LMFDB - Embedded newform 729.2.a.a.1.2

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			1.2
Root			$0.584534$ 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)

$f(q)$	$=$	$q-2.11662 q^{2} +2.48009 q^{4} -2.68310 q^{5} +0.972333 q^{7} -1.01617 q^{8} +O(q^{10})$	\(q-2.11662 q^{2} +2.48009 q^{4} -2.68310 q^{5} +0.972333 q^{7} -1.01617 q^{8} +5.67911 q^{10} -0.316902 q^{11} -1.51403 q^{13} -2.05806 q^{14} -2.80933 q^{16} +1.17468 q^{17} +6.22080 q^{19} -6.65433 q^{20} +0.670762 q^{22} +2.16274 q^{23} +2.19901 q^{25} +3.20463 q^{26} +2.41147 q^{28} -4.40491 q^{29} -8.67323 q^{31} +7.97863 q^{32} -2.48636 q^{34} -2.60886 q^{35} -4.46665 q^{37} -13.1671 q^{38} +2.72649 q^{40} +5.84518 q^{41} +5.59099 q^{43} -0.785946 q^{44} -4.57771 q^{46} -2.47607 q^{47} -6.05457 q^{49} -4.65448 q^{50} -3.75493 q^{52} -10.8920 q^{53} +0.850279 q^{55} -0.988058 q^{56} +9.32353 q^{58} -1.72421 q^{59} +1.01478 q^{61} +18.3580 q^{62} -11.2691 q^{64} +4.06229 q^{65} +0.856551 q^{67} +2.91332 q^{68} +5.52198 q^{70} -9.59577 q^{71} -15.2418 q^{73} +9.45420 q^{74} +15.4282 q^{76} -0.308134 q^{77} -11.2138 q^{79} +7.53771 q^{80} -12.3721 q^{82} +4.68493 q^{83} -3.15179 q^{85} -11.8340 q^{86} +0.322027 q^{88} -15.4995 q^{89} -1.47214 q^{91} +5.36380 q^{92} +5.24090 q^{94} -16.6910 q^{95} -5.54949 q^{97} +12.8152 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.11662	−1.49668	−0.748339	−	0.663316i	$-0.769148\pi$
$2$	−2.11662	−1.49668	−0.748339	+	0.663316i	$0.769148\pi$
$3$	0	0
$3$	0	0
$4$	2.48009	1.24005
$5$	−2.68310	−1.19992	−0.599959	−	0.800031i	$-0.704816\pi$
$5$	−2.68310	−1.19992	−0.599959	+	0.800031i	$0.704816\pi$
$6$	0	0
$7$	0.972333	0.367507	0.183754	−	0.982972i	$-0.441175\pi$
$7$	0.972333	0.367507	0.183754	+	0.982972i	$0.441175\pi$
$8$	−1.01617	−0.359271
$9$	0	0
$10$	5.67911	1.79589
$11$	−0.316902	−0.0955496	−0.0477748	−	0.998858i	$-0.515213\pi$
$11$	−0.316902	−0.0955496	−0.0477748	+	0.998858i	$0.515213\pi$
$12$	0	0
$13$	−1.51403	−0.419916	−0.209958	−	0.977710i	$-0.567333\pi$
$13$	−1.51403	−0.419916	−0.209958	+	0.977710i	$0.567333\pi$
$14$	−2.05806	−0.550040
$15$	0	0
$16$	−2.80933	−0.702332
$17$	1.17468	0.284903	0.142451	−	0.989802i	$-0.454502\pi$
$17$	1.17468	0.284903	0.142451	+	0.989802i	$0.454502\pi$
$18$	0	0
$19$	6.22080	1.42715	0.713575	−	0.700579i	$-0.247075\pi$
$19$	6.22080	1.42715	0.713575	+	0.700579i	$0.247075\pi$
$20$	−6.65433	−1.48795
$21$	0	0
$22$	0.670762	0.143007
$23$	2.16274	0.450963	0.225481	−	0.974247i	$-0.427605\pi$
$23$	2.16274	0.450963	0.225481	+	0.974247i	$0.427605\pi$
$24$	0	0
$25$	2.19901	0.439803
$26$	3.20463	0.628480
$27$	0	0
$28$	2.41147	0.455726
$29$	−4.40491	−0.817971	−0.408986	−	0.912541i	$-0.634117\pi$
$29$	−4.40491	−0.817971	−0.408986	+	0.912541i	$0.634117\pi$
$30$	0	0
$31$	−8.67323	−1.55776	−0.778879	−	0.627174i	$-0.784211\pi$
$31$	−8.67323	−1.55776	−0.778879	+	0.627174i	$0.784211\pi$
$32$	7.97863	1.41044
$33$	0	0
$34$	−2.48636	−0.426408
$35$	−2.60886	−0.440978
$36$	0	0
$37$	−4.46665	−0.734312	−0.367156	−	0.930159i	$-0.619668\pi$
$37$	−4.46665	−0.734312	−0.367156	+	0.930159i	$0.619668\pi$
$38$	−13.1671	−2.13598
$39$	0	0
$40$	2.72649	0.431096
$41$	5.84518	0.912864	0.456432	−	0.889758i	$-0.349127\pi$
$41$	5.84518	0.912864	0.456432	+	0.889758i	$0.349127\pi$
$42$	0	0
$43$	5.59099	0.852618	0.426309	−	0.904578i	$-0.359814\pi$
$43$	5.59099	0.852618	0.426309	+	0.904578i	$0.359814\pi$
$44$	−0.785946	−0.118486
$45$	0	0
$46$	−4.57771	−0.674946
$47$	−2.47607	−0.361172	−0.180586	−	0.983559i	$-0.557799\pi$
$47$	−2.47607	−0.361172	−0.180586	+	0.983559i	$0.557799\pi$
$48$	0	0
$49$	−6.05457	−0.864938
$50$	−4.65448	−0.658243
$51$	0	0
$52$	−3.75493	−0.520716
$53$	−10.8920	−1.49613	−0.748063	−	0.663628i	$-0.769016\pi$
$53$	−10.8920	−1.49613	−0.748063	+	0.663628i	$0.769016\pi$
$54$	0	0
$55$	0.850279	0.114652
$56$	−0.988058	−0.132035
$57$	0	0
$58$	9.32353	1.22424
$59$	−1.72421	−0.224473	−0.112237	−	0.993682i	$-0.535801\pi$
$59$	−1.72421	−0.224473	−0.112237	+	0.993682i	$0.535801\pi$
$60$	0	0
$61$	1.01478	0.129929	0.0649645	−	0.997888i	$-0.479307\pi$
$61$	1.01478	0.129929	0.0649645	+	0.997888i	$0.479307\pi$
$62$	18.3580	2.33146
$63$	0	0
$64$	−11.2691	−1.40864
$65$	4.06229	0.503865
$66$	0	0
$67$	0.856551	0.104644	0.0523222	−	0.998630i	$-0.483338\pi$
$67$	0.856551	0.104644	0.0523222	+	0.998630i	$0.483338\pi$
$68$	2.91332	0.353292
$69$	0	0
$70$	5.52198	0.660003
$71$	−9.59577	−1.13881	−0.569404	−	0.822058i	$-0.692826\pi$
$71$	−9.59577	−1.13881	−0.569404	+	0.822058i	$0.692826\pi$
$72$	0	0
$73$	−15.2418	−1.78392	−0.891960	−	0.452113i	$-0.850670\pi$
$73$	−15.2418	−1.78392	−0.891960	+	0.452113i	$0.850670\pi$
$74$	9.45420	1.09903
$75$	0	0
$76$	15.4282	1.76973
$77$	−0.308134	−0.0351152
$78$	0	0
$79$	−11.2138	−1.26165	−0.630824	−	0.775926i	$-0.717283\pi$
$79$	−11.2138	−1.26165	−0.630824	+	0.775926i	$0.717283\pi$
$80$	7.53771	0.842741
$81$	0	0
$82$	−12.3721	−1.36626
$83$	4.68493	0.514238	0.257119	−	0.966380i	$-0.417227\pi$
$83$	4.68493	0.514238	0.257119	+	0.966380i	$0.417227\pi$
