LMFDB - Embedded newform 6561.2.a.d.1.55

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6561,2,Mod(1,6561)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6561, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6561.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6561 = 3^{8} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6561.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:	$52.3898487662$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.55
Character	$\chi$	$=$	6561.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.71009 q^{2} +0.924414 q^{4} -1.37341 q^{5} -0.310223 q^{7} -1.83935 q^{8} +O(q^{10})$	\(q+1.71009 q^{2} +0.924414 q^{4} -1.37341 q^{5} -0.310223 q^{7} -1.83935 q^{8} -2.34865 q^{10} +2.41088 q^{11} -4.61706 q^{13} -0.530509 q^{14} -4.99429 q^{16} -0.210571 q^{17} +3.48497 q^{19} -1.26960 q^{20} +4.12283 q^{22} +8.00996 q^{23} -3.11375 q^{25} -7.89560 q^{26} -0.286774 q^{28} -8.31375 q^{29} +0.801699 q^{31} -4.86199 q^{32} -0.360095 q^{34} +0.426062 q^{35} +10.8717 q^{37} +5.95961 q^{38} +2.52618 q^{40} -4.69755 q^{41} +7.43824 q^{43} +2.22865 q^{44} +13.6978 q^{46} -6.22438 q^{47} -6.90376 q^{49} -5.32480 q^{50} -4.26807 q^{52} +8.54006 q^{53} -3.31112 q^{55} +0.570609 q^{56} -14.2173 q^{58} +7.06338 q^{59} +10.0170 q^{61} +1.37098 q^{62} +1.67413 q^{64} +6.34111 q^{65} -1.25496 q^{67} -0.194654 q^{68} +0.728606 q^{70} -1.78789 q^{71} +13.4974 q^{73} +18.5917 q^{74} +3.22155 q^{76} -0.747910 q^{77} -3.16728 q^{79} +6.85919 q^{80} -8.03324 q^{82} -0.802261 q^{83} +0.289200 q^{85} +12.7201 q^{86} -4.43446 q^{88} +13.0496 q^{89} +1.43232 q^{91} +7.40452 q^{92} -10.6443 q^{94} -4.78628 q^{95} -6.67041 q^{97} -11.8061 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) $ 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8	$ 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38} + 72 q^{41} + 90 q^{44} + 90 q^{47} + 18 q^{49} + 45 q^{50} + 45 q^{53} + 9 q^{55} + 108 q^{56} + 18 q^{58} + 108 q^{59} + 72 q^{62} + 9 q^{64} + 72 q^{65} + 108 q^{68} + 126 q^{71} + 90 q^{74} + 72 q^{77} + 144 q^{80} - 18 q^{82} + 108 q^{83} + 90 q^{86} + 108 q^{89} + 72 q^{92} + 144 q^{95} + 81 q^{98}+O(q^{100}) $ 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8 + 36 * q^11 + 36 * q^14 + 45 * q^16 + 36 * q^17 + 54 * q^20 + 54 * q^23 + 36 * q^25 + 45 * q^26 + 9 * q^28 + 54 * q^29 + 63 * q^32 + 72 * q^35 + 54 * q^38 + 72 * q^41 + 90 * q^44 + 90 * q^47 + 18 * q^49 + 45 * q^50 + 45 * q^53 + 9 * q^55 + 108 * q^56 + 18 * q^58 + 108 * q^59 + 72 * q^62 + 9 * q^64 + 72 * q^65 + 108 * q^68 + 126 * q^71 + 90 * q^74 + 72 * q^77 + 144 * q^80 - 18 * q^82 + 108 * q^83 + 90 * q^86 + 108 * q^89 + 72 * q^92 + 144 * q^95 + 81 * 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 $

$2$	1.71009	1.20922	0.604609	−	0.796523i	$-0.293329\pi$
$2$	1.71009	1.20922	0.604609	+	0.796523i	$0.293329\pi$
$3$	0	0
$3$	0	0
$4$	0.924414	0.462207
$5$	−1.37341	−0.614207	−0.307103	−	0.951676i	$-0.599360\pi$
$5$	−1.37341	−0.614207	−0.307103	+	0.951676i	$0.599360\pi$
$6$	0	0
$7$	−0.310223	−0.117253	−0.0586266	−	0.998280i	$-0.518672\pi$
$7$	−0.310223	−0.117253	−0.0586266	+	0.998280i	$0.518672\pi$
$8$	−1.83935	−0.650309
$9$	0	0
$10$	−2.34865	−0.742709
$11$	2.41088	0.726908	0.363454	−	0.931612i	$-0.381597\pi$
$11$	2.41088	0.726908	0.363454	+	0.931612i	$0.381597\pi$
$12$	0	0
$13$	−4.61706	−1.28054	−0.640271	−	0.768149i	$-0.721178\pi$
$13$	−4.61706	−1.28054	−0.640271	+	0.768149i	$0.721178\pi$
$14$	−0.530509	−0.141785
$15$	0	0
$16$	−4.99429	−1.24857
$17$	−0.210571	−0.0510709	−0.0255355	−	0.999674i	$-0.508129\pi$
$17$	−0.210571	−0.0510709	−0.0255355	+	0.999674i	$0.508129\pi$
$18$	0	0
$19$	3.48497	0.799506	0.399753	−	0.916623i	$-0.369096\pi$
$19$	3.48497	0.799506	0.399753	+	0.916623i	$0.369096\pi$
$20$	−1.26960	−0.283890
$21$	0	0
$22$	4.12283	0.878990
$23$	8.00996	1.67019	0.835096	−	0.550104i	$-0.185412\pi$
$23$	8.00996	1.67019	0.835096	+	0.550104i	$0.185412\pi$
$24$	0	0
$25$	−3.11375	−0.622750
$26$	−7.89560	−1.54845
$27$	0	0
$28$	−0.286774	−0.0541952
$29$	−8.31375	−1.54382	−0.771912	−	0.635729i	$-0.780700\pi$
$29$	−8.31375	−1.54382	−0.771912	+	0.635729i	$0.780700\pi$
$30$	0	0
$31$	0.801699	0.143989	0.0719947	−	0.997405i	$-0.477064\pi$
$31$	0.801699	0.143989	0.0719947	+	0.997405i	$0.477064\pi$
$32$	−4.86199	−0.859486
$33$	0	0
$34$	−0.360095	−0.0617558
$35$	0.426062	0.0720177
$36$	0	0
$37$	10.8717	1.78730	0.893651	−	0.448763i	$-0.148135\pi$
$37$	10.8717	1.78730	0.893651	+	0.448763i	$0.148135\pi$
$38$	5.95961	0.966777
$39$	0	0
$40$	2.52618	0.399424
$41$	−4.69755	−0.733634	−0.366817	−	0.930293i	$-0.619552\pi$
$41$	−4.69755	−0.733634	−0.366817	+	0.930293i	$0.619552\pi$
$42$	0	0
$43$	7.43824	1.13432	0.567160	−	0.823608i	$-0.308042\pi$
$43$	7.43824	1.13432	0.567160	+	0.823608i	$0.308042\pi$
$44$	2.22865	0.335982
$45$	0	0
$46$	13.6978	2.01963
$47$	−6.22438	−0.907919	−0.453960	−	0.891022i	$-0.649989\pi$
$47$	−6.22438	−0.907919	−0.453960	+	0.891022i	$0.649989\pi$
$48$	0	0
$49$	−6.90376	−0.986252
$50$	−5.32480	−0.753040
$51$	0	0
$52$	−4.26807	−0.591875
$53$	8.54006	1.17307	0.586533	−	0.809925i	$-0.300492\pi$
$53$	8.54006	1.17307	0.586533	+	0.809925i	$0.300492\pi$
$54$	0	0
$55$	−3.31112	−0.446472
