LMFDB - Embedded newform 3871.2.a.e.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3871,2,Mod(1,3871)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3871.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3871, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3871 = 7^{2} \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3871.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,3,-3,7,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$30.9100906224$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 7x^{6} + 24x^{5} + 6x^{4} - 40x^{3} + 6x^{2} + 8x + 1 $ x^8 - 3x^7 - 7x^6 + 24x^5 + 6x^4 - 40x^3 + 6x^2 + 8*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 553)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.58125$ of defining polynomial
Character	$\chi$	$=$	3871.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.58125 q^{2} -1.34420 q^{3} +4.66284 q^{4} +0.561042 q^{5} -3.46973 q^{6} +6.87345 q^{8} -1.19311 q^{9} +1.44819 q^{10} -4.22461 q^{11} -6.26781 q^{12} -2.71008 q^{13} -0.754156 q^{15} +8.41639 q^{16} -0.917226 q^{17} -3.07972 q^{18} -6.85309 q^{19} +2.61605 q^{20} -10.9048 q^{22} +0.979795 q^{23} -9.23932 q^{24} -4.68523 q^{25} -6.99538 q^{26} +5.63640 q^{27} -1.52445 q^{29} -1.94666 q^{30} +8.40354 q^{31} +7.97789 q^{32} +5.67874 q^{33} -2.36759 q^{34} -5.56330 q^{36} -9.29422 q^{37} -17.6895 q^{38} +3.64290 q^{39} +3.85629 q^{40} -2.23217 q^{41} -4.72010 q^{43} -19.6987 q^{44} -0.669387 q^{45} +2.52909 q^{46} -10.1995 q^{47} -11.3133 q^{48} -12.0937 q^{50} +1.23294 q^{51} -12.6367 q^{52} +10.1168 q^{53} +14.5490 q^{54} -2.37018 q^{55} +9.21196 q^{57} -3.93499 q^{58} -5.81954 q^{59} -3.51651 q^{60} -1.21993 q^{61} +21.6916 q^{62} +3.76013 q^{64} -1.52047 q^{65} +14.6582 q^{66} +12.6580 q^{67} -4.27688 q^{68} -1.31704 q^{69} -0.171320 q^{71} -8.20080 q^{72} +8.12529 q^{73} -23.9907 q^{74} +6.29791 q^{75} -31.9549 q^{76} +9.40323 q^{78} +1.00000 q^{79} +4.72195 q^{80} -3.99714 q^{81} -5.76178 q^{82} +1.55215 q^{83} -0.514603 q^{85} -12.1837 q^{86} +2.04917 q^{87} -29.0376 q^{88} -12.3567 q^{89} -1.72785 q^{90} +4.56862 q^{92} -11.2961 q^{93} -26.3274 q^{94} -3.84487 q^{95} -10.7239 q^{96} +6.82458 q^{97} +5.04043 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{2} - 3 q^{3} + 7 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{8} - q^{9} - 10 q^{10} + 3 q^{11} - 10 q^{12} - 3 q^{13} + 11 q^{15} + 13 q^{16} - 14 q^{17} + 4 q^{18} - 12 q^{19} - 12 q^{20} - 15 q^{22}+ \cdots - 25 q^{99}+O(q^{100}) $ 8 * q + 3 * q^2 - 3 * q^3 + 7 * q^4 - 4 * q^5 - 6 * q^6 + 6 * q^8 - q^9 - 10 * q^10 + 3 * q^11 - 10 * q^12 - 3 * q^13 + 11 * q^15 + 13 * q^16 - 14 * q^17 + 4 * q^18 - 12 * q^19 - 12 * q^20 - 15 * q^22 + 17 * q^23 - 14 * q^24 - 6 * q^25 + 13 * q^26 + 6 * q^27 - 6 * q^29 + 17 * q^30 - 15 * q^31 + 5 * q^32 - 2 * q^33 + 2 * q^34 - 33 * q^36 - 2 * q^37 - 19 * q^38 + 6 * q^39 + 13 * q^40 - 8 * q^41 + 9 * q^43 + 7 * q^44 - 5 * q^45 - 24 * q^46 - 34 * q^47 + 9 * q^48 - 3 * q^50 - 11 * q^51 - 17 * q^52 + 2 * q^53 + 10 * q^54 - 16 * q^55 + 6 * q^57 - 14 * q^58 - 21 * q^59 - 14 * q^60 - 11 * q^61 + 15 * q^62 - 16 * q^64 - 5 * q^65 + 18 * q^66 + 14 * q^67 - 7 * q^68 + 8 * q^69 - 2 * q^71 - 18 * q^72 - 7 * q^73 - 8 * q^74 - 28 * q^75 + 23 * q^76 - 8 * q^78 + 8 * q^79 - 6 * q^80 - 28 * q^81 + 10 * q^82 - 32 * q^83 - 9 * q^85 - 6 * q^86 - 43 * q^87 - 48 * q^88 - 6 * q^89 - q^90 - 3 * q^92 - 12 * q^93 - 33 * q^94 - 15 * q^95 + 9 * q^96 + 39 * q^97 - 25 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.58125	1.82522	0.912609	−	0.408834i	$-0.134064\pi$
$2$	2.58125	1.82522	0.912609	+	0.408834i	$0.134064\pi$
