LMFDB - Embedded newform 6561.2.a.c.1.41

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

Embedding invariants

Embedding label			1.41
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)

$f(q)$	$=$	\(q+0.241745 q^{2} -1.94156 q^{4} -4.24792 q^{5} -1.15215 q^{7} -0.952852 q^{8} -1.02691 q^{10} +1.97459 q^{11} -4.00518 q^{13} -0.278527 q^{14} +3.65277 q^{16} -4.21350 q^{17} +3.02364 q^{19} +8.24758 q^{20} +0.477346 q^{22} -1.21995 q^{23} +13.0448 q^{25} -0.968232 q^{26} +2.23697 q^{28} -0.596851 q^{29} -1.94546 q^{31} +2.78874 q^{32} -1.01859 q^{34} +4.89425 q^{35} +5.27123 q^{37} +0.730948 q^{38} +4.04763 q^{40} +5.72850 q^{41} +7.29895 q^{43} -3.83378 q^{44} -0.294917 q^{46} -0.436661 q^{47} -5.67254 q^{49} +3.15351 q^{50} +7.77630 q^{52} +4.47389 q^{53} -8.38788 q^{55} +1.09783 q^{56} -0.144286 q^{58} +8.33298 q^{59} -10.5469 q^{61} -0.470306 q^{62} -6.63138 q^{64} +17.0137 q^{65} +7.32577 q^{67} +8.18075 q^{68} +1.18316 q^{70} -4.78942 q^{71} +7.20843 q^{73} +1.27429 q^{74} -5.87057 q^{76} -2.27503 q^{77} +12.6029 q^{79} -15.5167 q^{80} +1.38483 q^{82} -11.5763 q^{83} +17.8986 q^{85} +1.76448 q^{86} -1.88149 q^{88} +11.0101 q^{89} +4.61458 q^{91} +2.36861 q^{92} -0.105560 q^{94} -12.8442 q^{95} +6.28775 q^{97} -1.37131 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.241745	0.170939	0.0854697	−	0.996341i	$-0.472761\pi$
$2$	0.241745	0.170939	0.0854697	+	0.996341i	$0.472761\pi$
$3$	0	0
$3$	0	0
$4$	−1.94156	−0.970780
$5$	−4.24792	−1.89973	−0.949863	−	0.312666i	$-0.898778\pi$
$5$	−4.24792	−1.89973	−0.949863	+	0.312666i	$0.898778\pi$
$6$	0	0
$7$	−1.15215	−0.435473	−0.217736	−	0.976008i	$-0.569867\pi$
$7$	−1.15215	−0.435473	−0.217736	+	0.976008i	$0.569867\pi$
$8$	−0.952852	−0.336884
$9$	0	0
$10$	−1.02691	−0.324738
$11$	1.97459	0.595360	0.297680	−	0.954666i	$-0.403787\pi$
$11$	1.97459	0.595360	0.297680	+	0.954666i	$0.403787\pi$
$12$	0	0
$13$	−4.00518	−1.11084	−0.555419	−	0.831571i	$-0.687442\pi$
$13$	−4.00518	−1.11084	−0.555419	+	0.831571i	$0.687442\pi$
$14$	−0.278527	−0.0744395
$15$	0	0
$16$	3.65277	0.913193
$17$	−4.21350	−1.02192	−0.510961	−	0.859604i	$-0.670711\pi$
$17$	−4.21350	−1.02192	−0.510961	+	0.859604i	$0.670711\pi$
$18$	0	0
$19$	3.02364	0.693669	0.346835	−	0.937926i	$-0.387256\pi$
$19$	3.02364	0.693669	0.346835	+	0.937926i	$0.387256\pi$
$20$	8.24758	1.84422
$21$	0	0
$22$	0.477346	0.101771
$23$	−1.21995	−0.254378	−0.127189	−	0.991879i	$-0.540595\pi$
$23$	−1.21995	−0.254378	−0.127189	+	0.991879i	$0.540595\pi$
$24$	0	0
$25$	13.0448	2.60896
$26$	−0.968232	−0.189886
$27$	0	0
$28$	2.23697	0.422748
$29$	−0.596851	−0.110832	−0.0554162	−	0.998463i	$-0.517649\pi$
$29$	−0.596851	−0.110832	−0.0554162	+	0.998463i	$0.517649\pi$
$30$	0	0
$31$	−1.94546	−0.349416	−0.174708	−	0.984620i	$-0.555898\pi$
$31$	−1.94546	−0.349416	−0.174708	+	0.984620i	$0.555898\pi$
$32$	2.78874	0.492985
$33$	0	0
$34$	−1.01859	−0.174687
$35$	4.89425	0.827279
$36$	0	0
$37$	5.27123	0.866585	0.433292	−	0.901253i	$-0.357352\pi$
$37$	5.27123	0.866585	0.433292	+	0.901253i	$0.357352\pi$
$38$	0.730948	0.118575
$39$	0	0
$40$	4.04763	0.639987
$41$	5.72850	0.894641	0.447321	−	0.894374i	$-0.352378\pi$
$41$	5.72850	0.894641	0.447321	+	0.894374i	$0.352378\pi$
$42$	0	0
$43$	7.29895	1.11308	0.556540	−	0.830821i	$-0.312129\pi$
$43$	7.29895	1.11308	0.556540	+	0.830821i	$0.312129\pi$
$44$	−3.83378	−0.577964
$45$	0	0
$46$	−0.294917	−0.0434832
$47$	−0.436661	−0.0636935	−0.0318468	−	0.999493i	$-0.510139\pi$
$47$	−0.436661	−0.0636935	−0.0318468	+	0.999493i	$0.510139\pi$
$48$	0	0
$49$	−5.67254	−0.810363
$50$	3.15351	0.445974
$51$	0	0
$52$	7.77630	1.07838
$53$	4.47389	0.614536	0.307268	−	0.951623i	$-0.400585\pi$
$53$	4.47389	0.614536	0.307268	+	0.951623i	$0.400585\pi$
$54$	0	0
$55$	−8.38788	−1.13102
$56$	1.09783	0.146704
$57$	0	0
$58$	−0.144286	−0.0189456
$59$	8.33298	1.08486	0.542431	−	0.840100i	$-0.317504\pi$
$59$	8.33298	1.08486	0.542431	+	0.840100i	$0.317504\pi$
$60$	0	0
$61$	−10.5469	−1.35039	−0.675195	−	0.737639i	$-0.735941\pi$
$61$	−10.5469	−1.35039	−0.675195	+	0.737639i	$0.735941\pi$
$62$	−0.470306	−0.0597289
$63$	0	0
$64$	−6.63138	−0.828923
$65$	17.0137	2.11029
$66$	0	0
$67$	7.32577	0.894985	0.447493	−	0.894288i	$-0.352317\pi$
$67$	7.32577	0.894985	0.447493	+	0.894288i	$0.352317\pi$
$68$	8.18075	0.992062
$69$	0	0
$70$	1.18316	0.141415
$71$	−4.78942	−0.568400	−0.284200	−	0.958765i	$-0.591728\pi$
$71$	−4.78942	−0.568400	−0.284200	+	0.958765i	$0.591728\pi$
$72$	0	0
$73$	7.20843	0.843683	0.421841	−	0.906670i	$-0.361384\pi$
$73$	7.20843	0.843683	0.421841	+	0.906670i	$0.361384\pi$
$74$	1.27429	0.148133
$75$	0	0
$76$	−5.87057	−0.673400
$77$	−2.27503	−0.259263
$78$	0	0
$79$	12.6029	1.41793	0.708967	−	0.705241i	$-0.249161\pi$
$79$	12.6029	1.41793	0.708967	+	0.705241i	$0.249161\pi$
$80$	−15.5167	−1.73482
$81$	0	0
$82$	1.38483	0.152929
$83$	−11.5763	−1.27066	−0.635332	−	0.772239i	$-0.719137\pi$
