LMFDB - Embedded newform 6561.2.a.d.1.4

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			1.4
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)

$f(q)$	$=$	$q-2.52847 q^{2} +4.39319 q^{4} +2.84004 q^{5} +0.550190 q^{7} -6.05111 q^{8} +O(q^{10})$	\(q-2.52847 q^{2} +4.39319 q^{4} +2.84004 q^{5} +0.550190 q^{7} -6.05111 q^{8} -7.18097 q^{10} +3.06752 q^{11} +4.64432 q^{13} -1.39114 q^{14} +6.51371 q^{16} +3.01691 q^{17} -5.03957 q^{19} +12.4768 q^{20} -7.75615 q^{22} +8.32434 q^{23} +3.06583 q^{25} -11.7430 q^{26} +2.41709 q^{28} +1.58964 q^{29} +4.93198 q^{31} -4.36753 q^{32} -7.62817 q^{34} +1.56256 q^{35} +3.58383 q^{37} +12.7424 q^{38} -17.1854 q^{40} +1.14317 q^{41} -0.360535 q^{43} +13.4762 q^{44} -21.0479 q^{46} +8.12827 q^{47} -6.69729 q^{49} -7.75188 q^{50} +20.4034 q^{52} -0.691183 q^{53} +8.71189 q^{55} -3.32926 q^{56} -4.01936 q^{58} +0.591777 q^{59} +3.48236 q^{61} -12.4704 q^{62} -1.98423 q^{64} +13.1901 q^{65} -10.0523 q^{67} +13.2538 q^{68} -3.95090 q^{70} -2.44663 q^{71} +5.17893 q^{73} -9.06162 q^{74} -22.1398 q^{76} +1.68772 q^{77} -2.24989 q^{79} +18.4992 q^{80} -2.89048 q^{82} -3.04597 q^{83} +8.56813 q^{85} +0.911605 q^{86} -18.5619 q^{88} +11.5498 q^{89} +2.55526 q^{91} +36.5704 q^{92} -20.5521 q^{94} -14.3126 q^{95} -4.20340 q^{97} +16.9339 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) $ 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8	$ 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38} + 72 q^{41} + 90 q^{44} + 90 q^{47} + 18 q^{49} + 45 q^{50} + 45 q^{53} + 9 q^{55} + 108 q^{56} + 18 q^{58} + 108 q^{59} + 72 q^{62} + 9 q^{64} + 72 q^{65} + 108 q^{68} + 126 q^{71} + 90 q^{74} + 72 q^{77} + 144 q^{80} - 18 q^{82} + 108 q^{83} + 90 q^{86} + 108 q^{89} + 72 q^{92} + 144 q^{95} + 81 q^{98}+O(q^{100}) $ 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8 + 36 * q^11 + 36 * q^14 + 45 * q^16 + 36 * q^17 + 54 * q^20 + 54 * q^23 + 36 * q^25 + 45 * q^26 + 9 * q^28 + 54 * q^29 + 63 * q^32 + 72 * q^35 + 54 * q^38 + 72 * q^41 + 90 * q^44 + 90 * q^47 + 18 * q^49 + 45 * q^50 + 45 * q^53 + 9 * q^55 + 108 * q^56 + 18 * q^58 + 108 * q^59 + 72 * q^62 + 9 * q^64 + 72 * q^65 + 108 * q^68 + 126 * q^71 + 90 * q^74 + 72 * q^77 + 144 * q^80 - 18 * q^82 + 108 * q^83 + 90 * q^86 + 108 * q^89 + 72 * q^92 + 144 * q^95 + 81 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.52847	−1.78790	−0.893951	−	0.448165i	$-0.852078\pi$
$2$	−2.52847	−1.78790	−0.893951	+	0.448165i	$0.852078\pi$
$3$	0	0
$3$	0	0
$4$	4.39319	2.19659
$5$	2.84004	1.27010	0.635052	−	0.772469i	$-0.280979\pi$
$5$	2.84004	1.27010	0.635052	+	0.772469i	$0.280979\pi$
$6$	0	0
$7$	0.550190	0.207952	0.103976	−	0.994580i	$-0.466843\pi$
$7$	0.550190	0.207952	0.103976	+	0.994580i	$0.466843\pi$
$8$	−6.05111	−2.13939
$9$	0	0
$10$	−7.18097	−2.27082
$11$	3.06752	0.924893	0.462446	−	0.886647i	$-0.346972\pi$
$11$	3.06752	0.924893	0.462446	+	0.886647i	$0.346972\pi$
$12$	0	0
$13$	4.64432	1.28810	0.644051	−	0.764983i	$-0.277252\pi$
$13$	4.64432	1.28810	0.644051	+	0.764983i	$0.277252\pi$
$14$	−1.39114	−0.371798
$15$	0	0
$16$	6.51371	1.62843
$17$	3.01691	0.731707	0.365853	−	0.930672i	$-0.380777\pi$
$17$	3.01691	0.731707	0.365853	+	0.930672i	$0.380777\pi$
$18$	0	0
$19$	−5.03957	−1.15616	−0.578078	−	0.815981i	$-0.696197\pi$
$19$	−5.03957	−1.15616	−0.578078	+	0.815981i	$0.696197\pi$
$20$	12.4768	2.78990
$21$	0	0
$22$	−7.75615	−1.65362
$23$	8.32434	1.73575	0.867873	−	0.496786i	$-0.165487\pi$
$23$	8.32434	1.73575	0.867873	+	0.496786i	$0.165487\pi$
$24$	0	0
$25$	3.06583	0.613166
$26$	−11.7430	−2.30300
$27$	0	0
$28$	2.41709	0.456786
$29$	1.58964	0.295189	0.147594	−	0.989048i	$-0.452847\pi$
$29$	1.58964	0.295189	0.147594	+	0.989048i	$0.452847\pi$
$30$	0	0
$31$	4.93198	0.885809	0.442905	−	0.896569i	$-0.353948\pi$
$31$	4.93198	0.885809	0.442905	+	0.896569i	$0.353948\pi$
$32$	−4.36753	−0.772077
$33$	0	0
$34$	−7.62817	−1.30822
$35$	1.56256	0.264121
$36$	0	0
$37$	3.58383	0.589178	0.294589	−	0.955624i	$-0.404817\pi$
$37$	3.58383	0.589178	0.294589	+	0.955624i	$0.404817\pi$
$38$	12.7424	2.06709
$39$	0	0
$40$	−17.1854	−2.71725
$41$	1.14317	0.178533	0.0892667	−	0.996008i	$-0.471548\pi$
$41$	1.14317	0.178533	0.0892667	+	0.996008i	$0.471548\pi$
$42$	0	0
$43$	−0.360535	−0.0549811	−0.0274906	−	0.999622i	$-0.508752\pi$
$43$	−0.360535	−0.0549811	−0.0274906	+	0.999622i	$0.508752\pi$
$44$	13.4762	2.03161
$45$	0	0
$46$	−21.0479	−3.10334
$47$	8.12827	1.18563	0.592815	−	0.805339i	$-0.298016\pi$
$47$	8.12827	1.18563	0.592815	+	0.805339i	$0.298016\pi$
$48$	0	0
$49$	−6.69729	−0.956756
$50$	−7.75188	−1.09628
$51$	0	0
$52$	20.4034	2.82944
$53$	−0.691183	−0.0949413	−0.0474706	−	0.998873i	$-0.515116\pi$
$53$	−0.691183	−0.0949413	−0.0474706	+	0.998873i	$0.515116\pi$
$54$	0	0
$55$	8.71189	1.17471
$56$	−3.32926	−0.444891
$57$	0	0
$58$	−4.01936	−0.527768
$59$	0.591777	0.0770428	0.0385214	−	0.999258i	$-0.487735\pi$
$59$	0.591777	0.0770428	0.0385214	+	0.999258i	$0.487735\pi$
$60$	0	0
$61$	3.48236	0.445870	0.222935	−	0.974833i	$-0.428436\pi$
$61$	3.48236	0.445870	0.222935	+	0.974833i	$0.428436\pi$
$62$	−12.4704	−1.58374
$63$	0	0
