LMFDB - Embedded newform 6561.2.a.d.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6561, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6561.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$52.3898487662$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q-1.54468 q^{2} +0.386029 q^{4} +0.508923 q^{5} +3.98845 q^{7} +2.49307 q^{8} +O(q^{10})$	\(q-1.54468 q^{2} +0.386029 q^{4} +0.508923 q^{5} +3.98845 q^{7} +2.49307 q^{8} -0.786122 q^{10} +3.82857 q^{11} +3.11326 q^{13} -6.16087 q^{14} -4.62304 q^{16} -1.50186 q^{17} -6.70690 q^{19} +0.196459 q^{20} -5.91391 q^{22} +1.18646 q^{23} -4.74100 q^{25} -4.80898 q^{26} +1.53966 q^{28} -6.91545 q^{29} -0.0463529 q^{31} +2.15498 q^{32} +2.31990 q^{34} +2.02982 q^{35} -3.05341 q^{37} +10.3600 q^{38} +1.26878 q^{40} +5.94130 q^{41} +0.238190 q^{43} +1.47794 q^{44} -1.83269 q^{46} +4.73760 q^{47} +8.90776 q^{49} +7.32331 q^{50} +1.20181 q^{52} +13.6235 q^{53} +1.94845 q^{55} +9.94347 q^{56} +10.6821 q^{58} +1.39679 q^{59} -0.313987 q^{61} +0.0716003 q^{62} +5.91734 q^{64} +1.58441 q^{65} +5.49248 q^{67} -0.579763 q^{68} -3.13541 q^{70} +11.7474 q^{71} +15.5606 q^{73} +4.71653 q^{74} -2.58906 q^{76} +15.2701 q^{77} -13.9968 q^{79} -2.35277 q^{80} -9.17740 q^{82} -0.702998 q^{83} -0.764334 q^{85} -0.367927 q^{86} +9.54488 q^{88} -11.5706 q^{89} +12.4171 q^{91} +0.458006 q^{92} -7.31806 q^{94} -3.41329 q^{95} -13.0100 q^{97} -13.7596 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.54468	−1.09225	−0.546126	−	0.837703i	$-0.683898\pi$
$2$	−1.54468	−1.09225	−0.546126	+	0.837703i	$0.683898\pi$
$3$	0	0
$3$	0	0
$4$	0.386029	0.193014
$5$	0.508923	0.227597	0.113799	−	0.993504i	$-0.463698\pi$
$5$	0.508923	0.227597	0.113799	+	0.993504i	$0.463698\pi$
$6$	0	0
$7$	3.98845	1.50749	0.753747	−	0.657165i	$-0.228245\pi$
$7$	3.98845	1.50749	0.753747	+	0.657165i	$0.228245\pi$
$8$	2.49307	0.881432
$9$	0	0
$10$	−0.786122	−0.248594
$11$	3.82857	1.15436	0.577179	−	0.816618i	$-0.304154\pi$
$11$	3.82857	1.15436	0.577179	+	0.816618i	$0.304154\pi$
$12$	0	0
$13$	3.11326	0.863462	0.431731	−	0.902002i	$-0.357903\pi$
$13$	3.11326	0.863462	0.431731	+	0.902002i	$0.357903\pi$
$14$	−6.16087	−1.64656
$15$	0	0
$16$	−4.62304	−1.15576
$17$	−1.50186	−0.364256	−0.182128	−	0.983275i	$-0.558298\pi$
$17$	−1.50186	−0.364256	−0.182128	+	0.983275i	$0.558298\pi$
$18$	0	0
$19$	−6.70690	−1.53867	−0.769334	−	0.638847i	$-0.779412\pi$
$19$	−6.70690	−1.53867	−0.769334	+	0.638847i	$0.779412\pi$
$20$	0.196459	0.0439296
$21$	0	0
$22$	−5.91391	−1.26085
$23$	1.18646	0.247393	0.123697	−	0.992320i	$-0.460525\pi$
$23$	1.18646	0.247393	0.123697	+	0.992320i	$0.460525\pi$
$24$	0	0
$25$	−4.74100	−0.948199
$26$	−4.80898	−0.943119
$27$	0	0
$28$	1.53966	0.290968
$29$	−6.91545	−1.28417	−0.642084	−	0.766634i	$-0.721930\pi$
$29$	−6.91545	−1.28417	−0.642084	+	0.766634i	$0.721930\pi$
$30$	0	0
$31$	−0.0463529	−0.00832522	−0.00416261	−	0.999991i	$-0.501325\pi$
$31$	−0.0463529	−0.00832522	−0.00416261	+	0.999991i	$0.501325\pi$
$32$	2.15498	0.380949
$33$	0	0
$34$	2.31990	0.397859
$35$	2.02982	0.343102
$36$	0	0
$37$	−3.05341	−0.501977	−0.250989	−	0.967990i	$-0.580756\pi$
$37$	−3.05341	−0.501977	−0.250989	+	0.967990i	$0.580756\pi$
$38$	10.3600	1.68061
$39$	0	0
$40$	1.26878	0.200611
$41$	5.94130	0.927876	0.463938	−	0.885868i	$-0.346436\pi$
$41$	5.94130	0.927876	0.463938	+	0.885868i	$0.346436\pi$
$42$	0	0
$43$	0.238190	0.0363236	0.0181618	−	0.999835i	$-0.494219\pi$
$43$	0.238190	0.0363236	0.0181618	+	0.999835i	$0.494219\pi$
$44$	1.47794	0.222808
$45$	0	0
$46$	−1.83269	−0.270216
$47$	4.73760	0.691050	0.345525	−	0.938410i	$-0.387701\pi$
$47$	4.73760	0.691050	0.345525	+	0.938410i	$0.387701\pi$
$48$	0	0
$49$	8.90776	1.27254
$50$	7.32331	1.03567
$51$	0	0
$52$	1.20181	0.166661
$53$	13.6235	1.87133	0.935663	−	0.352895i	$-0.114803\pi$
$53$	13.6235	1.87133	0.935663	+	0.352895i	$0.114803\pi$
$54$	0	0
$55$	1.94845	0.262729
$56$	9.94347	1.32875
$57$	0	0
$58$	10.6821	1.40263
$59$	1.39679	0.181847	0.0909234	−	0.995858i	$-0.471018\pi$
$59$	1.39679	0.181847	0.0909234	+	0.995858i	$0.471018\pi$
$60$	0	0
$61$	−0.313987	−0.0402019	−0.0201010	−	0.999798i	$-0.506399\pi$
$61$	−0.313987	−0.0402019	−0.0201010	+	0.999798i	$0.506399\pi$
$62$	0.0716003	0.00909324
$63$	0	0
$64$	5.91734	0.739667
$65$	1.58441	0.196522
$66$	0	0
$67$	5.49248	0.671013	0.335506	−	0.942038i	$-0.391093\pi$
$67$	5.49248	0.671013	0.335506	+	0.942038i	$0.391093\pi$
$68$	−0.579763	−0.0703066
$69$	0	0
$70$	−3.13541	−0.374753
$71$	11.7474	1.39416	0.697080	−	0.716994i	$-0.254482\pi$
$71$	11.7474	1.39416	0.697080	+	0.716994i	$0.254482\pi$
$72$	0	0
$73$	15.5606	1.82123	0.910614	−	0.413257i	$-0.135609\pi$
$73$	15.5606	1.82123	0.910614	+	0.413257i	$0.135609\pi$
$74$	4.71653	0.548286
$75$	0	0
$76$	−2.58906	−0.296985
$77$	15.2701	1.74019
$78$	0	0
$79$	−13.9968	−1.57476	−0.787381	−	0.616467i	$-0.788564\pi$
$79$	−13.9968	−1.57476	−0.787381	+	0.616467i	$0.788564\pi$
$80$	−2.35277	−0.263048
