LMFDB - Embedded newform 6561.2.a.d.1.18

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			1.18
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)

$f(q)$	$=$	$q-1.51921 q^{2} +0.307998 q^{4} +0.0796060 q^{5} -0.854507 q^{7} +2.57051 q^{8} +O(q^{10})$	\(q-1.51921 q^{2} +0.307998 q^{4} +0.0796060 q^{5} -0.854507 q^{7} +2.57051 q^{8} -0.120938 q^{10} -3.99297 q^{11} -2.02854 q^{13} +1.29818 q^{14} -4.52113 q^{16} -2.04970 q^{17} -5.59791 q^{19} +0.0245185 q^{20} +6.06615 q^{22} -7.50386 q^{23} -4.99366 q^{25} +3.08178 q^{26} -0.263186 q^{28} -1.11131 q^{29} +5.99250 q^{31} +1.72754 q^{32} +3.11392 q^{34} -0.0680239 q^{35} -1.02119 q^{37} +8.50441 q^{38} +0.204628 q^{40} +4.37748 q^{41} -12.0083 q^{43} -1.22982 q^{44} +11.3999 q^{46} -5.14530 q^{47} -6.26982 q^{49} +7.58642 q^{50} -0.624787 q^{52} -10.3219 q^{53} -0.317864 q^{55} -2.19652 q^{56} +1.68832 q^{58} +10.9547 q^{59} -10.6314 q^{61} -9.10386 q^{62} +6.41778 q^{64} -0.161484 q^{65} -4.59504 q^{67} -0.631302 q^{68} +0.103343 q^{70} +5.54928 q^{71} +1.54027 q^{73} +1.55140 q^{74} -1.72415 q^{76} +3.41202 q^{77} +13.6529 q^{79} -0.359909 q^{80} -6.65031 q^{82} +6.28343 q^{83} -0.163168 q^{85} +18.2431 q^{86} -10.2639 q^{88} +10.3381 q^{89} +1.73340 q^{91} -2.31117 q^{92} +7.81679 q^{94} -0.445628 q^{95} -13.7384 q^{97} +9.52517 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.51921	−1.07424	−0.537122	−	0.843505i	$-0.680488\pi$
$2$	−1.51921	−1.07424	−0.537122	+	0.843505i	$0.680488\pi$
$3$	0	0
$3$	0	0
$4$	0.307998	0.153999
$5$	0.0796060	0.0356009	0.0178005	−	0.999842i	$-0.494334\pi$
$5$	0.0796060	0.0356009	0.0178005	+	0.999842i	$0.494334\pi$
$6$	0	0
$7$	−0.854507	−0.322973	−0.161487	−	0.986875i	$-0.551629\pi$
$7$	−0.854507	−0.322973	−0.161487	+	0.986875i	$0.551629\pi$
$8$	2.57051	0.908811
$9$	0	0
$10$	−0.120938	−0.0382440
$11$	−3.99297	−1.20392	−0.601962	−	0.798525i	$-0.705614\pi$
$11$	−3.99297	−1.20392	−0.601962	+	0.798525i	$0.705614\pi$
$12$	0	0
$13$	−2.02854	−0.562616	−0.281308	−	0.959617i	$-0.590768\pi$
$13$	−2.02854	−0.562616	−0.281308	+	0.959617i	$0.590768\pi$
$14$	1.29818	0.346952
$15$	0	0
$16$	−4.52113	−1.13028
$17$	−2.04970	−0.497125	−0.248562	−	0.968616i	$-0.579958\pi$
$17$	−2.04970	−0.497125	−0.248562	+	0.968616i	$0.579958\pi$
$18$	0	0
$19$	−5.59791	−1.28425	−0.642125	−	0.766600i	$-0.721947\pi$
$19$	−5.59791	−1.28425	−0.642125	+	0.766600i	$0.721947\pi$
$20$	0.0245185	0.00548250
$21$	0	0
$22$	6.06615	1.29331
$23$	−7.50386	−1.56466	−0.782332	−	0.622862i	$-0.785970\pi$
$23$	−7.50386	−1.56466	−0.782332	+	0.622862i	$0.785970\pi$
$24$	0	0
$25$	−4.99366	−0.998733
$26$	3.08178	0.604387
$27$	0	0
$28$	−0.263186	−0.0497376
$29$	−1.11131	−0.206366	−0.103183	−	0.994662i	$-0.532903\pi$
$29$	−1.11131	−0.206366	−0.103183	+	0.994662i	$0.532903\pi$
$30$	0	0
$31$	5.99250	1.07628	0.538142	−	0.842854i	$-0.319126\pi$
$31$	5.99250	1.07628	0.538142	+	0.842854i	$0.319126\pi$
$32$	1.72754	0.305388
$33$	0	0
$34$	3.11392	0.534033
$35$	−0.0680239	−0.0114981
$36$	0	0
$37$	−1.02119	−0.167882	−0.0839410	−	0.996471i	$-0.526751\pi$
$37$	−1.02119	−0.167882	−0.0839410	+	0.996471i	$0.526751\pi$
$38$	8.50441	1.37960
$39$	0	0
$40$	0.204628	0.0323545
$41$	4.37748	0.683648	0.341824	−	0.939764i	$-0.388955\pi$
$41$	4.37748	0.683648	0.341824	+	0.939764i	$0.388955\pi$
$42$	0	0
$43$	−12.0083	−1.83125	−0.915623	−	0.402037i	$-0.868302\pi$
$43$	−12.0083	−1.83125	−0.915623	+	0.402037i	$0.868302\pi$
$44$	−1.22982	−0.185403
$45$	0	0
$46$	11.3999	1.68083
$47$	−5.14530	−0.750519	−0.375259	−	0.926920i	$-0.622446\pi$
$47$	−5.14530	−0.750519	−0.375259	+	0.926920i	$0.622446\pi$
$48$	0	0
$49$	−6.26982	−0.895688
$50$	7.58642	1.07288
$51$	0	0
$52$	−0.624787	−0.0866423
$53$	−10.3219	−1.41783	−0.708913	−	0.705296i	$-0.750814\pi$
$53$	−10.3219	−1.41783	−0.708913	+	0.705296i	$0.750814\pi$
$54$	0	0
$55$	−0.317864	−0.0428608
$56$	−2.19652	−0.293522
$57$	0	0
$58$	1.68832	0.221687
$59$	10.9547	1.42618	0.713091	−	0.701071i	$-0.247295\pi$
$59$	10.9547	1.42618	0.713091	+	0.701071i	$0.247295\pi$
$60$	0	0
$61$	−10.6314	−1.36121	−0.680607	−	0.732649i	$-0.738284\pi$
$61$	−10.6314	−1.36121	−0.680607	+	0.732649i	$0.738284\pi$
$62$	−9.10386	−1.15619
$63$	0	0
$64$	6.41778	0.802222
$65$	−0.161484	−0.0200296
$66$	0	0
$67$	−4.59504	−0.561374	−0.280687	−	0.959799i	$-0.590562\pi$
$67$	−4.59504	−0.561374	−0.280687	+	0.959799i	$0.590562\pi$
$68$	−0.631302	−0.0765566
$69$	0	0
$70$	0.103343	0.0123518
$71$	5.54928	0.658578	0.329289	−	0.944229i	$-0.393191\pi$
$71$	5.54928	0.658578	0.329289	+	0.944229i	$0.393191\pi$
$72$	0	0
$73$	1.54027	0.180274	0.0901372	−	0.995929i	$-0.471269\pi$
$73$	1.54027	0.180274	0.0901372	+	0.995929i	$0.471269\pi$
$74$	1.55140	0.180346
$75$	0	0
$76$	−1.72415	−0.197773
$77$	3.41202	0.388836
$78$	0	0
$79$	13.6529	1.53607	0.768036	−	0.640407i	$-0.221234\pi$
$79$	13.6529	1.53607	0.768036	+	0.640407i	$0.221234\pi$
$80$	−0.359909	−0.0402391
$81$	0	0
$82$	−6.65031	−0.734404
