LMFDB - Embedded newform 539.2.a.l.1.5

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			1.5
Root			$-2.32267$ 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)

$f(q)$	$=$	$q+0.566092 q^{2} -2.32267 q^{3} -1.67954 q^{4} -3.58219 q^{5} -1.31484 q^{6} -2.08296 q^{8} +2.39479 q^{9} +O(q^{10})$	\(q+0.566092 q^{2} -2.32267 q^{3} -1.67954 q^{4} -3.58219 q^{5} -1.31484 q^{6} -2.08296 q^{8} +2.39479 q^{9} -2.02785 q^{10} +1.00000 q^{11} +3.90101 q^{12} +2.11542 q^{13} +8.32025 q^{15} +2.17993 q^{16} -7.87739 q^{17} +1.35567 q^{18} +5.56323 q^{19} +6.01643 q^{20} +0.566092 q^{22} -1.25426 q^{23} +4.83802 q^{24} +7.83210 q^{25} +1.19752 q^{26} +1.40570 q^{27} -0.991656 q^{29} +4.71003 q^{30} +6.06899 q^{31} +5.39996 q^{32} -2.32267 q^{33} -4.45933 q^{34} -4.02214 q^{36} -1.67120 q^{37} +3.14930 q^{38} -4.91342 q^{39} +7.46156 q^{40} +5.35834 q^{41} -9.21855 q^{43} -1.67954 q^{44} -8.57860 q^{45} -0.710028 q^{46} -9.40225 q^{47} -5.06326 q^{48} +4.43369 q^{50} +18.2966 q^{51} -3.55293 q^{52} +12.2102 q^{53} +0.795757 q^{54} -3.58219 q^{55} -12.9216 q^{57} -0.561369 q^{58} -3.88070 q^{59} -13.9742 q^{60} +2.52992 q^{61} +3.43561 q^{62} -1.30299 q^{64} -7.57784 q^{65} -1.31484 q^{66} -3.51863 q^{67} +13.2304 q^{68} +2.91323 q^{69} +0.481369 q^{71} -4.98825 q^{72} +14.9367 q^{73} -0.946051 q^{74} -18.1914 q^{75} -9.34367 q^{76} -2.78145 q^{78} -8.00813 q^{79} -7.80894 q^{80} -10.4494 q^{81} +3.03332 q^{82} +4.23084 q^{83} +28.2183 q^{85} -5.21855 q^{86} +2.30329 q^{87} -2.08296 q^{88} -3.58219 q^{89} -4.85628 q^{90} +2.10658 q^{92} -14.0963 q^{93} -5.32254 q^{94} -19.9286 q^{95} -12.5423 q^{96} -0.164132 q^{97} +2.39479 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.566092	0.400288	0.200144	−	0.979767i	$-0.435859\pi$
$2$	0.566092	0.400288	0.200144	+	0.979767i	$0.435859\pi$
$3$	−2.32267	−1.34099	−0.670497	−	0.741913i	$-0.733919\pi$
$3$	−2.32267	−1.34099	−0.670497	+	0.741913i	$0.733919\pi$
$4$	−1.67954	−0.839770
$5$	−3.58219	−1.60201	−0.801003	−	0.598661i	$-0.795700\pi$
$5$	−3.58219	−1.60201	−0.801003	+	0.598661i	$0.795700\pi$
$6$	−1.31484	−0.536783
$7$	0	0
$7$	0	0
$8$	−2.08296	−0.736437
$9$	2.39479	0.798263
$10$	−2.02785	−0.641263
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	3.90101	1.12613
$13$	2.11542	0.586712	0.293356	−	0.956003i	$-0.405228\pi$
$13$	2.11542	0.586712	0.293356	+	0.956003i	$0.405228\pi$
$14$	0	0
$15$	8.32025	2.14828
$16$	2.17993	0.544983
$17$	−7.87739	−1.91055	−0.955274	−	0.295722i	$-0.904440\pi$
$17$	−7.87739	−1.91055	−0.955274	+	0.295722i	$0.904440\pi$
$18$	1.35567	0.319535
$19$	5.56323	1.27629	0.638147	−	0.769915i	$-0.279701\pi$
$19$	5.56323	1.27629	0.638147	+	0.769915i	$0.279701\pi$
$20$	6.01643	1.34532
$21$	0	0
$22$	0.566092	0.120691
$23$	−1.25426	−0.261532	−0.130766	−	0.991413i	$-0.541744\pi$
$23$	−1.25426	−0.261532	−0.130766	+	0.991413i	$0.541744\pi$
$24$	4.83802	0.987557
$25$	7.83210	1.56642
$26$	1.19752	0.234854
$27$	1.40570	0.270528
$28$	0	0
$29$	−0.991656	−0.184146	−0.0920730	−	0.995752i	$-0.529349\pi$
$29$	−0.991656	−0.184146	−0.0920730	+	0.995752i	$0.529349\pi$
$30$	4.71003	0.859929
$31$	6.06899	1.09002	0.545012	−	0.838428i	$-0.316525\pi$
$31$	6.06899	1.09002	0.545012	+	0.838428i	$0.316525\pi$
$32$	5.39996	0.954587
$33$	−2.32267	−0.404325
$34$	−4.45933	−0.764769
$35$	0	0
$36$	−4.02214	−0.670357
$37$	−1.67120	−0.274743	−0.137371	−	0.990520i	$-0.543865\pi$
$37$	−1.67120	−0.274743	−0.137371	+	0.990520i	$0.543865\pi$
$38$	3.14930	0.510885
$39$	−4.91342	−0.786777
$40$	7.46156	1.17978
$41$	5.35834	0.836833	0.418416	−	0.908255i	$-0.362585\pi$
$41$	5.35834	0.836833	0.418416	+	0.908255i	$0.362585\pi$
$42$	0	0
$43$	−9.21855	−1.40582	−0.702908	−	0.711281i	$-0.748115\pi$
$43$	−9.21855	−1.40582	−0.702908	+	0.711281i	$0.748115\pi$
$44$	−1.67954	−0.253200
$45$	−8.57860	−1.27882
$46$	−0.710028	−0.104688
$47$	−9.40225	−1.37146	−0.685729	−	0.727857i	$-0.740517\pi$
$47$	−9.40225	−1.37146	−0.685729	+	0.727857i	$0.740517\pi$
$48$	−5.06326	−0.730819
$49$	0	0
$50$	4.43369	0.627019
$51$	18.2966	2.56203
$52$	−3.55293	−0.492703
$53$	12.2102	1.67720	0.838600	−	0.544747i	$-0.183374\pi$
$53$	12.2102	1.67720	0.838600	+	0.544747i	$0.183374\pi$
$54$	0.795757	0.108289
$55$	−3.58219	−0.483023
$56$	0	0
$57$	−12.9216	−1.71150
$58$	−0.561369	−0.0737113
$59$	−3.88070	−0.505224	−0.252612	−	0.967568i	$-0.581290\pi$
$59$	−3.88070	−0.505224	−0.252612	+	0.967568i	$0.581290\pi$
$60$	−13.9742	−1.80406
$61$	2.52992	0.323923	0.161961	−	0.986797i	$-0.448218\pi$
$61$	2.52992	0.323923	0.161961	+	0.986797i	$0.448218\pi$
$62$	3.43561	0.436323
$63$	0	0
$64$	−1.30299	−0.162873
$65$	−7.57784	−0.939916
$66$	−1.31484	−0.161846
$67$	−3.51863	−0.429869	−0.214935	−	0.976628i	$-0.568954\pi$
$67$	−3.51863	−0.429869	−0.214935	+	0.976628i	$0.568954\pi$
$68$	13.2304	1.60442
$69$	2.91323	0.350712
$70$	0	0
$71$	0.481369	0.0571280	0.0285640	−	0.999592i	$-0.490907\pi$
$71$	0.481369	0.0571280	0.0285640	+	0.999592i	$0.490907\pi$
$72$	−4.98825	−0.587871
$73$	14.9367	1.74820	0.874102	−	0.485743i	$-0.161451\pi$
$73$	14.9367	1.74820	0.874102	+	0.485743i	$0.161451\pi$
$74$	−0.946051	−0.109976
$75$	−18.1914	−2.10056
$76$	−9.34367	−1.07179
$77$	0	0
$78$	−2.78145	−0.314937
$79$	−8.00813	−0.900985	−0.450493	−	0.892780i	$-0.648752\pi$
$79$	−8.00813	−0.900985	−0.450493	+	0.892780i	$0.648752\pi$