$84$	0	0
$85$	−3.15179	−0.341860
$86$	−11.8340	−1.27610
$87$	0	0
$88$	0.322027	0.0343282
$89$	−15.4995	−1.64295	−0.821473	−	0.570248i	$-0.806847\pi$
$89$	−15.4995	−1.64295	−0.821473	+	0.570248i	$0.806847\pi$
$90$	0	0
$91$	−1.47214	−0.154322
$92$	5.36380	0.559215
$93$	0	0
$94$	5.24090	0.540558
$95$	−16.6910	−1.71246
$96$	0	0
$97$	−5.54949	−0.563465	−0.281732	−	0.959493i	$-0.590909\pi$
$97$	−5.54949	−0.563465	−0.281732	+	0.959493i	$0.590909\pi$
$98$	12.8152	1.29453
$99$	0	0
$100$	5.45376	0.545376
$101$	−10.1390	−1.00887	−0.504436	−	0.863449i	$-0.668300\pi$
$101$	−10.1390	−1.00887	−0.504436	+	0.863449i	$0.668300\pi$
$102$	0	0
$103$	−9.85329	−0.970874	−0.485437	−	0.874272i	$-0.661339\pi$
$103$	−9.85329	−0.970874	−0.485437	+	0.874272i	$0.661339\pi$
$104$	1.53852	0.150864
$105$	0	0
$106$	23.0542	2.23922
$107$	5.17080	0.499880	0.249940	−	0.968261i	$-0.419589\pi$
$107$	5.17080	0.499880	0.249940	+	0.968261i	$0.419589\pi$
$108$	0	0
$109$	−7.31065	−0.700234	−0.350117	−	0.936706i	$-0.613858\pi$
$109$	−7.31065	−0.700234	−0.350117	+	0.936706i	$0.613858\pi$
$110$	−1.79972	−0.171597
$111$	0	0
$112$	−2.73160	−0.258112
$113$	10.3756	0.976058	0.488029	−	0.872827i	$-0.337716\pi$
$113$	10.3756	0.976058	0.488029	+	0.872827i	$0.337716\pi$
$114$	0	0
$115$	−5.80285	−0.541118
$116$	−10.9246	−1.01432
$117$	0	0
$118$	3.64950	0.335964
$119$	1.14218	0.104704
$120$	0	0
$121$	−10.8996	−0.990870
$122$	−2.14790	−0.194462
$123$	0	0
$124$	−21.5104	−1.93169
$125$	7.51532	0.672191
$126$	0	0
$127$	5.22743	0.463860	0.231930	−	0.972733i	$-0.425496\pi$
$127$	5.22743	0.463860	0.231930	+	0.972733i	$0.425496\pi$
$128$	7.89516	0.697841
$129$	0	0
$130$	−8.59834	−0.754124
$131$	−7.23430	−0.632064	−0.316032	−	0.948749i	$-0.602351\pi$
$131$	−7.23430	−0.632064	−0.316032	+	0.948749i	$0.602351\pi$
$132$	0	0
$133$	6.04869	0.524488
$134$	−1.81300	−0.156619
$135$	0	0
$136$	−1.19368	−0.102357
$137$	11.2493	0.961094	0.480547	−	0.876969i	$-0.340438\pi$
$137$	11.2493	0.961094	0.480547	+	0.876969i	$0.340438\pi$
$138$	0	0
$139$	9.38016	0.795615	0.397808	−	0.917469i	$-0.369771\pi$
$139$	9.38016	0.795615	0.397808	+	0.917469i	$0.369771\pi$
$140$	−6.47022	−0.546833
$141$	0	0
$142$	20.3106	1.70443
$143$	0.479799	0.0401228
$144$	0	0
$145$	11.8188	0.981498
$146$	32.2612	2.66996
$147$	0	0
$148$	−11.0777	−0.910581
$149$	19.0512	1.56074	0.780369	−	0.625319i	$-0.215031\pi$
$149$	19.0512	1.56074	0.780369	+	0.625319i	$0.215031\pi$
$150$	0	0
$151$	4.01400	0.326655	0.163327	−	0.986572i	$-0.447777\pi$
$151$	4.01400	0.326655	0.163327	+	0.986572i	$0.447777\pi$
$152$	−6.32141	−0.512734
$153$	0	0
$154$	0.652204	0.0525561
$155$	23.2711	1.86918
$156$	0	0
$157$	7.27592	0.580681	0.290341	−	0.956923i	$-0.406231\pi$
$157$	7.27592	0.580681	0.290341	+	0.956923i	$0.406231\pi$
$158$	23.7353	1.88828
$159$	0	0
$160$	−21.4075	−1.69241
$161$	2.10290	0.165732
$162$	0	0
$163$	12.4492	0.975094	0.487547	−	0.873097i	$-0.337892\pi$
$163$	12.4492	0.975094	0.487547	+	0.873097i	$0.337892\pi$
$164$	14.4966	1.13199
$165$	0	0
$166$	−9.91623	−0.769649
$167$	−2.33189	−0.180447	−0.0902234	−	0.995922i	$-0.528758\pi$
$167$	−2.33189	−0.180447	−0.0902234	+	0.995922i	$0.528758\pi$
$168$	0	0
$169$	−10.7077	−0.823670
$170$	6.67116	0.511654
$171$	0	0
$172$	13.8662	1.05729
$173$	−3.58227	−0.272355	−0.136177	−	0.990684i	$-0.543482\pi$
$173$	−3.58227	−0.272355	−0.136177	+	0.990684i	$0.543482\pi$
$174$	0	0
$175$	2.13817	0.161631
$176$	0.890282	0.0671076
$177$	0	0
$178$	32.8066	2.45896
$179$	−19.9957	−1.49455	−0.747275	−	0.664515i	$-0.768638\pi$
$179$	−19.9957	−1.49455	−0.747275	+	0.664515i	$0.768638\pi$
$180$	0	0
$181$	9.73232	0.723398	0.361699	−	0.932295i	$-0.382197\pi$
$181$	9.73232	0.723398	0.361699	+	0.932295i	$0.382197\pi$
$182$	3.11597	0.230971
$183$	0	0
$184$	−2.19772	−0.162018
$185$	11.9844	0.881114
$186$	0	0
$187$	−0.372260	−0.0272223
$188$	−6.14088	−0.447869
$189$	0	0
$190$	35.3286	2.56301
$191$	−17.7565	−1.28482	−0.642409	−	0.766362i	$-0.722065\pi$
$191$	−17.7565	−1.28482	−0.642409	+	0.766362i	$0.722065\pi$
$192$	0	0
$193$	10.5843	0.761877	0.380939	−	0.924600i	$-0.375601\pi$
$193$	10.5843	0.761877	0.380939	+	0.924600i	$0.375601\pi$
$194$	11.7462	0.843326
$195$	0	0
$196$	−15.0159	−1.07256
$197$	14.1589	1.00878	0.504390	−	0.863476i	$-0.331718\pi$
$197$	14.1589	1.00878	0.504390	+	0.863476i	$0.331718\pi$
$198$	0	0
$199$	7.54019	0.534510	0.267255	−	0.963626i	$-0.413883\pi$
$199$	7.54019	0.534510	0.267255	+	0.963626i	$0.413883\pi$
$200$	−2.23458	−0.158009
$201$	0	0
$202$	21.4605	1.50996
$203$	−4.28304	−0.300610
$204$	0	0
$205$	−15.6832	−1.09536
$206$	20.8557	1.45309
$207$	0	0
$208$	4.25341	0.294921
$209$	−1.97139	−0.136364
$210$	0	0
$211$	5.21364	0.358922	0.179461	−	0.983765i	$-0.442565\pi$
$211$	5.21364	0.358922	0.179461	+	0.983765i	$0.442565\pi$
$212$	−27.0131	−1.85526
$213$	0	0
$214$	−10.9446	−0.748160
$215$	−15.0012	−1.02307
$216$	0	0
$217$	−8.43326	−0.572487
$218$	15.4739	1.04802
$219$	0	0
$220$	2.10877	0.142173
$221$	−1.77851	−0.119635
