LMFDB - Embedded newform 6561.2.a.d.1.25

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6561,2,Mod(1,6561)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [72,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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			1.25
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)

$f(q)$	$=$	\(q-0.787501 q^{2} -1.37984 q^{4} +2.78580 q^{5} +4.98233 q^{7} +2.66163 q^{8} -2.19382 q^{10} -1.73708 q^{11} +2.17101 q^{13} -3.92359 q^{14} +0.663647 q^{16} +4.43144 q^{17} -2.13181 q^{19} -3.84396 q^{20} +1.36795 q^{22} +0.678102 q^{23} +2.76067 q^{25} -1.70967 q^{26} -6.87483 q^{28} +5.75413 q^{29} -5.88063 q^{31} -5.84588 q^{32} -3.48977 q^{34} +13.8798 q^{35} +0.759084 q^{37} +1.67880 q^{38} +7.41476 q^{40} -0.123466 q^{41} +2.54638 q^{43} +2.39689 q^{44} -0.534006 q^{46} -1.69849 q^{47} +17.8236 q^{49} -2.17403 q^{50} -2.99564 q^{52} -10.0438 q^{53} -4.83914 q^{55} +13.2611 q^{56} -4.53138 q^{58} +11.6301 q^{59} +7.92007 q^{61} +4.63100 q^{62} +3.27635 q^{64} +6.04798 q^{65} -0.465429 q^{67} -6.11469 q^{68} -10.9303 q^{70} +12.1942 q^{71} +2.14069 q^{73} -0.597780 q^{74} +2.94156 q^{76} -8.65469 q^{77} +7.01317 q^{79} +1.84879 q^{80} +0.0972297 q^{82} +3.30991 q^{83} +12.3451 q^{85} -2.00527 q^{86} -4.62345 q^{88} -5.02485 q^{89} +10.8167 q^{91} -0.935674 q^{92} +1.33756 q^{94} -5.93879 q^{95} -7.50723 q^{97} -14.0361 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.787501	−0.556847	−0.278424	−	0.960458i	$-0.589812\pi$
$2$	−0.787501	−0.556847	−0.278424	+	0.960458i	$0.589812\pi$
$3$	0	0
$3$	0	0
$4$	−1.37984	−0.689921
$5$	2.78580	1.24585	0.622923	−	0.782283i	$-0.285945\pi$
$5$	2.78580	1.24585	0.622923	+	0.782283i	$0.285945\pi$
$6$	0	0
$7$	4.98233	1.88314	0.941572	−	0.336812i	$-0.109349\pi$
$7$	4.98233	1.88314	0.941572	+	0.336812i	$0.109349\pi$
$8$	2.66163	0.941028
$9$	0	0
$10$	−2.19382	−0.693747
$11$	−1.73708	−0.523748	−0.261874	−	0.965102i	$-0.584341\pi$
$11$	−1.73708	−0.523748	−0.261874	+	0.965102i	$0.584341\pi$
$12$	0	0
$13$	2.17101	0.602128	0.301064	−	0.953604i	$-0.402658\pi$
$13$	2.17101	0.602128	0.301064	+	0.953604i	$0.402658\pi$
$14$	−3.92359	−1.04862
$15$	0	0
$16$	0.663647	0.165912
$17$	4.43144	1.07478	0.537391	−	0.843333i	$-0.319410\pi$
$17$	4.43144	1.07478	0.537391	+	0.843333i	$0.319410\pi$
$18$	0	0
$19$	−2.13181	−0.489071	−0.244535	−	0.969640i	$-0.578635\pi$
$19$	−2.13181	−0.489071	−0.244535	+	0.969640i	$0.578635\pi$
$20$	−3.84396	−0.859536
$21$	0	0
$22$	1.36795	0.291648
$23$	0.678102	0.141394	0.0706970	−	0.997498i	$-0.477478\pi$
$23$	0.678102	0.141394	0.0706970	+	0.997498i	$0.477478\pi$
$24$	0	0
$25$	2.76067	0.552134
$26$	−1.70967	−0.335294
$27$	0	0
$28$	−6.87483	−1.29922
$29$	5.75413	1.06851	0.534257	−	0.845322i	$-0.320591\pi$
$29$	5.75413	1.06851	0.534257	+	0.845322i	$0.320591\pi$
$30$	0	0
$31$	−5.88063	−1.05619	−0.528096	−	0.849185i	$-0.677094\pi$
$31$	−5.88063	−1.05619	−0.528096	+	0.849185i	$0.677094\pi$
$32$	−5.84588	−1.03342
$33$	0	0
$34$	−3.48977	−0.598490
$35$	13.8798	2.34611
$36$	0	0
$37$	0.759084	0.124793	0.0623964	−	0.998051i	$-0.480126\pi$
$37$	0.759084	0.124793	0.0623964	+	0.998051i	$0.480126\pi$
$38$	1.67880	0.272338
$39$	0	0
$40$	7.41476	1.17238
$41$	−0.123466	−0.0192822	−0.00964108	−	0.999954i	$-0.503069\pi$
$41$	−0.123466	−0.0192822	−0.00964108	+	0.999954i	$0.503069\pi$
$42$	0	0
$43$	2.54638	0.388319	0.194159	−	0.980970i	$-0.437802\pi$
$43$	2.54638	0.388319	0.194159	+	0.980970i	$0.437802\pi$
$44$	2.39689	0.361345
$45$	0	0
$46$	−0.534006	−0.0787349
$47$	−1.69849	−0.247750	−0.123875	−	0.992298i	$-0.539532\pi$
$47$	−1.69849	−0.247750	−0.123875	+	0.992298i	$0.539532\pi$
$48$	0	0
$49$	17.8236	2.54623
$50$	−2.17403	−0.307454
$51$	0	0
$52$	−2.99564	−0.415421
$53$	−10.0438	−1.37963	−0.689814	−	0.723987i	$-0.742308\pi$
$53$	−10.0438	−1.37963	−0.689814	+	0.723987i	$0.742308\pi$
$54$	0	0
$55$	−4.83914	−0.652510
$56$	13.2611	1.77209
$57$	0	0
$58$	−4.53138	−0.595000
$59$	11.6301	1.51411	0.757057	−	0.653349i	$-0.226636\pi$
$59$	11.6301	1.51411	0.757057	+	0.653349i	$0.226636\pi$
$60$	0	0
$61$	7.92007	1.01406	0.507030	−	0.861928i	$-0.330743\pi$
$61$	7.92007	1.01406	0.507030	+	0.861928i	$0.330743\pi$
$62$	4.63100	0.588138
$63$	0	0
$64$	3.27635	0.409543
$65$	6.04798	0.750160
$66$	0	0
$67$	−0.465429	−0.0568612	−0.0284306	−	0.999596i	$-0.509051\pi$
$67$	−0.465429	−0.0568612	−0.0284306	+	0.999596i	$0.509051\pi$
$68$	−6.11469	−0.741515
$69$	0	0
$70$	−10.9303	−1.30642
$71$	12.1942	1.44719	0.723594	−	0.690226i	$-0.242489\pi$
$71$	12.1942	1.44719	0.723594	+	0.690226i	$0.242489\pi$
$72$	0	0
$73$	2.14069	0.250549	0.125275	−	0.992122i	$-0.460019\pi$
$73$	2.14069	0.250549	0.125275	+	0.992122i	$0.460019\pi$
$74$	−0.597780	−0.0694905
$75$	0	0
$76$	2.94156	0.337420
$77$	−8.65469	−0.986293
$78$	0	0
$79$	7.01317	0.789043	0.394521	−	0.918887i	$-0.370910\pi$
$79$	7.01317	0.789043	0.394521	+	0.918887i	$0.370910\pi$
$80$	1.84879	0.206701
