LMFDB - Embedded newform 6561.2.a.c.1.20

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:	$1$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q-1.50881 q^{2} +0.276511 q^{4} -2.79694 q^{5} -3.90283 q^{7} +2.60042 q^{8} +O(q^{10})$	\(q-1.50881 q^{2} +0.276511 q^{4} -2.79694 q^{5} -3.90283 q^{7} +2.60042 q^{8} +4.22006 q^{10} -2.32588 q^{11} -5.69419 q^{13} +5.88864 q^{14} -4.47656 q^{16} -1.39672 q^{17} +2.90140 q^{19} -0.773384 q^{20} +3.50931 q^{22} +3.96104 q^{23} +2.82289 q^{25} +8.59146 q^{26} -1.07917 q^{28} -6.30329 q^{29} +5.64083 q^{31} +1.55345 q^{32} +2.10739 q^{34} +10.9160 q^{35} +2.55507 q^{37} -4.37766 q^{38} -7.27322 q^{40} -3.32810 q^{41} +5.01165 q^{43} -0.643130 q^{44} -5.97646 q^{46} -6.53067 q^{47} +8.23211 q^{49} -4.25920 q^{50} -1.57450 q^{52} +3.32910 q^{53} +6.50535 q^{55} -10.1490 q^{56} +9.51047 q^{58} +2.41024 q^{59} +6.76622 q^{61} -8.51095 q^{62} +6.60927 q^{64} +15.9263 q^{65} +11.3002 q^{67} -0.386209 q^{68} -16.4702 q^{70} -0.473648 q^{71} -1.19236 q^{73} -3.85512 q^{74} +0.802268 q^{76} +9.07752 q^{77} +6.24106 q^{79} +12.5207 q^{80} +5.02147 q^{82} +6.71797 q^{83} +3.90655 q^{85} -7.56163 q^{86} -6.04826 q^{88} -9.71608 q^{89} +22.2235 q^{91} +1.09527 q^{92} +9.85355 q^{94} -8.11505 q^{95} +5.27429 q^{97} -12.4207 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.50881	−1.06689	−0.533445	−	0.845835i	$-0.679103\pi$
$2$	−1.50881	−1.06689	−0.533445	+	0.845835i	$0.679103\pi$
$3$	0	0
$3$	0	0
$4$	0.276511	0.138255
$5$	−2.79694	−1.25083	−0.625415	−	0.780292i	$-0.715070\pi$
$5$	−2.79694	−1.25083	−0.625415	+	0.780292i	$0.715070\pi$
$6$	0	0
$7$	−3.90283	−1.47513	−0.737566	−	0.675275i	$-0.764025\pi$
$7$	−3.90283	−1.47513	−0.737566	+	0.675275i	$0.764025\pi$
$8$	2.60042	0.919387
$9$	0	0
$10$	4.22006	1.33450
$11$	−2.32588	−0.701279	−0.350639	−	0.936511i	$-0.614036\pi$
$11$	−2.32588	−0.701279	−0.350639	+	0.936511i	$0.614036\pi$
$12$	0	0
$13$	−5.69419	−1.57928	−0.789642	−	0.613567i	$-0.789734\pi$
$13$	−5.69419	−1.57928	−0.789642	+	0.613567i	$0.789734\pi$
$14$	5.88864	1.57380
$15$	0	0
$16$	−4.47656	−1.11914
$17$	−1.39672	−0.338755	−0.169378	−	0.985551i	$-0.554176\pi$
$17$	−1.39672	−0.338755	−0.169378	+	0.985551i	$0.554176\pi$
$18$	0	0
$19$	2.90140	0.665627	0.332813	−	0.942993i	$-0.392002\pi$
$19$	2.90140	0.665627	0.332813	+	0.942993i	$0.392002\pi$
$20$	−0.773384	−0.172934
$21$	0	0
$22$	3.50931	0.748188
$23$	3.96104	0.825934	0.412967	−	0.910746i	$-0.364493\pi$
$23$	3.96104	0.825934	0.412967	+	0.910746i	$0.364493\pi$
$24$	0	0
$25$	2.82289	0.564577
$26$	8.59146	1.68492
$27$	0	0
$28$	−1.07917	−0.203945
$29$	−6.30329	−1.17049	−0.585245	−	0.810856i	$-0.699002\pi$
$29$	−6.30329	−1.17049	−0.585245	+	0.810856i	$0.699002\pi$
$30$	0	0
$31$	5.64083	1.01312	0.506562	−	0.862204i	$-0.330916\pi$
$31$	5.64083	1.01312	0.506562	+	0.862204i	$0.330916\pi$
$32$	1.55345	0.274613
$33$	0	0
$34$	2.10739	0.361415
$35$	10.9160	1.84514
$36$	0	0
$37$	2.55507	0.420051	0.210026	−	0.977696i	$-0.432645\pi$
$37$	2.55507	0.420051	0.210026	+	0.977696i	$0.432645\pi$
$38$	−4.37766	−0.710151
$39$	0	0
$40$	−7.27322	−1.15000
$41$	−3.32810	−0.519762	−0.259881	−	0.965641i	$-0.583683\pi$
$41$	−3.32810	−0.519762	−0.259881	+	0.965641i	$0.583683\pi$
$42$	0	0
$43$	5.01165	0.764269	0.382134	−	0.924107i	$-0.375189\pi$
$43$	5.01165	0.764269	0.382134	+	0.924107i	$0.375189\pi$
$44$	−0.643130	−0.0969555
$45$	0	0
$46$	−5.97646	−0.881181
$47$	−6.53067	−0.952597	−0.476298	−	0.879284i	$-0.658022\pi$
$47$	−6.53067	−0.952597	−0.476298	+	0.879284i	$0.658022\pi$
$48$	0	0
$49$	8.23211	1.17602
$50$	−4.25920	−0.602342
$51$	0	0
$52$	−1.57450	−0.218345
$53$	3.32910	0.457287	0.228643	−	0.973510i	$-0.426571\pi$
$53$	3.32910	0.457287	0.228643	+	0.973510i	$0.426571\pi$
$54$	0	0
$55$	6.50535	0.877181
$56$	−10.1490	−1.35622
$57$	0	0
$58$	9.51047	1.24879
$59$	2.41024	0.313787	0.156893	−	0.987616i	$-0.449852\pi$
$59$	2.41024	0.313787	0.156893	+	0.987616i	$0.449852\pi$
$60$	0	0
$61$	6.76622	0.866326	0.433163	−	0.901316i	$-0.357398\pi$
$61$	6.76622	0.866326	0.433163	+	0.901316i	$0.357398\pi$
$62$	−8.51095	−1.08089
$63$	0	0
$64$	6.60927	0.826158
$65$	15.9263	1.97542
$66$	0	0
$67$	11.3002	1.38054	0.690269	−	0.723553i	$-0.257492\pi$
$67$	11.3002	1.38054	0.690269	+	0.723553i	$0.257492\pi$
$68$	−0.386209	−0.0468347
$69$	0	0
$70$	−16.4702	−1.96856
$71$	−0.473648	−0.0562116	−0.0281058	−	0.999605i	$-0.508948\pi$
$71$	−0.473648	−0.0562116	−0.0281058	+	0.999605i	$0.508948\pi$
$72$	0	0
$73$	−1.19236	−0.139556	−0.0697778	−	0.997563i	$-0.522229\pi$
$73$	−1.19236	−0.139556	−0.0697778	+	0.997563i	$0.522229\pi$
$74$	−3.85512	−0.448148
$75$	0	0
$76$	0.802268	0.0920264
$77$	9.07752	1.03448
$78$	0	0
$79$	6.24106	0.702174	0.351087	−	0.936343i	$-0.385812\pi$
$79$	6.24106	0.702174	0.351087	+	0.936343i	$0.385812\pi$
$80$	12.5207	1.39986
$81$	0	0
$82$	5.02147	0.554529
$83$	6.71797	0.737393	0.368696	−	0.929550i	$-0.379804\pi$
