LMFDB - Embedded newform 6561.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6561,2,Mod(1,6561)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6561, 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("6561.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6561 = 3^{8} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6561.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:	$52.3898487662$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6561.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.63957 q^{2} +4.96734 q^{4} +1.13561 q^{5} -0.795312 q^{7} -7.83251 q^{8} +O(q^{10})$	\(q-2.63957 q^{2} +4.96734 q^{4} +1.13561 q^{5} -0.795312 q^{7} -7.83251 q^{8} -2.99751 q^{10} +5.92390 q^{11} -3.02523 q^{13} +2.09928 q^{14} +10.7398 q^{16} +3.80927 q^{17} +0.491229 q^{19} +5.64094 q^{20} -15.6366 q^{22} -5.66922 q^{23} -3.71040 q^{25} +7.98530 q^{26} -3.95059 q^{28} -1.58542 q^{29} -2.36244 q^{31} -12.6834 q^{32} -10.0548 q^{34} -0.903161 q^{35} +5.04574 q^{37} -1.29664 q^{38} -8.89464 q^{40} +4.43235 q^{41} +3.12286 q^{43} +29.4260 q^{44} +14.9643 q^{46} -1.60957 q^{47} -6.36748 q^{49} +9.79387 q^{50} -15.0273 q^{52} +2.44341 q^{53} +6.72722 q^{55} +6.22929 q^{56} +4.18482 q^{58} +10.8082 q^{59} +1.22718 q^{61} +6.23583 q^{62} +11.9992 q^{64} -3.43546 q^{65} -8.67290 q^{67} +18.9219 q^{68} +2.38396 q^{70} +14.6801 q^{71} -5.82873 q^{73} -13.3186 q^{74} +2.44010 q^{76} -4.71135 q^{77} -8.09887 q^{79} +12.1962 q^{80} -11.6995 q^{82} +3.36376 q^{83} +4.32582 q^{85} -8.24302 q^{86} -46.3990 q^{88} -1.00958 q^{89} +2.40600 q^{91} -28.1610 q^{92} +4.24859 q^{94} +0.557843 q^{95} -5.34521 q^{97} +16.8074 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) $ 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8	$ 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38} + 72 q^{41} + 90 q^{44} + 90 q^{47} + 18 q^{49} + 45 q^{50} + 45 q^{53} + 9 q^{55} + 108 q^{56} + 18 q^{58} + 108 q^{59} + 72 q^{62} + 9 q^{64} + 72 q^{65} + 108 q^{68} + 126 q^{71} + 90 q^{74} + 72 q^{77} + 144 q^{80} - 18 q^{82} + 108 q^{83} + 90 q^{86} + 108 q^{89} + 72 q^{92} + 144 q^{95} + 81 q^{98}+O(q^{100}) $ 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8 + 36 * q^11 + 36 * q^14 + 45 * q^16 + 36 * q^17 + 54 * q^20 + 54 * q^23 + 36 * q^25 + 45 * q^26 + 9 * q^28 + 54 * q^29 + 63 * q^32 + 72 * q^35 + 54 * q^38 + 72 * q^41 + 90 * q^44 + 90 * q^47 + 18 * q^49 + 45 * q^50 + 45 * q^53 + 9 * q^55 + 108 * q^56 + 18 * q^58 + 108 * q^59 + 72 * q^62 + 9 * q^64 + 72 * q^65 + 108 * q^68 + 126 * q^71 + 90 * q^74 + 72 * q^77 + 144 * q^80 - 18 * q^82 + 108 * q^83 + 90 * q^86 + 108 * q^89 + 72 * q^92 + 144 * q^95 + 81 * 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 $

$2$	−2.63957	−1.86646	−0.933230	−	0.359281i	$-0.883022\pi$
$2$	−2.63957	−1.86646	−0.933230	+	0.359281i	$0.883022\pi$
$3$	0	0
$3$	0	0
$4$	4.96734	2.48367
$5$	1.13561	0.507858	0.253929	−	0.967223i	$-0.418277\pi$
$5$	1.13561	0.507858	0.253929	+	0.967223i	$0.418277\pi$
$6$	0	0
$7$	−0.795312	−0.300600	−0.150300	−	0.988640i	$-0.548024\pi$
$7$	−0.795312	−0.300600	−0.150300	+	0.988640i	$0.548024\pi$
$8$	−7.83251	−2.76921
$9$	0	0
$10$	−2.99751	−0.947897
$11$	5.92390	1.78612	0.893062	−	0.449933i	$-0.148552\pi$
$11$	5.92390	1.78612	0.893062	+	0.449933i	$0.148552\pi$
$12$	0	0
$13$	−3.02523	−0.839047	−0.419523	−	0.907745i	$-0.637803\pi$
$13$	−3.02523	−0.839047	−0.419523	+	0.907745i	$0.637803\pi$
$14$	2.09928	0.561057
$15$	0	0
$16$	10.7398	2.68495
$17$	3.80927	0.923883	0.461941	−	0.886910i	$-0.347153\pi$
$17$	3.80927	0.923883	0.461941	+	0.886910i	$0.347153\pi$
$18$	0	0
$19$	0.491229	0.112696	0.0563479	−	0.998411i	$-0.482054\pi$
$19$	0.491229	0.112696	0.0563479	+	0.998411i	$0.482054\pi$
$20$	5.64094	1.26135
$21$	0	0
$22$	−15.6366	−3.33373
$23$	−5.66922	−1.18211	−0.591057	−	0.806630i	$-0.701289\pi$
$23$	−5.66922	−1.18211	−0.591057	+	0.806630i	$0.701289\pi$
$24$	0	0
$25$	−3.71040	−0.742080
$26$	7.98530	1.56605
$27$	0	0
$28$	−3.95059	−0.746590
$29$	−1.58542	−0.294405	−0.147202	−	0.989106i	$-0.547027\pi$
$29$	−1.58542	−0.294405	−0.147202	+	0.989106i	$0.547027\pi$
$30$	0	0
$31$	−2.36244	−0.424307	−0.212153	−	0.977236i	$-0.568048\pi$
$31$	−2.36244	−0.424307	−0.212153	+	0.977236i	$0.568048\pi$
$32$	−12.6834	−2.24213
$33$	0	0
$34$	−10.0548	−1.72439
$35$	−0.903161	−0.152662
$36$	0	0
$37$	5.04574	0.829514	0.414757	−	0.909932i	$-0.363867\pi$
$37$	5.04574	0.829514	0.414757	+	0.909932i	$0.363867\pi$
$38$	−1.29664	−0.210342
$39$	0	0
$40$	−8.89464	−1.40637
$41$	4.43235	0.692216	0.346108	−	0.938195i	$-0.387503\pi$
$41$	4.43235	0.692216	0.346108	+	0.938195i	$0.387503\pi$
$42$	0	0
$43$	3.12286	0.476232	0.238116	−	0.971237i	$-0.423470\pi$
$43$	3.12286	0.476232	0.238116	+	0.971237i	$0.423470\pi$
$44$	29.4260	4.43614
$45$	0	0
$46$	14.9643	2.20637
$47$	−1.60957	−0.234781	−0.117390	−	0.993086i	$-0.537453\pi$
$47$	−1.60957	−0.234781	−0.117390	+	0.993086i	$0.537453\pi$
$48$	0	0
$49$	−6.36748	−0.909640
$50$	9.79387	1.38506
$51$	0	0
$52$	−15.0273	−2.08392
$53$	2.44341	0.335628	0.167814	−	0.985819i	$-0.446329\pi$
$53$	2.44341	0.335628	0.167814	+	0.985819i	$0.446329\pi$
$54$	0	0
$55$	6.72722	0.907098
