LMFDB - Embedded newform 6534.2.a.cw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6534,2,Mod(1,6534)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6534, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6534.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6534 = 2 \cdot 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6534.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:	$52.1742526807$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.493090625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 20x^{4} + 15x^{3} + 115x^{2} + 73x - 11 $ x^6 - 2x^5 - 20x^4 + 15x^3 + 115x^2 + 73*x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 594)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.88962$ of defining polynomial
Character	$\chi$	$=$	6534.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.88962 q^{5} -3.72393 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -3.88962 q^{5} -3.72393 q^{7} +1.00000 q^{8} -3.88962 q^{10} -5.50981 q^{13} -3.72393 q^{14} +1.00000 q^{16} -4.37189 q^{17} -4.37400 q^{19} -3.88962 q^{20} -0.169185 q^{23} +10.1292 q^{25} -5.50981 q^{26} -3.72393 q^{28} -8.39728 q^{29} -0.337037 q^{31} +1.00000 q^{32} -4.37189 q^{34} +14.4847 q^{35} +1.89975 q^{37} -4.37400 q^{38} -3.88962 q^{40} -4.97703 q^{41} +3.84121 q^{43} -0.169185 q^{46} +1.03768 q^{47} +6.86764 q^{49} +10.1292 q^{50} -5.50981 q^{52} +6.91374 q^{53} -3.72393 q^{56} -8.39728 q^{58} +2.69189 q^{59} -10.6852 q^{61} -0.337037 q^{62} +1.00000 q^{64} +21.4311 q^{65} -11.4875 q^{67} -4.37189 q^{68} +14.4847 q^{70} -12.7472 q^{71} +5.27159 q^{73} +1.89975 q^{74} -4.37400 q^{76} +10.2281 q^{79} -3.88962 q^{80} -4.97703 q^{82} -6.65182 q^{83} +17.0050 q^{85} +3.84121 q^{86} +12.2110 q^{89} +20.5182 q^{91} -0.169185 q^{92} +1.03768 q^{94} +17.0132 q^{95} +17.5359 q^{97} +6.86764 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} - 2 q^{5} + q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 6 * q^2 + 6 * q^4 - 2 * q^5 + q^7 + 6 * q^8	$ 6 q + 6 q^{2} + 6 q^{4} - 2 q^{5} + q^{7} + 6 q^{8} - 2 q^{10} - q^{13} + q^{14} + 6 q^{16} - 7 q^{19} - 2 q^{20} + 9 q^{23} + 14 q^{25} - q^{26} + q^{28} - q^{29} + 10 q^{31} + 6 q^{32} + 18 q^{35} + 12 q^{37} - 7 q^{38} - 2 q^{40} + 11 q^{41} - 5 q^{43} + 9 q^{46} + q^{47} + 29 q^{49} + 14 q^{50} - q^{52} - 6 q^{53} + q^{56} - q^{58} - 18 q^{59} + 17 q^{61} + 10 q^{62} + 6 q^{64} + 2 q^{65} + 14 q^{67} + 18 q^{70} - 10 q^{71} + 17 q^{73} + 12 q^{74} - 7 q^{76} + 22 q^{79} - 2 q^{80} + 11 q^{82} + 33 q^{83} - 35 q^{85} - 5 q^{86} + 12 q^{89} + 59 q^{91} + 9 q^{92} + q^{94} + 39 q^{95} + 29 q^{97} + 29 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 - 2 * q^5 + q^7 + 6 * q^8 - 2 * q^10 - q^13 + q^14 + 6 * q^16 - 7 * q^19 - 2 * q^20 + 9 * q^23 + 14 * q^25 - q^26 + q^28 - q^29 + 10 * q^31 + 6 * q^32 + 18 * q^35 + 12 * q^37 - 7 * q^38 - 2 * q^40 + 11 * q^41 - 5 * q^43 + 9 * q^46 + q^47 + 29 * q^49 + 14 * q^50 - q^52 - 6 * q^53 + q^56 - q^58 - 18 * q^59 + 17 * q^61 + 10 * q^62 + 6 * q^64 + 2 * q^65 + 14 * q^67 + 18 * q^70 - 10 * q^71 + 17 * q^73 + 12 * q^74 - 7 * q^76 + 22 * q^79 - 2 * q^80 + 11 * q^82 + 33 * q^83 - 35 * q^85 - 5 * q^86 + 12 * q^89 + 59 * q^91 + 9 * q^92 + q^94 + 39 * q^95 + 29 * q^97 + 29 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.88962	−1.73949	−0.869747	−	0.493499i	$-0.835718\pi$
$5$	−3.88962	−1.73949	−0.869747	+	0.493499i	$0.835718\pi$
$6$	0	0
$7$	−3.72393	−1.40751	−0.703756	−	0.710441i	$-0.748495\pi$
$7$	−3.72393	−1.40751	−0.703756	+	0.710441i	$0.748495\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.88962	−1.23001
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.50981	−1.52815	−0.764074	−	0.645129i	$-0.776804\pi$
$13$	−5.50981	−1.52815	−0.764074	+	0.645129i	$0.776804\pi$
$14$	−3.72393	−0.995262
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.37189	−1.06034	−0.530169	−	0.847892i	$-0.677872\pi$
$17$	−4.37189	−1.06034	−0.530169	+	0.847892i	$0.677872\pi$
$18$	0	0
$19$	−4.37400	−1.00346	−0.501732	−	0.865023i	$-0.667304\pi$
$19$	−4.37400	−1.00346	−0.501732	+	0.865023i	$0.667304\pi$
$20$	−3.88962	−0.869747
$21$	0	0
$22$	0	0
$23$	−0.169185	−0.0352775	−0.0176387	−	0.999844i	$-0.505615\pi$
$23$	−0.169185	−0.0352775	−0.0176387	+	0.999844i	$0.505615\pi$
$24$	0	0
$25$	10.1292	2.02584
$26$	−5.50981	−1.08056
$27$	0	0
$28$	−3.72393	−0.703756
$29$	−8.39728	−1.55934	−0.779668	−	0.626193i	$-0.784612\pi$
$29$	−8.39728	−1.55934	−0.779668	+	0.626193i	$0.784612\pi$
$30$	0	0
$31$	−0.337037	−0.0605337	−0.0302668	−	0.999542i	$-0.509636\pi$
$31$	−0.337037	−0.0605337	−0.0302668	+	0.999542i	$0.509636\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.37189	−0.749772
$35$	14.4847	2.44836
$36$	0	0
$37$	1.89975	0.312317	0.156158	−	0.987732i	$-0.450089\pi$
$37$	1.89975	0.312317	0.156158	+	0.987732i	$0.450089\pi$
$38$	−4.37400	−0.709556
$39$	0	0
$40$	−3.88962	−0.615004
$41$	−4.97703	−0.777281	−0.388640	−	0.921390i	$-0.627055\pi$
$41$	−4.97703	−0.777281	−0.388640	+	0.921390i	$0.627055\pi$
$42$	0	0
$43$	3.84121	0.585779	0.292889	−	0.956146i	$-0.405383\pi$
$43$	3.84121	0.585779	0.292889	+	0.956146i	$0.405383\pi$
$44$	0	0
