LMFDB - Embedded newform 539.2.a.l.1.4

Error: no document with id 239787623 found in table mf_hecke_traces.

Error: table True does not exist

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [539,2,Mod(1,539)] 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("539.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(539, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	539.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.30393666895$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 26x^{8} + 245x^{6} - 1038x^{4} + 1884x^{2} - 968 $ x^10 - 26x^8 + 245x^6 - 1038x^4 + 1884x^2 - 968
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.15293$ of defining polynomial
Character	$\chi$	$=$	539.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.14898 q^{2} +2.15293 q^{3} -0.679834 q^{4} -3.87589 q^{5} -2.47369 q^{6} +3.07909 q^{8} +1.63513 q^{9} +4.45334 q^{10} +1.00000 q^{11} -1.46364 q^{12} +4.09860 q^{13} -8.34453 q^{15} -2.17816 q^{16} -0.824752 q^{17} -1.87873 q^{18} +4.50196 q^{19} +2.63496 q^{20} -1.14898 q^{22} +4.86320 q^{23} +6.62907 q^{24} +10.0225 q^{25} -4.70923 q^{26} -2.93849 q^{27} +7.79629 q^{29} +9.58774 q^{30} +3.82646 q^{31} -3.65551 q^{32} +2.15293 q^{33} +0.947627 q^{34} -1.11161 q^{36} +8.11646 q^{37} -5.17268 q^{38} +8.82401 q^{39} -11.9342 q^{40} -11.2329 q^{41} -1.86134 q^{43} -0.679834 q^{44} -6.33756 q^{45} -5.58774 q^{46} +7.69973 q^{47} -4.68943 q^{48} -11.5157 q^{50} -1.77564 q^{51} -2.78637 q^{52} -3.93495 q^{53} +3.37627 q^{54} -3.87589 q^{55} +9.69242 q^{57} -8.95782 q^{58} -9.45193 q^{59} +5.67290 q^{60} -8.40447 q^{61} -4.39655 q^{62} +8.55644 q^{64} -15.8857 q^{65} -2.47369 q^{66} +9.45914 q^{67} +0.560694 q^{68} +10.4701 q^{69} +13.4591 q^{71} +5.03470 q^{72} +6.19352 q^{73} -9.32569 q^{74} +21.5778 q^{75} -3.06058 q^{76} -10.1387 q^{78} +0.868405 q^{79} +8.44230 q^{80} -11.2317 q^{81} +12.9064 q^{82} +8.19720 q^{83} +3.19665 q^{85} +2.13866 q^{86} +16.7849 q^{87} +3.07909 q^{88} -3.87589 q^{89} +7.28176 q^{90} -3.30617 q^{92} +8.23813 q^{93} -8.84687 q^{94} -17.4491 q^{95} -7.87007 q^{96} -2.10351 q^{97} +1.63513 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9} + 10 q^{11} + 8 q^{15} + 42 q^{16} + 6 q^{18} + 2 q^{22} + 4 q^{23} + 18 q^{25} + 12 q^{29} - 4 q^{30} - 30 q^{32} - 2 q^{36} + 40 q^{37} - 16 q^{39} - 8 q^{43}+ \cdots + 22 q^{99}+O(q^{100}) $ 10 * q + 2 * q^2 + 18 * q^4 - 6 * q^8 + 22 * q^9 + 10 * q^11 + 8 * q^15 + 42 * q^16 + 6 * q^18 + 2 * q^22 + 4 * q^23 + 18 * q^25 + 12 * q^29 - 4 * q^30 - 30 * q^32 - 2 * q^36 + 40 * q^37 - 16 * q^39 - 8 * q^43 + 18 * q^44 + 44 * q^46 - 62 * q^50 + 16 * q^53 - 8 * q^57 - 28 * q^58 + 36 * q^60 + 106 * q^64 - 32 * q^65 - 4 * q^67 + 36 * q^71 - 90 * q^72 - 28 * q^74 - 112 * q^78 + 8 * q^79 - 6 * q^81 + 88 * q^85 + 32 * q^86 - 6 * q^88 - 52 * q^92 + 44 * q^93 - 64 * q^95 + 22 * q^99

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 $

$2$	−1.14898	−0.812455	−0.406227	−	0.913772i	$-0.633156\pi$
$2$	−1.14898	−0.812455	−0.406227	+	0.913772i	$0.633156\pi$
$3$	2.15293	1.24300	0.621499	−	0.783415i	$-0.286524\pi$
$3$	2.15293	1.24300	0.621499	+	0.783415i	$0.286524\pi$
$4$	−0.679834	−0.339917
$5$	−3.87589	−1.73335	−0.866675	−	0.498873i	$-0.833747\pi$
$5$	−3.87589	−1.73335	−0.866675	+	0.498873i	$0.833747\pi$
$6$	−2.47369	−1.00988
$7$	0	0
$7$	0	0
$8$	3.07909	1.08862
$9$	1.63513	0.545042
$10$	4.45334	1.40827
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.46364	−0.422516
$13$	4.09860	1.13675	0.568373	−	0.822771i	$-0.307573\pi$
$13$	4.09860	1.13675	0.568373	+	0.822771i	$0.307573\pi$
$14$	0	0
$15$	−8.34453	−2.15455
$16$	−2.17816	−0.544539
$17$	−0.824752	−0.200032	−0.100016	−	0.994986i	$-0.531889\pi$
$17$	−0.824752	−0.200032	−0.100016	+	0.994986i	$0.531889\pi$
$18$	−1.87873	−0.442822
$19$	4.50196	1.03282	0.516410	−	0.856342i	$-0.327268\pi$
$19$	4.50196	1.03282	0.516410	+	0.856342i	$0.327268\pi$
$20$	2.63496	0.589195
$21$	0	0
$22$	−1.14898	−0.244964
$23$	4.86320	1.01405	0.507023	−	0.861932i	$-0.330746\pi$
$23$	4.86320	1.01405	0.507023	+	0.861932i	$0.330746\pi$
$24$	6.62907	1.35315
$25$	10.0225	2.00450
$26$	−4.70923	−0.923555
$27$	−2.93849	−0.565512
$28$	0	0
$29$	7.79629	1.44774	0.723868	−	0.689939i	$-0.242363\pi$
$29$	7.79629	1.44774	0.723868	+	0.689939i	$0.242363\pi$
$30$	9.58774	1.75047
$31$	3.82646	0.687253	0.343627	−	0.939106i	$-0.388345\pi$
$31$	3.82646	0.687253	0.343627	+	0.939106i	$0.388345\pi$
$32$	−3.65551	−0.646208
$33$	2.15293	0.374778
$34$	0.947627	0.162517
$35$	0	0
$36$	−1.11161	−0.185269
$37$	8.11646	1.33434	0.667169	−	0.744907i	$-0.267506\pi$
$37$	8.11646	1.33434	0.667169	+	0.744907i	$0.267506\pi$
$38$	−5.17268	−0.839120
$39$	8.82401	1.41297
$40$	−11.9342	−1.88696
$41$	−11.2329	−1.75428	−0.877142	−	0.480232i	$-0.840553\pi$
$41$	−11.2329	−1.75428	−0.877142	+	0.480232i	$0.840553\pi$
$42$	0	0
$43$	−1.86134	−0.283852	−0.141926	−	0.989877i	$-0.545330\pi$
$43$	−1.86134	−0.283852	−0.141926	+	0.989877i	$0.545330\pi$
$44$	−0.679834	−0.102489
$45$	−6.33756	−0.944748
$46$	−5.58774	−0.823867
$47$	7.69973	1.12312	0.561561	−	0.827436i	$-0.310201\pi$
$47$	7.69973	1.12312	0.561561	+	0.827436i	$0.310201\pi$
$48$	−4.68943	−0.676861
$49$	0	0
$50$	−11.5157	−1.62857
$51$	−1.77564	−0.248639
$52$	−2.78637	−0.386399
$53$	−3.93495	−0.540507	−0.270253	−	0.962789i	$-0.587107\pi$
$53$	−3.93495	−0.540507	−0.270253	+	0.962789i	$0.587107\pi$
$54$	3.37627	0.459453
$55$	−3.87589	−0.522625
$56$	0	0
$57$	9.69242	1.28379
$58$	−8.95782	−1.17622
$59$	−9.45193	−1.23054	−0.615268	−	0.788318i	$-0.710952\pi$