$3$	−1.34420	−0.776077	−0.388039	−	0.921643i	$-0.626847\pi$
$3$	−1.34420	−0.776077	−0.388039	+	0.921643i	$0.626847\pi$
$4$	4.66284	2.33142
$5$	0.561042	0.250906	0.125453	−	0.992100i	$-0.459962\pi$
$5$	0.561042	0.250906	0.125453	+	0.992100i	$0.459962\pi$
$6$	−3.46973	−1.41651
$7$	0	0
$7$	0	0
$8$	6.87345	2.43013
$9$	−1.19311	−0.397704
$10$	1.44819	0.457958
$11$	−4.22461	−1.27377	−0.636883	−	0.770960i	$-0.719777\pi$
$11$	−4.22461	−1.27377	−0.636883	+	0.770960i	$0.719777\pi$
$12$	−6.26781	−1.80936
$13$	−2.71008	−0.751640	−0.375820	−	0.926693i	$-0.622639\pi$
$13$	−2.71008	−0.751640	−0.375820	+	0.926693i	$0.622639\pi$
$14$	0	0
$15$	−0.754156	−0.194722
$16$	8.41639	2.10410
$17$	−0.917226	−0.222460	−0.111230	−	0.993795i	$-0.535479\pi$
$17$	−0.917226	−0.222460	−0.111230	+	0.993795i	$0.535479\pi$
$18$	−3.07972	−0.725897
$19$	−6.85309	−1.57221	−0.786104	−	0.618095i	$-0.787905\pi$
$19$	−6.85309	−1.57221	−0.786104	+	0.618095i	$0.787905\pi$
$20$	2.61605	0.584966
$21$	0	0
$22$	−10.9048	−2.32490
$23$	0.979795	0.204301	0.102151	−	0.994769i	$-0.467428\pi$
$23$	0.979795	0.204301	0.102151	+	0.994769i	$0.467428\pi$
$24$	−9.23932	−1.88597
$25$	−4.68523	−0.937046
$26$	−6.99538	−1.37191
$27$	5.63640	1.08473
$28$	0	0
$29$	−1.52445	−0.283083	−0.141542	−	0.989932i	$-0.545206\pi$
$29$	−1.52445	−0.283083	−0.141542	+	0.989932i	$0.545206\pi$
$30$	−1.94666	−0.355410
$31$	8.40354	1.50932	0.754660	−	0.656116i	$-0.227802\pi$
$31$	8.40354	1.50932	0.754660	+	0.656116i	$0.227802\pi$
$32$	7.97789	1.41030
$33$	5.67874	0.988541
$34$	−2.36759	−0.406038
$35$	0	0
$36$	−5.56330	−0.927216
$37$	−9.29422	−1.52796	−0.763980	−	0.645240i	$-0.776757\pi$
$37$	−9.29422	−1.52796	−0.763980	+	0.645240i	$0.776757\pi$
$38$	−17.6895	−2.86962
$39$	3.64290	0.583331
$40$	3.85629	0.609734
$41$	−2.23217	−0.348606	−0.174303	−	0.984692i	$-0.555767\pi$
$41$	−2.23217	−0.348606	−0.174303	+	0.984692i	$0.555767\pi$
$42$	0	0
$43$	−4.72010	−0.719809	−0.359904	−	0.932989i	$-0.617191\pi$
$43$	−4.72010	−0.719809	−0.359904	+	0.932989i	$0.617191\pi$
$44$	−19.6987	−2.96968
$45$	−0.669387	−0.0997863
$46$	2.52909	0.372894
$47$	−10.1995	−1.48775	−0.743874	−	0.668319i	$-0.767014\pi$
$47$	−10.1995	−1.48775	−0.743874	+	0.668319i	$0.767014\pi$
$48$	−11.3133	−1.63294
$49$	0	0
$50$	−12.0937	−1.71031
$51$	1.23294	0.172646
$52$	−12.6367	−1.75239
$53$	10.1168	1.38965	0.694823	−	0.719181i	$-0.255483\pi$
$53$	10.1168	1.38965	0.694823	+	0.719181i	$0.255483\pi$
$54$	14.5490	1.97986
$55$	−2.37018	−0.319595
$56$	0	0
$57$	9.21196	1.22015
$58$	−3.93499	−0.516689
$59$	−5.81954	−0.757640	−0.378820	−	0.925470i	$-0.623670\pi$
$59$	−5.81954	−0.757640	−0.378820	+	0.925470i	$0.623670\pi$
$60$	−3.51651	−0.453979
$61$	−1.21993	−0.156196	−0.0780978	−	0.996946i	$-0.524885\pi$
$61$	−1.21993	−0.156196	−0.0780978	+	0.996946i	$0.524885\pi$
$62$	21.6916	2.75484
$63$	0	0
$64$	3.76013	0.470016
$65$	−1.52047	−0.188591
$66$	14.6582	1.80430
$67$	12.6580	1.54643	0.773213	−	0.634146i	$-0.218648\pi$
$67$	12.6580	1.54643	0.773213	+	0.634146i	$0.218648\pi$
$68$	−4.27688	−0.518648
$69$	−1.31704	−0.158554
$70$	0	0
$71$	−0.171320	−0.0203319	−0.0101660	−	0.999948i	$-0.503236\pi$
$71$	−0.171320	−0.0203319	−0.0101660	+	0.999948i	$0.503236\pi$
$72$	−8.20080	−0.966474
$73$	8.12529	0.950994	0.475497	−	0.879717i	$-0.342268\pi$
$73$	8.12529	0.950994	0.475497	+	0.879717i	$0.342268\pi$
$74$	−23.9907	−2.78886
$75$	6.29791	0.727220
$76$	−31.9549	−3.66547
$77$	0	0
$78$	9.40323	1.06471
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	4.72195	0.527930
$81$	−3.99714	−0.444127
$82$	−5.76178	−0.636282
$83$	1.55215	0.170371	0.0851853	−	0.996365i	$-0.472852\pi$
