LMFDB - Embedded newform 6561.2.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6561.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-2.54050 q^{2} +4.45416 q^{4} +3.18683 q^{5} -1.66066 q^{7} -6.23481 q^{8} -8.09616 q^{10} -1.96180 q^{11} -3.88044 q^{13} +4.21891 q^{14} +6.93125 q^{16} -4.35949 q^{17} +1.41661 q^{19} +14.1947 q^{20} +4.98397 q^{22} -2.26830 q^{23} +5.15590 q^{25} +9.85829 q^{26} -7.39684 q^{28} +9.44157 q^{29} -4.17854 q^{31} -5.13924 q^{32} +11.0753 q^{34} -5.29223 q^{35} -6.84762 q^{37} -3.59890 q^{38} -19.8693 q^{40} +3.56777 q^{41} +7.45581 q^{43} -8.73818 q^{44} +5.76263 q^{46} +6.93565 q^{47} -4.24222 q^{49} -13.0986 q^{50} -17.2841 q^{52} +2.70633 q^{53} -6.25193 q^{55} +10.3539 q^{56} -23.9864 q^{58} +3.53534 q^{59} -2.78995 q^{61} +10.6156 q^{62} -0.806236 q^{64} -12.3663 q^{65} -1.84415 q^{67} -19.4179 q^{68} +13.4449 q^{70} -2.29664 q^{71} -11.4402 q^{73} +17.3964 q^{74} +6.30981 q^{76} +3.25788 q^{77} +6.96987 q^{79} +22.0887 q^{80} -9.06394 q^{82} +0.107298 q^{83} -13.8930 q^{85} -18.9415 q^{86} +12.2315 q^{88} -5.28583 q^{89} +6.44408 q^{91} -10.1034 q^{92} -17.6201 q^{94} +4.51449 q^{95} -4.35627 q^{97} +10.7774 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38}+ \cdots + 81 q^{98}+O(q^{100})$ 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

)

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

$2$	−2.54050	−1.79641	−0.898204	−	0.439579i	$-0.855128\pi$
$2$	−2.54050	−1.79641	−0.898204	+	0.439579i	$0.855128\pi$
$3$	0	0
$3$	0	0
$4$	4.45416	2.22708
$5$	3.18683	1.42519	0.712597	−	0.701573i	$-0.247519\pi$
$5$	3.18683	1.42519	0.712597	+	0.701573i	$0.247519\pi$
$6$	0	0
$7$	−1.66066	−0.627669	−0.313835	−	0.949478i	$-0.601614\pi$
$7$	−1.66066	−0.627669	−0.313835	+	0.949478i	$0.601614\pi$
$8$	−6.23481	−2.20434
$9$	0	0
$10$	−8.09616	−2.56023
$11$	−1.96180	−0.591505	−0.295753	−	0.955265i	$-0.595570\pi$
$11$	−1.96180	−0.591505	−0.295753	+	0.955265i	$0.595570\pi$
$12$	0	0
$13$	−3.88044	−1.07624	−0.538121	−	0.842868i	$-0.680866\pi$
$13$	−3.88044	−1.07624	−0.538121	+	0.842868i	$0.680866\pi$
$14$	4.21891	1.12755
$15$	0	0
$16$	6.93125	1.73281
$17$	−4.35949	−1.05733	−0.528666	−	0.848830i	$-0.677308\pi$
$17$	−4.35949	−1.05733	−0.528666	+	0.848830i	$0.677308\pi$
$18$	0	0
$19$	1.41661	0.324992	0.162496	−	0.986709i	$-0.448045\pi$
$19$	1.41661	0.324992	0.162496	+	0.986709i	$0.448045\pi$
$20$	14.1947	3.17403
$21$	0	0
$22$	4.98397	1.06258
$23$	−2.26830	−0.472973	−0.236487	−	0.971635i	$-0.575996\pi$
$23$	−2.26830	−0.472973	−0.236487	+	0.971635i	$0.575996\pi$
$24$	0	0
$25$	5.15590	1.03118
$26$	9.85829	1.93337
$27$	0	0
$28$	−7.39684	−1.39787
$29$	9.44157	1.75326	0.876628	−	0.481169i	$-0.159788\pi$
$29$	9.44157	1.75326	0.876628	+	0.481169i	$0.159788\pi$
$30$	0	0
$31$	−4.17854	−0.750487	−0.375244	−	0.926926i	$-0.622441\pi$
$31$	−4.17854	−0.750487	−0.375244	+	0.926926i	$0.622441\pi$
$32$	−5.13924	−0.908498
$33$	0	0
$34$	11.0753	1.89940
$35$	−5.29223	−0.894551
$36$	0	0
$37$	−6.84762	−1.12574	−0.562871	−	0.826545i	$-0.690303\pi$
$37$	−6.84762	−1.12574	−0.562871	+	0.826545i	$0.690303\pi$
$38$	−3.59890	−0.583819
$39$	0	0
$40$	−19.8693	−3.14161
$41$	3.56777	0.557192	0.278596	−	0.960408i	$-0.410131\pi$
$41$	3.56777	0.557192	0.278596	+	0.960408i	$0.410131\pi$
$42$	0	0
$43$	7.45581	1.13700	0.568500	−	0.822683i	$-0.307524\pi$
$43$	7.45581	1.13700	0.568500	+	0.822683i	$0.307524\pi$
$44$	−8.73818	−1.31733
$45$	0	0
$46$	5.76263	0.849653
$47$	6.93565	1.01167	0.505835	−	0.862631i	$-0.331185\pi$
$47$	6.93565	1.01167	0.505835	+	0.862631i	$0.331185\pi$
$48$	0	0
$49$	−4.24222	−0.606032
$50$	−13.0986	−1.85242
$51$	0	0
$52$	−17.2841	−2.39688
$53$	2.70633	0.371743	0.185872	−	0.982574i	$-0.440489\pi$
$53$	2.70633	0.371743	0.185872	+	0.982574i	$0.440489\pi$
$54$	0	0
$55$	−6.25193	−0.843010
$56$	10.3539	1.38360
$57$	0	0
$58$	−23.9864	−3.14956
$59$	3.53534	0.460262	0.230131	−	0.973160i	$-0.426085\pi$
$59$	3.53534	0.460262	0.230131	+	0.973160i	$0.426085\pi$
$60$	0	0
$61$	−2.78995	−0.357217	−0.178609	−	0.983920i	$-0.557160\pi$
$61$	−2.78995	−0.357217	−0.178609	+	0.983920i	$0.557160\pi$
$62$	10.6156	1.34818
$63$	0	0
$64$	−0.806236	−0.100779
$65$	−12.3663	−1.53385
$66$	0	0
$67$	−1.84415	−0.225299	−0.112650	−	0.993635i	$-0.535934\pi$
$67$	−1.84415	−0.225299	−0.112650	+	0.993635i	$0.535934\pi$
$68$	−19.4179	−2.35477
$69$	0	0
$70$	13.4449	1.60698
$71$	−2.29664	−0.272561	−0.136280	−	0.990670i	$-0.543515\pi$
$71$	−2.29664	−0.272561	−0.136280	+	0.990670i	$0.543515\pi$
$72$	0	0
$73$	−11.4402	−1.33898	−0.669489	−	0.742822i	$-0.733487\pi$
$73$	−11.4402	−1.33898	−0.669489	+	0.742822i	$0.733487\pi$
$74$	17.3964	2.02229
$75$	0	0
$76$	6.30981	0.723785
$77$	3.25788	0.371270
$78$	0	0
$79$	6.96987	0.784172	0.392086	−	0.919929i	$-0.371754\pi$