$45$	0	0
$46$	−0.169185	−0.0249449
$47$	1.03768	0.151361	0.0756805	−	0.997132i	$-0.475887\pi$
$47$	1.03768	0.151361	0.0756805	+	0.997132i	$0.475887\pi$
$48$	0	0
$49$	6.86764	0.981092
$50$	10.1292	1.43248
$51$	0	0
$52$	−5.50981	−0.764074
$53$	6.91374	0.949675	0.474837	−	0.880074i	$-0.342507\pi$
$53$	6.91374	0.949675	0.474837	+	0.880074i	$0.342507\pi$
$54$	0	0
$55$	0	0
$56$	−3.72393	−0.497631
$57$	0	0
$58$	−8.39728	−1.10262
$59$	2.69189	0.350455	0.175227	−	0.984528i	$-0.443934\pi$
$59$	2.69189	0.350455	0.175227	+	0.984528i	$0.443934\pi$
$60$	0	0
$61$	−10.6852	−1.36810	−0.684052	−	0.729434i	$-0.739784\pi$
$61$	−10.6852	−1.36810	−0.684052	+	0.729434i	$0.739784\pi$
$62$	−0.337037	−0.0428038
$63$	0	0
$64$	1.00000	0.125000
$65$	21.4311	2.65820
$66$	0	0
$67$	−11.4875	−1.40342	−0.701708	−	0.712465i	$-0.747579\pi$
$67$	−11.4875	−1.40342	−0.701708	+	0.712465i	$0.747579\pi$
$68$	−4.37189	−0.530169
$69$	0	0
$70$	14.4847	1.73125
$71$	−12.7472	−1.51281	−0.756406	−	0.654102i	$-0.773047\pi$
$71$	−12.7472	−1.51281	−0.756406	+	0.654102i	$0.773047\pi$
$72$	0	0
$73$	5.27159	0.616993	0.308497	−	0.951225i	$-0.400174\pi$
$73$	5.27159	0.616993	0.308497	+	0.951225i	$0.400174\pi$
$74$	1.89975	0.220841
$75$	0	0
$76$	−4.37400	−0.501732
$77$	0	0
$78$	0	0
$79$	10.2281	1.15075	0.575376	−	0.817889i	$-0.304856\pi$
$79$	10.2281	1.15075	0.575376	+	0.817889i	$0.304856\pi$
$80$	−3.88962	−0.434873
$81$	0	0
$82$	−4.97703	−0.549621
$83$	−6.65182	−0.730132	−0.365066	−	0.930982i	$-0.618954\pi$
$83$	−6.65182	−0.730132	−0.365066	+	0.930982i	$0.618954\pi$
$84$	0	0
$85$	17.0050	1.84445
$86$	3.84121	0.414208
$87$	0	0
$88$	0	0
$89$	12.2110	1.29436	0.647181	−	0.762337i	$-0.275948\pi$
$89$	12.2110	1.29436	0.647181	+	0.762337i	$0.275948\pi$
$90$	0	0
$91$	20.5182	2.15089
$92$	−0.169185	−0.0176387
$93$	0	0
$94$	1.03768	0.107028
$95$	17.0132	1.74552
$96$	0	0
$97$	17.5359	1.78050	0.890249	−	0.455474i	$-0.150530\pi$
$97$	17.5359	1.78050	0.890249	+	0.455474i	$0.150530\pi$
$98$	6.86764	0.693737
$99$	0	0
$100$	10.1292	1.01292
$101$	−6.73880	−0.670536	−0.335268	−	0.942123i	$-0.608827\pi$
$101$	−6.73880	−0.670536	−0.335268	+	0.942123i	$0.608827\pi$
$102$	0	0
$103$	−14.9510	−1.47317	−0.736583	−	0.676347i	$-0.763562\pi$
$103$	−14.9510	−1.47317	−0.736583	+	0.676347i	$0.763562\pi$
$104$	−5.50981	−0.540282
$105$	0	0
$106$	6.91374	0.671521
$107$	4.06329	0.392813	0.196407	−	0.980523i	$-0.437073\pi$
$107$	4.06329	0.392813	0.196407	+	0.980523i	$0.437073\pi$
$108$	0	0
$109$	15.7612	1.50965	0.754825	−	0.655927i	$-0.227722\pi$
$109$	15.7612	1.50965	0.754825	+	0.655927i	$0.227722\pi$
$110$	0	0
$111$	0	0
$112$	−3.72393	−0.351878
$113$	−0.634344	−0.0596741	−0.0298370	−	0.999555i	$-0.509499\pi$
$113$	−0.634344	−0.0596741	−0.0298370	+	0.999555i	$0.509499\pi$
$114$	0	0
$115$	0.658065	0.0613649
$116$	−8.39728	−0.779668
$117$	0	0
$118$	2.69189	0.247809
$119$	16.2806	1.49244
$120$	0	0
$121$	0	0
$122$	−10.6852	−0.967395
$123$	0	0
$124$	−0.337037	−0.0302668
$125$	−19.9506	−1.78444
$126$	0	0
$127$	8.64282	0.766926	0.383463	−	0.923556i	$-0.374731\pi$
$127$	8.64282	0.766926	0.383463	+	0.923556i	$0.374731\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	21.4311	1.87963
$131$	0.692672	0.0605191	0.0302595	−	0.999542i	$-0.490367\pi$
$131$	0.692672	0.0605191	0.0302595	+	0.999542i	$0.490367\pi$
$132$	0	0
$133$	16.2885	1.41239
$134$	−11.4875	−0.992365
$135$	0	0
$136$	−4.37189	−0.374886
$137$	−9.77714	−0.835317	−0.417659	−	0.908604i	$-0.637149\pi$
$137$	−9.77714	−0.835317	−0.417659	+	0.908604i	$0.637149\pi$
$138$	0	0
$139$	3.00759	0.255100	0.127550	−	0.991832i	$-0.459289\pi$
$139$	3.00759	0.255100	0.127550	+	0.991832i	$0.459289\pi$
$140$	14.4847	1.22418
$141$	0	0
$142$	−12.7472	−1.06972
$143$	0	0
$144$	0	0
$145$	32.6623	2.71245
$146$	5.27159	0.436280
$147$	0	0
$148$	1.89975	0.156158
$149$	−9.16847	−0.751110	−0.375555	−	0.926800i	$-0.622548\pi$
$149$	−9.16847	−0.751110	−0.375555	+	0.926800i	$0.622548\pi$
$150$	0	0
$151$	1.84056	0.149783	0.0748914	−	0.997192i	$-0.476139\pi$
$151$	1.84056	0.149783	0.0748914	+	0.997192i	$0.476139\pi$
$152$	−4.37400	−0.354778
$153$	0	0
$154$	0	0
$155$	1.31095	0.105298
$156$	0	0
$157$	−7.95486	−0.634867	−0.317433	−	0.948281i	$-0.602821\pi$
$157$	−7.95486	−0.634867	−0.317433	+	0.948281i	$0.602821\pi$
$158$	10.2281	0.813704
$159$	0	0
$160$	−3.88962	−0.307502
$161$	0.630032	0.0496535
$162$	0	0
$163$	−1.10949	−0.0869017	−0.0434509	−	0.999056i	$-0.513835\pi$
$163$	−1.10949	−0.0869017	−0.0434509	+	0.999056i	$0.513835\pi$
$164$	−4.97703	−0.388640
$165$	0	0
$166$	−6.65182	−0.516281
$167$	−5.03898	−0.389928	−0.194964	−	0.980810i	$-0.562459\pi$
$167$	−5.03898	−0.389928	−0.194964	+	0.980810i	$0.562459\pi$
$168$	0	0
$169$	17.3581	1.33524
$170$	17.0050	1.30422
$171$	0	0
$172$	3.84121	0.292889
$173$	−2.21200	−0.168175	−0.0840877	−	0.996458i	$-0.526798\pi$
$173$	−2.21200	−0.168175	−0.0840877	+	0.996458i	$0.526798\pi$
$174$	0	0
$175$	−37.7203	−2.85139
$176$	0	0
$177$	0	0
$178$	12.2110	0.915252
$179$	−20.1845	−1.50866	−0.754332	−	0.656493i	$-0.772039\pi$
$179$	−20.1845	−1.50866	−0.754332	+	0.656493i	$0.772039\pi$
$180$	0	0
$181$	18.3099	1.36097	0.680483	−	0.732764i	$-0.261770\pi$
