LMFDB - Embedded newform 6561.2.a.c.1.12

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			1.12
Character	$\chi$	$=$	6561.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.11576 q^{2} +2.47642 q^{4} +3.89294 q^{5} -3.33519 q^{7} -1.00799 q^{8} -8.23651 q^{10} +0.756735 q^{11} +2.48623 q^{13} +7.05645 q^{14} -2.82018 q^{16} +0.745917 q^{17} -4.48802 q^{19} +9.64056 q^{20} -1.60107 q^{22} -2.28672 q^{23} +10.1550 q^{25} -5.26026 q^{26} -8.25934 q^{28} +0.463993 q^{29} -3.94211 q^{31} +7.98279 q^{32} -1.57818 q^{34} -12.9837 q^{35} -4.75658 q^{37} +9.49556 q^{38} -3.92405 q^{40} +4.89250 q^{41} +5.94108 q^{43} +1.87399 q^{44} +4.83814 q^{46} -10.7663 q^{47} +4.12352 q^{49} -21.4854 q^{50} +6.15696 q^{52} -11.5059 q^{53} +2.94592 q^{55} +3.36184 q^{56} -0.981696 q^{58} -4.14919 q^{59} +0.267217 q^{61} +8.34055 q^{62} -11.2493 q^{64} +9.67875 q^{65} -1.91755 q^{67} +1.84720 q^{68} +27.4703 q^{70} +6.75379 q^{71} +7.93114 q^{73} +10.0638 q^{74} -11.1142 q^{76} -2.52386 q^{77} +4.31679 q^{79} -10.9788 q^{80} -10.3513 q^{82} -2.54689 q^{83} +2.90381 q^{85} -12.5699 q^{86} -0.762781 q^{88} -5.38761 q^{89} -8.29206 q^{91} -5.66288 q^{92} +22.7790 q^{94} -17.4716 q^{95} +9.48838 q^{97} -8.72436 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38}+ \cdots - 81 q^{98}+O(q^{100}) $ 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8 - 36 * q^11 - 36 * q^14 + 45 * q^16 - 36 * q^17 - 54 * q^20 - 54 * q^23 + 36 * q^25 - 45 * q^26 + 9 * q^28 - 54 * q^29 - 63 * q^32 - 72 * q^35 - 54 * q^38 - 72 * q^41 - 90 * q^44 - 90 * q^47 + 18 * q^49 - 45 * q^50 - 45 * q^53 + 9 * q^55 - 108 * q^56 + 18 * q^58 - 108 * q^59 - 72 * q^62 + 9 * q^64 - 72 * q^65 - 108 * q^68 - 126 * q^71 - 90 * q^74 - 72 * q^77 - 144 * q^80 - 18 * q^82 - 108 * q^83 - 90 * q^86 - 108 * q^89 - 72 * q^92 - 144 * q^95 - 81 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.11576	−1.49607	−0.748033	−	0.663662i	$-0.769001\pi$
$2$	−2.11576	−1.49607	−0.748033	+	0.663662i	$0.769001\pi$
$3$	0	0
$3$	0	0
$4$	2.47642	1.23821
$5$	3.89294	1.74098	0.870488	−	0.492190i	$-0.163804\pi$
$5$	3.89294	1.74098	0.870488	+	0.492190i	$0.163804\pi$
$6$	0	0
$7$	−3.33519	−1.26058	−0.630292	−	0.776358i	$-0.717065\pi$
$7$	−3.33519	−1.26058	−0.630292	+	0.776358i	$0.717065\pi$
$8$	−1.00799	−0.356378
$9$	0	0
$10$	−8.23651	−2.60461
$11$	0.756735	0.228164	0.114082	−	0.993471i	$-0.463607\pi$
$11$	0.756735	0.228164	0.114082	+	0.993471i	$0.463607\pi$
$12$	0	0
$13$	2.48623	0.689557	0.344778	−	0.938684i	$-0.387954\pi$
$13$	2.48623	0.689557	0.344778	+	0.938684i	$0.387954\pi$
$14$	7.05645	1.88592
$15$	0	0
$16$	−2.82018	−0.705045
$17$	0.745917	0.180911	0.0904557	−	0.995900i	$-0.471168\pi$
$17$	0.745917	0.180911	0.0904557	+	0.995900i	$0.471168\pi$
$18$	0	0
$19$	−4.48802	−1.02962	−0.514812	−	0.857303i	$-0.672138\pi$
$19$	−4.48802	−1.02962	−0.514812	+	0.857303i	$0.672138\pi$
$20$	9.64056	2.15569
$21$	0	0
$22$	−1.60107	−0.341348
$23$	−2.28672	−0.476814	−0.238407	−	0.971165i	$-0.576625\pi$
$23$	−2.28672	−0.476814	−0.238407	+	0.971165i	$0.576625\pi$
$24$	0	0
$25$	10.1550	2.03100
$26$	−5.26026	−1.03162
$27$	0	0
$28$	−8.25934	−1.56087
$29$	0.463993	0.0861614	0.0430807	−	0.999072i	$-0.486283\pi$
$29$	0.463993	0.0861614	0.0430807	+	0.999072i	$0.486283\pi$
$30$	0	0
$31$	−3.94211	−0.708025	−0.354012	−	0.935241i	$-0.615183\pi$
$31$	−3.94211	−0.708025	−0.354012	+	0.935241i	$0.615183\pi$
$32$	7.98279	1.41117
$33$	0	0
$34$	−1.57818	−0.270655
$35$	−12.9837	−2.19465
$36$	0	0
$37$	−4.75658	−0.781977	−0.390989	−	0.920395i	$-0.627867\pi$
$37$	−4.75658	−0.781977	−0.390989	+	0.920395i	$0.627867\pi$
$38$	9.49556	1.54038
$39$	0	0
$40$	−3.92405	−0.620446
$41$	4.89250	0.764081	0.382040	−	0.924146i	$-0.375222\pi$
$41$	4.89250	0.764081	0.382040	+	0.924146i	$0.375222\pi$
$42$	0	0
$43$	5.94108	0.906006	0.453003	−	0.891509i	$-0.350353\pi$
$43$	5.94108	0.906006	0.453003	+	0.891509i	$0.350353\pi$
$44$	1.87399	0.282515
$45$	0	0
$46$	4.83814	0.713345
$47$	−10.7663	−1.57043	−0.785217	−	0.619221i	$-0.787448\pi$
$47$	−10.7663	−1.57043	−0.785217	+	0.619221i	$0.787448\pi$
$48$	0	0
$49$	4.12352	0.589074
$50$	−21.4854	−3.03850
$51$	0	0
$52$	6.15696	0.853816
$53$	−11.5059	−1.58045	−0.790227	−	0.612815i	$-0.790037\pi$
$53$	−11.5059	−1.58045	−0.790227	+	0.612815i	$0.790037\pi$
$54$	0	0
$55$	2.94592	0.397228
$56$	3.36184	0.449245
$57$	0	0
$58$	−0.981696	−0.128903
$59$	−4.14919	−0.540178	−0.270089	−	0.962835i	$-0.587053\pi$
$59$	−4.14919	−0.540178	−0.270089	+	0.962835i	$0.587053\pi$
$60$	0	0
$61$	0.267217	0.0342136	0.0171068	−	0.999854i	$-0.494554\pi$
$61$	0.267217	0.0342136	0.0171068	+	0.999854i	$0.494554\pi$
$62$	8.34055	1.05925
$63$	0	0
$64$	−11.2493	−1.40616
$65$	9.67875	1.20050
$66$	0	0
$67$	−1.91755	−0.234266	−0.117133	−	0.993116i	$-0.537370\pi$
$67$	−1.91755	−0.234266	−0.117133	+	0.993116i	$0.537370\pi$
$68$	1.84720	0.224006
$69$	0	0
$70$	27.4703	3.28333
$71$	6.75379	0.801527	0.400764	−	0.916182i	$-0.368745\pi$
$71$	6.75379	0.801527	0.400764	+	0.916182i	$0.368745\pi$
$72$	0	0