$83$	6.28343	0.689696	0.344848	−	0.938659i	$-0.387931\pi$
$83$	6.28343	0.689696	0.344848	+	0.938659i	$0.387931\pi$
$84$	0	0
$85$	−0.163168	−0.0176981
$86$	18.2431	1.96720
$87$	0	0
$88$	−10.2639	−1.09414
$89$	10.3381	1.09584	0.547920	−	0.836531i	$-0.315420\pi$
$89$	10.3381	1.09584	0.547920	+	0.836531i	$0.315420\pi$
$90$	0	0
$91$	1.73340	0.181710
$92$	−2.31117	−0.240956
$93$	0	0
$94$	7.81679	0.806240
$95$	−0.445628	−0.0457204
$96$	0	0
$97$	−13.7384	−1.39492	−0.697462	−	0.716622i	$-0.745687\pi$
$97$	−13.7384	−1.39492	−0.697462	+	0.716622i	$0.745687\pi$
$98$	9.52517	0.962187
$99$	0	0
$100$	−1.53804	−0.153804
$101$	8.53700	0.849463	0.424732	−	0.905319i	$-0.360368\pi$
$101$	8.53700	0.849463	0.424732	+	0.905319i	$0.360368\pi$
$102$	0	0
$103$	8.22972	0.810898	0.405449	−	0.914118i	$-0.367115\pi$
$103$	8.22972	0.810898	0.405449	+	0.914118i	$0.367115\pi$
$104$	−5.21438	−0.511312
$105$	0	0
$106$	15.6812	1.52309
$107$	5.87510	0.567967	0.283984	−	0.958829i	$-0.408344\pi$
$107$	5.87510	0.567967	0.283984	+	0.958829i	$0.408344\pi$
$108$	0	0
$109$	10.4259	0.998616	0.499308	−	0.866425i	$-0.333588\pi$
$109$	10.4259	0.998616	0.499308	+	0.866425i	$0.333588\pi$
$110$	0.482902	0.0460429
$111$	0	0
$112$	3.86334	0.365051
$113$	2.77340	0.260899	0.130450	−	0.991455i	$-0.458358\pi$
$113$	2.77340	0.260899	0.130450	+	0.991455i	$0.458358\pi$
$114$	0	0
$115$	−0.597353	−0.0557034
$116$	−0.342282	−0.0317801
$117$	0	0
$118$	−16.6425	−1.53207
$119$	1.75148	0.160558
$120$	0	0
$121$	4.94378	0.449434
$122$	16.1514	1.46228
$123$	0	0
$124$	1.84568	0.165747
$125$	−0.795556	−0.0711567
$126$	0	0
$127$	−11.1268	−0.987343	−0.493671	−	0.869648i	$-0.664345\pi$
$127$	−11.1268	−0.987343	−0.493671	+	0.869648i	$0.664345\pi$
$128$	−13.2050	−1.16717
$129$	0	0
$130$	0.245328	0.0215167
$131$	−9.93072	−0.867651	−0.433826	−	0.900997i	$-0.642837\pi$
$131$	−9.93072	−0.867651	−0.433826	+	0.900997i	$0.642837\pi$
$132$	0	0
$133$	4.78346	0.414779
$134$	6.98084	0.603052
$135$	0	0
$136$	−5.26876	−0.451792
$137$	16.0795	1.37376	0.686881	−	0.726770i	$-0.258979\pi$
$137$	16.0795	1.37376	0.686881	+	0.726770i	$0.258979\pi$
$138$	0	0
$139$	−20.2951	−1.72140	−0.860702	−	0.509109i	$-0.829975\pi$
$139$	−20.2951	−1.72140	−0.860702	+	0.509109i	$0.829975\pi$
$140$	−0.0209512	−0.00177070
$141$	0	0
$142$	−8.43052	−0.707473
$143$	8.09990	0.677348
$144$	0	0
$145$	−0.0884673	−0.00734681
$146$	−2.33999	−0.193659
$147$	0	0
$148$	−0.314523	−0.0258537
$149$	5.11267	0.418846	0.209423	−	0.977825i	$-0.432841\pi$
$149$	5.11267	0.418846	0.209423	+	0.977825i	$0.432841\pi$
$150$	0	0
$151$	−17.7641	−1.44562	−0.722809	−	0.691047i	$-0.757150\pi$
$151$	−17.7641	−1.44562	−0.722809	+	0.691047i	$0.757150\pi$
$152$	−14.3895	−1.16714
$153$	0	0
$154$	−5.18357	−0.417704
$155$	0.477039	0.0383167
$156$	0	0
$157$	−7.21703	−0.575981	−0.287991	−	0.957633i	$-0.592987\pi$
$157$	−7.21703	−0.575981	−0.287991	+	0.957633i	$0.592987\pi$
$158$	−20.7416	−1.65012
$159$	0	0
$160$	0.137522	0.0108721
$161$	6.41210	0.505345
$162$	0	0
$163$	4.94597	0.387398	0.193699	−	0.981061i	$-0.437951\pi$
$163$	4.94597	0.387398	0.193699	+	0.981061i	$0.437951\pi$
$164$	1.34826	0.105281
$165$	0	0
$166$	−9.54584	−0.740901
$167$	17.9221	1.38685	0.693427	−	0.720527i	$-0.256100\pi$
$167$	17.9221	1.38685	0.693427	+	0.720527i	$0.256100\pi$
$168$	0	0
$169$	−8.88502	−0.683463
$170$	0.247887	0.0190120
$171$	0	0
$172$	−3.69853	−0.282010
$173$	−1.72304	−0.131000	−0.0655001	−	0.997853i	$-0.520864\pi$
$173$	−1.72304	−0.131000	−0.0655001	+	0.997853i	$0.520864\pi$
$174$	0	0
$175$	4.26712	0.322564
$176$	18.0527	1.36078
$177$	0	0
$178$	−15.7058	−1.17720
$179$	12.3521	0.923236	0.461618	−	0.887079i	$-0.347269\pi$
$179$	12.3521	0.923236	0.461618	+	0.887079i	$0.347269\pi$
$180$	0	0
$181$	0.0362973	0.00269796	0.00134898	−	0.999999i	$-0.499571\pi$
$181$	0.0362973	0.00269796	0.00134898	+	0.999999i	$0.499571\pi$
$182$	−2.63340	−0.195201
$183$	0	0
$184$	−19.2887	−1.42198
$185$	−0.0812926	−0.00597675
$186$	0	0
$187$	8.18437	0.598500
$188$	−1.58474	−0.115579
$189$	0	0
$190$	0.677002	0.0491149
$191$	−26.5564	−1.92155	−0.960776	−	0.277324i	$-0.910552\pi$
$191$	−26.5564	−1.92155	−0.960776	+	0.277324i	$0.910552\pi$
$192$	0	0
$193$	−0.0452311	−0.00325581	−0.00162790	−	0.999999i	$-0.500518\pi$
$193$	−0.0452311	−0.00325581	−0.00162790	+	0.999999i	$0.500518\pi$
$194$	20.8715	1.49849
$195$	0	0
$196$	−1.93109	−0.137935
$197$	−12.1822	−0.867946	−0.433973	−	0.900926i	$-0.642889\pi$
$197$	−12.1822	−0.867946	−0.433973	+	0.900926i	$0.642889\pi$
$198$	0	0
$199$	12.5732	0.891293	0.445647	−	0.895209i	$-0.352974\pi$
$199$	12.5732	0.891293	0.445647	+	0.895209i	$0.352974\pi$
$200$	−12.8362	−0.907659
$201$	0	0
$202$	−12.9695	−0.912530
$203$	0.949626	0.0666507
$204$	0	0
$205$	0.348474	0.0243385
$206$	−12.5027	−0.871102
$207$	0	0
$208$	9.17131	0.635916
$209$	22.3523	1.54614
$210$	0	0
$211$	13.7253	0.944887	0.472444	−	0.881361i	$-0.343372\pi$
$211$	13.7253	0.944887	0.472444	+	0.881361i	$0.343372\pi$
$212$	−3.17913	−0.218344
