LMFDB - Embedded newform 6561.2.a.c.1.2

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:	$1$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.66973 q^{2} +5.12748 q^{4} -2.53528 q^{5} +0.510929 q^{7} -8.34954 q^{8} +O(q^{10})$	\(q-2.66973 q^{2} +5.12748 q^{4} -2.53528 q^{5} +0.510929 q^{7} -8.34954 q^{8} +6.76852 q^{10} +1.53684 q^{11} +5.64132 q^{13} -1.36405 q^{14} +12.0361 q^{16} +1.41627 q^{17} -1.05032 q^{19} -12.9996 q^{20} -4.10294 q^{22} +0.260365 q^{23} +1.42764 q^{25} -15.0608 q^{26} +2.61978 q^{28} +1.77853 q^{29} +5.16674 q^{31} -15.4341 q^{32} -3.78107 q^{34} -1.29535 q^{35} -6.59230 q^{37} +2.80407 q^{38} +21.1684 q^{40} -6.75743 q^{41} -6.10982 q^{43} +7.88009 q^{44} -0.695104 q^{46} +7.31149 q^{47} -6.73895 q^{49} -3.81142 q^{50} +28.9258 q^{52} -9.48880 q^{53} -3.89631 q^{55} -4.26603 q^{56} -4.74819 q^{58} -4.95500 q^{59} -12.9008 q^{61} -13.7938 q^{62} +17.1328 q^{64} -14.3023 q^{65} +6.41900 q^{67} +7.26190 q^{68} +3.45824 q^{70} -5.16398 q^{71} -5.96195 q^{73} +17.5997 q^{74} -5.38549 q^{76} +0.785214 q^{77} +9.94206 q^{79} -30.5149 q^{80} +18.0405 q^{82} +12.9521 q^{83} -3.59064 q^{85} +16.3116 q^{86} -12.8319 q^{88} -9.86247 q^{89} +2.88231 q^{91} +1.33501 q^{92} -19.5197 q^{94} +2.66285 q^{95} -3.59700 q^{97} +17.9912 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.66973	−1.88779	−0.943894	−	0.330250i	$-0.892867\pi$
$2$	−2.66973	−1.88779	−0.943894	+	0.330250i	$0.892867\pi$
$3$	0	0
$3$	0	0
$4$	5.12748	2.56374
$5$	−2.53528	−1.13381	−0.566906	−	0.823783i	$-0.691860\pi$
$5$	−2.53528	−1.13381	−0.566906	+	0.823783i	$0.691860\pi$
$6$	0	0
$7$	0.510929	0.193113	0.0965565	−	0.995328i	$-0.469217\pi$
$7$	0.510929	0.193113	0.0965565	+	0.995328i	$0.469217\pi$
$8$	−8.34954	−2.95201
$9$	0	0
$10$	6.76852	2.14039
$11$	1.53684	0.463373	0.231687	−	0.972790i	$-0.425576\pi$
$11$	1.53684	0.463373	0.231687	+	0.972790i	$0.425576\pi$
$12$	0	0
$13$	5.64132	1.56462	0.782310	−	0.622889i	$-0.214041\pi$
$13$	5.64132	1.56462	0.782310	+	0.622889i	$0.214041\pi$
$14$	−1.36405	−0.364556
$15$	0	0
$16$	12.0361	3.00903
$17$	1.41627	0.343496	0.171748	−	0.985141i	$-0.445059\pi$
$17$	1.41627	0.343496	0.171748	+	0.985141i	$0.445059\pi$
$18$	0	0
$19$	−1.05032	−0.240960	−0.120480	−	0.992716i	$-0.538443\pi$
$19$	−1.05032	−0.240960	−0.120480	+	0.992716i	$0.538443\pi$
$20$	−12.9996	−2.90680
$21$	0	0
$22$	−4.10294	−0.874750
$23$	0.260365	0.0542898	0.0271449	−	0.999632i	$-0.491358\pi$
$23$	0.260365	0.0542898	0.0271449	+	0.999632i	$0.491358\pi$
$24$	0	0
$25$	1.42764	0.285528
$26$	−15.0608	−2.95367
$27$	0	0
$28$	2.61978	0.495092
$29$	1.77853	0.330264	0.165132	−	0.986271i	$-0.447195\pi$
$29$	1.77853	0.330264	0.165132	+	0.986271i	$0.447195\pi$
$30$	0	0
$31$	5.16674	0.927975	0.463987	−	0.885842i	$-0.346418\pi$
$31$	5.16674	0.927975	0.463987	+	0.885842i	$0.346418\pi$
$32$	−15.4341	−2.72839
$33$	0	0
$34$	−3.78107	−0.648447
$35$	−1.29535	−0.218954
$36$	0	0
$37$	−6.59230	−1.08377	−0.541884	−	0.840454i	$-0.682289\pi$
$37$	−6.59230	−1.08377	−0.541884	+	0.840454i	$0.682289\pi$
$38$	2.80407	0.454881
$39$	0	0
$40$	21.1684	3.34702
$41$	−6.75743	−1.05533	−0.527667	−	0.849451i	$-0.676933\pi$
$41$	−6.75743	−1.05533	−0.527667	+	0.849451i	$0.676933\pi$
$42$	0	0
$43$	−6.10982	−0.931739	−0.465870	−	0.884853i	$-0.654258\pi$
$43$	−6.10982	−0.931739	−0.465870	+	0.884853i	$0.654258\pi$
$44$	7.88009	1.18797
$45$	0	0
$46$	−0.695104	−0.102488
$47$	7.31149	1.06649	0.533246	−	0.845961i	$-0.320972\pi$
$47$	7.31149	1.06649	0.533246	+	0.845961i	$0.320972\pi$
$48$	0	0
$49$	−6.73895	−0.962707
$50$	−3.81142	−0.539017
$51$	0	0
$52$	28.9258	4.01128
$53$	−9.48880	−1.30339	−0.651694	−	0.758482i	$-0.725941\pi$
$53$	−9.48880	−1.30339	−0.651694	+	0.758482i	$0.725941\pi$
$54$	0	0
$55$	−3.89631	−0.525378
$56$	−4.26603	−0.570072
$57$	0	0
$58$	−4.74819	−0.623468
$59$	−4.95500	−0.645087	−0.322543	−	0.946555i	$-0.604538\pi$
$59$	−4.95500	−0.645087	−0.322543	+	0.946555i	$0.604538\pi$
$60$	0	0
$61$	−12.9008	−1.65178	−0.825888	−	0.563834i	$-0.809326\pi$
$61$	−12.9008	−1.65178	−0.825888	+	0.563834i	$0.809326\pi$
$62$	−13.7938	−1.75182
$63$	0	0
$64$	17.1328	2.14159
$65$	−14.3023	−1.77398
$66$	0	0
$67$	6.41900	0.784205	0.392103	−	0.919921i	$-0.371748\pi$
$67$	6.41900	0.784205	0.392103	+	0.919921i	$0.371748\pi$
$68$	7.26190	0.880635
$69$	0	0
$70$	3.45824	0.413338
$71$	−5.16398	−0.612852	−0.306426	−	0.951895i	$-0.599133\pi$
$71$	−5.16398	−0.612852	−0.306426	+	0.951895i	$0.599133\pi$
$72$	0	0
$73$	−5.96195	−0.697793	−0.348897	−	0.937161i	$-0.613444\pi$
$73$	−5.96195	−0.697793	−0.348897	+	0.937161i	$0.613444\pi$
$74$	17.5997	2.04592
$75$	0	0
$76$	−5.38549	−0.617758
$77$	0.785214	0.0894834
$78$	0	0
$79$	9.94206	1.11857	0.559284	−	0.828976i	$-0.311076\pi$
$79$	9.94206	1.11857	0.559284	+	0.828976i	$0.311076\pi$
$80$	−30.5149	−3.41167
$81$	0	0
$82$	18.0405	1.99224
$83$	12.9521	1.42168	0.710839	−	0.703355i	$-0.248315\pi$