$56$	6.22929	0.832424
$57$	0	0
$58$	4.18482	0.549494
$59$	10.8082	1.40710	0.703552	−	0.710644i	$-0.251596\pi$
$59$	10.8082	1.40710	0.703552	+	0.710644i	$0.251596\pi$
$60$	0	0
$61$	1.22718	0.157124	0.0785620	−	0.996909i	$-0.474967\pi$
$61$	1.22718	0.157124	0.0785620	+	0.996909i	$0.474967\pi$
$62$	6.23583	0.791952
$63$	0	0
$64$	11.9992	1.49991
$65$	−3.43546	−0.426117
$66$	0	0
$67$	−8.67290	−1.05956	−0.529782	−	0.848134i	$-0.677726\pi$
$67$	−8.67290	−1.05956	−0.529782	+	0.848134i	$0.677726\pi$
$68$	18.9219	2.29462
$69$	0	0
$70$	2.38396	0.284937
$71$	14.6801	1.74221	0.871103	−	0.491100i	$-0.163405\pi$
$71$	14.6801	1.74221	0.871103	+	0.491100i	$0.163405\pi$
$72$	0	0
$73$	−5.82873	−0.682202	−0.341101	−	0.940027i	$-0.610800\pi$
$73$	−5.82873	−0.682202	−0.341101	+	0.940027i	$0.610800\pi$
$74$	−13.3186	−1.54825
$75$	0	0
$76$	2.44010	0.279899
$77$	−4.71135	−0.536908
$78$	0	0
$79$	−8.09887	−0.911194	−0.455597	−	0.890186i	$-0.650574\pi$
$79$	−8.09887	−0.911194	−0.455597	+	0.890186i	$0.650574\pi$
$80$	12.1962	1.36357
$81$	0	0
$82$	−11.6995	−1.29199
$83$	3.36376	0.369221	0.184610	−	0.982812i	$-0.440898\pi$
$83$	3.36376	0.369221	0.184610	+	0.982812i	$0.440898\pi$
$84$	0	0
$85$	4.32582	0.469202
$86$	−8.24302	−0.888868
$87$	0	0
$88$	−46.3990	−4.94615
$89$	−1.00958	−0.107015	−0.0535077	−	0.998567i	$-0.517040\pi$
$89$	−1.00958	−0.107015	−0.0535077	+	0.998567i	$0.517040\pi$
$90$	0	0
$91$	2.40600	0.252217
$92$	−28.1610	−2.93598
$93$	0	0
$94$	4.24859	0.438208
$95$	0.557843	0.0572335
$96$	0	0
$97$	−5.34521	−0.542724	−0.271362	−	0.962477i	$-0.587474\pi$
$97$	−5.34521	−0.542724	−0.271362	+	0.962477i	$0.587474\pi$
$98$	16.8074	1.69781
$99$	0	0
$100$	−18.4308	−1.84308
$101$	9.42263	0.937587	0.468793	−	0.883308i	$-0.344689\pi$
$101$	9.42263	0.937587	0.468793	+	0.883308i	$0.344689\pi$
$102$	0	0
$103$	4.35867	0.429472	0.214736	−	0.976672i	$-0.431111\pi$
$103$	4.35867	0.429472	0.214736	+	0.976672i	$0.431111\pi$
$104$	23.6951	2.32350
$105$	0	0
$106$	−6.44955	−0.626436
$107$	3.32891	0.321818	0.160909	−	0.986969i	$-0.448557\pi$
$107$	3.32891	0.321818	0.160909	+	0.986969i	$0.448557\pi$
$108$	0	0
$109$	15.8470	1.51787	0.758936	−	0.651166i	$-0.225720\pi$
$109$	15.8470	1.51787	0.758936	+	0.651166i	$0.225720\pi$
$110$	−17.7570	−1.69306
$111$	0	0
$112$	−8.54148	−0.807094
$113$	2.82185	0.265458	0.132729	−	0.991152i	$-0.457626\pi$
$113$	2.82185	0.265458	0.132729	+	0.991152i	$0.457626\pi$
$114$	0	0
$115$	−6.43800	−0.600347
$116$	−7.87531	−0.731204
$117$	0	0
$118$	−28.5289	−2.62630
$119$	−3.02956	−0.277719
$120$	0	0
$121$	24.0926	2.19024
$122$	−3.23923	−0.293266
$123$	0	0
$124$	−11.7350	−1.05384
$125$	−9.89158	−0.884730
$126$	0	0
$127$	4.16619	0.369689	0.184845	−	0.982768i	$-0.440822\pi$
$127$	4.16619	0.369689	0.184845	+	0.982768i	$0.440822\pi$
$128$	−6.30601	−0.557378
$129$	0	0
$130$	9.06815	0.795330
$131$	−0.670960	−0.0586221	−0.0293110	−	0.999570i	$-0.509331\pi$
$131$	−0.670960	−0.0586221	−0.0293110	+	0.999570i	$0.509331\pi$
$132$	0	0
$133$	−0.390681	−0.0338763
$134$	22.8927	1.97763
$135$	0	0
$136$	−29.8361	−2.55843
$137$	9.68700	0.827616	0.413808	−	0.910364i	$-0.364198\pi$
$137$	9.68700	0.827616	0.413808	+	0.910364i	$0.364198\pi$
$138$	0	0
$139$	−19.0343	−1.61447	−0.807236	−	0.590229i	$-0.799037\pi$
$139$	−19.0343	−1.61447	−0.807236	+	0.590229i	$0.799037\pi$
$140$	−4.48631	−0.379162
$141$	0	0
$142$	−38.7492	−3.25176
$143$	−17.9211	−1.49864
$144$	0	0
$145$	−1.80041	−0.149516
$146$	15.3854	1.27330
$147$	0	0
$148$	25.0639	2.06024
$149$	−6.33570	−0.519041	−0.259521	−	0.965738i	$-0.583565\pi$
$149$	−6.33570	−0.519041	−0.259521	+	0.965738i	$0.583565\pi$
$150$	0	0
$151$	17.8865	1.45558	0.727792	−	0.685798i	$-0.240547\pi$
$151$	17.8865	1.45558	0.727792	+	0.685798i	$0.240547\pi$
$152$	−3.84756	−0.312078
$153$	0	0
$154$	12.4360	1.00212
$155$	−2.68280	−0.215488
$156$	0	0
$157$	13.4725	1.07522	0.537612	−	0.843192i	$-0.319326\pi$
$157$	13.4725	1.07522	0.537612	+	0.843192i	$0.319326\pi$
$158$	21.3776	1.70071
$159$	0	0
$160$	−14.4034	−1.13869
$161$	4.50880	0.355343
$162$	0	0
$163$	14.8895	1.16624	0.583119	−	0.812387i	$-0.301832\pi$
$163$	14.8895	1.16624	0.583119	+	0.812387i	$0.301832\pi$
$164$	22.0170	1.71924
$165$	0	0
$166$	−8.87888	−0.689135
$167$	−14.9323	−1.15550	−0.577749	−	0.816215i	$-0.696069\pi$
$167$	−14.9323	−1.15550	−0.577749	+	0.816215i	$0.696069\pi$
$168$	0	0
$169$	−3.84801	−0.296001
$170$	−11.4183	−0.875746
$171$	0	0
$172$	15.5123	1.18280
$173$	24.5388	1.86565	0.932825	−	0.360330i	$-0.117336\pi$
$173$	24.5388	1.86565	0.932825	+	0.360330i	$0.117336\pi$
$174$	0	0
$175$	2.95093	0.223069
$176$	63.6215	4.79565
$177$	0	0
$178$	2.66486	0.199740
$179$	−13.2120	−0.987511	−0.493755	−	0.869601i	$-0.664376\pi$
$179$	−13.2120	−0.987511	−0.493755	+	0.869601i	$0.664376\pi$
$180$	0	0
$181$	24.5800	1.82702	0.913508	−	0.406821i	$-0.133363\pi$
$181$	24.5800	1.82702	0.913508	+	0.406821i	$0.133363\pi$
$182$	−6.35081	−0.470753
$183$	0	0
$184$	44.4042	3.27352
$185$	5.72997	0.421276
$186$	0	0
$187$	22.5657	1.65017
$188$	−7.99530	−0.583117
$189$	0	0
$190$	−1.47247	−0.106824
$191$	2.49246	0.180348	0.0901739	−	0.995926i	$-0.471258\pi$
$191$	2.49246	0.180348	0.0901739	+	0.995926i	$0.471258\pi$