$222$	0	0
$223$	17.6938	1.18486	0.592432	−	0.805620i	$-0.298168\pi$
$223$	17.6938	1.18486	0.592432	+	0.805620i	$0.298168\pi$
$224$	7.75789	0.518346
$225$	0	0
$226$	−21.9613	−1.46085
$227$	−15.7720	−1.04682	−0.523412	−	0.852080i	$-0.675341\pi$
$227$	−15.7720	−1.04682	−0.523412	+	0.852080i	$0.675341\pi$
$228$	0	0
$229$	−1.76686	−0.116758	−0.0583788	−	0.998295i	$-0.518593\pi$
$229$	−1.76686	−0.116758	−0.0583788	+	0.998295i	$0.518593\pi$
$230$	12.2824	0.809880
$231$	0	0
$232$	4.47615	0.293874
$233$	−13.8984	−0.910514	−0.455257	−	0.890360i	$-0.650453\pi$
$233$	−13.8984	−0.910514	−0.455257	+	0.890360i	$0.650453\pi$
$234$	0	0
$235$	6.64354	0.433376
$236$	−4.27620	−0.278357
$237$	0	0
$238$	−2.41757	−0.156708
$239$	−19.8327	−1.28287	−0.641435	−	0.767177i	$-0.721661\pi$
$239$	−19.8327	−1.28287	−0.641435	+	0.767177i	$0.721661\pi$
$240$	0	0
$241$	19.3747	1.24803	0.624017	−	0.781411i	$-0.285500\pi$
$241$	19.3747	1.24803	0.624017	+	0.781411i	$0.285500\pi$
$242$	23.0703	1.48301
$243$	0	0
$244$	2.51674	0.161118
$245$	16.2450	1.03786
$246$	0	0
$247$	−9.41849	−0.599284
$248$	8.81349	0.559657
$249$	0	0
$250$	−15.9071	−1.00605
$251$	−5.47572	−0.345625	−0.172812	−	0.984955i	$-0.555285\pi$
$251$	−5.47572	−0.345625	−0.172812	+	0.984955i	$0.555285\pi$
$252$	0	0
$253$	−0.685377	−0.0430893
$254$	−11.0645	−0.694248
$255$	0	0
$256$	5.82712	0.364195
$257$	−11.5652	−0.721415	−0.360708	−	0.932679i	$-0.617465\pi$
$257$	−11.5652	−0.721415	−0.360708	+	0.932679i	$0.617465\pi$
$258$	0	0
$259$	−4.34307	−0.269865
$260$	10.0749	0.624816
$261$	0	0
$262$	15.3123	0.945996
$263$	6.47794	0.399447	0.199723	−	0.979852i	$-0.435996\pi$
$263$	6.47794	0.399447	0.199723	+	0.979852i	$0.435996\pi$
$264$	0	0
$265$	29.2242	1.79523
$266$	−12.8028	−0.784990
$267$	0	0
$268$	2.12432	0.129764
$269$	−13.8387	−0.843758	−0.421879	−	0.906652i	$-0.638629\pi$
$269$	−13.8387	−0.843758	−0.421879	+	0.906652i	$0.638629\pi$
$270$	0	0
$271$	1.94536	0.118172	0.0590860	−	0.998253i	$-0.481181\pi$
$271$	1.94536	0.118172	0.0590860	+	0.998253i	$0.481181\pi$
$272$	−3.30007	−0.200096
$273$	0	0
$274$	−23.8106	−1.43845
$275$	−0.696872	−0.0420230
$276$	0	0
$277$	12.4727	0.749411	0.374706	−	0.927144i	$-0.377744\pi$
$277$	12.4727	0.749411	0.374706	+	0.927144i	$0.377744\pi$
$278$	−19.8543	−1.19078
$279$	0	0
$280$	2.65106	0.158431
$281$	9.75587	0.581986	0.290993	−	0.956725i	$-0.406014\pi$
$281$	9.75587	0.581986	0.290993	+	0.956725i	$0.406014\pi$
$282$	0	0
$283$	26.5712	1.57950	0.789748	−	0.613431i	$-0.210211\pi$
$283$	26.5712	1.57950	0.789748	+	0.613431i	$0.210211\pi$
$284$	−23.7984	−1.41217
$285$	0	0
$286$	−1.01555	−0.0600510
$287$	5.68346	0.335484
$288$	0	0
$289$	−15.6201	−0.918830
$290$	−25.0160	−1.46899
$291$	0	0
$292$	−37.8011	−2.21214
$293$	12.2575	0.716089	0.358044	−	0.933705i	$-0.383444\pi$
$293$	12.2575	0.716089	0.358044	+	0.933705i	$0.383444\pi$
$294$	0	0
$295$	4.62623	0.269349
$296$	4.53888	0.263817
$297$	0	0
$298$	−40.3243	−2.33592
$299$	−3.27446	−0.189367
$300$	0	0
$301$	5.43630	0.313343
$302$	−8.49613	−0.488897
$303$	0	0
$304$	−17.4763	−1.00233
$305$	−2.72275	−0.155904
$306$	0	0
$307$	−26.4740	−1.51095	−0.755475	−	0.655178i	$-0.772594\pi$
$307$	−26.4740	−1.51095	−0.755475	+	0.655178i	$0.772594\pi$
$308$	−0.764201	−0.0435444
$309$	0	0
$310$	−49.2562	−2.79756
$311$	−17.6595	−1.00138	−0.500689	−	0.865627i	$-0.666920\pi$
$311$	−17.6595	−1.00138	−0.500689	+	0.865627i	$0.666920\pi$
$312$	0	0
$313$	9.64721	0.545292	0.272646	−	0.962114i	$-0.412101\pi$
$313$	9.64721	0.545292	0.272646	+	0.962114i	$0.412101\pi$
$314$	−15.4004	−0.869093
$315$	0	0
$316$	−27.8112	−1.56450
$317$	−3.71049	−0.208402	−0.104201	−	0.994556i	$-0.533229\pi$
$317$	−3.71049	−0.208402	−0.104201	+	0.994556i	$0.533229\pi$
$318$	0	0
$319$	1.39593	0.0781568
$320$	30.2361	1.69025
$321$	0	0
$322$	−4.45106	−0.248048
$323$	7.30748	0.406599
$324$	0	0
$325$	−3.32937	−0.184680
$326$	−26.3502	−1.45940
$327$	0	0
$328$	−5.93972	−0.327966
$329$	−2.40756	−0.132733
$330$	0	0
$331$	−1.41280	−0.0776543	−0.0388271	−	0.999246i	$-0.512362\pi$
$331$	−1.41280	−0.0776543	−0.0388271	+	0.999246i	$0.512362\pi$
$332$	11.6191	0.637678
$333$	0	0
$334$	4.93573	0.270071
$335$	−2.29821	−0.125565
$336$	0	0
$337$	−12.9901	−0.707614	−0.353807	−	0.935318i	$-0.615113\pi$
$337$	−12.9901	−0.707614	−0.353807	+	0.935318i	$0.615113\pi$
$338$	22.6642	1.23277
$339$	0	0
$340$	−7.81673	−0.423922
$341$	2.74856	0.148843
$342$	0	0
$343$	−12.6934	−0.685378
$344$	−5.68141	−0.306321
$345$	0	0
$346$	7.58231	0.407628
$347$	−4.80046	−0.257702	−0.128851	−	0.991664i	$-0.541129\pi$
$347$	−4.80046	−0.257702	−0.128851	+	0.991664i	$0.541129\pi$
$348$	0	0
$349$	−22.5776	−1.20855	−0.604275	−	0.796776i	$-0.706537\pi$
$349$	−22.5776	−1.20855	−0.604275	+	0.796776i	$0.706537\pi$
$350$	−4.52571	−0.241909
$351$	0	0
$352$	−2.52845	−0.134767
$353$	−29.7066	−1.58112	−0.790560	−	0.612384i	$-0.790211\pi$
$353$	−29.7066	−1.58112	−0.790560	+	0.612384i	$0.790211\pi$
$354$	0	0
$355$	25.7464	1.36648
$356$	−38.4402	−2.03733
$357$	0	0