$213$	0	0
$214$	−8.92550	−0.610135
$215$	−0.955932	−0.0651940
$216$	0	0
$217$	−5.12063	−0.347611
$218$	−15.8391	−1.07276
$219$	0	0
$220$	−0.0979015	−0.00660052
$221$	4.15790	0.279690
$222$	0	0
$223$	−2.93727	−0.196694	−0.0983471	−	0.995152i	$-0.531356\pi$
$223$	−2.93727	−0.196694	−0.0983471	+	0.995152i	$0.531356\pi$
$224$	−1.47619	−0.0986323
$225$	0	0
$226$	−4.21337	−0.280269
$227$	−3.96602	−0.263234	−0.131617	−	0.991301i	$-0.542017\pi$
$227$	−3.96602	−0.263234	−0.131617	+	0.991301i	$0.542017\pi$
$228$	0	0
$229$	−10.2178	−0.675208	−0.337604	−	0.941288i	$-0.609616\pi$
$229$	−10.2178	−0.675208	−0.337604	+	0.941288i	$0.609616\pi$
$230$	0.907504	0.0598390
$231$	0	0
$232$	−2.85664	−0.187548
$233$	−2.55357	−0.167290	−0.0836450	−	0.996496i	$-0.526656\pi$
$233$	−2.55357	−0.167290	−0.0836450	+	0.996496i	$0.526656\pi$
$234$	0	0
$235$	−0.409597	−0.0267191
$236$	3.37403	0.219630
$237$	0	0
$238$	−2.66087	−0.172478
$239$	4.66176	0.301544	0.150772	−	0.988569i	$-0.451824\pi$
$239$	4.66176	0.301544	0.150772	+	0.988569i	$0.451824\pi$
$240$	0	0
$241$	1.51827	0.0978005	0.0489003	−	0.998804i	$-0.484428\pi$
$241$	1.51827	0.0978005	0.0489003	+	0.998804i	$0.484428\pi$
$242$	−7.51064	−0.482802
$243$	0	0
$244$	−3.27446	−0.209626
$245$	−0.499115	−0.0318873
$246$	0	0
$247$	11.3556	0.722540
$248$	15.4038	0.978139
$249$	0	0
$250$	1.20862	0.0764396
$251$	−21.8833	−1.38126	−0.690632	−	0.723207i	$-0.742667\pi$
$251$	−21.8833	−1.38126	−0.690632	+	0.723207i	$0.742667\pi$
$252$	0	0
$253$	29.9627	1.88374
$254$	16.9039	1.06065
$255$	0	0
$256$	7.22564	0.451603
$257$	7.11792	0.444004	0.222002	−	0.975046i	$-0.428741\pi$
$257$	7.11792	0.444004	0.222002	+	0.975046i	$0.428741\pi$
$258$	0	0
$259$	0.872612	0.0542214
$260$	−0.0497368	−0.00308454
$261$	0	0
$262$	15.0868	0.932069
$263$	16.7899	1.03531	0.517654	−	0.855590i	$-0.326805\pi$
$263$	16.7899	1.03531	0.517654	+	0.855590i	$0.326805\pi$
$264$	0	0
$265$	−0.821687	−0.0504759
$266$	−7.26708	−0.445573
$267$	0	0
$268$	−1.41526	−0.0864510
$269$	−20.6000	−1.25601	−0.628003	−	0.778211i	$-0.716128\pi$
$269$	−20.6000	−1.25601	−0.628003	+	0.778211i	$0.716128\pi$
$270$	0	0
$271$	3.50749	0.213065	0.106533	−	0.994309i	$-0.466025\pi$
$271$	3.50749	0.213065	0.106533	+	0.994309i	$0.466025\pi$
$272$	9.26695	0.561892
$273$	0	0
$274$	−24.4281	−1.47576
$275$	19.9395	1.20240
$276$	0	0
$277$	−16.2435	−0.975980	−0.487990	−	0.872849i	$-0.662270\pi$
$277$	−16.2435	−0.975980	−0.487990	+	0.872849i	$0.662270\pi$
$278$	30.8324	1.84921
$279$	0	0
$280$	−0.174856	−0.0104496
$281$	0.407030	0.0242814	0.0121407	−	0.999926i	$-0.496135\pi$
$281$	0.407030	0.0242814	0.0121407	+	0.999926i	$0.496135\pi$
$282$	0	0
$283$	18.6905	1.11104	0.555519	−	0.831504i	$-0.312520\pi$
$283$	18.6905	1.11104	0.555519	+	0.831504i	$0.312520\pi$
$284$	1.70917	0.101420
$285$	0	0
$286$	−12.3054	−0.727636
$287$	−3.74059	−0.220800
$288$	0	0
$289$	−12.7987	−0.752867
$290$	0.134400	0.00789227
$291$	0	0
$292$	0.474398	0.0277621
$293$	−12.8780	−0.752340	−0.376170	−	0.926551i	$-0.622759\pi$
$293$	−12.8780	−0.752340	−0.376170	+	0.926551i	$0.622759\pi$
$294$	0	0
$295$	0.872061	0.0507734
$296$	−2.62497	−0.152573
$297$	0	0
$298$	−7.76722	−0.449943
$299$	15.2219	0.880305
$300$	0	0
$301$	10.2612	0.591444
$302$	26.9873	1.55295
$303$	0	0
$304$	25.3089	1.45157
$305$	−0.846325	−0.0484605
$306$	0	0
$307$	−8.73659	−0.498623	−0.249312	−	0.968423i	$-0.580204\pi$
$307$	−8.73659	−0.498623	−0.249312	+	0.968423i	$0.580204\pi$
$308$	1.05089	0.0598803
$309$	0	0
$310$	−0.724722	−0.0411615
$311$	18.5889	1.05408	0.527041	−	0.849840i	$-0.323302\pi$
$311$	18.5889	1.05408	0.527041	+	0.849840i	$0.323302\pi$
$312$	0	0
$313$	−21.0388	−1.18918	−0.594590	−	0.804029i	$-0.702686\pi$
$313$	−21.0388	−1.18918	−0.594590	+	0.804029i	$0.702686\pi$
$314$	10.9642	0.618744
$315$	0	0
$316$	4.20507	0.236553
$317$	−16.2013	−0.909958	−0.454979	−	0.890502i	$-0.650353\pi$
$317$	−16.2013	−0.909958	−0.454979	+	0.890502i	$0.650353\pi$
$318$	0	0
$319$	4.43744	0.248449
$320$	0.510894	0.0285598
$321$	0	0
$322$	−9.74133	−0.542863
$323$	11.4740	0.638432
$324$	0	0
$325$	10.1299	0.561903
$326$	−7.51397	−0.416160
$327$	0	0
$328$	11.2523	0.621307
$329$	4.39669	0.242398
$330$	0	0
$331$	19.0542	1.04732	0.523658	−	0.851929i	$-0.324567\pi$
$331$	19.0542	1.04732	0.523658	+	0.851929i	$0.324567\pi$
$332$	1.93528	0.106212
$333$	0	0
$334$	−27.2274	−1.48982
$335$	−0.365793	−0.0199854
$336$	0	0
$337$	−10.9312	−0.595458	−0.297729	−	0.954650i	$-0.596229\pi$
$337$	−10.9312	−0.595458	−0.297729	+	0.954650i	$0.596229\pi$
$338$	13.4982	0.734205
$339$	0	0
$340$	−0.0502555	−0.00272549
$341$	−23.9278	−1.29577
$342$	0	0
$343$	11.3392	0.612257
$344$	−30.8674	−1.66426
$345$	0	0
$346$	2.61766	0.140726
$347$	22.3168	1.19803	0.599014	−	0.800738i	$-0.295559\pi$
$347$	22.3168	1.19803	0.599014	+	0.800738i	$0.295559\pi$
$348$	0	0
$349$	−23.3775	−1.25137	−0.625684	−	0.780077i	$-0.715180\pi$
$349$	−23.3775	−1.25137	−0.625684	+	0.780077i	$0.715180\pi$