$81$	0	0
$82$	0.0972297	0.0107372
$83$	3.30991	0.363309	0.181655	−	0.983362i	$-0.441855\pi$
$83$	3.30991	0.363309	0.181655	+	0.983362i	$0.441855\pi$
$84$	0	0
$85$	12.3451	1.33901
$86$	−2.00527	−0.216234
$87$	0	0
$88$	−4.62345	−0.492862
$89$	−5.02485	−0.532634	−0.266317	−	0.963886i	$-0.585807\pi$
$89$	−5.02485	−0.532634	−0.266317	+	0.963886i	$0.585807\pi$
$90$	0	0
$91$	10.8167	1.13389
$92$	−0.935674	−0.0975507
$93$	0	0
$94$	1.33756	0.137959
$95$	−5.93879	−0.609307
$96$	0	0
$97$	−7.50723	−0.762244	−0.381122	−	0.924525i	$-0.624462\pi$
$97$	−7.50723	−0.762244	−0.381122	+	0.924525i	$0.624462\pi$
$98$	−14.0361	−1.41786
$99$	0	0
$100$	−3.80929	−0.380929
$101$	−2.25979	−0.224858	−0.112429	−	0.993660i	$-0.535863\pi$
$101$	−2.25979	−0.224858	−0.112429	+	0.993660i	$0.535863\pi$
$102$	0	0
$103$	−0.267831	−0.0263902	−0.0131951	−	0.999913i	$-0.504200\pi$
$103$	−0.267831	−0.0263902	−0.0131951	+	0.999913i	$0.504200\pi$
$104$	5.77841	0.566620
$105$	0	0
$106$	7.90954	0.768242
$107$	9.95975	0.962845	0.481423	−	0.876489i	$-0.340120\pi$
$107$	9.95975	0.962845	0.481423	+	0.876489i	$0.340120\pi$
$108$	0	0
$109$	−13.4145	−1.28488	−0.642438	−	0.766338i	$-0.722077\pi$
$109$	−13.4145	−1.28488	−0.642438	+	0.766338i	$0.722077\pi$
$110$	3.81083	0.363349
$111$	0	0
$112$	3.30651	0.312436
$113$	−0.401110	−0.0377332	−0.0188666	−	0.999822i	$-0.506006\pi$
$113$	−0.401110	−0.0377332	−0.0188666	+	0.999822i	$0.506006\pi$
$114$	0	0
$115$	1.88906	0.176155
$116$	−7.93979	−0.737191
$117$	0	0
$118$	−9.15874	−0.843131
$119$	22.0789	2.02397
$120$	0	0
$121$	−7.98257	−0.725688
$122$	−6.23706	−0.564677
$123$	0	0
$124$	8.11434	0.728689
$125$	−6.23832	−0.557972
$126$	0	0
$127$	−10.4504	−0.927327	−0.463664	−	0.886011i	$-0.653465\pi$
$127$	−10.4504	−0.927327	−0.463664	+	0.886011i	$0.653465\pi$
$128$	9.11164	0.805363
$129$	0	0
$130$	−4.76279	−0.417725
$131$	−13.4508	−1.17520	−0.587602	−	0.809150i	$-0.699928\pi$
$131$	−13.4508	−1.17520	−0.587602	+	0.809150i	$0.699928\pi$
$132$	0	0
$133$	−10.6214	−0.920990
$134$	0.366526	0.0316630
$135$	0	0
$136$	11.7949	1.01140
$137$	14.2986	1.22161	0.610807	−	0.791779i	$-0.290845\pi$
$137$	14.2986	1.22161	0.610807	+	0.791779i	$0.290845\pi$
$138$	0	0
$139$	−0.323818	−0.0274659	−0.0137329	−	0.999906i	$-0.504371\pi$
$139$	−0.323818	−0.0274659	−0.0137329	+	0.999906i	$0.504371\pi$
$140$	−19.1519	−1.61863
$141$	0	0
$142$	−9.60297	−0.805863
$143$	−3.77120	−0.315364
$144$	0	0
$145$	16.0298	1.33121
$146$	−1.68580	−0.139518
$147$	0	0
$148$	−1.04742	−0.0860971
$149$	3.07797	0.252157	0.126079	−	0.992020i	$-0.459761\pi$
$149$	3.07797	0.252157	0.126079	+	0.992020i	$0.459761\pi$
$150$	0	0
$151$	12.5260	1.01935	0.509677	−	0.860366i	$-0.329765\pi$
$151$	12.5260	1.01935	0.509677	+	0.860366i	$0.329765\pi$
$152$	−5.67409	−0.460229
$153$	0	0
$154$	6.81558	0.549215
$155$	−16.3822	−1.31585
$156$	0	0
$157$	5.82121	0.464583	0.232291	−	0.972646i	$-0.425378\pi$
$157$	5.82121	0.464583	0.232291	+	0.972646i	$0.425378\pi$
$158$	−5.52288	−0.439377
$159$	0	0
$160$	−16.2854	−1.28748
$161$	3.37853	0.266265
$162$	0	0
$163$	−6.75084	−0.528767	−0.264383	−	0.964418i	$-0.585168\pi$
$163$	−6.75084	−0.528767	−0.264383	+	0.964418i	$0.585168\pi$
$164$	0.170364	0.0133032
$165$	0	0
$166$	−2.60655	−0.202308
$167$	17.3882	1.34554	0.672770	−	0.739851i	$-0.265104\pi$
$167$	17.3882	1.34554	0.672770	+	0.739851i	$0.265104\pi$
$168$	0	0
$169$	−8.28674	−0.637441
$170$	−9.72178	−0.745627
$171$	0	0
$172$	−3.51359	−0.267909
$173$	13.7451	1.04502	0.522509	−	0.852634i	$-0.324996\pi$
$173$	13.7451	1.04502	0.522509	+	0.852634i	$0.324996\pi$
$174$	0	0
$175$	13.7546	1.03975
$176$	−1.15280	−0.0868959
$177$	0	0
$178$	3.95708	0.296596
$179$	−2.16767	−0.162019	−0.0810095	−	0.996713i	$-0.525814\pi$
$179$	−2.16767	−0.162019	−0.0810095	+	0.996713i	$0.525814\pi$
$180$	0	0
$181$	−23.1282	−1.71911	−0.859554	−	0.511045i	$-0.829259\pi$
$181$	−23.1282	−1.71911	−0.859554	+	0.511045i	$0.829259\pi$
$182$	−8.51814	−0.631406
$183$	0	0
$184$	1.80486	0.133056
$185$	2.11466	0.155473
$186$	0	0
$187$	−7.69775	−0.562915
$188$	2.34365	0.170928
$189$	0	0
$190$	4.67681	0.339291
$191$	21.8871	1.58370	0.791849	−	0.610717i	$-0.209119\pi$
$191$	21.8871	1.58370	0.791849	+	0.610717i	$0.209119\pi$
$192$	0	0
$193$	−7.21462	−0.519320	−0.259660	−	0.965700i	$-0.583610\pi$
$193$	−7.21462	−0.519320	−0.259660	+	0.965700i	$0.583610\pi$
$194$	5.91196	0.424454
$195$	0	0
$196$	−24.5938	−1.75670
$197$	−16.4030	−1.16866	−0.584332	−	0.811515i	$-0.698643\pi$
$197$	−16.4030	−1.16866	−0.584332	+	0.811515i	$0.698643\pi$
$198$	0	0
$199$	−8.06845	−0.571957	−0.285979	−	0.958236i	$-0.592319\pi$
$199$	−8.06845	−0.571957	−0.285979	+	0.958236i	$0.592319\pi$
$200$	7.34788	0.519574
$201$	0	0
$202$	1.77959	0.125211
$203$	28.6690	2.01217
$204$	0	0
$205$	−0.343952	−0.0240226
$206$	0.210918	0.0146953
$207$	0	0
$208$	1.44078	0.0999001
$209$	3.70312	0.256150
$210$	0	0
$211$	−23.9225	−1.64690	−0.823448	−	0.567392i	$-0.807952\pi$
