LMFDB - Embedded newform 539.2.a.l.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,2,Mod(1,539)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	539.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:	$4.30393666895$
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			1.3
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)

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

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 $

$2$	−1.14898	−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	$-0.713476\pi$
$3$	−2.15293	−1.24300	−0.621499	+	0.783415i	$0.713476\pi$
$4$	−0.679834	−0.339917
$5$	3.87589	1.73335	0.866675	−	0.498873i	$-0.166253\pi$
$5$	3.87589	1.73335	0.866675	+	0.498873i	$0.166253\pi$
$6$	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	$-0.692427\pi$
$13$	−4.09860	−1.13675	−0.568373	+	0.822771i	$0.692427\pi$
$14$	0	0
$15$	−8.34453	−2.15455
$16$	−2.17816	−0.544539
$17$	0.824752	0.200032	0.100016	−	0.994986i	$-0.468111\pi$
$17$	0.824752	0.200032	0.100016	+	0.994986i	$0.468111\pi$
$18$	−1.87873	−0.442822
$19$	−4.50196	−1.03282	−0.516410	−	0.856342i	$-0.672732\pi$
$19$	−4.50196	−1.03282	−0.516410	+	0.856342i	$0.672732\pi$
$20$	−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	$-0.611655\pi$
$31$	−3.82646	−0.687253	−0.343627	+	0.939106i	$0.611655\pi$
$32$	−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	$-0.159447\pi$
$41$	11.2329	1.75428	0.877142	+	0.480232i	$0.159447\pi$
$42$	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	$-0.689799\pi$
$47$	−7.69973	−1.12312	−0.561561	+	0.827436i	$0.689799\pi$
$48$	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	$-0.289048\pi$
$59$	9.45193	1.23054	0.615268	+	0.788318i	$0.289048\pi$
$60$	5.67290	0.732368
$61$	8.40447	1.07608	0.538041	−	0.842919i	$-0.319165\pi$
$61$	8.40447	1.07608	0.538041	+	0.842919i	$0.319165\pi$
$62$	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	$-0.618059\pi$
$73$	−6.19352	−0.724897	−0.362448	+	0.932004i	$0.618059\pi$
$74$	−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	$-0.648533\pi$
$83$	−8.19720	−0.899759	−0.449880	+	0.893089i	$0.648533\pi$
$84$	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	$-0.434143\pi$
$89$	3.87589	0.410843	0.205422	+	0.978674i	$0.434143\pi$
$90$	−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	$-0.465943\pi$
$97$	2.10351	0.213579	0.106790	+	0.994282i	$0.465943\pi$
$98$	0	0
$99$	1.63513	0.164336
$100$	−6.81364	−0.681364
$101$	12.8092	1.27456	0.637281	−	0.770632i	$-0.280059\pi$
$101$	12.8092	1.27456	0.637281	+	0.770632i	$0.280059\pi$
$102$	2.04018	0.202008
$103$	−2.23897	−0.220612	−0.110306	−	0.993898i	$-0.535183\pi$
$103$	−2.23897	−0.220612	−0.110306	+	0.993898i	$0.535183\pi$
$104$	−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	$-0.499666\pi$
$131$	0.0240271	0.00209926	0.00104963	+	0.999999i	$0.499666\pi$
$132$	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	$-0.548739\pi$
$139$	−3.59639	−0.305042	−0.152521	+	0.988300i	$0.548739\pi$
$140$	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	$-0.291283\pi$
$157$	15.2795	1.21943	0.609717	+	0.792619i	$0.291283\pi$
$158$	−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	$-0.443034\pi$
$167$	4.60081	0.356021	0.178011	+	0.984029i	$0.443034\pi$
$168$	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	$-0.740729\pi$
$173$	−18.0515	−1.37243	−0.686215	+	0.727399i	$0.740729\pi$
