LMFDB - Embedded newform 6561.2.a.c.1.8

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			1.8
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)

$f(q)$	$=$	$q-2.43593 q^{2} +3.93374 q^{4} +2.95122 q^{5} +2.28987 q^{7} -4.71046 q^{8} +O(q^{10})$	\(q-2.43593 q^{2} +3.93374 q^{4} +2.95122 q^{5} +2.28987 q^{7} -4.71046 q^{8} -7.18896 q^{10} +2.36834 q^{11} +0.613773 q^{13} -5.57797 q^{14} +3.60685 q^{16} -7.23393 q^{17} -4.50769 q^{19} +11.6093 q^{20} -5.76911 q^{22} -7.48901 q^{23} +3.70969 q^{25} -1.49511 q^{26} +9.00778 q^{28} -2.17184 q^{29} +3.10480 q^{31} +0.634886 q^{32} +17.6213 q^{34} +6.75792 q^{35} +4.26307 q^{37} +10.9804 q^{38} -13.9016 q^{40} -6.43861 q^{41} -4.06733 q^{43} +9.31644 q^{44} +18.2427 q^{46} -0.865244 q^{47} -1.75648 q^{49} -9.03654 q^{50} +2.41443 q^{52} +3.10647 q^{53} +6.98949 q^{55} -10.7864 q^{56} +5.29044 q^{58} +0.613657 q^{59} -5.46697 q^{61} -7.56306 q^{62} -8.76024 q^{64} +1.81138 q^{65} +0.282307 q^{67} -28.4564 q^{68} -16.4618 q^{70} -8.21471 q^{71} +15.2438 q^{73} -10.3845 q^{74} -17.7321 q^{76} +5.42320 q^{77} +7.45468 q^{79} +10.6446 q^{80} +15.6840 q^{82} -5.47724 q^{83} -21.3489 q^{85} +9.90772 q^{86} -11.1560 q^{88} -9.19539 q^{89} +1.40546 q^{91} -29.4599 q^{92} +2.10767 q^{94} -13.3032 q^{95} -1.71157 q^{97} +4.27866 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.43593	−1.72246	−0.861231	−	0.508215i	$-0.830306\pi$
$2$	−2.43593	−1.72246	−0.861231	+	0.508215i	$0.830306\pi$
$3$	0	0
$3$	0	0
$4$	3.93374	1.96687
$5$	2.95122	1.31983	0.659913	−	0.751342i	$-0.270593\pi$
$5$	2.95122	1.31983	0.659913	+	0.751342i	$0.270593\pi$
$6$	0	0
$7$	2.28987	0.865491	0.432745	−	0.901516i	$-0.357545\pi$
$7$	2.28987	0.865491	0.432745	+	0.901516i	$0.357545\pi$
$8$	−4.71046	−1.66540
$9$	0	0
$10$	−7.18896	−2.27335
$11$	2.36834	0.714081	0.357041	−	0.934089i	$-0.383786\pi$
$11$	2.36834	0.714081	0.357041	+	0.934089i	$0.383786\pi$
$12$	0	0
$13$	0.613773	0.170230	0.0851150	−	0.996371i	$-0.472874\pi$
$13$	0.613773	0.170230	0.0851150	+	0.996371i	$0.472874\pi$
$14$	−5.57797	−1.49077
$15$	0	0
$16$	3.60685	0.901713
$17$	−7.23393	−1.75449	−0.877243	−	0.480046i	$-0.840620\pi$
$17$	−7.23393	−1.75449	−0.877243	+	0.480046i	$0.840620\pi$
$18$	0	0
$19$	−4.50769	−1.03413	−0.517067	−	0.855945i	$-0.672976\pi$
$19$	−4.50769	−1.03413	−0.517067	+	0.855945i	$0.672976\pi$
$20$	11.6093	2.59593
$21$	0	0
$22$	−5.76911	−1.22998
$23$	−7.48901	−1.56157	−0.780783	−	0.624802i	$-0.785180\pi$
$23$	−7.48901	−1.56157	−0.780783	+	0.624802i	$0.785180\pi$
$24$	0	0
$25$	3.70969	0.741938
$26$	−1.49511	−0.293215
$27$	0	0
$28$	9.00778	1.70231
$29$	−2.17184	−0.403300	−0.201650	−	0.979458i	$-0.564630\pi$
$29$	−2.17184	−0.403300	−0.201650	+	0.979458i	$0.564630\pi$
$30$	0	0
$31$	3.10480	0.557638	0.278819	−	0.960344i	$-0.410057\pi$
$31$	3.10480	0.557638	0.278819	+	0.960344i	$0.410057\pi$
$32$	0.634886	0.112233
$33$	0	0
$34$	17.6213	3.02203
$35$	6.75792	1.14230
$36$	0	0
$37$	4.26307	0.700844	0.350422	−	0.936592i	$-0.386038\pi$
$37$	4.26307	0.700844	0.350422	+	0.936592i	$0.386038\pi$
$38$	10.9804	1.78126
$39$	0	0
$40$	−13.9016	−2.19804
$41$	−6.43861	−1.00554	−0.502771	−	0.864420i	$-0.667686\pi$
$41$	−6.43861	−1.00554	−0.502771	+	0.864420i	$0.667686\pi$
$42$	0	0
$43$	−4.06733	−0.620262	−0.310131	−	0.950694i	$-0.600373\pi$
$43$	−4.06733	−0.620262	−0.310131	+	0.950694i	$0.600373\pi$
$44$	9.31644	1.40451
$45$	0	0
$46$	18.2427	2.68974
$47$	−0.865244	−0.126209	−0.0631044	−	0.998007i	$-0.520100\pi$
$47$	−0.865244	−0.126209	−0.0631044	+	0.998007i	$0.520100\pi$
$48$	0	0
$49$	−1.75648	−0.250926
$50$	−9.03654	−1.27796
$51$	0	0
$52$	2.41443	0.334821
$53$	3.10647	0.426706	0.213353	−	0.976975i	$-0.431562\pi$
$53$	3.10647	0.426706	0.213353	+	0.976975i	$0.431562\pi$
$54$	0	0
$55$	6.98949	0.942463
$56$	−10.7864	−1.44139
$57$	0	0
$58$	5.29044	0.694668
$59$	0.613657	0.0798914	0.0399457	−	0.999202i	$-0.487282\pi$
$59$	0.613657	0.0798914	0.0399457	+	0.999202i	$0.487282\pi$
$60$	0	0
$61$	−5.46697	−0.699973	−0.349987	−	0.936755i	$-0.613814\pi$
$61$	−5.46697	−0.699973	−0.349987	+	0.936755i	$0.613814\pi$
$62$	−7.56306	−0.960510
$63$	0	0
$64$	−8.76024	−1.09503
$65$	1.81138	0.224674
$66$	0	0
$67$	0.282307	0.0344893	0.0172447	−	0.999851i	$-0.494511\pi$
$67$	0.282307	0.0344893	0.0172447	+	0.999851i	$0.494511\pi$
$68$	−28.4564	−3.45085
$69$	0	0
$70$	−16.4618	−1.96756
$71$	−8.21471	−0.974907	−0.487453	−	0.873149i	$-0.662074\pi$
$71$	−8.21471	−0.974907	−0.487453	+	0.873149i	$0.662074\pi$
$72$	0	0
$73$	15.2438	1.78415	0.892076	−	0.451885i	$-0.149248\pi$
$73$	15.2438	1.78415	0.892076	+	0.451885i	$0.149248\pi$
$74$	−10.3845	−1.20718
$75$	0	0
$76$	−17.7321	−2.03401
$77$	5.42320	0.618031
$78$	0	0
$79$	7.45468	0.838717	0.419358	−	0.907821i	$-0.362255\pi$
$79$	7.45468	0.838717	0.419358	+	0.907821i	$0.362255\pi$
$80$	10.6446	1.19010
$81$	0	0
$82$	15.6840	1.73201
$83$	−5.47724	−0.601205	−0.300603	−	0.953749i	$-0.597188\pi$