$59$	−9.45193	−1.23054	−0.615268	+	0.788318i	$0.710952\pi$
$60$	5.67290	0.732368
$61$	−8.40447	−1.07608	−0.538041	−	0.842919i	$-0.680835\pi$
$61$	−8.40447	−1.07608	−0.538041	+	0.842919i	$0.680835\pi$
$62$	−4.39655	−0.558362
$63$	0	0
$64$	8.55644	1.06955
$65$	−15.8857	−1.97038
$66$	−2.47369	−0.304490
$67$	9.45914	1.15562	0.577809	−	0.816172i	$-0.303908\pi$
$67$	9.45914	1.15562	0.577809	+	0.816172i	$0.303908\pi$
$68$	0.560694	0.0679942
$69$	10.4701	1.26046
$70$	0	0
$71$	13.4591	1.59731	0.798653	−	0.601792i	$-0.205546\pi$
$71$	13.4591	1.59731	0.798653	+	0.601792i	$0.205546\pi$
$72$	5.03470	0.593345
$73$	6.19352	0.724897	0.362448	−	0.932004i	$-0.381941\pi$
$73$	6.19352	0.724897	0.362448	+	0.932004i	$0.381941\pi$
$74$	−9.32569	−1.08409
$75$	21.5778	2.49159
$76$	−3.06058	−0.351073
$77$	0	0
$78$	−10.1387	−1.14798
$79$	0.868405	0.0977032	0.0488516	−	0.998806i	$-0.484444\pi$
$79$	0.868405	0.0977032	0.0488516	+	0.998806i	$0.484444\pi$
$80$	8.44230	0.943878
$81$	−11.2317	−1.24797
$82$	12.9064	1.42528
$83$	8.19720	0.899759	0.449880	−	0.893089i	$-0.351467\pi$
$83$	8.19720	0.899759	0.449880	+	0.893089i	$0.351467\pi$
$84$	0	0
$85$	3.19665	0.346725
$86$	2.13866	0.230617
$87$	16.7849	1.79953
$88$	3.07909	0.328232
$89$	−3.87589	−0.410843	−0.205422	−	0.978674i	$-0.565857\pi$
$89$	−3.87589	−0.410843	−0.205422	+	0.978674i	$0.565857\pi$
$90$	7.28176	0.767565
$91$	0	0
$92$	−3.30617	−0.344692
$93$	8.23813	0.854254
$94$	−8.84687	−0.912485
$95$	−17.4491	−1.79024
$96$	−7.87007	−0.803235
$97$	−2.10351	−0.213579	−0.106790	−	0.994282i	$-0.534057\pi$
$97$	−2.10351	−0.213579	−0.106790	+	0.994282i	$0.534057\pi$
$98$	0	0
$99$	1.63513	0.164336
$100$	−6.81364	−0.681364
$101$	−12.8092	−1.27456	−0.637281	−	0.770632i	$-0.719941\pi$
$101$	−12.8092	−1.27456	−0.637281	+	0.770632i	$0.719941\pi$
$102$	2.04018	0.202008
$103$	2.23897	0.220612	0.110306	−	0.993898i	$-0.464817\pi$
$103$	2.23897	0.220612	0.110306	+	0.993898i	$0.464817\pi$
$104$	12.6199	1.23749
$105$	0	0
$106$	4.52120	0.439137
$107$	−7.60300	−0.735010	−0.367505	−	0.930022i	$-0.619788\pi$
$107$	−7.60300	−0.735010	−0.367505	+	0.930022i	$0.619788\pi$
$108$	1.99768	0.192227
$109$	7.56337	0.724440	0.362220	−	0.932093i	$-0.382019\pi$
$109$	7.56337	0.724440	0.362220	+	0.932093i	$0.382019\pi$
$110$	4.45334	0.424609
$111$	17.4742	1.65858
$112$	0	0
$113$	−3.17816	−0.298976	−0.149488	−	0.988764i	$-0.547763\pi$
$113$	−3.17816	−0.298976	−0.149488	+	0.988764i	$0.547763\pi$
$114$	−11.1364	−1.04302
$115$	−18.8492	−1.75770
$116$	−5.30019	−0.492110
$117$	6.70172	0.619574
$118$	10.8601	0.999755
$119$	0	0
$120$	−25.6936	−2.34549
$121$	1.00000	0.0909091
$122$	9.65660	0.874268
$123$	−24.1837	−2.18057
$124$	−2.60136	−0.233609
$125$	−19.4667	−1.74115
$126$	0	0
$127$	5.72640	0.508136	0.254068	−	0.967186i	$-0.418231\pi$
$127$	5.72640	0.508136	0.254068	+	0.967186i	$0.418231\pi$
$128$	−2.52020	−0.222757
$129$	−4.00735	−0.352828
$130$	18.2524	1.60084
$131$	−0.0240271	−0.00209926	−0.00104963	−	0.999999i	$-0.500334\pi$
$131$	−0.0240271	−0.00209926	−0.00104963	+	0.999999i	$0.500334\pi$
$132$	−1.46364	−0.127393
$133$	0	0
$134$	−10.8684	−0.938887
$135$	11.3892	0.980230
$136$	−2.53948	−0.217759
$137$	3.83915	0.328001	0.164000	−	0.986460i	$-0.447560\pi$
$137$	3.83915	0.328001	0.164000	+	0.986460i	$0.447560\pi$
$138$	−12.0300	−1.02406
$139$	3.59639	0.305042	0.152521	−	0.988300i	$-0.451261\pi$
$139$	3.59639	0.305042	0.152521	+	0.988300i	$0.451261\pi$
$140$	0	0
$141$	16.5770	1.39604
$142$	−15.4643	−1.29774
$143$	4.09860	0.342742
$144$	−3.56156	−0.296797
$145$	−30.2176	−2.50943
$146$	−7.11626	−0.588946
$147$	0	0
$148$	−5.51784	−0.453564
$149$	−12.5456	−1.02777	−0.513886	−	0.857858i	$-0.671795\pi$
$149$	−12.5456	−1.02777	−0.513886	+	0.857858i	$0.671795\pi$
$150$	−24.7926	−2.02431
$151$	−0.591093	−0.0481025	−0.0240512	−	0.999711i	$-0.507656\pi$
$151$	−0.591093	−0.0481025	−0.0240512	+	0.999711i	$0.507656\pi$
$152$	13.8619	1.12435
$153$	−1.34857	−0.109026
$154$	0	0
$155$	−14.8309	−1.19125
$156$	−5.99886	−0.480293
$157$	−15.2795	−1.21943	−0.609717	−	0.792619i	$-0.708717\pi$
$157$	−15.2795	−1.21943	−0.609717	+	0.792619i	$0.708717\pi$
$158$	−0.997784	−0.0793795
$159$	−8.47169	−0.671848
$160$	14.1683	1.12011
$161$	0	0
$162$	12.9051	1.01392
$163$	−2.30282	−0.180370	−0.0901852	−	0.995925i	$-0.528746\pi$
$163$	−2.30282	−0.180370	−0.0901852	+	0.995925i	$0.528746\pi$
$164$	7.63650	0.596311
$165$	−8.34453	−0.649621
$166$	−9.41845	−0.731014
$167$	−4.60081	−0.356021	−0.178011	−	0.984029i	$-0.556966\pi$
$167$	−4.60081	−0.356021	−0.178011	+	0.984029i	$0.556966\pi$
$168$	0	0
$169$	3.79851	0.292193
$170$	−3.67290	−0.281698
$171$	7.36126	0.562930
$172$	1.26540	0.0964862
$173$	18.0515	1.37243	0.686215	−	0.727399i	$-0.259271\pi$
$173$	18.0515	1.37243	0.686215	+	0.727399i	$0.259271\pi$
$174$	−19.2856	−1.46204
$175$	0	0
$176$	−2.17816	−0.164185
$177$	−20.3494	−1.52955
$178$	4.45334	0.333792
$179$	15.5109	1.15934	0.579670	−	0.814851i	$-0.303182\pi$
$179$	15.5109	1.15934	0.579670	+	0.814851i	$0.303182\pi$
$180$	4.30849	0.321136
$181$	6.64371	0.493823	0.246912	−	0.969038i	$-0.420584\pi$
$181$	6.64371	0.493823	0.246912	+	0.969038i	$0.420584\pi$
$182$	0	0
$183$	−18.0943	−1.33757
$184$	14.9742	1.10391
$185$	−31.4585	−2.31287
$186$	−9.46548	−0.694043
$187$	−0.824752	−0.0603118
$188$	−5.23454	−0.381768
$189$	0	0
$190$	20.0487	1.45449
$191$	4.94524	0.357825	0.178912	−	0.983865i	$-0.442742\pi$
$191$	4.94524	0.357825	0.178912	+	0.983865i	$0.442742\pi$
$192$	18.4214	1.32945