$80$	−7.80894	−0.873066
$81$	−10.4494	−1.16104
$82$	3.03332	0.334974
$83$	4.23084	0.464395	0.232198	−	0.972669i	$-0.425408\pi$
$83$	4.23084	0.464395	0.232198	+	0.972669i	$0.425408\pi$
$84$	0	0
$85$	28.2183	3.06071
$86$	−5.21855	−0.562731
$87$	2.30329	0.246938
$88$	−2.08296	−0.222044
$89$	−3.58219	−0.379712	−0.189856	−	0.981812i	$-0.560802\pi$
$89$	−3.58219	−0.379712	−0.189856	+	0.981812i	$0.560802\pi$
$90$	−4.85628	−0.511897
$91$	0	0
$92$	2.10658	0.219626
$93$	−14.0963	−1.46171
$94$	−5.32254	−0.548978
$95$	−19.9286	−2.04463
$96$	−12.5423	−1.28010
$97$	−0.164132	−0.0166650	−0.00833252	−	0.999965i	$-0.502652\pi$
$97$	−0.164132	−0.0166650	−0.00833252	+	0.999965i	$0.502652\pi$
$98$	0	0
$99$	2.39479	0.240685
$100$	−13.1543	−1.31543
$101$	12.1489	1.20886	0.604428	−	0.796660i	$-0.293402\pi$
$101$	12.1489	1.20886	0.604428	+	0.796660i	$0.293402\pi$
$102$	10.3575	1.02555
$103$	6.46318	0.636836	0.318418	−	0.947950i	$-0.396848\pi$
$103$	6.46318	0.636836	0.318418	+	0.947950i	$0.396848\pi$
$104$	−4.40633	−0.432077
$105$	0	0
$106$	6.91210	0.671363
$107$	15.4910	1.49758	0.748788	−	0.662809i	$-0.230636\pi$
$107$	15.4910	1.49758	0.748788	+	0.662809i	$0.230636\pi$
$108$	−2.36093	−0.227181
$109$	18.3507	1.75768	0.878841	−	0.477115i	$-0.158318\pi$
$109$	18.3507	1.75768	0.878841	+	0.477115i	$0.158318\pi$
$110$	−2.02785	−0.193348
$111$	3.88163	0.368428
$112$	0	0
$113$	1.17993	0.110999	0.0554993	−	0.998459i	$-0.482325\pi$
$113$	1.17993	0.110999	0.0554993	+	0.998459i	$0.482325\pi$
$114$	−7.31479	−0.685093
$115$	4.49301	0.418975
$116$	1.66553	0.154640
$117$	5.06599	0.468351
$118$	−2.19684	−0.202235
$119$	0	0
$120$	−17.3307	−1.58207
$121$	1.00000	0.0909091
$122$	1.43217	0.129662
$123$	−12.4457	−1.12219
$124$	−10.1931	−0.915369
$125$	−10.1451	−0.907409
$126$	0	0
$127$	−6.50852	−0.577538	−0.288769	−	0.957399i	$-0.593246\pi$
$127$	−6.50852	−0.577538	−0.288769	+	0.957399i	$0.593246\pi$
$128$	−11.5375	−1.01978
$129$	21.4116	1.88519
$130$	−4.28976	−0.376237
$131$	7.36312	0.643319	0.321660	−	0.946855i	$-0.395759\pi$
$131$	7.36312	0.643319	0.321660	+	0.946855i	$0.395759\pi$
$132$	3.90101	0.339540
$133$	0	0
$134$	−1.99187	−0.172071
$135$	−5.03550	−0.433387
$136$	16.4083	1.40700
$137$	8.76591	0.748922	0.374461	−	0.927243i	$-0.377828\pi$
$137$	8.76591	0.748922	0.374461	+	0.927243i	$0.377828\pi$
$138$	1.64916	0.140386
$139$	3.64121	0.308843	0.154422	−	0.988005i	$-0.450649\pi$
$139$	3.64121	0.308843	0.154422	+	0.988005i	$0.450649\pi$
$140$	0	0
$141$	21.8383	1.83912
$142$	0.272500	0.0228677
$143$	2.11542	0.176900
$144$	5.22048	0.435040
$145$	3.55230	0.295003
$146$	8.45553	0.699784
$147$	0	0
$148$	2.80684	0.230721
$149$	0.728603	0.0596895	0.0298448	−	0.999555i	$-0.490499\pi$
$149$	0.728603	0.0596895	0.0298448	+	0.999555i	$0.490499\pi$
$150$	−10.2980	−0.840828
$151$	−6.42897	−0.523182	−0.261591	−	0.965179i	$-0.584247\pi$
$151$	−6.42897	−0.523182	−0.261591	+	0.965179i	$0.584247\pi$
$152$	−11.5880	−0.939910
$153$	−18.8647	−1.52512
$154$	0	0
$155$	−21.7403	−1.74622
$156$	8.25228	0.660711
$157$	−3.55610	−0.283808	−0.141904	−	0.989880i	$-0.545322\pi$
$157$	−3.55610	−0.283808	−0.141904	+	0.989880i	$0.545322\pi$
$158$	−4.53334	−0.360653
$159$	−28.3603	−2.24912
$160$	−19.3437	−1.52925
$161$	0	0
$162$	−5.91530	−0.464750
$163$	13.8255	1.08290	0.541449	−	0.840733i	$-0.317876\pi$
$163$	13.8255	1.08290	0.541449	+	0.840733i	$0.317876\pi$
$164$	−8.99955	−0.702747
$165$	8.32025	0.647730
$166$	2.39505	0.185892
$167$	−0.589634	−0.0456272	−0.0228136	−	0.999740i	$-0.507262\pi$
$167$	−0.589634	−0.0456272	−0.0228136	+	0.999740i	$0.507262\pi$
$168$	0	0
$169$	−8.52500	−0.655769
$170$	15.9742	1.22516
$171$	13.3228	1.01882
$172$	15.4829	1.18056
$173$	−4.28419	−0.325721	−0.162861	−	0.986649i	$-0.552072\pi$
$173$	−4.28419	−0.325721	−0.162861	+	0.986649i	$0.552072\pi$
$174$	1.30387	0.0988464
$175$	0	0
$176$	2.17993	0.164319
$177$	9.01359	0.677503
$178$	−2.02785	−0.151994
$179$	−13.4606	−1.00609	−0.503045	−	0.864260i	$-0.667787\pi$
$179$	−13.4606	−1.00609	−0.503045	+	0.864260i	$0.667787\pi$
$180$	14.4081	1.07392
$181$	20.2099	1.50219	0.751095	−	0.660194i	$-0.229526\pi$
$181$	20.2099	1.50219	0.751095	+	0.660194i	$0.229526\pi$
$182$	0	0
$183$	−5.87616	−0.434378
$184$	2.61258	0.192602
$185$	5.98654	0.440140
$186$	−7.97978	−0.585106
$187$	−7.87739	−0.576052
$188$	15.7914	1.15171
$189$	0	0
$190$	−11.2814	−0.818440
$191$	20.1625	1.45890	0.729452	−	0.684032i	$-0.239775\pi$
$191$	20.1625	1.45890	0.729452	+	0.684032i	$0.239775\pi$
$192$	3.02641	0.218412
$193$	1.85134	0.133262	0.0666312	−	0.997778i	$-0.478775\pi$
$193$	1.85134	0.133262	0.0666312	+	0.997778i	$0.478775\pi$
$194$	−0.0929136	−0.00667081
$195$	17.6008	1.26042
$196$	0	0
$197$	9.28141	0.661273	0.330637	−	0.943758i	$-0.392737\pi$
$197$	9.28141	0.661273	0.330637	+	0.943758i	$0.392737\pi$
$198$	1.35567	0.0963434
$199$	−0.993564	−0.0704319	−0.0352159	−	0.999380i	$-0.511212\pi$
$199$	−0.993564	−0.0704319	−0.0352159	+	0.999380i	$0.511212\pi$
$200$	−16.3140	−1.15357
$201$	8.17261	0.576452
$202$	6.87737	0.483890
$203$	0	0
$204$	−30.7298	−2.15152
$205$	−19.1946	−1.34061
$206$	3.65876	0.254918
$207$	−3.00369	−0.208771
$208$	4.61147	0.319748
$209$	5.56323	0.384817
$210$	0	0
$211$	16.4840	1.13481	0.567403	−	0.823440i	$-0.307948\pi$
$211$	16.4840	1.13481	0.567403	+	0.823440i	$0.307948\pi$