$64$	−1.98423	−0.248029
$65$	13.1901	1.63602
$66$	0	0
$67$	−10.0523	−1.22808	−0.614042	−	0.789273i	$-0.710458\pi$
$67$	−10.0523	−1.22808	−0.614042	+	0.789273i	$0.710458\pi$
$68$	13.2538	1.60726
$69$	0	0
$70$	−3.95090	−0.472223
$71$	−2.44663	−0.290361	−0.145181	−	0.989405i	$-0.546376\pi$
$71$	−2.44663	−0.290361	−0.145181	+	0.989405i	$0.546376\pi$
$72$	0	0
$73$	5.17893	0.606148	0.303074	−	0.952967i	$-0.401987\pi$
$73$	5.17893	0.606148	0.303074	+	0.952967i	$0.401987\pi$
$74$	−9.06162	−1.05339
$75$	0	0
$76$	−22.1398	−2.53961
$77$	1.68772	0.192333
$78$	0	0
$79$	−2.24989	−0.253132	−0.126566	−	0.991958i	$-0.540396\pi$
$79$	−2.24989	−0.253132	−0.126566	+	0.991958i	$0.540396\pi$
$80$	18.4992	2.06827
$81$	0	0
$82$	−2.89048	−0.319200
$83$	−3.04597	−0.334338	−0.167169	−	0.985928i	$-0.553463\pi$
$83$	−3.04597	−0.334338	−0.167169	+	0.985928i	$0.553463\pi$
$84$	0	0
$85$	8.56813	0.929344
$86$	0.911605	0.0983009
$87$	0	0
$88$	−18.5619	−1.97871
$89$	11.5498	1.22427	0.612136	−	0.790753i	$-0.290311\pi$
$89$	11.5498	1.22427	0.612136	+	0.790753i	$0.290311\pi$
$90$	0	0
$91$	2.55526	0.267864
$92$	36.5704	3.81273
$93$	0	0
$94$	−20.5521	−2.11979
$95$	−14.3126	−1.46844
$96$	0	0
$97$	−4.20340	−0.426790	−0.213395	−	0.976966i	$-0.568452\pi$
$97$	−4.20340	−0.426790	−0.213395	+	0.976966i	$0.568452\pi$
$98$	16.9339	1.71059
$99$	0	0
$100$	13.4688	1.34688
$101$	−3.85059	−0.383148	−0.191574	−	0.981478i	$-0.561359\pi$
$101$	−3.85059	−0.383148	−0.191574	+	0.981478i	$0.561359\pi$
$102$	0	0
$103$	−1.80797	−0.178144	−0.0890721	−	0.996025i	$-0.528390\pi$
$103$	−1.80797	−0.178144	−0.0890721	+	0.996025i	$0.528390\pi$
$104$	−28.1033	−2.75575
$105$	0	0
$106$	1.74764	0.169746
$107$	12.4050	1.19924	0.599619	−	0.800285i	$-0.295319\pi$
$107$	12.4050	1.19924	0.599619	+	0.800285i	$0.295319\pi$
$108$	0	0
$109$	−9.70008	−0.929099	−0.464550	−	0.885547i	$-0.653784\pi$
$109$	−9.70008	−0.929099	−0.464550	+	0.885547i	$0.653784\pi$
$110$	−22.0278	−2.10027
$111$	0	0
$112$	3.58378	0.338635
$113$	−14.7234	−1.38507	−0.692533	−	0.721387i	$-0.743505\pi$
$113$	−14.7234	−1.38507	−0.692533	+	0.721387i	$0.743505\pi$
$114$	0	0
$115$	23.6415	2.20458
$116$	6.98358	0.648409
$117$	0	0
$118$	−1.49629	−0.137745
$119$	1.65987	0.152160
$120$	0	0
$121$	−1.59031	−0.144574
$122$	−8.80506	−0.797173
$123$	0	0
$124$	21.6671	1.94576
$125$	−5.49312	−0.491320
$126$	0	0
$127$	16.1239	1.43076	0.715382	−	0.698733i	$-0.246253\pi$
$127$	16.1239	1.43076	0.715382	+	0.698733i	$0.246253\pi$
$128$	13.7521	1.21553
$129$	0	0
$130$	−33.3507	−2.92505
$131$	18.1576	1.58644	0.793218	−	0.608938i	$-0.208404\pi$
$131$	18.1576	1.58644	0.793218	+	0.608938i	$0.208404\pi$
$132$	0	0
$133$	−2.77272	−0.240425
$134$	25.4170	2.19569
$135$	0	0
$136$	−18.2556	−1.56541
$137$	−4.92150	−0.420472	−0.210236	−	0.977651i	$-0.567423\pi$
$137$	−4.92150	−0.420472	−0.210236	+	0.977651i	$0.567423\pi$
$138$	0	0
$139$	−8.49579	−0.720603	−0.360302	−	0.932836i	$-0.617326\pi$
$139$	−8.49579	−0.720603	−0.360302	+	0.932836i	$0.617326\pi$
$140$	6.86462	0.580166
$141$	0	0
$142$	6.18624	0.519138
$143$	14.2465	1.19136
$144$	0	0
$145$	4.51464	0.374920
$146$	−13.0948	−1.08373
$147$	0	0
$148$	15.7444	1.29418
$149$	10.5940	0.867891	0.433945	−	0.900939i	$-0.357121\pi$
$149$	10.5940	0.867891	0.433945	+	0.900939i	$0.357121\pi$
$150$	0	0
$151$	13.0407	1.06123	0.530617	−	0.847611i	$-0.321960\pi$
$151$	13.0407	1.06123	0.530617	+	0.847611i	$0.321960\pi$
$152$	30.4950	2.47347
$153$	0	0
$154$	−4.26736	−0.343873
$155$	14.0070	1.12507
$156$	0	0
$157$	−22.1480	−1.76760	−0.883800	−	0.467864i	$-0.845024\pi$
$157$	−22.1480	−1.76760	−0.883800	+	0.467864i	$0.845024\pi$
$158$	5.68879	0.452576
$159$	0	0
$160$	−12.4040	−0.980619
$161$	4.57997	0.360952
$162$	0	0
$163$	−17.2006	−1.34725	−0.673626	−	0.739072i	$-0.735264\pi$
$163$	−17.2006	−1.34725	−0.673626	+	0.739072i	$0.735264\pi$
$164$	5.02216	0.392165
$165$	0	0
$166$	7.70165	0.597764
$167$	11.7738	0.911087	0.455544	−	0.890213i	$-0.349445\pi$
$167$	11.7738	0.911087	0.455544	+	0.890213i	$0.349445\pi$
$168$	0	0
$169$	8.56969	0.659207
$170$	−21.6643	−1.66158
$171$	0	0
$172$	−1.58390	−0.120771
$173$	−11.9612	−0.909391	−0.454696	−	0.890647i	$-0.650252\pi$
$173$	−11.9612	−0.909391	−0.454696	+	0.890647i	$0.650252\pi$
$174$	0	0
$175$	1.68679	0.127509
$176$	19.9809	1.50612
$177$	0	0
$178$	−29.2033	−2.18888
$179$	−3.50870	−0.262253	−0.131126	−	0.991366i	$-0.541859\pi$
$179$	−3.50870	−0.262253	−0.131126	+	0.991366i	$0.541859\pi$
$180$	0	0
$181$	22.1580	1.64699	0.823495	−	0.567323i	$-0.192021\pi$
$181$	22.1580	1.64699	0.823495	+	0.567323i	$0.192021\pi$
$182$	−6.46090	−0.478914
$183$	0	0
$184$	−50.3715	−3.71344
$185$	10.1782	0.748318
$186$	0	0
$187$	9.25442	0.676750
$188$	35.7090	2.60435
$189$	0	0
$190$	36.1890	2.62543
$191$	2.61665	0.189334	0.0946670	−	0.995509i	$-0.469821\pi$
$191$	2.61665	0.189334	0.0946670	+	0.995509i	$0.469821\pi$
$192$	0	0
$193$	12.3237	0.887079	0.443540	−	0.896255i	$-0.353723\pi$
$193$	12.3237	0.887079	0.443540	+	0.896255i	$0.353723\pi$
$194$	10.6282	0.763059
$195$	0	0
$196$	−29.4224	−2.10160
$197$	−27.2753	−1.94329	−0.971643	−	0.236452i	$-0.924015\pi$
