LMFDB - Embedded newform 6561.2.a.d.1.27

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6561,2,Mod(1,6561)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6561.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6561, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [72,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

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

Embedding invariants

Embedding label			1.27
Character	$\chi$	$=$	6561.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.676638 q^{2} -1.54216 q^{4} -0.943174 q^{5} -1.46065 q^{7} +2.39676 q^{8} +0.638188 q^{10} -6.12074 q^{11} -5.24673 q^{13} +0.988331 q^{14} +1.46258 q^{16} +0.0308950 q^{17} +3.65660 q^{19} +1.45453 q^{20} +4.14153 q^{22} -0.104547 q^{23} -4.11042 q^{25} +3.55014 q^{26} +2.25255 q^{28} -0.512704 q^{29} -8.16933 q^{31} -5.78316 q^{32} -0.0209047 q^{34} +1.37765 q^{35} -5.77133 q^{37} -2.47420 q^{38} -2.26056 q^{40} -4.38597 q^{41} +1.37668 q^{43} +9.43916 q^{44} +0.0707407 q^{46} -5.60972 q^{47} -4.86651 q^{49} +2.78127 q^{50} +8.09130 q^{52} -9.78213 q^{53} +5.77293 q^{55} -3.50083 q^{56} +0.346915 q^{58} +5.64940 q^{59} +1.71779 q^{61} +5.52768 q^{62} +0.987951 q^{64} +4.94858 q^{65} +2.73766 q^{67} -0.0476450 q^{68} -0.932169 q^{70} -2.17282 q^{71} -8.33116 q^{73} +3.90511 q^{74} -5.63907 q^{76} +8.94025 q^{77} -12.6128 q^{79} -1.37947 q^{80} +2.96771 q^{82} -17.8244 q^{83} -0.0291393 q^{85} -0.931512 q^{86} -14.6700 q^{88} +13.9487 q^{89} +7.66363 q^{91} +0.161229 q^{92} +3.79575 q^{94} -3.44881 q^{95} -7.01137 q^{97} +3.29286 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38}+ \cdots + 81 q^{98}+O(q^{100}) $ 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8 + 36 * q^11 + 36 * q^14 + 45 * q^16 + 36 * q^17 + 54 * q^20 + 54 * q^23 + 36 * q^25 + 45 * q^26 + 9 * q^28 + 54 * q^29 + 63 * q^32 + 72 * q^35 + 54 * q^38 + 72 * q^41 + 90 * q^44 + 90 * q^47 + 18 * q^49 + 45 * q^50 + 45 * q^53 + 9 * q^55 + 108 * q^56 + 18 * q^58 + 108 * q^59 + 72 * q^62 + 9 * q^64 + 72 * q^65 + 108 * q^68 + 126 * q^71 + 90 * q^74 + 72 * q^77 + 144 * q^80 - 18 * q^82 + 108 * q^83 + 90 * q^86 + 108 * q^89 + 72 * q^92 + 144 * q^95 + 81 * q^98

Coefficient data

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

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

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

$2$	−0.676638	−0.478456	−0.239228	−	0.970963i	$-0.576894\pi$
$2$	−0.676638	−0.478456	−0.239228	+	0.970963i	$0.576894\pi$
$3$	0	0
$3$	0	0
$4$	−1.54216	−0.771080
$5$	−0.943174	−0.421800	−0.210900	−	0.977508i	$-0.567639\pi$
$5$	−0.943174	−0.421800	−0.210900	+	0.977508i	$0.567639\pi$
$6$	0	0
$7$	−1.46065	−0.552073	−0.276037	−	0.961147i	$-0.589021\pi$
$7$	−1.46065	−0.552073	−0.276037	+	0.961147i	$0.589021\pi$
$8$	2.39676	0.847383
$9$	0	0
$10$	0.638188	0.201813
$11$	−6.12074	−1.84547	−0.922737	−	0.385431i	$-0.874053\pi$
$11$	−6.12074	−1.84547	−0.922737	+	0.385431i	$0.874053\pi$
$12$	0	0
$13$	−5.24673	−1.45518	−0.727591	−	0.686011i	$-0.759360\pi$
$13$	−5.24673	−1.45518	−0.727591	+	0.686011i	$0.759360\pi$
$14$	0.988331	0.264143
$15$	0	0
$16$	1.46258	0.365645
$17$	0.0308950	0.00749313	0.00374656	−	0.999993i	$-0.498807\pi$
$17$	0.0308950	0.00749313	0.00374656	+	0.999993i	$0.498807\pi$
$18$	0	0
$19$	3.65660	0.838883	0.419441	−	0.907782i	$-0.362226\pi$
$19$	3.65660	0.838883	0.419441	+	0.907782i	$0.362226\pi$
$20$	1.45453	0.325242
$21$	0	0
$22$	4.14153	0.882977
$23$	−0.104547	−0.0217996	−0.0108998	−	0.999941i	$-0.503470\pi$
$23$	−0.104547	−0.0217996	−0.0108998	+	0.999941i	$0.503470\pi$
$24$	0	0
$25$	−4.11042	−0.822084
$26$	3.55014	0.696240
$27$	0	0
$28$	2.25255	0.425693
$29$	−0.512704	−0.0952068	−0.0476034	−	0.998866i	$-0.515158\pi$
$29$	−0.512704	−0.0952068	−0.0476034	+	0.998866i	$0.515158\pi$
$30$	0	0
$31$	−8.16933	−1.46725	−0.733627	−	0.679552i	$-0.762174\pi$
$31$	−8.16933	−1.46725	−0.733627	+	0.679552i	$0.762174\pi$
$32$	−5.78316	−1.02233
$33$	0	0
$34$	−0.0209047	−0.00358513
$35$	1.37765	0.232865
$36$	0	0
$37$	−5.77133	−0.948801	−0.474401	−	0.880309i	$-0.657335\pi$
$37$	−5.77133	−0.948801	−0.474401	+	0.880309i	$0.657335\pi$
$38$	−2.47420	−0.401368
$39$	0	0
$40$	−2.26056	−0.357427
$41$	−4.38597	−0.684973	−0.342486	−	0.939523i	$-0.611269\pi$
$41$	−4.38597	−0.684973	−0.342486	+	0.939523i	$0.611269\pi$
$42$	0	0
$43$	1.37668	0.209941	0.104971	−	0.994475i	$-0.466525\pi$
$43$	1.37668	0.209941	0.104971	+	0.994475i	$0.466525\pi$
$44$	9.43916	1.42301
$45$	0	0
$46$	0.0707407	0.0104301
$47$	−5.60972	−0.818262	−0.409131	−	0.912476i	$-0.634168\pi$
$47$	−5.60972	−0.818262	−0.409131	+	0.912476i	$0.634168\pi$
$48$	0	0
$49$	−4.86651	−0.695215
$50$	2.78127	0.393331
$51$	0	0
$52$	8.09130	1.12206
$53$	−9.78213	−1.34368	−0.671839	−	0.740697i	$-0.734495\pi$
$53$	−9.78213	−1.34368	−0.671839	+	0.740697i	$0.734495\pi$
$54$	0	0
$55$	5.77293	0.778421
$56$	−3.50083	−0.467818
$57$	0	0
$58$	0.346915	0.0455522
$59$	5.64940	0.735490	0.367745	−	0.929927i	$-0.380130\pi$
$59$	5.64940	0.735490	0.367745	+	0.929927i	$0.380130\pi$
$60$	0	0
$61$	1.71779	0.219940	0.109970	−	0.993935i	$-0.464924\pi$