$192$	0	0
$193$	7.11893	0.512432	0.256216	−	0.966620i	$-0.417524\pi$
$193$	7.11893	0.512432	0.256216	+	0.966620i	$0.417524\pi$
$194$	14.1091	1.01297
$195$	0	0
$196$	−31.6294	−2.25925
$197$	−3.23443	−0.230444	−0.115222	−	0.993340i	$-0.536758\pi$
$197$	−3.23443	−0.230444	−0.115222	+	0.993340i	$0.536758\pi$
$198$	0	0
$199$	19.8727	1.40874	0.704368	−	0.709835i	$-0.251230\pi$
$199$	19.8727	1.40874	0.704368	+	0.709835i	$0.251230\pi$
$200$	29.0617	2.05497
$201$	0	0
$202$	−24.8717	−1.74997
$203$	1.26090	0.0884979
$204$	0	0
$205$	5.03340	0.351548
$206$	−11.5050	−0.801593
$207$	0	0
$208$	−32.4903	−2.25280
$209$	2.91000	0.201289
$210$	0	0
$211$	19.2096	1.32244	0.661222	−	0.750191i	$-0.270038\pi$
$211$	19.2096	1.32244	0.661222	+	0.750191i	$0.270038\pi$
$212$	12.1372	0.833589
$213$	0	0
$214$	−8.78690	−0.600660
$215$	3.54634	0.241858
$216$	0	0
$217$	1.87888	0.127547
$218$	−41.8294	−2.83304
$219$	0	0
$220$	33.4164	2.25293
$221$	−11.5239	−0.775181
$222$	0	0
$223$	−4.43210	−0.296796	−0.148398	−	0.988928i	$-0.547412\pi$
$223$	−4.43210	−0.296796	−0.148398	+	0.988928i	$0.547412\pi$
$224$	10.0873	0.673985
$225$	0	0
$226$	−7.44848	−0.495466
$227$	−3.16203	−0.209871	−0.104936	−	0.994479i	$-0.533464\pi$
$227$	−3.16203	−0.209871	−0.104936	+	0.994479i	$0.533464\pi$
$228$	0	0
$229$	10.4424	0.690056	0.345028	−	0.938592i	$-0.387869\pi$
$229$	10.4424	0.690056	0.345028	+	0.938592i	$0.387869\pi$
$230$	16.9936	1.12052
$231$	0	0
$232$	12.4178	0.815268
$233$	1.65164	0.108203	0.0541013	−	0.998535i	$-0.482771\pi$
$233$	1.65164	0.108203	0.0541013	+	0.998535i	$0.482771\pi$
$234$	0	0
$235$	−1.82784	−0.119235
$236$	53.6878	3.49478
$237$	0	0
$238$	7.99673	0.518351
$239$	−14.6419	−0.947107	−0.473554	−	0.880765i	$-0.657029\pi$
$239$	−14.6419	−0.947107	−0.473554	+	0.880765i	$0.657029\pi$
$240$	0	0
$241$	−22.3764	−1.44139	−0.720696	−	0.693251i	$-0.756178\pi$
$241$	−22.3764	−1.44139	−0.720696	+	0.693251i	$0.756178\pi$
$242$	−63.5942	−4.08799
$243$	0	0
$244$	6.09581	0.390244
$245$	−7.23095	−0.461968
$246$	0	0
$247$	−1.48608	−0.0945570
$248$	18.5038	1.17500
$249$	0	0
$250$	26.1095	1.65131
$251$	23.8600	1.50603	0.753015	−	0.658003i	$-0.228599\pi$
$251$	23.8600	1.50603	0.753015	+	0.658003i	$0.228599\pi$
$252$	0	0
$253$	−33.5839	−2.11140
$254$	−10.9969	−0.690010
$255$	0	0
$256$	−7.35331	−0.459582
$257$	−23.2702	−1.45155	−0.725776	−	0.687931i	$-0.758519\pi$
$257$	−23.2702	−1.45155	−0.725776	+	0.687931i	$0.758519\pi$
$258$	0	0
$259$	−4.01294	−0.249352
$260$	−17.0651	−1.05833
$261$	0	0
$262$	1.77105	0.109416
$263$	−16.7258	−1.03135	−0.515677	−	0.856783i	$-0.672460\pi$
$263$	−16.7258	−1.03135	−0.515677	+	0.856783i	$0.672460\pi$
$264$	0	0
$265$	2.77475	0.170451
$266$	1.03123	0.0632287
$267$	0	0
$268$	−43.0812	−2.63161
$269$	−13.6833	−0.834286	−0.417143	−	0.908841i	$-0.636969\pi$
$269$	−13.6833	−0.834286	−0.417143	+	0.908841i	$0.636969\pi$
$270$	0	0
$271$	25.9397	1.57573	0.787863	−	0.615851i	$-0.211187\pi$
$271$	25.9397	1.57573	0.787863	+	0.615851i	$0.211187\pi$
$272$	40.9107	2.48058
$273$	0	0
$274$	−25.5695	−1.54471
$275$	−21.9801	−1.32545
$276$	0	0
$277$	−17.9384	−1.07781	−0.538907	−	0.842365i	$-0.681163\pi$
$277$	−17.9384	−1.07781	−0.538907	+	0.842365i	$0.681163\pi$
$278$	50.2425	3.01335
$279$	0	0
$280$	7.07401	0.422753
$281$	19.6849	1.17430	0.587151	−	0.809478i	$-0.300250\pi$
$281$	19.6849	1.17430	0.587151	+	0.809478i	$0.300250\pi$
$282$	0	0
$283$	−15.7507	−0.936283	−0.468141	−	0.883654i	$-0.655076\pi$
$283$	−15.7507	−0.936283	−0.468141	+	0.883654i	$0.655076\pi$
$284$	72.9210	4.32707
$285$	0	0
$286$	47.3042	2.79715
$287$	−3.52510	−0.208080
$288$	0	0
$289$	−2.48949	−0.146441
$290$	4.75231	0.279065
$291$	0	0
$292$	−28.9533	−1.69436
$293$	27.4871	1.60582	0.802908	−	0.596103i	$-0.203285\pi$
$293$	27.4871	1.60582	0.802908	+	0.596103i	$0.203285\pi$
$294$	0	0
$295$	12.2738	0.714609
$296$	−39.5208	−2.29710
$297$	0	0
$298$	16.7235	0.968769
$299$	17.1507	0.991849
$300$	0	0
$301$	−2.48365	−0.143155
$302$	−47.2127	−2.71679
$303$	0	0
$304$	5.27570	0.302582
$305$	1.39359	0.0797968
$306$	0	0
$307$	−11.9126	−0.679885	−0.339943	−	0.940446i	$-0.610408\pi$
$307$	−11.9126	−0.679885	−0.339943	+	0.940446i	$0.610408\pi$
$308$	−23.4029	−1.33350
$309$	0	0
$310$	7.08145	0.402199
$311$	−23.0598	−1.30760	−0.653801	−	0.756667i	$-0.726827\pi$
$311$	−23.0598	−1.30760	−0.653801	+	0.756667i	$0.726827\pi$
$312$	0	0
$313$	−14.2083	−0.803101	−0.401550	−	0.915837i	$-0.631528\pi$
$313$	−14.2083	−0.803101	−0.401550	+	0.915837i	$0.631528\pi$
$314$	−35.5617	−2.00686
$315$	0	0
$316$	−40.2299	−2.26311
$317$	−6.86841	−0.385768	−0.192884	−	0.981222i	$-0.561784\pi$
$317$	−6.86841	−0.385768	−0.192884	+	0.981222i	$0.561784\pi$
$318$	0	0
$319$	−9.39186	−0.525843
$320$	13.6264	0.761739
$321$	0	0
$322$	−11.9013	−0.663234
$323$	1.87122	0.104118
$324$	0	0
$325$	11.2248	0.622640
$326$	−39.3020	−2.17674
$327$	0	0
$328$	−34.7164	−1.91689
$329$	1.28011	0.0705749
$330$	0	0
$331$	7.72827	0.424784	0.212392	−	0.977185i	$-0.431875\pi$
$331$	7.72827	0.424784	0.212392	+	0.977185i	$0.431875\pi$
$332$	16.7089	0.917022
$333$	0	0
$334$	39.4149	2.15669
$335$	−9.84900	−0.538108
$336$	0	0
$337$	14.6772	0.799516	0.399758	−	0.916621i	$-0.369094\pi$