$83$	−11.5763	−1.27066	−0.635332	+	0.772239i	$0.719137\pi$
$84$	0	0
$85$	17.8986	1.94137
$86$	1.76448	0.190269
$87$	0	0
$88$	−1.88149	−0.200567
$89$	11.0101	1.16706	0.583532	−	0.812090i	$-0.301670\pi$
$89$	11.0101	1.16706	0.583532	+	0.812090i	$0.301670\pi$
$90$	0	0
$91$	4.61458	0.483740
$92$	2.36861	0.246945
$93$	0	0
$94$	−0.105560	−0.0108877
$95$	−12.8442	−1.31778
$96$	0	0
$97$	6.28775	0.638424	0.319212	−	0.947683i	$-0.396582\pi$
$97$	6.28775	0.638424	0.319212	+	0.947683i	$0.396582\pi$
$98$	−1.37131	−0.138523
$99$	0	0
$100$	−25.3272	−2.53272
$101$	−4.04781	−0.402772	−0.201386	−	0.979512i	$-0.564545\pi$
$101$	−4.04781	−0.402772	−0.201386	+	0.979512i	$0.564545\pi$
$102$	0	0
$103$	8.28479	0.816325	0.408162	−	0.912909i	$-0.366170\pi$
$103$	8.28479	0.816325	0.408162	+	0.912909i	$0.366170\pi$
$104$	3.81634	0.374223
$105$	0	0
$106$	1.08154	0.105048
$107$	−15.9074	−1.53782	−0.768912	−	0.639355i	$-0.779202\pi$
$107$	−15.9074	−1.53782	−0.768912	+	0.639355i	$0.779202\pi$
$108$	0	0
$109$	−3.62111	−0.346840	−0.173420	−	0.984848i	$-0.555482\pi$
$109$	−3.62111	−0.346840	−0.173420	+	0.984848i	$0.555482\pi$
$110$	−2.02773	−0.193336
$111$	0	0
$112$	−4.20855	−0.397671
$113$	−9.49512	−0.893226	−0.446613	−	0.894727i	$-0.647370\pi$
$113$	−9.49512	−0.893226	−0.446613	+	0.894727i	$0.647370\pi$
$114$	0	0
$115$	5.18226	0.483248
$116$	1.15882	0.107594
$117$	0	0
$118$	2.01446	0.185446
$119$	4.85459	0.445020
$120$	0	0
$121$	−7.10101	−0.645546
$122$	−2.54966	−0.230835
$123$	0	0
$124$	3.77723	0.339206
$125$	−34.1736	−3.05658
$126$	0	0
$127$	11.1476	0.989187	0.494594	−	0.869124i	$-0.335317\pi$
$127$	11.1476	0.989187	0.494594	+	0.869124i	$0.335317\pi$
$128$	−7.18059	−0.634680
$129$	0	0
$130$	4.11297	0.360731
$131$	12.1313	1.05991	0.529957	−	0.848024i	$-0.322208\pi$
$131$	12.1313	1.05991	0.529957	+	0.848024i	$0.322208\pi$
$132$	0	0
$133$	−3.48369	−0.302074
$134$	1.77097	0.152988
$135$	0	0
$136$	4.01484	0.344269
$137$	−0.299217	−0.0255639	−0.0127819	−	0.999918i	$-0.504069\pi$
$137$	−0.299217	−0.0255639	−0.0127819	+	0.999918i	$0.504069\pi$
$138$	0	0
$139$	−5.60959	−0.475799	−0.237899	−	0.971290i	$-0.576459\pi$
$139$	−5.60959	−0.475799	−0.237899	+	0.971290i	$0.576459\pi$
$140$	−9.50247	−0.803106
$141$	0	0
$142$	−1.15782	−0.0971620
$143$	−7.90858	−0.661349
$144$	0	0
$145$	2.53537	0.210551
$146$	1.74260	0.144219
$147$	0	0
$148$	−10.2344	−0.841263
$149$	−13.6263	−1.11631	−0.558154	−	0.829737i	$-0.688490\pi$
$149$	−13.6263	−1.11631	−0.558154	+	0.829737i	$0.688490\pi$
$150$	0	0
$151$	−9.45329	−0.769298	−0.384649	−	0.923063i	$-0.625678\pi$
$151$	−9.45329	−0.769298	−0.384649	+	0.923063i	$0.625678\pi$
$152$	−2.88108	−0.233686
$153$	0	0
$154$	−0.549976	−0.0443183
$155$	8.26417	0.663794
$156$	0	0
$157$	−9.12227	−0.728036	−0.364018	−	0.931392i	$-0.618595\pi$
$157$	−9.12227	−0.728036	−0.364018	+	0.931392i	$0.618595\pi$
$158$	3.04668	0.242381
$159$	0	0
$160$	−11.8463	−0.936536
$161$	1.40557	0.110775
$162$	0	0
$163$	7.94158	0.622033	0.311016	−	0.950405i	$-0.399331\pi$
$163$	7.94158	0.622033	0.311016	+	0.950405i	$0.399331\pi$
$164$	−11.1222	−0.868500
$165$	0	0
$166$	−2.79851	−0.217207
$167$	−4.44493	−0.343959	−0.171980	−	0.985101i	$-0.555016\pi$
$167$	−4.44493	−0.343959	−0.171980	+	0.985101i	$0.555016\pi$
$168$	0	0
$169$	3.04148	0.233960
$170$	4.32689	0.331857
$171$	0	0
$172$	−14.1713	−1.08055
$173$	3.11848	0.237094	0.118547	−	0.992948i	$-0.462176\pi$
$173$	3.11848	0.237094	0.118547	+	0.992948i	$0.462176\pi$
$174$	0	0
$175$	−15.0296	−1.13613
$176$	7.21271	0.543679
$177$	0	0
$178$	2.66163	0.199497
$179$	−14.2357	−1.06403	−0.532013	−	0.846736i	$-0.678564\pi$
$179$	−14.2357	−1.06403	−0.532013	+	0.846736i	$0.678564\pi$
$180$	0	0
$181$	15.8923	1.18126	0.590632	−	0.806941i	$-0.298879\pi$
$181$	15.8923	1.18126	0.590632	+	0.806941i	$0.298879\pi$
$182$	1.11555	0.0826901
$183$	0	0
$184$	1.16243	0.0856958
$185$	−22.3917	−1.64627
$186$	0	0
$187$	−8.31991	−0.608412
$188$	0.847803	0.0618324
$189$	0	0
$190$	−3.10501	−0.225261
$191$	20.3075	1.46940	0.734700	−	0.678392i	$-0.237323\pi$
$191$	20.3075	1.46940	0.734700	+	0.678392i	$0.237323\pi$
$192$	0	0
$193$	−2.67245	−0.192367	−0.0961836	−	0.995364i	$-0.530664\pi$
$193$	−2.67245	−0.192367	−0.0961836	+	0.995364i	$0.530664\pi$
$194$	1.52003	0.109132
$195$	0	0
$196$	11.0136	0.786684
$197$	16.7217	1.19137	0.595685	−	0.803218i	$-0.296881\pi$
$197$	16.7217	1.19137	0.595685	+	0.803218i	$0.296881\pi$
$198$	0	0
$199$	1.95609	0.138663	0.0693317	−	0.997594i	$-0.477913\pi$
$199$	1.95609	0.138663	0.0693317	+	0.997594i	$0.477913\pi$
$200$	−12.4298	−0.878916
$201$	0	0
$202$	−0.978536	−0.0688496
$203$	0.687664	0.0482645
$204$	0	0
$205$	−24.3342	−1.69957
$206$	2.00281	0.139542
$207$	0	0
$208$	−14.6300	−1.01441
$209$	5.97043	0.412983
$210$	0	0
$211$	10.3643	0.713509	0.356754	−	0.934198i	$-0.383883\pi$
$211$	10.3643	0.713509	0.356754	+	0.934198i	$0.383883\pi$
$212$	−8.68632	−0.596579
$213$	0	0
$214$	−3.84553	−0.262875