$83$	1.55215	0.170371	0.0851853	+	0.996365i	$0.472852\pi$
$84$	0	0
$85$	−0.514603	−0.0558165
$86$	−12.1837	−1.31381
$87$	2.04917	0.219695
$88$	−29.0376	−3.09542
$89$	−12.3567	−1.30981	−0.654904	−	0.755712i	$-0.727291\pi$
$89$	−12.3567	−1.30981	−0.654904	+	0.755712i	$0.727291\pi$
$90$	−1.72785	−0.182132
$91$	0	0
$92$	4.56862	0.476312
$93$	−11.2961	−1.17135
$94$	−26.3274	−2.71547
$95$	−3.84487	−0.394476
$96$	−10.7239	−1.09451
$97$	6.82458	0.692931	0.346465	−	0.938063i	$-0.387382\pi$
$97$	6.82458	0.692931	0.346465	+	0.938063i	$0.387382\pi$
$98$	0	0
$99$	5.04043	0.506583
$100$	−21.8465	−2.18465
$101$	8.82730	0.878349	0.439175	−	0.898402i	$-0.355271\pi$
$101$	8.82730	0.878349	0.439175	+	0.898402i	$0.355271\pi$
$102$	3.18252	0.315117
$103$	−7.94651	−0.782993	−0.391497	−	0.920180i	$-0.628043\pi$
$103$	−7.94651	−0.782993	−0.391497	+	0.920180i	$0.628043\pi$
$104$	−18.6276	−1.82658
$105$	0	0
$106$	26.1139	2.53641
$107$	5.60066	0.541436	0.270718	−	0.962659i	$-0.412739\pi$
$107$	5.60066	0.541436	0.270718	+	0.962659i	$0.412739\pi$
$108$	26.2816	2.52895
$109$	5.86221	0.561498	0.280749	−	0.959781i	$-0.409417\pi$
$109$	5.86221	0.561498	0.280749	+	0.959781i	$0.409417\pi$
$110$	−6.11803	−0.583331
$111$	12.4933	1.18581
$112$	0	0
$113$	16.3186	1.53513	0.767563	−	0.640974i	$-0.221469\pi$
$113$	16.3186	1.53513	0.767563	+	0.640974i	$0.221469\pi$
$114$	23.7783	2.22705
$115$	0.549706	0.0512604
$116$	−7.10827	−0.659986
$117$	3.23343	0.298931
$118$	−15.0217	−1.38286
$119$	0	0
$120$	−5.18365	−0.473200
$121$	6.84729	0.622481
$122$	−3.14893	−0.285091
$123$	3.00049	0.270545
$124$	39.1843	3.51886
$125$	−5.43382	−0.486016
$126$	0	0
$127$	−13.6220	−1.20876	−0.604378	−	0.796698i	$-0.706578\pi$
$127$	−13.6220	−1.20876	−0.604378	+	0.796698i	$0.706578\pi$
$128$	−6.24995	−0.552423
$129$	6.34478	0.558627
$130$	−3.92470	−0.344219
$131$	−11.4611	−1.00136	−0.500681	−	0.865632i	$-0.666917\pi$
$131$	−11.4611	−1.00136	−0.500681	+	0.865632i	$0.666917\pi$
$132$	26.4790	2.30470
$133$	0	0
$134$	32.6735	2.82256
$135$	3.16226	0.272164
$136$	−6.30450	−0.540607
$137$	−22.5078	−1.92297	−0.961487	−	0.274849i	$-0.911372\pi$
$137$	−22.5078	−1.92297	−0.961487	+	0.274849i	$0.911372\pi$
$138$	−3.39962	−0.289395
$139$	11.7916	1.00015	0.500076	−	0.865982i	$-0.333306\pi$
$139$	11.7916	1.00015	0.500076	+	0.865982i	$0.333306\pi$
$140$	0	0
$141$	13.7102	1.15461
$142$	−0.442219	−0.0371102
$143$	11.4490	0.957414
$144$	−10.0417	−0.836809
$145$	−0.855281	−0.0710273
$146$	20.9734	1.73577
$147$	0	0
$148$	−43.3374	−3.56232
$149$	14.1468	1.15895	0.579475	−	0.814990i	$-0.303257\pi$
$149$	14.1468	1.15895	0.579475	+	0.814990i	$0.303257\pi$
$150$	16.2565	1.32733
$151$	12.7365	1.03648	0.518240	−	0.855235i	$-0.326588\pi$
$151$	12.7365	1.03648	0.518240	+	0.855235i	$0.326588\pi$
$152$	−47.1044	−3.82067
$153$	1.09435	0.0884733
$154$	0	0
$155$	4.71474	0.378697
$156$	16.9863	1.35999
$157$	−13.1344	−1.04824	−0.524121	−	0.851644i	$-0.675606\pi$
$157$	−13.1344	−1.04824	−0.524121	+	0.851644i	$0.675606\pi$
$158$	2.58125	0.205353
$159$	−13.5990	−1.07847
$160$	4.47593	0.353854
$161$	0	0
$162$	−10.3176	−0.810628
$163$	−11.3164	−0.886367	−0.443184	−	0.896431i	$-0.646151\pi$
$163$	−11.3164	−0.886367	−0.443184	+	0.896431i	$0.646151\pi$
$164$	−10.4082	−0.812747
$165$	3.18601	0.248031
$166$	4.00648	0.310963
$167$	3.38556	0.261982	0.130991	−	0.991384i	$-0.458184\pi$
$167$	3.38556	0.261982	0.130991	+	0.991384i	$0.458184\pi$
$168$	0	0
$169$	−5.65548	−0.435037
$170$	−1.32832	−0.101877
$171$	8.17652	0.625274
$172$	−22.0091	−1.67818
$173$	4.83074	0.367274	0.183637	−	0.982994i	$-0.441213\pi$
$173$	4.83074	0.367274	0.183637	+	0.982994i	$0.441213\pi$