$56$	0.570609	0.0762508
$57$	0	0
$58$	−14.2173	−1.86682
$59$	7.06338	0.919574	0.459787	−	0.888029i	$-0.347926\pi$
$59$	7.06338	0.919574	0.459787	+	0.888029i	$0.347926\pi$
$60$	0	0
$61$	10.0170	1.28254	0.641272	−	0.767313i	$-0.278407\pi$
$61$	10.0170	1.28254	0.641272	+	0.767313i	$0.278407\pi$
$62$	1.37098	0.174114
$63$	0	0
$64$	1.67413	0.209267
$65$	6.34111	0.786518
$66$	0	0
$67$	−1.25496	−0.153317	−0.0766587	−	0.997057i	$-0.524425\pi$
$67$	−1.25496	−0.153317	−0.0766587	+	0.997057i	$0.524425\pi$
$68$	−0.194654	−0.0236053
$69$	0	0
$70$	0.728606	0.0870850
$71$	−1.78789	−0.212183	−0.106092	−	0.994356i	$-0.533834\pi$
$71$	−1.78789	−0.212183	−0.106092	+	0.994356i	$0.533834\pi$
$72$	0	0
$73$	13.4974	1.57976	0.789878	−	0.613264i	$-0.210144\pi$
$73$	13.4974	1.57976	0.789878	+	0.613264i	$0.210144\pi$
$74$	18.5917	2.16124
$75$	0	0
$76$	3.22155	0.369537
$77$	−0.747910	−0.0852323
$78$	0	0
$79$	−3.16728	−0.356346	−0.178173	−	0.983999i	$-0.557019\pi$
$79$	−3.16728	−0.356346	−0.178173	+	0.983999i	$0.557019\pi$
$80$	6.85919	0.766881
$81$	0	0
$82$	−8.03324	−0.887123
$83$	−0.802261	−0.0880596	−0.0440298	−	0.999030i	$-0.514020\pi$
$83$	−0.802261	−0.0880596	−0.0440298	+	0.999030i	$0.514020\pi$
$84$	0	0
$85$	0.289200	0.0313681
$86$	12.7201	1.37164
$87$	0	0
$88$	−4.43446	−0.472715
$89$	13.0496	1.38325	0.691627	−	0.722255i	$-0.256894\pi$
$89$	13.0496	1.38325	0.691627	+	0.722255i	$0.256894\pi$
$90$	0	0
$91$	1.43232	0.150148
$92$	7.40452	0.771974
$93$	0	0
$94$	−10.6443	−1.09787
$95$	−4.78628	−0.491062
$96$	0	0
$97$	−6.67041	−0.677278	−0.338639	−	0.940916i	$-0.609966\pi$
$97$	−6.67041	−0.677278	−0.338639	+	0.940916i	$0.609966\pi$
$98$	−11.8061	−1.19259
$99$	0	0
$100$	−2.87839	−0.287839
$101$	9.32010	0.927385	0.463692	−	0.885996i	$-0.346524\pi$
$101$	9.32010	0.927385	0.463692	+	0.885996i	$0.346524\pi$
$102$	0	0
$103$	−1.31303	−0.129377	−0.0646883	−	0.997906i	$-0.520605\pi$
$103$	−1.31303	−0.129377	−0.0646883	+	0.997906i	$0.520605\pi$
$104$	8.49240	0.832748
$105$	0	0
$106$	14.6043	1.41849
$107$	0.773710	0.0747973	0.0373987	−	0.999300i	$-0.488093\pi$
$107$	0.773710	0.0747973	0.0373987	+	0.999300i	$0.488093\pi$
$108$	0	0
$109$	−2.06208	−0.197512	−0.0987559	−	0.995112i	$-0.531486\pi$
$109$	−2.06208	−0.197512	−0.0987559	+	0.995112i	$0.531486\pi$
$110$	−5.66232	−0.539881
$111$	0	0
$112$	1.54934	0.146399
$113$	5.87773	0.552930	0.276465	−	0.961024i	$-0.410837\pi$
$113$	5.87773	0.552930	0.276465	+	0.961024i	$0.410837\pi$
$114$	0	0
$115$	−11.0009	−1.02584
$116$	−7.68534	−0.713566
$117$	0	0
$118$	12.0790	1.11197
$119$	0.0653238	0.00598823
$120$	0	0
$121$	−5.18765	−0.471605
$122$	17.1300	1.55088
$123$	0	0
$124$	0.741101	0.0665529
$125$	11.1435	0.996704
$126$	0	0
$127$	10.4543	0.927673	0.463837	−	0.885921i	$-0.346472\pi$
$127$	10.4543	0.927673	0.463837	+	0.885921i	$0.346472\pi$
$128$	12.5869	1.11253
$129$	0	0
$130$	10.8439	0.951071
$131$	5.98958	0.523312	0.261656	−	0.965161i	$-0.415731\pi$
$131$	5.98958	0.523312	0.261656	+	0.965161i	$0.415731\pi$
$132$	0	0
$133$	−1.08112	−0.0937446
$134$	−2.14609	−0.185394
$135$	0	0
$136$	0.387314	0.0332119
$137$	−1.16170	−0.0992504	−0.0496252	−	0.998768i	$-0.515803\pi$
$137$	−1.16170	−0.0992504	−0.0496252	+	0.998768i	$0.515803\pi$
$138$	0	0
$139$	16.3083	1.38326	0.691628	−	0.722254i	$-0.256894\pi$
$139$	16.3083	1.38326	0.691628	+	0.722254i	$0.256894\pi$
$140$	0.393858	0.0332871
$141$	0	0
$142$	−3.05746	−0.256576
$143$	−11.1312	−0.930837
$144$	0	0
$145$	11.4182	0.948227
$146$	23.0819	1.91027
$147$	0	0
$148$	10.0500	0.826103
$149$	8.92857	0.731457	0.365728	−	0.930722i	$-0.380820\pi$
$149$	8.92857	0.731457	0.365728	+	0.930722i	$0.380820\pi$
$150$	0	0
$151$	−15.5945	−1.26906	−0.634531	−	0.772897i	$-0.718807\pi$
$151$	−15.5945	−1.26906	−0.634531	+	0.772897i	$0.718807\pi$
$152$	−6.41008	−0.519926
$153$	0	0
$154$	−1.27899	−0.103064
$155$	−1.10106	−0.0884392
$156$	0	0
$157$	12.2390	0.976775	0.488387	−	0.872627i	$-0.337585\pi$
$157$	12.2390	0.976775	0.488387	+	0.872627i	$0.337585\pi$
$158$	−5.41633	−0.430900
$159$	0	0
$160$	6.67749	0.527902
$161$	−2.48487	−0.195835
$162$	0	0
$163$	−25.0816	−1.96454	−0.982271	−	0.187465i	$-0.939973\pi$
$163$	−25.0816	−1.96454	−0.982271	+	0.187465i	$0.939973\pi$
$164$	−4.34248	−0.339090
$165$	0	0
$166$	−1.37194	−0.106483
$167$	−8.05344	−0.623194	−0.311597	−	0.950214i	$-0.600864\pi$
$167$	−8.05344	−0.623194	−0.311597	+	0.950214i	$0.600864\pi$
$168$	0	0
$169$	8.31726	0.639789
$170$	0.494558	0.0379308
$171$	0	0
$172$	6.87601	0.524291
$173$	17.7868	1.35230	0.676152	−	0.736762i	$-0.263646\pi$
$173$	17.7868	1.35230	0.676152	+	0.736762i	$0.263646\pi$
$174$	0	0
$175$	0.965956	0.0730194
$176$	−12.0406	−0.907597
$177$	0	0
$178$	22.3160	1.67265
$179$	17.7190	1.32438	0.662190	−	0.749336i	$-0.269627\pi$
$179$	17.7190	1.32438	0.662190	+	0.749336i	$0.269627\pi$
$180$	0	0
$181$	26.7778	1.99038	0.995188	−	0.0979819i	$-0.0312387\pi$
$181$	26.7778	1.99038	0.995188	+	0.0979819i	$0.0312387\pi$
$182$	2.44939	0.181561
$183$	0	0
$184$	−14.7331	−1.08614
$185$	−14.9313	−1.09777
$186$	0	0
$187$	−0.507661	−0.0371239
$188$	−5.75390	−0.419646
$189$	0	0
$190$	−8.18498	−0.593801
$191$	3.13526	0.226860	0.113430	−	0.993546i	$-0.463816\pi$
$191$	3.13526	0.226860	0.113430	+	0.993546i	$0.463816\pi$