$81$	0	0
$82$	−9.17740	−1.01347
$83$	−0.702998	−0.0771641	−0.0385820	−	0.999255i	$-0.512284\pi$
$83$	−0.702998	−0.0771641	−0.0385820	+	0.999255i	$0.512284\pi$
$84$	0	0
$85$	−0.764334	−0.0829036
$86$	−0.367927	−0.0396745
$87$	0	0
$88$	9.54488	1.01749
$89$	−11.5706	−1.22648	−0.613241	−	0.789896i	$-0.710134\pi$
$89$	−11.5706	−1.22648	−0.613241	+	0.789896i	$0.710134\pi$
$90$	0	0
$91$	12.4171	1.30166
$92$	0.458006	0.0477505
$93$	0	0
$94$	−7.31806	−0.754800
$95$	−3.41329	−0.350197
$96$	0	0
$97$	−13.0100	−1.32096	−0.660482	−	0.750842i	$-0.729648\pi$
$97$	−13.0100	−1.32096	−0.660482	+	0.750842i	$0.729648\pi$
$98$	−13.7596	−1.38993
$99$	0	0
$100$	−1.83016	−0.183016
$101$	12.3640	1.23026	0.615132	−	0.788424i	$-0.289103\pi$
$101$	12.3640	1.23026	0.615132	+	0.788424i	$0.289103\pi$
$102$	0	0
$103$	−2.09015	−0.205948	−0.102974	−	0.994684i	$-0.532836\pi$
$103$	−2.09015	−0.205948	−0.102974	+	0.994684i	$0.532836\pi$
$104$	7.76156	0.761083
$105$	0	0
$106$	−21.0439	−2.04396
$107$	1.66273	0.160742	0.0803709	−	0.996765i	$-0.474390\pi$
$107$	1.66273	0.160742	0.0803709	+	0.996765i	$0.474390\pi$
$108$	0	0
$109$	4.29961	0.411828	0.205914	−	0.978570i	$-0.433983\pi$
$109$	4.29961	0.411828	0.205914	+	0.978570i	$0.433983\pi$
$110$	−3.00972	−0.286966
$111$	0	0
$112$	−18.4388	−1.74230
$113$	14.2345	1.33907	0.669534	−	0.742781i	$-0.266494\pi$
$113$	14.2345	1.33907	0.669534	+	0.742781i	$0.266494\pi$
$114$	0	0
$115$	0.603815	0.0563060
$116$	−2.66956	−0.247863
$117$	0	0
$118$	−2.15759	−0.198623
$119$	−5.99012	−0.549113
$120$	0	0
$121$	3.65796	0.332542
$122$	0.485009	0.0439106
$123$	0	0
$124$	−0.0178935	−0.00160689
$125$	−4.95742	−0.443405
$126$	0	0
$127$	18.5047	1.64203	0.821014	−	0.570907i	$-0.193408\pi$
$127$	18.5047	1.64203	0.821014	+	0.570907i	$0.193408\pi$
$128$	−13.4503	−1.18885
$129$	0	0
$130$	−2.44740	−0.214651
$131$	5.55320	0.485185	0.242592	−	0.970128i	$-0.422002\pi$
$131$	5.55320	0.485185	0.242592	+	0.970128i	$0.422002\pi$
$132$	0	0
$133$	−26.7501	−2.31953
$134$	−8.48411	−0.732915
$135$	0	0
$136$	−3.74425	−0.321066
$137$	−2.98360	−0.254906	−0.127453	−	0.991845i	$-0.540680\pi$
$137$	−2.98360	−0.254906	−0.127453	+	0.991845i	$0.540680\pi$
$138$	0	0
$139$	−5.89370	−0.499897	−0.249949	−	0.968259i	$-0.580414\pi$
$139$	−5.89370	−0.499897	−0.249949	+	0.968259i	$0.580414\pi$
$140$	0.783568	0.0662235
$141$	0	0
$142$	−18.1459	−1.52277
$143$	11.9193	0.996745
$144$	0	0
$145$	−3.51943	−0.292273
$146$	−24.0361	−1.98924
$147$	0	0
$148$	−1.17870	−0.0968889
$149$	−5.44589	−0.446145	−0.223072	−	0.974802i	$-0.571609\pi$
$149$	−5.44589	−0.446145	−0.223072	+	0.974802i	$0.571609\pi$
$150$	0	0
$151$	−14.9117	−1.21350	−0.606749	−	0.794894i	$-0.707527\pi$
$151$	−14.9117	−1.21350	−0.606749	+	0.794894i	$0.707527\pi$
$152$	−16.7207	−1.35623
$153$	0	0
$154$	−23.5873	−1.90072
$155$	−0.0235901	−0.00189480
$156$	0	0
$157$	8.38451	0.669156	0.334578	−	0.942368i	$-0.391406\pi$
$157$	8.38451	0.669156	0.334578	+	0.942368i	$0.391406\pi$
$158$	21.6205	1.72004
$159$	0	0
$160$	1.09672	0.0867031
$161$	4.73212	0.372944
$162$	0	0
$163$	3.04537	0.238531	0.119266	−	0.992862i	$-0.461946\pi$
$163$	3.04537	0.238531	0.119266	+	0.992862i	$0.461946\pi$
$164$	2.29351	0.179093
$165$	0	0
$166$	1.08591	0.0842826
$167$	21.1038	1.63306	0.816530	−	0.577303i	$-0.195895\pi$
$167$	21.1038	1.63306	0.816530	+	0.577303i	$0.195895\pi$
$168$	0	0
$169$	−3.30762	−0.254433
$170$	1.18065	0.0905517
$171$	0	0
$172$	0.0919482	0.00701098
$173$	−1.69474	−0.128849	−0.0644245	−	0.997923i	$-0.520521\pi$
$173$	−1.69474	−0.128849	−0.0644245	+	0.997923i	$0.520521\pi$
$174$	0	0
$175$	−18.9092	−1.42940
$176$	−17.6996	−1.33416
$177$	0	0
$178$	17.8728	1.33963
$179$	12.0637	0.901686	0.450843	−	0.892603i	$-0.351123\pi$
$179$	12.0637	0.901686	0.450843	+	0.892603i	$0.351123\pi$
$180$	0	0
$181$	7.55105	0.561265	0.280633	−	0.959815i	$-0.409456\pi$
$181$	7.55105	0.561265	0.280633	+	0.959815i	$0.409456\pi$
$182$	−19.1804	−1.42175
$183$	0	0
$184$	2.95791	0.218060
$185$	−1.55395	−0.114249
$186$	0	0
$187$	−5.75000	−0.420481
$188$	1.82885	0.133383
$189$	0	0
$190$	5.27244	0.382503
$191$	12.9507	0.937079	0.468539	−	0.883443i	$-0.344780\pi$
$191$	12.9507	0.937079	0.468539	+	0.883443i	$0.344780\pi$
$192$	0	0
$193$	18.0791	1.30136	0.650680	−	0.759352i	$-0.274484\pi$
$193$	18.0791	1.30136	0.650680	+	0.759352i	$0.274484\pi$
$194$	20.0962	1.44283
$195$	0	0
$196$	3.43865	0.245618
$197$	−9.75805	−0.695232	−0.347616	−	0.937637i	$-0.613009\pi$
$197$	−9.75805	−0.695232	−0.347616	+	0.937637i	$0.613009\pi$
$198$	0	0
$199$	−8.29534	−0.588041	−0.294020	−	0.955799i	$-0.594993\pi$
$199$	−8.29534	−0.588041	−0.294020	+	0.955799i	$0.594993\pi$
$200$	−11.8196	−0.835773
$201$	0	0
$202$	−19.0984	−1.34376
$203$	−27.5820	−1.93587
$204$	0	0
$205$	3.02367	0.211182
$206$	3.22860	0.224947
$207$	0	0
$208$	−14.3927	−0.997955
$209$	−25.6778	−1.77617
$210$	0	0
$211$	−11.3391	−0.780619	−0.390309	−	0.920684i	$-0.627632\pi$