$83$	6.71797	0.737393	0.368696	+	0.929550i	$0.379804\pi$
$84$	0	0
$85$	3.90655	0.423725
$86$	−7.56163	−0.815391
$87$	0	0
$88$	−6.04826	−0.644747
$89$	−9.71608	−1.02990	−0.514951	−	0.857220i	$-0.672190\pi$
$89$	−9.71608	−1.02990	−0.514951	+	0.857220i	$0.672190\pi$
$90$	0	0
$91$	22.2235	2.32965
$92$	1.09527	0.114190
$93$	0	0
$94$	9.85355	1.01632
$95$	−8.11505	−0.832586
$96$	0	0
$97$	5.27429	0.535524	0.267762	−	0.963485i	$-0.413716\pi$
$97$	5.27429	0.535524	0.267762	+	0.963485i	$0.413716\pi$
$98$	−12.4207	−1.25468
$99$	0	0
$100$	0.780558	0.0780558
$101$	−13.2203	−1.31547	−0.657737	−	0.753248i	$-0.728486\pi$
$101$	−13.2203	−1.31547	−0.657737	+	0.753248i	$0.728486\pi$
$102$	0	0
$103$	4.19318	0.413167	0.206583	−	0.978429i	$-0.433766\pi$
$103$	4.19318	0.413167	0.206583	+	0.978429i	$0.433766\pi$
$104$	−14.8073	−1.45197
$105$	0	0
$106$	−5.02298	−0.487875
$107$	−7.58299	−0.733075	−0.366538	−	0.930403i	$-0.619457\pi$
$107$	−7.58299	−0.733075	−0.366538	+	0.930403i	$0.619457\pi$
$108$	0	0
$109$	9.29451	0.890252	0.445126	−	0.895468i	$-0.353159\pi$
$109$	9.29451	0.890252	0.445126	+	0.895468i	$0.353159\pi$
$110$	−9.81534	−0.935856
$111$	0	0
$112$	17.4713	1.65088
$113$	7.74246	0.728350	0.364175	−	0.931331i	$-0.381351\pi$
$113$	7.74246	0.728350	0.364175	+	0.931331i	$0.381351\pi$
$114$	0	0
$115$	−11.0788	−1.03310
$116$	−1.74293	−0.161827
$117$	0	0
$118$	−3.63660	−0.334776
$119$	5.45118	0.499708
$120$	0	0
$121$	−5.59029	−0.508208
$122$	−10.2089	−0.924275
$123$	0	0
$124$	1.55975	0.140070
$125$	6.08926	0.544640
$126$	0	0
$127$	−19.6555	−1.74415	−0.872073	−	0.489376i	$-0.837225\pi$
$127$	−19.6555	−1.74415	−0.872073	+	0.489376i	$0.837225\pi$
$128$	−13.0790	−1.15603
$129$	0	0
$130$	−24.0298	−2.10755
$131$	4.63786	0.405211	0.202606	−	0.979260i	$-0.435059\pi$
$131$	4.63786	0.405211	0.202606	+	0.979260i	$0.435059\pi$
$132$	0	0
$133$	−11.3237	−0.981887
$134$	−17.0498	−1.47288
$135$	0	0
$136$	−3.63207	−0.311447
$137$	18.0680	1.54366	0.771828	−	0.635831i	$-0.219343\pi$
$137$	18.0680	1.54366	0.771828	+	0.635831i	$0.219343\pi$
$138$	0	0
$139$	−7.01447	−0.594960	−0.297480	−	0.954728i	$-0.596146\pi$
$139$	−7.01447	−0.594960	−0.297480	+	0.954728i	$0.596146\pi$
$140$	3.01839	0.255100
$141$	0	0
$142$	0.714645	0.0599717
$143$	13.2440	1.10752
$144$	0	0
$145$	17.6299	1.46409
$146$	1.79905	0.148891
$147$	0	0
$148$	0.706504	0.0580743
$149$	−16.0627	−1.31590	−0.657952	−	0.753060i	$-0.728577\pi$
$149$	−16.0627	−1.31590	−0.657952	+	0.753060i	$0.728577\pi$
$150$	0	0
$151$	22.4119	1.82386	0.911928	−	0.410350i	$-0.134593\pi$
$151$	22.4119	1.82386	0.911928	+	0.410350i	$0.134593\pi$
$152$	7.54486	0.611969
$153$	0	0
$154$	−13.6963	−1.10368
$155$	−15.7771	−1.26725
$156$	0	0
$157$	1.99486	0.159207	0.0796035	−	0.996827i	$-0.474635\pi$
$157$	1.99486	0.159207	0.0796035	+	0.996827i	$0.474635\pi$
$158$	−9.41658	−0.749143
$159$	0	0
$160$	−4.34490	−0.343495
$161$	−15.4593	−1.21836
$162$	0	0
$163$	24.2605	1.90023	0.950115	−	0.311900i	$-0.100966\pi$
$163$	24.2605	1.90023	0.950115	+	0.311900i	$0.100966\pi$
$164$	−0.920255	−0.0718598
$165$	0	0
$166$	−10.1361	−0.786717
$167$	−6.34171	−0.490736	−0.245368	−	0.969430i	$-0.578909\pi$
$167$	−6.34171	−0.490736	−0.245368	+	0.969430i	$0.578909\pi$
$168$	0	0
$169$	19.4238	1.49414
$170$	−5.89425	−0.452068
$171$	0	0
$172$	1.38577	0.105664
$173$	13.8662	1.05423	0.527113	−	0.849795i	$-0.323275\pi$
$173$	13.8662	1.05423	0.527113	+	0.849795i	$0.323275\pi$
$174$	0	0
$175$	−11.0173	−0.832826
$176$	10.4119	0.784830
$177$	0	0
$178$	14.6597	1.09879
$179$	13.4270	1.00358	0.501791	−	0.864989i	$-0.332675\pi$
$179$	13.4270	1.00358	0.501791	+	0.864989i	$0.332675\pi$
$180$	0	0
$181$	−11.1989	−0.832405	−0.416202	−	0.909272i	$-0.636639\pi$
$181$	−11.1989	−0.832405	−0.416202	+	0.909272i	$0.636639\pi$
$182$	−33.5310	−2.48549
$183$	0	0
$184$	10.3004	0.759353
$185$	−7.14639	−0.525413
$186$	0	0
$187$	3.24861	0.237562
$188$	−1.80580	−0.131702
$189$	0	0
$190$	12.2441	0.888278
$191$	−8.18590	−0.592311	−0.296155	−	0.955140i	$-0.595705\pi$
$191$	−8.18590	−0.592311	−0.296155	+	0.955140i	$0.595705\pi$
$192$	0	0
$193$	−22.4213	−1.61392	−0.806959	−	0.590607i	$-0.798888\pi$
$193$	−22.4213	−1.61392	−0.806959	+	0.590607i	$0.798888\pi$
$194$	−7.95791	−0.571345
$195$	0	0
$196$	2.27626	0.162590
$197$	−13.5098	−0.962534	−0.481267	−	0.876574i	$-0.659823\pi$
$197$	−13.5098	−0.962534	−0.481267	+	0.876574i	$0.659823\pi$
$198$	0	0
$199$	18.5672	1.31619	0.658097	−	0.752933i	$-0.271362\pi$
$199$	18.5672	1.31619	0.658097	+	0.752933i	$0.271362\pi$
$200$	7.34069	0.519065
$201$	0	0
$202$	19.9470	1.40347
$203$	24.6007	1.72663
$204$	0	0
$205$	9.30850	0.650134
$206$	−6.32672	−0.440804
$207$	0	0
$208$	25.4904	1.76744
$209$	−6.74830	−0.466790
$210$	0	0
$211$	2.30964	0.159002	0.0795010	−	0.996835i	$-0.474667\pi$
$211$	2.30964	0.159002	0.0795010	+	0.996835i	$0.474667\pi$
$212$	0.920531	0.0632223
$213$	0	0
$214$	11.4413	0.782111