$211$	−23.9225	−1.64690	−0.823448	+	0.567392i	$0.807952\pi$
$212$	13.8589	0.951834
$213$	0	0
$214$	−7.84331	−0.536158
$215$	7.09369	0.483785
$216$	0	0
$217$	−29.2992	−1.98896
$218$	10.5639	0.715480
$219$	0	0
$220$	6.67725	0.450180
$221$	9.62068	0.647157
$222$	0	0
$223$	11.9658	0.801286	0.400643	−	0.916234i	$-0.368787\pi$
$223$	11.9658	0.801286	0.400643	+	0.916234i	$0.368787\pi$
$224$	−29.1261	−1.94607
$225$	0	0
$226$	0.315874	0.0210117
$227$	25.1129	1.66680	0.833400	−	0.552671i	$-0.186391\pi$
$227$	25.1129	1.66680	0.833400	+	0.552671i	$0.186391\pi$
$228$	0	0
$229$	−25.7737	−1.70317	−0.851587	−	0.524214i	$-0.824359\pi$
$229$	−25.7737	−1.70317	−0.851587	+	0.524214i	$0.824359\pi$
$230$	−1.48763	−0.0980917
$231$	0	0
$232$	15.3154	1.00550
$233$	−9.13341	−0.598350	−0.299175	−	0.954198i	$-0.596711\pi$
$233$	−9.13341	−0.598350	−0.299175	+	0.954198i	$0.596711\pi$
$234$	0	0
$235$	−4.73165	−0.308659
$236$	−16.0477	−1.04462
$237$	0	0
$238$	−17.3872	−1.12704
$239$	21.7595	1.40751	0.703754	−	0.710444i	$-0.251506\pi$
$239$	21.7595	1.40751	0.703754	+	0.710444i	$0.251506\pi$
$240$	0	0
$241$	13.8647	0.893106	0.446553	−	0.894757i	$-0.352651\pi$
$241$	13.8647	0.893106	0.446553	+	0.894757i	$0.352651\pi$
$242$	6.28628	0.404097
$243$	0	0
$244$	−10.9284	−0.699622
$245$	49.6530	3.17221
$246$	0	0
$247$	−4.62817	−0.294483
$248$	−15.6521	−0.993907
$249$	0	0
$250$	4.91269	0.310705
$251$	−10.6570	−0.672665	−0.336333	−	0.941743i	$-0.609187\pi$
$251$	−10.6570	−0.672665	−0.336333	+	0.941743i	$0.609187\pi$
$252$	0	0
$253$	−1.17792	−0.0740549
$254$	8.22974	0.516380
$255$	0	0
$256$	−13.7281	−0.858007
$257$	1.17554	0.0733281	0.0366640	−	0.999328i	$-0.488327\pi$
$257$	1.17554	0.0733281	0.0366640	+	0.999328i	$0.488327\pi$
$258$	0	0
$259$	3.78201	0.235003
$260$	−8.34526	−0.517551
$261$	0	0
$262$	10.5925	0.654409
$263$	−9.33281	−0.575486	−0.287743	−	0.957708i	$-0.592905\pi$
$263$	−9.33281	−0.575486	−0.287743	+	0.957708i	$0.592905\pi$
$264$	0	0
$265$	−27.9801	−1.71880
$266$	8.36435	0.512851
$267$	0	0
$268$	0.642218	0.0392297
$269$	−2.50232	−0.152569	−0.0762845	−	0.997086i	$-0.524306\pi$
$269$	−2.50232	−0.152569	−0.0762845	+	0.997086i	$0.524306\pi$
$270$	0	0
$271$	5.52487	0.335612	0.167806	−	0.985820i	$-0.446332\pi$
$271$	5.52487	0.335612	0.167806	+	0.985820i	$0.446332\pi$
$272$	2.94091	0.178319
$273$	0	0
$274$	−11.2602	−0.680253
$275$	−4.79549	−0.289179
$276$	0	0
$277$	26.7133	1.60505	0.802524	−	0.596620i	$-0.203490\pi$
$277$	26.7133	1.60505	0.802524	+	0.596620i	$0.203490\pi$
$278$	0.255007	0.0152943
$279$	0	0
$280$	36.9428	2.20775
$281$	−31.4811	−1.87800	−0.939001	−	0.343914i	$-0.888247\pi$
$281$	−31.4811	−1.87800	−0.939001	+	0.343914i	$0.888247\pi$
$282$	0	0
$283$	6.69129	0.397756	0.198878	−	0.980024i	$-0.436270\pi$
$283$	6.69129	0.397756	0.198878	+	0.980024i	$0.436270\pi$
$284$	−16.8261	−0.998445
$285$	0	0
$286$	2.96983	0.175609
$287$	−0.615149	−0.0363111
$288$	0	0
$289$	2.63767	0.155157
$290$	−12.6235	−0.741278
$291$	0	0
$292$	−2.95382	−0.172859
$293$	−3.75383	−0.219301	−0.109651	−	0.993970i	$-0.534973\pi$
$293$	−3.75383	−0.219301	−0.109651	+	0.993970i	$0.534973\pi$
$294$	0	0
$295$	32.3992	1.88635
$296$	2.02040	0.117433
$297$	0	0
$298$	−2.42391	−0.140413
$299$	1.47216	0.0851374
$300$	0	0
$301$	12.6869	0.731260
$302$	−9.86427	−0.567625
$303$	0	0
$304$	−1.41477	−0.0811425
$305$	22.0637	1.26336
$306$	0	0
$307$	−29.9579	−1.70979	−0.854893	−	0.518804i	$-0.826377\pi$
$307$	−29.9579	−1.70979	−0.854893	+	0.518804i	$0.826377\pi$
$308$	11.9421	0.680464
$309$	0	0
$310$	12.9010	0.732730
$311$	−4.44826	−0.252238	−0.126119	−	0.992015i	$-0.540252\pi$
$311$	−4.44826	−0.252238	−0.126119	+	0.992015i	$0.540252\pi$
$312$	0	0
$313$	−31.2230	−1.76483	−0.882414	−	0.470473i	$-0.844083\pi$
$313$	−31.2230	−1.76483	−0.882414	+	0.470473i	$0.844083\pi$
$314$	−4.58421	−0.258702
$315$	0	0
$316$	−9.67706	−0.544377
$317$	27.8291	1.56304	0.781518	−	0.623883i	$-0.214446\pi$
$317$	27.8291	1.56304	0.781518	+	0.623883i	$0.214446\pi$
$318$	0	0
$319$	−9.99536	−0.559633
$320$	9.12724	0.510228
$321$	0	0
$322$	−2.66060	−0.148269
$323$	−9.44699	−0.525644
$324$	0	0
$325$	5.99343	0.332456
$326$	5.31630	0.294443
$327$	0	0
$328$	−0.328621	−0.0181451
$329$	−8.46244	−0.466549
$330$	0	0
$331$	0.197085	0.0108327	0.00541637	−	0.999985i	$-0.498276\pi$
$331$	0.197085	0.0108327	0.00541637	+	0.999985i	$0.498276\pi$
$332$	−4.56715	−0.250655
$333$	0	0
$334$	−13.6932	−0.749261
$335$	−1.29659	−0.0708403
$336$	0	0
$337$	−17.9046	−0.975325	−0.487662	−	0.873032i	$-0.662150\pi$
$337$	−17.9046	−0.975325	−0.487662	+	0.873032i	$0.662150\pi$
$338$	6.52582	0.354958
$339$	0	0
$340$	−17.0343	−0.923814
$341$	10.2151	0.553179
$342$	0	0
$343$	53.9268	2.91177
$344$	6.77751	0.365419
$345$	0	0
$346$	−10.8243	−0.581916
$347$	−22.5317	−1.20956	−0.604781	−	0.796392i	$-0.706739\pi$
$347$	−22.5317	−1.20956	−0.604781	+	0.796392i	$0.706739\pi$
$348$	0	0
$349$	−11.8561	−0.634641	−0.317320	−	0.948318i	$-0.602783\pi$