$83$	12.9521	1.42168	0.710839	+	0.703355i	$0.248315\pi$
$84$	0	0
$85$	−3.59064	−0.389460
$86$	16.3116	1.75893
$87$	0	0
$88$	−12.8319	−1.36788
$89$	−9.86247	−1.04542	−0.522710	−	0.852511i	$-0.675079\pi$
$89$	−9.86247	−1.04542	−0.522710	+	0.852511i	$0.675079\pi$
$90$	0	0
$91$	2.88231	0.302149
$92$	1.33501	0.139185
$93$	0	0
$94$	−19.5197	−2.01331
$95$	2.66285	0.273203
$96$	0	0
$97$	−3.59700	−0.365220	−0.182610	−	0.983185i	$-0.558455\pi$
$97$	−3.59700	−0.365220	−0.182610	+	0.983185i	$0.558455\pi$
$98$	17.9912	1.81739
$99$	0	0
$100$	7.32021	0.732021
$101$	17.8234	1.77349	0.886746	−	0.462257i	$-0.152960\pi$
$101$	17.8234	1.77349	0.886746	+	0.462257i	$0.152960\pi$
$102$	0	0
$103$	4.31751	0.425416	0.212708	−	0.977116i	$-0.431772\pi$
$103$	4.31751	0.425416	0.212708	+	0.977116i	$0.431772\pi$
$104$	−47.1024	−4.61877
$105$	0	0
$106$	25.3326	2.46052
$107$	−7.97881	−0.771340	−0.385670	−	0.922637i	$-0.626030\pi$
$107$	−7.97881	−0.771340	−0.385670	+	0.922637i	$0.626030\pi$
$108$	0	0
$109$	−0.212864	−0.0203886	−0.0101943	−	0.999948i	$-0.503245\pi$
$109$	−0.212864	−0.0203886	−0.0101943	+	0.999948i	$0.503245\pi$
$110$	10.4021	0.991802
$111$	0	0
$112$	6.14959	0.581082
$113$	5.54629	0.521751	0.260875	−	0.965372i	$-0.415989\pi$
$113$	5.54629	0.521751	0.260875	+	0.965372i	$0.415989\pi$
$114$	0	0
$115$	−0.660097	−0.0615543
$116$	9.11936	0.846711
$117$	0	0
$118$	13.2285	1.21779
$119$	0.723614	0.0663336
$120$	0	0
$121$	−8.63814	−0.785285
$122$	34.4417	3.11820
$123$	0	0
$124$	26.4924	2.37909
$125$	9.05693	0.810076
$126$	0	0
$127$	−18.2587	−1.62020	−0.810101	−	0.586291i	$-0.800588\pi$
$127$	−18.2587	−1.62020	−0.810101	+	0.586291i	$0.800588\pi$
$128$	−14.8717	−1.31448
$129$	0	0
$130$	38.1834	3.34891
$131$	8.04034	0.702488	0.351244	−	0.936284i	$-0.385759\pi$
$131$	8.04034	0.702488	0.351244	+	0.936284i	$0.385759\pi$
$132$	0	0
$133$	−0.536639	−0.0465325
$134$	−17.1370	−1.48041
$135$	0	0
$136$	−11.8252	−1.01400
$137$	8.53254	0.728984	0.364492	−	0.931206i	$-0.381243\pi$
$137$	8.53254	0.728984	0.364492	+	0.931206i	$0.381243\pi$
$138$	0	0
$139$	13.0494	1.10683	0.553417	−	0.832904i	$-0.313323\pi$
$139$	13.0494	1.10683	0.553417	+	0.832904i	$0.313323\pi$
$140$	−6.64187	−0.561341
$141$	0	0
$142$	13.7865	1.15693
$143$	8.66978	0.725003
$144$	0	0
$145$	−4.50906	−0.374457
$146$	15.9168	1.31729
$147$	0	0
$148$	−33.8019	−2.77850
$149$	−16.3147	−1.33655	−0.668275	−	0.743915i	$-0.732967\pi$
$149$	−16.3147	−1.33655	−0.668275	+	0.743915i	$0.732967\pi$
$150$	0	0
$151$	21.6564	1.76237	0.881186	−	0.472770i	$-0.156746\pi$
$151$	21.6564	1.76237	0.881186	+	0.472770i	$0.156746\pi$
$152$	8.76968	0.711315
$153$	0	0
$154$	−2.09631	−0.168926
$155$	−13.0991	−1.05215
$156$	0	0
$157$	−2.52400	−0.201437	−0.100718	−	0.994915i	$-0.532114\pi$
$157$	−2.52400	−0.201437	−0.100718	+	0.994915i	$0.532114\pi$
$158$	−26.5427	−2.11162
$159$	0	0
$160$	39.1298	3.09348
$161$	0.133028	0.0104841
$162$	0	0
$163$	7.30888	0.572476	0.286238	−	0.958159i	$-0.407595\pi$
$163$	7.30888	0.572476	0.286238	+	0.958159i	$0.407595\pi$
$164$	−34.6486	−2.70560
$165$	0	0
$166$	−34.5787	−2.68383
$167$	−6.41180	−0.496160	−0.248080	−	0.968740i	$-0.579800\pi$
$167$	−6.41180	−0.496160	−0.248080	+	0.968740i	$0.579800\pi$
$168$	0	0
$169$	18.8245	1.44804
$170$	9.58606	0.735217
$171$	0	0
$172$	−31.3280	−2.38874
$173$	−4.88855	−0.371669	−0.185835	−	0.982581i	$-0.559499\pi$
$173$	−4.88855	−0.371669	−0.185835	+	0.982581i	$0.559499\pi$
$174$	0	0
$175$	0.729424	0.0551393
$176$	18.4975	1.39430
$177$	0	0
$178$	26.3302	1.97353
$179$	−0.512591	−0.0383129	−0.0191564	−	0.999816i	$-0.506098\pi$
$179$	−0.512591	−0.0383129	−0.0191564	+	0.999816i	$0.506098\pi$
$180$	0	0
$181$	−3.77832	−0.280841	−0.140420	−	0.990092i	$-0.544845\pi$
$181$	−3.77832	−0.280841	−0.140420	+	0.990092i	$0.544845\pi$
$182$	−7.69501	−0.570392
$183$	0	0
$184$	−2.17392	−0.160264
$185$	16.7133	1.22879
$186$	0	0
$187$	2.17657	0.159167
$188$	37.4895	2.73421
$189$	0	0
$190$	−7.10911	−0.515749
$191$	−22.6612	−1.63971	−0.819854	−	0.572572i	$-0.805946\pi$
$191$	−22.6612	−1.63971	−0.819854	+	0.572572i	$0.805946\pi$
$192$	0	0
$193$	−5.72352	−0.411988	−0.205994	−	0.978553i	$-0.566043\pi$
$193$	−5.72352	−0.411988	−0.205994	+	0.978553i	$0.566043\pi$
$194$	9.60303	0.689458
$195$	0	0
$196$	−34.5538	−2.46813
$197$	−2.41997	−0.172416	−0.0862079	−	0.996277i	$-0.527475\pi$
$197$	−2.41997	−0.172416	−0.0862079	+	0.996277i	$0.527475\pi$
$198$	0	0
$199$	−9.78546	−0.693673	−0.346837	−	0.937926i	$-0.612744\pi$
$199$	−9.78546	−0.693673	−0.346837	+	0.937926i	$0.612744\pi$
$200$	−11.9202	−0.842883
$201$	0	0
$202$	−47.5837	−3.34798
$203$	0.908701	0.0637783
$204$	0	0
$205$	17.1320	1.19655
$206$	−11.5266	−0.803096
$207$	0	0
$208$	67.8995	4.70798
$209$	−1.61417	−0.111654
$210$	0	0
$211$	−19.6118	−1.35013	−0.675067	−	0.737756i	$-0.735885\pi$
$211$	−19.6118	−1.35013	−0.675067	+	0.737756i	$0.735885\pi$
$212$	−48.6537	−3.34155
$213$	0	0
