LMFDB - Embedded newform 729.2.a.a.1.6

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:	6.6.1397493.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 $ x^6 - 3x^5 - 3x^4 + 10x^3 + 3x^2 - 6*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.198473$ 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)

$f(q)$	$=$	$q+1.68091 q^{2} +0.825466 q^{4} -1.12954 q^{5} -3.90892 q^{7} -1.97429 q^{8} +O(q^{10})$	\(q+1.68091 q^{2} +0.825466 q^{4} -1.12954 q^{5} -3.90892 q^{7} -1.97429 q^{8} -1.89866 q^{10} -1.87046 q^{11} -0.732598 q^{13} -6.57056 q^{14} -4.96954 q^{16} +1.88964 q^{17} +2.74286 q^{19} -0.932400 q^{20} -3.14407 q^{22} -5.82770 q^{23} -3.72413 q^{25} -1.23143 q^{26} -3.22668 q^{28} -5.31994 q^{29} +1.34060 q^{31} -4.40478 q^{32} +3.17632 q^{34} +4.41530 q^{35} +3.39611 q^{37} +4.61050 q^{38} +2.23005 q^{40} +1.79633 q^{41} +5.03015 q^{43} -1.54400 q^{44} -9.79585 q^{46} +1.70868 q^{47} +8.27968 q^{49} -6.25994 q^{50} -0.604734 q^{52} +2.84494 q^{53} +2.11276 q^{55} +7.71734 q^{56} -8.94234 q^{58} -11.2600 q^{59} +5.23089 q^{61} +2.25343 q^{62} +2.53503 q^{64} +0.827502 q^{65} -1.88903 q^{67} +1.55984 q^{68} +7.42173 q^{70} -12.1839 q^{71} +9.88768 q^{73} +5.70857 q^{74} +2.26413 q^{76} +7.31147 q^{77} +12.3529 q^{79} +5.61331 q^{80} +3.01948 q^{82} -11.6832 q^{83} -2.13444 q^{85} +8.45525 q^{86} +3.69282 q^{88} -5.72873 q^{89} +2.86367 q^{91} -4.81057 q^{92} +2.87214 q^{94} -3.09818 q^{95} -0.343378 q^{97} +13.9174 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10}) $ 6 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 6 * q^8	$ 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8} + 3 q^{10} - 12 q^{11} - 6 q^{14} - 3 q^{16} - 9 q^{17} + 3 q^{19} - 6 q^{20} + 6 q^{22} - 15 q^{23} - 6 q^{25} - 15 q^{26} - 6 q^{28} - 12 q^{29} - 12 q^{35} + 3 q^{37} + 3 q^{38} + 6 q^{40} - 15 q^{41} - 3 q^{44} + 3 q^{46} - 21 q^{47} - 12 q^{49} - 3 q^{50} + 12 q^{52} - 9 q^{53} - 6 q^{55} + 6 q^{56} - 12 q^{58} - 24 q^{59} - 9 q^{61} + 12 q^{62} - 12 q^{64} + 6 q^{65} - 9 q^{67} + 9 q^{68} + 15 q^{70} - 27 q^{71} - 6 q^{73} + 12 q^{74} + 6 q^{76} + 12 q^{77} + 21 q^{80} - 6 q^{82} - 12 q^{83} + 21 q^{86} + 12 q^{88} - 9 q^{89} - 6 q^{91} - 6 q^{92} + 6 q^{94} - 12 q^{95} + 45 q^{98}+O(q^{100}) $ 6 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 6 * q^8 + 3 * q^10 - 12 * q^11 - 6 * q^14 - 3 * q^16 - 9 * q^17 + 3 * q^19 - 6 * q^20 + 6 * q^22 - 15 * q^23 - 6 * q^25 - 15 * q^26 - 6 * q^28 - 12 * q^29 - 12 * q^35 + 3 * q^37 + 3 * q^38 + 6 * q^40 - 15 * q^41 - 3 * q^44 + 3 * q^46 - 21 * q^47 - 12 * q^49 - 3 * q^50 + 12 * q^52 - 9 * q^53 - 6 * q^55 + 6 * q^56 - 12 * q^58 - 24 * q^59 - 9 * q^61 + 12 * q^62 - 12 * q^64 + 6 * q^65 - 9 * q^67 + 9 * q^68 + 15 * q^70 - 27 * q^71 - 6 * q^73 + 12 * q^74 + 6 * q^76 + 12 * q^77 + 21 * q^80 - 6 * q^82 - 12 * q^83 + 21 * q^86 + 12 * q^88 - 9 * q^89 - 6 * q^91 - 6 * q^92 + 6 * q^94 - 12 * q^95 + 45 * 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 $

$2$	1.68091	1.18858	0.594292	−	0.804249i	$-0.297432\pi$
$2$	1.68091	1.18858	0.594292	+	0.804249i	$0.297432\pi$
$3$	0	0
$3$	0	0
$4$	0.825466	0.412733
$5$	−1.12954	−0.505147	−0.252574	−	0.967578i	$-0.581277\pi$
$5$	−1.12954	−0.505147	−0.252574	+	0.967578i	$0.581277\pi$
$6$	0	0
$7$	−3.90892	−1.47743	−0.738717	−	0.674016i	$-0.764568\pi$
$7$	−3.90892	−1.47743	−0.738717	+	0.674016i	$0.764568\pi$
$8$	−1.97429	−0.698017
$9$	0	0
$10$	−1.89866	−0.600410
$11$	−1.87046	−0.563964	−0.281982	−	0.959420i	$-0.590992\pi$
$11$	−1.87046	−0.563964	−0.281982	+	0.959420i	$0.590992\pi$
$12$	0	0
$13$	−0.732598	−0.203186	−0.101593	−	0.994826i	$-0.532394\pi$
$13$	−0.732598	−0.203186	−0.101593	+	0.994826i	$0.532394\pi$
$14$	−6.57056	−1.75605
$15$	0	0
$16$	−4.96954	−1.24238
$17$	1.88964	0.458306	0.229153	−	0.973390i	$-0.426404\pi$
$17$	1.88964	0.458306	0.229153	+	0.973390i	$0.426404\pi$
$18$	0	0
$19$	2.74286	0.629254	0.314627	−	0.949215i	$-0.398121\pi$
$19$	2.74286	0.629254	0.314627	+	0.949215i	$0.398121\pi$
$20$	−0.932400	−0.208491
$21$	0	0
$22$	−3.14407	−0.670318
$23$	−5.82770	−1.21516	−0.607580	−	0.794259i	$-0.707859\pi$
$23$	−5.82770	−1.21516	−0.607580	+	0.794259i	$0.707859\pi$
$24$	0	0
$25$	−3.72413	−0.744826
$26$	−1.23143	−0.241504
$27$	0	0
$28$	−3.22668	−0.609786
$29$	−5.31994	−0.987887	−0.493944	−	0.869494i	$-0.664445\pi$
$29$	−5.31994	−0.987887	−0.493944	+	0.869494i	$0.664445\pi$
$30$	0	0
$31$	1.34060	0.240779	0.120390	−	0.992727i	$-0.461586\pi$
$31$	1.34060	0.240779	0.120390	+	0.992727i	$0.461586\pi$
$32$	−4.40478	−0.778662
$33$	0	0
$34$	3.17632	0.544735
$35$	4.41530	0.746322
$36$	0	0
$37$	3.39611	0.558318	0.279159	−	0.960245i	$-0.409944\pi$
$37$	3.39611	0.558318	0.279159	+	0.960245i	$0.409944\pi$
$38$	4.61050	0.747922
$39$	0	0
$40$	2.23005	0.352601
$41$	1.79633	0.280540	0.140270	−	0.990113i	$-0.455203\pi$
$41$	1.79633	0.280540	0.140270	+	0.990113i	$0.455203\pi$
$42$	0	0
$43$	5.03015	0.767091	0.383546	−	0.923522i	$-0.374703\pi$
$43$	5.03015	0.767091	0.383546	+	0.923522i	$0.374703\pi$
$44$	−1.54400	−0.232766
$45$	0	0
$46$	−9.79585	−1.44432
$47$	1.70868	0.249236	0.124618	−	0.992205i	$-0.460229\pi$
$47$	1.70868	0.249236	0.124618	+	0.992205i	$0.460229\pi$
$48$	0	0
$49$	8.27968	1.18281
$50$	−6.25994	−0.885289
$51$	0	0
$52$	−0.604734	−0.0838616
$53$	2.84494	0.390783	0.195391	−	0.980725i	$-0.437402\pi$
$53$	2.84494	0.390783	0.195391	+	0.980725i	$0.437402\pi$
