LMFDB - Embedded newform 729.2.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(1,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, 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("729.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	729.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:	$5.82109430735$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{36})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 6x^{4} + 9x^{2} - 3 $ x^6 - 6x^4 + 9x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.684040$ of defining polynomial
Character	$\chi$	$=$	729.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.684040 q^{2} -1.53209 q^{4} -1.04801 q^{5} -0.120615 q^{7} +2.41609 q^{8} +0.716881 q^{10} +5.43372 q^{11} -4.57398 q^{13} +0.0825054 q^{14} +1.41147 q^{16} -4.77833 q^{17} -0.588526 q^{19} +1.60565 q^{20} -3.71688 q^{22} +7.79596 q^{23} -3.90167 q^{25} +3.12879 q^{26} +0.184793 q^{28} -5.06975 q^{29} -8.75877 q^{31} -5.79769 q^{32} +3.26857 q^{34} +0.126406 q^{35} -2.18479 q^{37} +0.402575 q^{38} -2.53209 q^{40} -7.55839 q^{41} -1.30541 q^{43} -8.32494 q^{44} -5.33275 q^{46} -2.41609 q^{47} -6.98545 q^{49} +2.66890 q^{50} +7.00774 q^{52} +3.04628 q^{53} -5.69459 q^{55} -0.291416 q^{56} +3.46791 q^{58} +0.0439002 q^{59} -10.2121 q^{61} +5.99135 q^{62} +1.14290 q^{64} +4.79358 q^{65} -1.85978 q^{67} +7.32083 q^{68} -0.0864665 q^{70} +6.51038 q^{71} +12.2344 q^{73} +1.49449 q^{74} +0.901674 q^{76} -0.655386 q^{77} +0.702333 q^{79} -1.47924 q^{80} +5.17024 q^{82} -6.77660 q^{83} +5.00774 q^{85} +0.892951 q^{86} +13.1284 q^{88} -6.85565 q^{89} +0.551689 q^{91} -11.9441 q^{92} +1.65270 q^{94} +0.616781 q^{95} -9.02734 q^{97} +4.77833 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64}+ \cdots - 12 q^{97}+O(q^{100}) $ 6 * q - 12 * q^7 - 12 * q^10 - 12 * q^13 - 12 * q^16 - 24 * q^19 - 6 * q^22 - 6 * q^28 - 30 * q^31 - 6 * q^37 - 6 * q^40 - 12 * q^43 + 6 * q^46 - 6 * q^49 - 6 * q^52 - 30 * q^55 + 30 * q^58 - 12 * q^61 + 6 * q^64 + 6 * q^67 + 30 * q^70 + 12 * q^73 - 18 * q^76 - 48 * q^79 - 12 * q^82 - 18 * q^85 + 42 * q^88 + 12 * q^94 - 12 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.684040	−0.483690	−0.241845	−	0.970315i	$-0.577752\pi$
$2$	−0.684040	−0.483690	−0.241845	+	0.970315i	$0.577752\pi$
$3$	0	0
$3$	0	0
$4$	−1.53209	−0.766044
$5$	−1.04801	−0.468685	−0.234342	−	0.972154i	$-0.575294\pi$
$5$	−1.04801	−0.468685	−0.234342	+	0.972154i	$0.575294\pi$
$6$	0	0
$7$	−0.120615	−0.0455881	−0.0227940	−	0.999740i	$-0.507256\pi$
$7$	−0.120615	−0.0455881	−0.0227940	+	0.999740i	$0.507256\pi$
$8$	2.41609	0.854217
$9$	0	0
$10$	0.716881	0.226698
$11$	5.43372	1.63833	0.819164	−	0.573560i	$-0.194438\pi$
$11$	5.43372	1.63833	0.819164	+	0.573560i	$0.194438\pi$
$12$	0	0
$13$	−4.57398	−1.26859	−0.634297	−	0.773090i	$-0.718710\pi$
$13$	−4.57398	−1.26859	−0.634297	+	0.773090i	$0.718710\pi$
$14$	0.0825054	0.0220505
$15$	0	0
$16$	1.41147	0.352869
$17$	−4.77833	−1.15892	−0.579458	−	0.815002i	$-0.696736\pi$
$17$	−4.77833	−1.15892	−0.579458	+	0.815002i	$0.696736\pi$
$18$	0	0
$19$	−0.588526	−0.135017	−0.0675085	−	0.997719i	$-0.521505\pi$
$19$	−0.588526	−0.135017	−0.0675085	+	0.997719i	$0.521505\pi$
$20$	1.60565	0.359033
$21$	0	0
$22$	−3.71688	−0.792442
$23$	7.79596	1.62557	0.812785	−	0.582564i	$-0.197951\pi$
$23$	7.79596	1.62557	0.812785	+	0.582564i	$0.197951\pi$
$24$	0	0
$25$	−3.90167	−0.780335
$26$	3.12879	0.613605
$27$	0	0
$28$	0.184793	0.0349225
$29$	−5.06975	−0.941428	−0.470714	−	0.882286i	$-0.656004\pi$
$29$	−5.06975	−0.941428	−0.470714	+	0.882286i	$0.656004\pi$
$30$	0	0
$31$	−8.75877	−1.57312	−0.786561	−	0.617513i	$-0.788140\pi$
$31$	−8.75877	−1.57312	−0.786561	+	0.617513i	$0.788140\pi$
$32$	−5.79769	−1.02490
$33$	0	0
$34$	3.26857	0.560555
$35$	0.126406	0.0213664
$36$	0	0
$37$	−2.18479	−0.359178	−0.179589	−	0.983742i	$-0.557477\pi$
$37$	−2.18479	−0.359178	−0.179589	+	0.983742i	$0.557477\pi$
$38$	0.402575	0.0653064
$39$	0	0
$40$	−2.53209	−0.400358
$41$	−7.55839	−1.18042	−0.590211	−	0.807249i	$-0.700956\pi$
$41$	−7.55839	−1.18042	−0.590211	+	0.807249i	$0.700956\pi$
$42$	0	0
$43$	−1.30541	−0.199073	−0.0995364	−	0.995034i	$-0.531736\pi$
$43$	−1.30541	−0.199073	−0.0995364	+	0.995034i	$0.531736\pi$
$44$	−8.32494	−1.25503
$45$	0	0
$46$	−5.33275	−0.786271
$47$	−2.41609	−0.352423	−0.176212	−	0.984352i	$-0.556384\pi$
$47$	−2.41609	−0.352423	−0.176212	+	0.984352i	$0.556384\pi$
$48$	0	0
$49$	−6.98545	−0.997922
$50$	2.66890	0.377440
$51$	0	0
$52$	7.00774	0.971799
$53$	3.04628	0.418439	0.209219	−	0.977869i	$-0.432908\pi$
$53$	3.04628	0.418439	0.209219	+	0.977869i	$0.432908\pi$
$54$	0	0
$55$	−5.69459	−0.767859
$56$	−0.291416	−0.0389421
$57$	0	0
$58$	3.46791	0.455359
$59$	0.0439002	0.00571532	0.00285766	−	0.999996i	$-0.499090\pi$
$59$	0.0439002	0.00571532	0.00285766	+	0.999996i	$0.499090\pi$
$60$	0	0
$61$	−10.2121	−1.30753	−0.653765	−	0.756698i	$-0.726811\pi$
$61$	−10.2121	−1.30753	−0.653765	+	0.756698i	$0.726811\pi$
$62$	5.99135	0.760902
$63$	0	0
$64$	1.14290	0.142863
$65$	4.79358	0.594570
$66$	0	0
$67$	−1.85978	−0.227209	−0.113604	−	0.993526i	$-0.536240\pi$