$214$	21.3013	1.45613
$215$	15.4901	1.05642
$216$	0	0
$217$	2.63984	0.179204
$218$	0.568290	0.0384894
$219$	0	0
$220$	−19.9782	−1.34693
$221$	7.98963	0.537441
$222$	0	0
$223$	2.62006	0.175452	0.0877260	−	0.996145i	$-0.472040\pi$
$223$	2.62006	0.175452	0.0877260	+	0.996145i	$0.472040\pi$
$224$	−7.88573	−0.526888
$225$	0	0
$226$	−14.8071	−0.984955
$227$	1.23513	0.0819784	0.0409892	−	0.999160i	$-0.486949\pi$
$227$	1.23513	0.0819784	0.0409892	+	0.999160i	$0.486949\pi$
$228$	0	0
$229$	4.89914	0.323744	0.161872	−	0.986812i	$-0.448247\pi$
$229$	4.89914	0.323744	0.161872	+	0.986812i	$0.448247\pi$
$230$	1.76228	0.116202
$231$	0	0
$232$	−14.8499	−0.974943
$233$	−3.82410	−0.250525	−0.125263	−	0.992124i	$-0.539977\pi$
$233$	−3.82410	−0.250525	−0.125263	+	0.992124i	$0.539977\pi$
$234$	0	0
$235$	−18.5367	−1.20920
$236$	−25.4067	−1.65383
$237$	0	0
$238$	−1.93186	−0.125224
$239$	−9.04710	−0.585208	−0.292604	−	0.956234i	$-0.594522\pi$
$239$	−9.04710	−0.585208	−0.292604	+	0.956234i	$0.594522\pi$
$240$	0	0
$241$	−23.4260	−1.50900	−0.754501	−	0.656298i	$-0.772121\pi$
$241$	−23.4260	−1.50900	−0.754501	+	0.656298i	$0.772121\pi$
$242$	23.0615	1.48245
$243$	0	0
$244$	−66.1485	−4.23473
$245$	17.0851	1.09153
$246$	0	0
$247$	−5.92518	−0.377010
$248$	−43.1400	−2.73939
$249$	0	0
$250$	−24.1796	−1.52925
$251$	−12.3315	−0.778355	−0.389177	−	0.921163i	$-0.627241\pi$
$251$	−12.3315	−0.778355	−0.389177	+	0.921163i	$0.627241\pi$
$252$	0	0
$253$	0.400137	0.0251564
$254$	48.7460	3.05860
$255$	0	0
$256$	5.43796	0.339873
$257$	29.4800	1.83891	0.919456	−	0.393192i	$-0.128629\pi$
$257$	29.4800	1.83891	0.919456	+	0.393192i	$0.128629\pi$
$258$	0	0
$259$	−3.36820	−0.209290
$260$	−73.3349	−4.54804
$261$	0	0
$262$	−21.4656	−1.32615
$263$	−20.5893	−1.26959	−0.634794	−	0.772681i	$-0.718915\pi$
$263$	−20.5893	−1.26959	−0.634794	+	0.772681i	$0.718915\pi$
$264$	0	0
$265$	24.0568	1.47780
$266$	1.43268	0.0878434
$267$	0	0
$268$	32.9133	2.01050
$269$	12.1719	0.742132	0.371066	−	0.928607i	$-0.378992\pi$
$269$	12.1719	0.742132	0.371066	+	0.928607i	$0.378992\pi$
$270$	0	0
$271$	23.1153	1.40416	0.702078	−	0.712100i	$-0.252256\pi$
$271$	23.1153	1.40416	0.702078	+	0.712100i	$0.252256\pi$
$272$	17.0464	1.03359
$273$	0	0
$274$	−22.7796	−1.37617
$275$	2.19405	0.132306
$276$	0	0
$277$	1.36875	0.0822401	0.0411200	−	0.999154i	$-0.486907\pi$
$277$	1.36875	0.0822401	0.0411200	+	0.999154i	$0.486907\pi$
$278$	−34.8384	−2.08947
$279$	0	0
$280$	10.8156	0.646354
$281$	−17.9046	−1.06810	−0.534048	−	0.845454i	$-0.679330\pi$
$281$	−17.9046	−1.06810	−0.534048	+	0.845454i	$0.679330\pi$
$282$	0	0
$283$	9.75661	0.579970	0.289985	−	0.957031i	$-0.406350\pi$
$283$	9.75661	0.579970	0.289985	+	0.957031i	$0.406350\pi$
$284$	−26.4782	−1.57119
$285$	0	0
$286$	−23.1460	−1.36865
$287$	−3.45257	−0.203799
$288$	0	0
$289$	−14.9942	−0.882010
$290$	12.0380	0.706895
$291$	0	0
$292$	−30.5698	−1.78896
$293$	3.07192	0.179464	0.0897318	−	0.995966i	$-0.471399\pi$
$293$	3.07192	0.179464	0.0897318	+	0.995966i	$0.471399\pi$
$294$	0	0
$295$	12.5623	0.731407
$296$	55.0427	3.19929
$297$	0	0
$298$	43.5558	2.52312
$299$	1.46880	0.0849428
$300$	0	0
$301$	−3.12169	−0.179931
$302$	−57.8168	−3.32698
$303$	0	0
$304$	−12.6417	−0.725054
$305$	32.7071	1.87280
$306$	0	0
$307$	−5.69149	−0.324830	−0.162415	−	0.986723i	$-0.551928\pi$
$307$	−5.69149	−0.324830	−0.162415	+	0.986723i	$0.551928\pi$
$308$	4.02617	0.229412
$309$	0	0
$310$	34.9712	1.98623
$311$	−3.25150	−0.184376	−0.0921878	−	0.995742i	$-0.529386\pi$
$311$	−3.25150	−0.184376	−0.0921878	+	0.995742i	$0.529386\pi$
$312$	0	0
$313$	−9.45939	−0.534676	−0.267338	−	0.963603i	$-0.586144\pi$
$313$	−9.45939	−0.534676	−0.267338	+	0.963603i	$0.586144\pi$
$314$	6.73840	0.380270
$315$	0	0
$316$	50.9777	2.86772
$317$	20.5097	1.15194	0.575969	−	0.817472i	$-0.304625\pi$
$317$	20.5097	1.15194	0.575969	+	0.817472i	$0.304625\pi$
$318$	0	0
$319$	2.73330	0.153036
$320$	−43.4363	−2.42816
$321$	0	0
$322$	−0.355149	−0.0197917
$323$	−1.48754	−0.0827687
$324$	0	0
$325$	8.05378	0.446744
$326$	−19.5128	−1.08071
$327$	0	0
$328$	56.4215	3.11535
$329$	3.73566	0.205953
$330$	0	0
$331$	−6.90158	−0.379345	−0.189673	−	0.981847i	$-0.560743\pi$
$331$	−6.90158	−0.379345	−0.189673	+	0.981847i	$0.560743\pi$
$332$	66.4117	3.64481
$333$	0	0
$334$	17.1178	0.936644
$335$	−16.2739	−0.889141
$336$	0	0
$337$	27.1198	1.47731	0.738656	−	0.674082i	$-0.235461\pi$
$337$	27.1198	1.47731	0.738656	+	0.674082i	$0.235461\pi$
$338$	−50.2563	−2.73359
$339$	0	0
$340$	−18.4109	−0.998474
$341$	7.94044	0.429999
$342$	0	0
$343$	−7.01963	−0.379024
$344$	51.0142	2.75050
$345$	0	0
$346$	13.0511	0.701633
$347$	23.2132	1.24615	0.623074	−	0.782163i	$-0.285883\pi$
$347$	23.2132	1.24615	0.623074	+	0.782163i	$0.285883\pi$
$348$	0	0
$349$	3.06982	0.164324	0.0821619	−	0.996619i	$-0.473818\pi$
$349$	3.06982	0.164324	0.0821619	+	0.996619i	$0.473818\pi$
$350$	−1.94737	−0.104091
$351$	0	0
$352$	−23.7197	−1.26426