$211$	−11.3391	−0.780619	−0.390309	+	0.920684i	$0.627632\pi$
$212$	5.25905	0.361193
$213$	0	0
$214$	−2.56838	−0.175571
$215$	0.121220	0.00826716
$216$	0	0
$217$	−0.184876	−0.0125502
$218$	−6.64151	−0.449820
$219$	0	0
$220$	0.752157	0.0507104
$221$	−4.67569	−0.314521
$222$	0	0
$223$	18.9254	1.26734	0.633670	−	0.773603i	$-0.281548\pi$
$223$	18.9254	1.26734	0.633670	+	0.773603i	$0.281548\pi$
$224$	8.59502	0.574279
$225$	0	0
$226$	−21.9877	−1.46260
$227$	11.0987	0.736646	0.368323	−	0.929698i	$-0.379932\pi$
$227$	11.0987	0.736646	0.368323	+	0.929698i	$0.379932\pi$
$228$	0	0
$229$	3.04514	0.201229	0.100614	−	0.994925i	$-0.467919\pi$
$229$	3.04514	0.201229	0.100614	+	0.994925i	$0.467919\pi$
$230$	−0.932699	−0.0615004
$231$	0	0
$232$	−17.2407	−1.13191
$233$	14.2455	0.933252	0.466626	−	0.884455i	$-0.345469\pi$
$233$	14.2455	0.933252	0.466626	+	0.884455i	$0.345469\pi$
$234$	0	0
$235$	2.41107	0.157281
$236$	0.539202	0.0350991
$237$	0	0
$238$	9.25280	0.599770
$239$	7.94630	0.514004	0.257002	−	0.966411i	$-0.417265\pi$
$239$	7.94630	0.514004	0.257002	+	0.966411i	$0.417265\pi$
$240$	0	0
$241$	3.62377	0.233427	0.116714	−	0.993166i	$-0.462764\pi$
$241$	3.62377	0.233427	0.116714	+	0.993166i	$0.462764\pi$
$242$	−5.65036	−0.363219
$243$	0	0
$244$	−0.121208	−0.00775955
$245$	4.53336	0.289626
$246$	0	0
$247$	−20.8803	−1.32858
$248$	−0.115561	−0.00733811
$249$	0	0
$250$	7.65761	0.484310
$251$	2.54164	0.160427	0.0802134	−	0.996778i	$-0.474440\pi$
$251$	2.54164	0.160427	0.0802134	+	0.996778i	$0.474440\pi$
$252$	0	0
$253$	4.54243	0.285580
$254$	−28.5838	−1.79351
$255$	0	0
$256$	8.94175	0.558859
$257$	27.9997	1.74657	0.873286	−	0.487207i	$-0.161984\pi$
$257$	27.9997	1.74657	0.873286	+	0.487207i	$0.161984\pi$
$258$	0	0
$259$	−12.1784	−0.756728
$260$	0.611628	0.0379315
$261$	0	0
$262$	−8.57790	−0.529944
$263$	2.46064	0.151730	0.0758648	−	0.997118i	$-0.475828\pi$
$263$	2.46064	0.151730	0.0758648	+	0.997118i	$0.475828\pi$
$264$	0	0
$265$	6.93329	0.425909
$266$	41.3203	2.53351
$267$	0	0
$268$	2.12025	0.129515
$269$	−11.3054	−0.689303	−0.344652	−	0.938731i	$-0.612003\pi$
$269$	−11.3054	−0.689303	−0.344652	+	0.938731i	$0.612003\pi$
$270$	0	0
$271$	−4.28168	−0.260093	−0.130047	−	0.991508i	$-0.541513\pi$
$271$	−4.28168	−0.260093	−0.130047	+	0.991508i	$0.541513\pi$
$272$	6.94318	0.420992
$273$	0	0
$274$	4.60871	0.278422
$275$	−18.1512	−1.09456
$276$	0	0
$277$	18.8225	1.13094	0.565469	−	0.824770i	$-0.308695\pi$
$277$	18.8225	1.13094	0.565469	+	0.824770i	$0.308695\pi$
$278$	9.10387	0.546014
$279$	0	0
$280$	5.06046	0.302421
$281$	13.6036	0.811522	0.405761	−	0.913979i	$-0.367007\pi$
$281$	13.6036	0.811522	0.405761	+	0.913979i	$0.367007\pi$
$282$	0	0
$283$	16.6903	0.992135	0.496068	−	0.868284i	$-0.334777\pi$
$283$	16.6903	0.992135	0.496068	+	0.868284i	$0.334777\pi$
$284$	4.53483	0.269093
$285$	0	0
$286$	−18.4115	−1.08870
$287$	23.6966	1.39877
$288$	0	0
$289$	−14.7444	−0.867318
$290$	5.43639	0.319236
$291$	0	0
$292$	6.00683	0.351523
$293$	25.4095	1.48444	0.742218	−	0.670158i	$-0.233774\pi$
$293$	25.4095	1.48444	0.742218	+	0.670158i	$0.233774\pi$
$294$	0	0
$295$	0.710860	0.0413879
$296$	−7.61235	−0.442459
$297$	0	0
$298$	8.41215	0.487302
$299$	3.69374	0.213615
$300$	0	0
$301$	0.950009	0.0547576
$302$	23.0338	1.32545
$303$	0	0
$304$	31.0062	1.77833
$305$	−0.159795	−0.00914985
$306$	0	0
$307$	10.0361	0.572788	0.286394	−	0.958112i	$-0.407543\pi$
$307$	10.0361	0.572788	0.286394	+	0.958112i	$0.407543\pi$
$308$	5.89469	0.335881
$309$	0	0
$310$	0.0364390	0.00206960
$311$	−26.9511	−1.52826	−0.764128	−	0.645065i	$-0.776830\pi$
$311$	−26.9511	−1.52826	−0.764128	+	0.645065i	$0.776830\pi$
$312$	0	0
$313$	13.0313	0.736575	0.368287	−	0.929712i	$-0.379944\pi$
$313$	13.0313	0.736575	0.368287	+	0.929712i	$0.379944\pi$
$314$	−12.9514	−0.730887
$315$	0	0
$316$	−5.40317	−0.303952
$317$	−9.49374	−0.533221	−0.266611	−	0.963804i	$-0.585904\pi$
$317$	−9.49374	−0.533221	−0.266611	+	0.963804i	$0.585904\pi$
$318$	0	0
$319$	−26.4763	−1.48239
$320$	3.01147	0.168346
$321$	0	0
$322$	−7.30961	−0.407348
$323$	10.0729	0.560468
$324$	0	0
$325$	−14.7599	−0.818735
$326$	−4.70411	−0.260536
$327$	0	0
$328$	14.8121	0.817859
$329$	18.8957	1.04175
$330$	0	0
$331$	−23.5141	−1.29245	−0.646226	−	0.763146i	$-0.723654\pi$
$331$	−23.5141	−1.29245	−0.646226	+	0.763146i	$0.723654\pi$
$332$	−0.271378	−0.0148938
$333$	0	0
$334$	−32.5986	−1.78371
$335$	2.79525	0.152721
$336$	0	0
$337$	−3.10813	−0.169310	−0.0846552	−	0.996410i	$-0.526979\pi$
$337$	−3.10813	−0.169310	−0.0846552	+	0.996410i	$0.526979\pi$
$338$	5.10921	0.277904
$339$	0	0
$340$	−0.295055	−0.0160016
$341$	−0.177465	−0.00961028
$342$	0	0
$343$	7.60900	0.410847
$344$	0.593823	0.0320168
$345$	0	0
$346$	2.61783	0.140736
$347$	22.2947	1.19684	0.598421	−	0.801182i	$-0.295795\pi$
$347$	22.2947	1.19684	0.598421	+	0.801182i	$0.295795\pi$
$348$	0	0
$349$	3.01277	0.161270	0.0806349	−	0.996744i	$-0.474305\pi$
$349$	3.01277	0.161270	0.0806349	+	0.996744i	$0.474305\pi$