$193$	5.37008	0.386547	0.193273	−	0.981145i	$-0.438090\pi$
$193$	5.37008	0.386547	0.193273	+	0.981145i	$0.438090\pi$
$194$	2.41690	0.173523
$195$	−34.2009	−2.44918
$196$	0	0
$197$	−22.0487	−1.57091	−0.785454	−	0.618921i	$-0.787570\pi$
$197$	−22.0487	−1.57091	−0.785454	+	0.618921i	$0.787570\pi$
$198$	−1.87873	−0.133516
$199$	12.4218	0.880558	0.440279	−	0.897861i	$-0.354879\pi$
$199$	12.4218	0.880558	0.440279	+	0.897861i	$0.354879\pi$
$200$	30.8602	2.18215
$201$	20.3649	1.43643
$202$	14.7176	1.03552
$203$	0	0
$204$	1.20714	0.0845165
$205$	43.5374	3.04079
$206$	−2.57254	−0.179237
$207$	7.95194	0.552698
$208$	−8.92739	−0.619003
$209$	4.50196	0.311407
$210$	0	0
$211$	−26.7445	−1.84117	−0.920584	−	0.390545i	$-0.872287\pi$
$211$	−26.7445	−1.84117	−0.920584	+	0.390545i	$0.872287\pi$
$212$	2.67511	0.183727
$213$	28.9766	1.98545
$214$	8.73573	0.597162
$215$	7.21436	0.492015
$216$	−9.04786	−0.615629
$217$	0	0
$218$	−8.69020	−0.588575
$219$	13.3342	0.901045
$220$	2.63496	0.177649
$221$	−3.38033	−0.227385
$222$	−20.0776	−1.34752
$223$	1.79215	0.120011	0.0600055	−	0.998198i	$-0.480888\pi$
$223$	1.79215	0.120011	0.0600055	+	0.998198i	$0.480888\pi$
$224$	0	0
$225$	16.3881	1.09254
$226$	3.65166	0.242904
$227$	17.6173	1.16930	0.584649	−	0.811286i	$-0.301232\pi$
$227$	17.6173	1.16930	0.584649	+	0.811286i	$0.301232\pi$
$228$	−6.58923	−0.436383
$229$	5.25609	0.347332	0.173666	−	0.984805i	$-0.444439\pi$
$229$	5.25609	0.347332	0.173666	+	0.984805i	$0.444439\pi$
$230$	21.6575	1.42805
$231$	0	0
$232$	24.0055	1.57604
$233$	−13.7069	−0.897967	−0.448984	−	0.893540i	$-0.648214\pi$
$233$	−13.7069	−0.897967	−0.448984	+	0.893540i	$0.648214\pi$
$234$	−7.70018	−0.503376
$235$	−29.8433	−1.94676
$236$	6.42574	0.418280
$237$	1.86962	0.121445
$238$	0	0
$239$	8.17883	0.529045	0.264522	−	0.964380i	$-0.414786\pi$
$239$	8.17883	0.529045	0.264522	+	0.964380i	$0.414786\pi$
$240$	18.1757	1.17324
$241$	−11.3061	−0.728290	−0.364145	−	0.931342i	$-0.618639\pi$
$241$	−11.3061	−0.728290	−0.364145	+	0.931342i	$0.618639\pi$
$242$	−1.14898	−0.0738595
$243$	−15.3657	−0.985713
$244$	5.71364	0.365778
$245$	0	0
$246$	27.7867	1.77161
$247$	18.4517	1.17405
$248$	11.7820	0.748159
$249$	17.6480	1.11840
$250$	22.3669	1.41461
$251$	−19.9903	−1.26178	−0.630889	−	0.775873i	$-0.717309\pi$
$251$	−19.9903	−1.26178	−0.630889	+	0.775873i	$0.717309\pi$
$252$	0	0
$253$	4.86320	0.305747
$254$	−6.57954	−0.412837
$255$	6.88217	0.430978
$256$	−14.2172	−0.888575
$257$	−6.13629	−0.382771	−0.191386	−	0.981515i	$-0.561298\pi$
$257$	−6.13629	−0.382771	−0.191386	+	0.981515i	$0.561298\pi$
$258$	4.60439	0.286657
$259$	0	0
$260$	10.7996	0.669766
$261$	12.7479	0.789076
$262$	0.0276068	0.00170555
$263$	−22.3223	−1.37645	−0.688227	−	0.725495i	$-0.741611\pi$
$263$	−22.3223	−1.37645	−0.688227	+	0.725495i	$0.741611\pi$
$264$	6.62907	0.407991
$265$	15.2514	0.936888
$266$	0	0
$267$	−8.34453	−0.510677
$268$	−6.43064	−0.392814
$269$	3.43592	0.209492	0.104746	−	0.994499i	$-0.466597\pi$
$269$	3.43592	0.209492	0.104746	+	0.994499i	$0.466597\pi$
$270$	−13.0861	−0.796393
$271$	17.8410	1.08376	0.541880	−	0.840456i	$-0.317713\pi$
$271$	17.8410	1.08376	0.541880	+	0.840456i	$0.317713\pi$
$272$	1.79644	0.108925
$273$	0	0
$274$	−4.41112	−0.266486
$275$	10.0225	0.604380
$276$	−7.11796	−0.428451
$277$	16.9426	1.01798	0.508990	−	0.860772i	$-0.330019\pi$
$277$	16.9426	1.01798	0.508990	+	0.860772i	$0.330019\pi$
$278$	−4.13220	−0.247833
$279$	6.25675	0.374582
$280$	0	0
$281$	24.2080	1.44413	0.722066	−	0.691825i	$-0.243193\pi$
$281$	24.2080	1.44413	0.722066	+	0.691825i	$0.243193\pi$
$282$	−19.0467	−1.13422
$283$	−2.24541	−0.133475	−0.0667377	−	0.997771i	$-0.521259\pi$
$283$	−2.24541	−0.133475	−0.0667377	+	0.997771i	$0.521259\pi$
$284$	−9.14998	−0.542951
$285$	−37.5667	−2.22526
$286$	−4.70923	−0.278462
$287$	0	0
$288$	−5.97721	−0.352211
$289$	−16.3198	−0.959987
$290$	34.7195	2.03880
$291$	−4.52872	−0.265478
$292$	−4.21057	−0.246405
$293$	−0.635515	−0.0371272	−0.0185636	−	0.999828i	$-0.505909\pi$
$293$	−0.635515	−0.0371272	−0.0185636	+	0.999828i	$0.505909\pi$
$294$	0	0
$295$	36.6346	2.13295
$296$	24.9913	1.45259
$297$	−2.93849	−0.170508
$298$	14.4147	0.835019
$299$	19.9323	1.15271
$300$	−14.6693	−0.846934
$301$	0	0
$302$	0.679157	0.0390811
$303$	−27.5773	−1.58428
$304$	−9.80597	−0.562411
$305$	32.5748	1.86523
$306$	1.54949	0.0885784
$307$	−17.7490	−1.01299	−0.506493	−	0.862244i	$-0.669059\pi$
$307$	−17.7490	−1.01299	−0.506493	+	0.862244i	$0.669059\pi$
$308$	0	0
$309$	4.82034	0.274220
$310$	17.0405	0.967837
$311$	11.2429	0.637525	0.318763	−	0.947835i	$-0.396733\pi$
$311$	11.2429	0.637525	0.318763	+	0.947835i	$0.396733\pi$
$312$	27.1699	1.53819
$313$	−4.25546	−0.240533	−0.120266	−	0.992742i	$-0.538375\pi$
$313$	−4.25546	−0.240533	−0.120266	+	0.992742i	$0.538375\pi$
$314$	17.5559	0.990736
$315$	0	0
$316$	−0.590371	−0.0332110
$317$	21.2336	1.19260	0.596299	−	0.802763i	$-0.296637\pi$
$317$	21.2336	1.19260	0.596299	+	0.802763i	$0.296637\pi$
$318$	9.73384	0.545847
$319$	7.79629	0.436509
$320$	−33.1638	−1.85391
$321$	−16.3688	−0.913615
$322$	0	0
$323$	−3.71300	−0.206597
$324$	7.63572	0.424207
$325$	41.0783	2.27861
$326$	2.64590	0.146543
$327$	16.2834	0.900477
$328$	−34.5871	−1.90975
$329$	0	0
$330$	9.58774	0.527788
$331$	23.4568	1.28930	0.644652	−	0.764476i	$-0.277002\pi$
$331$	23.4568	1.28930	0.644652	+	0.764476i	$0.277002\pi$
$332$	−5.57273	−0.305843
$333$	13.2714	0.727270
$334$	5.28626	0.289251
$335$	−36.6626	−2.00309
$336$	0	0
$337$	−34.7640	−1.89372	−0.946859	−	0.321650i	$-0.895762\pi$
