LMFDB - Embedded newform 6561.2.a.c.1.14

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			1.14
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)

$f(q)$	$=$	\(q-2.01939 q^{2} +2.07795 q^{4} -2.76814 q^{5} -0.611277 q^{7} -0.157420 q^{8} +5.58998 q^{10} -3.74484 q^{11} +3.41992 q^{13} +1.23441 q^{14} -3.83801 q^{16} -7.57368 q^{17} +0.763428 q^{19} -5.75208 q^{20} +7.56230 q^{22} +2.17394 q^{23} +2.66263 q^{25} -6.90616 q^{26} -1.27021 q^{28} -4.54272 q^{29} +8.18558 q^{31} +8.06531 q^{32} +15.2942 q^{34} +1.69210 q^{35} -4.22381 q^{37} -1.54166 q^{38} +0.435762 q^{40} +9.04205 q^{41} +1.75473 q^{43} -7.78160 q^{44} -4.39004 q^{46} +11.7637 q^{47} -6.62634 q^{49} -5.37689 q^{50} +7.10643 q^{52} +5.60125 q^{53} +10.3663 q^{55} +0.0962275 q^{56} +9.17355 q^{58} +1.51320 q^{59} +3.02355 q^{61} -16.5299 q^{62} -8.61101 q^{64} -9.46682 q^{65} -7.65215 q^{67} -15.7378 q^{68} -3.41703 q^{70} -6.78616 q^{71} +0.572436 q^{73} +8.52954 q^{74} +1.58637 q^{76} +2.28913 q^{77} +9.21512 q^{79} +10.6242 q^{80} -18.2595 q^{82} -6.18843 q^{83} +20.9650 q^{85} -3.54350 q^{86} +0.589514 q^{88} -4.60472 q^{89} -2.09052 q^{91} +4.51734 q^{92} -23.7556 q^{94} -2.11328 q^{95} -10.5169 q^{97} +13.3812 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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 $

$2$	−2.01939	−1.42793	−0.713964	−	0.700183i	$-0.753102\pi$
$2$	−2.01939	−1.42793	−0.713964	+	0.700183i	$0.753102\pi$
$3$	0	0
$3$	0	0
$4$	2.07795	1.03898
$5$	−2.76814	−1.23795	−0.618976	−	0.785410i	$-0.712452\pi$
$5$	−2.76814	−1.23795	−0.618976	+	0.785410i	$0.712452\pi$
$6$	0	0
$7$	−0.611277	−0.231041	−0.115521	−	0.993305i	$-0.536854\pi$
$7$	−0.611277	−0.231041	−0.115521	+	0.993305i	$0.536854\pi$
$8$	−0.157420	−0.0556565
$9$	0	0
$10$	5.58998	1.76771
$11$	−3.74484	−1.12911	−0.564556	−	0.825395i	$-0.690952\pi$
$11$	−3.74484	−1.12911	−0.564556	+	0.825395i	$0.690952\pi$
$12$	0	0
$13$	3.41992	0.948514	0.474257	−	0.880386i	$-0.342717\pi$
$13$	3.41992	0.948514	0.474257	+	0.880386i	$0.342717\pi$
$14$	1.23441	0.329910
$15$	0	0
$16$	−3.83801	−0.959504
$17$	−7.57368	−1.83689	−0.918443	−	0.395553i	$-0.870553\pi$
$17$	−7.57368	−1.83689	−0.918443	+	0.395553i	$0.870553\pi$
$18$	0	0
$19$	0.763428	0.175142	0.0875712	−	0.996158i	$-0.472089\pi$
$19$	0.763428	0.175142	0.0875712	+	0.996158i	$0.472089\pi$
$20$	−5.75208	−1.28620
$21$	0	0
$22$	7.56230	1.61229
$23$	2.17394	0.453297	0.226649	−	0.973977i	$-0.427223\pi$
$23$	2.17394	0.453297	0.226649	+	0.973977i	$0.427223\pi$
$24$	0	0
$25$	2.66263	0.532525
$26$	−6.90616	−1.35441
$27$	0	0
$28$	−1.27021	−0.240046
$29$	−4.54272	−0.843563	−0.421781	−	0.906698i	$-0.638595\pi$
$29$	−4.54272	−0.843563	−0.421781	+	0.906698i	$0.638595\pi$
$30$	0	0
$31$	8.18558	1.47017	0.735087	−	0.677973i	$-0.237142\pi$
$31$	8.18558	1.47017	0.735087	+	0.677973i	$0.237142\pi$
$32$	8.06531	1.42576
$33$	0	0
$34$	15.2942	2.62294
$35$	1.69210	0.286018
$36$	0	0
$37$	−4.22381	−0.694390	−0.347195	−	0.937793i	$-0.612866\pi$
$37$	−4.22381	−0.694390	−0.347195	+	0.937793i	$0.612866\pi$
$38$	−1.54166	−0.250091
$39$	0	0
$40$	0.435762	0.0689001
$41$	9.04205	1.41213	0.706065	−	0.708147i	$-0.250469\pi$
$41$	9.04205	1.41213	0.706065	+	0.708147i	$0.250469\pi$
$42$	0	0
$43$	1.75473	0.267594	0.133797	−	0.991009i	$-0.457283\pi$
$43$	1.75473	0.267594	0.133797	+	0.991009i	$0.457283\pi$
$44$	−7.78160	−1.17312
$45$	0	0
$46$	−4.39004	−0.647276
$47$	11.7637	1.71592	0.857959	−	0.513718i	$-0.171732\pi$
$47$	11.7637	1.71592	0.857959	+	0.513718i	$0.171732\pi$
$48$	0	0
$49$	−6.62634	−0.946620
$50$	−5.37689	−0.760407
$51$	0	0
$52$	7.10643	0.985484
$53$	5.60125	0.769391	0.384695	−	0.923044i	$-0.374307\pi$
$53$	5.60125	0.769391	0.384695	+	0.923044i	$0.374307\pi$
$54$	0	0
$55$	10.3663	1.39779
$56$	0.0962275	0.0128589
$57$	0	0
$58$	9.17355	1.20455
$59$	1.51320	0.197002	0.0985008	−	0.995137i	$-0.468595\pi$
$59$	1.51320	0.197002	0.0985008	+	0.995137i	$0.468595\pi$
$60$	0	0
$61$	3.02355	0.387126	0.193563	−	0.981088i	$-0.437996\pi$
$61$	3.02355	0.387126	0.193563	+	0.981088i	$0.437996\pi$
$62$	−16.5299	−2.09930
$63$	0	0
$64$	−8.61101	−1.07638
$65$	−9.46682	−1.17421
$66$	0	0
$67$	−7.65215	−0.934859	−0.467430	−	0.884030i	$-0.654820\pi$
$67$	−7.65215	−0.934859	−0.467430	+	0.884030i	$0.654820\pi$
$68$	−15.7378	−1.90848
$69$	0	0
$70$	−3.41703	−0.408413
$71$	−6.78616	−0.805368	−0.402684	−	0.915339i	$-0.631923\pi$
$71$	−6.78616	−0.805368	−0.402684	+	0.915339i	$0.631923\pi$
$72$	0	0
$73$	0.572436	0.0669985	0.0334993	−	0.999439i	$-0.489335\pi$
$73$	0.572436	0.0669985	0.0334993	+	0.999439i	$0.489335\pi$
$74$	8.52954	0.991539
$75$	0	0
$76$	1.58637	0.181969
$77$	2.28913	0.260871
$78$	0	0
$79$	9.21512	1.03678	0.518391	−	0.855144i	$-0.326531\pi$
$79$	9.21512	1.03678	0.518391	+	0.855144i	$0.326531\pi$
$80$	10.6242	1.18782
$81$	0	0
$82$	−18.2595	−2.01642
$83$	−6.18843	−0.679269	−0.339634	−	0.940558i	$-0.610303\pi$