$212$	−20.5075	−1.40846
$213$	−1.11806	−0.0766083
$214$	8.76936	0.599461
$215$	33.0226	2.25212
$216$	−2.92802	−0.199227
$217$	0	0
$218$	10.3882	0.703578
$219$	−34.6929	−2.34433
$220$	6.01643	0.405628
$221$	−16.6640	−1.12094
$222$	2.19736	0.147477
$223$	−26.2084	−1.75504	−0.877521	−	0.479538i	$-0.840804\pi$
$223$	−26.2084	−1.75504	−0.877521	+	0.479538i	$0.840804\pi$
$224$	0	0
$225$	18.7562	1.25042
$226$	0.667950	0.0444314
$227$	−18.3200	−1.21594	−0.607970	−	0.793960i	$-0.708016\pi$
$227$	−18.3200	−1.21594	−0.607970	+	0.793960i	$0.708016\pi$
$228$	21.7023	1.43727
$229$	−13.4169	−0.886613	−0.443307	−	0.896370i	$-0.646195\pi$
$229$	−13.4169	−0.886613	−0.443307	+	0.896370i	$0.646195\pi$
$230$	2.54346	0.167711
$231$	0	0
$232$	2.06558	0.135612
$233$	−4.43884	−0.290798	−0.145399	−	0.989373i	$-0.546447\pi$
$233$	−4.43884	−0.290798	−0.145399	+	0.989373i	$0.546447\pi$
$234$	2.86782	0.187475
$235$	33.6807	2.19708
$236$	6.51779	0.424272
$237$	18.6002	1.20822
$238$	0	0
$239$	9.13900	0.591153	0.295576	−	0.955319i	$-0.404488\pi$
$239$	9.13900	0.591153	0.295576	+	0.955319i	$0.404488\pi$
$240$	18.1376	1.17078
$241$	−17.1870	−1.10711	−0.553554	−	0.832813i	$-0.686729\pi$
$241$	−17.1870	−1.10711	−0.553554	+	0.832813i	$0.686729\pi$
$242$	0.566092	0.0363898
$243$	20.0533	1.28642
$244$	−4.24910	−0.272020
$245$	0	0
$246$	−7.04539	−0.449198
$247$	11.7686	0.748817
$248$	−12.6415	−0.802734
$249$	−9.82684	−0.622751
$250$	−5.74309	−0.363225
$251$	17.7972	1.12335	0.561675	−	0.827358i	$-0.310157\pi$
$251$	17.7972	1.12335	0.561675	+	0.827358i	$0.310157\pi$
$252$	0	0
$253$	−1.25426	−0.0788547
$254$	−3.68442	−0.231181
$255$	−65.5418	−4.10439
$256$	−3.92533	−0.245333
$257$	−6.99447	−0.436303	−0.218152	−	0.975915i	$-0.570003\pi$
$257$	−6.99447	−0.436303	−0.218152	+	0.975915i	$0.570003\pi$
$258$	12.1210	0.754618
$259$	0	0
$260$	12.7273	0.789313
$261$	−2.37481	−0.146997
$262$	4.16821	0.257513
$263$	−3.22711	−0.198992	−0.0994960	−	0.995038i	$-0.531723\pi$
$263$	−3.22711	−0.198992	−0.0994960	+	0.995038i	$0.531723\pi$
$264$	4.83802	0.297760
$265$	−43.7393	−2.68688
$266$	0	0
$267$	8.32025	0.509191
$268$	5.90968	0.360991
$269$	−11.1383	−0.679112	−0.339556	−	0.940586i	$-0.610277\pi$
$269$	−11.1383	−0.679112	−0.339556	+	0.940586i	$0.610277\pi$
$270$	−2.85056	−0.173479
$271$	−3.94320	−0.239532	−0.119766	−	0.992802i	$-0.538214\pi$
$271$	−3.94320	−0.239532	−0.119766	+	0.992802i	$0.538214\pi$
$272$	−17.1722	−1.04122
$273$	0	0
$274$	4.96231	0.299784
$275$	7.83210	0.472294
$276$	−4.89289	−0.294517
$277$	26.7624	1.60800	0.804000	−	0.594630i	$-0.202701\pi$
$277$	26.7624	1.60800	0.804000	+	0.594630i	$0.202701\pi$
$278$	2.06126	0.123626
$279$	14.5340	0.870126
$280$	0	0
$281$	−3.19505	−0.190601	−0.0953003	−	0.995449i	$-0.530381\pi$
$281$	−3.19505	−0.190601	−0.0953003	+	0.995449i	$0.530381\pi$
$282$	12.3625	0.736176
$283$	−4.65272	−0.276576	−0.138288	−	0.990392i	$-0.544160\pi$
$283$	−4.65272	−0.276576	−0.138288	+	0.990392i	$0.544160\pi$
$284$	−0.808479	−0.0479744
$285$	46.2875	2.74183
$286$	1.19752	0.0708110
$287$	0	0
$288$	12.9318	0.762012
$289$	45.0533	2.65019
$290$	2.01093	0.118086
$291$	0.381223	0.0223477
$292$	−25.0867	−1.46809
$293$	−15.6199	−0.912525	−0.456263	−	0.889845i	$-0.650812\pi$
$293$	−15.6199	−0.912525	−0.456263	+	0.889845i	$0.650812\pi$
$294$	0	0
$295$	13.9014	0.809372
$296$	3.48103	0.202331
$297$	1.40570	0.0815671
$298$	0.412457	0.0238930
$299$	−2.65329	−0.153444
$300$	30.5531	1.76399
$301$	0	0
$302$	−3.63939	−0.209423
$303$	−28.2178	−1.62107
$304$	12.1275	0.695558
$305$	−9.06265	−0.518926
$306$	−10.6792	−0.610487
$307$	−18.9815	−1.08333	−0.541666	−	0.840594i	$-0.682206\pi$
$307$	−18.9815	−1.08333	−0.541666	+	0.840594i	$0.682206\pi$
$308$	0	0
$309$	−15.0118	−0.853993
$310$	−12.3070	−0.698992
$311$	17.5896	0.997417	0.498709	−	0.866770i	$-0.333808\pi$
$311$	17.5896	0.997417	0.498709	+	0.866770i	$0.333808\pi$
$312$	10.2345	0.579412
$313$	18.0021	1.01754	0.508769	−	0.860903i	$-0.330101\pi$
$313$	18.0021	1.01754	0.508769	+	0.860903i	$0.330101\pi$
$314$	−2.01308	−0.113605
$315$	0	0
$316$	13.4500	0.756620
$317$	6.97654	0.391841	0.195921	−	0.980620i	$-0.437230\pi$
$317$	6.97654	0.391841	0.195921	+	0.980620i	$0.437230\pi$
$318$	−16.0545	−0.900293
$319$	−0.991656	−0.0555221
$320$	4.66755	0.260924
$321$	−35.9806	−2.00824
$322$	0	0
$323$	−43.8238	−2.43842
$324$	17.5501	0.975005
$325$	16.5682	0.919038
$326$	7.82652	0.433471
$327$	−42.6227	−2.35704
$328$	−11.1612	−0.616275
$329$	0	0
$330$	4.71003	0.259278
$331$	15.0208	0.825619	0.412810	−	0.910817i	$-0.364547\pi$
$331$	15.0208	0.825619	0.412810	+	0.910817i	$0.364547\pi$
$332$	−7.10586	−0.389985
$333$	−4.00216	−0.219317
$334$	−0.333787	−0.0182640
$335$	12.6044	0.688653
$336$	0	0
$337$	11.4314	0.622708	0.311354	−	0.950294i	$-0.399218\pi$
$337$	11.4314	0.622708	0.311354	+	0.950294i	$0.399218\pi$
$338$	−4.82594	−0.262496
$339$	−2.74059	−0.148848
$340$	−47.3938	−2.57029
$341$	6.06899	0.328654
$342$	7.54192	0.407820
$343$	0	0
$344$	19.2019	1.03530
$345$	−10.4358	−0.561843
$346$	−2.42525	−0.130382
$347$	9.14713	0.491044	0.245522	−	0.969391i	$-0.421041\pi$
$347$	9.14713	0.491044	0.245522	+	0.969391i	$0.421041\pi$
$348$	−3.86846	−0.207371
$349$	28.2369	1.51149	0.755743	−	0.654868i	$-0.227276\pi$
$349$	28.2369	1.51149	0.755743	+	0.654868i	$0.227276\pi$