$350$	29.2087	1.56127
$351$	0	0
$352$	8.25048	0.439752
$353$	−2.16426	−0.115192	−0.0575961	−	0.998340i	$-0.518344\pi$
$353$	−2.16426	−0.115192	−0.0575961	+	0.998340i	$0.518344\pi$
$354$	0	0
$355$	5.97852	0.317307
$356$	−4.46659	−0.236729
$357$	0	0
$358$	−18.6346	−0.984869
$359$	0.648519	0.0342275	0.0171138	−	0.999854i	$-0.494552\pi$
$359$	0.648519	0.0342275	0.0171138	+	0.999854i	$0.494552\pi$
$360$	0	0
$361$	25.9825	1.36750
$362$	−11.6639	−0.613043
$363$	0	0
$364$	4.79335	0.251240
$365$	7.91914	0.414507
$366$	0	0
$367$	−9.46927	−0.494292	−0.247146	−	0.968978i	$-0.579493\pi$
$367$	−9.46927	−0.494292	−0.247146	+	0.968978i	$0.579493\pi$
$368$	−5.48503	−0.285927
$369$	0	0
$370$	2.40035	0.124788
$371$	54.3365	2.82101
$372$	0	0
$373$	−4.88169	−0.252765	−0.126382	−	0.991982i	$-0.540337\pi$
$373$	−4.88169	−0.252765	−0.126382	+	0.991982i	$0.540337\pi$
$374$	8.88189	0.459272
$375$	0	0
$376$	11.8111	0.609113
$377$	−21.5296	−1.10883
$378$	0	0
$379$	−26.0379	−1.33748	−0.668739	−	0.743497i	$-0.733166\pi$
$379$	−26.0379	−1.33748	−0.668739	+	0.743497i	$0.733166\pi$
$380$	−1.31763	−0.0675930
$381$	0	0
$382$	−20.0046	−1.02353
$383$	8.03098	0.410364	0.205182	−	0.978724i	$-0.434221\pi$
$383$	8.03098	0.410364	0.205182	+	0.978724i	$0.434221\pi$
$384$	0	0
$385$	7.77129	0.396062
$386$	−27.9263	−1.42141
$387$	0	0
$388$	−5.02223	−0.254965
$389$	22.7460	1.15327	0.576635	−	0.817002i	$-0.304365\pi$
$389$	22.7460	1.15327	0.576635	+	0.817002i	$0.304365\pi$
$390$	0	0
$391$	−1.78190	−0.0901144
$392$	22.2076	1.12165
$393$	0	0
$394$	15.0730	0.759369
$395$	−7.12329	−0.358412
$396$	0	0
$397$	24.7220	1.24076	0.620382	−	0.784300i	$-0.286978\pi$
$397$	24.7220	1.24076	0.620382	+	0.784300i	$0.286978\pi$
$398$	12.8136	0.642289
$399$	0	0
$400$	21.9178	1.09589
$401$	16.9764	0.847763	0.423882	−	0.905718i	$-0.360667\pi$
$401$	16.9764	0.847763	0.423882	+	0.905718i	$0.360667\pi$
$402$	0	0
$403$	−0.144308	−0.00718852
$404$	4.77286	0.237459
$405$	0	0
$406$	42.6052	2.11446
$407$	−11.6902	−0.579462
$408$	0	0
$409$	11.2986	0.558681	0.279341	−	0.960192i	$-0.409884\pi$
$409$	11.2986	0.558681	0.279341	+	0.960192i	$0.409884\pi$
$410$	−4.67059	−0.230664
$411$	0	0
$412$	−0.806856	−0.0397510
$413$	5.57104	0.274133
$414$	0	0
$415$	−0.357772	−0.0175623
$416$	6.70899	0.328936
$417$	0	0
$418$	39.6640	1.94003
$419$	−32.4310	−1.58436	−0.792180	−	0.610288i	$-0.791054\pi$
$419$	−32.4310	−1.58436	−0.792180	+	0.610288i	$0.791054\pi$
$420$	0	0
$421$	−0.664405	−0.0323811	−0.0161906	−	0.999869i	$-0.505154\pi$
$421$	−0.664405	−0.0323811	−0.0161906	+	0.999869i	$0.505154\pi$
$422$	17.5153	0.852632
$423$	0	0
$424$	33.9642	1.64945
$425$	7.12034	0.345387
$426$	0	0
$427$	−1.25232	−0.0606042
$428$	0.641861	0.0310255
$429$	0	0
$430$	−0.187246	−0.00902982
$431$	−30.3722	−1.46298	−0.731488	−	0.681854i	$-0.761174\pi$
$431$	−30.3722	−1.46298	−0.731488	+	0.681854i	$0.761174\pi$
$432$	0	0
$433$	−11.3660	−0.546214	−0.273107	−	0.961984i	$-0.588051\pi$
$433$	−11.3660	−0.546214	−0.273107	+	0.961984i	$0.588051\pi$
$434$	0.285574	0.0137080
$435$	0	0
$436$	1.65977	0.0794888
$437$	−7.95744	−0.380656
$438$	0	0
$439$	−7.17141	−0.342273	−0.171136	−	0.985247i	$-0.554744\pi$
$439$	−7.17141	−0.342273	−0.171136	+	0.985247i	$0.554744\pi$
$440$	4.85761	0.231577
$441$	0	0
$442$	7.22244	0.343536
$443$	1.74661	0.0829839	0.0414919	−	0.999139i	$-0.486789\pi$
$443$	1.74661	0.0829839	0.0414919	+	0.999139i	$0.486789\pi$
$444$	0	0
$445$	−5.88855	−0.279144
$446$	−29.2337	−1.38426
$447$	0	0
$448$	23.6010	1.11504
$449$	−33.5091	−1.58139	−0.790695	−	0.612210i	$-0.790281\pi$
$449$	−33.5091	−1.58139	−0.790695	+	0.612210i	$0.790281\pi$
$450$	0	0
$451$	22.7467	1.07110
$452$	5.49492	0.258459
$453$	0	0
$454$	−17.1439	−0.804603
$455$	6.31934	0.296255
$456$	0	0
$457$	11.4271	0.534536	0.267268	−	0.963622i	$-0.413879\pi$
$457$	11.4271	0.534536	0.267268	+	0.963622i	$0.413879\pi$
$458$	−4.70377	−0.219793
$459$	0	0
$460$	0.233090	0.0108679
$461$	−16.6478	−0.775363	−0.387682	−	0.921793i	$-0.626724\pi$
$461$	−16.6478	−0.775363	−0.387682	+	0.921793i	$0.626724\pi$
$462$	0	0
$463$	−4.57640	−0.212683	−0.106342	−	0.994330i	$-0.533914\pi$
$463$	−4.57640	−0.212683	−0.106342	+	0.994330i	$0.533914\pi$
$464$	31.9704	1.48419
$465$	0	0
$466$	−22.0047	−1.01935
$467$	−33.8973	−1.56858	−0.784291	−	0.620394i	$-0.786973\pi$
$467$	−33.8973	−1.56858	−0.784291	+	0.620394i	$0.786973\pi$
$468$	0	0
$469$	21.9065	1.01155
$470$	−3.72433	−0.171791
$471$	0	0
$472$	3.48229	0.160286
$473$	0.911927	0.0419304
$474$	0	0
$475$	31.7974	1.45896
$476$	−2.31236	−0.105987
$477$	0	0
$478$	−12.2745	−0.561421
$479$	−8.68324	−0.396747	−0.198374	−	0.980126i	$-0.563566\pi$
$479$	−8.68324	−0.396747	−0.198374	+	0.980126i	$0.563566\pi$
$480$	0	0
$481$	−9.50605	−0.433439
$482$	−5.59755	−0.254961
$483$	0	0
$484$	1.41208	0.0641853
$485$	−6.62108	−0.300648
$486$	0	0
$487$	−21.3432	−0.967154	−0.483577	−	0.875302i	$-0.660663\pi$