$73$	7.93114	0.928269	0.464135	−	0.885765i	$-0.346365\pi$
$73$	7.93114	0.928269	0.464135	+	0.885765i	$0.346365\pi$
$74$	10.0638	1.16989
$75$	0	0
$76$	−11.1142	−1.27489
$77$	−2.52386	−0.287620
$78$	0	0
$79$	4.31679	0.485677	0.242838	−	0.970067i	$-0.421922\pi$
$79$	4.31679	0.485677	0.242838	+	0.970067i	$0.421922\pi$
$80$	−10.9788	−1.22747
$81$	0	0
$82$	−10.3513	−1.14311
$83$	−2.54689	−0.279558	−0.139779	−	0.990183i	$-0.544639\pi$
$83$	−2.54689	−0.279558	−0.139779	+	0.990183i	$0.544639\pi$
$84$	0	0
$85$	2.90381	0.314962
$86$	−12.5699	−1.35544
$87$	0	0
$88$	−0.762781	−0.0813128
$89$	−5.38761	−0.571086	−0.285543	−	0.958366i	$-0.592174\pi$
$89$	−5.38761	−0.571086	−0.285543	+	0.958366i	$0.592174\pi$
$90$	0	0
$91$	−8.29206	−0.869245
$92$	−5.66288	−0.590396
$93$	0	0
$94$	22.7790	2.34947
$95$	−17.4716	−1.79255
$96$	0	0
$97$	9.48838	0.963399	0.481700	−	0.876336i	$-0.340020\pi$
$97$	9.48838	0.963399	0.481700	+	0.876336i	$0.340020\pi$
$98$	−8.72436	−0.881293
$99$	0	0
$100$	25.1480	2.51480
$101$	−12.3077	−1.22466	−0.612329	−	0.790603i	$-0.709767\pi$
$101$	−12.3077	−1.22466	−0.612329	+	0.790603i	$0.709767\pi$
$102$	0	0
$103$	16.6703	1.64257	0.821286	−	0.570516i	$-0.193257\pi$
$103$	16.6703	1.64257	0.821286	+	0.570516i	$0.193257\pi$
$104$	−2.50610	−0.245743
$105$	0	0
$106$	24.3436	2.36446
$107$	−3.69388	−0.357101	−0.178550	−	0.983931i	$-0.557141\pi$
$107$	−3.69388	−0.357101	−0.178550	+	0.983931i	$0.557141\pi$
$108$	0	0
$109$	−17.3392	−1.66080	−0.830398	−	0.557170i	$-0.811887\pi$
$109$	−17.3392	−1.66080	−0.830398	+	0.557170i	$0.811887\pi$
$110$	−6.23285	−0.594279
$111$	0	0
$112$	9.40585	0.888769
$113$	−10.3090	−0.969792	−0.484896	−	0.874572i	$-0.661143\pi$
$113$	−10.3090	−0.969792	−0.484896	+	0.874572i	$0.661143\pi$
$114$	0	0
$115$	−8.90206	−0.830121
$116$	1.14904	0.106686
$117$	0	0
$118$	8.77867	0.808142
$119$	−2.48778	−0.228054
$120$	0	0
$121$	−10.4274	−0.947941
$122$	−0.565366	−0.0511858
$123$	0	0
$124$	−9.76233	−0.876683
$125$	20.0680	1.79494
$126$	0	0
$127$	−13.2130	−1.17246	−0.586232	−	0.810143i	$-0.699389\pi$
$127$	−13.2130	−1.17246	−0.586232	+	0.810143i	$0.699389\pi$
$128$	7.83513	0.692534
$129$	0	0
$130$	−20.4779	−1.79603
$131$	10.0613	0.879064	0.439532	−	0.898227i	$-0.355144\pi$
$131$	10.0613	0.879064	0.439532	+	0.898227i	$0.355144\pi$
$132$	0	0
$133$	14.9684	1.29793
$134$	4.05706	0.350477
$135$	0	0
$136$	−0.751877	−0.0644729
$137$	−9.46863	−0.808960	−0.404480	−	0.914547i	$-0.632547\pi$
$137$	−9.46863	−0.808960	−0.404480	+	0.914547i	$0.632547\pi$
$138$	0	0
$139$	−10.5003	−0.890627	−0.445314	−	0.895375i	$-0.646908\pi$
$139$	−10.5003	−0.890627	−0.445314	+	0.895375i	$0.646908\pi$
$140$	−32.1531	−2.71744
$141$	0	0
$142$	−14.2894	−1.19914
$143$	1.88142	0.157332
$144$	0	0
$145$	1.80630	0.150005
$146$	−16.7803	−1.38875
$147$	0	0
$148$	−11.7793	−0.968253
$149$	12.4294	1.01826	0.509128	−	0.860691i	$-0.329968\pi$
$149$	12.4294	1.01826	0.509128	+	0.860691i	$0.329968\pi$
$150$	0	0
$151$	−18.2270	−1.48329	−0.741647	−	0.670791i	$-0.765955\pi$
$151$	−18.2270	−1.48329	−0.741647	+	0.670791i	$0.765955\pi$
$152$	4.52389	0.366936
$153$	0	0
$154$	5.33986	0.430299
$155$	−15.3464	−1.23265
$156$	0	0
$157$	23.3719	1.86528	0.932640	−	0.360809i	$-0.117499\pi$
$157$	23.3719	1.86528	0.932640	+	0.360809i	$0.117499\pi$
$158$	−9.13327	−0.726604
$159$	0	0
$160$	31.0765	2.45682
$161$	7.62665	0.601064
$162$	0	0
$163$	11.7238	0.918278	0.459139	−	0.888364i	$-0.348158\pi$
$163$	11.7238	0.918278	0.459139	+	0.888364i	$0.348158\pi$
$164$	12.1159	0.946093
$165$	0	0
$166$	5.38860	0.418237
$167$	−24.2131	−1.87366	−0.936832	−	0.349780i	$-0.886256\pi$
$167$	−24.2131	−1.87366	−0.936832	+	0.349780i	$0.886256\pi$
$168$	0	0
$169$	−6.81865	−0.524512
$170$	−6.14375	−0.471204
$171$	0	0
$172$	14.7126	1.12183
$173$	−15.5125	−1.17939	−0.589695	−	0.807626i	$-0.700752\pi$
$173$	−15.5125	−1.17939	−0.589695	+	0.807626i	$0.700752\pi$
$174$	0	0
$175$	−33.8688	−2.56024
$176$	−2.13413	−0.160866
$177$	0	0
$178$	11.3989	0.854382
$179$	−7.18825	−0.537275	−0.268637	−	0.963241i	$-0.586573\pi$
$179$	−7.18825	−0.537275	−0.268637	+	0.963241i	$0.586573\pi$
$180$	0	0
$181$	−7.73776	−0.575143	−0.287572	−	0.957759i	$-0.592848\pi$
$181$	−7.73776	−0.575143	−0.287572	+	0.957759i	$0.592848\pi$
$182$	17.5440	1.30045
$183$	0	0
$184$	2.30499	0.169926
$185$	−18.5171	−1.36140
$186$	0	0
$187$	0.564461	0.0412775
$188$	−26.6620	−1.94453
$189$	0	0
$190$	36.9657	2.68177
$191$	−10.0782	−0.729230	−0.364615	−	0.931158i	$-0.618799\pi$
$191$	−10.0782	−0.729230	−0.364615	+	0.931158i	$0.618799\pi$
$192$	0	0
$193$	−15.9923	−1.15115	−0.575575	−	0.817749i	$-0.695222\pi$
$193$	−15.9923	−1.15115	−0.575575	+	0.817749i	$0.695222\pi$
$194$	−20.0751	−1.44131
$195$	0	0
$196$	10.2116	0.729398
$197$	10.9700	0.781578	0.390789	−	0.920480i	$-0.372202\pi$
$197$	10.9700	0.781578	0.390789	+	0.920480i	$0.372202\pi$
$198$	0	0
$199$	3.51072	0.248869	0.124434	−	0.992228i	$-0.460288\pi$
$199$	3.51072	0.248869	0.124434	+	0.992228i	$0.460288\pi$
$200$	−10.2361	−0.723803
$201$	0	0
$202$	26.0400	1.83217