$349$	−11.8561	−0.634641	−0.317320	+	0.948318i	$0.602783\pi$
$350$	−10.8317	−0.578981
$351$	0	0
$352$	10.1547	0.541250
$353$	18.8041	1.00084	0.500421	−	0.865782i	$-0.333179\pi$
$353$	18.8041	1.00084	0.500421	+	0.865782i	$0.333179\pi$
$354$	0	0
$355$	33.9707	1.80297
$356$	6.93350	0.367475
$357$	0	0
$358$	1.70704	0.0902199
$359$	9.28472	0.490029	0.245014	−	0.969519i	$-0.421207\pi$
$359$	9.28472	0.490029	0.245014	+	0.969519i	$0.421207\pi$
$360$	0	0
$361$	−14.4554	−0.760810
$362$	18.2135	0.957281
$363$	0	0
$364$	−14.9253	−0.782298
$365$	5.96354	0.312146
$366$	0	0
$367$	−29.2198	−1.52526	−0.762631	−	0.646834i	$-0.776093\pi$
$367$	−29.2198	−1.52526	−0.762631	+	0.646834i	$0.776093\pi$
$368$	0.450020	0.0234589
$369$	0	0
$370$	−1.66529	−0.0865745
$371$	−50.0417	−2.59804
$372$	0	0
$373$	5.56465	0.288127	0.144063	−	0.989568i	$-0.453983\pi$
$373$	5.56465	0.288127	0.144063	+	0.989568i	$0.453983\pi$
$374$	6.06199	0.313458
$375$	0	0
$376$	−4.52075	−0.233140
$377$	12.4922	0.643383
$378$	0	0
$379$	29.5838	1.51962	0.759808	−	0.650147i	$-0.225293\pi$
$379$	29.5838	1.51962	0.759808	+	0.650147i	$0.225293\pi$
$380$	8.19459	0.420374
$381$	0	0
$382$	−17.2361	−0.881878
$383$	12.3868	0.632936	0.316468	−	0.948603i	$-0.397503\pi$
$383$	12.3868	0.632936	0.316468	+	0.948603i	$0.397503\pi$
$384$	0	0
$385$	−24.1102	−1.22877
$386$	5.68153	0.289182
$387$	0	0
$388$	10.3588	0.525888
$389$	−18.5787	−0.941979	−0.470990	−	0.882139i	$-0.656103\pi$
$389$	−18.5787	−0.941979	−0.470990	+	0.882139i	$0.656103\pi$
$390$	0	0
$391$	3.00497	0.151968
$392$	47.4399	2.39607
$393$	0	0
$394$	12.9174	0.650768
$395$	19.5373	0.983026
$396$	0	0
$397$	−10.2129	−0.512568	−0.256284	−	0.966601i	$-0.582498\pi$
$397$	−10.2129	−0.512568	−0.256284	+	0.966601i	$0.582498\pi$
$398$	6.35391	0.318493
$399$	0	0
$400$	1.83211	0.0916055
$401$	−2.01887	−0.100817	−0.0504087	−	0.998729i	$-0.516052\pi$
$401$	−2.01887	−0.100817	−0.0504087	+	0.998729i	$0.516052\pi$
$402$	0	0
$403$	−12.7669	−0.635963
$404$	3.11815	0.155134
$405$	0	0
$406$	−22.5768	−1.12047
$407$	−1.31859	−0.0653600
$408$	0	0
$409$	−0.869695	−0.0430036	−0.0215018	−	0.999769i	$-0.506845\pi$
$409$	−0.869695	−0.0430036	−0.0215018	+	0.999769i	$0.506845\pi$
$410$	0.270862	0.0133769
$411$	0	0
$412$	0.369565	0.0182072
$413$	57.9452	2.85130
$414$	0	0
$415$	9.22073	0.452628
$416$	−12.6914	−0.622249
$417$	0	0
$418$	−2.91621	−0.142636
$419$	25.2444	1.23327	0.616636	−	0.787248i	$-0.288495\pi$
$419$	25.2444	1.23327	0.616636	+	0.787248i	$0.288495\pi$
$420$	0	0
$421$	16.7527	0.816478	0.408239	−	0.912875i	$-0.366143\pi$
$421$	16.7527	0.816478	0.408239	+	0.912875i	$0.366143\pi$
$422$	18.8390	0.917070
$423$	0	0
$424$	−26.7330	−1.29827
$425$	12.2337	0.593424
$426$	0	0
$427$	39.4604	1.90962
$428$	−13.7429	−0.664287
$429$	0	0
$430$	−5.58629	−0.269395
$431$	−26.3622	−1.26982	−0.634911	−	0.772585i	$-0.718963\pi$
$431$	−26.3622	−1.26982	−0.634911	+	0.772585i	$0.718963\pi$
$432$	0	0
$433$	−12.5869	−0.604888	−0.302444	−	0.953167i	$-0.597802\pi$
$433$	−12.5869	−0.604888	−0.302444	+	0.953167i	$0.597802\pi$
$434$	23.0732	1.10755
$435$	0	0
$436$	18.5099	0.886462
$437$	−1.44558	−0.0691517
$438$	0	0
$439$	8.79873	0.419940	0.209970	−	0.977708i	$-0.432663\pi$
$439$	8.79873	0.419940	0.209970	+	0.977708i	$0.432663\pi$
$440$	−12.8800	−0.614030
$441$	0	0
$442$	−7.57630	−0.360368
$443$	15.2224	0.723240	0.361620	−	0.932326i	$-0.382224\pi$
$443$	15.2224	0.723240	0.361620	+	0.932326i	$0.382224\pi$
$444$	0	0
$445$	−13.9982	−0.663580
$446$	−9.42305	−0.446194
$447$	0	0
$448$	16.3238	0.771229
$449$	1.23690	0.0583730	0.0291865	−	0.999574i	$-0.490708\pi$
$449$	1.23690	0.0583730	0.0291865	+	0.999574i	$0.490708\pi$
$450$	0	0
$451$	0.214470	0.0100990
$452$	0.553468	0.0260329
$453$	0	0
$454$	−19.7764	−0.928153
$455$	30.1330	1.41266
$456$	0	0
$457$	30.9854	1.44943	0.724717	−	0.689046i	$-0.241970\pi$
$457$	30.9854	1.44943	0.724717	+	0.689046i	$0.241970\pi$
$458$	20.2968	0.948408
$459$	0	0
$460$	−2.60660	−0.121533
$461$	16.9415	0.789046	0.394523	−	0.918886i	$-0.370910\pi$
$461$	16.9415	0.789046	0.394523	+	0.918886i	$0.370910\pi$
$462$	0	0
$463$	35.6873	1.65853	0.829264	−	0.558857i	$-0.188760\pi$
$463$	35.6873	1.65853	0.829264	+	0.558857i	$0.188760\pi$
$464$	3.81871	0.177279
$465$	0	0
$466$	7.19258	0.333190
$467$	42.0203	1.94447	0.972233	−	0.234013i	$-0.0751859\pi$
$467$	42.0203	1.94447	0.972233	+	0.234013i	$0.0751859\pi$
$468$	0	0
$469$	−2.31892	−0.107078
$470$	3.72618	0.171876
$471$	0	0
$472$	30.9551	1.42482
$473$	−4.42325	−0.203381
$474$	0	0
$475$	−5.88522	−0.270033
$476$	−30.4654	−1.39638
$477$	0	0
$478$	−17.1357	−0.783767
$479$	−36.0910	−1.64904	−0.824520	−	0.565833i	$-0.808555\pi$
$479$	−36.0910	−1.64904	−0.824520	+	0.565833i	$0.808555\pi$
$480$	0	0
$481$	1.64798	0.0751412
$482$	−10.9185	−0.497324
$483$	0	0
$484$	11.0147	0.500667
$485$	−20.9136	−0.949639
$486$	0	0
$487$	27.8890	1.26377	0.631885	−	0.775062i	$-0.282282\pi$
$487$	27.8890	1.26377	0.631885	+	0.775062i	$0.282282\pi$