$215$	−14.0173	−0.955971
$216$	0	0
$217$	−22.0152	−1.49449
$218$	−14.0237	−0.949802
$219$	0	0
$220$	1.79880	0.121275
$221$	7.95321	0.534991
$222$	0	0
$223$	11.4866	0.769197	0.384598	−	0.923084i	$-0.374340\pi$
$223$	11.4866	0.769197	0.384598	+	0.923084i	$0.374340\pi$
$224$	−6.06285	−0.405091
$225$	0	0
$226$	−11.6819	−0.777069
$227$	−17.8254	−1.18312	−0.591558	−	0.806262i	$-0.701487\pi$
$227$	−17.8254	−1.18312	−0.591558	+	0.806262i	$0.701487\pi$
$228$	0	0
$229$	−29.3977	−1.94265	−0.971326	−	0.237750i	$-0.923590\pi$
$229$	−29.3977	−1.94265	−0.971326	+	0.237750i	$0.923590\pi$
$230$	16.7158	1.10221
$231$	0	0
$232$	−16.3912	−1.07613
$233$	10.3759	0.679750	0.339875	−	0.940471i	$-0.389615\pi$
$233$	10.3759	0.679750	0.339875	+	0.940471i	$0.389615\pi$
$234$	0	0
$235$	18.2659	1.19154
$236$	0.666458	0.0433827
$237$	0	0
$238$	−8.22479	−0.533134
$239$	21.7218	1.40506	0.702532	−	0.711652i	$-0.252053\pi$
$239$	21.7218	1.40506	0.702532	+	0.711652i	$0.252053\pi$
$240$	0	0
$241$	1.04089	0.0670497	0.0335248	−	0.999438i	$-0.489327\pi$
$241$	1.04089	0.0670497	0.0335248	+	0.999438i	$0.489327\pi$
$242$	8.43469	0.542202
$243$	0	0
$244$	1.87093	0.119774
$245$	−23.0247	−1.47100
$246$	0	0
$247$	−16.5211	−1.05121
$248$	14.6685	0.931453
$249$	0	0
$250$	−9.18754	−0.581071
$251$	13.5724	0.856685	0.428342	−	0.903616i	$-0.359098\pi$
$251$	13.5724	0.856685	0.428342	+	0.903616i	$0.359098\pi$
$252$	0	0
$253$	−9.21290	−0.579210
$254$	29.6565	1.86081
$255$	0	0
$256$	6.51525	0.407203
$257$	14.1483	0.882545	0.441272	−	0.897373i	$-0.354527\pi$
$257$	14.1483	0.882545	0.441272	+	0.897373i	$0.354527\pi$
$258$	0	0
$259$	−9.97202	−0.619631
$260$	4.40380	0.273112
$261$	0	0
$262$	−6.99765	−0.432316
$263$	−7.58280	−0.467576	−0.233788	−	0.972288i	$-0.575112\pi$
$263$	−7.58280	−0.467576	−0.233788	+	0.972288i	$0.575112\pi$
$264$	0	0
$265$	−9.31130	−0.571988
$266$	17.0853	1.04757
$267$	0	0
$268$	3.12462	0.190867
$269$	−29.5407	−1.80113	−0.900564	−	0.434724i	$-0.856846\pi$
$269$	−29.5407	−1.80113	−0.900564	+	0.434724i	$0.856846\pi$
$270$	0	0
$271$	−9.99943	−0.607422	−0.303711	−	0.952764i	$-0.598226\pi$
$271$	−9.99943	−0.607422	−0.303711	+	0.952764i	$0.598226\pi$
$272$	6.25252	0.379115
$273$	0	0
$274$	−27.2612	−1.64691
$275$	−6.56569	−0.395926
$276$	0	0
$277$	11.4143	0.685817	0.342909	−	0.939369i	$-0.388588\pi$
$277$	11.4143	0.685817	0.342909	+	0.939369i	$0.388588\pi$
$278$	10.5835	0.634757
$279$	0	0
$280$	28.3862	1.69640
$281$	−6.68369	−0.398715	−0.199358	−	0.979927i	$-0.563886\pi$
$281$	−6.68369	−0.398715	−0.199358	+	0.979927i	$0.563886\pi$
$282$	0	0
$283$	−9.67447	−0.575087	−0.287544	−	0.957768i	$-0.592839\pi$
$283$	−9.67447	−0.575087	−0.287544	+	0.957768i	$0.592839\pi$
$284$	−0.130969	−0.00777156
$285$	0	0
$286$	−19.9827	−1.18160
$287$	12.9890	0.766718
$288$	0	0
$289$	−15.0492	−0.885245
$290$	−26.6002	−1.56202
$291$	0	0
$292$	−0.329701	−0.0192943
$293$	18.6679	1.09059	0.545295	−	0.838244i	$-0.316418\pi$
$293$	18.6679	1.09059	0.545295	+	0.838244i	$0.316418\pi$
$294$	0	0
$295$	−6.74131	−0.392494
$296$	6.64426	0.386190
$297$	0	0
$298$	24.2355	1.40393
$299$	−22.5549	−1.30438
$300$	0	0
$301$	−19.5596	−1.12740
$302$	−33.8154	−1.94586
$303$	0	0
$304$	−12.9883	−0.744930
$305$	−18.9247	−1.08363
$306$	0	0
$307$	14.8642	0.848345	0.424173	−	0.905581i	$-0.360565\pi$
$307$	14.8642	0.848345	0.424173	+	0.905581i	$0.360565\pi$
$308$	2.51003	0.143022
$309$	0	0
$310$	23.8046	1.35201
$311$	−8.66966	−0.491611	−0.245806	−	0.969319i	$-0.579052\pi$
$311$	−8.66966	−0.491611	−0.245806	+	0.969319i	$0.579052\pi$
$312$	0	0
$313$	5.67093	0.320540	0.160270	−	0.987073i	$-0.448764\pi$
$313$	5.67093	0.320540	0.160270	+	0.987073i	$0.448764\pi$
$314$	−3.00987	−0.169857
$315$	0	0
$316$	1.72572	0.0970793
$317$	−7.67728	−0.431199	−0.215600	−	0.976482i	$-0.569171\pi$
$317$	−7.67728	−0.431199	−0.215600	+	0.976482i	$0.569171\pi$
$318$	0	0
$319$	14.6607	0.820841
$320$	−18.4857	−1.03338
$321$	0	0
$322$	23.3251	1.29986
$323$	−4.05245	−0.225484
$324$	0	0
$325$	−16.0741	−0.891629
$326$	−36.6045	−2.02734
$327$	0	0
$328$	−8.65446	−0.477862
$329$	25.4881	1.40521
$330$	0	0
$331$	3.36312	0.184854	0.0924270	−	0.995719i	$-0.470538\pi$
$331$	3.36312	0.184854	0.0924270	+	0.995719i	$0.470538\pi$
$332$	1.85759	0.101948
$333$	0	0
$334$	9.56844	0.523562
$335$	−31.6060	−1.72682
$336$	0	0
$337$	3.79788	0.206884	0.103442	−	0.994635i	$-0.467014\pi$
$337$	3.79788	0.206884	0.103442	+	0.994635i	$0.467014\pi$
$338$	−29.3069	−1.59408
$339$	0	0
$340$	1.08020	0.0585823
$341$	−13.1199	−0.710482
$342$	0	0
$343$	−4.80871	−0.259646
$344$	13.0324	0.702659
$345$	0	0
$346$	−20.9214	−1.12474
$347$	−1.72488	−0.0925965	−0.0462983	−	0.998928i	$-0.514742\pi$
$347$	−1.72488	−0.0925965	−0.0462983	+	0.998928i	$0.514742\pi$
$348$	0	0
$349$	−18.5771	−0.994412	−0.497206	−	0.867633i	$-0.665641\pi$
$349$	−18.5771	−0.994412	−0.497206	+	0.867633i	$0.665641\pi$
$350$	16.6230	0.888535
$351$	0	0
$352$	−3.61313	−0.192581