$79$	6.96987	0.784172	0.392086	+	0.919929i	$0.371754\pi$
$80$	22.0887	2.46959
$81$	0	0
$82$	−9.06394	−1.00094
$83$	0.107298	0.0117775	0.00588873	−	0.999983i	$-0.498126\pi$
$83$	0.107298	0.0117775	0.00588873	+	0.999983i	$0.498126\pi$
$84$	0	0
$85$	−13.8930	−1.50690
$86$	−18.9415	−2.04252
$87$	0	0
$88$	12.2315	1.30388
$89$	−5.28583	−0.560297	−0.280149	−	0.959957i	$-0.590384\pi$
$89$	−5.28583	−0.560297	−0.280149	+	0.959957i	$0.590384\pi$
$90$	0	0
$91$	6.44408	0.675524
$92$	−10.1034	−1.05335
$93$	0	0
$94$	−17.6201	−1.81737
$95$	4.51449	0.463177
$96$	0	0
$97$	−4.35627	−0.442312	−0.221156	−	0.975238i	$-0.570983\pi$
$97$	−4.35627	−0.442312	−0.221156	+	0.975238i	$0.570983\pi$
$98$	10.7774	1.08868
$99$	0	0
$100$	22.9652	2.29652
$101$	11.2030	1.11474	0.557369	−	0.830265i	$-0.311811\pi$
$101$	11.2030	1.11474	0.557369	+	0.830265i	$0.311811\pi$
$102$	0	0
$103$	−4.02820	−0.396910	−0.198455	−	0.980110i	$-0.563592\pi$
$103$	−4.02820	−0.396910	−0.198455	+	0.980110i	$0.563592\pi$
$104$	24.1939	2.37240
$105$	0	0
$106$	−6.87545	−0.667803
$107$	−16.2002	−1.56614	−0.783068	−	0.621936i	$-0.786346\pi$
$107$	−16.2002	−1.56614	−0.783068	+	0.621936i	$0.786346\pi$
$108$	0	0
$109$	12.8910	1.23473	0.617365	−	0.786677i	$-0.288200\pi$
$109$	12.8910	1.23473	0.617365	+	0.786677i	$0.288200\pi$
$110$	15.8831	1.51439
$111$	0	0
$112$	−11.5104	−1.08763
$113$	16.1325	1.51761	0.758807	−	0.651315i	$-0.225782\pi$
$113$	16.1325	1.51761	0.758807	+	0.651315i	$0.225782\pi$
$114$	0	0
$115$	−7.22869	−0.674079
$116$	42.0543	3.90464
$117$	0	0
$118$	−8.98154	−0.826818
$119$	7.23962	0.663655
$120$	0	0
$121$	−7.15134	−0.650121
$122$	7.08789	0.641708
$123$	0	0
$124$	−18.6119	−1.67140
$125$	0.496823	0.0444372
$126$	0	0
$127$	−3.99204	−0.354236	−0.177118	−	0.984190i	$-0.556677\pi$
$127$	−3.99204	−0.354236	−0.177118	+	0.984190i	$0.556677\pi$
$128$	12.3267	1.08954
$129$	0	0
$130$	31.4167	2.75543
$131$	9.76431	0.853111	0.426556	−	0.904461i	$-0.359727\pi$
$131$	9.76431	0.853111	0.426556	+	0.904461i	$0.359727\pi$
$132$	0	0
$133$	−2.35250	−0.203988
$134$	4.68508	0.404729
$135$	0	0
$136$	27.1806	2.33072
$137$	7.42951	0.634745	0.317373	−	0.948301i	$-0.397199\pi$
$137$	7.42951	0.634745	0.317373	+	0.948301i	$0.397199\pi$
$138$	0	0
$139$	17.7537	1.50585	0.752924	−	0.658108i	$-0.228643\pi$
$139$	17.7537	1.50585	0.752924	+	0.658108i	$0.228643\pi$
$140$	−23.5725	−1.99224
$141$	0	0
$142$	5.83462	0.489630
$143$	7.61266	0.636603
$144$	0	0
$145$	30.0887	2.49873
$146$	29.0640	2.40535
$147$	0	0
$148$	−30.5004	−2.50712
$149$	14.0449	1.15061	0.575303	−	0.817940i	$-0.304884\pi$
$149$	14.0449	1.15061	0.575303	+	0.817940i	$0.304884\pi$
$150$	0	0
$151$	2.76578	0.225076	0.112538	−	0.993647i	$-0.464102\pi$
$151$	2.76578	0.225076	0.112538	+	0.993647i	$0.464102\pi$
$152$	−8.83229	−0.716394
$153$	0	0
$154$	−8.27665	−0.666952
$155$	−13.3163	−1.06959
$156$	0	0
$157$	4.99735	0.398832	0.199416	−	0.979915i	$-0.436096\pi$
$157$	4.99735	0.398832	0.199416	+	0.979915i	$0.436096\pi$
$158$	−17.7070	−1.40869
$159$	0	0
$160$	−16.3779	−1.29479
$161$	3.76687	0.296871
$162$	0	0
$163$	12.7147	0.995894	0.497947	−	0.867208i	$-0.334087\pi$
$163$	12.7147	0.995894	0.497947	+	0.867208i	$0.334087\pi$
$164$	15.8914	1.24091
$165$	0	0
$166$	−0.272591	−0.0211571
$167$	20.2459	1.56668	0.783338	−	0.621596i	$-0.213515\pi$
$167$	20.2459	1.56668	0.783338	+	0.621596i	$0.213515\pi$
$168$	0	0
$169$	2.05785	0.158296
$170$	35.2952	2.70701
$171$	0	0
$172$	33.2094	2.53219
$173$	6.79458	0.516582	0.258291	−	0.966067i	$-0.416841\pi$
$173$	6.79458	0.516582	0.258291	+	0.966067i	$0.416841\pi$
$174$	0	0
$175$	−8.56218	−0.647240
$176$	−13.5977	−1.02497
$177$	0	0
$178$	13.4287	1.00652
$179$	22.4033	1.67450	0.837252	−	0.546817i	$-0.184161\pi$
$179$	22.4033	1.67450	0.837252	+	0.546817i	$0.184161\pi$
$180$	0	0
$181$	−22.7104	−1.68805	−0.844024	−	0.536306i	$-0.819819\pi$
$181$	−22.7104	−1.68805	−0.844024	+	0.536306i	$0.819819\pi$
$182$	−16.3712	−1.21352
$183$	0	0
$184$	14.1424	1.04259
$185$	−21.8222	−1.60440
$186$	0	0
$187$	8.55246	0.625418
$188$	30.8925	2.25307
$189$	0	0
$190$	−11.4691	−0.832056
$191$	−18.2867	−1.32318	−0.661591	−	0.749865i	$-0.730118\pi$
$191$	−18.2867	−1.32318	−0.661591	+	0.749865i	$0.730118\pi$
$192$	0	0
$193$	8.79042	0.632748	0.316374	−	0.948634i	$-0.397534\pi$
$193$	8.79042	0.632748	0.316374	+	0.948634i	$0.397534\pi$
$194$	11.0671	0.794573
$195$	0	0
$196$	−18.8955	−1.34968
$197$	−1.86015	−0.132530	−0.0662649	−	0.997802i	$-0.521108\pi$
$197$	−1.86015	−0.132530	−0.0662649	+	0.997802i	$0.521108\pi$
$198$	0	0
$199$	13.3896	0.949160	0.474580	−	0.880212i	$-0.342600\pi$
$199$	13.3896	0.949160	0.474580	+	0.880212i	$0.342600\pi$
$200$	−32.1461	−2.27307
$201$	0	0
$202$	−28.4612	−2.00252
$203$	−15.6792	−1.10046
$204$	0	0