$181$	18.3099	1.36097	0.680483	+	0.732764i	$0.261770\pi$
$182$	20.5182	1.52091
$183$	0	0
$184$	−0.169185	−0.0124725
$185$	−7.38931	−0.543273
$186$	0	0
$187$	0	0
$188$	1.03768	0.0756805
$189$	0	0
$190$	17.0132	1.23427
$191$	5.91014	0.427643	0.213821	−	0.976873i	$-0.431409\pi$
$191$	5.91014	0.427643	0.213821	+	0.976873i	$0.431409\pi$
$192$	0	0
$193$	−7.16369	−0.515654	−0.257827	−	0.966191i	$-0.583006\pi$
$193$	−7.16369	−0.515654	−0.257827	+	0.966191i	$0.583006\pi$
$194$	17.5359	1.25900
$195$	0	0
$196$	6.86764	0.490546
$197$	−18.0381	−1.28516	−0.642581	−	0.766218i	$-0.722136\pi$
$197$	−18.0381	−1.28516	−0.642581	+	0.766218i	$0.722136\pi$
$198$	0	0
$199$	−12.5714	−0.891166	−0.445583	−	0.895241i	$-0.647004\pi$
$199$	−12.5714	−0.891166	−0.445583	+	0.895241i	$0.647004\pi$
$200$	10.1292	0.716241
$201$	0	0
$202$	−6.73880	−0.474140
$203$	31.2709	2.19479
$204$	0	0
$205$	19.3588	1.35207
$206$	−14.9510	−1.04169
$207$	0	0
$208$	−5.50981	−0.382037
$209$	0	0
$210$	0	0
$211$	22.5656	1.55348	0.776741	−	0.629820i	$-0.216871\pi$
$211$	22.5656	1.55348	0.776741	+	0.629820i	$0.216871\pi$
$212$	6.91374	0.474837
$213$	0	0
$214$	4.06329	0.277761
$215$	−14.9409	−1.01896
$216$	0	0
$217$	1.25510	0.0852019
$218$	15.7612	1.06748
$219$	0	0
$220$	0	0
$221$	24.0883	1.62035
$222$	0	0
$223$	25.6187	1.71556	0.857778	−	0.514020i	$-0.171844\pi$
$223$	25.6187	1.71556	0.857778	+	0.514020i	$0.171844\pi$
$224$	−3.72393	−0.248815
$225$	0	0
$226$	−0.634344	−0.0421959
$227$	9.61057	0.637876	0.318938	−	0.947776i	$-0.396674\pi$
$227$	9.61057	0.637876	0.318938	+	0.947776i	$0.396674\pi$
$228$	0	0
$229$	−3.14702	−0.207961	−0.103980	−	0.994579i	$-0.533158\pi$
$229$	−3.14702	−0.207961	−0.103980	+	0.994579i	$0.533158\pi$
$230$	0.658065	0.0433915
$231$	0	0
$232$	−8.39728	−0.551309
$233$	−3.37892	−0.221361	−0.110680	−	0.993856i	$-0.535303\pi$
$233$	−3.37892	−0.221361	−0.110680	+	0.993856i	$0.535303\pi$
$234$	0	0
$235$	−4.03618	−0.263291
$236$	2.69189	0.175227
$237$	0	0
$238$	16.2806	1.05531
$239$	2.96663	0.191896	0.0959478	−	0.995386i	$-0.469412\pi$
$239$	2.96663	0.191896	0.0959478	+	0.995386i	$0.469412\pi$
$240$	0	0
$241$	22.0533	1.42058	0.710289	−	0.703910i	$-0.248564\pi$
$241$	22.0533	1.42058	0.710289	+	0.703910i	$0.248564\pi$
$242$	0	0
$243$	0	0
$244$	−10.6852	−0.684052
$245$	−26.7126	−1.70660
$246$	0	0
$247$	24.0999	1.53344
$248$	−0.337037	−0.0214019
$249$	0	0
$250$	−19.9506	−1.26179
$251$	−15.6905	−0.990378	−0.495189	−	0.868785i	$-0.664901\pi$
$251$	−15.6905	−0.990378	−0.495189	+	0.868785i	$0.664901\pi$
$252$	0	0
$253$	0	0
$254$	8.64282	0.542299
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.3955	0.897969	0.448985	−	0.893539i	$-0.351786\pi$
$257$	14.3955	0.897969	0.448985	+	0.893539i	$0.351786\pi$
$258$	0	0
$259$	−7.07453	−0.439590
$260$	21.4311	1.32910
$261$	0	0
$262$	0.692672	0.0427934
$263$	−19.2655	−1.18796	−0.593980	−	0.804480i	$-0.702444\pi$
$263$	−19.2655	−1.18796	−0.593980	+	0.804480i	$0.702444\pi$
$264$	0	0
$265$	−26.8918	−1.65195
$266$	16.2885	0.998709
$267$	0	0
$268$	−11.4875	−0.701708
$269$	−28.9688	−1.76626	−0.883130	−	0.469128i	$-0.844568\pi$
$269$	−28.9688	−1.76626	−0.883130	+	0.469128i	$0.844568\pi$
$270$	0	0
$271$	−19.8648	−1.20670	−0.603351	−	0.797476i	$-0.706168\pi$
$271$	−19.8648	−1.20670	−0.603351	+	0.797476i	$0.706168\pi$
$272$	−4.37189	−0.265085
$273$	0	0
$274$	−9.77714	−0.590659
$275$	0	0
$276$	0	0
$277$	15.9415	0.957829	0.478915	−	0.877861i	$-0.341030\pi$
$277$	15.9415	0.957829	0.478915	+	0.877861i	$0.341030\pi$
$278$	3.00759	0.180383
$279$	0	0
$280$	14.4847	0.865625
$281$	8.07105	0.481478	0.240739	−	0.970590i	$-0.422610\pi$
$281$	8.07105	0.481478	0.240739	+	0.970590i	$0.422610\pi$
$282$	0	0
$283$	−21.0217	−1.24961	−0.624806	−	0.780780i	$-0.714822\pi$
$283$	−21.0217	−1.24961	−0.624806	+	0.780780i	$0.714822\pi$
$284$	−12.7472	−0.756406
$285$	0	0
$286$	0	0
$287$	18.5341	1.09403
$288$	0	0
$289$	2.11339	0.124317
$290$	32.6623	1.91800
$291$	0	0
$292$	5.27159	0.308497
$293$	−10.8568	−0.634259	−0.317130	−	0.948382i	$-0.602719\pi$
$293$	−10.8568	−0.634259	−0.317130	+	0.948382i	$0.602719\pi$
$294$	0	0
$295$	−10.4705	−0.609614
$296$	1.89975	0.110421
$297$	0	0
$298$	−9.16847	−0.531115
$299$	0.932177	0.0539092
$300$	0	0
$301$	−14.3044	−0.824491
$302$	1.84056	0.105912
$303$	0	0
$304$	−4.37400	−0.250866
$305$	41.5615	2.37981
$306$	0	0
$307$	−11.3580	−0.648235	−0.324118	−	0.946017i	$-0.605067\pi$
$307$	−11.3580	−0.648235	−0.324118	+	0.946017i	$0.605067\pi$
$308$	0	0
$309$	0	0
$310$	1.31095	0.0744568
$311$	0.830666	0.0471027	0.0235514	−	0.999723i	$-0.492503\pi$
$311$	0.830666	0.0471027	0.0235514	+	0.999723i	$0.492503\pi$
$312$	0	0
$313$	1.68426	0.0951999	0.0476000	−	0.998866i	$-0.484843\pi$
$313$	1.68426	0.0951999	0.0476000	+	0.998866i	$0.484843\pi$
$314$	−7.95486	−0.448919
$315$	0	0
$316$	10.2281	0.575376
$317$	18.1677	1.02040	0.510199	−	0.860056i	$-0.329572\pi$
$317$	18.1677	1.02040	0.510199	+	0.860056i	$0.329572\pi$
$318$	0	0
$319$	0	0
$320$	−3.88962	−0.217437
$321$	0	0
$322$	0.630032	0.0351103
$323$	19.1226	1.06401
$324$	0	0
$325$	−55.8099	−3.09578
$326$	−1.10949	−0.0614488
$327$	0	0
$328$	−4.97703	−0.274810