$83$	−6.18843	−0.679269	−0.339634	+	0.940558i	$0.610303\pi$
$84$	0	0
$85$	20.9650	2.27398
$86$	−3.54350	−0.382105
$87$	0	0
$88$	0.589514	0.0628424
$89$	−4.60472	−0.488099	−0.244050	−	0.969763i	$-0.578476\pi$
$89$	−4.60472	−0.488099	−0.244050	+	0.969763i	$0.578476\pi$
$90$	0	0
$91$	−2.09052	−0.219146
$92$	4.51734	0.470965
$93$	0	0
$94$	−23.7556	−2.45021
$95$	−2.11328	−0.216818
$96$	0	0
$97$	−10.5169	−1.06783	−0.533913	−	0.845540i	$-0.679279\pi$
$97$	−10.5169	−1.06783	−0.533913	+	0.845540i	$0.679279\pi$
$98$	13.3812	1.35170
$99$	0	0
$100$	5.53281	0.553281
$101$	2.44854	0.243639	0.121819	−	0.992552i	$-0.461127\pi$
$101$	2.44854	0.243639	0.121819	+	0.992552i	$0.461127\pi$
$102$	0	0
$103$	17.0579	1.68076	0.840381	−	0.541996i	$-0.182331\pi$
$103$	17.0579	1.68076	0.840381	+	0.541996i	$0.182331\pi$
$104$	−0.538364	−0.0527910
$105$	0	0
$106$	−11.3111	−1.09863
$107$	−0.804111	−0.0777364	−0.0388682	−	0.999244i	$-0.512375\pi$
$107$	−0.804111	−0.0777364	−0.0388682	+	0.999244i	$0.512375\pi$
$108$	0	0
$109$	−4.22270	−0.404461	−0.202230	−	0.979338i	$-0.564819\pi$
$109$	−4.22270	−0.404461	−0.202230	+	0.979338i	$0.564819\pi$
$110$	−20.9336	−1.99594
$111$	0	0
$112$	2.34609	0.221685
$113$	−12.3432	−1.16115	−0.580575	−	0.814207i	$-0.697172\pi$
$113$	−12.3432	−1.16115	−0.580575	+	0.814207i	$0.697172\pi$
$114$	0	0
$115$	−6.01777	−0.561160
$116$	−9.43957	−0.876442
$117$	0	0
$118$	−3.05574	−0.281304
$119$	4.62962	0.424396
$120$	0	0
$121$	3.02381	0.274892
$122$	−6.10574	−0.552788
$123$	0	0
$124$	17.0093	1.52748
$125$	6.47019	0.578711
$126$	0	0
$127$	9.97518	0.885154	0.442577	−	0.896731i	$-0.354064\pi$
$127$	9.97518	0.885154	0.442577	+	0.896731i	$0.354064\pi$
$128$	1.25841	0.111228
$129$	0	0
$130$	19.1172	1.67669
$131$	5.49315	0.479939	0.239969	−	0.970780i	$-0.422863\pi$
$131$	5.49315	0.479939	0.239969	+	0.970780i	$0.422863\pi$
$132$	0	0
$133$	−0.466666	−0.0404651
$134$	15.4527	1.33491
$135$	0	0
$136$	1.19225	0.102235
$137$	−15.2306	−1.30124	−0.650618	−	0.759405i	$-0.725490\pi$
$137$	−15.2306	−1.30124	−0.650618	+	0.759405i	$0.725490\pi$
$138$	0	0
$139$	16.0163	1.35848	0.679241	−	0.733915i	$-0.262309\pi$
$139$	16.0163	1.35848	0.679241	+	0.733915i	$0.262309\pi$
$140$	3.51612	0.297166
$141$	0	0
$142$	13.7039	1.15001
$143$	−12.8070	−1.07098
$144$	0	0
$145$	12.5749	1.04429
$146$	−1.15597	−0.0956690
$147$	0	0
$148$	−8.77688	−0.721455
$149$	8.84036	0.724231	0.362115	−	0.932133i	$-0.382055\pi$
$149$	8.84036	0.724231	0.362115	+	0.932133i	$0.382055\pi$
$150$	0	0
$151$	0.987846	0.0803898	0.0401949	−	0.999192i	$-0.487202\pi$
$151$	0.987846	0.0803898	0.0401949	+	0.999192i	$0.487202\pi$
$152$	−0.120179	−0.00974781
$153$	0	0
$154$	−4.62267	−0.372505
$155$	−22.6589	−1.82000
$156$	0	0
$157$	18.1985	1.45240	0.726201	−	0.687483i	$-0.241284\pi$
$157$	18.1985	1.45240	0.726201	+	0.687483i	$0.241284\pi$
$158$	−18.6090	−1.48045
$159$	0	0
$160$	−22.3259	−1.76502
$161$	−1.32888	−0.104730
$162$	0	0
$163$	17.1469	1.34305	0.671524	−	0.740983i	$-0.265640\pi$
$163$	17.1469	1.34305	0.671524	+	0.740983i	$0.265640\pi$
$164$	18.7890	1.46717
$165$	0	0
$166$	12.4969	0.969947
$167$	5.90116	0.456646	0.228323	−	0.973585i	$-0.426676\pi$
$167$	5.90116	0.456646	0.228323	+	0.973585i	$0.426676\pi$
$168$	0	0
$169$	−1.30418	−0.100321
$170$	−42.3367	−3.24707
$171$	0	0
$172$	3.64625	0.278024
$173$	8.99272	0.683704	0.341852	−	0.939754i	$-0.388946\pi$
$173$	8.99272	0.683704	0.341852	+	0.939754i	$0.388946\pi$
$174$	0	0
$175$	−1.62760	−0.123035
$176$	14.3727	1.08339
$177$	0	0
$178$	9.29875	0.696971
$179$	1.28893	0.0963390	0.0481695	−	0.998839i	$-0.484661\pi$
$179$	1.28893	0.0963390	0.0481695	+	0.998839i	$0.484661\pi$
$180$	0	0
$181$	3.71663	0.276255	0.138128	−	0.990414i	$-0.455892\pi$
$181$	3.71663	0.276255	0.138128	+	0.990414i	$0.455892\pi$
$182$	4.22158	0.312924
$183$	0	0
$184$	−0.342222	−0.0252289
$185$	11.6921	0.859622
$186$	0	0
$187$	28.3622	2.07405
$188$	24.4445	1.78280
$189$	0	0
$190$	4.26755	0.309600
$191$	7.29004	0.527489	0.263744	−	0.964593i	$-0.415042\pi$
$191$	7.29004	0.527489	0.263744	+	0.964593i	$0.415042\pi$
$192$	0	0
$193$	−9.62715	−0.692978	−0.346489	−	0.938054i	$-0.612626\pi$
$193$	−9.62715	−0.692978	−0.346489	+	0.938054i	$0.612626\pi$
$194$	21.2377	1.52478
$195$	0	0
$196$	−13.7692	−0.983517
$197$	−4.93670	−0.351725	−0.175862	−	0.984415i	$-0.556271\pi$
$197$	−4.93670	−0.351725	−0.175862	+	0.984415i	$0.556271\pi$
$198$	0	0
$199$	2.51732	0.178448	0.0892239	−	0.996012i	$-0.471561\pi$
$199$	2.51732	0.178448	0.0892239	+	0.996012i	$0.471561\pi$
$200$	−0.419151	−0.0296385
$201$	0	0
$202$	−4.94457	−0.347899
$203$	2.77686	0.194898
$204$	0	0
$205$	−25.0297	−1.74815
$206$	−34.4466	−2.40001
$207$	0	0
$208$	−13.1257	−0.910103
$209$	−2.85891	−0.197755
$210$	0	0
$211$	6.98781	0.481061	0.240530	−	0.970642i	$-0.422679\pi$
$211$	6.98781	0.481061	0.240530	+	0.970642i	$0.422679\pi$
$212$	11.6391	0.799379
$213$	0	0
$214$	1.62382	0.111002
$215$	−4.85735	−0.331269