$203$	−1.54751	−0.108614
$204$	0	0
$205$	19.0462	1.33025
$206$	−35.2703	−2.45740
$207$	0	0
$208$	−7.01162	−0.486169
$209$	−3.39624	−0.234923
$210$	0	0
$211$	−7.68374	−0.528971	−0.264485	−	0.964390i	$-0.585202\pi$
$211$	−7.68374	−0.528971	−0.264485	+	0.964390i	$0.585202\pi$
$212$	−28.4934	−1.95693
$213$	0	0
$214$	7.81534	0.534246
$215$	23.1283	1.57733
$216$	0	0
$217$	13.1477	0.892525
$218$	36.6856	2.48466
$219$	0	0
$220$	7.29534	0.491852
$221$	1.85452	0.124749
$222$	0	0
$223$	10.8443	0.726186	0.363093	−	0.931753i	$-0.381721\pi$
$223$	10.8443	0.726186	0.363093	+	0.931753i	$0.381721\pi$
$224$	−26.6242	−1.77890
$225$	0	0
$226$	21.8114	1.45087
$227$	−10.8613	−0.720893	−0.360446	−	0.932780i	$-0.617376\pi$
$227$	−10.8613	−0.720893	−0.360446	+	0.932780i	$0.617376\pi$
$228$	0	0
$229$	2.28946	0.151292	0.0756460	−	0.997135i	$-0.475898\pi$
$229$	2.28946	0.151292	0.0756460	+	0.997135i	$0.475898\pi$
$230$	18.8346	1.24192
$231$	0	0
$232$	−0.467701	−0.0307060
$233$	17.8111	1.16684	0.583422	−	0.812169i	$-0.301714\pi$
$233$	17.8111	1.16684	0.583422	+	0.812169i	$0.301714\pi$
$234$	0	0
$235$	−41.9127	−2.73409
$236$	−10.2751	−0.668854
$237$	0	0
$238$	5.26353	0.341184
$239$	−12.2663	−0.793438	−0.396719	−	0.917940i	$-0.629851\pi$
$239$	−12.2663	−0.793438	−0.396719	+	0.917940i	$0.629851\pi$
$240$	0	0
$241$	17.4995	1.12724	0.563620	−	0.826034i	$-0.309408\pi$
$241$	17.4995	1.12724	0.563620	+	0.826034i	$0.309408\pi$
$242$	22.0617	1.41818
$243$	0	0
$244$	0.661742	0.0423637
$245$	16.0526	1.02556
$246$	0	0
$247$	−11.1583	−0.709984
$248$	3.97361	0.252325
$249$	0	0
$250$	−42.4590	−2.68534
$251$	18.8202	1.18792	0.593962	−	0.804493i	$-0.297563\pi$
$251$	18.8202	1.18792	0.593962	+	0.804493i	$0.297563\pi$
$252$	0	0
$253$	−1.73044	−0.108792
$254$	27.9555	1.75408
$255$	0	0
$256$	5.92133	0.370083
$257$	17.2431	1.07559	0.537797	−	0.843074i	$-0.319257\pi$
$257$	17.2431	1.07559	0.537797	+	0.843074i	$0.319257\pi$
$258$	0	0
$259$	15.8641	0.985749
$260$	23.9687	1.48647
$261$	0	0
$262$	−21.2873	−1.31514
$263$	−24.6115	−1.51761	−0.758806	−	0.651317i	$-0.774217\pi$
$263$	−24.6115	−1.51761	−0.758806	+	0.651317i	$0.774217\pi$
$264$	0	0
$265$	−44.7917	−2.75153
$266$	−31.6695	−1.94178
$267$	0	0
$268$	−4.74866	−0.290070
$269$	10.6490	0.649279	0.324639	−	0.945838i	$-0.394757\pi$
$269$	10.6490	0.649279	0.324639	+	0.945838i	$0.394757\pi$
$270$	0	0
$271$	−4.71634	−0.286498	−0.143249	−	0.989687i	$-0.545755\pi$
$271$	−4.71634	−0.286498	−0.143249	+	0.989687i	$0.545755\pi$
$272$	−2.10362	−0.127551
$273$	0	0
$274$	20.0333	1.21026
$275$	7.68462	0.463400
$276$	0	0
$277$	−6.61161	−0.397253	−0.198627	−	0.980075i	$-0.563648\pi$
$277$	−6.61161	−0.397253	−0.198627	+	0.980075i	$0.563648\pi$
$278$	22.2162	1.33244
$279$	0	0
$280$	13.0875	0.782125
$281$	−2.47619	−0.147717	−0.0738586	−	0.997269i	$-0.523531\pi$
$281$	−2.47619	−0.147717	−0.0738586	+	0.997269i	$0.523531\pi$
$282$	0	0
$283$	0.927155	0.0551137	0.0275568	−	0.999620i	$-0.491227\pi$
$283$	0.927155	0.0551137	0.0275568	+	0.999620i	$0.491227\pi$
$284$	16.7252	0.992459
$285$	0	0
$286$	−3.98062	−0.235379
$287$	−16.3174	−0.963189
$288$	0	0
$289$	−16.4436	−0.967271
$290$	−3.82168	−0.224417
$291$	0	0
$292$	19.6408	1.14939
$293$	2.34506	0.137000	0.0684999	−	0.997651i	$-0.478179\pi$
$293$	2.34506	0.137000	0.0684999	+	0.997651i	$0.478179\pi$
$294$	0	0
$295$	−16.1525	−0.940437
$296$	4.79459	0.278680
$297$	0	0
$298$	−26.2976	−1.52338
$299$	−5.68531	−0.328790
$300$	0	0
$301$	−19.8147	−1.14210
$302$	38.5639	2.21910
$303$	0	0
$304$	12.6570	0.725931
$305$	1.04026	0.0595651
$306$	0	0
$307$	−15.2577	−0.870805	−0.435403	−	0.900236i	$-0.643394\pi$
$307$	−15.2577	−0.870805	−0.435403	+	0.900236i	$0.643394\pi$
$308$	−6.25013	−0.356134
$309$	0	0
$310$	32.4692	1.84413
$311$	−16.7436	−0.949445	−0.474722	−	0.880136i	$-0.657451\pi$
$311$	−16.7436	−0.949445	−0.474722	+	0.880136i	$0.657451\pi$
$312$	0	0
$313$	32.5733	1.84115	0.920577	−	0.390562i	$-0.127719\pi$
$313$	32.5733	1.84115	0.920577	+	0.390562i	$0.127719\pi$
$314$	−49.4492	−2.79058
$315$	0	0
$316$	10.6902	0.601370
$317$	21.6708	1.21716	0.608578	−	0.793494i	$-0.291740\pi$
$317$	21.6708	1.21716	0.608578	+	0.793494i	$0.291740\pi$
$318$	0	0
$319$	0.351120	0.0196589
$320$	−43.7928	−2.44809
$321$	0	0
$322$	−16.1361	−0.899231
$323$	−3.34769	−0.186271
$324$	0	0
$325$	25.2476	1.40049
$326$	−24.8047	−1.37380
$327$	0	0
$328$	−4.93160	−0.272302
$329$	35.9079	1.97966
$330$	0	0
$331$	23.8122	1.30884	0.654419	−	0.756132i	$-0.272913\pi$
$331$	23.8122	1.30884	0.654419	+	0.756132i	$0.272913\pi$
$332$	−6.30718	−0.346151
$333$	0	0
$334$	51.2289	2.80312
$335$	−7.46490	−0.407851
$336$	0	0
$337$	−18.3953	−1.00205	−0.501027	−	0.865432i	$-0.667044\pi$
$337$	−18.3953	−1.00205	−0.501027	+	0.865432i	$0.667044\pi$
$338$	14.4266	0.784704
$339$	0	0
$340$	7.19105	0.389990
$341$	−2.98313	−0.161546
$342$	0	0
$343$	9.59362	0.518007
$344$	−5.98855	−0.322881
$345$	0	0
$346$	32.8206	1.76444
$347$	−16.9111	−0.907836	−0.453918	−	0.891043i	$-0.649974\pi$
$347$	−16.9111	−0.907836	−0.453918	+	0.891043i	$0.649974\pi$