$61$	1.71779	0.219940	0.109970	+	0.993935i	$0.464924\pi$
$62$	5.52768	0.702016
$63$	0	0
$64$	0.987951	0.123494
$65$	4.94858	0.613796
$66$	0	0
$67$	2.73766	0.334459	0.167229	−	0.985918i	$-0.446518\pi$
$67$	2.73766	0.334459	0.167229	+	0.985918i	$0.446518\pi$
$68$	−0.0476450	−0.00577780
$69$	0	0
$70$	−0.932169	−0.111415
$71$	−2.17282	−0.257866	−0.128933	−	0.991653i	$-0.541155\pi$
$71$	−2.17282	−0.257866	−0.128933	+	0.991653i	$0.541155\pi$
$72$	0	0
$73$	−8.33116	−0.975088	−0.487544	−	0.873098i	$-0.662107\pi$
$73$	−8.33116	−0.975088	−0.487544	+	0.873098i	$0.662107\pi$
$74$	3.90511	0.453959
$75$	0	0
$76$	−5.63907	−0.646846
$77$	8.94025	1.01884
$78$	0	0
$79$	−12.6128	−1.41905	−0.709526	−	0.704679i	$-0.751091\pi$
$79$	−12.6128	−1.41905	−0.709526	+	0.704679i	$0.751091\pi$
$80$	−1.37947	−0.154229
$81$	0	0
$82$	2.96771	0.327729
$83$	−17.8244	−1.95648	−0.978242	−	0.207466i	$-0.933479\pi$
$83$	−17.8244	−1.95648	−0.978242	+	0.207466i	$0.933479\pi$
$84$	0	0
$85$	−0.0291393	−0.00316060
$86$	−0.931512	−0.100448
$87$	0	0
$88$	−14.6700	−1.56382
$89$	13.9487	1.47856	0.739279	−	0.673399i	$-0.235167\pi$
$89$	13.9487	1.47856	0.739279	+	0.673399i	$0.235167\pi$
$90$	0	0
$91$	7.66363	0.803367
$92$	0.161229	0.0168093
$93$	0	0
$94$	3.79575	0.391502
$95$	−3.44881	−0.353841
$96$	0	0
$97$	−7.01137	−0.711896	−0.355948	−	0.934506i	$-0.615842\pi$
$97$	−7.01137	−0.711896	−0.355948	+	0.934506i	$0.615842\pi$
$98$	3.29286	0.332630
$99$	0	0
$100$	6.33893	0.633893
$101$	8.69078	0.864765	0.432383	−	0.901690i	$-0.357673\pi$
$101$	8.69078	0.864765	0.432383	+	0.901690i	$0.357673\pi$
$102$	0	0
$103$	−5.12769	−0.505247	−0.252623	−	0.967565i	$-0.581293\pi$
$103$	−5.12769	−0.505247	−0.252623	+	0.967565i	$0.581293\pi$
$104$	−12.5752	−1.23310
$105$	0	0
$106$	6.61896	0.642891
$107$	−9.95284	−0.962177	−0.481089	−	0.876672i	$-0.659758\pi$
$107$	−9.95284	−0.962177	−0.481089	+	0.876672i	$0.659758\pi$
$108$	0	0
$109$	−12.7371	−1.21999	−0.609997	−	0.792404i	$-0.708829\pi$
$109$	−12.7371	−1.21999	−0.609997	+	0.792404i	$0.708829\pi$
$110$	−3.90618	−0.372440
$111$	0	0
$112$	−2.13631	−0.201863
$113$	−5.74689	−0.540622	−0.270311	−	0.962773i	$-0.587126\pi$
$113$	−5.74689	−0.540622	−0.270311	+	0.962773i	$0.587126\pi$
$114$	0	0
$115$	0.0986063	0.00919508
$116$	0.790672	0.0734121
$117$	0	0
$118$	−3.82260	−0.351899
$119$	−0.0451267	−0.00413676
$120$	0	0
$121$	26.4635	2.40577
$122$	−1.16232	−0.105232
$123$	0	0
$124$	12.5984	1.13137
$125$	8.59272	0.768556
$126$	0	0
$127$	10.6013	0.940709	0.470354	−	0.882478i	$-0.344126\pi$
$127$	10.6013	0.940709	0.470354	+	0.882478i	$0.344126\pi$
$128$	10.8978	0.963242
$129$	0	0
$130$	−3.34840	−0.293674
$131$	−14.2638	−1.24623	−0.623116	−	0.782130i	$-0.714133\pi$
$131$	−14.2638	−1.24623	−0.623116	+	0.782130i	$0.714133\pi$
$132$	0	0
$133$	−5.34101	−0.463125
$134$	−1.85241	−0.160024
$135$	0	0
$136$	0.0740479	0.00634955
$137$	−5.33618	−0.455900	−0.227950	−	0.973673i	$-0.573202\pi$
$137$	−5.33618	−0.455900	−0.227950	+	0.973673i	$0.573202\pi$
$138$	0	0
$139$	−4.09415	−0.347261	−0.173631	−	0.984811i	$-0.555550\pi$
$139$	−4.09415	−0.347261	−0.173631	+	0.984811i	$0.555550\pi$
$140$	−2.12455	−0.179557
$141$	0	0
$142$	1.47021	0.123378
$143$	32.1139	2.68550
$144$	0	0
$145$	0.483569	0.0401583
$146$	5.63718	0.466537
$147$	0	0
$148$	8.90032	0.731602
$149$	18.8765	1.54643	0.773213	−	0.634146i	$-0.218648\pi$
$149$	18.8765	1.54643	0.773213	+	0.634146i	$0.218648\pi$
$150$	0	0
$151$	−3.08043	−0.250682	−0.125341	−	0.992114i	$-0.540003\pi$
$151$	−3.08043	−0.250682	−0.125341	+	0.992114i	$0.540003\pi$
$152$	8.76401	0.710855
$153$	0	0
$154$	−6.04932	−0.487468
$155$	7.70510	0.618888
$156$	0	0
$157$	−3.42861	−0.273633	−0.136816	−	0.990596i	$-0.543687\pi$
$157$	−3.42861	−0.273633	−0.136816	+	0.990596i	$0.543687\pi$
$158$	8.53431	0.678954
$159$	0	0
$160$	5.45453	0.431218
$161$	0.152707	0.0120350
$162$	0	0
$163$	−19.7151	−1.54420	−0.772101	−	0.635500i	$-0.780794\pi$
$163$	−19.7151	−1.54420	−0.772101	+	0.635500i	$0.780794\pi$
$164$	6.76386	0.528169
$165$	0	0
$166$	12.0607	0.936091
$167$	19.3425	1.49677	0.748384	−	0.663265i	$-0.230830\pi$
$167$	19.3425	1.49677	0.748384	+	0.663265i	$0.230830\pi$
$168$	0	0
$169$	14.5282	1.11755
$170$	0.0197168	0.00151221
$171$	0	0
$172$	−2.12306	−0.161881
$173$	6.16733	0.468893	0.234447	−	0.972129i	$-0.424672\pi$
$173$	6.16733	0.468893	0.234447	+	0.972129i	$0.424672\pi$
$174$	0	0
$175$	6.00388	0.453851
$176$	−8.95207	−0.674788
$177$	0	0
$178$	−9.43822	−0.707424
$179$	3.46177	0.258745	0.129372	−	0.991596i	$-0.458704\pi$
$179$	3.46177	0.258745	0.129372	+	0.991596i	$0.458704\pi$
$180$	0	0
$181$	−19.0803	−1.41823	−0.709115	−	0.705093i	$-0.750905\pi$
$181$	−19.0803	−1.41823	−0.709115	+	0.705093i	$0.750905\pi$
$182$	−5.18551	−0.384376
$183$	0	0
$184$	−0.250575	−0.0184726
$185$	5.44337	0.400205
$186$	0	0
$187$	−0.189100	−0.0138284
$188$	8.65109	0.630945
$189$	0	0
$190$	2.33360	0.169297
$191$	19.9278	1.44192	0.720962	−	0.692975i	$-0.243700\pi$
$191$	19.9278	1.44192	0.720962	+	0.692975i	$0.243700\pi$
$192$	0	0
$193$	−10.7174	−0.771454	−0.385727	−	0.922613i	$-0.626049\pi$
$193$	−10.7174	−0.771454	−0.385727	+	0.922613i	$0.626049\pi$