$192$	0	0
$193$	9.88041	0.711207	0.355604	−	0.934637i	$-0.384275\pi$
$193$	9.88041	0.711207	0.355604	+	0.934637i	$0.384275\pi$
$194$	−11.4070	−0.818976
$195$	0	0
$196$	−6.38193	−0.455852
$197$	−2.03866	−0.145249	−0.0726243	−	0.997359i	$-0.523137\pi$
$197$	−2.03866	−0.145249	−0.0726243	+	0.997359i	$0.523137\pi$
$198$	0	0
$199$	−13.0695	−0.926471	−0.463235	−	0.886235i	$-0.653312\pi$
$199$	−13.0695	−0.926471	−0.463235	+	0.886235i	$0.653312\pi$
$200$	5.72728	0.404980
$201$	0	0
$202$	15.9382	1.12141
$203$	2.57911	0.181018
$204$	0	0
$205$	6.45165	0.450603
$206$	−2.24540	−0.156445
$207$	0	0
$208$	23.0589	1.59885
$209$	8.40184	0.581167
$210$	0	0
$211$	−15.8736	−1.09278	−0.546392	−	0.837530i	$-0.683999\pi$
$211$	−15.8736	−1.09278	−0.546392	+	0.837530i	$0.683999\pi$
$212$	7.89454	0.542199
$213$	0	0
$214$	1.32311	0.0904463
$215$	−10.2157	−0.696707
$216$	0	0
$217$	−0.248705	−0.0168832
$218$	−3.52635	−0.238835
$219$	0	0
$220$	−3.06085	−0.206362
$221$	0.972218	0.0653985
$222$	0	0
$223$	−1.89563	−0.126941	−0.0634705	−	0.997984i	$-0.520217\pi$
$223$	−1.89563	−0.126941	−0.0634705	+	0.997984i	$0.520217\pi$
$224$	1.50830	0.100777
$225$	0	0
$226$	10.0514	0.668612
$227$	25.5574	1.69631	0.848154	−	0.529750i	$-0.177714\pi$
$227$	25.5574	1.69631	0.848154	+	0.529750i	$0.177714\pi$
$228$	0	0
$229$	−4.69156	−0.310027	−0.155014	−	0.987912i	$-0.549542\pi$
$229$	−4.69156	−0.310027	−0.155014	+	0.987912i	$0.549542\pi$
$230$	−18.8126	−1.24047
$231$	0	0
$232$	15.2919	1.00396
$233$	−11.4618	−0.750890	−0.375445	−	0.926845i	$-0.622510\pi$
$233$	−11.4618	−0.750890	−0.375445	+	0.926845i	$0.622510\pi$
$234$	0	0
$235$	8.54861	0.557650
$236$	6.52949	0.425033
$237$	0	0
$238$	0.111710	0.00724107
$239$	11.7465	0.759817	0.379908	−	0.925024i	$-0.375956\pi$
$239$	11.7465	0.759817	0.379908	+	0.925024i	$0.375956\pi$
$240$	0	0
$241$	4.41893	0.284648	0.142324	−	0.989820i	$-0.454543\pi$
$241$	4.41893	0.284648	0.142324	+	0.989820i	$0.454543\pi$
$242$	−8.87136	−0.570273
$243$	0	0
$244$	9.25985	0.592801
$245$	9.48168	0.605762
$246$	0	0
$247$	−16.0903	−1.02380
$248$	−1.47461	−0.0936376
$249$	0	0
$250$	19.0564	1.20523
$251$	−6.27946	−0.396356	−0.198178	−	0.980166i	$-0.563502\pi$
$251$	−6.27946	−0.396356	−0.198178	+	0.980166i	$0.563502\pi$
$252$	0	0
$253$	19.3111	1.21408
$254$	17.8779	1.12176
$255$	0	0
$256$	18.1765	1.13603
$257$	2.36392	0.147457	0.0737287	−	0.997278i	$-0.476510\pi$
$257$	2.36392	0.147457	0.0737287	+	0.997278i	$0.476510\pi$
$258$	0	0
$259$	−3.37266	−0.209567
$260$	5.86181	0.363534
$261$	0	0
$262$	10.2427	0.632798
$263$	13.9963	0.863050	0.431525	−	0.902101i	$-0.357976\pi$
$263$	13.9963	0.863050	0.431525	+	0.902101i	$0.357976\pi$
$264$	0	0
$265$	−11.7290	−0.720505
$266$	−1.84881	−0.113358
$267$	0	0
$268$	−1.16010	−0.0708643
$269$	25.7259	1.56854	0.784268	−	0.620422i	$-0.213039\pi$
$269$	25.7259	1.56854	0.784268	+	0.620422i	$0.213039\pi$
$270$	0	0
$271$	13.0450	0.792425	0.396212	−	0.918159i	$-0.370324\pi$
$271$	13.0450	0.792425	0.396212	+	0.918159i	$0.370324\pi$
$272$	1.05165	0.0637657
$273$	0	0
$274$	−1.98661	−0.120015
$275$	−7.50688	−0.452682
$276$	0	0
$277$	5.21823	0.313533	0.156767	−	0.987636i	$-0.449893\pi$
$277$	5.21823	0.313533	0.156767	+	0.987636i	$0.449893\pi$
$278$	27.8888	1.67266
$279$	0	0
$280$	−0.783678	−0.0468337
$281$	5.67994	0.338837	0.169418	−	0.985544i	$-0.445811\pi$
$281$	5.67994	0.338837	0.169418	+	0.985544i	$0.445811\pi$
$282$	0	0
$283$	−11.3546	−0.674960	−0.337480	−	0.941333i	$-0.609574\pi$
$283$	−11.3546	−0.674960	−0.337480	+	0.941333i	$0.609574\pi$
$284$	−1.65275	−0.0980726
$285$	0	0
$286$	−19.0353	−1.12558
$287$	1.45729	0.0860209
$288$	0	0
$289$	−16.9557	−0.997392
$290$	19.5261	1.14661
$291$	0	0
$292$	12.4772	0.730174
$293$	−17.2808	−1.00955	−0.504777	−	0.863250i	$-0.668425\pi$
$293$	−17.2808	−1.00955	−0.504777	+	0.863250i	$0.668425\pi$
$294$	0	0
$295$	−9.70090	−0.564808
$296$	−19.9969	−1.16230
$297$	0	0
$298$	15.2687	0.884490
$299$	−36.9825	−2.13875
$300$	0	0
$301$	−2.30751	−0.133003
$302$	−26.6680	−1.53457
$303$	0	0
$304$	−17.4049	−0.998240
$305$	−13.7574	−0.787748
$306$	0	0
$307$	−8.41282	−0.480145	−0.240073	−	0.970755i	$-0.577171\pi$
$307$	−8.41282	−0.480145	−0.240073	+	0.970755i	$0.577171\pi$
$308$	−0.691378	−0.0393949
$309$	0	0
$310$	−1.88291	−0.106942
$311$	−6.55540	−0.371723	−0.185861	−	0.982576i	$-0.559508\pi$
$311$	−6.55540	−0.371723	−0.185861	+	0.982576i	$0.559508\pi$
$312$	0	0
$313$	5.45209	0.308171	0.154085	−	0.988058i	$-0.450757\pi$
$313$	5.45209	0.308171	0.154085	+	0.988058i	$0.450757\pi$
$314$	20.9297	1.18113
$315$	0	0
$316$	−2.92787	−0.164706
$317$	3.59532	0.201933	0.100967	−	0.994890i	$-0.467806\pi$
$317$	3.59532	0.201933	0.100967	+	0.994890i	$0.467806\pi$
$318$	0	0
$319$	−20.0435	−1.12222
$320$	−2.29927	−0.128533
$321$	0	0
$322$	−4.24936	−0.236808
$323$	−0.733832	−0.0408315
$324$	0	0
$325$	14.3764	0.797458
$326$	−42.8918	−2.37556
$327$	0	0
$328$	8.64044	0.477089
$329$	1.93094	0.106456
$330$	0	0
$331$	−26.8071	−1.47345	−0.736726	−	0.676191i	$-0.763629\pi$
$331$	−26.8071	−1.47345	−0.736726	+	0.676191i	$0.763629\pi$
$332$	−0.741621	−0.0407018
$333$	0	0
$334$	−13.7721	−0.753577
$335$	1.72357	0.0941685
$336$	0	0
$337$	21.6174	1.17757	0.588787	−	0.808288i	$-0.299606\pi$