$197$	−27.2753	−1.94329	−0.971643	+	0.236452i	$0.924015\pi$
$198$	0	0
$199$	0.765067	0.0542341	0.0271171	−	0.999632i	$-0.491367\pi$
$199$	0.765067	0.0542341	0.0271171	+	0.999632i	$0.491367\pi$
$200$	−18.5517	−1.31180
$201$	0	0
$202$	9.73612	0.685031
$203$	0.874603	0.0613851
$204$	0	0
$205$	3.24665	0.226756
$206$	4.57140	0.318504
$207$	0	0
$208$	30.2517	2.09758
$209$	−15.4590	−1.06932
$210$	0	0
$211$	18.7418	1.29024	0.645119	−	0.764082i	$-0.276808\pi$
$211$	18.7418	1.29024	0.645119	+	0.764082i	$0.276808\pi$
$212$	−3.03649	−0.208547
$213$	0	0
$214$	−31.3658	−2.14412
$215$	−1.02394	−0.0698318
$216$	0	0
$217$	2.71352	0.184206
$218$	24.5264	1.66114
$219$	0	0
$220$	38.2729	2.58036
$221$	14.0115	0.942513
$222$	0	0
$223$	−2.76382	−0.185079	−0.0925396	−	0.995709i	$-0.529498\pi$
$223$	−2.76382	−0.185079	−0.0925396	+	0.995709i	$0.529498\pi$
$224$	−2.40297	−0.160555
$225$	0	0
$226$	37.2279	2.47636
$227$	6.69321	0.444244	0.222122	−	0.975019i	$-0.428702\pi$
$227$	6.69321	0.444244	0.222122	+	0.975019i	$0.428702\pi$
$228$	0	0
$229$	−13.1309	−0.867715	−0.433857	−	0.900982i	$-0.642848\pi$
$229$	−13.1309	−0.867715	−0.433857	+	0.900982i	$0.642848\pi$
$230$	−59.7769	−3.94157
$231$	0	0
$232$	−9.61908	−0.631523
$233$	−13.4631	−0.881998	−0.440999	−	0.897508i	$-0.645376\pi$
$233$	−13.4631	−0.881998	−0.440999	+	0.897508i	$0.645376\pi$
$234$	0	0
$235$	23.0846	1.50587
$236$	2.59979	0.169232
$237$	0	0
$238$	−4.19694	−0.272047
$239$	14.9432	0.966598	0.483299	−	0.875455i	$-0.339438\pi$
$239$	14.9432	0.966598	0.483299	+	0.875455i	$0.339438\pi$
$240$	0	0
$241$	19.2444	1.23964	0.619820	−	0.784744i	$-0.287205\pi$
$241$	19.2444	1.23964	0.619820	+	0.784744i	$0.287205\pi$
$242$	4.02106	0.258484
$243$	0	0
$244$	15.2987	0.979396
$245$	−19.0206	−1.21518
$246$	0	0
$247$	−23.4054	−1.48925
$248$	−29.8439	−1.89509
$249$	0	0
$250$	13.8892	0.878431
$251$	−16.1946	−1.02220	−0.511098	−	0.859523i	$-0.670761\pi$
$251$	−16.1946	−1.02220	−0.511098	+	0.859523i	$0.670761\pi$
$252$	0	0
$253$	25.5351	1.60538
$254$	−40.7689	−2.55807
$255$	0	0
$256$	−30.8035	−1.92522
$257$	22.1023	1.37870	0.689352	−	0.724427i	$-0.257896\pi$
$257$	22.1023	1.37870	0.689352	+	0.724427i	$0.257896\pi$
$258$	0	0
$259$	1.97179	0.122521
$260$	57.9463	3.59368
$261$	0	0
$262$	−45.9110	−2.83639
$263$	−15.9024	−0.980585	−0.490292	−	0.871558i	$-0.663110\pi$
$263$	−15.9024	−0.980585	−0.490292	+	0.871558i	$0.663110\pi$
$264$	0	0
$265$	−1.96299	−0.120585
$266$	7.01075	0.429857
$267$	0	0
$268$	−44.1616	−2.69760
$269$	−19.6776	−1.19976	−0.599882	−	0.800088i	$-0.704786\pi$
$269$	−19.6776	−1.19976	−0.599882	+	0.800088i	$0.704786\pi$
$270$	0	0
$271$	−9.22624	−0.560454	−0.280227	−	0.959934i	$-0.590410\pi$
$271$	−9.22624	−0.560454	−0.280227	+	0.959934i	$0.590410\pi$
$272$	19.6512	1.19153
$273$	0	0
$274$	12.4439	0.751762
$275$	9.40450	0.567113
$276$	0	0
$277$	30.0595	1.80610	0.903049	−	0.429537i	$-0.141323\pi$
$277$	30.0595	1.80610	0.903049	+	0.429537i	$0.141323\pi$
$278$	21.4814	1.28837
$279$	0	0
$280$	−9.45523	−0.565058
$281$	−8.60112	−0.513100	−0.256550	−	0.966531i	$-0.582586\pi$
$281$	−8.60112	−0.513100	−0.256550	+	0.966531i	$0.582586\pi$
$282$	0	0
$283$	−28.8175	−1.71302	−0.856510	−	0.516130i	$-0.827372\pi$
$283$	−28.8175	−1.71302	−0.856510	+	0.516130i	$0.827372\pi$
$284$	−10.7485	−0.637806
$285$	0	0
$286$	−36.0220	−2.13003
$287$	0.628961	0.0371264
$288$	0	0
$289$	−7.89828	−0.464605
$290$	−11.4152	−0.670321
$291$	0	0
$292$	22.7520	1.33146
$293$	−9.13341	−0.533580	−0.266790	−	0.963755i	$-0.585963\pi$
$293$	−9.13341	−0.533580	−0.266790	+	0.963755i	$0.585963\pi$
$294$	0	0
$295$	1.68067	0.0978524
$296$	−21.6861	−1.26048
$297$	0	0
$298$	−26.7865	−1.55170
$299$	38.6609	2.23582
$300$	0	0
$301$	−0.198363	−0.0114334
$302$	−32.9730	−1.89738
$303$	0	0
$304$	−32.8263	−1.88272
$305$	9.89004	0.566302
$306$	0	0
$307$	−33.6161	−1.91857	−0.959287	−	0.282433i	$-0.908858\pi$
$307$	−33.6161	−1.91857	−0.959287	+	0.282433i	$0.908858\pi$
$308$	7.41446	0.422478
$309$	0	0
$310$	−35.4164	−2.01152
$311$	20.6243	1.16950	0.584748	−	0.811215i	$-0.301193\pi$
$311$	20.6243	1.16950	0.584748	+	0.811215i	$0.301193\pi$
$312$	0	0
$313$	−33.8153	−1.91135	−0.955677	−	0.294417i	$-0.904874\pi$
$313$	−33.8153	−1.91135	−0.955677	+	0.294417i	$0.904874\pi$
$314$	56.0006	3.16030
$315$	0	0
$316$	−9.88418	−0.556028
$317$	−8.41152	−0.472438	−0.236219	−	0.971700i	$-0.575908\pi$
$317$	−8.41152	−0.472438	−0.236219	+	0.971700i	$0.575908\pi$
$318$	0	0
$319$	4.87625	0.273018
$320$	−5.63530	−0.315023
$321$	0	0
$322$	−11.5803	−0.645347
$323$	−15.2039	−0.845968
$324$	0	0
$325$	14.2387	0.789820
$326$	43.4912	2.40876
$327$	0	0
$328$	−6.91745	−0.381953
$329$	4.47209	0.246554
$330$	0	0
$331$	6.50635	0.357621	0.178811	−	0.983884i	$-0.442775\pi$
$331$	6.50635	0.357621	0.178811	+	0.983884i	$0.442775\pi$
$332$	−13.3815	−0.734405
$333$	0	0
$334$	−29.7699	−1.62893
$335$	−28.5490	−1.55980
$336$	0	0
$337$	3.42394	0.186514	0.0932570	−	0.995642i	$-0.470272\pi$
$337$	3.42394	0.186514	0.0932570	+	0.995642i	$0.470272\pi$
$338$	−21.6682	−1.17860
$339$	0	0
$340$	37.6414	2.04139
$341$	15.1289	0.819278
$342$	0	0
$343$	−7.53611	−0.406912
$344$	2.18164	0.117626