$67$	−1.85978	−0.227209	−0.113604	+	0.993526i	$0.536240\pi$
$68$	7.32083	0.887781
$69$	0	0
$70$	−0.0864665	−0.0103347
$71$	6.51038	0.772640	0.386320	−	0.922365i	$-0.373746\pi$
$71$	6.51038	0.772640	0.386320	+	0.922365i	$0.373746\pi$
$72$	0	0
$73$	12.2344	1.43193	0.715965	−	0.698136i	$-0.245987\pi$
$73$	12.2344	1.43193	0.715965	+	0.698136i	$0.245987\pi$
$74$	1.49449	0.173730
$75$	0	0
$76$	0.901674	0.103429
$77$	−0.655386	−0.0746882
$78$	0	0
$79$	0.702333	0.0790187	0.0395093	−	0.999219i	$-0.487421\pi$
$79$	0.702333	0.0790187	0.0395093	+	0.999219i	$0.487421\pi$
$80$	−1.47924	−0.165384
$81$	0	0
$82$	5.17024	0.570958
$83$	−6.77660	−0.743828	−0.371914	−	0.928267i	$-0.621298\pi$
$83$	−6.77660	−0.743828	−0.371914	+	0.928267i	$0.621298\pi$
$84$	0	0
$85$	5.00774	0.543166
$86$	0.892951	0.0962894
$87$	0	0
$88$	13.1284	1.39949
$89$	−6.85565	−0.726697	−0.363349	−	0.931653i	$-0.618367\pi$
$89$	−6.85565	−0.726697	−0.363349	+	0.931653i	$0.618367\pi$
$90$	0	0
$91$	0.551689	0.0578327
$92$	−11.9441	−1.24526
$93$	0	0
$94$	1.65270	0.170463
$95$	0.616781	0.0632804
$96$	0	0
$97$	−9.02734	−0.916588	−0.458294	−	0.888801i	$-0.651539\pi$
$97$	−9.02734	−0.916588	−0.458294	+	0.888801i	$0.651539\pi$
$98$	4.77833	0.482684
$99$	0	0
$100$	5.97771	0.597771
$101$	−12.9921	−1.29276	−0.646382	−	0.763014i	$-0.723719\pi$
$101$	−12.9921	−1.29276	−0.646382	+	0.763014i	$0.723719\pi$
$102$	0	0
$103$	9.07192	0.893883	0.446941	−	0.894563i	$-0.352513\pi$
$103$	9.07192	0.893883	0.446941	+	0.894563i	$0.352513\pi$
$104$	−11.0511	−1.08365
$105$	0	0
$106$	−2.08378	−0.202394
$107$	−11.3865	−1.10077	−0.550386	−	0.834911i	$-0.685519\pi$
$107$	−11.3865	−1.10077	−0.550386	+	0.834911i	$0.685519\pi$
$108$	0	0
$109$	2.89899	0.277672	0.138836	−	0.990315i	$-0.455664\pi$
$109$	2.89899	0.277672	0.138836	+	0.990315i	$0.455664\pi$
$110$	3.89533	0.371405
$111$	0	0
$112$	−0.170245	−0.0160866
$113$	11.5801	1.08937	0.544683	−	0.838642i	$-0.316650\pi$
$113$	11.5801	1.08937	0.544683	+	0.838642i	$0.316650\pi$
$114$	0	0
$115$	−8.17024	−0.761879
$116$	7.76730	0.721176
$117$	0	0
$118$	−0.0300295	−0.00276444
$119$	0.576337	0.0528327
$120$	0	0
$121$	18.5253	1.68412
$122$	6.98551	0.632438
$123$	0	0
$124$	13.4192	1.20508
$125$	9.32905	0.834415
$126$	0	0
$127$	−15.4192	−1.36823	−0.684117	−	0.729372i	$-0.739812\pi$
$127$	−15.4192	−1.36823	−0.684117	+	0.729372i	$0.739812\pi$
$128$	10.8136	0.955795
$129$	0	0
$130$	−3.27900	−0.287587
$131$	−13.3561	−1.16693	−0.583463	−	0.812140i	$-0.698303\pi$
$131$	−13.3561	−1.16693	−0.583463	+	0.812140i	$0.698303\pi$
$132$	0	0
$133$	0.0709849	0.00615517
$134$	1.27217	0.109899
$135$	0	0
$136$	−11.5449	−0.989965
$137$	19.6236	1.67656	0.838279	−	0.545242i	$-0.183562\pi$
$137$	19.6236	1.67656	0.838279	+	0.545242i	$0.183562\pi$
$138$	0	0
$139$	16.3063	1.38309	0.691543	−	0.722335i	$-0.256931\pi$
$139$	16.3063	1.38309	0.691543	+	0.722335i	$0.256931\pi$
$140$	−0.193665	−0.0163676
$141$	0	0
$142$	−4.45336	−0.373718
$143$	−24.8537	−2.07837
$144$	0	0
$145$	5.31315	0.441233
$146$	−8.36884	−0.692610
$147$	0	0
$148$	3.34730	0.275146
$149$	−11.0477	−0.905062	−0.452531	−	0.891749i	$-0.649479\pi$
$149$	−11.0477	−0.905062	−0.452531	+	0.891749i	$0.649479\pi$
$150$	0	0
$151$	−6.80066	−0.553430	−0.276715	−	0.960952i	$-0.589246\pi$
$151$	−6.80066	−0.553430	−0.276715	+	0.960952i	$0.589246\pi$
$152$	−1.42193	−0.115334
$153$	0	0
$154$	0.448311	0.0361259
$155$	9.17928	0.737298
$156$	0	0
$157$	16.8871	1.34774	0.673870	−	0.738850i	$-0.264631\pi$
$157$	16.8871	1.34774	0.673870	+	0.738850i	$0.264631\pi$
$158$	−0.480424	−0.0382205
$159$	0	0
$160$	6.07604	0.480353
$161$	−0.940307	−0.0741066
$162$	0	0
$163$	8.41147	0.658838	0.329419	−	0.944184i	$-0.393147\pi$
$163$	8.41147	0.658838	0.329419	+	0.944184i	$0.393147\pi$
$164$	11.5801	0.904256
$165$	0	0
$166$	4.63547	0.359782
$167$	2.62500	0.203129	0.101564	−	0.994829i	$-0.467615\pi$
$167$	2.62500	0.203129	0.101564	+	0.994829i	$0.467615\pi$
$168$	0	0
$169$	7.92127	0.609329
$170$	−3.42550	−0.262724
$171$	0	0
$172$	2.00000	0.152499
$173$	−13.1033	−0.996223	−0.498112	−	0.867113i	$-0.665973\pi$
$173$	−13.1033	−0.996223	−0.498112	+	0.867113i	$0.665973\pi$
$174$	0	0
$175$	0.470599	0.0355740
$176$	7.66955	0.578114
$177$	0	0
$178$	4.68954	0.351496
$179$	22.0988	1.65174	0.825872	−	0.563857i	$-0.190683\pi$
$179$	22.0988	1.65174	0.825872	+	0.563857i	$0.190683\pi$
$180$	0	0
$181$	−2.05913	−0.153054	−0.0765268	−	0.997068i	$-0.524383\pi$
$181$	−2.05913	−0.153054	−0.0765268	+	0.997068i	$0.524383\pi$
$182$	−0.377378	−0.0279731
$183$	0	0
$184$	18.8357	1.38859
$185$	2.28969	0.168341
$186$	0	0
$187$	−25.9641	−1.89868
$188$	3.70167	0.269972
$189$	0	0
$190$	−0.421903	−0.0306081
$191$	3.94918	0.285753	0.142876	−	0.989741i	$-0.454365\pi$
$191$	3.94918	0.285753	0.142876	+	0.989741i	$0.454365\pi$
$192$	0	0
$193$	−8.44656	−0.607996	−0.303998	−	0.952673i	$-0.598322\pi$
$193$	−8.44656	−0.607996	−0.303998	+	0.952673i	$0.598322\pi$
$194$	6.17507	0.443344
$195$	0	0
$196$	10.7023	0.764452
$197$	2.29498	0.163511	0.0817553	−	0.996652i	$-0.473947\pi$
$197$	2.29498	0.163511	0.0817553	+	0.996652i	$0.473947\pi$
$198$	0	0