$205$	11.3699	0.794107
$206$	10.2336	0.713012
$207$	0	0
$208$	−26.8963	−1.86492
$209$	−2.77911	−0.192235
$210$	0	0
$211$	24.3694	1.67766	0.838831	−	0.544392i	$-0.183240\pi$
$211$	24.3694	1.67766	0.838831	+	0.544392i	$0.183240\pi$
$212$	12.0545	0.827903
$213$	0	0
$214$	41.1568	2.81342
$215$	23.7604	1.62045
$216$	0	0
$217$	6.93911	0.471058
$218$	−32.7495	−2.21808
$219$	0	0
$220$	−27.8471	−1.87745
$221$	16.9168	1.13794
$222$	0	0
$223$	−28.9499	−1.93863	−0.969315	−	0.245821i	$-0.920943\pi$
$223$	−28.9499	−1.93863	−0.969315	+	0.245821i	$0.920943\pi$
$224$	8.53451	0.570236
$225$	0	0
$226$	−40.9846	−2.72625
$227$	−5.76469	−0.382616	−0.191308	−	0.981530i	$-0.561273\pi$
$227$	−5.76469	−0.382616	−0.191308	+	0.981530i	$0.561273\pi$
$228$	0	0
$229$	−22.8731	−1.51150	−0.755748	−	0.654863i	$-0.772726\pi$
$229$	−22.8731	−1.51150	−0.755748	+	0.654863i	$0.772726\pi$
$230$	18.3645	1.21092
$231$	0	0
$232$	−58.8665	−3.86477
$233$	−20.4379	−1.33893	−0.669464	−	0.742844i	$-0.733476\pi$
$233$	−20.4379	−1.33893	−0.669464	+	0.742844i	$0.733476\pi$
$234$	0	0
$235$	22.1028	1.44183
$236$	15.7470	1.02504
$237$	0	0
$238$	−18.3923	−1.19219
$239$	11.6116	0.751090	0.375545	−	0.926804i	$-0.377455\pi$
$239$	11.6116	0.751090	0.375545	+	0.926804i	$0.377455\pi$
$240$	0	0
$241$	13.2687	0.854710	0.427355	−	0.904084i	$-0.359446\pi$
$241$	13.2687	0.854710	0.427355	+	0.904084i	$0.359446\pi$
$242$	18.1680	1.16788
$243$	0	0
$244$	−12.4269	−0.795552
$245$	−13.5192	−0.863713
$246$	0	0
$247$	−5.49707	−0.349770
$248$	26.0524	1.65433
$249$	0	0
$250$	−1.26218	−0.0798274
$251$	7.29619	0.460532	0.230266	−	0.973128i	$-0.426040\pi$
$251$	7.29619	0.460532	0.230266	+	0.973128i	$0.426040\pi$
$252$	0	0
$253$	4.44996	0.279766
$254$	10.1418	0.636352
$255$	0	0
$256$	−29.7036	−1.85648
$257$	13.6877	0.853818	0.426909	−	0.904295i	$-0.359603\pi$
$257$	13.6877	0.853818	0.426909	+	0.904295i	$0.359603\pi$
$258$	0	0
$259$	11.3715	0.706594
$260$	−55.0816	−3.41602
$261$	0	0
$262$	−24.8063	−1.53254
$263$	0.175949	0.0108495	0.00542473	−	0.999985i	$-0.498273\pi$
$263$	0.175949	0.0108495	0.00542473	+	0.999985i	$0.498273\pi$
$264$	0	0
$265$	8.62463	0.529807
$266$	5.97654	0.366445
$267$	0	0
$268$	−8.21416	−0.501760
$269$	8.05183	0.490929	0.245464	−	0.969406i	$-0.421060\pi$
$269$	8.05183	0.490929	0.245464	+	0.969406i	$0.421060\pi$
$270$	0	0
$271$	19.1488	1.16321	0.581605	−	0.813471i	$-0.302425\pi$
$271$	19.1488	1.16321	0.581605	+	0.813471i	$0.302425\pi$
$272$	−30.2167	−1.83216
$273$	0	0
$274$	−18.8747	−1.14026
$275$	−10.1148	−0.609948
$276$	0	0
$277$	21.2874	1.27904	0.639518	−	0.768776i	$-0.279134\pi$
$277$	21.2874	1.27904	0.639518	+	0.768776i	$0.279134\pi$
$278$	−45.1033	−2.70512
$279$	0	0
$280$	32.9961	1.97189
$281$	−10.0072	−0.596977	−0.298489	−	0.954413i	$-0.596483\pi$
$281$	−10.0072	−0.596977	−0.298489	+	0.954413i	$0.596483\pi$
$282$	0	0
$283$	27.5972	1.64048	0.820241	−	0.572019i	$-0.193840\pi$
$283$	27.5972	1.64048	0.820241	+	0.572019i	$0.193840\pi$
$284$	−10.2296	−0.607015
$285$	0	0
$286$	−19.3400	−1.14360
$287$	−5.92484	−0.349732
$288$	0	0
$289$	2.00517	0.117951
$290$	−76.4405	−4.48874
$291$	0	0
$292$	−50.9567	−2.98202
$293$	7.33127	0.428298	0.214149	−	0.976801i	$-0.431302\pi$
$293$	7.33127	0.428298	0.214149	+	0.976801i	$0.431302\pi$
$294$	0	0
$295$	11.2665	0.655963
$296$	42.6937	2.48152
$297$	0	0
$298$	−35.6812	−2.06696
$299$	8.80201	0.509034
$300$	0	0
$301$	−12.3815	−0.713660
$302$	−7.02648	−0.404328
$303$	0	0
$304$	9.81887	0.563151
$305$	−8.89112	−0.509104
$306$	0	0
$307$	5.82836	0.332642	0.166321	−	0.986072i	$-0.446811\pi$
$307$	5.82836	0.332642	0.166321	+	0.986072i	$0.446811\pi$
$308$	14.5111	0.826848
$309$	0	0
$310$	33.8301	1.92142
$311$	−9.12122	−0.517217	−0.258608	−	0.965982i	$-0.583264\pi$
$311$	−9.12122	−0.517217	−0.258608	+	0.965982i	$0.583264\pi$
$312$	0	0
$313$	6.39310	0.361359	0.180680	−	0.983542i	$-0.442170\pi$
$313$	6.39310	0.361359	0.180680	+	0.983542i	$0.442170\pi$
$314$	−12.6958	−0.716465
$315$	0	0
$316$	31.0450	1.74642
$317$	−24.7081	−1.38775	−0.693873	−	0.720097i	$-0.744097\pi$
$317$	−24.7081	−1.38775	−0.693873	+	0.720097i	$0.744097\pi$
$318$	0	0
$319$	−18.5225	−1.03706
$320$	−2.56934	−0.143630
$321$	0	0
$322$	−9.56975	−0.533301
$323$	−6.17570	−0.343625
$324$	0	0
$325$	−20.0072	−1.10980
$326$	−32.3018	−1.78903
$327$	0	0
$328$	−22.2444	−1.22824
$329$	−11.5177	−0.634993
$330$	0	0
$331$	−7.06870	−0.388531	−0.194265	−	0.980949i	$-0.562232\pi$
$331$	−7.06870	−0.388531	−0.194265	+	0.980949i	$0.562232\pi$
$332$	0.477922	0.0262294
$333$	0	0
$334$	−51.4349	−2.81439
$335$	−5.87701	−0.321095
$336$	0	0
$337$	0.489921	0.0266877	0.0133439	−	0.999911i	$-0.495752\pi$
$337$	0.489921	0.0266877	0.0133439	+	0.999911i	$0.495752\pi$
$338$	−5.22797	−0.284364
$339$	0	0
$340$	−61.8816	−3.35600