$215$	−31.0053	−2.11455
$216$	0	0
$217$	2.24147	0.152161
$218$	−0.875385	−0.0592886
$219$	0	0
$220$	16.2856	1.09797
$221$	16.8758	1.13519
$222$	0	0
$223$	7.32809	0.490725	0.245363	−	0.969431i	$-0.421093\pi$
$223$	7.32809	0.490725	0.245363	+	0.969431i	$0.421093\pi$
$224$	−3.21306	−0.214681
$225$	0	0
$226$	−2.29540	−0.152688
$227$	−23.2777	−1.54500	−0.772499	−	0.635016i	$-0.780993\pi$
$227$	−23.2777	−1.54500	−0.772499	+	0.635016i	$0.780993\pi$
$228$	0	0
$229$	−0.0499144	−0.00329844	−0.00164922	−	0.999999i	$-0.500525\pi$
$229$	−0.0499144	−0.00329844	−0.00164922	+	0.999999i	$0.500525\pi$
$230$	1.25278	0.0826061
$231$	0	0
$232$	0.568711	0.0373377
$233$	−22.4092	−1.46807	−0.734037	−	0.679109i	$-0.762366\pi$
$233$	−22.4092	−1.46807	−0.734037	+	0.679109i	$0.762366\pi$
$234$	0	0
$235$	1.85490	0.121000
$236$	−16.1790	−1.05316
$237$	0	0
$238$	1.17357	0.0760714
$239$	−20.5331	−1.32817	−0.664087	−	0.747655i	$-0.731180\pi$
$239$	−20.5331	−1.32817	−0.664087	+	0.747655i	$0.731180\pi$
$240$	0	0
$241$	−28.7869	−1.85433	−0.927165	−	0.374654i	$-0.877762\pi$
$241$	−28.7869	−1.85433	−0.927165	+	0.374654i	$0.877762\pi$
$242$	−1.71663	−0.110349
$243$	0	0
$244$	20.4774	1.31093
$245$	24.0965	1.53947
$246$	0	0
$247$	−12.1102	−0.770554
$248$	1.85374	0.117712
$249$	0	0
$250$	−8.26130	−0.522490
$251$	−10.1930	−0.643373	−0.321687	−	0.946846i	$-0.604250\pi$
$251$	−10.1930	−0.643373	−0.321687	+	0.946846i	$0.604250\pi$
$252$	0	0
$253$	−2.40890	−0.151446
$254$	2.69487	0.169091
$255$	0	0
$256$	11.5269	0.720431
$257$	6.06049	0.378043	0.189021	−	0.981973i	$-0.439468\pi$
$257$	6.06049	0.378043	0.189021	+	0.981973i	$0.439468\pi$
$258$	0	0
$259$	−6.07326	−0.377374
$260$	−33.0331	−2.04862
$261$	0	0
$262$	2.93267	0.181181
$263$	13.4179	0.827383	0.413691	−	0.910417i	$-0.364239\pi$
$263$	13.4179	0.827383	0.413691	+	0.910417i	$0.364239\pi$
$264$	0	0
$265$	−19.0047	−1.16745
$266$	−0.842164	−0.0516364
$267$	0	0
$268$	−14.2234	−0.868834
$269$	−2.88240	−0.175743	−0.0878715	−	0.996132i	$-0.528006\pi$
$269$	−2.88240	−0.175743	−0.0878715	+	0.996132i	$0.528006\pi$
$270$	0	0
$271$	3.21146	0.195082	0.0975410	−	0.995232i	$-0.468902\pi$
$271$	3.21146	0.195082	0.0975410	+	0.995232i	$0.468902\pi$
$272$	−15.3909	−0.933213
$273$	0	0
$274$	−0.0723343	−0.00436987
$275$	25.7581	1.55327
$276$	0	0
$277$	−3.40628	−0.204664	−0.102332	−	0.994750i	$-0.532630\pi$
$277$	−3.40628	−0.204664	−0.102332	+	0.994750i	$0.532630\pi$
$278$	−1.35609	−0.0813328
$279$	0	0
$280$	−4.66349	−0.278697
$281$	−11.4849	−0.685130	−0.342565	−	0.939494i	$-0.611296\pi$
$281$	−11.4849	−0.685130	−0.342565	+	0.939494i	$0.611296\pi$
$282$	0	0
$283$	25.3416	1.50640	0.753202	−	0.657790i	$-0.228508\pi$
$283$	25.3416	1.50640	0.753202	+	0.657790i	$0.228508\pi$
$284$	9.29895	0.551791
$285$	0	0
$286$	−1.91186	−0.113051
$287$	−6.60011	−0.389592
$288$	0	0
$289$	0.753549	0.0443264
$290$	0.612914	0.0359915
$291$	0	0
$292$	−13.9956	−0.819030
$293$	−4.01614	−0.234625	−0.117313	−	0.993095i	$-0.537428\pi$
$293$	−4.01614	−0.234625	−0.117313	+	0.993095i	$0.537428\pi$
$294$	0	0
$295$	−35.3978	−2.06094
$296$	−5.02270	−0.291938
$297$	0	0
$298$	−3.29408	−0.190821
$299$	4.88613	0.282572
$300$	0	0
$301$	−8.40950	−0.484716
$302$	−2.28528	−0.131503
$303$	0	0
$304$	11.0446	0.633454
$305$	44.8023	2.56537
$306$	0	0
$307$	19.8796	1.13459	0.567295	−	0.823514i	$-0.307990\pi$
$307$	19.8796	1.13459	0.567295	+	0.823514i	$0.307990\pi$
$308$	4.41710	0.251687
$309$	0	0
$310$	1.99782	0.113469
$311$	1.54419	0.0875629	0.0437814	−	0.999041i	$-0.486059\pi$
$311$	1.54419	0.0875629	0.0437814	+	0.999041i	$0.486059\pi$
$312$	0	0
$313$	−17.4992	−0.989111	−0.494555	−	0.869146i	$-0.664669\pi$
$313$	−17.4992	−0.989111	−0.494555	+	0.869146i	$0.664669\pi$
$314$	−2.20526	−0.124450
$315$	0	0
$316$	−24.4692	−1.37650
$317$	1.96450	0.110337	0.0551685	−	0.998477i	$-0.482430\pi$
$317$	1.96450	0.110337	0.0551685	+	0.998477i	$0.482430\pi$
$318$	0	0
$319$	−1.17853	−0.0659853
$320$	28.1696	1.57473
$321$	0	0
$322$	0.339790	0.0189357
$323$	−12.7401	−0.708877
$324$	0	0
$325$	−52.2468	−2.89813
$326$	1.91984	0.106330
$327$	0	0
$328$	−5.45841	−0.301390
$329$	0.503100	0.0277368
$330$	0	0
$331$	34.4822	1.89532	0.947658	−	0.319288i	$-0.103444\pi$
$331$	34.4822	1.89532	0.947658	+	0.319288i	$0.103444\pi$
$332$	22.4761	1.23354
$333$	0	0
$334$	−1.07454	−0.0587961
$335$	−31.1193	−1.70023
$336$	0	0
$337$	16.1761	0.881167	0.440584	−	0.897712i	$-0.354772\pi$
$337$	16.1761	0.881167	0.440584	+	0.897712i	$0.354772\pi$
$338$	0.735263	0.0399930
$339$	0	0
$340$	−34.7512	−1.88465
$341$	−3.84149	−0.208028
$342$	0	0
$343$	14.6007	0.788364
$344$	−6.95481	−0.374978
$345$	0	0
$346$	0.753876	0.0405286
$347$	31.8148	1.70791	0.853954	−	0.520348i	$-0.174198\pi$
$347$	31.8148	1.70791	0.853954	+	0.520348i	$0.174198\pi$
$348$	0	0
$349$	−17.3353	−0.927937	−0.463969	−	0.885852i	$-0.653575\pi$
$349$	−17.3353	−0.927937	−0.463969	+	0.885852i	$0.653575\pi$