$199$	−23.5030	−1.66608	−0.833042	−	0.553210i	$-0.813403\pi$
$199$	−23.5030	−1.66608	−0.833042	+	0.553210i	$0.813403\pi$
$200$	−9.42680	−0.666575
$201$	0	0
$202$	8.88713	0.625296
$203$	0.611486	0.0429179
$204$	0	0
$205$	7.92127	0.553246
$206$	−6.20556	−0.432362
$207$	0	0
$208$	−6.45605	−0.447647
$209$	−3.19788	−0.221202
$210$	0	0
$211$	−1.66725	−0.114778	−0.0573892	−	0.998352i	$-0.518278\pi$
$211$	−1.66725	−0.114778	−0.0573892	+	0.998352i	$0.518278\pi$
$212$	−4.66717	−0.320543
$213$	0	0
$214$	7.78880	0.532431
$215$	1.36808	0.0933023
$216$	0	0
$217$	1.05644	0.0717156
$218$	−1.98302	−0.134307
$219$	0	0
$220$	8.72462	0.588214
$221$	21.8560	1.47019
$222$	0	0
$223$	−9.80066	−0.656301	−0.328150	−	0.944626i	$-0.606425\pi$
$223$	−9.80066	−0.656301	−0.328150	+	0.944626i	$0.606425\pi$
$224$	0.699287	0.0467231
$225$	0	0
$226$	−7.92127	−0.526915
$227$	15.4808	1.02749	0.513747	−	0.857942i	$-0.328257\pi$
$227$	15.4808	1.02749	0.513747	+	0.857942i	$0.328257\pi$
$228$	0	0
$229$	−11.2540	−0.743687	−0.371843	−	0.928295i	$-0.621274\pi$
$229$	−11.2540	−0.743687	−0.371843	+	0.928295i	$0.621274\pi$
$230$	5.58878	0.368513
$231$	0	0
$232$	−12.2490	−0.804184
$233$	5.23476	0.342940	0.171470	−	0.985189i	$-0.445148\pi$
$233$	5.23476	0.342940	0.171470	+	0.985189i	$0.445148\pi$
$234$	0	0
$235$	2.53209	0.165175
$236$	−0.0672590	−0.00437819
$237$	0	0
$238$	−0.394238	−0.0255546
$239$	−10.3098	−0.666885	−0.333443	−	0.942770i	$-0.608210\pi$
$239$	−10.3098	−0.666885	−0.333443	+	0.942770i	$0.608210\pi$
$240$	0	0
$241$	10.7442	0.692096	0.346048	−	0.938217i	$-0.387523\pi$
$241$	10.7442	0.692096	0.346048	+	0.938217i	$0.387523\pi$
$242$	−12.6720	−0.814590
$243$	0	0
$244$	15.6459	1.00163
$245$	7.32083	0.467710
$246$	0	0
$247$	2.69190	0.171282
$248$	−21.1620	−1.34379
$249$	0	0
$250$	−6.38144	−0.403598
$251$	15.0729	0.951392	0.475696	−	0.879610i	$-0.342196\pi$
$251$	15.0729	0.951392	0.475696	+	0.879610i	$0.342196\pi$
$252$	0	0
$253$	42.3610	2.66321
$254$	10.5474	0.661800
$255$	0	0
$256$	−9.68273	−0.605171
$257$	−3.32245	−0.207249	−0.103624	−	0.994617i	$-0.533044\pi$
$257$	−3.32245	−0.207249	−0.103624	+	0.994617i	$0.533044\pi$
$258$	0	0
$259$	0.263518	0.0163742
$260$	−7.34419	−0.455467
$261$	0	0
$262$	9.13610	0.564430
$263$	8.83867	0.545016	0.272508	−	0.962154i	$-0.412147\pi$
$263$	8.83867	0.545016	0.272508	+	0.962154i	$0.412147\pi$
$264$	0	0
$265$	−3.19253	−0.196116
$266$	−0.0485565	−0.00297719
$267$	0	0
$268$	2.84936	0.174052
$269$	8.09267	0.493419	0.246709	−	0.969090i	$-0.420651\pi$
$269$	8.09267	0.493419	0.246709	+	0.969090i	$0.420651\pi$
$270$	0	0
$271$	−19.0000	−1.15417	−0.577084	−	0.816685i	$-0.695809\pi$
$271$	−19.0000	−1.15417	−0.577084	+	0.816685i	$0.695809\pi$
$272$	−6.74449	−0.408945
$273$	0	0
$274$	−13.4233	−0.810933
$275$	−21.2006	−1.27844
$276$	0	0
$277$	−4.22937	−0.254118	−0.127059	−	0.991895i	$-0.540554\pi$
$277$	−4.22937	−0.254118	−0.127059	+	0.991895i	$0.540554\pi$
$278$	−11.1542	−0.668984
$279$	0	0
$280$	0.305407	0.0182516
$281$	7.30212	0.435608	0.217804	−	0.975993i	$-0.430111\pi$
$281$	7.30212	0.435608	0.217804	+	0.975993i	$0.430111\pi$
$282$	0	0
$283$	16.0205	0.952322	0.476161	−	0.879358i	$-0.342028\pi$
$283$	16.0205	0.952322	0.476161	+	0.879358i	$0.342028\pi$
$284$	−9.97448	−0.591877
$285$	0	0
$286$	17.0009	1.00529
$287$	0.911654	0.0538132
$288$	0	0
$289$	5.83244	0.343085
$290$	−3.63441	−0.213420
$291$	0	0
$292$	−18.7442	−1.09692
$293$	18.0045	1.05184	0.525918	−	0.850535i	$-0.323722\pi$
$293$	18.0045	1.05184	0.525918	+	0.850535i	$0.323722\pi$
$294$	0	0
$295$	−0.0460079	−0.00267868
$296$	−5.27866	−0.306816
$297$	0	0
$298$	7.55707	0.437769
$299$	−35.6585	−2.06219
$300$	0	0
$301$	0.157451	0.00907535
$302$	4.65193	0.267688
$303$	0	0
$304$	−0.830689	−0.0476433
$305$	10.7024	0.612819
$306$	0	0
$307$	−13.5107	−0.771098	−0.385549	−	0.922687i	$-0.625988\pi$
$307$	−13.5107	−0.771098	−0.385549	+	0.922687i	$0.625988\pi$
$308$	1.00411	0.0572145
$309$	0	0
$310$	−6.27900	−0.356623
$311$	16.4223	0.931221	0.465611	−	0.884990i	$-0.345835\pi$
$311$	16.4223	0.931221	0.465611	+	0.884990i	$0.345835\pi$
$312$	0	0
$313$	−19.5648	−1.10587	−0.552934	−	0.833225i	$-0.686492\pi$
$313$	−19.5648	−1.10587	−0.552934	+	0.833225i	$0.686492\pi$
$314$	−11.5515	−0.651887
$315$	0	0
$316$	−1.07604	−0.0605318
$317$	−24.3786	−1.36924	−0.684619	−	0.728902i	$-0.740031\pi$
$317$	−24.3786	−1.36924	−0.684619	+	0.728902i	$0.740031\pi$
$318$	0	0
$319$	−27.5476	−1.54237
$320$	−1.19777	−0.0669576
$321$	0	0
$322$	0.643208	0.0358446
$323$	2.81217	0.156473
$324$	0	0
$325$	17.8462	0.989927
$326$	−5.75379	−0.318673
$327$	0	0
$328$	−18.2618	−1.00834
$329$	0.291416	0.0160663
$330$	0	0
$331$	28.4989	1.56644	0.783220	−	0.621745i	$-0.213576\pi$
$331$	28.4989	1.56644	0.783220	+	0.621745i	$0.213576\pi$
$332$	10.3824	0.569806
$333$	0	0
$334$	−1.79561	−0.0982512
$335$	1.94907	0.106489
$336$	0	0
$337$	−17.4466	−0.950374	−0.475187	−	0.879885i	$-0.657620\pi$
$337$	−17.4466	−0.950374	−0.475187	+	0.879885i	$0.657620\pi$
$338$	−5.41847	−0.294726
$339$	0	0
$340$	−7.67230	−0.416089
$341$	−47.5927	−2.57729
$342$	0	0
$343$	1.68685	0.0910814
$344$	−3.15398	−0.170051
$345$	0	0
