LMFDB - Embedded newform 729.2.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [729,2,Mod(1,729)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [6,3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$5.82109430735$
Analytic rank:	$0$
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			1.2
Root			$-1.40162$ 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)

$f(q)$	$=$	\(q-1.05432 q^{2} -0.888399 q^{4} +1.74579 q^{5} +2.45925 q^{7} +3.04531 q^{8} -1.84063 q^{10} +1.25421 q^{11} -4.54903 q^{13} -2.59284 q^{14} -1.43395 q^{16} +6.64717 q^{17} +0.249156 q^{19} -1.55096 q^{20} -1.32235 q^{22} -0.842001 q^{23} -1.95223 q^{25} +4.79615 q^{26} -2.18479 q^{28} +0.512383 q^{29} -0.820004 q^{31} -4.57877 q^{32} -7.00828 q^{34} +4.29332 q^{35} +2.60806 q^{37} -0.262692 q^{38} +5.31646 q^{40} +8.15281 q^{41} +4.32714 q^{43} -1.11424 q^{44} +0.887743 q^{46} +5.30233 q^{47} -0.952106 q^{49} +2.05828 q^{50} +4.04135 q^{52} +10.4841 q^{53} +2.18959 q^{55} +7.48917 q^{56} -0.540218 q^{58} +3.00620 q^{59} +2.88317 q^{61} +0.864550 q^{62} +7.69541 q^{64} -7.94164 q^{65} +10.0863 q^{67} -5.90534 q^{68} -4.52655 q^{70} -0.0894756 q^{71} -5.32114 q^{73} -2.74974 q^{74} -0.221350 q^{76} +3.08442 q^{77} +4.77692 q^{79} -2.50337 q^{80} -8.59571 q^{82} -8.04066 q^{83} +11.6045 q^{85} -4.56222 q^{86} +3.81947 q^{88} -6.70377 q^{89} -11.1872 q^{91} +0.748033 q^{92} -5.59038 q^{94} +0.434974 q^{95} +5.49058 q^{97} +1.00383 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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 $

$2$	−1.05432	−0.745520	−0.372760	−	0.927928i	$-0.621589\pi$
$2$	−1.05432	−0.745520	−0.372760	+	0.927928i	$0.621589\pi$
$3$	0	0
$3$	0	0
$4$	−0.888399	−0.444200
$5$	1.74579	0.780740	0.390370	−	0.920658i	$-0.372347\pi$
$5$	1.74579	0.780740	0.390370	+	0.920658i	$0.372347\pi$
$6$	0	0
$7$	2.45925	0.929508	0.464754	−	0.885440i	$-0.346143\pi$
$7$	2.45925	0.929508	0.464754	+	0.885440i	$0.346143\pi$
$8$	3.04531	1.07668
$9$	0	0
$10$	−1.84063	−0.582057
$11$	1.25421	0.378159	0.189080	−	0.981962i	$-0.439450\pi$
$11$	1.25421	0.378159	0.189080	+	0.981962i	$0.439450\pi$
$12$	0	0
$13$	−4.54903	−1.26167	−0.630837	−	0.775915i	$-0.717288\pi$
$13$	−4.54903	−1.26167	−0.630837	+	0.775915i	$0.717288\pi$
$14$	−2.59284	−0.692967
$15$	0	0
$16$	−1.43395	−0.358487
$17$	6.64717	1.61218	0.806088	−	0.591796i	$-0.201581\pi$
$17$	6.64717	1.61218	0.806088	+	0.591796i	$0.201581\pi$
$18$	0	0
$19$	0.249156	0.0571604	0.0285802	−	0.999592i	$-0.490901\pi$
$19$	0.249156	0.0571604	0.0285802	+	0.999592i	$0.490901\pi$
$20$	−1.55096	−0.346804
$21$	0	0
$22$	−1.32235	−0.281926
$23$	−0.842001	−0.175569	−0.0877847	−	0.996139i	$-0.527979\pi$
$23$	−0.842001	−0.175569	−0.0877847	+	0.996139i	$0.527979\pi$
$24$	0	0
$25$	−1.95223	−0.390446
$26$	4.79615	0.940603
$27$	0	0
$28$	−2.18479	−0.412887
$29$	0.512383	0.0951471	0.0475736	−	0.998868i	$-0.484851\pi$
$29$	0.512383	0.0951471	0.0475736	+	0.998868i	$0.484851\pi$
$30$	0	0
$31$	−0.820004	−0.147277	−0.0736385	−	0.997285i	$-0.523461\pi$
$31$	−0.820004	−0.147277	−0.0736385	+	0.997285i	$0.523461\pi$
$32$	−4.57877	−0.809421
$33$	0	0
$34$	−7.00828	−1.20191
$35$	4.29332	0.725704
$36$	0	0
$37$	2.60806	0.428763	0.214381	−	0.976750i	$-0.431226\pi$
$37$	2.60806	0.428763	0.214381	+	0.976750i	$0.431226\pi$
$38$	−0.262692	−0.0426142
$39$	0	0
$40$	5.31646	0.840607
$41$	8.15281	1.27326	0.636628	−	0.771171i	$-0.280329\pi$
$41$	8.15281	1.27326	0.636628	+	0.771171i	$0.280329\pi$
$42$	0	0
$43$	4.32714	0.659883	0.329942	−	0.944001i	$-0.392971\pi$
$43$	4.32714	0.659883	0.329942	+	0.944001i	$0.392971\pi$
$44$	−1.11424	−0.167978
$45$	0	0
$46$	0.887743	0.130890
$47$	5.30233	0.773425	0.386713	−	0.922200i	$-0.373611\pi$
$47$	5.30233	0.773425	0.386713	+	0.922200i	$0.373611\pi$
$48$	0	0
$49$	−0.952106	−0.136015
$50$	2.05828	0.291085
$51$	0	0
$52$	4.04135	0.560435
$53$	10.4841	1.44010	0.720052	−	0.693920i	$-0.244118\pi$
$53$	10.4841	1.44010	0.720052	+	0.693920i	$0.244118\pi$
$54$	0	0
$55$	2.18959	0.295244
$56$	7.48917	1.00078
$57$	0	0
$58$	−0.540218	−0.0709341
$59$	3.00620	0.391374	0.195687	−	0.980666i	$-0.437306\pi$
$59$	3.00620	0.391374	0.195687	+	0.980666i	$0.437306\pi$
$60$	0	0
$61$	2.88317	0.369152	0.184576	−	0.982818i	$-0.440909\pi$
$61$	2.88317	0.369152	0.184576	+	0.982818i	$0.440909\pi$
$62$	0.864550	0.109798
$63$	0	0
$64$	7.69541	0.961927
$65$	−7.94164	−0.985039
$66$	0	0
$67$	10.0863	1.23223	0.616117	−	0.787655i	$-0.288705\pi$
$67$	10.0863	1.23223	0.616117	+	0.787655i	$0.288705\pi$
$68$	−5.90534	−0.716128
$69$	0	0
$70$	−4.52655	−0.541027
$71$	−0.0894756	−0.0106188	−0.00530940	−	0.999986i	$-0.501690\pi$
$71$	−0.0894756	−0.0106188	−0.00530940	+	0.999986i	$0.501690\pi$
$72$	0	0
$73$	−5.32114	−0.622792	−0.311396	−	0.950280i	$-0.600797\pi$
$73$	−5.32114	−0.622792	−0.311396	+	0.950280i	$0.600797\pi$
$74$	−2.74974	−0.319651
$75$	0	0
$76$	−0.221350	−0.0253906
$77$	3.08442	0.351502
$78$	0	0
$79$	4.77692	0.537445	0.268723	−	0.963218i	$-0.413398\pi$
$79$	4.77692	0.537445	0.268723	+	0.963218i	$0.413398\pi$
$80$	−2.50337	−0.279885
$81$	0	0