$345$	0	0
$346$	30.2435	1.62590
$347$	−25.3434	−1.36051	−0.680253	−	0.732977i	$-0.738130\pi$
$347$	−25.3434	−1.36051	−0.680253	+	0.732977i	$0.738130\pi$
$348$	0	0
$349$	10.4880	0.561411	0.280706	−	0.959794i	$-0.409432\pi$
$349$	10.4880	0.561411	0.280706	+	0.959794i	$0.409432\pi$
$350$	−4.26500	−0.227974
$351$	0	0
$352$	−13.3975	−0.714088
$353$	11.3927	0.606374	0.303187	−	0.952931i	$-0.401949\pi$
$353$	11.3927	0.606374	0.303187	+	0.952931i	$0.401949\pi$
$354$	0	0
$355$	−6.94853	−0.368789
$356$	50.7402	2.68923
$357$	0	0
$358$	8.87166	0.468882
$359$	−16.7169	−0.882283	−0.441142	−	0.897438i	$-0.645426\pi$
$359$	−16.7169	−0.882283	−0.441142	+	0.897438i	$0.645426\pi$
$360$	0	0
$361$	6.39728	0.336699
$362$	−56.0259	−2.94466
$363$	0	0
$364$	11.2257	0.588387
$365$	14.7084	0.769872
$366$	0	0
$367$	−22.0233	−1.14961	−0.574803	−	0.818292i	$-0.694921\pi$
$367$	−22.0233	−1.14961	−0.574803	+	0.818292i	$0.694921\pi$
$368$	54.2223	2.82653
$369$	0	0
$370$	−25.7354	−1.33792
$371$	−0.380282	−0.0197432
$372$	0	0
$373$	−30.4128	−1.57472	−0.787358	−	0.616496i	$-0.788552\pi$
$373$	−30.4128	−1.57472	−0.787358	+	0.616496i	$0.788552\pi$
$374$	−23.3996	−1.20996
$375$	0	0
$376$	−49.1851	−2.53653
$377$	7.38279	0.380233
$378$	0	0
$379$	−12.3006	−0.631839	−0.315919	−	0.948786i	$-0.602313\pi$
$379$	−12.3006	−0.631839	−0.315919	+	0.948786i	$0.602313\pi$
$380$	−62.8778	−3.22557
$381$	0	0
$382$	−6.61613	−0.338511
$383$	−7.26622	−0.371287	−0.185643	−	0.982617i	$-0.559437\pi$
$383$	−7.26622	−0.371287	−0.185643	+	0.982617i	$0.559437\pi$
$384$	0	0
$385$	4.79319	0.244284
$386$	−31.1602	−1.58601
$387$	0	0
$388$	−18.4663	−0.937484
$389$	−19.1896	−0.972954	−0.486477	−	0.873693i	$-0.661718\pi$
$389$	−19.1896	−0.972954	−0.486477	+	0.873693i	$0.661718\pi$
$390$	0	0
$391$	25.1138	1.27006
$392$	40.5260	2.04687
$393$	0	0
$394$	68.9650	3.47440
$395$	−6.38978	−0.321504
$396$	0	0
$397$	−26.8051	−1.34531	−0.672655	−	0.739956i	$-0.734846\pi$
$397$	−26.8051	−1.34531	−0.672655	+	0.739956i	$0.734846\pi$
$398$	−1.93445	−0.0969653
$399$	0	0
$400$	19.9699	0.998496
$401$	−20.7899	−1.03820	−0.519100	−	0.854714i	$-0.673733\pi$
$401$	−20.7899	−1.03820	−0.519100	+	0.854714i	$0.673733\pi$
$402$	0	0
$403$	22.9057	1.14101
$404$	−16.9164	−0.841620
$405$	0	0
$406$	−2.21141	−0.109751
$407$	10.9935	0.544926
$408$	0	0
$409$	−1.68411	−0.0832740	−0.0416370	−	0.999133i	$-0.513257\pi$
$409$	−1.68411	−0.0832740	−0.0416370	+	0.999133i	$0.513257\pi$
$410$	−8.20908	−0.405418
$411$	0	0
$412$	−7.94273	−0.391310
$413$	0.325590	0.0160212
$414$	0	0
$415$	−8.65067	−0.424645
$416$	−20.2842	−0.994514
$417$	0	0
$418$	39.0877	1.91184
$419$	7.62622	0.372565	0.186283	−	0.982496i	$-0.440356\pi$
$419$	7.62622	0.372565	0.186283	+	0.982496i	$0.440356\pi$
$420$	0	0
$421$	−32.4866	−1.58330	−0.791651	−	0.610973i	$-0.790778\pi$
$421$	−32.4866	−1.58330	−0.791651	+	0.610973i	$0.790778\pi$
$422$	−47.3881	−2.30682
$423$	0	0
$424$	4.18242	0.203116
$425$	9.24932	0.448658
$426$	0	0
$427$	1.91596	0.0927198
$428$	54.4975	2.63424
$429$	0	0
$430$	2.58899	0.124852
$431$	9.25023	0.445568	0.222784	−	0.974868i	$-0.428486\pi$
$431$	9.25023	0.445568	0.222784	+	0.974868i	$0.428486\pi$
$432$	0	0
$433$	−7.56102	−0.363359	−0.181680	−	0.983358i	$-0.558153\pi$
$433$	−7.56102	−0.363359	−0.181680	+	0.983358i	$0.558153\pi$
$434$	−6.86108	−0.329342
$435$	0	0
$436$	−42.6143	−2.04085
$437$	−41.9511	−2.00679
$438$	0	0
$439$	21.5772	1.02982	0.514912	−	0.857243i	$-0.327824\pi$
$439$	21.5772	1.02982	0.514912	+	0.857243i	$0.327824\pi$
$440$	−52.7166	−2.51316
$441$	0	0
$442$	−35.4276	−1.68512
$443$	13.3036	0.632072	0.316036	−	0.948747i	$-0.397648\pi$
$443$	13.3036	0.632072	0.316036	+	0.948747i	$0.397648\pi$
$444$	0	0
$445$	32.8018	1.55495
$446$	6.98825	0.330903
$447$	0	0
$448$	−1.09170	−0.0515782
$449$	18.5529	0.875566	0.437783	−	0.899081i	$-0.355764\pi$
$449$	18.5529	0.875566	0.437783	+	0.899081i	$0.355764\pi$
$450$	0	0
$451$	3.50670	0.165124
$452$	−64.6828	−3.04242
$453$	0	0
$454$	−16.9236	−0.794265
$455$	7.25703	0.340215
$456$	0	0
$457$	33.3760	1.56126	0.780632	−	0.624991i	$-0.214898\pi$
$457$	33.3760	1.56126	0.780632	+	0.624991i	$0.214898\pi$
$458$	33.2012	1.55139
$459$	0	0
$460$	103.861	4.84256
$461$	−24.9957	−1.16417	−0.582083	−	0.813129i	$-0.697762\pi$
$461$	−24.9957	−1.16417	−0.582083	+	0.813129i	$0.697762\pi$
$462$	0	0
$463$	11.0985	0.515790	0.257895	−	0.966173i	$-0.416971\pi$
$463$	11.0985	0.515790	0.257895	+	0.966173i	$0.416971\pi$
$464$	10.3544	0.480693
$465$	0	0
$466$	34.0412	1.57693
$467$	−14.8391	−0.686674	−0.343337	−	0.939212i	$-0.611557\pi$
$467$	−14.8391	−0.686674	−0.343337	+	0.939212i	$0.611557\pi$
$468$	0	0
$469$	−5.53068	−0.255383
$470$	−58.3689	−2.69236
$471$	0	0
$472$	−3.58091	−0.164825
$473$	−1.10595	−0.0508516
$474$	0	0
$475$	−15.4505	−0.708916
$476$	7.29212	0.334234
$477$	0	0
$478$	−37.7836	−1.72818
$479$	−26.9534	−1.23153	−0.615767	−	0.787928i	$-0.711154\pi$
$479$	−26.9534	−1.23153	−0.615767	+	0.787928i	$0.711154\pi$
$480$	0	0
$481$	16.6444	0.758921
$482$	−48.6590	−2.21636
$483$	0	0
$484$	−6.98653	−0.317570
$485$	−11.9378	−0.542068
$486$	0	0
$487$	−0.903717	−0.0409513	−0.0204757	−	0.999790i	$-0.506518\pi$