$329$	−3.86424	−0.213043
$330$	0	0
$331$	24.1155	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$331$	24.1155	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$332$	−6.65182	−0.365066
$333$	0	0
$334$	−5.03898	−0.275721
$335$	44.6819	2.44123
$336$	0	0
$337$	7.15948	0.390002	0.195001	−	0.980803i	$-0.437529\pi$
$337$	7.15948	0.390002	0.195001	+	0.980803i	$0.437529\pi$
$338$	17.3581	0.944154
$339$	0	0
$340$	17.0050	0.922225
$341$	0	0
$342$	0	0
$343$	0.492889	0.0266135
$344$	3.84121	0.207104
$345$	0	0
$346$	−2.21200	−0.118918
$347$	1.98859	0.106753	0.0533765	−	0.998574i	$-0.483002\pi$
$347$	1.98859	0.106753	0.0533765	+	0.998574i	$0.483002\pi$
$348$	0	0
$349$	13.3795	0.716187	0.358093	−	0.933686i	$-0.383427\pi$
$349$	13.3795	0.716187	0.358093	+	0.933686i	$0.383427\pi$
$350$	−37.7203	−2.01624
$351$	0	0
$352$	0	0
$353$	0.543932	0.0289506	0.0144753	−	0.999895i	$-0.495392\pi$
$353$	0.543932	0.0289506	0.0144753	+	0.999895i	$0.495392\pi$
$354$	0	0
$355$	49.5818	2.63153
$356$	12.2110	0.647181
$357$	0	0
$358$	−20.1845	−1.06679
$359$	18.6625	0.984967	0.492483	−	0.870322i	$-0.336089\pi$
$359$	18.6625	0.984967	0.492483	+	0.870322i	$0.336089\pi$
$360$	0	0
$361$	0.131848	0.00693939
$362$	18.3099	0.962348
$363$	0	0
$364$	20.5182	1.07544
$365$	−20.5045	−1.07326
$366$	0	0
$367$	5.70810	0.297960	0.148980	−	0.988840i	$-0.452401\pi$
$367$	5.70810	0.297960	0.148980	+	0.988840i	$0.452401\pi$
$368$	−0.169185	−0.00881937
$369$	0	0
$370$	−7.38931	−0.384152
$371$	−25.7463	−1.33668
$372$	0	0
$373$	−37.6045	−1.94709	−0.973544	−	0.228498i	$-0.926619\pi$
$373$	−37.6045	−1.94709	−0.973544	+	0.228498i	$0.926619\pi$
$374$	0	0
$375$	0	0
$376$	1.03768	0.0535142
$377$	46.2675	2.38290
$378$	0	0
$379$	21.4100	1.09976	0.549878	−	0.835245i	$-0.314674\pi$
$379$	21.4100	1.09976	0.549878	+	0.835245i	$0.314674\pi$
$380$	17.0132	0.872759
$381$	0	0
$382$	5.91014	0.302389
$383$	−34.7444	−1.77536	−0.887678	−	0.460465i	$-0.847683\pi$
$383$	−34.7444	−1.77536	−0.887678	+	0.460465i	$0.847683\pi$
$384$	0	0
$385$	0	0
$386$	−7.16369	−0.364622
$387$	0	0
$388$	17.5359	0.890249
$389$	−8.66923	−0.439547	−0.219774	−	0.975551i	$-0.570532\pi$
$389$	−8.66923	−0.439547	−0.219774	+	0.975551i	$0.570532\pi$
$390$	0	0
$391$	0.739657	0.0374060
$392$	6.86764	0.346868
$393$	0	0
$394$	−18.0381	−0.908746
$395$	−39.7835	−2.00172
$396$	0	0
$397$	13.0635	0.655640	0.327820	−	0.944740i	$-0.393686\pi$
$397$	13.0635	0.655640	0.327820	+	0.944740i	$0.393686\pi$
$398$	−12.5714	−0.630149
$399$	0	0
$400$	10.1292	0.506459
$401$	−3.29858	−0.164723	−0.0823616	−	0.996603i	$-0.526246\pi$
$401$	−3.29858	−0.164723	−0.0823616	+	0.996603i	$0.526246\pi$
$402$	0	0
$403$	1.85701	0.0925044
$404$	−6.73880	−0.335268
$405$	0	0
$406$	31.2709	1.55195
$407$	0	0
$408$	0	0
$409$	−26.0301	−1.28710	−0.643552	−	0.765402i	$-0.722540\pi$
$409$	−26.0301	−1.28710	−0.643552	+	0.765402i	$0.722540\pi$
$410$	19.3588	0.956061
$411$	0	0
$412$	−14.9510	−0.736583
$413$	−10.0244	−0.493269
$414$	0	0
$415$	25.8731	1.27006
$416$	−5.50981	−0.270141
$417$	0	0
$418$	0	0
$419$	22.7916	1.11344	0.556722	−	0.830699i	$-0.312059\pi$
$419$	22.7916	1.11344	0.556722	+	0.830699i	$0.312059\pi$
$420$	0	0
$421$	−6.59051	−0.321202	−0.160601	−	0.987019i	$-0.551343\pi$
$421$	−6.59051	−0.321202	−0.160601	+	0.987019i	$0.551343\pi$
$422$	22.5656	1.09848
$423$	0	0
$424$	6.91374	0.335761
$425$	−44.2836	−2.14807
$426$	0	0
$427$	39.7910	1.92562
$428$	4.06329	0.196407
$429$	0	0
$430$	−14.9409	−0.720512
$431$	3.62968	0.174836	0.0874178	−	0.996172i	$-0.472138\pi$
$431$	3.62968	0.174836	0.0874178	+	0.996172i	$0.472138\pi$
$432$	0	0
$433$	1.05071	0.0504938	0.0252469	−	0.999681i	$-0.491963\pi$
$433$	1.05071	0.0504938	0.0252469	+	0.999681i	$0.491963\pi$
$434$	1.25510	0.0602468
$435$	0	0
$436$	15.7612	0.754825
$437$	0.740014	0.0353997
$438$	0	0
$439$	−35.1100	−1.67571	−0.837854	−	0.545894i	$-0.816190\pi$
$439$	−35.1100	−1.67571	−0.837854	+	0.545894i	$0.816190\pi$
$440$	0	0
$441$	0	0
$442$	24.0883	1.14576
$443$	−24.6020	−1.16888	−0.584438	−	0.811439i	$-0.698685\pi$
$443$	−24.6020	−1.16888	−0.584438	+	0.811439i	$0.698685\pi$
$444$	0	0
$445$	−47.4961	−2.25153
$446$	25.6187	1.21308
$447$	0	0
$448$	−3.72393	−0.175939
$449$	22.6302	1.06799	0.533993	−	0.845489i	$-0.320691\pi$
$449$	22.6302	1.06799	0.533993	+	0.845489i	$0.320691\pi$
$450$	0	0
$451$	0	0
$452$	−0.634344	−0.0298370
$453$	0	0
$454$	9.61057	0.451046
$455$	−79.8079	−3.74145
$456$	0	0
$457$	−1.46787	−0.0686640	−0.0343320	−	0.999410i	$-0.510930\pi$
$457$	−1.46787	−0.0686640	−0.0343320	+	0.999410i	$0.510930\pi$
$458$	−3.14702	−0.147050
$459$	0	0
$460$	0.658065	0.0306825
$461$	−3.89222	−0.181279	−0.0906393	−	0.995884i	$-0.528891\pi$
$461$	−3.89222	−0.181279	−0.0906393	+	0.995884i	$0.528891\pi$
$462$	0	0
$463$	0.722010	0.0335547	0.0167773	−	0.999859i	$-0.494659\pi$
$463$	0.722010	0.0335547	0.0167773	+	0.999859i	$0.494659\pi$
$464$	−8.39728	−0.389834
$465$	0	0
$466$	−3.37892	−0.156526
$467$	10.4531	0.483713	0.241857	−	0.970312i	$-0.422244\pi$
$467$	10.4531	0.483713	0.241857	+	0.970312i	$0.422244\pi$
$468$	0	0
$469$	42.7785	1.97533
$470$	−4.03618	−0.186175
$471$	0	0
$472$	2.69189	0.123904