$174$	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	$-0.579416\pi$
$181$	−6.64371	−0.493823	−0.246912	+	0.969038i	$0.579416\pi$
$182$	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	$-0.645121\pi$
$199$	−12.4218	−0.880558	−0.440279	+	0.897861i	$0.645121\pi$
$200$	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	$-0.519112\pi$
$223$	−1.79215	−0.120011	−0.0600055	+	0.998198i	$0.519112\pi$
$224$	0	0
$225$	16.3881	1.09254
$226$	3.65166	0.242904
$227$	−17.6173	−1.16930	−0.584649	−	0.811286i	$-0.698768\pi$
$227$	−17.6173	−1.16930	−0.584649	+	0.811286i	$0.698768\pi$
$228$	−6.58923	−0.436383
$229$	−5.25609	−0.347332	−0.173666	−	0.984805i	$-0.555561\pi$
$229$	−5.25609	−0.347332	−0.173666	+	0.984805i	$0.555561\pi$
$230$	−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	$-0.381361\pi$
$241$	11.3061	0.728290	0.364145	+	0.931342i	$0.381361\pi$
$242$	−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	$-0.282691\pi$
$251$	19.9903	1.26178	0.630889	+	0.775873i	$0.282691\pi$
$252$	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	$-0.438702\pi$
$257$	6.13629	0.382771	0.191386	+	0.981515i	$0.438702\pi$
$258$	−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	$-0.533403\pi$
$269$	−3.43592	−0.209492	−0.104746	+	0.994499i	$0.533403\pi$
$270$	−13.0861	−0.796393
$271$	−17.8410	−1.08376	−0.541880	−	0.840456i	$-0.682287\pi$
$271$	−17.8410	−1.08376	−0.541880	+	0.840456i	$0.682287\pi$
$272$	−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	$-0.478741\pi$
$283$	2.24541	0.133475	0.0667377	+	0.997771i	$0.478741\pi$
$284$	−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	$-0.494091\pi$
$293$	0.635515	0.0371272	0.0185636	+	0.999828i	$0.494091\pi$
$294$	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	$-0.330941\pi$
$307$	17.7490	1.01299	0.506493	+	0.862244i	$0.330941\pi$
$308$	0	0
$309$	4.82034	0.274220
$310$	17.0405	0.967837
$311$	−11.2429	−0.637525	−0.318763	−	0.947835i	$-0.603267\pi$
$311$	−11.2429	−0.637525	−0.318763	+	0.947835i	$0.603267\pi$
$312$	27.1699	1.53819
$313$	4.25546	0.240533	0.120266	−	0.992742i	$-0.461625\pi$
$313$	4.25546	0.240533	0.120266	+	0.992742i	$0.461625\pi$
$314$	−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	$-0.679792\pi$
$349$	−19.9995	−1.07055	−0.535275	+	0.844678i	$0.679792\pi$
$350$	0	0
$351$	−12.0437	−0.642844
$352$	−3.65551	−0.194839
$353$	−3.57359	−0.190203	−0.0951014	−	0.995468i	$-0.530318\pi$
$353$	−3.57359	−0.190203	−0.0951014	+	0.995468i	$0.530318\pi$
$354$	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	$-0.347620\pi$
$367$	17.6491	0.921277	0.460638	+	0.887588i	$0.347620\pi$
$368$	−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	$-0.422304\pi$
$383$	9.45929	0.483347	0.241674	+	0.970358i	$0.422304\pi$
$384$	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	$-0.786549\pi$
$397$	−31.2208	−1.56693	−0.783464	+	0.621437i	$0.786549\pi$
$398$	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	$-0.609546\pi$
$409$	−13.6468	−0.674790	−0.337395	+	0.941363i	$0.609546\pi$
$410$	−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	$-0.603992\pi$
$419$	−13.1381	−0.641840	−0.320920	+	0.947106i	$0.603992\pi$
$420$	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	$-0.788536\pi$
$433$	−32.7665	−1.57466	−0.787329	+	0.616534i	$0.788536\pi$
$434$	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	$-0.129068\pi$
$439$	38.5068	1.83783	0.918914	+	0.394459i	$0.129068\pi$
