LMFDB - Embedded newform 539.2.a.l.1.6

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			1.6
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	$-0.266081\pi$
$3$	2.32267	1.34099	0.670497	+	0.741913i	$0.266081\pi$
$4$	−1.67954	−0.839770
$5$	3.58219	1.60201	0.801003	−	0.598661i	$-0.204300\pi$
$5$	3.58219	1.60201	0.801003	+	0.598661i	$0.204300\pi$
$6$	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	$-0.594772\pi$
$13$	−2.11542	−0.586712	−0.293356	+	0.956003i	$0.594772\pi$
$14$	0	0
$15$	8.32025	2.14828
$16$	2.17993	0.544983
$17$	7.87739	1.91055	0.955274	−	0.295722i	$-0.0955601\pi$
$17$	7.87739	1.91055	0.955274	+	0.295722i	$0.0955601\pi$
$18$	1.35567	0.319535
$19$	−5.56323	−1.27629	−0.638147	−	0.769915i	$-0.720299\pi$
$19$	−5.56323	−1.27629	−0.638147	+	0.769915i	$0.720299\pi$
$20$	−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	$-0.683475\pi$
$31$	−6.06899	−1.09002	−0.545012	+	0.838428i	$0.683475\pi$
$32$	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	$-0.637415\pi$
$41$	−5.35834	−0.836833	−0.418416	+	0.908255i	$0.637415\pi$
$42$	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	$-0.259483\pi$
$47$	9.40225	1.37146	0.685729	+	0.727857i	$0.259483\pi$
$48$	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	$-0.418710\pi$
$59$	3.88070	0.505224	0.252612	+	0.967568i	$0.418710\pi$
$60$	−13.9742	−1.80406
$61$	−2.52992	−0.323923	−0.161961	−	0.986797i	$-0.551782\pi$
$61$	−2.52992	−0.323923	−0.161961	+	0.986797i	$0.551782\pi$
$62$	−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	$-0.838549\pi$
$73$	−14.9367	−1.74820	−0.874102	+	0.485743i	$0.838549\pi$
$74$	−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	$-0.574592\pi$
$83$	−4.23084	−0.464395	−0.232198	+	0.972669i	$0.574592\pi$
$84$	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	$-0.439198\pi$
$89$	3.58219	0.379712	0.189856	+	0.981812i	$0.439198\pi$
$90$	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	$-0.497348\pi$
$97$	0.164132	0.0166650	0.00833252	+	0.999965i	$0.497348\pi$
$98$	0	0
$99$	2.39479	0.240685
$100$	−13.1543	−1.31543
$101$	−12.1489	−1.20886	−0.604428	−	0.796660i	$-0.706598\pi$
$101$	−12.1489	−1.20886	−0.604428	+	0.796660i	$0.706598\pi$
$102$	10.3575	1.02555
$103$	−6.46318	−0.636836	−0.318418	−	0.947950i	$-0.603152\pi$
$103$	−6.46318	−0.636836	−0.318418	+	0.947950i	$0.603152\pi$
$104$	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	$-0.604241\pi$
$131$	−7.36312	−0.643319	−0.321660	+	0.946855i	$0.604241\pi$
$132$	−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	$-0.549351\pi$
$139$	−3.64121	−0.308843	−0.154422	+	0.988005i	$0.549351\pi$
$140$	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	$-0.454678\pi$
$157$	3.55610	0.283808	0.141904	+	0.989880i	$0.454678\pi$
$158$	−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	$-0.492738\pi$
$167$	0.589634	0.0456272	0.0228136	+	0.999740i	$0.492738\pi$
$168$	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	$-0.447928\pi$
$173$	4.28419	0.325721	0.162861	+	0.986649i	$0.447928\pi$
$174$	−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	$-0.770474\pi$
$181$	−20.2099	−1.50219	−0.751095	+	0.660194i	$0.770474\pi$
$182$	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	$-0.488788\pi$
$199$	0.993564	0.0704319	0.0352159	+	0.999380i	$0.488788\pi$
$200$	−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	$-0.159196\pi$
$223$	26.2084	1.75504	0.877521	+	0.479538i	$0.159196\pi$
$224$	0	0
$225$	18.7562	1.25042
$226$	0.667950	0.0444314
$227$	18.3200	1.21594	0.607970	−	0.793960i	$-0.291984\pi$
$227$	18.3200	1.21594	0.607970	+	0.793960i	$0.291984\pi$
$228$	21.7023	1.43727
$229$	13.4169	0.886613	0.443307	−	0.896370i	$-0.353805\pi$
$229$	13.4169	0.886613	0.443307	+	0.896370i	$0.353805\pi$
$230$	−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	$-0.313271\pi$
$241$	17.1870	1.10711	0.553554	+	0.832813i	$0.313271\pi$
$242$	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	$-0.689843\pi$
$251$	−17.7972	−1.12335	−0.561675	+	0.827358i	$0.689843\pi$
$252$	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	$-0.429997\pi$
$257$	6.99447	0.436303	0.218152	+	0.975915i	$0.429997\pi$
$258$	−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	$-0.389723\pi$
$269$	11.1383	0.679112	0.339556	+	0.940586i	$0.389723\pi$