$337$	21.6174	1.17757	0.588787	+	0.808288i	$0.299606\pi$
$338$	14.2233	0.773644
$339$	0	0
$340$	0.267340	0.0144985
$341$	1.93280	0.104667
$342$	0	0
$343$	4.31326	0.232894
$344$	−13.6815	−0.737659
$345$	0	0
$346$	30.4170	1.63523
$347$	−16.8422	−0.904139	−0.452070	−	0.891983i	$-0.649314\pi$
$347$	−16.8422	−0.904139	−0.452070	+	0.891983i	$0.649314\pi$
$348$	0	0
$349$	−1.97127	−0.105519	−0.0527597	−	0.998607i	$-0.516802\pi$
$349$	−1.97127	−0.105519	−0.0527597	+	0.998607i	$0.516802\pi$
$350$	1.65187	0.0882964
$351$	0	0
$352$	−11.7217	−0.624767
$353$	−5.81304	−0.309397	−0.154698	−	0.987962i	$-0.549441\pi$
$353$	−5.81304	−0.309397	−0.154698	+	0.987962i	$0.549441\pi$
$354$	0	0
$355$	2.45550	0.130324
$356$	12.0632	0.639349
$357$	0	0
$358$	30.3011	1.60146
$359$	−19.4999	−1.02916	−0.514581	−	0.857442i	$-0.672053\pi$
$359$	−19.4999	−1.02916	−0.514581	+	0.857442i	$0.672053\pi$
$360$	0	0
$361$	−6.85501	−0.360790
$362$	45.7925	2.40680
$363$	0	0
$364$	1.32405	0.0693993
$365$	−18.5375	−0.970296
$366$	0	0
$367$	−18.8912	−0.986113	−0.493056	−	0.869997i	$-0.664120\pi$
$367$	−18.8912	−0.986113	−0.493056	+	0.869997i	$0.664120\pi$
$368$	−40.0041	−2.08536
$369$	0	0
$370$	−25.5339	−1.32745
$371$	−2.64932	−0.137546
$372$	0	0
$373$	−22.2954	−1.15441	−0.577205	−	0.816599i	$-0.695857\pi$
$373$	−22.2954	−1.15441	−0.577205	+	0.816599i	$0.695857\pi$
$374$	−0.868147	−0.0448908
$375$	0	0
$376$	11.4488	0.590428
$377$	38.3851	1.97693
$378$	0	0
$379$	−11.9421	−0.613426	−0.306713	−	0.951802i	$-0.599229\pi$
$379$	−11.9421	−0.613426	−0.306713	+	0.951802i	$0.599229\pi$
$380$	−4.42450	−0.226972
$381$	0	0
$382$	5.36159	0.274323
$383$	−20.9096	−1.06843	−0.534216	−	0.845348i	$-0.679393\pi$
$383$	−20.9096	−1.06843	−0.534216	+	0.845348i	$0.679393\pi$
$384$	0	0
$385$	1.02719	0.0523502
$386$	16.8964	0.860004
$387$	0	0
$388$	−6.16622	−0.313042
$389$	−9.60176	−0.486828	−0.243414	−	0.969922i	$-0.578267\pi$
$389$	−9.60176	−0.486828	−0.243414	+	0.969922i	$0.578267\pi$
$390$	0	0
$391$	−1.68666	−0.0852983
$392$	12.6984	0.641368
$393$	0	0
$394$	−3.48630	−0.175637
$395$	4.34996	0.218870
$396$	0	0
$397$	−2.05681	−0.103228	−0.0516142	−	0.998667i	$-0.516437\pi$
$397$	−2.05681	−0.103228	−0.0516142	+	0.998667i	$0.516437\pi$
$398$	−22.3500	−1.12030
$399$	0	0
$400$	15.5510	0.777548
$401$	−0.676919	−0.0338037	−0.0169019	−	0.999857i	$-0.505380\pi$
$401$	−0.676919	−0.0338037	−0.0169019	+	0.999857i	$0.505380\pi$
$402$	0	0
$403$	−3.70149	−0.184385
$404$	8.61563	0.428644
$405$	0	0
$406$	4.41052	0.218891
$407$	26.2105	1.29920
$408$	0	0
$409$	17.1534	0.848182	0.424091	−	0.905619i	$-0.360594\pi$
$409$	17.1534	0.848182	0.424091	+	0.905619i	$0.360594\pi$
$410$	11.0329	0.544877
$411$	0	0
$412$	−1.21378	−0.0597988
$413$	−2.19122	−0.107823
$414$	0	0
$415$	1.10183	0.0540868
$416$	22.4481	1.10061
$417$	0	0
$418$	14.3679	0.702758
$419$	28.0875	1.37216	0.686081	−	0.727525i	$-0.259329\pi$
$419$	28.0875	1.37216	0.686081	+	0.727525i	$0.259329\pi$
$420$	0	0
$421$	−10.0492	−0.489766	−0.244883	−	0.969553i	$-0.578750\pi$
$421$	−10.0492	−0.489766	−0.244883	+	0.969553i	$0.578750\pi$
$422$	−27.1453	−1.32141
$423$	0	0
$424$	−15.7082	−0.762856
$425$	0.655665	0.0318044
$426$	0	0
$427$	−3.10750	−0.150382
$428$	0.715228	0.0345718
$429$	0	0
$430$	−17.4698	−0.842470
$431$	−5.34760	−0.257585	−0.128792	−	0.991672i	$-0.541110\pi$
$431$	−5.34760	−0.257585	−0.128792	+	0.991672i	$0.541110\pi$
$432$	0	0
$433$	−6.89372	−0.331291	−0.165646	−	0.986185i	$-0.552971\pi$
$433$	−6.89372	−0.331291	−0.165646	+	0.986185i	$0.552971\pi$
$434$	−0.425309	−0.0204155
$435$	0	0
$436$	−1.90622	−0.0912913
$437$	27.9144	1.33533
$438$	0	0
$439$	15.5725	0.743235	0.371618	−	0.928386i	$-0.378803\pi$
$439$	15.5725	0.743235	0.371618	+	0.928386i	$0.378803\pi$
$440$	6.09032	0.290345
$441$	0	0
$442$	1.66258	0.0790810
$443$	27.9543	1.32815	0.664074	−	0.747667i	$-0.268826\pi$
$443$	27.9543	1.32815	0.664074	+	0.747667i	$0.268826\pi$
$444$	0	0
$445$	−17.9224	−0.849604
$446$	−3.24170	−0.153499
$447$	0	0
$448$	−0.519354	−0.0245372
$449$	10.9354	0.516073	0.258037	−	0.966135i	$-0.416924\pi$
$449$	10.9354	0.516073	0.258037	+	0.966135i	$0.416924\pi$
$450$	0	0
$451$	−11.3252	−0.533284
$452$	5.43345	0.255568
$453$	0	0
$454$	43.7056	2.05120
$455$	−1.96716	−0.0922217
$456$	0	0
$457$	−16.4308	−0.768599	−0.384300	−	0.923208i	$-0.625557\pi$
$457$	−16.4308	−0.768599	−0.384300	+	0.923208i	$0.625557\pi$
$458$	−8.02300	−0.374890
$459$	0	0
$460$	−10.1694	−0.474152
$461$	25.0016	1.16444	0.582220	−	0.813032i	$-0.302184\pi$
$461$	25.0016	1.16444	0.582220	+	0.813032i	$0.302184\pi$
$462$	0	0
$463$	13.5651	0.630422	0.315211	−	0.949022i	$-0.397925\pi$
$463$	13.5651	0.630422	0.315211	+	0.949022i	$0.397925\pi$
$464$	41.5213	1.92758
$465$	0	0
$466$	−19.6008	−0.907989
$467$	34.3124	1.58779	0.793895	−	0.608055i	$-0.208050\pi$
$467$	34.3124	1.58779	0.793895	+	0.608055i	$0.208050\pi$
$468$	0	0
$469$	0.389316	0.0179769
$470$	14.6189	0.674320
$471$	0	0
$472$	−12.9920	−0.598007
$473$	17.9327	0.824546
$474$	0	0
$475$	−10.8513	−0.497893
$476$	0.0603862	0.00276780
$477$	0	0
$478$	20.0876	0.918783
$479$	21.6619	0.989758	0.494879	−	0.868962i	$-0.335212\pi$
$479$	21.6619	0.989758	0.494879	+	0.868962i	$0.335212\pi$
$480$	0	0