$473$	0	0
$474$	0	0
$475$	−44.3050	−2.03285
$476$	16.2806	0.746220
$477$	0	0
$478$	2.96663	0.135691
$479$	21.5459	0.984459	0.492230	−	0.870465i	$-0.336182\pi$
$479$	21.5459	0.984459	0.492230	+	0.870465i	$0.336182\pi$
$480$	0	0
$481$	−10.4673	−0.477266
$482$	22.0533	1.00450
$483$	0	0
$484$	0	0
$485$	−68.2080	−3.09716
$486$	0	0
$487$	2.32816	0.105499	0.0527496	−	0.998608i	$-0.483201\pi$
$487$	2.32816	0.105499	0.0527496	+	0.998608i	$0.483201\pi$
$488$	−10.6852	−0.483698
$489$	0	0
$490$	−26.7126	−1.20675
$491$	−13.4826	−0.608462	−0.304231	−	0.952598i	$-0.598399\pi$
$491$	−13.4826	−0.608462	−0.304231	+	0.952598i	$0.598399\pi$
$492$	0	0
$493$	36.7120	1.65342
$494$	24.0999	1.08431
$495$	0	0
$496$	−0.337037	−0.0151334
$497$	47.4696	2.12930
$498$	0	0
$499$	42.6919	1.91115	0.955575	−	0.294747i	$-0.0952355\pi$
$499$	42.6919	1.91115	0.955575	+	0.294747i	$0.0952355\pi$
$500$	−19.9506	−0.892218
$501$	0	0
$502$	−15.6905	−0.700303
$503$	29.1100	1.29795	0.648976	−	0.760809i	$-0.275198\pi$
$503$	29.1100	1.29795	0.648976	+	0.760809i	$0.275198\pi$
$504$	0	0
$505$	26.2114	1.16639
$506$	0	0
$507$	0	0
$508$	8.64282	0.383463
$509$	−19.2578	−0.853586	−0.426793	−	0.904349i	$-0.640357\pi$
$509$	−19.2578	−0.853586	−0.426793	+	0.904349i	$0.640357\pi$
$510$	0	0
$511$	−19.6310	−0.868425
$512$	1.00000	0.0441942
$513$	0	0
$514$	14.3955	0.634960
$515$	58.1538	2.56256
$516$	0	0
$517$	0	0
$518$	−7.07453	−0.310837
$519$	0	0
$520$	21.4311	0.939816
$521$	−29.7366	−1.30279	−0.651393	−	0.758740i	$-0.725815\pi$
$521$	−29.7366	−1.30279	−0.651393	+	0.758740i	$0.725815\pi$
$522$	0	0
$523$	−13.4944	−0.590070	−0.295035	−	0.955486i	$-0.595331\pi$
$523$	−13.4944	−0.590070	−0.295035	+	0.955486i	$0.595331\pi$
$524$	0.692672	0.0302595
$525$	0	0
$526$	−19.2655	−0.840015
$527$	1.47349	0.0641861
$528$	0	0
$529$	−22.9714	−0.998756
$530$	−26.8918	−1.16811
$531$	0	0
$532$	16.2885	0.706194
$533$	27.4225	1.18780
$534$	0	0
$535$	−15.8047	−0.683296
$536$	−11.4875	−0.496182
$537$	0	0
$538$	−28.9688	−1.24893
$539$	0	0
$540$	0	0
$541$	13.5154	0.581074	0.290537	−	0.956864i	$-0.406166\pi$
$541$	13.5154	0.581074	0.290537	+	0.956864i	$0.406166\pi$
$542$	−19.8648	−0.853267
$543$	0	0
$544$	−4.37189	−0.187443
$545$	−61.3052	−2.62602
$546$	0	0
$547$	−22.7611	−0.973193	−0.486596	−	0.873627i	$-0.661762\pi$
$547$	−22.7611	−0.973193	−0.486596	+	0.873627i	$0.661762\pi$
$548$	−9.77714	−0.417659
$549$	0	0
$550$	0	0
$551$	36.7297	1.56474
$552$	0	0
$553$	−38.0887	−1.61970
$554$	15.9415	0.677288
$555$	0	0
$556$	3.00759	0.127550
$557$	6.41046	0.271620	0.135810	−	0.990735i	$-0.456636\pi$
$557$	6.41046	0.271620	0.135810	+	0.990735i	$0.456636\pi$
$558$	0	0
$559$	−21.1643	−0.895156
$560$	14.4847	0.612090
$561$	0	0
$562$	8.07105	0.340457
$563$	6.52972	0.275195	0.137597	−	0.990488i	$-0.456062\pi$
$563$	6.52972	0.275195	0.137597	+	0.990488i	$0.456062\pi$
$564$	0	0
$565$	2.46736	0.103803
$566$	−21.0217	−0.883610
$567$	0	0
$568$	−12.7472	−0.534860
$569$	−32.1193	−1.34651	−0.673255	−	0.739411i	$-0.735104\pi$
$569$	−32.1193	−1.34651	−0.673255	+	0.739411i	$0.735104\pi$
$570$	0	0
$571$	0.178273	0.00746047	0.00373024	−	0.999993i	$-0.498813\pi$
$571$	0.178273	0.00746047	0.00373024	+	0.999993i	$0.498813\pi$
$572$	0	0
$573$	0	0
$574$	18.5341	0.773598
$575$	−1.71370	−0.0714664
$576$	0	0
$577$	−37.2692	−1.55154	−0.775769	−	0.631017i	$-0.782638\pi$
$577$	−37.2692	−1.55154	−0.775769	+	0.631017i	$0.782638\pi$
$578$	2.11339	0.0879053
$579$	0	0
$580$	32.6623	1.35623
$581$	24.7709	1.02767
$582$	0	0
$583$	0	0
$584$	5.27159	0.218140
$585$	0	0
$586$	−10.8568	−0.448489
$587$	−17.4735	−0.721208	−0.360604	−	0.932719i	$-0.617429\pi$
$587$	−17.4735	−0.721208	−0.360604	+	0.932719i	$0.617429\pi$
$588$	0	0
$589$	1.47420	0.0607433
$590$	−10.4705	−0.431062
$591$	0	0
$592$	1.89975	0.0780792
$593$	33.4398	1.37321	0.686603	−	0.727032i	$-0.259101\pi$
$593$	33.4398	1.37321	0.686603	+	0.727032i	$0.259101\pi$
$594$	0	0
$595$	−63.3254	−2.59609
$596$	−9.16847	−0.375555
$597$	0	0
$598$	0.932177	0.0381195
$599$	−22.4464	−0.917136	−0.458568	−	0.888659i	$-0.651637\pi$
$599$	−22.4464	−0.917136	−0.458568	+	0.888659i	$0.651637\pi$
$600$	0	0
$601$	25.1280	1.02499	0.512497	−	0.858689i	$-0.328721\pi$
$601$	25.1280	1.02499	0.512497	+	0.858689i	$0.328721\pi$
$602$	−14.3044	−0.583003
$603$	0	0
$604$	1.84056	0.0748914
$605$	0	0
$606$	0	0
$607$	11.7513	0.476971	0.238485	−	0.971146i	$-0.423349\pi$
$607$	11.7513	0.476971	0.238485	+	0.971146i	$0.423349\pi$
$608$	−4.37400	−0.177389
$609$	0	0
$610$	41.5615	1.68278
$611$	−5.71742	−0.231302
$612$	0	0
$613$	16.8488	0.680517	0.340258	−	0.940332i	$-0.389486\pi$
$613$	16.8488	0.680517	0.340258	+	0.940332i	$0.389486\pi$
$614$	−11.3580	−0.458372
$615$	0	0
$616$	0	0
$617$	−29.4952	−1.18743	−0.593716	−	0.804675i	$-0.702340\pi$
$617$	−29.4952	−1.18743	−0.593716	+	0.804675i	$0.702340\pi$
$618$	0	0
$619$	−19.0347	−0.765068	−0.382534	−	0.923941i	$-0.624948\pi$
$619$	−19.0347	−0.765068	−0.382534	+	0.923941i	$0.624948\pi$
$620$	1.31095	0.0526489
$621$	0	0
$622$	0.830666	0.0333067
$623$	−45.4728	−1.82183
$624$	0	0
$625$	26.9544	1.07818
$626$	1.68426	0.0673165
$627$	0	0
$628$	−7.95486	−0.317433