$487$	−21.3432	−0.967154	−0.483577	+	0.875302i	$0.660663\pi$
$488$	−0.782790	−0.0354353
$489$	0	0
$490$	−7.00259	−0.316345
$491$	−4.21516	−0.190227	−0.0951137	−	0.995466i	$-0.530321\pi$
$491$	−4.21516	−0.190227	−0.0951137	+	0.995466i	$0.530321\pi$
$492$	0	0
$493$	10.3861	0.467765
$494$	32.2533	1.45115
$495$	0	0
$496$	0.214291	0.00962196
$497$	46.8539	2.10169
$498$	0	0
$499$	30.4249	1.36201	0.681003	−	0.732280i	$-0.261544\pi$
$499$	30.4249	1.36201	0.681003	+	0.732280i	$0.261544\pi$
$500$	−1.91371	−0.0855836
$501$	0	0
$502$	−3.92601	−0.175226
$503$	−12.6755	−0.565173	−0.282586	−	0.959242i	$-0.591192\pi$
$503$	−12.6755	−0.565173	−0.282586	+	0.959242i	$0.591192\pi$
$504$	0	0
$505$	6.29233	0.280005
$506$	−7.01659	−0.311926
$507$	0	0
$508$	7.14336	0.316935
$509$	3.51223	0.155677	0.0778383	−	0.996966i	$-0.475198\pi$
$509$	3.51223	0.155677	0.0778383	+	0.996966i	$0.475198\pi$
$510$	0	0
$511$	62.0627	2.74549
$512$	13.0885	0.578437
$513$	0	0
$514$	−43.2505	−1.90770
$515$	−1.06372	−0.0468733
$516$	0	0
$517$	18.1382	0.797719
$518$	18.8117	0.826537
$519$	0	0
$520$	3.95004	0.173220
$521$	−8.49809	−0.372308	−0.186154	−	0.982521i	$-0.559602\pi$
$521$	−8.49809	−0.372308	−0.186154	+	0.982521i	$0.559602\pi$
$522$	0	0
$523$	−12.6918	−0.554974	−0.277487	−	0.960729i	$-0.589502\pi$
$523$	−12.6918	−0.554974	−0.277487	+	0.960729i	$0.589502\pi$
$524$	2.14369	0.0936477
$525$	0	0
$526$	−3.80090	−0.165727
$527$	0.0696158	0.00303251
$528$	0	0
$529$	−21.5923	−0.938797
$530$	−10.7097	−0.465200
$531$	0	0
$532$	−10.3263	−0.447703
$533$	18.4968	0.801186
$534$	0	0
$535$	0.846200	0.0365844
$536$	13.6931	0.591452
$537$	0	0
$538$	17.4632	0.752893
$539$	34.1040	1.46896
$540$	0	0
$541$	12.2904	0.528405	0.264203	−	0.964467i	$-0.414891\pi$
$541$	12.2904	0.528405	0.264203	+	0.964467i	$0.414891\pi$
$542$	6.61381	0.284088
$543$	0	0
$544$	−3.23648	−0.138763
$545$	2.18817	0.0937310
$546$	0	0
$547$	−26.8939	−1.14990	−0.574950	−	0.818188i	$-0.694979\pi$
$547$	−26.8939	−1.14990	−0.574950	+	0.818188i	$0.694979\pi$
$548$	−1.15176	−0.0492006
$549$	0	0
$550$	28.0378	1.19554
$551$	46.3812	1.97591
$552$	0	0
$553$	−55.8255	−2.37394
$554$	−29.0748	−1.23527
$555$	0	0
$556$	−2.27514	−0.0964874
$557$	−21.9915	−0.931811	−0.465905	−	0.884835i	$-0.654271\pi$
$557$	−21.9915	−0.931811	−0.465905	+	0.884835i	$0.654271\pi$
$558$	0	0
$559$	0.741547	0.0313641
$560$	−9.38392	−0.396543
$561$	0	0
$562$	−21.0132	−0.886386
$563$	−47.3481	−1.99548	−0.997741	−	0.0671735i	$-0.978602\pi$
$563$	−47.3481	−1.99548	−0.997741	+	0.0671735i	$0.978602\pi$
$564$	0	0
$565$	7.24426	0.304768
$566$	−25.7811	−1.08366
$567$	0	0
$568$	29.2870	1.22886
$569$	17.3331	0.726641	0.363320	−	0.931664i	$-0.381643\pi$
$569$	17.3331	0.726641	0.363320	+	0.931664i	$0.381643\pi$
$570$	0	0
$571$	35.6536	1.49206	0.746028	−	0.665914i	$-0.231958\pi$
$571$	35.6536	1.49206	0.746028	+	0.665914i	$0.231958\pi$
$572$	4.60121	0.192386
$573$	0	0
$574$	−36.6036	−1.52781
$575$	−5.62498	−0.234578
$576$	0	0
$577$	−1.31809	−0.0548729	−0.0274365	−	0.999624i	$-0.508734\pi$
$577$	−1.31809	−0.0548729	−0.0274365	+	0.999624i	$0.508734\pi$
$578$	22.7753	0.947330
$579$	0	0
$580$	−1.35860	−0.0564129
$581$	−2.80387	−0.116324
$582$	0	0
$583$	52.1584	2.16018
$584$	38.7935	1.60529
$585$	0	0
$586$	−39.2494	−1.62138
$587$	−5.03783	−0.207933	−0.103967	−	0.994581i	$-0.533154\pi$
$587$	−5.03783	−0.207933	−0.103967	+	0.994581i	$0.533154\pi$
$588$	0	0
$589$	0.310884	0.0128098
$590$	−1.09805	−0.0452060
$591$	0	0
$592$	14.1160	0.580165
$593$	26.0024	1.06779	0.533895	−	0.845551i	$-0.320728\pi$
$593$	26.0024	1.06779	0.533895	+	0.845551i	$0.320728\pi$
$594$	0	0
$595$	−3.04851	−0.124977
$596$	−2.10227	−0.0861124
$597$	0	0
$598$	−5.70564	−0.233321
$599$	17.9424	0.733105	0.366552	−	0.930397i	$-0.380538\pi$
$599$	17.9424	0.733105	0.366552	+	0.930397i	$0.380538\pi$
$600$	0	0
$601$	−34.3012	−1.39918	−0.699588	−	0.714546i	$-0.746633\pi$
$601$	−34.3012	−1.39918	−0.699588	+	0.714546i	$0.746633\pi$
$602$	−1.46746	−0.0598091
$603$	0	0
$604$	−5.75635	−0.234223
$605$	1.86162	0.0756856
$606$	0	0
$607$	−1.96473	−0.0797458	−0.0398729	−	0.999205i	$-0.512695\pi$
$607$	−1.96473	−0.0797458	−0.0398729	+	0.999205i	$0.512695\pi$
$608$	−14.4532	−0.586155
$609$	0	0
$610$	0.246832	0.00999395
$611$	14.7494	0.596696
$612$	0	0
$613$	−2.08953	−0.0843953	−0.0421976	−	0.999109i	$-0.513436\pi$
$613$	−2.08953	−0.0843953	−0.0421976	+	0.999109i	$0.513436\pi$
$614$	−15.5025	−0.625629
$615$	0	0
$616$	38.0693	1.53386
$617$	41.5669	1.67342	0.836710	−	0.547646i	$-0.184476\pi$
$617$	41.5669	1.67342	0.836710	+	0.547646i	$0.184476\pi$
$618$	0	0
$619$	−7.03173	−0.282629	−0.141315	−	0.989965i	$-0.545133\pi$
$619$	−7.03173	−0.282629	−0.141315	+	0.989965i	$0.545133\pi$
$620$	−0.00910644	−0.000365723 0
$621$	0	0
$622$	41.6307	1.66924
$623$	−46.1488	−1.84891
$624$	0	0
$625$	21.1820	0.847282
$626$	−20.1292	−0.804525
$627$	0	0
$628$	3.23666	0.129157
$629$	4.58581	0.182848
$630$	0	0