$216$	0	0
$217$	−5.00366	−0.339670
$218$	8.52729	0.577541
$219$	0	0
$220$	21.5406	1.45227
$221$	−25.9013	−1.74231
$222$	0	0
$223$	−17.9982	−1.20525	−0.602625	−	0.798025i	$-0.705878\pi$
$223$	−17.9982	−1.20525	−0.602625	+	0.798025i	$0.705878\pi$
$224$	−4.93014	−0.329409
$225$	0	0
$226$	24.9258	1.65804
$227$	−0.613511	−0.0407201	−0.0203601	−	0.999793i	$-0.506481\pi$
$227$	−0.613511	−0.0407201	−0.0203601	+	0.999793i	$0.506481\pi$
$228$	0	0
$229$	−4.99523	−0.330094	−0.165047	−	0.986286i	$-0.552778\pi$
$229$	−4.99523	−0.330094	−0.165047	+	0.986286i	$0.552778\pi$
$230$	12.1523	0.801296
$231$	0	0
$232$	0.715117	0.0469497
$233$	13.1575	0.861975	0.430988	−	0.902358i	$-0.358165\pi$
$233$	13.1575	0.861975	0.430988	+	0.902358i	$0.358165\pi$
$234$	0	0
$235$	−32.5637	−2.12422
$236$	3.14436	0.204680
$237$	0	0
$238$	−9.34902	−0.606007
$239$	−9.09818	−0.588512	−0.294256	−	0.955727i	$-0.595072\pi$
$239$	−9.09818	−0.588512	−0.294256	+	0.955727i	$0.595072\pi$
$240$	0	0
$241$	−14.2501	−0.917931	−0.458965	−	0.888454i	$-0.651780\pi$
$241$	−14.2501	−0.917931	−0.458965	+	0.888454i	$0.651780\pi$
$242$	−6.10626	−0.392526
$243$	0	0
$244$	6.28280	0.402215
$245$	18.3427	1.17187
$246$	0	0
$247$	2.61086	0.166125
$248$	−1.28858	−0.0818247
$249$	0	0
$250$	−13.0659	−0.826358
$251$	8.59452	0.542482	0.271241	−	0.962512i	$-0.412566\pi$
$251$	8.59452	0.542482	0.271241	+	0.962512i	$0.412566\pi$
$252$	0	0
$253$	−8.14104	−0.511823
$254$	−20.1438	−1.26394
$255$	0	0
$256$	14.6808	0.917550
$257$	−22.1967	−1.38459	−0.692295	−	0.721614i	$-0.743401\pi$
$257$	−22.1967	−1.38459	−0.692295	+	0.721614i	$0.743401\pi$
$258$	0	0
$259$	2.58192	0.160433
$260$	−19.6716	−1.21998
$261$	0	0
$262$	−11.0928	−0.685318
$263$	5.26252	0.324501	0.162251	−	0.986750i	$-0.448125\pi$
$263$	5.26252	0.324501	0.162251	+	0.986750i	$0.448125\pi$
$264$	0	0
$265$	−15.5051	−0.952469
$266$	0.942383	0.0577812
$267$	0	0
$268$	−15.9008	−0.971297
$269$	6.22846	0.379756	0.189878	−	0.981808i	$-0.439191\pi$
$269$	6.22846	0.379756	0.189878	+	0.981808i	$0.439191\pi$
$270$	0	0
$271$	7.31870	0.444579	0.222290	−	0.974981i	$-0.428647\pi$
$271$	7.31870	0.444579	0.222290	+	0.974981i	$0.428647\pi$
$272$	29.0679	1.76250
$273$	0	0
$274$	30.7565	1.85807
$275$	−9.97110	−0.601280
$276$	0	0
$277$	−18.6314	−1.11945	−0.559725	−	0.828678i	$-0.689093\pi$
$277$	−18.6314	−1.11945	−0.559725	+	0.828678i	$0.689093\pi$
$278$	−32.3432	−1.93981
$279$	0	0
$280$	−0.266372	−0.0159187
$281$	−1.02194	−0.0609639	−0.0304819	−	0.999535i	$-0.509704\pi$
$281$	−1.02194	−0.0609639	−0.0304819	+	0.999535i	$0.509704\pi$
$282$	0	0
$283$	−10.9633	−0.651701	−0.325850	−	0.945421i	$-0.605651\pi$
$283$	−10.9633	−0.651701	−0.325850	+	0.945421i	$0.605651\pi$
$284$	−14.1013	−0.836759
$285$	0	0
$286$	25.8624	1.52928
$287$	−5.52720	−0.326260
$288$	0	0
$289$	40.3606	2.37415
$290$	−25.3937	−1.49117
$291$	0	0
$292$	1.18950	0.0696099
$293$	24.2795	1.41843	0.709213	−	0.704994i	$-0.249051\pi$
$293$	24.2795	1.41843	0.709213	+	0.704994i	$0.249051\pi$
$294$	0	0
$295$	−4.18875	−0.243878
$296$	0.664914	0.0386473
$297$	0	0
$298$	−17.8522	−1.03415
$299$	7.43468	0.429959
$300$	0	0
$301$	−1.07263	−0.0618253
$302$	−1.99485	−0.114791
$303$	0	0
$304$	−2.93005	−0.168050
$305$	−8.36963	−0.479244
$306$	0	0
$307$	−24.8301	−1.41713	−0.708563	−	0.705648i	$-0.750656\pi$
$307$	−24.8301	−1.41713	−0.708563	+	0.705648i	$0.750656\pi$
$308$	4.75672	0.271039
$309$	0	0
$310$	45.7572	2.59883
$311$	1.85071	0.104944	0.0524720	−	0.998622i	$-0.483290\pi$
$311$	1.85071	0.104944	0.0524720	+	0.998622i	$0.483290\pi$
$312$	0	0
$313$	−6.95154	−0.392924	−0.196462	−	0.980511i	$-0.562945\pi$
$313$	−6.95154	−0.392924	−0.196462	+	0.980511i	$0.562945\pi$
$314$	−36.7500	−2.07392
$315$	0	0
$316$	19.1486	1.07719
$317$	26.0373	1.46240	0.731200	−	0.682163i	$-0.238961\pi$
$317$	26.0373	1.46240	0.731200	+	0.682163i	$0.238961\pi$
$318$	0	0
$319$	17.0118	0.952476
$320$	23.8365	1.33250
$321$	0	0
$322$	2.68353	0.149547
$323$	−5.78196	−0.321717
$324$	0	0
$325$	9.10595	0.505107
$326$	−34.6263	−1.91777
$327$	0	0
$328$	−1.42340	−0.0785942
$329$	−7.19091	−0.396448
$330$	0	0
$331$	−26.0355	−1.43104	−0.715519	−	0.698593i	$-0.753810\pi$
$331$	−26.0355	−1.43104	−0.715519	+	0.698593i	$0.753810\pi$
$332$	−12.8593	−0.705745
$333$	0	0
$334$	−11.9168	−0.652057
$335$	21.1823	1.15731
$336$	0	0
$337$	−15.0311	−0.818797	−0.409398	−	0.912356i	$-0.634261\pi$
$337$	−15.0311	−0.818797	−0.409398	+	0.912356i	$0.634261\pi$
$338$	2.63365	0.143252
$339$	0	0
$340$	43.5644	2.36261
$341$	−30.6537	−1.65999
$342$	0	0
$343$	8.32947	0.449749
$344$	−0.276231	−0.0148934
$345$	0	0
$346$	−18.1599	−0.976280
$347$	−25.4221	−1.36473	−0.682365	−	0.731012i	$-0.739049\pi$
$347$	−25.4221	−1.36473	−0.682365	+	0.731012i	$0.739049\pi$
$348$	0	0
$349$	−29.2656	−1.56655	−0.783275	−	0.621676i	$-0.786452\pi$
$349$	−29.2656	−1.56655	−0.783275	+	0.621676i	$0.786452\pi$
$350$	3.28677	0.175685
$351$	0	0
$352$	−30.2033	−1.60984