$174$	5.28943	0.400990
$175$	0	0
$176$	−35.5559	−2.68013
$177$	7.82266	0.587987
$178$	−31.8957	−2.39069
$179$	−4.75159	−0.355151	−0.177575	−	0.984107i	$-0.556825\pi$
$179$	−4.75159	−0.355151	−0.177575	+	0.984107i	$0.556825\pi$
$180$	−3.12124	−0.232644
$181$	−10.3047	−0.765942	−0.382971	−	0.923760i	$-0.625099\pi$
$181$	−10.3047	−0.765942	−0.382971	+	0.923760i	$0.625099\pi$
$182$	0	0
$183$	1.63983	0.121220
$184$	6.73457	0.496479
$185$	−5.21445	−0.383374
$186$	−29.1580	−2.13797
$187$	3.87492	0.283362
$188$	−47.5586	−3.46857
$189$	0	0
$190$	−9.92457	−0.720004
$191$	17.4521	1.26279	0.631393	−	0.775463i	$-0.282483\pi$
$191$	17.4521	1.26279	0.631393	+	0.775463i	$0.282483\pi$
$192$	−5.05438	−0.364769
$193$	−15.5061	−1.11616	−0.558078	−	0.829789i	$-0.688461\pi$
$193$	−15.5061	−1.11616	−0.558078	+	0.829789i	$0.688461\pi$
$194$	17.6159	1.26475
$195$	2.04382	0.146361
$196$	0	0
$197$	14.6117	1.04104	0.520519	−	0.853850i	$-0.325738\pi$
$197$	14.6117	1.04104	0.520519	+	0.853850i	$0.325738\pi$
$198$	13.0106	0.924624
$199$	−5.15119	−0.365158	−0.182579	−	0.983191i	$-0.558445\pi$
$199$	−5.15119	−0.365158	−0.182579	+	0.983191i	$0.558445\pi$
$200$	−32.2037	−2.27714
$201$	−17.0150	−1.20015
$202$	22.7854	1.60318
$203$	0	0
$204$	5.74900	0.402510
$205$	−1.25234	−0.0874672
$206$	−20.5119	−1.42913
$207$	−1.16901	−0.0812516
$208$	−22.8091	−1.58152
$209$	28.9516	2.00262
$210$	0	0
$211$	7.02148	0.483378	0.241689	−	0.970354i	$-0.422299\pi$
$211$	7.02148	0.483378	0.241689	+	0.970354i	$0.422299\pi$
$212$	47.1729	3.23985
$213$	0.230289	0.0157791
$214$	14.4567	0.988239
$215$	−2.64818	−0.180604
$216$	38.7415	2.63603
$217$	0	0
$218$	15.1318	1.02486
$219$	−10.9221	−0.738044
$220$	−11.0518	−0.745111
$221$	2.48575	0.167210
$222$	32.2484	2.16437
$223$	0.413867	0.0277146	0.0138573	−	0.999904i	$-0.495589\pi$
$223$	0.413867	0.0277146	0.0138573	+	0.999904i	$0.495589\pi$
$224$	0	0
$225$	5.59001	0.372668
$226$	42.1224	2.80194
$227$	−10.8012	−0.716900	−0.358450	−	0.933549i	$-0.616695\pi$
$227$	−10.8012	−0.716900	−0.358450	+	0.933549i	$0.616695\pi$
$228$	42.9539	2.84469
$229$	14.8786	0.983206	0.491603	−	0.870820i	$-0.336411\pi$
$229$	14.8786	0.983206	0.491603	+	0.870820i	$0.336411\pi$
$230$	1.41893	0.0935613
$231$	0	0
$232$	−10.4782	−0.687930
$233$	−14.4241	−0.944957	−0.472479	−	0.881342i	$-0.656641\pi$
$233$	−14.4241	−0.944957	−0.472479	+	0.881342i	$0.656641\pi$
$234$	8.34628	0.545614
$235$	−5.72235	−0.373285
$236$	−27.1356	−1.76638
$237$	−1.34420	−0.0873155
$238$	0	0
$239$	−18.8811	−1.22132	−0.610658	−	0.791894i	$-0.709095\pi$
$239$	−18.8811	−1.22132	−0.610658	+	0.791894i	$0.709095\pi$
$240$	−6.34727	−0.409714
$241$	−24.8942	−1.60357	−0.801787	−	0.597610i	$-0.796117\pi$
$241$	−24.8942	−1.60357	−0.801787	+	0.597610i	$0.796117\pi$
$242$	17.6746	1.13616
$243$	−11.5362	−0.740050
$244$	−5.68832	−0.364157
$245$	0	0
$246$	7.74501	0.493804
$247$	18.5724	1.18173
$248$	57.7612	3.66784
$249$	−2.08641	−0.132221
$250$	−14.0260	−0.887085
$251$	23.4440	1.47977	0.739887	−	0.672731i	$-0.234879\pi$
$251$	23.4440	1.47977	0.739887	+	0.672731i	$0.234879\pi$
$252$	0	0
$253$	−4.13925	−0.260232
$254$	−35.1617	−2.20624
$255$	0.691731	0.0433179
$256$	−23.6529	−1.47831
$257$	−15.0310	−0.937611	−0.468805	−	0.883302i	$-0.655315\pi$
$257$	−15.0310	−0.937611	−0.468805	+	0.883302i	$0.655315\pi$
$258$	16.3775	1.01962
$259$	0	0
$260$	−7.08970	−0.439684
$261$	1.81884	0.112584
$262$	−29.5840	−1.82771
$263$	−9.96764	−0.614631	−0.307316	−	0.951608i	$-0.599431\pi$
$263$	−9.96764	−0.614631	−0.307316	+	0.951608i	$0.599431\pi$
$264$	39.0325	2.40228
$265$	5.67594	0.348670
$266$	0	0
$267$	16.6100	1.01651