$488$	21.0803	0.954260
$489$	0	0
$490$	−39.1018	−1.76644
$491$	−21.7519	−0.981648	−0.490824	−	0.871259i	$-0.663304\pi$
$491$	−21.7519	−0.981648	−0.490824	+	0.871259i	$0.663304\pi$
$492$	0	0
$493$	25.4991	1.14842
$494$	3.64469	0.163982
$495$	0	0
$496$	−3.90266	−0.175235
$497$	60.7557	2.72526
$498$	0	0
$499$	−2.98436	−0.133598	−0.0667992	−	0.997766i	$-0.521279\pi$
$499$	−2.98436	−0.133598	−0.0667992	+	0.997766i	$0.521279\pi$
$500$	8.60789	0.384957
$501$	0	0
$502$	8.39242	0.374572
$503$	39.8441	1.77656	0.888280	−	0.459303i	$-0.151901\pi$
$503$	39.8441	1.77656	0.888280	+	0.459303i	$0.151901\pi$
$504$	0	0
$505$	−6.29532	−0.280138
$506$	0.927610	0.0412373
$507$	0	0
$508$	14.4200	0.639782
$509$	19.9278	0.883285	0.441643	−	0.897191i	$-0.354396\pi$
$509$	19.9278	0.883285	0.441643	+	0.897191i	$0.354396\pi$
$510$	0	0
$511$	10.6656	0.471820
$512$	−7.41236	−0.327583
$513$	0	0
$514$	−0.925738	−0.0408326
$515$	−0.746124	−0.0328782
$516$	0	0
$517$	2.95041	0.129759
$518$	−2.97834	−0.130861
$519$	0	0
$520$	16.0975	0.705922
$521$	35.5422	1.55713	0.778565	−	0.627563i	$-0.215948\pi$
$521$	35.5422	1.55713	0.778565	+	0.627563i	$0.215948\pi$
$522$	0	0
$523$	24.0645	1.05227	0.526133	−	0.850403i	$-0.323642\pi$
$523$	24.0645	1.05227	0.526133	+	0.850403i	$0.323642\pi$
$524$	18.5600	0.810797
$525$	0	0
$526$	7.34960	0.320458
$527$	−26.0597	−1.13518
$528$	0	0
$529$	−22.5402	−0.980008
$530$	22.0344	0.957112
$531$	0	0
$532$	14.6558	0.635410
$533$	−0.268046	−0.0116103
$534$	0	0
$535$	27.7458	1.19956
$536$	−1.23880	−0.0535080
$537$	0	0
$538$	1.97058	0.0849577
$539$	−30.9610	−1.33358
$540$	0	0
$541$	−10.6529	−0.458003	−0.229002	−	0.973426i	$-0.573546\pi$
$541$	−10.6529	−0.458003	−0.229002	+	0.973426i	$0.573546\pi$
$542$	−4.35084	−0.186885
$543$	0	0
$544$	−25.9057	−1.11070
$545$	−37.3701	−1.60076
$546$	0	0
$547$	13.6003	0.581505	0.290752	−	0.956798i	$-0.406094\pi$
$547$	13.6003	0.581505	0.290752	+	0.956798i	$0.406094\pi$
$548$	−19.7299	−0.842818
$549$	0	0
$550$	3.77646	0.161029
$551$	−12.2667	−0.522579
$552$	0	0
$553$	34.9419	1.48588
$554$	−21.0368	−0.893766
$555$	0	0
$556$	0.446818	0.0189493
$557$	4.59040	0.194502	0.0972508	−	0.995260i	$-0.468995\pi$
$557$	4.59040	0.194502	0.0972508	+	0.995260i	$0.468995\pi$
$558$	0	0
$559$	5.52819	0.233818
$560$	9.21126	0.389247
$561$	0	0
$562$	24.7914	1.04576
$563$	−0.991100	−0.0417699	−0.0208849	−	0.999782i	$-0.506648\pi$
$563$	−0.991100	−0.0417699	−0.0208849	+	0.999782i	$0.506648\pi$
$564$	0	0
$565$	−1.11741	−0.0470098
$566$	−5.26940	−0.221489
$567$	0	0
$568$	32.4565	1.36185
$569$	7.76921	0.325702	0.162851	−	0.986651i	$-0.447931\pi$
$569$	7.76921	0.325702	0.162851	+	0.986651i	$0.447931\pi$
$570$	0	0
$571$	21.3973	0.895449	0.447724	−	0.894172i	$-0.352235\pi$
$571$	21.3973	0.895449	0.447724	+	0.894172i	$0.352235\pi$
$572$	5.20366	0.217576
$573$	0	0
$574$	0.484430	0.0202197
$575$	1.87202	0.0780685
$576$	0	0
$577$	8.66161	0.360587	0.180294	−	0.983613i	$-0.442295\pi$
$577$	8.66161	0.360587	0.180294	+	0.983613i	$0.442295\pi$
$578$	−2.07717	−0.0863987
$579$	0	0
$580$	−22.1186	−0.918426
$581$	16.4910	0.684164
$582$	0	0
$583$	17.4469	0.722578
$584$	5.69773	0.235774
$585$	0	0
$586$	2.95614	0.122117
$587$	−10.5137	−0.433947	−0.216973	−	0.976178i	$-0.569618\pi$
$587$	−10.5137	−0.433947	−0.216973	+	0.976178i	$0.569618\pi$
$588$	0	0
$589$	12.5364	0.516553
$590$	−25.5144	−1.05041
$591$	0	0
$592$	0.503764	0.0207046
$593$	28.0350	1.15126	0.575630	−	0.817710i	$-0.304757\pi$
$593$	28.0350	1.15126	0.575630	+	0.817710i	$0.304757\pi$
$594$	0	0
$595$	61.5074	2.52156
$596$	−4.24711	−0.173969
$597$	0	0
$598$	−1.15933	−0.0474086
$599$	23.2712	0.950837	0.475419	−	0.879760i	$-0.342297\pi$
$599$	23.2712	0.950837	0.475419	+	0.879760i	$0.342297\pi$
$600$	0	0
$601$	21.4911	0.876639	0.438319	−	0.898819i	$-0.355574\pi$
$601$	21.4911	0.876639	0.438319	+	0.898819i	$0.355574\pi$
$602$	−9.99094	−0.407200
$603$	0	0
$604$	−17.2839	−0.703274
$605$	−22.2378	−0.904096
$606$	0	0
$607$	4.84398	0.196611	0.0983056	−	0.995156i	$-0.468658\pi$
$607$	4.84398	0.196611	0.0983056	+	0.995156i	$0.468658\pi$
$608$	12.4623	0.505413
$609$	0	0
$610$	−17.3752	−0.703501
$611$	−3.68743	−0.149178
$612$	0	0
$613$	13.9582	0.563766	0.281883	−	0.959449i	$-0.409041\pi$
$613$	13.9582	0.563766	0.281883	+	0.959449i	$0.409041\pi$
$614$	23.5919	0.952090
$615$	0	0
$616$	−23.0356	−0.928130
$617$	−20.2103	−0.813636	−0.406818	−	0.913509i	$-0.633362\pi$
$617$	−20.2103	−0.813636	−0.406818	+	0.913509i	$0.633362\pi$
$618$	0	0
$619$	−28.8625	−1.16008	−0.580041	−	0.814587i	$-0.696964\pi$
$619$	−28.8625	−1.16008	−0.580041	+	0.814587i	$0.696964\pi$
$620$	22.6049	0.907835
$621$	0	0
$622$	3.50301	0.140458
$623$	−25.0355	−1.00303
$624$	0	0
$625$	−31.1821	−1.24728
$626$	24.5882	0.982740
$627$	0	0
$628$	−8.03234	−0.320525
$629$	3.36384	0.134125
$630$	0	0
$631$	−46.1919	−1.83887	−0.919436	−	0.393241i	$-0.871354\pi$
$631$	−46.1919	−1.83887	−0.919436	+	0.393241i	$0.871354\pi$