$350$	0	0
$351$	2.97365	0.158722
$352$	5.39996	0.287819
$353$	−5.54734	−0.295255	−0.147627	−	0.989043i	$-0.547164\pi$
$353$	−5.54734	−0.295255	−0.147627	+	0.989043i	$0.547164\pi$
$354$	5.10252	0.271196
$355$	−1.72436	−0.0915194
$356$	6.01643	0.318870
$357$	0	0
$358$	−7.61992	−0.402725
$359$	−0.333903	−0.0176227	−0.00881137	−	0.999961i	$-0.502805\pi$
$359$	−0.333903	−0.0176227	−0.00881137	+	0.999961i	$0.502805\pi$
$360$	17.8689	0.941772
$361$	11.9496	0.628925
$362$	11.4407	0.601308
$363$	−2.32267	−0.121908
$364$	0	0
$365$	−53.5060	−2.80063
$366$	−3.32645	−0.173876
$367$	−8.11154	−0.423419	−0.211710	−	0.977333i	$-0.567903\pi$
$367$	−8.11154	−0.423419	−0.211710	+	0.977333i	$0.567903\pi$
$368$	−2.73420	−0.142530
$369$	12.8321	0.668013
$370$	3.38894	0.176182
$371$	0	0
$372$	23.6752	1.22750
$373$	34.2270	1.77221	0.886104	−	0.463486i	$-0.153401\pi$
$373$	34.2270	1.77221	0.886104	+	0.463486i	$0.153401\pi$
$374$	−4.45933	−0.230587
$375$	23.5638	1.21683
$376$	19.5845	1.00999
$377$	−2.09777	−0.108041
$378$	0	0
$379$	−21.9557	−1.12779	−0.563895	−	0.825846i	$-0.690698\pi$
$379$	−21.9557	−1.12779	−0.563895	+	0.825846i	$0.690698\pi$
$380$	33.4708	1.71702
$381$	15.1171	0.774475
$382$	11.4138	0.583981
$383$	−7.77526	−0.397297	−0.198649	−	0.980071i	$-0.563655\pi$
$383$	−7.77526	−0.397297	−0.198649	+	0.980071i	$0.563655\pi$
$384$	26.7979	1.36752
$385$	0	0
$386$	1.04803	0.0533433
$387$	−22.0765	−1.12221
$388$	0.275665	0.0139948
$389$	5.77816	0.292964	0.146482	−	0.989213i	$-0.453205\pi$
$389$	5.77816	0.292964	0.146482	+	0.989213i	$0.453205\pi$
$390$	9.96369	0.504531
$391$	9.88031	0.499669
$392$	0	0
$393$	−17.1021	−0.862687
$394$	5.25414	0.264700
$395$	28.6867	1.44338
$396$	−4.02214	−0.202120
$397$	20.0863	1.00810	0.504052	−	0.863673i	$-0.331842\pi$
$397$	20.0863	1.00810	0.504052	+	0.863673i	$0.331842\pi$
$398$	−0.562449	−0.0281930
$399$	0	0
$400$	17.0735	0.853673
$401$	−13.8268	−0.690479	−0.345240	−	0.938515i	$-0.612202\pi$
$401$	−13.8268	−0.690479	−0.345240	+	0.938515i	$0.612202\pi$
$402$	4.62645	0.230747
$403$	12.8385	0.639530
$404$	−20.4045	−1.01516
$405$	37.4316	1.85999
$406$	0	0
$407$	−1.67120	−0.0828381
$408$	−38.1110	−1.88678
$409$	5.33474	0.263786	0.131893	−	0.991264i	$-0.457894\pi$
$409$	5.33474	0.263786	0.131893	+	0.991264i	$0.457894\pi$
$410$	−10.8659	−0.536630
$411$	−20.3603	−1.00430
$412$	−10.8552	−0.534796
$413$	0	0
$414$	−1.70037	−0.0835685
$415$	−15.1557	−0.743963
$416$	11.4232	0.560068
$417$	−8.45732	−0.414157
$418$	3.14930	0.154038
$419$	8.67099	0.423605	0.211803	−	0.977312i	$-0.432067\pi$
$419$	8.67099	0.423605	0.211803	+	0.977312i	$0.432067\pi$
$420$	0	0
$421$	29.7894	1.45184	0.725922	−	0.687777i	$-0.241413\pi$
$421$	29.7894	1.45184	0.725922	+	0.687777i	$0.241413\pi$
$422$	9.33148	0.454249
$423$	−22.5164	−1.09479
$424$	−25.4334	−1.23515
$425$	−61.6966	−2.99272
$426$	−0.632926	−0.0306654
$427$	0	0
$428$	−26.0178	−1.25762
$429$	−4.91342	−0.237222
$430$	18.6939	0.901498
$431$	12.6486	0.609263	0.304631	−	0.952470i	$-0.401467\pi$
$431$	12.6486	0.609263	0.304631	+	0.952470i	$0.401467\pi$
$432$	3.06434	0.147433
$433$	−6.32172	−0.303802	−0.151901	−	0.988396i	$-0.548540\pi$
$433$	−6.32172	−0.303802	−0.151901	+	0.988396i	$0.548540\pi$
$434$	0	0
$435$	−8.25082	−0.395597
$436$	−30.8208	−1.47605
$437$	−6.97775	−0.333791
$438$	−19.6394	−0.938406
$439$	22.2462	1.06175	0.530876	−	0.847450i	$-0.321863\pi$
$439$	22.2462	1.06175	0.530876	+	0.847450i	$0.321863\pi$
$440$	7.46156	0.355716
$441$	0	0
$442$	−9.43336	−0.448699
$443$	−14.3049	−0.679645	−0.339823	−	0.940490i	$-0.610367\pi$
$443$	−14.3049	−0.679645	−0.339823	+	0.940490i	$0.610367\pi$
$444$	−6.51936	−0.309395
$445$	12.8321	0.608300
$446$	−14.8364	−0.702522
$447$	−1.69230	−0.0800432
$448$	0	0
$449$	21.9196	1.03445	0.517224	−	0.855850i	$-0.326965\pi$
$449$	21.9196	1.03445	0.517224	+	0.855850i	$0.326965\pi$
$450$	10.6178	0.500526
$451$	5.35834	0.252315
$452$	−1.98174	−0.0932133
$453$	14.9324	0.701584
$454$	−10.3708	−0.486726
$455$	0	0
$456$	26.9151	1.26041
$457$	−25.9527	−1.21401	−0.607007	−	0.794697i	$-0.707630\pi$
$457$	−25.9527	−1.21401	−0.607007	+	0.794697i	$0.707630\pi$
$458$	−7.59520	−0.354900
$459$	−11.0733	−0.516856
$460$	−7.54618	−0.351843
$461$	−18.5060	−0.861910	−0.430955	−	0.902374i	$-0.641823\pi$
$461$	−18.5060	−0.861910	−0.430955	+	0.902374i	$0.641823\pi$
$462$	0	0
$463$	−8.76243	−0.407224	−0.203612	−	0.979052i	$-0.565268\pi$
$463$	−8.76243	−0.407224	−0.203612	+	0.979052i	$0.565268\pi$
$464$	−2.16174	−0.100356
$465$	50.4955	2.34167
$466$	−2.51280	−0.116403
$467$	−24.3313	−1.12592	−0.562960	−	0.826484i	$-0.690337\pi$
$467$	−24.3313	−1.12592	−0.562960	+	0.826484i	$0.690337\pi$
$468$	−8.50853	−0.393307
$469$	0	0
$470$	19.0664	0.879466
$471$	8.25964	0.380584
$472$	8.08334	0.372066
$473$	−9.21855	−0.423869
$474$	10.5294	0.483634
$475$	43.5718	1.99921
$476$	0	0
$477$	29.2409	1.33885
$478$	5.17352	0.236631
$479$	−3.43695	−0.157038	−0.0785190	−	0.996913i	$-0.525019\pi$
$479$	−3.43695	−0.157038	−0.0785190	+	0.996913i	$0.525019\pi$
$480$	44.9290	2.05072
$481$	−3.53528	−0.161195
$482$	−9.72940	−0.443162
$483$	0	0
$484$	−1.67954	−0.0763427
$485$	0.587951	0.0266975
$486$	11.3520	0.514937
$487$	−29.6366	−1.34296	−0.671482	−	0.741021i	$-0.734342\pi$
$487$	−29.6366	−1.34296	−0.671482	+	0.741021i	$0.734342\pi$