$341$	8.19746	0.443917
$342$	0	0
$343$	18.6695	1.00806
$344$	−46.4856	−2.50634
$345$	0	0
$346$	−17.2617	−0.927993
$347$	18.8536	1.01211	0.506057	−	0.862500i	$-0.331103\pi$
$347$	18.8536	1.01211	0.506057	+	0.862500i	$0.331103\pi$
$348$	0	0
$349$	35.4610	1.89818	0.949092	−	0.314999i	$-0.102004\pi$
$349$	35.4610	1.89818	0.949092	+	0.314999i	$0.102004\pi$
$350$	21.7522	1.16271
$351$	0	0
$352$	10.0822	0.537381
$353$	−1.97621	−0.105183	−0.0525914	−	0.998616i	$-0.516748\pi$
$353$	−1.97621	−0.105183	−0.0525914	+	0.998616i	$0.516748\pi$
$354$	0	0
$355$	−7.31900	−0.388452
$356$	−23.5440	−1.24783
$357$	0	0
$358$	−56.9158	−3.00809
$359$	−8.05736	−0.425251	−0.212626	−	0.977134i	$-0.568201\pi$
$359$	−8.05736	−0.425251	−0.212626	+	0.977134i	$0.568201\pi$
$360$	0	0
$361$	−16.9932	−0.894380
$362$	57.6958	3.03242
$363$	0	0
$364$	28.7030	1.50445
$365$	−36.4581	−1.90831
$366$	0	0
$367$	17.8111	0.929730	0.464865	−	0.885382i	$-0.346103\pi$
$367$	17.8111	0.929730	0.464865	+	0.885382i	$0.346103\pi$
$368$	−15.7222	−0.819574
$369$	0	0
$370$	55.4395	2.88216
$371$	−4.49429	−0.233332
$372$	0	0
$373$	2.35142	0.121752	0.0608760	−	0.998145i	$-0.480611\pi$
$373$	2.35142	0.121752	0.0608760	+	0.998145i	$0.480611\pi$
$374$	−21.7276	−1.12351
$375$	0	0
$376$	−43.2425	−2.23006
$377$	−36.6375	−1.88693
$378$	0	0
$379$	11.6298	0.597382	0.298691	−	0.954350i	$-0.403450\pi$
$379$	11.6298	0.597382	0.298691	+	0.954350i	$0.403450\pi$
$380$	20.1083	1.03153
$381$	0	0
$382$	46.4575	2.37697
$383$	−29.9411	−1.52992	−0.764958	−	0.644080i	$-0.777240\pi$
$383$	−29.9411	−1.52992	−0.764958	+	0.644080i	$0.777240\pi$
$384$	0	0
$385$	10.3823	0.529131
$386$	−22.3321	−1.13667
$387$	0	0
$388$	−19.4035	−0.985065
$389$	3.41170	0.172980	0.0864900	−	0.996253i	$-0.472435\pi$
$389$	3.41170	0.172980	0.0864900	+	0.996253i	$0.472435\pi$
$390$	0	0
$391$	9.88864	0.500090
$392$	26.4495	1.33590
$393$	0	0
$394$	4.72571	0.238078
$395$	22.2118	1.11760
$396$	0	0
$397$	−2.85437	−0.143257	−0.0716285	−	0.997431i	$-0.522820\pi$
$397$	−2.85437	−0.143257	−0.0716285	+	0.997431i	$0.522820\pi$
$398$	−34.0162	−1.70508
$399$	0	0
$400$	35.7368	1.78684
$401$	24.8247	1.23969	0.619843	−	0.784726i	$-0.287196\pi$
$401$	24.8247	1.23969	0.619843	+	0.784726i	$0.287196\pi$
$402$	0	0
$403$	16.2146	0.807705
$404$	49.8999	2.48261
$405$	0	0
$406$	39.8331	1.97688
$407$	13.4337	0.665883
$408$	0	0
$409$	−14.3691	−0.710506	−0.355253	−	0.934770i	$-0.615605\pi$
$409$	−14.3691	−0.710506	−0.355253	+	0.934770i	$0.615605\pi$
$410$	−28.8852	−1.42654
$411$	0	0
$412$	−17.9422	−0.883951
$413$	−5.87098	−0.288892
$414$	0	0
$415$	0.341940	0.0167852
$416$	19.9425	0.977763
$417$	0	0
$418$	7.06033	0.345332
$419$	−18.4515	−0.901415	−0.450707	−	0.892672i	$-0.648828\pi$
$419$	−18.4515	−0.901415	−0.450707	+	0.892672i	$0.648828\pi$
$420$	0	0
$421$	17.7932	0.867188	0.433594	−	0.901108i	$-0.357245\pi$
$421$	17.7932	0.867188	0.433594	+	0.901108i	$0.357245\pi$
$422$	−61.9107	−3.01377
$423$	0	0
$424$	−16.8735	−0.819449
$425$	−22.4771	−1.09030
$426$	0	0
$427$	4.63316	0.224214
$428$	−72.1585	−3.48791
$429$	0	0
$430$	−60.3634	−2.91098
$431$	−10.8330	−0.521808	−0.260904	−	0.965365i	$-0.584021\pi$
$431$	−10.8330	−0.521808	−0.260904	+	0.965365i	$0.584021\pi$
$432$	0	0
$433$	−30.1293	−1.44792	−0.723962	−	0.689840i	$-0.757681\pi$
$433$	−30.1293	−1.44792	−0.723962	+	0.689840i	$0.757681\pi$
$434$	−17.6288	−0.846212
$435$	0	0
$436$	57.4184	2.74984
$437$	−3.21330	−0.153713
$438$	0	0
$439$	28.0736	1.33988	0.669940	−	0.742415i	$-0.266320\pi$
$439$	28.0736	1.33988	0.669940	+	0.742415i	$0.266320\pi$
$440$	38.9796	1.85828
$441$	0	0
$442$	−42.9771	−2.04421
$443$	14.5133	0.689546	0.344773	−	0.938686i	$-0.387956\pi$
$443$	14.5133	0.689546	0.344773	+	0.938686i	$0.387956\pi$
$444$	0	0
$445$	−16.8451	−0.798532
$446$	73.5474	3.48257
$447$	0	0
$448$	1.33888	0.0632562
$449$	7.86431	0.371140	0.185570	−	0.982631i	$-0.440587\pi$
$449$	7.86431	0.371140	0.185570	+	0.982631i	$0.440587\pi$
$450$	0	0
$451$	−6.99926	−0.329582
$452$	71.8566	3.37985
$453$	0	0
$454$	14.6452	0.687335
$455$	20.5362	0.962753
$456$	0	0
$457$	25.4437	1.19021	0.595103	−	0.803649i	$-0.297111\pi$
$457$	25.4437	1.19021	0.595103	+	0.803649i	$0.297111\pi$
$458$	58.1092	2.71526
$459$	0	0
$460$	−32.1978	−1.50123
$461$	−20.4311	−0.951570	−0.475785	−	0.879562i	$-0.657836\pi$
$461$	−20.4311	−0.951570	−0.475785	+	0.879562i	$0.657836\pi$
$462$	0	0
$463$	27.6385	1.28447	0.642236	−	0.766507i	$-0.278007\pi$
$463$	27.6385	1.28447	0.642236	+	0.766507i	$0.278007\pi$
$464$	65.4419	3.03806
$465$	0	0
$466$	51.9225	2.40526
$467$	9.16496	0.424104	0.212052	−	0.977258i	$-0.431985\pi$
$467$	9.16496	0.424104	0.212052	+	0.977258i	$0.431985\pi$
$468$	0	0
$469$	3.06250	0.141413
$470$	−56.1522	−2.59011
$471$	0	0
$472$	−22.0422	−1.01457