$350$	−6.48265	−0.346512
$351$	0	0
$352$	−6.89800	−0.367664
$353$	31.8818	1.69690	0.848449	−	0.529277i	$-0.177537\pi$
$353$	31.8818	1.69690	0.848449	+	0.529277i	$0.177537\pi$
$354$	0	0
$355$	0.441756	0.0234460
$356$	3.18412	0.168758
$357$	0	0
$358$	−18.7654	−0.991780
$359$	−11.3148	−0.597170	−0.298585	−	0.954383i	$-0.596515\pi$
$359$	−11.3148	−0.597170	−0.298585	+	0.954383i	$0.596515\pi$
$360$	0	0
$361$	12.3366	0.649297
$362$	−0.0551433	−0.00289827
$363$	0	0
$364$	0.533885	0.0279832
$365$	0.122614	0.00641793
$366$	0	0
$367$	−0.678380	−0.0354111	−0.0177056	−	0.999843i	$-0.505636\pi$
$367$	−0.678380	−0.0354111	−0.0177056	+	0.999843i	$0.505636\pi$
$368$	33.9260	1.76851
$369$	0	0
$370$	0.123501	0.00642049
$371$	8.82016	0.457920
$372$	0	0
$373$	−11.1535	−0.577506	−0.288753	−	0.957404i	$-0.593241\pi$
$373$	−11.1535	−0.577506	−0.288753	+	0.957404i	$0.593241\pi$
$374$	−12.4338	−0.642935
$375$	0	0
$376$	−13.2260	−0.682080
$377$	2.25435	0.116105
$378$	0	0
$379$	19.9822	1.02642	0.513209	−	0.858263i	$-0.328456\pi$
$379$	19.9822	1.02642	0.513209	+	0.858263i	$0.328456\pi$
$380$	−0.137252	−0.00704090
$381$	0	0
$382$	40.3447	2.06422
$383$	23.5461	1.20315	0.601575	−	0.798816i	$-0.294540\pi$
$383$	23.5461	1.20315	0.601575	+	0.798816i	$0.294540\pi$
$384$	0	0
$385$	0.271617	0.0138429
$386$	0.0687156	0.00349753
$387$	0	0
$388$	−4.23140	−0.214817
$389$	6.24514	0.316641	0.158321	−	0.987388i	$-0.449392\pi$
$389$	6.24514	0.316641	0.158321	+	0.987388i	$0.449392\pi$
$390$	0	0
$391$	15.3806	0.777833
$392$	−16.1166	−0.814011
$393$	0	0
$394$	18.5073	0.932385
$395$	1.08685	0.0546855
$396$	0	0
$397$	−1.12202	−0.0563123	−0.0281562	−	0.999604i	$-0.508964\pi$
$397$	−1.12202	−0.0563123	−0.0281562	+	0.999604i	$0.508964\pi$
$398$	−19.1014	−0.957466
$399$	0	0
$400$	22.5770	1.12885
$401$	−4.23856	−0.211664	−0.105832	−	0.994384i	$-0.533750\pi$
$401$	−4.23856	−0.211664	−0.105832	+	0.994384i	$0.533750\pi$
$402$	0	0
$403$	−12.1560	−0.605535
$404$	2.62938	0.130816
$405$	0	0
$406$	−1.44268	−0.0715991
$407$	4.07756	0.202117
$408$	0	0
$409$	−1.91094	−0.0944898	−0.0472449	−	0.998883i	$-0.515044\pi$
$409$	−1.91094	−0.0944898	−0.0472449	+	0.998883i	$0.515044\pi$
$410$	−0.529405	−0.0261455
$411$	0	0
$412$	2.53474	0.124877
$413$	−9.36088	−0.460619
$414$	0	0
$415$	0.500199	0.0245538
$416$	−3.50438	−0.171816
$417$	0	0
$418$	−33.9578	−1.66093
$419$	−2.48186	−0.121247	−0.0606235	−	0.998161i	$-0.519309\pi$
$419$	−2.48186	−0.121247	−0.0606235	+	0.998161i	$0.519309\pi$
$420$	0	0
$421$	17.4651	0.851199	0.425600	−	0.904912i	$-0.360063\pi$
$421$	17.4651	0.851199	0.425600	+	0.904912i	$0.360063\pi$
$422$	−20.8516	−1.01504
$423$	0	0
$424$	−26.5326	−1.28854
$425$	10.2355	0.496494
$426$	0	0
$427$	9.08463	0.439636
$428$	1.80952	0.0874663
$429$	0	0
$430$	1.45226	0.0700343
$431$	23.3699	1.12569	0.562845	−	0.826562i	$-0.309707\pi$
$431$	23.3699	1.12569	0.562845	+	0.826562i	$0.309707\pi$
$432$	0	0
$433$	−30.5747	−1.46933	−0.734664	−	0.678432i	$-0.762660\pi$
$433$	−30.5747	−1.46933	−0.734664	+	0.678432i	$0.762660\pi$
$434$	7.77931	0.373419
$435$	0	0
$436$	3.21114	0.153786
$437$	42.0060	2.00942
$438$	0	0
$439$	12.4082	0.592211	0.296105	−	0.955155i	$-0.404312\pi$
$439$	12.4082	0.592211	0.296105	+	0.955155i	$0.404312\pi$
$440$	−0.817072	−0.0389524
$441$	0	0
$442$	−6.31672	−0.300456
$443$	−3.46846	−0.164792	−0.0823959	−	0.996600i	$-0.526257\pi$
$443$	−3.46846	−0.164792	−0.0823959	+	0.996600i	$0.526257\pi$
$444$	0	0
$445$	0.822978	0.0390129
$446$	4.46233	0.211298
$447$	0	0
$448$	−5.48404	−0.259096
$449$	−3.10375	−0.146475	−0.0732375	−	0.997315i	$-0.523333\pi$
$449$	−3.10375	−0.146475	−0.0732375	+	0.997315i	$0.523333\pi$
$450$	0	0
$451$	−17.4791	−0.823061
$452$	0.854200	0.0401782
$453$	0	0
$454$	6.02521	0.282777
$455$	0.137989	0.00646904
$456$	0	0
$457$	35.4792	1.65965	0.829824	−	0.558026i	$-0.188441\pi$
$457$	35.4792	1.65965	0.829824	+	0.558026i	$0.188441\pi$
$458$	15.5229	0.725338
$459$	0	0
$460$	−0.183983	−0.00857827
$461$	−25.4874	−1.18707	−0.593533	−	0.804810i	$-0.702267\pi$
$461$	−25.4874	−1.18707	−0.593533	+	0.804810i	$0.702267\pi$
$462$	0	0
$463$	−9.31677	−0.432987	−0.216493	−	0.976284i	$-0.569462\pi$
$463$	−9.31677	−0.432987	−0.216493	+	0.976284i	$0.569462\pi$
$464$	5.02440	0.233252
$465$	0	0
$466$	3.87941	0.179710
$467$	12.4096	0.574248	0.287124	−	0.957893i	$-0.407301\pi$
$467$	12.4096	0.574248	0.287124	+	0.957893i	$0.407301\pi$
$468$	0	0
$469$	3.92650	0.181309
$470$	0.622263	0.0287029
$471$	0	0
$472$	28.1592	1.29613
$473$	47.9487	2.20468
$474$	0	0
$475$	27.9541	1.28262
$476$	0.539452	0.0247258
$477$	0	0
$478$	−7.08219	−0.323932
$479$	34.7646	1.58844	0.794218	−	0.607633i	$-0.207881\pi$
$479$	34.7646	1.58844	0.794218	+	0.607633i	$0.207881\pi$
$480$	0	0
$481$	2.07152	0.0944532
$482$	−2.30657	−0.105062
$483$	0	0
$484$	1.52267	0.0692124
$485$	−1.09366	−0.0496605
$486$	0	0
$487$	4.53564	0.205529	0.102765	−	0.994706i	$-0.467231\pi$
$487$	4.53564	0.205529	0.102765	+	0.994706i	$0.467231\pi$