$54$	0	0
$55$	2.11276	0.284885
$56$	7.71734	1.03127
$57$	0	0
$58$	−8.94234	−1.17419
$59$	−11.2600	−1.46593	−0.732967	−	0.680265i	$-0.761865\pi$
$59$	−11.2600	−1.46593	−0.732967	+	0.680265i	$0.761865\pi$
$60$	0	0
$61$	5.23089	0.669747	0.334874	−	0.942263i	$-0.391306\pi$
$61$	5.23089	0.669747	0.334874	+	0.942263i	$0.391306\pi$
$62$	2.25343	0.286186
$63$	0	0
$64$	2.53503	0.316879
$65$	0.827502	0.102639
$66$	0	0
$67$	−1.88903	−0.230781	−0.115391	−	0.993320i	$-0.536812\pi$
$67$	−1.88903	−0.230781	−0.115391	+	0.993320i	$0.536812\pi$
$68$	1.55984	0.189158
$69$	0	0
$70$	7.42173	0.887067
$71$	−12.1839	−1.44596	−0.722980	−	0.690869i	$-0.757228\pi$
$71$	−12.1839	−1.44596	−0.722980	+	0.690869i	$0.757228\pi$
$72$	0	0
$73$	9.88768	1.15727	0.578633	−	0.815588i	$-0.303587\pi$
$73$	9.88768	1.15727	0.578633	+	0.815588i	$0.303587\pi$
$74$	5.70857	0.663608
$75$	0	0
$76$	2.26413	0.259714
$77$	7.31147	0.833219
$78$	0	0
$79$	12.3529	1.38981	0.694904	−	0.719103i	$-0.255447\pi$
$79$	12.3529	1.38981	0.694904	+	0.719103i	$0.255447\pi$
$80$	5.61331	0.627587
$81$	0	0
$82$	3.01948	0.333445
$83$	−11.6832	−1.28239	−0.641197	−	0.767376i	$-0.721562\pi$
$83$	−11.6832	−1.28239	−0.641197	+	0.767376i	$0.721562\pi$
$84$	0	0
$85$	−2.13444	−0.231512
$86$	8.45525	0.911753
$87$	0	0
$88$	3.69282	0.393656
$89$	−5.72873	−0.607244	−0.303622	−	0.952793i	$-0.598196\pi$
$89$	−5.72873	−0.607244	−0.303622	+	0.952793i	$0.598196\pi$
$90$	0	0
$91$	2.86367	0.300194
$92$	−4.81057	−0.501536
$93$	0	0
$94$	2.87214	0.296238
$95$	−3.09818	−0.317866
$96$	0	0
$97$	−0.343378	−0.0348648	−0.0174324	−	0.999848i	$-0.505549\pi$
$97$	−0.343378	−0.0348648	−0.0174324	+	0.999848i	$0.505549\pi$
$98$	13.9174	1.40587
$99$	0	0
$100$	−3.07414	−0.307414
$101$	−17.4023	−1.73159	−0.865794	−	0.500400i	$-0.833186\pi$
$101$	−17.4023	−1.73159	−0.865794	+	0.500400i	$0.833186\pi$
$102$	0	0
$103$	−15.8130	−1.55810	−0.779051	−	0.626961i	$-0.784299\pi$
$103$	−15.8130	−1.55810	−0.779051	+	0.626961i	$0.784299\pi$
$104$	1.44636	0.141827
$105$	0	0
$106$	4.78210	0.464478
$107$	−16.5298	−1.59800	−0.798999	−	0.601332i	$-0.794637\pi$
$107$	−16.5298	−1.59800	−0.798999	+	0.601332i	$0.794637\pi$
$108$	0	0
$109$	−4.71844	−0.451945	−0.225972	−	0.974134i	$-0.572556\pi$
$109$	−4.71844	−0.451945	−0.225972	+	0.974134i	$0.572556\pi$
$110$	3.55137	0.338610
$111$	0	0
$112$	19.4255	1.83554
$113$	19.9448	1.87625	0.938125	−	0.346296i	$-0.112561\pi$
$113$	19.9448	1.87625	0.938125	+	0.346296i	$0.112561\pi$
$114$	0	0
$115$	6.58264	0.613835
$116$	−4.39142	−0.407733
$117$	0	0
$118$	−18.9271	−1.74239
$119$	−7.38647	−0.677117
$120$	0	0
$121$	−7.50139	−0.681945
$122$	8.79267	0.796051
$123$	0	0
$124$	1.10662	0.0993774
$125$	9.85429	0.881394
$126$	0	0
$127$	1.06946	0.0948989	0.0474495	−	0.998874i	$-0.484891\pi$
$127$	1.06946	0.0948989	0.0474495	+	0.998874i	$0.484891\pi$
$128$	13.0707	1.15530
$129$	0	0
$130$	1.39096	0.121995
$131$	7.64030	0.667536	0.333768	−	0.942655i	$-0.391680\pi$
$131$	7.64030	0.667536	0.333768	+	0.942655i	$0.391680\pi$
$132$	0	0
$133$	−10.7216	−0.929682
$134$	−3.17529	−0.274303
$135$	0	0
$136$	−3.73070	−0.319905
$137$	15.6505	1.33711	0.668556	−	0.743662i	$-0.266913\pi$
$137$	15.6505	1.33711	0.668556	+	0.743662i	$0.266913\pi$
$138$	0	0
$139$	8.65693	0.734272	0.367136	−	0.930167i	$-0.380338\pi$
$139$	8.65693	0.734272	0.367136	+	0.930167i	$0.380338\pi$
$140$	3.64468	0.308032
$141$	0	0
$142$	−20.4800	−1.71864
$143$	1.37029	0.114590
$144$	0	0
$145$	6.00910	0.499029
$146$	16.6203	1.37551
$147$	0	0
$148$	2.80337	0.230436
$149$	−2.46382	−0.201844	−0.100922	−	0.994894i	$-0.532179\pi$
$149$	−2.46382	−0.201844	−0.100922	+	0.994894i	$0.532179\pi$
$150$	0	0
$151$	−10.5134	−0.855569	−0.427785	−	0.903881i	$-0.640706\pi$
$151$	−10.5134	−0.855569	−0.427785	+	0.903881i	$0.640706\pi$
$152$	−5.41519	−0.439230
$153$	0	0
$154$	12.2899	0.990351
$155$	−1.51427	−0.121629
$156$	0	0
$157$	0.359949	0.0287270	0.0143635	−	0.999897i	$-0.495428\pi$
$157$	0.359949	0.0287270	0.0143635	+	0.999897i	$0.495428\pi$
$158$	20.7641	1.65190
$159$	0	0
$160$	4.97539	0.393339
$161$	22.7800	1.79532
$162$	0	0
$163$	14.6186	1.14502	0.572508	−	0.819899i	$-0.305971\pi$
$163$	14.6186	1.14502	0.572508	+	0.819899i	$0.305971\pi$
$164$	1.48281	0.115788
$165$	0	0
$166$	−19.6384	−1.52423
$167$	2.17585	0.168372	0.0841861	−	0.996450i	$-0.473171\pi$
$167$	2.17585	0.168372	0.0841861	+	0.996450i	$0.473171\pi$
$168$	0	0
$169$	−12.4633	−0.958715
$170$	−3.58780	−0.275172
$171$	0	0
$172$	4.15222	0.316604
$173$	−17.5670	−1.33560	−0.667798	−	0.744343i	$-0.732763\pi$
$173$	−17.5670	−1.33560	−0.667798	+	0.744343i	$0.732763\pi$
$174$	0	0
$175$	14.5573	1.10043
$176$	9.29530	0.700660
$177$	0	0
$178$	−9.62950	−0.721761
$179$	1.00447	0.0750777	0.0375388	−	0.999295i	$-0.488048\pi$
$179$	1.00447	0.0750777	0.0375388	+	0.999295i	$0.488048\pi$
$180$	0	0
$181$	−21.1732	−1.57380	−0.786898	−	0.617084i	$-0.788314\pi$
$181$	−21.1732	−1.57380	−0.786898	+	0.617084i	$0.788314\pi$
$182$	4.81358	0.356806
$183$	0	0
$184$	11.5056	0.848201
$185$	−3.83606	−0.282033
$186$	0	0
$187$	−3.53449	−0.258468
$188$	1.41045	0.102868
$189$	0	0
$190$	−5.20776	−0.377811
$191$	−9.82838	−0.711157	−0.355578	−	0.934646i	$-0.615716\pi$
$191$	−9.82838	−0.711157	−0.355578	+	0.934646i	$0.615716\pi$
$192$	0	0
$193$	−11.1561	−0.803033	−0.401516	−	0.915852i	$-0.631517\pi$
$193$	−11.1561	−0.803033	−0.401516	+	0.915852i	$0.631517\pi$