$194$	4.74416	0.340611
$195$	0	0
$196$	7.50493	0.536067
$197$	23.7019	1.68869	0.844344	−	0.535802i	$-0.179991\pi$
$197$	23.7019	1.68869	0.844344	+	0.535802i	$0.179991\pi$
$198$	0	0
$199$	8.73512	0.619217	0.309608	−	0.950864i	$-0.399802\pi$
$199$	8.73512	0.619217	0.309608	+	0.950864i	$0.399802\pi$
$200$	−9.85170	−0.696621
$201$	0	0
$202$	−5.88052	−0.413752
$203$	0.748881	0.0525611
$204$	0	0
$205$	4.13673	0.288922
$206$	3.46960	0.241738
$207$	0	0
$208$	−7.67376	−0.532080
$209$	−22.3811	−1.54814
$210$	0	0
$211$	−7.27806	−0.501043	−0.250521	−	0.968111i	$-0.580602\pi$
$211$	−7.27806	−0.501043	−0.250521	+	0.968111i	$0.580602\pi$
$212$	15.0856	1.03608
$213$	0	0
$214$	6.73447	0.460359
$215$	−1.29845	−0.0885532
$216$	0	0
$217$	11.9325	0.810032
$218$	8.61842	0.583713
$219$	0	0
$220$	−8.90278	−0.600225
$221$	−0.162098	−0.0109039
$222$	0	0
$223$	0.748474	0.0501215	0.0250608	−	0.999686i	$-0.492022\pi$
$223$	0.748474	0.0501215	0.0250608	+	0.999686i	$0.492022\pi$
$224$	8.44717	0.564400
$225$	0	0
$226$	3.88857	0.258664
$227$	−7.60596	−0.504825	−0.252413	−	0.967620i	$-0.581224\pi$
$227$	−7.60596	−0.504825	−0.252413	+	0.967620i	$0.581224\pi$
$228$	0	0
$229$	−15.6018	−1.03100	−0.515499	−	0.856890i	$-0.672394\pi$
$229$	−15.6018	−1.03100	−0.515499	+	0.856890i	$0.672394\pi$
$230$	−0.0667208	−0.00439944
$231$	0	0
$232$	−1.22883	−0.0806766
$233$	16.4777	1.07949	0.539746	−	0.841828i	$-0.318520\pi$
$233$	16.4777	1.07949	0.539746	+	0.841828i	$0.318520\pi$
$234$	0	0
$235$	5.29094	0.345143
$236$	−8.71229	−0.567122
$237$	0	0
$238$	0.0305345	0.00197925
$239$	−19.3469	−1.25145	−0.625724	−	0.780044i	$-0.715196\pi$
$239$	−19.3469	−1.25145	−0.625724	+	0.780044i	$0.715196\pi$
$240$	0	0
$241$	5.76490	0.371350	0.185675	−	0.982611i	$-0.440553\pi$
$241$	5.76490	0.371350	0.185675	+	0.982611i	$0.440553\pi$
$242$	−17.9062	−1.15105
$243$	0	0
$244$	−2.64910	−0.169591
$245$	4.58996	0.293242
$246$	0	0
$247$	−19.1852	−1.22073
$248$	−19.5799	−1.24333
$249$	0	0
$250$	−5.81416	−0.367720
$251$	18.8961	1.19271	0.596355	−	0.802721i	$-0.296615\pi$
$251$	18.8961	1.19271	0.596355	+	0.802721i	$0.296615\pi$
$252$	0	0
$253$	0.639907	0.0402306
$254$	−7.17321	−0.450087
$255$	0	0
$256$	−9.34980	−0.584362
$257$	18.1207	1.13034	0.565169	−	0.824975i	$-0.308811\pi$
$257$	18.1207	1.13034	0.565169	+	0.824975i	$0.308811\pi$
$258$	0	0
$259$	8.42989	0.523808
$260$	−7.63151	−0.473286
$261$	0	0
$262$	9.65142	0.596267
$263$	−8.66915	−0.534563	−0.267281	−	0.963619i	$-0.586125\pi$
$263$	−8.66915	−0.534563	−0.267281	+	0.963619i	$0.586125\pi$
$264$	0	0
$265$	9.22625	0.566764
$266$	3.61394	0.221585
$267$	0	0
$268$	−4.22192	−0.257895
$269$	−11.1861	−0.682029	−0.341014	−	0.940058i	$-0.610770\pi$
$269$	−11.1861	−0.682029	−0.341014	+	0.940058i	$0.610770\pi$
$270$	0	0
$271$	−10.1741	−0.618033	−0.309016	−	0.951057i	$-0.600000\pi$
$271$	−10.1741	−0.618033	−0.309016	+	0.951057i	$0.600000\pi$
$272$	0.0451863	0.00273982
$273$	0	0
$274$	3.61066	0.218128
$275$	25.1588	1.51713
$276$	0	0
$277$	−1.30204	−0.0782319	−0.0391160	−	0.999235i	$-0.512454\pi$
$277$	−1.30204	−0.0782319	−0.0391160	+	0.999235i	$0.512454\pi$
$278$	2.77026	0.166149
$279$	0	0
$280$	3.30189	0.197326
$281$	6.40982	0.382378	0.191189	−	0.981553i	$-0.438766\pi$
$281$	6.40982	0.382378	0.191189	+	0.981553i	$0.438766\pi$
$282$	0	0
$283$	−19.1171	−1.13640	−0.568198	−	0.822892i	$-0.692359\pi$
$283$	−19.1171	−1.13640	−0.568198	+	0.822892i	$0.692359\pi$
$284$	3.35083	0.198835
$285$	0	0
$286$	−21.7295	−1.28489
$287$	6.40636	0.378155
$288$	0	0
$289$	−16.9990	−0.999944
$290$	−0.327202	−0.0192139
$291$	0	0
$292$	12.8480	0.751871
$293$	1.82606	0.106680	0.0533398	−	0.998576i	$-0.483013\pi$
$293$	1.82606	0.106680	0.0533398	+	0.998576i	$0.483013\pi$
$294$	0	0
$295$	−5.32837	−0.310230
$296$	−13.8325	−0.803998
$297$	0	0
$298$	−12.7726	−0.739896
$299$	0.548532	0.0317224
$300$	0	0
$301$	−2.01084	−0.115903
$302$	2.08434	0.119940
$303$	0	0
$304$	5.34807	0.306733
$305$	−1.62017	−0.0927708
$306$	0	0
$307$	−14.9847	−0.855225	−0.427612	−	0.903962i	$-0.640645\pi$
$307$	−14.9847	−0.855225	−0.427612	+	0.903962i	$0.640645\pi$
$308$	−13.7873	−0.785605
$309$	0	0
$310$	−5.21357	−0.296111
$311$	−20.6598	−1.17151	−0.585754	−	0.810489i	$-0.699201\pi$
$311$	−20.6598	−1.17151	−0.585754	+	0.810489i	$0.699201\pi$
$312$	0	0
$313$	22.3234	1.26180	0.630898	−	0.775866i	$-0.282687\pi$
$313$	22.3234	1.26180	0.630898	+	0.775866i	$0.282687\pi$
$314$	2.31993	0.130921
$315$	0	0
$316$	19.4510	1.09420
$317$	−11.3601	−0.638049	−0.319024	−	0.947747i	$-0.603355\pi$
$317$	−11.3601	−0.638049	−0.319024	+	0.947747i	$0.603355\pi$
$318$	0	0
$319$	3.13813	0.175702
$320$	−0.931810	−0.0520898
$321$	0	0
$322$	−0.103327	−0.00575821
$323$	0.112971	0.00628586
$324$	0	0
$325$	21.5663	1.19628
$326$	13.3400	0.738832
$327$	0	0
$328$	−10.5121	−0.580435
$329$	8.19383	0.451740
$330$	0	0
$331$	14.4072	0.791893	0.395947	−	0.918274i	$-0.370417\pi$
$331$	14.4072	0.791893	0.395947	+	0.918274i	$0.370417\pi$
$332$	27.4881	1.50861
$333$	0	0
$334$	−13.0879	−0.716137
$335$	−2.58209	−0.141075
$336$	0	0
$337$	−9.90761	−0.539702	−0.269851	−	0.962902i	$-0.586974\pi$
$337$	−9.90761	−0.539702	−0.269851	+	0.962902i	$0.586974\pi$