$488$	−27.3281	−1.23709
$489$	0	0
$490$	0.758261	0.0342547
$491$	−29.9534	−1.35178	−0.675890	−	0.737003i	$-0.736241\pi$
$491$	−29.9534	−1.35178	−0.675890	+	0.737003i	$0.736241\pi$
$492$	0	0
$493$	2.27786	0.102590
$494$	−17.2515	−0.776184
$495$	0	0
$496$	−27.0929	−1.21651
$497$	−4.74190	−0.212703
$498$	0	0
$499$	6.82458	0.305510	0.152755	−	0.988264i	$-0.451185\pi$
$499$	6.82458	0.305510	0.152755	+	0.988264i	$0.451185\pi$
$500$	−0.245029	−0.0109581
$501$	0	0
$502$	33.2454	1.48381
$503$	−5.33541	−0.237894	−0.118947	−	0.992901i	$-0.537952\pi$
$503$	−5.33541	−0.237894	−0.118947	+	0.992901i	$0.537952\pi$
$504$	0	0
$505$	0.679597	0.0302417
$506$	−45.5196	−2.02359
$507$	0	0
$508$	−3.42703	−0.152050
$509$	−44.7102	−1.98174	−0.990872	−	0.134806i	$-0.956959\pi$
$509$	−44.7102	−1.98174	−0.990872	+	0.134806i	$0.956959\pi$
$510$	0	0
$511$	−1.31617	−0.0582238
$512$	15.4328	0.682039
$513$	0	0
$514$	−10.8136	−0.476968
$515$	0.655135	0.0288687
$516$	0	0
$517$	20.5450	0.903568
$518$	−1.32568	−0.0582470
$519$	0	0
$520$	−0.415096	−0.0182032
$521$	−9.87541	−0.432649	−0.216325	−	0.976321i	$-0.569407\pi$
$521$	−9.87541	−0.432649	−0.216325	+	0.976321i	$0.569407\pi$
$522$	0	0
$523$	37.9022	1.65735	0.828673	−	0.559733i	$-0.189096\pi$
$523$	37.9022	1.65735	0.828673	+	0.559733i	$0.189096\pi$
$524$	−3.05864	−0.133617
$525$	0	0
$526$	−25.5073	−1.11217
$527$	−12.2828	−0.535047
$528$	0	0
$529$	33.3079	1.44817
$530$	1.24832	0.0542234
$531$	0	0
$532$	1.47330	0.0638754
$533$	−8.87991	−0.384632
$534$	0	0
$535$	0.467693	0.0202201
$536$	−11.8116	−0.510183
$537$	0	0
$538$	31.2958	1.34926
$539$	25.0352	1.07834
$540$	0	0
$541$	44.3843	1.90823	0.954116	−	0.299438i	$-0.0967990\pi$
$541$	44.3843	1.90823	0.954116	+	0.299438i	$0.0967990\pi$
$542$	−5.32862	−0.228884
$543$	0	0
$544$	−3.54093	−0.151816
$545$	0.829961	0.0355516
$546$	0	0
$547$	3.19107	0.136440	0.0682202	−	0.997670i	$-0.478268\pi$
$547$	3.19107	0.136440	0.0682202	+	0.997670i	$0.478268\pi$
$548$	4.95244	0.211558
$549$	0	0
$550$	−30.2923	−1.29167
$551$	6.22104	0.265025
$552$	0	0
$553$	−11.6665	−0.496110
$554$	24.6773	1.04844
$555$	0	0
$556$	−6.25083	−0.265094
$557$	25.6054	1.08493	0.542467	−	0.840077i	$-0.317490\pi$
$557$	25.6054	1.08493	0.542467	+	0.840077i	$0.317490\pi$
$558$	0	0
$559$	24.3593	1.03029
$560$	0.307545	0.0129962
$561$	0	0
$562$	−0.618364	−0.0260841
$563$	−15.2767	−0.643834	−0.321917	−	0.946768i	$-0.604327\pi$
$563$	−15.2767	−0.643834	−0.321917	+	0.946768i	$0.604327\pi$
$564$	0	0
$565$	0.220779	0.00928825
$566$	−28.3949	−1.19352
$567$	0	0
$568$	14.2645	0.598523
$569$	16.0869	0.674397	0.337199	−	0.941433i	$-0.390521\pi$
$569$	16.0869	0.674397	0.337199	+	0.941433i	$0.390521\pi$
$570$	0	0
$571$	−18.2871	−0.765293	−0.382647	−	0.923895i	$-0.624987\pi$
$571$	−18.2871	−0.765293	−0.382647	+	0.923895i	$0.624987\pi$
$572$	2.49475	0.104311
$573$	0	0
$574$	5.68274	0.237193
$575$	37.4718	1.56268
$576$	0	0
$577$	10.9688	0.456639	0.228320	−	0.973586i	$-0.426677\pi$
$577$	10.9688	0.456639	0.228320	+	0.973586i	$0.426677\pi$
$578$	19.4440	0.808763
$579$	0	0
$580$	−0.0272478	−0.00113140
$581$	−5.36923	−0.222753
$582$	0	0
$583$	41.2151	1.70695
$584$	3.95926	0.163835
$585$	0	0
$586$	19.5644	0.808196
$587$	15.6852	0.647396	0.323698	−	0.946160i	$-0.395074\pi$
$587$	15.6852	0.647396	0.323698	+	0.946160i	$0.395074\pi$
$588$	0	0
$589$	−33.5455	−1.38222
$590$	−1.32484	−0.0545430
$591$	0	0
$592$	4.61692	0.189754
$593$	−40.8831	−1.67887	−0.839434	−	0.543462i	$-0.817113\pi$
$593$	−40.8831	−1.67887	−0.839434	+	0.543462i	$0.817113\pi$
$594$	0	0
$595$	0.139428	0.00571601
$596$	1.57469	0.0645019
$597$	0	0
$598$	−23.1253	−0.945662
$599$	−27.0481	−1.10516	−0.552578	−	0.833461i	$-0.686356\pi$
$599$	−27.0481	−1.10516	−0.552578	+	0.833461i	$0.686356\pi$
$600$	0	0
$601$	16.6985	0.681145	0.340572	−	0.940218i	$-0.389379\pi$
$601$	16.6985	0.681145	0.340572	+	0.940218i	$0.389379\pi$
$602$	−15.5889	−0.635355
$603$	0	0
$604$	−5.47129	−0.222624
$605$	0.393555	0.0160003
$606$	0	0
$607$	−45.6461	−1.85272	−0.926359	−	0.376643i	$-0.877078\pi$
$607$	−45.6461	−1.85272	−0.926359	+	0.376643i	$0.877078\pi$
$608$	−9.67060	−0.392195
$609$	0	0
$610$	1.28575	0.0520583
$611$	10.4375	0.422254
$612$	0	0
$613$	39.2861	1.58675	0.793376	−	0.608732i	$-0.208322\pi$
$613$	39.2861	1.58675	0.793376	+	0.608732i	$0.208322\pi$
$614$	13.2727	0.535643
$615$	0	0
$616$	8.77061	0.353378
$617$	2.53763	0.102161	0.0510806	−	0.998695i	$-0.483733\pi$
$617$	2.53763	0.102161	0.0510806	+	0.998695i	$0.483733\pi$
$618$	0	0
$619$	−12.2641	−0.492935	−0.246468	−	0.969151i	$-0.579270\pi$
$619$	−12.2641	−0.492935	−0.246468	+	0.969151i	$0.579270\pi$
$620$	0.146927	0.00590073
$621$	0	0
$622$	−28.2405	−1.13234
$623$	−8.83402	−0.353927
$624$	0	0
$625$	24.9050	0.996199
$626$	31.9623	1.27747
$627$	0	0
$628$	−2.22283	−0.0887005
$629$	2.09312	0.0834583
$630$	0	0
$631$	40.5456	1.61410	0.807048	−	0.590486i	$-0.201064\pi$
$631$	40.5456	1.61410	0.807048	+	0.590486i	$0.201064\pi$