$194$	−0.577189	−0.0414397
$195$	0	0
$196$	6.83459	0.488185
$197$	9.08994	0.647631	0.323816	−	0.946120i	$-0.395034\pi$
$197$	9.08994	0.647631	0.323816	+	0.946120i	$0.395034\pi$
$198$	0	0
$199$	−14.6939	−1.04162	−0.520811	−	0.853672i	$-0.674370\pi$
$199$	−14.6939	−1.04162	−0.520811	+	0.853672i	$0.674370\pi$
$200$	7.35251	0.519901
$201$	0	0
$202$	−29.2517	−2.05814
$203$	20.7952	1.45954
$204$	0	0
$205$	−2.02904	−0.141714
$206$	−26.5803	−1.85194
$207$	0	0
$208$	3.64067	0.252435
$209$	−5.13039	−0.354877
$210$	0	0
$211$	7.85830	0.540988	0.270494	−	0.962722i	$-0.412813\pi$
$211$	7.85830	0.540988	0.270494	+	0.962722i	$0.412813\pi$
$212$	2.34840	0.161289
$213$	0	0
$214$	−27.7852	−1.89936
$215$	−5.68178	−0.387494
$216$	0	0
$217$	−5.24030	−0.355735
$218$	−7.93128	−0.537174
$219$	0	0
$220$	1.74401	0.117581
$221$	−1.38435	−0.0931214
$222$	0	0
$223$	−8.74089	−0.585333	−0.292666	−	0.956215i	$-0.594543\pi$
$223$	−8.74089	−0.585333	−0.292666	+	0.956215i	$0.594543\pi$
$224$	17.2179	1.15042
$225$	0	0
$226$	33.5255	2.23008
$227$	4.06842	0.270030	0.135015	−	0.990844i	$-0.456892\pi$
$227$	4.06842	0.270030	0.135015	+	0.990844i	$0.456892\pi$
$228$	0	0
$229$	16.1559	1.06761	0.533805	−	0.845608i	$-0.320762\pi$
$229$	16.1559	1.06761	0.533805	+	0.845608i	$0.320762\pi$
$230$	11.0648	0.729594
$231$	0	0
$232$	10.5031	0.689562
$233$	17.2132	1.12767	0.563836	−	0.825887i	$-0.309325\pi$
$233$	17.2132	1.12767	0.563836	+	0.825887i	$0.309325\pi$
$234$	0	0
$235$	−1.93003	−0.125901
$236$	−9.29478	−0.605039
$237$	0	0
$238$	−12.4160	−0.804810
$239$	−1.53544	−0.0993193	−0.0496597	−	0.998766i	$-0.515814\pi$
$239$	−1.53544	−0.0993193	−0.0496597	+	0.998766i	$0.515814\pi$
$240$	0	0
$241$	−5.52167	−0.355682	−0.177841	−	0.984059i	$-0.556911\pi$
$241$	−5.52167	−0.355682	−0.177841	+	0.984059i	$0.556911\pi$
$242$	−12.6092	−0.810549
$243$	0	0
$244$	4.31792	0.276427
$245$	−9.35226	−0.597494
$246$	0	0
$247$	−2.00941	−0.127856
$248$	−2.64673	−0.168068
$249$	0	0
$250$	16.5642	1.04761
$251$	−21.4409	−1.35334	−0.676668	−	0.736288i	$-0.736577\pi$
$251$	−21.4409	−1.35334	−0.676668	+	0.736288i	$0.736577\pi$
$252$	0	0
$253$	10.9005	0.685306
$254$	1.79766	0.112795
$255$	0	0
$256$	16.9007	1.05629
$257$	−14.7865	−0.922355	−0.461177	−	0.887308i	$-0.652573\pi$
$257$	−14.7865	−0.922355	−0.461177	+	0.887308i	$0.652573\pi$
$258$	0	0
$259$	−13.2751	−0.824877
$260$	0.683074	0.0423625
$261$	0	0
$262$	12.8427	0.793423
$263$	2.80276	0.172825	0.0864127	−	0.996259i	$-0.472460\pi$
$263$	2.80276	0.172825	0.0864127	+	0.996259i	$0.472460\pi$
$264$	0	0
$265$	−3.21349	−0.197403
$266$	−18.0221	−1.10501
$267$	0	0
$268$	−1.55933	−0.0952511
$269$	−0.356528	−0.0217379	−0.0108689	−	0.999941i	$-0.503460\pi$
$269$	−0.356528	−0.0217379	−0.0108689	+	0.999941i	$0.503460\pi$
$270$	0	0
$271$	−12.1467	−0.737857	−0.368928	−	0.929458i	$-0.620275\pi$
$271$	−12.1467	−0.737857	−0.368928	+	0.929458i	$0.620275\pi$
$272$	−9.39065	−0.569392
$273$	0	0
$274$	26.3071	1.58927
$275$	6.96582	0.420055
$276$	0	0
$277$	24.9900	1.50150	0.750751	−	0.660586i	$-0.229692\pi$
$277$	24.9900	1.50150	0.750751	+	0.660586i	$0.229692\pi$
$278$	14.5515	0.872744
$279$	0	0
$280$	−8.71708	−0.520945
$281$	7.32287	0.436846	0.218423	−	0.975854i	$-0.429909\pi$
$281$	7.32287	0.436846	0.218423	+	0.975854i	$0.429909\pi$
$282$	0	0
$283$	−14.3881	−0.855283	−0.427641	−	0.903948i	$-0.640655\pi$
$283$	−14.3881	−0.855283	−0.427641	+	0.903948i	$0.640655\pi$
$284$	−10.0574	−0.596795
$285$	0	0
$286$	2.30334	0.136199
$287$	−7.02172	−0.414479
$288$	0	0
$289$	−13.4292	−0.789956
$290$	10.1008	0.593138
$291$	0	0
$292$	8.16194	0.477641
$293$	−14.4155	−0.842163	−0.421082	−	0.907023i	$-0.638349\pi$
$293$	−14.4155	−0.842163	−0.421082	+	0.907023i	$0.638349\pi$
$294$	0	0
$295$	12.7187	0.740512
$296$	−6.70491	−0.389715
$297$	0	0
$298$	−4.14147	−0.239909
$299$	4.26936	0.246904
$300$	0	0
$301$	−19.6625	−1.13333
$302$	−17.6721	−1.01692
$303$	0	0
$304$	−13.6307	−0.781776
$305$	−5.90853	−0.338321
$306$	0	0
$307$	−30.4326	−1.73688	−0.868440	−	0.495795i	$-0.834877\pi$
$307$	−30.4326	−1.73688	−0.868440	+	0.495795i	$0.834877\pi$
$308$	6.03537	0.343897
$309$	0	0
$310$	−2.54535	−0.144566
$311$	−13.9956	−0.793618	−0.396809	−	0.917901i	$-0.629882\pi$
$311$	−13.9956	−0.793618	−0.396809	+	0.917901i	$0.629882\pi$
$312$	0	0
$313$	22.0876	1.24846	0.624231	−	0.781239i	$-0.285412\pi$
$313$	22.0876	1.24846	0.624231	+	0.781239i	$0.285412\pi$
$314$	0.605042	0.0341445
$315$	0	0
$316$	10.1969	0.573619
$317$	17.3887	0.976648	0.488324	−	0.872662i	$-0.337608\pi$
$317$	17.3887	0.976648	0.488324	+	0.872662i	$0.337608\pi$
$318$	0	0
$319$	9.95070	0.557132
$320$	−2.86343	−0.160070
$321$	0	0
$322$	38.2912	2.13389
$323$	5.18302	0.288391
$324$	0	0
$325$	2.72829	0.151338
$326$	24.5725	1.36095
$327$	0	0
$328$	−3.54648	−0.195822
$329$	−6.67909	−0.368230
$330$	0	0
$331$	0.864147	0.0474978	0.0237489	−	0.999718i	$-0.492440\pi$
$331$	0.864147	0.0474978	0.0237489	+	0.999718i	$0.492440\pi$
$332$	−9.64405	−0.529286
$333$	0	0
$334$	3.65741	0.200124
$335$	2.13374	0.116579
$336$	0	0
$337$	0.418529	0.0227987	0.0113994	−	0.999935i	$-0.496371\pi$
$337$	0.418529	0.0227987	0.0113994	+	0.999935i	$0.496371\pi$
$338$	−20.9497	−1.13951
$339$	0	0
$340$	−1.76190	−0.0955526
$341$	−2.50753	−0.135791
$342$	0	0
$343$	−5.00216	−0.270091
$344$	−9.93098	−0.535442
$345$	0	0
$346$	−29.5286	−1.58747
$347$	23.4097	1.25670	0.628351	−	0.777930i	$-0.283730\pi$