$488$	−5.26971	−0.238549
$489$	−32.1121	−1.45216
$490$	0	0
$491$	−38.5364	−1.73913	−0.869563	−	0.493823i	$-0.835599\pi$
$491$	−38.5364	−1.73913	−0.869563	+	0.493823i	$0.835599\pi$
$492$	20.9030	0.942379
$493$	7.81166	0.351820
$494$	6.66210	0.299742
$495$	−8.57860	−0.385579
$496$	13.2300	0.594044
$497$	0	0
$498$	−5.56290	−0.249279
$499$	−40.8755	−1.82984	−0.914920	−	0.403636i	$-0.867746\pi$
$499$	−40.8755	−1.82984	−0.914920	+	0.403636i	$0.867746\pi$
$500$	17.0392	0.762015
$501$	1.36952	0.0611858
$502$	10.0749	0.449663
$503$	−26.9049	−1.19963	−0.599814	−	0.800139i	$-0.704759\pi$
$503$	−26.9049	−1.19963	−0.599814	+	0.800139i	$0.704759\pi$
$504$	0	0
$505$	−43.5195	−1.93659
$506$	−0.710028	−0.0315646
$507$	19.8007	0.879382
$508$	10.9313	0.484999
$509$	39.9683	1.77156	0.885781	−	0.464103i	$-0.153623\pi$
$509$	39.9683	1.77156	0.885781	+	0.464103i	$0.153623\pi$
$510$	−37.1027	−1.64294
$511$	0	0
$512$	20.8530	0.921580
$513$	7.82025	0.345273
$514$	−3.95952	−0.174647
$515$	−23.1524	−1.02021
$516$	−35.9617	−1.58313
$517$	−9.40225	−0.413510
$518$	0	0
$519$	9.95076	0.436790
$520$	15.7843	0.692189
$521$	−35.9052	−1.57303	−0.786517	−	0.617568i	$-0.788118\pi$
$521$	−35.9052	−1.57303	−0.786517	+	0.617568i	$0.788118\pi$
$522$	−1.34436	−0.0588411
$523$	12.1191	0.529931	0.264966	−	0.964258i	$-0.414639\pi$
$523$	12.1191	0.529931	0.264966	+	0.964258i	$0.414639\pi$
$524$	−12.3667	−0.540240
$525$	0	0
$526$	−1.82684	−0.0796541
$527$	−47.8078	−2.08254
$528$	−5.06326	−0.220350
$529$	−21.4268	−0.931601
$530$	−24.7605	−1.07553
$531$	−9.29347	−0.403302
$532$	0	0
$533$	11.3352	0.490980
$534$	4.71003	0.203823
$535$	−55.4919	−2.39913
$536$	7.32916	0.316572
$537$	31.2644	1.34916
$538$	−6.30529	−0.271840
$539$	0	0
$540$	8.45732	0.363945
$541$	38.0198	1.63460	0.817299	−	0.576214i	$-0.195470\pi$
$541$	38.0198	1.63460	0.817299	+	0.576214i	$0.195470\pi$
$542$	−2.23222	−0.0958818
$543$	−46.9409	−2.01443
$544$	−42.5376	−1.82378
$545$	−65.7359	−2.81582
$546$	0	0
$547$	−21.8268	−0.933248	−0.466624	−	0.884456i	$-0.654530\pi$
$547$	−21.8268	−0.933248	−0.466624	+	0.884456i	$0.654530\pi$
$548$	−14.7227	−0.628922
$549$	6.05862	0.258576
$550$	4.43369	0.189053
$551$	−5.51682	−0.235024
$552$	−6.06815	−0.258277
$553$	0	0
$554$	15.1500	0.643662
$555$	−13.9048	−0.590224
$556$	−6.11555	−0.259357
$557$	−21.0982	−0.893961	−0.446981	−	0.894544i	$-0.647501\pi$
$557$	−21.0982	−0.893961	−0.446981	+	0.894544i	$0.647501\pi$
$558$	8.22756	0.348301
$559$	−19.5011	−0.824809
$560$	0	0
$561$	18.2966	0.772482
$562$	−1.80869	−0.0762951
$563$	36.7160	1.54739	0.773696	−	0.633557i	$-0.218406\pi$
$563$	36.7160	1.54739	0.773696	+	0.633557i	$0.218406\pi$
$564$	−36.6783	−1.54443
$565$	−4.22674	−0.177820
$566$	−2.63387	−0.110710
$567$	0	0
$568$	−1.00267	−0.0420712
$569$	−21.9167	−0.918797	−0.459399	−	0.888230i	$-0.651935\pi$
$569$	−21.9167	−0.918797	−0.459399	+	0.888230i	$0.651935\pi$
$570$	26.2030	1.09752
$571$	−11.8342	−0.495247	−0.247623	−	0.968856i	$-0.579650\pi$
$571$	−11.8342	−0.495247	−0.247623	+	0.968856i	$0.579650\pi$
$572$	−3.55293	−0.148556
$573$	−46.8307	−1.95638
$574$	0	0
$575$	−9.82351	−0.409669
$576$	−3.12038	−0.130016
$577$	30.6197	1.27471	0.637357	−	0.770568i	$-0.280028\pi$
$577$	30.6197	1.27471	0.637357	+	0.770568i	$0.280028\pi$
$578$	25.5043	1.06084
$579$	−4.30005	−0.178704
$580$	−5.96623	−0.247734
$581$	0	0
$582$	0.215808	0.00894551
$583$	12.2102	0.505695
$584$	−31.1125	−1.28744
$585$	−18.1473	−0.750300
$586$	−8.84231	−0.365273
$587$	40.9866	1.69170	0.845848	−	0.533424i	$-0.179095\pi$
$587$	40.9866	1.69170	0.845848	+	0.533424i	$0.179095\pi$
$588$	0	0
$589$	33.7632	1.39119
$590$	7.86949	0.323982
$591$	−21.5577	−0.886763
$592$	−3.64309	−0.149730
$593$	6.69626	0.274983	0.137491	−	0.990503i	$-0.456096\pi$
$593$	6.69626	0.274983	0.137491	+	0.990503i	$0.456096\pi$
$594$	0.795757	0.0326503
$595$	0	0
$596$	−1.22372	−0.0501255
$597$	2.30772	0.0944487
$598$	−1.50201	−0.0614216
$599$	17.9097	0.731770	0.365885	−	0.930660i	$-0.380766\pi$
$599$	17.9097	0.731770	0.365885	+	0.930660i	$0.380766\pi$
$600$	37.8919	1.54693
$601$	21.3193	0.869630	0.434815	−	0.900520i	$-0.356814\pi$
$601$	21.3193	0.869630	0.434815	+	0.900520i	$0.356814\pi$
$602$	0	0
$603$	−8.42638	−0.343149
$604$	10.7977	0.439353
$605$	−3.58219	−0.145637
$606$	−15.9739	−0.648894
$607$	6.67801	0.271052	0.135526	−	0.990774i	$-0.456728\pi$
$607$	6.67801	0.271052	0.135526	+	0.990774i	$0.456728\pi$
$608$	30.0412	1.21833
$609$	0	0
$610$	−5.13030	−0.207720
$611$	−19.8897	−0.804651
$612$	31.6840	1.28075
$613$	24.7905	1.00128	0.500639	−	0.865656i	$-0.333098\pi$
$613$	24.7905	1.00128	0.500639	+	0.865656i	$0.333098\pi$
$614$	−10.7453	−0.433644
$615$	44.5827	1.79775
$616$	0	0
$617$	23.0708	0.928795	0.464397	−	0.885627i	$-0.346271\pi$
$617$	23.0708	0.928795	0.464397	+	0.885627i	$0.346271\pi$
$618$	−8.49808	−0.341843
$619$	−27.1443	−1.09102	−0.545512	−	0.838103i	$-0.683665\pi$
$619$	−27.1443	−1.09102	−0.545512	+	0.838103i	$0.683665\pi$
$620$	36.5137	1.46643
$621$	−1.76312	−0.0707515
$622$	9.95736	0.399254
$623$	0	0
$624$	−10.7109	−0.428780
$625$	−2.81866	−0.112746
$626$	10.1908	0.407308
$627$	−12.9216	−0.516037
$628$	5.97261	0.238333
$629$	13.1647	0.524909
$630$	0	0
$631$	0.850302	0.0338500	0.0169250	−	0.999857i	$-0.494612\pi$