$629$	−8.30549	−0.331162
$630$	0	0
$631$	0.464051	0.0184736	0.00923679	−	0.999957i	$-0.497060\pi$
$631$	0.464051	0.0184736	0.00923679	+	0.999957i	$0.497060\pi$
$632$	10.2281	0.406852
$633$	0	0
$634$	18.1677	0.721531
$635$	−33.6173	−1.33406
$636$	0	0
$637$	−37.8394	−1.49925
$638$	0	0
$639$	0	0
$640$	−3.88962	−0.153751
$641$	48.0978	1.89975	0.949874	−	0.312633i	$-0.101211\pi$
$641$	48.0978	1.89975	0.949874	+	0.312633i	$0.101211\pi$
$642$	0	0
$643$	9.91559	0.391033	0.195516	−	0.980700i	$-0.437362\pi$
$643$	9.91559	0.391033	0.195516	+	0.980700i	$0.437362\pi$
$644$	0.630032	0.0248267
$645$	0	0
$646$	19.1226	0.752369
$647$	−45.0843	−1.77245	−0.886224	−	0.463257i	$-0.846681\pi$
$647$	−45.0843	−1.77245	−0.886224	+	0.463257i	$0.846681\pi$
$648$	0	0
$649$	0	0
$650$	−55.8099	−2.18904
$651$	0	0
$652$	−1.10949	−0.0434509
$653$	−7.67438	−0.300322	−0.150161	−	0.988662i	$-0.547979\pi$
$653$	−7.67438	−0.300322	−0.150161	+	0.988662i	$0.547979\pi$
$654$	0	0
$655$	−2.69424	−0.105273
$656$	−4.97703	−0.194320
$657$	0	0
$658$	−3.86424	−0.150644
$659$	21.4800	0.836743	0.418372	−	0.908276i	$-0.362601\pi$
$659$	21.4800	0.836743	0.418372	+	0.908276i	$0.362601\pi$
$660$	0	0
$661$	35.0869	1.36472	0.682362	−	0.731015i	$-0.260953\pi$
$661$	35.0869	1.36472	0.682362	+	0.731015i	$0.260953\pi$
$662$	24.1155	0.937276
$663$	0	0
$664$	−6.65182	−0.258141
$665$	−63.3560	−2.45684
$666$	0	0
$667$	1.42069	0.0550094
$668$	−5.03898	−0.194964
$669$	0	0
$670$	44.6819	1.72621
$671$	0	0
$672$	0	0
$673$	−8.43001	−0.324953	−0.162476	−	0.986712i	$-0.551948\pi$
$673$	−8.43001	−0.324953	−0.162476	+	0.986712i	$0.551948\pi$
$674$	7.15948	0.275773
$675$	0	0
$676$	17.3581	0.667618
$677$	−14.4454	−0.555184	−0.277592	−	0.960699i	$-0.589536\pi$
$677$	−14.4454	−0.555184	−0.277592	+	0.960699i	$0.589536\pi$
$678$	0	0
$679$	−65.3023	−2.50607
$680$	17.0050	0.652112
$681$	0	0
$682$	0	0
$683$	−36.5728	−1.39942	−0.699710	−	0.714427i	$-0.746688\pi$
$683$	−36.5728	−1.39942	−0.699710	+	0.714427i	$0.746688\pi$
$684$	0	0
$685$	38.0294	1.45303
$686$	0.492889	0.0188186
$687$	0	0
$688$	3.84121	0.146445
$689$	−38.0934	−1.45124
$690$	0	0
$691$	19.1400	0.728120	0.364060	−	0.931376i	$-0.381390\pi$
$691$	19.1400	0.728120	0.364060	+	0.931376i	$0.381390\pi$
$692$	−2.21200	−0.0840877
$693$	0	0
$694$	1.98859	0.0754857
$695$	−11.6984	−0.443745
$696$	0	0
$697$	21.7590	0.824180
$698$	13.3795	0.506421
$699$	0	0
$700$	−37.7203	−1.42570
$701$	−23.1812	−0.875543	−0.437771	−	0.899086i	$-0.644232\pi$
$701$	−23.1812	−0.875543	−0.437771	+	0.899086i	$0.644232\pi$
$702$	0	0
$703$	−8.30950	−0.313399
$704$	0	0
$705$	0	0
$706$	0.543932	0.0204712
$707$	25.0948	0.943788
$708$	0	0
$709$	45.3651	1.70372	0.851860	−	0.523769i	$-0.175475\pi$
$709$	45.3651	1.70372	0.851860	+	0.523769i	$0.175475\pi$
$710$	49.5818	1.86077
$711$	0	0
$712$	12.2110	0.457626
$713$	0.0570216	0.00213547
$714$	0	0
$715$	0	0
$716$	−20.1845	−0.754332
$717$	0	0
$718$	18.6625	0.696477
$719$	−0.559793	−0.0208767	−0.0104384	−	0.999946i	$-0.503323\pi$
$719$	−0.559793	−0.0208767	−0.0104384	+	0.999946i	$0.503323\pi$
$720$	0	0
$721$	55.6765	2.07350
$722$	0.131848	0.00490689
$723$	0	0
$724$	18.3099	0.680483
$725$	−85.0576	−3.15896
$726$	0	0
$727$	−21.5446	−0.799045	−0.399523	−	0.916723i	$-0.630824\pi$
$727$	−21.5446	−0.799045	−0.399523	+	0.916723i	$0.630824\pi$
$728$	20.5182	0.760453
$729$	0	0
$730$	−20.5045	−0.758906
$731$	−16.7933	−0.621123
$732$	0	0
$733$	−32.2240	−1.19022	−0.595111	−	0.803643i	$-0.702892\pi$
$733$	−32.2240	−1.19022	−0.595111	+	0.803643i	$0.702892\pi$
$734$	5.70810	0.210690
$735$	0	0
$736$	−0.169185	−0.00623623
$737$	0	0
$738$	0	0
$739$	−1.79513	−0.0660350	−0.0330175	−	0.999455i	$-0.510512\pi$
$739$	−1.79513	−0.0660350	−0.0330175	+	0.999455i	$0.510512\pi$
$740$	−7.38931	−0.271637
$741$	0	0
$742$	−25.7463	−0.945175
$743$	−42.3582	−1.55397	−0.776986	−	0.629518i	$-0.783253\pi$
$743$	−42.3582	−1.55397	−0.776986	+	0.629518i	$0.783253\pi$
$744$	0	0
$745$	35.6619	1.30655
$746$	−37.6045	−1.37680
$747$	0	0
$748$	0	0
$749$	−15.1314	−0.552889
$750$	0	0
$751$	29.2172	1.06615	0.533075	−	0.846068i	$-0.321036\pi$
$751$	29.2172	1.06615	0.533075	+	0.846068i	$0.321036\pi$
$752$	1.03768	0.0378403
$753$	0	0
$754$	46.2675	1.68496
$755$	−7.15909	−0.260546
$756$	0	0
$757$	29.7290	1.08052	0.540259	−	0.841499i	$-0.318326\pi$
$757$	29.7290	1.08052	0.540259	+	0.841499i	$0.318326\pi$
$758$	21.4100	0.777645
$759$	0	0
$760$	17.0132	0.617134
$761$	15.8628	0.575024	0.287512	−	0.957777i	$-0.407172\pi$
$761$	15.8628	0.575024	0.287512	+	0.957777i	$0.407172\pi$
$762$	0	0
$763$	−58.6936	−2.12485
$764$	5.91014	0.213821
$765$	0	0
$766$	−34.7444	−1.25537
$767$	−14.8318	−0.535547
$768$	0	0
$769$	20.0451	0.722845	0.361423	−	0.932402i	$-0.382291\pi$
$769$	20.0451	0.722845	0.361423	+	0.932402i	$0.382291\pi$
$770$	0	0
$771$	0	0
$772$	−7.16369	−0.257827
$773$	−12.0655	−0.433967	−0.216984	−	0.976175i	$-0.569622\pi$
$773$	−12.0655	−0.433967	−0.216984	+	0.976175i	$0.569622\pi$
$774$	0	0
$775$	−3.41391	−0.122631
$776$	17.5359	0.629501
$777$	0	0
$778$	−8.66923	−0.310807
$779$	21.7695	0.779973
$780$	0	0
$781$	0	0
$782$	0.739657	0.0264501
$783$	0	0
$784$	6.86764	0.245273