$346$	8.96316	0.481863
$347$	25.1890	1.35222	0.676109	−	0.736802i	$-0.263665\pi$
$347$	25.1890	1.35222	0.676109	+	0.736802i	$0.263665\pi$
$348$	0	0
$349$	−11.8520	−0.634425	−0.317213	−	0.948354i	$-0.602747\pi$
$349$	−11.8520	−0.634425	−0.317213	+	0.948354i	$0.602747\pi$
$350$	−0.321909	−0.0172068
$351$	0	0
$352$	−31.5030	−1.67912
$353$	−8.39749	−0.446953	−0.223477	−	0.974709i	$-0.571741\pi$
$353$	−8.39749	−0.446953	−0.223477	+	0.974709i	$0.571741\pi$
$354$	0	0
$355$	−6.82295	−0.362124
$356$	10.5035	0.556682
$357$	0	0
$358$	−15.1165	−0.798932
$359$	2.64025	0.139347	0.0696735	−	0.997570i	$-0.477804\pi$
$359$	2.64025	0.139347	0.0696735	+	0.997570i	$0.477804\pi$
$360$	0	0
$361$	−18.6536	−0.981770
$362$	1.40852	0.0740304
$363$	0	0
$364$	−0.845237	−0.0443025
$365$	−12.8218	−0.671124
$366$	0	0
$367$	9.19759	0.480110	0.240055	−	0.970759i	$-0.422835\pi$
$367$	9.19759	0.480110	0.240055	+	0.970759i	$0.422835\pi$
$368$	11.0038	0.573612
$369$	0	0
$370$	−1.56624	−0.0814248
$371$	−0.367426	−0.0190758
$372$	0	0
$373$	−26.7912	−1.38719	−0.693597	−	0.720363i	$-0.743975\pi$
$373$	−26.7912	−1.38719	−0.693597	+	0.720363i	$0.743975\pi$
$374$	17.7605	0.918373
$375$	0	0
$376$	−5.83750	−0.301046
$377$	23.1889	1.19429
$378$	0	0
$379$	−27.1242	−1.39328	−0.696639	−	0.717422i	$-0.745322\pi$
$379$	−27.1242	−1.39328	−0.696639	+	0.717422i	$0.745322\pi$
$380$	−0.944964	−0.0484756
$381$	0	0
$382$	−2.70140	−0.138216
$383$	−34.4268	−1.75913	−0.879564	−	0.475781i	$-0.842166\pi$
$383$	−34.4268	−1.75913	−0.879564	+	0.475781i	$0.842166\pi$
$384$	0	0
$385$	0.686852	0.0350052
$386$	5.77778	0.294081
$387$	0	0
$388$	13.8307	0.702147
$389$	−0.756594	−0.0383609	−0.0191804	−	0.999816i	$-0.506106\pi$
$389$	−0.756594	−0.0383609	−0.0191804	+	0.999816i	$0.506106\pi$
$390$	0	0
$391$	−37.2517	−1.88390
$392$	−16.8775	−0.852442
$393$	0	0
$394$	−1.56986	−0.0790884
$395$	−0.736053	−0.0370348
$396$	0	0
$397$	7.00774	0.351708	0.175854	−	0.984416i	$-0.443731\pi$
$397$	7.00774	0.351708	0.175854	+	0.984416i	$0.443731\pi$
$398$	16.0770	0.805867
$399$	0	0
$400$	−5.50711	−0.275356
$401$	9.45080	0.471950	0.235975	−	0.971759i	$-0.424172\pi$
$401$	9.45080	0.471950	0.235975	+	0.971759i	$0.424172\pi$
$402$	0	0
$403$	40.0624	1.99565
$404$	19.9051	0.990314
$405$	0	0
$406$	−0.418281	−0.0207590
$407$	−11.8715	−0.588451
$408$	0	0
$409$	5.51754	0.272825	0.136412	−	0.990652i	$-0.456443\pi$
$409$	5.51754	0.272825	0.136412	+	0.990652i	$0.456443\pi$
$410$	−5.41847	−0.267599
$411$	0	0
$412$	−13.8990	−0.684754
$413$	−0.00529501	−0.000260550 0
$414$	0	0
$415$	7.10195	0.348621
$416$	26.5185	1.30018
$417$	0	0
$418$	2.18748	0.106993
$419$	0.466378	0.0227841	0.0113920	−	0.999935i	$-0.496374\pi$
$419$	0.466378	0.0227841	0.0113920	+	0.999935i	$0.496374\pi$
$420$	0	0
$421$	−2.41416	−0.117659	−0.0588295	−	0.998268i	$-0.518737\pi$
$421$	−2.41416	−0.117659	−0.0588295	+	0.998268i	$0.518737\pi$
$422$	1.14047	0.0555171
$423$	0	0
$424$	7.36009	0.357438
$425$	18.6435	0.904342
$426$	0	0
$427$	1.23173	0.0596078
$428$	17.4451	0.843240
$429$	0	0
$430$	−0.935822	−0.0451294
$431$	−12.0992	−0.582796	−0.291398	−	0.956602i	$-0.594120\pi$
$431$	−12.0992	−0.582796	−0.291398	+	0.956602i	$0.594120\pi$
$432$	0	0
$433$	33.3509	1.60274	0.801371	−	0.598167i	$-0.204104\pi$
$433$	33.3509	1.60274	0.801371	+	0.598167i	$0.204104\pi$
$434$	−0.722645	−0.0346881
$435$	0	0
$436$	−4.44150	−0.212709
$437$	−4.58812	−0.219480
$438$	0	0
$439$	−9.04458	−0.431674	−0.215837	−	0.976429i	$-0.569248\pi$
$439$	−9.04458	−0.431674	−0.215837	+	0.976429i	$0.569248\pi$
$440$	−13.7587	−0.655918
$441$	0	0
$442$	−14.9504	−0.711117
$443$	−4.67712	−0.222217	−0.111108	−	0.993808i	$-0.535440\pi$
$443$	−4.67712	−0.222217	−0.111108	+	0.993808i	$0.535440\pi$
$444$	0	0
$445$	7.18479	0.340592
$446$	6.70405	0.317446
$447$	0	0
$448$	−0.137851	−0.00651285
$449$	−26.7069	−1.26037	−0.630187	−	0.776443i	$-0.717022\pi$
$449$	−26.7069	−1.26037	−0.630187	+	0.776443i	$0.717022\pi$
$450$	0	0
$451$	−41.0702	−1.93392
$452$	−17.7418	−0.834503
$453$	0	0
$454$	−10.5895	−0.496988
$455$	−0.578176	−0.0271053
$456$	0	0
$457$	−0.657756	−0.0307685	−0.0153843	−	0.999882i	$-0.504897\pi$
$457$	−0.657756	−0.0307685	−0.0153843	+	0.999882i	$0.504897\pi$
$458$	7.69820	0.359713
$459$	0	0
$460$	12.5175	0.583633
$461$	20.7037	0.964268	0.482134	−	0.876097i	$-0.339862\pi$
$461$	20.7037	0.964268	0.482134	+	0.876097i	$0.339862\pi$
$462$	0	0
$463$	−24.8239	−1.15366	−0.576832	−	0.816863i	$-0.695711\pi$
$463$	−24.8239	−1.15366	−0.576832	+	0.816863i	$0.695711\pi$
$464$	−7.15582	−0.332200
$465$	0	0
$466$	−3.58079	−0.165877
$467$	26.6729	1.23428	0.617138	−	0.786855i	$-0.288292\pi$
$467$	26.6729	1.23428	0.617138	+	0.786855i	$0.288292\pi$
$468$	0	0
$469$	0.224318	0.0103580
$470$	−1.73205	−0.0798935
$471$	0	0
$472$	0.106067	0.00488212
$473$	−7.09321	−0.326146
$474$	0	0
$475$	2.29624	0.105359
$476$	−0.883000	−0.0404722
$477$	0	0
$478$	7.05232	0.322566
$479$	−3.22654	−0.147424	−0.0737121	−	0.997280i	$-0.523485\pi$
$479$	−3.22654	−0.147424	−0.0737121	+	0.997280i	$0.523485\pi$
$480$	0	0
$481$	9.99319	0.455650
$482$	−7.34948	−0.334760
$483$	0	0
$484$	−28.3824	−1.29011