$83$	−5.47724	−0.601205	−0.300603	+	0.953749i	$0.597188\pi$
$84$	0	0
$85$	−21.3489	−2.31562
$86$	9.90772	1.06838
$87$	0	0
$88$	−11.1560	−1.18923
$89$	−9.19539	−0.974709	−0.487355	−	0.873204i	$-0.662038\pi$
$89$	−9.19539	−0.974709	−0.487355	+	0.873204i	$0.662038\pi$
$90$	0	0
$91$	1.40546	0.147333
$92$	−29.4599	−3.07140
$93$	0	0
$94$	2.10767	0.217390
$95$	−13.3032	−1.36488
$96$	0	0
$97$	−1.71157	−0.173784	−0.0868918	−	0.996218i	$-0.527693\pi$
$97$	−1.71157	−0.173784	−0.0868918	+	0.996218i	$0.527693\pi$
$98$	4.27866	0.432210
$99$	0	0
$100$	14.5930	1.45930
$101$	−8.47564	−0.843358	−0.421679	−	0.906745i	$-0.638559\pi$
$101$	−8.47564	−0.843358	−0.421679	+	0.906745i	$0.638559\pi$
$102$	0	0
$103$	−1.81410	−0.178748	−0.0893741	−	0.995998i	$-0.528487\pi$
$103$	−1.81410	−0.178748	−0.0893741	+	0.995998i	$0.528487\pi$
$104$	−2.89115	−0.283501
$105$	0	0
$106$	−7.56713	−0.734985
$107$	−4.53337	−0.438257	−0.219129	−	0.975696i	$-0.570321\pi$
$107$	−4.53337	−0.438257	−0.219129	+	0.975696i	$0.570321\pi$
$108$	0	0
$109$	13.2526	1.26937	0.634684	−	0.772772i	$-0.281130\pi$
$109$	13.2526	1.26937	0.634684	+	0.772772i	$0.281130\pi$
$110$	−17.0259	−1.62336
$111$	0	0
$112$	8.25924	0.780425
$113$	−4.29241	−0.403796	−0.201898	−	0.979407i	$-0.564711\pi$
$113$	−4.29241	−0.403796	−0.201898	+	0.979407i	$0.564711\pi$
$114$	0	0
$115$	−22.1017	−2.06100
$116$	−8.54345	−0.793239
$117$	0	0
$118$	−1.49482	−0.137610
$119$	−16.5648	−1.51849
$120$	0	0
$121$	−5.39096	−0.490088
$122$	13.3171	1.20568
$123$	0	0
$124$	12.2135	1.09680
$125$	−3.80798	−0.340596
$126$	0	0
$127$	−5.88721	−0.522405	−0.261203	−	0.965284i	$-0.584119\pi$
$127$	−5.88721	−0.522405	−0.261203	+	0.965284i	$0.584119\pi$
$128$	20.0695	1.77391
$129$	0	0
$130$	−4.41239	−0.386992
$131$	−13.0594	−1.14101	−0.570504	−	0.821294i	$-0.693252\pi$
$131$	−13.0594	−1.14101	−0.570504	+	0.821294i	$0.693252\pi$
$132$	0	0
$133$	−10.3220	−0.895034
$134$	−0.687680	−0.0594065
$135$	0	0
$136$	34.0752	2.92192
$137$	4.52721	0.386785	0.193393	−	0.981121i	$-0.438051\pi$
$137$	4.52721	0.386785	0.193393	+	0.981121i	$0.438051\pi$
$138$	0	0
$139$	−4.79271	−0.406512	−0.203256	−	0.979126i	$-0.565152\pi$
$139$	−4.79271	−0.406512	−0.203256	+	0.979126i	$0.565152\pi$
$140$	26.5839	2.24675
$141$	0	0
$142$	20.0104	1.67924
$143$	1.45362	0.121558
$144$	0	0
$145$	−6.40956	−0.532285
$146$	−37.1328	−3.07313
$147$	0	0
$148$	16.7698	1.37847
$149$	−9.81560	−0.804125	−0.402062	−	0.915612i	$-0.631707\pi$
$149$	−9.81560	−0.804125	−0.402062	+	0.915612i	$0.631707\pi$
$150$	0	0
$151$	13.3533	1.08667	0.543337	−	0.839515i	$-0.317161\pi$
$151$	13.3533	1.08667	0.543337	+	0.839515i	$0.317161\pi$
$152$	21.2333	1.72225
$153$	0	0
$154$	−13.2105	−1.06453
$155$	9.16294	0.735985
$156$	0	0
$157$	−2.99911	−0.239355	−0.119677	−	0.992813i	$-0.538186\pi$
$157$	−2.99911	−0.239355	−0.119677	+	0.992813i	$0.538186\pi$
$158$	−18.1591	−1.44466
$159$	0	0
$160$	1.87369	0.148128
$161$	−17.1489	−1.35152
$162$	0	0
$163$	17.7302	1.38874	0.694370	−	0.719618i	$-0.255683\pi$
$163$	17.7302	1.38874	0.694370	+	0.719618i	$0.255683\pi$
$164$	−25.3278	−1.97777
$165$	0	0
$166$	13.3422	1.03555
$167$	−21.9914	−1.70174	−0.850872	−	0.525373i	$-0.823926\pi$
$167$	−21.9914	−1.70174	−0.850872	+	0.525373i	$0.823926\pi$
$168$	0	0
$169$	−12.6233	−0.971022
$170$	52.0044	3.98856
$171$	0	0
$172$	−15.9998	−1.21998
$173$	10.8250	0.823008	0.411504	−	0.911408i	$-0.365004\pi$
$173$	10.8250	0.823008	0.411504	+	0.911408i	$0.365004\pi$
$174$	0	0
$175$	8.49473	0.642141
$176$	8.54226	0.643897
$177$	0	0
$178$	22.3993	1.67890
$179$	−8.94169	−0.668333	−0.334167	−	0.942514i	$-0.608455\pi$
$179$	−8.94169	−0.668333	−0.334167	+	0.942514i	$0.608455\pi$
$180$	0	0
$181$	4.32494	0.321470	0.160735	−	0.986998i	$-0.448613\pi$
$181$	4.32494	0.321470	0.160735	+	0.986998i	$0.448613\pi$
$182$	−3.42361	−0.253775
$183$	0	0
$184$	35.2767	2.60063
$185$	12.5813	0.924992
$186$	0	0
$187$	−17.1324	−1.25285
$188$	−3.40365	−0.248237
$189$	0	0
$190$	32.4056	2.35095
$191$	21.2125	1.53489	0.767443	−	0.641117i	$-0.221529\pi$
$191$	21.2125	1.53489	0.767443	+	0.641117i	$0.221529\pi$
$192$	0	0
$193$	−18.1549	−1.30682	−0.653408	−	0.757006i	$-0.726662\pi$
$193$	−18.1549	−1.30682	−0.653408	+	0.757006i	$0.726662\pi$
$194$	4.16926	0.299336
$195$	0	0
$196$	−6.90954	−0.493539
$197$	7.13839	0.508589	0.254294	−	0.967127i	$-0.418157\pi$
$197$	7.13839	0.508589	0.254294	+	0.967127i	$0.418157\pi$
$198$	0	0
$199$	−3.65601	−0.259168	−0.129584	−	0.991568i	$-0.541364\pi$
$199$	−3.65601	−0.259168	−0.129584	+	0.991568i	$0.541364\pi$
$200$	−17.4744	−1.23562
$201$	0	0
$202$	20.6461	1.45265
$203$	−4.97323	−0.349052
$204$	0	0
$205$	−19.0017	−1.32714
$206$	4.41901	0.307887
$207$	0	0
$208$	2.21379	0.153499
$209$	−10.6757	−0.738456
$210$	0	0
$211$	−17.8521	−1.22899	−0.614495	−	0.788921i	$-0.710640\pi$
$211$	−17.8521	−1.22899	−0.614495	+	0.788921i	$0.710640\pi$
$212$	12.2200	0.839276