$631$	−6.73381	−0.268069	−0.134034	−	0.990977i	$-0.542793\pi$
$631$	−6.73381	−0.268069	−0.134034	+	0.990977i	$0.542793\pi$
$632$	−34.8949	−1.38805
$633$	0	0
$634$	14.6648	0.582412
$635$	9.41748	0.373721
$636$	0	0
$637$	27.7321	1.09879
$638$	40.8974	1.61914
$639$	0	0
$640$	−6.84518	−0.270580
$641$	−13.0834	−0.516764	−0.258382	−	0.966043i	$-0.583189\pi$
$641$	−13.0834	−0.516764	−0.258382	+	0.966043i	$0.583189\pi$
$642$	0	0
$643$	−2.62280	−0.103433	−0.0517165	−	0.998662i	$-0.516469\pi$
$643$	−2.62280	−0.103433	−0.0517165	+	0.998662i	$0.516469\pi$
$644$	1.82674	0.0719835
$645$	0	0
$646$	−15.5593	−0.612173
$647$	−2.94663	−0.115844	−0.0579219	−	0.998321i	$-0.518447\pi$
$647$	−2.94663	−0.115844	−0.0579219	+	0.998321i	$0.518447\pi$
$648$	0	0
$649$	5.34772	0.209916
$650$	22.7994	0.894265
$651$	0	0
$652$	1.17560	0.0460400
$653$	−35.2292	−1.37863	−0.689313	−	0.724463i	$-0.742088\pi$
$653$	−35.2292	−1.37863	−0.689313	+	0.724463i	$0.742088\pi$
$654$	0	0
$655$	2.82615	0.110427
$656$	−27.4669	−1.07240
$657$	0	0
$658$	−29.1877	−1.13786
$659$	−16.8039	−0.654586	−0.327293	−	0.944923i	$-0.606136\pi$
$659$	−16.8039	−0.654586	−0.327293	+	0.944923i	$0.606136\pi$
$660$	0	0
$661$	−2.91032	−0.113198	−0.0565992	−	0.998397i	$-0.518026\pi$
$661$	−2.91032	−0.113198	−0.0565992	+	0.998397i	$0.518026\pi$
$662$	36.3217	1.41168
$663$	0	0
$664$	−1.75262	−0.0680148
$665$	−13.6138	−0.527919
$666$	0	0
$667$	−8.20488	−0.317694
$668$	8.14667	0.315204
$669$	0	0
$670$	−4.31776	−0.166810
$671$	−1.20212	−0.0464074
$672$	0	0
$673$	3.04251	0.117280	0.0586400	−	0.998279i	$-0.481324\pi$
$673$	3.04251	0.117280	0.0586400	+	0.998279i	$0.481324\pi$
$674$	4.80105	0.184930
$675$	0	0
$676$	−1.27684	−0.0491091
$677$	−37.4337	−1.43869	−0.719346	−	0.694652i	$-0.755559\pi$
$677$	−37.4337	−1.43869	−0.719346	+	0.694652i	$0.755559\pi$
$678$	0	0
$679$	−51.8897	−1.99134
$680$	−1.90553	−0.0730739
$681$	0	0
$682$	0.274127	0.0104969
$683$	31.1355	1.19137	0.595684	−	0.803219i	$-0.296881\pi$
$683$	31.1355	1.19137	0.595684	+	0.803219i	$0.296881\pi$
$684$	0	0
$685$	−1.51843	−0.0580160
$686$	−11.7535	−0.448749
$687$	0	0
$688$	−1.10116	−0.0419814
$689$	42.4133	1.61582
$690$	0	0
$691$	10.7146	0.407602	0.203801	−	0.979012i	$-0.434670\pi$
$691$	10.7146	0.407602	0.203801	+	0.979012i	$0.434670\pi$
$692$	−0.654220	−0.0248697
$693$	0	0
$694$	−34.4381	−1.30725
$695$	−2.99944	−0.113775
$696$	0	0
$697$	−8.92303	−0.337984
$698$	−4.65376	−0.176147
$699$	0	0
$700$	−7.29951	−0.275896
$701$	−20.7160	−0.782433	−0.391216	−	0.920299i	$-0.627946\pi$
$701$	−20.7160	−0.782433	−0.391216	+	0.920299i	$0.627946\pi$
$702$	0	0
$703$	20.4789	0.772376
$704$	22.6549	0.853840
$705$	0	0
$706$	3.34309	0.125819
$707$	49.3132	1.85462
$708$	0	0
$709$	−18.1896	−0.683123	−0.341562	−	0.939859i	$-0.610956\pi$
$709$	−18.1896	−0.683123	−0.341562	+	0.939859i	$0.610956\pi$
$710$	−9.23489	−0.346579
$711$	0	0
$712$	−28.8463	−1.08106
$713$	−0.0549957	−0.00205960
$714$	0	0
$715$	6.06602	0.226856
$716$	4.65695	0.174038
$717$	0	0
$718$	−1.00175	−0.0373851
$719$	31.8550	1.18799	0.593995	−	0.804469i	$-0.297550\pi$
$719$	31.8550	1.18799	0.593995	+	0.804469i	$0.297550\pi$
$720$	0	0
$721$	−8.33645	−0.310466
$722$	−40.1345	−1.49365
$723$	0	0
$724$	2.91492	0.108332
$725$	32.7861	1.21765
$726$	0	0
$727$	9.47883	0.351550	0.175775	−	0.984430i	$-0.443757\pi$
$727$	9.47883	0.351550	0.175775	+	0.984430i	$0.443757\pi$
$728$	30.9566	1.14733
$729$	0	0
$730$	−12.2325	−0.452746
$731$	−0.357729	−0.0132311
$732$	0	0
$733$	−42.1434	−1.55660	−0.778301	−	0.627892i	$-0.783918\pi$
$733$	−42.1434	−1.55660	−0.778301	+	0.627892i	$0.783918\pi$
$734$	14.6270	0.539891
$735$	0	0
$736$	2.55678	0.0942443
$737$	21.0283	0.774589
$738$	0	0
$739$	47.9455	1.76370	0.881852	−	0.471527i	$-0.156297\pi$
$739$	47.9455	1.76370	0.881852	+	0.471527i	$0.156297\pi$
$740$	−0.599870	−0.0220517
$741$	0	0
$742$	−83.9324	−3.08126
$743$	10.3104	0.378252	0.189126	−	0.981953i	$-0.439435\pi$
$743$	10.3104	0.378252	0.189126	+	0.981953i	$0.439435\pi$
$744$	0	0
$745$	−2.77154	−0.101541
$746$	7.54064	0.276083
$747$	0	0
$748$	−2.21966	−0.0811590
$749$	6.63171	0.242317
$750$	0	0
$751$	25.0549	0.914267	0.457133	−	0.889398i	$-0.348876\pi$
$751$	25.0549	0.914267	0.457133	+	0.889398i	$0.348876\pi$
$752$	−21.9021	−0.798688
$753$	0	0
$754$	33.2563	1.21112
$755$	−7.58891	−0.276189
$756$	0	0
$757$	−6.11750	−0.222344	−0.111172	−	0.993801i	$-0.535460\pi$
$757$	−6.11750	−0.222344	−0.111172	+	0.993801i	$0.535460\pi$
$758$	40.2202	1.46086
$759$	0	0
$760$	−8.50957	−0.308674
$761$	−44.8545	−1.62598	−0.812988	−	0.582281i	$-0.802160\pi$
$761$	−44.8545	−1.62598	−0.812988	+	0.582281i	$0.802160\pi$
$762$	0	0
$763$	17.1488	0.620828
$764$	4.99934	0.180870
$765$	0	0
$766$	−12.4053	−0.448221
$767$	4.34857	0.157018
$768$	0	0
$769$	37.9306	1.36781	0.683906	−	0.729570i	$-0.260280\pi$
$769$	37.9306	1.36781	0.683906	+	0.729570i	$0.260280\pi$
$770$	−12.0041	−0.432599
$771$	0	0
$772$	6.97904	0.251181