$631$	0.850302	0.0338500	0.0169250	+	0.999857i	$0.494612\pi$
$632$	16.6806	0.663519
$633$	−38.2869	−1.52177
$634$	3.94937	0.156849
$635$	23.3148	0.925219
$636$	47.6322	1.88874
$637$	0	0
$638$	−0.561369	−0.0222248
$639$	1.15278	0.0456032
$640$	41.3297	1.63370
$641$	40.6878	1.60707	0.803536	−	0.595256i	$-0.202949\pi$
$641$	40.6878	1.60707	0.803536	+	0.595256i	$0.202949\pi$
$642$	−20.3683	−0.803874
$643$	3.89995	0.153799	0.0768994	−	0.997039i	$-0.475498\pi$
$643$	3.89995	0.153799	0.0768994	+	0.997039i	$0.475498\pi$
$644$	0	0
$645$	−76.7006	−3.02008
$646$	−24.8083	−0.976070
$647$	21.3898	0.840918	0.420459	−	0.907311i	$-0.361869\pi$
$647$	21.3898	0.840918	0.420459	+	0.907311i	$0.361869\pi$
$648$	21.7656	0.855032
$649$	−3.88070	−0.152331
$650$	9.37913	0.367880
$651$	0	0
$652$	−23.2205	−0.909386
$653$	−9.70937	−0.379957	−0.189978	−	0.981788i	$-0.560842\pi$
$653$	−9.70937	−0.379957	−0.189978	+	0.981788i	$0.560842\pi$
$654$	−24.1284	−0.943494
$655$	−26.3761	−1.03060
$656$	11.6808	0.456060
$657$	35.7702	1.39553
$658$	0	0
$659$	45.3120	1.76511	0.882553	−	0.470212i	$-0.155823\pi$
$659$	45.3120	1.76511	0.882553	+	0.470212i	$0.155823\pi$
$660$	−13.9742	−0.543944
$661$	−25.1885	−0.979719	−0.489859	−	0.871802i	$-0.662952\pi$
$661$	−25.1885	−0.979719	−0.489859	+	0.871802i	$0.662952\pi$
$662$	8.50317	0.330485
$663$	38.7049	1.50318
$664$	−8.81267	−0.341998
$665$	0	0
$666$	−2.26559	−0.0877900
$667$	1.24380	0.0481600
$668$	0.990314	0.0383164
$669$	60.8734	2.35350
$670$	7.13526	0.275659
$671$	2.52992	0.0976664
$672$	0	0
$673$	−24.5134	−0.944924	−0.472462	−	0.881351i	$-0.656635\pi$
$673$	−24.5134	−0.944924	−0.472462	+	0.881351i	$0.656635\pi$
$674$	6.47122	0.249262
$675$	11.0096	0.423760
$676$	14.3181	0.550695
$677$	−9.46644	−0.363825	−0.181912	−	0.983315i	$-0.558229\pi$
$677$	−9.46644	−0.363825	−0.181912	+	0.983315i	$0.558229\pi$
$678$	−1.55143	−0.0595822
$679$	0	0
$680$	−58.7776	−2.25402
$681$	42.5512	1.63057
$682$	3.43561	0.131556
$683$	−1.90836	−0.0730215	−0.0365107	−	0.999333i	$-0.511624\pi$
$683$	−1.90836	−0.0730215	−0.0365107	+	0.999333i	$0.511624\pi$
$684$	−22.3761	−0.855573
$685$	−31.4012	−1.19978
$686$	0	0
$687$	31.1630	1.18894
$688$	−20.0958	−0.766146
$689$	25.8297	0.984034
$690$	−5.90761	−0.224899
$691$	5.38145	0.204720	0.102360	−	0.994747i	$-0.467361\pi$
$691$	5.38145	0.204720	0.102360	+	0.994747i	$0.467361\pi$
$692$	7.19547	0.273531
$693$	0	0
$694$	5.17812	0.196559
$695$	−13.0435	−0.494768
$696$	−4.79766	−0.181855
$697$	−42.2098	−1.59881
$698$	15.9847	0.605029
$699$	10.3100	0.389959
$700$	0	0
$701$	41.3569	1.56203	0.781014	−	0.624513i	$-0.214703\pi$
$701$	41.3569	1.56203	0.781014	+	0.624513i	$0.214703\pi$
$702$	1.68336	0.0635344
$703$	−9.29725	−0.350653
$704$	−1.30299	−0.0491082
$705$	−78.2290	−2.94628
$706$	−3.14031	−0.118187
$707$	0	0
$708$	−15.1387	−0.568946
$709$	−11.3771	−0.427276	−0.213638	−	0.976913i	$-0.568531\pi$
$709$	−11.3771	−0.427276	−0.213638	+	0.976913i	$0.568531\pi$
$710$	−0.976146	−0.0366341
$711$	−19.1778	−0.719223
$712$	7.46156	0.279634
$713$	−7.61210	−0.285076
$714$	0	0
$715$	−7.57784	−0.283395
$716$	22.6075	0.844884
$717$	−21.2269	−0.792732
$718$	−0.189020	−0.00705416
$719$	26.5668	0.990776	0.495388	−	0.868672i	$-0.335026\pi$
$719$	26.5668	0.990776	0.495388	+	0.868672i	$0.335026\pi$
$720$	−18.7008	−0.696936
$721$	0	0
$722$	6.76456	0.251751
$723$	39.9196	1.48463
$724$	−33.9433	−1.26149
$725$	−7.76675	−0.288450
$726$	−1.31484	−0.0487985
$727$	20.1922	0.748889	0.374444	−	0.927249i	$-0.377833\pi$
$727$	20.1922	0.748889	0.374444	+	0.927249i	$0.377833\pi$
$728$	0	0
$729$	−15.2291	−0.564039
$730$	−30.2893	−1.12106
$731$	72.6181	2.68588
$732$	9.86924	0.364778
$733$	33.9684	1.25465	0.627326	−	0.778757i	$-0.284149\pi$
$733$	33.9684	1.25465	0.627326	+	0.778757i	$0.284149\pi$
$734$	−4.59188	−0.169489
$735$	0	0
$736$	−6.77296	−0.249655
$737$	−3.51863	−0.129610
$738$	7.26416	0.267397
$739$	−3.64752	−0.134176	−0.0670881	−	0.997747i	$-0.521371\pi$
$739$	−3.64752	−0.134176	−0.0670881	+	0.997747i	$0.521371\pi$
$740$	−10.0546	−0.369616
$741$	−27.3345	−1.00416
$742$	0	0
$743$	13.0411	0.478432	0.239216	−	0.970966i	$-0.423110\pi$
$743$	13.0411	0.478432	0.239216	+	0.970966i	$0.423110\pi$
$744$	29.3619	1.07646
$745$	−2.61000	−0.0956229
$746$	19.3757	0.709393
$747$	10.1320	0.370710
$748$	13.2304	0.483751
$749$	0	0
$750$	13.3393	0.487082
$751$	28.6269	1.04461	0.522305	−	0.852759i	$-0.325072\pi$
$751$	28.6269	1.04461	0.522305	+	0.852759i	$0.325072\pi$
$752$	−20.4963	−0.747422
$753$	−41.3370	−1.50640
$754$	−1.18753	−0.0432473
$755$	23.0298	0.838141
$756$	0	0
$757$	−41.8901	−1.52252	−0.761261	−	0.648446i	$-0.775419\pi$
$757$	−41.8901	−1.52252	−0.761261	+	0.648446i	$0.775419\pi$
$758$	−12.4290	−0.451441
$759$	2.91323	0.105744
$760$	41.5104	1.50574
$761$	−22.7774	−0.825682	−0.412841	−	0.910803i	$-0.635463\pi$
$761$	−22.7774	−0.825682	−0.412841	+	0.910803i	$0.635463\pi$
$762$	8.55770	0.310013
$763$	0	0
$764$	−33.8636	−1.22514
$765$	67.5770	2.44325
$766$	−4.40152	−0.159033
$767$	−8.20932	−0.296421
$768$	9.11725	0.328990
$769$	13.3517	0.481475	0.240738	−	0.970590i	$-0.422611\pi$
$769$	13.3517	0.481475	0.240738	+	0.970590i	$0.422611\pi$
$770$	0	0
$771$	16.2458	0.585080
$772$	−3.10940	−0.111910
$773$	−17.9214	−0.644588	−0.322294	−	0.946640i	$-0.604454\pi$
$773$	−17.9214	−0.644588	−0.322294	+	0.946640i	$0.604454\pi$