$487$	−0.903717	−0.0409513	−0.0204757	+	0.999790i	$0.506518\pi$
$488$	−21.0721	−0.953891
$489$	0	0
$490$	48.0931	2.17262
$491$	−11.6661	−0.526483	−0.263241	−	0.964730i	$-0.584792\pi$
$491$	−11.6661	−0.526483	−0.263241	+	0.964730i	$0.584792\pi$
$492$	0	0
$493$	4.79579	0.215991
$494$	59.1799	2.66263
$495$	0	0
$496$	32.1255	1.44248
$497$	−1.34611	−0.0603813
$498$	0	0
$499$	27.2454	1.21967	0.609837	−	0.792527i	$-0.291235\pi$
$499$	27.2454	1.21967	0.609837	+	0.792527i	$0.291235\pi$
$500$	−24.1323	−1.07923
$501$	0	0
$502$	40.9477	1.82758
$503$	−15.2173	−0.678505	−0.339253	−	0.940695i	$-0.610174\pi$
$503$	−15.2173	−0.678505	−0.339253	+	0.940695i	$0.610174\pi$
$504$	0	0
$505$	−10.9358	−0.486638
$506$	−64.5649	−2.87026
$507$	0	0
$508$	70.8353	3.14281
$509$	−5.33127	−0.236305	−0.118152	−	0.992995i	$-0.537697\pi$
$509$	−5.33127	−0.236305	−0.118152	+	0.992995i	$0.537697\pi$
$510$	0	0
$511$	2.84940	0.126050
$512$	50.3815	2.22657
$513$	0	0
$514$	−55.8851	−2.46499
$515$	−5.13470	−0.226262
$516$	0	0
$517$	24.9336	1.09658
$518$	−4.98561	−0.219055
$519$	0	0
$520$	−79.8144	−3.50009
$521$	30.6752	1.34390	0.671952	−	0.740595i	$-0.265456\pi$
$521$	30.6752	1.34390	0.671952	+	0.740595i	$0.265456\pi$
$522$	0	0
$523$	11.1862	0.489140	0.244570	−	0.969632i	$-0.421353\pi$
$523$	11.1862	0.489140	0.244570	+	0.969632i	$0.421353\pi$
$524$	79.7697	3.48475
$525$	0	0
$526$	40.2089	1.75319
$527$	14.8793	0.648153
$528$	0	0
$529$	46.2947	2.01281
$530$	4.96336	0.215595
$531$	0	0
$532$	−12.1811	−0.528117
$533$	5.30925	0.229969
$534$	0	0
$535$	35.2308	1.52316
$536$	60.8276	2.62735
$537$	0	0
$538$	49.7543	2.14506
$539$	−20.5441	−0.884896
$540$	0	0
$541$	31.7161	1.36358	0.681792	−	0.731547i	$-0.261201\pi$
$541$	31.7161	1.36358	0.681792	+	0.731547i	$0.261201\pi$
$542$	23.3283	1.00204
$543$	0	0
$544$	−13.1764	−0.564934
$545$	−27.5486	−1.18005
$546$	0	0
$547$	40.9072	1.74907	0.874533	−	0.484966i	$-0.161168\pi$
$547$	40.9072	1.74907	0.874533	+	0.484966i	$0.161168\pi$
$548$	−21.6210	−0.923605
$549$	0	0
$550$	−23.7790	−1.01394
$551$	−8.01110	−0.341284
$552$	0	0
$553$	−1.23787	−0.0526394
$554$	−76.0046	−3.22913
$555$	0	0
$556$	−37.3236	−1.58287
$557$	13.8402	0.586426	0.293213	−	0.956047i	$-0.405275\pi$
$557$	13.8402	0.586426	0.293213	+	0.956047i	$0.405275\pi$
$558$	0	0
$559$	−1.67444	−0.0708213
$560$	10.1781	0.430102
$561$	0	0
$562$	21.7477	0.917372
$563$	26.3018	1.10849	0.554245	−	0.832353i	$-0.313007\pi$
$563$	26.3018	1.10849	0.554245	+	0.832353i	$0.313007\pi$
$564$	0	0
$565$	−41.8152	−1.75918
$566$	72.8642	3.06271
$567$	0	0
$568$	14.8048	0.621196
$569$	5.72257	0.239902	0.119951	−	0.992780i	$-0.461726\pi$
$569$	5.72257	0.239902	0.119951	+	0.992780i	$0.461726\pi$
$570$	0	0
$571$	3.17472	0.132858	0.0664290	−	0.997791i	$-0.478839\pi$
$571$	3.17472	0.132858	0.0664290	+	0.997791i	$0.478839\pi$
$572$	62.5877	2.61692
$573$	0	0
$574$	−1.59031	−0.0663784
$575$	25.5210	1.06430
$576$	0	0
$577$	20.0789	0.835894	0.417947	−	0.908471i	$-0.362750\pi$
$577$	20.0789	0.835894	0.417947	+	0.908471i	$0.362750\pi$
$578$	19.9706	0.830668
$579$	0	0
$580$	19.8336	0.823547
$581$	−1.67586	−0.0695264
$582$	0	0
$583$	−2.12022	−0.0878105
$584$	−31.3383	−1.29679
$585$	0	0
$586$	23.0936	0.953988
$587$	32.2489	1.33105	0.665526	−	0.746374i	$-0.268207\pi$
$587$	32.2489	1.33105	0.665526	+	0.746374i	$0.268207\pi$
$588$	0	0
$589$	−24.8551	−1.02413
$590$	−4.24953	−0.174951
$591$	0	0
$592$	23.3440	0.959433
$593$	−14.7593	−0.606092	−0.303046	−	0.952976i	$-0.598004\pi$
$593$	−14.7593	−0.606092	−0.303046	+	0.952976i	$0.598004\pi$
$594$	0	0
$595$	4.71410	0.193259
$596$	46.5412	1.90640
$597$	0	0
$598$	−97.7531	−3.99742
$599$	−0.317913	−0.0129896	−0.00649479	−	0.999979i	$-0.502067\pi$
$599$	−0.317913	−0.0129896	−0.00649479	+	0.999979i	$0.502067\pi$
$600$	0	0
$601$	−9.67941	−0.394831	−0.197416	−	0.980320i	$-0.563255\pi$
$601$	−9.67941	−0.394831	−0.197416	+	0.980320i	$0.563255\pi$
$602$	0.501556	0.0204419
$603$	0	0
$604$	57.2901	2.33110
$605$	−4.51655	−0.183624
$606$	0	0
$607$	6.40039	0.259784	0.129892	−	0.991528i	$-0.458537\pi$
$607$	6.40039	0.259784	0.129892	+	0.991528i	$0.458537\pi$
$608$	22.0105	0.892642
$609$	0	0
$610$	−25.0067	−1.01249
$611$	37.7503	1.52721
$612$	0	0
$613$	−4.63967	−0.187395	−0.0936973	−	0.995601i	$-0.529869\pi$
$613$	−4.63967	−0.187395	−0.0936973	+	0.995601i	$0.529869\pi$
$614$	84.9975	3.43022
$615$	0	0
$616$	−10.2126	−0.411476
$617$	−19.7650	−0.795710	−0.397855	−	0.917448i	$-0.630245\pi$
$617$	−19.7650	−0.795710	−0.397855	+	0.917448i	$0.630245\pi$
$618$	0	0
$619$	0.403822	0.0162310	0.00811549	−	0.999967i	$-0.497417\pi$
$619$	0.403822	0.0162310	0.00811549	+	0.999967i	$0.497417\pi$
$620$	61.5354	2.47132
$621$	0	0
$622$	−52.1480	−2.09094
$623$	6.35456	0.254590
$624$	0	0
$625$	−30.9298	−1.23719
$626$	85.5011	3.41731
$627$	0	0
$628$	−97.3002	−3.88270
$629$	10.8121	0.431106
$630$	0	0
$631$	34.1222	1.35838	0.679192	−	0.733960i	$-0.262330\pi$
$631$	34.1222	1.35838	0.679192	+	0.733960i	$0.262330\pi$
$632$	13.6143	0.541549
$633$	0	0
$634$	21.2683	0.844673
$635$	45.7925	1.81722
$636$	0	0
$637$	−31.1044	−1.23240