$353$	28.6917	1.52710	0.763552	−	0.645747i	$-0.223454\pi$
$353$	28.6917	1.52710	0.763552	+	0.645747i	$0.223454\pi$
$354$	0	0
$355$	18.7851	0.997007
$356$	−9.56840	−0.507124
$357$	0	0
$358$	−2.60285	−0.137565
$359$	−20.8561	−1.10075	−0.550373	−	0.834919i	$-0.685514\pi$
$359$	−20.8561	−1.10075	−0.550373	+	0.834919i	$0.685514\pi$
$360$	0	0
$361$	−18.4172	−0.969325
$362$	−7.50535	−0.394472
$363$	0	0
$364$	−4.34400	−0.227687
$365$	−1.58458	−0.0829410
$366$	0	0
$367$	−34.5668	−1.80437	−0.902185	−	0.431348i	$-0.858038\pi$
$367$	−34.5668	−1.80437	−0.902185	+	0.431348i	$0.858038\pi$
$368$	−8.34360	−0.434940
$369$	0	0
$370$	−23.6110	−1.22748
$371$	−3.42392	−0.177761
$372$	0	0
$373$	−9.36943	−0.485131	−0.242565	−	0.970135i	$-0.577989\pi$
$373$	−9.36943	−0.485131	−0.242565	+	0.970135i	$0.577989\pi$
$374$	−57.2744	−2.96159
$375$	0	0
$376$	−1.85185	−0.0955020
$377$	−15.5357	−0.800131
$378$	0	0
$379$	−1.92754	−0.0990111	−0.0495056	−	0.998774i	$-0.515765\pi$
$379$	−1.92754	−0.0990111	−0.0495056	+	0.998774i	$0.515765\pi$
$380$	−4.39130	−0.225269
$381$	0	0
$382$	−14.7215	−0.753215
$383$	19.4920	0.995997	0.497999	−	0.867178i	$-0.334068\pi$
$383$	19.4920	0.995997	0.497999	+	0.867178i	$0.334068\pi$
$384$	0	0
$385$	−6.33666	−0.322946
$386$	19.4410	0.989522
$387$	0	0
$388$	−21.8536	−1.10945
$389$	−9.13991	−0.463412	−0.231706	−	0.972786i	$-0.574431\pi$
$389$	−9.13991	−0.463412	−0.231706	+	0.972786i	$0.574431\pi$
$390$	0	0
$391$	−16.4647	−0.832655
$392$	1.04312	0.0526855
$393$	0	0
$394$	9.96914	0.502238
$395$	−25.5088	−1.28349
$396$	0	0
$397$	33.3487	1.67372	0.836862	−	0.547414i	$-0.184388\pi$
$397$	33.3487	1.67372	0.836862	+	0.547414i	$0.184388\pi$
$398$	−5.08345	−0.254811
$399$	0	0
$400$	−10.2192	−0.510960
$401$	−3.54872	−0.177214	−0.0886072	−	0.996067i	$-0.528242\pi$
$401$	−3.54872	−0.177214	−0.0886072	+	0.996067i	$0.528242\pi$
$402$	0	0
$403$	27.9940	1.39448
$404$	5.08795	0.253135
$405$	0	0
$406$	−5.60758	−0.278300
$407$	15.8175	0.784044
$408$	0	0
$409$	−32.0349	−1.58402	−0.792011	−	0.610507i	$-0.790966\pi$
$409$	−32.0349	−1.58402	−0.792011	+	0.610507i	$0.790966\pi$
$410$	50.5448	2.49623
$411$	0	0
$412$	35.4455	1.74627
$413$	−0.924984	−0.0455155
$414$	0	0
$415$	17.1305	0.840902
$416$	27.5827	1.35235
$417$	0	0
$418$	5.77328	0.282380
$419$	−14.8348	−0.724727	−0.362363	−	0.932037i	$-0.618030\pi$
$419$	−14.8348	−0.724727	−0.362363	+	0.932037i	$0.618030\pi$
$420$	0	0
$421$	−15.5797	−0.759310	−0.379655	−	0.925128i	$-0.623957\pi$
$421$	−15.5797	−0.759310	−0.379655	+	0.925128i	$0.623957\pi$
$422$	−14.1111	−0.686920
$423$	0	0
$424$	−0.881750	−0.0428216
$425$	−20.1659	−0.978188
$426$	0	0
$427$	−1.84823	−0.0894421
$428$	−1.67091	−0.0807663
$429$	0	0
$430$	9.80891	0.473028
$431$	1.29743	0.0624948	0.0312474	−	0.999512i	$-0.490052\pi$
$431$	1.29743	0.0624948	0.0312474	+	0.999512i	$0.490052\pi$
$432$	0	0
$433$	17.5301	0.842444	0.421222	−	0.906958i	$-0.361601\pi$
$433$	17.5301	0.842444	0.421222	+	0.906958i	$0.361601\pi$
$434$	10.1044	0.485025
$435$	0	0
$436$	−8.77457	−0.420226
$437$	1.65964	0.0793916
$438$	0	0
$439$	21.7182	1.03655	0.518277	−	0.855213i	$-0.326574\pi$
$439$	21.7182	1.03655	0.518277	+	0.855213i	$0.326574\pi$
$440$	−1.63186	−0.0777958
$441$	0	0
$442$	52.3050	2.48790
$443$	9.14365	0.434428	0.217214	−	0.976124i	$-0.430303\pi$
$443$	9.14365	0.434428	0.217214	+	0.976124i	$0.430303\pi$
$444$	0	0
$445$	12.7465	0.604244
$446$	36.3455	1.72101
$447$	0	0
$448$	5.26371	0.248687
$449$	12.4224	0.586248	0.293124	−	0.956074i	$-0.405305\pi$
$449$	12.4224	0.586248	0.293124	+	0.956074i	$0.405305\pi$
$450$	0	0
$451$	−33.8610	−1.59445
$452$	−25.6486	−1.20641
$453$	0	0
$454$	1.23892	0.0581454
$455$	5.78685	0.271292
$456$	0	0
$457$	−18.6949	−0.874510	−0.437255	−	0.899338i	$-0.644049\pi$
$457$	−18.6949	−0.874510	−0.437255	+	0.899338i	$0.644049\pi$
$458$	10.0873	0.471350
$459$	0	0
$460$	−12.5047	−0.583033
$461$	41.0819	1.91337	0.956687	−	0.291117i	$-0.0940269\pi$
$461$	41.0819	1.91337	0.956687	+	0.291117i	$0.0940269\pi$
$462$	0	0
$463$	−4.12981	−0.191929	−0.0959643	−	0.995385i	$-0.530593\pi$
$463$	−4.12981	−0.191929	−0.0959643	+	0.995385i	$0.530593\pi$
$464$	17.4350	0.809401
$465$	0	0
$466$	−26.5702	−1.23084
$467$	−25.2765	−1.16966	−0.584828	−	0.811157i	$-0.698838\pi$
$467$	−25.2765	−1.16966	−0.584828	+	0.811157i	$0.698838\pi$
$468$	0	0
$469$	4.67759	0.215991
$470$	65.7591	3.03324
$471$	0	0
$472$	−0.238208	−0.0109644
$473$	−6.57119	−0.302144
$474$	0	0
$475$	2.03272	0.0932677
$476$	9.62013	0.440938
$477$	0	0
$478$	18.3728	0.840353
$479$	−20.8021	−0.950471	−0.475236	−	0.879859i	$-0.657637\pi$
$479$	−20.8021	−0.950471	−0.475236	+	0.879859i	$0.657637\pi$
$480$	0	0
$481$	−14.4451	−0.658639
$482$	28.7766	1.31074
$483$	0	0
$484$	6.28334	0.285606
$485$	29.1122	1.32192
$486$	0	0
$487$	−6.60060	−0.299102	−0.149551	−	0.988754i	$-0.547783\pi$
$487$	−6.60060	−0.299102	−0.149551	+	0.988754i	$0.547783\pi$
$488$	−0.475968	−0.0215461