$347$	23.4097	1.25670	0.628351	+	0.777930i	$0.283730\pi$
$348$	0	0
$349$	21.1764	1.13355	0.566773	−	0.823874i	$-0.308192\pi$
$349$	21.1764	1.13355	0.566773	+	0.823874i	$0.308192\pi$
$350$	24.4696	1.30796
$351$	0	0
$352$	8.23894	0.439137
$353$	23.5445	1.25315	0.626573	−	0.779363i	$-0.284457\pi$
$353$	23.5445	1.25315	0.626573	+	0.779363i	$0.284457\pi$
$354$	0	0
$355$	13.7622	0.730423
$356$	−4.72887	−0.250630
$357$	0	0
$358$	1.68843	0.0892362
$359$	−10.4609	−0.552107	−0.276053	−	0.961142i	$-0.589027\pi$
$359$	−10.4609	−0.552107	−0.276053	+	0.961142i	$0.589027\pi$
$360$	0	0
$361$	−11.4767	−0.604039
$362$	−35.5904	−1.87059
$363$	0	0
$364$	2.36386	0.123900
$365$	−11.1686	−0.584590
$366$	0	0
$367$	21.2583	1.10967	0.554836	−	0.831959i	$-0.312781\pi$
$367$	21.2583	1.10967	0.554836	+	0.831959i	$0.312781\pi$
$368$	28.9610	1.50970
$369$	0	0
$370$	−6.44808	−0.335220
$371$	−11.1207	−0.577356
$372$	0	0
$373$	2.31009	0.119612	0.0598059	−	0.998210i	$-0.480952\pi$
$373$	2.31009	0.119612	0.0598059	+	0.998210i	$0.480952\pi$
$374$	−5.94118	−0.307211
$375$	0	0
$376$	−3.37342	−0.173971
$377$	3.89737	0.200725
$378$	0	0
$379$	12.5539	0.644850	0.322425	−	0.946595i	$-0.395502\pi$
$379$	12.5539	0.644850	0.322425	+	0.946595i	$0.395502\pi$
$380$	−2.55744	−0.131194
$381$	0	0
$382$	−16.5206	−0.845270
$383$	−19.6545	−1.00430	−0.502150	−	0.864781i	$-0.667457\pi$
$383$	−19.6545	−1.00430	−0.502150	+	0.864781i	$0.667457\pi$
$384$	0	0
$385$	−8.25862	−0.420898
$386$	−18.7524	−0.954472
$387$	0	0
$388$	−0.283447	−0.0143898
$389$	−18.8426	−0.955357	−0.477679	−	0.878535i	$-0.658522\pi$
$389$	−18.8426	−0.955357	−0.477679	+	0.878535i	$0.658522\pi$
$390$	0	0
$391$	−11.0123	−0.556915
$392$	−16.3465	−0.825622
$393$	0	0
$394$	15.2794	0.769764
$395$	−13.9531	−0.702058
$396$	0	0
$397$	20.1178	1.00968	0.504841	−	0.863212i	$-0.331551\pi$
$397$	20.1178	1.00968	0.504841	+	0.863212i	$0.331551\pi$
$398$	−24.6991	−1.23806
$399$	0	0
$400$	18.5072	0.925360
$401$	−30.8057	−1.53836	−0.769181	−	0.639031i	$-0.779336\pi$
$401$	−30.8057	−1.53836	−0.769181	+	0.639031i	$0.779336\pi$
$402$	0	0
$403$	−0.982121	−0.0489230
$404$	−14.3650	−0.714684
$405$	0	0
$406$	34.9549	1.73478
$407$	−6.35228	−0.314871
$408$	0	0
$409$	39.6921	1.96265	0.981323	−	0.192367i	$-0.0616164\pi$
$409$	39.6921	1.96265	0.981323	+	0.192367i	$0.0616164\pi$
$410$	−3.41063	−0.168439
$411$	0	0
$412$	−13.0531	−0.643080
$413$	44.0146	2.16582
$414$	0	0
$415$	13.1966	0.647798
$416$	3.22693	0.158213
$417$	0	0
$418$	−8.62374	−0.421801
$419$	15.4411	0.754349	0.377174	−	0.926142i	$-0.376896\pi$
$419$	15.4411	0.754349	0.377174	+	0.926142i	$0.376896\pi$
$420$	0	0
$421$	−18.0936	−0.881830	−0.440915	−	0.897549i	$-0.645346\pi$
$421$	−18.0936	−0.881830	−0.440915	+	0.897549i	$0.645346\pi$
$422$	13.2091	0.643009
$423$	0	0
$424$	−5.61674	−0.272773
$425$	−7.03728	−0.341358
$426$	0	0
$427$	−20.4472	−0.989507
$428$	−13.6448	−0.659546
$429$	0	0
$430$	−9.55057	−0.460569
$431$	−28.9683	−1.39535	−0.697677	−	0.716412i	$-0.745783\pi$
$431$	−28.9683	−1.39535	−0.697677	+	0.716412i	$0.745783\pi$
$432$	0	0
$433$	−37.5902	−1.80647	−0.903235	−	0.429146i	$-0.858815\pi$
$433$	−37.5902	−1.80647	−0.903235	+	0.429146i	$0.858815\pi$
$434$	−8.80849	−0.422821
$435$	0	0
$436$	−3.89491	−0.186532
$437$	−15.9845	−0.764644
$438$	0	0
$439$	10.2635	0.489852	0.244926	−	0.969542i	$-0.421236\pi$
$439$	10.2635	0.489852	0.244926	+	0.969542i	$0.421236\pi$
$440$	−4.17120	−0.198854
$441$	0	0
$442$	−2.32697	−0.110683
$443$	21.9085	1.04090	0.520452	−	0.853891i	$-0.325763\pi$
$443$	21.9085	1.04090	0.520452	+	0.853891i	$0.325763\pi$
$444$	0	0
$445$	6.47086	0.306748
$446$	−14.6927	−0.695718
$447$	0	0
$448$	−9.90924	−0.468167
$449$	9.97130	0.470575	0.235287	−	0.971926i	$-0.424397\pi$
$449$	9.97130	0.470575	0.235287	+	0.971926i	$0.424397\pi$
$450$	0	0
$451$	−3.35996	−0.158214
$452$	16.4638	0.774390
$453$	0	0
$454$	6.83865	0.320954
$455$	−3.23464	−0.151642
$456$	0	0
$457$	−7.61810	−0.356360	−0.178180	−	0.983998i	$-0.557021\pi$
$457$	−7.61810	−0.356360	−0.178180	+	0.983998i	$0.557021\pi$
$458$	27.1566	1.26894
$459$	0	0
$460$	5.43375	0.253350
$461$	−22.8005	−1.06193	−0.530963	−	0.847395i	$-0.678170\pi$
$461$	−22.8005	−1.06193	−0.530963	+	0.847395i	$0.678170\pi$
$462$	0	0
$463$	−8.79573	−0.408772	−0.204386	−	0.978890i	$-0.565520\pi$
$463$	−8.79573	−0.408772	−0.204386	+	0.978890i	$0.565520\pi$
$464$	26.4376	1.22734
$465$	0	0
$466$	28.9338	1.34033
$467$	−10.9976	−0.508906	−0.254453	−	0.967085i	$-0.581895\pi$
$467$	−10.9976	−0.508906	−0.254453	+	0.967085i	$0.581895\pi$
$468$	0	0
$469$	7.38406	0.340964
$470$	−3.24420	−0.149644
$471$	0	0
$472$	22.2306	1.02325
$473$	−9.40868	−0.432612
$474$	0	0
$475$	−10.2148	−0.468685
$476$	−6.09728	−0.279468
$477$	0	0
$478$	−2.58094	−0.118049
$479$	25.6925	1.17392	0.586961	−	0.809616i	$-0.300324\pi$
$479$	25.6925	1.17392	0.586961	+	0.809616i	$0.300324\pi$
$480$	0	0
$481$	−2.48799	−0.113442
$482$	−9.28145	−0.422758
$483$	0	0
$484$	−6.19214	−0.281461
$485$	0.387861	0.0176119
$486$	0	0
$487$	−30.3800	−1.37665	−0.688325	−	0.725402i	$-0.741654\pi$
$487$	−30.3800	−1.37665	−0.688325	+	0.725402i	$0.741654\pi$
$488$	−10.3273	−0.467495
$489$	0	0
$490$	−15.7203	−0.710172
$491$	8.70827	0.392999	0.196499	−	0.980504i	$-0.437043\pi$
$491$	8.70827	0.392999	0.196499	+	0.980504i	$0.437043\pi$
$492$	0	0
$493$	−10.0528	−0.452754
$494$	−3.37764	−0.151967
$495$	0	0
$496$	−6.66217	−0.299140