$213$	0	0
$214$	11.0430	0.754881
$215$	−12.0036	−0.818637
$216$	0	0
$217$	7.10959	0.482631
$218$	−32.2824	−2.18644
$219$	0	0
$220$	27.4949	1.85370
$221$	−4.43999	−0.298666
$222$	0	0
$223$	−14.1063	−0.944627	−0.472313	−	0.881431i	$-0.656581\pi$
$223$	−14.1063	−0.944627	−0.472313	+	0.881431i	$0.656581\pi$
$224$	1.45381	0.0971367
$225$	0	0
$226$	10.4560	0.695523
$227$	−5.75305	−0.381843	−0.190922	−	0.981605i	$-0.561148\pi$
$227$	−5.75305	−0.381843	−0.190922	+	0.981605i	$0.561148\pi$
$228$	0	0
$229$	−24.5752	−1.62398	−0.811988	−	0.583674i	$-0.801615\pi$
$229$	−24.5752	−1.62398	−0.811988	+	0.583674i	$0.801615\pi$
$230$	53.8382	3.54998
$231$	0	0
$232$	10.2303	0.671655
$233$	−12.7491	−0.835222	−0.417611	−	0.908626i	$-0.637133\pi$
$233$	−12.7491	−0.835222	−0.417611	+	0.908626i	$0.637133\pi$
$234$	0	0
$235$	−2.55352	−0.166574
$236$	2.41397	0.157136
$237$	0	0
$238$	40.3506	2.61554
$239$	24.3522	1.57521	0.787607	−	0.616178i	$-0.211320\pi$
$239$	24.3522	1.57521	0.787607	+	0.616178i	$0.211320\pi$
$240$	0	0
$241$	−29.0716	−1.87267	−0.936333	−	0.351114i	$-0.885803\pi$
$241$	−29.0716	−1.87267	−0.936333	+	0.351114i	$0.885803\pi$
$242$	13.1320	0.844157
$243$	0	0
$244$	−21.5057	−1.37676
$245$	−5.18375	−0.331178
$246$	0	0
$247$	−2.76670	−0.176041
$248$	−14.6250	−0.928690
$249$	0	0
$250$	9.27597	0.586664
$251$	−23.8428	−1.50494	−0.752471	−	0.658625i	$-0.771138\pi$
$251$	−23.8428	−1.50494	−0.752471	+	0.658625i	$0.771138\pi$
$252$	0	0
$253$	−17.7365	−1.11509
$254$	14.3408	0.899822
$255$	0	0
$256$	−31.3675	−1.96047
$257$	12.1373	0.757104	0.378552	−	0.925580i	$-0.376422\pi$
$257$	12.1373	0.757104	0.378552	+	0.925580i	$0.376422\pi$
$258$	0	0
$259$	9.76189	0.606574
$260$	7.12550	0.441905
$261$	0	0
$262$	31.8119	1.96534
$263$	8.00169	0.493405	0.246703	−	0.969091i	$-0.420653\pi$
$263$	8.00169	0.493405	0.246703	+	0.969091i	$0.420653\pi$
$264$	0	0
$265$	9.16787	0.563177
$266$	25.1437	1.54166
$267$	0	0
$268$	1.11052	0.0678361
$269$	22.5155	1.37279	0.686397	−	0.727227i	$-0.259191\pi$
$269$	22.5155	1.37279	0.686397	+	0.727227i	$0.259191\pi$
$270$	0	0
$271$	−30.3028	−1.84076	−0.920381	−	0.391022i	$-0.872121\pi$
$271$	−30.3028	−1.84076	−0.920381	+	0.391022i	$0.872121\pi$
$272$	−26.0917	−1.58204
$273$	0	0
$274$	−11.0280	−0.666223
$275$	8.78581	0.529804
$276$	0	0
$277$	−4.42373	−0.265796	−0.132898	−	0.991130i	$-0.542428\pi$
$277$	−4.42373	−0.265796	−0.132898	+	0.991130i	$0.542428\pi$
$278$	11.6747	0.700201
$279$	0	0
$280$	−31.8329	−1.90238
$281$	5.66809	0.338130	0.169065	−	0.985605i	$-0.445925\pi$
$281$	5.66809	0.338130	0.169065	+	0.985605i	$0.445925\pi$
$282$	0	0
$283$	28.8234	1.71337	0.856685	−	0.515839i	$-0.172520\pi$
$283$	28.8234	1.71337	0.856685	+	0.515839i	$0.172520\pi$
$284$	−32.3146	−1.91752
$285$	0	0
$286$	−3.54092	−0.209379
$287$	−14.7436	−0.870287
$288$	0	0
$289$	35.3298	2.07822
$290$	15.6132	0.916841
$291$	0	0
$292$	59.9652	3.50920
$293$	2.16487	0.126473	0.0632365	−	0.997999i	$-0.479858\pi$
$293$	2.16487	0.126473	0.0632365	+	0.997999i	$0.479858\pi$
$294$	0	0
$295$	1.81104	0.105443
$296$	−20.0810	−1.16719
$297$	0	0
$298$	23.9101	1.38507
$299$	−4.59656	−0.265826
$300$	0	0
$301$	−9.31367	−0.536831
$302$	−32.5276	−1.87175
$303$	0	0
$304$	−16.2586	−0.932493
$305$	−16.1342	−0.923843
$306$	0	0
$307$	4.74470	0.270795	0.135397	−	0.990791i	$-0.456769\pi$
$307$	4.74470	0.270795	0.135397	+	0.990791i	$0.456769\pi$
$308$	21.3335	1.21559
$309$	0	0
$310$	−22.3203	−1.26771
$311$	23.6045	1.33849	0.669245	−	0.743042i	$-0.266618\pi$
$311$	23.6045	1.33849	0.669245	+	0.743042i	$0.266618\pi$
$312$	0	0
$313$	23.6302	1.33566	0.667828	−	0.744315i	$-0.267224\pi$
$313$	23.6302	1.33566	0.667828	+	0.744315i	$0.267224\pi$
$314$	7.30561	0.412279
$315$	0	0
$316$	29.3248	1.64965
$317$	34.0742	1.91380	0.956899	−	0.290422i	$-0.0937958\pi$
$317$	34.0742	1.91380	0.956899	+	0.290422i	$0.0937958\pi$
$318$	0	0
$319$	−5.14365	−0.287989
$320$	−25.8534	−1.44525
$321$	0	0
$322$	41.7735	2.32794
$323$	32.6083	1.81437
$324$	0	0
$325$	2.27691	0.126300
$326$	−43.1896	−2.39205
$327$	0	0
$328$	30.3288	1.67463
$329$	−1.98130	−0.109233
$330$	0	0
$331$	25.9270	1.42508	0.712538	−	0.701633i	$-0.247545\pi$
$331$	25.9270	1.42508	0.712538	+	0.701633i	$0.247545\pi$
$332$	−21.5461	−1.18249
$333$	0	0
$334$	53.5694	2.93119
$335$	0.833151	0.0455199
$336$	0	0
$337$	22.1514	1.20667	0.603333	−	0.797490i	$-0.293839\pi$
$337$	22.1514	1.20667	0.603333	+	0.797490i	$0.293839\pi$
$338$	30.7494	1.67255
$339$	0	0
$340$	−83.9812	−4.55452
$341$	7.35322	0.398199
$342$	0	0
$343$	−20.0512	−1.08266
$344$	19.1590	1.03298
$345$	0	0
$346$	−26.3689	−1.41760
$347$	1.79340	0.0962748	0.0481374	−	0.998841i	$-0.484671\pi$
$347$	1.79340	0.0962748	0.0481374	+	0.998841i	$0.484671\pi$
$348$	0	0
$349$	0.175877	0.00941448	0.00470724	−	0.999989i	$-0.498502\pi$
$349$	0.175877	0.00941448	0.00470724	+	0.999989i	$0.498502\pi$
$350$	−20.6925	−1.10606
$351$	0	0
$352$	1.50363	0.0801436
