LMFDB - Embedded newform 729.2.a.d.1.5

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:	$0$
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			1.5
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)

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

$2$	2.11662	1.49668	0.748339	−	0.663316i	$-0.230852\pi$
$2$	2.11662	1.49668	0.748339	+	0.663316i	$0.230852\pi$
$3$	0	0
$3$	0	0
$4$	2.48009	1.24005
$5$	2.68310	1.19992	0.599959	−	0.800031i	$-0.295184\pi$
$5$	2.68310	1.19992	0.599959	+	0.800031i	$0.295184\pi$
$6$	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	$-0.484787\pi$
$11$	0.316902	0.0955496	0.0477748	+	0.998858i	$0.484787\pi$
$12$	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	$-0.545498\pi$
$17$	−1.17468	−0.284903	−0.142451	+	0.989802i	$0.545498\pi$
$18$	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	$-0.572395\pi$
$23$	−2.16274	−0.450963	−0.225481	+	0.974247i	$0.572395\pi$
$24$	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	$-0.365883\pi$
$29$	4.40491	0.817971	0.408986	+	0.912541i	$0.365883\pi$
$30$	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	$-0.650873\pi$
$41$	−5.84518	−0.912864	−0.456432	+	0.889758i	$0.650873\pi$
$42$	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	$-0.442201\pi$
$47$	2.47607	0.361172	0.180586	+	0.983559i	$0.442201\pi$
$48$	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	$-0.230984\pi$
$53$	10.8920	1.49613	0.748063	+	0.663628i	$0.230984\pi$
$54$	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	$-0.464199\pi$
$59$	1.72421	0.224473	0.112237	+	0.993682i	$0.464199\pi$
$60$	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	$-0.307174\pi$
$71$	9.59577	1.13881	0.569404	+	0.822058i	$0.307174\pi$
$72$	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	$-0.582773\pi$
$83$	−4.68493	−0.514238	−0.257119	+	0.966380i	$0.582773\pi$
$84$	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	$-0.193153\pi$
$89$	15.4995	1.64295	0.821473	+	0.570248i	$0.193153\pi$
$90$	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	$-0.331700\pi$
$101$	10.1390	1.00887	0.504436	+	0.863449i	$0.331700\pi$
$102$	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	$-0.580411\pi$
$107$	−5.17080	−0.499880	−0.249940	+	0.968261i	$0.580411\pi$
$108$	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	$-0.662284\pi$
$113$	−10.3756	−0.976058	−0.488029	+	0.872827i	$0.662284\pi$
$114$	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	$-0.397649\pi$
$131$	7.23430	0.632064	0.316032	+	0.948749i	$0.397649\pi$
$132$	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	$-0.659562\pi$
$137$	−11.2493	−0.961094	−0.480547	+	0.876969i	$0.659562\pi$
$138$	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	$-0.784969\pi$
$149$	−19.0512	−1.56074	−0.780369	+	0.625319i	$0.784969\pi$
$150$	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	$-0.471242\pi$
$167$	2.33189	0.180447	0.0902234	+	0.995922i	$0.471242\pi$
$168$	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	$-0.456518\pi$
$173$	3.58227	0.272355	0.136177	+	0.990684i	$0.456518\pi$
$174$	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	$-0.231362\pi$
$179$	19.9957	1.49455	0.747275	+	0.664515i	$0.231362\pi$
$180$	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	$-0.277935\pi$
$191$	17.7565	1.28482	0.642409	+	0.766362i	$0.277935\pi$
$192$	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	$-0.668282\pi$
$197$	−14.1589	−1.00878	−0.504390	+	0.863476i	$0.668282\pi$
$198$	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	$-0.324659\pi$
$227$	15.7720	1.04682	0.523412	+	0.852080i	$0.324659\pi$
$228$	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	$-0.349547\pi$
$233$	13.8984	0.910514	0.455257	+	0.890360i	$0.349547\pi$
$234$	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	$-0.278339\pi$