$350$	−3.63333	−0.194210
$351$	0	0
$352$	5.50661	0.293503
$353$	−3.39981	−0.180954	−0.0904768	−	0.995899i	$-0.528839\pi$
$353$	−3.39981	−0.180954	−0.0904768	+	0.995899i	$0.528839\pi$
$354$	0	0
$355$	20.3451	1.07980
$356$	−21.3767	−1.13296
$357$	0	0
$358$	−3.44141	−0.181884
$359$	−10.1468	−0.535528	−0.267764	−	0.963484i	$-0.586285\pi$
$359$	−10.1468	−0.535528	−0.267764	+	0.963484i	$0.586285\pi$
$360$	0	0
$361$	−9.85763	−0.518823
$362$	3.84188	0.201924
$363$	0	0
$364$	−8.95948	−0.469605
$365$	−30.6208	−1.60277
$366$	0	0
$367$	14.8640	0.775893	0.387946	−	0.921682i	$-0.373185\pi$
$367$	14.8640	0.775893	0.387946	+	0.921682i	$0.373185\pi$
$368$	−4.45621	−0.232296
$369$	0	0
$370$	−5.41309	−0.281413
$371$	−5.15461	−0.267614
$372$	0	0
$373$	−21.0809	−1.09153	−0.545763	−	0.837940i	$-0.683760\pi$
$373$	−21.0809	−1.09153	−0.545763	+	0.837940i	$0.683760\pi$
$374$	−2.01130	−0.104002
$375$	0	0
$376$	0.416073	0.0214573
$377$	2.39050	0.123117
$378$	0	0
$379$	6.94925	0.356959	0.178479	−	0.983944i	$-0.442882\pi$
$379$	6.94925	0.356959	0.178479	+	0.983944i	$0.442882\pi$
$380$	24.9377	1.27928
$381$	0	0
$382$	4.90924	0.251179
$383$	−35.3323	−1.80540	−0.902699	−	0.430273i	$-0.858417\pi$
$383$	−35.3323	−1.80540	−0.902699	+	0.430273i	$0.858417\pi$
$384$	0	0
$385$	9.66412	0.492529
$386$	−0.646051	−0.0328831
$387$	0	0
$388$	−12.2080	−0.619769
$389$	−25.7236	−1.30424	−0.652119	−	0.758116i	$-0.726120\pi$
$389$	−25.7236	−1.30424	−0.652119	+	0.758116i	$0.726120\pi$
$390$	0	0
$391$	5.14027	0.259954
$392$	5.40509	0.272998
$393$	0	0
$394$	4.04238	0.203652
$395$	−53.5360	−2.69369
$396$	0	0
$397$	18.5982	0.933416	0.466708	−	0.884411i	$-0.345440\pi$
$397$	18.5982	0.933416	0.466708	+	0.884411i	$0.345440\pi$
$398$	0.472874	0.0237030
$399$	0	0
$400$	47.6497	2.38248
$401$	−4.62371	−0.230897	−0.115448	−	0.993313i	$-0.536831\pi$
$401$	−4.62371	−0.230897	−0.115448	+	0.993313i	$0.536831\pi$
$402$	0	0
$403$	7.79194	0.388144
$404$	7.85906	0.391003
$405$	0	0
$406$	0.166239	0.00825031
$407$	10.4085	0.515930
$408$	0	0
$409$	−10.9390	−0.540900	−0.270450	−	0.962734i	$-0.587173\pi$
$409$	−10.9390	−0.540900	−0.270450	+	0.962734i	$0.587173\pi$
$410$	−5.88266	−0.290524
$411$	0	0
$412$	−16.0854	−0.792472
$413$	−9.60087	−0.472428
$414$	0	0
$415$	49.1752	2.41392
$416$	−11.1694	−0.547626
$417$	0	0
$418$	1.44332	0.0705951
$419$	−33.5412	−1.63859	−0.819297	−	0.573369i	$-0.805636\pi$
$419$	−33.5412	−1.63859	−0.819297	+	0.573369i	$0.805636\pi$
$420$	0	0
$421$	1.65864	0.0808374	0.0404187	−	0.999183i	$-0.487131\pi$
$421$	1.65864	0.0808374	0.0404187	+	0.999183i	$0.487131\pi$
$422$	2.50552	0.121967
$423$	0	0
$424$	−4.26295	−0.207027
$425$	−54.9642	−2.66616
$426$	0	0
$427$	12.1516	0.588058
$428$	30.8851	1.49289
$429$	0	0
$430$	−7.49538	−0.361459
$431$	−15.1085	−0.727751	−0.363876	−	0.931448i	$-0.618547\pi$
$431$	−15.1085	−0.727751	−0.363876	+	0.931448i	$0.618547\pi$
$432$	0	0
$433$	4.93309	0.237069	0.118535	−	0.992950i	$-0.462180\pi$
$433$	4.93309	0.237069	0.118535	+	0.992950i	$0.462180\pi$
$434$	0.541864	0.0260103
$435$	0	0
$436$	7.03061	0.336705
$437$	−3.68869	−0.176454
$438$	0	0
$439$	−20.1806	−0.963169	−0.481584	−	0.876400i	$-0.659939\pi$
$439$	−20.1806	−0.963169	−0.481584	+	0.876400i	$0.659939\pi$
$440$	7.99240	0.381023
$441$	0	0
$442$	4.07964	0.194049
$443$	28.1863	1.33917	0.669586	−	0.742734i	$-0.266471\pi$
$443$	28.1863	1.33917	0.669586	+	0.742734i	$0.266471\pi$
$444$	0	0
$445$	−46.7698	−2.21710
$446$	1.77153	0.0838843
$447$	0	0
$448$	7.64036	0.360973
$449$	0.434117	0.0204873	0.0102436	−	0.999948i	$-0.496739\pi$
$449$	0.434117	0.0204873	0.0102436	+	0.999948i	$0.496739\pi$
$450$	0	0
$451$	11.3114	0.532634
$452$	18.4353	0.867126
$453$	0	0
$454$	−5.62727	−0.264101
$455$	−19.6024	−0.918973
$456$	0	0
$457$	27.1019	1.26778	0.633888	−	0.773425i	$-0.281458\pi$
$457$	27.1019	1.26778	0.633888	+	0.773425i	$0.281458\pi$
$458$	−0.0120666	−0.000563833 0
$459$	0	0
$460$	−10.0617	−0.469127
$461$	−42.7262	−1.98996	−0.994980	−	0.100077i	$-0.968091\pi$
$461$	−42.7262	−1.98996	−0.994980	+	0.100077i	$0.968091\pi$
$462$	0	0
$463$	−30.3416	−1.41009	−0.705047	−	0.709161i	$-0.749074\pi$
$463$	−30.3416	−1.41009	−0.705047	+	0.709161i	$0.749074\pi$
$464$	−2.18016	−0.101211
$465$	0	0
$466$	−5.41730	−0.250952
$467$	13.4560	0.622672	0.311336	−	0.950300i	$-0.399224\pi$
$467$	13.4560	0.622672	0.311336	+	0.950300i	$0.399224\pi$
$468$	0	0
$469$	−8.44041	−0.389742
$470$	0.448412	0.0206837
$471$	0	0
$472$	−7.94010	−0.365473
$473$	14.4124	0.662683
$474$	0	0
$475$	39.4427	1.80976
$476$	−9.42548	−0.432016
$477$	0	0
$478$	−4.96377	−0.227037
$479$	−25.9765	−1.18689	−0.593447	−	0.804873i	$-0.702234\pi$
$479$	−25.9765	−1.18689	−0.593447	+	0.804873i	$0.702234\pi$
$480$	0	0
$481$	−21.1122	−0.962635
$482$	−6.95909	−0.316978
$483$	0	0
$484$	13.7870	0.626683
$485$	−26.7098	−1.21283
$486$	0	0
$487$	2.70578	0.122611	0.0613053	−	0.998119i	$-0.480474\pi$