$358$	42.3234	2.23686
$359$	13.4198	0.708271	0.354136	−	0.935194i	$-0.384775\pi$
$359$	13.4198	0.708271	0.354136	+	0.935194i	$0.384775\pi$
$360$	0	0
$361$	19.6984	1.03676
$362$	−20.5997	−1.08269
$363$	0	0
$364$	−3.65105	−0.191367
$365$	40.8953	2.14056
$366$	0	0
$367$	−7.94998	−0.414985	−0.207493	−	0.978237i	$-0.566530\pi$
$367$	−7.94998	−0.414985	−0.207493	+	0.978237i	$0.566530\pi$
$368$	−6.07585	−0.316726
$369$	0	0
$370$	−25.3666	−1.31874
$371$	−10.5906	−0.549837
$372$	0	0
$373$	−11.4205	−0.591332	−0.295666	−	0.955291i	$-0.595541\pi$
$373$	−11.4205	−0.591332	−0.295666	+	0.955291i	$0.595541\pi$
$374$	0.787934	0.0407431
$375$	0	0
$376$	2.51611	0.129759
$377$	6.66917	0.343480
$378$	0	0
$379$	−24.1705	−1.24155	−0.620777	−	0.783987i	$-0.713183\pi$
$379$	−24.1705	−1.24155	−0.620777	+	0.783987i	$0.713183\pi$
$380$	−41.3953	−2.12353
$381$	0	0
$382$	37.5839	1.92296
$383$	9.44328	0.482529	0.241265	−	0.970459i	$-0.422438\pi$
$383$	9.44328	0.482529	0.241265	+	0.970459i	$0.422438\pi$
$384$	0	0
$385$	0.826754	0.0421353
$386$	−22.4030	−1.14029
$387$	0	0
$388$	−13.7632	−0.698722
$389$	−2.54771	−0.129174	−0.0645869	−	0.997912i	$-0.520573\pi$
$389$	−2.54771	−0.129174	−0.0645869	+	0.997912i	$0.520573\pi$
$390$	0	0
$391$	2.54054	0.128481
$392$	6.15249	0.310747
$393$	0	0
$394$	−29.9690	−1.50982
$395$	30.0876	1.51387
$396$	0	0
$397$	3.67517	0.184452	0.0922258	−	0.995738i	$-0.470602\pi$
$397$	3.67517	0.184452	0.0922258	+	0.995738i	$0.470602\pi$
$398$	−15.9597	−0.799990
$399$	0	0
$400$	−6.17776	−0.308888
$401$	16.1487	0.806429	0.403214	−	0.915106i	$-0.367893\pi$
$401$	16.1487	0.806429	0.403214	+	0.915106i	$0.367893\pi$
$402$	0	0
$403$	13.1315	0.654128
$404$	−25.1458	−1.25105
$405$	0	0
$406$	9.06558	0.449917
$407$	1.41549	0.0701632
$408$	0	0
$409$	9.18197	0.454019	0.227010	−	0.973893i	$-0.427105\pi$
$409$	9.18197	0.454019	0.227010	+	0.973893i	$0.427105\pi$
$410$	33.1954	1.63941
$411$	0	0
$412$	−24.4371	−1.20393
$413$	−1.67651	−0.0824955
$414$	0	0
$415$	−12.5701	−0.617043
$416$	−12.0799	−0.592266
$417$	0	0
$418$	4.17268	0.204092
$419$	−6.97888	−0.340940	−0.170470	−	0.985363i	$-0.554529\pi$
$419$	−6.97888	−0.340940	−0.170470	+	0.985363i	$0.554529\pi$
$420$	0	0
$421$	30.8106	1.50162	0.750809	−	0.660520i	$-0.229664\pi$
$421$	30.8106	1.50162	0.750809	+	0.660520i	$0.229664\pi$
$422$	−11.0353	−0.537190
$423$	0	0
$424$	11.0681	0.537515
$425$	2.58315	0.125301
$426$	0	0
$427$	0.986702	0.0477498
$428$	12.8241	0.619874
$429$	0	0
$430$	31.7518	1.53121
$431$	27.8971	1.34376	0.671879	−	0.740661i	$-0.265487\pi$
$431$	27.8971	1.34376	0.671879	+	0.740661i	$0.265487\pi$
$432$	0	0
$433$	19.1706	0.921278	0.460639	−	0.887588i	$-0.347620\pi$
$433$	19.1706	0.921278	0.460639	+	0.887588i	$0.347620\pi$
$434$	17.8500	0.856829
$435$	0	0
$436$	−18.1311	−0.868322
$437$	13.4540	0.643592
$438$	0	0
$439$	−23.7490	−1.13348	−0.566739	−	0.823898i	$-0.691795\pi$
$439$	−23.7490	−1.13348	−0.566739	+	0.823898i	$0.691795\pi$
$440$	−0.864030	−0.0411910
$441$	0	0
$442$	3.76443	0.179056
$443$	23.3583	1.10978	0.554892	−	0.831922i	$-0.312759\pi$
$443$	23.3583	1.10978	0.554892	+	0.831922i	$0.312759\pi$
$444$	0	0
$445$	41.5867	1.97140
$446$	−37.4511	−1.77336
$447$	0	0
$448$	−10.9573	−0.517685
$449$	4.81906	0.227426	0.113713	−	0.993514i	$-0.463726\pi$
$449$	4.81906	0.227426	0.113713	+	0.993514i	$0.463726\pi$
$450$	0	0
$451$	−1.85235	−0.0872238
$452$	25.7325	1.21036
$453$	0	0
$454$	33.3833	1.56676
$455$	3.94990	0.185174
$456$	0	0
$457$	−4.89360	−0.228913	−0.114456	−	0.993428i	$-0.536513\pi$
$457$	−4.89360	−0.228913	−0.114456	+	0.993428i	$0.536513\pi$
$458$	3.73978	0.174749
$459$	0	0
$460$	−14.3916	−0.671012
$461$	27.9905	1.30365	0.651823	−	0.758371i	$-0.274005\pi$
$461$	27.9905	1.30365	0.651823	+	0.758371i	$0.274005\pi$
$462$	0	0
$463$	−27.4753	−1.27689	−0.638444	−	0.769669i	$-0.720421\pi$
$463$	−27.4753	−1.27689	−0.638444	+	0.769669i	$0.720421\pi$
$464$	12.3748	0.574488
$465$	0	0
$466$	29.4177	1.36275
$467$	−21.2465	−0.983170	−0.491585	−	0.870830i	$-0.663582\pi$
$467$	−21.2465	−0.983170	−0.491585	+	0.870830i	$0.663582\pi$
$468$	0	0
$469$	0.832853	0.0384576
$470$	−14.0619	−0.648625
$471$	0	0
$472$	1.75210	0.0806467
$473$	−1.77180	−0.0814673
$474$	0	0
$475$	13.6796	0.627665
$476$	2.83272	0.129838
$477$	0	0
$478$	41.9783	1.92004
$479$	−41.6788	−1.90435	−0.952177	−	0.305546i	$-0.901161\pi$
$479$	−41.6788	−1.90435	−0.952177	+	0.305546i	$0.901161\pi$
$480$	0	0
$481$	6.76264	0.308350
$482$	−41.0089	−1.86791
$483$	0	0
$484$	−27.0319	−1.22872
$485$	14.8898	0.676112
$486$	0	0
$487$	4.02801	0.182527	0.0912634	−	0.995827i	$-0.470909\pi$
$487$	4.02801	0.182527	0.0912634	+	0.995827i	$0.470909\pi$
$488$	−1.03119	−0.0466797
$489$	0	0
$490$	−34.3845	−1.55334
$491$	−38.6214	−1.74296	−0.871479	−	0.490433i	$-0.836838\pi$
$491$	−38.6214	−1.74296	−0.871479	+	0.490433i	$0.836838\pi$
$492$	0	0
$493$	−5.17438	−0.233042
$494$	19.9354	0.896935
$495$	0	0
$496$	24.3660	1.09406
$497$	−9.33028	−0.418520
$498$	0	0
$499$	−4.07186	−0.182281	−0.0911407	−	0.995838i	$-0.529051\pi$