$338$	−9.83034	−0.534700
$339$	0	0
$340$	0.0449375	0.00243708
$341$	50.0023	2.70778
$342$	0	0
$343$	17.3328	0.935883
$344$	3.29957	0.177901
$345$	0	0
$346$	−4.17305	−0.224345
$347$	3.37824	0.181353	0.0906766	−	0.995880i	$-0.471097\pi$
$347$	3.37824	0.181353	0.0906766	+	0.995880i	$0.471097\pi$
$348$	0	0
$349$	−1.32993	−0.0711897	−0.0355949	−	0.999366i	$-0.511333\pi$
$349$	−1.32993	−0.0711897	−0.0355949	+	0.999366i	$0.511333\pi$
$350$	−4.06246	−0.217148
$351$	0	0
$352$	35.3972	1.88668
$353$	−11.7936	−0.627709	−0.313854	−	0.949471i	$-0.601620\pi$
$353$	−11.7936	−0.627709	−0.313854	+	0.949471i	$0.601620\pi$
$354$	0	0
$355$	2.04935	0.108768
$356$	−21.5111	−1.14009
$357$	0	0
$358$	−2.34236	−0.123798
$359$	29.7113	1.56810	0.784052	−	0.620696i	$-0.213150\pi$
$359$	29.7113	1.56810	0.784052	+	0.620696i	$0.213150\pi$
$360$	0	0
$361$	−5.62924	−0.296276
$362$	12.9105	0.678560
$363$	0	0
$364$	−11.8186	−0.619460
$365$	7.85773	0.411293
$366$	0	0
$367$	−8.02701	−0.419006	−0.209503	−	0.977808i	$-0.567185\pi$
$367$	−8.02701	−0.419006	−0.209503	+	0.977808i	$0.567185\pi$
$368$	−0.152909	−0.00797092
$369$	0	0
$370$	−3.68319	−0.191480
$371$	14.2883	0.741809
$372$	0	0
$373$	−2.99382	−0.155014	−0.0775072	−	0.996992i	$-0.524696\pi$
$373$	−2.99382	−0.155014	−0.0775072	+	0.996992i	$0.524696\pi$
$374$	0.127952	0.00661626
$375$	0	0
$376$	−13.4452	−0.693381
$377$	2.69002	0.138543
$378$	0	0
$379$	−22.1075	−1.13558	−0.567792	−	0.823172i	$-0.692202\pi$
$379$	−22.1075	−1.13558	−0.567792	+	0.823172i	$0.692202\pi$
$380$	5.31863	0.272840
$381$	0	0
$382$	−13.4839	−0.689896
$383$	−14.3576	−0.733641	−0.366820	−	0.930292i	$-0.619554\pi$
$383$	−14.3576	−0.733641	−0.366820	+	0.930292i	$0.619554\pi$
$384$	0	0
$385$	−8.43222	−0.429746
$386$	7.25179	0.369106
$387$	0	0
$388$	10.8127	0.548929
$389$	18.2300	0.924298	0.462149	−	0.886802i	$-0.347078\pi$
$389$	18.2300	0.924298	0.462149	+	0.886802i	$0.347078\pi$
$390$	0	0
$391$	−0.00322998	−0.000163347 0
$392$	−11.6639	−0.589114
$393$	0	0
$394$	−16.0376	−0.807962
$395$	11.8961	0.598557
$396$	0	0
$397$	1.38700	0.0696115	0.0348057	−	0.999394i	$-0.488919\pi$
$397$	1.38700	0.0696115	0.0348057	+	0.999394i	$0.488919\pi$
$398$	−5.91052	−0.296268
$399$	0	0
$400$	−6.01182	−0.300591
$401$	−22.0863	−1.10294	−0.551468	−	0.834196i	$-0.685932\pi$
$401$	−22.0863	−1.10294	−0.551468	+	0.834196i	$0.685932\pi$
$402$	0	0
$403$	42.8623	2.13512
$404$	−13.4026	−0.666803
$405$	0	0
$406$	−0.506722	−0.0251482
$407$	35.3248	1.75099
$408$	0	0
$409$	0.577381	0.0285497	0.0142748	−	0.999898i	$-0.495456\pi$
$409$	0.577381	0.0285497	0.0142748	+	0.999898i	$0.495456\pi$
$410$	−2.79907	−0.138236
$411$	0	0
$412$	7.90773	0.389586
$413$	−8.25179	−0.406044
$414$	0	0
$415$	16.8115	0.825246
$416$	30.3427	1.48767
$417$	0	0
$418$	15.1439	0.740714
$419$	12.6315	0.617091	0.308546	−	0.951210i	$-0.400158\pi$
$419$	12.6315	0.617091	0.308546	+	0.951210i	$0.400158\pi$
$420$	0	0
$421$	21.4260	1.04424	0.522121	−	0.852872i	$-0.325141\pi$
$421$	21.4260	1.04424	0.522121	+	0.852872i	$0.325141\pi$
$422$	4.92462	0.239727
$423$	0	0
$424$	−23.4454	−1.13861
$425$	−0.126991	−0.00615999
$426$	0	0
$427$	−2.50908	−0.121423
$428$	15.3489	0.741916
$429$	0	0
$430$	0.878578	0.0423688
$431$	−4.47131	−0.215376	−0.107688	−	0.994185i	$-0.534345\pi$
$431$	−4.47131	−0.215376	−0.107688	+	0.994185i	$0.534345\pi$
$432$	0	0
$433$	9.13341	0.438924	0.219462	−	0.975621i	$-0.429570\pi$
$433$	9.13341	0.438924	0.219462	+	0.975621i	$0.429570\pi$
$434$	−8.07400	−0.387564
$435$	0	0
$436$	19.6427	0.940713
$437$	−0.382288	−0.0182873
$438$	0	0
$439$	2.84864	0.135958	0.0679790	−	0.997687i	$-0.478345\pi$
$439$	2.84864	0.135958	0.0679790	+	0.997687i	$0.478345\pi$
$440$	13.8363	0.659621
$441$	0	0
$442$	0.109681	0.00521702
$443$	12.3716	0.587791	0.293895	−	0.955838i	$-0.405048\pi$
$443$	12.3716	0.587791	0.293895	+	0.955838i	$0.405048\pi$
$444$	0	0
$445$	−13.1560	−0.623656
$446$	−0.506446	−0.0239809
$447$	0	0
$448$	−1.44305	−0.0681777
$449$	−17.1746	−0.810521	−0.405260	−	0.914201i	$-0.632819\pi$
$449$	−17.1746	−0.810521	−0.405260	+	0.914201i	$0.632819\pi$
$450$	0	0
$451$	26.8454	1.26410
$452$	8.86262	0.416863
$453$	0	0
$454$	5.14649	0.241537
$455$	−7.22814	−0.338860
$456$	0	0
$457$	0.884764	0.0413875	0.0206938	−	0.999786i	$-0.493413\pi$
$457$	0.884764	0.0413875	0.0206938	+	0.999786i	$0.493413\pi$
$458$	10.5568	0.493287
$459$	0	0
$460$	−0.152067	−0.00709015
$461$	15.2318	0.709416	0.354708	−	0.934977i	$-0.384580\pi$
$461$	15.2318	0.709416	0.354708	+	0.934977i	$0.384580\pi$
$462$	0	0
$463$	33.6370	1.56324	0.781621	−	0.623753i	$-0.214393\pi$
$463$	33.6370	1.56324	0.781621	+	0.623753i	$0.214393\pi$
$464$	−0.749871	−0.0348119
$465$	0	0
$466$	−11.1495	−0.516489
$467$	−11.0292	−0.510370	−0.255185	−	0.966892i	$-0.582136\pi$
$467$	−11.0292	−0.510370	−0.255185	+	0.966892i	$0.582136\pi$
$468$	0	0
$469$	−3.99877	−0.184646
$470$	−3.58005	−0.165136
$471$	0	0
$472$	13.5403	0.623242
$473$	−8.42628	−0.387441
$474$	0	0
$475$	−15.0302	−0.689632
$476$	0.0695926	0.00318977
$477$	0	0
$478$	13.0909	0.598763
$479$	11.9228	0.544768	0.272384	−	0.962189i	$-0.412188\pi$