$337$	14.6772	0.799516	0.399758	+	0.916621i	$0.369094\pi$
$338$	10.1571	0.552473
$339$	0	0
$340$	21.4878	1.16534
$341$	−13.9949	−0.757865
$342$	0	0
$343$	10.6313	0.574037
$344$	−24.4598	−1.31879
$345$	0	0
$346$	−64.7719	−3.48216
$347$	6.85095	0.367779	0.183889	−	0.982947i	$-0.441131\pi$
$347$	6.85095	0.367779	0.183889	+	0.982947i	$0.441131\pi$
$348$	0	0
$349$	8.43616	0.451578	0.225789	−	0.974176i	$-0.427504\pi$
$349$	8.43616	0.451578	0.225789	+	0.974176i	$0.427504\pi$
$350$	−7.78918	−0.416349
$351$	0	0
$352$	−75.1354	−4.00473
$353$	−7.86760	−0.418750	−0.209375	−	0.977835i	$-0.567143\pi$
$353$	−7.86760	−0.418750	−0.209375	+	0.977835i	$0.567143\pi$
$354$	0	0
$355$	16.6708	0.884794
$356$	−5.01494	−0.265791
$357$	0	0
$358$	34.8740	1.84315
$359$	14.4360	0.761905	0.380953	−	0.924595i	$-0.375596\pi$
$359$	14.4360	0.761905	0.380953	+	0.924595i	$0.375596\pi$
$360$	0	0
$361$	−18.7587	−0.987300
$362$	−64.8806	−3.41005
$363$	0	0
$364$	11.9514	0.626424
$365$	−6.61914	−0.346462
$366$	0	0
$367$	−30.6529	−1.60007	−0.800035	−	0.599953i	$-0.795186\pi$
$367$	−30.6529	−1.60007	−0.800035	+	0.599953i	$0.795186\pi$
$368$	−60.8862	−3.17392
$369$	0	0
$370$	−15.1247	−0.786294
$371$	−1.94327	−0.100890
$372$	0	0
$373$	−21.6087	−1.11886	−0.559429	−	0.828878i	$-0.688980\pi$
$373$	−21.6087	−1.11886	−0.559429	+	0.828878i	$0.688980\pi$
$374$	−59.5639	−3.07997
$375$	0	0
$376$	12.6070	0.650156
$377$	4.79625	0.247019
$378$	0	0
$379$	28.4756	1.46269	0.731347	−	0.682005i	$-0.238892\pi$
$379$	28.4756	1.46269	0.731347	+	0.682005i	$0.238892\pi$
$380$	2.77100	0.142149
$381$	0	0
$382$	−6.57902	−0.336612
$383$	23.0787	1.17926	0.589632	−	0.807672i	$-0.299273\pi$
$383$	23.0787	1.17926	0.589632	+	0.807672i	$0.299273\pi$
$384$	0	0
$385$	−5.35024	−0.272673
$386$	−18.7909	−0.956433
$387$	0	0
$388$	−26.5515	−1.34795
$389$	−5.10149	−0.258656	−0.129328	−	0.991602i	$-0.541282\pi$
$389$	−5.10149	−0.258656	−0.129328	+	0.991602i	$0.541282\pi$
$390$	0	0
$391$	−21.5956	−1.09214
$392$	49.8733	2.51898
$393$	0	0
$394$	8.53752	0.430114
$395$	−9.19713	−0.462758
$396$	0	0
$397$	24.1002	1.20955	0.604776	−	0.796396i	$-0.293263\pi$
$397$	24.1002	1.20955	0.604776	+	0.796396i	$0.293263\pi$
$398$	−52.4553	−2.62935
$399$	0	0
$400$	−39.8489	−1.99245
$401$	−10.2158	−0.510154	−0.255077	−	0.966921i	$-0.582101\pi$
$401$	−10.2158	−0.510154	−0.255077	+	0.966921i	$0.582101\pi$
$402$	0	0
$403$	7.14692	0.356013
$404$	46.8054	2.32866
$405$	0	0
$406$	−3.32824	−0.165178
$407$	29.8905	1.48162
$408$	0	0
$409$	31.7173	1.56832	0.784161	−	0.620558i	$-0.213094\pi$
$409$	31.7173	1.56832	0.784161	+	0.620558i	$0.213094\pi$
$410$	−13.2860	−0.656149
$411$	0	0
$412$	21.6510	1.06667
$413$	−8.59586	−0.422975
$414$	0	0
$415$	3.81990	0.187512
$416$	38.3702	1.88126
$417$	0	0
$418$	−7.68114	−0.375697
$419$	21.0907	1.03035	0.515175	−	0.857085i	$-0.327727\pi$
$419$	21.0907	1.03035	0.515175	+	0.857085i	$0.327727\pi$
$420$	0	0
$421$	9.08577	0.442813	0.221407	−	0.975182i	$-0.428935\pi$
$421$	9.08577	0.442813	0.221407	+	0.975182i	$0.428935\pi$
$422$	−50.7051	−2.46829
$423$	0	0
$424$	−19.1380	−0.929424
$425$	−14.1339	−0.685595
$426$	0	0
$427$	−0.975990	−0.0472314
$428$	16.5358	0.799289
$429$	0	0
$430$	−9.36082	−0.451419
$431$	−23.8314	−1.14792	−0.573960	−	0.818883i	$-0.694593\pi$
$431$	−23.8314	−1.14792	−0.573960	+	0.818883i	$0.694593\pi$
$432$	0	0
$433$	−7.16608	−0.344380	−0.172190	−	0.985064i	$-0.555084\pi$
$433$	−7.16608	−0.344380	−0.172190	+	0.985064i	$0.555084\pi$
$434$	−4.95943	−0.238060
$435$	0	0
$436$	78.7176	3.76989
$437$	−2.78489	−0.133219
$438$	0	0
$439$	17.4017	0.830537	0.415269	−	0.909699i	$-0.363688\pi$
$439$	17.4017	0.830537	0.415269	+	0.909699i	$0.363688\pi$
$440$	−52.6910	−2.51194
$441$	0	0
$442$	30.4181	1.44684
$443$	−6.60947	−0.314026	−0.157013	−	0.987597i	$-0.550186\pi$
$443$	−6.60947	−0.314026	−0.157013	+	0.987597i	$0.550186\pi$
$444$	0	0
$445$	−1.14649	−0.0543487
$446$	11.6989	0.553957
$447$	0	0
$448$	−9.54314	−0.450871
$449$	−23.3239	−1.10072	−0.550360	−	0.834927i	$-0.685510\pi$
$449$	−23.3239	−1.10072	−0.550360	+	0.834927i	$0.685510\pi$
$450$	0	0
$451$	26.2568	1.23638
$452$	14.0171	0.659309
$453$	0	0
$454$	8.34641	0.391717
$455$	2.73227	0.128091
$456$	0	0
$457$	−23.7145	−1.10932	−0.554659	−	0.832078i	$-0.687151\pi$
$457$	−23.7145	−1.10932	−0.554659	+	0.832078i	$0.687151\pi$
$458$	−27.5636	−1.28796
$459$	0	0
$460$	−31.9797	−1.49106
$461$	37.0019	1.72335	0.861676	−	0.507458i	$-0.169415\pi$
$461$	37.0019	1.72335	0.861676	+	0.507458i	$0.169415\pi$
$462$	0	0
$463$	−2.30429	−0.107089	−0.0535446	−	0.998565i	$-0.517052\pi$
$463$	−2.30429	−0.107089	−0.0535446	+	0.998565i	$0.517052\pi$
$464$	−17.0270	−0.790461
$465$	0	0
$466$	−4.35963	−0.201956
$467$	17.8661	0.826745	0.413373	−	0.910562i	$-0.364351\pi$
$467$	17.8661	0.826745	0.413373	+	0.910562i	$0.364351\pi$
$468$	0	0
$469$	6.89766	0.318504
$470$	4.82472	0.222548
$471$	0	0
$472$	−84.6550	−3.89656
$473$	18.4995	0.850610
$474$	0	0
$475$	−1.82266	−0.0836292
$476$	−15.0488	−0.689762
$477$	0	0
$478$	38.6484	1.76774
$479$	5.98599	0.273507	0.136754	−	0.990605i	$-0.456333\pi$
$479$	5.98599	0.273507	0.136754	+	0.990605i	$0.456333\pi$
$480$	0	0