$353$	10.5013	0.558929	0.279465	−	0.960156i	$-0.409843\pi$
$353$	10.5013	0.558929	0.279465	+	0.960156i	$0.409843\pi$
$354$	0	0
$355$	13.0921	0.694858
$356$	−50.5696	−2.68018
$357$	0	0
$358$	1.36848	0.0723265
$359$	30.8781	1.62969	0.814843	−	0.579682i	$-0.196823\pi$
$359$	30.8781	1.62969	0.814843	+	0.579682i	$0.196823\pi$
$360$	0	0
$361$	−17.8968	−0.941938
$362$	10.0871	0.530167
$363$	0	0
$364$	14.7790	0.774631
$365$	15.1152	0.791166
$366$	0	0
$367$	15.2239	0.794680	0.397340	−	0.917671i	$-0.369933\pi$
$367$	15.2239	0.794680	0.397340	+	0.917671i	$0.369933\pi$
$368$	3.13377	0.163359
$369$	0	0
$370$	−44.6201	−2.31969
$371$	−4.84811	−0.251701
$372$	0	0
$373$	−0.933334	−0.0483262	−0.0241631	−	0.999708i	$-0.507692\pi$
$373$	−0.933334	−0.0483262	−0.0241631	+	0.999708i	$0.507692\pi$
$374$	−5.81088	−0.300473
$375$	0	0
$376$	−61.0476	−3.14829
$377$	10.0332	0.516738
$378$	0	0
$379$	−38.4657	−1.97585	−0.987924	−	0.154939i	$-0.950482\pi$
$379$	−38.4657	−1.97585	−0.987924	+	0.154939i	$0.950482\pi$
$380$	13.6537	0.700421
$381$	0	0
$382$	60.4994	3.09542
$383$	14.7526	0.753824	0.376912	−	0.926249i	$-0.376986\pi$
$383$	14.7526	0.753824	0.376912	+	0.926249i	$0.376986\pi$
$384$	0	0
$385$	−1.99074	−0.101457
$386$	15.2803	0.777745
$387$	0	0
$388$	−18.4435	−0.936329
$389$	−17.0888	−0.866434	−0.433217	−	0.901290i	$-0.642622\pi$
$389$	−17.0888	−0.866434	−0.433217	+	0.901290i	$0.642622\pi$
$390$	0	0
$391$	0.368747	0.0186483
$392$	56.2672	2.84192
$393$	0	0
$394$	6.46068	0.325484
$395$	−25.2059	−1.26825
$396$	0	0
$397$	−25.1897	−1.26423	−0.632117	−	0.774873i	$-0.717814\pi$
$397$	−25.1897	−1.26423	−0.632117	+	0.774873i	$0.717814\pi$
$398$	26.1246	1.30951
$399$	0	0
$400$	17.1832	0.859162
$401$	−20.6304	−1.03023	−0.515116	−	0.857120i	$-0.672251\pi$
$401$	−20.6304	−1.03023	−0.515116	+	0.857120i	$0.672251\pi$
$402$	0	0
$403$	29.1473	1.45193
$404$	91.3890	4.54677
$405$	0	0
$406$	−2.42599	−0.120400
$407$	−10.1313	−0.502189
$408$	0	0
$409$	35.1805	1.73956	0.869781	−	0.493437i	$-0.164260\pi$
$409$	35.1805	1.73956	0.869781	+	0.493437i	$0.164260\pi$
$410$	−45.7378	−2.25883
$411$	0	0
$412$	22.1379	1.09066
$413$	−2.53166	−0.124575
$414$	0	0
$415$	−32.8372	−1.61191
$416$	−87.0687	−4.26889
$417$	0	0
$418$	4.30940	0.210780
$419$	−27.1057	−1.32420	−0.662099	−	0.749416i	$-0.730334\pi$
$419$	−27.1057	−1.32420	−0.662099	+	0.749416i	$0.730334\pi$
$420$	0	0
$421$	−30.4383	−1.48347	−0.741736	−	0.670692i	$-0.765997\pi$
$421$	−30.4383	−1.48347	−0.741736	+	0.670692i	$0.765997\pi$
$422$	52.3584	2.54877
$423$	0	0
$424$	79.2272	3.84761
$425$	2.02193	0.0980779
$426$	0	0
$427$	−6.59139	−0.318980
$428$	−40.9112	−1.97752
$429$	0	0
$430$	−41.3545	−1.99429
$431$	9.01867	0.434414	0.217207	−	0.976126i	$-0.430305\pi$
$431$	9.01867	0.434414	0.217207	+	0.976126i	$0.430305\pi$
$432$	0	0
$433$	23.7045	1.13917	0.569584	−	0.821933i	$-0.307104\pi$
$433$	23.7045	1.13917	0.569584	+	0.821933i	$0.307104\pi$
$434$	−7.04767	−0.338299
$435$	0	0
$436$	−1.09145	−0.0522712
$437$	−0.273466	−0.0130816
$438$	0	0
$439$	4.02472	0.192090	0.0960448	−	0.995377i	$-0.469381\pi$
$439$	4.02472	0.192090	0.0960448	+	0.995377i	$0.469381\pi$
$440$	32.5324	1.55092
$441$	0	0
$442$	−21.3302	−1.01457
$443$	7.95764	0.378079	0.189039	−	0.981970i	$-0.439463\pi$
$443$	7.95764	0.378079	0.189039	+	0.981970i	$0.439463\pi$
$444$	0	0
$445$	25.0041	1.18531
$446$	−6.99486	−0.331216
$447$	0	0
$448$	8.75362	0.413570
$449$	−14.7366	−0.695465	−0.347733	−	0.937594i	$-0.613048\pi$
$449$	−14.7366	−0.695465	−0.347733	+	0.937594i	$0.613048\pi$
$450$	0	0
$451$	−10.3851	−0.489013
$452$	28.4385	1.33763
$453$	0	0
$454$	−3.29747	−0.154758
$455$	−7.30747	−0.342580
$456$	0	0
$457$	−7.86066	−0.367706	−0.183853	−	0.982954i	$-0.558857\pi$
$457$	−7.86066	−0.367706	−0.183853	+	0.982954i	$0.558857\pi$
$458$	−13.0794	−0.611160
$459$	0	0
$460$	−3.38463	−0.157809
$461$	3.30382	0.153874	0.0769371	−	0.997036i	$-0.475486\pi$
$461$	3.30382	0.153874	0.0769371	+	0.997036i	$0.475486\pi$
$462$	0	0
$463$	−17.7015	−0.822657	−0.411329	−	0.911487i	$-0.634935\pi$
$463$	−17.7015	−0.822657	−0.411329	+	0.911487i	$0.634935\pi$
$464$	21.4065	0.993773
$465$	0	0
$466$	10.2093	0.472939
$467$	−39.8582	−1.84442	−0.922210	−	0.386690i	$-0.873618\pi$
$467$	−39.8582	−1.84442	−0.922210	+	0.386690i	$0.873618\pi$
$468$	0	0
$469$	3.27965	0.151440
$470$	49.4880	2.28271
$471$	0	0
$472$	41.3720	1.90430
$473$	−9.38979	−0.431743
$474$	0	0
$475$	−1.49948	−0.0688008
$476$	3.71032	0.170062
$477$	0	0
$478$	24.1534	1.10475
$479$	−39.1908	−1.79067	−0.895337	−	0.445389i	$-0.853065\pi$
$479$	−39.1908	−1.79067	−0.895337	+	0.445389i	$0.853065\pi$
$480$	0	0
$481$	−37.1893	−1.69568
$482$	62.5413	2.84868
$483$	0	0
$484$	−44.2919	−2.01327
$485$	9.11940	0.414091
$486$	0	0
$487$	35.0579	1.58863	0.794313	−	0.607509i	$-0.207831\pi$
$487$	35.0579	1.58863	0.794313	+	0.607509i	$0.207831\pi$
$488$	107.716	4.87606
$489$	0	0
$490$	−45.6127	−2.06057