$337$	−34.7640	−1.89372	−0.946859	+	0.321650i	$0.895762\pi$
$338$	−4.36443	−0.237394
$339$	−6.84236	−0.371626
$340$	−2.17319	−0.117858
$341$	3.82646	0.207215
$342$	−8.45798	−0.457355
$343$	0	0
$344$	−5.73124	−0.309008
$345$	−40.5811	−2.18481
$346$	−20.7409	−1.11504
$347$	−0.689571	−0.0370181	−0.0185090	−	0.999829i	$-0.505892\pi$
$347$	−0.689571	−0.0370181	−0.0185090	+	0.999829i	$0.505892\pi$
$348$	−11.4109	−0.611691
$349$	19.9995	1.07055	0.535275	−	0.844678i	$-0.320208\pi$
$349$	19.9995	1.07055	0.535275	+	0.844678i	$0.320208\pi$
$350$	0	0
$351$	−12.0437	−0.642844
$352$	−3.65551	−0.194839
$353$	3.57359	0.190203	0.0951014	−	0.995468i	$-0.469682\pi$
$353$	3.57359	0.190203	0.0951014	+	0.995468i	$0.469682\pi$
$354$	23.3811	1.24269
$355$	−52.1661	−2.76869
$356$	2.63496	0.139653
$357$	0	0
$358$	−17.8218	−0.941911
$359$	−31.6809	−1.67205	−0.836026	−	0.548690i	$-0.815127\pi$
$359$	−31.6809	−1.67205	−0.836026	+	0.548690i	$0.815127\pi$
$360$	−19.5139	−1.02847
$361$	1.26762	0.0667169
$362$	−7.63352	−0.401209
$363$	2.15293	0.113000
$364$	0	0
$365$	−24.0054	−1.25650
$366$	20.7900	1.08671
$367$	−17.6491	−0.921277	−0.460638	−	0.887588i	$-0.652380\pi$
$367$	−17.6491	−0.921277	−0.460638	+	0.887588i	$0.652380\pi$
$368$	−10.5928	−0.552189
$369$	−18.3672	−0.956158
$370$	36.1453	1.87911
$371$	0	0
$372$	−5.60056	−0.290375
$373$	−24.0524	−1.24539	−0.622694	−	0.782465i	$-0.713962\pi$
$373$	−24.0524	−1.24539	−0.622694	+	0.782465i	$0.713962\pi$
$374$	0.947627	0.0490006
$375$	−41.9105	−2.16425
$376$	23.7082	1.22265
$377$	31.9539	1.64571
$378$	0	0
$379$	5.73645	0.294662	0.147331	−	0.989087i	$-0.452932\pi$
$379$	5.73645	0.294662	0.147331	+	0.989087i	$0.452932\pi$
$380$	11.8625	0.608532
$381$	12.3286	0.631611
$382$	−5.68200	−0.290717
$383$	−9.45929	−0.483347	−0.241674	−	0.970358i	$-0.577696\pi$
$383$	−9.45929	−0.483347	−0.241674	+	0.970358i	$0.577696\pi$
$384$	−5.42583	−0.276886
$385$	0	0
$386$	−6.17014	−0.314052
$387$	−3.04353	−0.154711
$388$	1.43004	0.0725992
$389$	16.3482	0.828887	0.414443	−	0.910075i	$-0.363976\pi$
$389$	16.3482	0.828887	0.414443	+	0.910075i	$0.363976\pi$
$390$	39.2963	1.98985
$391$	−4.01093	−0.202841
$392$	0	0
$393$	−0.0517288	−0.00260937
$394$	25.3337	1.27629
$395$	−3.36584	−0.169354
$396$	−1.11161	−0.0558607
$397$	31.2208	1.56693	0.783464	−	0.621437i	$-0.213451\pi$
$397$	31.2208	1.56693	0.783464	+	0.621437i	$0.213451\pi$
$398$	−14.2725	−0.715414
$399$	0	0
$400$	−21.8306	−1.09153
$401$	13.6480	0.681550	0.340775	−	0.940145i	$-0.389311\pi$
$401$	13.6480	0.681550	0.340775	+	0.940145i	$0.389311\pi$
$402$	−23.3990	−1.16703
$403$	15.6831	0.781233
$404$	8.70812	0.433245
$405$	43.5330	2.16317
$406$	0	0
$407$	8.11646	0.402318
$408$	−5.46734	−0.270674
$409$	13.6468	0.674790	0.337395	−	0.941363i	$-0.390454\pi$
$409$	13.6468	0.674790	0.337395	+	0.941363i	$0.390454\pi$
$410$	−50.0239	−2.47050
$411$	8.26543	0.407704
$412$	−1.52212	−0.0749897
$413$	0	0
$414$	−9.13666	−0.449042
$415$	−31.7714	−1.55960
$416$	−14.9825	−0.734575
$417$	7.74279	0.379166
$418$	−5.17268	−0.253004
$419$	13.1381	0.641840	0.320920	−	0.947106i	$-0.396008\pi$
$419$	13.1381	0.641840	0.320920	+	0.947106i	$0.396008\pi$
$420$	0	0
$421$	10.6056	0.516883	0.258441	−	0.966027i	$-0.416791\pi$
$421$	10.6056	0.516883	0.258441	+	0.966027i	$0.416791\pi$
$422$	30.7290	1.49587
$423$	12.5900	0.612148
$424$	−12.1161	−0.588408
$425$	−8.26608	−0.400964
$426$	−33.2937	−1.61309
$427$	0	0
$428$	5.16878	0.249842
$429$	8.82401	0.426027
$430$	−8.28919	−0.399740
$431$	−29.5575	−1.42373	−0.711867	−	0.702315i	$-0.752150\pi$
$431$	−29.5575	−1.42373	−0.711867	+	0.702315i	$0.752150\pi$
$432$	6.40048	0.307943
$433$	32.7665	1.57466	0.787329	−	0.616534i	$-0.211464\pi$
$433$	32.7665	1.57466	0.787329	+	0.616534i	$0.211464\pi$
$434$	0	0
$435$	−65.0564	−3.11922
$436$	−5.14184	−0.246249
$437$	21.8939	1.04733
$438$	−15.3208	−0.732058
$439$	−38.5068	−1.83783	−0.918914	−	0.394459i	$-0.870932\pi$
$439$	−38.5068	−1.83783	−0.918914	+	0.394459i	$0.870932\pi$
$440$	−11.9342	−0.568941
$441$	0	0
$442$	3.88394	0.184740
$443$	31.2823	1.48627	0.743134	−	0.669142i	$-0.233338\pi$
$443$	31.2823	1.48627	0.743134	+	0.669142i	$0.233338\pi$
$444$	−11.8796	−0.563779
$445$	15.0225	0.712135
$446$	−2.05915	−0.0975035
$447$	−27.0098	−1.27752
$448$	0	0
$449$	−8.82808	−0.416623	−0.208311	−	0.978063i	$-0.566797\pi$
$449$	−8.82808	−0.416623	−0.208311	+	0.978063i	$0.566797\pi$
$450$	−18.8296	−0.887638
$451$	−11.2329	−0.528936
$452$	2.16062	0.101627
$453$	−1.27259	−0.0597913
$454$	−20.2420	−0.950002
$455$	0	0
$456$	29.8438	1.39756
$457$	−19.1378	−0.895228	−0.447614	−	0.894227i	$-0.647726\pi$
$457$	−19.1378	−0.895228	−0.447614	+	0.894227i	$0.647726\pi$
$458$	−6.03917	−0.282192
$459$	2.42352	0.113120
$460$	12.8143	0.597472
$461$	20.9755	0.976926	0.488463	−	0.872584i	$-0.337558\pi$
$461$	20.9755	0.976926	0.488463	+	0.872584i	$0.337558\pi$
$462$	0	0
$463$	−32.4558	−1.50835	−0.754174	−	0.656674i	$-0.771963\pi$
$463$	−32.4558	−1.50835	−0.754174	+	0.656674i	$0.771963\pi$
$464$	−16.9816	−0.788349
$465$	−31.9301	−1.48072
$466$	15.7490	0.729558
$467$	−4.99179	−0.230993	−0.115496	−	0.993308i	$-0.536846\pi$
$467$	−4.99179	−0.230993	−0.115496	+	0.993308i	$0.536846\pi$
$468$	−4.55606	−0.210604
$469$	0	0
$470$	34.2895	1.58166
$471$	−32.8957	−1.51575
$472$	−29.1033	−1.33959
$473$	−1.86134	−0.0855847
$474$	−2.14816	−0.0986685
$475$	45.1209	2.07029
$476$	0	0
$477$	−6.43414	−0.294599
$478$	−9.39736	−0.429825
$479$	−20.8655	−0.953368	−0.476684	−	0.879075i	$-0.658161\pi$