$785$	30.9414	1.10435
$786$	0	0
$787$	−23.8997	−0.851933	−0.425967	−	0.904739i	$-0.640066\pi$
$787$	−23.8997	−0.851933	−0.425967	+	0.904739i	$0.640066\pi$
$788$	−18.0381	−0.642581
$789$	0	0
$790$	−39.7835	−1.41543
$791$	2.36225	0.0839920
$792$	0	0
$793$	58.8736	2.09066
$794$	13.0635	0.463608
$795$	0	0
$796$	−12.5714	−0.445583
$797$	−39.4934	−1.39893	−0.699463	−	0.714669i	$-0.746578\pi$
$797$	−39.4934	−1.39893	−0.699463	+	0.714669i	$0.746578\pi$
$798$	0	0
$799$	−4.53661	−0.160494
$800$	10.1292	0.358121
$801$	0	0
$802$	−3.29858	−0.116477
$803$	0	0
$804$	0	0
$805$	−2.45059	−0.0863719
$806$	1.85701	0.0654105
$807$	0	0
$808$	−6.73880	−0.237070
$809$	−15.8885	−0.558609	−0.279304	−	0.960203i	$-0.590104\pi$
$809$	−15.8885	−0.558609	−0.279304	+	0.960203i	$0.590104\pi$
$810$	0	0
$811$	−10.0176	−0.351765	−0.175882	−	0.984411i	$-0.556278\pi$
$811$	−10.0176	−0.351765	−0.175882	+	0.984411i	$0.556278\pi$
$812$	31.2709	1.09739
$813$	0	0
$814$	0	0
$815$	4.31549	0.151165
$816$	0	0
$817$	−16.8014	−0.587808
$818$	−26.0301	−0.910120
$819$	0	0
$820$	19.3588	0.676037
$821$	−43.6797	−1.52443	−0.762217	−	0.647322i	$-0.775889\pi$
$821$	−43.6797	−1.52443	−0.762217	+	0.647322i	$0.775889\pi$
$822$	0	0
$823$	−32.5753	−1.13550	−0.567751	−	0.823200i	$-0.692186\pi$
$823$	−32.5753	−1.13550	−0.567751	+	0.823200i	$0.692186\pi$
$824$	−14.9510	−0.520843
$825$	0	0
$826$	−10.0244	−0.348794
$827$	25.8332	0.898308	0.449154	−	0.893454i	$-0.351725\pi$
$827$	25.8332	0.898308	0.449154	+	0.893454i	$0.351725\pi$
$828$	0	0
$829$	−25.4608	−0.884289	−0.442145	−	0.896944i	$-0.645782\pi$
$829$	−25.4608	−0.884289	−0.442145	+	0.896944i	$0.645782\pi$
$830$	25.8731	0.898068
$831$	0	0
$832$	−5.50981	−0.191018
$833$	−30.0246	−1.04029
$834$	0	0
$835$	19.5998	0.678277
$836$	0	0
$837$	0	0
$838$	22.7916	0.787323
$839$	−17.1512	−0.592125	−0.296063	−	0.955169i	$-0.595674\pi$
$839$	−17.1512	−0.592125	−0.296063	+	0.955169i	$0.595674\pi$
$840$	0	0
$841$	41.5144	1.43153
$842$	−6.59051	−0.227124
$843$	0	0
$844$	22.5656	0.776741
$845$	−67.5163	−2.32263
$846$	0	0
$847$	0	0
$848$	6.91374	0.237419
$849$	0	0
$850$	−44.2836	−1.51892
$851$	−0.321409	−0.0110178
$852$	0	0
$853$	5.71717	0.195752	0.0978760	−	0.995199i	$-0.468795\pi$
$853$	5.71717	0.195752	0.0978760	+	0.995199i	$0.468795\pi$
$854$	39.7910	1.36162
$855$	0	0
$856$	4.06329	0.138880
$857$	13.0129	0.444511	0.222256	−	0.974988i	$-0.428658\pi$
$857$	13.0129	0.444511	0.222256	+	0.974988i	$0.428658\pi$
$858$	0	0
$859$	−36.5028	−1.24546	−0.622729	−	0.782437i	$-0.713976\pi$
$859$	−36.5028	−1.24546	−0.622729	+	0.782437i	$0.713976\pi$
$860$	−14.9409	−0.509479
$861$	0	0
$862$	3.62968	0.123627
$863$	7.72911	0.263102	0.131551	−	0.991309i	$-0.458004\pi$
$863$	7.72911	0.263102	0.131551	+	0.991309i	$0.458004\pi$
$864$	0	0
$865$	8.60386	0.292540
$866$	1.05071	0.0357045
$867$	0	0
$868$	1.25510	0.0426009
$869$	0	0
$870$	0	0
$871$	63.2937	2.14463
$872$	15.7612	0.533742
$873$	0	0
$874$	0.740014	0.0250313
$875$	74.2946	2.51161
$876$	0	0
$877$	−47.5327	−1.60507	−0.802533	−	0.596607i	$-0.796515\pi$
$877$	−47.5327	−1.60507	−0.802533	+	0.596607i	$0.796515\pi$
$878$	−35.1100	−1.18490
$879$	0	0
$880$	0	0
$881$	45.3262	1.52708	0.763539	−	0.645761i	$-0.223460\pi$
$881$	45.3262	1.52708	0.763539	+	0.645761i	$0.223460\pi$
$882$	0	0
$883$	−44.8274	−1.50856	−0.754281	−	0.656552i	$-0.772014\pi$
$883$	−44.8274	−1.50856	−0.754281	+	0.656552i	$0.772014\pi$
$884$	24.0883	0.810177
$885$	0	0
$886$	−24.6020	−0.826519
$887$	−12.7328	−0.427526	−0.213763	−	0.976886i	$-0.568572\pi$
$887$	−12.7328	−0.427526	−0.213763	+	0.976886i	$0.568572\pi$
$888$	0	0
$889$	−32.1852	−1.07946
$890$	−47.4961	−1.59207
$891$	0	0
$892$	25.6187	0.857778
$893$	−4.53880	−0.151885
$894$	0	0
$895$	78.5103	2.62431
$896$	−3.72393	−0.124408
$897$	0	0
$898$	22.6302	0.755181
$899$	2.83020	0.0943923
$900$	0	0
$901$	−30.2261	−1.00698
$902$	0	0
$903$	0	0
$904$	−0.634344	−0.0210980
$905$	−71.2187	−2.36739
$906$	0	0
$907$	7.83861	0.260277	0.130138	−	0.991496i	$-0.458458\pi$
$907$	7.83861	0.260277	0.130138	+	0.991496i	$0.458458\pi$
$908$	9.61057	0.318938
$909$	0	0
$910$	−79.8079	−2.64561
$911$	23.6254	0.782746	0.391373	−	0.920232i	$-0.372000\pi$
$911$	23.6254	0.782746	0.391373	+	0.920232i	$0.372000\pi$
$912$	0	0
$913$	0	0
$914$	−1.46787	−0.0485528
$915$	0	0
$916$	−3.14702	−0.103980
$917$	−2.57946	−0.0851814
$918$	0	0
$919$	56.7167	1.87091	0.935456	−	0.353442i	$-0.114989\pi$
$919$	56.7167	1.87091	0.935456	+	0.353442i	$0.114989\pi$
$920$	0.658065	0.0216958
$921$	0	0
$922$	−3.89222	−0.128183
$923$	70.2346	2.31180
$924$	0	0
$925$	19.2429	0.632703
$926$	0.722010	0.0237267
$927$	0	0
$928$	−8.39728	−0.275654
$929$	7.36655	0.241689	0.120844	−	0.992671i	$-0.461440\pi$
$929$	7.36655	0.241689	0.120844	+	0.992671i	$0.461440\pi$
$930$	0	0
$931$	−30.0390	−0.984490
$932$	−3.37892	−0.110680
$933$	0	0
$934$	10.4531	0.342037
$935$	0	0
$936$	0	0
$937$	−27.5554	−0.900194	−0.450097	−	0.892980i	$-0.648611\pi$
$937$	−27.5554	−0.900194	−0.450097	+	0.892980i	$0.648611\pi$
$938$	42.7785	1.39677
$939$	0	0
$940$	−4.03618	−0.131646
$941$	25.8859	0.843856	0.421928	−	0.906629i	$-0.361354\pi$