$497$	47.6258	2.13631
$498$	0	0
$499$	11.6355	0.520875	0.260438	−	0.965491i	$-0.416133\pi$
$499$	11.6355	0.520875	0.260438	+	0.965491i	$0.416133\pi$
$500$	8.13438	0.363780
$501$	0	0
$502$	−36.0402	−1.60855
$503$	−37.7991	−1.68538	−0.842689	−	0.538400i	$-0.819029\pi$
$503$	−37.7991	−1.68538	−0.842689	+	0.538400i	$0.819029\pi$
$504$	0	0
$505$	19.6566	0.874708
$506$	18.3227	0.814544
$507$	0	0
$508$	0.882800	0.0391679
$509$	23.5736	1.04488	0.522442	−	0.852675i	$-0.325021\pi$
$509$	23.5736	1.04488	0.522442	+	0.852675i	$0.325021\pi$
$510$	0	0
$511$	−38.6502	−1.70978
$512$	2.26711	0.100193
$513$	0	0
$514$	−24.8548	−1.09630
$515$	17.8615	0.787071
$516$	0	0
$517$	−3.19601	−0.140560
$518$	−22.3143	−0.980436
$519$	0	0
$520$	−1.63373	−0.0716437
$521$	7.86948	0.344768	0.172384	−	0.985030i	$-0.444853\pi$
$521$	7.86948	0.344768	0.172384	+	0.985030i	$0.444853\pi$
$522$	0	0
$523$	33.2935	1.45582	0.727911	−	0.685672i	$-0.240491\pi$
$523$	33.2935	1.45582	0.727911	+	0.685672i	$0.240491\pi$
$524$	6.30680	0.275514
$525$	0	0
$526$	4.71119	0.205418
$527$	2.53326	0.110350
$528$	0	0
$529$	10.9621	0.476612
$530$	−5.40159	−0.234630
$531$	0	0
$532$	−8.85032	−0.383710
$533$	−1.31599	−0.0570018
$534$	0	0
$535$	18.6712	0.807225
$536$	3.72949	0.161089
$537$	0	0
$538$	−0.599292	−0.0258373
$539$	−15.4868	−0.667062
$540$	0	0
$541$	22.4283	0.964266	0.482133	−	0.876098i	$-0.339862\pi$
$541$	22.4283	0.964266	0.482133	+	0.876098i	$0.339862\pi$
$542$	−20.4175	−0.877005
$543$	0	0
$544$	−8.32346	−0.356865
$545$	5.32969	0.228299
$546$	0	0
$547$	−20.8912	−0.893241	−0.446621	−	0.894723i	$-0.647373\pi$
$547$	−20.8912	−0.893241	−0.446621	+	0.894723i	$0.647373\pi$
$548$	12.9190	0.551870
$549$	0	0
$550$	11.7089	0.499271
$551$	−14.5918	−0.621632
$552$	0	0
$553$	−48.2864	−2.05335
$554$	42.0059	1.78466
$555$	0	0
$556$	7.14600	0.303058
$557$	8.57840	0.363478	0.181739	−	0.983347i	$-0.441827\pi$
$557$	8.57840	0.363478	0.181739	+	0.983347i	$0.441827\pi$
$558$	0	0
$559$	−3.68508	−0.155862
$560$	−21.9420	−0.927219
$561$	0	0
$562$	12.3091	0.519228
$563$	−15.7149	−0.662303	−0.331152	−	0.943578i	$-0.607437\pi$
$563$	−15.7149	−0.662303	−0.331152	+	0.943578i	$0.607437\pi$
$564$	0	0
$565$	−22.5285	−0.947783
$566$	−24.1851	−1.01658
$567$	0	0
$568$	24.0545	1.00930
$569$	12.6728	0.531271	0.265635	−	0.964074i	$-0.414418\pi$
$569$	12.6728	0.531271	0.265635	+	0.964074i	$0.414418\pi$
$570$	0	0
$571$	−26.3227	−1.10157	−0.550785	−	0.834647i	$-0.685672\pi$
$571$	−26.3227	−1.10157	−0.550785	+	0.834647i	$0.685672\pi$
$572$	1.13113	0.0472949
$573$	0	0
$574$	−11.8029	−0.492644
$575$	21.7031	0.905082
$576$	0	0
$577$	−22.1154	−0.920676	−0.460338	−	0.887744i	$-0.652272\pi$
$577$	−22.1154	−0.920676	−0.460338	+	0.887744i	$0.652272\pi$
$578$	−22.5734	−0.938929
$579$	0	0
$580$	4.96031	0.205966
$581$	45.6686	1.89465
$582$	0	0
$583$	−5.32134	−0.220387
$584$	−19.5211	−0.807790
$585$	0	0
$586$	−24.2312	−1.00098
$587$	14.3031	0.590353	0.295177	−	0.955443i	$-0.404621\pi$
$587$	14.3031	0.590353	0.295177	+	0.955443i	$0.404621\pi$
$588$	0	0
$589$	3.67708	0.151511
$590$	21.3790	0.880161
$591$	0	0
$592$	−16.8771	−0.693645
$593$	47.7300	1.96004	0.980018	−	0.198908i	$-0.0637397\pi$
$593$	47.7300	1.96004	0.980018	+	0.198908i	$0.0637397\pi$
$594$	0	0
$595$	8.34334	0.342044
$596$	−2.03380	−0.0833077
$597$	0	0
$598$	7.17642	0.293466
$599$	−0.499570	−0.0204119	−0.0102059	−	0.999948i	$-0.503249\pi$
$599$	−0.499570	−0.0204119	−0.0102059	+	0.999948i	$0.503249\pi$
$600$	0	0
$601$	−16.9271	−0.690470	−0.345235	−	0.938516i	$-0.612201\pi$
$601$	−16.9271	−0.690470	−0.345235	+	0.938516i	$0.612201\pi$
$602$	−33.0509	−1.34705
$603$	0	0
$604$	−8.67846	−0.353121
$605$	8.47316	0.344483
$606$	0	0
$607$	0.694633	0.0281943	0.0140971	−	0.999901i	$-0.495513\pi$
$607$	0.694633	0.0281943	0.0140971	+	0.999901i	$0.495513\pi$
$608$	−12.0817	−0.489977
$609$	0	0
$610$	−9.93171	−0.402123
$611$	−1.25177	−0.0506413
$612$	0	0
$613$	−32.6633	−1.31926	−0.659630	−	0.751590i	$-0.729287\pi$
$613$	−32.6633	−1.31926	−0.659630	+	0.751590i	$0.729287\pi$
$614$	−51.1545	−2.06443
$615$	0	0
$616$	−14.4349	−0.581601
$617$	22.8844	0.921292	0.460646	−	0.887584i	$-0.347618\pi$
$617$	22.8844	0.921292	0.460646	+	0.887584i	$0.347618\pi$
$618$	0	0
$619$	8.48574	0.341071	0.170535	−	0.985352i	$-0.445450\pi$
$619$	8.48574	0.341071	0.170535	+	0.985352i	$0.445450\pi$
$620$	−1.24998	−0.0502002
$621$	0	0
$622$	−23.5254	−0.943282
$623$	22.3932	0.897163
$624$	0	0
$625$	7.48980	0.299592
$626$	37.1273	1.48390
$627$	0	0
$628$	0.297125	0.0118566
$629$	6.41744	0.255880
$630$	0	0
$631$	1.59173	0.0633657	0.0316829	−	0.999498i	$-0.489913\pi$
$631$	1.59173	0.0633657	0.0316829	+	0.999498i	$0.489913\pi$
$632$	−24.3881	−0.970108
$633$	0	0
$634$	29.2289	1.16083
$635$	−1.20800	−0.0479379
$636$	0	0
$637$	−6.06567	−0.240331
$638$	16.7263	0.662199
$639$	0	0
$640$	−14.7640	−0.583597
$641$	21.6747	0.856098	0.428049	−	0.903756i	$-0.359201\pi$
$641$	21.6747	0.856098	0.428049	+	0.903756i	$0.359201\pi$
$642$	0	0
$643$	−6.90206	−0.272191	−0.136095	−	0.990696i	$-0.543455\pi$
$643$	−6.90206	−0.272191	−0.136095	+	0.990696i	$0.543455\pi$
$644$	18.8041	0.740987
$645$	0	0
$646$	8.71220	0.342777
$647$	−6.18972	−0.243343	−0.121671	−	0.992570i	$-0.538825\pi$
$647$	−6.18972	−0.243343	−0.121671	+	0.992570i	$0.538825\pi$
$648$	0	0
$649$	21.0614	0.826733
$650$	4.58602	0.179878
$651$	0	0
$652$	12.0671	0.472585