$491$	20.5821	0.928858	0.464429	−	0.885610i	$-0.346260\pi$
$491$	20.5821	0.928858	0.464429	+	0.885610i	$0.346260\pi$
$492$	0	0
$493$	2.51887	0.113444
$494$	15.8187	0.711715
$495$	0	0
$496$	62.1875	2.79230
$497$	−2.63843	−0.118350
$498$	0	0
$499$	8.24527	0.369109	0.184554	−	0.982822i	$-0.440916\pi$
$499$	8.24527	0.369109	0.184554	+	0.982822i	$0.440916\pi$
$500$	46.4392	2.07682
$501$	0	0
$502$	32.9217	1.46937
$503$	−17.0166	−0.758735	−0.379367	−	0.925246i	$-0.623858\pi$
$503$	−17.0166	−0.758735	−0.379367	+	0.925246i	$0.623858\pi$
$504$	0	0
$505$	−45.1872	−2.01081
$506$	−1.06826	−0.0474900
$507$	0	0
$508$	−93.6214	−4.15378
$509$	−16.4052	−0.727149	−0.363574	−	0.931565i	$-0.618444\pi$
$509$	−16.4052	−0.727149	−0.363574	+	0.931565i	$0.618444\pi$
$510$	0	0
$511$	−3.04613	−0.134753
$512$	15.2255	0.672877
$513$	0	0
$514$	−78.7038	−3.47148
$515$	−10.9461	−0.482342
$516$	0	0
$517$	11.2366	0.494183
$518$	8.99219	0.395094
$519$	0	0
$520$	119.418	5.23682
$521$	−12.0781	−0.529152	−0.264576	−	0.964365i	$-0.585232\pi$
$521$	−12.0781	−0.529152	−0.264576	+	0.964365i	$0.585232\pi$
$522$	0	0
$523$	−7.22034	−0.315723	−0.157862	−	0.987461i	$-0.550460\pi$
$523$	−7.22034	−0.315723	−0.157862	+	0.987461i	$0.550460\pi$
$524$	41.2267	1.80100
$525$	0	0
$526$	54.9678	2.39671
$527$	7.31751	0.318756
$528$	0	0
$529$	−22.9322	−0.997053
$530$	−64.2252	−2.78976
$531$	0	0
$532$	−2.75160	−0.119297
$533$	−38.1208	−1.65120
$534$	0	0
$535$	20.2285	0.874555
$536$	−53.5957	−2.31498
$537$	0	0
$538$	−32.4956	−1.40099
$539$	−10.3567	−0.446093
$540$	0	0
$541$	38.1612	1.64068	0.820339	−	0.571878i	$-0.193785\pi$
$541$	38.1612	1.64068	0.820339	+	0.571878i	$0.193785\pi$
$542$	−61.7118	−2.65075
$543$	0	0
$544$	−21.8589	−0.937191
$545$	0.539669	0.0231169
$546$	0	0
$547$	−37.1831	−1.58984	−0.794918	−	0.606717i	$-0.792486\pi$
$547$	−37.1831	−1.58984	−0.794918	+	0.606717i	$0.792486\pi$
$548$	43.7505	1.86893
$549$	0	0
$550$	−5.85753	−0.249766
$551$	−1.86802	−0.0795803
$552$	0	0
$553$	5.07969	0.216010
$554$	−3.65419	−0.155252
$555$	0	0
$556$	66.9105	2.83764
$557$	24.5448	1.04000	0.519998	−	0.854168i	$-0.325933\pi$
$557$	24.5448	1.04000	0.519998	+	0.854168i	$0.325933\pi$
$558$	0	0
$559$	−34.4675	−1.45782
$560$	−15.5909	−0.658838
$561$	0	0
$562$	47.8004	2.01634
$563$	7.06005	0.297546	0.148773	−	0.988871i	$-0.452468\pi$
$563$	7.06005	0.297546	0.148773	+	0.988871i	$0.452468\pi$
$564$	0	0
$565$	−14.0614	−0.591567
$566$	−26.0476	−1.09486
$567$	0	0
$568$	43.1169	1.80914
$569$	−8.46120	−0.354712	−0.177356	−	0.984147i	$-0.556754\pi$
$569$	−8.46120	−0.354712	−0.177356	+	0.984147i	$0.556754\pi$
$570$	0	0
$571$	−0.441876	−0.0184919	−0.00924597	−	0.999957i	$-0.502943\pi$
$571$	−0.441876	−0.0184919	−0.00924597	+	0.999957i	$0.502943\pi$
$572$	44.4541	1.85872
$573$	0	0
$574$	9.21744	0.384729
$575$	0.371707	0.0155013
$576$	0	0
$577$	10.4125	0.433480	0.216740	−	0.976229i	$-0.430458\pi$
$577$	10.4125	0.433480	0.216740	+	0.976229i	$0.430458\pi$
$578$	40.0305	1.66505
$579$	0	0
$580$	−23.1201	−0.960011
$581$	6.61761	0.274545
$582$	0	0
$583$	−14.5827	−0.603955
$584$	49.7796	2.05989
$585$	0	0
$586$	−8.20122	−0.338789
$587$	−36.9284	−1.52420	−0.762100	−	0.647459i	$-0.775832\pi$
$587$	−36.9284	−1.52420	−0.762100	+	0.647459i	$0.775832\pi$
$588$	0	0
$589$	−5.42673	−0.223604
$590$	−33.5381	−1.38074
$591$	0	0
$592$	−79.3456	−3.26108
$593$	−2.31384	−0.0950181	−0.0475091	−	0.998871i	$-0.515128\pi$
$593$	−2.31384	−0.0950181	−0.0475091	+	0.998871i	$0.515128\pi$
$594$	0	0
$595$	−1.83456	−0.0752098
$596$	−83.6531	−3.42657
$597$	0	0
$598$	−3.92130	−0.160354
$599$	8.07013	0.329737	0.164868	−	0.986316i	$-0.447280\pi$
$599$	8.07013	0.329737	0.164868	+	0.986316i	$0.447280\pi$
$600$	0	0
$601$	15.2375	0.621549	0.310775	−	0.950484i	$-0.399412\pi$
$601$	15.2375	0.621549	0.310775	+	0.950484i	$0.399412\pi$
$602$	8.33407	0.339672
$603$	0	0
$604$	111.043	4.51826
$605$	21.9001	0.890365
$606$	0	0
$607$	−29.2880	−1.18876	−0.594382	−	0.804183i	$-0.702603\pi$
$607$	−29.2880	−1.18876	−0.594382	+	0.804183i	$0.702603\pi$
$608$	16.2107	0.657432
$609$	0	0
$610$	−87.3193	−3.53545
$611$	41.2465	1.66865
$612$	0	0
$613$	6.74102	0.272267	0.136134	−	0.990690i	$-0.456532\pi$
$613$	6.74102	0.272267	0.136134	+	0.990690i	$0.456532\pi$
$614$	15.1948	0.613211
$615$	0	0
$616$	−6.55618	−0.264156
$617$	12.0439	0.484870	0.242435	−	0.970168i	$-0.422054\pi$
$617$	12.0439	0.484870	0.242435	+	0.970168i	$0.422054\pi$
$618$	0	0
$619$	−13.4353	−0.540010	−0.270005	−	0.962859i	$-0.587025\pi$
$619$	−13.4353	−0.540010	−0.270005	+	0.962859i	$0.587025\pi$
$620$	−67.1656	−2.69744
$621$	0	0
$622$	8.68064	0.348062
$623$	−5.03902	−0.201884
$624$	0	0
$625$	−30.1000	−1.20400
$626$	25.2540	1.00935
$627$	0	0
$628$	−12.9418	−0.516432
$629$	−9.33648	−0.372270
$630$	0	0
$631$	0.607670	0.0241909	0.0120955	−	0.999927i	$-0.496150\pi$
$631$	0.607670	0.0241909	0.0120955	+	0.999927i	$0.496150\pi$