$479$	11.9228	0.544768	0.272384	+	0.962189i	$0.412188\pi$
$480$	0	0
$481$	30.2806	1.38068
$482$	−3.90075	−0.177674
$483$	0	0
$484$	−40.8109	−1.85504
$485$	6.61294	0.300278
$486$	0	0
$487$	38.2435	1.73298	0.866489	−	0.499196i	$-0.166371\pi$
$487$	38.2435	1.73298	0.866489	+	0.499196i	$0.166371\pi$
$488$	4.11713	0.186374
$489$	0	0
$490$	−3.10575	−0.140303
$491$	−4.23956	−0.191329	−0.0956644	−	0.995414i	$-0.530498\pi$
$491$	−4.23956	−0.191329	−0.0956644	+	0.995414i	$0.530498\pi$
$492$	0	0
$493$	−0.0158400	−0.000713397 0
$494$	12.9815	0.584064
$495$	0	0
$496$	−11.9483	−0.536494
$497$	3.17373	0.142361
$498$	0	0
$499$	37.4040	1.67443	0.837216	−	0.546873i	$-0.184182\pi$
$499$	37.4040	1.67443	0.837216	+	0.546873i	$0.184182\pi$
$500$	−13.2513	−0.592618
$501$	0	0
$502$	−12.7858	−0.570659
$503$	−0.342107	−0.0152538	−0.00762689	−	0.999971i	$-0.502428\pi$
$503$	−0.342107	−0.0152538	−0.00762689	+	0.999971i	$0.502428\pi$
$504$	0	0
$505$	−8.19692	−0.364758
$506$	−0.432986	−0.0192486
$507$	0	0
$508$	−16.3488	−0.725362
$509$	6.25390	0.277199	0.138600	−	0.990348i	$-0.455740\pi$
$509$	6.25390	0.277199	0.138600	+	0.990348i	$0.455740\pi$
$510$	0	0
$511$	12.1689	0.538320
$512$	−15.4692	−0.683650
$513$	0	0
$514$	−12.2612	−0.540817
$515$	4.83631	0.213113
$516$	0	0
$517$	34.3356	1.51008
$518$	−5.70399	−0.250619
$519$	0	0
$520$	11.8606	0.520121
$521$	31.4377	1.37731	0.688655	−	0.725089i	$-0.258201\pi$
$521$	31.4377	1.37731	0.688655	+	0.725089i	$0.258201\pi$
$522$	0	0
$523$	21.4197	0.936619	0.468309	−	0.883565i	$-0.344863\pi$
$523$	21.4197	0.936619	0.468309	+	0.883565i	$0.344863\pi$
$524$	21.9970	0.960944
$525$	0	0
$526$	5.86588	0.255765
$527$	−0.252391	−0.0109943
$528$	0	0
$529$	−22.9891	−0.999525
$530$	−6.24284	−0.271171
$531$	0	0
$532$	8.23670	0.357106
$533$	23.0120	0.996760
$534$	0	0
$535$	9.38726	0.405847
$536$	6.56153	0.283415
$537$	0	0
$538$	7.56895	0.326320
$539$	29.7866	1.28300
$540$	0	0
$541$	−0.376066	−0.0161683	−0.00808416	−	0.999967i	$-0.502573\pi$
$541$	−0.376066	−0.0161683	−0.00808416	+	0.999967i	$0.502573\pi$
$542$	6.88419	0.295701
$543$	0	0
$544$	−0.178671	−0.00766044
$545$	12.0133	0.514594
$546$	0	0
$547$	−6.67720	−0.285496	−0.142748	−	0.989759i	$-0.545594\pi$
$547$	−6.67720	−0.285496	−0.142748	+	0.989759i	$0.545594\pi$
$548$	8.22924	0.351536
$549$	0	0
$550$	−17.0234	−0.725882
$551$	−1.87476	−0.0798673
$552$	0	0
$553$	18.4229	0.783421
$554$	0.881010	0.0374305
$555$	0	0
$556$	6.31384	0.267766
$557$	−3.00742	−0.127429	−0.0637143	−	0.997968i	$-0.520295\pi$
$557$	−3.00742	−0.127429	−0.0637143	+	0.997968i	$0.520295\pi$
$558$	0	0
$559$	−7.22305	−0.305503
$560$	2.01492	0.0851458
$561$	0	0
$562$	−4.33713	−0.182951
$563$	32.8109	1.38282	0.691408	−	0.722465i	$-0.256991\pi$
$563$	32.8109	1.38282	0.691408	+	0.722465i	$0.256991\pi$
$564$	0	0
$565$	5.42032	0.228034
$566$	12.9354	0.543715
$567$	0	0
$568$	−5.20773	−0.218511
$569$	−26.1470	−1.09614	−0.548070	−	0.836433i	$-0.684637\pi$
$569$	−26.1470	−1.09614	−0.548070	+	0.836433i	$0.684637\pi$
$570$	0	0
$571$	−0.800753	−0.0335105	−0.0167552	−	0.999860i	$-0.505334\pi$
$571$	−0.800753	−0.0335105	−0.0167552	+	0.999860i	$0.505334\pi$
$572$	−49.5248	−2.07073
$573$	0	0
$574$	−4.33479	−0.180931
$575$	0.429733	0.0179211
$576$	0	0
$577$	−42.2910	−1.76060	−0.880298	−	0.474422i	$-0.842657\pi$
$577$	−42.2910	−1.76060	−0.880298	+	0.474422i	$0.842657\pi$
$578$	11.5022	0.478429
$579$	0	0
$580$	−0.745742	−0.0309652
$581$	26.0352	1.08012
$582$	0	0
$583$	59.8739	2.47972
$584$	−19.9678	−0.826274
$585$	0	0
$586$	−1.23558	−0.0510415
$587$	−13.3448	−0.550800	−0.275400	−	0.961330i	$-0.588810\pi$
$587$	−13.3448	−0.550800	−0.275400	+	0.961330i	$0.588810\pi$
$588$	0	0
$589$	−29.8720	−1.23085
$590$	3.60538	0.148431
$591$	0	0
$592$	−8.44103	−0.346924
$593$	−34.4089	−1.41301	−0.706503	−	0.707710i	$-0.749728\pi$
$593$	−34.4089	−1.41301	−0.706503	+	0.707710i	$0.749728\pi$
$594$	0	0
$595$	0.0425623	0.00174489
$596$	−29.1106	−1.19242
$597$	0	0
$598$	−0.371158	−0.0151778
$599$	40.0605	1.63683	0.818413	−	0.574630i	$-0.194854\pi$
$599$	40.0605	1.63683	0.818413	+	0.574630i	$0.194854\pi$
$600$	0	0
$601$	−43.1861	−1.76160	−0.880798	−	0.473492i	$-0.842993\pi$
$601$	−43.1861	−1.76160	−0.880798	+	0.473492i	$0.842993\pi$
$602$	1.36061	0.0554544
$603$	0	0
$604$	4.75052	0.193296
$605$	−24.9597	−1.01475
$606$	0	0
$607$	17.6200	0.715174	0.357587	−	0.933880i	$-0.383600\pi$
$607$	17.6200	0.715174	0.357587	+	0.933880i	$0.383600\pi$
$608$	−21.1467	−0.857613
$609$	0	0
$610$	1.09627	0.0443867
$611$	29.4327	1.19072
$612$	0	0
$613$	−27.6357	−1.11620	−0.558098	−	0.829775i	$-0.688469\pi$
$613$	−27.6357	−1.11620	−0.558098	+	0.829775i	$0.688469\pi$
$614$	10.1393	0.409187
$615$	0	0
$616$	21.4277	0.863345
$617$	−4.40957	−0.177523	−0.0887613	−	0.996053i	$-0.528291\pi$
$617$	−4.40957	−0.177523	−0.0887613	+	0.996053i	$0.528291\pi$
$618$	0	0
$619$	−25.0160	−1.00548	−0.502739	−	0.864438i	$-0.667674\pi$
$619$	−25.0160	−1.00548	−0.502739	+	0.864438i	$0.667674\pi$
$620$	−11.8825	−0.477213
$621$	0	0
$622$	13.9792	0.560514
$623$	−20.3741	−0.816272
$624$	0	0
$625$	12.4477	0.497907
$626$	−15.1049	−0.603713
$627$	0	0
$628$	5.28746	0.210993