$489$	0	0
$490$	−37.0411	−1.67335
$491$	−36.5453	−1.64927	−0.824633	−	0.565668i	$-0.808618\pi$
$491$	−36.5453	−1.64927	−0.824633	+	0.565668i	$0.808618\pi$
$492$	0	0
$493$	34.4051	1.54953
$494$	−5.27236	−0.237214
$495$	0	0
$496$	−31.4164	−1.41064
$497$	4.14822	0.186073
$498$	0	0
$499$	36.3939	1.62922	0.814608	−	0.580012i	$-0.196952\pi$
$499$	36.3939	1.62922	0.814608	+	0.580012i	$0.196952\pi$
$500$	13.4448	0.601268
$501$	0	0
$502$	−17.3557	−0.774624
$503$	7.96032	0.354933	0.177467	−	0.984127i	$-0.443210\pi$
$503$	7.96032	0.354933	0.177467	+	0.984127i	$0.443210\pi$
$504$	0	0
$505$	−6.77791	−0.301613
$506$	16.4400	0.730846
$507$	0	0
$508$	20.7280	0.919654
$509$	−1.66881	−0.0739687	−0.0369843	−	0.999316i	$-0.511775\pi$
$509$	−1.66881	−0.0739687	−0.0369843	+	0.999316i	$0.511775\pi$
$510$	0	0
$511$	−0.349917	−0.0154794
$512$	−32.1631	−1.42142
$513$	0	0
$514$	44.8239	1.97710
$515$	−47.2187	−2.08070
$516$	0	0
$517$	−44.0533	−1.93746
$518$	−5.21391	−0.229086
$519$	0	0
$520$	1.49027	0.0653527
$521$	−35.0886	−1.53726	−0.768629	−	0.639695i	$-0.779061\pi$
$521$	−35.0886	−1.53726	−0.768629	+	0.639695i	$0.779061\pi$
$522$	0	0
$523$	−12.0344	−0.526226	−0.263113	−	0.964765i	$-0.584749\pi$
$523$	−12.0344	−0.526226	−0.263113	+	0.964765i	$0.584749\pi$
$524$	11.4145	0.498645
$525$	0	0
$526$	−10.6271	−0.463364
$527$	−61.9949	−2.70054
$528$	0	0
$529$	−18.2740	−0.794522
$530$	31.3109	1.36006
$531$	0	0
$532$	−0.969711	−0.0420423
$533$	30.9230	1.33943
$534$	0	0
$535$	2.22590	0.0962339
$536$	1.20460	0.0520310
$537$	0	0
$538$	−12.5777	−0.542264
$539$	24.8146	1.06884
$540$	0	0
$541$	19.6261	0.843793	0.421896	−	0.906644i	$-0.361365\pi$
$541$	19.6261	0.843793	0.421896	+	0.906644i	$0.361365\pi$
$542$	−14.7793	−0.634827
$543$	0	0
$544$	−61.0840	−2.61896
$545$	11.6890	0.500703
$546$	0	0
$547$	8.72390	0.373007	0.186504	−	0.982454i	$-0.440284\pi$
$547$	8.72390	0.373007	0.186504	+	0.982454i	$0.440284\pi$
$548$	−31.6484	−1.35195
$549$	0	0
$550$	20.1356	0.858584
$551$	−3.46804	−0.147744
$552$	0	0
$553$	−5.63300	−0.239539
$554$	37.6241	1.59849
$555$	0	0
$556$	33.2811	1.41143
$557$	19.8028	0.839070	0.419535	−	0.907739i	$-0.362193\pi$
$557$	19.8028	0.839070	0.419535	+	0.907739i	$0.362193\pi$
$558$	0	0
$559$	6.00104	0.253817
$560$	−6.49432	−0.274435
$561$	0	0
$562$	2.06370	0.0870520
$563$	33.9565	1.43110	0.715548	−	0.698564i	$-0.246177\pi$
$563$	33.9565	1.43110	0.715548	+	0.698564i	$0.246177\pi$
$564$	0	0
$565$	34.1677	1.43745
$566$	22.1392	0.930581
$567$	0	0
$568$	1.06828	0.0448240
$569$	−19.2392	−0.806547	−0.403274	−	0.915079i	$-0.632128\pi$
$569$	−19.2392	−0.806547	−0.403274	+	0.915079i	$0.632128\pi$
$570$	0	0
$571$	−2.48692	−0.104074	−0.0520371	−	0.998645i	$-0.516571\pi$
$571$	−2.48692	−0.104074	−0.0520371	+	0.998645i	$0.516571\pi$
$572$	−26.6124	−1.11272
$573$	0	0
$574$	11.1616	0.465876
$575$	5.78838	0.241392
$576$	0	0
$577$	25.4917	1.06123	0.530616	−	0.847612i	$-0.321961\pi$
$577$	25.4917	1.06123	0.530616	+	0.847612i	$0.321961\pi$
$578$	−81.5039	−3.39012
$579$	0	0
$580$	26.1301	1.08499
$581$	3.78285	0.156939
$582$	0	0
$583$	−20.9758	−0.868728
$584$	−0.0901130	−0.00372890
$585$	0	0
$586$	−49.0300	−2.02541
$587$	0.895381	0.0369563	0.0184782	−	0.999829i	$-0.494118\pi$
$587$	0.895381	0.0369563	0.0184782	+	0.999829i	$0.494118\pi$
$588$	0	0
$589$	6.24910	0.257490
$590$	8.45874	0.348241
$591$	0	0
$592$	16.2110	0.666270
$593$	32.4289	1.33170	0.665848	−	0.746088i	$-0.268070\pi$
$593$	32.4289	1.33170	0.665848	+	0.746088i	$0.268070\pi$
$594$	0	0
$595$	−12.8155	−0.525382
$596$	18.3699	0.752459
$597$	0	0
$598$	−15.0136	−0.613950
$599$	−42.0938	−1.71991	−0.859954	−	0.510372i	$-0.829508\pi$
$599$	−42.0938	−1.71991	−0.859954	+	0.510372i	$0.829508\pi$
$600$	0	0
$601$	−30.9396	−1.26205	−0.631026	−	0.775762i	$-0.717366\pi$
$601$	−30.9396	−1.26205	−0.631026	+	0.775762i	$0.717366\pi$
$602$	2.16606	0.0882820
$603$	0	0
$604$	2.05270	0.0835231
$605$	−8.37034	−0.340303
$606$	0	0
$607$	−14.9743	−0.607789	−0.303894	−	0.952706i	$-0.598287\pi$
$607$	−14.9743	−0.607789	−0.303894	+	0.952706i	$0.598287\pi$
$608$	6.15728	0.249711
$609$	0	0
$610$	16.9016	0.684325
$611$	40.2310	1.62757
$612$	0	0
$613$	−18.3536	−0.741295	−0.370647	−	0.928774i	$-0.620864\pi$
$613$	−18.3536	−0.741295	−0.370647	+	0.928774i	$0.620864\pi$
$614$	50.1417	2.02355
$615$	0	0
$616$	−0.360356	−0.0145192
$617$	−32.7753	−1.31949	−0.659743	−	0.751492i	$-0.729335\pi$
$617$	−32.7753	−1.31949	−0.659743	+	0.751492i	$0.729335\pi$
$618$	0	0
$619$	−6.00233	−0.241254	−0.120627	−	0.992698i	$-0.538491\pi$
$619$	−6.00233	−0.241254	−0.120627	+	0.992698i	$0.538491\pi$
$620$	−47.0841	−1.89094
$621$	0	0
$622$	−3.73731	−0.149852
$623$	2.81476	0.112771
$624$	0	0
$625$	−31.2236	−1.24894
$626$	14.0379	0.561067
$627$	0	0
$628$	37.8157	1.50901
$629$	31.9898	1.27552
$630$	0	0
$631$	−24.4380	−0.972860	−0.486430	−	0.873720i	$-0.661701\pi$
$631$	−24.4380	−0.972860	−0.486430	+	0.873720i	$0.661701\pi$