$941$	25.8859	0.843856	0.421928	+	0.906629i	$0.361354\pi$
$942$	0	0
$943$	0.842037	0.0274205
$944$	2.69189	0.0876137
$945$	0	0
$946$	0	0
$947$	40.0180	1.30041	0.650206	−	0.759758i	$-0.274683\pi$
$947$	40.0180	1.30041	0.650206	+	0.759758i	$0.274683\pi$
$948$	0	0
$949$	−29.0455	−0.942856
$950$	−44.3050	−1.43744
$951$	0	0
$952$	16.2806	0.527657
$953$	14.8324	0.480468	0.240234	−	0.970715i	$-0.422776\pi$
$953$	14.8324	0.480468	0.240234	+	0.970715i	$0.422776\pi$
$954$	0	0
$955$	−22.9882	−0.743882
$956$	2.96663	0.0959478
$957$	0	0
$958$	21.5459	0.696118
$959$	36.4094	1.17572
$960$	0	0
$961$	−30.8864	−0.996336
$962$	−10.4673	−0.337478
$963$	0	0
$964$	22.0533	0.710289
$965$	27.8641	0.896976
$966$	0	0
$967$	53.6333	1.72473	0.862366	−	0.506286i	$-0.168982\pi$
$967$	53.6333	1.72473	0.862366	+	0.506286i	$0.168982\pi$
$968$	0	0
$969$	0	0
$970$	−68.2080	−2.19003
$971$	8.30534	0.266531	0.133265	−	0.991080i	$-0.457454\pi$
$971$	8.30534	0.266531	0.133265	+	0.991080i	$0.457454\pi$
$972$	0	0
$973$	−11.2001	−0.359057
$974$	2.32816	0.0745992
$975$	0	0
$976$	−10.6852	−0.342026
$977$	1.11029	0.0355214	0.0177607	−	0.999842i	$-0.494346\pi$
$977$	1.11029	0.0355214	0.0177607	+	0.999842i	$0.494346\pi$
$978$	0	0
$979$	0	0
$980$	−26.7126	−0.853301
$981$	0	0
$982$	−13.4826	−0.430248
$983$	41.7968	1.33311	0.666556	−	0.745455i	$-0.267768\pi$
$983$	41.7968	1.33311	0.666556	+	0.745455i	$0.267768\pi$
$984$	0	0
$985$	70.1614	2.23553
$986$	36.7120	1.16915
$987$	0	0
$988$	24.0999	0.766720
$989$	−0.649874	−0.0206648
$990$	0	0
$991$	45.5023	1.44543	0.722715	−	0.691147i	$-0.242894\pi$
$991$	45.5023	1.44543	0.722715	+	0.691147i	$0.242894\pi$
$992$	−0.337037	−0.0107009
$993$	0	0
$994$	47.4696	1.50564
$995$	48.8982	1.55018
$996$	0	0
$997$	14.2855	0.452427	0.226213	−	0.974078i	$-0.427365\pi$
$997$	14.2855	0.452427	0.226213	+	0.974078i	$0.427365\pi$
$998$	42.6919	1.35139
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6534.2.a.cw.1.2		6
3.2	odd	2		6534.2.a.cv.1.5		6
11.2	odd	10		594.2.f.l.433.3	yes	12
11.6	odd	10		594.2.f.l.487.3	yes	12
11.10	odd	2		6534.2.a.cu.1.2		6
33.2	even	10		594.2.f.k.433.1	✓	12
33.17	even	10		594.2.f.k.487.1	yes	12
33.32	even	2		6534.2.a.cx.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
594.2.f.k.433.1	✓	12	33.2	even	10
594.2.f.k.487.1	yes	12	33.17	even	10
594.2.f.l.433.3	yes	12	11.2	odd	10
594.2.f.l.487.3	yes	12	11.6	odd	10
6534.2.a.cu.1.2		6	11.10	odd	2
6534.2.a.cv.1.5		6	3.2	odd	2
6534.2.a.cw.1.2		6	1.1	even	1	trivial
6534.2.a.cx.1.5		6	33.32	even	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:	\( N \)	\(=\)	\( 6534 = 2 \cdot 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6534.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.88962 q^{5} -3.72393 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.88962 q^{5} -3.72393 q^{7} +1.00000 q^{8} -3.88962 q^{10} -5.50981 q^{13} -3.72393 q^{14} +1.00000 q^{16} -4.37189 q^{17} -4.37400 q^{19} -3.88962 q^{20} -0.169185 q^{23} +10.1292 q^{25} -5.50981 q^{26} -3.72393 q^{28} -8.39728 q^{29} -0.337037 q^{31} +1.00000 q^{32} -4.37189 q^{34} +14.4847 q^{35} +1.89975 q^{37} -4.37400 q^{38} -3.88962 q^{40} -4.97703 q^{41} +3.84121 q^{43} -0.169185 q^{46} +1.03768 q^{47} +6.86764 q^{49} +10.1292 q^{50} -5.50981 q^{52} +6.91374 q^{53} -3.72393 q^{56} -8.39728 q^{58} +2.69189 q^{59} -10.6852 q^{61} -0.337037 q^{62} +1.00000 q^{64} +21.4311 q^{65} -11.4875 q^{67} -4.37189 q^{68} +14.4847 q^{70} -12.7472 q^{71} +5.27159 q^{73} +1.89975 q^{74} -4.37400 q^{76} +10.2281 q^{79} -3.88962 q^{80} -4.97703 q^{82} -6.65182 q^{83} +17.0050 q^{85} +3.84121 q^{86} +12.2110 q^{89} +20.5182 q^{91} -0.169185 q^{92} +1.03768 q^{94} +17.0132 q^{95} +17.5359 q^{97} +6.86764 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} - 2 q^{5} + q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^4 - 2 * q^5 + q^7 + 6 * q^8	\( 6 q + 6 q^{2} + 6 q^{4} - 2 q^{5} + q^{7} + 6 q^{8} - 2 q^{10} - q^{13} + q^{14} + 6 q^{16} - 7 q^{19} - 2 q^{20} + 9 q^{23} + 14 q^{25} - q^{26} + q^{28} - q^{29} + 10 q^{31} + 6 q^{32} + 18 q^{35} + 12 q^{37} - 7 q^{38} - 2 q^{40} + 11 q^{41} - 5 q^{43} + 9 q^{46} + q^{47} + 29 q^{49} + 14 q^{50} - q^{52} - 6 q^{53} + q^{56} - q^{58} - 18 q^{59} + 17 q^{61} + 10 q^{62} + 6 q^{64} + 2 q^{65} + 14 q^{67} + 18 q^{70} - 10 q^{71} + 17 q^{73} + 12 q^{74} - 7 q^{76} + 22 q^{79} - 2 q^{80} + 11 q^{82} + 33 q^{83} - 35 q^{85} - 5 q^{86} + 12 q^{89} + 59 q^{91} + 9 q^{92} + q^{94} + 39 q^{95} + 29 q^{97} + 29 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 - 2 * q^5 + q^7 + 6 * q^8 - 2 * q^10 - q^13 + q^14 + 6 * q^16 - 7 * q^19 - 2 * q^20 + 9 * q^23 + 14 * q^25 - q^26 + q^28 - q^29 + 10 * q^31 + 6 * q^32 + 18 * q^35 + 12 * q^37 - 7 * q^38 - 2 * q^40 + 11 * q^41 - 5 * q^43 + 9 * q^46 + q^47 + 29 * q^49 + 14 * q^50 - q^52 - 6 * q^53 + q^56 - q^58 - 18 * q^59 + 17 * q^61 + 10 * q^62 + 6 * q^64 + 2 * q^65 + 14 * q^67 + 18 * q^70 - 10 * q^71 + 17 * q^73 + 12 * q^74 - 7 * q^76 + 22 * q^79 - 2 * q^80 + 11 * q^82 + 33 * q^83 - 35 * q^85 - 5 * q^86 + 12 * q^89 + 59 * q^91 + 9 * q^92 + q^94 + 39 * q^95 + 29 * q^97 + 29 * q^98

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