$479$	−20.8655	−0.953368	−0.476684	+	0.879075i	$0.658161\pi$
$480$	30.5035	1.39229
$481$	33.2661	1.51680
$482$	12.9905	0.591703
$483$	0	0
$484$	−0.679834	−0.0309015
$485$	8.15297	0.370207
$486$	17.6550	0.800847
$487$	−23.9012	−1.08307	−0.541533	−	0.840680i	$-0.682156\pi$
$487$	−23.9012	−1.08307	−0.541533	+	0.840680i	$0.682156\pi$
$488$	−25.8781	−1.17145
$489$	−4.95781	−0.224200
$490$	0	0
$491$	19.3897	0.875044	0.437522	−	0.899208i	$-0.355856\pi$
$491$	19.3897	0.875044	0.437522	+	0.899208i	$0.355856\pi$
$492$	16.4409	0.741212
$493$	−6.43001	−0.289593
$494$	−21.2007	−0.953866
$495$	−6.33756	−0.284852
$496$	−8.33464	−0.374237
$497$	0	0
$498$	−20.2773	−0.908648
$499$	−0.100167	−0.00448408	−0.00224204	−	0.999997i	$-0.500714\pi$
$499$	−0.100167	−0.00448408	−0.00224204	+	0.999997i	$0.500714\pi$
$500$	13.2341	0.591848
$501$	−9.90523	−0.442533
$502$	22.9686	1.02514
$503$	14.2263	0.634317	0.317159	−	0.948372i	$-0.397271\pi$
$503$	14.2263	0.634317	0.317159	+	0.948372i	$0.397271\pi$
$504$	0	0
$505$	49.6470	2.20926
$506$	−5.58774	−0.248405
$507$	8.17794	0.363195
$508$	−3.89300	−0.172724
$509$	8.67585	0.384550	0.192275	−	0.981341i	$-0.438413\pi$
$509$	8.67585	0.384550	0.192275	+	0.981341i	$0.438413\pi$
$510$	−7.90750	−0.350150
$511$	0	0
$512$	21.3758	0.944684
$513$	−13.2289	−0.584072
$514$	7.05050	0.310984
$515$	−8.67798	−0.382397
$516$	2.72433	0.119932
$517$	7.69973	0.338634
$518$	0	0
$519$	38.8637	1.70593
$520$	−48.9135	−2.14500
$521$	−11.0313	−0.483288	−0.241644	−	0.970365i	$-0.577687\pi$
$521$	−11.0313	−0.483288	−0.241644	+	0.970365i	$0.577687\pi$
$522$	−14.6472	−0.641089
$523$	−11.4402	−0.500243	−0.250122	−	0.968214i	$-0.580471\pi$
$523$	−11.4402	−0.500243	−0.250122	+	0.968214i	$0.580471\pi$
$524$	0.0163344	0.000713573 0
$525$	0	0
$526$	25.6480	1.11831
$527$	−3.15588	−0.137472
$528$	−4.68943	−0.204081
$529$	0.650700	0.0282913
$530$	−17.5237	−0.761179
$531$	−15.4551	−0.670694
$532$	0	0
$533$	−46.0391	−1.99418
$534$	9.58774	0.414902
$535$	29.4684	1.27403
$536$	29.1255	1.25803
$537$	33.3940	1.44106
$538$	−3.94782	−0.170203
$539$	0	0
$540$	−7.74279	−0.333197
$541$	1.20512	0.0518122	0.0259061	−	0.999664i	$-0.491753\pi$
$541$	1.20512	0.0518122	0.0259061	+	0.999664i	$0.491753\pi$
$542$	−20.4990	−0.880507
$543$	14.3035	0.613821
$544$	3.01488	0.129262
$545$	−29.3148	−1.25571
$546$	0	0
$547$	5.64802	0.241492	0.120746	−	0.992683i	$-0.461471\pi$
$547$	5.64802	0.241492	0.120746	+	0.992683i	$0.461471\pi$
$548$	−2.60998	−0.111493
$549$	−13.7424	−0.586509
$550$	−11.5157	−0.491032
$551$	35.0986	1.49525
$552$	32.2385	1.37216
$553$	0	0
$554$	−19.4667	−0.827063
$555$	−67.7281	−2.87490
$556$	−2.44495	−0.103689
$557$	−6.89754	−0.292258	−0.146129	−	0.989266i	$-0.546681\pi$
$557$	−6.89754	−0.292258	−0.146129	+	0.989266i	$0.546681\pi$
$558$	−7.18891	−0.304331
$559$	−7.62890	−0.322668
$560$	0	0
$561$	−1.77564	−0.0749674
$562$	−27.8147	−1.17329
$563$	1.52144	0.0641211	0.0320605	−	0.999486i	$-0.489793\pi$
$563$	1.52144	0.0641211	0.0320605	+	0.999486i	$0.489793\pi$
$564$	−11.2696	−0.474536
$565$	12.3182	0.518230
$566$	2.57994	0.108443
$567$	0	0
$568$	41.4419	1.73886
$569$	28.5468	1.19674	0.598372	−	0.801219i	$-0.295815\pi$
$569$	28.5468	1.19674	0.598372	+	0.801219i	$0.295815\pi$
$570$	43.1636	1.80792
$571$	37.5519	1.57150	0.785749	−	0.618545i	$-0.212278\pi$
$571$	37.5519	1.57150	0.785749	+	0.618545i	$0.212278\pi$
$572$	−2.78637	−0.116504
$573$	10.6468	0.444775
$574$	0	0
$575$	48.7415	2.03266
$576$	13.9908	0.582952
$577$	12.5401	0.522051	0.261025	−	0.965332i	$-0.415939\pi$
$577$	12.5401	0.522051	0.261025	+	0.965332i	$0.415939\pi$
$578$	18.7512	0.779946
$579$	11.5614	0.480477
$580$	20.5429	0.852999
$581$	0	0
$582$	5.20343	0.215689
$583$	−3.93495	−0.162969
$584$	19.0704	0.789139
$585$	−25.9751	−1.07394
$586$	0.730198	0.0301642
$587$	−33.6526	−1.38899	−0.694496	−	0.719496i	$-0.744373\pi$
$587$	−33.6526	−1.38899	−0.694496	+	0.719496i	$0.744373\pi$
$588$	0	0
$589$	17.2266	0.709809
$590$	−42.0926	−1.73293
$591$	−47.4695	−1.95263
$592$	−17.6789	−0.726600
$593$	−24.4214	−1.00287	−0.501434	−	0.865196i	$-0.667194\pi$
$593$	−24.4214	−1.00287	−0.501434	+	0.865196i	$0.667194\pi$
$594$	3.37627	0.138530
$595$	0	0
$596$	8.52890	0.349357
$597$	26.7433	1.09453
$598$	−22.9019	−0.936529
$599$	5.56936	0.227558	0.113779	−	0.993506i	$-0.463704\pi$
$599$	5.56936	0.227558	0.113779	+	0.993506i	$0.463704\pi$
$600$	66.4400	2.71240
$601$	−10.8079	−0.440864	−0.220432	−	0.975402i	$-0.570747\pi$
$601$	−10.8079	−0.440864	−0.220432	+	0.975402i	$0.570747\pi$
$602$	0	0
$603$	15.4669	0.629860
$604$	0.401845	0.0163509
$605$	−3.87589	−0.157577
$606$	31.6859	1.28715
$607$	−10.5106	−0.426611	−0.213305	−	0.976986i	$-0.568423\pi$
$607$	−10.5106	−0.426611	−0.213305	+	0.976986i	$0.568423\pi$
$608$	−16.4569	−0.667417
$609$	0	0
$610$	−37.4279	−1.51541
$611$	31.5581	1.27670
$612$	0.916805	0.0370597
$613$	−23.4044	−0.945293	−0.472647	−	0.881252i	$-0.656701\pi$
$613$	−23.4044	−0.945293	−0.472647	+	0.881252i	$0.656701\pi$
$614$	20.3933	0.823006
$615$	93.7332	3.77969
$616$	0	0
$617$	−27.4432	−1.10482	−0.552410	−	0.833572i	$-0.686292\pi$
$617$	−27.4432	−1.10482	−0.552410	+	0.833572i	$0.686292\pi$
$618$	−5.53850	−0.222791
$619$	8.37072	0.336448	0.168224	−	0.985749i	$-0.446197\pi$
$619$	8.37072	0.336448	0.168224	+	0.985749i	$0.446197\pi$
$620$	10.0826	0.404926
$621$	−14.2904	−0.573455
$622$	−12.9179	−0.517961
$623$	0	0
$624$	−19.2201	−0.769419
$625$	25.3382	1.01353
$626$	4.88946	0.195422
$627$	9.69242	0.387078
$628$	10.3875	0.414507
$629$	−6.69406	−0.266910
$630$	0	0