$239$	19.8327	1.28287	0.641435	+	0.767177i	$0.278339\pi$
$240$	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	$-0.444715\pi$
$251$	5.47572	0.345625	0.172812	+	0.984955i	$0.444715\pi$
$252$	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	$-0.382535\pi$
$257$	11.5652	0.721415	0.360708	+	0.932679i	$0.382535\pi$
$258$	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	$-0.564004\pi$
$263$	−6.47794	−0.399447	−0.199723	+	0.979852i	$0.564004\pi$
$264$	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	$-0.361371\pi$
$269$	13.8387	0.843758	0.421879	+	0.906652i	$0.361371\pi$
$270$	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	$-0.593986\pi$
$281$	−9.75587	−0.581986	−0.290993	+	0.956725i	$0.593986\pi$
$282$	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	$-0.616556\pi$
$293$	−12.2575	−0.716089	−0.358044	+	0.933705i	$0.616556\pi$
$294$	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	$-0.333080\pi$
$311$	17.6595	1.00138	0.500689	+	0.865627i	$0.333080\pi$
$312$	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	$-0.466771\pi$
$317$	3.71049	0.208402	0.104201	+	0.994556i	$0.466771\pi$
$318$	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	$-0.458871\pi$
$347$	4.80046	0.257702	0.128851	+	0.991664i	$0.458871\pi$
$348$	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	$-0.209789\pi$
$353$	29.7066	1.58112	0.790560	+	0.612384i	$0.209789\pi$
$354$	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	$-0.615225\pi$
$359$	−13.4198	−0.708271	−0.354136	+	0.935194i	$0.615225\pi$
$360$	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	$-0.577562\pi$
$383$	−9.44328	−0.482529	−0.241265	+	0.970459i	$0.577562\pi$
$384$	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	$-0.479427\pi$
$389$	2.54771	0.129174	0.0645869	+	0.997912i	$0.479427\pi$
$390$	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	$-0.632107\pi$
$401$	−16.1487	−0.806429	−0.403214	+	0.915106i	$0.632107\pi$
$402$	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	$-0.445471\pi$
$419$	6.97888	0.340940	0.170470	+	0.985363i	$0.445471\pi$
$420$	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	$-0.734513\pi$
$431$	−27.8971	−1.34376	−0.671879	+	0.740661i	$0.734513\pi$
$432$	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	$-0.687241\pi$
$443$	−23.3583	−1.10978	−0.554892	+	0.831922i	$0.687241\pi$
$444$	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	$-0.536274\pi$
$449$	−4.81906	−0.227426	−0.113713	+	0.993514i	$0.536274\pi$
$450$	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	$-0.725995\pi$
$461$	−27.9905	−1.30365	−0.651823	+	0.758371i	$0.725995\pi$
$462$	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	$-0.336418\pi$
$467$	21.2465	0.983170	0.491585	+	0.870830i	$0.336418\pi$
$468$	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	$-0.0988389\pi$
$479$	41.6788	1.90435	0.952177	+	0.305546i	$0.0988389\pi$
$480$	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	$-0.163162\pi$
$491$	38.6214	1.74296	0.871479	+	0.490433i	$0.163162\pi$
$492$	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	$-0.475665\pi$
$503$	3.42594	0.152755	0.0763775	+	0.997079i	$0.475665\pi$
$504$	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	$-0.412387\pi$
$509$	12.2634	0.543566	0.271783	+	0.962359i	$0.412387\pi$
$510$	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	$-0.599820\pi$
$521$	−14.0823	−0.616959	−0.308479	+	0.951231i	$0.599820\pi$
$522$	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	$-0.766290\pi$
$557$	−35.0403	−1.48470	−0.742352	+	0.670010i	$0.766290\pi$
$558$	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	$-0.195633\pi$
$563$	38.7712	1.63401	0.817005	+	0.576630i	$0.195633\pi$
$564$	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	$-0.247784\pi$