$653$	−27.2145	−1.06498	−0.532492	−	0.846435i	$-0.678744\pi$
$653$	−27.2145	−1.06498	−0.532492	+	0.846435i	$0.678744\pi$
$654$	0	0
$655$	−8.63005	−0.337204
$656$	−8.92694	−0.348539
$657$	0	0
$658$	−11.2270	−0.437672
$659$	5.28200	0.205757	0.102879	−	0.994694i	$-0.467195\pi$
$659$	5.28200	0.205757	0.102879	+	0.994694i	$0.467195\pi$
$660$	0	0
$661$	11.6650	0.453717	0.226859	−	0.973928i	$-0.427154\pi$
$661$	11.6650	0.453717	0.226859	+	0.973928i	$0.427154\pi$
$662$	1.45256	0.0564552
$663$	0	0
$664$	23.0659	0.895132
$665$	12.1105	0.469626
$666$	0	0
$667$	31.0030	1.20044
$668$	1.79609	0.0694927
$669$	0	0
$670$	3.58663	0.138564
$671$	−9.78416	−0.377713
$672$	0	0
$673$	49.1024	1.89276	0.946380	−	0.323057i	$-0.104710\pi$
$673$	49.1024	1.89276	0.946380	+	0.323057i	$0.104710\pi$
$674$	0.703510	0.0270982
$675$	0	0
$676$	−10.2880	−0.395693
$677$	−22.6190	−0.869321	−0.434660	−	0.900594i	$-0.643132\pi$
$677$	−22.6190	−0.869321	−0.434660	+	0.900594i	$0.643132\pi$
$678$	0	0
$679$	1.34224	0.0515104
$680$	4.21399	0.161599
$681$	0	0
$682$	−4.21495	−0.161399
$683$	−17.1386	−0.655791	−0.327896	−	0.944714i	$-0.606339\pi$
$683$	−17.1386	−0.655791	−0.327896	+	0.944714i	$0.606339\pi$
$684$	0	0
$685$	−17.6779	−0.675439
$686$	−8.40818	−0.321026
$687$	0	0
$688$	−24.9975	−0.953022
$689$	−2.08420	−0.0794017
$690$	0	0
$691$	18.2559	0.694488	0.347244	−	0.937775i	$-0.387118\pi$
$691$	18.2559	0.694488	0.347244	+	0.937775i	$0.387118\pi$
$692$	−14.5010	−0.551244
$693$	0	0
$694$	39.3497	1.49370
$695$	−9.77839	−0.370915
$696$	0	0
$697$	3.39443	0.128573
$698$	35.5957	1.34732
$699$	0	0
$700$	12.0166	0.454184
$701$	−2.92075	−0.110315	−0.0551575	−	0.998478i	$-0.517566\pi$
$701$	−2.92075	−0.110315	−0.0551575	+	0.998478i	$0.517566\pi$
$702$	0	0
$703$	9.31505	0.351324
$704$	−4.74166	−0.178708
$705$	0	0
$706$	39.5762	1.48947
$707$	68.0241	2.55831
$708$	0	0
$709$	30.0273	1.12770	0.563850	−	0.825877i	$-0.309320\pi$
$709$	30.0273	1.12770	0.563850	+	0.825877i	$0.309320\pi$
$710$	23.1331	0.868169
$711$	0	0
$712$	11.3102	0.423867
$713$	−7.81262	−0.292585
$714$	0	0
$715$	−1.54781	−0.0578846
$716$	0.829156	0.0309870
$717$	0	0
$718$	−17.5839	−0.656226
$719$	40.0569	1.49387	0.746936	−	0.664896i	$-0.231524\pi$
$719$	40.0569	1.49387	0.746936	+	0.664896i	$0.231524\pi$
$720$	0	0
$721$	61.8118	2.30199
$722$	−19.2914	−0.717951
$723$	0	0
$724$	−17.4778	−0.649557
$725$	19.8121	0.735804
$726$	0	0
$727$	−1.53878	−0.0570702	−0.0285351	−	0.999593i	$-0.509084\pi$
$727$	−1.53878	−0.0570702	−0.0285351	+	0.999593i	$0.509084\pi$
$728$	−5.65371	−0.209540
$729$	0	0
$730$	−18.7734	−0.694834
$731$	9.50520	0.351562
$732$	0	0
$733$	−1.51189	−0.0558431	−0.0279215	−	0.999610i	$-0.508889\pi$
$733$	−1.51189	−0.0558431	−0.0279215	+	0.999610i	$0.508889\pi$
$734$	35.7333	1.31894
$735$	0	0
$736$	25.6697	0.946199
$737$	3.53334	0.130152
$738$	0	0
$739$	21.3558	0.785584	0.392792	−	0.919627i	$-0.371509\pi$
$739$	21.3558	0.785584	0.392792	+	0.919627i	$0.371509\pi$
$740$	−3.16654	−0.116404
$741$	0	0
$742$	−18.6929	−0.686236
$743$	−20.3287	−0.745789	−0.372895	−	0.927874i	$-0.621635\pi$
$743$	−20.3287	−0.745789	−0.372895	+	0.927874i	$0.621635\pi$
$744$	0	0
$745$	2.78300	0.101961
$746$	3.88305	0.142169
$747$	0	0
$748$	−2.91760	−0.106678
$749$	64.6138	2.36094
$750$	0	0
$751$	4.26316	0.155565	0.0777824	−	0.996970i	$-0.475216\pi$
$751$	4.26316	0.155565	0.0777824	+	0.996970i	$0.475216\pi$
$752$	−8.49134	−0.309647
$753$	0	0
$754$	6.55114	0.238579
$755$	11.8754	0.432189
$756$	0	0
$757$	54.3419	1.97509	0.987546	−	0.157332i	$-0.0502892\pi$
$757$	54.3419	1.97509	0.987546	+	0.157332i	$0.0502892\pi$
$758$	21.1020	0.766458
$759$	0	0
$760$	6.11670	0.221876
$761$	18.1611	0.658340	0.329170	−	0.944271i	$-0.393231\pi$
$761$	18.1611	0.658340	0.329170	+	0.944271i	$0.393231\pi$
$762$	0	0
$763$	18.4440	0.667718
$764$	−8.11299	−0.293518
$765$	0	0
$766$	−33.0375	−1.19369
$767$	8.24909	0.297857
$768$	0	0
$769$	24.8547	0.896285	0.448142	−	0.893962i	$-0.352086\pi$
$769$	24.8547	0.896285	0.448142	+	0.893962i	$0.352086\pi$
$770$	−13.8820	−0.500273
$771$	0	0
$772$	−9.20897	−0.331438
$773$	38.4832	1.38414	0.692071	−	0.721829i	$-0.256698\pi$
$773$	38.4832	1.38414	0.692071	+	0.721829i	$0.256698\pi$
$774$	0	0
$775$	−4.99257	−0.179338
$776$	0.677928	0.0243362
$777$	0	0
$778$	−31.6727	−1.13552
$779$	4.92708	0.176531
$780$	0	0
$781$	22.7894	0.815468
$782$	−18.5107	−0.661940
$783$	0	0
$784$	−41.1462	−1.46951
$785$	−0.406578	−0.0145114
$786$	0	0
$787$	−1.07248	−0.0382297	−0.0191149	−	0.999817i	$-0.506085\pi$
$787$	−1.07248	−0.0382297	−0.0191149	+	0.999817i	$0.506085\pi$
$788$	7.50343	0.267299
$789$	0	0
$790$	−23.4540	−0.834455
$791$	−77.9627	−2.77204
$792$	0	0
$793$	−3.83214	−0.136083
$794$	33.8162	1.20009
$795$	0	0
$796$	−12.1293	−0.429912
$797$	25.2472	0.894302	0.447151	−	0.894458i	$-0.352439\pi$
$797$	25.2472	0.894302	0.447151	+	0.894458i	$0.352439\pi$
$798$	0	0
$799$	3.22879	0.114226
$800$	16.4040	0.579968
$801$	0	0
$802$	−51.7817	−1.82847
$803$	−18.4945	−0.652656
$804$	0	0
$805$	−25.7310	−0.906900
$806$	−1.65086	−0.0581491
$807$	0	0
$808$	34.3571	1.20868
$809$	−11.7337	−0.412536	−0.206268	−	0.978495i	$-0.566132\pi$
$809$	−11.7337	−0.412536	−0.206268	+	0.978495i	$0.566132\pi$
$810$	0	0
$811$	−38.4085	−1.34871	−0.674353	−	0.738409i	$-0.735577\pi$
$811$	−38.4085	−1.34871	−0.674353	+	0.738409i	$0.735577\pi$
$812$	17.1657	0.602399
$813$	0	0
$814$	−10.6776	−0.374251