$268$	59.0224	3.60537
$269$	15.9710	0.973770	0.486885	−	0.873466i	$-0.338133\pi$
$269$	15.9710	0.973770	0.486885	+	0.873466i	$0.338133\pi$
$270$	8.16258	0.496759
$271$	2.33529	0.141859	0.0709294	−	0.997481i	$-0.477403\pi$
$271$	2.33529	0.141859	0.0709294	+	0.997481i	$0.477403\pi$
$272$	−7.71973	−0.468077
$273$	0	0
$274$	−58.0983	−3.50985
$275$	19.7933	1.19358
$276$	−6.14117	−0.369655
$277$	−18.4986	−1.11147	−0.555737	−	0.831358i	$-0.687564\pi$
$277$	−18.4986	−1.11147	−0.555737	+	0.831358i	$0.687564\pi$
$278$	30.4371	1.82549
$279$	−10.0264	−0.600263
$280$	0	0
$281$	−28.1409	−1.67875	−0.839374	−	0.543554i	$-0.817078\pi$
$281$	−28.1409	−1.67875	−0.839374	+	0.543554i	$0.817078\pi$
$282$	35.3894	2.10741
$283$	28.6826	1.70500	0.852501	−	0.522725i	$-0.175085\pi$
$283$	28.6826	1.70500	0.852501	+	0.522725i	$0.175085\pi$
$284$	−0.798837	−0.0474023
$285$	5.16830	0.306144
$286$	29.5527	1.74749
$287$	0	0
$288$	−9.51853	−0.560885
$289$	−16.1587	−0.950512
$290$	−2.20769	−0.129640
$291$	−9.17363	−0.537768
$292$	37.8869	2.21716
$293$	8.20183	0.479156	0.239578	−	0.970877i	$-0.422991\pi$
$293$	8.20183	0.479156	0.239578	+	0.970877i	$0.422991\pi$
$294$	0	0
$295$	−3.26501	−0.190096
$296$	−63.8833	−3.71314
$297$	−23.8116	−1.38169
$298$	36.5164	2.11534
$299$	−2.65532	−0.153561
$300$	29.3661	1.69546
$301$	0	0
$302$	32.8760	1.89180
$303$	−11.8657	−0.681667
$304$	−57.6783	−3.30808
$305$	−0.684430	−0.0391904
$306$	2.82480	0.161483
$307$	−12.6167	−0.720074	−0.360037	−	0.932938i	$-0.617236\pi$
$307$	−12.6167	−0.720074	−0.360037	+	0.932938i	$0.617236\pi$
$308$	0	0
$309$	10.6817	0.607663
$310$	12.1699	0.691204
$311$	−14.0887	−0.798899	−0.399450	−	0.916755i	$-0.630799\pi$
$311$	−14.0887	−0.798899	−0.399450	+	0.916755i	$0.630799\pi$
$312$	25.0393	1.41757
$313$	−29.1629	−1.64839	−0.824193	−	0.566310i	$-0.808371\pi$
$313$	−29.1629	−1.64839	−0.824193	+	0.566310i	$0.808371\pi$
$314$	−33.9032	−1.91327
$315$	0	0
$316$	4.66284	0.262305
$317$	−5.78458	−0.324895	−0.162447	−	0.986717i	$-0.551939\pi$
$317$	−5.78458	−0.324895	−0.162447	+	0.986717i	$0.551939\pi$
$318$	−35.1024	−1.96845
$319$	6.44020	0.360582
$320$	2.10959	0.117930
$321$	−7.52844	−0.420196
$322$	0	0
$323$	6.28583	0.349753
$324$	−18.6380	−1.03545
$325$	12.6973	0.704322
$326$	−29.2104	−1.61781
$327$	−7.88001	−0.435766
$328$	−15.3427	−0.847158
$329$	0	0
$330$	8.22388	0.452710
$331$	15.2466	0.838028	0.419014	−	0.907980i	$-0.362376\pi$
$331$	15.2466	0.838028	0.419014	+	0.907980i	$0.362376\pi$
$332$	7.23742	0.397205
$333$	11.0891	0.607677
$334$	8.73896	0.478174
$335$	7.10170	0.388007
$336$	0	0
$337$	−10.1400	−0.552364	−0.276182	−	0.961105i	$-0.589069\pi$
$337$	−10.1400	−0.552364	−0.276182	+	0.961105i	$0.589069\pi$
$338$	−14.5982	−0.794037
$339$	−21.9356	−1.19138
$340$	−2.39951	−0.130132
$341$	−35.5016	−1.92252
$342$	21.1056	1.14126
$343$	0	0
$344$	−32.4434	−1.74923
$345$	−0.738918	−0.0397820
$346$	12.4693	0.670355
$347$	7.34908	0.394519	0.197260	−	0.980351i	$-0.436796\pi$
$347$	7.34908	0.394519	0.197260	+	0.980351i	$0.436796\pi$
$348$	9.55497	0.512200
$349$	28.2606	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$349$	28.2606	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$350$	0	0
$351$	−15.2751	−0.815324
$352$	−33.7034	−1.79640
$353$	33.4363	1.77963	0.889817	−	0.456318i	$-0.150832\pi$
$353$	33.4363	1.77963	0.889817	+	0.456318i	$0.150832\pi$
$354$	20.1922	1.07320
$355$	−0.0961177	−0.00510140
$356$	−57.6174	−3.05371
$357$	0	0
$358$	−12.2650	−0.648227
$359$	32.0780	1.69301	0.846505	−	0.532381i	$-0.178702\pi$
$359$	32.0780	1.69301	0.846505	+	0.532381i	$0.178702\pi$
$360$	−4.60100	−0.242494
$361$	27.9649	1.47184