$632$	35.0949	1.39600
$633$	0	0
$634$	24.6132	0.977516
$635$	−0.885760	−0.0351503
$636$	0	0
$637$	12.7186	0.503929
$638$	−6.74140	−0.266895
$639$	0	0
$640$	−1.05120	−0.0415523
$641$	30.1929	1.19255	0.596274	−	0.802781i	$-0.296647\pi$
$641$	30.1929	1.19255	0.596274	+	0.802781i	$0.296647\pi$
$642$	0	0
$643$	−0.444530	−0.0175306	−0.00876528	−	0.999962i	$-0.502790\pi$
$643$	−0.444530	−0.0175306	−0.00876528	+	0.999962i	$0.502790\pi$
$644$	1.97491	0.0778225
$645$	0	0
$646$	−17.4315	−0.685831
$647$	−38.0107	−1.49436	−0.747178	−	0.664624i	$-0.768592\pi$
$647$	−38.0107	−1.49436	−0.747178	+	0.664624i	$0.768592\pi$
$648$	0	0
$649$	−43.7418	−1.71702
$650$	−15.3894	−0.603621
$651$	0	0
$652$	1.52335	0.0596589
$653$	13.5185	0.529022	0.264511	−	0.964383i	$-0.414789\pi$
$653$	13.5185	0.529022	0.264511	+	0.964383i	$0.414789\pi$
$654$	0	0
$655$	−0.790545	−0.0308892
$656$	−19.7912	−0.772716
$657$	0	0
$658$	−6.67950	−0.260394
$659$	−26.0448	−1.01456	−0.507281	−	0.861780i	$-0.669350\pi$
$659$	−26.0448	−1.01456	−0.507281	+	0.861780i	$0.669350\pi$
$660$	0	0
$661$	2.40565	0.0935689	0.0467844	−	0.998905i	$-0.485103\pi$
$661$	2.40565	0.0935689	0.0467844	+	0.998905i	$0.485103\pi$
$662$	−28.9474	−1.12507
$663$	0	0
$664$	16.1516	0.626803
$665$	0.380792	0.0147665
$666$	0	0
$667$	8.33915	0.322893
$668$	5.51997	0.213574
$669$	0	0
$670$	0.555717	0.0214692
$671$	42.4509	1.63880
$672$	0	0
$673$	−13.1864	−0.508296	−0.254148	−	0.967165i	$-0.581795\pi$
$673$	−13.1864	−0.508296	−0.254148	+	0.967165i	$0.581795\pi$
$674$	16.6067	0.639667
$675$	0	0
$676$	−2.73657	−0.105253
$677$	11.5404	0.443534	0.221767	−	0.975100i	$-0.428818\pi$
$677$	11.5404	0.443534	0.221767	+	0.975100i	$0.428818\pi$
$678$	0	0
$679$	11.7396	0.450523
$680$	−0.419425	−0.0160842
$681$	0	0
$682$	36.3514	1.39197
$683$	9.35498	0.357959	0.178979	−	0.983853i	$-0.442720\pi$
$683$	9.35498	0.357959	0.178979	+	0.983853i	$0.442720\pi$
$684$	0	0
$685$	1.28002	0.0489072
$686$	−17.2266	−0.657713
$687$	0	0
$688$	54.2911	2.06983
$689$	20.9385	0.797692
$690$	0	0
$691$	15.0232	0.571508	0.285754	−	0.958303i	$-0.407756\pi$
$691$	15.0232	0.571508	0.285754	+	0.958303i	$0.407756\pi$
$692$	−0.530692	−0.0201739
$693$	0	0
$694$	−33.9039	−1.28697
$695$	−1.61561	−0.0612835
$696$	0	0
$697$	−8.97251	−0.339858
$698$	35.5153	1.34427
$699$	0	0
$700$	1.31426	0.0496745
$701$	−16.8266	−0.635530	−0.317765	−	0.948169i	$-0.602932\pi$
$701$	−16.8266	−0.635530	−0.317765	+	0.948169i	$0.602932\pi$
$702$	0	0
$703$	5.71652	0.215602
$704$	−25.6260	−0.965815
$705$	0	0
$706$	−48.4352	−1.82288
$707$	−7.29493	−0.274354
$708$	0	0
$709$	9.61808	0.361214	0.180607	−	0.983555i	$-0.442194\pi$
$709$	9.61808	0.361214	0.180607	+	0.983555i	$0.442194\pi$
$710$	−0.671120	−0.0251867
$711$	0	0
$712$	26.5742	0.995912
$713$	−44.9669	−1.68402
$714$	0	0
$715$	0.644801	0.0241142
$716$	3.80441	0.142177
$717$	0	0
$718$	17.1895	0.641506
$719$	−22.3383	−0.833078	−0.416539	−	0.909118i	$-0.636757\pi$
$719$	−22.3383	−0.833078	−0.416539	+	0.909118i	$0.636757\pi$
$720$	0	0
$721$	−7.03236	−0.261899
$722$	−18.7420	−0.697503
$723$	0	0
$724$	0.0111795	0.000415483 0
$725$	5.54953	0.206104
$726$	0	0
$727$	−40.8546	−1.51521	−0.757606	−	0.652712i	$-0.773631\pi$
$727$	−40.8546	−1.51521	−0.757606	+	0.652712i	$0.773631\pi$
$728$	4.45573	0.165140
$729$	0	0
$730$	−0.186277	−0.00689442
$731$	24.6134	0.910358
$732$	0	0
$733$	−27.3459	−1.01004	−0.505022	−	0.863106i	$-0.668516\pi$
$733$	−27.3459	−1.01004	−0.505022	+	0.863106i	$0.668516\pi$
$734$	1.03060	0.0380402
$735$	0	0
$736$	−12.9632	−0.477830
$737$	18.3479	0.675852
$738$	0	0
$739$	−8.92139	−0.328179	−0.164089	−	0.986445i	$-0.552469\pi$
$739$	−8.92139	−0.328179	−0.164089	+	0.986445i	$0.552469\pi$
$740$	−0.0250380	−0.000920413 0
$741$	0	0
$742$	−13.3997	−0.491917
$743$	−15.4260	−0.565923	−0.282962	−	0.959131i	$-0.591317\pi$
$743$	−15.4260	−0.565923	−0.282962	+	0.959131i	$0.591317\pi$
$744$	0	0
$745$	0.407000	0.0149113
$746$	16.9445	0.620382
$747$	0	0
$748$	2.52077	0.0921684
$749$	−5.02031	−0.183438
$750$	0	0
$751$	29.2849	1.06862	0.534310	−	0.845289i	$-0.320571\pi$
$751$	29.2849	1.06862	0.534310	+	0.845289i	$0.320571\pi$
$752$	23.2626	0.848299
$753$	0	0
$754$	−3.42483	−0.124725
$755$	−1.41413	−0.0514653
$756$	0	0
$757$	33.2659	1.20907	0.604536	−	0.796578i	$-0.293359\pi$
$757$	33.2659	1.20907	0.604536	+	0.796578i	$0.293359\pi$
$758$	−30.3572	−1.10262
$759$	0	0
$760$	−1.14549	−0.0415512
$761$	−0.320033	−0.0116012	−0.00580059	−	0.999983i	$-0.501846\pi$
$761$	−0.320033	−0.0116012	−0.00580059	+	0.999983i	$0.501846\pi$
$762$	0	0
$763$	−8.90897	−0.322526
$764$	−8.17931	−0.295917
$765$	0	0
$766$	−35.7715	−1.29248
$767$	−22.2221	−0.802393
$768$	0	0
$769$	2.07834	0.0749470	0.0374735	−	0.999298i	$-0.488069\pi$
$769$	2.07834	0.0749470	0.0374735	+	0.999298i	$0.488069\pi$
$770$	−0.412644	−0.0148706
$771$	0	0
$772$	−0.0139311	−0.000501391 0
$773$	−6.60785	−0.237668	−0.118834	−	0.992914i	$-0.537916\pi$