$82$	−8.59571	−0.949238
$83$	−8.04066	−0.882577	−0.441289	−	0.897365i	$-0.645478\pi$
$83$	−8.04066	−0.882577	−0.441289	+	0.897365i	$0.645478\pi$
$84$	0	0
$85$	11.6045	1.25869
$86$	−4.56222	−0.491956
$87$	0	0
$88$	3.81947	0.407157
$89$	−6.70377	−0.710598	−0.355299	−	0.934753i	$-0.615621\pi$
$89$	−6.70377	−0.710598	−0.355299	+	0.934753i	$0.615621\pi$
$90$	0	0
$91$	−11.1872	−1.17274
$92$	0.748033	0.0779878
$93$	0	0
$94$	−5.59038	−0.576604
$95$	0.434974	0.0446274
$96$	0	0
$97$	5.49058	0.557484	0.278742	−	0.960366i	$-0.410083\pi$
$97$	5.49058	0.557484	0.278742	+	0.960366i	$0.410083\pi$
$98$	1.00383	0.101402
$99$	0	0
$100$	1.73436	0.173436
$101$	−5.00546	−0.498062	−0.249031	−	0.968495i	$-0.580112\pi$
$101$	−5.00546	−0.498062	−0.249031	+	0.968495i	$0.580112\pi$
$102$	0	0
$103$	11.6192	1.14487	0.572435	−	0.819950i	$-0.305999\pi$
$103$	11.6192	1.14487	0.572435	+	0.819950i	$0.305999\pi$
$104$	−13.8532	−1.35842
$105$	0	0
$106$	−11.0537	−1.07363
$107$	−19.4581	−1.88109	−0.940544	−	0.339673i	$-0.889684\pi$
$107$	−19.4581	−1.88109	−0.940544	+	0.339673i	$0.889684\pi$
$108$	0	0
$109$	6.31515	0.604881	0.302441	−	0.953168i	$-0.402199\pi$
$109$	6.31515	0.604881	0.302441	+	0.953168i	$0.402199\pi$
$110$	−2.30854	−0.220110
$111$	0	0
$112$	−3.52643	−0.333217
$113$	6.91572	0.650577	0.325288	−	0.945615i	$-0.394539\pi$
$113$	6.91572	0.650577	0.325288	+	0.945615i	$0.394539\pi$
$114$	0	0
$115$	−1.46995	−0.137074
$116$	−0.455201	−0.0422643
$117$	0	0
$118$	−3.16951	−0.291777
$119$	16.3470	1.49853
$120$	0	0
$121$	−9.42695	−0.856995
$122$	−3.03980	−0.275211
$123$	0	0
$124$	0.728491	0.0654204
$125$	−12.1371	−1.08558
$126$	0	0
$127$	12.0232	1.06689	0.533445	−	0.845835i	$-0.320897\pi$
$127$	12.0232	1.06689	0.533445	+	0.845835i	$0.320897\pi$
$128$	1.04408	0.0922848
$129$	0	0
$130$	8.37306	0.734366
$131$	14.0848	1.23059	0.615297	−	0.788295i	$-0.289036\pi$
$131$	14.0848	1.23059	0.615297	+	0.788295i	$0.289036\pi$
$132$	0	0
$133$	0.612737	0.0531310
$134$	−10.6342	−0.918655
$135$	0	0
$136$	20.2427	1.73580
$137$	−2.25814	−0.192926	−0.0964630	−	0.995337i	$-0.530753\pi$
$137$	−2.25814	−0.192926	−0.0964630	+	0.995337i	$0.530753\pi$
$138$	0	0
$139$	−7.97509	−0.676439	−0.338219	−	0.941067i	$-0.609825\pi$
$139$	−7.97509	−0.676439	−0.338219	+	0.941067i	$0.609825\pi$
$140$	−3.81418	−0.322357
$141$	0	0
$142$	0.0943364	0.00791653
$143$	−5.70545	−0.477114
$144$	0	0
$145$	0.894512	0.0742851
$146$	5.61021	0.464304
$147$	0	0
$148$	−2.31700	−0.190456
$149$	0.106938	0.00876074	0.00438037	−	0.999990i	$-0.498606\pi$
$149$	0.106938	0.00876074	0.00438037	+	0.999990i	$0.498606\pi$
$150$	0	0
$151$	−20.2594	−1.64869	−0.824344	−	0.566090i	$-0.808456\pi$
$151$	−20.2594	−1.64869	−0.824344	+	0.566090i	$0.808456\pi$
$152$	0.758758	0.0615434
$153$	0	0
$154$	−3.25198	−0.262052
$155$	−1.43155	−0.114985
$156$	0	0
$157$	−20.7498	−1.65602	−0.828009	−	0.560715i	$-0.810526\pi$
$157$	−20.7498	−1.65602	−0.828009	+	0.560715i	$0.810526\pi$
$158$	−5.03642	−0.400676
$159$	0	0
$160$	−7.99356	−0.631947
$161$	−2.07069	−0.163193
$162$	0	0
$163$	−20.1346	−1.57706	−0.788531	−	0.614995i	$-0.789158\pi$
$163$	−20.1346	−1.57706	−0.788531	+	0.614995i	$0.789158\pi$
$164$	−7.24295	−0.565580
$165$	0	0
$166$	8.47747	0.657979
$167$	19.8626	1.53701	0.768507	−	0.639841i	$-0.221000\pi$
$167$	19.8626	1.53701	0.768507	+	0.639841i	$0.221000\pi$
$168$	0	0
$169$	7.69367	0.591821
$170$	−12.2350	−0.938378
$171$	0	0
$172$	−3.84423	−0.293120
$173$	−18.9251	−1.43885	−0.719425	−	0.694570i	$-0.755595\pi$
$173$	−18.9251	−1.43885	−0.719425	+	0.694570i	$0.755595\pi$
$174$	0	0
$175$	−4.80101	−0.362922
$176$	−1.79848	−0.135565
$177$	0	0
$178$	7.06795	0.529765
$179$	−10.9137	−0.815725	−0.407863	−	0.913043i	$-0.633726\pi$
$179$	−10.9137	−0.815725	−0.407863	+	0.913043i	$0.633726\pi$
$180$	0	0
$181$	−17.9479	−1.33405	−0.667027	−	0.745033i	$-0.732433\pi$
$181$	−17.9479	−1.33405	−0.667027	+	0.745033i	$0.732433\pi$
$182$	11.7949	0.874298
$183$	0	0
$184$	−2.56415	−0.189032
$185$	4.55312	0.334752
$186$	0	0
$187$	8.33697	0.609659
$188$	−4.71059	−0.343555
$189$	0	0
$190$	−0.458604	−0.0332706
$191$	26.9661	1.95120	0.975598	−	0.219566i	$-0.0704640\pi$
$191$	26.9661	1.95120	0.975598	+	0.219566i	$0.0704640\pi$
$192$	0	0
$193$	−17.1548	−1.23483	−0.617415	−	0.786638i	$-0.711820\pi$
$193$	−17.1548	−1.23483	−0.617415	+	0.786638i	$0.711820\pi$
$194$	−5.78885	−0.415615
$195$	0	0
$196$	0.845850	0.0604179
$197$	−2.51225	−0.178990	−0.0894951	−	0.995987i	$-0.528525\pi$
$197$	−2.51225	−0.178990	−0.0894951	+	0.995987i	$0.528525\pi$
$198$	0	0
$199$	18.5388	1.31418	0.657092	−	0.753810i	$-0.271786\pi$
$199$	18.5388	1.31418	0.657092	+	0.753810i	$0.271786\pi$
$200$	−5.94514	−0.420385
$201$	0	0
$202$	5.27739	0.371316
$203$	1.26008	0.0884400
$204$	0	0
$205$	14.2331	0.994081
$206$	−12.2504	−0.853524
$207$	0	0
$208$	6.52307	0.452294
$209$	0.312495	0.0216157
$210$	0	0
$211$	−3.69118	−0.254111	−0.127056	−	0.991896i	$-0.540553\pi$
$211$	−3.69118	−0.254111	−0.127056	+	0.991896i	$0.540553\pi$
$212$	−9.31408	−0.639693
$213$	0	0
$214$	20.5152	1.40239
$215$	7.55427	0.515197