$632$	−83.0116	−3.30203
$633$	0	0
$634$	−54.7554	−2.17461
$635$	46.2910	1.83700
$636$	0	0
$637$	−38.0166	−1.50627
$638$	−7.29719	−0.288899
$639$	0	0
$640$	37.7039	1.49038
$641$	−22.6533	−0.894750	−0.447375	−	0.894347i	$-0.647641\pi$
$641$	−22.6533	−0.894750	−0.447375	+	0.894347i	$0.647641\pi$
$642$	0	0
$643$	48.7378	1.92203	0.961016	−	0.276494i	$-0.0891727\pi$
$643$	48.7378	1.92203	0.961016	+	0.276494i	$0.0891727\pi$
$644$	0.682098	0.0268784
$645$	0	0
$646$	3.97133	0.156250
$647$	−20.1953	−0.793960	−0.396980	−	0.917827i	$-0.629942\pi$
$647$	−20.1953	−0.793960	−0.396980	+	0.917827i	$0.629942\pi$
$648$	0	0
$649$	−7.61503	−0.298916
$650$	−21.5015	−0.843357
$651$	0	0
$652$	37.4761	1.46768
$653$	19.1440	0.749163	0.374581	−	0.927194i	$-0.377786\pi$
$653$	19.1440	0.749163	0.374581	+	0.927194i	$0.377786\pi$
$654$	0	0
$655$	−20.3845	−0.796488
$656$	−81.3331	−3.17552
$657$	0	0
$658$	−9.97321	−0.388796
$659$	−41.1553	−1.60318	−0.801591	−	0.597873i	$-0.796013\pi$
$659$	−41.1553	−1.60318	−0.801591	+	0.597873i	$0.796013\pi$
$660$	0	0
$661$	−28.4786	−1.10769	−0.553845	−	0.832620i	$-0.686840\pi$
$661$	−28.4786	−1.10769	−0.553845	+	0.832620i	$0.686840\pi$
$662$	18.4254	0.716123
$663$	0	0
$664$	−108.144	−4.19681
$665$	1.36053	0.0527590
$666$	0	0
$667$	0.463065	0.0179300
$668$	−32.8764	−1.27203
$669$	0	0
$670$	43.4471	1.67851
$671$	−19.8264	−0.765389
$672$	0	0
$673$	−36.5124	−1.40745	−0.703724	−	0.710473i	$-0.748481\pi$
$673$	−36.5124	−1.40745	−0.703724	+	0.710473i	$0.748481\pi$
$674$	−72.4028	−2.78885
$675$	0	0
$676$	96.5221	3.71239
$677$	1.77005	0.0680287	0.0340143	−	0.999421i	$-0.489171\pi$
$677$	1.77005	0.0680287	0.0340143	+	0.999421i	$0.489171\pi$
$678$	0	0
$679$	−1.83781	−0.0705287
$680$	29.9802	1.14969
$681$	0	0
$682$	−21.1989	−0.811746
$683$	22.8486	0.874276	0.437138	−	0.899395i	$-0.355992\pi$
$683$	22.8486	0.874276	0.437138	+	0.899395i	$0.355992\pi$
$684$	0	0
$685$	−21.6324	−0.826531
$686$	18.7405	0.715517
$687$	0	0
$688$	−73.5385	−2.80363
$689$	−53.5294	−2.03931
$690$	0	0
$691$	5.65686	0.215197	0.107599	−	0.994194i	$-0.465684\pi$
$691$	5.65686	0.215197	0.107599	+	0.994194i	$0.465684\pi$
$692$	−25.0659	−0.952864
$693$	0	0
$694$	−61.9730	−2.35246
$695$	−33.0839	−1.25494
$696$	0	0
$697$	−9.57035	−0.362503
$698$	−8.19561	−0.310208
$699$	0	0
$700$	3.74011	0.141363
$701$	25.7311	0.971851	0.485926	−	0.874000i	$-0.338483\pi$
$701$	25.7311	0.971851	0.485926	+	0.874000i	$0.338483\pi$
$702$	0	0
$703$	6.92401	0.261144
$704$	26.3302	0.992358
$705$	0	0
$706$	−28.0357	−1.05514
$707$	9.10648	0.342485
$708$	0	0
$709$	33.5079	1.25842	0.629208	−	0.777237i	$-0.283380\pi$
$709$	33.5079	1.25842	0.629208	+	0.777237i	$0.283380\pi$
$710$	−34.9525	−1.31174
$711$	0	0
$712$	82.3471	3.08609
$713$	1.34524	0.0503795
$714$	0	0
$715$	−21.9803	−0.822017
$716$	−2.62830	−0.0982242
$717$	0	0
$718$	−82.4364	−3.07650
$719$	−2.14859	−0.0801290	−0.0400645	−	0.999197i	$-0.512756\pi$
$719$	−2.14859	−0.0801290	−0.0400645	+	0.999197i	$0.512756\pi$
$720$	0	0
$721$	2.20594	0.0821535
$722$	47.7798	1.77818
$723$	0	0
$724$	−19.3733	−0.720002
$725$	2.53910	0.0942998
$726$	0	0
$727$	11.0633	0.410313	0.205157	−	0.978729i	$-0.434230\pi$
$727$	11.0633	0.410313	0.205157	+	0.978729i	$0.434230\pi$
$728$	−24.0660	−0.891946
$729$	0	0
$730$	−40.3536	−1.49355
$731$	−8.65316	−0.320049
$732$	0	0
$733$	−17.0031	−0.628025	−0.314012	−	0.949419i	$-0.601673\pi$
$733$	−17.0031	−0.628025	−0.314012	+	0.949419i	$0.601673\pi$
$734$	−40.6437	−1.50019
$735$	0	0
$736$	−4.01849	−0.148124
$737$	9.86494	0.363380
$738$	0	0
$739$	−24.3996	−0.897553	−0.448776	−	0.893644i	$-0.648140\pi$
$739$	−24.3996	−0.897553	−0.448776	+	0.893644i	$0.648140\pi$
$740$	85.6972	3.15029
$741$	0	0
$742$	12.9432	0.475158
$743$	−5.90356	−0.216581	−0.108290	−	0.994119i	$-0.534538\pi$
$743$	−5.90356	−0.216581	−0.108290	+	0.994119i	$0.534538\pi$
$744$	0	0
$745$	41.3622	1.51539
$746$	2.49175	0.0912295
$747$	0	0
$748$	11.1603	0.408063
$749$	−4.07661	−0.148956
$750$	0	0
$751$	−34.5523	−1.26083	−0.630415	−	0.776258i	$-0.717115\pi$
$751$	−34.5523	−1.26083	−0.630415	+	0.776258i	$0.717115\pi$
$752$	88.0019	3.20910
$753$	0	0
$754$	−26.7861	−0.975491
$755$	−54.9050	−1.99820
$756$	0	0
$757$	−38.8183	−1.41087	−0.705437	−	0.708773i	$-0.749249\pi$
$757$	−38.8183	−1.41087	−0.705437	+	0.708773i	$0.749249\pi$
$758$	102.693	3.72998
$759$	0	0
$760$	−22.2336	−0.806497
$761$	−40.8721	−1.48161	−0.740806	−	0.671719i	$-0.765556\pi$
$761$	−40.8721	−1.48161	−0.740806	+	0.671719i	$0.765556\pi$
$762$	0	0
$763$	−0.108758	−0.00393731
$764$	−116.195	−4.20379
$765$	0	0
$766$	−39.3856	−1.42306
$767$	−27.9528	−1.00932
$768$	0	0
$769$	−30.9051	−1.11447	−0.557233	−	0.830357i	$-0.688137\pi$
$769$	−30.9051	−1.11447	−0.557233	+	0.830357i	$0.688137\pi$
$770$	5.31474	0.191530
$771$	0	0
$772$	−29.3472	−1.05623
$773$	29.4711	1.06000	0.530001	−	0.847997i	$-0.322191\pi$
$773$	29.4711	1.06000	0.530001	+	0.847997i	$0.322191\pi$