$348$	0	0
$349$	8.16721	0.437181	0.218590	−	0.975817i	$-0.429854\pi$
$349$	8.16721	0.437181	0.218590	+	0.975817i	$0.429854\pi$
$350$	71.6581	3.83029
$351$	0	0
$352$	6.04086	0.321979
$353$	24.6144	1.31009	0.655047	−	0.755588i	$-0.272649\pi$
$353$	24.6144	1.31009	0.655047	+	0.755588i	$0.272649\pi$
$354$	0	0
$355$	26.2921	1.39544
$356$	−13.3420	−0.707125
$357$	0	0
$358$	15.2086	0.803798
$359$	−12.6452	−0.667390	−0.333695	−	0.942681i	$-0.608296\pi$
$359$	−12.6452	−0.667390	−0.333695	+	0.942681i	$0.608296\pi$
$360$	0	0
$361$	1.14237	0.0601247
$362$	16.3712	0.860452
$363$	0	0
$364$	−20.5346	−1.07631
$365$	30.8754	1.61609
$366$	0	0
$367$	−10.4942	−0.547790	−0.273895	−	0.961760i	$-0.588312\pi$
$367$	−10.4942	−0.547790	−0.273895	+	0.961760i	$0.588312\pi$
$368$	6.44896	0.336175
$369$	0	0
$370$	39.1776	2.03675
$371$	38.3743	1.99230
$372$	0	0
$373$	11.7964	0.610794	0.305397	−	0.952225i	$-0.401211\pi$
$373$	11.7964	0.610794	0.305397	+	0.952225i	$0.401211\pi$
$374$	−1.19426	−0.0617538
$375$	0	0
$376$	10.8524	0.559669
$377$	1.15359	0.0594131
$378$	0	0
$379$	18.1371	0.931639	0.465819	−	0.884880i	$-0.345760\pi$
$379$	18.1371	0.931639	0.465819	+	0.884880i	$0.345760\pi$
$380$	−43.2671	−2.21955
$381$	0	0
$382$	21.3229	1.09098
$383$	15.5176	0.792914	0.396457	−	0.918053i	$-0.370240\pi$
$383$	15.5176	0.792914	0.396457	+	0.918053i	$0.370240\pi$
$384$	0	0
$385$	−9.82522	−0.500740
$386$	33.8358	1.72219
$387$	0	0
$388$	23.4972	1.19289
$389$	−15.4343	−0.782551	−0.391275	−	0.920274i	$-0.627966\pi$
$389$	−15.4343	−0.782551	−0.391275	+	0.920274i	$0.627966\pi$
$390$	0	0
$391$	−1.70570	−0.0862611
$392$	−4.15647	−0.209933
$393$	0	0
$394$	−23.2098	−1.16929
$395$	16.8050	0.845551
$396$	0	0
$397$	2.04261	0.102515	0.0512577	−	0.998685i	$-0.483677\pi$
$397$	2.04261	0.102515	0.0512577	+	0.998685i	$0.483677\pi$
$398$	−7.42783	−0.372324
$399$	0	0
$400$	−28.6389	−1.43194
$401$	14.9541	0.746774	0.373387	−	0.927676i	$-0.378196\pi$
$401$	14.9541	0.746774	0.373387	+	0.927676i	$0.378196\pi$
$402$	0	0
$403$	−9.80101	−0.488223
$404$	−30.4790	−1.51639
$405$	0	0
$406$	3.27415	0.162493
$407$	−3.59947	−0.178419
$408$	0	0
$409$	10.3886	0.513681	0.256841	−	0.966454i	$-0.417318\pi$
$409$	10.3886	0.513681	0.256841	+	0.966454i	$0.417318\pi$
$410$	−40.2971	−1.99013
$411$	0	0
$412$	41.2827	2.03385
$413$	13.8383	0.680940
$414$	0	0
$415$	−9.91490	−0.486703
$416$	19.8471	0.973083
$417$	0	0
$418$	7.18562	0.351460
$419$	0.530105	0.0258973	0.0129487	−	0.999916i	$-0.495878\pi$
$419$	0.530105	0.0258973	0.0129487	+	0.999916i	$0.495878\pi$
$420$	0	0
$421$	−32.0657	−1.56279	−0.781393	−	0.624039i	$-0.785490\pi$
$421$	−32.0657	−1.56279	−0.781393	+	0.624039i	$0.785490\pi$
$422$	16.2569	0.791374
$423$	0	0
$424$	11.5978	0.563239
$425$	7.57477	0.367430
$426$	0	0
$427$	−0.891221	−0.0431292
$428$	−9.14760	−0.442166
$429$	0	0
$430$	−48.9337	−2.35979
$431$	5.38293	0.259287	0.129643	−	0.991561i	$-0.458617\pi$
$431$	5.38293	0.259287	0.129643	+	0.991561i	$0.458617\pi$
$432$	0	0
$433$	33.2929	1.59996	0.799978	−	0.600029i	$-0.204844\pi$
$433$	33.2929	1.59996	0.799978	+	0.600029i	$0.204844\pi$
$434$	−27.8173	−1.33528
$435$	0	0
$436$	−42.9392	−2.05642
$437$	10.2629	0.490939
$438$	0	0
$439$	−18.6820	−0.891641	−0.445820	−	0.895122i	$-0.647088\pi$
$439$	−18.6820	−0.891641	−0.445820	+	0.895122i	$0.647088\pi$
$440$	−2.96946	−0.141564
$441$	0	0
$442$	−3.92371	−0.186632
$443$	−18.8705	−0.896565	−0.448282	−	0.893892i	$-0.647964\pi$
$443$	−18.8705	−0.896565	−0.448282	+	0.893892i	$0.647964\pi$
$444$	0	0
$445$	−20.9737	−0.994247
$446$	−22.9438	−1.08642
$447$	0	0
$448$	37.5185	1.77258
$449$	−6.96621	−0.328756	−0.164378	−	0.986397i	$-0.552562\pi$
$449$	−6.96621	−0.328756	−0.164378	+	0.986397i	$0.552562\pi$
$450$	0	0
$451$	3.70233	0.174336
$452$	−25.5295	−1.20081
$453$	0	0
$454$	22.9800	1.07850
$455$	−32.2805	−1.51333
$456$	0	0
$457$	37.5977	1.75875	0.879373	−	0.476133i	$-0.157962\pi$
$457$	37.5977	1.75875	0.879373	+	0.476133i	$0.157962\pi$
$458$	−4.84394	−0.226343
$459$	0	0
$460$	−22.0452	−1.02786
$461$	−27.6370	−1.28719	−0.643593	−	0.765368i	$-0.722557\pi$
$461$	−27.6370	−1.28719	−0.643593	+	0.765368i	$0.722557\pi$
$462$	0	0
$463$	24.5673	1.14174	0.570870	−	0.821040i	$-0.306606\pi$
$463$	24.5673	1.14174	0.570870	+	0.821040i	$0.306606\pi$
$464$	−1.30854	−0.0607477
$465$	0	0
$466$	−37.6839	−1.74567
$467$	7.86084	0.363756	0.181878	−	0.983321i	$-0.441782\pi$
$467$	7.86084	0.363756	0.181878	+	0.983321i	$0.441782\pi$
$468$	0	0
$469$	6.39540	0.295312
$470$	88.6771	4.09037
$471$	0	0
$472$	4.18234	0.192508
$473$	4.49582	0.206718
$474$	0	0
$475$	−45.5758	−2.09116
$476$	−6.16078	−0.282379
$477$	0	0
$478$	25.9524	1.18704
$479$	−17.7116	−0.809263	−0.404632	−	0.914480i	$-0.632600\pi$
$479$	−17.7116	−0.809263	−0.404632	+	0.914480i	$0.632600\pi$
$480$	0	0
$481$	−11.8260	−0.539218
$482$	−37.0246	−1.68642
$483$	0	0
$484$	−25.8225	−1.17375
$485$	36.9377	1.67725
$486$	0	0
$487$	−42.1146	−1.90840	−0.954198	−	0.299175i	$-0.903289\pi$
$487$	−42.1146	−1.90840	−0.954198	+	0.299175i	$0.903289\pi$