$362$	−26.5990	−1.39801
$363$	−9.20416	−0.483093
$364$	0	0
$365$	4.55863	0.238610
$366$	4.23281	0.221252
$367$	−6.42446	−0.335354	−0.167677	−	0.985842i	$-0.553627\pi$
$367$	−6.42446	−0.335354	−0.167677	+	0.985842i	$0.553627\pi$
$368$	8.24633	0.429870
$369$	2.66323	0.138642
$370$	−13.4598	−0.699741
$371$	0	0
$372$	−52.6718	−2.73090
$373$	−4.79795	−0.248428	−0.124214	−	0.992255i	$-0.539641\pi$
$373$	−4.79795	−0.248428	−0.124214	+	0.992255i	$0.539641\pi$
$374$	10.0021	0.517197
$375$	7.30417	0.377186
$376$	−70.1057	−3.61542
$377$	4.13138	0.212777
$378$	0	0
$379$	20.4106	1.04842	0.524211	−	0.851588i	$-0.324360\pi$
$379$	20.4106	1.04842	0.524211	+	0.851588i	$0.324360\pi$
$380$	−17.9280	−0.919688
$381$	18.3107	0.938088
$382$	45.0481	2.30486
$383$	−17.1778	−0.877743	−0.438871	−	0.898550i	$-0.644622\pi$
$383$	−17.1778	−0.877743	−0.438871	+	0.898550i	$0.644622\pi$
$384$	8.40122	0.428723
$385$	0	0
$386$	−40.0252	−2.03723
$387$	5.63162	0.286271
$388$	31.8219	1.61551
$389$	−3.82464	−0.193917	−0.0969584	−	0.995288i	$-0.530911\pi$
$389$	−3.82464	−0.193917	−0.0969584	+	0.995288i	$0.530911\pi$
$390$	5.27561	0.267141
$391$	−0.898693	−0.0454489
$392$	0	0
$393$	15.4061	0.777135
$394$	37.7164	1.90012
$395$	0.561042	0.0282291
$396$	23.5027	1.18106
$397$	−3.13287	−0.157234	−0.0786170	−	0.996905i	$-0.525050\pi$
$397$	−3.13287	−0.157234	−0.0786170	+	0.996905i	$0.525050\pi$
$398$	−13.2965	−0.666493
$399$	0	0
$400$	−39.4327	−1.97164
$401$	−15.2722	−0.762660	−0.381330	−	0.924439i	$-0.624534\pi$
$401$	−15.2722	−0.762660	−0.381330	+	0.924439i	$0.624534\pi$
$402$	−43.9199	−2.19053
$403$	−22.7742	−1.13447
$404$	41.1603	2.04780
$405$	−2.24256	−0.111434
$406$	0	0
$407$	39.2644	1.94626
$408$	8.47454	0.419552
$409$	19.6550	0.971878	0.485939	−	0.873993i	$-0.338478\pi$
$409$	19.6550	0.971878	0.485939	+	0.873993i	$0.338478\pi$
$410$	−3.23260	−0.159647
$411$	30.2551	1.49238
$412$	−37.0533	−1.82549
$413$	0	0
$414$	−3.01749	−0.148302
$415$	0.870822	0.0427469
$416$	−21.6207	−1.06004
$417$	−15.8503	−0.776195
$418$	74.7313	3.65523
$419$	−7.11168	−0.347428	−0.173714	−	0.984796i	$-0.555577\pi$
$419$	−7.11168	−0.347428	−0.173714	+	0.984796i	$0.555577\pi$
$420$	0	0
$421$	−33.0992	−1.61316	−0.806578	−	0.591128i	$-0.798683\pi$
$421$	−33.0992	−1.61316	−0.806578	+	0.591128i	$0.798683\pi$
$422$	18.1242	0.882271
$423$	12.1692	0.591684
$424$	69.5371	3.37702
$425$	4.29742	0.208455
$426$	0.594433	0.0288004
$427$	0	0
$428$	26.1150	1.26232
$429$	−15.3898	−0.743027
$430$	−6.83560	−0.329642
$431$	6.49998	0.313093	0.156547	−	0.987671i	$-0.449964\pi$
$431$	6.49998	0.313093	0.156547	+	0.987671i	$0.449964\pi$
$432$	47.4382	2.28237
$433$	28.0215	1.34663	0.673313	−	0.739357i	$-0.264871\pi$
$433$	28.0215	1.34663	0.673313	+	0.739357i	$0.264871\pi$
$434$	0	0
$435$	1.14967	0.0551226
$436$	27.3345	1.30909
$437$	−6.71462	−0.321204
$438$	−28.1925	−1.34709
$439$	4.64973	0.221919	0.110960	−	0.993825i	$-0.464608\pi$
$439$	4.64973	0.221919	0.110960	+	0.993825i	$0.464608\pi$
$440$	−16.2913	−0.776658
$441$	0	0
$442$	6.41635	0.305194
$443$	−4.82223	−0.229111	−0.114555	−	0.993417i	$-0.536544\pi$
$443$	−4.82223	−0.229111	−0.114555	+	0.993417i	$0.536544\pi$
$444$	58.2544	2.76463
$445$	−6.93264	−0.328639
$446$	1.06829	0.0505852
$447$	−19.0162	−0.899435
$448$	0	0
$449$	8.41292	0.397030	0.198515	−	0.980098i	$-0.436388\pi$
$449$	8.41292	0.397030	0.198515	+	0.980098i	$0.436388\pi$
$450$	14.4292	0.680199
$451$	9.43002	0.444043
$452$	76.0911	3.57902
$453$	−17.1204	−0.804388
$454$	−27.8805	−1.30850
$455$	0	0
$456$	63.3179	2.96513
$457$	−11.9418	−0.558615	−0.279307	−	0.960202i	$-0.590105\pi$