$638$	−12.3295	−0.488129
$639$	0	0
$640$	39.0566	1.54385
$641$	28.2095	1.11421	0.557105	−	0.830442i	$-0.311912\pi$
$641$	28.2095	1.11421	0.557105	+	0.830442i	$0.311912\pi$
$642$	0	0
$643$	32.3993	1.27770	0.638851	−	0.769330i	$-0.279410\pi$
$643$	32.3993	1.27770	0.638851	+	0.769330i	$0.279410\pi$
$644$	20.1207	0.792865
$645$	0	0
$646$	38.4427	1.51251
$647$	3.64502	0.143300	0.0716502	−	0.997430i	$-0.477173\pi$
$647$	3.64502	0.143300	0.0716502	+	0.997430i	$0.477173\pi$
$648$	0	0
$649$	1.81529	0.0712563
$650$	−36.0022	−1.41212
$651$	0	0
$652$	−75.5653	−2.95937
$653$	−39.5912	−1.54932	−0.774662	−	0.632375i	$-0.782080\pi$
$653$	−39.5912	−1.54932	−0.774662	+	0.632375i	$0.782080\pi$
$654$	0	0
$655$	51.5683	2.01494
$656$	7.44628	0.290729
$657$	0	0
$658$	−11.3076	−0.440815
$659$	20.9765	0.817130	0.408565	−	0.912729i	$-0.366029\pi$
$659$	20.9765	0.817130	0.408565	+	0.912729i	$0.366029\pi$
$660$	0	0
$661$	−32.3787	−1.25939	−0.629693	−	0.776844i	$-0.716819\pi$
$661$	−32.3787	−1.25939	−0.629693	+	0.776844i	$0.716819\pi$
$662$	−16.4511	−0.639392
$663$	0	0
$664$	18.4315	0.715280
$665$	−7.87464	−0.305365
$666$	0	0
$667$	13.2327	0.512372
$668$	51.7247	2.00129
$669$	0	0
$670$	72.1853	2.78876
$671$	10.6822	0.412382
$672$	0	0
$673$	42.0885	1.62239	0.811197	−	0.584773i	$-0.198817\pi$
$673$	42.0885	1.62239	0.811197	+	0.584773i	$0.198817\pi$
$674$	−8.65735	−0.333469
$675$	0	0
$676$	37.6482	1.44801
$677$	6.47653	0.248913	0.124457	−	0.992225i	$-0.460281\pi$
$677$	6.47653	0.248913	0.124457	+	0.992225i	$0.460281\pi$
$678$	0	0
$679$	−2.31267	−0.0887520
$680$	−51.8467	−1.98823
$681$	0	0
$682$	−38.2532	−1.46479
$683$	7.95118	0.304243	0.152122	−	0.988362i	$-0.451389\pi$
$683$	7.95118	0.304243	0.152122	+	0.988362i	$0.451389\pi$
$684$	0	0
$685$	−13.9772	−0.534043
$686$	19.0549	0.727518
$687$	0	0
$688$	−2.34842	−0.0895328
$689$	−3.21007	−0.122294
$690$	0	0
$691$	−23.4880	−0.893524	−0.446762	−	0.894653i	$-0.647423\pi$
$691$	−23.4880	−0.893524	−0.446762	+	0.894653i	$0.647423\pi$
$692$	−52.5476	−1.99756
$693$	0	0
$694$	64.0802	2.43245
$695$	−24.1284	−0.915242
$696$	0	0
$697$	3.44884	0.130634
$698$	−26.5187	−1.00375
$699$	0	0
$700$	7.41038	0.280086
$701$	−2.85803	−0.107946	−0.0539731	−	0.998542i	$-0.517189\pi$
$701$	−2.85803	−0.107946	−0.0539731	+	0.998542i	$0.517189\pi$
$702$	0	0
$703$	−18.0610	−0.681182
$704$	−6.08667	−0.229400
$705$	0	0
$706$	−28.8063	−1.08414
$707$	−2.11856	−0.0796765
$708$	0	0
$709$	44.5592	1.67346	0.836728	−	0.547618i	$-0.184465\pi$
$709$	44.5592	1.67346	0.836728	+	0.547618i	$0.184465\pi$
$710$	17.5692	0.659359
$711$	0	0
$712$	−69.8888	−2.61919
$713$	41.0555	1.53754
$714$	0	0
$715$	40.4608	1.51315
$716$	−15.4144	−0.576062
$717$	0	0
$718$	42.2682	1.57744
$719$	−17.1281	−0.638771	−0.319386	−	0.947625i	$-0.603477\pi$
$719$	−17.1281	−0.638771	−0.319386	+	0.947625i	$0.603477\pi$
$720$	0	0
$721$	−0.994724	−0.0370455
$722$	−16.1754	−0.601984
$723$	0	0
$724$	97.3441	3.61777
$725$	4.87356	0.181000
$726$	0	0
$727$	7.87959	0.292238	0.146119	−	0.989267i	$-0.453322\pi$
$727$	7.87959	0.292238	0.146119	+	0.989267i	$0.453322\pi$
$728$	−15.4621	−0.573065
$729$	0	0
$730$	−37.1898	−1.37645
$731$	−1.08770	−0.0402301
$732$	0	0
$733$	13.9307	0.514541	0.257270	−	0.966339i	$-0.417177\pi$
$733$	13.9307	0.514541	0.257270	+	0.966339i	$0.417177\pi$
$734$	55.6853	2.05538
$735$	0	0
$736$	−36.3568	−1.34013
$737$	−30.8357	−1.13585
$738$	0	0
$739$	−18.3313	−0.674329	−0.337165	−	0.941446i	$-0.609468\pi$
$739$	−18.3313	−0.674329	−0.337165	+	0.941446i	$0.609468\pi$
$740$	44.7148	1.64375
$741$	0	0
$742$	0.961533	0.0352990
$743$	4.33514	0.159041	0.0795205	−	0.996833i	$-0.474661\pi$
$743$	4.33514	0.159041	0.0795205	+	0.996833i	$0.474661\pi$
$744$	0	0
$745$	30.0873	1.10231
$746$	76.8980	2.81544
$747$	0	0
$748$	40.6564	1.48654
$749$	6.82512	0.249384
$750$	0	0
$751$	18.2856	0.667251	0.333626	−	0.942706i	$-0.391728\pi$
$751$	18.2856	0.667251	0.333626	+	0.942706i	$0.391728\pi$
$752$	52.9452	1.93071
$753$	0	0
$754$	−18.6672	−0.679819
$755$	37.0360	1.34788
$756$	0	0
$757$	−6.48188	−0.235588	−0.117794	−	0.993038i	$-0.537582\pi$
$757$	−6.48188	−0.235588	−0.117794	+	0.993038i	$0.537582\pi$
$758$	31.1017	1.12967
$759$	0	0
$760$	86.6070	3.14157
$761$	30.3539	1.10033	0.550164	−	0.835056i	$-0.314565\pi$
$761$	30.3539	1.10033	0.550164	+	0.835056i	$0.314565\pi$
$762$	0	0
$763$	−5.33689	−0.193208
$764$	11.4954	0.415890
$765$	0	0
$766$	18.3725	0.663824
$767$	2.74840	0.0992390
$768$	0	0
$769$	−49.0428	−1.76853	−0.884264	−	0.466987i	$-0.845340\pi$
$769$	−49.0428	−1.76853	−0.884264	+	0.466987i	$0.845340\pi$
$770$	−12.1195	−0.436755
$771$	0	0
$772$	54.1403	1.94855
$773$	−39.4293	−1.41817	−0.709087	−	0.705121i	$-0.750893\pi$
$773$	−39.4293	−1.41817	−0.709087	+	0.705121i	$0.750893\pi$
$774$	0	0
$775$	15.1206	0.543148
$776$	25.4352	0.913071
$777$	0	0
$778$	48.5205	1.73955
$779$	−5.76109	−0.206413
$780$	0	0
$781$	−7.50509	−0.268553
$782$	−63.4995	−2.27074
$783$	0	0
$784$	−43.6242	−1.55801
$785$	−62.9011	−2.24504
$786$	0	0
$787$	−1.23744	−0.0441101	−0.0220551	−	0.999757i	$-0.507021\pi$
$787$	−1.23744	−0.0441101	−0.0220551	+	0.999757i	$0.507021\pi$