$481$	−15.2645	−0.696001
$482$	59.0642	2.69030
$483$	0	0
$484$	119.676	5.43983
$485$	−6.07005	−0.275627
$486$	0	0
$487$	2.99308	0.135630	0.0678148	−	0.997698i	$-0.478397\pi$
$487$	2.99308	0.135630	0.0678148	+	0.997698i	$0.478397\pi$
$488$	−9.61188	−0.435109
$489$	0	0
$490$	19.0866	0.862245
$491$	22.0986	0.997296	0.498648	−	0.866805i	$-0.333830\pi$
$491$	22.0986	0.997296	0.498648	+	0.866805i	$0.333830\pi$
$492$	0	0
$493$	−6.03928	−0.271995
$494$	3.92261	0.176487
$495$	0	0
$496$	−25.3721	−1.13924
$497$	−11.6753	−0.523707
$498$	0	0
$499$	−23.9284	−1.07118	−0.535590	−	0.844478i	$-0.679911\pi$
$499$	−23.9284	−1.07118	−0.535590	+	0.844478i	$0.679911\pi$
$500$	−49.1348	−2.19738
$501$	0	0
$502$	−62.9802	−2.81094
$503$	28.3295	1.26315	0.631576	−	0.775314i	$-0.282408\pi$
$503$	28.3295	1.26315	0.631576	+	0.775314i	$0.282408\pi$
$504$	0	0
$505$	10.7004	0.476161
$506$	88.6472	3.94085
$507$	0	0
$508$	20.6949	0.918186
$509$	−16.0643	−0.712036	−0.356018	−	0.934479i	$-0.615866\pi$
$509$	−16.0643	−0.712036	−0.356018	+	0.934479i	$0.615866\pi$
$510$	0	0
$511$	4.63566	0.205070
$512$	32.0216	1.41517
$513$	0	0
$514$	61.4232	2.70926
$515$	4.94973	0.218111
$516$	0	0
$517$	−9.53496	−0.419347
$518$	10.5924	0.465405
$519$	0	0
$520$	26.9083	1.18001
$521$	−35.3254	−1.54764	−0.773818	−	0.633408i	$-0.781655\pi$
$521$	−35.3254	−1.54764	−0.773818	+	0.633408i	$0.781655\pi$
$522$	0	0
$523$	−40.1517	−1.75571	−0.877856	−	0.478924i	$-0.841027\pi$
$523$	−40.1517	−1.75571	−0.877856	+	0.478924i	$0.841027\pi$
$524$	−3.33289	−0.145598
$525$	0	0
$526$	44.1489	1.92498
$527$	−8.99917	−0.392010
$528$	0	0
$529$	9.14008	0.397395
$530$	−7.32414	−0.318141
$531$	0	0
$532$	−1.94064	−0.0841376
$533$	−13.4088	−0.580802
$534$	0	0
$535$	3.78033	0.163438
$536$	67.9306	2.93415
$537$	0	0
$538$	36.1181	1.55716
$539$	−37.7203	−1.62473
$540$	0	0
$541$	15.7877	0.678765	0.339383	−	0.940648i	$-0.389782\pi$
$541$	15.7877	0.678765	0.339383	+	0.940648i	$0.389782\pi$
$542$	−68.4698	−2.94103
$543$	0	0
$544$	−48.3146	−2.07147
$545$	17.9960	0.770863
$546$	0	0
$547$	30.6753	1.31158	0.655790	−	0.754943i	$-0.272336\pi$
$547$	30.6753	1.31158	0.655790	+	0.754943i	$0.272336\pi$
$548$	48.1186	2.05553
$549$	0	0
$550$	58.0179	2.47389
$551$	−0.778804	−0.0331781
$552$	0	0
$553$	6.44113	0.273905
$554$	47.3497	2.01170
$555$	0	0
$556$	−94.5500	−4.00981
$557$	14.5022	0.614478	0.307239	−	0.951632i	$-0.400595\pi$
$557$	14.5022	0.614478	0.307239	+	0.951632i	$0.400595\pi$
$558$	0	0
$559$	−9.44737	−0.399581
$560$	−9.69976	−0.409890
$561$	0	0
$562$	−51.9597	−2.19179
$563$	2.94003	0.123907	0.0619537	−	0.998079i	$-0.480267\pi$
$563$	2.94003	0.123907	0.0619537	+	0.998079i	$0.480267\pi$
$564$	0	0
$565$	3.20451	0.134815
$566$	41.5751	1.74753
$567$	0	0
$568$	−114.982	−4.82454
$569$	−9.60775	−0.402778	−0.201389	−	0.979511i	$-0.564546\pi$
$569$	−9.60775	−0.402778	−0.201389	+	0.979511i	$0.564546\pi$
$570$	0	0
$571$	−43.0702	−1.80243	−0.901215	−	0.433371i	$-0.857324\pi$
$571$	−43.0702	−1.80243	−0.901215	+	0.433371i	$0.857324\pi$
$572$	−89.0204	−3.72213
$573$	0	0
$574$	9.30475	0.388373
$575$	21.0351	0.877224
$576$	0	0
$577$	−36.9956	−1.54015	−0.770074	−	0.637955i	$-0.779781\pi$
$577$	−36.9956	−1.54015	−0.770074	+	0.637955i	$0.779781\pi$
$578$	6.57119	0.273325
$579$	0	0
$580$	−8.94324	−0.371348
$581$	−2.67524	−0.110988
$582$	0	0
$583$	14.4745	0.599473
$584$	45.6536	1.88916
$585$	0	0
$586$	−72.5543	−2.99719
$587$	10.0684	0.415567	0.207784	−	0.978175i	$-0.433375\pi$
$587$	10.0684	0.415567	0.207784	+	0.978175i	$0.433375\pi$
$588$	0	0
$589$	−1.16050	−0.0478176
$590$	−32.3976	−1.33379
$591$	0	0
$592$	54.1902	2.22720
$593$	26.7640	1.09907	0.549534	−	0.835472i	$-0.314805\pi$
$593$	26.7640	1.09907	0.549534	+	0.835472i	$0.314805\pi$
$594$	0	0
$595$	−3.44038	−0.141042
$596$	−31.4716	−1.28913
$597$	0	0
$598$	−45.2705	−1.85125
$599$	35.5595	1.45292	0.726462	−	0.687207i	$-0.241164\pi$
$599$	35.5595	1.45292	0.726462	+	0.687207i	$0.241164\pi$
$600$	0	0
$601$	17.5697	0.716685	0.358342	−	0.933590i	$-0.383342\pi$
$601$	17.5697	0.716685	0.358342	+	0.933590i	$0.383342\pi$
$602$	6.55577	0.267193
$603$	0	0
$604$	88.8484	3.61519
$605$	27.3597	1.11233
$606$	0	0
$607$	−2.95017	−0.119744	−0.0598719	−	0.998206i	$-0.519069\pi$
$607$	−2.95017	−0.119744	−0.0598719	+	0.998206i	$0.519069\pi$
$608$	−6.23047	−0.252679
$609$	0	0
$610$	−3.67848	−0.148937
$611$	4.86933	0.196992
$612$	0	0
$613$	27.8774	1.12596	0.562978	−	0.826472i	$-0.309655\pi$
$613$	27.8774	1.12596	0.562978	+	0.826472i	$0.309655\pi$
$614$	31.4440	1.26898
$615$	0	0
$616$	36.9017	1.48681
$617$	−11.3858	−0.458376	−0.229188	−	0.973382i	$-0.573607\pi$
$617$	−11.3858	−0.458376	−0.229188	+	0.973382i	$0.573607\pi$
$618$	0	0
$619$	−34.8705	−1.40156	−0.700781	−	0.713376i	$-0.747165\pi$
$619$	−34.8705	−1.40156	−0.700781	+	0.713376i	$0.747165\pi$
$620$	−13.3264	−0.535201
$621$	0	0
$622$	60.8680	2.44059
$623$	0.802932	0.0321688
$624$	0	0
$625$	7.31906	0.292763
$626$	37.5038	1.49895
$627$	0	0
$628$	66.9226	2.67050
$629$	19.2206	0.766374
$630$	0	0
$631$	−34.8447	−1.38715	−0.693573	−	0.720387i	$-0.743964\pi$
$631$	−34.8447	−1.38715	−0.693573	+	0.720387i	$0.743964\pi$
$632$	63.4345	2.52329
$633$	0	0