$457$	−11.9418	−0.558615	−0.279307	+	0.960202i	$0.590105\pi$
$458$	38.4054	1.79456
$459$	−5.16986	−0.241308
$460$	2.56319	0.119509
$461$	−37.0289	−1.72461	−0.862304	−	0.506391i	$-0.830979\pi$
$461$	−37.0289	−1.72461	−0.862304	+	0.506391i	$0.830979\pi$
$462$	0	0
$463$	28.8634	1.34140	0.670698	−	0.741731i	$-0.265995\pi$
$463$	28.8634	1.34140	0.670698	+	0.741731i	$0.265995\pi$
$464$	−12.8304	−0.595635
$465$	−6.33757	−0.293898
$466$	−37.2323	−1.72475
$467$	20.8245	0.963641	0.481820	−	0.876270i	$-0.339976\pi$
$467$	20.8245	0.963641	0.481820	+	0.876270i	$0.339976\pi$
$468$	15.0770	0.696933
$469$	0	0
$470$	−14.7708	−0.681326
$471$	17.6554	0.813517
$472$	−40.0003	−1.84116
$473$	19.9406	0.916868
$474$	−3.46973	−0.159370
$475$	32.1083	1.47323
$476$	0	0
$477$	−12.0705	−0.552668
$478$	−48.7368	−2.22917
$479$	−11.0241	−0.503702	−0.251851	−	0.967766i	$-0.581039\pi$
$479$	−11.0241	−0.503702	−0.251851	+	0.967766i	$0.581039\pi$
$480$	−6.01657	−0.274618
$481$	25.1881	1.14848
$482$	−64.2580	−2.92687
$483$	0	0
$484$	31.9278	1.45126
$485$	3.82888	0.173860
$486$	−29.7779	−1.35075
$487$	10.3815	0.470431	0.235215	−	0.971943i	$-0.424420\pi$
$487$	10.3815	0.470431	0.235215	+	0.971943i	$0.424420\pi$
$488$	−8.38510	−0.379576
$489$	15.2115	0.687889
$490$	0	0
$491$	36.7965	1.66060	0.830301	−	0.557316i	$-0.188169\pi$
$491$	36.7965	1.66060	0.830301	+	0.557316i	$0.188169\pi$
$492$	13.9908	0.630754
$493$	1.39827	0.0629747
$494$	47.9400	2.15692
$495$	2.82790	0.127104
$496$	70.7274	3.17575
$497$	0	0
$498$	−5.38553	−0.241332
$499$	39.9364	1.78780	0.893899	−	0.448269i	$-0.147959\pi$
$499$	39.9364	1.78780	0.893899	+	0.448269i	$0.147959\pi$
$500$	−25.3370	−1.13311
$501$	−4.55088	−0.203318
$502$	60.5149	2.70091
$503$	−34.2923	−1.52902	−0.764508	−	0.644614i	$-0.777018\pi$
$503$	−34.2923	−1.52902	−0.764508	+	0.644614i	$0.777018\pi$
$504$	0	0
$505$	4.95249	0.220383
$506$	−10.6844	−0.474980
$507$	7.60212	0.337622
$508$	−63.5171	−2.81812
$509$	5.88004	0.260628	0.130314	−	0.991473i	$-0.458401\pi$
$509$	5.88004	0.260628	0.130314	+	0.991473i	$0.458401\pi$
$510$	1.78553	0.0790646
$511$	0	0
$512$	−48.5542	−2.14581
$513$	−38.6268	−1.70541
$514$	−38.7988	−1.71134
$515$	−4.45833	−0.196457
$516$	29.5847	1.30239
$517$	43.0888	1.89504
$518$	0	0
$519$	−6.49350	−0.285033
$520$	−10.4509	−0.458300
$521$	29.6440	1.29873	0.649363	−	0.760479i	$-0.275036\pi$
$521$	29.6440	1.29873	0.649363	+	0.760479i	$0.275036\pi$
$522$	4.69488	0.205489
$523$	41.9543	1.83454	0.917268	−	0.398271i	$-0.130390\pi$
$523$	41.9543	1.83454	0.917268	+	0.398271i	$0.130390\pi$
$524$	−53.4414	−2.33460
$525$	0	0
$526$	−25.7290	−1.12184
$527$	−7.70794	−0.335763
$528$	47.7944	2.07999
$529$	−22.0400	−0.958261
$530$	14.6510	0.636399
$531$	6.94338	0.301317
$532$	0	0
$533$	6.04935	0.262026
$534$	42.8744	1.85536
$535$	3.14221	0.135850
$536$	87.0044	3.75802
$537$	6.38711	0.275624
$538$	41.2252	1.77734
$539$	0	0
$540$	14.7451	0.634529
$541$	−21.0921	−0.906822	−0.453411	−	0.891302i	$-0.649793\pi$
$541$	−21.0921	−0.906822	−0.453411	+	0.891302i	$0.649793\pi$
$542$	6.02796	0.258923
$543$	13.8516	0.594430
$544$	−7.31753	−0.313736
$545$	3.28895	0.140883
$546$	0	0
$547$	29.7466	1.27187	0.635936	−	0.771742i	$-0.280614\pi$
$547$	29.7466	1.27187	0.635936	+	0.771742i	$0.280614\pi$
$548$	−104.950	−4.48326
$549$	1.45551	0.0621197
$550$	51.0913	2.17854
$551$	10.4472	0.445066
$552$	−9.05264	−0.385306
$553$	0	0
$554$	−47.7495	−2.02868
$555$	7.00929	0.297528
$556$	54.9824	2.33177
$557$	−0.784565	−0.0332431	−0.0166215	−	0.999862i	$-0.505291\pi$
$557$	−0.784565	−0.0332431	−0.0166215	+	0.999862i	$0.505291\pi$
$558$	−25.8805	−1.09561
$559$	12.7918	0.541037
$560$	0	0
$561$	−5.20868	−0.219911