$629$	−0.178305	−0.00710949
$630$	0	0
$631$	14.1166	0.561974	0.280987	−	0.959712i	$-0.409338\pi$
$631$	14.1166	0.561974	0.280987	+	0.959712i	$0.409338\pi$
$632$	−30.2299	−1.20248
$633$	0	0
$634$	7.68670	0.305278
$635$	−9.99883	−0.396791
$636$	0	0
$637$	25.5332	1.01166
$638$	−2.12338	−0.0840654
$639$	0	0
$640$	−10.2786	−0.406296
$641$	−34.2091	−1.35118	−0.675588	−	0.737279i	$-0.736110\pi$
$641$	−34.2091	−1.35118	−0.675588	+	0.737279i	$0.736110\pi$
$642$	0	0
$643$	−13.4865	−0.531856	−0.265928	−	0.963993i	$-0.585678\pi$
$643$	−13.4865	−0.531856	−0.265928	+	0.963993i	$0.585678\pi$
$644$	−0.235498	−0.00927994
$645$	0	0
$646$	−0.0764403	−0.00300750
$647$	−3.19249	−0.125510	−0.0627548	−	0.998029i	$-0.519989\pi$
$647$	−3.19249	−0.125510	−0.0627548	+	0.998029i	$0.519989\pi$
$648$	0	0
$649$	−34.5785	−1.35733
$650$	−14.5926	−0.572368
$651$	0	0
$652$	30.4038	1.19070
$653$	32.1675	1.25881	0.629405	−	0.777077i	$-0.283299\pi$
$653$	32.1675	1.25881	0.629405	+	0.777077i	$0.283299\pi$
$654$	0	0
$655$	13.4532	0.525661
$656$	−6.41482	−0.250457
$657$	0	0
$658$	−5.54426	−0.216138
$659$	−14.5942	−0.568510	−0.284255	−	0.958749i	$-0.591746\pi$
$659$	−14.5942	−0.568510	−0.284255	+	0.958749i	$0.591746\pi$
$660$	0	0
$661$	−7.79173	−0.303063	−0.151532	−	0.988452i	$-0.548421\pi$
$661$	−7.79173	−0.303063	−0.151532	+	0.988452i	$0.548421\pi$
$662$	−9.74849	−0.378886
$663$	0	0
$664$	−42.7209	−1.65789
$665$	5.03751	0.195346
$666$	0	0
$667$	0.0536018	0.00207547
$668$	−29.8293	−1.15413
$669$	0	0
$670$	1.74714	0.0674981
$671$	−10.5141	−0.405894
$672$	0	0
$673$	8.01766	0.309058	0.154529	−	0.987988i	$-0.450614\pi$
$673$	8.01766	0.309058	0.154529	+	0.987988i	$0.450614\pi$
$674$	6.70387	0.258223
$675$	0	0
$676$	−22.4048	−0.861723
$677$	−8.51903	−0.327413	−0.163707	−	0.986509i	$-0.552345\pi$
$677$	−8.51903	−0.327413	−0.163707	+	0.986509i	$0.552345\pi$
$678$	0	0
$679$	10.2411	0.393019
$680$	−0.0698401	−0.00267824
$681$	0	0
$682$	−33.8335	−1.29555
$683$	−14.4153	−0.551588	−0.275794	−	0.961217i	$-0.588941\pi$
$683$	−14.4153	−0.551588	−0.275794	+	0.961217i	$0.588941\pi$
$684$	0	0
$685$	5.03295	0.192299
$686$	−11.7280	−0.447779
$687$	0	0
$688$	2.01350	0.0767639
$689$	51.3242	1.95530
$690$	0	0
$691$	36.4694	1.38736	0.693681	−	0.720283i	$-0.255988\pi$
$691$	36.4694	1.38736	0.693681	+	0.720283i	$0.255988\pi$
$692$	−9.51101	−0.361554
$693$	0	0
$694$	−2.28584	−0.0867695
$695$	3.86150	0.146475
$696$	0	0
$697$	−0.135504	−0.00513259
$698$	0.899885	0.0340611
$699$	0	0
$700$	−9.25895	−0.349955
$701$	−9.52020	−0.359573	−0.179787	−	0.983706i	$-0.557541\pi$
$701$	−9.52020	−0.359573	−0.179787	+	0.983706i	$0.557541\pi$
$702$	0	0
$703$	−21.1035	−0.795933
$704$	−6.04699	−0.227905
$705$	0	0
$706$	7.97999	0.300331
$707$	−12.6942	−0.477414
$708$	0	0
$709$	11.9597	0.449157	0.224579	−	0.974456i	$-0.427899\pi$
$709$	11.9597	0.449157	0.224579	+	0.974456i	$0.427899\pi$
$710$	−1.38667	−0.0520407
$711$	0	0
$712$	33.4317	1.25291
$713$	0.854081	0.0319856
$714$	0	0
$715$	−30.2890	−1.13274
$716$	−5.33860	−0.199513
$717$	0	0
$718$	−20.1038	−0.750268
$719$	21.3781	0.797269	0.398634	−	0.917110i	$-0.369484\pi$
$719$	21.3781	0.797269	0.398634	+	0.917110i	$0.369484\pi$
$720$	0	0
$721$	7.48976	0.278933
$722$	3.80896	0.141755
$723$	0	0
$724$	29.4249	1.09357
$725$	2.10743	0.0782680
$726$	0	0
$727$	−7.88115	−0.292296	−0.146148	−	0.989263i	$-0.546688\pi$
$727$	−7.88115	−0.292296	−0.146148	+	0.989263i	$0.546688\pi$
$728$	18.3679	0.680760
$729$	0	0
$730$	−5.31685	−0.196785
$731$	0.0425324	0.00157312
$732$	0	0
$733$	34.8233	1.28623	0.643114	−	0.765771i	$-0.277642\pi$
$733$	34.8233	1.28623	0.643114	+	0.765771i	$0.277642\pi$
$734$	5.43138	0.200476
$735$	0	0
$736$	0.604614	0.0222864
$737$	−16.7565	−0.617235
$738$	0	0
$739$	43.8485	1.61299	0.806497	−	0.591238i	$-0.201361\pi$
$739$	43.8485	1.61299	0.806497	+	0.591238i	$0.201361\pi$
$740$	−8.39455	−0.308590
$741$	0	0
$742$	−9.66798	−0.354923
$743$	−21.3524	−0.783344	−0.391672	−	0.920105i	$-0.628103\pi$
$743$	−21.3524	−0.783344	−0.391672	+	0.920105i	$0.628103\pi$
$744$	0	0
$745$	−17.8039	−0.652283
$746$	2.02574	0.0741675
$747$	0	0
$748$	0.291623	0.0106628
$749$	14.5376	0.531192
$750$	0	0
$751$	4.52817	0.165235	0.0826177	−	0.996581i	$-0.473672\pi$
$751$	4.52817	0.165235	0.0826177	+	0.996581i	$0.473672\pi$
$752$	−8.20466	−0.299193
$753$	0	0
$754$	−1.82017	−0.0662868
$755$	2.90539	0.105738
$756$	0	0
$757$	30.7938	1.11922	0.559610	−	0.828756i	$-0.310951\pi$
$757$	30.7938	1.11922	0.559610	+	0.828756i	$0.310951\pi$
$758$	14.9588	0.543327
$759$	0	0
$760$	−8.26599	−0.299839
$761$	3.57881	0.129732	0.0648658	−	0.997894i	$-0.479338\pi$
$761$	3.57881	0.129732	0.0648658	+	0.997894i	$0.479338\pi$
$762$	0	0
$763$	18.6045	0.673526
$764$	−30.7318	−1.11184
$765$	0	0
$766$	9.71493	0.351015
$767$	−29.6409	−1.07027
$768$	0	0
$769$	−14.6050	−0.526669	−0.263335	−	0.964705i	$-0.584822\pi$
$769$	−14.6050	−0.526669	−0.263335	+	0.964705i	$0.584822\pi$
$770$	5.70556	0.205614
$771$	0	0
$772$	16.5279	0.594853
$773$	−20.0048	−0.719521	−0.359760	−	0.933045i	$-0.617142\pi$
$773$	−20.0048	−0.719521	−0.359760	+	0.933045i	$0.617142\pi$
$774$	0	0
$775$	33.5794	1.20621