$487$	2.70578	0.122611	0.0613053	+	0.998119i	$0.480474\pi$
$488$	10.0496	0.454925
$489$	0	0
$490$	5.82520	0.263156
$491$	39.8076	1.79649	0.898246	−	0.439493i	$-0.144842\pi$
$491$	39.8076	1.79649	0.898246	+	0.439493i	$0.144842\pi$
$492$	0	0
$493$	2.51483	0.113262
$494$	−2.92758	−0.131718
$495$	0	0
$496$	−7.10633	−0.319084
$497$	5.51815	0.247523
$498$	0	0
$499$	2.48537	0.111260	0.0556302	−	0.998451i	$-0.482283\pi$
$499$	2.48537	0.111260	0.0556302	+	0.998451i	$0.482283\pi$
$500$	66.3501	2.96727
$501$	0	0
$502$	−2.46409	−0.109978
$503$	−14.7845	−0.659206	−0.329603	−	0.944120i	$-0.606915\pi$
$503$	−14.7845	−0.659206	−0.329603	+	0.944120i	$0.606915\pi$
$504$	0	0
$505$	17.1947	0.765156
$506$	−0.582340	−0.0258882
$507$	0	0
$508$	−21.6437	−0.960283
$509$	−24.0089	−1.06418	−0.532089	−	0.846689i	$-0.678593\pi$
$509$	−24.0089	−1.06418	−0.532089	+	0.846689i	$0.678593\pi$
$510$	0	0
$511$	−8.30521	−0.367401
$512$	17.1477	0.757830
$513$	0	0
$514$	1.46509	0.0646224
$515$	−35.1931	−1.55079
$516$	0	0
$517$	−0.862224	−0.0379206
$518$	−1.46818	−0.0645081
$519$	0	0
$520$	−16.2115	−0.710922
$521$	−19.7851	−0.866801	−0.433400	−	0.901201i	$-0.642686\pi$
$521$	−19.7851	−0.866801	−0.433400	+	0.901201i	$0.642686\pi$
$522$	0	0
$523$	−34.3703	−1.50291	−0.751455	−	0.659785i	$-0.770648\pi$
$523$	−34.3703	−1.50291	−0.751455	+	0.659785i	$0.770648\pi$
$524$	−23.5536	−1.02894
$525$	0	0
$526$	3.24371	0.141432
$527$	8.19720	0.357076
$528$	0	0
$529$	−21.5117	−0.935292
$530$	−4.59429	−0.199563
$531$	0	0
$532$	6.76379	0.293247
$533$	−22.9437	−0.993801
$534$	0	0
$535$	67.5732	2.92144
$536$	−6.98037	−0.301506
$537$	0	0
$538$	−0.696805	−0.0300414
$539$	−11.2009	−0.482458
$540$	0	0
$541$	36.0540	1.55008	0.775042	−	0.631909i	$-0.217729\pi$
$541$	36.0540	1.55008	0.775042	+	0.631909i	$0.217729\pi$
$542$	0.776353	0.0333472
$543$	0	0
$544$	−11.7504	−0.503792
$545$	15.3822	0.658900
$546$	0	0
$547$	8.64015	0.369426	0.184713	−	0.982793i	$-0.440864\pi$
$547$	8.64015	0.369426	0.184713	+	0.982793i	$0.440864\pi$
$548$	0.580948	0.0248169
$549$	0	0
$550$	6.22688	0.265515
$551$	−1.80466	−0.0768811
$552$	0	0
$553$	−14.5204	−0.617472
$554$	−0.823451	−0.0349851
$555$	0	0
$556$	10.8913	0.461896
$557$	21.3648	0.905255	0.452628	−	0.891700i	$-0.350487\pi$
$557$	21.3648	0.905255	0.452628	+	0.891700i	$0.350487\pi$
$558$	0	0
$559$	−29.2336	−1.23645
$560$	17.8776	0.755465
$561$	0	0
$562$	−2.77641	−0.117116
$563$	27.6618	1.16581	0.582903	−	0.812541i	$-0.301917\pi$
$563$	27.6618	1.16581	0.582903	+	0.812541i	$0.301917\pi$
$564$	0	0
$565$	40.3345	1.69689
$566$	6.12621	0.257504
$567$	0	0
$568$	4.56361	0.191485
$569$	22.1064	0.926750	0.463375	−	0.886162i	$-0.346638\pi$
$569$	22.1064	0.926750	0.463375	+	0.886162i	$0.346638\pi$
$570$	0	0
$571$	2.12819	0.0890622	0.0445311	−	0.999008i	$-0.485821\pi$
$571$	2.12819	0.0890622	0.0445311	+	0.999008i	$0.485821\pi$
$572$	15.3550	0.642024
$573$	0	0
$574$	−1.59554	−0.0665966
$575$	−15.9140	−0.663661
$576$	0	0
$577$	−20.7499	−0.863832	−0.431916	−	0.901914i	$-0.642162\pi$
$577$	−20.7499	−0.863832	−0.431916	+	0.901914i	$0.642162\pi$
$578$	0.182167	0.00757713
$579$	0	0
$580$	−4.92258	−0.204399
$581$	13.3377	0.553340
$582$	0	0
$583$	8.83408	0.365870
$584$	−6.86856	−0.284223
$585$	0	0
$586$	−0.970881	−0.0401067
$587$	11.7448	0.484759	0.242379	−	0.970182i	$-0.422072\pi$
$587$	11.7448	0.484759	0.242379	+	0.970182i	$0.422072\pi$
$588$	0	0
$589$	−5.88237	−0.242379
$590$	−8.55724	−0.352296
$591$	0	0
$592$	19.2546	0.791359
$593$	−21.7680	−0.893904	−0.446952	−	0.894558i	$-0.647491\pi$
$593$	−21.7680	−0.893904	−0.446952	+	0.894558i	$0.647491\pi$
$594$	0	0
$595$	−20.6219	−0.845415
$596$	26.4562	1.08369
$597$	0	0
$598$	1.18120	0.0483027
$599$	10.2440	0.418559	0.209279	−	0.977856i	$-0.432888\pi$
$599$	10.2440	0.418559	0.209279	+	0.977856i	$0.432888\pi$
$600$	0	0
$601$	2.82256	0.115135	0.0575673	−	0.998342i	$-0.481666\pi$
$601$	2.82256	0.115135	0.0575673	+	0.998342i	$0.481666\pi$
$602$	−2.03295	−0.0828570
$603$	0	0
$604$	18.3541	0.746819
$605$	30.1645	1.22636
$606$	0	0
$607$	13.6040	0.552169	0.276084	−	0.961133i	$-0.410963\pi$
$607$	13.6040	0.552169	0.276084	+	0.961133i	$0.410963\pi$
$608$	8.43214	0.341968
$609$	0	0
$610$	10.8307	0.438523
$611$	1.74891	0.0707532
$612$	0	0
$613$	−16.2937	−0.658098	−0.329049	−	0.944313i	$-0.606728\pi$
$613$	−16.2937	−0.658098	−0.329049	+	0.944313i	$0.606728\pi$
$614$	4.80580	0.193946
$615$	0	0
$616$	2.16776	0.0873416
$617$	−7.76105	−0.312448	−0.156224	−	0.987722i	$-0.549932\pi$
$617$	−7.76105	−0.312448	−0.156224	+	0.987722i	$0.549932\pi$
$618$	0	0
$619$	43.8234	1.76141	0.880705	−	0.473666i	$-0.157069\pi$
$619$	43.8234	1.76141	0.880705	+	0.473666i	$0.157069\pi$
$620$	−16.0454	−0.644398
$621$	0	0
$622$	0.373299	0.0149679
$623$	−12.6853	−0.508225
$624$	0	0
$625$	79.9427	3.19771
$626$	−4.23033	−0.169078
$627$	0	0
$628$	17.7114	0.706763
$629$	−22.2103	−0.885583
$630$	0	0
$631$	20.4859	0.815530	0.407765	−	0.913087i	$-0.366308\pi$