$481$	−50.1955	−2.28872
$482$	7.55677	0.344201
$483$	0	0
$484$	−4.79554	−0.217979
$485$	9.16119	0.415988
$486$	0	0
$487$	38.8502	1.76047	0.880235	−	0.474538i	$-0.157385\pi$
$487$	38.8502	1.76047	0.880235	+	0.474538i	$0.157385\pi$
$488$	−18.4248	−0.834050
$489$	0	0
$490$	16.2145	0.732498
$491$	−29.2827	−1.32151	−0.660756	−	0.750601i	$-0.729764\pi$
$491$	−29.2827	−1.32151	−0.660756	+	0.750601i	$0.729764\pi$
$492$	0	0
$493$	1.75063	0.0788445
$494$	−27.5159	−1.23800
$495$	0	0
$496$	−4.00391	−0.179781
$497$	0.554644	0.0248792
$498$	0	0
$499$	−27.4739	−1.22990	−0.614950	−	0.788566i	$-0.710824\pi$
$499$	−27.4739	−1.22990	−0.614950	+	0.788566i	$0.710824\pi$
$500$	10.3012	0.460683
$501$	0	0
$502$	−10.7385	−0.479281
$503$	32.9193	1.46780	0.733901	−	0.679257i	$-0.237698\pi$
$503$	32.9193	1.46780	0.733901	+	0.679257i	$0.237698\pi$
$504$	0	0
$505$	−12.8003	−0.569606
$506$	33.0237	1.46808
$507$	0	0
$508$	9.66414	0.428777
$509$	−5.13391	−0.227556	−0.113778	−	0.993506i	$-0.536295\pi$
$509$	−5.13391	−0.227556	−0.113778	+	0.993506i	$0.536295\pi$
$510$	0	0
$511$	−4.18721	−0.185231
$512$	5.90965	0.261172
$513$	0	0
$514$	4.04253	0.178308
$515$	1.80333	0.0794640
$516$	0	0
$517$	−15.0062	−0.659974
$518$	−5.76755	−0.253412
$519$	0	0
$520$	−11.6635	−0.511479
$521$	7.62191	0.333922	0.166961	−	0.985964i	$-0.446605\pi$
$521$	7.62191	0.333922	0.166961	+	0.985964i	$0.446605\pi$
$522$	0	0
$523$	−11.1585	−0.487928	−0.243964	−	0.969784i	$-0.578448\pi$
$523$	−11.1585	−0.487928	−0.243964	+	0.969784i	$0.578448\pi$
$524$	5.53685	0.241879
$525$	0	0
$526$	23.9350	1.04362
$527$	−0.168814	−0.00735367
$528$	0	0
$529$	41.1595	1.78954
$530$	−20.0576	−0.871248
$531$	0	0
$532$	−0.999398	−0.0433294
$533$	21.6889	0.939449
$534$	0	0
$535$	−1.06262	−0.0459410
$536$	2.30831	0.0997036
$537$	0	0
$538$	43.9937	1.89670
$539$	−16.6441	−0.716914
$540$	0	0
$541$	−25.7992	−1.10920	−0.554598	−	0.832119i	$-0.687128\pi$
$541$	−25.7992	−1.10920	−0.554598	+	0.832119i	$0.687128\pi$
$542$	22.3081	0.958214
$543$	0	0
$544$	1.02379	0.0438947
$545$	2.83208	0.121313
$546$	0	0
$547$	36.1493	1.54563	0.772815	−	0.634631i	$-0.218848\pi$
$547$	36.1493	1.54563	0.772815	+	0.634631i	$0.218848\pi$
$548$	−1.07389	−0.0458742
$549$	0	0
$550$	−12.8375	−0.547391
$551$	−28.9731	−1.23430
$552$	0	0
$553$	0.982561	0.0417827
$554$	8.92365	0.379130
$555$	0	0
$556$	15.0757	0.639350
$557$	33.3273	1.41212	0.706061	−	0.708151i	$-0.250470\pi$
$557$	33.3273	1.41212	0.706061	+	0.708151i	$0.250470\pi$
$558$	0	0
$559$	−34.3428	−1.45255
$560$	−2.12788	−0.0899192
$561$	0	0
$562$	9.71321	0.409727
$563$	−34.6506	−1.46035	−0.730175	−	0.683260i	$-0.760562\pi$
$563$	−34.6506	−1.46035	−0.730175	+	0.683260i	$0.760562\pi$
$564$	0	0
$565$	−8.07251	−0.339613
$566$	−19.4174	−0.816173
$567$	0	0
$568$	3.28856	0.137985
$569$	−32.8623	−1.37766	−0.688829	−	0.724924i	$-0.741875\pi$
$569$	−32.8623	−1.37766	−0.688829	+	0.724924i	$0.741875\pi$
$570$	0	0
$571$	−15.2636	−0.638763	−0.319382	−	0.947626i	$-0.603475\pi$
$571$	−15.2636	−0.638763	−0.319382	+	0.947626i	$0.603475\pi$
$572$	−10.2898	−0.430239
$573$	0	0
$574$	2.49209	0.104018
$575$	−24.9410	−1.04011
$576$	0	0
$577$	−9.58031	−0.398834	−0.199417	−	0.979915i	$-0.563905\pi$
$577$	−9.58031	−0.398834	−0.199417	+	0.979915i	$0.563905\pi$
$578$	−28.9957	−1.20606
$579$	0	0
$580$	10.5551	0.438277
$581$	0.248880	0.0103253
$582$	0	0
$583$	20.5891	0.852712
$584$	−24.8265	−1.02733
$585$	0	0
$586$	−29.5517	−1.22077
$587$	28.5211	1.17719	0.588596	−	0.808428i	$-0.299681\pi$
$587$	28.5211	1.17719	0.588596	+	0.808428i	$0.299681\pi$
$588$	0	0
$589$	2.79389	0.115120
$590$	−16.5894	−0.682976
$591$	0	0
$592$	−54.2965	−2.23157
$593$	−42.2461	−1.73484	−0.867419	−	0.497578i	$-0.834223\pi$
$593$	−42.2461	−1.73484	−0.867419	+	0.497578i	$0.834223\pi$
$594$	0	0
$595$	−0.0897163	−0.00367801
$596$	8.25369	0.338084
$597$	0	0
$598$	−63.2435	−2.58622
$599$	17.9048	0.731572	0.365786	−	0.930699i	$-0.380800\pi$
$599$	17.9048	0.731572	0.365786	+	0.930699i	$0.380800\pi$
$600$	0	0
$601$	−10.0265	−0.408991	−0.204496	−	0.978867i	$-0.565555\pi$
$601$	−10.0265	−0.408991	−0.204496	+	0.978867i	$0.565555\pi$
$602$	−3.94605	−0.160829
$603$	0	0
$604$	−14.4158	−0.586569
$605$	7.12476	0.289663
$606$	0	0
$607$	−11.9514	−0.485091	−0.242545	−	0.970140i	$-0.577982\pi$
$607$	−11.9514	−0.485091	−0.242545	+	0.970140i	$0.577982\pi$
$608$	−16.9439	−0.687164
$609$	0	0
$610$	−23.5264	−0.952558
$611$	28.7384	1.16263
$612$	0	0
$613$	−39.2924	−1.58700	−0.793502	−	0.608567i	$-0.791745\pi$
$613$	−39.2924	−1.58700	−0.793502	+	0.608567i	$0.791745\pi$
$614$	−14.3867	−0.580600
$615$	0	0
$616$	1.37567	0.0554273
$617$	40.8753	1.64558	0.822788	−	0.568348i	$-0.192417\pi$
$617$	40.8753	1.64558	0.822788	+	0.568348i	$0.192417\pi$
$618$	0	0
$619$	25.5493	1.02691	0.513457	−	0.858116i	$-0.328365\pi$
$619$	25.5493	1.02691	0.513457	+	0.858116i	$0.328365\pi$
$620$	−1.01783	−0.0408772
$621$	0	0
$622$	−11.2103	−0.449493
$623$	−4.04828	−0.162191
$624$	0	0
$625$	0.264202	0.0105681
$626$	9.32358	0.372645
$627$	0	0
$628$	11.3139	0.451472
$629$	−2.28927	−0.0912791
$630$	0	0
$631$	−4.04144	−0.160887	−0.0804436	−	0.996759i	$-0.525634\pi$
$631$	−4.04144	−0.160887	−0.0804436	+	0.996759i	$0.525634\pi$
$632$	5.82573	0.231735
