{19} -0.133571 q^{20} +1.41779 q^{22} -8.10534 q^{23} -4.88780 q^{25} -7.45578 q^{26} -1.69325 q^{28} +6.45221 q^{29} +10.7804 q^{31} +2.22414 q^{32} -8.34668 q^{34} +1.42232 q^{35} +5.16908 q^{37} +6.73415 q^{38} -0.830699 q^{40} +0.612344 q^{41} -1.85213 q^{43} -0.365036 q^{44} +12.5535 q^{46} +2.21116 q^{47} +11.0305 q^{49} +7.57020 q^{50} +1.91962 q^{52} -0.00846193 q^{53} +0.306629 q^{55} -10.5306 q^{56} -9.99315 q^{58} -8.85799 q^{59} +3.78644 q^{61} -16.6967 q^{62} +5.83230 q^{64} -1.61248 q^{65} +4.67684 q^{67} +2.14900 q^{68} -2.20289 q^{70} -1.48694 q^{71} -10.3630 q^{73} -8.00585 q^{74} -1.73383 q^{76} +3.88708 q^{77} -17.6786 q^{79} +1.55372 q^{80} -0.948396 q^{82} -7.45731 q^{83} -1.80516 q^{85} +2.86857 q^{86} -2.27022 q^{88} +0.520713 q^{89} -20.4410 q^{91} -3.23212 q^{92} -3.42463 q^{94} +1.45641 q^{95} -6.33494 q^{97} -17.0840 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8} - 7 q^{10} - 22 q^{11} - 5 q^{13} - 6 q^{14} + 15 q^{16} + 4 q^{17} - 6 q^{19} - 10 q^{20} - 12 q^{22} - 32 q^{23} + 4 q^{25} - 8 q^{26} - 11 q^{28} - 6 q^{29} + 8 q^{31} - q^{32} - 19 q^{34} - 15 q^{35} - 8 q^{37} + 10 q^{38} - 52 q^{40} + q^{41} - 2 q^{43} - 42 q^{44} - 25 q^{46} - 34 q^{47} - 9 q^{49} + 27 q^{50} - 41 q^{52} - 5 q^{53} - 3 q^{55} - q^{56} - 33 q^{58} - 26 q^{59} - 26 q^{61} + 17 q^{62} + 13 q^{64} + 25 q^{65} + 6 q^{67} + 35 q^{68} - 4 q^{70} - 94 q^{71} - 22 q^{73} - 26 q^{74} - 20 q^{76} + 7 q^{77} + 9 q^{79} - 19 q^{80} + 15 q^{82} + 8 q^{83} + 4 q^{85} - 9 q^{86} + 6 q^{88} + 3 q^{89} - 20 q^{91} - 36 q^{92} + 48 q^{94} - 33 q^{95} - 29 q^{97} - 28 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8 - 7 * q^10 - 22 * q^11 - 5 * q^13 - 6 * q^14 + 15 * q^16 + 4 * q^17 - 6 * q^19 - 10 * q^20 - 12 * q^22 - 32 * q^23 + 4 * q^25 - 8 * q^26 - 11 * q^28 - 6 * q^29 + 8 * q^31 - q^32 - 19 * q^34 - 15 * q^35 - 8 * q^37 + 10 * q^38 - 52 * q^40 + q^41 - 2 * q^43 - 42 * q^44 - 25 * q^46 - 34 * q^47 - 9 * q^49 + 27 * q^50 - 41 * q^52 - 5 * q^53 - 3 * q^55 - q^56 - 33 * q^58 - 26 * q^59 - 26 * q^61 + 17 * q^62 + 13 * q^64 + 25 * q^65 + 6 * q^67 + 35 * q^68 - 4 * q^70 - 94 * q^71 - 22 * q^73 - 26 * q^74 - 20 * q^76 + 7 * q^77 + 9 * q^79 - 19 * q^80 + 15 * q^82 + 8 * q^83 + 4 * q^85 - 9 * q^86 + 6 * q^88 + 3 * q^89 - 20 * q^91 - 36 * q^92 + 48 * q^94 - 33 * q^95 - 29 * q^97 - 28 * q^98

Introduction

L-functions

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Varieties

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Database

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