$569$	33.9683	1.42403	0.712013	+	0.702166i	$0.247784\pi$
$570$	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	$-0.727422\pi$
$587$	−31.7492	−1.31043	−0.655215	+	0.755443i	$0.727422\pi$
$588$	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	$-0.589339\pi$
$593$	−13.4906	−0.553993	−0.276996	+	0.960871i	$0.589339\pi$
$594$	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	$-0.166154\pi$
$599$	42.4304	1.73366	0.866829	+	0.498606i	$0.166154\pi$
$600$	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	$-0.952066\pi$
$617$	−49.1167	−1.97737	−0.988683	+	0.150022i	$0.952066\pi$
$618$	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	$-0.770645\pi$
$641$	−38.0504	−1.50290	−0.751450	+	0.659790i	$0.770645\pi$
$642$	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	$-0.312725\pi$
$647$	28.2333	1.10997	0.554983	+	0.831862i	$0.312725\pi$
$648$	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	$-0.742759\pi$
$653$	−35.3072	−1.38168	−0.690839	+	0.723009i	$0.742759\pi$
$654$	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	$-0.801664\pi$
$659$	−41.6937	−1.62416	−0.812078	+	0.583548i	$0.801664\pi$
$660$	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	$-0.612438\pi$
$677$	−18.0019	−0.691869	−0.345934	+	0.938259i	$0.612438\pi$
$678$	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	$-0.775151\pi$
$683$	−39.7614	−1.52143	−0.760715	+	0.649087i	$0.775151\pi$
$684$	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	$-0.445823\pi$
$701$	8.96921	0.338762	0.169381	+	0.985551i	$0.445823\pi$
$702$	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	$-0.700367\pi$
$719$	−31.5720	−1.17744	−0.588718	+	0.808339i	$0.700367\pi$
$720$	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	$-0.731730\pi$
$743$	−36.2738	−1.33076	−0.665378	+	0.746506i	$0.731730\pi$
$744$	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	$-0.632102\pi$
$761$	−22.2456	−0.806401	−0.403200	+	0.915112i	$0.632102\pi$
$762$	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	$-0.378866\pi$
$773$	20.6540	0.742871	0.371436	+	0.928459i	$0.378866\pi$
$774$	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	$-0.317914\pi$
$797$	30.5658	1.08270	0.541348	+	0.840799i	$0.317914\pi$
$798$	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	$-0.808124\pi$
$809$	−46.8599	−1.64751	−0.823753	+	0.566949i	$0.808124\pi$
$810$	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	$-0.283909\pi$
$821$	35.9833	1.25583	0.627913	+	0.778283i	$0.283909\pi$
$822$	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	$-0.412507\pi$
$827$	15.6107	0.542836	0.271418	+	0.962462i	$0.412507\pi$
$828$	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	$-0.496289\pi$
$839$	0.675324	0.0233148	0.0116574	+	0.999932i	$0.496289\pi$
$840$	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	$-0.688377\pi$
$857$	−32.6621	−1.11572	−0.557858	+	0.829936i	$0.688377\pi$
$858$	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	$-0.626148\pi$
$863$	−22.6796	−0.772024	−0.386012	+	0.922494i	$0.626148\pi$
$864$	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	$-0.541878\pi$
$881$	−7.78755	−0.262369	−0.131185	+	0.991358i	$0.541878\pi$
$882$	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	$-0.307622\pi$
$887$	33.8477	1.13649	0.568247	+	0.822858i	$0.307622\pi$
$888$	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	$-0.566637\pi$
$911$	−12.5452	−0.415640	−0.207820	+	0.978167i	$0.566637\pi$
$912$	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	$-0.447372\pi$
$929$	10.0328	0.329165	0.164582	+	0.986363i	$0.447372\pi$
$930$	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	$-0.341518\pi$
$941$	29.2996	0.955138	0.477569	+	0.878594i	$0.341518\pi$
$942$	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	$-0.837105\pi$
$947$	−53.6619	−1.74378	−0.871889	+	0.489704i	$0.837105\pi$