$633$	0	0
$634$	6.14832	0.244181
$635$	−14.3581	−0.569783
$636$	0	0
$637$	31.8751	1.26294
$638$	−34.2762	−1.35701
$639$	0	0
$640$	−17.2869	−0.683326
$641$	−11.8251	−0.467062	−0.233531	−	0.972349i	$-0.575028\pi$
$641$	−11.8251	−0.467062	−0.233531	+	0.972349i	$0.575028\pi$
$642$	0	0
$643$	−20.2547	−0.798767	−0.399384	−	0.916784i	$-0.630776\pi$
$643$	−20.2547	−0.798767	−0.399384	+	0.916784i	$0.630776\pi$
$644$	−2.29705	−0.0905164
$645$	0	0
$646$	−1.25492	−0.0493742
$647$	−25.4173	−0.999257	−0.499629	−	0.866240i	$-0.666530\pi$
$647$	−25.4173	−0.999257	−0.499629	+	0.866240i	$0.666530\pi$
$648$	0	0
$649$	17.0290	0.668446
$650$	24.5849	0.964300
$651$	0	0
$652$	−23.1858	−0.908025
$653$	−6.53831	−0.255864	−0.127932	−	0.991783i	$-0.540834\pi$
$653$	−6.53831	−0.255864	−0.127932	+	0.991783i	$0.540834\pi$
$654$	0	0
$655$	−8.22614	−0.321422
$656$	23.4609	0.915994
$657$	0	0
$658$	3.30209	0.128729
$659$	24.9420	0.971602	0.485801	−	0.874070i	$-0.338528\pi$
$659$	24.9420	0.971602	0.485801	+	0.874070i	$0.338528\pi$
$660$	0	0
$661$	−45.0257	−1.75130	−0.875649	−	0.482948i	$-0.839566\pi$
$661$	−45.0257	−1.75130	−0.875649	+	0.482948i	$0.839566\pi$
$662$	−45.8426	−1.78172
$663$	0	0
$664$	1.47564	0.0572660
$665$	1.48481	0.0575786
$666$	0	0
$667$	−66.5928	−2.57849
$668$	−7.44471	−0.288044
$669$	0	0
$670$	2.94746	0.113870
$671$	24.1498	0.932292
$672$	0	0
$673$	12.1318	0.467647	0.233824	−	0.972279i	$-0.424876\pi$
$673$	12.1318	0.467647	0.233824	+	0.972279i	$0.424876\pi$
$674$	36.9677	1.42394
$675$	0	0
$676$	7.68859	0.295715
$677$	5.98168	0.229895	0.114947	−	0.993372i	$-0.463330\pi$
$677$	5.98168	0.229895	0.114947	+	0.993372i	$0.463330\pi$
$678$	0	0
$679$	2.06931	0.0794129
$680$	−0.531940	−0.0203990
$681$	0	0
$682$	3.30527	0.126565
$683$	−27.6032	−1.05621	−0.528103	−	0.849180i	$-0.677096\pi$
$683$	−27.6032	−1.05621	−0.528103	+	0.849180i	$0.677096\pi$
$684$	0	0
$685$	1.59548	0.0609602
$686$	7.37607	0.281620
$687$	0	0
$688$	−37.1487	−1.41628
$689$	−39.4300	−1.50216
$690$	0	0
$691$	−29.3314	−1.11582	−0.557910	−	0.829901i	$-0.688397\pi$
$691$	−29.3314	−1.11582	−0.557910	+	0.829901i	$0.688397\pi$
$692$	16.4424	0.625044
$693$	0	0
$694$	−28.8018	−1.09330
$695$	−22.3980	−0.849605
$696$	0	0
$697$	0.989166	0.0374673
$698$	−3.37105	−0.127596
$699$	0	0
$700$	0.892943	0.0337501
$701$	−2.90452	−0.109702	−0.0548512	−	0.998495i	$-0.517468\pi$
$701$	−2.90452	−0.109702	−0.0548512	+	0.998495i	$0.517468\pi$
$702$	0	0
$703$	37.8876	1.42896
$704$	4.03614	0.152118
$705$	0	0
$706$	−9.94083	−0.374128
$707$	−2.89131	−0.108739
$708$	0	0
$709$	−24.7136	−0.928137	−0.464069	−	0.885799i	$-0.653611\pi$
$709$	−24.7136	−0.928137	−0.464069	+	0.885799i	$0.653611\pi$
$710$	4.19913	0.157591
$711$	0	0
$712$	−24.0028	−0.899542
$713$	6.42158	0.240490
$714$	0	0
$715$	15.2877	0.571726
$716$	16.3797	0.612138
$717$	0	0
$718$	−33.3465	−1.24448
$719$	25.3727	0.946240	0.473120	−	0.880998i	$-0.343128\pi$
$719$	25.3727	0.946240	0.473120	+	0.880998i	$0.343128\pi$
$720$	0	0
$721$	0.407332	0.0151698
$722$	−11.7227	−0.436274
$723$	0	0
$724$	24.7537	0.919965
$725$	25.8870	0.961417
$726$	0	0
$727$	17.7616	0.658743	0.329371	−	0.944200i	$-0.393163\pi$
$727$	17.7616	0.658743	0.329371	+	0.944200i	$0.393163\pi$
$728$	−2.63453	−0.0976424
$729$	0	0
$730$	−31.7008	−1.17330
$731$	−1.56627	−0.0579308
$732$	0	0
$733$	9.26167	0.342087	0.171044	−	0.985263i	$-0.445286\pi$
$733$	9.26167	0.342087	0.171044	+	0.985263i	$0.445286\pi$
$734$	−32.3057	−1.19242
$735$	0	0
$736$	−38.9443	−1.43551
$737$	−3.02555	−0.111448
$738$	0	0
$739$	−5.34745	−0.196709	−0.0983546	−	0.995151i	$-0.531358\pi$
$739$	−5.34745	−0.196709	−0.0983546	+	0.995151i	$0.531358\pi$
$740$	−13.8027	−0.507398
$741$	0	0
$742$	−4.53058	−0.166323
$743$	26.0235	0.954709	0.477355	−	0.878711i	$-0.341596\pi$
$743$	26.0235	0.954709	0.477355	+	0.878711i	$0.341596\pi$
$744$	0	0
$745$	−12.2626	−0.449266
$746$	−38.1271	−1.39593
$747$	0	0
$748$	−0.469289	−0.0171589
$749$	−0.240022	−0.00877023
$750$	0	0
$751$	−3.64556	−0.133028	−0.0665141	−	0.997785i	$-0.521188\pi$
$751$	−3.64556	−0.133028	−0.0665141	+	0.997785i	$0.521188\pi$
$752$	31.0863	1.13360
$753$	0	0
$754$	65.6420	2.39054
$755$	21.4176	0.779467
$756$	0	0
$757$	10.8839	0.395583	0.197792	−	0.980244i	$-0.436623\pi$
$757$	10.8839	0.395583	0.197792	+	0.980244i	$0.436623\pi$
$758$	−20.4221	−0.741766
$759$	0	0
$760$	8.80365	0.319342
$761$	−24.6861	−0.894872	−0.447436	−	0.894316i	$-0.647663\pi$
$761$	−24.6861	−0.894872	−0.447436	+	0.894316i	$0.647663\pi$
$762$	0	0
$763$	0.639705	0.0231589
$764$	2.89828	0.104856
$765$	0	0
$766$	−35.7574	−1.29197
$767$	−32.6121	−1.17755
$768$	0	0
$769$	31.2935	1.12847	0.564236	−	0.825614i	$-0.309171\pi$
$769$	31.2935	1.12847	0.564236	+	0.825614i	$0.309171\pi$
$770$	1.75658	0.0633028
$771$	0	0
$772$	9.13358	0.328725
$773$	−13.4091	−0.482291	−0.241146	−	0.970489i	$-0.577523\pi$
$773$	−13.4091	−0.482291	−0.241146	+	0.970489i	$0.577523\pi$
$774$	0	0
$775$	−2.49629	−0.0896694
$776$	12.2692	0.440440
$777$	0	0
$778$	−16.4199	−0.588681
$779$	−16.3708	−0.586545
$780$	0	0
$781$	−4.31039	−0.154238
$782$	−2.88435	−0.103144
$783$	0	0
$784$	34.4794	1.23141
$785$	−16.8091	−0.599942
$786$	0	0
$787$	20.8803	0.744302	0.372151	−	0.928172i	$-0.378620\pi$