$634$	18.1297	0.720021
$635$	4.73114	0.187750
$636$	0	0
$637$	19.2631	0.763230
$638$	24.7905	0.981465
$639$	0	0
$640$	−7.16114	−0.283069
$641$	22.2357	0.878257	0.439128	−	0.898424i	$-0.355287\pi$
$641$	22.2357	0.878257	0.439128	+	0.898424i	$0.355287\pi$
$642$	0	0
$643$	24.7314	0.975311	0.487655	−	0.873036i	$-0.337852\pi$
$643$	24.7314	0.975311	0.487655	+	0.873036i	$0.337852\pi$
$644$	22.3967	0.882555
$645$	0	0
$646$	−4.93923	−0.194331
$647$	9.10943	0.358129	0.179064	−	0.983837i	$-0.442693\pi$
$647$	9.10943	0.358129	0.179064	+	0.983837i	$0.442693\pi$
$648$	0	0
$649$	64.0265	2.51326
$650$	−29.6287	−1.16213
$651$	0	0
$652$	73.9613	2.89655
$653$	11.4076	0.446415	0.223208	−	0.974771i	$-0.428347\pi$
$653$	11.4076	0.446415	0.223208	+	0.974771i	$0.428347\pi$
$654$	0	0
$655$	−0.761946	−0.0297717
$656$	47.6025	1.85856
$657$	0	0
$658$	−3.37895	−0.131725
$659$	28.5344	1.11154	0.555772	−	0.831335i	$-0.312423\pi$
$659$	28.5344	1.11154	0.555772	+	0.831335i	$0.312423\pi$
$660$	0	0
$661$	38.7410	1.50685	0.753425	−	0.657534i	$-0.228400\pi$
$661$	38.7410	1.50685	0.753425	+	0.657534i	$0.228400\pi$
$662$	−20.3993	−0.792842
$663$	0	0
$664$	−26.3467	−1.02245
$665$	−0.443659	−0.0172044
$666$	0	0
$667$	8.98808	0.348020
$668$	−74.1739	−2.86987
$669$	0	0
$670$	25.9971	1.00436
$671$	7.26969	0.280643
$672$	0	0
$673$	−17.4793	−0.673778	−0.336889	−	0.941544i	$-0.609375\pi$
$673$	−17.4793	−0.673778	−0.336889	+	0.941544i	$0.609375\pi$
$674$	−38.7414	−1.49226
$675$	0	0
$676$	−19.1144	−0.735168
$677$	51.2481	1.96962	0.984812	−	0.173625i	$-0.0555481\pi$
$677$	51.2481	1.96962	0.984812	+	0.173625i	$0.0555481\pi$
$678$	0	0
$679$	4.25111	0.163143
$680$	−33.8821	−1.29932
$681$	0	0
$682$	36.9405	1.41452
$683$	−8.81002	−0.337106	−0.168553	−	0.985693i	$-0.553909\pi$
$683$	−8.81002	−0.337106	−0.168553	+	0.985693i	$0.553909\pi$
$684$	0	0
$685$	11.0006	0.420312
$686$	−28.0621	−1.07142
$687$	0	0
$688$	33.5389	1.27866
$689$	−7.39186	−0.281607
$690$	0	0
$691$	−35.2069	−1.33933	−0.669667	−	0.742662i	$-0.733563\pi$
$691$	−35.2069	−1.33933	−0.669667	+	0.742662i	$0.733563\pi$
$692$	121.893	4.63366
$693$	0	0
$694$	−18.0836	−0.686444
$695$	−21.6155	−0.819923
$696$	0	0
$697$	16.8840	0.639527
$698$	−22.2679	−0.842851
$699$	0	0
$700$	14.6583	0.554030
$701$	27.6589	1.04466	0.522331	−	0.852743i	$-0.325063\pi$
$701$	27.6589	1.04466	0.522331	+	0.852743i	$0.325063\pi$
$702$	0	0
$703$	2.47861	0.0934827
$704$	71.0824	2.67902
$705$	0	0
$706$	20.7671	0.781580
$707$	−7.49393	−0.281838
$708$	0	0
$709$	13.6521	0.512714	0.256357	−	0.966582i	$-0.417478\pi$
$709$	13.6521	0.512714	0.256357	+	0.966582i	$0.417478\pi$
$710$	−44.0038	−1.65143
$711$	0	0
$712$	7.90756	0.296348
$713$	13.3932	0.501579
$714$	0	0
$715$	−20.3514	−0.761098
$716$	−65.6285	−2.45265
$717$	0	0
$718$	−38.1050	−1.42207
$719$	−28.2641	−1.05407	−0.527036	−	0.849843i	$-0.676697\pi$
$719$	−28.2641	−1.05407	−0.527036	+	0.849843i	$0.676697\pi$
$720$	0	0
$721$	−3.46650	−0.129099
$722$	49.5149	1.84275
$723$	0	0
$724$	122.097	4.53771
$725$	5.88253	0.218472
$726$	0	0
$727$	24.9384	0.924913	0.462456	−	0.886642i	$-0.346968\pi$
$727$	24.9384	0.924913	0.462456	+	0.886642i	$0.346968\pi$
$728$	−18.8450	−0.698442
$729$	0	0
$730$	17.4717	0.646657
$731$	11.8958	0.439983
$732$	0	0
$733$	8.32824	0.307611	0.153805	−	0.988101i	$-0.450847\pi$
$733$	8.32824	0.307611	0.153805	+	0.988101i	$0.450847\pi$
$734$	80.9107	2.98647
$735$	0	0
$736$	71.9052	2.65046
$737$	−51.3774	−1.89251
$738$	0	0
$739$	19.1438	0.704215	0.352107	−	0.935960i	$-0.385465\pi$
$739$	19.1438	0.704215	0.352107	+	0.935960i	$0.385465\pi$
$740$	28.4627	1.04631
$741$	0	0
$742$	5.12940	0.188306
$743$	50.0102	1.83470	0.917348	−	0.398086i	$-0.130325\pi$
$743$	50.0102	1.83470	0.917348	+	0.398086i	$0.130325\pi$
$744$	0	0
$745$	−7.19486	−0.263599
$746$	57.0378	2.08830
$747$	0	0
$748$	112.092	4.09848
$749$	−2.64752	−0.0967383
$750$	0	0
$751$	21.9876	0.802340	0.401170	−	0.916004i	$-0.368604\pi$
$751$	21.9876	0.802340	0.401170	+	0.916004i	$0.368604\pi$
$752$	−17.2865	−0.630373
$753$	0	0
$754$	−12.6600	−0.461051
$755$	20.3120	0.739230
$756$	0	0
$757$	−3.90470	−0.141919	−0.0709594	−	0.997479i	$-0.522606\pi$
$757$	−3.90470	−0.141919	−0.0709594	+	0.997479i	$0.522606\pi$
$758$	−75.1635	−2.73006
$759$	0	0
$760$	−4.36931	−0.158491
$761$	3.56728	0.129314	0.0646568	−	0.997908i	$-0.479405\pi$
$761$	3.56728	0.129314	0.0646568	+	0.997908i	$0.479405\pi$
$762$	0	0
$763$	−12.6033	−0.456272
$764$	12.3809	0.447925
$765$	0	0
$766$	−60.9178	−2.20105
$767$	−32.6971	−1.18063
$768$	0	0
$769$	−13.8514	−0.499493	−0.249747	−	0.968311i	$-0.580347\pi$
$769$	−13.8514	−0.499493	−0.249747	+	0.968311i	$0.580347\pi$
$770$	14.1223	0.508934
$771$	0	0
$772$	35.3621	1.27271
$773$	32.1527	1.15645	0.578226	−	0.815877i	$-0.303745\pi$
$773$	32.1527	1.15645	0.578226	+	0.815877i	$0.303745\pi$
$774$	0	0
$775$	8.76560	0.314870
$776$	41.8664	1.50292
$777$	0	0
$778$	13.4658	0.482771
$779$	2.17730	0.0780098
$780$	0	0
$781$	86.9635	3.11180
$782$	57.0031	2.03843
$783$	0	0
$784$	−68.3854	−2.44233
$785$	15.2995	0.546062
$786$	0	0
$787$	−12.3938	−0.441792	−0.220896	−	0.975297i	$-0.570898\pi$
$787$	−12.3938	−0.441792	−0.220896	+	0.975297i	$0.570898\pi$