$632$	18.6665	0.742512
$633$	0	0
$634$	−21.9154	−0.870372
$635$	−29.1128	−1.15531
$636$	0	0
$637$	38.6952	1.53316
$638$	7.87136	0.311630
$639$	0	0
$640$	25.3832	1.00336
$641$	24.8921	0.983178	0.491589	−	0.870827i	$-0.336416\pi$
$641$	24.8921	0.983178	0.491589	+	0.870827i	$0.336416\pi$
$642$	0	0
$643$	2.21419	0.0873190	0.0436595	−	0.999046i	$-0.486098\pi$
$643$	2.21419	0.0873190	0.0436595	+	0.999046i	$0.486098\pi$
$644$	−4.66184	−0.183702
$645$	0	0
$646$	7.43951	0.292704
$647$	−26.5378	−1.04331	−0.521654	−	0.853157i	$-0.674685\pi$
$647$	−26.5378	−1.04331	−0.521654	+	0.853157i	$0.674685\pi$
$648$	0	0
$649$	−20.2024	−0.793015
$650$	−4.71983	−0.185127
$651$	0	0
$652$	9.31510	0.364807
$653$	−0.212239	−0.00830557	−0.00415278	−	0.999991i	$-0.501322\pi$
$653$	−0.212239	−0.00830557	−0.00415278	+	0.999991i	$0.501322\pi$
$654$	0	0
$655$	−37.4713	−1.46412
$656$	−0.0819379	−0.00319914
$657$	0	0
$658$	6.66418	0.259797
$659$	27.0749	1.05469	0.527345	−	0.849651i	$-0.323188\pi$
$659$	27.0749	1.05469	0.527345	+	0.849651i	$0.323188\pi$
$660$	0	0
$661$	35.7776	1.39159	0.695793	−	0.718242i	$-0.255053\pi$
$661$	35.7776	1.39159	0.695793	+	0.718242i	$0.255053\pi$
$662$	−0.155204	−0.00603219
$663$	0	0
$664$	8.80974	0.341884
$665$	−29.5890	−1.14741
$666$	0	0
$667$	3.90189	0.151082
$668$	−23.9930	−0.928316
$669$	0	0
$670$	1.02107	0.0394472
$671$	−13.7578	−0.531112
$672$	0	0
$673$	−28.5524	−1.10061	−0.550306	−	0.834963i	$-0.685489\pi$
$673$	−28.5524	−1.10061	−0.550306	+	0.834963i	$0.685489\pi$
$674$	14.0999	0.543107
$675$	0	0
$676$	11.4344	0.439784
$677$	−14.7249	−0.565923	−0.282962	−	0.959131i	$-0.591317\pi$
$677$	−14.7249	−0.565923	−0.282962	+	0.959131i	$0.591317\pi$
$678$	0	0
$679$	−37.4035	−1.43542
$680$	32.8581	1.26005
$681$	0	0
$682$	−8.04440	−0.308036
$683$	−4.40051	−0.168381	−0.0841903	−	0.996450i	$-0.526830\pi$
$683$	−4.40051	−0.168381	−0.0841903	+	0.996450i	$0.526830\pi$
$684$	0	0
$685$	39.8331	1.52195
$686$	−42.4674	−1.62141
$687$	0	0
$688$	1.68989	0.0644266
$689$	−21.8052	−0.830713
$690$	0	0
$691$	3.02847	0.115208	0.0576042	−	0.998339i	$-0.481654\pi$
$691$	3.02847	0.115208	0.0576042	+	0.998339i	$0.481654\pi$
$692$	−18.9660	−0.720980
$693$	0	0
$694$	17.7437	0.673542
$695$	−0.902092	−0.0342183
$696$	0	0
$697$	−0.547133	−0.0207241
$698$	9.33667	0.353398
$699$	0	0
$700$	−18.9791	−0.717344
$701$	−43.2294	−1.63275	−0.816377	−	0.577520i	$-0.804021\pi$
$701$	−43.2294	−1.63275	−0.816377	+	0.577520i	$0.804021\pi$
$702$	0	0
$703$	−1.61822	−0.0610325
$704$	−5.69126	−0.214498
$705$	0	0
$706$	−14.8083	−0.557316
$707$	−11.2590	−0.423439
$708$	0	0
$709$	−32.1764	−1.20841	−0.604205	−	0.796829i	$-0.706509\pi$
$709$	−32.1764	−1.20841	−0.604205	+	0.796829i	$0.706509\pi$
$710$	−26.7519	−1.00398
$711$	0	0
$712$	−13.3743	−0.501223
$713$	−3.98767	−0.149339
$714$	0	0
$715$	−10.5058	−0.392895
$716$	2.99104	0.111780
$717$	0	0
$718$	−7.31173	−0.272871
$719$	−10.1586	−0.378853	−0.189427	−	0.981895i	$-0.560663\pi$
$719$	−10.1586	−0.378853	−0.189427	+	0.981895i	$0.560663\pi$
$720$	0	0
$721$	−1.33442	−0.0496966
$722$	11.3836	0.423655
$723$	0	0
$724$	31.9133	1.18605
$725$	15.8852	0.589963
$726$	0	0
$727$	14.0946	0.522740	0.261370	−	0.965239i	$-0.415826\pi$
$727$	14.0946	0.522740	0.261370	+	0.965239i	$0.415826\pi$
$728$	28.7900	1.06703
$729$	0	0
$730$	−4.69629	−0.173818
$731$	11.2841	0.417358
$732$	0	0
$733$	−15.6983	−0.579830	−0.289915	−	0.957052i	$-0.593627\pi$
$733$	−15.6983	−0.579830	−0.289915	+	0.957052i	$0.593627\pi$
$734$	23.0106	0.849338
$735$	0	0
$736$	−3.96411	−0.146119
$737$	0.808485	0.0297809
$738$	0	0
$739$	3.63976	0.133891	0.0669453	−	0.997757i	$-0.478675\pi$
$739$	3.63976	0.133891	0.0669453	+	0.997757i	$0.478675\pi$
$740$	−2.91789	−0.107264
$741$	0	0
$742$	39.4079	1.44671
$743$	−23.2813	−0.854109	−0.427055	−	0.904226i	$-0.640449\pi$
$743$	−23.2813	−0.854109	−0.427055	+	0.904226i	$0.640449\pi$
$744$	0	0
$745$	8.57461	0.314149
$746$	−4.38217	−0.160443
$747$	0	0
$748$	10.6217	0.388367
$749$	49.6228	1.81318
$750$	0	0
$751$	7.89915	0.288244	0.144122	−	0.989560i	$-0.453964\pi$
$751$	7.89915	0.288244	0.144122	+	0.989560i	$0.453964\pi$
$752$	−1.12720	−0.0411047
$753$	0	0
$754$	−9.83766	−0.358266
$755$	34.8950	1.26996
$756$	0	0
$757$	4.24148	0.154159	0.0770795	−	0.997025i	$-0.475440\pi$
$757$	4.24148	0.154159	0.0770795	+	0.997025i	$0.475440\pi$
$758$	−23.2973	−0.846195
$759$	0	0
$760$	−15.8069	−0.573375
$761$	−40.8881	−1.48219	−0.741096	−	0.671399i	$-0.765694\pi$
$761$	−40.8881	−1.48219	−0.741096	+	0.671399i	$0.765694\pi$
$762$	0	0
$763$	−66.8354	−2.41961
$764$	−30.2008	−1.09263
$765$	0	0
$766$	−9.75461	−0.352449
$767$	25.2491	0.911691
$768$	0	0
$769$	32.6954	1.17903	0.589514	−	0.807758i	$-0.299319\pi$
$769$	32.6954	1.17903	0.589514	+	0.807758i	$0.299319\pi$
$770$	18.9868	0.684238
$771$	0	0
$772$	9.95504	0.358290
$773$	38.8600	1.39770	0.698848	−	0.715270i	$-0.253696\pi$
$773$	38.8600	1.39770	0.698848	+	0.715270i	$0.253696\pi$