$631$	20.4859	0.815530	0.407765	+	0.913087i	$0.366308\pi$
$632$	−12.0087	−0.477679
$633$	0	0
$634$	0.474907	0.0188609
$635$	−47.3540	−1.87918
$636$	0	0
$637$	22.7196	0.900182
$638$	−0.284905	−0.0112795
$639$	0	0
$640$	30.5025	1.20572
$641$	34.5586	1.36498	0.682492	−	0.730893i	$-0.260896\pi$
$641$	34.5586	1.36498	0.682492	+	0.730893i	$0.260896\pi$
$642$	0	0
$643$	29.7415	1.17289	0.586445	−	0.809989i	$-0.300527\pi$
$643$	29.7415	1.17289	0.586445	+	0.809989i	$0.300527\pi$
$644$	−2.72900	−0.107538
$645$	0	0
$646$	−3.07985	−0.121175
$647$	43.1443	1.69618	0.848089	−	0.529853i	$-0.177753\pi$
$647$	43.1443	1.69618	0.848089	+	0.529853i	$0.177753\pi$
$648$	0	0
$649$	16.4542	0.645884
$650$	−12.6304	−0.495405
$651$	0	0
$652$	−15.4191	−0.603857
$653$	−31.1897	−1.22055	−0.610273	−	0.792191i	$-0.708940\pi$
$653$	−31.1897	−1.22055	−0.610273	+	0.792191i	$0.708940\pi$
$654$	0	0
$655$	−51.5326	−2.01355
$656$	20.9249	0.816980
$657$	0	0
$658$	0.121622	0.00474131
$659$	40.6668	1.58415	0.792077	−	0.610421i	$-0.209000\pi$
$659$	40.6668	1.58415	0.792077	+	0.610421i	$0.209000\pi$
$660$	0	0
$661$	17.3627	0.675329	0.337665	−	0.941266i	$-0.390363\pi$
$661$	17.3627	0.675329	0.337665	+	0.941266i	$0.390363\pi$
$662$	8.33590	0.323984
$663$	0	0
$664$	11.0305	0.428067
$665$	14.7984	0.573858
$666$	0	0
$667$	0.728130	0.0281933
$668$	8.63010	0.333908
$669$	0	0
$670$	−7.52292	−0.290636
$671$	−20.8257	−0.803969
$672$	0	0
$673$	44.0885	1.69949	0.849743	−	0.527198i	$-0.176757\pi$
$673$	44.0885	1.69949	0.849743	+	0.527198i	$0.176757\pi$
$674$	3.91048	0.150626
$675$	0	0
$676$	−5.90522	−0.227124
$677$	34.6936	1.33338	0.666691	−	0.745334i	$-0.267710\pi$
$677$	34.6936	1.33338	0.666691	+	0.745334i	$0.267710\pi$
$678$	0	0
$679$	−7.24445	−0.278016
$680$	−17.0547	−0.654018
$681$	0	0
$682$	−0.928659	−0.0355602
$683$	42.7336	1.63515	0.817577	−	0.575819i	$-0.195317\pi$
$683$	42.7336	1.63515	0.817577	+	0.575819i	$0.195317\pi$
$684$	0	0
$685$	1.27105	0.0485644
$686$	3.52964	0.134762
$687$	0	0
$688$	26.6614	1.01646
$689$	−17.9187	−0.682650
$690$	0	0
$691$	−16.0321	−0.609888	−0.304944	−	0.952370i	$-0.598638\pi$
$691$	−16.0321	−0.609888	−0.304944	+	0.952370i	$0.598638\pi$
$692$	−6.05471	−0.230166
$693$	0	0
$694$	7.69106	0.291949
$695$	23.8291	0.903888
$696$	0	0
$697$	−24.1370	−0.914254
$698$	−4.19072	−0.158621
$699$	0	0
$700$	29.1809	1.10293
$701$	−41.7837	−1.57815	−0.789075	−	0.614297i	$-0.789440\pi$
$701$	−41.7837	−1.57815	−0.789075	+	0.614297i	$0.789440\pi$
$702$	0	0
$703$	15.9383	0.601123
$704$	−13.0942	−0.493507
$705$	0	0
$706$	−0.821886	−0.0309321
$707$	4.66369	0.175396
$708$	0	0
$709$	−16.8956	−0.634527	−0.317263	−	0.948337i	$-0.602764\pi$
$709$	−16.8956	−0.634527	−0.317263	+	0.948337i	$0.602764\pi$
$710$	4.91832	0.184581
$711$	0	0
$712$	−10.4910	−0.393165
$713$	2.37337	0.0888835
$714$	0	0
$715$	33.5950	1.25638
$716$	27.6395	1.03294
$717$	0	0
$718$	−2.45294	−0.0915429
$719$	−16.7005	−0.622825	−0.311412	−	0.950275i	$-0.600802\pi$
$719$	−16.7005	−0.622825	−0.311412	+	0.950275i	$0.600802\pi$
$720$	0	0
$721$	−9.54535	−0.355487
$722$	−2.38303	−0.0886872
$723$	0	0
$724$	−30.8558	−1.14675
$725$	−7.78580	−0.289157
$726$	0	0
$727$	22.1192	0.820356	0.410178	−	0.912005i	$-0.365466\pi$
$727$	22.1192	0.820356	0.410178	+	0.912005i	$0.365466\pi$
$728$	−4.39701	−0.162964
$729$	0	0
$730$	−7.40242	−0.273976
$731$	−30.7541	−1.13748
$732$	0	0
$733$	−13.1367	−0.485217	−0.242608	−	0.970124i	$-0.578003\pi$
$733$	−13.1367	−0.485217	−0.242608	+	0.970124i	$0.578003\pi$
$734$	3.59329	0.132631
$735$	0	0
$736$	−3.40213	−0.125404
$737$	14.4654	0.532839
$738$	0	0
$739$	−44.9292	−1.65275	−0.826373	−	0.563123i	$-0.809600\pi$
$739$	−44.9292	−1.65275	−0.826373	+	0.563123i	$0.809600\pi$
$740$	43.4749	1.59817
$741$	0	0
$742$	−1.24610	−0.0457457
$743$	−22.3697	−0.820666	−0.410333	−	0.911936i	$-0.634587\pi$
$743$	−22.3697	−0.820666	−0.410333	+	0.911936i	$0.634587\pi$
$744$	0	0
$745$	57.8833	2.12068
$746$	−5.09619	−0.186585
$747$	0	0
$748$	16.1536	0.590634
$749$	18.3277	0.669680
$750$	0	0
$751$	11.8860	0.433728	0.216864	−	0.976202i	$-0.430417\pi$
$751$	11.8860	0.433728	0.216864	+	0.976202i	$0.430417\pi$
$752$	−1.59502	−0.0581645
$753$	0	0
$754$	0.577890	0.0210455
$755$	40.1568	1.46146
$756$	0	0
$757$	5.21780	0.189644	0.0948221	−	0.995494i	$-0.469772\pi$
$757$	5.21780	0.189644	0.0948221	+	0.995494i	$0.469772\pi$
$758$	1.67994	0.0610183
$759$	0	0
$760$	12.2386	0.443940
$761$	−0.698407	−0.0253172	−0.0126586	−	0.999920i	$-0.504029\pi$
$761$	−0.698407	−0.0253172	−0.0126586	+	0.999920i	$0.504029\pi$
$762$	0	0
$763$	4.17207	0.151039
$764$	−39.4283	−1.42646
$765$	0	0
$766$	−8.54141	−0.308614
$767$	−33.3751	−1.20511
$768$	0	0
$769$	11.1266	0.401235	0.200617	−	0.979670i	$-0.435705\pi$
$769$	11.1266	0.401235	0.200617	+	0.979670i	$0.435705\pi$
$770$	2.33625	0.0841926
$771$	0	0
$772$	5.18872	0.186746
$773$	15.4895	0.557120	0.278560	−	0.960419i	$-0.410143\pi$