$774$	0	0
$775$	7.37626	0.264963
$776$	30.0333	1.07813
$777$	0	0
$778$	45.6224	1.63564
$779$	7.09746	0.254293
$780$	0	0
$781$	−7.93619	−0.283979
$782$	−0.984455	−0.0352041
$783$	0	0
$784$	−81.1107	−2.89681
$785$	6.39904	0.228392
$786$	0	0
$787$	16.4317	0.585726	0.292863	−	0.956154i	$-0.405392\pi$
$787$	16.4317	0.585726	0.292863	+	0.956154i	$0.405392\pi$
$788$	−12.4084	−0.442029
$789$	0	0
$790$	67.2930	2.39418
$791$	2.83376	0.100757
$792$	0	0
$793$	−72.7774	−2.58440
$794$	67.2497	2.38660
$795$	0	0
$796$	−50.1748	−1.77840
$797$	−8.40955	−0.297882	−0.148941	−	0.988846i	$-0.547586\pi$
$797$	−8.40955	−0.297882	−0.148941	+	0.988846i	$0.547586\pi$
$798$	0	0
$799$	10.3551	0.366335
$800$	−22.0344	−0.779033
$801$	0	0
$802$	55.0777	1.94486
$803$	−9.16253	−0.323339
$804$	0	0
$805$	−0.337263	−0.0118869
$806$	−77.8154	−2.74093
$807$	0	0
$808$	−148.817	−5.23537
$809$	−1.83823	−0.0646289	−0.0323144	−	0.999478i	$-0.510288\pi$
$809$	−1.83823	−0.0646289	−0.0323144	+	0.999478i	$0.510288\pi$
$810$	0	0
$811$	−8.25761	−0.289964	−0.144982	−	0.989434i	$-0.546312\pi$
$811$	−8.25761	−0.289964	−0.144982	+	0.989434i	$0.546312\pi$
$812$	4.65935	0.163511
$813$	0	0
$814$	27.0478	0.948025
$815$	−18.5301	−0.649079
$816$	0	0
$817$	6.41726	0.224512
$818$	−93.9225	−3.28392
$819$	0	0
$820$	87.8439	3.06764
$821$	−25.5312	−0.891045	−0.445522	−	0.895271i	$-0.646982\pi$
$821$	−25.5312	−0.891045	−0.445522	+	0.895271i	$0.646982\pi$
$822$	0	0
$823$	−3.51179	−0.122413	−0.0612066	−	0.998125i	$-0.519495\pi$
$823$	−3.51179	−0.122413	−0.0612066	+	0.998125i	$0.519495\pi$
$824$	−36.0492	−1.25583
$825$	0	0
$826$	6.75885	0.235170
$827$	−31.2577	−1.08694	−0.543468	−	0.839430i	$-0.682889\pi$
$827$	−31.2577	−1.08694	−0.543468	+	0.839430i	$0.682889\pi$
$828$	0	0
$829$	−47.4186	−1.64692	−0.823458	−	0.567377i	$-0.807958\pi$
$829$	−47.4186	−1.64692	−0.823458	+	0.567377i	$0.807958\pi$
$830$	87.6666	3.04295
$831$	0	0
$832$	96.6513	3.35078
$833$	−9.54418	−0.330686
$834$	0	0
$835$	16.2557	0.562552
$836$	−8.27661	−0.286253
$837$	0	0
$838$	72.3649	2.49980
$839$	29.7213	1.02609	0.513046	−	0.858361i	$-0.328517\pi$
$839$	29.7213	1.02609	0.513046	+	0.858361i	$0.328517\pi$
$840$	0	0
$841$	−25.8368	−0.890926
$842$	81.2622	2.80048
$843$	0	0
$844$	−100.559	−3.46139
$845$	−47.7253	−1.64180
$846$	0	0
$847$	−4.41348	−0.151649
$848$	−114.208	−3.92193
$849$	0	0
$850$	−5.39801	−0.185150
$851$	−1.71640	−0.0588374
$852$	0	0
$853$	10.5440	0.361018	0.180509	−	0.983573i	$-0.442225\pi$
$853$	10.5440	0.361018	0.180509	+	0.983573i	$0.442225\pi$
$854$	17.5973	0.602166
$855$	0	0
$856$	66.6194	2.27700
$857$	14.8301	0.506587	0.253293	−	0.967389i	$-0.418486\pi$
$857$	14.8301	0.506587	0.253293	+	0.967389i	$0.418486\pi$
$858$	0	0
$859$	31.6103	1.07853	0.539264	−	0.842137i	$-0.318702\pi$
$859$	31.6103	1.07853	0.539264	+	0.842137i	$0.318702\pi$
$860$	79.4252	2.70838
$861$	0	0
$862$	−24.0774	−0.820081
$863$	−3.29125	−0.112035	−0.0560177	−	0.998430i	$-0.517840\pi$
$863$	−3.29125	−0.112035	−0.0560177	+	0.998430i	$0.517840\pi$
$864$	0	0
$865$	12.3938	0.421403
$866$	−63.2848	−2.15051
$867$	0	0
$868$	13.5357	0.459433
$869$	15.2793	0.518315
$870$	0	0
$871$	36.2116	1.22698
$872$	1.77731	0.0601875
$873$	0	0
$874$	0.730081	0.0246954
$875$	4.62745	0.156436
$876$	0	0
$877$	−36.9355	−1.24722	−0.623612	−	0.781734i	$-0.714336\pi$
$877$	−36.9355	−1.24722	−0.623612	+	0.781734i	$0.714336\pi$
$878$	−10.7449	−0.362624
$879$	0	0
$880$	−46.8963	−1.58088
$881$	29.1311	0.981452	0.490726	−	0.871314i	$-0.336732\pi$
$881$	29.1311	0.981452	0.490726	+	0.871314i	$0.336732\pi$
$882$	0	0
$883$	−18.8832	−0.635471	−0.317735	−	0.948179i	$-0.602922\pi$
$883$	−18.8832	−0.635471	−0.317735	+	0.948179i	$0.602922\pi$
$884$	40.9667	1.37786
$885$	0	0
$886$	−21.2448	−0.713732
$887$	24.2897	0.815570	0.407785	−	0.913078i	$-0.366301\pi$
$887$	24.2897	0.815570	0.407785	+	0.913078i	$0.366301\pi$
$888$	0	0
$889$	−9.32892	−0.312882
$890$	−66.7543	−2.23761
$891$	0	0
$892$	13.4343	0.449814
$893$	−7.67940	−0.256981
$894$	0	0
$895$	1.29956	0.0434396
$896$	−7.59838	−0.253844
$897$	0	0
$898$	39.3429	1.31289
$899$	9.18919	0.306477
$900$	0	0
$901$	−13.4387	−0.447708
$902$	27.7253	0.923153
$903$	0	0
$904$	−46.3090	−1.54021
$905$	9.57910	0.318420
$906$	0	0
$907$	39.4257	1.30911	0.654554	−	0.756015i	$-0.272856\pi$
$907$	39.4257	1.30911	0.654554	+	0.756015i	$0.272856\pi$
$908$	6.33310	0.210171
$909$	0	0
$910$	19.5090	0.646717
$911$	41.3811	1.37102	0.685509	−	0.728064i	$-0.259580\pi$
$911$	41.3811	1.37102	0.685509	+	0.728064i	$0.259580\pi$
$912$	0	0
$913$	19.9052	0.658768
$914$	20.9859	0.694151
$915$	0	0
$916$	25.1202	0.829996
$917$	4.10804	0.135660
$918$	0	0
$919$	−1.35408	−0.0446668	−0.0223334	−	0.999751i	$-0.507110\pi$
$919$	−1.35408	−0.0446668	−0.0223334	+	0.999751i	$0.507110\pi$
$920$	5.51151	0.181709
$921$	0	0
$922$	−8.82032	−0.290482
$923$	−29.1317	−0.958880
$924$	0	0
$925$	−9.41144	−0.309446