$353$	−30.4137	−1.61876	−0.809380	−	0.587286i	$-0.800197\pi$
$353$	−30.4137	−1.61876	−0.809380	+	0.587286i	$0.800197\pi$
$354$	0	0
$355$	−24.2434	−1.28671
$356$	−36.1723	−1.91713
$357$	0	0
$358$	21.7813	1.15118
$359$	−6.44928	−0.340380	−0.170190	−	0.985411i	$-0.554438\pi$
$359$	−6.44928	−0.340380	−0.170190	+	0.985411i	$0.554438\pi$
$360$	0	0
$361$	1.31925	0.0694341
$362$	−10.5352	−0.553720
$363$	0	0
$364$	5.52873	0.289784
$365$	44.9878	2.35477
$366$	0	0
$367$	18.4047	0.960716	0.480358	−	0.877072i	$-0.340507\pi$
$367$	18.4047	0.960716	0.480358	+	0.877072i	$0.340507\pi$
$368$	−27.0118	−1.40809
$369$	0	0
$370$	−30.6470	−1.59326
$371$	7.11342	0.369310
$372$	0	0
$373$	−32.2846	−1.67163	−0.835816	−	0.549010i	$-0.815005\pi$
$373$	−32.2846	−1.67163	−0.835816	+	0.549010i	$0.815005\pi$
$374$	41.7333	2.15798
$375$	0	0
$376$	4.07570	0.210188
$377$	−1.33302	−0.0686538
$378$	0	0
$379$	−22.0326	−1.13174	−0.565870	−	0.824494i	$-0.691460\pi$
$379$	−22.0326	−1.13174	−0.565870	+	0.824494i	$0.691460\pi$
$380$	−52.3313	−2.68454
$381$	0	0
$382$	−51.6722	−2.64378
$383$	−14.2260	−0.726917	−0.363458	−	0.931610i	$-0.618404\pi$
$383$	−14.2260	−0.726917	−0.363458	+	0.931610i	$0.618404\pi$
$384$	0	0
$385$	16.0050	0.815693
$386$	44.2240	2.25094
$387$	0	0
$388$	−6.73288	−0.341810
$389$	12.2661	0.621915	0.310958	−	0.950424i	$-0.399350\pi$
$389$	12.2661	0.621915	0.310958	+	0.950424i	$0.399350\pi$
$390$	0	0
$391$	54.1750	2.73975
$392$	8.27383	0.417891
$393$	0	0
$394$	−17.3886	−0.876025
$395$	22.0004	1.10696
$396$	0	0
$397$	9.71444	0.487554	0.243777	−	0.969831i	$-0.421614\pi$
$397$	9.71444	0.487554	0.243777	+	0.969831i	$0.421614\pi$
$398$	8.90579	0.446407
$399$	0	0
$400$	13.3803	0.669016
$401$	1.16926	0.0583902	0.0291951	−	0.999574i	$-0.490706\pi$
$401$	1.16926	0.0583902	0.0291951	+	0.999574i	$0.490706\pi$
$402$	0	0
$403$	1.90564	0.0949268
$404$	−33.3410	−1.65878
$405$	0	0
$406$	12.1144	0.601229
$407$	10.0964	0.500460
$408$	0	0
$409$	−27.4609	−1.35785	−0.678927	−	0.734206i	$-0.737555\pi$
$409$	−27.4609	−1.35785	−0.678927	+	0.734206i	$0.737555\pi$
$410$	46.2869	2.28595
$411$	0	0
$412$	−7.13619	−0.351575
$413$	1.40520	0.0691452
$414$	0	0
$415$	−16.1645	−0.793486
$416$	0.389676	0.0191054
$417$	0	0
$418$	26.0053	1.27196
$419$	24.5975	1.20167	0.600834	−	0.799374i	$-0.294835\pi$
$419$	24.5975	1.20167	0.600834	+	0.799374i	$0.294835\pi$
$420$	0	0
$421$	−12.3022	−0.599573	−0.299787	−	0.954006i	$-0.596915\pi$
$421$	−12.3022	−0.599573	−0.299787	+	0.954006i	$0.596915\pi$
$422$	43.4864	2.11689
$423$	0	0
$424$	−14.6329	−0.710636
$425$	−26.8357	−1.30172
$426$	0	0
$427$	−12.5187	−0.605821
$428$	−17.8331	−0.861996
$429$	0	0
$430$	29.2399	1.41007
$431$	−7.95662	−0.383257	−0.191628	−	0.981468i	$-0.561377\pi$
$431$	−7.95662	−0.383257	−0.191628	+	0.981468i	$0.561377\pi$
$432$	0	0
$433$	11.9147	0.572584	0.286292	−	0.958142i	$-0.407577\pi$
$433$	11.9147	0.572584	0.286292	+	0.958142i	$0.407577\pi$
$434$	−17.3185	−0.831313
$435$	0	0
$436$	52.1323	2.49669
$437$	33.7581	1.61487
$438$	0	0
$439$	6.90592	0.329602	0.164801	−	0.986327i	$-0.447302\pi$
$439$	6.90592	0.329602	0.164801	+	0.986327i	$0.447302\pi$
$440$	−32.9237	−1.56958
$441$	0	0
$442$	10.8155	0.514441
$443$	7.24509	0.344225	0.172112	−	0.985077i	$-0.444941\pi$
$443$	7.24509	0.344225	0.172112	+	0.985077i	$0.444941\pi$
$444$	0	0
$445$	−27.1376	−1.28645
$446$	34.3619	1.62708
$447$	0	0
$448$	−20.0599	−0.947739
$449$	34.3854	1.62275	0.811373	−	0.584529i	$-0.198721\pi$
$449$	34.3854	1.62275	0.811373	+	0.584529i	$0.198721\pi$
$450$	0	0
$451$	−15.2488	−0.718039
$452$	−16.8852	−0.794215
$453$	0	0
$454$	14.0140	0.657710
$455$	4.14783	0.194453
$456$	0	0
$457$	16.6537	0.779028	0.389514	−	0.921021i	$-0.372643\pi$
$457$	16.6537	0.779028	0.389514	+	0.921021i	$0.372643\pi$
$458$	59.8635	2.79724
$459$	0	0
$460$	−86.9425	−4.05371
$461$	13.7835	0.641960	0.320980	−	0.947086i	$-0.395988\pi$
$461$	13.7835	0.641960	0.320980	+	0.947086i	$0.395988\pi$
$462$	0	0
$463$	12.2429	0.568974	0.284487	−	0.958680i	$-0.408177\pi$
$463$	12.2429	0.568974	0.284487	+	0.958680i	$0.408177\pi$
$464$	−7.83350	−0.363661
$465$	0	0
$466$	31.0559	1.43864
$467$	−15.3940	−0.712349	−0.356175	−	0.934419i	$-0.615919\pi$
$467$	−15.3940	−0.712349	−0.356175	+	0.934419i	$0.615919\pi$
$468$	0	0
$469$	0.646448	0.0298502
$470$	6.22020	0.286916
$471$	0	0
$472$	−2.89061	−0.133051
$473$	−9.63282	−0.442918
$474$	0	0
$475$	−16.7221	−0.767264
$476$	−65.1616	−2.98668
$477$	0	0
$478$	−59.3203	−2.71325
$479$	−15.1504	−0.692239	−0.346119	−	0.938191i	$-0.612501\pi$
$479$	−15.1504	−0.692239	−0.346119	+	0.938191i	$0.612501\pi$
$480$	0	0
$481$	2.61656	0.119305
$482$	70.8163	3.22559
$483$	0	0
$484$	−21.2067	−0.963940
$485$	−5.05122	−0.229364
$486$	0	0
$487$	−9.74343	−0.441517	−0.220758	−	0.975329i	$-0.570853\pi$
$487$	−9.74343	−0.441517	−0.220758	+	0.975329i	$0.570853\pi$
$488$	25.7519	1.16574
$489$	0	0
$490$	12.6273	0.570441