$774$	−12.4973	−0.449207
$775$	47.5330	1.70744
$776$	0.341879	0.0122728
$777$	0	0
$778$	3.27097	0.117270
$779$	29.8097	1.06804
$780$	−29.5613	−1.05846
$781$	0.481369	0.0172248
$782$	5.59317	0.200011
$783$	−1.39397	−0.0498165
$784$	0	0
$785$	12.7386	0.454661
$786$	−9.68136	−0.345323
$787$	8.61899	0.307234	0.153617	−	0.988130i	$-0.450908\pi$
$787$	8.61899	0.307234	0.153617	+	0.988130i	$0.450908\pi$
$788$	−15.5885	−0.555317
$789$	7.49550	0.266847
$790$	16.2393	0.577768
$791$	0	0
$792$	−4.98825	−0.177250
$793$	5.35184	0.190049
$794$	11.3707	0.403532
$795$	101.592	3.60309
$796$	1.66873	0.0591466
$797$	−8.15199	−0.288758	−0.144379	−	0.989522i	$-0.546118\pi$
$797$	−8.15199	−0.288758	−0.144379	+	0.989522i	$0.546118\pi$
$798$	0	0
$799$	74.0652	2.62024
$800$	42.2931	1.49529
$801$	−8.57860	−0.303110
$802$	−7.82727	−0.276390
$803$	14.9367	0.527103
$804$	−13.7262	−0.484087
$805$	0	0
$806$	7.26776	0.255996
$807$	25.8705	0.910685
$808$	−25.3056	−0.890247
$809$	17.8331	0.626978	0.313489	−	0.949592i	$-0.398502\pi$
$809$	17.8331	0.626978	0.313489	+	0.949592i	$0.398502\pi$
$810$	21.1897	0.744531
$811$	33.8231	1.18769	0.593845	−	0.804580i	$-0.297609\pi$
$811$	33.8231	1.18769	0.593845	+	0.804580i	$0.297609\pi$
$812$	0	0
$813$	9.15875	0.321211
$814$	−0.946051	−0.0331591
$815$	−49.5257	−1.73481
$816$	39.8853	1.39626
$817$	−51.2850	−1.79423
$818$	3.01996	0.105590
$819$	0	0
$820$	32.2381	1.12580
$821$	18.0163	0.628772	0.314386	−	0.949295i	$-0.398201\pi$
$821$	18.0163	0.628772	0.314386	+	0.949295i	$0.398201\pi$
$822$	−11.5258	−0.402009
$823$	−6.20675	−0.216354	−0.108177	−	0.994132i	$-0.534501\pi$
$823$	−6.20675	−0.216354	−0.108177	+	0.994132i	$0.534501\pi$
$824$	−13.4625	−0.468990
$825$	−18.1914	−0.633343
$826$	0	0
$827$	14.4737	0.503299	0.251650	−	0.967818i	$-0.419027\pi$
$827$	14.4737	0.503299	0.251650	+	0.967818i	$0.419027\pi$
$828$	5.04482	0.175320
$829$	−17.3725	−0.603371	−0.301685	−	0.953408i	$-0.597549\pi$
$829$	−17.3725	−0.603371	−0.301685	+	0.953408i	$0.597549\pi$
$830$	−8.57952	−0.297799
$831$	−62.1603	−2.15632
$832$	−2.75637	−0.0955598
$833$	0	0
$834$	−4.78762	−0.165782
$835$	2.11218	0.0730951
$836$	−9.34367	−0.323158
$837$	8.53120	0.294881
$838$	4.90858	0.169564
$839$	1.93117	0.0666713	0.0333357	−	0.999444i	$-0.489387\pi$
$839$	1.93117	0.0666713	0.0333357	+	0.999444i	$0.489387\pi$
$840$	0	0
$841$	−28.0166	−0.966090
$842$	16.8635	0.581156
$843$	7.42104	0.255594
$844$	−27.6856	−0.952977
$845$	30.5382	1.05055
$846$	−12.7464	−0.438229
$847$	0	0
$848$	26.6174	0.914046
$849$	10.8067	0.370886
$850$	−34.9259	−1.19795
$851$	2.09612	0.0718539
$852$	1.87783	0.0643334
$853$	−1.10204	−0.0377332	−0.0188666	−	0.999822i	$-0.506006\pi$
$853$	−1.10204	−0.0377332	−0.0188666	+	0.999822i	$0.506006\pi$
$854$	0	0
$855$	−47.7248	−1.63215
$856$	−32.2672	−1.10287
$857$	21.7149	0.741766	0.370883	−	0.928680i	$-0.379055\pi$
$857$	21.7149	0.741766	0.370883	+	0.928680i	$0.379055\pi$
$858$	−2.78145	−0.0949571
$859$	39.6186	1.35177	0.675885	−	0.737007i	$-0.263762\pi$
$859$	39.6186	1.35177	0.675885	+	0.737007i	$0.263762\pi$
$860$	−55.4628	−1.89127
$861$	0	0
$862$	7.16029	0.243880
$863$	2.29774	0.0782161	0.0391081	−	0.999235i	$-0.487548\pi$
$863$	2.29774	0.0782161	0.0391081	+	0.999235i	$0.487548\pi$
$864$	7.59074	0.258242
$865$	15.3468	0.521807
$866$	−3.57867	−0.121608
$867$	−104.644	−3.55389
$868$	0	0
$869$	−8.00813	−0.271657
$870$	−4.67073	−0.158352
$871$	−7.44338	−0.252209
$872$	−38.2238	−1.29442
$873$	−0.393061	−0.0133031
$874$	−3.95005	−0.133612
$875$	0	0
$876$	58.2681	1.96870
$877$	16.2876	0.549992	0.274996	−	0.961445i	$-0.411324\pi$
$877$	16.2876	0.549992	0.274996	+	0.961445i	$0.411324\pi$
$878$	12.5934	0.425006
$879$	36.2799	1.22369
$880$	−7.80894	−0.263239
$881$	23.8815	0.804588	0.402294	−	0.915511i	$-0.368213\pi$
$881$	23.8815	0.804588	0.402294	+	0.915511i	$0.368213\pi$
$882$	0	0
$883$	34.7760	1.17031	0.585153	−	0.810923i	$-0.301035\pi$
$883$	34.7760	1.17031	0.585153	+	0.810923i	$0.301035\pi$
$884$	27.9878	0.941333
$885$	−32.2884	−1.08536
$886$	−8.09788	−0.272054
$887$	20.9659	0.703965	0.351983	−	0.936006i	$-0.385508\pi$
$887$	20.9659	0.703965	0.351983	+	0.936006i	$0.385508\pi$
$888$	−8.08528	−0.271324
$889$	0	0
$890$	7.26416	0.243495
$891$	−10.4494	−0.350066
$892$	44.0180	1.47383
$893$	−52.3069	−1.75038
$894$	−0.958000	−0.0320403
$895$	48.2183	1.61176
$896$	0	0
$897$	6.16271	0.205767
$898$	12.4085	0.414077
$899$	−6.01835	−0.200723
$900$	−31.5019	−1.05006
$901$	−96.1846	−3.20437
$902$	3.03332	0.100998
$903$	0	0
$904$	−2.45775	−0.0817435
$905$	−72.3957	−2.40652
$906$	8.45310	0.280835
$907$	−32.4206	−1.07651	−0.538254	−	0.842783i	$-0.680916\pi$
$907$	−32.4206	−1.07651	−0.538254	+	0.842783i	$0.680916\pi$
$908$	30.7691	1.02111
$909$	29.0940	0.964986
$910$	0	0
$911$	31.2801	1.03636	0.518178	−	0.855273i	$-0.326610\pi$
$911$	31.2801	1.03636	0.518178	+	0.855273i	$0.326610\pi$
$912$	−28.1681	−0.932739
$913$	4.23084	0.140020
$914$	−14.6916	−0.485955
$915$	21.0495	0.695876
$916$	22.5342	0.744551
$917$	0	0
$918$	−6.26849	−0.206891
$919$	−12.0241	−0.396638	−0.198319	−	0.980138i	$-0.563548\pi$
$919$	−12.0241	−0.396638	−0.198319	+	0.980138i	$0.563548\pi$
$920$	−9.35875	−0.308549
$921$	44.0877	1.45274
$922$	−10.4761	−0.345012
$923$	1.01830	0.0335177
$924$	0	0