$488$	−0.269352	−0.0121930
$489$	0	0
$490$	−33.9634	−1.53431
$491$	−7.27755	−0.328431	−0.164216	−	0.986424i	$-0.552509\pi$
$491$	−7.27755	−0.328431	−0.164216	+	0.986424i	$0.552509\pi$
$492$	0	0
$493$	0.346100	0.0155876
$494$	23.6082	1.06218
$495$	0	0
$496$	11.1175	0.499189
$497$	−22.5252	−1.01039
$498$	0	0
$499$	16.1300	0.722078	0.361039	−	0.932551i	$-0.382422\pi$
$499$	16.1300	0.722078	0.361039	+	0.932551i	$0.382422\pi$
$500$	49.6968	2.22251
$501$	0	0
$502$	−39.8190	−1.77721
$503$	−36.9391	−1.64703	−0.823517	−	0.567292i	$-0.807991\pi$
$503$	−36.9391	−1.64703	−0.823517	+	0.567292i	$0.807991\pi$
$504$	0	0
$505$	−47.9130	−2.13210
$506$	3.66119	0.162760
$507$	0	0
$508$	−32.7209	−1.45176
$509$	11.6819	0.517792	0.258896	−	0.965905i	$-0.416641\pi$
$509$	11.6819	0.517792	0.258896	+	0.965905i	$0.416641\pi$
$510$	0	0
$511$	−26.4519	−1.17016
$512$	−28.1984	−1.24620
$513$	0	0
$514$	−36.4822	−1.60916
$515$	64.8964	2.85968
$516$	0	0
$517$	−8.14727	−0.358317
$518$	−33.5646	−1.47474
$519$	0	0
$520$	−9.75609	−0.427833
$521$	−0.861490	−0.0377426	−0.0188713	−	0.999822i	$-0.506007\pi$
$521$	−0.861490	−0.0377426	−0.0188713	+	0.999822i	$0.506007\pi$
$522$	0	0
$523$	−7.92246	−0.346425	−0.173212	−	0.984884i	$-0.555415\pi$
$523$	−7.92246	−0.346425	−0.173212	+	0.984884i	$0.555415\pi$
$524$	24.9161	1.08847
$525$	0	0
$526$	52.0720	2.27045
$527$	−2.94049	−0.128090
$528$	0	0
$529$	−17.7709	−0.772649
$530$	94.7682	4.11647
$531$	0	0
$532$	37.0681	1.60711
$533$	12.1639	0.526877
$534$	0	0
$535$	−14.3800	−0.621703
$536$	1.93287	0.0834873
$537$	0	0
$538$	−22.5306	−0.971363
$539$	3.12041	0.134406
$540$	0	0
$541$	−12.0292	−0.517176	−0.258588	−	0.965988i	$-0.583257\pi$
$541$	−12.0292	−0.517176	−0.258588	+	0.965988i	$0.583257\pi$
$542$	9.97863	0.428619
$543$	0	0
$544$	5.95450	0.255297
$545$	−67.5006	−2.89141
$546$	0	0
$547$	−6.38107	−0.272835	−0.136417	−	0.990651i	$-0.543559\pi$
$547$	−6.38107	−0.272835	−0.136417	+	0.990651i	$0.543559\pi$
$548$	−23.4483	−1.00166
$549$	0	0
$550$	−16.2588	−0.693277
$551$	−2.08241	−0.0887138
$552$	0	0
$553$	−14.3973	−0.612237
$554$	13.9886	0.594317
$555$	0	0
$556$	−26.0033	−1.10278
$557$	−40.2925	−1.70725	−0.853623	−	0.520891i	$-0.825600\pi$
$557$	−40.2925	−1.70725	−0.853623	+	0.520891i	$0.825600\pi$
$558$	0	0
$559$	14.7709	0.624742
$560$	36.6164	1.54733
$561$	0	0
$562$	5.23902	0.220995
$563$	−1.58567	−0.0668279	−0.0334139	−	0.999442i	$-0.510638\pi$
$563$	−1.58567	−0.0668279	−0.0334139	+	0.999442i	$0.510638\pi$
$564$	0	0
$565$	−40.1324	−1.68838
$566$	−1.96163	−0.0824536
$567$	0	0
$568$	−6.80775	−0.285647
$569$	11.0442	0.462996	0.231498	−	0.972835i	$-0.425637\pi$
$569$	11.0442	0.462996	0.231498	+	0.972835i	$0.425637\pi$
$570$	0	0
$571$	7.90795	0.330937	0.165469	−	0.986215i	$-0.447086\pi$
$571$	7.90795	0.330937	0.165469	+	0.986215i	$0.447086\pi$
$572$	4.65918	0.194810
$573$	0	0
$574$	34.5237	1.44099
$575$	−23.2216	−0.968407
$576$	0	0
$577$	−46.1655	−1.92190	−0.960948	−	0.276729i	$-0.910749\pi$
$577$	−46.1655	−1.92190	−0.960948	+	0.276729i	$0.910749\pi$
$578$	34.7907	1.44710
$579$	0	0
$580$	4.47315	0.185738
$581$	8.49438	0.352406
$582$	0	0
$583$	−8.70690	−0.360603
$584$	−7.99451	−0.330815
$585$	0	0
$586$	−4.96157	−0.204961
$587$	36.7605	1.51727	0.758633	−	0.651518i	$-0.225867\pi$
$587$	36.7605	1.51727	0.758633	+	0.651518i	$0.225867\pi$
$588$	0	0
$589$	17.6923	0.728999
$590$	34.1748	1.40695
$591$	0	0
$592$	13.4144	0.551329
$593$	14.3481	0.589207	0.294604	−	0.955620i	$-0.404812\pi$
$593$	14.3481	0.589207	0.294604	+	0.955620i	$0.404812\pi$
$594$	0	0
$595$	−9.68477	−0.397037
$596$	30.7804	1.26082
$597$	0	0
$598$	12.0287	0.491891
$599$	−13.0159	−0.531814	−0.265907	−	0.963999i	$-0.585671\pi$
$599$	−13.0159	−0.531814	−0.265907	+	0.963999i	$0.585671\pi$
$600$	0	0
$601$	−6.29784	−0.256894	−0.128447	−	0.991716i	$-0.540999\pi$
$601$	−6.29784	−0.256894	−0.128447	+	0.991716i	$0.540999\pi$
$602$	41.9230	1.70865
$603$	0	0
$604$	−45.1378	−1.83663
$605$	−40.5931	−1.65034
$606$	0	0
$607$	13.1496	0.533726	0.266863	−	0.963734i	$-0.414013\pi$
$607$	13.1496	0.533726	0.266863	+	0.963734i	$0.414013\pi$
$608$	−35.8270	−1.45298
$609$	0	0
$610$	−2.20094	−0.0891133
$611$	−26.7676	−1.08290
$612$	0	0
$613$	−36.9849	−1.49381	−0.746903	−	0.664933i	$-0.768460\pi$
$613$	−36.9849	−1.49381	−0.746903	+	0.664933i	$0.768460\pi$
$614$	32.2817	1.30278
$615$	0	0
$616$	2.54402	0.102502
$617$	32.4778	1.30751	0.653753	−	0.756708i	$-0.273193\pi$
$617$	32.4778	1.30751	0.653753	+	0.756708i	$0.273193\pi$
$618$	0	0
$619$	−42.9211	−1.72514	−0.862572	−	0.505934i	$-0.831148\pi$
$619$	−42.9211	−1.72514	−0.862572	+	0.505934i	$0.831148\pi$
$620$	−38.0042	−1.52628
$621$	0	0
$622$	35.4254	1.42043
$623$	17.9687	0.719902
$624$	0	0
$625$	27.3487	1.09395
$626$	−68.9172	−2.75449
$627$	0	0
$628$	57.8786	2.30961
$629$	−3.54801	−0.141469
$630$	0	0
$631$	−13.9019	−0.553424	−0.276712	−	0.960953i	$-0.589245\pi$
$631$	−13.9019	−0.553424	−0.276712	+	0.960953i	$0.589245\pi$
$632$	−4.35128	−0.173085
$633$	0	0
$634$	−45.8502	−1.82094
$635$	−51.4374	−2.04123