$631$	−40.8816	−1.62747	−0.813736	−	0.581235i	$-0.802570\pi$
$631$	−40.8816	−1.62747	−0.813736	+	0.581235i	$0.802570\pi$
$632$	2.67390	0.106362
$633$	−57.5792	−2.28857
$634$	−24.3971	−0.968932
$635$	−22.1949	−0.880777
$636$	5.75934	0.228373
$637$	0	0
$638$	−8.95782	−0.354644
$639$	22.0074	0.870598
$640$	9.76803	0.386115
$641$	−28.8987	−1.14143	−0.570714	−	0.821149i	$-0.693334\pi$
$641$	−28.8987	−1.14143	−0.570714	+	0.821149i	$0.693334\pi$
$642$	18.8075	0.742271
$643$	−44.9484	−1.77259	−0.886296	−	0.463120i	$-0.846730\pi$
$643$	−44.9484	−1.77259	−0.886296	+	0.463120i	$0.846730\pi$
$644$	0	0
$645$	15.5320	0.611574
$646$	4.26618	0.167850
$647$	1.45440	0.0571784	0.0285892	−	0.999591i	$-0.490899\pi$
$647$	1.45440	0.0571784	0.0285892	+	0.999591i	$0.490899\pi$
$648$	−34.5835	−1.35857
$649$	−9.45193	−0.371021
$650$	−47.1983	−1.85127
$651$	0	0
$652$	1.56553	0.0613110
$653$	4.89313	0.191483	0.0957415	−	0.995406i	$-0.469478\pi$
$653$	4.89313	0.191483	0.0957415	+	0.995406i	$0.469478\pi$
$654$	−18.7094	−0.731597
$655$	0.0931263	0.00363875
$656$	24.4670	0.955276
$657$	10.1272	0.395099
$658$	0	0
$659$	14.2035	0.553291	0.276646	−	0.960972i	$-0.410777\pi$
$659$	14.2035	0.553291	0.276646	+	0.960972i	$0.410777\pi$
$660$	5.67290	0.220817
$661$	−33.4970	−1.30288	−0.651442	−	0.758698i	$-0.725836\pi$
$661$	−33.4970	−1.30288	−0.651442	+	0.758698i	$0.725836\pi$
$662$	−26.9515	−1.04750
$663$	−7.27762	−0.282639
$664$	25.2399	0.979498
$665$	0	0
$666$	−15.2487	−0.590874
$667$	37.9149	1.46807
$668$	3.12778	0.121018
$669$	3.85837	0.149173
$670$	42.1247	1.62742
$671$	−8.40447	−0.324451
$672$	0	0
$673$	−50.4763	−1.94572	−0.972859	−	0.231398i	$-0.925670\pi$
$673$	−50.4763	−1.94572	−0.972859	+	0.231398i	$0.925670\pi$
$674$	39.9433	1.53856
$675$	−29.4510	−1.13357
$676$	−2.58236	−0.0993214
$677$	−41.4961	−1.59482	−0.797412	−	0.603436i	$-0.793798\pi$
$677$	−41.4961	−1.59482	−0.797412	+	0.603436i	$0.793798\pi$
$678$	7.86177	0.301930
$679$	0	0
$680$	9.84275	0.377452
$681$	37.9288	1.45343
$682$	−4.39655	−0.168353
$683$	−0.914566	−0.0349949	−0.0174975	−	0.999847i	$-0.505570\pi$
$683$	−0.914566	−0.0349949	−0.0174975	+	0.999847i	$0.505570\pi$
$684$	−5.00444	−0.191349
$685$	−14.8801	−0.568540
$686$	0	0
$687$	11.3160	0.431733
$688$	4.05430	0.154569
$689$	−16.1278	−0.614419
$690$	46.6271	1.77506
$691$	−33.6540	−1.28026	−0.640130	−	0.768267i	$-0.721119\pi$
$691$	−33.6540	−1.28026	−0.640130	+	0.768267i	$0.721119\pi$
$692$	−12.2720	−0.466512
$693$	0	0
$694$	0.792306	0.0300755
$695$	−13.9392	−0.528744
$696$	51.6822	1.95901
$697$	9.26435	0.350912
$698$	−22.9791	−0.869773
$699$	−29.5100	−1.11617
$700$	0	0
$701$	−19.9539	−0.753649	−0.376825	−	0.926285i	$-0.622984\pi$
$701$	−19.9539	−0.753649	−0.376825	+	0.926285i	$0.622984\pi$
$702$	13.8380	0.522281
$703$	36.5400	1.37813
$704$	8.55644	0.322483
$705$	−64.2507	−2.41982
$706$	−4.10600	−0.154531
$707$	0	0
$708$	13.8342	0.519921
$709$	17.9062	0.672481	0.336240	−	0.941776i	$-0.390845\pi$
$709$	17.9062	0.672481	0.336240	+	0.941776i	$0.390845\pi$
$710$	59.9381	2.24944
$711$	1.41995	0.0532523
$712$	−11.9342	−0.447253
$713$	18.6089	0.696907
$714$	0	0
$715$	−15.8857	−0.594092
$716$	−10.5448	−0.394079
$717$	17.6085	0.657601
$718$	36.4008	1.35847
$719$	−2.99817	−0.111813	−0.0559064	−	0.998436i	$-0.517805\pi$
$719$	−2.99817	−0.111813	−0.0559064	+	0.998436i	$0.517805\pi$
$720$	13.8042	0.514453
$721$	0	0
$722$	−1.45648	−0.0542044
$723$	−24.3413	−0.905263
$724$	−4.51662	−0.167859
$725$	78.1384	2.90199
$726$	−2.47369	−0.0918072
$727$	−29.4088	−1.09071	−0.545355	−	0.838205i	$-0.683605\pi$
$727$	−29.4088	−1.09071	−0.545355	+	0.838205i	$0.683605\pi$
$728$	0	0
$729$	0.613791	0.0227330
$730$	27.5818	1.02085
$731$	1.53515	0.0567794
$732$	12.3011	0.454662
$733$	−1.39138	−0.0513918	−0.0256959	−	0.999670i	$-0.508180\pi$
$733$	−1.39138	−0.0513918	−0.0256959	+	0.999670i	$0.508180\pi$
$734$	20.2786	0.748496
$735$	0	0
$736$	−17.7775	−0.655286
$737$	9.45914	0.348432
$738$	21.1036	0.776835
$739$	9.54756	0.351213	0.175606	−	0.984460i	$-0.443811\pi$
$739$	9.54756	0.351213	0.175606	+	0.984460i	$0.443811\pi$
$740$	21.3866	0.786185
$741$	39.7253	1.45935
$742$	0	0
$743$	−15.7641	−0.578328	−0.289164	−	0.957280i	$-0.593377\pi$
$743$	−15.7641	−0.578328	−0.289164	+	0.957280i	$0.593377\pi$
$744$	25.3659	0.929960
$745$	48.6252	1.78149
$746$	27.6359	1.01182
$747$	13.4034	0.490406
$748$	0.560694	0.0205010
$749$	0	0
$750$	48.1545	1.75836
$751$	−43.1559	−1.57478	−0.787391	−	0.616455i	$-0.788568\pi$
$751$	−43.1559	−1.57478	−0.787391	+	0.616455i	$0.788568\pi$
$752$	−16.7712	−0.611584
$753$	−43.0378	−1.56839
$754$	−36.7145	−1.33706
$755$	2.29101	0.0833785
$756$	0	0
$757$	−36.7539	−1.33584	−0.667922	−	0.744231i	$-0.732816\pi$
$757$	−36.7539	−1.33584	−0.667922	+	0.744231i	$0.732816\pi$
$758$	−6.59109	−0.239399
$759$	10.4701	0.380042
$760$	−53.7273	−1.94889
$761$	34.6094	1.25459	0.627296	−	0.778781i	$-0.284162\pi$
$761$	34.6094	1.25459	0.627296	+	0.778781i	$0.284162\pi$
$762$	−14.1653	−0.513156
$763$	0	0
$764$	−3.36194	−0.121631
$765$	5.22692	0.188980
$766$	10.8686	0.392698
$767$	−38.7397	−1.39881
$768$	−30.6087	−1.10450
$769$	−28.4872	−1.02728	−0.513638	−	0.858007i	$-0.671703\pi$
$769$	−28.4872	−1.02728	−0.513638	+	0.858007i	$0.671703\pi$
$770$	0	0
$771$	−13.2110	−0.475783
$772$	−3.65076	−0.131394
$773$	11.5635	0.415910	0.207955	−	0.978138i	$-0.433319\pi$
$773$	11.5635	0.415910	0.207955	+	0.978138i	$0.433319\pi$
$774$	3.49697	0.125696
$775$	38.3508	1.37760
$776$	−6.47689	−0.232507
$777$	0	0
$778$	−18.7838	−0.673433