$485$	9.46075	0.429590
$486$	0	0
$487$	−7.77930	−0.352514	−0.176257	−	0.984344i	$-0.556399\pi$
$487$	−7.77930	−0.352514	−0.176257	+	0.984344i	$0.556399\pi$
$488$	−24.6734	−1.11691
$489$	0	0
$490$	−5.00774	−0.226227
$491$	17.0460	0.769273	0.384637	−	0.923068i	$-0.374327\pi$
$491$	17.0460	0.769273	0.384637	+	0.923068i	$0.374327\pi$
$492$	0	0
$493$	24.2249	1.09104
$494$	−1.84137	−0.0828472
$495$	0	0
$496$	−12.3628	−0.555105
$497$	−0.785248	−0.0352232
$498$	0	0
$499$	7.93077	0.355030	0.177515	−	0.984118i	$-0.443194\pi$
$499$	7.93077	0.355030	0.177515	+	0.984118i	$0.443194\pi$
$500$	−14.2929	−0.639199
$501$	0	0
$502$	−10.3105	−0.460178
$503$	24.9496	1.11245	0.556224	−	0.831032i	$-0.312250\pi$
$503$	24.9496	1.11245	0.556224	+	0.831032i	$0.312250\pi$
$504$	0	0
$505$	13.6159	0.605898
$506$	−28.9766	−1.28817
$507$	0	0
$508$	23.6236	1.04813
$509$	−40.4690	−1.79376	−0.896878	−	0.442278i	$-0.854170\pi$
$509$	−40.4690	−1.79376	−0.896878	+	0.442278i	$0.854170\pi$
$510$	0	0
$511$	−1.47565	−0.0652790
$512$	−15.0038	−0.663080
$513$	0	0
$514$	2.27269	0.100244
$515$	−9.50747	−0.418949
$516$	0	0
$517$	−13.1284	−0.577384
$518$	−0.180257	−0.00792004
$519$	0	0
$520$	11.5817	0.507892
$521$	25.3674	1.11137	0.555684	−	0.831394i	$-0.312457\pi$
$521$	25.3674	1.11137	0.555684	+	0.831394i	$0.312457\pi$
$522$	0	0
$523$	12.7219	0.556291	0.278146	−	0.960539i	$-0.410280\pi$
$523$	12.7219	0.556291	0.278146	+	0.960539i	$0.410280\pi$
$524$	20.4627	0.893917
$525$	0	0
$526$	−6.04601	−0.263618
$527$	41.8523	1.82311
$528$	0	0
$529$	37.7769	1.64248
$530$	2.18382	0.0948591
$531$	0	0
$532$	−0.108755	−0.00471514
$533$	34.5719	1.49748
$534$	0	0
$535$	11.9331	0.515914
$536$	−4.49341	−0.194086
$537$	0	0
$538$	−5.53571	−0.238661
$539$	−37.9570	−1.63492
$540$	0	0
$541$	2.09327	0.0899969	0.0449984	−	0.998987i	$-0.485672\pi$
$541$	2.09327	0.0899969	0.0449984	+	0.998987i	$0.485672\pi$
$542$	12.9968	0.558259
$543$	0	0
$544$	27.7033	1.18777
$545$	−3.03817	−0.130141
$546$	0	0
$547$	3.45100	0.147554	0.0737770	−	0.997275i	$-0.476495\pi$
$547$	3.45100	0.147554	0.0737770	+	0.997275i	$0.476495\pi$
$548$	−30.0651	−1.28432
$549$	0	0
$550$	14.5021	0.618370
$551$	2.98368	0.127109
$552$	0	0
$553$	−0.0847118	−0.00360231
$554$	2.89306	0.122914
$555$	0	0
$556$	−24.9828	−1.05951
$557$	−11.1003	−0.470337	−0.235168	−	0.971955i	$-0.575564\pi$
$557$	−11.1003	−0.470337	−0.235168	+	0.971955i	$0.575564\pi$
$558$	0	0
$559$	5.97090	0.252542
$560$	0.178418	0.00753954
$561$	0	0
$562$	−4.99495	−0.210699
$563$	24.3107	1.02457	0.512286	−	0.858815i	$-0.328799\pi$
$563$	24.3107	1.02457	0.512286	+	0.858815i	$0.328799\pi$
$564$	0	0
$565$	−12.1361	−0.510569
$566$	−10.9587	−0.460628
$567$	0	0
$568$	15.7297	0.660002
$569$	42.7640	1.79276	0.896379	−	0.443288i	$-0.146188\pi$
$569$	42.7640	1.79276	0.896379	+	0.443288i	$0.146188\pi$
$570$	0	0
$571$	18.0283	0.754460	0.377230	−	0.926120i	$-0.376877\pi$
$571$	18.0283	0.754460	0.377230	+	0.926120i	$0.376877\pi$
$572$	38.0781	1.59212
$573$	0	0
$574$	−0.623608	−0.0260289
$575$	−30.4173	−1.26849
$576$	0	0
$577$	25.2763	1.05227	0.526133	−	0.850402i	$-0.323641\pi$
$577$	25.2763	1.05227	0.526133	+	0.850402i	$0.323641\pi$
$578$	−3.98963	−0.165947
$579$	0	0
$580$	−8.14022	−0.338004
$581$	0.817358	0.0339097
$582$	0	0
$583$	16.5526	0.685540
$584$	29.5595	1.22318
$585$	0	0
$586$	−12.3158	−0.508763
$587$	28.5653	1.17902	0.589508	−	0.807762i	$-0.299321\pi$
$587$	28.5653	1.17902	0.589508	+	0.807762i	$0.299321\pi$
$588$	0	0
$589$	5.15476	0.212398
$590$	0.0314712	0.00129565
$591$	0	0
$592$	−3.08378	−0.126743
$593$	20.6009	0.845977	0.422989	−	0.906135i	$-0.360981\pi$
$593$	20.6009	0.845977	0.422989	+	0.906135i	$0.360981\pi$
$594$	0	0
$595$	−0.604007	−0.0247619
$596$	16.9260	0.693318
$597$	0	0
$598$	24.3919	0.997458
$599$	−14.4748	−0.591424	−0.295712	−	0.955277i	$-0.595557\pi$
$599$	−14.4748	−0.591424	−0.295712	+	0.955277i	$0.595557\pi$
$600$	0	0
$601$	−13.3550	−0.544763	−0.272382	−	0.962189i	$-0.587811\pi$
$601$	−13.3550	−0.544763	−0.272382	+	0.962189i	$0.587811\pi$
$602$	−0.107703	−0.00438965
$603$	0	0
$604$	10.4192	0.423952
$605$	−19.4147	−0.789319
$606$	0	0
$607$	31.2131	1.26690	0.633450	−	0.773784i	$-0.281638\pi$
$607$	31.2131	1.26690	0.633450	+	0.773784i	$0.281638\pi$
$608$	3.41209	0.138378
$609$	0	0
$610$	−7.32089	−0.296414
$611$	11.0511	0.447082
$612$	0	0
$613$	30.0651	1.21432	0.607159	−	0.794580i	$-0.292309\pi$
$613$	30.0651	1.21432	0.607159	+	0.794580i	$0.292309\pi$
$614$	9.24189	0.372972
$615$	0	0
$616$	−1.58347	−0.0638000
$617$	−42.1537	−1.69704	−0.848521	−	0.529161i	$-0.822507\pi$
$617$	−42.1537	−1.69704	−0.848521	+	0.529161i	$0.822507\pi$
$618$	0	0
$619$	11.6108	0.466678	0.233339	−	0.972395i	$-0.425035\pi$
$619$	11.6108	0.466678	0.233339	+	0.972395i	$0.425035\pi$
$620$	−14.0635	−0.564803
$621$	0	0
$622$	−11.2335	−0.450422
$623$	0.826892	0.0331287
$624$	0	0
$625$	9.73143	0.389257
$626$	13.3831	0.534897
$627$	0	0
$628$	−25.8726	−1.03243
$629$	10.4397	0.416257
$630$	0	0
$631$	−29.3105	−1.16683	−0.583415	−	0.812174i	$-0.698284\pi$
$631$	−29.3105	−1.16683	−0.583415	+	0.812174i	$0.698284\pi$
$632$	1.69690	0.0674991