$773$	−6.60785	−0.237668	−0.118834	+	0.992914i	$0.537916\pi$
$774$	0	0
$775$	−29.9245	−1.07492
$776$	−35.3146	−1.26772
$777$	0	0
$778$	−9.48768	−0.340150
$779$	−24.5048	−0.877975
$780$	0	0
$781$	−22.1581	−0.792878
$782$	−23.3664	−0.835581
$783$	0	0
$784$	28.3467	1.01238
$785$	−0.574519	−0.0205055
$786$	0	0
$787$	3.15923	0.112614	0.0563072	−	0.998413i	$-0.482067\pi$
$787$	3.15923	0.112614	0.0563072	+	0.998413i	$0.482067\pi$
$788$	−3.75209	−0.133663
$789$	0	0
$790$	−1.65116	−0.0587456
$791$	−2.36989	−0.0842635
$792$	0	0
$793$	21.5663	0.765841
$794$	1.70458	0.0604932
$795$	0	0
$796$	3.87253	0.137258
$797$	−46.8213	−1.65849	−0.829247	−	0.558882i	$-0.811231\pi$
$797$	−46.8213	−1.65849	−0.829247	+	0.558882i	$0.811231\pi$
$798$	0	0
$799$	10.5463	0.373101
$800$	−8.62674	−0.305001
$801$	0	0
$802$	6.43926	0.227378
$803$	−6.15023	−0.217037
$804$	0	0
$805$	0.510442	0.0179907
$806$	18.4676	0.650492
$807$	0	0
$808$	21.9444	0.772001
$809$	47.8251	1.68144	0.840721	−	0.541468i	$-0.182131\pi$
$809$	47.8251	1.68144	0.840721	+	0.541468i	$0.182131\pi$
$810$	0	0
$811$	11.3948	0.400124	0.200062	−	0.979783i	$-0.435886\pi$
$811$	11.3948	0.400124	0.200062	+	0.979783i	$0.435886\pi$
$812$	0.292483	0.0102641
$813$	0	0
$814$	−6.19467	−0.217123
$815$	0.393729	0.0137917
$816$	0	0
$817$	67.2214	2.35178
$818$	2.90312	0.101505
$819$	0	0
$820$	0.107329	0.00374810
$821$	15.9428	0.556407	0.278204	−	0.960522i	$-0.410261\pi$
$821$	15.9428	0.556407	0.278204	+	0.960522i	$0.410261\pi$
$822$	0	0
$823$	38.6151	1.34604	0.673018	−	0.739626i	$-0.264998\pi$
$823$	38.6151	1.34604	0.673018	+	0.739626i	$0.264998\pi$
$824$	21.1545	0.736953
$825$	0	0
$826$	14.2211	0.494817
$827$	−34.8835	−1.21302	−0.606509	−	0.795076i	$-0.707431\pi$
$827$	−34.8835	−1.21302	−0.606509	+	0.795076i	$0.707431\pi$
$828$	0	0
$829$	−27.4831	−0.954526	−0.477263	−	0.878761i	$-0.658371\pi$
$829$	−27.4831	−0.954526	−0.477263	+	0.878761i	$0.658371\pi$
$830$	−0.759907	−0.0263767
$831$	0	0
$832$	−13.0187	−0.451343
$833$	12.8512	0.445269
$834$	0	0
$835$	1.42671	0.0493733
$836$	6.88446	0.238104
$837$	0	0
$838$	3.77047	0.130249
$839$	22.8775	0.789819	0.394910	−	0.918720i	$-0.370776\pi$
$839$	22.8775	0.789819	0.394910	+	0.918720i	$0.370776\pi$
$840$	0	0
$841$	−27.7650	−0.957413
$842$	−26.5332	−0.914395
$843$	0	0
$844$	4.22736	0.145512
$845$	−0.707301	−0.0243319
$846$	0	0
$847$	−4.22450	−0.145155
$848$	46.6668	1.60254
$849$	0	0
$850$	−15.5499	−0.533356
$851$	7.66284	0.262679
$852$	0	0
$853$	52.9362	1.81250	0.906251	−	0.422740i	$-0.138932\pi$
$853$	52.9362	1.81250	0.906251	+	0.422740i	$0.138932\pi$
$854$	−13.8015	−0.472276
$855$	0	0
$856$	15.1020	0.516175
$857$	15.7366	0.537550	0.268775	−	0.963203i	$-0.413381\pi$
$857$	15.7366	0.537550	0.268775	+	0.963203i	$0.413381\pi$
$858$	0	0
$859$	−17.9389	−0.612068	−0.306034	−	0.952021i	$-0.599002\pi$
$859$	−17.9389	−0.612068	−0.306034	+	0.952021i	$0.599002\pi$
$860$	−0.294425	−0.0100398
$861$	0	0
$862$	−35.5038	−1.20927
$863$	49.9633	1.70077	0.850385	−	0.526161i	$-0.176369\pi$
$863$	49.9633	1.70077	0.850385	+	0.526161i	$0.176369\pi$
$864$	0	0
$865$	−0.137164	−0.00466373
$866$	46.4494	1.57841
$867$	0	0
$868$	−1.57714	−0.0535318
$869$	−54.5156	−1.84931
$870$	0	0
$871$	9.32124	0.315838
$872$	26.7997	0.907553
$873$	0	0
$874$	−63.8159	−2.15860
$875$	0.679808	0.0229817
$876$	0	0
$877$	−8.19013	−0.276561	−0.138280	−	0.990393i	$-0.544158\pi$
$877$	−8.19013	−0.276561	−0.138280	+	0.990393i	$0.544158\pi$
$878$	−18.8506	−0.636178
$879$	0	0
$880$	1.43711	0.0484448
$881$	15.2372	0.513354	0.256677	−	0.966497i	$-0.417372\pi$
$881$	15.2372	0.513354	0.256677	+	0.966497i	$0.417372\pi$
$882$	0	0
$883$	−30.6804	−1.03248	−0.516239	−	0.856444i	$-0.672669\pi$
$883$	−30.6804	−1.03248	−0.516239	+	0.856444i	$0.672669\pi$
$884$	1.28062	0.0430720
$885$	0	0
$886$	5.26932	0.177026
$887$	−23.7737	−0.798242	−0.399121	−	0.916898i	$-0.630685\pi$
$887$	−23.7737	−0.798242	−0.399121	+	0.916898i	$0.630685\pi$
$888$	0	0
$889$	9.50792	0.318886
$890$	−1.25028	−0.0419094
$891$	0	0
$892$	−0.904673	−0.0302907
$893$	28.8029	0.963853
$894$	0	0
$895$	0.983298	0.0328680
$896$	11.2838	0.376965
$897$	0	0
$898$	4.71525	0.157350
$899$	−6.65955	−0.222108
$900$	0	0
$901$	21.1568	0.704836
$902$	26.5545	0.884167
$903$	0	0
$904$	7.12903	0.237108
$905$	0.00288949	9.60498e−5 0
$906$	0	0
$907$	−34.5490	−1.14718	−0.573591	−	0.819142i	$-0.694450\pi$
$907$	−34.5490	−1.14718	−0.573591	+	0.819142i	$0.694450\pi$
$908$	−1.22152	−0.0405377
$909$	0	0
$910$	−0.209635	−0.00694933
$911$	50.0854	1.65940	0.829702	−	0.558207i	$-0.188510\pi$
$911$	50.0854	1.65940	0.829702	+	0.558207i	$0.188510\pi$
$912$	0	0
$913$	−25.0895	−0.830342
$914$	−53.9004	−1.78287
$915$	0	0
$916$	−3.14705	−0.103981
$917$	8.48587	0.280228
$918$	0	0
$919$	−2.82886	−0.0933155	−0.0466577	−	0.998911i	$-0.514857\pi$
$919$	−2.82886	−0.0933155	−0.0466577	+	0.998911i	$0.514857\pi$
$920$	−1.53550	−0.0506239
$921$	0	0
$922$	38.7207	1.27520
$923$	−11.2569	−0.370527
$924$	0	0
$925$	5.09946	0.167669