$788$	−16.0665	−0.572346
$789$	0	0
$790$	24.2765	0.863718
$791$	−2.24425	−0.0797965
$792$	0	0
$793$	−3.71249	−0.131834
$794$	−63.6141	−2.25758
$795$	0	0
$796$	98.7143	3.49884
$797$	−4.92168	−0.174335	−0.0871675	−	0.996194i	$-0.527782\pi$
$797$	−4.92168	−0.174335	−0.0871675	+	0.996194i	$0.527782\pi$
$798$	0	0
$799$	−6.13130	−0.216910
$800$	47.0606	1.66384
$801$	0	0
$802$	26.9654	0.952182
$803$	−34.5289	−1.21850
$804$	0	0
$805$	5.12022	0.180464
$806$	−18.8648	−0.664484
$807$	0	0
$808$	−73.8028	−2.59637
$809$	−6.46429	−0.227272	−0.113636	−	0.993522i	$-0.536250\pi$
$809$	−6.46429	−0.227272	−0.113636	+	0.993522i	$0.536250\pi$
$810$	0	0
$811$	25.3520	0.890228	0.445114	−	0.895474i	$-0.353163\pi$
$811$	25.3520	0.890228	0.445114	+	0.895474i	$0.353163\pi$
$812$	6.26333	0.219800
$813$	0	0
$814$	−78.8980	−2.76538
$815$	16.9086	0.592284
$816$	0	0
$817$	1.53404	0.0536693
$818$	−83.7202	−2.92721
$819$	0	0
$820$	25.0026	0.873129
$821$	−31.1240	−1.08624	−0.543118	−	0.839656i	$-0.682757\pi$
$821$	−31.1240	−1.08624	−0.543118	+	0.839656i	$0.682757\pi$
$822$	0	0
$823$	10.1368	0.353347	0.176673	−	0.984270i	$-0.443466\pi$
$823$	10.1368	0.353347	0.176673	+	0.984270i	$0.443466\pi$
$824$	−34.1393	−1.18930
$825$	0	0
$826$	22.6894	0.789465
$827$	29.4106	1.02271	0.511353	−	0.859370i	$-0.329144\pi$
$827$	29.4106	1.02271	0.511353	+	0.859370i	$0.329144\pi$
$828$	0	0
$829$	6.33820	0.220135	0.110067	−	0.993924i	$-0.464893\pi$
$829$	6.33820	0.220135	0.110067	+	0.993924i	$0.464893\pi$
$830$	−10.0829	−0.349983
$831$	0	0
$832$	−36.3004	−1.25849
$833$	−24.2554	−0.840401
$834$	0	0
$835$	−16.9572	−0.586829
$836$	14.4549	0.499934
$837$	0	0
$838$	−55.6705	−1.92311
$839$	30.2704	1.04505	0.522525	−	0.852624i	$-0.324990\pi$
$839$	30.2704	1.04505	0.522525	+	0.852624i	$0.324990\pi$
$840$	0	0
$841$	−26.4865	−0.913326
$842$	−23.9825	−0.826493
$843$	0	0
$844$	95.4206	3.28451
$845$	−4.36982	−0.150326
$846$	0	0
$847$	−19.1612	−0.658385
$848$	26.2417	0.901143
$849$	0	0
$850$	37.3074	1.27963
$851$	−28.6054	−0.980581
$852$	0	0
$853$	21.6343	0.740746	0.370373	−	0.928883i	$-0.379230\pi$
$853$	21.6343	0.740746	0.370373	+	0.928883i	$0.379230\pi$
$854$	2.57619	0.0881556
$855$	0	0
$856$	−26.0737	−0.891181
$857$	−48.6548	−1.66202	−0.831008	−	0.556261i	$-0.812236\pi$
$857$	−48.6548	−1.66202	−0.831008	+	0.556261i	$0.812236\pi$
$858$	0	0
$859$	−9.81063	−0.334734	−0.167367	−	0.985895i	$-0.553527\pi$
$859$	−9.81063	−0.334734	−0.167367	+	0.985895i	$0.553527\pi$
$860$	17.6159	0.600697
$861$	0	0
$862$	62.9048	2.14255
$863$	9.64646	0.328369	0.164185	−	0.986430i	$-0.447501\pi$
$863$	9.64646	0.328369	0.164185	+	0.986430i	$0.447501\pi$
$864$	0	0
$865$	27.8664	0.947486
$866$	18.9154	0.642771
$867$	0	0
$868$	9.33303	0.316784
$869$	−47.9769	−1.62751
$870$	0	0
$871$	26.2375	0.889023
$872$	−124.122	−4.20330
$873$	0	0
$874$	7.35091	0.248648
$875$	7.86689	0.265949
$876$	0	0
$877$	16.2751	0.549570	0.274785	−	0.961506i	$-0.411393\pi$
$877$	16.2751	0.549570	0.274785	+	0.961506i	$0.411393\pi$
$878$	−45.9330	−1.55016
$879$	0	0
$880$	72.2489	2.43551
$881$	25.4844	0.858590	0.429295	−	0.903164i	$-0.358762\pi$
$881$	25.4844	0.858590	0.429295	+	0.903164i	$0.358762\pi$
$882$	0	0
$883$	39.7750	1.33854	0.669268	−	0.743021i	$-0.266608\pi$
$883$	39.7750	1.33854	0.669268	+	0.743021i	$0.266608\pi$
$884$	−57.2431	−1.92529
$885$	0	0
$886$	17.4462	0.586116
$887$	4.79234	0.160911	0.0804555	−	0.996758i	$-0.474363\pi$
$887$	4.79234	0.160911	0.0804555	+	0.996758i	$0.474363\pi$
$888$	0	0
$889$	−3.31342	−0.111128
$890$	3.02623	0.101440
$891$	0	0
$892$	−22.0158	−0.737143
$893$	−0.790670	−0.0264588
$894$	0	0
$895$	−15.0036	−0.501516
$896$	5.01525	0.167548
$897$	0	0
$898$	61.5650	2.05445
$899$	3.74546	0.124918
$900$	0	0
$901$	9.30759	0.310081
$902$	−69.3067	−2.30766
$903$	0	0
$904$	−22.1022	−0.735108
$905$	27.9132	0.927865
$906$	0	0
$907$	21.1070	0.700847	0.350424	−	0.936591i	$-0.386038\pi$
$907$	21.1070	0.700847	0.350424	+	0.936591i	$0.386038\pi$
$908$	−15.7069	−0.521251
$909$	0	0
$910$	−7.21201	−0.239076
$911$	−22.0145	−0.729374	−0.364687	−	0.931130i	$-0.618824\pi$
$911$	−22.0145	−0.729374	−0.364687	+	0.931130i	$0.618824\pi$
$912$	0	0
$913$	19.9266	0.659474
$914$	62.5961	2.07050
$915$	0	0
$916$	51.8712	1.71387
$917$	0.533623	0.0176218
$918$	0	0
$919$	8.57624	0.282904	0.141452	−	0.989945i	$-0.454823\pi$
$919$	8.57624	0.282904	0.141452	+	0.989945i	$0.454823\pi$
$920$	50.4257	1.66249
$921$	0	0
$922$	−97.6693	−3.21657
$923$	−44.4106	−1.46179
$924$	0	0
$925$	−18.7217	−0.615566
$926$	6.08233	0.199878
$927$	0	0
$928$	20.1085	0.660095
$929$	36.0995	1.18439	0.592194	−	0.805796i	$-0.298262\pi$
$929$	36.0995	1.18439	0.592194	+	0.805796i	$0.298262\pi$
$930$	0	0
$931$	−3.12789	−0.102513
$932$	8.20427	0.268740
$933$	0	0
$934$	−47.1589	−1.54309
$935$	25.6258	0.838052
$936$	0	0
$937$	−35.7735	−1.16867	−0.584335	−	0.811513i	$-0.698644\pi$
$937$	−35.7735	−1.16867	−0.584335	+	0.811513i	$0.698644\pi$
$938$	−18.2069	−0.594476
$939$	0	0
$940$	−9.07951	−0.296141
$941$	−17.9920	−0.586524	−0.293262	−	0.956032i	$-0.594741\pi$
$941$	−17.9920	−0.586524	−0.293262	+	0.956032i	$0.594741\pi$
$942$	0	0
$943$	−25.1280	−0.818279
$944$	116.077	3.77800
$945$	0	0