$473$	−14.6268	−0.672542
$474$	0	0
$475$	7.30389	0.335126
$476$	32.2464	1.47801
$477$	0	0
$478$	−29.4993	−1.34926
$479$	6.75559	0.308671	0.154335	−	0.988019i	$-0.450676\pi$
$479$	6.75559	0.308671	0.154335	+	0.988019i	$0.450676\pi$
$480$	0	0
$481$	26.5718	1.21157
$482$	−33.7091	−1.53541
$483$	0	0
$484$	−31.8532	−1.44787
$485$	−13.8827	−0.630381
$486$	0	0
$487$	−19.6769	−0.891646	−0.445823	−	0.895121i	$-0.647089\pi$
$487$	−19.6769	−0.891646	−0.445823	+	0.895121i	$0.647089\pi$
$488$	17.3949	0.787428
$489$	0	0
$490$	34.3457	1.55158
$491$	−33.2134	−1.49890	−0.749449	−	0.662062i	$-0.769681\pi$
$491$	−33.2134	−1.49890	−0.749449	+	0.662062i	$0.769681\pi$
$492$	0	0
$493$	−41.1605	−1.85377
$494$	13.9653	0.628330
$495$	0	0
$496$	−28.9625	−1.30045
$497$	3.81393	0.171078
$498$	0	0
$499$	32.1585	1.43961	0.719805	−	0.694176i	$-0.244231\pi$
$499$	32.1585	1.43961	0.719805	+	0.694176i	$0.244231\pi$
$500$	2.21293	0.0989654
$501$	0	0
$502$	−18.5360	−0.827303
$503$	17.2428	0.768819	0.384410	−	0.923163i	$-0.374405\pi$
$503$	17.2428	0.768819	0.384410	+	0.923163i	$0.374405\pi$
$504$	0	0
$505$	35.7020	1.58872
$506$	−11.3051	−0.502574
$507$	0	0
$508$	−17.7812	−0.788912
$509$	32.1700	1.42591	0.712954	−	0.701211i	$-0.247357\pi$
$509$	32.1700	1.42591	0.712954	+	0.701211i	$0.247357\pi$
$510$	0	0
$511$	18.9983	0.840436
$512$	50.8088	2.24545
$513$	0	0
$514$	−34.7738	−1.53380
$515$	−12.8372	−0.565674
$516$	0	0
$517$	−13.6064	−0.598408
$518$	−28.8895	−1.26933
$519$	0	0
$520$	77.1017	3.38113
$521$	24.6284	1.07899	0.539495	−	0.841989i	$-0.318615\pi$
$521$	24.6284	1.07899	0.539495	+	0.841989i	$0.318615\pi$
$522$	0	0
$523$	5.84292	0.255493	0.127747	−	0.991807i	$-0.459226\pi$
$523$	5.84292	0.255493	0.127747	+	0.991807i	$0.459226\pi$
$524$	43.4918	1.89995
$525$	0	0
$526$	−0.446998	−0.0194900
$527$	18.2163	0.793514
$528$	0	0
$529$	−17.8548	−0.776296
$530$	−21.9109	−0.951749
$531$	0	0
$532$	−10.4784	−0.454297
$533$	−13.8445	−0.599673
$534$	0	0
$535$	−51.6274	−2.23205
$536$	11.4980	0.496636
$537$	0	0
$538$	−20.4557	−0.881908
$539$	8.32239	0.358471
$540$	0	0
$541$	−8.50279	−0.365564	−0.182782	−	0.983154i	$-0.558510\pi$
$541$	−8.50279	−0.365564	−0.182782	+	0.983154i	$0.558510\pi$
$542$	−48.6477	−2.08960
$543$	0	0
$544$	22.4045	0.960584
$545$	41.0813	1.75973
$546$	0	0
$547$	−44.9602	−1.92236	−0.961179	−	0.275925i	$-0.911016\pi$
$547$	−44.9602	−1.92236	−0.961179	+	0.275925i	$0.911016\pi$
$548$	33.0922	1.41363
$549$	0	0
$550$	25.6968	1.09572
$551$	13.3750	0.569795
$552$	0	0
$553$	−11.5746	−0.492201
$554$	−54.0807	−2.29767
$555$	0	0
$556$	79.0778	3.35365
$557$	−24.4636	−1.03656	−0.518278	−	0.855212i	$-0.673427\pi$
$557$	−24.4636	−1.03656	−0.518278	+	0.855212i	$0.673427\pi$
$558$	0	0
$559$	−28.9319	−1.22369
$560$	−36.6818	−1.55009
$561$	0	0
$562$	25.4233	1.07242
$563$	6.06992	0.255817	0.127908	−	0.991786i	$-0.459174\pi$
$563$	6.06992	0.255817	0.127908	+	0.991786i	$0.459174\pi$
$564$	0	0
$565$	51.4114	2.16290
$566$	−70.1107	−2.94697
$567$	0	0
$568$	14.3191	0.600817
$569$	10.6297	0.445620	0.222810	−	0.974862i	$-0.428477\pi$
$569$	10.6297	0.445620	0.222810	+	0.974862i	$0.428477\pi$
$570$	0	0
$571$	27.4864	1.15027	0.575134	−	0.818059i	$-0.304950\pi$
$571$	27.4864	1.15027	0.575134	+	0.818059i	$0.304950\pi$
$572$	33.9080	1.41777
$573$	0	0
$574$	15.0521	0.628262
$575$	−11.6951	−0.487721
$576$	0	0
$577$	−5.93714	−0.247166	−0.123583	−	0.992334i	$-0.539439\pi$
$577$	−5.93714	−0.247166	−0.123583	+	0.992334i	$0.539439\pi$
$578$	−5.09415	−0.211889
$579$	0	0
$580$	134.020	5.56488
$581$	−0.178185	−0.00739235
$582$	0	0
$583$	−5.30929	−0.219888
$584$	71.3278	2.95156
$585$	0	0
$586$	−18.6251	−0.769397
$587$	−11.1071	−0.458440	−0.229220	−	0.973375i	$-0.573617\pi$
$587$	−11.1071	−0.458440	−0.229220	+	0.973375i	$0.573617\pi$
$588$	0	0
$589$	−5.91935	−0.243903
$590$	−28.6227	−1.17838
$591$	0	0
$592$	−47.4626	−1.95070
$593$	2.05072	0.0842128	0.0421064	−	0.999113i	$-0.486593\pi$
$593$	2.05072	0.0842128	0.0421064	+	0.999113i	$0.486593\pi$
$594$	0	0
$595$	23.0714	0.945837
$596$	62.5584	2.56249
$597$	0	0
$598$	−22.3616	−0.914432
$599$	2.80439	0.114584	0.0572921	−	0.998357i	$-0.481753\pi$
$599$	2.80439	0.114584	0.0572921	+	0.998357i	$0.481753\pi$
$600$	0	0
$601$	−36.0402	−1.47011	−0.735055	−	0.678007i	$-0.762844\pi$
$601$	−36.0402	−1.47011	−0.735055	+	0.678007i	$0.762844\pi$
$602$	31.4554	1.28202
$603$	0	0
$604$	12.3192	0.501263
$605$	−22.7901	−0.926550
$606$	0	0
$607$	38.5142	1.56324	0.781621	−	0.623754i	$-0.214393\pi$
$607$	38.5142	1.56324	0.781621	+	0.623754i	$0.214393\pi$
$608$	−7.28029	−0.295255
$609$	0	0
$610$	22.5879	0.914558
$611$	−26.9134	−1.08880
$612$	0	0
$613$	1.45973	0.0589580	0.0294790	−	0.999565i	$-0.490615\pi$
$613$	1.45973	0.0589580	0.0294790	+	0.999565i	$0.490615\pi$