$353$	−23.2646	−1.23825	−0.619124	−	0.785294i	$-0.712512\pi$
$353$	−23.2646	−1.23825	−0.619124	+	0.785294i	$0.712512\pi$
$354$	0	0
$355$	1.32477	0.0703112
$356$	−2.68660	−0.142389
$357$	0	0
$358$	−20.2588	−1.07071
$359$	−31.6306	−1.66940	−0.834699	−	0.550707i	$-0.814358\pi$
$359$	−31.6306	−1.66940	−0.834699	+	0.550707i	$0.814358\pi$
$360$	0	0
$361$	−10.5819	−0.556941
$362$	16.8970	0.888085
$363$	0	0
$364$	6.14503	0.322087
$365$	3.33497	0.174560
$366$	0	0
$367$	−2.54102	−0.132640	−0.0663201	−	0.997798i	$-0.521126\pi$
$367$	−2.54102	−0.132640	−0.0663201	+	0.997798i	$0.521126\pi$
$368$	−17.7318	−0.924336
$369$	0	0
$370$	10.7825	0.560558
$371$	−12.9929	−0.674558
$372$	0	0
$373$	−3.91912	−0.202924	−0.101462	−	0.994839i	$-0.532352\pi$
$373$	−3.91912	−0.202924	−0.101462	+	0.994839i	$0.532352\pi$
$374$	−4.90154	−0.253452
$375$	0	0
$376$	−16.9825	−0.875805
$377$	35.8921	1.84854
$378$	0	0
$379$	25.8921	1.32999	0.664994	−	0.746848i	$-0.268434\pi$
$379$	25.8921	1.32999	0.664994	+	0.746848i	$0.268434\pi$
$380$	−2.24390	−0.115109
$381$	0	0
$382$	12.3510	0.631931
$383$	5.37654	0.274728	0.137364	−	0.990521i	$-0.456137\pi$
$383$	5.37654	0.274728	0.137364	+	0.990521i	$0.456137\pi$
$384$	0	0
$385$	−25.3893	−1.29396
$386$	33.8295	1.72187
$387$	0	0
$388$	1.45840	0.0740390
$389$	36.6948	1.86050	0.930251	−	0.366924i	$-0.119589\pi$
$389$	36.6948	1.86050	0.930251	+	0.366924i	$0.119589\pi$
$390$	0	0
$391$	−5.53247	−0.279789
$392$	21.4069	1.08121
$393$	0	0
$394$	20.3837	1.02692
$395$	−17.4559	−0.878301
$396$	0	0
$397$	13.0151	0.653207	0.326603	−	0.945161i	$-0.394096\pi$
$397$	13.0151	0.653207	0.326603	+	0.945161i	$0.394096\pi$
$398$	−28.0144	−1.40423
$399$	0	0
$400$	−12.6368	−0.631842
$401$	−11.6578	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$401$	−11.6578	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$402$	0	0
$403$	−32.1200	−1.60001
$404$	−3.65557	−0.181871
$405$	0	0
$406$	−37.1178	−1.84212
$407$	−5.94279	−0.294573
$408$	0	0
$409$	19.9079	0.984383	0.492192	−	0.870487i	$-0.336196\pi$
$409$	19.9079	0.984383	0.492192	+	0.870487i	$0.336196\pi$
$410$	−14.0448	−0.693622
$411$	0	0
$412$	1.15946	0.0571225
$413$	−9.40678	−0.462877
$414$	0	0
$415$	−18.7898	−0.922354
$416$	−8.84563	−0.433693
$417$	0	0
$418$	10.1819	0.498014
$419$	−2.61963	−0.127977	−0.0639887	−	0.997951i	$-0.520382\pi$
$419$	−2.61963	−0.127977	−0.0639887	+	0.997951i	$0.520382\pi$
$420$	0	0
$421$	−13.5201	−0.658928	−0.329464	−	0.944168i	$-0.606868\pi$
$421$	−13.5201	−0.658928	−0.329464	+	0.944168i	$0.606868\pi$
$422$	−3.48481	−0.169638
$423$	0	0
$424$	8.65705	0.420424
$425$	−3.94279	−0.191253
$426$	0	0
$427$	−26.4074	−1.27794
$428$	−2.09678	−0.101352
$429$	0	0
$430$	21.1494	1.01992
$431$	16.1584	0.778323	0.389162	−	0.921169i	$-0.372765\pi$
$431$	16.1584	0.778323	0.389162	+	0.921169i	$0.372765\pi$
$432$	0	0
$433$	10.5696	0.507943	0.253972	−	0.967212i	$-0.418263\pi$
$433$	10.5696	0.507943	0.253972	+	0.967212i	$0.418263\pi$
$434$	33.2168	1.59446
$435$	0	0
$436$	2.57003	0.123082
$437$	11.4926	0.549764
$438$	0	0
$439$	10.0810	0.481140	0.240570	−	0.970632i	$-0.422666\pi$
$439$	10.0810	0.481140	0.240570	+	0.970632i	$0.422666\pi$
$440$	16.9166	0.806469
$441$	0	0
$442$	−11.9999	−0.570777
$443$	14.7985	0.703097	0.351549	−	0.936170i	$-0.385655\pi$
$443$	14.7985	0.703097	0.351549	+	0.936170i	$0.385655\pi$
$444$	0	0
$445$	27.1753	1.28823
$446$	−17.3310	−0.820649
$447$	0	0
$448$	−25.7949	−1.21869
$449$	−24.5916	−1.16055	−0.580275	−	0.814421i	$-0.697055\pi$
$449$	−24.5916	−1.16055	−0.580275	+	0.814421i	$0.697055\pi$
$450$	0	0
$451$	7.74076	0.364498
$452$	2.14087	0.100698
$453$	0	0
$454$	26.8952	1.26226
$455$	−62.1578	−2.91400
$456$	0	0
$457$	31.2086	1.45988	0.729939	−	0.683513i	$-0.239549\pi$
$457$	31.2086	1.45988	0.729939	+	0.683513i	$0.239549\pi$
$458$	44.3555	2.07260
$459$	0	0
$460$	−3.06341	−0.142832
$461$	−37.3730	−1.74064	−0.870318	−	0.492491i	$-0.836086\pi$
$461$	−37.3730	−1.74064	−0.870318	+	0.492491i	$0.836086\pi$
$462$	0	0
$463$	−2.93839	−0.136558	−0.0682792	−	0.997666i	$-0.521751\pi$
$463$	−2.93839	−0.136558	−0.0682792	+	0.997666i	$0.521751\pi$
$464$	28.2171	1.30994
$465$	0	0
$466$	−15.6553	−0.725218
$467$	−0.538424	−0.0249153	−0.0124577	−	0.999922i	$-0.503965\pi$
$467$	−0.538424	−0.0249153	−0.0124577	+	0.999922i	$0.503965\pi$
$468$	0	0
$469$	−44.1027	−2.03648
$470$	−27.5598	−1.27124
$471$	0	0
$472$	6.26764	0.288492
$473$	−11.6565	−0.535966
$474$	0	0
$475$	8.19032	0.375798
$476$	1.50731	0.0690873
$477$	0	0
$478$	−32.7740	−1.49905
$479$	−33.6098	−1.53567	−0.767837	−	0.640646i	$-0.778667\pi$
$479$	−33.6098	−1.53567	−0.767837	+	0.640646i	$0.778667\pi$
$480$	0	0
$481$	−14.5491	−0.663380
$482$	−1.57051	−0.0715347
$483$	0	0
$484$	−1.54577	−0.0702624
$485$	−14.7519	−0.669849
$486$	0	0
$487$	−0.524362	−0.0237611	−0.0118805	−	0.999929i	$-0.503782\pi$
$487$	−0.524362	−0.0237611	−0.0118805	+	0.999929i	$0.503782\pi$