$636$	0	0
$637$	10.2520	0.406200
$638$	−0.742883	−0.0294110
$639$	0	0
$640$	30.5017	1.20569
$641$	−2.61962	−0.103469	−0.0517345	−	0.998661i	$-0.516475\pi$
$641$	−2.61962	−0.103469	−0.0517345	+	0.998661i	$0.516475\pi$
$642$	0	0
$643$	−0.102506	−0.00404244	−0.00202122	−	0.999998i	$-0.500643\pi$
$643$	−0.102506	−0.00404244	−0.00202122	+	0.999998i	$0.500643\pi$
$644$	18.8868	0.744244
$645$	0	0
$646$	7.08290	0.278673
$647$	−25.1564	−0.988998	−0.494499	−	0.869178i	$-0.664649\pi$
$647$	−25.1564	−0.988998	−0.494499	+	0.869178i	$0.664649\pi$
$648$	0	0
$649$	−3.13983	−0.123249
$650$	−53.4178	−2.09522
$651$	0	0
$652$	29.0330	1.13702
$653$	−7.45951	−0.291913	−0.145957	−	0.989291i	$-0.546626\pi$
$653$	−7.45951	−0.291913	−0.145957	+	0.989291i	$0.546626\pi$
$654$	0	0
$655$	39.1682	1.53043
$656$	−13.7977	−0.538711
$657$	0	0
$658$	−75.9723	−2.96171
$659$	16.2600	0.633399	0.316699	−	0.948526i	$-0.397425\pi$
$659$	16.2600	0.633399	0.316699	+	0.948526i	$0.397425\pi$
$660$	0	0
$661$	18.8700	0.733960	0.366980	−	0.930229i	$-0.380392\pi$
$661$	18.8700	0.733960	0.366980	+	0.930229i	$0.380392\pi$
$662$	−50.3808	−1.95811
$663$	0	0
$664$	2.56724	0.0996284
$665$	58.2712	2.25966
$666$	0	0
$667$	−1.06102	−0.0410829
$668$	−59.9618	−2.31999
$669$	0	0
$670$	15.7939	0.610172
$671$	0.202212	0.00780632
$672$	0	0
$673$	−8.18112	−0.315359	−0.157679	−	0.987490i	$-0.550401\pi$
$673$	−8.18112	−0.315359	−0.157679	+	0.987490i	$0.550401\pi$
$674$	38.9199	1.49914
$675$	0	0
$676$	−16.8859	−0.649456
$677$	−23.0920	−0.887497	−0.443748	−	0.896151i	$-0.646352\pi$
$677$	−23.0920	−0.887497	−0.443748	+	0.896151i	$0.646352\pi$
$678$	0	0
$679$	−31.6456	−1.21445
$680$	−2.92701	−0.112246
$681$	0	0
$682$	6.31158	0.241683
$683$	−18.7412	−0.717112	−0.358556	−	0.933508i	$-0.616731\pi$
$683$	−18.7412	−0.717112	−0.358556	+	0.933508i	$0.616731\pi$
$684$	0	0
$685$	−36.8608	−1.40838
$686$	−20.2978	−0.774972
$687$	0	0
$688$	−16.7549	−0.638775
$689$	−28.6063	−1.08981
$690$	0	0
$691$	−6.64930	−0.252951	−0.126476	−	0.991970i	$-0.540367\pi$
$691$	−6.64930	−0.252951	−0.126476	+	0.991970i	$0.540367\pi$
$692$	−38.4154	−1.46033
$693$	0	0
$694$	35.7798	1.35818
$695$	−40.8772	−1.55056
$696$	0	0
$697$	3.64940	0.138231
$698$	−17.2798	−0.654051
$699$	0	0
$700$	−83.8735	−3.17012
$701$	−24.2954	−0.917625	−0.458813	−	0.888533i	$-0.651725\pi$
$701$	−24.2954	−0.917625	−0.458813	+	0.888533i	$0.651725\pi$
$702$	0	0
$703$	21.3477	0.805142
$704$	−8.51272	−0.320835
$705$	0	0
$706$	−52.0781	−1.95999
$707$	41.0485	1.54379
$708$	0	0
$709$	29.9220	1.12375	0.561873	−	0.827224i	$-0.310081\pi$
$709$	29.9220	1.12375	0.561873	+	0.827224i	$0.310081\pi$
$710$	−55.6276	−2.08767
$711$	0	0
$712$	5.43066	0.203523
$713$	9.01451	0.337596
$714$	0	0
$715$	7.32425	0.273911
$716$	−17.8011	−0.665259
$717$	0	0
$718$	26.7542	0.998460
$719$	−12.2598	−0.457213	−0.228607	−	0.973519i	$-0.573417\pi$
$719$	−12.2598	−0.457213	−0.228607	+	0.973519i	$0.573417\pi$
$720$	0	0
$721$	−55.5987	−2.07060
$722$	−2.41697	−0.0899504
$723$	0	0
$724$	−19.1620	−0.712149
$725$	4.71184	0.174993
$726$	0	0
$727$	−49.0199	−1.81805	−0.909023	−	0.416745i	$-0.863171\pi$
$727$	−49.0199	−1.81805	−0.909023	+	0.416745i	$0.863171\pi$
$728$	8.35832	0.309780
$729$	0	0
$730$	−65.3249	−2.41778
$731$	4.43155	0.163907
$732$	0	0
$733$	−20.1993	−0.746080	−0.373040	−	0.927815i	$-0.621685\pi$
$733$	−20.1993	−0.746080	−0.373040	+	0.927815i	$0.621685\pi$
$734$	22.2031	0.819530
$735$	0	0
$736$	−18.2544	−0.672866
$737$	−1.45108	−0.0534511
$738$	0	0
$739$	2.01212	0.0740170	0.0370085	−	0.999315i	$-0.488217\pi$
$739$	2.01212	0.0740170	0.0370085	+	0.999315i	$0.488217\pi$
$740$	−45.8561	−1.68570
$741$	0	0
$742$	−81.1907	−2.98060
$743$	21.5721	0.791402	0.395701	−	0.918379i	$-0.370502\pi$
$743$	21.5721	0.791402	0.395701	+	0.918379i	$0.370502\pi$
$744$	0	0
$745$	48.3869	1.77276
$746$	−24.9583	−0.913788
$747$	0	0
$748$	1.39784	0.0511102
$749$	12.3198	0.450156
$750$	0	0
$751$	−14.4775	−0.528293	−0.264147	−	0.964483i	$-0.585090\pi$
$751$	−14.4775	−0.528293	−0.264147	+	0.964483i	$0.585090\pi$
$752$	30.3631	1.10723
$753$	0	0
$754$	−2.44072	−0.0888859
$755$	−70.9567	−2.58238
$756$	0	0
$757$	10.5290	0.382684	0.191342	−	0.981523i	$-0.438716\pi$
$757$	10.5290	0.382684	0.191342	+	0.981523i	$0.438716\pi$
$758$	−38.3736	−1.39379
$759$	0	0
$760$	17.6112	0.638826
$761$	−42.2753	−1.53248	−0.766240	−	0.642555i	$-0.777875\pi$
$761$	−42.2753	−1.53248	−0.766240	+	0.642555i	$0.777875\pi$
$762$	0	0
$763$	57.8297	2.09357
$764$	−24.9578	−0.902941
$765$	0	0
$766$	−32.8315	−1.18625
$767$	−10.3158	−0.372483
$768$	0	0
$769$	−9.01925	−0.325242	−0.162621	−	0.986689i	$-0.551995\pi$
$769$	−9.01925	−0.325242	−0.162621	+	0.986689i	$0.551995\pi$
$770$	20.7878	0.749139
$771$	0	0
$772$	−39.6036	−1.42537
$773$	0.719581	0.0258815	0.0129408	−	0.999916i	$-0.495881\pi$
$773$	0.719581	0.0258815	0.0129408	+	0.999916i	$0.495881\pi$
$774$	0	0
$775$	−40.0321	−1.43799
$776$	−9.56420	−0.343335
$777$	0	0
$778$	32.6552	1.17075
$779$	−21.9577	−0.786716
$780$	0	0
$781$	5.11083	0.182880