$614$	−14.8070	−0.597561
$615$	0	0
$616$	−20.3123	−0.818404
$617$	−12.0248	−0.484102	−0.242051	−	0.970264i	$-0.577820\pi$
$617$	−12.0248	−0.484102	−0.242051	+	0.970264i	$0.577820\pi$
$618$	0	0
$619$	31.3697	1.26085	0.630427	−	0.776249i	$-0.282880\pi$
$619$	31.3697	1.26085	0.630427	+	0.776249i	$0.282880\pi$
$620$	−59.3129	−2.38206
$621$	0	0
$622$	23.1725	0.929132
$623$	8.77795	0.351681
$624$	0	0
$625$	−24.1962	−0.967848
$626$	−16.2417	−0.649149
$627$	0	0
$628$	22.2590	0.888231
$629$	29.8522	1.19028
$630$	0	0
$631$	−17.4483	−0.694605	−0.347303	−	0.937753i	$-0.612902\pi$
$631$	−17.4483	−0.694605	−0.347303	+	0.937753i	$0.612902\pi$
$632$	−43.4559	−1.72858
$633$	0	0
$634$	62.7711	2.49296
$635$	−12.7219	−0.504855
$636$	0	0
$637$	16.4617	0.652236
$638$	47.0565	1.86298
$639$	0	0
$640$	39.2832	1.55280
$641$	9.22820	0.364492	0.182246	−	0.983253i	$-0.441663\pi$
$641$	9.22820	0.364492	0.182246	+	0.983253i	$0.441663\pi$
$642$	0	0
$643$	7.30126	0.287934	0.143967	−	0.989583i	$-0.454014\pi$
$643$	7.30126	0.287934	0.143967	+	0.989583i	$0.454014\pi$
$644$	16.7782	0.661156
$645$	0	0
$646$	15.6894	0.617291
$647$	34.3969	1.35228	0.676142	−	0.736772i	$-0.263651\pi$
$647$	34.3969	1.35228	0.676142	+	0.736772i	$0.263651\pi$
$648$	0	0
$649$	−6.93563	−0.272247
$650$	50.8283	1.99365
$651$	0	0
$652$	56.6335	2.21794
$653$	14.2491	0.557612	0.278806	−	0.960347i	$-0.410061\pi$
$653$	14.2491	0.557612	0.278806	+	0.960347i	$0.410061\pi$
$654$	0	0
$655$	31.1172	1.21585
$656$	24.7291	0.965509
$657$	0	0
$658$	29.2609	1.14071
$659$	44.0929	1.71761	0.858807	−	0.512300i	$-0.171206\pi$
$659$	44.0929	1.71761	0.858807	+	0.512300i	$0.171206\pi$
$660$	0	0
$661$	−2.63277	−0.102403	−0.0512015	−	0.998688i	$-0.516305\pi$
$661$	−2.63277	−0.102403	−0.0512015	+	0.998688i	$0.516305\pi$
$662$	17.9581	0.697960
$663$	0	0
$664$	−0.668982	−0.0259615
$665$	−7.49702	−0.290722
$666$	0	0
$667$	−21.4163	−0.829244
$668$	90.1787	3.48912
$669$	0	0
$670$	14.9306	0.576818
$671$	5.47334	0.211296
$672$	0	0
$673$	−3.41710	−0.131720	−0.0658598	−	0.997829i	$-0.520979\pi$
$673$	−3.41710	−0.131720	−0.0658598	+	0.997829i	$0.520979\pi$
$674$	−1.24465	−0.0479420
$675$	0	0
$676$	9.16598	0.352538
$677$	38.8868	1.49454	0.747271	−	0.664520i	$-0.231364\pi$
$677$	38.8868	1.49454	0.747271	+	0.664520i	$0.231364\pi$
$678$	0	0
$679$	7.23427	0.277626
$680$	86.6201	3.32173
$681$	0	0
$682$	−20.8257	−0.797456
$683$	23.1782	0.886891	0.443445	−	0.896301i	$-0.353756\pi$
$683$	23.1782	0.886891	0.443445	+	0.896301i	$0.353756\pi$
$684$	0	0
$685$	23.6766	0.904636
$686$	−47.4299	−1.81088
$687$	0	0
$688$	51.6781	1.97021
$689$	−10.5018	−0.400086
$690$	0	0
$691$	17.3931	0.661665	0.330833	−	0.943689i	$-0.392670\pi$
$691$	17.3931	0.661665	0.330833	+	0.943689i	$0.392670\pi$
$692$	30.2642	1.15047
$693$	0	0
$694$	−47.8976	−1.81817
$695$	56.5780	2.14613
$696$	0	0
$697$	−15.5537	−0.589137
$698$	−90.0888	−3.40991
$699$	0	0
$700$	−38.1373	−1.44146
$701$	−18.9249	−0.714785	−0.357393	−	0.933954i	$-0.616334\pi$
$701$	−18.9249	−0.714785	−0.357393	+	0.933954i	$0.616334\pi$
$702$	0	0
$703$	−9.70041	−0.365858
$704$	1.58167	0.0596116
$705$	0	0
$706$	5.02056	0.188951
$707$	−18.6043	−0.699687
$708$	0	0
$709$	42.7604	1.60590	0.802951	−	0.596045i	$-0.203262\pi$
$709$	42.7604	1.60590	0.802951	+	0.596045i	$0.203262\pi$
$710$	18.5940	0.697819
$711$	0	0
$712$	32.9562	1.23509
$713$	9.47818	0.354960
$714$	0	0
$715$	24.2603	0.907283
$716$	99.7882	3.72926
$717$	0	0
$718$	20.4698	0.763924
$719$	13.0136	0.485326	0.242663	−	0.970111i	$-0.421979\pi$
$719$	13.0136	0.485326	0.242663	+	0.970111i	$0.421979\pi$
$720$	0	0
$721$	6.68945	0.249128
$722$	43.1714	1.60667
$723$	0	0
$724$	−101.156	−3.75942
$725$	48.6798	1.80792
$726$	0	0
$727$	−50.4565	−1.87133	−0.935664	−	0.352891i	$-0.885199\pi$
$727$	−50.4565	−1.87133	−0.935664	+	0.352891i	$0.885199\pi$
$728$	−40.1777	−1.48908
$729$	0	0
$730$	92.6220	3.42810
$731$	−32.5035	−1.20219
$732$	0	0
$733$	−15.9879	−0.590527	−0.295263	−	0.955416i	$-0.595407\pi$
$733$	−15.9879	−0.590527	−0.295263	+	0.955416i	$0.595407\pi$
$734$	−45.2491	−1.67017
$735$	0	0
$736$	11.6573	0.429695
$737$	3.61786	0.133266
$738$	0	0
$739$	−7.13598	−0.262501	−0.131251	−	0.991349i	$-0.541899\pi$
$739$	−7.13598	−0.262501	−0.131251	+	0.991349i	$0.541899\pi$
$740$	−97.1998	−3.57313
$741$	0	0
$742$	11.4178	0.419159
$743$	−33.9109	−1.24407	−0.622036	−	0.782989i	$-0.713694\pi$
$743$	−33.9109	−1.24407	−0.622036	+	0.782989i	$0.713694\pi$
$744$	0	0
$745$	44.7589	1.63984
$746$	−5.97380	−0.218716
$747$	0	0
$748$	38.0940	1.39286
$749$	26.9030	0.983015
$750$	0	0
$751$	−41.9333	−1.53017	−0.765085	−	0.643930i	$-0.777303\pi$
$751$	−41.9333	−1.53017	−0.765085	+	0.643930i	$0.777303\pi$
$752$	48.0727	1.75303
$753$	0	0
$754$	93.0777	3.38969
$755$	8.81408	0.320777