$632$	−1.45065	−0.0577037
$633$	0	0
$634$	−52.5796	−2.08820
$635$	−27.6127	−1.09578
$636$	0	0
$637$	−22.6615	−0.897882
$638$	−34.3535	−1.36007
$639$	0	0
$640$	−3.48345	−0.137695
$641$	10.3151	0.407423	0.203712	−	0.979031i	$-0.434699\pi$
$641$	10.3151	0.407423	0.203712	+	0.979031i	$0.434699\pi$
$642$	0	0
$643$	−43.3068	−1.70785	−0.853926	−	0.520394i	$-0.825785\pi$
$643$	−43.3068	−1.70785	−0.853926	+	0.520394i	$0.825785\pi$
$644$	−2.76135	−0.108812
$645$	0	0
$646$	11.6761	0.459388
$647$	6.50021	0.255550	0.127775	−	0.991803i	$-0.459217\pi$
$647$	6.50021	0.255550	0.127775	+	0.991803i	$0.459217\pi$
$648$	0	0
$649$	−5.66668	−0.222437
$650$	−18.3885	−0.721257
$651$	0	0
$652$	35.6304	1.39540
$653$	2.97710	0.116503	0.0582514	−	0.998302i	$-0.481448\pi$
$653$	2.97710	0.116503	0.0582514	+	0.998302i	$0.481448\pi$
$654$	0	0
$655$	−15.2058	−0.594141
$656$	−34.7035	−1.35494
$657$	0	0
$658$	14.5213	0.566099
$659$	−28.1099	−1.09501	−0.547503	−	0.836804i	$-0.684422\pi$
$659$	−28.1099	−1.09501	−0.547503	+	0.836804i	$0.684422\pi$
$660$	0	0
$661$	−3.25065	−0.126435	−0.0632177	−	0.998000i	$-0.520136\pi$
$661$	−3.25065	−0.126435	−0.0632177	+	0.998000i	$0.520136\pi$
$662$	52.5759	2.04342
$663$	0	0
$664$	0.974185	0.0378057
$665$	1.29180	0.0500939
$666$	0	0
$667$	−9.87560	−0.382385
$668$	12.2623	0.474444
$669$	0	0
$670$	−42.7753	−1.65256
$671$	−11.3227	−0.437108
$672$	0	0
$673$	13.8944	0.535589	0.267794	−	0.963476i	$-0.413705\pi$
$673$	13.8944	0.535589	0.267794	+	0.963476i	$0.413705\pi$
$674$	30.3537	1.16918
$675$	0	0
$676$	−2.71002	−0.104232
$677$	−10.8567	−0.417256	−0.208628	−	0.977995i	$-0.566900\pi$
$677$	−10.8567	−0.417256	−0.208628	+	0.977995i	$0.566900\pi$
$678$	0	0
$679$	6.42872	0.246712
$680$	−3.30032	−0.126562
$681$	0	0
$682$	61.9018	2.37034
$683$	33.6129	1.28616	0.643081	−	0.765798i	$-0.277656\pi$
$683$	33.6129	1.28616	0.643081	+	0.765798i	$0.277656\pi$
$684$	0	0
$685$	42.1604	1.61087
$686$	−16.8205	−0.642209
$687$	0	0
$688$	−6.73469	−0.256758
$689$	19.1558	0.729778
$690$	0	0
$691$	−0.447664	−0.0170299	−0.00851497	−	0.999964i	$-0.502710\pi$
$691$	−0.447664	−0.0170299	−0.00851497	+	0.999964i	$0.502710\pi$
$692$	18.6865	0.710353
$693$	0	0
$694$	51.3373	1.94874
$695$	−44.3354	−1.68174
$696$	0	0
$697$	−68.4815	−2.59392
$698$	59.0987	2.23692
$699$	0	0
$700$	−3.38208	−0.127831
$701$	−3.88664	−0.146796	−0.0733982	−	0.997303i	$-0.523384\pi$
$701$	−3.88664	−0.146796	−0.0733982	+	0.997303i	$0.523384\pi$
$702$	0	0
$703$	−3.22458	−0.121617
$704$	32.2468	1.21535
$705$	0	0
$706$	−57.9398	−2.18059
$707$	−1.49674	−0.0562906
$708$	0	0
$709$	−14.7400	−0.553572	−0.276786	−	0.960932i	$-0.589269\pi$
$709$	−14.7400	−0.553572	−0.276786	+	0.960932i	$0.589269\pi$
$710$	−37.9344	−1.42365
$711$	0	0
$712$	0.724877	0.0271659
$713$	17.7949	0.666425
$714$	0	0
$715$	35.4517	1.32582
$716$	2.67833	0.100094
$717$	0	0
$718$	42.1168	1.57178
$719$	32.3508	1.20648	0.603240	−	0.797559i	$-0.293876\pi$
$719$	32.3508	1.20648	0.603240	+	0.797559i	$0.293876\pi$
$720$	0	0
$721$	−10.4271	−0.388325
$722$	37.1915	1.38413
$723$	0	0
$724$	7.72299	0.287023
$725$	−12.0956	−0.449218
$726$	0	0
$727$	−9.23020	−0.342329	−0.171165	−	0.985242i	$-0.554753\pi$
$727$	−9.23020	−0.342329	−0.171165	+	0.985242i	$0.554753\pi$
$728$	0.329090	0.0121969
$729$	0	0
$730$	3.19990	0.118434
$731$	−13.2898	−0.491540
$732$	0	0
$733$	−52.9937	−1.95737	−0.978683	−	0.205379i	$-0.934157\pi$
$733$	−52.9937	−1.95737	−0.978683	+	0.205379i	$0.934157\pi$
$734$	69.8040	2.57651
$735$	0	0
$736$	17.5335	0.646292
$737$	28.6561	1.05556
$738$	0	0
$739$	2.84991	0.104836	0.0524178	−	0.998625i	$-0.483307\pi$
$739$	2.84991	0.104836	0.0524178	+	0.998625i	$0.483307\pi$
$740$	24.2957	0.893127
$741$	0	0
$742$	6.91424	0.253830
$743$	−29.4126	−1.07904	−0.539522	−	0.841971i	$-0.681395\pi$
$743$	−29.4126	−1.07904	−0.539522	+	0.841971i	$0.681395\pi$
$744$	0	0
$745$	−24.4714	−0.896563
$746$	18.9206	0.692732
$747$	0	0
$748$	58.9353	2.15489
$749$	0.491535	0.0179603
$750$	0	0
$751$	−27.4298	−1.00093	−0.500463	−	0.865758i	$-0.666837\pi$
$751$	−27.4298	−1.00093	−0.500463	+	0.865758i	$0.666837\pi$
$752$	−45.1494	−1.64643
$753$	0	0
$754$	31.3728	1.14253
$755$	−2.73450	−0.0995187
$756$	0	0
$757$	−28.4646	−1.03456	−0.517282	−	0.855815i	$-0.673056\pi$
$757$	−28.4646	−1.03456	−0.517282	+	0.855815i	$0.673056\pi$
$758$	3.89247	0.141381
$759$	0	0
$760$	0.332673	0.0120673
$761$	27.9722	1.01399	0.506996	−	0.861948i	$-0.330756\pi$
$761$	27.9722	1.01399	0.506996	+	0.861948i	$0.330756\pi$
$762$	0	0
$763$	2.58124	0.0934471
$764$	15.1484	0.548049
$765$	0	0
$766$	−39.3621	−1.42221
$767$	5.17501	0.186859
$768$	0	0
$769$	46.9637	1.69355	0.846777	−	0.531947i	$-0.178540\pi$
$769$	46.9637	1.69355	0.846777	+	0.531947i	$0.178540\pi$
$770$	12.7962	0.461143
$771$	0	0
$772$	−20.0048	−0.719988
$773$	−44.2173	−1.59039	−0.795193	−	0.606356i	$-0.792631\pi$
$773$	−44.2173	−1.59039	−0.795193	+	0.606356i	$0.792631\pi$