$773$	15.4895	0.557120	0.278560	+	0.960419i	$0.410143\pi$
$774$	0	0
$775$	−25.3782	−0.911611
$776$	−5.99129	−0.215075
$777$	0	0
$778$	−6.21855	−0.222946
$779$	17.3209	0.620585
$780$	0	0
$781$	−9.45713	−0.338403
$782$	1.24263	0.0444365
$783$	0	0
$784$	−20.7205	−0.740018
$785$	38.7506	1.38307
$786$	0	0
$787$	−35.2358	−1.25602	−0.628011	−	0.778204i	$-0.716131\pi$
$787$	−35.2358	−1.25602	−0.628011	+	0.778204i	$0.716131\pi$
$788$	−32.4661	−1.15656
$789$	0	0
$790$	−12.9420	−0.460457
$791$	10.9398	0.388976
$792$	0	0
$793$	42.2422	1.50006
$794$	4.49601	0.159558
$795$	0	0
$796$	−3.79786	−0.134612
$797$	−26.5128	−0.939131	−0.469565	−	0.882898i	$-0.655589\pi$
$797$	−26.5128	−0.939131	−0.469565	+	0.882898i	$0.655589\pi$
$798$	0	0
$799$	1.83987	0.0650899
$800$	36.3786	1.28618
$801$	0	0
$802$	−1.11776	−0.0394694
$803$	14.2337	0.502295
$804$	0	0
$805$	−5.97075	−0.210441
$806$	1.88366	0.0663491
$807$	0	0
$808$	3.85696	0.135687
$809$	−4.72350	−0.166070	−0.0830348	−	0.996547i	$-0.526461\pi$
$809$	−4.72350	−0.166070	−0.0830348	+	0.996547i	$0.526461\pi$
$810$	0	0
$811$	−14.2884	−0.501733	−0.250866	−	0.968022i	$-0.580716\pi$
$811$	−14.2884	−0.501733	−0.250866	+	0.968022i	$0.580716\pi$
$812$	−1.33514	−0.0468542
$813$	0	0
$814$	2.51620	0.0881928
$815$	−33.7352	−1.18169
$816$	0	0
$817$	22.0694	0.772109
$818$	−2.64445	−0.0924612
$819$	0	0
$820$	47.2463	1.64991
$821$	−21.8703	−0.763279	−0.381639	−	0.924311i	$-0.624640\pi$
$821$	−21.8703	−0.763279	−0.381639	+	0.924311i	$0.624640\pi$
$822$	0	0
$823$	23.5079	0.819434	0.409717	−	0.912213i	$-0.365628\pi$
$823$	23.5079	0.819434	0.409717	+	0.912213i	$0.365628\pi$
$824$	−7.89418	−0.275007
$825$	0	0
$826$	−2.32096	−0.0807565
$827$	17.8136	0.619441	0.309720	−	0.950828i	$-0.399765\pi$
$827$	17.8136	0.619441	0.309720	+	0.950828i	$0.399765\pi$
$828$	0	0
$829$	−53.8459	−1.87014	−0.935072	−	0.354458i	$-0.884665\pi$
$829$	−53.8459	−1.87014	−0.935072	+	0.354458i	$0.884665\pi$
$830$	11.8878	0.412633
$831$	0	0
$832$	26.5599	0.920798
$833$	23.9012	0.828129
$834$	0	0
$835$	18.8817	0.653428
$836$	−11.5919	−0.400916
$837$	0	0
$838$	−8.10841	−0.280100
$839$	−52.0831	−1.79811	−0.899055	−	0.437837i	$-0.855745\pi$
$839$	−52.0831	−1.79811	−0.899055	+	0.437837i	$0.855745\pi$
$840$	0	0
$841$	−28.6438	−0.987716
$842$	0.400969	0.0138183
$843$	0	0
$844$	−20.1229	−0.692660
$845$	−12.9200	−0.444460
$846$	0	0
$847$	8.18145	0.281118
$848$	16.3421	0.561190
$849$	0	0
$850$	−13.2873	−0.455751
$851$	−6.43065	−0.220440
$852$	0	0
$853$	48.5628	1.66276	0.831379	−	0.555706i	$-0.187552\pi$
$853$	48.5628	1.66276	0.831379	+	0.555706i	$0.187552\pi$
$854$	2.93759	0.100522
$855$	0	0
$856$	15.1574	0.518068
$857$	43.5996	1.48933	0.744667	−	0.667436i	$-0.232608\pi$
$857$	43.5996	1.48933	0.744667	+	0.667436i	$0.232608\pi$
$858$	0	0
$859$	16.3040	0.556285	0.278143	−	0.960540i	$-0.410281\pi$
$859$	16.3040	0.556285	0.278143	+	0.960540i	$0.410281\pi$
$860$	60.1987	2.05276
$861$	0	0
$862$	−3.65240	−0.124401
$863$	50.1840	1.70828	0.854142	−	0.520039i	$-0.174083\pi$
$863$	50.1840	1.70828	0.854142	+	0.520039i	$0.174083\pi$
$864$	0	0
$865$	−13.2470	−0.450413
$866$	1.19255	0.0405245
$867$	0	0
$868$	−4.35195	−0.147715
$869$	24.8855	0.844182
$870$	0	0
$871$	−29.3410	−0.994183
$872$	3.45038	0.116845
$873$	0	0
$874$	−0.891722	−0.0301630
$875$	39.3732	1.33106
$876$	0	0
$877$	−44.8657	−1.51501	−0.757503	−	0.652831i	$-0.773581\pi$
$877$	−44.8657	−1.51501	−0.757503	+	0.652831i	$0.773581\pi$
$878$	−4.87856	−0.164643
$879$	0	0
$880$	−30.6390	−1.03284
$881$	18.2315	0.614235	0.307117	−	0.951672i	$-0.400636\pi$
$881$	18.2315	0.614235	0.307117	+	0.951672i	$0.400636\pi$
$882$	0	0
$883$	−17.5838	−0.591741	−0.295871	−	0.955228i	$-0.595610\pi$
$883$	−17.5838	−0.591741	−0.295871	+	0.955228i	$0.595610\pi$
$884$	−32.7654	−1.10202
$885$	0	0
$886$	6.81390	0.228917
$887$	−32.9670	−1.10692	−0.553461	−	0.832875i	$-0.686693\pi$
$887$	−32.9670	−1.10692	−0.553461	+	0.832875i	$0.686693\pi$
$888$	0	0
$889$	−12.8437	−0.430764
$890$	−11.3064	−0.378990
$891$	0	0
$892$	−14.2279	−0.476386
$893$	−1.32030	−0.0441823
$894$	0	0
$895$	60.4721	2.02136
$896$	8.27313	0.276386
$897$	0	0
$898$	0.104946	0.00350208
$899$	1.16115	0.0387266
$900$	0	0
$901$	−18.8507	−0.628009
$902$	2.73448	0.0910481
$903$	0	0
$904$	9.04744	0.300914
$905$	−67.5091	−2.24408
$906$	0	0
$907$	−39.2560	−1.30347	−0.651737	−	0.758445i	$-0.725959\pi$
$907$	−39.2560	−1.30347	−0.651737	+	0.758445i	$0.725959\pi$
$908$	45.1951	1.49985
$909$	0	0
$910$	−4.73877	−0.157089
$911$	15.2296	0.504581	0.252290	−	0.967652i	$-0.418816\pi$
$911$	15.2296	0.504581	0.252290	+	0.967652i	$0.418816\pi$
$912$	0	0
$913$	−22.8584	−0.756503
$914$	6.55175	0.216713
$915$	0	0
$916$	0.0969119	0.00320206
$917$	−13.9771	−0.461564
$918$	0	0
$919$	14.7991	0.488176	0.244088	−	0.969753i	$-0.421511\pi$
$919$	14.7991	0.488176	0.244088	+	0.969753i	$0.421511\pi$
$920$	−4.93792	−0.162798
$921$	0	0
$922$	−10.3288	−0.340162