$491$	−19.1343	−0.863521	−0.431760	−	0.901988i	$-0.642107\pi$
$491$	−19.1343	−0.863521	−0.431760	+	0.901988i	$0.642107\pi$
$492$	0	0
$493$	15.7109	0.707584
$494$	6.73948	0.303223
$495$	0	0
$496$	11.1986	0.502830
$497$	−18.8106	−0.843773
$498$	0	0
$499$	−26.1237	−1.16946	−0.584729	−	0.811229i	$-0.698799\pi$
$499$	−26.1237	−1.16946	−0.584729	+	0.811229i	$0.698799\pi$
$500$	−14.9796	−0.669909
$501$	0	0
$502$	58.0793	2.59220
$503$	1.10887	0.0494421	0.0247210	−	0.999694i	$-0.492130\pi$
$503$	1.10887	0.0494421	0.0247210	+	0.999694i	$0.492130\pi$
$504$	0	0
$505$	−25.0135	−1.11309
$506$	43.2049	1.92069
$507$	0	0
$508$	−23.1588	−1.02750
$509$	−16.9747	−0.752390	−0.376195	−	0.926541i	$-0.622768\pi$
$509$	−16.9747	−0.752390	−0.376195	+	0.926541i	$0.622768\pi$
$510$	0	0
$511$	34.9064	1.54417
$512$	36.2698	1.60292
$513$	0	0
$514$	−29.5656	−1.30408
$515$	−5.35380	−0.235917
$516$	0	0
$517$	−2.04919	−0.0901234
$518$	−23.7793	−1.04480
$519$	0	0
$520$	−8.53243	−0.374172
$521$	−31.2481	−1.36901	−0.684503	−	0.729010i	$-0.739981\pi$
$521$	−31.2481	−1.36901	−0.684503	+	0.729010i	$0.739981\pi$
$522$	0	0
$523$	−12.8885	−0.563575	−0.281787	−	0.959477i	$-0.590927\pi$
$523$	−12.8885	−0.563575	−0.281787	+	0.959477i	$0.590927\pi$
$524$	−51.3725	−2.24422
$525$	0	0
$526$	−19.4915	−0.849872
$527$	−22.4599	−0.978369
$528$	0	0
$529$	33.0853	1.43849
$530$	−22.3323	−0.970051
$531$	0	0
$532$	−40.6042	−1.76042
$533$	−3.95185	−0.171173
$534$	0	0
$535$	−13.3790	−0.578423
$536$	−1.32980	−0.0574385
$537$	0	0
$538$	−54.8461	−2.36459
$539$	−4.15994	−0.179181
$540$	0	0
$541$	7.94206	0.341456	0.170728	−	0.985318i	$-0.445388\pi$
$541$	7.94206	0.341456	0.170728	+	0.985318i	$0.445388\pi$
$542$	73.8154	3.17064
$543$	0	0
$544$	−4.59272	−0.196911
$545$	39.1113	1.67534
$546$	0	0
$547$	−10.8326	−0.463166	−0.231583	−	0.972815i	$-0.574391\pi$
$547$	−10.8326	−0.463166	−0.231583	+	0.972815i	$0.574391\pi$
$548$	17.8089	0.760757
$549$	0	0
$550$	−21.4016	−0.912568
$551$	9.78996	0.417066
$552$	0	0
$553$	17.0703	0.725902
$554$	10.7759	0.457824
$555$	0	0
$556$	−18.8533	−0.799557
$557$	−1.76797	−0.0749114	−0.0374557	−	0.999298i	$-0.511925\pi$
$557$	−1.76797	−0.0749114	−0.0374557	+	0.999298i	$0.511925\pi$
$558$	0	0
$559$	−2.49642	−0.105587
$560$	24.3748	1.03002
$561$	0	0
$562$	−13.8071	−0.582415
$563$	−7.65073	−0.322440	−0.161220	−	0.986919i	$-0.551543\pi$
$563$	−7.65073	−0.322440	−0.161220	+	0.986919i	$0.551543\pi$
$564$	0	0
$565$	−12.6678	−0.532940
$566$	−70.2116	−2.95121
$567$	0	0
$568$	38.6951	1.62361
$569$	19.8983	0.834181	0.417091	−	0.908865i	$-0.363050\pi$
$569$	19.8983	0.834181	0.417091	+	0.908865i	$0.363050\pi$
$570$	0	0
$571$	−46.0688	−1.92792	−0.963959	−	0.266052i	$-0.914281\pi$
$571$	−46.0688	−1.92792	−0.963959	+	0.266052i	$0.914281\pi$
$572$	5.71818	0.239089
$573$	0	0
$574$	35.9143	1.49904
$575$	−27.7819	−1.15859
$576$	0	0
$577$	25.9478	1.08022	0.540110	−	0.841595i	$-0.318383\pi$
$577$	25.9478	1.08022	0.540110	+	0.841595i	$0.318383\pi$
$578$	−86.0608	−3.57966
$579$	0	0
$580$	−25.2136	−1.04694
$581$	−12.5422	−0.520338
$582$	0	0
$583$	7.35717	0.304703
$584$	−71.8054	−2.97133
$585$	0	0
$586$	−5.27347	−0.217845
$587$	21.0149	0.867378	0.433689	−	0.901063i	$-0.357212\pi$
$587$	21.0149	0.867378	0.433689	+	0.901063i	$0.357212\pi$
$588$	0	0
$589$	−13.9955	−0.576673
$590$	−4.41155	−0.181621
$591$	0	0
$592$	15.3763	0.631961
$593$	25.8048	1.05967	0.529837	−	0.848099i	$-0.322253\pi$
$593$	25.8048	1.05967	0.529837	+	0.848099i	$0.322253\pi$
$594$	0	0
$595$	−48.8863	−2.00414
$596$	−38.6120	−1.58161
$597$	0	0
$598$	11.1969	0.457874
$599$	−27.4549	−1.12178	−0.560888	−	0.827891i	$-0.689540\pi$
$599$	−27.4549	−1.12178	−0.560888	+	0.827891i	$0.689540\pi$
$600$	0	0
$601$	0.381462	0.0155602	0.00778009	−	0.999970i	$-0.497523\pi$
$601$	0.381462	0.0155602	0.00778009	+	0.999970i	$0.497523\pi$
$602$	22.6874	0.924671
$603$	0	0
$604$	52.5283	2.13735
$605$	−15.9099	−0.646830
$606$	0	0
$607$	6.16050	0.250047	0.125024	−	0.992154i	$-0.460099\pi$
$607$	6.16050	0.250047	0.125024	+	0.992154i	$0.460099\pi$
$608$	−2.86187	−0.116064
$609$	0	0
$610$	39.3018	1.59128
$611$	−0.531064	−0.0214845
$612$	0	0
$613$	−28.0321	−1.13220	−0.566102	−	0.824335i	$-0.691549\pi$
$613$	−28.0321	−1.13220	−0.566102	+	0.824335i	$0.691549\pi$
$614$	−11.5578	−0.466433
$615$	0	0
$616$	−25.5458	−1.02927
$617$	−14.9225	−0.600756	−0.300378	−	0.953820i	$-0.597113\pi$
$617$	−14.9225	−0.600756	−0.300378	+	0.953820i	$0.597113\pi$
$618$	0	0
$619$	−4.17453	−0.167788	−0.0838942	−	0.996475i	$-0.526736\pi$
$619$	−4.17453	−0.167788	−0.0838942	+	0.996475i	$0.526736\pi$
$620$	36.0447	1.44759
$621$	0	0
$622$	−57.4989	−2.30550
$623$	−21.0563	−0.843602
$624$	0	0
$625$	−29.7866	−1.19147
$626$	−57.5614	−2.30062
$627$	0	0
$628$	−11.7977	−0.470780
$629$	−30.8388	−1.22962
$630$	0	0
$631$	32.0854	1.27730	0.638651	−	0.769497i	$-0.279493\pi$
$631$	32.0854	1.27730	0.638651	+	0.769497i	$0.279493\pi$