$815$	−16.5123	−0.578401
$816$	0	0
$817$	13.7970	0.482695
$818$	66.7189	2.33277
$819$	0	0
$820$	−1.67490	−0.0584901
$821$	26.7321	0.932955	0.466477	−	0.884533i	$-0.345523\pi$
$821$	26.7321	0.932955	0.466477	+	0.884533i	$0.345523\pi$
$822$	0	0
$823$	4.36426	0.152129	0.0760643	−	0.997103i	$-0.475765\pi$
$823$	4.36426	0.152129	0.0760643	+	0.997103i	$0.475765\pi$
$824$	31.2194	1.08758
$825$	0	0
$826$	73.9848	2.57426
$827$	−39.1454	−1.36122	−0.680610	−	0.732646i	$-0.738285\pi$
$827$	−39.1454	−1.36122	−0.680610	+	0.732646i	$0.738285\pi$
$828$	0	0
$829$	−32.8867	−1.14220	−0.571101	−	0.820880i	$-0.693484\pi$
$829$	−32.8867	−1.14220	−0.571101	+	0.820880i	$0.693484\pi$
$830$	22.1824	0.769963
$831$	0	0
$832$	−1.85716	−0.0643854
$833$	15.6456	0.542089
$834$	0	0
$835$	−2.45772	−0.0850527
$836$	−4.23496	−0.146469
$837$	0	0
$838$	25.9552	0.896607
$839$	31.6403	1.09235	0.546173	−	0.837673i	$-0.316084\pi$
$839$	31.6403	1.09235	0.546173	+	0.837673i	$0.316084\pi$
$840$	0	0
$841$	−0.698289	−0.0240789
$842$	−30.4138	−1.04813
$843$	0	0
$844$	6.48676	0.223283
$845$	14.0778	0.484293
$846$	0	0
$847$	29.3224	1.00753
$848$	−14.1380	−0.485503
$849$	0	0
$850$	−11.8290	−0.405733
$851$	−19.7915	−0.678445
$852$	0	0
$853$	−5.65765	−0.193714	−0.0968571	−	0.995298i	$-0.530879\pi$
$853$	−5.65765	−0.193714	−0.0968571	+	0.995298i	$0.530879\pi$
$854$	−34.3699	−1.17611
$855$	0	0
$856$	32.6346	1.11543
$857$	−41.2104	−1.40772	−0.703859	−	0.710339i	$-0.748542\pi$
$857$	−41.2104	−1.40772	−0.703859	+	0.710339i	$0.748542\pi$
$858$	0	0
$859$	35.8544	1.22334	0.611668	−	0.791114i	$-0.290499\pi$
$859$	35.8544	1.22334	0.611668	+	0.791114i	$0.290499\pi$
$860$	−4.69011	−0.159932
$861$	0	0
$862$	−48.6932	−1.65850
$863$	20.9694	0.713806	0.356903	−	0.934142i	$-0.383833\pi$
$863$	20.9694	0.713806	0.356903	+	0.934142i	$0.383833\pi$
$864$	0	0
$865$	19.8427	0.674673
$866$	−63.1858	−2.14714
$867$	0	0
$868$	−4.32569	−0.146824
$869$	−23.1055	−0.783801
$870$	0	0
$871$	1.38390	0.0468916
$872$	9.31556	0.315465
$873$	0	0
$874$	−26.8686	−0.908844
$875$	−38.5197	−1.30220
$876$	0	0
$877$	20.5104	0.692588	0.346294	−	0.938126i	$-0.387440\pi$
$877$	20.5104	0.692588	0.346294	+	0.938126i	$0.387440\pi$
$878$	17.2521	0.582230
$879$	0	0
$880$	−10.4995	−0.353936
$881$	−38.2877	−1.28994	−0.644972	−	0.764206i	$-0.723131\pi$
$881$	−38.2877	−1.28994	−0.644972	+	0.764206i	$0.723131\pi$
$882$	0	0
$883$	−17.4465	−0.587122	−0.293561	−	0.955940i	$-0.594840\pi$
$883$	−17.4465	−0.587122	−0.293561	+	0.955940i	$0.594840\pi$
$884$	−1.14273	−0.0384343
$885$	0	0
$886$	36.8263	1.23720
$887$	−54.2169	−1.82043	−0.910213	−	0.414141i	$-0.864082\pi$
$887$	−54.2169	−1.82043	−0.910213	+	0.414141i	$0.864082\pi$
$888$	0	0
$889$	−4.18042	−0.140207
$890$	10.8769	0.364596
$891$	0	0
$892$	−7.21530	−0.241586
$893$	4.68666	0.156833
$894$	0	0
$895$	−1.13459	−0.0379253
$896$	−51.0924	−1.70688
$897$	0	0
$898$	16.7609	0.559318
$899$	−7.13191	−0.237862
$900$	0	0
$901$	5.37593	0.179098
$902$	−5.64780	−0.188051
$903$	0	0
$904$	−39.3768	−1.30965
$905$	23.9161	0.794999
$906$	0	0
$907$	28.6832	0.952409	0.476204	−	0.879335i	$-0.342012\pi$
$907$	28.6832	0.952409	0.476204	+	0.879335i	$0.342012\pi$
$908$	3.35834	0.111450
$909$	0	0
$910$	−5.43715	−0.180240
$911$	−28.3816	−0.940324	−0.470162	−	0.882580i	$-0.655805\pi$
$911$	−28.3816	−0.940324	−0.470162	+	0.882580i	$0.655805\pi$
$912$	0	0
$913$	21.8528	0.723224
$914$	−12.8054	−0.423564
$915$	0	0
$916$	13.3361	0.440638
$917$	−29.8653	−0.986240
$918$	0	0
$919$	37.7786	1.24620	0.623101	−	0.782141i	$-0.285873\pi$
$919$	37.7786	1.24620	0.623101	+	0.782141i	$0.285873\pi$
$920$	−12.9960	−0.428467
$921$	0	0
$922$	−38.3257	−1.26219
$923$	8.92588	0.293799
$924$	0	0
$925$	−12.6476	−0.415849
$926$	−14.7849	−0.485861
$927$	0	0
$928$	23.4331	0.769230
$929$	−34.5769	−1.13443	−0.567215	−	0.823570i	$-0.691979\pi$
$929$	−34.5769	−1.13443	−0.567215	+	0.823570i	$0.691979\pi$
$930$	0	0
$931$	22.7100	0.744289
$932$	14.2089	0.465427
$933$	0	0
$934$	−18.4859	−0.604878
$935$	3.99237	0.130564
$936$	0	0
$937$	−19.4368	−0.634972	−0.317486	−	0.948263i	$-0.602839\pi$
$937$	−19.4368	−0.634972	−0.317486	+	0.948263i	$0.602839\pi$
$938$	12.4120	0.405265
$939$	0	0
$940$	−1.59317	−0.0519635
$941$	9.93846	0.323984	0.161992	−	0.986792i	$-0.448208\pi$
$941$	9.93846	0.323984	0.161992	+	0.986792i	$0.448208\pi$
$942$	0	0
$943$	−10.4685	−0.340901
$944$	55.9572	1.82125
$945$	0	0
$946$	−15.8152	−0.514195
$947$	16.9107	0.549524	0.274762	−	0.961512i	$-0.411401\pi$
$947$	16.9107	0.549524	0.274762	+	0.961512i	$0.411401\pi$
$948$	0	0
$949$	−7.24369	−0.235140
$950$	−17.1701	−0.557072
$951$	0	0
$952$	14.5830	0.472639
$953$	17.3573	0.562259	0.281129	−	0.959670i	$-0.409291\pi$
$953$	17.3573	0.562259	0.281129	+	0.959670i	$0.409291\pi$
$954$	0	0
$955$	11.1016	0.359239
$956$	−1.26745	−0.0409923
$957$	0	0
$958$	43.1869	1.39530
$959$	−61.1766	−1.97550
$960$	0	0
$961$	−29.2028	−0.942025
$962$	−4.18208	−0.134836
$963$	0	0
$964$	−4.55795	−0.146802
$965$	12.6013	0.405650
$966$	0	0
$967$	−16.7834	−0.539716	−0.269858	−	0.962900i	$-0.586977\pi$
$967$	−16.7834	−0.539716	−0.269858	+	0.962900i	$0.586977\pi$
$968$	14.8099	0.476009
$969$	0	0
$970$	0.651960	0.0209332
$971$	−33.4811	−1.07446	−0.537230	−	0.843436i	$-0.680529\pi$
$971$	−33.4811	−1.07446	−0.537230	+	0.843436i	$0.680529\pi$
$972$	0	0
$973$	−33.8393	−1.08484
$974$	−51.0662	−1.63627
$975$	0	0
$976$	−25.9951	−0.832084