$216$	0	0
$217$	−2.01659	−0.136895
$218$	−6.65822	−0.450951
$219$	0	0
$220$	−1.94523	−0.131147
$221$	−30.2382	−2.03404
$222$	0	0
$223$	−21.2410	−1.42241	−0.711203	−	0.702987i	$-0.751849\pi$
$223$	−21.2410	−1.42241	−0.711203	+	0.702987i	$0.751849\pi$
$224$	−11.2603	−0.752363
$225$	0	0
$226$	−7.29142	−0.485018
$227$	14.3400	0.951783	0.475891	−	0.879504i	$-0.342126\pi$
$227$	14.3400	0.951783	0.475891	+	0.879504i	$0.342126\pi$
$228$	0	0
$229$	16.8858	1.11585	0.557923	−	0.829893i	$-0.311598\pi$
$229$	16.8858	1.11585	0.557923	+	0.829893i	$0.311598\pi$
$230$	1.54981	0.102191
$231$	0	0
$232$	1.56037	0.102443
$233$	−5.59945	−0.366832	−0.183416	−	0.983035i	$-0.558716\pi$
$233$	−5.59945	−0.366832	−0.183416	+	0.983035i	$0.558716\pi$
$234$	0	0
$235$	9.25675	0.603844
$236$	−2.67071	−0.173848
$237$	0	0
$238$	−17.2351	−1.11718
$239$	5.27427	0.341164	0.170582	−	0.985343i	$-0.445435\pi$
$239$	5.27427	0.341164	0.170582	+	0.985343i	$0.445435\pi$
$240$	0	0
$241$	−8.90248	−0.573459	−0.286730	−	0.958012i	$-0.592568\pi$
$241$	−8.90248	−0.573459	−0.286730	+	0.958012i	$0.592568\pi$
$242$	9.93907	0.638907
$243$	0	0
$244$	−2.56141	−0.163977
$245$	−1.66217	−0.106192
$246$	0	0
$247$	−1.13342	−0.0721177
$248$	−2.49717	−0.158570
$249$	0	0
$250$	12.7965	0.809319
$251$	7.78021	0.491082	0.245541	−	0.969386i	$-0.421034\pi$
$251$	7.78021	0.491082	0.245541	+	0.969386i	$0.421034\pi$
$252$	0	0
$253$	−1.05605	−0.0663932
$254$	−12.6764	−0.795388
$255$	0	0
$256$	−16.4916	−1.03073
$257$	20.4366	1.27480	0.637399	−	0.770534i	$-0.280011\pi$
$257$	20.4366	1.27480	0.637399	+	0.770534i	$0.280011\pi$
$258$	0	0
$259$	6.41387	0.398538
$260$	7.05534	0.437554
$261$	0	0
$262$	−14.8500	−0.917433
$263$	−11.2798	−0.695543	−0.347771	−	0.937579i	$-0.613061\pi$
$263$	−11.2798	−0.695543	−0.347771	+	0.937579i	$0.613061\pi$
$264$	0	0
$265$	18.3030	1.12435
$266$	−0.646023	−0.0396102
$267$	0	0
$268$	−8.96063	−0.547357
$269$	0.307761	0.0187645	0.00938226	−	0.999956i	$-0.497013\pi$
$269$	0.307761	0.0187645	0.00938226	+	0.999956i	$0.497013\pi$
$270$	0	0
$271$	−2.22251	−0.135008	−0.0675040	−	0.997719i	$-0.521504\pi$
$271$	−2.22251	−0.135008	−0.0675040	+	0.997719i	$0.521504\pi$
$272$	−9.53170	−0.577944
$273$	0	0
$274$	2.38081	0.143830
$275$	−2.44851	−0.147651
$276$	0	0
$277$	−23.3297	−1.40175	−0.700874	−	0.713285i	$-0.747206\pi$
$277$	−23.3297	−1.40175	−0.700874	+	0.713285i	$0.747206\pi$
$278$	8.40834	0.504299
$279$	0	0
$280$	13.0745	0.781351
$281$	−7.22546	−0.431035	−0.215517	−	0.976500i	$-0.569144\pi$
$281$	−7.22546	−0.431035	−0.215517	+	0.976500i	$0.569144\pi$
$282$	0	0
$283$	7.12029	0.423257	0.211629	−	0.977350i	$-0.432123\pi$
$283$	7.12029	0.423257	0.211629	+	0.977350i	$0.432123\pi$
$284$	0.0794901	0.00471687
$285$	0	0
$286$	6.01540	0.355698
$287$	20.0498	1.18350
$288$	0	0
$289$	27.1849	1.59911
$290$	−0.943106	−0.0553811
$291$	0	0
$292$	4.72730	0.276644
$293$	−0.552485	−0.0322765	−0.0161383	−	0.999870i	$-0.505137\pi$
$293$	−0.552485	−0.0322765	−0.0161383	+	0.999870i	$0.505137\pi$
$294$	0	0
$295$	5.24819	0.305561
$296$	7.94236	0.461640
$297$	0	0
$298$	−0.112748	−0.00653131
$299$	3.83029	0.221511
$300$	0	0
$301$	10.6415	0.613367
$302$	21.3600	1.22913
$303$	0	0
$304$	−0.357277	−0.0204913
$305$	5.03340	0.288212
$306$	0	0
$307$	6.72876	0.384031	0.192015	−	0.981392i	$-0.438498\pi$
$307$	6.72876	0.384031	0.192015	+	0.981392i	$0.438498\pi$
$308$	−2.74020	−0.156137
$309$	0	0
$310$	1.50932	0.0857236
$311$	−15.4235	−0.874584	−0.437292	−	0.899320i	$-0.644062\pi$
$311$	−15.4235	−0.874584	−0.437292	+	0.899320i	$0.644062\pi$
$312$	0	0
$313$	−23.5609	−1.33174	−0.665870	−	0.746068i	$-0.731939\pi$
$313$	−23.5609	−1.33174	−0.665870	+	0.746068i	$0.731939\pi$
$314$	21.8771	1.23459
$315$	0	0
$316$	−4.24381	−0.238733
$317$	−7.25204	−0.407315	−0.203658	−	0.979042i	$-0.565283\pi$
$317$	−7.25204	−0.407315	−0.203658	+	0.979042i	$0.565283\pi$
$318$	0	0
$319$	0.642637	0.0359808
$320$	13.4346	0.751014
$321$	0	0
$322$	2.18318	0.121664
$323$	1.65618	0.0921525
$324$	0	0
$325$	8.88074	0.492615
$326$	21.2284	1.17573
$327$	0	0
$328$	24.8279	1.37089
$329$	13.0397	0.718905
$330$	0	0
$331$	29.0345	1.59588	0.797939	−	0.602738i	$-0.205924\pi$
$331$	29.0345	1.59588	0.797939	+	0.602738i	$0.205924\pi$
$332$	7.14332	0.392040
$333$	0	0
$334$	−20.9416	−1.14588
$335$	17.6085	0.962053
$336$	0	0
$337$	−1.15910	−0.0631400	−0.0315700	−	0.999502i	$-0.510051\pi$
$337$	−1.15910	−0.0631400	−0.0315700	+	0.999502i	$0.510051\pi$
$338$	−8.11163	−0.441214
$339$	0	0
$340$	−10.3095	−0.559109
$341$	−1.02846	−0.0556942
$342$	0	0
$343$	−19.5562	−1.05594
$344$	13.1775	0.710483
$345$	0	0
$346$	19.9532	1.07269
$347$	−5.88971	−0.316176	−0.158088	−	0.987425i	$-0.550533\pi$
$347$	−5.88971	−0.316176	−0.158088	+	0.987425i	$0.550533\pi$
$348$	0	0
$349$	−30.5927	−1.63759	−0.818795	−	0.574087i	$-0.805357\pi$
$349$	−30.5927	−1.63759	−0.818795	+	0.574087i	$0.805357\pi$
$350$	5.06182	0.270566
$351$	0	0
$352$	−5.74276	−0.306090
$353$	−36.9613	−1.96725	−0.983625	−	0.180227i	$-0.942317\pi$
$353$	−36.9613	−1.96725	−0.983625	+	0.180227i	$0.942317\pi$
$354$	0	0