$488$	17.5950	0.796489
$489$	0	0
$490$	34.7400	1.56939
$491$	12.0921	0.545709	0.272854	−	0.962055i	$-0.412032\pi$
$491$	12.0921	0.545709	0.272854	+	0.962055i	$0.412032\pi$
$492$	0	0
$493$	8.80394	0.396510
$494$	24.9273	1.12153
$495$	0	0
$496$	−25.2515	−1.13383
$497$	1.84857	0.0829196
$498$	0	0
$499$	−5.64372	−0.252648	−0.126324	−	0.991989i	$-0.540318\pi$
$499$	−5.64372	−0.252648	−0.126324	+	0.991989i	$0.540318\pi$
$500$	1.68374	0.0752993
$501$	0	0
$502$	−20.4783	−0.913989
$503$	31.6621	1.41174	0.705871	−	0.708340i	$-0.250556\pi$
$503$	31.6621	1.41174	0.705871	+	0.708340i	$0.250556\pi$
$504$	0	0
$505$	36.9766	1.64544
$506$	13.9005	0.617954
$507$	0	0
$508$	−5.43496	−0.241137
$509$	31.4025	1.39189	0.695946	−	0.718095i	$-0.254985\pi$
$509$	31.4025	1.39189	0.695946	+	0.718095i	$0.254985\pi$
$510$	0	0
$511$	4.65360	0.205863
$512$	16.3278	0.721593
$513$	0	0
$514$	−21.3471	−0.941579
$515$	−11.7281	−0.516802
$516$	0	0
$517$	15.1896	0.668036
$518$	15.0459	0.661078
$519$	0	0
$520$	41.4151	1.81617
$521$	−11.7425	−0.514447	−0.257223	−	0.966352i	$-0.582808\pi$
$521$	−11.7425	−0.514447	−0.257223	+	0.966352i	$0.582808\pi$
$522$	0	0
$523$	−30.3917	−1.32894	−0.664469	−	0.747316i	$-0.731342\pi$
$523$	−30.3917	−1.32894	−0.664469	+	0.747316i	$0.731342\pi$
$524$	1.28242	0.0560226
$525$	0	0
$526$	11.4410	0.498852
$527$	−7.87868	−0.343201
$528$	0	0
$529$	−7.31017	−0.317833
$530$	14.0490	0.610249
$531$	0	0
$532$	−3.13112	−0.135751
$533$	18.9508	0.820852
$534$	0	0
$535$	21.2092	0.916953
$536$	29.3852	1.26925
$537$	0	0
$538$	44.5713	1.92161
$539$	−19.1469	−0.824715
$540$	0	0
$541$	−20.2071	−0.868773	−0.434387	−	0.900727i	$-0.643035\pi$
$541$	−20.2071	−0.868773	−0.434387	+	0.900727i	$0.643035\pi$
$542$	15.0873	0.648053
$543$	0	0
$544$	−2.16974	−0.0930267
$545$	−25.9962	−1.11355
$546$	0	0
$547$	9.31633	0.398338	0.199169	−	0.979965i	$-0.436176\pi$
$547$	9.31633	0.398338	0.199169	+	0.979965i	$0.436176\pi$
$548$	4.99600	0.213419
$549$	0	0
$550$	9.90639	0.422410
$551$	−18.2883	−0.779110
$552$	0	0
$553$	−24.3578	−1.03580
$554$	−17.2220	−0.731692
$555$	0	0
$556$	−1.93958	−0.0822563
$557$	−29.6730	−1.25728	−0.628642	−	0.777694i	$-0.716389\pi$
$557$	−29.6730	−1.25728	−0.628642	+	0.777694i	$0.716389\pi$
$558$	0	0
$559$	−28.5373	−1.20700
$560$	−48.8662	−2.06497
$561$	0	0
$562$	10.0844	0.425386
$563$	−23.2453	−0.979674	−0.489837	−	0.871814i	$-0.662944\pi$
$563$	−23.2453	−0.979674	−0.489837	+	0.871814i	$0.662944\pi$
$564$	0	0
$565$	−21.6552	−0.911042
$566$	14.5969	0.613555
$567$	0	0
$568$	−1.23168	−0.0516803
$569$	10.5462	0.442121	0.221061	−	0.975260i	$-0.429048\pi$
$569$	10.5462	0.442121	0.221061	+	0.975260i	$0.429048\pi$
$570$	0	0
$571$	1.09183	0.0456916	0.0228458	−	0.999739i	$-0.492727\pi$
$571$	1.09183	0.0456916	0.0228458	+	0.999739i	$0.492727\pi$
$572$	3.66211	0.153120
$573$	0	0
$574$	−19.5980	−0.818004
$575$	11.1816	0.466304
$576$	0	0
$577$	−13.5374	−0.563568	−0.281784	−	0.959478i	$-0.590926\pi$
$577$	−13.5374	−0.563568	−0.281784	+	0.959478i	$0.590926\pi$
$578$	22.7063	0.944459
$579$	0	0
$580$	4.87486	0.202418
$581$	−26.2191	−1.08775
$582$	0	0
$583$	−7.74308	−0.320686
$584$	−3.10065	−0.128306
$585$	0	0
$586$	−28.1663	−1.16354
$587$	−34.2048	−1.41178	−0.705892	−	0.708319i	$-0.749454\pi$
$587$	−34.2048	−1.41178	−0.705892	+	0.708319i	$0.749454\pi$
$588$	0	0
$589$	16.3663	0.674362
$590$	10.1714	0.418748
$591$	0	0
$592$	−11.4379	−0.470096
$593$	31.7942	1.30563	0.652815	−	0.757517i	$-0.273588\pi$
$593$	31.7942	1.30563	0.652815	+	0.757517i	$0.273588\pi$
$594$	0	0
$595$	−15.2466	−0.625051
$596$	−4.44149	−0.181931
$597$	0	0
$598$	34.0311	1.39164
$599$	−36.6390	−1.49703	−0.748514	−	0.663119i	$-0.769232\pi$
$599$	−36.6390	−1.49703	−0.748514	+	0.663119i	$0.769232\pi$
$600$	0	0
$601$	−20.1000	−0.819896	−0.409948	−	0.912109i	$-0.634453\pi$
$601$	−20.1000	−0.819896	−0.409948	+	0.912109i	$0.634453\pi$
$602$	29.5118	1.20281
$603$	0	0
$604$	6.19713	0.252158
$605$	15.6357	0.635682
$606$	0	0
$607$	−45.7280	−1.85604	−0.928021	−	0.372527i	$-0.878491\pi$
$607$	−45.7280	−1.85604	−0.928021	+	0.372527i	$0.878491\pi$
$608$	4.50717	0.182790
$609$	0	0
$610$	28.5538	1.15611
$611$	37.1869	1.50442
$612$	0	0
$613$	43.0323	1.73806	0.869030	−	0.494760i	$-0.164744\pi$
$613$	43.0323	1.73806	0.869030	+	0.494760i	$0.164744\pi$
$614$	−22.4273	−0.905092
$615$	0	0
$616$	23.6054	0.951087
$617$	18.0405	0.726283	0.363142	−	0.931734i	$-0.381704\pi$
$617$	18.0405	0.726283	0.363142	+	0.931734i	$0.381704\pi$
$618$	0	0
$619$	−8.52399	−0.342608	−0.171304	−	0.985218i	$-0.554798\pi$
$619$	−8.52399	−0.342608	−0.171304	+	0.985218i	$0.554798\pi$
$620$	−4.36253	−0.175203
$621$	0	0
$622$	13.0809	0.524495
$623$	37.9202	1.51924
$624$	0	0
$625$	−31.1457	−1.24583
$626$	−8.55636	−0.341981
$627$	0	0
$628$	0.551600	0.0220112
$629$	−3.56873	−0.142294
$630$	0	0
$631$	15.4320	0.614339	0.307169	−	0.951655i	$-0.400618\pi$
$631$	15.4320	0.614339	0.307169	+	0.951655i	$0.400618\pi$