$787$	20.8803	0.744302	0.372151	+	0.928172i	$0.378620\pi$
$788$	−1.88457	−0.0671349
$789$	0	0
$790$	7.43883	0.264662
$791$	−1.82340	−0.0648328
$792$	0	0
$793$	−46.2491	−1.64235
$794$	−3.51734	−0.124826
$795$	0	0
$796$	−12.0816	−0.428221
$797$	−52.0745	−1.84457	−0.922286	−	0.386508i	$-0.873681\pi$
$797$	−52.0745	−1.84457	−0.922286	+	0.386508i	$0.873681\pi$
$798$	0	0
$799$	1.31067	0.0463683
$800$	15.1390	0.535245
$801$	0	0
$802$	−1.15759	−0.0408761
$803$	32.5407	1.14834
$804$	0	0
$805$	3.41274	0.120283
$806$	−6.32989	−0.222961
$807$	0	0
$808$	−17.1429	−0.603087
$809$	41.7858	1.46911	0.734554	−	0.678550i	$-0.237391\pi$
$809$	41.7858	1.46911	0.734554	+	0.678550i	$0.237391\pi$
$810$	0	0
$811$	5.44750	0.191287	0.0956437	−	0.995416i	$-0.469509\pi$
$811$	5.44750	0.191287	0.0956437	+	0.995416i	$0.469509\pi$
$812$	2.38417	0.0836679
$813$	0	0
$814$	44.8223	1.57102
$815$	34.4473	1.20664
$816$	0	0
$817$	25.9220	0.906896
$818$	29.3339	1.02564
$819$	0	0
$820$	5.96399	0.208272
$821$	−30.7796	−1.07421	−0.537107	−	0.843514i	$-0.680483\pi$
$821$	−30.7796	−1.07421	−0.537107	+	0.843514i	$0.680483\pi$
$822$	0	0
$823$	18.6824	0.651228	0.325614	−	0.945503i	$-0.394429\pi$
$823$	18.6824	0.651228	0.325614	+	0.945503i	$0.394429\pi$
$824$	2.41512	0.0841348
$825$	0	0
$826$	−3.74719	−0.130381
$827$	1.67688	0.0583107	0.0291553	−	0.999575i	$-0.490718\pi$
$827$	1.67688	0.0583107	0.0291553	+	0.999575i	$0.490718\pi$
$828$	0	0
$829$	30.7359	1.06750	0.533752	−	0.845641i	$-0.320782\pi$
$829$	30.7359	1.06750	0.533752	+	0.845641i	$0.320782\pi$
$830$	1.88423	0.0654027
$831$	0	0
$832$	−7.72958	−0.267975
$833$	1.45373	0.0503688
$834$	0	0
$835$	11.0607	0.382770
$836$	7.76677	0.268619
$837$	0	0
$838$	48.0322	1.65924
$839$	12.1881	0.420779	0.210389	−	0.977618i	$-0.432527\pi$
$839$	12.1881	0.420779	0.210389	+	0.977618i	$0.432527\pi$
$840$	0	0
$841$	40.1185	1.38340
$842$	−17.1850	−0.592234
$843$	0	0
$844$	−14.6738	−0.505092
$845$	−11.4230	−0.392963
$846$	0	0
$847$	1.60933	0.0552971
$848$	−42.6515	−1.46466
$849$	0	0
$850$	1.12125	0.0384585
$851$	87.0822	2.98514
$852$	0	0
$853$	−9.07567	−0.310745	−0.155372	−	0.987856i	$-0.549658\pi$
$853$	−9.07567	−0.310745	−0.155372	+	0.987856i	$0.549658\pi$
$854$	−5.31411	−0.181845
$855$	0	0
$856$	−1.42312	−0.0486414
$857$	−25.3920	−0.867373	−0.433687	−	0.901064i	$-0.642788\pi$
$857$	−25.3920	−0.867373	−0.433687	+	0.901064i	$0.642788\pi$
$858$	0	0
$859$	−0.428033	−0.0146043	−0.00730215	−	0.999973i	$-0.502324\pi$
$859$	−0.428033	−0.0146043	−0.00730215	+	0.999973i	$0.502324\pi$
$860$	−9.44356	−0.322023
$861$	0	0
$862$	−9.14488	−0.311476
$863$	41.2232	1.40326	0.701628	−	0.712544i	$-0.252457\pi$
$863$	41.2232	1.40326	0.701628	+	0.712544i	$0.252457\pi$
$864$	0	0
$865$	−24.4285	−0.830595
$866$	−11.7889	−0.400603
$867$	0	0
$868$	−0.229906	−0.00780353
$869$	−7.63593	−0.259031
$870$	0	0
$871$	5.79421	0.196329
$872$	3.79290	0.128444
$873$	0	0
$874$	47.7363	1.61470
$875$	−3.45696	−0.116867
$876$	0	0
$877$	−37.6729	−1.27212	−0.636061	−	0.771639i	$-0.719437\pi$
$877$	−37.6729	−1.27212	−0.636061	+	0.771639i	$0.719437\pi$
$878$	26.6304	0.898733
$879$	0	0
$880$	16.5367	0.557452
$881$	3.12965	0.105441	0.0527203	−	0.998609i	$-0.483211\pi$
$881$	3.12965	0.105441	0.0527203	+	0.998609i	$0.483211\pi$
$882$	0	0
$883$	−49.2375	−1.65697	−0.828487	−	0.560008i	$-0.810798\pi$
$883$	−49.2375	−1.65697	−0.828487	+	0.560008i	$0.810798\pi$
$884$	0.898732	0.0302276
$885$	0	0
$886$	47.8044	1.60602
$887$	54.3157	1.82374	0.911871	−	0.410477i	$-0.134638\pi$
$887$	54.3157	1.82374	0.911871	+	0.410477i	$0.134638\pi$
$888$	0	0
$889$	−3.24318	−0.108773
$890$	−30.6490	−1.02736
$891$	0	0
$892$	−1.75235	−0.0586730
$893$	−21.6918	−0.725887
$894$	0	0
$895$	−24.3354	−0.813443
$896$	−3.90474	−0.130448
$897$	0	0
$898$	18.7005	0.624045
$899$	−6.66512	−0.222294
$900$	0	0
$901$	−1.79829	−0.0599096
$902$	−19.3672	−0.644857
$903$	0	0
$904$	−10.8112	−0.359575
$905$	−36.7768	−1.22250
$906$	0	0
$907$	47.0681	1.56287	0.781435	−	0.623987i	$-0.214488\pi$
$907$	47.0681	1.56287	0.781435	+	0.623987i	$0.214488\pi$
$908$	23.6257	0.784045
$909$	0	0
$910$	−3.36402	−0.111516
$911$	−51.7621	−1.71495	−0.857477	−	0.514522i	$-0.827970\pi$
$911$	−51.7621	−1.71495	−0.857477	+	0.514522i	$0.827970\pi$
$912$	0	0
$913$	−1.93416	−0.0640112
$914$	−28.0981	−0.929404
$915$	0	0
$916$	−4.33694	−0.143297
$917$	−1.85810	−0.0613600
$918$	0	0
$919$	13.1610	0.434141	0.217070	−	0.976156i	$-0.430350\pi$
$919$	13.1610	0.434141	0.217070	+	0.976156i	$0.430350\pi$
$920$	20.2346	0.667115
$921$	0	0
$922$	42.7550	1.40806
$923$	8.25480	0.271710
$924$	0	0
$925$	−33.8519	−1.11304
$926$	23.1975	0.762317
$927$	0	0
$928$	40.4213	1.32690
$929$	−17.2261	−0.565168	−0.282584	−	0.959242i	$-0.591192\pi$
$929$	−17.2261	−0.565168	−0.282584	+	0.959242i	$0.591192\pi$
$930$	0	0
$931$	−24.0594	−0.788514
$932$	−10.5955	−0.347066
$933$	0	0
$934$	58.6774	1.91998
$935$	0.697226	0.0228017
$936$	0	0
$937$	38.1691	1.24693	0.623465	−	0.781852i	$-0.285725\pi$
$937$	38.1691	1.24693	0.623465	+	0.781852i	$0.285725\pi$
$938$	0.665766	0.0217380
$939$	0	0
$940$	7.90245	0.257750
$941$	21.2801	0.693710	0.346855	−	0.937919i	$-0.387250\pi$
$941$	21.2801	0.693710	0.346855	+	0.937919i	$0.387250\pi$
$942$	0	0