$756$	0	0
$757$	26.1805	0.951548	0.475774	−	0.879568i	$-0.342168\pi$
$757$	26.1805	0.951548	0.475774	+	0.879568i	$0.342168\pi$
$758$	−29.5455	−1.07314
$759$	0	0
$760$	−28.1470	−1.02100
$761$	−46.8751	−1.69922	−0.849610	−	0.527411i	$-0.823163\pi$
$761$	−46.8751	−1.69922	−0.849610	+	0.527411i	$0.823163\pi$
$762$	0	0
$763$	−21.4074	−0.775001
$764$	−81.4521	−2.94683
$765$	0	0
$766$	76.0654	2.74835
$767$	−13.7187	−0.495353
$768$	0	0
$769$	−13.7103	−0.494407	−0.247203	−	0.968964i	$-0.579512\pi$
$769$	−13.7103	−0.494407	−0.247203	+	0.968964i	$0.579512\pi$
$770$	−26.3763	−0.950536
$771$	0	0
$772$	39.1540	1.40918
$773$	15.2517	0.548566	0.274283	−	0.961649i	$-0.411559\pi$
$773$	15.2517	0.548566	0.274283	+	0.961649i	$0.411559\pi$
$774$	0	0
$775$	−21.5441	−0.773887
$776$	27.1605	0.975006
$777$	0	0
$778$	−8.66743	−0.310743
$779$	5.05414	0.181083
$780$	0	0
$781$	4.50555	0.161221
$782$	−25.1221	−0.898366
$783$	0	0
$784$	−29.4039	−1.05014
$785$	15.9257	0.568413
$786$	0	0
$787$	−21.0003	−0.748580	−0.374290	−	0.927312i	$-0.622114\pi$
$787$	−21.0003	−0.748580	−0.374290	+	0.927312i	$0.622114\pi$
$788$	−8.28539	−0.295155
$789$	0	0
$790$	−56.4292	−2.00766
$791$	−26.7905	−0.952560
$792$	0	0
$793$	10.8263	0.384452
$794$	7.25155	0.257348
$795$	0	0
$796$	59.6393	2.11386
$797$	46.1575	1.63498	0.817491	−	0.575941i	$-0.195364\pi$
$797$	46.1575	1.63498	0.817491	+	0.575941i	$0.195364\pi$
$798$	0	0
$799$	−30.2359	−1.06967
$800$	−26.4974	−0.936824
$801$	0	0
$802$	−63.0672	−2.22698
$803$	22.4435	0.792013
$804$	0	0
$805$	12.0044	0.423099
$806$	−41.1932	−1.45097
$807$	0	0
$808$	−69.8485	−2.45726
$809$	15.0049	0.527545	0.263772	−	0.964585i	$-0.415033\pi$
$809$	15.0049	0.527545	0.263772	+	0.964585i	$0.415033\pi$
$810$	0	0
$811$	47.0448	1.65197	0.825983	−	0.563695i	$-0.190621\pi$
$811$	47.0448	1.65197	0.825983	+	0.563695i	$0.190621\pi$
$812$	−69.8378	−2.45082
$813$	0	0
$814$	−34.1283	−1.19620
$815$	40.5197	1.41934
$816$	0	0
$817$	10.5620	0.369516
$818$	36.5047	1.27636
$819$	0	0
$820$	50.6433	1.76854
$821$	1.52951	0.0533801	0.0266901	−	0.999644i	$-0.491503\pi$
$821$	1.52951	0.0533801	0.0266901	+	0.999644i	$0.491503\pi$
$822$	0	0
$823$	2.16425	0.0754410	0.0377205	−	0.999288i	$-0.487990\pi$
$823$	2.16425	0.0754410	0.0377205	+	0.999288i	$0.487990\pi$
$824$	25.1151	0.874924
$825$	0	0
$826$	14.9153	0.518968
$827$	−40.8370	−1.42004	−0.710020	−	0.704181i	$-0.751314\pi$
$827$	−40.8370	−1.42004	−0.710020	+	0.704181i	$0.751314\pi$
$828$	0	0
$829$	17.1117	0.594316	0.297158	−	0.954828i	$-0.403961\pi$
$829$	17.1117	0.594316	0.297158	+	0.954828i	$0.403961\pi$
$830$	−0.868701	−0.0301530
$831$	0	0
$832$	3.12855	0.108463
$833$	18.4939	0.640777
$834$	0	0
$835$	64.5204	2.23282
$836$	−12.3786	−0.428122
$837$	0	0
$838$	46.8761	1.61931
$839$	−18.3623	−0.633937	−0.316969	−	0.948436i	$-0.602665\pi$
$839$	−18.3623	−0.633937	−0.316969	+	0.948436i	$0.602665\pi$
$840$	0	0
$841$	60.1433	2.07391
$842$	−45.2037	−1.55782
$843$	0	0
$844$	108.546	3.73629
$845$	6.55801	0.225602
$846$	0	0
$847$	11.8759	0.408061
$848$	18.7583	0.644162
$849$	0	0
$850$	57.1032	1.95862
$851$	15.5325	0.532446
$852$	0	0
$853$	15.5086	0.531005	0.265503	−	0.964110i	$-0.414462\pi$
$853$	15.5086	0.531005	0.265503	+	0.964110i	$0.414462\pi$
$854$	−11.7706	−0.402780
$855$	0	0
$856$	101.005	3.45229
$857$	19.0971	0.652344	0.326172	−	0.945310i	$-0.394241\pi$
$857$	19.0971	0.652344	0.326172	+	0.945310i	$0.394241\pi$
$858$	0	0
$859$	2.83131	0.0966032	0.0483016	−	0.998833i	$-0.484619\pi$
$859$	2.83131	0.0966032	0.0483016	+	0.998833i	$0.484619\pi$
$860$	105.833	3.60887
$861$	0	0
$862$	27.5213	0.937380
$863$	3.01216	0.102535	0.0512675	−	0.998685i	$-0.483674\pi$
$863$	3.01216	0.102535	0.0512675	+	0.998685i	$0.483674\pi$
$864$	0	0
$865$	21.6532	0.736231
$866$	76.5437	2.60106
$867$	0	0
$868$	30.9079	1.04908
$869$	−13.6735	−0.463842
$870$	0	0
$871$	7.15613	0.242476
$872$	−80.3727	−2.72176
$873$	0	0
$874$	8.16339	0.276131
$875$	−0.825053	−0.0278919
$876$	0	0
$877$	14.5173	0.490214	0.245107	−	0.969496i	$-0.421177\pi$
$877$	14.5173	0.490214	0.245107	+	0.969496i	$0.421177\pi$
$878$	−71.3212	−2.40697
$879$	0	0
$880$	−43.3337	−1.46078
$881$	6.48384	0.218446	0.109223	−	0.994017i	$-0.465164\pi$
$881$	6.48384	0.218446	0.109223	+	0.994017i	$0.465164\pi$
$882$	0	0
$883$	25.3933	0.854554	0.427277	−	0.904121i	$-0.359473\pi$
$883$	25.3933	0.854554	0.427277	+	0.904121i	$0.359473\pi$
$884$	75.3500	2.53430
$885$	0	0
$886$	−36.8710	−1.23871
$887$	0.107021	0.00359341	0.00179671	−	0.999998i	$-0.499428\pi$
$887$	0.107021	0.00359341	0.00179671	+	0.999998i	$0.499428\pi$
$888$	0	0
$889$	6.62940	0.222343
$890$	42.7950	1.43449
$891$	0	0
$892$	−128.948	−4.31749
$893$	9.82511	0.328785
$894$	0	0
$895$	71.3957	2.38650
$896$	−20.4705	−0.683870