$779$	−50.5700	−1.81186
$780$	23.2509	0.832517
$781$	13.4591	0.481606
$782$	4.60850	0.164800
$783$	−22.9093	−0.818711
$784$	0	0
$785$	59.2215	2.11371
$786$	0.0594355	0.00212000
$787$	−4.02977	−0.143646	−0.0718229	−	0.997417i	$-0.522882\pi$
$787$	−4.02977	−0.143646	−0.0718229	+	0.997417i	$0.522882\pi$
$788$	14.9895	0.533978
$789$	−48.0585	−1.71093
$790$	3.86730	0.137592
$791$	0	0
$792$	5.03470	0.178900
$793$	−34.4465	−1.22323
$794$	−35.8722	−1.27306
$795$	32.8353	1.16455
$796$	−8.44476	−0.299317
$797$	5.65757	0.200401	0.100201	−	0.994967i	$-0.468052\pi$
$797$	5.65757	0.200401	0.100201	+	0.994967i	$0.468052\pi$
$798$	0	0
$799$	−6.35036	−0.224660
$800$	−36.6374	−1.29533
$801$	−6.33756	−0.223927
$802$	−15.6814	−0.553729
$803$	6.19352	0.218565
$804$	−13.8448	−0.488267
$805$	0	0
$806$	−18.0197	−0.634717
$807$	7.39732	0.260398
$808$	−39.4406	−1.38752
$809$	17.2095	0.605052	0.302526	−	0.953141i	$-0.402170\pi$
$809$	17.2095	0.605052	0.302526	+	0.953141i	$0.402170\pi$
$810$	−50.0187	−1.75748
$811$	−33.3821	−1.17221	−0.586103	−	0.810237i	$-0.699338\pi$
$811$	−33.3821	−1.17221	−0.586103	+	0.810237i	$0.699338\pi$
$812$	0	0
$813$	38.4104	1.34711
$814$	−9.32569	−0.326865
$815$	8.92546	0.312645
$816$	3.86762	0.135394
$817$	−8.37969	−0.293168
$818$	−15.6799	−0.548236
$819$	0	0
$820$	−29.5982	−1.03362
$821$	0.263189	0.00918537	0.00459269	−	0.999989i	$-0.498538\pi$
$821$	0.263189	0.00918537	0.00459269	+	0.999989i	$0.498538\pi$
$822$	−9.49686	−0.331241
$823$	14.8315	0.516995	0.258497	−	0.966012i	$-0.416773\pi$
$823$	14.8315	0.516995	0.258497	+	0.966012i	$0.416773\pi$
$824$	6.89397	0.240163
$825$	21.5778	0.751243
$826$	0	0
$827$	38.1437	1.32639	0.663194	−	0.748448i	$-0.269201\pi$
$827$	38.1437	1.32639	0.663194	+	0.748448i	$0.269201\pi$
$828$	−5.40600	−0.187871
$829$	−10.4064	−0.361429	−0.180715	−	0.983536i	$-0.557841\pi$
$829$	−10.4064	−0.361429	−0.180715	+	0.983536i	$0.557841\pi$
$830$	36.5049	1.26710
$831$	36.4762	1.26535
$832$	35.0694	1.21581
$833$	0	0
$834$	−8.89635	−0.308055
$835$	17.8322	0.617109
$836$	−3.06058	−0.105852
$837$	−11.2440	−0.388650
$838$	−15.0955	−0.521466
$839$	35.4338	1.22331	0.611655	−	0.791125i	$-0.290504\pi$
$839$	35.4338	1.22331	0.611655	+	0.791125i	$0.290504\pi$
$840$	0	0
$841$	31.7822	1.09594
$842$	−12.1856	−0.419944
$843$	52.1183	1.79505
$844$	18.1818	0.625844
$845$	−14.7226	−0.506473
$846$	−14.4657	−0.497343
$847$	0	0
$848$	8.57094	0.294327
$849$	−4.83421	−0.165910
$850$	9.49760	0.325765
$851$	39.4720	1.35308
$852$	−19.6993	−0.674887
$853$	10.5950	0.362765	0.181382	−	0.983413i	$-0.441943\pi$
$853$	10.5950	0.362765	0.181382	+	0.983413i	$0.441943\pi$
$854$	0	0
$855$	−28.5314	−0.975755
$856$	−23.4103	−0.800148
$857$	−42.4041	−1.44850	−0.724248	−	0.689540i	$-0.757813\pi$
$857$	−42.4041	−1.44850	−0.724248	+	0.689540i	$0.757813\pi$
$858$	−10.1387	−0.346128
$859$	7.47204	0.254943	0.127471	−	0.991842i	$-0.459314\pi$
$859$	7.47204	0.254943	0.127471	+	0.991842i	$0.459314\pi$
$860$	−4.90457	−0.167244
$861$	0	0
$862$	33.9611	1.15672
$863$	30.5892	1.04127	0.520635	−	0.853779i	$-0.325695\pi$
$863$	30.5892	1.04127	0.520635	+	0.853779i	$0.325695\pi$
$864$	10.7417	0.365438
$865$	−69.9656	−2.37890
$866$	−37.6482	−1.27934
$867$	−35.1354	−1.19326
$868$	0	0
$869$	0.868405	0.0294586
$870$	74.7488	2.53422
$871$	38.7692	1.31364
$872$	23.2883	0.788641
$873$	−3.43950	−0.116410
$874$	−25.1558	−0.850907
$875$	0	0
$876$	−9.06507	−0.306280
$877$	51.7674	1.74806	0.874030	−	0.485872i	$-0.161498\pi$
$877$	51.7674	1.74806	0.874030	+	0.485872i	$0.161498\pi$
$878$	44.2437	1.49315
$879$	−1.36822	−0.0461490
$880$	8.44230	0.284590
$881$	41.8750	1.41081	0.705403	−	0.708807i	$-0.250766\pi$
$881$	41.8750	1.41081	0.705403	+	0.708807i	$0.250766\pi$
$882$	0	0
$883$	−38.7098	−1.30269	−0.651344	−	0.758782i	$-0.725795\pi$
$883$	−38.7098	−1.30269	−0.651344	+	0.758782i	$0.725795\pi$
$884$	2.29806	0.0772921
$885$	78.8719	2.65125
$886$	−35.9429	−1.20753
$887$	−44.1008	−1.48076	−0.740380	−	0.672188i	$-0.765355\pi$
$887$	−44.1008	−1.48076	−0.740380	+	0.672188i	$0.765355\pi$
$888$	53.8046	1.80556
$889$	0	0
$890$	−17.2606	−0.578578
$891$	−11.2317	−0.376277
$892$	−1.21836	−0.0407938
$893$	34.6639	1.15998
$894$	31.0338	1.03793
$895$	−60.1185	−2.00954
$896$	0	0
$897$	42.9129	1.43282
$898$	10.1433	0.338487
$899$	29.8322	0.994961
$900$	−11.1412	−0.371372
$901$	3.24536	0.108118
$902$	12.9064	0.429737
$903$	0	0
$904$	−9.78583	−0.325472
$905$	−25.7503	−0.855968
$906$	1.46218	0.0485777
$907$	−37.5463	−1.24670	−0.623351	−	0.781942i	$-0.714229\pi$
$907$	−37.5463	−1.24670	−0.623351	+	0.781942i	$0.714229\pi$
$908$	−11.9768	−0.397464
$909$	−20.9446	−0.694689
$910$	0	0
$911$	11.2965	0.374269	0.187134	−	0.982334i	$-0.440080\pi$
$911$	11.2965	0.374269	0.187134	+	0.982334i	$0.440080\pi$
$912$	−21.1116	−0.699075
$913$	8.19720	0.271288
$914$	21.9890	0.727333
$915$	70.1313	2.31847
$916$	−3.57327	−0.118064
$917$	0	0
$918$	−2.78459	−0.0919051
$919$	−7.68871	−0.253627	−0.126814	−	0.991927i	$-0.540475\pi$
$919$	−7.68871	−0.253627	−0.126814	+	0.991927i	$0.540475\pi$
$920$	−58.0384	−1.91347
$921$	−38.2123	−1.25914
$922$	−24.1005	−0.793709
$923$	55.1636	1.81573
$924$	0	0
$925$	81.3473	2.67468
$926$	37.2912	1.22546
$927$	3.66099	0.120243
$928$	−28.4994	−0.935539
$929$	39.5573	1.29783	0.648916	−	0.760860i	$-0.275223\pi$
$929$	39.5573	1.29783	0.648916	+	0.760860i	$0.275223\pi$
$930$	36.6872	1.20302
$931$	0	0
$932$	9.31840	0.305234
$933$	24.2052	0.792442
$934$	5.73550	0.187671
$935$	3.19665	0.104541
$936$	20.6352	0.674482