$270$	−2.85056	−0.173479
$271$	3.94320	0.239532	0.119766	−	0.992802i	$-0.461786\pi$
$271$	3.94320	0.239532	0.119766	+	0.992802i	$0.461786\pi$
$272$	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	$-0.455840\pi$
$283$	4.65272	0.276576	0.138288	+	0.990392i	$0.455840\pi$
$284$	−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	$-0.349188\pi$
$293$	15.6199	0.912525	0.456263	+	0.889845i	$0.349188\pi$
$294$	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	$-0.317794\pi$
$307$	18.9815	1.08333	0.541666	+	0.840594i	$0.317794\pi$
$308$	0	0
$309$	−15.0118	−0.853993
$310$	−12.3070	−0.698992
$311$	−17.5896	−0.997417	−0.498709	−	0.866770i	$-0.666192\pi$
$311$	−17.5896	−0.997417	−0.498709	+	0.866770i	$0.666192\pi$
$312$	10.2345	0.579412
$313$	−18.0021	−1.01754	−0.508769	−	0.860903i	$-0.669899\pi$
$313$	−18.0021	−1.01754	−0.508769	+	0.860903i	$0.669899\pi$
$314$	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	$-0.772724\pi$
$349$	−28.2369	−1.51149	−0.755743	+	0.654868i	$0.772724\pi$
$350$	0	0
$351$	2.97365	0.158722
$352$	5.39996	0.287819
$353$	5.54734	0.295255	0.147627	−	0.989043i	$-0.452836\pi$
$353$	5.54734	0.295255	0.147627	+	0.989043i	$0.452836\pi$
$354$	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	$-0.432097\pi$
$367$	8.11154	0.423419	0.211710	+	0.977333i	$0.432097\pi$
$368$	−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	$-0.436345\pi$
$383$	7.77526	0.397297	0.198649	+	0.980071i	$0.436345\pi$
$384$	−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	$-0.668158\pi$
$397$	−20.0863	−1.00810	−0.504052	+	0.863673i	$0.668158\pi$
$398$	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	$-0.542106\pi$
$409$	−5.33474	−0.263786	−0.131893	+	0.991264i	$0.542106\pi$
$410$	−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	$-0.567933\pi$
$419$	−8.67099	−0.423605	−0.211803	+	0.977312i	$0.567933\pi$
$420$	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	$-0.451460\pi$
$433$	6.32172	0.303802	0.151901	+	0.988396i	$0.451460\pi$
$434$	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	$-0.678137\pi$
$439$	−22.2462	−1.06175	−0.530876	+	0.847450i	$0.678137\pi$
$440$	−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	$-0.358177\pi$
$461$	18.5060	0.861910	0.430955	+	0.902374i	$0.358177\pi$
$462$	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	$-0.309663\pi$
$467$	24.3313	1.12592	0.562960	+	0.826484i	$0.309663\pi$
$468$	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	$-0.474981\pi$
$479$	3.43695	0.157038	0.0785190	+	0.996913i	$0.474981\pi$
$480$	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	$-0.295241\pi$
$503$	26.9049	1.19963	0.599814	+	0.800139i	$0.295241\pi$
$504$	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	$-0.846377\pi$
$509$	−39.9683	−1.77156	−0.885781	+	0.464103i	$0.846377\pi$
$510$	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	$-0.211882\pi$
$521$	35.9052	1.57303	0.786517	+	0.617568i	$0.211882\pi$
$522$	−1.34436	−0.0588411
$523$	−12.1191	−0.529931	−0.264966	−	0.964258i	$-0.585361\pi$
$523$	−12.1191	−0.529931	−0.264966	+	0.964258i	$0.585361\pi$
$524$	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	$-0.781594\pi$
$563$	−36.7160	−1.54739	−0.773696	+	0.633557i	$0.781594\pi$
$564$	−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	$-0.719972\pi$
$577$	−30.6197	−1.27471	−0.637357	+	0.770568i	$0.719972\pi$
$578$	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	$-0.820905\pi$
$587$	−40.9866	−1.69170	−0.845848	+	0.533424i	$0.820905\pi$
$588$	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	$-0.543904\pi$
$593$	−6.69626	−0.274983	−0.137491	+	0.990503i	$0.543904\pi$
$594$	−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	$-0.643186\pi$
$601$	−21.3193	−0.869630	−0.434815	+	0.900520i	$0.643186\pi$
$602$	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	$-0.543272\pi$
$607$	−6.67801	−0.271052	−0.135526	+	0.990774i	$0.543272\pi$
$608$	−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	$-0.316335\pi$
$619$	27.1443	1.09102	0.545512	+	0.838103i	$0.316335\pi$
$620$	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	$-0.524502\pi$
$643$	−3.89995	−0.153799	−0.0768994	+	0.997039i	$0.524502\pi$
$644$	0	0
$645$	−76.7006	−3.02008
$646$	−24.8083	−0.976070