$943$	−37.6272	−1.22531
$944$	−35.2766	−1.14815
$945$	0	0
$946$	30.6666	0.997056
$947$	7.78625	0.253019	0.126510	−	0.991965i	$-0.459623\pi$
$947$	7.78625	0.253019	0.126510	+	0.991965i	$0.459623\pi$
$948$	0	0
$949$	−62.3185	−2.02294
$950$	−18.5567	−0.602060
$951$	0	0
$952$	−0.120153	−0.00389420
$953$	−37.9733	−1.23008	−0.615038	−	0.788498i	$-0.710859\pi$
$953$	−37.9733	−1.23008	−0.615038	+	0.788498i	$0.710859\pi$
$954$	0	0
$955$	−4.30599	−0.139339
$956$	10.8586	0.351192
$957$	0	0
$958$	37.0439	1.19683
$959$	0.360384	0.0116374
$960$	0	0
$961$	−30.3573	−0.979267
$962$	−85.8388	−2.76756
$963$	0	0
$964$	4.08491	0.131566
$965$	−13.5698	−0.436828
$966$	0	0
$967$	−5.41726	−0.174207	−0.0871037	−	0.996199i	$-0.527761\pi$
$967$	−5.41726	−0.174207	−0.0871037	+	0.996199i	$0.527761\pi$
$968$	9.54192	0.306689
$969$	0	0
$970$	15.6665	0.503020
$971$	45.8419	1.47114	0.735568	−	0.677451i	$-0.236915\pi$
$971$	45.8419	1.47114	0.735568	+	0.677451i	$0.236915\pi$
$972$	0	0
$973$	−5.05922	−0.162191
$974$	66.4374	2.12879
$975$	0	0
$976$	−50.0277	−1.60135
$977$	50.7281	1.62294	0.811468	−	0.584397i	$-0.198669\pi$
$977$	50.7281	1.62294	0.811468	+	0.584397i	$0.198669\pi$
$978$	0	0
$979$	31.4610	1.00550
$980$	8.76499	0.279987
$981$	0	0
$982$	−50.0762	−1.59799
$983$	35.3293	1.12683	0.563415	−	0.826174i	$-0.309487\pi$
$983$	35.3293	1.12683	0.563415	+	0.826174i	$0.309487\pi$
$984$	0	0
$985$	2.79991	0.0892126
$986$	2.99374	0.0953402
$987$	0	0
$988$	−14.8741	−0.473208
$989$	59.5800	1.89453
$990$	0	0
$991$	−4.20256	−0.133499	−0.0667494	−	0.997770i	$-0.521263\pi$
$991$	−4.20256	−0.133499	−0.0667494	+	0.997770i	$0.521263\pi$
$992$	−3.89785	−0.123757
$993$	0	0
$994$	0.948492	0.0300843
$995$	17.9497	0.569044
$996$	0	0
$997$	6.88471	0.218041	0.109020	−	0.994040i	$-0.465229\pi$
$997$	6.88471	0.218041	0.109020	+	0.994040i	$0.465229\pi$
$998$	−46.9829	−1.48722
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.55		72
3.2	odd	2		6561.2.a.c.1.18		72
81.2	odd	54		81.2.g.a.4.7	✓	144
81.13	even	27		729.2.g.b.541.7		144
81.14	odd	54		729.2.g.d.55.2		144
81.25	even	27		729.2.g.b.190.7		144
81.29	odd	54		729.2.g.d.676.2		144
81.40	even	27		243.2.g.a.19.2		144
81.41	odd	54		81.2.g.a.61.7	yes	144
81.52	even	27		729.2.g.a.676.7		144
81.56	odd	54		729.2.g.c.190.2		144
81.67	even	27		729.2.g.a.55.7		144
81.68	odd	54		729.2.g.c.541.2		144
81.79	even	27		243.2.g.a.64.2		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.4.7	✓	144	81.2	odd	54
81.2.g.a.61.7	yes	144	81.41	odd	54
243.2.g.a.19.2		144	81.40	even	27
243.2.g.a.64.2		144	81.79	even	27
729.2.g.a.55.7		144	81.67	even	27
729.2.g.a.676.7		144	81.52	even	27
729.2.g.b.190.7		144	81.25	even	27
729.2.g.b.541.7		144	81.13	even	27
729.2.g.c.190.2		144	81.56	odd	54
729.2.g.c.541.2		144	81.68	odd	54
729.2.g.d.55.2		144	81.14	odd	54
729.2.g.d.676.2		144	81.29	odd	54
6561.2.a.c.1.18		72	3.2	odd	2
6561.2.a.d.1.55		72	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:	\( N \)	\(=\)	\( 6561 = 3^{8} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6561.a (trivial)

\(f(q)\)	\(=\)	\(q+1.71009 q^{2} +0.924414 q^{4} -1.37341 q^{5} -0.310223 q^{7} -1.83935 q^{8} +O(q^{10})\)	\(q+1.71009 q^{2} +0.924414 q^{4} -1.37341 q^{5} -0.310223 q^{7} -1.83935 q^{8} -2.34865 q^{10} +2.41088 q^{11} -4.61706 q^{13} -0.530509 q^{14} -4.99429 q^{16} -0.210571 q^{17} +3.48497 q^{19} -1.26960 q^{20} +4.12283 q^{22} +8.00996 q^{23} -3.11375 q^{25} -7.89560 q^{26} -0.286774 q^{28} -8.31375 q^{29} +0.801699 q^{31} -4.86199 q^{32} -0.360095 q^{34} +0.426062 q^{35} +10.8717 q^{37} +5.95961 q^{38} +2.52618 q^{40} -4.69755 q^{41} +7.43824 q^{43} +2.22865 q^{44} +13.6978 q^{46} -6.22438 q^{47} -6.90376 q^{49} -5.32480 q^{50} -4.26807 q^{52} +8.54006 q^{53} -3.31112 q^{55} +0.570609 q^{56} -14.2173 q^{58} +7.06338 q^{59} +10.0170 q^{61} +1.37098 q^{62} +1.67413 q^{64} +6.34111 q^{65} -1.25496 q^{67} -0.194654 q^{68} +0.728606 q^{70} -1.78789 q^{71} +13.4974 q^{73} +18.5917 q^{74} +3.22155 q^{76} -0.747910 q^{77} -3.16728 q^{79} +6.85919 q^{80} -8.03324 q^{82} -0.802261 q^{83} +0.289200 q^{85} +12.7201 q^{86} -4.43446 q^{88} +13.0496 q^{89} +1.43232 q^{91} +7.40452 q^{92} -10.6443 q^{94} -4.78628 q^{95} -6.67041 q^{97} -11.8061 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) \) 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38} + 72 q^{41} + 90 q^{44} + 90 q^{47} + 18 q^{49} + 45 q^{50} + 45 q^{53} + 9 q^{55} + 108 q^{56} + 18 q^{58} + 108 q^{59} + 72 q^{62} + 9 q^{64} + 72 q^{65} + 108 q^{68} + 126 q^{71} + 90 q^{74} + 72 q^{77} + 144 q^{80} - 18 q^{82} + 108 q^{83} + 90 q^{86} + 108 q^{89} + 72 q^{92} + 144 q^{95} + 81 q^{98}+O(q^{100}) \) 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8 + 36 * q^11 + 36 * q^14 + 45 * q^16 + 36 * q^17 + 54 * q^20 + 54 * q^23 + 36 * q^25 + 45 * q^26 + 9 * q^28 + 54 * q^29 + 63 * q^32 + 72 * q^35 + 54 * q^38 + 72 * q^41 + 90 * q^44 + 90 * q^47 + 18 * q^49 + 45 * q^50 + 45 * q^53 + 9 * q^55 + 108 * q^56 + 18 * q^58 + 108 * q^59 + 72 * q^62 + 9 * q^64 + 72 * q^65 + 108 * q^68 + 126 * q^71 + 90 * q^74 + 72 * q^77 + 144 * q^80 - 18 * q^82 + 108 * q^83 + 90 * q^86 + 108 * q^89 + 72 * q^92 + 144 * q^95 + 81 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more