$937$	−52.4348	−1.71297	−0.856485	−	0.516173i	$-0.827356\pi$
$937$	−52.4348	−1.71297	−0.856485	+	0.516173i	$0.827356\pi$
$938$	0	0
$939$	−9.16172	−0.298981
$940$	20.2885	0.661737
$941$	49.1101	1.60094	0.800472	−	0.599370i	$-0.204582\pi$
$941$	49.1101	1.60094	0.800472	+	0.599370i	$0.204582\pi$
$942$	37.7967	1.23148
$943$	−54.6278	−1.77893
$944$	20.5878	0.670076
$945$	0	0
$946$	2.13866	0.0695337
$947$	−3.82953	−0.124443	−0.0622216	−	0.998062i	$-0.519819\pi$
$947$	−3.82953	−0.124443	−0.0622216	+	0.998062i	$0.519819\pi$
$948$	−1.27103	−0.0412812
$949$	25.3848	0.824024
$950$	−51.8433	−1.68202
$951$	45.7145	1.48240
$952$	0	0
$953$	−45.7248	−1.48117	−0.740585	−	0.671963i	$-0.765452\pi$
$953$	−45.7248	−1.48117	−0.740585	+	0.671963i	$0.765452\pi$
$954$	7.39272	0.239348
$955$	−19.1672	−0.620236
$956$	−5.56025	−0.179831
$957$	16.7849	0.542579
$958$	23.9741	0.774568
$959$	0	0
$960$	−71.3995	−2.30441
$961$	−16.3582	−0.527683
$962$	−38.2223	−1.23233
$963$	−12.4319	−0.400611
$964$	7.68628	0.247558
$965$	−20.8138	−0.670021
$966$	0	0
$967$	40.7847	1.31155	0.655773	−	0.754958i	$-0.272343\pi$
$967$	40.7847	1.31155	0.655773	+	0.754958i	$0.272343\pi$
$968$	3.07909	0.0989657
$969$	−7.99384	−0.256799
$970$	−9.36764	−0.300777
$971$	47.1520	1.51318	0.756589	−	0.653890i	$-0.226864\pi$
$971$	47.1520	1.51318	0.756589	+	0.653890i	$0.226864\pi$
$972$	10.4462	0.335061
$973$	0	0
$974$	27.4621	0.879942
$975$	88.4388	2.83231
$976$	18.3063	0.585969
$977$	−25.3516	−0.811068	−0.405534	−	0.914080i	$-0.632914\pi$
$977$	−25.3516	−0.811068	−0.405534	+	0.914080i	$0.632914\pi$
$978$	5.69645	0.182152
$979$	−3.87589	−0.123874
$980$	0	0
$981$	12.3671	0.394850
$982$	−22.2784	−0.710934
$983$	61.6984	1.96787	0.983936	−	0.178520i	$-0.0571308\pi$
$983$	61.6984	1.96787	0.983936	+	0.178520i	$0.0571308\pi$
$984$	−74.4637	−2.37382
$985$	85.4584	2.72293
$986$	7.38798	0.235281
$987$	0	0
$988$	−12.5441	−0.399081
$989$	−9.05209	−0.287840
$990$	7.28176	0.231430
$991$	43.8014	1.39140	0.695698	−	0.718334i	$-0.255095\pi$
$991$	43.8014	1.39140	0.695698	+	0.718334i	$0.255095\pi$
$992$	−13.9877	−0.444109
$993$	50.5010	1.60260
$994$	0	0
$995$	−48.1455	−1.52632
$996$	−11.9977	−0.380163
$997$	−29.6657	−0.939522	−0.469761	−	0.882794i	$-0.655660\pi$
$997$	−29.6657	−0.939522	−0.469761	+	0.882794i	$0.655660\pi$
$998$	0.115090	0.00364311
$999$	−23.8501	−0.754584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.a.l.1.4	yes	10
3.2	odd	2		4851.2.a.cg.1.8		10
4.3	odd	2		8624.2.a.df.1.3		10
7.2	even	3		539.2.e.o.67.7		20
7.3	odd	6		539.2.e.o.177.8		20
7.4	even	3		539.2.e.o.177.7		20
7.5	odd	6		539.2.e.o.67.8		20
7.6	odd	2	inner	539.2.a.l.1.3	✓	10
11.10	odd	2		5929.2.a.bv.1.8		10
21.20	even	2		4851.2.a.cg.1.7		10
28.27	even	2		8624.2.a.df.1.8		10
77.76	even	2		5929.2.a.bv.1.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.a.l.1.3	✓	10	7.6	odd	2	inner
539.2.a.l.1.4	yes	10	1.1	even	1	trivial
539.2.e.o.67.7		20	7.2	even	3
539.2.e.o.67.8		20	7.5	odd	6
539.2.e.o.177.7		20	7.4	even	3
539.2.e.o.177.8		20	7.3	odd	6
4851.2.a.cg.1.7		10	21.20	even	2
4851.2.a.cg.1.8		10	3.2	odd	2
5929.2.a.bv.1.7		10	77.76	even	2
5929.2.a.bv.1.8		10	11.10	odd	2
8624.2.a.df.1.3		10	4.3	odd	2
8624.2.a.df.1.8		10	28.27	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

\(f(q)\)	\(=\)	\(q-1.14898 q^{2} +2.15293 q^{3} -0.679834 q^{4} -3.87589 q^{5} -2.47369 q^{6} +3.07909 q^{8} +1.63513 q^{9} +4.45334 q^{10} +1.00000 q^{11} -1.46364 q^{12} +4.09860 q^{13} -8.34453 q^{15} -2.17816 q^{16} -0.824752 q^{17} -1.87873 q^{18} +4.50196 q^{19} +2.63496 q^{20} -1.14898 q^{22} +4.86320 q^{23} +6.62907 q^{24} +10.0225 q^{25} -4.70923 q^{26} -2.93849 q^{27} +7.79629 q^{29} +9.58774 q^{30} +3.82646 q^{31} -3.65551 q^{32} +2.15293 q^{33} +0.947627 q^{34} -1.11161 q^{36} +8.11646 q^{37} -5.17268 q^{38} +8.82401 q^{39} -11.9342 q^{40} -11.2329 q^{41} -1.86134 q^{43} -0.679834 q^{44} -6.33756 q^{45} -5.58774 q^{46} +7.69973 q^{47} -4.68943 q^{48} -11.5157 q^{50} -1.77564 q^{51} -2.78637 q^{52} -3.93495 q^{53} +3.37627 q^{54} -3.87589 q^{55} +9.69242 q^{57} -8.95782 q^{58} -9.45193 q^{59} +5.67290 q^{60} -8.40447 q^{61} -4.39655 q^{62} +8.55644 q^{64} -15.8857 q^{65} -2.47369 q^{66} +9.45914 q^{67} +0.560694 q^{68} +10.4701 q^{69} +13.4591 q^{71} +5.03470 q^{72} +6.19352 q^{73} -9.32569 q^{74} +21.5778 q^{75} -3.06058 q^{76} -10.1387 q^{78} +0.868405 q^{79} +8.44230 q^{80} -11.2317 q^{81} +12.9064 q^{82} +8.19720 q^{83} +3.19665 q^{85} +2.13866 q^{86} +16.7849 q^{87} +3.07909 q^{88} -3.87589 q^{89} +7.28176 q^{90} -3.30617 q^{92} +8.23813 q^{93} -8.84687 q^{94} -17.4491 q^{95} -7.87007 q^{96} -2.10351 q^{97} +1.63513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9} + 10 q^{11} + 8 q^{15} + 42 q^{16} + 6 q^{18} + 2 q^{22} + 4 q^{23} + 18 q^{25} + 12 q^{29} - 4 q^{30} - 30 q^{32} - 2 q^{36} + 40 q^{37} - 16 q^{39} - 8 q^{43}+ \cdots + 22 q^{99}+O(q^{100}) \) 10 * q + 2 * q^2 + 18 * q^4 - 6 * q^8 + 22 * q^9 + 10 * q^11 + 8 * q^15 + 42 * q^16 + 6 * q^18 + 2 * q^22 + 4 * q^23 + 18 * q^25 + 12 * q^29 - 4 * q^30 - 30 * q^32 - 2 * q^36 + 40 * q^37 - 16 * q^39 - 8 * q^43 + 18 * q^44 + 44 * q^46 - 62 * q^50 + 16 * q^53 - 8 * q^57 - 28 * q^58 + 36 * q^60 + 106 * q^64 - 32 * q^65 - 4 * q^67 + 36 * q^71 - 90 * q^72 - 28 * q^74 - 112 * q^78 + 8 * q^79 - 6 * q^81 + 88 * q^85 + 32 * q^86 - 6 * q^88 - 52 * q^92 + 44 * q^93 - 64 * q^95 + 22 * q^99

Introduction

L-functions

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Database

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