$926$	47.2582	1.55300
$927$	0	0
$928$	−27.4500	−0.901089
$929$	−26.7202	−0.876660	−0.438330	−	0.898814i	$-0.644430\pi$
$929$	−26.7202	−0.876660	−0.438330	+	0.898814i	$0.644430\pi$
$930$	0	0
$931$	7.07805	0.231974
$932$	−19.6080	−0.642282
$933$	0	0
$934$	106.411	3.48187
$935$	−5.51822	−0.180465
$936$	0	0
$937$	−2.72186	−0.0889194	−0.0444597	−	0.999011i	$-0.514157\pi$
$937$	−2.72186	−0.0889194	−0.0444597	+	0.999011i	$0.514157\pi$
$938$	−8.75580	−0.285887
$939$	0	0
$940$	−95.0465	−3.10007
$941$	22.8033	0.743367	0.371683	−	0.928360i	$-0.378781\pi$
$941$	22.8033	0.743367	0.371683	+	0.928360i	$0.378781\pi$
$942$	0	0
$943$	−1.75940	−0.0572938
$944$	−59.6389	−1.94108
$945$	0	0
$946$	25.0682	0.815039
$947$	16.3864	0.532487	0.266243	−	0.963906i	$-0.414218\pi$
$947$	16.3864	0.532487	0.266243	+	0.963906i	$0.414218\pi$
$948$	0	0
$949$	−33.6333	−1.09178
$950$	4.00321	0.129881
$951$	0	0
$952$	−6.04185	−0.195817
$953$	−35.6653	−1.15531	−0.577656	−	0.816281i	$-0.696032\pi$
$953$	−35.6653	−1.15531	−0.577656	+	0.816281i	$0.696032\pi$
$954$	0	0
$955$	57.4525	1.85912
$956$	−46.3888	−1.50032
$957$	0	0
$958$	104.629	3.38041
$959$	4.35953	0.140776
$960$	0	0
$961$	−4.30475	−0.138863
$962$	99.2854	3.20109
$963$	0	0
$964$	−120.116	−3.86869
$965$	14.5107	0.467116
$966$	0	0
$967$	−8.04701	−0.258775	−0.129387	−	0.991594i	$-0.541301\pi$
$967$	−8.04701	−0.258775	−0.129387	+	0.991594i	$0.541301\pi$
$968$	72.1245	2.31817
$969$	0	0
$970$	−24.3464	−0.781715
$971$	−44.9410	−1.44223	−0.721113	−	0.692818i	$-0.756369\pi$
$971$	−44.9410	−1.44223	−0.721113	+	0.692818i	$0.756369\pi$
$972$	0	0
$973$	6.66731	0.213744
$974$	−93.5953	−2.99899
$975$	0	0
$976$	−155.275	−4.97024
$977$	−27.7690	−0.888410	−0.444205	−	0.895925i	$-0.646514\pi$
$977$	−27.7690	−0.888410	−0.444205	+	0.895925i	$0.646514\pi$
$978$	0	0
$979$	−15.1570	−0.484420
$980$	87.6037	2.79840
$981$	0	0
$982$	−54.9488	−1.75349
$983$	−7.00210	−0.223332	−0.111666	−	0.993746i	$-0.535619\pi$
$983$	−7.00210	−0.223332	−0.111666	+	0.993746i	$0.535619\pi$
$984$	0	0
$985$	6.13530	0.195487
$986$	−6.72473	−0.214159
$987$	0	0
$988$	−30.3813	−0.966557
$989$	−1.59078	−0.0505839
$990$	0	0
$991$	−58.5551	−1.86006	−0.930032	−	0.367479i	$-0.880221\pi$
$991$	−58.5551	−1.86006	−0.930032	+	0.367479i	$0.880221\pi$
$992$	−79.7441	−2.53188
$993$	0	0
$994$	7.04390	0.223419
$995$	24.8089	0.786494
$996$	0	0
$997$	−38.4974	−1.21922	−0.609612	−	0.792700i	$-0.708675\pi$
$997$	−38.4974	−1.21922	−0.609612	+	0.792700i	$0.708675\pi$
$998$	−22.0127	−0.696799
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.2		72
3.2	odd	2		6561.2.a.d.1.71		72
81.5	odd	54		243.2.g.a.73.8		144
81.11	odd	54		729.2.g.a.514.1		144
81.16	even	27		81.2.g.a.13.1	✓	144
81.22	even	27		729.2.g.d.217.8		144
81.32	odd	54		729.2.g.b.703.1		144
81.38	odd	54		729.2.g.b.28.1		144
81.43	even	27		729.2.g.c.28.8		144
81.49	even	27		729.2.g.c.703.8		144
81.59	odd	54		729.2.g.a.217.1		144
81.65	odd	54		243.2.g.a.10.8		144
81.70	even	27		729.2.g.d.514.8		144
81.76	even	27		81.2.g.a.25.1	yes	144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.13.1	✓	144	81.16	even	27
81.2.g.a.25.1	yes	144	81.76	even	27
243.2.g.a.10.8		144	81.65	odd	54
243.2.g.a.73.8		144	81.5	odd	54
729.2.g.a.217.1		144	81.59	odd	54
729.2.g.a.514.1		144	81.11	odd	54
729.2.g.b.28.1		144	81.38	odd	54
729.2.g.b.703.1		144	81.32	odd	54
729.2.g.c.28.8		144	81.43	even	27
729.2.g.c.703.8		144	81.49	even	27
729.2.g.d.217.8		144	81.22	even	27
729.2.g.d.514.8		144	81.70	even	27
6561.2.a.c.1.2		72	1.1	even	1	trivial
6561.2.a.d.1.71		72	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-2.66973 q^{2} +5.12748 q^{4} -2.53528 q^{5} +0.510929 q^{7} -8.34954 q^{8} +O(q^{10})\)	\(q-2.66973 q^{2} +5.12748 q^{4} -2.53528 q^{5} +0.510929 q^{7} -8.34954 q^{8} +6.76852 q^{10} +1.53684 q^{11} +5.64132 q^{13} -1.36405 q^{14} +12.0361 q^{16} +1.41627 q^{17} -1.05032 q^{19} -12.9996 q^{20} -4.10294 q^{22} +0.260365 q^{23} +1.42764 q^{25} -15.0608 q^{26} +2.61978 q^{28} +1.77853 q^{29} +5.16674 q^{31} -15.4341 q^{32} -3.78107 q^{34} -1.29535 q^{35} -6.59230 q^{37} +2.80407 q^{38} +21.1684 q^{40} -6.75743 q^{41} -6.10982 q^{43} +7.88009 q^{44} -0.695104 q^{46} +7.31149 q^{47} -6.73895 q^{49} -3.81142 q^{50} +28.9258 q^{52} -9.48880 q^{53} -3.89631 q^{55} -4.26603 q^{56} -4.74819 q^{58} -4.95500 q^{59} -12.9008 q^{61} -13.7938 q^{62} +17.1328 q^{64} -14.3023 q^{65} +6.41900 q^{67} +7.26190 q^{68} +3.45824 q^{70} -5.16398 q^{71} -5.96195 q^{73} +17.5997 q^{74} -5.38549 q^{76} +0.785214 q^{77} +9.94206 q^{79} -30.5149 q^{80} +18.0405 q^{82} +12.9521 q^{83} -3.59064 q^{85} +16.3116 q^{86} -12.8319 q^{88} -9.86247 q^{89} +2.88231 q^{91} +1.33501 q^{92} -19.5197 q^{94} +2.66285 q^{95} -3.59700 q^{97} +17.9912 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8}+O(q^{10}) \) 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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