$776$	−16.8046	−0.603249
$777$	0	0
$778$	−12.3351	−0.442236
$779$	−16.0377	−0.574612
$780$	0	0
$781$	13.2993	0.475885
$782$	0.00218553	7.81545e−5 0
$783$	0	0
$784$	−7.11765	−0.254202
$785$	3.23378	0.115418
$786$	0	0
$787$	−13.5451	−0.482829	−0.241415	−	0.970422i	$-0.577611\pi$
$787$	−13.5451	−0.482829	−0.241415	+	0.970422i	$0.577611\pi$
$788$	−36.5521	−1.30211
$789$	0	0
$790$	−8.04935	−0.286383
$791$	8.39418	0.298463
$792$	0	0
$793$	−9.01277	−0.320053
$794$	−0.938497	−0.0333060
$795$	0	0
$796$	−13.4710	−0.477466
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321694\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321694\pi$
$798$	0	0
$799$	−0.173312	−0.00613134
$800$	23.7712	0.840440
$801$	0	0
$802$	14.9444	0.527706
$803$	50.9929	1.79950
$804$	0	0
$805$	−0.144029	−0.00507636
$806$	−29.0023	−1.02156
$807$	0	0
$808$	20.8297	0.732788
$809$	−48.8577	−1.71774	−0.858872	−	0.512190i	$-0.828834\pi$
$809$	−48.8577	−1.71774	−0.858872	+	0.512190i	$0.828834\pi$
$810$	0	0
$811$	−14.0558	−0.493566	−0.246783	−	0.969071i	$-0.579373\pi$
$811$	−14.0558	−0.493566	−0.246783	+	0.969071i	$0.579373\pi$
$812$	−1.15489	−0.0405288
$813$	0	0
$814$	−23.9021	−0.837770
$815$	18.5947	0.651345
$816$	0	0
$817$	5.03396	0.176116
$818$	−0.390679	−0.0136598
$819$	0	0
$820$	−6.37950	−0.222782
$821$	−32.2456	−1.12538	−0.562689	−	0.826668i	$-0.690233\pi$
$821$	−32.2456	−1.12538	−0.562689	+	0.826668i	$0.690233\pi$
$822$	0	0
$823$	−33.0814	−1.15315	−0.576573	−	0.817046i	$-0.695610\pi$
$823$	−33.0814	−1.15315	−0.576573	+	0.817046i	$0.695610\pi$
$824$	−12.2899	−0.428138
$825$	0	0
$826$	5.58348	0.194274
$827$	−39.9115	−1.38786	−0.693929	−	0.720043i	$-0.744122\pi$
$827$	−39.9115	−1.38786	−0.693929	+	0.720043i	$0.744122\pi$
$828$	0	0
$829$	−17.4600	−0.606410	−0.303205	−	0.952925i	$-0.598057\pi$
$829$	−17.4600	−0.606410	−0.303205	+	0.952925i	$0.598057\pi$
$830$	−11.3753	−0.394844
$831$	0	0
$832$	−5.18352	−0.179706
$833$	−0.150351	−0.00520934
$834$	0	0
$835$	−18.2434	−0.631337
$836$	34.5153	1.19374
$837$	0	0
$838$	−8.54698	−0.295251
$839$	39.4855	1.36319	0.681595	−	0.731730i	$-0.261287\pi$
$839$	39.4855	1.36319	0.681595	+	0.731730i	$0.261287\pi$
$840$	0	0
$841$	−28.7371	−0.990936
$842$	−14.4977	−0.499623
$843$	0	0
$844$	11.2239	0.386344
$845$	−13.7026	−0.471384
$846$	0	0
$847$	−38.6538	−1.32816
$848$	−14.3071	−0.491309
$849$	0	0
$850$	0.0859272	0.00294728
$851$	0.603377	0.0206835
$852$	0	0
$853$	−31.3190	−1.07234	−0.536171	−	0.844109i	$-0.680130\pi$
$853$	−31.3190	−1.07234	−0.536171	+	0.844109i	$0.680130\pi$
$854$	1.69774	0.0580956
$855$	0	0
$856$	−23.8546	−0.815333
$857$	−35.3472	−1.20744	−0.603719	−	0.797197i	$-0.706315\pi$
$857$	−35.3472	−1.20744	−0.603719	+	0.797197i	$0.706315\pi$
$858$	0	0
$859$	28.2526	0.963968	0.481984	−	0.876180i	$-0.339916\pi$
$859$	28.2526	0.963968	0.481984	+	0.876180i	$0.339916\pi$
$860$	2.00241	0.0682817
$861$	0	0
$862$	3.02546	0.103048
$863$	−3.82178	−0.130095	−0.0650475	−	0.997882i	$-0.520720\pi$
$863$	−3.82178	−0.130095	−0.0650475	+	0.997882i	$0.520720\pi$
$864$	0	0
$865$	−5.81686	−0.197779
$866$	−6.18002	−0.210006
$867$	0	0
$868$	−18.4019	−0.624600
$869$	77.1998	2.61882
$870$	0	0
$871$	−14.3638	−0.486698
$872$	−30.5278	−1.03380
$873$	0	0
$874$	0.258671	0.00874967
$875$	−12.5509	−0.424299
$876$	0	0
$877$	−34.2898	−1.15788	−0.578942	−	0.815369i	$-0.696534\pi$
$877$	−34.2898	−1.15788	−0.578942	+	0.815369i	$0.696534\pi$
$878$	−1.92750	−0.0650499
$879$	0	0
$880$	8.44336	0.284626
$881$	−51.0611	−1.72029	−0.860147	−	0.510047i	$-0.829628\pi$
$881$	−51.0611	−1.72029	−0.860147	+	0.510047i	$0.829628\pi$
$882$	0	0
$883$	−40.0329	−1.34721	−0.673606	−	0.739090i	$-0.735256\pi$
$883$	−40.0329	−1.34721	−0.673606	+	0.739090i	$0.735256\pi$
$884$	0.249981	0.00840775
$885$	0	0
$886$	−8.37107	−0.281232
$887$	−22.8986	−0.768861	−0.384430	−	0.923154i	$-0.625602\pi$
$887$	−22.8986	−0.768861	−0.384430	+	0.923154i	$0.625602\pi$
$888$	0	0
$889$	−15.4847	−0.519340
$890$	8.90188	0.298392
$891$	0	0
$892$	−1.15427	−0.0386477
$893$	−20.5125	−0.686425
$894$	0	0
$895$	−3.26505	−0.109139
$896$	−15.9179	−0.531780
$897$	0	0
$898$	11.6210	0.387798
$899$	4.18845	0.139693
$900$	0	0
$901$	−0.302218	−0.0100684
$902$	−18.1646	−0.604815
$903$	0	0
$904$	−13.7739	−0.458114
$905$	17.9961	0.598210
$906$	0	0
$907$	−23.9431	−0.795018	−0.397509	−	0.917598i	$-0.630125\pi$
$907$	−23.9431	−0.795018	−0.397509	+	0.917598i	$0.630125\pi$
$908$	11.7296	0.389261
$909$	0	0
$910$	4.89084	0.162130
$911$	30.7833	1.01990	0.509948	−	0.860206i	$-0.329665\pi$
$911$	30.7833	1.01990	0.509948	+	0.860206i	$0.329665\pi$
$912$	0	0
$913$	109.099	3.61064
$914$	−0.598665	−0.0198021
$915$	0	0
$916$	24.0605	0.794982
$917$	20.8344	0.688011
$918$	0	0
$919$	−43.9789	−1.45073	−0.725365	−	0.688365i	$-0.758329\pi$
$919$	−43.9789	−1.45073	−0.725365	+	0.688365i	$0.758329\pi$
$920$	0.236336	0.00779176
$921$	0	0
$922$	−10.3064	−0.339424
$923$	11.4002	0.375242
$924$	0	0
$925$	23.7226	0.779995
$926$	−22.7601	−0.747942
$927$	0	0
$928$	2.96505	0.0973326
$929$	28.3360	0.929672	0.464836	−	0.885397i	$-0.346113\pi$
$929$	28.3360	0.929672	0.464836	+	0.885397i	$0.346113\pi$
$930$	0	0