$633$	0	0
$634$	16.6759	0.662286
$635$	16.1595	0.641270
$636$	0	0
$637$	31.9513	1.26596
$638$	18.8436	0.746027
$639$	0	0
$640$	−11.3327	−0.447966
$641$	−31.0979	−1.22829	−0.614146	−	0.789192i	$-0.710499\pi$
$641$	−31.0979	−1.22829	−0.614146	+	0.789192i	$0.710499\pi$
$642$	0	0
$643$	42.0719	1.65915	0.829577	−	0.558392i	$-0.188581\pi$
$643$	42.0719	1.65915	0.829577	+	0.558392i	$0.188581\pi$
$644$	1.44063	0.0567690
$645$	0	0
$646$	−1.92364	−0.0756845
$647$	4.66717	0.183485	0.0917427	−	0.995783i	$-0.470756\pi$
$647$	4.66717	0.183485	0.0917427	+	0.995783i	$0.470756\pi$
$648$	0	0
$649$	0.238541	0.00936356
$650$	−12.2075	−0.478818
$651$	0	0
$652$	−12.8871	−0.504699
$653$	3.12413	0.122257	0.0611283	−	0.998130i	$-0.480530\pi$
$653$	3.12413	0.122257	0.0611283	+	0.998130i	$0.480530\pi$
$654$	0	0
$655$	13.9973	0.546920
$656$	−10.6685	−0.416534
$657$	0	0
$658$	−0.199340	−0.00777110
$659$	−37.4292	−1.45803	−0.729017	−	0.684495i	$-0.760023\pi$
$659$	−37.4292	−1.45803	−0.729017	+	0.684495i	$0.760023\pi$
$660$	0	0
$661$	26.5003	1.03074	0.515371	−	0.856967i	$-0.327654\pi$
$661$	26.5003	1.03074	0.515371	+	0.856967i	$0.327654\pi$
$662$	−19.4944	−0.757671
$663$	0	0
$664$	−16.3729	−0.635391
$665$	−0.0743929	−0.00288483
$666$	0	0
$667$	−39.5235	−1.53036
$668$	−4.02174	−0.155606
$669$	0	0
$670$	−1.33325	−0.0515078
$671$	−55.4898	−2.14216
$672$	0	0
$673$	1.72100	0.0663397	0.0331698	−	0.999450i	$-0.489440\pi$
$673$	1.72100	0.0663397	0.0331698	+	0.999450i	$0.489440\pi$
$674$	11.9341	0.459686
$675$	0	0
$676$	−12.1361	−0.466773
$677$	28.2792	1.08686	0.543429	−	0.839455i	$-0.317126\pi$
$677$	28.2792	1.08686	0.543429	+	0.839455i	$0.317126\pi$
$678$	0	0
$679$	1.08883	0.0417855
$680$	12.0992	0.463982
$681$	0	0
$682$	32.5553	1.24661
$683$	−24.7139	−0.945651	−0.472825	−	0.881156i	$-0.656766\pi$
$683$	−24.7139	−0.945651	−0.472825	+	0.881156i	$0.656766\pi$
$684$	0	0
$685$	−20.5657	−0.785777
$686$	−1.15387	−0.0440551
$687$	0	0
$688$	−1.84255	−0.0702465
$689$	−13.9336	−0.530829
$690$	0	0
$691$	−43.7725	−1.66518	−0.832592	−	0.553887i	$-0.813144\pi$
$691$	−43.7725	−1.66518	−0.832592	+	0.553887i	$0.813144\pi$
$692$	20.0754	0.763151
$693$	0	0
$694$	−17.2303	−0.654053
$695$	−17.0892	−0.648231
$696$	0	0
$697$	36.1165	1.36801
$698$	8.10728	0.306865
$699$	0	0
$700$	−0.721000	−0.0272512
$701$	−14.6504	−0.553338	−0.276669	−	0.960965i	$-0.589231\pi$
$701$	−14.6504	−0.553338	−0.276669	+	0.960965i	$0.589231\pi$
$702$	0	0
$703$	1.28581	0.0484951
$704$	6.21021	0.234056
$705$	0	0
$706$	5.74422	0.216187
$707$	1.56704	0.0589346
$708$	0	0
$709$	5.07367	0.190546	0.0952729	−	0.995451i	$-0.469628\pi$
$709$	5.07367	0.190546	0.0952729	+	0.995451i	$0.469628\pi$
$710$	4.66717	0.175156
$711$	0	0
$712$	−16.5639	−0.620757
$713$	−68.2830	−2.55722
$714$	0	0
$715$	26.0469	0.974100
$716$	−33.8574	−1.26531
$717$	0	0
$718$	−1.80604	−0.0674007
$719$	−5.33717	−0.199043	−0.0995213	−	0.995035i	$-0.531731\pi$
$719$	−5.33717	−0.199043	−0.0995213	+	0.995035i	$0.531731\pi$
$720$	0	0
$721$	−1.09421	−0.0407504
$722$	12.7598	0.474872
$723$	0	0
$724$	3.15476	0.117246
$725$	19.7805	0.734629
$726$	0	0
$727$	37.1198	1.37670	0.688348	−	0.725380i	$-0.258336\pi$
$727$	37.1198	1.37670	0.688348	+	0.725380i	$0.258336\pi$
$728$	1.33293	0.0494017
$729$	0	0
$730$	8.77063	0.324616
$731$	6.23767	0.230708
$732$	0	0
$733$	29.9463	1.10609	0.553045	−	0.833151i	$-0.313466\pi$
$733$	29.9463	1.10609	0.553045	+	0.833151i	$0.313466\pi$
$734$	−6.29152	−0.232224
$735$	0	0
$736$	−45.1985	−1.66604
$737$	−10.1055	−0.372243
$738$	0	0
$739$	−28.6100	−1.05244	−0.526218	−	0.850350i	$-0.676390\pi$
$739$	−28.6100	−1.05244	−0.526218	+	0.850350i	$0.676390\pi$
$740$	−3.50800	−0.128957
$741$	0	0
$742$	0.251334	0.00922678
$743$	49.6436	1.82125	0.910624	−	0.413237i	$-0.135602\pi$
$743$	49.6436	1.82125	0.910624	+	0.413237i	$0.135602\pi$
$744$	0	0
$745$	11.5781	0.424189
$746$	18.3262	0.670971
$747$	0	0
$748$	39.7793	1.45448
$749$	1.37338	0.0501821
$750$	0	0
$751$	−31.6236	−1.15396	−0.576981	−	0.816758i	$-0.695769\pi$
$751$	−31.6236	−1.15396	−0.576981	+	0.816758i	$0.695769\pi$
$752$	−3.41025	−0.124359
$753$	0	0
$754$	−15.8621	−0.577665
$755$	7.12716	0.259384
$756$	0	0
$757$	3.04189	0.110559	0.0552797	−	0.998471i	$-0.482395\pi$
$757$	3.04189	0.110559	0.0552797	+	0.998471i	$0.482395\pi$
$758$	18.5541	0.673914
$759$	0	0
$760$	1.49020	0.0540552
$761$	37.9283	1.37490	0.687450	−	0.726232i	$-0.258730\pi$
$761$	37.9283	1.37490	0.687450	+	0.726232i	$0.258730\pi$
$762$	0	0
$763$	−0.349660	−0.0126586
$764$	−6.05050	−0.218899
$765$	0	0
$766$	23.5493	0.850872
$767$	−0.200798	−0.00725041
$768$	0	0
$769$	−20.3878	−0.735201	−0.367601	−	0.929984i	$-0.619821\pi$
$769$	−20.3878	−0.735201	−0.367601	+	0.929984i	$0.619821\pi$
$770$	−0.469834	−0.0169317
$771$	0	0
$772$	12.9409	0.465752
$773$	24.4664	0.879994	0.439997	−	0.897999i	$-0.354979\pi$
$773$	24.4664	0.879994	0.439997	+	0.897999i	$0.354979\pi$
$774$	0	0
$775$	34.1739	1.22756
$776$	−21.8109	−0.782965
$777$	0	0
$778$	0.517541	0.0185547
$779$	4.44831	0.159377
$780$	0	0
$781$	35.3756	1.26584
$782$	25.4816	0.911221
$783$	0	0
$784$	−9.85978	−0.352135