$774$	0	0
$775$	−16.2345	−0.583160
$776$	−19.9815	−0.717293
$777$	0	0
$778$	14.6308	0.524539
$779$	0.263206	0.00943034
$780$	0	0
$781$	−21.1823	−0.757962
$782$	−2.36642	−0.0846229
$783$	0	0
$784$	11.8286	0.422449
$785$	16.2167	0.578799
$786$	0	0
$787$	2.28115	0.0813141	0.0406570	−	0.999173i	$-0.487055\pi$
$787$	2.28115	0.0813141	0.0406570	+	0.999173i	$0.487055\pi$
$788$	22.6335	0.806286
$789$	0	0
$790$	−15.3856	−0.547396
$791$	−1.99846	−0.0710571
$792$	0	0
$793$	17.1945	0.610595
$794$	8.04263	0.285422
$795$	0	0
$796$	11.1332	0.394605
$797$	−34.2053	−1.21161	−0.605807	−	0.795612i	$-0.707150\pi$
$797$	−34.2053	−1.21161	−0.605807	+	0.795612i	$0.707150\pi$
$798$	0	0
$799$	−7.52676	−0.266278
$800$	−16.1386	−0.570584
$801$	0	0
$802$	1.58986	0.0561399
$803$	−3.71855	−0.131225
$804$	0	0
$805$	9.41190	0.331726
$806$	10.0539	0.354135
$807$	0	0
$808$	−6.01472	−0.211597
$809$	22.5844	0.794027	0.397013	−	0.917813i	$-0.370047\pi$
$809$	22.5844	0.794027	0.397013	+	0.917813i	$0.370047\pi$
$810$	0	0
$811$	15.6809	0.550631	0.275315	−	0.961354i	$-0.411218\pi$
$811$	15.6809	0.550631	0.275315	+	0.961354i	$0.411218\pi$
$812$	−39.5586	−1.38824
$813$	0	0
$814$	1.03839	0.0363955
$815$	−18.8065	−0.658763
$816$	0	0
$817$	−5.42839	−0.189915
$818$	0.684886	0.0239465
$819$	0	0
$820$	0.474599	0.0165737
$821$	−51.1210	−1.78413	−0.892067	−	0.451902i	$-0.850746\pi$
$821$	−51.1210	−1.78413	−0.892067	+	0.451902i	$0.850746\pi$
$822$	0	0
$823$	3.29755	0.114945	0.0574727	−	0.998347i	$-0.481696\pi$
$823$	3.29755	0.114945	0.0574727	+	0.998347i	$0.481696\pi$
$824$	−0.712868	−0.0248339
$825$	0	0
$826$	−45.6319	−1.58774
$827$	−20.5907	−0.716008	−0.358004	−	0.933720i	$-0.616543\pi$
$827$	−20.5907	−0.716008	−0.358004	+	0.933720i	$0.616543\pi$
$828$	0	0
$829$	13.2264	0.459370	0.229685	−	0.973265i	$-0.426230\pi$
$829$	13.2264	0.459370	0.229685	+	0.973265i	$0.426230\pi$
$830$	−7.26133	−0.252045
$831$	0	0
$832$	7.11296	0.246598
$833$	78.9843	2.73664
$834$	0	0
$835$	48.4401	1.67634
$836$	−5.10971	−0.176723
$837$	0	0
$838$	−19.8800	−0.686744
$839$	−6.30912	−0.217815	−0.108908	−	0.994052i	$-0.534735\pi$
$839$	−6.30912	−0.217815	−0.108908	+	0.994052i	$0.534735\pi$
$840$	0	0
$841$	4.10999	0.141724
$842$	−13.1928	−0.454654
$843$	0	0
$844$	33.0093	1.13623
$845$	−23.0852	−0.794154
$846$	0	0
$847$	−39.7718	−1.36657
$848$	−6.66556	−0.228896
$849$	0	0
$850$	−9.63409	−0.330447
$851$	0.514737	0.0176450
$852$	0	0
$853$	18.1238	0.620546	0.310273	−	0.950647i	$-0.399579\pi$
$853$	18.1238	0.620546	0.310273	+	0.950647i	$0.399579\pi$
$854$	−31.0751	−1.06337
$855$	0	0
$856$	26.5092	0.906064
$857$	34.0073	1.16167	0.580834	−	0.814022i	$-0.302726\pi$
$857$	34.0073	1.16167	0.580834	+	0.814022i	$0.302726\pi$
$858$	0	0
$859$	47.5959	1.62395	0.811976	−	0.583691i	$-0.198392\pi$
$859$	47.5959	1.62395	0.811976	+	0.583691i	$0.198392\pi$
$860$	−9.78817	−0.333774
$861$	0	0
$862$	20.7603	0.707097
$863$	53.5703	1.82355	0.911777	−	0.410686i	$-0.134711\pi$
$863$	53.5703	1.82355	0.911777	+	0.410686i	$0.134711\pi$
$864$	0	0
$865$	38.2910	1.30193
$866$	9.91220	0.336830
$867$	0	0
$868$	40.4283	1.37223
$869$	−12.1824	−0.413260
$870$	0	0
$871$	−1.01045	−0.0342377
$872$	−35.7044	−1.20910
$873$	0	0
$874$	1.13840	0.0385069
$875$	−31.0814	−1.05074
$876$	0	0
$877$	−14.3667	−0.485129	−0.242564	−	0.970135i	$-0.577989\pi$
$877$	−14.3667	−0.485129	−0.242564	+	0.970135i	$0.577989\pi$
$878$	−6.92901	−0.233843
$879$	0	0
$880$	−3.21148	−0.108259
$881$	0.689547	0.0232314	0.0116157	−	0.999933i	$-0.496303\pi$
$881$	0.689547	0.0232314	0.0116157	+	0.999933i	$0.496303\pi$
$882$	0	0
$883$	25.3682	0.853706	0.426853	−	0.904321i	$-0.359622\pi$
$883$	25.3682	0.853706	0.426853	+	0.904321i	$0.359622\pi$
$884$	−13.2750	−0.446487
$885$	0	0
$886$	−11.9877	−0.402734
$887$	−41.7138	−1.40061	−0.700307	−	0.713842i	$-0.746953\pi$
$887$	−41.7138	−1.40061	−0.700307	+	0.713842i	$0.746953\pi$
$888$	0	0
$889$	−52.0676	−1.74629
$890$	11.0236	0.369513
$891$	0	0
$892$	−16.5109	−0.552824
$893$	3.62086	0.121167
$894$	0	0
$895$	−6.03868	−0.201851
$896$	45.3972	1.51661
$897$	0	0
$898$	−0.974061	−0.0325049
$899$	−33.8379	−1.12856
$900$	0	0
$901$	−44.5087	−1.48280
$902$	−0.168895	−0.00562360
$903$	0	0
$904$	−1.06761	−0.0355080
$905$	−64.4306	−2.14175
$906$	0	0
$907$	−39.7006	−1.31824	−0.659119	−	0.752039i	$-0.729071\pi$
$907$	−39.7006	−1.31824	−0.659119	+	0.752039i	$0.729071\pi$
$908$	−34.6518	−1.14996
$909$	0	0
$910$	−23.7298	−0.786636
$911$	−39.3933	−1.30516	−0.652578	−	0.757721i	$-0.726313\pi$
$911$	−39.3933	−1.30516	−0.652578	+	0.757721i	$0.726313\pi$
$912$	0	0
$913$	−5.74956	−0.190283
$914$	−24.4010	−0.807114
$915$	0	0
$916$	35.5636	1.17505
$917$	−67.0164	−2.21308
$918$	0	0
$919$	−15.9530	−0.526241	−0.263120	−	0.964763i	$-0.584752\pi$
$919$	−15.9530	−0.526241	−0.263120	+	0.964763i	$0.584752\pi$
$920$	5.02797	0.165767
$921$	0	0
$922$	−13.3415	−0.439378
$923$	26.4737	0.871393
$924$	0	0
$925$	2.09558	0.0689023