$632$	16.2294	0.645570
$633$	0	0
$634$	11.5836	0.460042
$635$	54.9754	2.18163
$636$	0	0
$637$	−46.8752	−1.85726
$638$	−22.1202	−0.875747
$639$	0	0
$640$	36.5813	1.44600
$641$	40.5764	1.60267	0.801336	−	0.598215i	$-0.204123\pi$
$641$	40.5764	1.60267	0.801336	+	0.598215i	$0.204123\pi$
$642$	0	0
$643$	3.89837	0.153737	0.0768684	−	0.997041i	$-0.475508\pi$
$643$	3.89837	0.153737	0.0768684	+	0.997041i	$0.475508\pi$
$644$	−4.27465	−0.168445
$645$	0	0
$646$	6.11438	0.240567
$647$	−33.8517	−1.33085	−0.665424	−	0.746465i	$-0.731749\pi$
$647$	−33.8517	−1.33085	−0.665424	+	0.746465i	$0.731749\pi$
$648$	0	0
$649$	−5.60593	−0.220052
$650$	24.2527	0.951270
$651$	0	0
$652$	6.70829	0.262717
$653$	−12.1974	−0.477320	−0.238660	−	0.971103i	$-0.576708\pi$
$653$	−12.1974	−0.477320	−0.238660	+	0.971103i	$0.576708\pi$
$654$	0	0
$655$	−12.9718	−0.506851
$656$	14.8984	0.581687
$657$	0	0
$658$	−38.4568	−1.49920
$659$	7.61910	0.296798	0.148399	−	0.988928i	$-0.452588\pi$
$659$	7.61910	0.296798	0.148399	+	0.988928i	$0.452588\pi$
$660$	0	0
$661$	−25.5255	−0.992825	−0.496413	−	0.868087i	$-0.665350\pi$
$661$	−25.5255	−0.992825	−0.496413	+	0.868087i	$0.665350\pi$
$662$	−5.07432	−0.197219
$663$	0	0
$664$	17.4695	0.677950
$665$	31.6717	1.22817
$666$	0	0
$667$	−24.9676	−0.966748
$668$	−1.75355	−0.0678469
$669$	0	0
$670$	47.6874	1.84233
$671$	−15.7374	−0.607536
$672$	0	0
$673$	16.0290	0.617873	0.308936	−	0.951083i	$-0.400027\pi$
$673$	16.0290	0.617873	0.308936	+	0.951083i	$0.400027\pi$
$674$	−5.73029	−0.220722
$675$	0	0
$676$	5.37090	0.206573
$677$	36.7257	1.41148	0.705742	−	0.708469i	$-0.250614\pi$
$677$	36.7257	1.41148	0.705742	+	0.708469i	$0.250614\pi$
$678$	0	0
$679$	−20.5847	−0.789968
$680$	10.1587	0.389568
$681$	0	0
$682$	19.7954	0.758007
$683$	6.96341	0.266448	0.133224	−	0.991086i	$-0.457467\pi$
$683$	6.96341	0.266448	0.133224	+	0.991086i	$0.457467\pi$
$684$	0	0
$685$	−50.5352	−1.93085
$686$	7.25543	0.277014
$687$	0	0
$688$	−22.4349	−0.855324
$689$	−18.9565	−0.722186
$690$	0	0
$691$	42.8061	1.62842	0.814210	−	0.580570i	$-0.197170\pi$
$691$	42.8061	1.62842	0.814210	+	0.580570i	$0.197170\pi$
$692$	3.83414	0.145752
$693$	0	0
$694$	2.60252	0.0987903
$695$	19.6191	0.744194
$696$	0	0
$697$	4.64843	0.176072
$698$	28.0294	1.06093
$699$	0	0
$700$	−3.04639	−0.115143
$701$	−38.2909	−1.44623	−0.723114	−	0.690729i	$-0.757290\pi$
$701$	−38.2909	−1.44623	−0.723114	+	0.690729i	$0.757290\pi$
$702$	0	0
$703$	7.41328	0.279597
$704$	−15.3724	−0.579367
$705$	0	0
$706$	35.1018	1.32107
$707$	51.5968	1.94050
$708$	0	0
$709$	26.9501	1.01213	0.506066	−	0.862495i	$-0.331099\pi$
$709$	26.9501	1.01213	0.506066	+	0.862495i	$0.331099\pi$
$710$	−1.99882	−0.0750144
$711$	0	0
$712$	−25.2659	−0.946879
$713$	22.3436	0.836773
$714$	0	0
$715$	−37.0427	−1.38532
$716$	3.71271	0.138750
$717$	0	0
$718$	47.7246	1.78106
$719$	1.55823	0.0581122	0.0290561	−	0.999578i	$-0.490750\pi$
$719$	1.55823	0.0581122	0.0290561	+	0.999578i	$0.490750\pi$
$720$	0	0
$721$	−16.3653	−0.609475
$722$	15.9661	0.594195
$723$	0	0
$724$	−3.09660	−0.115084
$725$	−17.7935	−0.660833
$726$	0	0
$727$	−18.8714	−0.699901	−0.349951	−	0.936768i	$-0.613802\pi$
$727$	−18.8714	−0.699901	−0.349951	+	0.936768i	$0.613802\pi$
$728$	57.7904	2.14185
$729$	0	0
$730$	−5.03184	−0.186237
$731$	−6.99988	−0.258900
$732$	0	0
$733$	45.9166	1.69597	0.847984	−	0.530022i	$-0.177816\pi$
$733$	45.9166	1.69597	0.847984	+	0.530022i	$0.177816\pi$
$734$	3.83392	0.141512
$735$	0	0
$736$	6.15327	0.226812
$737$	−26.2829	−0.968142
$738$	0	0
$739$	14.3419	0.527574	0.263787	−	0.964581i	$-0.415028\pi$
$739$	14.3419	0.527574	0.263787	+	0.964581i	$0.415028\pi$
$740$	−1.97605	−0.0726411
$741$	0	0
$742$	19.6039	0.719680
$743$	18.0652	0.662749	0.331374	−	0.943499i	$-0.392488\pi$
$743$	18.0652	0.662749	0.331374	+	0.943499i	$0.392488\pi$
$744$	0	0
$745$	44.9263	1.64597
$746$	5.91321	0.216498
$747$	0	0
$748$	0.898275	0.0328442
$749$	29.5951	1.08138
$750$	0	0
$751$	23.6787	0.864049	0.432024	−	0.901862i	$-0.357799\pi$
$751$	23.6787	0.864049	0.432024	+	0.901862i	$0.357799\pi$
$752$	29.2350	1.06609
$753$	0	0
$754$	−54.1544	−1.97219
$755$	−62.6849	−2.28134
$756$	0	0
$757$	−14.5723	−0.529641	−0.264820	−	0.964298i	$-0.585313\pi$
$757$	−14.5723	−0.529641	−0.264820	+	0.964298i	$0.585313\pi$
$758$	−39.0663	−1.41895
$759$	0	0
$760$	−21.1025	−0.765469
$761$	−33.1349	−1.20114	−0.600570	−	0.799572i	$-0.705060\pi$
$761$	−33.1349	−1.20114	−0.600570	+	0.799572i	$0.705060\pi$
$762$	0	0
$763$	−36.2749	−1.31324
$764$	−2.26349	−0.0818901
$765$	0	0
$766$	−8.11218	−0.293105
$767$	−13.7244	−0.495559
$768$	0	0
$769$	−1.59470	−0.0575063	−0.0287531	−	0.999587i	$-0.509154\pi$
$769$	−1.59470	−0.0575063	−0.0287531	+	0.999587i	$0.509154\pi$
$770$	38.3076	1.38051
$771$	0	0
$772$	−6.19972	−0.223133
$773$	−44.5890	−1.60376	−0.801878	−	0.597487i	$-0.796166\pi$
$773$	−44.5890	−1.60376	−0.801878	+	0.597487i	$0.796166\pi$