$440$	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	$-0.662442\pi$
$461$	−20.9755	−0.976926	−0.488463	+	0.872584i	$0.662442\pi$
$462$	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	$-0.463154\pi$
$467$	4.99179	0.230993	0.115496	+	0.993308i	$0.463154\pi$
$468$	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	$-0.341839\pi$
$479$	20.8655	0.953368	0.476684	+	0.879075i	$0.341839\pi$
$480$	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	$-0.602729\pi$
$503$	−14.2263	−0.634317	−0.317159	+	0.948372i	$0.602729\pi$
$504$	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	$-0.561587\pi$
$509$	−8.67585	−0.384550	−0.192275	+	0.981341i	$0.561587\pi$
$510$	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	$-0.422313\pi$
$521$	11.0313	0.483288	0.241644	+	0.970365i	$0.422313\pi$
$522$	−14.6472	−0.641089
$523$	11.4402	0.500243	0.250122	−	0.968214i	$-0.419529\pi$
$523$	11.4402	0.500243	0.250122	+	0.968214i	$0.419529\pi$
$524$	−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	$-0.510207\pi$
$563$	−1.52144	−0.0641211	−0.0320605	+	0.999486i	$0.510207\pi$
$564$	−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	$-0.584061\pi$
$577$	−12.5401	−0.522051	−0.261025	+	0.965332i	$0.584061\pi$
$578$	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	$-0.255627\pi$
$587$	33.6526	1.38899	0.694496	+	0.719496i	$0.255627\pi$
$588$	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	$-0.332806\pi$
$593$	24.4214	1.00287	0.501434	+	0.865196i	$0.332806\pi$
$594$	−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	$-0.429253\pi$
$601$	10.8079	0.440864	0.220432	+	0.975402i	$0.429253\pi$
$602$	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	$-0.431577\pi$
$607$	10.5106	0.426611	0.213305	+	0.976986i	$0.431577\pi$
$608$	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	$-0.553803\pi$
$619$	−8.37072	−0.336448	−0.168224	+	0.985749i	$0.553803\pi$
$620$	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	$-0.153270\pi$
$643$	44.9484	1.77259	0.886296	+	0.463120i	$0.153270\pi$
$644$	0	0
$645$	15.5320	0.611574
$646$	4.26618	0.167850
$647$	−1.45440	−0.0571784	−0.0285892	−	0.999591i	$-0.509101\pi$
$647$	−1.45440	−0.0571784	−0.0285892	+	0.999591i	$0.509101\pi$
$648$	−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	$-0.274164\pi$
$661$	33.4970	1.30288	0.651442	+	0.758698i	$0.274164\pi$
$662$	−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	$-0.206202\pi$
$677$	41.4961	1.59482	0.797412	+	0.603436i	$0.206202\pi$
$678$	−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	$-0.278881\pi$
$691$	33.6540	1.28026	0.640130	+	0.768267i	$0.278881\pi$
$692$	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	$-0.482195\pi$
$719$	2.99817	0.111813	0.0559064	+	0.998436i	$0.482195\pi$
$720$	−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	$-0.316395\pi$
$727$	29.4088	1.09071	0.545355	+	0.838205i	$0.316395\pi$
$728$	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	$-0.491820\pi$
$733$	1.39138	0.0513918	0.0256959	+	0.999670i	$0.491820\pi$
$734$	−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	$-0.715838\pi$
$761$	−34.6094	−1.25459	−0.627296	+	0.778781i	$0.715838\pi$
$762$	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	$-0.328297\pi$
$769$	28.4872	1.02728	0.513638	+	0.858007i	$0.328297\pi$
$770$	0	0
$771$	−13.2110	−0.475783
$772$	−3.65076	−0.131394