$925$	−13.0890	−0.430363
$926$	−4.96034	−0.163007
$927$	15.4780	0.508363
$928$	−5.35490	−0.175783
$929$	46.3451	1.52053	0.760267	−	0.649611i	$-0.225068\pi$
$929$	46.3451	1.52053	0.760267	+	0.649611i	$0.225068\pi$
$930$	28.5851	0.937343
$931$	0	0
$932$	7.45522	0.244204
$933$	−40.8549	−1.33753
$934$	−13.7738	−0.450692
$935$	28.2183	0.922838
$936$	−10.5522	−0.344911
$937$	47.5786	1.55433	0.777163	−	0.629299i	$-0.216658\pi$
$937$	47.5786	1.55433	0.777163	+	0.629299i	$0.216658\pi$
$938$	0	0
$939$	−41.8129	−1.36451
$940$	−56.5680	−1.84504
$941$	−51.5693	−1.68111	−0.840556	−	0.541725i	$-0.817771\pi$
$941$	−51.5693	−1.68111	−0.840556	+	0.541725i	$0.817771\pi$
$942$	4.67572	0.152343
$943$	−6.72076	−0.218858
$944$	−8.45967	−0.275339
$945$	0	0
$946$	−5.21855	−0.169670
$947$	17.2878	0.561779	0.280889	−	0.959740i	$-0.409371\pi$
$947$	17.2878	0.561779	0.280889	+	0.959740i	$0.409371\pi$
$948$	−31.2398	−1.01462
$949$	31.5973	1.02569
$950$	24.6657	0.800260
$951$	−16.2042	−0.525457
$952$	0	0
$953$	4.25417	0.137806	0.0689031	−	0.997623i	$-0.478050\pi$
$953$	4.25417	0.137806	0.0689031	+	0.997623i	$0.478050\pi$
$954$	16.5530	0.535924
$955$	−72.2258	−2.33717
$956$	−15.3493	−0.496432
$957$	2.30329	0.0744547
$958$	−1.94563	−0.0628604
$959$	0	0
$960$	−10.8412	−0.349898
$961$	5.83268	0.188151
$962$	−2.00130	−0.0645244
$963$	37.0978	1.19546
$964$	28.8662	0.929717
$965$	−6.63186	−0.213487
$966$	0	0
$967$	5.87353	0.188880	0.0944399	−	0.995531i	$-0.469894\pi$
$967$	5.87353	0.188880	0.0944399	+	0.995531i	$0.469894\pi$
$968$	−2.08296	−0.0669488
$969$	101.788	3.26991
$970$	0.332835	0.0106867
$971$	−25.2611	−0.810667	−0.405333	−	0.914169i	$-0.632845\pi$
$971$	−25.2611	−0.810667	−0.405333	+	0.914169i	$0.632845\pi$
$972$	−33.6803	−1.08029
$973$	0	0
$974$	−16.7771	−0.537572
$975$	−38.4824	−1.23242
$976$	5.51505	0.176532
$977$	−25.4971	−0.815724	−0.407862	−	0.913043i	$-0.633726\pi$
$977$	−25.4971	−0.815724	−0.407862	+	0.913043i	$0.633726\pi$
$978$	−18.1784	−0.581282
$979$	−3.58219	−0.114487
$980$	0	0
$981$	43.9462	1.40309
$982$	−21.8152	−0.696150
$983$	37.0719	1.18241	0.591205	−	0.806521i	$-0.298653\pi$
$983$	37.0719	1.18241	0.591205	+	0.806521i	$0.298653\pi$
$984$	25.9238	0.826420
$985$	−33.2478	−1.05936
$986$	4.42212	0.140829
$987$	0	0
$988$	−19.7658	−0.628834
$989$	11.5625	0.367665
$990$	−4.85628	−0.154343
$991$	18.7006	0.594044	0.297022	−	0.954871i	$-0.404007\pi$
$991$	18.7006	0.594044	0.297022	+	0.954871i	$0.404007\pi$
$992$	32.7723	1.04052
$993$	−34.8884	−1.10715
$994$	0	0
$995$	3.55914	0.112832
$996$	16.5046	0.522967
$997$	−32.8555	−1.04054	−0.520272	−	0.854001i	$-0.674169\pi$
$997$	−32.8555	−1.04054	−0.520272	+	0.854001i	$0.674169\pi$
$998$	−23.1393	−0.732462
$999$	−2.34920	−0.0743255

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.a.l.1.5	✓	10
3.2	odd	2		4851.2.a.cg.1.6		10
4.3	odd	2		8624.2.a.df.1.9		10
7.2	even	3		539.2.e.o.67.6		20
7.3	odd	6		539.2.e.o.177.5		20
7.4	even	3		539.2.e.o.177.6		20
7.5	odd	6		539.2.e.o.67.5		20
7.6	odd	2	inner	539.2.a.l.1.6	yes	10
11.10	odd	2		5929.2.a.bv.1.5		10
21.20	even	2		4851.2.a.cg.1.5		10
28.27	even	2		8624.2.a.df.1.2		10
77.76	even	2		5929.2.a.bv.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.a.l.1.5	✓	10	1.1	even	1	trivial
539.2.a.l.1.6	yes	10	7.6	odd	2	inner
539.2.e.o.67.5		20	7.5	odd	6
539.2.e.o.67.6		20	7.2	even	3
539.2.e.o.177.5		20	7.3	odd	6
539.2.e.o.177.6		20	7.4	even	3
4851.2.a.cg.1.5		10	21.20	even	2
4851.2.a.cg.1.6		10	3.2	odd	2
5929.2.a.bv.1.5		10	11.10	odd	2
5929.2.a.bv.1.6		10	77.76	even	2
8624.2.a.df.1.2		10	28.27	even	2
8624.2.a.df.1.9		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)

\(f(q)\)	\(=\)	\(q+0.566092 q^{2} -2.32267 q^{3} -1.67954 q^{4} -3.58219 q^{5} -1.31484 q^{6} -2.08296 q^{8} +2.39479 q^{9} +O(q^{10})\)	\(q+0.566092 q^{2} -2.32267 q^{3} -1.67954 q^{4} -3.58219 q^{5} -1.31484 q^{6} -2.08296 q^{8} +2.39479 q^{9} -2.02785 q^{10} +1.00000 q^{11} +3.90101 q^{12} +2.11542 q^{13} +8.32025 q^{15} +2.17993 q^{16} -7.87739 q^{17} +1.35567 q^{18} +5.56323 q^{19} +6.01643 q^{20} +0.566092 q^{22} -1.25426 q^{23} +4.83802 q^{24} +7.83210 q^{25} +1.19752 q^{26} +1.40570 q^{27} -0.991656 q^{29} +4.71003 q^{30} +6.06899 q^{31} +5.39996 q^{32} -2.32267 q^{33} -4.45933 q^{34} -4.02214 q^{36} -1.67120 q^{37} +3.14930 q^{38} -4.91342 q^{39} +7.46156 q^{40} +5.35834 q^{41} -9.21855 q^{43} -1.67954 q^{44} -8.57860 q^{45} -0.710028 q^{46} -9.40225 q^{47} -5.06326 q^{48} +4.43369 q^{50} +18.2966 q^{51} -3.55293 q^{52} +12.2102 q^{53} +0.795757 q^{54} -3.58219 q^{55} -12.9216 q^{57} -0.561369 q^{58} -3.88070 q^{59} -13.9742 q^{60} +2.52992 q^{61} +3.43561 q^{62} -1.30299 q^{64} -7.57784 q^{65} -1.31484 q^{66} -3.51863 q^{67} +13.2304 q^{68} +2.91323 q^{69} +0.481369 q^{71} -4.98825 q^{72} +14.9367 q^{73} -0.946051 q^{74} -18.1914 q^{75} -9.34367 q^{76} -2.78145 q^{78} -8.00813 q^{79} -7.80894 q^{80} -10.4494 q^{81} +3.03332 q^{82} +4.23084 q^{83} +28.2183 q^{85} -5.21855 q^{86} +2.30329 q^{87} -2.08296 q^{88} -3.58219 q^{89} -4.85628 q^{90} +2.10658 q^{92} -14.0963 q^{93} -5.32254 q^{94} -19.9286 q^{95} -12.5423 q^{96} -0.164132 q^{97} +2.39479 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9}+O(q^{10}) \) 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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