$897$	0	0
$898$	−19.9793	−0.666718
$899$	−39.4519	−1.31580
$900$	0	0
$901$	−11.7982	−0.393056
$902$	17.7816	0.592064
$903$	0	0
$904$	−100.583	−3.34534
$905$	−72.3741	−2.40580
$906$	0	0
$907$	43.5972	1.44762	0.723810	−	0.689999i	$-0.242389\pi$
$907$	43.5972	1.44762	0.723810	+	0.689999i	$0.242389\pi$
$908$	−25.6769	−0.852118
$909$	0	0
$910$	−52.1723	−1.72950
$911$	55.4124	1.83590	0.917948	−	0.396702i	$-0.129845\pi$
$911$	55.4124	1.83590	0.917948	+	0.396702i	$0.129845\pi$
$912$	0	0
$913$	−0.210497	−0.00696644
$914$	−64.6399	−2.13810
$915$	0	0
$916$	−101.880	−3.36622
$917$	−16.2152	−0.535472
$918$	0	0
$919$	−40.9441	−1.35062	−0.675311	−	0.737533i	$-0.735990\pi$
$919$	−40.9441	−1.35062	−0.675311	+	0.737533i	$0.735990\pi$
$920$	45.0696	1.48590
$921$	0	0
$922$	51.9053	1.70941
$923$	8.91198	0.293341
$924$	0	0
$925$	−35.3057	−1.16084
$926$	−70.2159	−2.30744
$927$	0	0
$928$	−48.5225	−1.59283
$929$	−32.3050	−1.05989	−0.529947	−	0.848031i	$-0.677788\pi$
$929$	−32.3050	−1.05989	−0.529947	+	0.848031i	$0.677788\pi$
$930$	0	0
$931$	−6.00957	−0.196956
$932$	−91.0336	−2.98190
$933$	0	0
$934$	−23.2836	−0.761863
$935$	27.2552	0.891342
$936$	0	0
$937$	−9.06200	−0.296043	−0.148021	−	0.988984i	$-0.547290\pi$
$937$	−9.06200	−0.296043	−0.148021	+	0.988984i	$0.547290\pi$
$938$	−7.78031	−0.254036
$939$	0	0
$940$	98.4493	3.21106
$941$	23.9044	0.779263	0.389631	−	0.920971i	$-0.372602\pi$
$941$	23.9044	0.779263	0.389631	+	0.920971i	$0.372602\pi$
$942$	0	0
$943$	−8.09278	−0.263537
$944$	24.5043	0.797547
$945$	0	0
$946$	37.1595	1.20816
$947$	8.23165	0.267493	0.133746	−	0.991016i	$-0.457299\pi$
$947$	8.23165	0.267493	0.133746	+	0.991016i	$0.457299\pi$
$948$	0	0
$949$	44.3932	1.44106
$950$	−18.5556	−0.602022
$951$	0	0
$952$	−45.1377	−1.46292
$953$	−17.0755	−0.553129	−0.276564	−	0.960995i	$-0.589196\pi$
$953$	−17.0755	−0.553129	−0.276564	+	0.960995i	$0.589196\pi$
$954$	0	0
$955$	−58.2768	−1.88579
$956$	51.7199	1.67274
$957$	0	0
$958$	−17.1626	−0.554498
$959$	−12.3379	−0.398410
$960$	0	0
$961$	−13.5398	−0.436769
$962$	−67.5058	−2.17648
$963$	0	0
$964$	59.1008	1.90351
$965$	28.0136	0.901790
$966$	0	0
$967$	15.3524	0.493698	0.246849	−	0.969054i	$-0.420605\pi$
$967$	15.3524	0.493698	0.246849	+	0.969054i	$0.420605\pi$
$968$	44.5873	1.43309
$969$	0	0
$970$	35.2691	1.13242
$971$	−10.5213	−0.337645	−0.168822	−	0.985646i	$-0.553996\pi$
$971$	−10.5213	−0.337645	−0.168822	+	0.985646i	$0.553996\pi$
$972$	0	0
$973$	−29.4828	−0.945174
$974$	49.9893	1.60176
$975$	0	0
$976$	−19.3379	−0.618990
$977$	23.0942	0.738850	0.369425	−	0.929261i	$-0.379555\pi$
$977$	23.0942	0.738850	0.369425	+	0.929261i	$0.379555\pi$
$978$	0	0
$979$	10.3698	0.331419
$980$	−60.2169	−1.92356
$981$	0	0
$982$	84.3787	2.69263
$983$	1.19718	0.0381843	0.0190921	−	0.999818i	$-0.493922\pi$
$983$	1.19718	0.0381843	0.0190921	+	0.999818i	$0.493922\pi$
$984$	0	0
$985$	−5.92797	−0.188881
$986$	104.568	3.33013
$987$	0	0
$988$	−24.4849	−0.778967
$989$	−16.9120	−0.537771
$990$	0	0
$991$	−21.7076	−0.689565	−0.344783	−	0.938683i	$-0.612047\pi$
$991$	−21.7076	−0.689565	−0.344783	+	0.938683i	$0.612047\pi$
$992$	21.4745	0.681816
$993$	0	0
$994$	−9.68930	−0.307326
$995$	42.6703	1.35274
$996$	0	0
$997$	−34.5063	−1.09282	−0.546412	−	0.837516i	$-0.684007\pi$
$997$	−34.5063	−1.09282	−0.546412	+	0.837516i	$0.684007\pi$
$998$	−81.6987	−2.58613
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.3		72
3.2	odd	2		6561.2.a.c.1.70		72
81.4	even	27		729.2.g.a.136.1		144
81.7	even	27		243.2.g.a.37.8		144
81.20	odd	54		729.2.g.d.595.8		144
81.23	odd	54		81.2.g.a.43.1	✓	144
81.31	even	27		729.2.g.b.622.8		144
81.34	even	27		729.2.g.b.109.8		144
81.47	odd	54		729.2.g.c.109.1		144
81.50	odd	54		729.2.g.c.622.1		144
81.58	even	27		243.2.g.a.46.8		144
81.61	even	27		729.2.g.a.595.1		144
81.74	odd	54		81.2.g.a.49.1	yes	144
81.77	odd	54		729.2.g.d.136.8		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.43.1	✓	144	81.23	odd	54
81.2.g.a.49.1	yes	144	81.74	odd	54
243.2.g.a.37.8		144	81.7	even	27
243.2.g.a.46.8		144	81.58	even	27
729.2.g.a.136.1		144	81.4	even	27
729.2.g.a.595.1		144	81.61	even	27
729.2.g.b.109.8		144	81.34	even	27
729.2.g.b.622.8		144	81.31	even	27
729.2.g.c.109.1		144	81.47	odd	54
729.2.g.c.622.1		144	81.50	odd	54
729.2.g.d.136.8		144	81.77	odd	54
729.2.g.d.595.8		144	81.20	odd	54
6561.2.a.c.1.70		72	3.2	odd	2
6561.2.a.d.1.3		72	1.1	even	1	trivial

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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

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Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	6561.2.a.d.1.3

Level	$6561$
Weight	$2$
Character	6561.1
Self dual	yes
Analytic conductor	$52.390$
Analytic rank	$0$
Dimension	$72$
CM	no
Inner twists	$1$