$632$	−35.1150	−1.39680
$633$	0	0
$634$	−83.0023	−3.29644
$635$	−17.3744	−0.689483
$636$	0	0
$637$	−1.07808	−0.0427151
$638$	12.5296	0.496050
$639$	0	0
$640$	59.2296	2.34126
$641$	−39.4060	−1.55644	−0.778222	−	0.627989i	$-0.783878\pi$
$641$	−39.4060	−1.55644	−0.778222	+	0.627989i	$0.783878\pi$
$642$	0	0
$643$	−28.5441	−1.12567	−0.562835	−	0.826569i	$-0.690289\pi$
$643$	−28.5441	−1.12567	−0.562835	+	0.826569i	$0.690289\pi$
$644$	−67.4593	−2.65827
$645$	0	0
$646$	−79.4315	−3.12519
$647$	3.61582	0.142153	0.0710764	−	0.997471i	$-0.477357\pi$
$647$	3.61582	0.142153	0.0710764	+	0.997471i	$0.477357\pi$
$648$	0	0
$649$	1.45335	0.0570489
$650$	−5.54639	−0.217547
$651$	0	0
$652$	69.7463	2.73147
$653$	11.8667	0.464381	0.232191	−	0.972670i	$-0.425411\pi$
$653$	11.8667	0.464381	0.232191	+	0.972670i	$0.425411\pi$
$654$	0	0
$655$	−38.5413	−1.50593
$656$	−23.2231	−0.906710
$657$	0	0
$658$	4.82630	0.188149
$659$	−2.63237	−0.102543	−0.0512713	−	0.998685i	$-0.516327\pi$
$659$	−2.63237	−0.102543	−0.0512713	+	0.998685i	$0.516327\pi$
$660$	0	0
$661$	25.8462	1.00530	0.502650	−	0.864490i	$-0.332358\pi$
$661$	25.8462	1.00530	0.502650	+	0.864490i	$0.332358\pi$
$662$	−63.1563	−2.45464
$663$	0	0
$664$	25.8003	1.00125
$665$	−30.4626	−1.18129
$666$	0	0
$667$	16.2649	0.629780
$668$	−86.5084	−3.34711
$669$	0	0
$670$	−2.02950	−0.0784062
$671$	−12.9476	−0.499838
$672$	0	0
$673$	1.20349	0.0463913	0.0231956	−	0.999731i	$-0.492616\pi$
$673$	1.20349	0.0463913	0.0231956	+	0.999731i	$0.492616\pi$
$674$	−53.9593	−2.07843
$675$	0	0
$676$	−49.6568	−1.90988
$677$	−39.2031	−1.50670	−0.753349	−	0.657621i	$-0.771563\pi$
$677$	−39.2031	−1.50670	−0.753349	+	0.657621i	$0.771563\pi$
$678$	0	0
$679$	−3.91928	−0.150408
$680$	100.563	3.85642
$681$	0	0
$682$	−17.9119	−0.685882
$683$	6.64773	0.254368	0.127184	−	0.991879i	$-0.459406\pi$
$683$	6.64773	0.254368	0.127184	+	0.991879i	$0.459406\pi$
$684$	0	0
$685$	13.3608	0.510489
$686$	48.8433	1.86485
$687$	0	0
$688$	−14.6703	−0.559299
$689$	1.90667	0.0726382
$690$	0	0
$691$	−11.3683	−0.432469	−0.216235	−	0.976341i	$-0.569378\pi$
$691$	−11.3683	−0.432469	−0.216235	+	0.976341i	$0.569378\pi$
$692$	42.5827	1.61875
$693$	0	0
$694$	−4.36859	−0.165830
$695$	−14.1443	−0.536525
$696$	0	0
$697$	46.5765	1.76421
$698$	−0.428424	−0.0162161
$699$	0	0
$700$	33.4161	1.26301
$701$	28.5496	1.07830	0.539152	−	0.842209i	$-0.318745\pi$
$701$	28.5496	1.07830	0.539152	+	0.842209i	$0.318745\pi$
$702$	0	0
$703$	−19.2166	−0.724767
$704$	−20.7472	−0.781941
$705$	0	0
$706$	74.0856	2.78825
$707$	−19.4082	−0.729919
$708$	0	0
$709$	−43.6522	−1.63939	−0.819696	−	0.572799i	$-0.805857\pi$
$709$	−43.6522	−1.63939	−0.819696	+	0.572799i	$0.805857\pi$
$710$	59.0552	2.21630
$711$	0	0
$712$	43.3145	1.62328
$713$	−23.2519	−0.870789
$714$	0	0
$715$	4.28996	0.160436
$716$	−35.1743	−1.31453
$717$	0	0
$718$	15.7100	0.586292
$719$	29.9882	1.11837	0.559186	−	0.829042i	$-0.311114\pi$
$719$	29.9882	1.11837	0.559186	+	0.829042i	$0.311114\pi$
$720$	0	0
$721$	−4.15405	−0.154705
$722$	−3.21359	−0.119597
$723$	0	0
$724$	17.0132	0.632291
$725$	−8.05684	−0.299224
$726$	0	0
$727$	40.0822	1.48657	0.743284	−	0.668976i	$-0.233267\pi$
$727$	40.0822	1.48657	0.743284	+	0.668976i	$0.233267\pi$
$728$	−6.62038	−0.245368
$729$	0	0
$730$	−109.587	−4.05600
$731$	29.4228	1.08824
$732$	0	0
$733$	27.4639	1.01440	0.507201	−	0.861828i	$-0.330680\pi$
$733$	27.4639	1.01440	0.507201	+	0.861828i	$0.330680\pi$
$734$	−44.8325	−1.65480
$735$	0	0
$736$	−4.75467	−0.175259
$737$	0.668600	0.0246282
$738$	0	0
$739$	31.0654	1.14276	0.571379	−	0.820687i	$-0.306409\pi$
$739$	31.0654	1.14276	0.571379	+	0.820687i	$0.306409\pi$
$740$	49.4914	1.81934
$741$	0	0
$742$	−17.3278	−0.636122
$743$	−6.47678	−0.237610	−0.118805	−	0.992918i	$-0.537906\pi$
$743$	−6.47678	−0.237610	−0.118805	+	0.992918i	$0.537906\pi$
$744$	0	0
$745$	−28.9680	−1.06130
$746$	78.6429	2.87932
$747$	0	0
$748$	−67.3945	−2.46419
$749$	−10.3808	−0.379308
$750$	0	0
$751$	−7.41189	−0.270464	−0.135232	−	0.990814i	$-0.543178\pi$
$751$	−7.41189	−0.270464	−0.135232	+	0.990814i	$0.543178\pi$
$752$	−3.12081	−0.113804
$753$	0	0
$754$	3.24713	0.118253
$755$	39.4084	1.43422
$756$	0	0
$757$	44.5682	1.61986	0.809930	−	0.586526i	$-0.199505\pi$
$757$	44.5682	1.61986	0.809930	+	0.586526i	$0.199505\pi$
$758$	53.6699	1.94938
$759$	0	0
$760$	62.6641	2.27306
$761$	−20.0195	−0.725708	−0.362854	−	0.931846i	$-0.618198\pi$
$761$	−20.0195	−0.725708	−0.362854	+	0.931846i	$0.618198\pi$
$762$	0	0
$763$	30.3468	1.09863
$764$	83.4447	3.01892
$765$	0	0
$766$	34.6536	1.25209
$767$	0.376646	0.0135999
$768$	0	0
$769$	−4.98844	−0.179888	−0.0899438	−	0.995947i	$-0.528669\pi$
$769$	−4.98844	−0.179888	−0.0899438	+	0.995947i	$0.528669\pi$
$770$	−38.9871	−1.40500
$771$	0	0
$772$	−71.4166	−2.57034
$773$	−23.4107	−0.842024	−0.421012	−	0.907055i	$-0.638325\pi$
$773$	−23.4107	−0.842024	−0.421012	+	0.907055i	$0.638325\pi$