$782$	3.60885	0.129052
$783$	0	0
$784$	−11.6291	−0.415324
$785$	90.9853	3.24741
$786$	0	0
$787$	15.4274	0.549929	0.274964	−	0.961454i	$-0.411334\pi$
$787$	15.4274	0.549929	0.274964	+	0.961454i	$0.411334\pi$
$788$	27.1663	0.967758
$789$	0	0
$790$	−35.5553	−1.26500
$791$	34.3826	1.22251
$792$	0	0
$793$	0.664364	0.0235922
$794$	−4.32165	−0.153370
$795$	0	0
$796$	8.69403	0.308152
$797$	13.9305	0.493444	0.246722	−	0.969086i	$-0.420647\pi$
$797$	13.9305	0.493444	0.246722	+	0.969086i	$0.420647\pi$
$798$	0	0
$799$	−8.03080	−0.284109
$800$	81.0651	2.86608
$801$	0	0
$802$	−31.6393	−1.11722
$803$	6.00177	0.211798
$804$	0	0
$805$	29.6901	1.04644
$806$	20.7365	0.730413
$807$	0	0
$808$	12.4060	0.436442
$809$	−40.8781	−1.43720	−0.718599	−	0.695424i	$-0.755216\pi$
$809$	−40.8781	−1.43720	−0.718599	+	0.695424i	$0.755216\pi$
$810$	0	0
$811$	51.1039	1.79450	0.897250	−	0.441524i	$-0.145562\pi$
$811$	51.1039	1.79450	0.897250	+	0.441524i	$0.145562\pi$
$812$	−3.83228	−0.134487
$813$	0	0
$814$	7.61560	0.266927
$815$	45.6400	1.59870
$816$	0	0
$817$	−26.6637	−0.932845
$818$	−21.9797	−0.768501
$819$	0	0
$820$	47.1665	1.64712
$821$	−48.5713	−1.69515	−0.847574	−	0.530677i	$-0.821938\pi$
$821$	−48.5713	−1.69515	−0.847574	+	0.530677i	$0.821938\pi$
$822$	0	0
$823$	27.0271	0.942107	0.471054	−	0.882105i	$-0.343874\pi$
$823$	27.0271	0.942107	0.471054	+	0.882105i	$0.343874\pi$
$824$	−16.8035	−0.585377
$825$	0	0
$826$	−29.2786	−1.01873
$827$	15.9406	0.554309	0.277155	−	0.960825i	$-0.410609\pi$
$827$	15.9406	0.554309	0.277155	+	0.960825i	$0.410609\pi$
$828$	0	0
$829$	−1.69947	−0.0590250	−0.0295125	−	0.999564i	$-0.509395\pi$
$829$	−1.69947	−0.0590250	−0.0295125	+	0.999564i	$0.509395\pi$
$830$	20.9775	0.728140
$831$	0	0
$832$	−27.9683	−0.969627
$833$	3.07580	0.106570
$834$	0	0
$835$	−94.2600	−3.26200
$836$	−8.41053	−0.290884
$837$	0	0
$838$	−1.12157	−0.0387441
$839$	−46.3515	−1.60023	−0.800116	−	0.599845i	$-0.795229\pi$
$839$	−46.3515	−1.60023	−0.800116	+	0.599845i	$0.795229\pi$
$840$	0	0
$841$	−28.7847	−0.992576
$842$	67.8431	2.33803
$843$	0	0
$844$	−19.0282	−0.654977
$845$	−26.5446	−0.913162
$846$	0	0
$847$	34.7772	1.19496
$848$	32.4486	1.11429
$849$	0	0
$850$	−16.0264	−0.549699
$851$	10.8770	0.372858
$852$	0	0
$853$	−24.1948	−0.828413	−0.414207	−	0.910183i	$-0.635941\pi$
$853$	−24.1948	−0.828413	−0.414207	+	0.910183i	$0.635941\pi$
$854$	1.88560	0.0645241
$855$	0	0
$856$	3.72339	0.127263
$857$	−12.8286	−0.438217	−0.219109	−	0.975700i	$-0.570315\pi$
$857$	−12.8286	−0.438217	−0.219109	+	0.975700i	$0.570315\pi$
$858$	0	0
$859$	20.5200	0.700134	0.350067	−	0.936725i	$-0.386159\pi$
$859$	20.5200	0.700134	0.350067	+	0.936725i	$0.386159\pi$
$860$	57.2753	1.95307
$861$	0	0
$862$	−11.3890	−0.387910
$863$	−37.0220	−1.26024	−0.630121	−	0.776497i	$-0.716995\pi$
$863$	−37.0220	−1.26024	−0.630121	+	0.776497i	$0.716995\pi$
$864$	0	0
$865$	−60.3890	−2.05329
$866$	−70.4397	−2.39364
$867$	0	0
$868$	32.5593	1.10513
$869$	3.26666	0.110814
$870$	0	0
$871$	−4.76747	−0.161540
$872$	17.4778	0.591872
$873$	0	0
$874$	−21.7137	−0.734476
$875$	−66.9307	−2.26267
$876$	0	0
$877$	43.3678	1.46443	0.732213	−	0.681076i	$-0.238488\pi$
$877$	43.3678	1.46443	0.732213	+	0.681076i	$0.238488\pi$
$878$	39.5264	1.33395
$879$	0	0
$880$	−8.30803	−0.280064
$881$	10.2812	0.346382	0.173191	−	0.984888i	$-0.444592\pi$
$881$	10.2812	0.346382	0.173191	+	0.984888i	$0.444592\pi$
$882$	0	0
$883$	7.85133	0.264218	0.132109	−	0.991235i	$-0.457825\pi$
$883$	7.85133	0.264218	0.132109	+	0.991235i	$0.457825\pi$
$884$	4.59258	0.154465
$885$	0	0
$886$	39.9254	1.34132
$887$	−34.9936	−1.17497	−0.587485	−	0.809235i	$-0.699882\pi$
$887$	−34.9936	−1.17497	−0.587485	+	0.809235i	$0.699882\pi$
$888$	0	0
$889$	44.0679	1.47799
$890$	44.3751	1.48746
$891$	0	0
$892$	26.8550	0.899172
$893$	48.3196	1.61696
$894$	0	0
$895$	−27.9834	−0.935382
$896$	−26.1317	−0.872998
$897$	0	0
$898$	14.7388	0.491840
$899$	−1.82911	−0.0610044
$900$	0	0
$901$	−8.58242	−0.285922
$902$	−7.83322	−0.260818
$903$	0	0
$904$	10.3914	0.345613
$905$	−30.1226	−1.00131
$906$	0	0
$907$	−50.2389	−1.66815	−0.834077	−	0.551648i	$-0.813999\pi$
$907$	−50.2389	−1.66815	−0.834077	+	0.551648i	$0.813999\pi$
$908$	−26.8973	−0.892617
$909$	0	0
$910$	68.2977	2.26405
$911$	−24.6615	−0.817073	−0.408537	−	0.912742i	$-0.633961\pi$
$911$	−24.6615	−0.817073	−0.408537	+	0.912742i	$0.633961\pi$
$912$	0	0
$913$	−1.92732	−0.0637851
$914$	−79.5475	−2.63120
$915$	0	0
$916$	5.66967	0.187331
$917$	−33.5565	−1.10813
$918$	0	0
$919$	8.25476	0.272300	0.136150	−	0.990688i	$-0.456527\pi$
$919$	8.25476	0.272300	0.136150	+	0.990688i	$0.456527\pi$
$920$	8.97319	0.295837
$921$	0	0
$922$	58.4732	1.92571
$923$	16.7915	0.552698
$924$	0	0
$925$	−48.3030	−1.58819
$926$	−51.9785	−1.70812
$927$	0	0
$928$	3.70396	0.121588
$929$	−9.80781	−0.321784	−0.160892	−	0.986972i	$-0.551437\pi$
$929$	−9.80781	−0.321784	−0.160892	+	0.986972i	$0.551437\pi$
$930$	0	0
$931$	−18.5065	−0.606525
$932$	44.1078	1.44480
$933$	0	0
$934$	−16.6316	−0.544203
$935$	2.19741	0.0718631