$647$	−21.3898	−0.840918	−0.420459	−	0.907311i	$-0.638131\pi$
$647$	−21.3898	−0.840918	−0.420459	+	0.907311i	$0.638131\pi$
$648$	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	$-0.337048\pi$
$661$	25.1885	0.979719	0.489859	+	0.871802i	$0.337048\pi$
$662$	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	$-0.441771\pi$
$677$	9.46644	0.363825	0.181912	+	0.983315i	$0.441771\pi$
$678$	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	$-0.532639\pi$
$691$	−5.38145	−0.204720	−0.102360	+	0.994747i	$0.532639\pi$
$692$	−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	$-0.664974\pi$
$719$	−26.5668	−0.990776	−0.495388	+	0.868672i	$0.664974\pi$
$720$	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	$-0.622167\pi$
$727$	−20.1922	−0.748889	−0.374444	+	0.927249i	$0.622167\pi$
$728$	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	$-0.715851\pi$
$733$	−33.9684	−1.25465	−0.627326	+	0.778757i	$0.715851\pi$
$734$	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	$-0.364537\pi$
$761$	22.7774	0.825682	0.412841	+	0.910803i	$0.364537\pi$
$762$	−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	$-0.577389\pi$
$769$	−13.3517	−0.481475	−0.240738	+	0.970590i	$0.577389\pi$
$770$	0	0
$771$	16.2458	0.585080
$772$	−3.10940	−0.111910
$773$	17.9214	0.644588	0.322294	−	0.946640i	$-0.395546\pi$
$773$	17.9214	0.644588	0.322294	+	0.946640i	$0.395546\pi$
$774$	−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	$-0.549092\pi$
$787$	−8.61899	−0.307234	−0.153617	+	0.988130i	$0.549092\pi$
$788$	−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	$-0.453882\pi$
$797$	8.15199	0.288758	0.144379	+	0.989522i	$0.453882\pi$
$798$	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	$-0.702391\pi$
$811$	−33.8231	−1.18769	−0.593845	+	0.804580i	$0.702391\pi$
$812$	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	$-0.402451\pi$
$829$	17.3725	0.603371	0.301685	+	0.953408i	$0.402451\pi$
$830$	−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	$-0.510613\pi$
$839$	−1.93117	−0.0666713	−0.0333357	+	0.999444i	$0.510613\pi$
$840$	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	$-0.493994\pi$
$853$	1.10204	0.0377332	0.0188666	+	0.999822i	$0.493994\pi$
$854$	0	0
$855$	−47.7248	−1.63215
$856$	−32.2672	−1.10287
$857$	−21.7149	−0.741766	−0.370883	−	0.928680i	$-0.620945\pi$
$857$	−21.7149	−0.741766	−0.370883	+	0.928680i	$0.620945\pi$
$858$	−2.78145	−0.0949571
$859$	−39.6186	−1.35177	−0.675885	−	0.737007i	$-0.736238\pi$
$859$	−39.6186	−1.35177	−0.675885	+	0.737007i	$0.736238\pi$
$860$	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	$-0.631787\pi$
$881$	−23.8815	−0.804588	−0.402294	+	0.915511i	$0.631787\pi$
$882$	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	$-0.614492\pi$
$887$	−20.9659	−0.703965	−0.351983	+	0.936006i	$0.614492\pi$
$888$	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	$-0.774932\pi$
$929$	−46.3451	−1.52053	−0.760267	+	0.649611i	$0.774932\pi$
$930$	−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	$-0.783342\pi$
$937$	−47.5786	−1.55433	−0.777163	+	0.629299i	$0.783342\pi$
$938$	0	0
$939$	−41.8129	−1.36451
$940$	−56.5680	−1.84504
$941$	51.5693	1.68111	0.840556	−	0.541725i	$-0.182229\pi$
$941$	51.5693	1.68111	0.840556	+	0.541725i	$0.182229\pi$
$942$	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	$-0.367155\pi$
$971$	25.2611	0.810667	0.405333	+	0.914169i	$0.367155\pi$
$972$	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	$-0.701347\pi$
$983$	−37.0719	−1.18241	−0.591205	+	0.806521i	$0.701347\pi$
$984$	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	$-0.325831\pi$
$997$	32.8555	1.04054	0.520272	+	0.854001i	$0.325831\pi$
$998$	−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.1.6	yes	10
3.2	odd	2		4851.2.a.cg.1.5		10
4.3	odd	2		8624.2.a.df.1.2		10
7.2	even	3		539.2.e.o.67.5		20
7.3	odd	6		539.2.e.o.177.6		20
7.4	even	3		539.2.e.o.177.5		20
7.5	odd	6		539.2.e.o.67.6		20
7.6	odd	2	inner	539.2.a.l.1.5	✓	10
11.10	odd	2		5929.2.a.bv.1.6		10
21.20	even	2		4851.2.a.cg.1.6		10
28.27	even	2		8624.2.a.df.1.9		10
77.76	even	2		5929.2.a.bv.1.5		10

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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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