$931$	−17.7949	−0.583204
$932$	−25.4113	−0.832376
$933$	0	0
$934$	7.46278	0.244190
$935$	0.178354	0.00583281
$936$	0	0
$937$	−13.2836	−0.433957	−0.216979	−	0.976176i	$-0.569620\pi$
$937$	−13.2836	−0.433957	−0.216979	+	0.976176i	$0.569620\pi$
$938$	2.70572	0.0883449
$939$	0	0
$940$	−8.15948	−0.266133
$941$	14.1023	0.459722	0.229861	−	0.973224i	$-0.426173\pi$
$941$	14.1023	0.459722	0.229861	+	0.973224i	$0.426173\pi$
$942$	0	0
$943$	0.458541	0.0149321
$944$	8.26270	0.268928
$945$	0	0
$946$	5.70154	0.185373
$947$	−24.2008	−0.786419	−0.393209	−	0.919449i	$-0.628635\pi$
$947$	−24.2008	−0.786419	−0.393209	+	0.919449i	$0.628635\pi$
$948$	0	0
$949$	43.7114	1.41893
$950$	10.1700	0.329959
$951$	0	0
$952$	−0.108158	−0.00350542
$953$	1.81703	0.0588592	0.0294296	−	0.999567i	$-0.490631\pi$
$953$	1.81703	0.0588592	0.0294296	+	0.999567i	$0.490631\pi$
$954$	0	0
$955$	−18.7954	−0.608204
$956$	29.8361	0.964967
$957$	0	0
$958$	−8.06744	−0.260647
$959$	7.79428	0.251691
$960$	0	0
$961$	35.7379	1.15284
$962$	−20.4890	−0.660593
$963$	0	0
$964$	−8.89040	−0.286340
$965$	10.1084	0.325400
$966$	0	0
$967$	−1.89675	−0.0609954	−0.0304977	−	0.999535i	$-0.509709\pi$
$967$	−1.89675	−0.0609954	−0.0304977	+	0.999535i	$0.509709\pi$
$968$	63.4267	2.03861
$969$	0	0
$970$	−4.47457	−0.143670
$971$	27.1629	0.871698	0.435849	−	0.900020i	$-0.356448\pi$
$971$	27.1629	0.871698	0.435849	+	0.900020i	$0.356448\pi$
$972$	0	0
$973$	5.98012	0.191714
$974$	−25.8770	−0.829153
$975$	0	0
$976$	2.51240	0.0804200
$977$	25.6090	0.819304	0.409652	−	0.912242i	$-0.365650\pi$
$977$	25.6090	0.819304	0.409652	+	0.912242i	$0.365650\pi$
$978$	0	0
$979$	−85.3763	−2.72864
$980$	−7.07846	−0.226113
$981$	0	0
$982$	2.86865	0.0915424
$983$	29.0429	0.926324	0.463162	−	0.886274i	$-0.346715\pi$
$983$	29.0429	0.926324	0.463162	+	0.886274i	$0.346715\pi$
$984$	0	0
$985$	−22.3550	−0.712289
$986$	0.0107179	0.000341329 0
$987$	0	0
$988$	29.5867	0.941278
$989$	−0.143928	−0.00457664
$990$	0	0
$991$	36.2171	1.15047	0.575237	−	0.817987i	$-0.304910\pi$
$991$	36.2171	1.15047	0.575237	+	0.817987i	$0.304910\pi$
$992$	47.2445	1.50002
$993$	0	0
$994$	−2.14746	−0.0681134
$995$	−8.23874	−0.261186
$996$	0	0
$997$	−15.1740	−0.480565	−0.240283	−	0.970703i	$-0.577240\pi$
$997$	−15.1740	−0.480565	−0.240283	+	0.970703i	$0.577240\pi$
$998$	−25.3090	−0.801141
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.27		72
3.2	odd	2		6561.2.a.c.1.46		72
81.4	even	27		243.2.g.a.208.3		144
81.7	even	27		729.2.g.b.352.6		144
81.20	odd	54		81.2.g.a.76.6	yes	144
81.23	odd	54		729.2.g.c.379.3		144
81.31	even	27		729.2.g.a.622.6		144
81.34	even	27		729.2.g.a.109.6		144
81.47	odd	54		729.2.g.d.109.3		144
81.50	odd	54		729.2.g.d.622.3		144
81.58	even	27		729.2.g.b.379.6		144
81.61	even	27		243.2.g.a.118.3		144
81.74	odd	54		729.2.g.c.352.3		144
81.77	odd	54		81.2.g.a.16.6	✓	144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.16.6	✓	144	81.77	odd	54
81.2.g.a.76.6	yes	144	81.20	odd	54
243.2.g.a.118.3		144	81.61	even	27
243.2.g.a.208.3		144	81.4	even	27
729.2.g.a.109.6		144	81.34	even	27
729.2.g.a.622.6		144	81.31	even	27
729.2.g.b.352.6		144	81.7	even	27
729.2.g.b.379.6		144	81.58	even	27
729.2.g.c.352.3		144	81.74	odd	54
729.2.g.c.379.3		144	81.23	odd	54
729.2.g.d.109.3		144	81.47	odd	54
729.2.g.d.622.3		144	81.50	odd	54
6561.2.a.c.1.46		72	3.2	odd	2
6561.2.a.d.1.27		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-0.676638 q^{2} -1.54216 q^{4} -0.943174 q^{5} -1.46065 q^{7} +2.39676 q^{8} +0.638188 q^{10} -6.12074 q^{11} -5.24673 q^{13} +0.988331 q^{14} +1.46258 q^{16} +0.0308950 q^{17} +3.65660 q^{19} +1.45453 q^{20} +4.14153 q^{22} -0.104547 q^{23} -4.11042 q^{25} +3.55014 q^{26} +2.25255 q^{28} -0.512704 q^{29} -8.16933 q^{31} -5.78316 q^{32} -0.0209047 q^{34} +1.37765 q^{35} -5.77133 q^{37} -2.47420 q^{38} -2.26056 q^{40} -4.38597 q^{41} +1.37668 q^{43} +9.43916 q^{44} +0.0707407 q^{46} -5.60972 q^{47} -4.86651 q^{49} +2.78127 q^{50} +8.09130 q^{52} -9.78213 q^{53} +5.77293 q^{55} -3.50083 q^{56} +0.346915 q^{58} +5.64940 q^{59} +1.71779 q^{61} +5.52768 q^{62} +0.987951 q^{64} +4.94858 q^{65} +2.73766 q^{67} -0.0476450 q^{68} -0.932169 q^{70} -2.17282 q^{71} -8.33116 q^{73} +3.90511 q^{74} -5.63907 q^{76} +8.94025 q^{77} -12.6128 q^{79} -1.37947 q^{80} +2.96771 q^{82} -17.8244 q^{83} -0.0291393 q^{85} -0.931512 q^{86} -14.6700 q^{88} +13.9487 q^{89} +7.66363 q^{91} +0.161229 q^{92} +3.79575 q^{94} -3.44881 q^{95} -7.01137 q^{97} +3.29286 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38}+ \cdots + 81 q^{98}+O(q^{100}) \) 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8 + 36 * q^11 + 36 * q^14 + 45 * q^16 + 36 * q^17 + 54 * q^20 + 54 * q^23 + 36 * q^25 + 45 * q^26 + 9 * q^28 + 54 * q^29 + 63 * q^32 + 72 * q^35 + 54 * q^38 + 72 * q^41 + 90 * q^44 + 90 * q^47 + 18 * q^49 + 45 * q^50 + 45 * q^53 + 9 * q^55 + 108 * q^56 + 18 * q^58 + 108 * q^59 + 72 * q^62 + 9 * q^64 + 72 * q^65 + 108 * q^68 + 126 * q^71 + 90 * q^74 + 72 * q^77 + 144 * q^80 - 18 * q^82 + 108 * q^83 + 90 * q^86 + 108 * q^89 + 72 * q^92 + 144 * q^95 + 81 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more