$355$	−0.156205	−0.00829052
$356$	5.95562	0.315647
$357$	0	0
$358$	11.5065	0.608140
$359$	26.3761	1.39207	0.696037	−	0.718006i	$-0.254945\pi$
$359$	26.3761	1.39207	0.696037	+	0.718006i	$0.254945\pi$
$360$	0	0
$361$	−18.9379	−0.996733
$362$	18.9229	0.994564
$363$	0	0
$364$	9.93869	0.520929
$365$	−9.28958	−0.486239
$366$	0	0
$367$	−11.3131	−0.590541	−0.295270	−	0.955414i	$-0.595410\pi$
$367$	−11.3131	−0.590541	−0.295270	+	0.955414i	$0.595410\pi$
$368$	1.20739	0.0629394
$369$	0	0
$370$	−4.80047	−0.249564
$371$	25.7830	1.33859
$372$	0	0
$373$	5.84408	0.302595	0.151297	−	0.988488i	$-0.451655\pi$
$373$	5.84408	0.302595	0.151297	+	0.988488i	$0.451655\pi$
$374$	−8.78987	−0.454513
$375$	0	0
$376$	16.1473	0.832731
$377$	−2.33085	−0.120045
$378$	0	0
$379$	24.3265	1.24957	0.624783	−	0.780798i	$-0.285187\pi$
$379$	24.3265	1.24957	0.624783	+	0.780798i	$0.285187\pi$
$380$	−0.386430	−0.0198235
$381$	0	0
$382$	−28.4310	−1.45466
$383$	3.81605	0.194991	0.0974955	−	0.995236i	$-0.468917\pi$
$383$	3.81605	0.194991	0.0974955	+	0.995236i	$0.468917\pi$
$384$	0	0
$385$	5.38474	0.274432
$386$	18.0867	0.920591
$387$	0	0
$388$	−4.87782	−0.247634
$389$	−10.8418	−0.549704	−0.274852	−	0.961487i	$-0.588629\pi$
$389$	−10.8418	−0.549704	−0.274852	+	0.961487i	$0.588629\pi$
$390$	0	0
$391$	−5.59692	−0.283049
$392$	−2.89946	−0.146445
$393$	0	0
$394$	2.64872	0.133441
$395$	8.33948	0.419605
$396$	0	0
$397$	−10.5092	−0.527442	−0.263721	−	0.964599i	$-0.584950\pi$
$397$	−10.5092	−0.527442	−0.263721	+	0.964599i	$0.584950\pi$
$398$	−19.5460	−0.979751
$399$	0	0
$400$	2.79939	0.139970
$401$	−14.3656	−0.717383	−0.358691	−	0.933456i	$-0.616777\pi$
$401$	−14.3656	−0.717383	−0.358691	+	0.933456i	$0.616777\pi$
$402$	0	0
$403$	3.73022	0.185816
$404$	4.44685	0.221239
$405$	0	0
$406$	−1.32853	−0.0659338
$407$	3.27106	0.162141
$408$	0	0
$409$	17.6823	0.874332	0.437166	−	0.899381i	$-0.355982\pi$
$409$	17.6823	0.874332	0.437166	+	0.899381i	$0.355982\pi$
$410$	−15.0063	−0.741108
$411$	0	0
$412$	−10.3225	−0.508551
$413$	7.39299	0.363785
$414$	0	0
$415$	−14.0373	−0.689063
$416$	20.8290	1.02122
$417$	0	0
$418$	−0.329471	−0.0161150
$419$	−9.13376	−0.446214	−0.223107	−	0.974794i	$-0.571620\pi$
$419$	−9.13376	−0.446214	−0.223107	+	0.974794i	$0.571620\pi$
$420$	0	0
$421$	24.1949	1.17919	0.589594	−	0.807700i	$-0.299288\pi$
$421$	24.1949	1.17919	0.589594	+	0.807700i	$0.299288\pi$
$422$	3.89170	0.189445
$423$	0	0
$424$	31.9274	1.55053
$425$	−12.9768	−0.629467
$426$	0	0
$427$	7.09043	0.343130
$428$	17.2866	0.835578
$429$	0	0
$430$	−7.96466	−0.384090
$431$	29.5332	1.42256	0.711282	−	0.702907i	$-0.248115\pi$
$431$	29.5332	1.42256	0.711282	+	0.702907i	$0.248115\pi$
$432$	0	0
$433$	0.669754	0.0321863	0.0160932	−	0.999870i	$-0.494877\pi$
$433$	0.669754	0.0321863	0.0160932	+	0.999870i	$0.494877\pi$
$434$	2.12614	0.102058
$435$	0	0
$436$	−5.61037	−0.268688
$437$	−0.209790	−0.0100356
$438$	0	0
$439$	−6.34887	−0.303015	−0.151507	−	0.988456i	$-0.548413\pi$
$439$	−6.34887	−0.303015	−0.151507	+	0.988456i	$0.548413\pi$
$440$	6.66798	0.317883
$441$	0	0
$442$	31.8809	1.51642
$443$	−15.3539	−0.729487	−0.364743	−	0.931108i	$-0.618843\pi$
$443$	−15.3539	−0.729487	−0.364743	+	0.931108i	$0.618843\pi$
$444$	0	0
$445$	−11.7034	−0.554792
$446$	22.3950	1.06043
$447$	0	0
$448$	18.9249	0.894118
$449$	32.0398	1.51205	0.756027	−	0.654541i	$-0.227138\pi$
$449$	32.0398	1.51205	0.756027	+	0.654541i	$0.227138\pi$
$450$	0	0
$451$	10.2254	0.481494
$452$	−6.14392	−0.288986
$453$	0	0
$454$	−15.1191	−0.709573
$455$	−19.5304	−0.915601
$456$	0	0
$457$	19.1680	0.896643	0.448321	−	0.893872i	$-0.352022\pi$
$457$	19.1680	0.896643	0.448321	+	0.893872i	$0.352022\pi$
$458$	−17.8031	−0.831886
$459$	0	0
$460$	1.30591	0.0608882
$461$	5.22029	0.243133	0.121567	−	0.992583i	$-0.461208\pi$
$461$	5.22029	0.243133	0.121567	+	0.992583i	$0.461208\pi$
$462$	0	0
$463$	1.69739	0.0788844	0.0394422	−	0.999222i	$-0.487442\pi$
$463$	1.69739	0.0788844	0.0394422	+	0.999222i	$0.487442\pi$
$464$	−0.734731	−0.0341090
$465$	0	0
$466$	5.90364	0.273481
$467$	−19.6827	−0.910808	−0.455404	−	0.890285i	$-0.650505\pi$
$467$	−19.6827	−0.910808	−0.455404	+	0.890285i	$0.650505\pi$
$468$	0	0
$469$	24.8046	1.14537
$470$	−9.75962	−0.450178
$471$	0	0
$472$	9.15482	0.421385
$473$	5.42716	0.249541
$474$	0	0
$475$	−0.486410	−0.0223180
$476$	−14.5227	−0.665646
$477$	0	0
$478$	−5.56080	−0.254345
$479$	29.2534	1.33662	0.668311	−	0.743882i	$-0.267017\pi$
$479$	29.2534	1.33662	0.668311	+	0.743882i	$0.267017\pi$
$480$	0	0
$481$	−11.8641	−0.540959
$482$	9.38610	0.427525
$483$	0	0
$484$	8.37489	0.380677
$485$	9.58538	0.435250
$486$	0	0
$487$	−20.5056	−0.929199	−0.464600	−	0.885521i	$-0.653802\pi$
$487$	−20.5056	−0.929199	−0.464600	+	0.885521i	$0.653802\pi$
$488$	8.78016	0.397459
$489$	0	0
$490$	1.75247	0.0791686
$491$	−17.5270	−0.790982	−0.395491	−	0.918470i	$-0.629426\pi$
$491$	−17.5270	−0.790982	−0.395491	+	0.918470i	$0.629426\pi$
$492$	0	0
$493$	3.40590	0.153394
$494$	1.19499	0.0537652
$495$	0	0
$496$	1.17584	0.0527969
$497$	−0.220043	−0.00987026
$498$	0	0