$948$	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	$-0.474729\pi$
$953$	4.89656	0.158615	0.0793076	+	0.996850i	$0.474729\pi$
$954$	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	$-0.486288\pi$
$971$	2.68374	0.0861253	0.0430627	+	0.999072i	$0.486288\pi$
$972$	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	$-0.556267\pi$
$977$	−10.9930	−0.351698	−0.175849	+	0.984417i	$0.556267\pi$
$978$	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	$-0.224265\pi$
$983$	47.7756	1.52381	0.761903	+	0.647691i	$0.224265\pi$
$984$	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.d.1.5		6
3.2	odd	2		729.2.a.a.1.2		6
9.2	odd	6		729.2.c.e.244.5		12
9.4	even	3		729.2.c.b.487.2		12
9.5	odd	6		729.2.c.e.487.5		12
9.7	even	3		729.2.c.b.244.2		12
27.2	odd	18		243.2.e.c.190.2		12
27.4	even	9		81.2.e.a.46.2		12
27.5	odd	18		243.2.e.d.217.1		12
27.7	even	9		81.2.e.a.37.2		12
27.11	odd	18		243.2.e.d.28.1		12
27.13	even	9		243.2.e.b.55.1		12
27.14	odd	18		243.2.e.c.55.2		12
27.16	even	9		243.2.e.a.28.2		12
27.20	odd	18		27.2.e.a.22.1	yes	12
27.22	even	9		243.2.e.a.217.2		12
27.23	odd	18		27.2.e.a.16.1	✓	12
27.25	even	9		243.2.e.b.190.1		12
108.23	even	18		432.2.u.c.97.2		12
108.47	even	18		432.2.u.c.49.2		12
135.23	even	36		675.2.u.b.124.1		24
135.47	even	36		675.2.u.b.49.1		24
135.74	odd	18		675.2.l.c.76.2		12
135.77	even	36		675.2.u.b.124.4		24
135.104	odd	18		675.2.l.c.151.2		12
135.128	even	36		675.2.u.b.49.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.16.1	✓	12	27.23	odd	18
27.2.e.a.22.1	yes	12	27.20	odd	18
81.2.e.a.37.2		12	27.7	even	9
81.2.e.a.46.2		12	27.4	even	9
243.2.e.a.28.2		12	27.16	even	9
243.2.e.a.217.2		12	27.22	even	9
243.2.e.b.55.1		12	27.13	even	9
243.2.e.b.190.1		12	27.25	even	9
243.2.e.c.55.2		12	27.14	odd	18
243.2.e.c.190.2		12	27.2	odd	18
243.2.e.d.28.1		12	27.11	odd	18
243.2.e.d.217.1		12	27.5	odd	18
432.2.u.c.49.2		12	108.47	even	18
432.2.u.c.97.2		12	108.23	even	18
675.2.l.c.76.2		12	135.74	odd	18
675.2.l.c.151.2		12	135.104	odd	18
675.2.u.b.49.1		24	135.47	even	36
675.2.u.b.49.4		24	135.128	even	36
675.2.u.b.124.1		24	135.23	even	36
675.2.u.b.124.4		24	135.77	even	36
729.2.a.a.1.2		6	3.2	odd	2
729.2.a.d.1.5		6	1.1	even	1	trivial
729.2.c.b.244.2		12	9.7	even	3
729.2.c.b.487.2		12	9.4	even	3
729.2.c.e.244.5		12	9.2	odd	6
729.2.c.e.487.5		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.11662 q^{2} +2.48009 q^{4} +2.68310 q^{5} +0.972333 q^{7} +1.01617 q^{8} +5.67911 q^{10} +0.316902 q^{11} -1.51403 q^{13} +2.05806 q^{14} -2.80933 q^{16} -1.17468 q^{17} +6.22080 q^{19} +6.65433 q^{20} +0.670762 q^{22} -2.16274 q^{23} +2.19901 q^{25} -3.20463 q^{26} +2.41147 q^{28} +4.40491 q^{29} -8.67323 q^{31} -7.97863 q^{32} -2.48636 q^{34} +2.60886 q^{35} -4.46665 q^{37} +13.1671 q^{38} +2.72649 q^{40} -5.84518 q^{41} +5.59099 q^{43} +0.785946 q^{44} -4.57771 q^{46} +2.47607 q^{47} -6.05457 q^{49} +4.65448 q^{50} -3.75493 q^{52} +10.8920 q^{53} +0.850279 q^{55} +0.988058 q^{56} +9.32353 q^{58} +1.72421 q^{59} +1.01478 q^{61} -18.3580 q^{62} -11.2691 q^{64} -4.06229 q^{65} +0.856551 q^{67} -2.91332 q^{68} +5.52198 q^{70} +9.59577 q^{71} -15.2418 q^{73} -9.45420 q^{74} +15.4282 q^{76} +0.308134 q^{77} -11.2138 q^{79} -7.53771 q^{80} -12.3721 q^{82} -4.68493 q^{83} -3.15179 q^{85} +11.8340 q^{86} +0.322027 q^{88} +15.4995 q^{89} -1.47214 q^{91} -5.36380 q^{92} +5.24090 q^{94} +16.6910 q^{95} -5.54949 q^{97} -12.8152 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8} + 3 q^{10} + 12 q^{11} + 6 q^{14} - 3 q^{16} + 9 q^{17} + 3 q^{19} + 6 q^{20} + 6 q^{22} + 15 q^{23} - 6 q^{25} + 15 q^{26} - 6 q^{28} + 12 q^{29} + 12 q^{35}+ \cdots - 45 q^{98}+O(q^{100}) \) 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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