$773$	−15.3661	−0.552679	−0.276339	−	0.961060i	$-0.589121\pi$
$773$	−15.3661	−0.552679	−0.276339	+	0.961060i	$0.589121\pi$
$774$	0	0
$775$	0.219759	0.00789397
$776$	−32.4347	−1.16434
$777$	0	0
$778$	−35.1353	−1.25966
$779$	−39.8477	−1.42769
$780$	0	0
$781$	44.9757	1.60936
$782$	2.75246	0.0984276
$783$	0	0
$784$	−41.1809	−1.47075
$785$	4.26707	0.152298
$786$	0	0
$787$	−49.3321	−1.75850	−0.879251	−	0.476360i	$-0.841956\pi$
$787$	−49.3321	−1.75850	−0.879251	+	0.476360i	$0.841956\pi$
$788$	−3.76689	−0.134190
$789$	0	0
$790$	11.0032	0.391476
$791$	56.7736	2.01864
$792$	0	0
$793$	−0.977523	−0.0347129
$794$	−38.1876	−1.35523
$795$	0	0
$796$	−3.20224	−0.113500
$797$	15.2567	0.540422	0.270211	−	0.962801i	$-0.412907\pi$
$797$	15.2567	0.540422	0.270211	+	0.962801i	$0.412907\pi$
$798$	0	0
$799$	−7.11523	−0.251719
$800$	−10.2167	−0.361216
$801$	0	0
$802$	−26.2231	−0.925971
$803$	59.5748	2.10235
$804$	0	0
$805$	2.40829	0.0848810
$806$	0.222910	0.00785167
$807$	0	0
$808$	30.8243	1.08439
$809$	−30.2451	−1.06336	−0.531681	−	0.846945i	$-0.678439\pi$
$809$	−30.2451	−1.06336	−0.531681	+	0.846945i	$0.678439\pi$
$810$	0	0
$811$	−20.7859	−0.729892	−0.364946	−	0.931029i	$-0.618913\pi$
$811$	−20.7859	−0.729892	−0.364946	+	0.931029i	$0.618913\pi$
$812$	−10.6474	−0.373652
$813$	0	0
$814$	18.0576	0.632918
$815$	1.54986	0.0542891
$816$	0	0
$817$	−1.59751	−0.0558900
$818$	−17.4527	−0.610221
$819$	0	0
$820$	1.16722	0.0407612
$821$	−43.8012	−1.52867	−0.764336	−	0.644818i	$-0.776933\pi$
$821$	−43.8012	−1.52867	−0.764336	+	0.644818i	$0.776933\pi$
$822$	0	0
$823$	7.47645	0.260613	0.130306	−	0.991474i	$-0.458404\pi$
$823$	7.47645	0.260613	0.130306	+	0.991474i	$0.458404\pi$
$824$	−5.21087	−0.181529
$825$	0	0
$826$	−8.60546	−0.299422
$827$	−38.8777	−1.35191	−0.675956	−	0.736942i	$-0.736269\pi$
$827$	−38.8777	−1.35191	−0.675956	+	0.736942i	$0.736269\pi$
$828$	0	0
$829$	34.9772	1.21481	0.607404	−	0.794393i	$-0.292211\pi$
$829$	34.9772	1.21481	0.607404	+	0.794393i	$0.292211\pi$
$830$	0.552642	0.0191825
$831$	0	0
$832$	18.4222	0.638675
$833$	−13.3782	−0.463529
$834$	0	0
$835$	10.7402	0.371680
$836$	−9.91238	−0.342827
$837$	0	0
$838$	50.0955	1.73052
$839$	−5.37996	−0.185737	−0.0928684	−	0.995678i	$-0.529604\pi$
$839$	−5.37996	−0.185737	−0.0928684	+	0.995678i	$0.529604\pi$
$840$	0	0
$841$	18.8235	0.649086
$842$	1.02629	0.0353684
$843$	0	0
$844$	−4.37724	−0.150671
$845$	−1.68333	−0.0579082
$846$	0	0
$847$	14.5896	0.501304
$848$	−62.9818	−2.16280
$849$	0	0
$850$	−10.9986	−0.377250
$851$	−3.62274	−0.124186
$852$	0	0
$853$	19.1707	0.656393	0.328197	−	0.944609i	$-0.393559\pi$
$853$	19.1707	0.656393	0.328197	+	0.944609i	$0.393559\pi$
$854$	1.93444	0.0661950
$855$	0	0
$856$	4.14529	0.141683
$857$	31.8617	1.08838	0.544188	−	0.838963i	$-0.316838\pi$
$857$	31.8617	1.08838	0.544188	+	0.838963i	$0.316838\pi$
$858$	0	0
$859$	15.7538	0.537512	0.268756	−	0.963208i	$-0.413387\pi$
$859$	15.7538	0.537512	0.268756	+	0.963208i	$0.413387\pi$
$860$	0.0467945	0.00159568
$861$	0	0
$862$	46.9152	1.59794
$863$	3.11237	0.105946	0.0529731	−	0.998596i	$-0.483130\pi$
$863$	3.11237	0.105946	0.0529731	+	0.998596i	$0.483130\pi$
$864$	0	0
$865$	−0.862495	−0.0293257
$866$	17.5568	0.596604
$867$	0	0
$868$	−0.0713676	−0.00242237
$869$	−53.5877	−1.81784
$870$	0	0
$871$	17.0995	0.579394
$872$	10.7192	0.362998
$873$	0	0
$874$	12.2917	0.415772
$875$	−19.7724	−0.668430
$876$	0	0
$877$	−7.68437	−0.259483	−0.129741	−	0.991548i	$-0.541415\pi$
$877$	−7.68437	−0.259483	−0.129741	+	0.991548i	$0.541415\pi$
$878$	11.0775	0.373848
$879$	0	0
$880$	−9.00775	−0.303651
$881$	−47.4858	−1.59984	−0.799918	−	0.600109i	$-0.795124\pi$
$881$	−47.4858	−1.59984	−0.799918	+	0.600109i	$0.795124\pi$
$882$	0	0
$883$	31.0096	1.04356	0.521778	−	0.853081i	$-0.325269\pi$
$883$	31.0096	1.04356	0.521778	+	0.853081i	$0.325269\pi$
$884$	−1.80495	−0.0607071
$885$	0	0
$886$	−2.69795	−0.0906393
$887$	−3.05033	−0.102420	−0.0512101	−	0.998688i	$-0.516308\pi$
$887$	−3.05033	−0.102420	−0.0512101	+	0.998688i	$0.516308\pi$
$888$	0	0
$889$	73.8052	2.47535
$890$	9.09590	0.304895
$891$	0	0
$892$	7.30576	0.244615
$893$	−31.7746	−1.06330
$894$	0	0
$895$	6.13952	0.205221
$896$	−53.6460	−1.79219
$897$	0	0
$898$	51.7607	1.72728
$899$	0.320551	0.0106910
$900$	0	0
$901$	−20.4606	−0.681641
$902$	−35.1363	−1.16991
$903$	0	0
$904$	35.4875	1.18030
$905$	3.84290	0.127742
$906$	0	0
$907$	14.7774	0.490677	0.245338	−	0.969437i	$-0.421101\pi$
$907$	14.7774	0.490677	0.245338	+	0.969437i	$0.421101\pi$
$908$	4.28442	0.142183
$909$	0	0
$910$	−9.76134	−0.323585
$911$	−30.0054	−0.994124	−0.497062	−	0.867715i	$-0.665588\pi$
$911$	−30.0054	−0.994124	−0.497062	+	0.867715i	$0.665588\pi$
$912$	0	0
$913$	−2.69148	−0.0890749
$914$	−17.6511	−0.583848
$915$	0	0
$916$	1.17551	0.0388401
$917$	22.1487	0.731413
$918$	0	0
$919$	27.2669	0.899451	0.449725	−	0.893167i	$-0.351522\pi$
$919$	27.2669	0.899451	0.449725	+	0.893167i	$0.351522\pi$
$920$	1.50535	0.0496299
$921$	0	0