$499$	−19.1060	−0.855301	−0.427651	−	0.903944i	$-0.640659\pi$
$499$	−19.1060	−0.855301	−0.427651	+	0.903944i	$0.640659\pi$
$500$	10.7826	0.482212
$501$	0	0
$502$	−8.20287	−0.366112
$503$	10.9676	0.489022	0.244511	−	0.969646i	$-0.421373\pi$
$503$	10.9676	0.489022	0.244511	+	0.969646i	$0.421373\pi$
$504$	0	0
$505$	−8.73847	−0.388857
$506$	1.11342	0.0494975
$507$	0	0
$508$	−10.6814	−0.473912
$509$	19.8994	0.882023	0.441012	−	0.897501i	$-0.354620\pi$
$509$	19.8994	0.882023	0.441012	+	0.897501i	$0.354620\pi$
$510$	0	0
$511$	−13.0860	−0.578890
$512$	15.2994	0.676143
$513$	0	0
$514$	−21.5468	−0.950387
$515$	20.2846	0.893845
$516$	0	0
$517$	6.65026	0.292478
$518$	−6.76230	−0.297118
$519$	0	0
$520$	−24.1848	−1.06057
$521$	−35.1167	−1.53849	−0.769244	−	0.638955i	$-0.779367\pi$
$521$	−35.1167	−1.53849	−0.769244	+	0.638955i	$0.779367\pi$
$522$	0	0
$523$	−14.2454	−0.622907	−0.311453	−	0.950261i	$-0.600816\pi$
$523$	−14.2454	−0.622907	−0.311453	+	0.950261i	$0.600816\pi$
$524$	−12.5129	−0.546629
$525$	0	0
$526$	11.8926	0.518541
$527$	−5.45070	−0.237436
$528$	0	0
$529$	−22.2910	−0.969175
$530$	−19.2973	−0.838223
$531$	0	0
$532$	−0.544355	−0.0236008
$533$	−37.0874	−1.60643
$534$	0	0
$535$	−33.9697	−1.46864
$536$	30.7158	1.32672
$537$	0	0
$538$	−0.324480	−0.0139893
$539$	−1.19414	−0.0514354
$540$	0	0
$541$	−13.2368	−0.569094	−0.284547	−	0.958662i	$-0.591843\pi$
$541$	−13.2368	−0.569094	−0.284547	+	0.958662i	$0.591843\pi$
$542$	2.34325	0.100651
$543$	0	0
$544$	−30.4359	−1.30493
$545$	11.0249	0.472255
$546$	0	0
$547$	−16.3049	−0.697148	−0.348574	−	0.937281i	$-0.613334\pi$
$547$	−16.3049	−0.697148	−0.348574	+	0.937281i	$0.613334\pi$
$548$	2.00613	0.0856976
$549$	0	0
$550$	2.58152	0.110077
$551$	0.127663	0.00543864
$552$	0	0
$553$	11.7476	0.499560
$554$	24.5971	1.04503
$555$	0	0
$556$	7.08507	0.300474
$557$	−30.8972	−1.30915	−0.654577	−	0.755995i	$-0.727153\pi$
$557$	−30.8972	−1.30915	−0.654577	+	0.755995i	$0.727153\pi$
$558$	0	0
$559$	−19.6843	−0.832557
$560$	−6.15640	−0.260155
$561$	0	0
$562$	7.61798	0.321345
$563$	−26.7759	−1.12847	−0.564236	−	0.825614i	$-0.690829\pi$
$563$	−26.7759	−1.12847	−0.564236	+	0.825614i	$0.690829\pi$
$564$	0	0
$565$	12.0734	0.507931
$566$	−7.50710	−0.315547
$567$	0	0
$568$	−0.272481	−0.0114331
$569$	−19.0606	−0.799064	−0.399532	−	0.916719i	$-0.630827\pi$
$569$	−19.0606	−0.799064	−0.399532	+	0.916719i	$0.630827\pi$
$570$	0	0
$571$	−18.7742	−0.785676	−0.392838	−	0.919608i	$-0.628507\pi$
$571$	−18.7742	−0.785676	−0.392838	+	0.919608i	$0.628507\pi$
$572$	5.06872	0.211934
$573$	0	0
$574$	−21.1390	−0.882324
$575$	1.64378	0.0685503
$576$	0	0
$577$	−4.85962	−0.202309	−0.101154	−	0.994871i	$-0.532254\pi$
$577$	−4.85962	−0.202309	−0.101154	+	0.994871i	$0.532254\pi$
$578$	−28.6617	−1.19217
$579$	0	0
$580$	−0.794683	−0.0329974
$581$	−19.7740	−0.820362
$582$	0	0
$583$	13.1493	0.544589
$584$	−16.2045	−0.670548
$585$	0	0
$586$	0.582499	0.0240628
$587$	32.7973	1.35369	0.676846	−	0.736125i	$-0.263346\pi$
$587$	32.7973	1.35369	0.676846	+	0.736125i	$0.263346\pi$
$588$	0	0
$589$	−0.204309	−0.00841841
$590$	−5.53330	−0.227802
$591$	0	0
$592$	−3.73983	−0.153706
$593$	−17.3446	−0.712258	−0.356129	−	0.934437i	$-0.615904\pi$
$593$	−17.3446	−0.712258	−0.356129	+	0.934437i	$0.615904\pi$
$594$	0	0
$595$	28.5384	1.16996
$596$	−0.0950040	−0.00389151
$597$	0	0
$598$	−4.03837	−0.165141
$599$	−24.4079	−0.997280	−0.498640	−	0.866809i	$-0.666167\pi$
$599$	−24.4079	−0.997280	−0.498640	+	0.866809i	$0.666167\pi$
$600$	0	0
$601$	7.87027	0.321035	0.160517	−	0.987033i	$-0.448684\pi$
$601$	7.87027	0.321035	0.160517	+	0.987033i	$0.448684\pi$
$602$	−11.2196	−0.457277
$603$	0	0
$604$	17.9984	0.732346
$605$	−16.4574	−0.669090
$606$	0	0
$607$	−10.2375	−0.415526	−0.207763	−	0.978179i	$-0.566618\pi$
$607$	−10.2375	−0.415526	−0.207763	+	0.978179i	$0.566618\pi$
$608$	−1.14083	−0.0462668
$609$	0	0
$610$	−5.30684	−0.214868
$611$	−24.1205	−0.975810
$612$	0	0
$613$	2.23507	0.0902736	0.0451368	−	0.998981i	$-0.485628\pi$
$613$	2.23507	0.0902736	0.0451368	+	0.998981i	$0.485628\pi$
$614$	−7.09430	−0.286303
$615$	0	0
$616$	9.39301	0.378455
$617$	−33.9757	−1.36781	−0.683905	−	0.729571i	$-0.739720\pi$
$617$	−33.9757	−1.36781	−0.683905	+	0.729571i	$0.739720\pi$
$618$	0	0
$619$	−28.8560	−1.15982	−0.579910	−	0.814681i	$-0.696912\pi$
$619$	−28.8560	−1.15982	−0.579910	+	0.814681i	$0.696912\pi$
$620$	1.27179	0.0510763
$621$	0	0
$622$	16.2613	0.652020
$623$	−16.4862	−0.660507
$624$	0	0
$625$	−11.4277	−0.457107
$626$	24.8408	0.992838
$627$	0	0
$628$	18.4341	0.735602
$629$	17.3362	0.691241
$630$	0	0
$631$	−3.14078	−0.125032	−0.0625162	−	0.998044i	$-0.519913\pi$
$631$	−3.14078	−0.125032	−0.0625162	+	0.998044i	$0.519913\pi$
$632$	14.5472	0.578657
$633$	0	0
$634$	7.64601	0.303662
$635$	20.9900	0.832964
$636$	0	0
$637$	4.33116	0.171607
$638$	−0.677549	−0.0268244
$639$	0	0
$640$	1.82275	0.0720504
$641$	31.8225	1.25691	0.628457	−	0.777844i	$-0.283687\pi$
$641$	31.8225	1.25691	0.628457	+	0.777844i	$0.283687\pi$
$642$	0	0
$643$	−12.8471	−0.506639	−0.253319	−	0.967383i	$-0.581522\pi$