$926$	−28.1038	−0.923547
$927$	0	0
$928$	−33.6380	−1.10422
$929$	25.1013	0.823547	0.411774	−	0.911286i	$-0.364909\pi$
$929$	25.1013	0.823547	0.411774	+	0.911286i	$0.364909\pi$
$930$	0	0
$931$	−37.9966	−1.24529
$932$	12.6027	0.412814
$933$	0	0
$934$	−33.0910	−1.08277
$935$	−21.4444	−0.701306
$936$	0	0
$937$	−38.9604	−1.27278	−0.636390	−	0.771367i	$-0.719573\pi$
$937$	−38.9604	−1.27278	−0.636390	+	0.771367i	$0.719573\pi$
$938$	1.82615	0.0596260
$939$	0	0
$940$	6.52893	0.212950
$941$	−33.0736	−1.07817	−0.539084	−	0.842252i	$-0.681230\pi$
$941$	−33.0736	−1.07817	−0.539084	+	0.842252i	$0.681230\pi$
$942$	0	0
$943$	−0.0837226	−0.00272638
$944$	7.71830	0.251209
$945$	0	0
$946$	3.48331	0.113252
$947$	−14.5042	−0.471325	−0.235662	−	0.971835i	$-0.575726\pi$
$947$	−14.5042	−0.471325	−0.235662	+	0.971835i	$0.575726\pi$
$948$	0	0
$949$	4.64745	0.150863
$950$	4.63462	0.150367
$951$	0	0
$952$	58.7659	1.90461
$953$	5.82253	0.188610	0.0943051	−	0.995543i	$-0.469937\pi$
$953$	5.82253	0.188610	0.0943051	+	0.995543i	$0.469937\pi$
$954$	0	0
$955$	60.9731	1.97304
$956$	−30.0247	−0.971069
$957$	0	0
$958$	28.4217	0.918263
$959$	71.2405	2.30048
$960$	0	0
$961$	3.58180	0.115542
$962$	−1.29778	−0.0418422
$963$	0	0
$964$	−19.1311	−0.616172
$965$	−20.0985	−0.646993
$966$	0	0
$967$	−45.6901	−1.46930	−0.734648	−	0.678449i	$-0.762652\pi$
$967$	−45.6901	−1.46930	−0.734648	+	0.678449i	$0.762652\pi$
$968$	−21.2466	−0.682893
$969$	0	0
$970$	16.4695	0.528804
$971$	−1.52462	−0.0489275	−0.0244638	−	0.999701i	$-0.507788\pi$
$971$	−1.52462	−0.0489275	−0.0244638	+	0.999701i	$0.507788\pi$
$972$	0	0
$973$	−1.61337	−0.0517222
$974$	−21.9626	−0.703727
$975$	0	0
$976$	5.25613	0.168245
$977$	−20.3945	−0.652477	−0.326238	−	0.945287i	$-0.605781\pi$
$977$	−20.3945	−0.652477	−0.326238	+	0.945287i	$0.605781\pi$
$978$	0	0
$979$	8.72856	0.278966
$980$	−68.5133	−2.18858
$981$	0	0
$982$	17.1296	0.546628
$983$	−42.1644	−1.34483	−0.672417	−	0.740172i	$-0.734744\pi$
$983$	−42.1644	−1.34483	−0.672417	+	0.740172i	$0.734744\pi$
$984$	0	0
$985$	−45.6954	−1.45598
$986$	−20.0806	−0.639495
$987$	0	0
$988$	6.38614	0.203170
$989$	1.72670	0.0549059
$990$	0	0
$991$	−1.55820	−0.0494980	−0.0247490	−	0.999694i	$-0.507879\pi$
$991$	−1.55820	−0.0494980	−0.0247490	+	0.999694i	$0.507879\pi$
$992$	34.3775	1.09149
$993$	0	0
$994$	−47.8452	−1.51756
$995$	−22.4771	−0.712571
$996$	0	0
$997$	−52.3365	−1.65751	−0.828757	−	0.559608i	$-0.810952\pi$
$997$	−52.3365	−1.65751	−0.828757	+	0.559608i	$0.810952\pi$
$998$	2.35019	0.0743939
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.25		72
3.2	odd	2		6561.2.a.c.1.48		72
81.2	odd	54		729.2.g.d.433.3		144
81.13	even	27		243.2.g.a.100.3		144
81.14	odd	54		729.2.g.c.55.6		144
81.25	even	27		243.2.g.a.226.3		144
81.29	odd	54		729.2.g.c.676.6		144
81.40	even	27		729.2.g.a.298.6		144
81.41	odd	54		729.2.g.d.298.3		144
81.52	even	27		729.2.g.b.676.3		144
81.56	odd	54		81.2.g.a.58.6	yes	144
81.67	even	27		729.2.g.b.55.3		144
81.68	odd	54		81.2.g.a.7.6	✓	144
81.79	even	27		729.2.g.a.433.6		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.7.6	✓	144	81.68	odd	54
81.2.g.a.58.6	yes	144	81.56	odd	54
243.2.g.a.100.3		144	81.13	even	27
243.2.g.a.226.3		144	81.25	even	27
729.2.g.a.298.6		144	81.40	even	27
729.2.g.a.433.6		144	81.79	even	27
729.2.g.b.55.3		144	81.67	even	27
729.2.g.b.676.3		144	81.52	even	27
729.2.g.c.55.6		144	81.14	odd	54
729.2.g.c.676.6		144	81.29	odd	54
729.2.g.d.298.3		144	81.41	odd	54
729.2.g.d.433.3		144	81.2	odd	54
6561.2.a.c.1.48		72	3.2	odd	2
6561.2.a.d.1.25		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-0.787501 q^{2} -1.37984 q^{4} +2.78580 q^{5} +4.98233 q^{7} +2.66163 q^{8} -2.19382 q^{10} -1.73708 q^{11} +2.17101 q^{13} -3.92359 q^{14} +0.663647 q^{16} +4.43144 q^{17} -2.13181 q^{19} -3.84396 q^{20} +1.36795 q^{22} +0.678102 q^{23} +2.76067 q^{25} -1.70967 q^{26} -6.87483 q^{28} +5.75413 q^{29} -5.88063 q^{31} -5.84588 q^{32} -3.48977 q^{34} +13.8798 q^{35} +0.759084 q^{37} +1.67880 q^{38} +7.41476 q^{40} -0.123466 q^{41} +2.54638 q^{43} +2.39689 q^{44} -0.534006 q^{46} -1.69849 q^{47} +17.8236 q^{49} -2.17403 q^{50} -2.99564 q^{52} -10.0438 q^{53} -4.83914 q^{55} +13.2611 q^{56} -4.53138 q^{58} +11.6301 q^{59} +7.92007 q^{61} +4.63100 q^{62} +3.27635 q^{64} +6.04798 q^{65} -0.465429 q^{67} -6.11469 q^{68} -10.9303 q^{70} +12.1942 q^{71} +2.14069 q^{73} -0.597780 q^{74} +2.94156 q^{76} -8.65469 q^{77} +7.01317 q^{79} +1.84879 q^{80} +0.0972297 q^{82} +3.30991 q^{83} +12.3451 q^{85} -2.00527 q^{86} -4.62345 q^{88} -5.02485 q^{89} +10.8167 q^{91} -0.935674 q^{92} +1.33756 q^{94} -5.93879 q^{95} -7.50723 q^{97} -14.0361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38}+ \cdots + 81 q^{98}+O(q^{100}) \) 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

L-functions

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Database

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