$773$	−11.5635	−0.415910	−0.207955	−	0.978138i	$-0.566681\pi$
$773$	−11.5635	−0.415910	−0.207955	+	0.978138i	$0.566681\pi$
$774$	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	$-0.477118\pi$
$787$	4.02977	0.143646	0.0718229	+	0.997417i	$0.477118\pi$
$788$	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	$-0.531948\pi$
$797$	−5.65757	−0.200401	−0.100201	+	0.994967i	$0.531948\pi$
$798$	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	$-0.300662\pi$
$811$	33.3821	1.17221	0.586103	+	0.810237i	$0.300662\pi$
$812$	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	$-0.442159\pi$
$829$	10.4064	0.361429	0.180715	+	0.983536i	$0.442159\pi$
$830$	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	$-0.709496\pi$
$839$	−35.4338	−1.22331	−0.611655	+	0.791125i	$0.709496\pi$
$840$	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	$-0.558057\pi$
$853$	−10.5950	−0.362765	−0.181382	+	0.983413i	$0.558057\pi$
$854$	0	0
$855$	−28.5314	−0.975755
$856$	−23.4103	−0.800148
$857$	42.4041	1.44850	0.724248	−	0.689540i	$-0.242187\pi$
$857$	42.4041	1.44850	0.724248	+	0.689540i	$0.242187\pi$
$858$	−10.1387	−0.346128
$859$	−7.47204	−0.254943	−0.127471	−	0.991842i	$-0.540686\pi$
$859$	−7.47204	−0.254943	−0.127471	+	0.991842i	$0.540686\pi$
$860$	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	$-0.749234\pi$
$881$	−41.8750	−1.41081	−0.705403	+	0.708807i	$0.749234\pi$
$882$	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	$-0.234645\pi$
$887$	44.1008	1.48076	0.740380	+	0.672188i	$0.234645\pi$
$888$	−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	$-0.724777\pi$
$929$	−39.5573	−1.29783	−0.648916	+	0.760860i	$0.724777\pi$
$930$	−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	$-0.172644\pi$
$937$	52.4348	1.71297	0.856485	+	0.516173i	$0.172644\pi$
$938$	0	0
$939$	−9.16172	−0.298981
$940$	20.2885	0.661737
$941$	−49.1101	−1.60094	−0.800472	−	0.599370i	$-0.795418\pi$
$941$	−49.1101	−1.60094	−0.800472	+	0.599370i	$0.795418\pi$
$942$	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	$-0.773136\pi$
$971$	−47.1520	−1.51318	−0.756589	+	0.653890i	$0.773136\pi$
$972$	−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	$-0.942869\pi$
$983$	−61.6984	−1.96787	−0.983936	+	0.178520i	$0.942869\pi$
$984$	−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	$-0.344340\pi$
$997$	29.6657	0.939522	0.469761	+	0.882794i	$0.344340\pi$
$998$	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.1.3	✓	10
3.2	odd	2		4851.2.a.cg.1.7		10
4.3	odd	2		8624.2.a.df.1.8		10
7.2	even	3		539.2.e.o.67.8		20
7.3	odd	6		539.2.e.o.177.7		20
7.4	even	3		539.2.e.o.177.8		20
7.5	odd	6		539.2.e.o.67.7		20
7.6	odd	2	inner	539.2.a.l.1.4	yes	10
11.10	odd	2		5929.2.a.bv.1.7		10
21.20	even	2		4851.2.a.cg.1.8		10
28.27	even	2		8624.2.a.df.1.3		10
77.76	even	2		5929.2.a.bv.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.a.l.1.3	✓	10	1.1	even	1	trivial
539.2.a.l.1.4	yes	10	7.6	odd	2	inner
539.2.e.o.67.7		20	7.5	odd	6
539.2.e.o.67.8		20	7.2	even	3
539.2.e.o.177.7		20	7.3	odd	6
539.2.e.o.177.8		20	7.4	even	3
4851.2.a.cg.1.7		10	3.2	odd	2
4851.2.a.cg.1.8		10	21.20	even	2
5929.2.a.bv.1.7		10	11.10	odd	2
5929.2.a.bv.1.8		10	77.76	even	2
8624.2.a.df.1.3		10	28.27	even	2
8624.2.a.df.1.8		10	4.3	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:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

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