$643$	−12.8471	−0.506639	−0.253319	+	0.967383i	$0.581522\pi$
$644$	1.83960	0.0724903
$645$	0	0
$646$	−1.74616	−0.0687016
$647$	−28.2444	−1.11040	−0.555200	−	0.831717i	$-0.687358\pi$
$647$	−28.2444	−1.11040	−0.555200	+	0.831717i	$0.687358\pi$
$648$	0	0
$649$	3.77042	0.148002
$650$	−9.36319	−0.367254
$651$	0	0
$652$	17.8875	0.700530
$653$	−25.9905	−1.01709	−0.508543	−	0.861036i	$-0.669816\pi$
$653$	−25.9905	−1.01709	−0.508543	+	0.861036i	$0.669816\pi$
$654$	0	0
$655$	24.5891	0.960774
$656$	−11.6907	−0.456446
$657$	0	0
$658$	−13.7481	−0.535958
$659$	47.6178	1.85493	0.927463	−	0.373914i	$-0.121985\pi$
$659$	47.6178	1.85493	0.927463	+	0.373914i	$0.121985\pi$
$660$	0	0
$661$	−0.876508	−0.0340922	−0.0170461	−	0.999855i	$-0.505426\pi$
$661$	−0.876508	−0.0340922	−0.0170461	+	0.999855i	$0.505426\pi$
$662$	−30.6118	−1.18976
$663$	0	0
$664$	−24.4863	−0.950253
$665$	1.06971	0.0414815
$666$	0	0
$667$	−0.431427	−0.0167049
$668$	−17.6459	−0.682741
$669$	0	0
$670$	−18.5650	−0.717230
$671$	3.61611	0.139598
$672$	0	0
$673$	37.3394	1.43933	0.719665	−	0.694321i	$-0.244295\pi$
$673$	37.3394	1.43933	0.719665	+	0.694321i	$0.244295\pi$
$674$	1.22206	0.0470721
$675$	0	0
$676$	−6.83505	−0.262886
$677$	13.7192	0.527270	0.263635	−	0.964622i	$-0.415078\pi$
$677$	13.7192	0.527270	0.263635	+	0.964622i	$0.415078\pi$
$678$	0	0
$679$	13.5027	0.518185
$680$	35.3394	1.35521
$681$	0	0
$682$	1.08433	0.0415211
$683$	−49.9887	−1.91276	−0.956381	−	0.292121i	$-0.905639\pi$
$683$	−49.9887	−1.91276	−0.956381	+	0.292121i	$0.905639\pi$
$684$	0	0
$685$	−3.94223	−0.150625
$686$	20.6186	0.787221
$687$	0	0
$688$	−6.20490	−0.236560
$689$	−47.6925	−1.81694
$690$	0	0
$691$	23.8151	0.905967	0.452984	−	0.891519i	$-0.350360\pi$
$691$	23.8151	0.905967	0.452984	+	0.891519i	$0.350360\pi$
$692$	16.8131	0.639137
$693$	0	0
$694$	6.20967	0.235716
$695$	−13.9228	−0.528122
$696$	0	0
$697$	54.1931	2.05271
$698$	32.2546	1.22086
$699$	0	0
$700$	4.26521	0.161210
$701$	−34.4493	−1.30113	−0.650565	−	0.759450i	$-0.725468\pi$
$701$	−34.4493	−1.30113	−0.650565	+	0.759450i	$0.725468\pi$
$702$	0	0
$703$	0.649815	0.0245082
$704$	9.65169	0.363762
$705$	0	0
$706$	38.9692	1.46662
$707$	−12.3097	−0.462953
$708$	0	0
$709$	15.5233	0.582989	0.291494	−	0.956573i	$-0.405848\pi$
$709$	15.5233	0.582989	0.291494	+	0.956573i	$0.405848\pi$
$710$	0.164691	0.00618075
$711$	0	0
$712$	−20.4151	−0.765087
$713$	0.690444	0.0258573
$714$	0	0
$715$	−9.96050	−0.372502
$716$	9.69569	0.362345
$717$	0	0
$718$	−27.8089	−1.03782
$719$	−12.0537	−0.449528	−0.224764	−	0.974413i	$-0.572161\pi$
$719$	−12.0537	−0.449528	−0.224764	+	0.974413i	$0.572161\pi$
$720$	0	0
$721$	28.5744	1.06417
$722$	19.9667	0.743084
$723$	0	0
$724$	15.9449	0.592586
$725$	−1.00029	−0.0371498
$726$	0	0
$727$	−31.6302	−1.17310	−0.586550	−	0.809913i	$-0.699514\pi$
$727$	−31.6302	−1.17310	−0.586550	+	0.809913i	$0.699514\pi$
$728$	−34.0685	−1.26266
$729$	0	0
$730$	9.79423	0.362501
$731$	28.7633	1.06385
$732$	0	0
$733$	−19.0308	−0.702918	−0.351459	−	0.936203i	$-0.614314\pi$
$733$	−19.0308	−0.702918	−0.351459	+	0.936203i	$0.614314\pi$
$734$	11.9277	0.440260
$735$	0	0
$736$	3.85533	0.142109
$737$	12.6503	0.465981
$738$	0	0
$739$	16.6007	0.610668	0.305334	−	0.952245i	$-0.401232\pi$
$739$	16.6007	0.610668	0.305334	+	0.952245i	$0.401232\pi$
$740$	−4.04499	−0.148697
$741$	0	0
$742$	−27.1837	−0.997944
$743$	33.3035	1.22179	0.610894	−	0.791712i	$-0.290810\pi$
$743$	33.3035	1.22179	0.610894	+	0.791712i	$0.290810\pi$
$744$	0	0
$745$	0.186692	0.00683985
$746$	−6.16156	−0.225591
$747$	0	0
$748$	−7.40655	−0.270810
$749$	−47.8523	−1.74849
$750$	0	0
$751$	27.7816	1.01376	0.506882	−	0.862015i	$-0.330798\pi$
$751$	27.7816	1.01376	0.506882	+	0.862015i	$0.330798\pi$
$752$	−7.60328	−0.277263
$753$	0	0
$754$	2.45747	0.0894957
$755$	−35.3686	−1.28720
$756$	0	0
$757$	−3.12036	−0.113411	−0.0567057	−	0.998391i	$-0.518060\pi$
$757$	−3.12036	−0.113411	−0.0567057	+	0.998391i	$0.518060\pi$
$758$	−25.6480	−0.931577
$759$	0	0
$760$	1.32463	0.0480494
$761$	43.9592	1.59352	0.796760	−	0.604296i	$-0.206546\pi$
$761$	43.9592	1.59352	0.796760	+	0.604296i	$0.206546\pi$
$762$	0	0
$763$	15.5305	0.562242
$764$	−23.9566	−0.866720
$765$	0	0
$766$	−4.02336	−0.145370
$767$	−13.6753	−0.493787
$768$	0	0
$769$	3.71286	0.133889	0.0669445	−	0.997757i	$-0.478675\pi$
$769$	3.71286	0.133889	0.0669445	+	0.997757i	$0.478675\pi$
$770$	−5.67726	−0.204594
$771$	0	0
$772$	15.2403	0.548511
$773$	8.96903	0.322594	0.161297	−	0.986906i	$-0.448432\pi$
$773$	8.96903	0.322594	0.161297	+	0.986906i	$0.448432\pi$
$774$	0	0
$775$	1.60083	0.0575036
$776$	16.7205	0.600232
$777$	0	0
$778$	11.4308	0.409815
$779$	2.03132	0.0727797
$780$	0	0
$781$	−0.112222	−0.00401560
$782$	5.90098	0.211018
$783$	0	0
$784$	1.36527	0.0487597
$785$	−36.2248	−1.29292
$786$	0	0
$787$	43.8514	1.56313	0.781567	−	0.623821i	$-0.214421\pi$
$787$	43.8514	1.56313	0.781567	+	0.623821i	$0.214421\pi$
$788$	2.23188	0.0795074
$789$	0	0
$790$	−8.79253	−0.312824
$791$	17.0075	0.604716
$792$	0	0
$793$	−13.1156	−0.465750
$794$	11.0801	0.393219
$795$	0	0
$796$	−16.4699	−0.583760