$788$	−119.826	−4.26861
$789$	0	0
$790$	16.1564	0.574818
$791$	−8.10069	−0.288027
$792$	0	0
$793$	16.1732	0.574327
$794$	67.7761	2.40528
$795$	0	0
$796$	3.36108	0.119130
$797$	26.4355	0.936392	0.468196	−	0.883625i	$-0.344904\pi$
$797$	26.4355	0.936392	0.468196	+	0.883625i	$0.344904\pi$
$798$	0	0
$799$	24.5222	0.867534
$800$	−13.3901	−0.473411
$801$	0	0
$802$	52.5668	1.85620
$803$	15.8865	0.560622
$804$	0	0
$805$	13.0073	0.458447
$806$	−57.9164	−2.04002
$807$	0	0
$808$	23.3003	0.819703
$809$	51.6224	1.81495	0.907473	−	0.420110i	$-0.138008\pi$
$809$	51.6224	1.81495	0.907473	+	0.420110i	$0.138008\pi$
$810$	0	0
$811$	29.2739	1.02795	0.513973	−	0.857806i	$-0.328173\pi$
$811$	29.2739	1.02795	0.513973	+	0.857806i	$0.328173\pi$
$812$	3.84229	0.134838
$813$	0	0
$814$	−27.7967	−0.974275
$815$	−48.8503	−1.71115
$816$	0	0
$817$	1.81694	0.0635668
$818$	4.25824	0.148886
$819$	0	0
$820$	14.2631	0.498091
$821$	14.5229	0.506853	0.253427	−	0.967355i	$-0.418442\pi$
$821$	14.5229	0.506853	0.253427	+	0.967355i	$0.418442\pi$
$822$	0	0
$823$	33.0689	1.15271	0.576355	−	0.817200i	$-0.304475\pi$
$823$	33.0689	1.15271	0.576355	+	0.817200i	$0.304475\pi$
$824$	10.9402	0.381120
$825$	0	0
$826$	−0.823245	−0.0286444
$827$	−11.2398	−0.390846	−0.195423	−	0.980719i	$-0.562608\pi$
$827$	−11.2398	−0.390846	−0.195423	+	0.980719i	$0.562608\pi$
$828$	0	0
$829$	24.4963	0.850792	0.425396	−	0.905007i	$-0.360135\pi$
$829$	24.4963	0.850792	0.425396	+	0.905007i	$0.360135\pi$
$830$	21.8730	0.759223
$831$	0	0
$832$	−9.21540	−0.319487
$833$	−20.2051	−0.700065
$834$	0	0
$835$	33.4382	1.15718
$836$	−67.9142	−2.34886
$837$	0	0
$838$	−19.2827	−0.666110
$839$	0.358632	0.0123814	0.00619068	−	0.999981i	$-0.498029\pi$
$839$	0.358632	0.0123814	0.00619068	+	0.999981i	$0.498029\pi$
$840$	0	0
$841$	−26.4730	−0.912864
$842$	82.1417	2.83079
$843$	0	0
$844$	82.3361	2.83413
$845$	24.3383	0.837262
$846$	0	0
$847$	−0.874973	−0.0300644
$848$	−4.50216	−0.154605
$849$	0	0
$850$	−23.3867	−0.802156
$851$	29.8330	1.02266
$852$	0	0
$853$	−51.7660	−1.77244	−0.886218	−	0.463269i	$-0.846676\pi$
$853$	−51.7660	−1.77244	−0.886218	+	0.463269i	$0.846676\pi$
$854$	−4.84445	−0.165774
$855$	0	0
$856$	−75.0641	−2.56564
$857$	−13.9456	−0.476373	−0.238187	−	0.971219i	$-0.576553\pi$
$857$	−13.9456	−0.476373	−0.238187	+	0.971219i	$0.576553\pi$
$858$	0	0
$859$	27.2568	0.929992	0.464996	−	0.885313i	$-0.346056\pi$
$859$	27.2568	0.929992	0.464996	+	0.885313i	$0.346056\pi$
$860$	−4.49834	−0.153392
$861$	0	0
$862$	−23.3890	−0.796631
$863$	−45.2804	−1.54136	−0.770682	−	0.637220i	$-0.780084\pi$
$863$	−45.2804	−1.54136	−0.770682	+	0.637220i	$0.780084\pi$
$864$	0	0
$865$	−33.9702	−1.15502
$866$	19.1178	0.649651
$867$	0	0
$868$	11.9210	0.404626
$869$	−6.90158	−0.234120
$870$	0	0
$871$	−46.6861	−1.58190
$872$	58.6962	1.98771
$873$	0	0
$874$	106.072	3.58795
$875$	−3.02226	−0.102171
$876$	0	0
$877$	−28.6764	−0.968334	−0.484167	−	0.874976i	$-0.660877\pi$
$877$	−28.6764	−0.968334	−0.484167	+	0.874976i	$0.660877\pi$
$878$	−54.5575	−1.84123
$879$	0	0
$880$	56.7467	1.91293
$881$	0.200745	0.00676326	0.00338163	−	0.999994i	$-0.498924\pi$
$881$	0.200745	0.00676326	0.00338163	+	0.999994i	$0.498924\pi$
$882$	0	0
$883$	−27.2864	−0.918261	−0.459130	−	0.888369i	$-0.651839\pi$
$883$	−27.2864	−0.918261	−0.459130	+	0.888369i	$0.651839\pi$
$884$	61.5550	2.07032
$885$	0	0
$886$	−33.6378	−1.13008
$887$	15.9177	0.534463	0.267232	−	0.963632i	$-0.413891\pi$
$887$	15.9177	0.534463	0.267232	+	0.963632i	$0.413891\pi$
$888$	0	0
$889$	8.87120	0.297531
$890$	−82.9385	−2.78010
$891$	0	0
$892$	−12.1420	−0.406543
$893$	−40.9630	−1.37077
$894$	0	0
$895$	−9.96485	−0.333088
$896$	7.56628	0.252772
$897$	0	0
$898$	−46.9106	−1.56543
$899$	7.84006	0.261481
$900$	0	0
$901$	−2.08523	−0.0694692
$902$	−8.86661	−0.295226
$903$	0	0
$904$	89.0932	2.96319
$905$	62.9296	2.09185
$906$	0	0
$907$	−5.95128	−0.197609	−0.0988045	−	0.995107i	$-0.531502\pi$
$907$	−5.95128	−0.197609	−0.0988045	+	0.995107i	$0.531502\pi$
$908$	29.4045	0.975824
$909$	0	0
$910$	−18.3492	−0.608271
$911$	15.8373	0.524712	0.262356	−	0.964971i	$-0.415501\pi$
$911$	15.8373	0.524712	0.262356	+	0.964971i	$0.415501\pi$
$912$	0	0
$913$	−9.34357	−0.309227
$914$	−84.3904	−2.79139
$915$	0	0
$916$	−57.6865	−1.90602
$917$	9.99012	0.329903
$918$	0	0
$919$	17.0990	0.564045	0.282023	−	0.959408i	$-0.408995\pi$
$919$	17.0990	0.564045	0.282023	+	0.959408i	$0.408995\pi$
$920$	−143.057	−4.71645
$921$	0	0
$922$	63.2010	2.08141
$923$	−11.3629	−0.374015
$924$	0	0
$925$	10.9874	0.361264
$926$	−28.0622	−0.922181
$927$	0	0
$928$	−6.94279	−0.227908
$929$	−28.4025	−0.931855	−0.465927	−	0.884823i	$-0.654279\pi$
$929$	−28.4025	−0.931855	−0.465927	+	0.884823i	$0.654279\pi$
$930$	0	0
$931$	33.7515	1.10616
$932$	−59.1460	−1.93739
$933$	0	0
$934$	37.5204	1.22771
$935$	26.2829	0.859544
$936$	0	0
$937$	30.4920	0.996130	0.498065	−	0.867140i	$-0.334044\pi$
$937$	30.4920	0.996130	0.498065	+	0.867140i	$0.334044\pi$
$938$	13.9842	0.456599
$939$	0	0
$940$	101.415	3.30779
$941$	−12.3962	−0.404105	−0.202052	−	0.979375i	$-0.564761\pi$
$941$	−12.3962	−0.404105	−0.202052	+	0.979375i	$0.564761\pi$