$936$	0	0
$937$	44.1867	1.44352	0.721759	−	0.692145i	$-0.243334\pi$
$937$	44.1867	1.44352	0.721759	+	0.692145i	$0.243334\pi$
$938$	−13.5311	−0.441806
$939$	0	0
$940$	−103.794	−3.38537
$941$	−10.6403	−0.346864	−0.173432	−	0.984846i	$-0.555486\pi$
$941$	−10.6403	−0.346864	−0.173432	+	0.984846i	$0.555486\pi$
$942$	0	0
$943$	−11.1878	−0.364324
$944$	11.7015	0.380850
$945$	0	0
$946$	−9.51206	−0.309264
$947$	−10.9942	−0.357263	−0.178632	−	0.983916i	$-0.557167\pi$
$947$	−10.9942	−0.357263	−0.178632	+	0.983916i	$0.557167\pi$
$948$	0	0
$949$	19.7186	0.640094
$950$	96.4272	3.12851
$951$	0	0
$952$	2.50766	0.0812736
$953$	−42.3248	−1.37103	−0.685517	−	0.728057i	$-0.740424\pi$
$953$	−42.3248	−1.37103	−0.685517	+	0.728057i	$0.740424\pi$
$954$	0	0
$955$	−39.2337	−1.26957
$956$	−30.3764	−0.982444
$957$	0	0
$958$	37.4734	1.21071
$959$	31.5797	1.01976
$960$	0	0
$961$	−15.4597	−0.498701
$962$	25.0209	0.806705
$963$	0	0
$964$	43.3361	1.39576
$965$	−62.2570	−2.00412
$966$	0	0
$967$	−8.31143	−0.267278	−0.133639	−	0.991030i	$-0.542666\pi$
$967$	−8.31143	−0.267278	−0.133639	+	0.991030i	$0.542666\pi$
$968$	10.5107	0.337826
$969$	0	0
$970$	−78.1511	−2.50928
$971$	−17.9539	−0.576169	−0.288084	−	0.957605i	$-0.593018\pi$
$971$	−17.9539	−0.576169	−0.288084	+	0.957605i	$0.593018\pi$
$972$	0	0
$973$	35.0207	1.12271
$974$	89.1043	2.85509
$975$	0	0
$976$	−0.753600	−0.0241222
$977$	−24.4102	−0.780952	−0.390476	−	0.920613i	$-0.627690\pi$
$977$	−24.4102	−0.780952	−0.390476	+	0.920613i	$0.627690\pi$
$978$	0	0
$979$	−4.07700	−0.130301
$980$	39.7530	1.26986
$981$	0	0
$982$	15.3975	0.491355
$983$	46.6132	1.48673	0.743365	−	0.668886i	$-0.233229\pi$
$983$	46.6132	1.48673	0.743365	+	0.668886i	$0.233229\pi$
$984$	0	0
$985$	42.7054	1.36071
$986$	−0.732263	−0.0233200
$987$	0	0
$988$	−27.6326	−0.879109
$989$	−13.5856	−0.431996
$990$	0	0
$991$	58.8851	1.87055	0.935273	−	0.353926i	$-0.115154\pi$
$991$	58.8851	1.87055	0.935273	+	0.353926i	$0.115154\pi$
$992$	−31.4691	−0.999144
$993$	0	0
$994$	47.6578	1.51161
$995$	13.6670	0.433274
$996$	0	0
$997$	60.2906	1.90942	0.954712	−	0.297532i	$-0.0961635\pi$
$997$	60.2906	1.90942	0.954712	+	0.297532i	$0.0961635\pi$
$998$	−34.1272	−1.08028
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.12		72
3.2	odd	2		6561.2.a.d.1.61		72
81.5	odd	54		243.2.g.a.73.7		144
81.11	odd	54		729.2.g.a.514.2		144
81.16	even	27		81.2.g.a.13.2	✓	144
81.22	even	27		729.2.g.d.217.7		144
81.32	odd	54		729.2.g.b.703.2		144
81.38	odd	54		729.2.g.b.28.2		144
81.43	even	27		729.2.g.c.28.7		144
81.49	even	27		729.2.g.c.703.7		144
81.59	odd	54		729.2.g.a.217.2		144
81.65	odd	54		243.2.g.a.10.7		144
81.70	even	27		729.2.g.d.514.7		144
81.76	even	27		81.2.g.a.25.2	yes	144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.13.2	✓	144	81.16	even	27
81.2.g.a.25.2	yes	144	81.76	even	27
243.2.g.a.10.7		144	81.65	odd	54
243.2.g.a.73.7		144	81.5	odd	54
729.2.g.a.217.2		144	81.59	odd	54
729.2.g.a.514.2		144	81.11	odd	54
729.2.g.b.28.2		144	81.38	odd	54
729.2.g.b.703.2		144	81.32	odd	54
729.2.g.c.28.7		144	81.43	even	27
729.2.g.c.703.7		144	81.49	even	27
729.2.g.d.217.7		144	81.22	even	27
729.2.g.d.514.7		144	81.70	even	27
6561.2.a.c.1.12		72	1.1	even	1	trivial
6561.2.a.d.1.61		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)

\(f(q)\)	\(=\)	\(q-2.11576 q^{2} +2.47642 q^{4} +3.89294 q^{5} -3.33519 q^{7} -1.00799 q^{8} -8.23651 q^{10} +0.756735 q^{11} +2.48623 q^{13} +7.05645 q^{14} -2.82018 q^{16} +0.745917 q^{17} -4.48802 q^{19} +9.64056 q^{20} -1.60107 q^{22} -2.28672 q^{23} +10.1550 q^{25} -5.26026 q^{26} -8.25934 q^{28} +0.463993 q^{29} -3.94211 q^{31} +7.98279 q^{32} -1.57818 q^{34} -12.9837 q^{35} -4.75658 q^{37} +9.49556 q^{38} -3.92405 q^{40} +4.89250 q^{41} +5.94108 q^{43} +1.87399 q^{44} +4.83814 q^{46} -10.7663 q^{47} +4.12352 q^{49} -21.4854 q^{50} +6.15696 q^{52} -11.5059 q^{53} +2.94592 q^{55} +3.36184 q^{56} -0.981696 q^{58} -4.14919 q^{59} +0.267217 q^{61} +8.34055 q^{62} -11.2493 q^{64} +9.67875 q^{65} -1.91755 q^{67} +1.84720 q^{68} +27.4703 q^{70} +6.75379 q^{71} +7.93114 q^{73} +10.0638 q^{74} -11.1142 q^{76} -2.52386 q^{77} +4.31679 q^{79} -10.9788 q^{80} -10.3513 q^{82} -2.54689 q^{83} +2.90381 q^{85} -12.5699 q^{86} -0.762781 q^{88} -5.38761 q^{89} -8.29206 q^{91} -5.66288 q^{92} +22.7790 q^{94} -17.4716 q^{95} +9.48838 q^{97} -8.72436 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38}+ \cdots - 81 q^{98}+O(q^{100}) \) 72 * q - 9 * q^2 + 63 * q^4 - 18 * q^5 - 27 * q^8 - 36 * q^11 - 36 * q^14 + 45 * q^16 - 36 * q^17 - 54 * q^20 - 54 * q^23 + 36 * q^25 - 45 * q^26 + 9 * q^28 - 54 * q^29 - 63 * q^32 - 72 * q^35 - 54 * q^38 - 72 * q^41 - 90 * q^44 - 90 * q^47 + 18 * q^49 - 45 * q^50 - 45 * q^53 + 9 * q^55 - 108 * q^56 + 18 * q^58 - 108 * q^59 - 72 * q^62 + 9 * q^64 - 72 * q^65 - 108 * q^68 - 126 * q^71 - 90 * q^74 - 72 * q^77 - 144 * q^80 - 18 * q^82 - 108 * q^83 - 90 * q^86 - 108 * q^89 - 72 * q^92 - 144 * q^95 - 81 * q^98

Introduction

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