$797$	−12.0160	−0.425629	−0.212815	−	0.977093i	$-0.568263\pi$
$797$	−12.0160	−0.425629	−0.212815	+	0.977093i	$0.568263\pi$
$798$	0	0
$799$	35.2455	1.24690
$800$	8.93881	0.316035
$801$	0	0
$802$	15.1460	0.534823
$803$	−6.67384	−0.235515
$804$	0	0
$805$	−3.61498	−0.127411
$806$	−3.93286	−0.138529
$807$	0	0
$808$	−15.2432	−0.536254
$809$	8.02937	0.282298	0.141149	−	0.989988i	$-0.454920\pi$
$809$	8.02937	0.282298	0.141149	+	0.989988i	$0.454920\pi$
$810$	0	0
$811$	−12.8345	−0.450681	−0.225341	−	0.974280i	$-0.572349\pi$
$811$	−12.8345	−0.450681	−0.225341	+	0.974280i	$0.572349\pi$
$812$	−1.11945	−0.0392850
$813$	0	0
$814$	−3.44876	−0.120879
$815$	−35.1507	−1.23127
$816$	0	0
$817$	1.07813	0.0377192
$818$	−18.6428	−0.651832
$819$	0	0
$820$	−12.6447	−0.441570
$821$	29.6654	1.03533	0.517665	−	0.855584i	$-0.326802\pi$
$821$	29.6654	1.03533	0.517665	+	0.855584i	$0.326802\pi$
$822$	0	0
$823$	49.5407	1.72688	0.863441	−	0.504451i	$-0.168305\pi$
$823$	49.5407	1.72688	0.863441	+	0.504451i	$0.168305\pi$
$824$	35.3840	1.23266
$825$	0	0
$826$	−7.79462	−0.271209
$827$	−40.8431	−1.42025	−0.710126	−	0.704074i	$-0.751362\pi$
$827$	−40.8431	−1.42025	−0.710126	+	0.704074i	$0.751362\pi$
$828$	0	0
$829$	−9.45276	−0.328308	−0.164154	−	0.986435i	$-0.552489\pi$
$829$	−9.45276	−0.328308	−0.164154	+	0.986435i	$0.552489\pi$
$830$	14.7999	0.513711
$831$	0	0
$832$	−35.0067	−1.21364
$833$	−6.32881	−0.219280
$834$	0	0
$835$	34.6759	1.20001
$836$	−0.277620	−0.00960170
$837$	0	0
$838$	9.62995	0.332661
$839$	12.2869	0.424191	0.212095	−	0.977249i	$-0.431971\pi$
$839$	12.2869	0.424191	0.212095	+	0.977249i	$0.431971\pi$
$840$	0	0
$841$	−28.7375	−0.990947
$842$	−25.5093	−0.879108
$843$	0	0
$844$	3.27924	0.112876
$845$	13.4315	0.462058
$846$	0	0
$847$	−23.1832	−0.796584
$848$	−15.0337	−0.516259
$849$	0	0
$850$	13.6817	0.469280
$851$	−2.19599	−0.0752776
$852$	0	0
$853$	−30.8659	−1.05683	−0.528413	−	0.848987i	$-0.677213\pi$
$853$	−30.8659	−1.05683	−0.528413	+	0.848987i	$0.677213\pi$
$854$	−7.47562	−0.255810
$855$	0	0
$856$	−59.2560	−2.02533
$857$	−11.1460	−0.380741	−0.190371	−	0.981712i	$-0.560969\pi$
$857$	−11.1460	−0.380741	−0.190371	+	0.981712i	$0.560969\pi$
$858$	0	0
$859$	4.14868	0.141551	0.0707755	−	0.997492i	$-0.477453\pi$
$859$	4.14868	0.141551	0.0707755	+	0.997492i	$0.477453\pi$
$860$	−6.71121	−0.228850
$861$	0	0
$862$	−31.1376	−1.06055
$863$	47.2534	1.60852	0.804262	−	0.594275i	$-0.202561\pi$
$863$	47.2534	1.60852	0.804262	+	0.594275i	$0.202561\pi$
$864$	0	0
$865$	−33.0392	−1.12337
$866$	−0.706138	−0.0239956
$867$	0	0
$868$	1.79154	0.0608088
$869$	5.99127	0.203240
$870$	0	0
$871$	−45.8827	−1.55468
$872$	19.2316	0.651264
$873$	0	0
$874$	0.221187	0.00748175
$875$	−29.8481	−1.00905
$876$	0	0
$877$	35.1504	1.18695	0.593473	−	0.804854i	$-0.297756\pi$
$877$	35.1504	1.18695	0.593473	+	0.804854i	$0.297756\pi$
$878$	6.69377	0.225904
$879$	0	0
$880$	−3.13976	−0.105841
$881$	19.3596	0.652242	0.326121	−	0.945328i	$-0.394258\pi$
$881$	19.3596	0.652242	0.326121	+	0.945328i	$0.394258\pi$
$882$	0	0
$883$	−13.7860	−0.463937	−0.231969	−	0.972723i	$-0.574517\pi$
$883$	−13.7860	−0.463937	−0.231969	+	0.972723i	$0.574517\pi$
$884$	26.8636	0.903519
$885$	0	0
$886$	16.1880	0.543847
$887$	29.8175	1.00117	0.500586	−	0.865687i	$-0.333118\pi$
$887$	29.8175	1.00117	0.500586	+	0.865687i	$0.333118\pi$
$888$	0	0
$889$	29.5681	0.991683
$890$	12.3391	0.413609
$891$	0	0
$892$	18.8705	0.631832
$893$	1.32111	0.0442092
$894$	0	0
$895$	−19.0529	−0.636869
$896$	2.56766	0.0857795
$897$	0	0
$898$	−33.7804	−1.12727
$899$	−0.420156	−0.0140130
$900$	0	0
$901$	69.6897	2.32170
$902$	−10.7809	−0.358963
$903$	0	0
$904$	21.0605	0.700463
$905$	−31.3331	−1.04155
$906$	0	0
$907$	−6.23157	−0.206916	−0.103458	−	0.994634i	$-0.532991\pi$
$907$	−6.23157	−0.206916	−0.103458	+	0.994634i	$0.532991\pi$
$908$	−12.7397	−0.422781
$909$	0	0
$910$	20.5914	0.682599
$911$	31.6395	1.04826	0.524131	−	0.851638i	$-0.324390\pi$
$911$	31.6395	1.04826	0.524131	+	0.851638i	$0.324390\pi$
$912$	0	0
$913$	−10.0847	−0.333755
$914$	−20.2093	−0.668465
$915$	0	0
$916$	−15.0013	−0.495658
$917$	34.6380	1.14385
$918$	0	0
$919$	36.0031	1.18763	0.593816	−	0.804601i	$-0.297621\pi$
$919$	36.0031	1.18763	0.593816	+	0.804601i	$0.297621\pi$
$920$	−4.47647	−0.147585
$921$	0	0
$922$	−5.50388	−0.181261
$923$	0.407027	0.0133975
$924$	0	0
$925$	−5.09153	−0.167408
$926$	−1.78960	−0.0588099
$927$	0	0
$928$	−2.34609	−0.0770141
$929$	−25.4477	−0.834913	−0.417456	−	0.908697i	$-0.637078\pi$
$929$	−25.4477	−0.834913	−0.417456	+	0.908697i	$0.637078\pi$
$930$	0	0
$931$	−0.237223	−0.00777468
$932$	4.97454	0.162947
$933$	0	0
$934$	20.7520	0.679026
$935$	14.5546	0.475985
$936$	0	0
$937$	28.3048	0.924677	0.462338	−	0.886704i	$-0.347011\pi$
$937$	28.3048	0.924677	0.462338	+	0.886704i	$0.347011\pi$
$938$	−26.1521	−0.853897
$939$	0	0
$940$	−8.22369	−0.268227
$941$	−8.29444	−0.270391	−0.135196	−	0.990819i	$-0.543166\pi$
$941$	−8.29444	−0.270391	−0.135196	+	0.990819i	$0.543166\pi$
$942$	0	0
$943$	−6.86468	−0.223545
$944$	−4.31074	−0.140303
$945$	0	0
$946$	−5.72199	−0.186038