$942$	0	0
$943$	9.51615	0.309889
$944$	3.85466	0.125459
$945$	0	0
$946$	2.79637	0.0909177
$947$	−5.10521	−0.165897	−0.0829485	−	0.996554i	$-0.526434\pi$
$947$	−5.10521	−0.165897	−0.0829485	+	0.996554i	$0.526434\pi$
$948$	0	0
$949$	24.0526	0.780781
$950$	39.0661	1.26747
$951$	0	0
$952$	−10.0441	−0.325530
$953$	−8.03136	−0.260161	−0.130081	−	0.991503i	$-0.541524\pi$
$953$	−8.03136	−0.260161	−0.130081	+	0.991503i	$0.541524\pi$
$954$	0	0
$955$	7.43139	0.240474
$956$	65.6484	2.12322
$957$	0	0
$958$	68.1511	2.20186
$959$	−2.70776	−0.0874380
$960$	0	0
$961$	−6.67560	−0.215342
$962$	−42.0851	−1.35688
$963$	0	0
$964$	84.5442	2.72299
$965$	34.9998	1.12668
$966$	0	0
$967$	1.57978	0.0508023	0.0254011	−	0.999677i	$-0.491914\pi$
$967$	1.57978	0.0508023	0.0254011	+	0.999677i	$0.491914\pi$
$968$	9.62314	0.309300
$969$	0	0
$970$	30.1845	0.969165
$971$	−2.19584	−0.0704678	−0.0352339	−	0.999379i	$-0.511218\pi$
$971$	−2.19584	−0.0704678	−0.0352339	+	0.999379i	$0.511218\pi$
$972$	0	0
$973$	−4.67430	−0.149851
$974$	2.28503	0.0732170
$975$	0	0
$976$	22.6831	0.726067
$977$	43.7264	1.39893	0.699466	−	0.714666i	$-0.253421\pi$
$977$	43.7264	1.39893	0.699466	+	0.714666i	$0.253421\pi$
$978$	0	0
$979$	35.4291	1.13232
$980$	−83.5609	−2.66926
$981$	0	0
$982$	29.4974	0.941299
$983$	50.6420	1.61523	0.807615	−	0.589710i	$-0.200758\pi$
$983$	50.6420	1.61523	0.807615	+	0.589710i	$0.200758\pi$
$984$	0	0
$985$	−77.4630	−2.46818
$986$	−12.1260	−0.386172
$987$	0	0
$988$	−102.824	−3.27127
$989$	−3.00122	−0.0954333
$990$	0	0
$991$	−53.1401	−1.68805	−0.844026	−	0.536303i	$-0.819820\pi$
$991$	−53.1401	−1.68805	−0.844026	+	0.536303i	$0.819820\pi$
$992$	−21.5405	−0.683913
$993$	0	0
$994$	3.40361	0.107956
$995$	2.17282	0.0688830
$996$	0	0
$997$	−25.2851	−0.800786	−0.400393	−	0.916343i	$-0.631126\pi$
$997$	−25.2851	−0.800786	−0.400393	+	0.916343i	$0.631126\pi$
$998$	−68.8894	−2.18066
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.4		72
3.2	odd	2		6561.2.a.c.1.69		72
81.4	even	27		729.2.g.b.136.1		144
81.7	even	27		729.2.g.a.352.8		144
81.20	odd	54		729.2.g.c.595.8		144
81.23	odd	54		729.2.g.d.379.1		144
81.31	even	27		243.2.g.a.127.8		144
81.34	even	27		243.2.g.a.199.8		144
81.47	odd	54		81.2.g.a.22.1	✓	144
81.50	odd	54		81.2.g.a.70.1	yes	144
81.58	even	27		729.2.g.a.379.8		144
81.61	even	27		729.2.g.b.595.1		144
81.74	odd	54		729.2.g.d.352.1		144
81.77	odd	54		729.2.g.c.136.8		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.22.1	✓	144	81.47	odd	54
81.2.g.a.70.1	yes	144	81.50	odd	54
243.2.g.a.127.8		144	81.31	even	27
243.2.g.a.199.8		144	81.34	even	27
729.2.g.a.352.8		144	81.7	even	27
729.2.g.a.379.8		144	81.58	even	27
729.2.g.b.136.1		144	81.4	even	27
729.2.g.b.595.1		144	81.61	even	27
729.2.g.c.136.8		144	81.77	odd	54
729.2.g.c.595.8		144	81.20	odd	54
729.2.g.d.352.1		144	81.74	odd	54
729.2.g.d.379.1		144	81.23	odd	54
6561.2.a.c.1.69		72	3.2	odd	2
6561.2.a.d.1.4		72	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.52847 q^{2} +4.39319 q^{4} +2.84004 q^{5} +0.550190 q^{7} -6.05111 q^{8} +O(q^{10})\)	\(q-2.52847 q^{2} +4.39319 q^{4} +2.84004 q^{5} +0.550190 q^{7} -6.05111 q^{8} -7.18097 q^{10} +3.06752 q^{11} +4.64432 q^{13} -1.39114 q^{14} +6.51371 q^{16} +3.01691 q^{17} -5.03957 q^{19} +12.4768 q^{20} -7.75615 q^{22} +8.32434 q^{23} +3.06583 q^{25} -11.7430 q^{26} +2.41709 q^{28} +1.58964 q^{29} +4.93198 q^{31} -4.36753 q^{32} -7.62817 q^{34} +1.56256 q^{35} +3.58383 q^{37} +12.7424 q^{38} -17.1854 q^{40} +1.14317 q^{41} -0.360535 q^{43} +13.4762 q^{44} -21.0479 q^{46} +8.12827 q^{47} -6.69729 q^{49} -7.75188 q^{50} +20.4034 q^{52} -0.691183 q^{53} +8.71189 q^{55} -3.32926 q^{56} -4.01936 q^{58} +0.591777 q^{59} +3.48236 q^{61} -12.4704 q^{62} -1.98423 q^{64} +13.1901 q^{65} -10.0523 q^{67} +13.2538 q^{68} -3.95090 q^{70} -2.44663 q^{71} +5.17893 q^{73} -9.06162 q^{74} -22.1398 q^{76} +1.68772 q^{77} -2.24989 q^{79} +18.4992 q^{80} -2.89048 q^{82} -3.04597 q^{83} +8.56813 q^{85} +0.911605 q^{86} -18.5619 q^{88} +11.5498 q^{89} +2.55526 q^{91} +36.5704 q^{92} -20.5521 q^{94} -14.3126 q^{95} -4.20340 q^{97} +16.9339 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) \) 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38} + 72 q^{41} + 90 q^{44} + 90 q^{47} + 18 q^{49} + 45 q^{50} + 45 q^{53} + 9 q^{55} + 108 q^{56} + 18 q^{58} + 108 q^{59} + 72 q^{62} + 9 q^{64} + 72 q^{65} + 108 q^{68} + 126 q^{71} + 90 q^{74} + 72 q^{77} + 144 q^{80} - 18 q^{82} + 108 q^{83} + 90 q^{86} + 108 q^{89} + 72 q^{92} + 144 q^{95} + 81 q^{98}+O(q^{100}) \) 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8 + 36 * q^11 + 36 * q^14 + 45 * q^16 + 36 * q^17 + 54 * q^20 + 54 * q^23 + 36 * q^25 + 45 * q^26 + 9 * q^28 + 54 * q^29 + 63 * q^32 + 72 * q^35 + 54 * q^38 + 72 * q^41 + 90 * q^44 + 90 * q^47 + 18 * q^49 + 45 * q^50 + 45 * q^53 + 9 * q^55 + 108 * q^56 + 18 * q^58 + 108 * q^59 + 72 * q^62 + 9 * q^64 + 72 * q^65 + 108 * q^68 + 126 * q^71 + 90 * q^74 + 72 * q^77 + 144 * q^80 - 18 * q^82 + 108 * q^83 + 90 * q^86 + 108 * q^89 + 72 * q^92 + 144 * q^95 + 81 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more