$774$	0	0
$775$	11.5178	0.413733
$776$	8.06228	0.289419
$777$	0	0
$778$	−29.8793	−1.07122
$779$	29.0232	1.03987
$780$	0	0
$781$	−19.4552	−0.696163
$782$	−131.966	−4.71911
$783$	0	0
$784$	−6.33536	−0.226263
$785$	−8.85102	−0.315907
$786$	0	0
$787$	−49.9163	−1.77932	−0.889661	−	0.456621i	$-0.849060\pi$
$787$	−49.9163	−1.77932	−0.889661	+	0.456621i	$0.849060\pi$
$788$	28.0806	1.00033
$789$	0	0
$790$	−53.5913	−1.90669
$791$	−9.82908	−0.349482
$792$	0	0
$793$	−3.35548	−0.119157
$794$	−23.6637	−0.839793
$795$	0	0
$796$	−14.3818	−0.509750
$797$	−40.2061	−1.42417	−0.712087	−	0.702091i	$-0.752250\pi$
$797$	−40.2061	−1.42417	−0.712087	+	0.702091i	$0.752250\pi$
$798$	0	0
$799$	6.25912	0.221432
$800$	2.35523	0.0832700
$801$	0	0
$802$	−2.84824	−0.100575
$803$	36.1025	1.27403
$804$	0	0
$805$	−50.6101	−1.78377
$806$	−4.64201	−0.163508
$807$	0	0
$808$	39.9242	1.40453
$809$	44.4490	1.56274	0.781371	−	0.624066i	$-0.214521\pi$
$809$	44.4490	1.56274	0.781371	+	0.624066i	$0.214521\pi$
$810$	0	0
$811$	−38.6494	−1.35716	−0.678582	−	0.734524i	$-0.737405\pi$
$811$	−38.6494	−1.35716	−0.678582	+	0.734524i	$0.737405\pi$
$812$	−19.5634	−0.686541
$813$	0	0
$814$	−24.5941	−0.862023
$815$	52.3258	1.83289
$816$	0	0
$817$	18.3343	0.641434
$818$	66.8928	2.33885
$819$	0	0
$820$	−74.7480	−2.61031
$821$	43.3001	1.51118	0.755591	−	0.655043i	$-0.227350\pi$
$821$	43.3001	1.51118	0.755591	+	0.655043i	$0.227350\pi$
$822$	0	0
$823$	−30.1042	−1.04937	−0.524683	−	0.851298i	$-0.675816\pi$
$823$	−30.1042	−1.04937	−0.524683	+	0.851298i	$0.675816\pi$
$824$	8.54523	0.297687
$825$	0	0
$826$	−3.42296	−0.119100
$827$	−16.4038	−0.570414	−0.285207	−	0.958466i	$-0.592062\pi$
$827$	−16.4038	−0.570414	−0.285207	+	0.958466i	$0.592062\pi$
$828$	0	0
$829$	29.0317	1.00831	0.504157	−	0.863612i	$-0.331803\pi$
$829$	29.0317	1.00831	0.504157	+	0.863612i	$0.331803\pi$
$830$	39.3756	1.36675
$831$	0	0
$832$	−5.37680	−0.186407
$833$	12.7063	0.440246
$834$	0	0
$835$	−64.9014	−2.24600
$836$	−41.9956	−1.45245
$837$	0	0
$838$	−59.9178	−2.06983
$839$	35.5031	1.22570	0.612852	−	0.790198i	$-0.290022\pi$
$839$	35.5031	1.22570	0.612852	+	0.790198i	$0.290022\pi$
$840$	0	0
$841$	−24.2831	−0.837349
$842$	29.9673	1.03274
$843$	0	0
$844$	−70.2256	−2.41727
$845$	−37.2541	−1.28158
$846$	0	0
$847$	−12.3446	−0.424166
$848$	11.2046	0.384767
$849$	0	0
$850$	65.3697	2.24216
$851$	−31.9262	−1.09442
$852$	0	0
$853$	−0.962392	−0.0329517	−0.0164758	−	0.999864i	$-0.505245\pi$
$853$	−0.962392	−0.0329517	−0.0164758	+	0.999864i	$0.505245\pi$
$854$	30.4946	1.04350
$855$	0	0
$856$	21.3543	0.729874
$857$	0.567569	0.0193878	0.00969390	−	0.999953i	$-0.496914\pi$
$857$	0.567569	0.0193878	0.00969390	+	0.999953i	$0.496914\pi$
$858$	0	0
$859$	20.1089	0.686107	0.343054	−	0.939316i	$-0.388539\pi$
$859$	20.1089	0.686107	0.343054	+	0.939316i	$0.388539\pi$
$860$	−47.2190	−1.61015
$861$	0	0
$862$	19.3818	0.660145
$863$	37.6306	1.28096	0.640481	−	0.767974i	$-0.278735\pi$
$863$	37.6306	1.28096	0.640481	+	0.767974i	$0.278735\pi$
$864$	0	0
$865$	31.9469	1.08623
$866$	−29.0234	−0.986254
$867$	0	0
$868$	27.9673	0.949273
$869$	17.6552	0.598912
$870$	0	0
$871$	0.173273	0.00587112
$872$	−62.4258	−2.11401
$873$	0	0
$874$	−82.2323	−2.78155
$875$	−8.71979	−0.294783
$876$	0	0
$877$	27.4060	0.925436	0.462718	−	0.886506i	$-0.346874\pi$
$877$	27.4060	0.925436	0.462718	+	0.886506i	$0.346874\pi$
$878$	−16.8223	−0.567726
$879$	0	0
$880$	25.2101	0.849831
$881$	33.6133	1.13246	0.566231	−	0.824247i	$-0.308401\pi$
$881$	33.6133	1.13246	0.566231	+	0.824247i	$0.308401\pi$
$882$	0	0
$883$	−42.9993	−1.44704	−0.723520	−	0.690303i	$-0.757477\pi$
$883$	−42.9993	−1.44704	−0.723520	+	0.690303i	$0.757477\pi$
$884$	−17.4658	−0.587438
$885$	0	0
$886$	−17.6485	−0.592913
$887$	−31.0012	−1.04092	−0.520459	−	0.853887i	$-0.674239\pi$
$887$	−31.0012	−1.04092	−0.520459	+	0.853887i	$0.674239\pi$
$888$	0	0
$889$	−13.4810	−0.452137
$890$	66.1052	2.21585
$891$	0	0
$892$	−55.4905	−1.85796
$893$	3.90025	0.130517
$894$	0	0
$895$	−26.3889	−0.882083
$896$	45.9567	1.53531
$897$	0	0
$898$	−83.7602	−2.79512
$899$	−6.74311	−0.224895
$900$	0	0
$901$	−22.4720	−0.748650
$902$	37.1450	1.23679
$903$	0	0
$904$	20.2192	0.672481
$905$	12.7639	0.424285
$906$	0	0
$907$	−3.91329	−0.129939	−0.0649693	−	0.997887i	$-0.520695\pi$
$907$	−3.91329	−0.129939	−0.0649693	+	0.997887i	$0.520695\pi$
$908$	−22.6310	−0.751037
$909$	0	0
$910$	−10.1038	−0.334938
$911$	38.9679	1.29107	0.645533	−	0.763733i	$-0.276635\pi$
$911$	38.9679	1.29107	0.645533	+	0.763733i	$0.276635\pi$
$912$	0	0
$913$	−12.9720	−0.429309
$914$	−40.5672	−1.34184
$915$	0	0
$916$	−96.6727	−3.19415
$917$	−29.9045	−0.987533
$918$	0	0
$919$	−21.2506	−0.700994	−0.350497	−	0.936564i	$-0.613987\pi$
$919$	−21.2506	−0.700994	−0.350497	+	0.936564i	$0.613987\pi$
$920$	104.109	3.43238
$921$	0	0
$922$	−33.5755	−1.10575
$923$	−5.04197	−0.165958
$924$	0	0
$925$	15.8147	0.519983