$499$	−4.07186	−0.182281	−0.0911407	+	0.995838i	$0.529051\pi$
$500$	18.6387	0.833547
$501$	0	0
$502$	11.5900	0.517289
$503$	−3.42594	−0.152755	−0.0763775	−	0.997079i	$-0.524335\pi$
$503$	−3.42594	−0.152755	−0.0763775	+	0.997079i	$0.524335\pi$
$504$	0	0
$505$	27.2041	1.21056
$506$	1.45069	0.0644908
$507$	0	0
$508$	12.9645	0.575207
$509$	−12.2634	−0.543566	−0.271783	−	0.962359i	$-0.587613\pi$
$509$	−12.2634	−0.543566	−0.271783	+	0.962359i	$0.587613\pi$
$510$	0	0
$511$	−14.8201	−0.655604
$512$	−28.1241	−1.24292
$513$	0	0
$514$	24.4791	1.07973
$515$	26.4373	1.16497
$516$	0	0
$517$	0.784671	0.0345098
$518$	9.19263	0.403901
$519$	0	0
$520$	−4.12799	−0.181024
$521$	14.0823	0.616959	0.308479	−	0.951231i	$-0.400180\pi$
$521$	14.0823	0.616959	0.308479	+	0.951231i	$0.400180\pi$
$522$	0	0
$523$	9.77912	0.427611	0.213806	−	0.976876i	$-0.431414\pi$
$523$	9.77912	0.427611	0.213806	+	0.976876i	$0.431414\pi$
$524$	−17.9417	−0.783788
$525$	0	0
$526$	−13.7113	−0.597843
$527$	−10.1883	−0.443809
$528$	0	0
$529$	−18.3225	−0.796632
$530$	−61.8566	−2.68688
$531$	0	0
$532$	15.0013	0.650389
$533$	−8.84979	−0.383327
$534$	0	0
$535$	−13.8738	−0.599815
$536$	−0.870404	−0.0375957
$537$	0	0
$538$	29.2912	1.26283
$539$	1.91871	0.0826445
$540$	0	0
$541$	40.9454	1.76038	0.880189	−	0.474623i	$-0.157416\pi$
$541$	40.9454	1.76038	0.880189	+	0.474623i	$0.157416\pi$
$542$	−4.11759	−0.176866
$543$	0	0
$544$	9.37238	0.401837
$545$	19.6152	0.840223
$546$	0	0
$547$	1.11028	0.0474720	0.0237360	−	0.999718i	$-0.492444\pi$
$547$	1.11028	0.0474720	0.0237360	+	0.999718i	$0.492444\pi$
$548$	27.8993	1.19180
$549$	0	0
$550$	1.47502	0.0628949
$551$	−27.4021	−1.16737
$552$	0	0
$553$	−10.9035	−0.463664
$554$	−26.4000	−1.12163
$555$	0	0
$556$	23.2637	0.986599
$557$	35.0403	1.48470	0.742352	−	0.670010i	$-0.233710\pi$
$557$	35.0403	1.48470	0.742352	+	0.670010i	$0.233710\pi$
$558$	0	0
$559$	−8.46493	−0.358028
$560$	7.32916	0.309713
$561$	0	0
$562$	−20.6495	−0.871046
$563$	−38.7712	−1.63401	−0.817005	−	0.576630i	$-0.804367\pi$
$563$	−38.7712	−1.63401	−0.817005	+	0.576630i	$0.804367\pi$
$564$	0	0
$565$	−27.8389	−1.17119
$566$	−56.2413	−2.36400
$567$	0	0
$568$	9.75095	0.409141
$569$	−33.9683	−1.42403	−0.712013	−	0.702166i	$-0.752216\pi$
$569$	−33.9683	−1.42403	−0.712013	+	0.702166i	$0.752216\pi$
$570$	0	0
$571$	10.0889	0.422206	0.211103	−	0.977464i	$-0.432294\pi$
$571$	10.0889	0.422206	0.211103	+	0.977464i	$0.432294\pi$
$572$	1.18995	0.0497542
$573$	0	0
$574$	−12.0297	−0.502112
$575$	4.75590	0.198335
$576$	0	0
$577$	−12.1323	−0.505074	−0.252537	−	0.967587i	$-0.581265\pi$
$577$	−12.1323	−0.505074	−0.252537	+	0.967587i	$0.581265\pi$
$578$	33.0619	1.37519
$579$	0	0
$580$	29.3117	1.21710
$581$	4.55531	0.188986
$582$	0	0
$583$	3.45168	0.142954
$584$	15.4883	0.640911
$585$	0	0
$586$	−25.9444	−1.07175
$587$	31.7492	1.31043	0.655215	−	0.755443i	$-0.272578\pi$
$587$	31.7492	1.31043	0.655215	+	0.755443i	$0.272578\pi$
$588$	0	0
$589$	−53.9544	−2.22315
$590$	−9.79197	−0.403129
$591$	0	0
$592$	12.5483	0.515731
$593$	13.4906	0.553993	0.276996	−	0.960871i	$-0.410661\pi$
$593$	13.4906	0.553993	0.276996	+	0.960871i	$0.410661\pi$
$594$	0	0
$595$	−3.06459	−0.125636
$596$	47.2488	1.93539
$597$	0	0
$598$	6.93079	0.283421
$599$	−42.4304	−1.73366	−0.866829	−	0.498606i	$-0.833846\pi$
$599$	−42.4304	−1.73366	−0.866829	+	0.498606i	$0.833846\pi$
$600$	0	0
$601$	−19.8047	−0.807852	−0.403926	−	0.914792i	$-0.632355\pi$
$601$	−19.8047	−0.807852	−0.403926	+	0.914792i	$0.632355\pi$
$602$	−11.5066	−0.468974
$603$	0	0
$604$	9.95509	0.405067
$605$	29.2446	1.18896
$606$	0	0
$607$	−35.9990	−1.46116	−0.730578	−	0.682830i	$-0.760749\pi$
$607$	−35.9990	−1.46116	−0.730578	+	0.682830i	$0.760749\pi$
$608$	49.6335	2.01291
$609$	0	0
$610$	5.76303	0.233338
$611$	3.74884	0.151662
$612$	0	0
$613$	26.4628	1.06882	0.534411	−	0.845225i	$-0.320533\pi$
$613$	26.4628	1.06882	0.534411	+	0.845225i	$0.320533\pi$
$614$	56.0354	2.26141
$615$	0	0
$616$	0.313118	0.0126159
$617$	49.1167	1.97737	0.988683	−	0.150022i	$-0.0479343\pi$
$617$	49.1167	1.97737	0.988683	+	0.150022i	$0.0479343\pi$
$618$	0	0
$619$	−24.2062	−0.972930	−0.486465	−	0.873700i	$-0.661714\pi$
$619$	−24.2062	−0.972930	−0.486465	+	0.873700i	$0.661714\pi$
$620$	57.7145	2.31787
$621$	0	0
$622$	37.3785	1.49874
$623$	−15.0707	−0.603794
$624$	0	0
$625$	−31.1594	−1.24638
$626$	−20.4195	−0.816127
$627$	0	0
$628$	18.0449	0.720072
$629$	−5.24690	−0.209208
$630$	0	0
$631$	17.6968	0.704500	0.352250	−	0.935906i	$-0.385417\pi$
$631$	17.6968	0.704500	0.352250	+	0.935906i	$0.385417\pi$
$632$	11.3951	0.453274
$633$	0	0
$634$	7.85371	0.311911
$635$	−14.0257	−0.556593
$636$	0	0
$637$	9.16680	0.363202
$638$	−2.95465	−0.116976
$639$	0	0
$640$	−21.1835	−0.837351
$641$	38.0504	1.50290	0.751450	−	0.659790i	$-0.229355\pi$
$641$	38.0504	1.50290	0.751450	+	0.659790i	$0.229355\pi$
$642$	0	0
$643$	46.9425	1.85123	0.925616	−	0.378465i	$-0.123548\pi$
$643$	46.9425	1.85123	0.925616	+	0.378465i	$0.123548\pi$
$644$	5.21540	0.205515