$926$	14.1541	0.465133
$927$	0	0
$928$	−1.91984	−0.0630217
$929$	−12.3494	−0.405172	−0.202586	−	0.979265i	$-0.564935\pi$
$929$	−12.3494	−0.405172	−0.202586	+	0.979265i	$0.564935\pi$
$930$	0	0
$931$	35.0979	1.15029
$932$	−0.786494	−0.0257625
$933$	0	0
$934$	−18.8528	−0.616882
$935$	0.651525	0.0213072
$936$	0	0
$937$	−6.45970	−0.211029	−0.105515	−	0.994418i	$-0.533649\pi$
$937$	−6.45970	−0.211029	−0.105515	+	0.994418i	$0.533649\pi$
$938$	−5.96518	−0.194770
$939$	0	0
$940$	−0.126155	−0.00411472
$941$	12.5117	0.407870	0.203935	−	0.978984i	$-0.434627\pi$
$941$	12.5117	0.407870	0.203935	+	0.978984i	$0.434627\pi$
$942$	0	0
$943$	−32.8480	−1.06968
$944$	−49.5277	−1.61199
$945$	0	0
$946$	−72.8441	−2.36837
$947$	48.1574	1.56490	0.782452	−	0.622711i	$-0.213969\pi$
$947$	48.1574	1.56490	0.782452	+	0.622711i	$0.213969\pi$
$948$	0	0
$949$	−3.12449	−0.101425
$950$	−42.4681	−1.37785
$951$	0	0
$952$	4.50219	0.145917
$953$	−25.1608	−0.815037	−0.407519	−	0.913197i	$-0.633606\pi$
$953$	−25.1608	−0.815037	−0.407519	+	0.913197i	$0.633606\pi$
$954$	0	0
$955$	−2.11405	−0.0684090
$956$	1.43581	0.0464374
$957$	0	0
$958$	−52.8148	−1.70637
$959$	−13.7400	−0.443689
$960$	0	0
$961$	4.91002	0.158388
$962$	−3.14707	−0.101466
$963$	0	0
$964$	0.467625	0.0150612
$965$	−0.00360067	−0.000115910 0
$966$	0	0
$967$	−47.1703	−1.51690	−0.758448	−	0.651734i	$-0.774042\pi$
$967$	−47.1703	−1.51690	−0.758448	+	0.651734i	$0.774042\pi$
$968$	12.7080	0.408451
$969$	0	0
$970$	1.66150	0.0533475
$971$	26.3271	0.844876	0.422438	−	0.906392i	$-0.361174\pi$
$971$	26.3271	0.844876	0.422438	+	0.906392i	$0.361174\pi$
$972$	0	0
$973$	17.3423	0.555968
$974$	−6.89059	−0.220789
$975$	0	0
$976$	48.0661	1.53856
$977$	−17.4772	−0.559144	−0.279572	−	0.960125i	$-0.590193\pi$
$977$	−17.4772	−0.559144	−0.279572	+	0.960125i	$0.590193\pi$
$978$	0	0
$979$	−41.2798	−1.31931
$980$	−0.153726	−0.00491061
$981$	0	0
$982$	45.5056	1.45214
$983$	−39.7347	−1.26734	−0.633670	−	0.773603i	$-0.718452\pi$
$983$	−39.7347	−1.26734	−0.633670	+	0.773603i	$0.718452\pi$
$984$	0	0
$985$	−0.969777	−0.0308997
$986$	−3.46054	−0.110206
$987$	0	0
$988$	3.49750	0.111270
$989$	90.1085	2.86528
$990$	0	0
$991$	12.2443	0.388953	0.194477	−	0.980907i	$-0.437699\pi$
$991$	12.2443	0.388953	0.194477	+	0.980907i	$0.437699\pi$
$992$	10.3523	0.328685
$993$	0	0
$994$	7.20394	0.228495
$995$	1.00091	0.0317308
$996$	0	0
$997$	−8.37616	−0.265276	−0.132638	−	0.991165i	$-0.542345\pi$
$997$	−8.37616	−0.265276	−0.132638	+	0.991165i	$0.542345\pi$
$998$	−10.3680	−0.328192
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.18		72
3.2	odd	2		6561.2.a.c.1.55		72
81.5	odd	54		729.2.g.c.460.6		144
81.11	odd	54		81.2.g.a.40.3	✓	144
81.16	even	27		729.2.g.b.271.3		144
81.22	even	27		243.2.g.a.235.6		144
81.32	odd	54		729.2.g.d.703.3		144
81.38	odd	54		729.2.g.d.28.3		144
81.43	even	27		729.2.g.a.28.6		144
81.49	even	27		729.2.g.a.703.6		144
81.59	odd	54		81.2.g.a.79.3	yes	144
81.65	odd	54		729.2.g.c.271.6		144
81.70	even	27		243.2.g.a.91.6		144
81.76	even	27		729.2.g.b.460.3		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.40.3	✓	144	81.11	odd	54
81.2.g.a.79.3	yes	144	81.59	odd	54
243.2.g.a.91.6		144	81.70	even	27
243.2.g.a.235.6		144	81.22	even	27
729.2.g.a.28.6		144	81.43	even	27
729.2.g.a.703.6		144	81.49	even	27
729.2.g.b.271.3		144	81.16	even	27
729.2.g.b.460.3		144	81.76	even	27
729.2.g.c.271.6		144	81.65	odd	54
729.2.g.c.460.6		144	81.5	odd	54
729.2.g.d.28.3		144	81.38	odd	54
729.2.g.d.703.3		144	81.32	odd	54
6561.2.a.c.1.55		72	3.2	odd	2
6561.2.a.d.1.18		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)

\(f(q)\)	\(=\)	\(q-1.51921 q^{2} +0.307998 q^{4} +0.0796060 q^{5} -0.854507 q^{7} +2.57051 q^{8} +O(q^{10})\)	\(q-1.51921 q^{2} +0.307998 q^{4} +0.0796060 q^{5} -0.854507 q^{7} +2.57051 q^{8} -0.120938 q^{10} -3.99297 q^{11} -2.02854 q^{13} +1.29818 q^{14} -4.52113 q^{16} -2.04970 q^{17} -5.59791 q^{19} +0.0245185 q^{20} +6.06615 q^{22} -7.50386 q^{23} -4.99366 q^{25} +3.08178 q^{26} -0.263186 q^{28} -1.11131 q^{29} +5.99250 q^{31} +1.72754 q^{32} +3.11392 q^{34} -0.0680239 q^{35} -1.02119 q^{37} +8.50441 q^{38} +0.204628 q^{40} +4.37748 q^{41} -12.0083 q^{43} -1.22982 q^{44} +11.3999 q^{46} -5.14530 q^{47} -6.26982 q^{49} +7.58642 q^{50} -0.624787 q^{52} -10.3219 q^{53} -0.317864 q^{55} -2.19652 q^{56} +1.68832 q^{58} +10.9547 q^{59} -10.6314 q^{61} -9.10386 q^{62} +6.41778 q^{64} -0.161484 q^{65} -4.59504 q^{67} -0.631302 q^{68} +0.103343 q^{70} +5.54928 q^{71} +1.54027 q^{73} +1.55140 q^{74} -1.72415 q^{76} +3.41202 q^{77} +13.6529 q^{79} -0.359909 q^{80} -6.65031 q^{82} +6.28343 q^{83} -0.163168 q^{85} +18.2431 q^{86} -10.2639 q^{88} +10.3381 q^{89} +1.73340 q^{91} -2.31117 q^{92} +7.81679 q^{94} -0.445628 q^{95} -13.7384 q^{97} +9.52517 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) \) 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

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