$774$	0	0
$775$	15.9234	0.571987
$776$	13.7154	0.492353
$777$	0	0
$778$	−55.3656	−1.98495
$779$	−9.65615	−0.345967
$780$	0	0
$781$	1.10165	0.0394200
$782$	8.34746	0.298504
$783$	0	0
$784$	−36.8515	−1.31613
$785$	−5.57951	−0.199141
$786$	0	0
$787$	17.8466	0.636162	0.318081	−	0.948063i	$-0.396962\pi$
$787$	17.8466	0.636162	0.318081	+	0.948063i	$0.396962\pi$
$788$	−3.73561	−0.133075
$789$	0	0
$790$	26.3376	0.937051
$791$	−30.2175	−1.07441
$792$	0	0
$793$	−38.5282	−1.36818
$794$	−19.6373	−0.696900
$795$	0	0
$796$	5.13403	0.181971
$797$	26.6461	0.943852	0.471926	−	0.881638i	$-0.343559\pi$
$797$	26.6461	0.943852	0.471926	+	0.881638i	$0.343559\pi$
$798$	0	0
$799$	9.12154	0.322697
$800$	4.38521	0.155041
$801$	0	0
$802$	17.5894	0.621103
$803$	2.77329	0.0978674
$804$	0	0
$805$	43.2387	1.52396
$806$	48.4630	1.70704
$807$	0	0
$808$	−34.3785	−1.20943
$809$	−25.2438	−0.887524	−0.443762	−	0.896145i	$-0.646356\pi$
$809$	−25.2438	−0.887524	−0.443762	+	0.896145i	$0.646356\pi$
$810$	0	0
$811$	−30.2069	−1.06071	−0.530353	−	0.847777i	$-0.677941\pi$
$811$	−30.2069	−1.06071	−0.530353	+	0.847777i	$0.677941\pi$
$812$	6.80235	0.238716
$813$	0	0
$814$	8.96654	0.314277
$815$	−67.8552	−2.37687
$816$	0	0
$817$	14.5408	0.508718
$818$	−30.0373	−1.05023
$819$	0	0
$820$	2.57390	0.0898845
$821$	−0.878546	−0.0306615	−0.0153307	−	0.999882i	$-0.504880\pi$
$821$	−0.878546	−0.0306615	−0.0153307	+	0.999882i	$0.504880\pi$
$822$	0	0
$823$	−39.6954	−1.38370	−0.691848	−	0.722044i	$-0.743203\pi$
$823$	−39.6954	−1.38370	−0.691848	+	0.722044i	$0.743203\pi$
$824$	10.9040	0.379860
$825$	0	0
$826$	14.1930	0.493839
$827$	−48.2665	−1.67839	−0.839196	−	0.543829i	$-0.816974\pi$
$827$	−48.2665	−1.67839	−0.839196	+	0.543829i	$0.816974\pi$
$828$	0	0
$829$	25.5272	0.886595	0.443297	−	0.896375i	$-0.353809\pi$
$829$	25.5272	0.886595	0.443297	+	0.896375i	$0.353809\pi$
$830$	28.3502	0.984050
$831$	0	0
$832$	−37.6344	−1.30474
$833$	−11.4980	−0.398381
$834$	0	0
$835$	17.7374	0.613828
$836$	−1.86598	−0.0645362
$837$	0	0
$838$	3.95253	0.136538
$839$	39.9678	1.37984	0.689921	−	0.723885i	$-0.257645\pi$
$839$	39.9678	1.37984	0.689921	+	0.723885i	$0.257645\pi$
$840$	0	0
$841$	10.7314	0.370049
$842$	20.3992	0.703004
$843$	0	0
$844$	0.638639	0.0219829
$845$	−54.3273	−1.86892
$846$	0	0
$847$	21.8180	0.749674
$848$	−14.9029	−0.511768
$849$	0	0
$850$	5.94893	0.204046
$851$	10.1207	0.346934
$852$	0	0
$853$	−12.6476	−0.433047	−0.216523	−	0.976277i	$-0.569472\pi$
$853$	−12.6476	−0.433047	−0.216523	+	0.976277i	$0.569472\pi$
$854$	39.8438	1.36343
$855$	0	0
$856$	−19.7190	−0.673980
$857$	−40.0590	−1.36839	−0.684195	−	0.729299i	$-0.739846\pi$
$857$	−40.0590	−1.36839	−0.684195	+	0.729299i	$0.739846\pi$
$858$	0	0
$859$	12.5413	0.427904	0.213952	−	0.976844i	$-0.431366\pi$
$859$	12.5413	0.427904	0.213952	+	0.976844i	$0.431366\pi$
$860$	−3.87593	−0.132168
$861$	0	0
$862$	−24.3800	−0.830386
$863$	−17.3781	−0.591559	−0.295780	−	0.955256i	$-0.595579\pi$
$863$	−17.3781	−0.591559	−0.295780	+	0.955256i	$0.595579\pi$
$864$	0	0
$865$	−38.7829	−1.31866
$866$	−15.9475	−0.541920
$867$	0	0
$868$	−6.08744	−0.206621
$869$	−14.5160	−0.492420
$870$	0	0
$871$	−64.3454	−2.18026
$872$	24.1696	0.818487
$873$	0	0
$874$	−17.3401	−0.586537
$875$	−23.7654	−0.803416
$876$	0	0
$877$	−15.6298	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$877$	−15.6298	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$878$	−15.2103	−0.513324
$879$	0	0
$880$	−29.1216	−0.981689
$881$	39.2575	1.32262	0.661309	−	0.750114i	$-0.270001\pi$
$881$	39.2575	1.32262	0.661309	+	0.750114i	$0.270001\pi$
$882$	0	0
$883$	31.5580	1.06201	0.531005	−	0.847369i	$-0.321814\pi$
$883$	31.5580	1.06201	0.531005	+	0.847369i	$0.321814\pi$
$884$	2.19915	0.0739653
$885$	0	0
$886$	−22.3281	−0.750128
$887$	38.4971	1.29261	0.646304	−	0.763080i	$-0.276314\pi$
$887$	38.4971	1.29261	0.646304	+	0.763080i	$0.276314\pi$
$888$	0	0
$889$	76.7122	2.57285
$890$	−41.0024	−1.37440
$891$	0	0
$892$	3.17615	0.106346
$893$	−18.9481	−0.634074
$894$	0	0
$895$	−37.5545	−1.25531
$896$	51.0453	1.70530
$897$	0	0
$898$	37.1041	1.23818
$899$	−35.5558	−1.18585
$900$	0	0
$901$	−4.64983	−0.154908
$902$	−11.6793	−0.388880
$903$	0	0
$904$	20.1337	0.669635
$905$	31.3226	1.04120
$906$	0	0
$907$	−45.9467	−1.52564	−0.762818	−	0.646614i	$-0.776185\pi$
$907$	−45.9467	−1.52564	−0.762818	+	0.646614i	$0.776185\pi$
$908$	−4.92892	−0.163572
$909$	0	0
$910$	93.7844	3.10892
$911$	−17.7844	−0.589225	−0.294612	−	0.955617i	$-0.595191\pi$
$911$	−17.7844	−0.589225	−0.294612	+	0.955617i	$0.595191\pi$
$912$	0	0
$913$	−15.6252	−0.517118
$914$	−47.0879	−1.55753
$915$	0	0
$916$	−8.12877	−0.268582
$917$	−18.1008	−0.597740
$918$	0	0
$919$	−11.6978	−0.385873	−0.192937	−	0.981211i	$-0.561801\pi$
$919$	−11.6978	−0.385873	−0.192937	+	0.981211i	$0.561801\pi$
$920$	−28.8095	−0.949822
$921$	0	0
$922$	56.3888	1.85707
$923$	2.69704	0.0887742
$924$	0	0
$925$	7.21268	0.237151