$645$	0	0
$646$	−15.4672	−0.608548
$647$	−28.2333	−1.10997	−0.554983	−	0.831862i	$-0.687275\pi$
$647$	−28.2333	−1.10997	−0.554983	+	0.831862i	$0.687275\pi$
$648$	0	0
$649$	0.546406	0.0214483
$650$	7.04703	0.276407
$651$	0	0
$652$	30.8751	1.20916
$653$	35.3072	1.38168	0.690839	−	0.723009i	$-0.257241\pi$
$653$	35.3072	1.38168	0.690839	+	0.723009i	$0.257241\pi$
$654$	0	0
$655$	19.4103	0.758425
$656$	−16.4210	−0.641134
$657$	0	0
$658$	5.09590	0.198659
$659$	41.6937	1.62416	0.812078	−	0.583548i	$-0.198336\pi$
$659$	41.6937	1.62416	0.812078	+	0.583548i	$0.198336\pi$
$660$	0	0
$661$	1.27391	0.0495495	0.0247747	−	0.999693i	$-0.492113\pi$
$661$	1.27391	0.0495495	0.0247747	+	0.999693i	$0.492113\pi$
$662$	2.99036	0.116223
$663$	0	0
$664$	−4.76070	−0.184751
$665$	−16.2292	−0.629343
$666$	0	0
$667$	−9.52668	−0.368875
$668$	−5.78329	−0.223762
$669$	0	0
$670$	4.86444	0.187930
$671$	−0.321585	−0.0124147
$672$	0	0
$673$	35.6328	1.37354	0.686771	−	0.726874i	$-0.259028\pi$
$673$	35.6328	1.37354	0.686771	+	0.726874i	$0.259028\pi$
$674$	27.4951	1.05907
$675$	0	0
$676$	−26.5561	−1.02139
$677$	18.0019	0.691869	0.345934	−	0.938259i	$-0.387562\pi$
$677$	18.0019	0.691869	0.345934	+	0.938259i	$0.387562\pi$
$678$	0	0
$679$	−5.39595	−0.207077
$680$	3.20276	0.122820
$681$	0	0
$682$	−5.81767	−0.222770
$683$	39.7614	1.52143	0.760715	−	0.649087i	$-0.224849\pi$
$683$	39.7614	1.52143	0.760715	+	0.649087i	$0.224849\pi$
$684$	0	0
$685$	−30.1830	−1.15323
$686$	26.8671	1.02579
$687$	0	0
$688$	−15.7069	−0.598821
$689$	16.4908	0.628248
$690$	0	0
$691$	−16.8407	−0.640651	−0.320325	−	0.947308i	$-0.603792\pi$
$691$	−16.8407	−0.640651	−0.320325	+	0.947308i	$0.603792\pi$
$692$	−8.88436	−0.337733
$693$	0	0
$694$	10.1608	0.385697
$695$	−25.1679	−0.954673
$696$	0	0
$697$	6.86625	0.260078
$698$	47.7882	1.80881
$699$	0	0
$700$	5.30287	0.200430
$701$	−8.96921	−0.338762	−0.169381	−	0.985551i	$-0.554177\pi$
$701$	−8.96921	−0.338762	−0.169381	+	0.985551i	$0.554177\pi$
$702$	0	0
$703$	−27.7861	−1.04797
$704$	3.57120	0.134595
$705$	0	0
$706$	62.8776	2.36643
$707$	−9.85853	−0.370768
$708$	0	0
$709$	18.1416	0.681323	0.340662	−	0.940186i	$-0.389349\pi$
$709$	18.1416	0.681323	0.340662	+	0.940186i	$0.389349\pi$
$710$	−54.4954	−2.04518
$711$	0	0
$712$	15.7502	0.590263
$713$	−18.7580	−0.702491
$714$	0	0
$715$	−1.28735	−0.0481441
$716$	−49.5912	−1.85331
$717$	0	0
$718$	−28.4047	−1.06005
$719$	31.5720	1.17744	0.588718	−	0.808339i	$-0.299633\pi$
$719$	31.5720	1.17744	0.588718	+	0.808339i	$0.299633\pi$
$720$	0	0
$721$	−9.58068	−0.356803
$722$	−41.6941	−1.55169
$723$	0	0
$724$	24.1370	0.897046
$725$	−9.68646	−0.359746
$726$	0	0
$727$	38.4093	1.42452	0.712260	−	0.701915i	$-0.247672\pi$
$727$	38.4093	1.42452	0.712260	+	0.701915i	$0.247672\pi$
$728$	1.49595	0.0554436
$729$	0	0
$730$	−86.5599	−3.20373
$731$	6.56765	0.242913
$732$	0	0
$733$	−30.3303	−1.12028	−0.560138	−	0.828400i	$-0.689252\pi$
$733$	−30.3303	−1.12028	−0.560138	+	0.828400i	$0.689252\pi$
$734$	16.8271	0.621100
$735$	0	0
$736$	17.2557	0.636055
$737$	−0.271443	−0.00999872
$738$	0	0
$739$	10.0025	0.367949	0.183975	−	0.982931i	$-0.441104\pi$
$739$	10.0025	0.367949	0.183975	+	0.982931i	$0.441104\pi$
$740$	29.7225	1.09262
$741$	0	0
$742$	22.4163	0.822929
$743$	36.2738	1.33076	0.665378	−	0.746506i	$-0.268270\pi$
$743$	36.2738	1.33076	0.665378	+	0.746506i	$0.268270\pi$
$744$	0	0
$745$	−51.1164	−1.87276
$746$	24.1729	0.885033
$747$	0	0
$748$	−0.923239	−0.0337569
$749$	5.02774	0.183710
$750$	0	0
$751$	−14.5544	−0.531096	−0.265548	−	0.964098i	$-0.585553\pi$
$751$	−14.5544	−0.531096	−0.265548	+	0.964098i	$0.585553\pi$
$752$	6.95609	0.253663
$753$	0	0
$754$	−14.1161	−0.514078
$755$	−10.7700	−0.391959
$756$	0	0
$757$	−45.5754	−1.65646	−0.828232	−	0.560385i	$-0.810653\pi$
$757$	−45.5754	−1.65646	−0.828232	+	0.560385i	$0.810653\pi$
$758$	51.1598	1.85821
$759$	0	0
$760$	16.9610	0.615239
$761$	22.2456	0.806401	0.403200	−	0.915112i	$-0.367898\pi$
$761$	22.2456	0.806401	0.403200	+	0.915112i	$0.367898\pi$
$762$	0	0
$763$	−7.10839	−0.257341
$764$	−44.0378	−1.59323
$765$	0	0
$766$	−19.9879	−0.722191
$767$	2.61051	0.0942600
$768$	0	0
$769$	13.6649	0.492770	0.246385	−	0.969172i	$-0.420757\pi$
$769$	13.6649	0.492770	0.246385	+	0.969172i	$0.420757\pi$
$770$	−1.74993	−0.0630630
$771$	0	0
$772$	26.2501	0.944763
$773$	−20.6540	−0.742871	−0.371436	−	0.928459i	$-0.621134\pi$
$773$	−20.6540	−0.742871	−0.371436	+	0.928459i	$0.621134\pi$
$774$	0	0
$775$	−19.0726	−0.685106
$776$	5.63923	0.202437
$777$	0	0
$778$	5.39253	0.193332
$779$	36.3617	1.30279
$780$	0	0
$781$	3.04092	0.108813
$782$	−5.37736	−0.192294
$783$	0	0
$784$	17.0093	0.607474
$785$	−19.5220	−0.696770
$786$	0	0
$787$	−24.7034	−0.880580	−0.440290	−	0.897856i	$-0.645124\pi$
$787$	−24.7034	−0.880580	−0.440290	+	0.897856i	$0.645124\pi$
$788$	35.1153	1.25093
$789$	0	0
$790$	−63.6842	−2.26578
$791$	10.0886	0.358708
$792$	0	0
$793$	−1.53640	−0.0545593
$794$	−7.77895	−0.276065
$795$	0	0
$796$	18.7004	0.662817