$977$	22.7348	0.727352	0.363676	−	0.931526i	$-0.381522\pi$
$977$	22.7348	0.727352	0.363676	+	0.931526i	$0.381522\pi$
$978$	0	0
$979$	10.7153	0.342464
$980$	−7.71997	−0.246605
$981$	0	0
$982$	14.6378	0.467112
$983$	47.1596	1.50416	0.752078	−	0.659074i	$-0.229051\pi$
$983$	47.1596	1.50416	0.752078	+	0.659074i	$0.229051\pi$
$984$	0	0
$985$	−10.2675	−0.327149
$986$	−16.8978	−0.538137
$987$	0	0
$988$	−1.65870	−0.0527703
$989$	−29.3142	−0.932138
$990$	0	0
$991$	−4.37673	−0.139032	−0.0695158	−	0.997581i	$-0.522145\pi$
$991$	−4.37673	−0.139032	−0.0695158	+	0.997581i	$0.522145\pi$
$992$	−5.90505	−0.187486
$993$	0	0
$994$	80.0548	2.53918
$995$	16.5974	0.526173
$996$	0	0
$997$	−2.14450	−0.0679170	−0.0339585	−	0.999423i	$-0.510811\pi$
$997$	−2.14450	−0.0679170	−0.0339585	+	0.999423i	$0.510811\pi$
$998$	19.5582	0.619104
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.a.a.1.6		6
3.2	odd	2		729.2.a.d.1.1		6
9.2	odd	6		729.2.c.b.244.6		12
9.4	even	3		729.2.c.e.487.1		12
9.5	odd	6		729.2.c.b.487.6		12
9.7	even	3		729.2.c.e.244.1		12
27.2	odd	18		81.2.e.a.64.2		12
27.4	even	9		243.2.e.d.136.2		12
27.5	odd	18		243.2.e.b.217.1		12
27.7	even	9		243.2.e.d.109.2		12
27.11	odd	18		243.2.e.b.28.1		12
27.13	even	9		27.2.e.a.7.1	yes	12
27.14	odd	18		81.2.e.a.19.2		12
27.16	even	9		243.2.e.c.28.2		12
27.20	odd	18		243.2.e.a.109.1		12
27.22	even	9		243.2.e.c.217.2		12
27.23	odd	18		243.2.e.a.136.1		12
27.25	even	9		27.2.e.a.4.1	✓	12
108.67	odd	18		432.2.u.c.385.1		12
108.79	odd	18		432.2.u.c.193.1		12
135.13	odd	36		675.2.u.b.574.4		24
135.52	odd	36		675.2.u.b.274.4		24
135.67	odd	36		675.2.u.b.574.1		24
135.79	even	18		675.2.l.c.301.2		12
135.94	even	18		675.2.l.c.601.2		12
135.133	odd	36		675.2.u.b.274.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.4.1	✓	12	27.25	even	9
27.2.e.a.7.1	yes	12	27.13	even	9
81.2.e.a.19.2		12	27.14	odd	18
81.2.e.a.64.2		12	27.2	odd	18
243.2.e.a.109.1		12	27.20	odd	18
243.2.e.a.136.1		12	27.23	odd	18
243.2.e.b.28.1		12	27.11	odd	18
243.2.e.b.217.1		12	27.5	odd	18
243.2.e.c.28.2		12	27.16	even	9
243.2.e.c.217.2		12	27.22	even	9
243.2.e.d.109.2		12	27.7	even	9
243.2.e.d.136.2		12	27.4	even	9
432.2.u.c.193.1		12	108.79	odd	18
432.2.u.c.385.1		12	108.67	odd	18
675.2.l.c.301.2		12	135.79	even	18
675.2.l.c.601.2		12	135.94	even	18
675.2.u.b.274.1		24	135.133	odd	36
675.2.u.b.274.4		24	135.52	odd	36
675.2.u.b.574.1		24	135.67	odd	36
675.2.u.b.574.4		24	135.13	odd	36
729.2.a.a.1.6		6	1.1	even	1	trivial
729.2.a.d.1.1		6	3.2	odd	2
729.2.c.b.244.6		12	9.2	odd	6
729.2.c.b.487.6		12	9.5	odd	6
729.2.c.e.244.1		12	9.7	even	3
729.2.c.e.487.1		12	9.4	even	3

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)

\(f(q)\)	\(=\)	\(q+1.68091 q^{2} +0.825466 q^{4} -1.12954 q^{5} -3.90892 q^{7} -1.97429 q^{8} +O(q^{10})\)	\(q+1.68091 q^{2} +0.825466 q^{4} -1.12954 q^{5} -3.90892 q^{7} -1.97429 q^{8} -1.89866 q^{10} -1.87046 q^{11} -0.732598 q^{13} -6.57056 q^{14} -4.96954 q^{16} +1.88964 q^{17} +2.74286 q^{19} -0.932400 q^{20} -3.14407 q^{22} -5.82770 q^{23} -3.72413 q^{25} -1.23143 q^{26} -3.22668 q^{28} -5.31994 q^{29} +1.34060 q^{31} -4.40478 q^{32} +3.17632 q^{34} +4.41530 q^{35} +3.39611 q^{37} +4.61050 q^{38} +2.23005 q^{40} +1.79633 q^{41} +5.03015 q^{43} -1.54400 q^{44} -9.79585 q^{46} +1.70868 q^{47} +8.27968 q^{49} -6.25994 q^{50} -0.604734 q^{52} +2.84494 q^{53} +2.11276 q^{55} +7.71734 q^{56} -8.94234 q^{58} -11.2600 q^{59} +5.23089 q^{61} +2.25343 q^{62} +2.53503 q^{64} +0.827502 q^{65} -1.88903 q^{67} +1.55984 q^{68} +7.42173 q^{70} -12.1839 q^{71} +9.88768 q^{73} +5.70857 q^{74} +2.26413 q^{76} +7.31147 q^{77} +12.3529 q^{79} +5.61331 q^{80} +3.01948 q^{82} -11.6832 q^{83} -2.13444 q^{85} +8.45525 q^{86} +3.69282 q^{88} -5.72873 q^{89} +2.86367 q^{91} -4.81057 q^{92} +2.87214 q^{94} -3.09818 q^{95} -0.343378 q^{97} +13.9174 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10}) \) 6 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 6 * q^8	\( 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8} + 3 q^{10} - 12 q^{11} - 6 q^{14} - 3 q^{16} - 9 q^{17} + 3 q^{19} - 6 q^{20} + 6 q^{22} - 15 q^{23} - 6 q^{25} - 15 q^{26} - 6 q^{28} - 12 q^{29} - 12 q^{35} + 3 q^{37} + 3 q^{38} + 6 q^{40} - 15 q^{41} - 3 q^{44} + 3 q^{46} - 21 q^{47} - 12 q^{49} - 3 q^{50} + 12 q^{52} - 9 q^{53} - 6 q^{55} + 6 q^{56} - 12 q^{58} - 24 q^{59} - 9 q^{61} + 12 q^{62} - 12 q^{64} + 6 q^{65} - 9 q^{67} + 9 q^{68} + 15 q^{70} - 27 q^{71} - 6 q^{73} + 12 q^{74} + 6 q^{76} + 12 q^{77} + 21 q^{80} - 6 q^{82} - 12 q^{83} + 21 q^{86} + 12 q^{88} - 9 q^{89} - 6 q^{91} - 6 q^{92} + 6 q^{94} - 12 q^{95} + 45 q^{98}+O(q^{100}) \) 6 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 6 * q^8 + 3 * q^10 - 12 * q^11 - 6 * q^14 - 3 * q^16 - 9 * q^17 + 3 * q^19 - 6 * q^20 + 6 * q^22 - 15 * q^23 - 6 * q^25 - 15 * q^26 - 6 * q^28 - 12 * q^29 - 12 * q^35 + 3 * q^37 + 3 * q^38 + 6 * q^40 - 15 * q^41 - 3 * q^44 + 3 * q^46 - 21 * q^47 - 12 * q^49 - 3 * q^50 + 12 * q^52 - 9 * q^53 - 6 * q^55 + 6 * q^56 - 12 * q^58 - 24 * q^59 - 9 * q^61 + 12 * q^62 - 12 * q^64 + 6 * q^65 - 9 * q^67 + 9 * q^68 + 15 * q^70 - 27 * q^71 - 6 * q^73 + 12 * q^74 + 6 * q^76 + 12 * q^77 + 21 * q^80 - 6 * q^82 - 12 * q^83 + 21 * q^86 + 12 * q^88 - 9 * q^89 - 6 * q^91 - 6 * q^92 + 6 * q^94 - 12 * q^95 + 45 * q^98

Introduction

L-functions

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