$926$	−29.8227	−0.980035
$927$	0	0
$928$	−1.37887	−0.0452636
$929$	16.3440	0.536228	0.268114	−	0.963387i	$-0.413600\pi$
$929$	16.3440	0.536228	0.268114	+	0.963387i	$0.413600\pi$
$930$	0	0
$931$	7.91766	0.259491
$932$	−50.1517	−1.64277
$933$	0	0
$934$	37.4987	1.22699
$935$	−50.5615	−1.65354
$936$	0	0
$937$	−22.7256	−0.742413	−0.371206	−	0.928550i	$-0.621056\pi$
$937$	−22.7256	−0.742413	−0.371206	+	0.928550i	$0.621056\pi$
$938$	−1.57470	−0.0514158
$939$	0	0
$940$	−10.0449	−0.327629
$941$	−13.1060	−0.427244	−0.213622	−	0.976916i	$-0.568526\pi$
$941$	−13.1060	−0.427244	−0.213622	+	0.976916i	$0.568526\pi$
$942$	0	0
$943$	48.2188	1.57022
$944$	2.21337	0.0720391
$945$	0	0
$946$	23.4649	0.762908
$947$	21.6269	0.702781	0.351391	−	0.936229i	$-0.385709\pi$
$947$	21.6269	0.702781	0.351391	+	0.936229i	$0.385709\pi$
$948$	0	0
$949$	9.35624	0.303716
$950$	40.7339	1.32158
$951$	0	0
$952$	78.0278	2.52890
$953$	21.9896	0.712314	0.356157	−	0.934426i	$-0.384087\pi$
$953$	21.9896	0.712314	0.356157	+	0.934426i	$0.384087\pi$
$954$	0	0
$955$	62.6029	2.02578
$956$	95.7954	3.09825
$957$	0	0
$958$	36.9052	1.19235
$959$	10.3667	0.334759
$960$	0	0
$961$	−21.3602	−0.689040
$962$	−6.37375	−0.205498
$963$	0	0
$964$	−114.360	−3.68329
$965$	−53.5790	−1.72477
$966$	0	0
$967$	23.7727	0.764479	0.382239	−	0.924063i	$-0.375153\pi$
$967$	23.7727	0.764479	0.382239	+	0.924063i	$0.375153\pi$
$968$	25.3939	0.816192
$969$	0	0
$970$	12.3044	0.395071
$971$	−16.4980	−0.529445	−0.264723	−	0.964325i	$-0.585280\pi$
$971$	−16.4980	−0.529445	−0.264723	+	0.964325i	$0.585280\pi$
$972$	0	0
$973$	−10.9747	−0.351833
$974$	23.7343	0.760496
$975$	0	0
$976$	−19.7186	−0.631175
$977$	−6.14980	−0.196750	−0.0983748	−	0.995149i	$-0.531364\pi$
$977$	−6.14980	−0.196750	−0.0983748	+	0.995149i	$0.531364\pi$
$978$	0	0
$979$	−21.7778	−0.696022
$980$	−20.3916	−0.651385
$981$	0	0
$982$	46.6099	1.48738
$983$	11.6559	0.371764	0.185882	−	0.982572i	$-0.440486\pi$
$983$	11.6559	0.371764	0.185882	+	0.982572i	$0.440486\pi$
$984$	0	0
$985$	21.0669	0.671249
$986$	−38.2707	−1.21879
$987$	0	0
$988$	−10.8835	−0.346250
$989$	30.4603	0.968581
$990$	0	0
$991$	−27.3623	−0.869193	−0.434596	−	0.900625i	$-0.643109\pi$
$991$	−27.3623	−0.869193	−0.434596	+	0.900625i	$0.643109\pi$
$992$	1.97119	0.0625854
$993$	0	0
$994$	45.8214	1.45337
$995$	−10.7897	−0.342056
$996$	0	0
$997$	54.6226	1.72991	0.864957	−	0.501846i	$-0.167346\pi$
$997$	54.6226	1.72991	0.864957	+	0.501846i	$0.167346\pi$
$998$	63.6354	2.01434
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.8		72
3.2	odd	2		6561.2.a.d.1.65		72
81.2	odd	54		729.2.g.a.433.1		144
81.13	even	27		81.2.g.a.7.1	✓	144
81.14	odd	54		729.2.g.b.55.8		144
81.25	even	27		81.2.g.a.58.1	yes	144
81.29	odd	54		729.2.g.b.676.8		144
81.40	even	27		729.2.g.d.298.8		144
81.41	odd	54		729.2.g.a.298.1		144
81.52	even	27		729.2.g.c.676.1		144
81.56	odd	54		243.2.g.a.226.8		144
81.67	even	27		729.2.g.c.55.1		144
81.68	odd	54		243.2.g.a.100.8		144
81.79	even	27		729.2.g.d.433.8		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.7.1	✓	144	81.13	even	27
81.2.g.a.58.1	yes	144	81.25	even	27
243.2.g.a.100.8		144	81.68	odd	54
243.2.g.a.226.8		144	81.56	odd	54
729.2.g.a.298.1		144	81.41	odd	54
729.2.g.a.433.1		144	81.2	odd	54
729.2.g.b.55.8		144	81.14	odd	54
729.2.g.b.676.8		144	81.29	odd	54
729.2.g.c.55.1		144	81.67	even	27
729.2.g.c.676.1		144	81.52	even	27
729.2.g.d.298.8		144	81.40	even	27
729.2.g.d.433.8		144	81.79	even	27
6561.2.a.c.1.8		72	1.1	even	1	trivial
6561.2.a.d.1.65		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)

\(f(q)\)	\(=\)	\(q-2.43593 q^{2} +3.93374 q^{4} +2.95122 q^{5} +2.28987 q^{7} -4.71046 q^{8} +O(q^{10})\)	\(q-2.43593 q^{2} +3.93374 q^{4} +2.95122 q^{5} +2.28987 q^{7} -4.71046 q^{8} -7.18896 q^{10} +2.36834 q^{11} +0.613773 q^{13} -5.57797 q^{14} +3.60685 q^{16} -7.23393 q^{17} -4.50769 q^{19} +11.6093 q^{20} -5.76911 q^{22} -7.48901 q^{23} +3.70969 q^{25} -1.49511 q^{26} +9.00778 q^{28} -2.17184 q^{29} +3.10480 q^{31} +0.634886 q^{32} +17.6213 q^{34} +6.75792 q^{35} +4.26307 q^{37} +10.9804 q^{38} -13.9016 q^{40} -6.43861 q^{41} -4.06733 q^{43} +9.31644 q^{44} +18.2427 q^{46} -0.865244 q^{47} -1.75648 q^{49} -9.03654 q^{50} +2.41443 q^{52} +3.10647 q^{53} +6.98949 q^{55} -10.7864 q^{56} +5.29044 q^{58} +0.613657 q^{59} -5.46697 q^{61} -7.56306 q^{62} -8.76024 q^{64} +1.81138 q^{65} +0.282307 q^{67} -28.4564 q^{68} -16.4618 q^{70} -8.21471 q^{71} +15.2438 q^{73} -10.3845 q^{74} -17.7321 q^{76} +5.42320 q^{77} +7.45468 q^{79} +10.6446 q^{80} +15.6840 q^{82} -5.47724 q^{83} -21.3489 q^{85} +9.90772 q^{86} -11.1560 q^{88} -9.19539 q^{89} +1.40546 q^{91} -29.4599 q^{92} +2.10767 q^{94} -13.3032 q^{95} -1.71157 q^{97} +4.27866 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8}+O(q^{10}) \) 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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