$797$	−30.5658	−1.08270	−0.541348	−	0.840799i	$-0.682086\pi$
$797$	−30.5658	−1.08270	−0.541348	+	0.840799i	$0.682086\pi$
$798$	0	0
$799$	−2.90860	−0.102899
$800$	17.5451	0.620314
$801$	0	0
$802$	−34.1808	−1.20696
$803$	4.83017	0.170453
$804$	0	0
$805$	−5.64230	−0.198865
$806$	−27.7945	−0.979019
$807$	0	0
$808$	10.3030	0.362459
$809$	46.8599	1.64751	0.823753	−	0.566949i	$-0.191876\pi$
$809$	46.8599	1.64751	0.823753	+	0.566949i	$0.191876\pi$
$810$	0	0
$811$	10.9984	0.386206	0.193103	−	0.981178i	$-0.438145\pi$
$811$	10.9984	0.386206	0.193103	+	0.981178i	$0.438145\pi$
$812$	−10.6223	−0.372771
$813$	0	0
$814$	−2.99606	−0.105012
$815$	−33.4023	−1.17003
$816$	0	0
$817$	34.7805	1.21681
$818$	−19.4348	−0.679521
$819$	0	0
$820$	−38.8958	−1.35830
$821$	−35.9833	−1.25583	−0.627913	−	0.778283i	$-0.716091\pi$
$821$	−35.9833	−1.25583	−0.627913	+	0.778283i	$0.716091\pi$
$822$	0	0
$823$	−48.9866	−1.70757	−0.853783	−	0.520628i	$-0.825698\pi$
$823$	−48.9866	−1.70757	−0.853783	+	0.520628i	$0.825698\pi$
$824$	10.0126	0.348807
$825$	0	0
$826$	3.54853	0.123469
$827$	−15.6107	−0.542836	−0.271418	−	0.962462i	$-0.587493\pi$
$827$	−15.6107	−0.542836	−0.271418	+	0.962462i	$0.587493\pi$
$828$	0	0
$829$	11.4708	0.398398	0.199199	−	0.979959i	$-0.436166\pi$
$829$	11.4708	0.398398	0.199199	+	0.979959i	$0.436166\pi$
$830$	26.6062	0.923515
$831$	0	0
$832$	17.0618	0.591510
$833$	−7.11221	−0.246423
$834$	0	0
$835$	6.25668	0.216521
$836$	−4.88922	−0.169097
$837$	0	0
$838$	14.7716	0.510278
$839$	−0.675324	−0.0233148	−0.0116574	−	0.999932i	$-0.503711\pi$
$839$	−0.675324	−0.0233148	−0.0116574	+	0.999932i	$0.503711\pi$
$840$	0	0
$841$	−9.59676	−0.330923
$842$	−65.2145	−2.24744
$843$	0	0
$844$	12.9303	0.445079
$845$	28.7298	0.988337
$846$	0	0
$847$	−10.5980	−0.364152
$848$	30.5991	1.05078
$849$	0	0
$850$	−5.46755	−0.187535
$851$	−9.66020	−0.331147
$852$	0	0
$853$	39.3558	1.34752	0.673758	−	0.738952i	$-0.264679\pi$
$853$	39.3558	1.34752	0.673758	+	0.738952i	$0.264679\pi$
$854$	−2.08848	−0.0714662
$855$	0	0
$856$	−5.25443	−0.179593
$857$	32.6621	1.11572	0.557858	−	0.829936i	$-0.311623\pi$
$857$	32.6621	1.11572	0.557858	+	0.829936i	$0.311623\pi$
$858$	0	0
$859$	−33.1966	−1.13265	−0.566327	−	0.824180i	$-0.691636\pi$
$859$	−33.1966	−1.13265	−0.566327	+	0.824180i	$0.691636\pi$
$860$	−37.2043	−1.26866
$861$	0	0
$862$	−59.0477	−2.01117
$863$	22.6796	0.772024	0.386012	−	0.922494i	$-0.373852\pi$
$863$	22.6796	0.772024	0.386012	+	0.922494i	$0.373852\pi$
$864$	0	0
$865$	9.61158	0.326804
$866$	−40.5768	−1.37886
$867$	0	0
$868$	−20.9153	−0.709910
$869$	3.55367	0.120550
$870$	0	0
$871$	−1.29684	−0.0439419
$872$	7.42888	0.251574
$873$	0	0
$874$	−28.4770	−0.963250
$875$	7.30739	0.247035
$876$	0	0
$877$	9.24376	0.312140	0.156070	−	0.987746i	$-0.450118\pi$
$877$	9.24376	0.312140	0.156070	+	0.987746i	$0.450118\pi$
$878$	50.2676	1.69645
$879$	0	0
$880$	−2.38871	−0.0805236
$881$	7.78755	0.262369	0.131185	−	0.991358i	$-0.458122\pi$
$881$	7.78755	0.262369	0.131185	+	0.991358i	$0.458122\pi$
$882$	0	0
$883$	−32.4618	−1.09243	−0.546213	−	0.837647i	$-0.683931\pi$
$883$	−32.4618	−1.09243	−0.546213	+	0.837647i	$0.683931\pi$
$884$	−4.41086	−0.148353
$885$	0	0
$886$	−49.4406	−1.66099
$887$	−33.8477	−1.13649	−0.568247	−	0.822858i	$-0.692378\pi$
$887$	−33.8477	−1.13649	−0.568247	+	0.822858i	$0.692378\pi$
$888$	0	0
$889$	5.08280	0.170472
$890$	−88.0234	−2.95055
$891$	0	0
$892$	43.8822	1.46929
$893$	−15.4031	−0.515446
$894$	0	0
$895$	53.6504	1.79334
$896$	7.67673	0.256461
$897$	0	0
$898$	−10.2001	−0.340383
$899$	38.2048	1.27420
$900$	0	0
$901$	−12.7946	−0.426250
$902$	3.92073	0.130546
$903$	0	0
$904$	−10.5434	−0.350670
$905$	−26.1128	−0.868018
$906$	0	0
$907$	−11.0598	−0.367236	−0.183618	−	0.982998i	$-0.558781\pi$
$907$	−11.0598	−0.367236	−0.183618	+	0.982998i	$0.558781\pi$
$908$	−39.1160	−1.29811
$909$	0	0
$910$	−8.36045	−0.277146
$911$	12.5452	0.415640	0.207820	−	0.978167i	$-0.433363\pi$
$911$	12.5452	0.415640	0.207820	+	0.978167i	$0.433363\pi$
$912$	0	0
$913$	−1.48466	−0.0491352
$914$	10.3579	0.342609
$915$	0	0
$916$	−4.38198	−0.144785
$917$	−7.03415	−0.232288
$918$	0	0
$919$	−5.92909	−0.195583	−0.0977913	−	0.995207i	$-0.531178\pi$
$919$	−5.92909	−0.195583	−0.0977913	+	0.995207i	$0.531178\pi$
$920$	5.89669	0.194408
$921$	0	0
$922$	−59.2452	−1.95114
$923$	14.5283	0.478204
$924$	0	0
$925$	−9.82222	−0.322953
$926$	58.1549	1.91109
$927$	0	0
$928$	−35.1452	−1.15370
$929$	−10.0328	−0.329165	−0.164582	−	0.986363i	$-0.552628\pi$
$929$	−10.0328	−0.329165	−0.164582	+	0.986363i	$0.552628\pi$
$930$	0	0
$931$	−37.6643	−1.23440
$932$	−34.4693	−1.12908
$933$	0	0
$934$	44.9708	1.47149
$935$	0.998810	0.0326646
$936$	0	0
$937$	−23.5341	−0.768826	−0.384413	−	0.923161i	$-0.625596\pi$
$937$	−23.5341	−0.768826	−0.384413	+	0.923161i	$0.625596\pi$
$938$	−1.76283	−0.0575586
$939$	0	0
$940$	16.4766	0.537407
$941$	−29.2996	−0.955138	−0.477569	−	0.878594i	$-0.658482\pi$
$941$	−29.2996	−0.955138	−0.477569	+	0.878594i	$0.658482\pi$
$942$	0	0
$943$	12.6416	0.411668