$922$	25.7154	0.846892
$923$	36.5727	1.20380
$924$	0	0
$925$	14.4762	0.475975
$926$	7.06906	0.232304
$927$	0	0
$928$	−14.9026	−0.489203
$929$	43.0167	1.41133	0.705666	−	0.708545i	$-0.250648\pi$
$929$	43.0167	1.41133	0.705666	+	0.708545i	$0.250648\pi$
$930$	0	0
$931$	−59.7434	−1.95801
$932$	5.49916	0.180131
$933$	0	0
$934$	52.3604	1.71329
$935$	−2.92631	−0.0957004
$936$	0	0
$937$	47.9842	1.56757	0.783787	−	0.621029i	$-0.213285\pi$
$937$	47.9842	1.56757	0.783787	+	0.621029i	$0.213285\pi$
$938$	−33.8385	−1.10486
$939$	0	0
$940$	0.930744	0.0303575
$941$	−29.1616	−0.950639	−0.475320	−	0.879813i	$-0.657668\pi$
$941$	−29.1616	−0.950639	−0.475320	+	0.879813i	$0.657668\pi$
$942$	0	0
$943$	7.04909	0.229550
$944$	−6.45742	−0.210171
$945$	0	0
$946$	−1.40863	−0.0457986
$947$	52.1827	1.69571	0.847856	−	0.530227i	$-0.177893\pi$
$947$	52.1827	1.69571	0.847856	+	0.530227i	$0.177893\pi$
$948$	0	0
$949$	48.4441	1.57256
$950$	−49.1167	−1.59356
$951$	0	0
$952$	−14.9338	−0.484006
$953$	11.1677	0.361756	0.180878	−	0.983506i	$-0.442106\pi$
$953$	11.1677	0.361756	0.180878	+	0.983506i	$0.442106\pi$
$954$	0	0
$955$	6.59090	0.213277
$956$	3.06750	0.0992101
$957$	0	0
$958$	13.4128	0.433348
$959$	−11.9000	−0.384270
$960$	0	0
$961$	−30.9979	−0.999931
$962$	14.6838	0.473424
$963$	0	0
$964$	1.39888	0.0450548
$965$	9.20085	0.296186
$966$	0	0
$967$	53.0468	1.70587	0.852935	−	0.522017i	$-0.174820\pi$
$967$	53.0468	1.70587	0.852935	+	0.522017i	$0.174820\pi$
$968$	9.11953	0.293113
$969$	0	0
$970$	10.2274	0.328383
$971$	45.0507	1.44575	0.722873	−	0.690981i	$-0.242821\pi$
$971$	45.0507	1.44575	0.722873	+	0.690981i	$0.242821\pi$
$972$	0	0
$973$	−23.5068	−0.753592
$974$	32.9684	1.05638
$975$	0	0
$976$	1.45157	0.0464638
$977$	55.1599	1.76472	0.882360	−	0.470575i	$-0.155953\pi$
$977$	55.1599	1.76472	0.882360	+	0.470575i	$0.155953\pi$
$978$	0	0
$979$	−44.2989	−1.41580
$980$	1.75001	0.0559020
$981$	0	0
$982$	6.51106	0.207776
$983$	38.9139	1.24116	0.620580	−	0.784143i	$-0.286897\pi$
$983$	38.9139	1.24116	0.620580	+	0.784143i	$0.286897\pi$
$984$	0	0
$985$	−4.96610	−0.158233
$986$	−16.0431	−0.510918
$987$	0	0
$988$	−8.06040	−0.256435
$989$	0.282602	0.00898621
$990$	0	0
$991$	8.41838	0.267419	0.133709	−	0.991021i	$-0.457311\pi$
$991$	8.41838	0.267419	0.133709	+	0.991021i	$0.457311\pi$
$992$	−0.0998893	−0.00317149
$993$	0	0
$994$	−72.3742	−2.29557
$995$	−4.22169	−0.133837
$996$	0	0
$997$	−48.8670	−1.54763	−0.773816	−	0.633410i	$-0.781655\pi$
$997$	−48.8670	−1.54763	−0.773816	+	0.633410i	$0.781655\pi$
$998$	−46.9967	−1.48765
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.17		72
3.2	odd	2		6561.2.a.c.1.56		72
81.5	odd	54		81.2.g.a.25.7	yes	144
81.11	odd	54		729.2.g.d.514.2		144
81.16	even	27		243.2.g.a.10.2		144
81.22	even	27		729.2.g.a.217.7		144
81.32	odd	54		729.2.g.c.703.2		144
81.38	odd	54		729.2.g.c.28.2		144
81.43	even	27		729.2.g.b.28.7		144
81.49	even	27		729.2.g.b.703.7		144
81.59	odd	54		729.2.g.d.217.2		144
81.65	odd	54		81.2.g.a.13.7	✓	144
81.70	even	27		729.2.g.a.514.7		144
81.76	even	27		243.2.g.a.73.2		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.13.7	✓	144	81.65	odd	54
81.2.g.a.25.7	yes	144	81.5	odd	54
243.2.g.a.10.2		144	81.16	even	27
243.2.g.a.73.2		144	81.76	even	27
729.2.g.a.217.7		144	81.22	even	27
729.2.g.a.514.7		144	81.70	even	27
729.2.g.b.28.7		144	81.43	even	27
729.2.g.b.703.7		144	81.49	even	27
729.2.g.c.28.2		144	81.38	odd	54
729.2.g.c.703.2		144	81.32	odd	54
729.2.g.d.217.2		144	81.59	odd	54
729.2.g.d.514.2		144	81.11	odd	54
6561.2.a.c.1.56		72	3.2	odd	2
6561.2.a.d.1.17		72	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.54468 q^{2} +0.386029 q^{4} +0.508923 q^{5} +3.98845 q^{7} +2.49307 q^{8} +O(q^{10})\)	\(q-1.54468 q^{2} +0.386029 q^{4} +0.508923 q^{5} +3.98845 q^{7} +2.49307 q^{8} -0.786122 q^{10} +3.82857 q^{11} +3.11326 q^{13} -6.16087 q^{14} -4.62304 q^{16} -1.50186 q^{17} -6.70690 q^{19} +0.196459 q^{20} -5.91391 q^{22} +1.18646 q^{23} -4.74100 q^{25} -4.80898 q^{26} +1.53966 q^{28} -6.91545 q^{29} -0.0463529 q^{31} +2.15498 q^{32} +2.31990 q^{34} +2.02982 q^{35} -3.05341 q^{37} +10.3600 q^{38} +1.26878 q^{40} +5.94130 q^{41} +0.238190 q^{43} +1.47794 q^{44} -1.83269 q^{46} +4.73760 q^{47} +8.90776 q^{49} +7.32331 q^{50} +1.20181 q^{52} +13.6235 q^{53} +1.94845 q^{55} +9.94347 q^{56} +10.6821 q^{58} +1.39679 q^{59} -0.313987 q^{61} +0.0716003 q^{62} +5.91734 q^{64} +1.58441 q^{65} +5.49248 q^{67} -0.579763 q^{68} -3.13541 q^{70} +11.7474 q^{71} +15.5606 q^{73} +4.71653 q^{74} -2.58906 q^{76} +15.2701 q^{77} -13.9968 q^{79} -2.35277 q^{80} -9.17740 q^{82} -0.702998 q^{83} -0.764334 q^{85} -0.367927 q^{86} +9.54488 q^{88} -11.5706 q^{89} +12.4171 q^{91} +0.458006 q^{92} -7.31806 q^{94} -3.41329 q^{95} -13.0100 q^{97} -13.7596 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) \) 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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