$947$	44.4506	1.44445	0.722225	−	0.691658i	$-0.243119\pi$
$947$	44.4506	1.44445	0.722225	+	0.691658i	$0.243119\pi$
$948$	0	0
$949$	24.2060	0.785761
$950$	0.512834	0.0166385
$951$	0	0
$952$	49.7818	1.61344
$953$	9.67149	0.313290	0.156645	−	0.987655i	$-0.449932\pi$
$953$	9.67149	0.313290	0.156645	+	0.987655i	$0.449932\pi$
$954$	0	0
$955$	47.0770	1.52338
$956$	−4.68566	−0.151545
$957$	0	0
$958$	−30.8426	−0.996479
$959$	−5.55332	−0.179326
$960$	0	0
$961$	−30.3276	−0.978309
$962$	12.5087	0.403296
$963$	0	0
$964$	7.90895	0.254730
$965$	−29.9486	−0.964081
$966$	0	0
$967$	−33.1151	−1.06491	−0.532455	−	0.846459i	$-0.678730\pi$
$967$	−33.1151	−1.06491	−0.532455	+	0.846459i	$0.678730\pi$
$968$	−28.7080	−0.922710
$969$	0	0
$970$	−10.1061	−0.324487
$971$	27.4309	0.880298	0.440149	−	0.897925i	$-0.354926\pi$
$971$	27.4309	0.880298	0.440149	+	0.897925i	$0.354926\pi$
$972$	0	0
$973$	−19.6127	−0.628755
$974$	21.6196	0.692737
$975$	0	0
$976$	−4.13432	−0.132336
$977$	−36.3776	−1.16382	−0.581911	−	0.813253i	$-0.697695\pi$
$977$	−36.3776	−1.16382	−0.581911	+	0.813253i	$0.697695\pi$
$978$	0	0
$979$	−8.40796	−0.268719
$980$	1.47667	0.0471706
$981$	0	0
$982$	18.4791	0.589693
$983$	−42.1372	−1.34397	−0.671984	−	0.740566i	$-0.734558\pi$
$983$	−42.1372	−1.34397	−0.671984	+	0.740566i	$0.734558\pi$
$984$	0	0
$985$	−4.38585	−0.139745
$986$	−3.59092	−0.114358
$987$	0	0
$988$	1.00693	0.0320347
$989$	−3.64346	−0.115855
$990$	0	0
$991$	25.5409	0.811333	0.405667	−	0.914021i	$-0.367039\pi$
$991$	25.5409	0.811333	0.405667	+	0.914021i	$0.367039\pi$
$992$	3.75461	0.119209
$993$	0	0
$994$	0.231996	0.00735848
$995$	32.3649	1.02604
$996$	0	0
$997$	23.5329	0.745294	0.372647	−	0.927973i	$-0.378450\pi$
$997$	23.5329	0.745294	0.372647	+	0.927973i	$0.378450\pi$
$998$	20.1439	0.637644
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.a.d.1.2		6
3.2	odd	2		729.2.a.a.1.5		6
9.2	odd	6		729.2.c.e.244.2		12
9.4	even	3		729.2.c.b.487.5		12
9.5	odd	6		729.2.c.e.487.2		12
9.7	even	3		729.2.c.b.244.5		12
27.2	odd	18		243.2.e.d.190.1		12
27.4	even	9		243.2.e.b.136.1		12
27.5	odd	18		27.2.e.a.25.2	yes	12
27.7	even	9		243.2.e.b.109.1		12
27.11	odd	18		27.2.e.a.13.2	✓	12
27.13	even	9		243.2.e.a.55.2		12
27.14	odd	18		243.2.e.d.55.1		12
27.16	even	9		81.2.e.a.10.1		12
27.20	odd	18		243.2.e.c.109.2		12
27.22	even	9		81.2.e.a.73.1		12
27.23	odd	18		243.2.e.c.136.2		12
27.25	even	9		243.2.e.a.190.2		12
108.11	even	18		432.2.u.c.337.2		12
108.59	even	18		432.2.u.c.241.2		12
135.32	even	36		675.2.u.b.349.3		24
135.38	even	36		675.2.u.b.499.3		24
135.59	odd	18		675.2.l.c.376.1		12
135.92	even	36		675.2.u.b.499.2		24
135.113	even	36		675.2.u.b.349.2		24
135.119	odd	18		675.2.l.c.526.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.13.2	✓	12	27.11	odd	18
27.2.e.a.25.2	yes	12	27.5	odd	18
81.2.e.a.10.1		12	27.16	even	9
81.2.e.a.73.1		12	27.22	even	9
243.2.e.a.55.2		12	27.13	even	9
243.2.e.a.190.2		12	27.25	even	9
243.2.e.b.109.1		12	27.7	even	9
243.2.e.b.136.1		12	27.4	even	9
243.2.e.c.109.2		12	27.20	odd	18
243.2.e.c.136.2		12	27.23	odd	18
243.2.e.d.55.1		12	27.14	odd	18
243.2.e.d.190.1		12	27.2	odd	18
432.2.u.c.241.2		12	108.59	even	18
432.2.u.c.337.2		12	108.11	even	18
675.2.l.c.376.1		12	135.59	odd	18
675.2.l.c.526.1		12	135.119	odd	18
675.2.u.b.349.2		24	135.113	even	36
675.2.u.b.349.3		24	135.32	even	36
675.2.u.b.499.2		24	135.92	even	36
675.2.u.b.499.3		24	135.38	even	36
729.2.a.a.1.5		6	3.2	odd	2
729.2.a.d.1.2		6	1.1	even	1	trivial
729.2.c.b.244.5		12	9.7	even	3
729.2.c.b.487.5		12	9.4	even	3
729.2.c.e.244.2		12	9.2	odd	6
729.2.c.e.487.2		12	9.5	odd	6

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.05432 q^{2} -0.888399 q^{4} +1.74579 q^{5} +2.45925 q^{7} +3.04531 q^{8} -1.84063 q^{10} +1.25421 q^{11} -4.54903 q^{13} -2.59284 q^{14} -1.43395 q^{16} +6.64717 q^{17} +0.249156 q^{19} -1.55096 q^{20} -1.32235 q^{22} -0.842001 q^{23} -1.95223 q^{25} +4.79615 q^{26} -2.18479 q^{28} +0.512383 q^{29} -0.820004 q^{31} -4.57877 q^{32} -7.00828 q^{34} +4.29332 q^{35} +2.60806 q^{37} -0.262692 q^{38} +5.31646 q^{40} +8.15281 q^{41} +4.32714 q^{43} -1.11424 q^{44} +0.887743 q^{46} +5.30233 q^{47} -0.952106 q^{49} +2.05828 q^{50} +4.04135 q^{52} +10.4841 q^{53} +2.18959 q^{55} +7.48917 q^{56} -0.540218 q^{58} +3.00620 q^{59} +2.88317 q^{61} +0.864550 q^{62} +7.69541 q^{64} -7.94164 q^{65} +10.0863 q^{67} -5.90534 q^{68} -4.52655 q^{70} -0.0894756 q^{71} -5.32114 q^{73} -2.74974 q^{74} -0.221350 q^{76} +3.08442 q^{77} +4.77692 q^{79} -2.50337 q^{80} -8.59571 q^{82} -8.04066 q^{83} +11.6045 q^{85} -4.56222 q^{86} +3.81947 q^{88} -6.70377 q^{89} -11.1872 q^{91} +0.748033 q^{92} -5.59038 q^{94} +0.434974 q^{95} +5.49058 q^{97} +1.00383 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8} + 3 q^{10} + 12 q^{11} + 6 q^{14} - 3 q^{16} + 9 q^{17} + 3 q^{19} + 6 q^{20} + 6 q^{22} + 15 q^{23} - 6 q^{25} + 15 q^{26} - 6 q^{28} + 12 q^{29} + 12 q^{35}+ \cdots - 45 q^{98}+O(q^{100}) \) 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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