$944$	4.84388	0.157655
$945$	0	0
$946$	3.75023	0.121930
$947$	53.6619	1.74378	0.871889	−	0.489704i	$-0.162895\pi$
$947$	53.6619	1.74378	0.871889	+	0.489704i	$0.162895\pi$
$948$	0	0
$949$	23.0766	0.749098
$950$	−28.9546	−0.939412
$951$	0	0
$952$	−1.16066	−0.0376171
$953$	−4.89656	−0.158615	−0.0793076	−	0.996850i	$-0.525271\pi$
$953$	−4.89656	−0.158615	−0.0793076	+	0.996850i	$0.525271\pi$
$954$	0	0
$955$	47.6425	1.54167
$956$	−49.1869	−1.59082
$957$	0	0
$958$	88.2184	2.85021
$959$	10.9381	0.353209
$960$	0	0
$961$	44.2249	1.42661
$962$	−14.3140	−0.461500
$963$	0	0
$964$	48.0510	1.54762
$965$	−28.3988	−0.914190
$966$	0	0
$967$	16.9931	0.546462	0.273231	−	0.961948i	$-0.411908\pi$
$967$	16.9931	0.546462	0.273231	+	0.961948i	$0.411908\pi$
$968$	11.0758	0.355991
$969$	0	0
$970$	−31.5161	−1.01192
$971$	−2.68374	−0.0861253	−0.0430627	−	0.999072i	$-0.513712\pi$
$971$	−2.68374	−0.0861253	−0.0430627	+	0.999072i	$0.513712\pi$
$972$	0	0
$973$	9.12064	0.292394
$974$	−8.52579	−0.273184
$975$	0	0
$976$	−2.85085	−0.0912533
$977$	10.9930	0.351698	0.175849	−	0.984417i	$-0.443733\pi$
$977$	10.9930	0.351698	0.175849	+	0.984417i	$0.443733\pi$
$978$	0	0
$979$	4.91183	0.156983
$980$	40.2891	1.28699
$981$	0	0
$982$	81.7468	2.60865
$983$	−47.7756	−1.52381	−0.761903	−	0.647691i	$-0.775735\pi$
$983$	−47.7756	−1.52381	−0.761903	+	0.647691i	$0.775735\pi$
$984$	0	0
$985$	−37.9897	−1.21045
$986$	10.9522	0.348789
$987$	0	0
$988$	−23.3587	−0.743139
$989$	12.0919	0.384499
$990$	0	0
$991$	55.5006	1.76303	0.881517	−	0.472152i	$-0.156523\pi$
$991$	55.5006	1.76303	0.881517	+	0.472152i	$0.156523\pi$
$992$	−69.2005	−2.19712
$993$	0	0
$994$	19.7487	0.626390
$995$	−20.2311	−0.641368
$996$	0	0
$997$	−45.0467	−1.42664	−0.713322	−	0.700837i	$-0.752810\pi$
$997$	−45.0467	−1.42664	−0.713322	+	0.700837i	$0.752810\pi$
$998$	8.61859	0.272817
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.a.a.1.2		6
3.2	odd	2		729.2.a.d.1.5		6
9.2	odd	6		729.2.c.b.244.2		12
9.4	even	3		729.2.c.e.487.5		12
9.5	odd	6		729.2.c.b.487.2		12
9.7	even	3		729.2.c.e.244.5		12
27.2	odd	18		243.2.e.b.190.1		12
27.4	even	9		27.2.e.a.16.1	✓	12
27.5	odd	18		243.2.e.a.217.2		12
27.7	even	9		27.2.e.a.22.1	yes	12
27.11	odd	18		243.2.e.a.28.2		12
27.13	even	9		243.2.e.c.55.2		12
27.14	odd	18		243.2.e.b.55.1		12
27.16	even	9		243.2.e.d.28.1		12
27.20	odd	18		81.2.e.a.37.2		12
27.22	even	9		243.2.e.d.217.1		12
27.23	odd	18		81.2.e.a.46.2		12
27.25	even	9		243.2.e.c.190.2		12
108.7	odd	18		432.2.u.c.49.2		12
108.31	odd	18		432.2.u.c.97.2		12
135.4	even	18		675.2.l.c.151.2		12
135.7	odd	36		675.2.u.b.49.1		24
135.34	even	18		675.2.l.c.76.2		12
135.58	odd	36		675.2.u.b.124.1		24
135.88	odd	36		675.2.u.b.49.4		24
135.112	odd	36		675.2.u.b.124.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.16.1	✓	12	27.4	even	9
27.2.e.a.22.1	yes	12	27.7	even	9
81.2.e.a.37.2		12	27.20	odd	18
81.2.e.a.46.2		12	27.23	odd	18
243.2.e.a.28.2		12	27.11	odd	18
243.2.e.a.217.2		12	27.5	odd	18
243.2.e.b.55.1		12	27.14	odd	18
243.2.e.b.190.1		12	27.2	odd	18
243.2.e.c.55.2		12	27.13	even	9
243.2.e.c.190.2		12	27.25	even	9
243.2.e.d.28.1		12	27.16	even	9
243.2.e.d.217.1		12	27.22	even	9
432.2.u.c.49.2		12	108.7	odd	18
432.2.u.c.97.2		12	108.31	odd	18
675.2.l.c.76.2		12	135.34	even	18
675.2.l.c.151.2		12	135.4	even	18
675.2.u.b.49.1		24	135.7	odd	36
675.2.u.b.49.4		24	135.88	odd	36
675.2.u.b.124.1		24	135.58	odd	36
675.2.u.b.124.4		24	135.112	odd	36
729.2.a.a.1.2		6	1.1	even	1	trivial
729.2.a.d.1.5		6	3.2	odd	2
729.2.c.b.244.2		12	9.2	odd	6
729.2.c.b.487.2		12	9.5	odd	6
729.2.c.e.244.5		12	9.7	even	3
729.2.c.e.487.5		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)

\(f(q)\)	\(=\)	\(q-2.11662 q^{2} +2.48009 q^{4} -2.68310 q^{5} +0.972333 q^{7} -1.01617 q^{8} +O(q^{10})\)	\(q-2.11662 q^{2} +2.48009 q^{4} -2.68310 q^{5} +0.972333 q^{7} -1.01617 q^{8} +5.67911 q^{10} -0.316902 q^{11} -1.51403 q^{13} -2.05806 q^{14} -2.80933 q^{16} +1.17468 q^{17} +6.22080 q^{19} -6.65433 q^{20} +0.670762 q^{22} +2.16274 q^{23} +2.19901 q^{25} +3.20463 q^{26} +2.41147 q^{28} -4.40491 q^{29} -8.67323 q^{31} +7.97863 q^{32} -2.48636 q^{34} -2.60886 q^{35} -4.46665 q^{37} -13.1671 q^{38} +2.72649 q^{40} +5.84518 q^{41} +5.59099 q^{43} -0.785946 q^{44} -4.57771 q^{46} -2.47607 q^{47} -6.05457 q^{49} -4.65448 q^{50} -3.75493 q^{52} -10.8920 q^{53} +0.850279 q^{55} -0.988058 q^{56} +9.32353 q^{58} -1.72421 q^{59} +1.01478 q^{61} +18.3580 q^{62} -11.2691 q^{64} +4.06229 q^{65} +0.856551 q^{67} +2.91332 q^{68} +5.52198 q^{70} -9.59577 q^{71} -15.2418 q^{73} +9.45420 q^{74} +15.4282 q^{76} -0.308134 q^{77} -11.2138 q^{79} +7.53771 q^{80} -12.3721 q^{82} +4.68493 q^{83} -3.15179 q^{85} -11.8340 q^{86} +0.322027 q^{88} -15.4995 q^{89} -1.47214 q^{91} +5.36380 q^{92} +5.24090 q^{94} -16.6910 q^{95} -5.54949 q^{97} +12.8152 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10}) \) 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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