$774$	0	0
$775$	21.7951	0.782904
$776$	1.65557	0.0594314
$777$	0	0
$778$	18.4571	0.661719
$779$	6.90295	0.247324
$780$	0	0
$781$	25.4131	0.909350
$782$	33.2487	1.18897
$783$	0	0
$784$	25.4320	0.908285
$785$	−50.3762	−1.79800
$786$	0	0
$787$	36.4827	1.30047	0.650234	−	0.759734i	$-0.274671\pi$
$787$	36.4827	1.30047	0.650234	+	0.759734i	$0.274671\pi$
$788$	−10.2582	−0.365434
$789$	0	0
$790$	51.5123	1.83273
$791$	7.54511	0.268273
$792$	0	0
$793$	10.3403	0.367194
$794$	−67.3442	−2.38996
$795$	0	0
$796$	5.23087	0.185403
$797$	−15.1172	−0.535480	−0.267740	−	0.963491i	$-0.586277\pi$
$797$	−15.1172	−0.535480	−0.267740	+	0.963491i	$0.586277\pi$
$798$	0	0
$799$	−89.0948	−3.15195
$800$	21.4749	0.759252
$801$	0	0
$802$	7.16626	0.253049
$803$	−2.14368	−0.0756488
$804$	0	0
$805$	3.67853	0.129651
$806$	−56.5309	−1.99122
$807$	0	0
$808$	−0.385450	−0.0135601
$809$	−49.6978	−1.74728	−0.873641	−	0.486571i	$-0.838247\pi$
$809$	−49.6978	−1.74728	−0.873641	+	0.486571i	$0.838247\pi$
$810$	0	0
$811$	−8.76133	−0.307652	−0.153826	−	0.988098i	$-0.549159\pi$
$811$	−8.76133	−0.307652	−0.153826	+	0.988098i	$0.549159\pi$
$812$	5.77020	0.202494
$813$	0	0
$814$	−31.9417	−1.11956
$815$	−47.4651	−1.66263
$816$	0	0
$817$	1.33961	0.0468671
$818$	64.6910	2.26187
$819$	0	0
$820$	−52.0106	−1.81629
$821$	−35.4641	−1.23771	−0.618853	−	0.785507i	$-0.712402\pi$
$821$	−35.4641	−1.23771	−0.618853	+	0.785507i	$0.712402\pi$
$822$	0	0
$823$	−1.87915	−0.0655029	−0.0327514	−	0.999464i	$-0.510427\pi$
$823$	−1.87915	−0.0655029	−0.0327514	+	0.999464i	$0.510427\pi$
$824$	−2.68526	−0.0935454
$825$	0	0
$826$	1.86791	0.0649928
$827$	9.61685	0.334411	0.167205	−	0.985922i	$-0.446526\pi$
$827$	9.61685	0.334411	0.167205	+	0.985922i	$0.446526\pi$
$828$	0	0
$829$	−7.59327	−0.263725	−0.131863	−	0.991268i	$-0.542096\pi$
$829$	−7.59327	−0.263725	−0.131863	+	0.991268i	$0.542096\pi$
$830$	−34.5932	−1.20075
$831$	0	0
$832$	−29.4489	−1.02096
$833$	50.1858	1.73883
$834$	0	0
$835$	−16.3353	−0.565305
$836$	−5.94069	−0.205463
$837$	0	0
$838$	29.9573	1.03486
$839$	−39.6226	−1.36792	−0.683962	−	0.729517i	$-0.739745\pi$
$839$	−39.6226	−1.36792	−0.683962	+	0.729517i	$0.739745\pi$
$840$	0	0
$841$	−8.36366	−0.288402
$842$	31.4616	1.08424
$843$	0	0
$844$	14.5203	0.499811
$845$	3.61015	0.124193
$846$	0	0
$847$	−1.84839	−0.0635113
$848$	−21.4977	−0.738233
$849$	0	0
$850$	40.7228	1.39678
$851$	−9.18230	−0.314765
$852$	0	0
$853$	3.44370	0.117910	0.0589550	−	0.998261i	$-0.481223\pi$
$853$	3.44370	0.117910	0.0589550	+	0.998261i	$0.481223\pi$
$854$	3.73230	0.127717
$855$	0	0
$856$	0.126583	0.00432653
$857$	−22.1464	−0.756508	−0.378254	−	0.925702i	$-0.623475\pi$
$857$	−22.1464	−0.756508	−0.378254	+	0.925702i	$0.623475\pi$
$858$	0	0
$859$	35.6226	1.21543	0.607714	−	0.794156i	$-0.292087\pi$
$859$	35.6226	1.21543	0.607714	+	0.794156i	$0.292087\pi$
$860$	−10.0934	−0.344181
$861$	0	0
$862$	−2.62002	−0.0892381
$863$	−20.9746	−0.713984	−0.356992	−	0.934108i	$-0.616198\pi$
$863$	−20.9746	−0.713984	−0.356992	+	0.934108i	$0.616198\pi$
$864$	0	0
$865$	−24.8932	−0.846393
$866$	−35.4002	−1.20295
$867$	0	0
$868$	−10.3974	−0.352910
$869$	−34.5091	−1.17064
$870$	0	0
$871$	−26.1697	−0.886727
$872$	0.664738	0.0225109
$873$	0	0
$874$	−3.35148	−0.113365
$875$	−3.95508	−0.133706
$876$	0	0
$877$	42.2135	1.42545	0.712724	−	0.701445i	$-0.247461\pi$
$877$	42.2135	1.42545	0.712724	+	0.701445i	$0.247461\pi$
$878$	−43.8576	−1.48012
$879$	0	0
$880$	−39.7858	−1.34118
$881$	−38.2370	−1.28824	−0.644119	−	0.764925i	$-0.722776\pi$
$881$	−38.2370	−1.28824	−0.644119	+	0.764925i	$0.722776\pi$
$882$	0	0
$883$	−28.8095	−0.969517	−0.484759	−	0.874648i	$-0.661093\pi$
$883$	−28.8095	−0.969517	−0.484759	+	0.874648i	$0.661093\pi$
$884$	−53.8218	−1.81022
$885$	0	0
$886$	−18.4646	−0.620332
$887$	−13.7183	−0.460615	−0.230307	−	0.973118i	$-0.573973\pi$
$887$	−13.7183	−0.460615	−0.230307	+	0.973118i	$0.573973\pi$
$888$	0	0
$889$	−6.09760	−0.204507
$890$	−25.7403	−0.862816
$891$	0	0
$892$	−37.3995	−1.25223
$893$	8.98077	0.300530
$894$	0	0
$895$	−3.56794	−0.119263
$896$	−0.769235	−0.0256983
$897$	0	0
$898$	−25.0857	−0.837120
$899$	−37.1848	−1.24018
$900$	0	0
$901$	−42.4220	−1.41328
$902$	68.3787	2.27676
$903$	0	0
$904$	1.94307	0.0646255
$905$	−10.2882	−0.341991
$906$	0	0
$907$	56.7371	1.88392	0.941962	−	0.335720i	$-0.108979\pi$
$907$	56.7371	1.88392	0.941962	+	0.335720i	$0.108979\pi$
$908$	−1.27485	−0.0423073
$909$	0	0
$910$	−11.6859	−0.387385
$911$	−0.713715	−0.0236464	−0.0118232	−	0.999930i	$-0.503764\pi$
$911$	−0.713715	−0.0236464	−0.0118232	+	0.999930i	$0.503764\pi$
$912$	0	0
$913$	23.1747	0.766970
$914$	37.7524	1.24874
$915$	0	0
$916$	−10.3799	−0.342960
$917$	−3.35784	−0.110886
$918$	0	0
$919$	−57.3212	−1.89085	−0.945425	−	0.325839i	$-0.894353\pi$
$919$	−57.3212	−1.89085	−0.945425	+	0.325839i	$0.894353\pi$
$920$	0.947320	0.0312322
$921$	0	0
$922$	−82.9606	−2.73216
$923$	−23.2081	−0.763903