$946$	−48.8309	−1.58763
$947$	−23.4672	−0.762582	−0.381291	−	0.924455i	$-0.624520\pi$
$947$	−23.4672	−0.762582	−0.381291	+	0.924455i	$0.624520\pi$
$948$	0	0
$949$	17.6332	0.572399
$950$	4.81103	0.156091
$951$	0	0
$952$	23.7290	0.769062
$953$	28.4813	0.922600	0.461300	−	0.887244i	$-0.347383\pi$
$953$	28.4813	0.922600	0.461300	+	0.887244i	$0.347383\pi$
$954$	0	0
$955$	2.83045	0.0915912
$956$	−72.7314	−2.35230
$957$	0	0
$958$	−15.8005	−0.510490
$959$	−7.70419	−0.248781
$960$	0	0
$961$	−25.4189	−0.819964
$962$	40.2917	1.29906
$963$	0	0
$964$	−111.151	−3.57994
$965$	8.08429	0.260243
$966$	0	0
$967$	−41.3878	−1.33094	−0.665472	−	0.746423i	$-0.731770\pi$
$967$	−41.3878	−1.33094	−0.665472	+	0.746423i	$0.731770\pi$
$968$	−188.706	−6.06523
$969$	0	0
$970$	16.0223	0.514446
$971$	28.8070	0.924460	0.462230	−	0.886760i	$-0.347049\pi$
$971$	28.8070	0.924460	0.462230	+	0.886760i	$0.347049\pi$
$972$	0	0
$973$	15.1382	0.485310
$974$	−7.90046	−0.253147
$975$	0	0
$976$	13.1796	0.421870
$977$	−24.8004	−0.793435	−0.396718	−	0.917941i	$-0.629851\pi$
$977$	−24.8004	−0.793435	−0.396718	+	0.917941i	$0.629851\pi$
$978$	0	0
$979$	−5.98066	−0.191143
$980$	−35.9186	−1.14738
$981$	0	0
$982$	−58.3309	−1.86141
$983$	0.0460931	0.00147014	0.000735070	−	1.00000i	$-0.499766\pi$
$983$	0.0460931	0.00147014	0.000735070		1.00000i	$0.499766\pi$
$984$	0	0
$985$	−3.67304	−0.117033
$986$	15.9411	0.507668
$987$	0	0
$988$	−7.38186	−0.234848
$989$	−17.7042	−0.562961
$990$	0	0
$991$	−10.5685	−0.335718	−0.167859	−	0.985811i	$-0.553685\pi$
$991$	−10.5685	−0.335718	−0.167859	+	0.985811i	$0.553685\pi$
$992$	29.9639	0.951353
$993$	0	0
$994$	30.8177	0.977477
$995$	22.5675	0.715438
$996$	0	0
$997$	6.43044	0.203654	0.101827	−	0.994802i	$-0.467531\pi$
$997$	6.43044	0.203654	0.101827	+	0.994802i	$0.467531\pi$
$998$	63.1606	1.99931
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.1		72
3.2	odd	2		6561.2.a.c.1.72		72
81.5	odd	54		729.2.g.d.460.8		144
81.11	odd	54		729.2.g.c.514.1		144
81.16	even	27		729.2.g.a.271.1		144
81.22	even	27		729.2.g.b.217.8		144
81.32	odd	54		81.2.g.a.52.1	✓	144
81.38	odd	54		81.2.g.a.67.1	yes	144
81.43	even	27		243.2.g.a.172.8		144
81.49	even	27		243.2.g.a.154.8		144
81.59	odd	54		729.2.g.c.217.1		144
81.65	odd	54		729.2.g.d.271.8		144
81.70	even	27		729.2.g.b.514.8		144
81.76	even	27		729.2.g.a.460.1		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.52.1	✓	144	81.32	odd	54
81.2.g.a.67.1	yes	144	81.38	odd	54
243.2.g.a.154.8		144	81.49	even	27
243.2.g.a.172.8		144	81.43	even	27
729.2.g.a.271.1		144	81.16	even	27
729.2.g.a.460.1		144	81.76	even	27
729.2.g.b.217.8		144	81.22	even	27
729.2.g.b.514.8		144	81.70	even	27
729.2.g.c.217.1		144	81.59	odd	54
729.2.g.c.514.1		144	81.11	odd	54
729.2.g.d.271.8		144	81.65	odd	54
729.2.g.d.460.8		144	81.5	odd	54
6561.2.a.c.1.72		72	3.2	odd	2
6561.2.a.d.1.1		72	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6561 = 3^{8} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6561.a (trivial)

\(f(q)\)	\(=\)	\(q-2.63957 q^{2} +4.96734 q^{4} +1.13561 q^{5} -0.795312 q^{7} -7.83251 q^{8} +O(q^{10})\)	\(q-2.63957 q^{2} +4.96734 q^{4} +1.13561 q^{5} -0.795312 q^{7} -7.83251 q^{8} -2.99751 q^{10} +5.92390 q^{11} -3.02523 q^{13} +2.09928 q^{14} +10.7398 q^{16} +3.80927 q^{17} +0.491229 q^{19} +5.64094 q^{20} -15.6366 q^{22} -5.66922 q^{23} -3.71040 q^{25} +7.98530 q^{26} -3.95059 q^{28} -1.58542 q^{29} -2.36244 q^{31} -12.6834 q^{32} -10.0548 q^{34} -0.903161 q^{35} +5.04574 q^{37} -1.29664 q^{38} -8.89464 q^{40} +4.43235 q^{41} +3.12286 q^{43} +29.4260 q^{44} +14.9643 q^{46} -1.60957 q^{47} -6.36748 q^{49} +9.79387 q^{50} -15.0273 q^{52} +2.44341 q^{53} +6.72722 q^{55} +6.22929 q^{56} +4.18482 q^{58} +10.8082 q^{59} +1.22718 q^{61} +6.23583 q^{62} +11.9992 q^{64} -3.43546 q^{65} -8.67290 q^{67} +18.9219 q^{68} +2.38396 q^{70} +14.6801 q^{71} -5.82873 q^{73} -13.3186 q^{74} +2.44010 q^{76} -4.71135 q^{77} -8.09887 q^{79} +12.1962 q^{80} -11.6995 q^{82} +3.36376 q^{83} +4.32582 q^{85} -8.24302 q^{86} -46.3990 q^{88} -1.00958 q^{89} +2.40600 q^{91} -28.1610 q^{92} +4.24859 q^{94} +0.557843 q^{95} -5.34521 q^{97} +16.8074 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) \) 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38} + 72 q^{41} + 90 q^{44} + 90 q^{47} + 18 q^{49} + 45 q^{50} + 45 q^{53} + 9 q^{55} + 108 q^{56} + 18 q^{58} + 108 q^{59} + 72 q^{62} + 9 q^{64} + 72 q^{65} + 108 q^{68} + 126 q^{71} + 90 q^{74} + 72 q^{77} + 144 q^{80} - 18 q^{82} + 108 q^{83} + 90 q^{86} + 108 q^{89} + 72 q^{92} + 144 q^{95} + 81 q^{98}+O(q^{100}) \) 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8 + 36 * q^11 + 36 * q^14 + 45 * q^16 + 36 * q^17 + 54 * q^20 + 54 * q^23 + 36 * q^25 + 45 * q^26 + 9 * q^28 + 54 * q^29 + 63 * q^32 + 72 * q^35 + 54 * q^38 + 72 * q^41 + 90 * q^44 + 90 * q^47 + 18 * q^49 + 45 * q^50 + 45 * q^53 + 9 * q^55 + 108 * q^56 + 18 * q^58 + 108 * q^59 + 72 * q^62 + 9 * q^64 + 72 * q^65 + 108 * q^68 + 126 * q^71 + 90 * q^74 + 72 * q^77 + 144 * q^80 - 18 * q^82 + 108 * q^83 + 90 * q^86 + 108 * q^89 + 72 * q^92 + 144 * q^95 + 81 * q^98

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