$785$	−17.6979	−0.631665
$786$	0	0
$787$	37.5749	1.33940	0.669700	−	0.742631i	$-0.266422\pi$
$787$	37.5749	1.33940	0.669700	+	0.742631i	$0.266422\pi$
$788$	−3.51611	−0.125256
$789$	0	0
$790$	0.503490	0.0179134
$791$	−1.39673	−0.0496622
$792$	0	0
$793$	46.7101	1.65872
$794$	−4.79358	−0.170118
$795$	0	0
$796$	36.0087	1.27629
$797$	2.30493	0.0816449	0.0408224	−	0.999166i	$-0.487002\pi$
$797$	2.30493	0.0816449	0.0408224	+	0.999166i	$0.487002\pi$
$798$	0	0
$799$	11.5449	0.408429
$800$	22.6207	0.799762
$801$	0	0
$802$	−6.46473	−0.228277
$803$	66.4784	2.34597
$804$	0	0
$805$	0.985452	0.0347326
$806$	−27.4043	−0.965276
$807$	0	0
$808$	−31.3901	−1.10430
$809$	−28.8614	−1.01471	−0.507356	−	0.861736i	$-0.669377\pi$
$809$	−28.8614	−1.01471	−0.507356	+	0.861736i	$0.669377\pi$
$810$	0	0
$811$	−51.8631	−1.82116	−0.910580	−	0.413334i	$-0.864364\pi$
$811$	−51.8631	−1.82116	−0.910580	+	0.413334i	$0.864364\pi$
$812$	−0.936851	−0.0328770
$813$	0	0
$814$	8.12061	0.284627
$815$	−8.81531	−0.308787
$816$	0	0
$817$	0.768266	0.0268782
$818$	−3.77422	−0.131963
$819$	0	0
$820$	−12.1361	−0.423811
$821$	18.4411	0.643598	0.321799	−	0.946808i	$-0.395712\pi$
$821$	18.4411	0.643598	0.321799	+	0.946808i	$0.395712\pi$
$822$	0	0
$823$	−34.7665	−1.21188	−0.605942	−	0.795509i	$-0.707204\pi$
$823$	−34.7665	−1.21188	−0.605942	+	0.795509i	$0.707204\pi$
$824$	21.9186	0.763570
$825$	0	0
$826$	0.00362200	0.000126025 0
$827$	−32.7773	−1.13978	−0.569889	−	0.821722i	$-0.693014\pi$
$827$	−32.7773	−1.13978	−0.569889	+	0.821722i	$0.693014\pi$
$828$	0	0
$829$	5.35267	0.185906	0.0929530	−	0.995670i	$-0.470369\pi$
$829$	5.35267	0.185906	0.0929530	+	0.995670i	$0.470369\pi$
$830$	−4.85802	−0.168624
$831$	0	0
$832$	−5.22762	−0.181235
$833$	33.3788	1.15651
$834$	0	0
$835$	−2.75103	−0.0952033
$836$	4.89944	0.169451
$837$	0	0
$838$	−0.319022	−0.0110204
$839$	5.43837	0.187754	0.0938768	−	0.995584i	$-0.470074\pi$
$839$	5.43837	0.187754	0.0938768	+	0.995584i	$0.470074\pi$
$840$	0	0
$841$	−3.29767	−0.113713
$842$	1.65138	0.0569105
$843$	0	0
$844$	2.55438	0.0879253
$845$	−8.30158	−0.285583
$846$	0	0
$847$	−2.23442	−0.0767757
$848$	4.29975	0.147654
$849$	0	0
$850$	−12.7529	−0.437421
$851$	−17.0325	−0.583868
$852$	0	0
$853$	−47.1171	−1.61326	−0.806629	−	0.591057i	$-0.798711\pi$
$853$	−47.1171	−1.61326	−0.806629	+	0.591057i	$0.798711\pi$
$854$	−0.842556	−0.0288317
$855$	0	0
$856$	−27.5107	−0.940298
$857$	−46.9560	−1.60399	−0.801993	−	0.597333i	$-0.796227\pi$
$857$	−46.9560	−1.60399	−0.801993	+	0.597333i	$0.796227\pi$
$858$	0	0
$859$	7.16344	0.244413	0.122207	−	0.992505i	$-0.461003\pi$
$859$	7.16344	0.244413	0.122207	+	0.992505i	$0.461003\pi$
$860$	−2.09602	−0.0714737
$861$	0	0
$862$	8.27631	0.281892
$863$	−35.4309	−1.20608	−0.603041	−	0.797710i	$-0.706045\pi$
$863$	−35.4309	−1.20608	−0.603041	+	0.797710i	$0.706045\pi$
$864$	0	0
$865$	13.7324	0.466914
$866$	−22.8134	−0.775230
$867$	0	0
$868$	−1.61856	−0.0549373
$869$	3.81628	0.129458
$870$	0	0
$871$	8.50662	0.288236
$872$	7.00421	0.237193
$873$	0	0
$874$	3.13846	0.106160
$875$	−1.12522	−0.0380394
$876$	0	0
$877$	8.17293	0.275980	0.137990	−	0.990434i	$-0.455936\pi$
$877$	8.17293	0.275980	0.137990	+	0.990434i	$0.455936\pi$
$878$	6.18686	0.208796
$879$	0	0
$880$	−8.03777	−0.270953
$881$	−33.2307	−1.11957	−0.559785	−	0.828638i	$-0.689116\pi$
$881$	−33.2307	−1.11957	−0.559785	+	0.828638i	$0.689116\pi$
$882$	0	0
$883$	33.0479	1.11215	0.556075	−	0.831132i	$-0.312307\pi$
$883$	33.0479	1.11215	0.556075	+	0.831132i	$0.312307\pi$
$884$	−33.4853	−1.12623
$885$	0	0
$886$	3.19934	0.107484
$887$	−49.7472	−1.67035	−0.835174	−	0.549986i	$-0.814633\pi$
$887$	−49.7472	−1.67035	−0.835174	+	0.549986i	$0.814633\pi$
$888$	0	0
$889$	1.85978	0.0623752
$890$	−4.91469	−0.164741
$891$	0	0
$892$	15.0155	0.502756
$893$	1.42193	0.0475831
$894$	0	0
$895$	−23.1598	−0.774147
$896$	−1.30428	−0.0435729
$897$	0	0
$898$	18.2686	0.609630
$899$	44.4047	1.48098
$900$	0	0
$901$	−14.5561	−0.484935
$902$	28.0936	0.935416
$903$	0	0
$904$	27.9786	0.930556
$905$	2.15799	0.0717338
$906$	0	0
$907$	34.5485	1.14716	0.573582	−	0.819148i	$-0.305553\pi$
$907$	34.5485	1.14716	0.573582	+	0.819148i	$0.305553\pi$
$908$	−23.7179	−0.787106
$909$	0	0
$910$	0.395496	0.0131106
$911$	−13.6369	−0.451811	−0.225905	−	0.974149i	$-0.572534\pi$
$911$	−13.6369	−0.451811	−0.225905	+	0.974149i	$0.572534\pi$
$912$	0	0
$913$	−36.8221	−1.21863
$914$	0.449932	0.0148824
$915$	0	0
$916$	17.2422	0.569697
$917$	1.61094	0.0531979
$918$	0	0
$919$	32.7701	1.08099	0.540493	−	0.841348i	$-0.318238\pi$
$919$	32.7701	1.08099	0.540493	+	0.841348i	$0.318238\pi$
$920$	−19.7401	−0.650810
$921$	0	0
$922$	−14.1622	−0.466407
$923$	−29.7783	−0.980166
$924$	0	0
$925$	8.52435	0.280279
$926$	16.9805	0.558015
$927$	0	0
$928$	29.3928	0.964866
$929$	5.53147	0.181482	0.0907408	−	0.995875i	$-0.471077\pi$
$929$	5.53147	0.181482	0.0907408	+	0.995875i	$0.471077\pi$
$930$	0	0
$931$	4.11112	0.134736
$932$	−8.02011	−0.262708
$933$	0	0
$934$	−18.2453	−0.597006
$935$	27.2106	0.889883
$936$	0	0
$937$	−0.994014	−0.0324730	−0.0162365	−	0.999868i	$-0.505168\pi$
$937$	−0.994014	−0.0324730	−0.0162365	+	0.999868i	$0.505168\pi$