$923$	19.1825	0.631400
$924$	0	0
$925$	68.7621	2.26088
$926$	−7.33493	−0.241041
$927$	0	0
$928$	−1.66446	−0.0546387
$929$	30.6071	1.00419	0.502093	−	0.864814i	$-0.332564\pi$
$929$	30.6071	1.00419	0.502093	+	0.864814i	$0.332564\pi$
$930$	0	0
$931$	−17.1517	−0.562124
$932$	43.5088	1.42518
$933$	0	0
$934$	3.25293	0.106439
$935$	35.3423	1.15582
$936$	0	0
$937$	−3.43602	−0.112250	−0.0561249	−	0.998424i	$-0.517875\pi$
$937$	−3.43602	−0.112250	−0.0561249	+	0.998424i	$0.517875\pi$
$938$	−2.04042	−0.0666222
$939$	0	0
$940$	−3.60140	−0.117465
$941$	−19.3715	−0.631492	−0.315746	−	0.948844i	$-0.602255\pi$
$941$	−19.3715	−0.631492	−0.315746	+	0.948844i	$0.602255\pi$
$942$	0	0
$943$	−6.98850	−0.227577
$944$	30.4385	0.990688
$945$	0	0
$946$	3.48412	0.113279
$947$	13.7891	0.448084	0.224042	−	0.974579i	$-0.428075\pi$
$947$	13.7891	0.448084	0.224042	+	0.974579i	$0.428075\pi$
$948$	0	0
$949$	−28.8711	−0.937195
$950$	9.53507	0.309359
$951$	0	0
$952$	−4.62570	−0.149920
$953$	−37.5436	−1.21616	−0.608078	−	0.793878i	$-0.708059\pi$
$953$	−37.5436	−1.21616	−0.608078	+	0.793878i	$0.708059\pi$
$954$	0	0
$955$	−86.2647	−2.79146
$956$	39.8662	1.28936
$957$	0	0
$958$	−6.27968	−0.202887
$959$	0.344744	0.0111324
$960$	0	0
$961$	−27.2152	−0.877909
$962$	−5.10377	−0.164552
$963$	0	0
$964$	55.8915	1.80015
$965$	11.3523	0.365445
$966$	0	0
$967$	24.5820	0.790505	0.395252	−	0.918573i	$-0.370657\pi$
$967$	24.5820	0.790505	0.395252	+	0.918573i	$0.370657\pi$
$968$	6.76621	0.217474
$969$	0	0
$970$	−6.45696	−0.207321
$971$	13.0415	0.418522	0.209261	−	0.977860i	$-0.432894\pi$
$971$	13.0415	0.418522	0.209261	+	0.977860i	$0.432894\pi$
$972$	0	0
$973$	6.46310	0.207197
$974$	0.654109	0.0209590
$975$	0	0
$976$	−38.5254	−1.23317
$977$	48.4395	1.54972	0.774859	−	0.632134i	$-0.217821\pi$
$977$	48.4395	1.54972	0.774859	+	0.632134i	$0.217821\pi$
$978$	0	0
$979$	21.7403	0.694824
$980$	−46.7848	−1.49448
$981$	0	0
$982$	9.62328	0.307091
$983$	−33.1022	−1.05580	−0.527898	−	0.849308i	$-0.677020\pi$
$983$	−33.1022	−1.05580	−0.527898	+	0.849308i	$0.677020\pi$
$984$	0	0
$985$	−71.0323	−2.26328
$986$	0.607947	0.0193610
$987$	0	0
$988$	23.5127	0.748038
$989$	−8.90437	−0.283143
$990$	0	0
$991$	−17.3919	−0.552471	−0.276235	−	0.961090i	$-0.589087\pi$
$991$	−17.3919	−0.552471	−0.276235	+	0.961090i	$0.589087\pi$
$992$	−5.42540	−0.172256
$993$	0	0
$994$	1.33398	0.0423114
$995$	−8.30930	−0.263422
$996$	0	0
$997$	−17.2834	−0.547370	−0.273685	−	0.961819i	$-0.588243\pi$
$997$	−17.2834	−0.547370	−0.273685	+	0.961819i	$0.588243\pi$
$998$	0.600826	0.0190188
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.41		72
3.2	odd	2		6561.2.a.d.1.32		72
81.4	even	27		729.2.g.c.136.5		144
81.7	even	27		729.2.g.d.352.4		144
81.20	odd	54		729.2.g.b.595.4		144
81.23	odd	54		729.2.g.a.379.5		144
81.31	even	27		81.2.g.a.70.4	yes	144
81.34	even	27		81.2.g.a.22.4	✓	144
81.47	odd	54		243.2.g.a.199.5		144
81.50	odd	54		243.2.g.a.127.5		144
81.58	even	27		729.2.g.d.379.4		144
81.61	even	27		729.2.g.c.595.5		144
81.74	odd	54		729.2.g.a.352.5		144
81.77	odd	54		729.2.g.b.136.4		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.22.4	✓	144	81.34	even	27
81.2.g.a.70.4	yes	144	81.31	even	27
243.2.g.a.127.5		144	81.50	odd	54
243.2.g.a.199.5		144	81.47	odd	54
729.2.g.a.352.5		144	81.74	odd	54
729.2.g.a.379.5		144	81.23	odd	54
729.2.g.b.136.4		144	81.77	odd	54
729.2.g.b.595.4		144	81.20	odd	54
729.2.g.c.136.5		144	81.4	even	27
729.2.g.c.595.5		144	81.61	even	27
729.2.g.d.352.4		144	81.7	even	27
729.2.g.d.379.4		144	81.58	even	27
6561.2.a.c.1.41		72	1.1	even	1	trivial
6561.2.a.d.1.32		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+0.241745 q^{2} -1.94156 q^{4} -4.24792 q^{5} -1.15215 q^{7} -0.952852 q^{8} -1.02691 q^{10} +1.97459 q^{11} -4.00518 q^{13} -0.278527 q^{14} +3.65277 q^{16} -4.21350 q^{17} +3.02364 q^{19} +8.24758 q^{20} +0.477346 q^{22} -1.21995 q^{23} +13.0448 q^{25} -0.968232 q^{26} +2.23697 q^{28} -0.596851 q^{29} -1.94546 q^{31} +2.78874 q^{32} -1.01859 q^{34} +4.89425 q^{35} +5.27123 q^{37} +0.730948 q^{38} +4.04763 q^{40} +5.72850 q^{41} +7.29895 q^{43} -3.83378 q^{44} -0.294917 q^{46} -0.436661 q^{47} -5.67254 q^{49} +3.15351 q^{50} +7.77630 q^{52} +4.47389 q^{53} -8.38788 q^{55} +1.09783 q^{56} -0.144286 q^{58} +8.33298 q^{59} -10.5469 q^{61} -0.470306 q^{62} -6.63138 q^{64} +17.0137 q^{65} +7.32577 q^{67} +8.18075 q^{68} +1.18316 q^{70} -4.78942 q^{71} +7.20843 q^{73} +1.27429 q^{74} -5.87057 q^{76} -2.27503 q^{77} +12.6029 q^{79} -15.5167 q^{80} +1.38483 q^{82} -11.5763 q^{83} +17.8986 q^{85} +1.76448 q^{86} -1.88149 q^{88} +11.0101 q^{89} +4.61458 q^{91} +2.36861 q^{92} -0.105560 q^{94} -12.8442 q^{95} +6.28775 q^{97} -1.37131 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38}+ \cdots - 81 q^{98}+O(q^{100}) \) 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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