$924$	0	0
$925$	−11.2464	−0.369780
$926$	8.33972	0.274060
$927$	0	0
$928$	−36.6385	−1.20272
$929$	14.9631	0.490924	0.245462	−	0.969406i	$-0.421060\pi$
$929$	14.9631	0.490924	0.245462	+	0.969406i	$0.421060\pi$
$930$	0	0
$931$	−5.05873	−0.165793
$932$	27.3406	0.895573
$933$	0	0
$934$	51.0432	1.67018
$935$	−78.5106	−2.56757
$936$	0	0
$937$	−24.6696	−0.805922	−0.402961	−	0.915217i	$-0.632019\pi$
$937$	−24.6696	−0.805922	−0.402961	+	0.915217i	$0.632019\pi$
$938$	−9.44589	−0.308419
$939$	0	0
$940$	−67.6660	−2.20702
$941$	−23.2604	−0.758267	−0.379134	−	0.925342i	$-0.623778\pi$
$941$	−23.2604	−0.758267	−0.379134	+	0.925342i	$0.623778\pi$
$942$	0	0
$943$	19.6568	0.640115
$944$	−5.80768	−0.189024
$945$	0	0
$946$	13.2698	0.431439
$947$	−19.1774	−0.623182	−0.311591	−	0.950216i	$-0.600862\pi$
$947$	−19.1774	−0.623182	−0.311591	+	0.950216i	$0.600862\pi$
$948$	0	0
$949$	1.95768	0.0635490
$950$	−4.10487	−0.133180
$951$	0	0
$952$	−0.728796	−0.0236204
$953$	30.9817	1.00359	0.501797	−	0.864985i	$-0.332672\pi$
$953$	30.9817	1.00359	0.501797	+	0.864985i	$0.332672\pi$
$954$	0	0
$955$	−20.1799	−0.653005
$956$	−18.9056	−0.611451
$957$	0	0
$958$	42.0076	1.35720
$959$	9.31010	0.300639
$960$	0	0
$961$	36.0037	1.16141
$962$	29.1703	0.940488
$963$	0	0
$964$	−29.6111	−0.953709
$965$	26.6494	0.857873
$966$	0	0
$967$	19.8577	0.638582	0.319291	−	0.947657i	$-0.396555\pi$
$967$	19.8577	0.638582	0.319291	+	0.947657i	$0.396555\pi$
$968$	−0.476009	−0.0152995
$969$	0	0
$970$	−58.7890	−1.88760
$971$	−3.39436	−0.108930	−0.0544651	−	0.998516i	$-0.517345\pi$
$971$	−3.39436	−0.108930	−0.0544651	+	0.998516i	$0.517345\pi$
$972$	0	0
$973$	−9.79039	−0.313865
$974$	13.3292	0.427096
$975$	0	0
$976$	−11.6044	−0.371449
$977$	−53.5676	−1.71378	−0.856889	−	0.515501i	$-0.827606\pi$
$977$	−53.5676	−1.71378	−0.856889	+	0.515501i	$0.827606\pi$
$978$	0	0
$979$	17.2439	0.551119
$980$	38.1152	1.21755
$981$	0	0
$982$	73.7994	2.35503
$983$	27.8229	0.887412	0.443706	−	0.896173i	$-0.353664\pi$
$983$	27.8229	0.887412	0.443706	+	0.896173i	$0.353664\pi$
$984$	0	0
$985$	13.6655	0.435419
$986$	−69.4775	−2.21261
$987$	0	0
$988$	5.42525	0.172600
$989$	3.81468	0.121300
$990$	0	0
$991$	14.5949	0.463622	0.231811	−	0.972761i	$-0.425535\pi$
$991$	14.5949	0.463622	0.231811	+	0.972761i	$0.425535\pi$
$992$	66.0192	2.09611
$993$	0	0
$994$	−8.37690	−0.265699
$995$	−6.96829	−0.220910
$996$	0	0
$997$	−2.43698	−0.0771799	−0.0385900	−	0.999255i	$-0.512287\pi$
$997$	−2.43698	−0.0771799	−0.0385900	+	0.999255i	$0.512287\pi$
$998$	−73.4937	−2.32640
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.14		72
3.2	odd	2		6561.2.a.d.1.59		72
81.4	even	27		81.2.g.a.16.2	✓	144
81.7	even	27		729.2.g.c.352.7		144
81.20	odd	54		243.2.g.a.118.7		144
81.23	odd	54		729.2.g.b.379.2		144
81.31	even	27		729.2.g.d.622.7		144
81.34	even	27		729.2.g.d.109.7		144
81.47	odd	54		729.2.g.a.109.2		144
81.50	odd	54		729.2.g.a.622.2		144
81.58	even	27		729.2.g.c.379.7		144
81.61	even	27		81.2.g.a.76.2	yes	144
81.74	odd	54		729.2.g.b.352.2		144
81.77	odd	54		243.2.g.a.208.7		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.16.2	✓	144	81.4	even	27
81.2.g.a.76.2	yes	144	81.61	even	27
243.2.g.a.118.7		144	81.20	odd	54
243.2.g.a.208.7		144	81.77	odd	54
729.2.g.a.109.2		144	81.47	odd	54
729.2.g.a.622.2		144	81.50	odd	54
729.2.g.b.352.2		144	81.74	odd	54
729.2.g.b.379.2		144	81.23	odd	54
729.2.g.c.352.7		144	81.7	even	27
729.2.g.c.379.7		144	81.58	even	27
729.2.g.d.109.7		144	81.34	even	27
729.2.g.d.622.7		144	81.31	even	27
6561.2.a.c.1.14		72	1.1	even	1	trivial
6561.2.a.d.1.59		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)

\(f(q)\)	\(=\)	\(q-2.01939 q^{2} +2.07795 q^{4} -2.76814 q^{5} -0.611277 q^{7} -0.157420 q^{8} +5.58998 q^{10} -3.74484 q^{11} +3.41992 q^{13} +1.23441 q^{14} -3.83801 q^{16} -7.57368 q^{17} +0.763428 q^{19} -5.75208 q^{20} +7.56230 q^{22} +2.17394 q^{23} +2.66263 q^{25} -6.90616 q^{26} -1.27021 q^{28} -4.54272 q^{29} +8.18558 q^{31} +8.06531 q^{32} +15.2942 q^{34} +1.69210 q^{35} -4.22381 q^{37} -1.54166 q^{38} +0.435762 q^{40} +9.04205 q^{41} +1.75473 q^{43} -7.78160 q^{44} -4.39004 q^{46} +11.7637 q^{47} -6.62634 q^{49} -5.37689 q^{50} +7.10643 q^{52} +5.60125 q^{53} +10.3663 q^{55} +0.0962275 q^{56} +9.17355 q^{58} +1.51320 q^{59} +3.02355 q^{61} -16.5299 q^{62} -8.61101 q^{64} -9.46682 q^{65} -7.65215 q^{67} -15.7378 q^{68} -3.41703 q^{70} -6.78616 q^{71} +0.572436 q^{73} +8.52954 q^{74} +1.58637 q^{76} +2.28913 q^{77} +9.21512 q^{79} +10.6242 q^{80} -18.2595 q^{82} -6.18843 q^{83} +20.9650 q^{85} -3.54350 q^{86} +0.589514 q^{88} -4.60472 q^{89} -2.09052 q^{91} +4.51734 q^{92} -23.7556 q^{94} -2.11328 q^{95} -10.5169 q^{97} +13.3812 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38}+ \cdots - 81 q^{98}+O(q^{100}) \) 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

Introduction

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