$938$	−0.153442	−0.00501007
$939$	0	0
$940$	−3.87939	−0.126532
$941$	11.4572	0.373493	0.186747	−	0.982408i	$-0.440206\pi$
$941$	11.4572	0.373493	0.186747	+	0.982408i	$0.440206\pi$
$942$	0	0
$943$	−58.9249	−1.91886
$944$	0.0619640	0.00201676
$945$	0	0
$946$	4.85204	0.157754
$947$	−45.6850	−1.48456	−0.742282	−	0.670088i	$-0.766257\pi$
$947$	−45.6850	−1.48456	−0.742282	+	0.670088i	$0.766257\pi$
$948$	0	0
$949$	−55.9600	−1.81654
$950$	−1.57072	−0.0509608
$951$	0	0
$952$	1.39248	0.0451306
$953$	−14.5053	−0.469873	−0.234936	−	0.972011i	$-0.575488\pi$
$953$	−14.5053	−0.469873	−0.234936	+	0.972011i	$0.575488\pi$
$954$	0	0
$955$	−4.13878	−0.133928
$956$	15.7955	0.510864
$957$	0	0
$958$	2.20708	0.0713076
$959$	−2.36690	−0.0764311
$960$	0	0
$961$	45.7161	1.47471
$962$	−6.83575	−0.220393
$963$	0	0
$964$	−16.4611	−0.530176
$965$	8.85208	0.284959
$966$	0	0
$967$	−25.0368	−0.805130	−0.402565	−	0.915391i	$-0.631881\pi$
$967$	−25.0368	−0.805130	−0.402565	+	0.915391i	$0.631881\pi$
$968$	44.7588	1.43860
$969$	0	0
$970$	−6.47153	−0.207788
$971$	27.0907	0.869383	0.434692	−	0.900579i	$-0.356857\pi$
$971$	27.0907	0.869383	0.434692	+	0.900579i	$0.356857\pi$
$972$	0	0
$973$	−1.96679	−0.0630522
$974$	5.32136	0.170507
$975$	0	0
$976$	−14.4142	−0.461386
$977$	−2.47340	−0.0791310	−0.0395655	−	0.999217i	$-0.512597\pi$
$977$	−2.47340	−0.0791310	−0.0395655	+	0.999217i	$0.512597\pi$
$978$	0	0
$979$	−37.2517	−1.19057
$980$	−11.2162	−0.358287
$981$	0	0
$982$	−11.6601	−0.372089
$983$	19.1080	0.609451	0.304726	−	0.952440i	$-0.401435\pi$
$983$	19.1080	0.609451	0.304726	+	0.952440i	$0.401435\pi$
$984$	0	0
$985$	−2.40516	−0.0766349
$986$	−16.5708	−0.527723
$987$	0	0
$988$	−4.12424	−0.131209
$989$	−10.1769	−0.323607
$990$	0	0
$991$	−38.3164	−1.21716	−0.608581	−	0.793492i	$-0.708261\pi$
$991$	−38.3164	−1.21716	−0.608581	+	0.793492i	$0.708261\pi$
$992$	50.7806	1.61229
$993$	0	0
$994$	0.537141	0.0170371
$995$	24.6314	0.780867
$996$	0	0
$997$	41.3746	1.31035	0.655174	−	0.755478i	$-0.272595\pi$
$997$	41.3746	1.31035	0.655174	+	0.755478i	$0.272595\pi$
$998$	−5.42497	−0.171724
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.a.c.1.3	✓	6
3.2	odd	2	inner	729.2.a.c.1.4	yes	6
9.2	odd	6		729.2.c.c.244.3		12
9.4	even	3		729.2.c.c.487.4		12
9.5	odd	6		729.2.c.c.487.3		12
9.7	even	3		729.2.c.c.244.4		12
27.2	odd	18		729.2.e.m.568.1		12
27.4	even	9		729.2.e.r.406.1		12
27.5	odd	18		729.2.e.q.649.2		12
27.7	even	9		729.2.e.r.325.1		12
27.11	odd	18		729.2.e.q.82.2		12
27.13	even	9		729.2.e.m.163.2		12
27.14	odd	18		729.2.e.m.163.1		12
27.16	even	9		729.2.e.q.82.1		12
27.20	odd	18		729.2.e.r.325.2		12
27.22	even	9		729.2.e.q.649.1		12
27.23	odd	18		729.2.e.r.406.2		12
27.25	even	9		729.2.e.m.568.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.2.a.c.1.3	✓	6	1.1	even	1	trivial
729.2.a.c.1.4	yes	6	3.2	odd	2	inner
729.2.c.c.244.3		12	9.2	odd	6
729.2.c.c.244.4		12	9.7	even	3
729.2.c.c.487.3		12	9.5	odd	6
729.2.c.c.487.4		12	9.4	even	3
729.2.e.m.163.1		12	27.14	odd	18
729.2.e.m.163.2		12	27.13	even	9
729.2.e.m.568.1		12	27.2	odd	18
729.2.e.m.568.2		12	27.25	even	9
729.2.e.q.82.1		12	27.16	even	9
729.2.e.q.82.2		12	27.11	odd	18
729.2.e.q.649.1		12	27.22	even	9
729.2.e.q.649.2		12	27.5	odd	18
729.2.e.r.325.1		12	27.7	even	9
729.2.e.r.325.2		12	27.20	odd	18
729.2.e.r.406.1		12	27.4	even	9
729.2.e.r.406.2		12	27.23	odd	18

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:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	729.a (trivial)

\(f(q)\)	\(=\)	\(q-0.684040 q^{2} -1.53209 q^{4} -1.04801 q^{5} -0.120615 q^{7} +2.41609 q^{8} +0.716881 q^{10} +5.43372 q^{11} -4.57398 q^{13} +0.0825054 q^{14} +1.41147 q^{16} -4.77833 q^{17} -0.588526 q^{19} +1.60565 q^{20} -3.71688 q^{22} +7.79596 q^{23} -3.90167 q^{25} +3.12879 q^{26} +0.184793 q^{28} -5.06975 q^{29} -8.75877 q^{31} -5.79769 q^{32} +3.26857 q^{34} +0.126406 q^{35} -2.18479 q^{37} +0.402575 q^{38} -2.53209 q^{40} -7.55839 q^{41} -1.30541 q^{43} -8.32494 q^{44} -5.33275 q^{46} -2.41609 q^{47} -6.98545 q^{49} +2.66890 q^{50} +7.00774 q^{52} +3.04628 q^{53} -5.69459 q^{55} -0.291416 q^{56} +3.46791 q^{58} +0.0439002 q^{59} -10.2121 q^{61} +5.99135 q^{62} +1.14290 q^{64} +4.79358 q^{65} -1.85978 q^{67} +7.32083 q^{68} -0.0864665 q^{70} +6.51038 q^{71} +12.2344 q^{73} +1.49449 q^{74} +0.901674 q^{76} -0.655386 q^{77} +0.702333 q^{79} -1.47924 q^{80} +5.17024 q^{82} -6.77660 q^{83} +5.00774 q^{85} +0.892951 q^{86} +13.1284 q^{88} -6.85565 q^{89} +0.551689 q^{91} -11.9441 q^{92} +1.65270 q^{94} +0.616781 q^{95} -9.02734 q^{97} +4.77833 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64}+ \cdots - 12 q^{97}+O(q^{100}) \) 6 * q - 12 * q^7 - 12 * q^10 - 12 * q^13 - 12 * q^16 - 24 * q^19 - 6 * q^22 - 6 * q^28 - 30 * q^31 - 6 * q^37 - 6 * q^40 - 12 * q^43 + 6 * q^46 - 6 * q^49 - 6 * q^52 - 30 * q^55 + 30 * q^58 - 12 * q^61 + 6 * q^64 + 6 * q^67 + 30 * q^70 + 12 * q^73 - 18 * q^76 - 48 * q^79 - 12 * q^82 - 18 * q^85 + 42 * q^88 + 12 * q^94 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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