$926$	4.43347	0.145693
$927$	0	0
$928$	−9.79183	−0.321432
$929$	10.1674	0.333582	0.166791	−	0.985992i	$-0.446659\pi$
$929$	10.1674	0.333582	0.166791	+	0.985992i	$0.446659\pi$
$930$	0	0
$931$	23.8846	0.782787
$932$	2.86905	0.0939790
$933$	0	0
$934$	0.812380	0.0265819
$935$	−9.08617	−0.297150
$936$	0	0
$937$	−5.18244	−0.169303	−0.0846515	−	0.996411i	$-0.526978\pi$
$937$	−5.18244	−0.169303	−0.0846515	+	0.996411i	$0.526978\pi$
$938$	66.5427	2.17270
$939$	0	0
$940$	5.05072	0.164736
$941$	−27.8698	−0.908529	−0.454264	−	0.890867i	$-0.650098\pi$
$941$	−27.8698	−0.908529	−0.454264	+	0.890867i	$0.650098\pi$
$942$	0	0
$943$	−13.1827	−0.429289
$944$	−10.7896	−0.351172
$945$	0	0
$946$	17.5874	0.571817
$947$	40.7064	1.32278	0.661389	−	0.750043i	$-0.269967\pi$
$947$	40.7064	1.32278	0.661389	+	0.750043i	$0.269967\pi$
$948$	0	0
$949$	6.78955	0.220398
$950$	−12.3576	−0.400935
$951$	0	0
$952$	14.1753	0.459426
$953$	7.18638	0.232790	0.116395	−	0.993203i	$-0.462866\pi$
$953$	7.18638	0.232790	0.116395	+	0.993203i	$0.462866\pi$
$954$	0	0
$955$	22.8955	0.740881
$956$	6.00630	0.194258
$957$	0	0
$958$	50.7109	1.63839
$959$	−70.5165	−2.27710
$960$	0	0
$961$	0.818997	0.0264193
$962$	21.9518	0.707754
$963$	0	0
$964$	0.287817	0.00926997
$965$	62.7110	2.01874
$966$	0	0
$967$	2.71003	0.0871488	0.0435744	−	0.999050i	$-0.486125\pi$
$967$	2.71003	0.0871488	0.0435744	+	0.999050i	$0.486125\pi$
$968$	−14.5371	−0.467240
$969$	0	0
$970$	22.2578	0.714656
$971$	26.8026	0.860136	0.430068	−	0.902796i	$-0.358490\pi$
$971$	26.8026	0.860136	0.430068	+	0.902796i	$0.358490\pi$
$972$	0	0
$973$	27.3763	0.877644
$974$	0.791163	0.0253505
$975$	0	0
$976$	−30.2894	−0.969540
$977$	3.54269	0.113341	0.0566703	−	0.998393i	$-0.481952\pi$
$977$	3.54269	0.113341	0.0566703	+	0.998393i	$0.481952\pi$
$978$	0	0
$979$	22.5984	0.722249
$980$	−6.36658	−0.203373
$981$	0	0
$982$	−18.2447	−0.582212
$983$	−42.7100	−1.36224	−0.681119	−	0.732173i	$-0.738506\pi$
$983$	−42.7100	−1.36224	−0.681119	+	0.732173i	$0.738506\pi$
$984$	0	0
$985$	37.7862	1.20397
$986$	−13.2835	−0.423032
$987$	0	0
$988$	−4.56827	−0.145336
$989$	19.8513	0.631235
$990$	0	0
$991$	−32.2710	−1.02512	−0.512562	−	0.858650i	$-0.671303\pi$
$991$	−32.2710	−1.02512	−0.512562	+	0.858650i	$0.671303\pi$
$992$	8.76274	0.278217
$993$	0	0
$994$	−2.78914	−0.0884661
$995$	−51.9314	−1.64634
$996$	0	0
$997$	33.1998	1.05145	0.525725	−	0.850655i	$-0.323794\pi$
$997$	33.1998	1.05145	0.525725	+	0.850655i	$0.323794\pi$
$998$	8.51531	0.269547
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.20		72
3.2	odd	2		6561.2.a.d.1.53		72
81.2	odd	54		729.2.g.b.433.3		144
81.13	even	27		729.2.g.d.541.3		144
81.14	odd	54		243.2.g.a.181.6		144
81.25	even	27		729.2.g.d.190.3		144
81.29	odd	54		243.2.g.a.145.6		144
81.40	even	27		729.2.g.c.298.6		144
81.41	odd	54		729.2.g.b.298.3		144
81.52	even	27		81.2.g.a.31.3	✓	144
81.56	odd	54		729.2.g.a.190.6		144
81.67	even	27		81.2.g.a.34.3	yes	144
81.68	odd	54		729.2.g.a.541.6		144
81.79	even	27		729.2.g.c.433.6		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.31.3	✓	144	81.52	even	27
81.2.g.a.34.3	yes	144	81.67	even	27
243.2.g.a.145.6		144	81.29	odd	54
243.2.g.a.181.6		144	81.14	odd	54
729.2.g.a.190.6		144	81.56	odd	54
729.2.g.a.541.6		144	81.68	odd	54
729.2.g.b.298.3		144	81.41	odd	54
729.2.g.b.433.3		144	81.2	odd	54
729.2.g.c.298.6		144	81.40	even	27
729.2.g.c.433.6		144	81.79	even	27
729.2.g.d.190.3		144	81.25	even	27
729.2.g.d.541.3		144	81.13	even	27
6561.2.a.c.1.20		72	1.1	even	1	trivial
6561.2.a.d.1.53		72	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.50881 q^{2} +0.276511 q^{4} -2.79694 q^{5} -3.90283 q^{7} +2.60042 q^{8} +O(q^{10})\)	\(q-1.50881 q^{2} +0.276511 q^{4} -2.79694 q^{5} -3.90283 q^{7} +2.60042 q^{8} +4.22006 q^{10} -2.32588 q^{11} -5.69419 q^{13} +5.88864 q^{14} -4.47656 q^{16} -1.39672 q^{17} +2.90140 q^{19} -0.773384 q^{20} +3.50931 q^{22} +3.96104 q^{23} +2.82289 q^{25} +8.59146 q^{26} -1.07917 q^{28} -6.30329 q^{29} +5.64083 q^{31} +1.55345 q^{32} +2.10739 q^{34} +10.9160 q^{35} +2.55507 q^{37} -4.37766 q^{38} -7.27322 q^{40} -3.32810 q^{41} +5.01165 q^{43} -0.643130 q^{44} -5.97646 q^{46} -6.53067 q^{47} +8.23211 q^{49} -4.25920 q^{50} -1.57450 q^{52} +3.32910 q^{53} +6.50535 q^{55} -10.1490 q^{56} +9.51047 q^{58} +2.41024 q^{59} +6.76622 q^{61} -8.51095 q^{62} +6.60927 q^{64} +15.9263 q^{65} +11.3002 q^{67} -0.386209 q^{68} -16.4702 q^{70} -0.473648 q^{71} -1.19236 q^{73} -3.85512 q^{74} +0.802268 q^{76} +9.07752 q^{77} +6.24106 q^{79} +12.5207 q^{80} +5.02147 q^{82} +6.71797 q^{83} +3.90655 q^{85} -7.56163 q^{86} -6.04826 q^{88} -9.71608 q^{89} +22.2235 q^{91} +1.09527 q^{92} +9.85355 q^{94} -8.11505 q^{95} +5.27429 q^{97} -12.4207 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8}+O(q^{10}) \) 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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