$562$	−72.6387	−3.06408
$563$	36.5778	1.54157	0.770786	−	0.637095i	$-0.219864\pi$
$563$	36.5778	1.54157	0.770786	+	0.637095i	$0.219864\pi$
$564$	63.9285	2.69187
$565$	9.15543	0.385172
$566$	74.0368	3.11200
$567$	0	0
$568$	−1.17756	−0.0494093
$569$	13.7388	0.575959	0.287979	−	0.957637i	$-0.407017\pi$
$569$	13.7388	0.575959	0.287979	+	0.957637i	$0.407017\pi$
$570$	13.3407	0.558779
$571$	−41.9699	−1.75639	−0.878194	−	0.478305i	$-0.841251\pi$
$571$	−41.9699	−1.75639	−0.878194	+	0.478305i	$0.841251\pi$
$572$	53.3849	2.23213
$573$	−23.4591	−0.980020
$574$	0	0
$575$	−4.59057	−0.191440
$576$	−4.48626	−0.186927
$577$	−14.7980	−0.616047	−0.308024	−	0.951379i	$-0.599668\pi$
$577$	−14.7980	−0.616047	−0.308024	+	0.951379i	$0.599668\pi$
$578$	−41.7096	−1.73489
$579$	20.8434	0.866223
$580$	−3.98804	−0.165594
$581$	0	0
$582$	−23.6794	−0.981543
$583$	−42.7394	−1.77008
$584$	55.8488	2.31104
$585$	1.81409	0.0750034
$586$	21.1710	0.874564
$587$	−9.60442	−0.396417	−0.198208	−	0.980160i	$-0.563512\pi$
$587$	−9.60442	−0.396417	−0.198208	+	0.980160i	$0.563512\pi$
$588$	0	0
$589$	−57.5902	−2.37296
$590$	−8.42780	−0.346967
$591$	−19.6411	−0.807926
$592$	−78.2237	−3.21498
$593$	20.1858	0.828931	0.414465	−	0.910065i	$-0.363969\pi$
$593$	20.1858	0.828931	0.414465	+	0.910065i	$0.363969\pi$
$594$	−61.4636	−2.52188
$595$	0	0
$596$	65.9642	2.70200
$597$	6.92425	0.283391
$598$	−6.85404	−0.280282
$599$	−5.62946	−0.230014	−0.115007	−	0.993365i	$-0.536689\pi$
$599$	−5.62946	−0.230014	−0.115007	+	0.993365i	$0.536689\pi$
$600$	43.2883	1.76724
$601$	−32.3588	−1.31994	−0.659971	−	0.751291i	$-0.729431\pi$
$601$	−32.3588	−1.31994	−0.659971	+	0.751291i	$0.729431\pi$
$602$	0	0
$603$	−15.1025	−0.615021
$604$	59.3881	2.41647
$605$	3.84162	0.156184
$606$	−30.6283	−1.24419
$607$	−20.5879	−0.835638	−0.417819	−	0.908530i	$-0.637205\pi$
$607$	−20.5879	−0.835638	−0.417819	+	0.908530i	$0.637205\pi$
$608$	−54.6732	−2.21729
$609$	0	0
$610$	−1.76668	−0.0715309
$611$	27.6414	1.11825
$612$	5.10280	0.206268
$613$	−16.5257	−0.667467	−0.333734	−	0.942667i	$-0.608309\pi$
$613$	−16.5257	−0.667467	−0.333734	+	0.942667i	$0.608309\pi$
$614$	−32.5669	−1.31429
$615$	1.68340	0.0678813
$616$	0	0
$617$	−32.6876	−1.31595	−0.657976	−	0.753039i	$-0.728587\pi$
$617$	−32.6876	−1.31595	−0.657976	+	0.753039i	$0.728587\pi$
$618$	27.5722	1.10912
$619$	29.2252	1.17466	0.587331	−	0.809347i	$-0.300179\pi$
$619$	29.2252	1.17466	0.587331	+	0.809347i	$0.300179\pi$
$620$	21.9841	0.882901
$621$	5.52252	0.221611
$622$	−36.3665	−1.45816
$623$	0	0
$624$	30.6601	1.22738
$625$	20.3776	0.815102
$626$	−75.2767	−3.00866
$627$	−38.9169	−1.55419
$628$	−61.2438	−2.44389
$629$	8.52490	0.339910
$630$	0	0
$631$	−30.7169	−1.22282	−0.611409	−	0.791314i	$-0.709397\pi$
$631$	−30.7169	−1.22282	−0.611409	+	0.791314i	$0.709397\pi$
$632$	6.87345	0.273411
$633$	−9.43830	−0.375139
$634$	−14.9314	−0.593003
$635$	−7.64251	−0.303284
$636$	−63.4100	−2.51437
$637$	0	0
$638$	16.6238	0.658141
$639$	0.204404	0.00808610
$640$	−3.50649	−0.138606
$641$	−24.0275	−0.949028	−0.474514	−	0.880248i	$-0.657376\pi$
$641$	−24.0275	−0.949028	−0.474514	+	0.880248i	$0.657376\pi$
$642$	−19.4328	−0.766950
$643$	43.9357	1.73266	0.866328	−	0.499476i	$-0.166474\pi$
$643$	43.9357	1.73266	0.866328	+	0.499476i	$0.166474\pi$
$644$	0	0
$645$	3.55969	0.140163
$646$	16.2253	0.638376
$647$	−15.5049	−0.609561	−0.304780	−	0.952423i	$-0.598583\pi$
$647$	−15.5049	−0.609561	−0.304780	+	0.952423i	$0.598583\pi$
$648$	−27.4741	−1.07929
$649$	24.5853	0.965057
$650$	32.7750	1.28554
$651$	0	0
$652$	−52.7664	−2.06649
$653$	6.34795	0.248415	0.124207	−	0.992256i	$-0.460361\pi$
$653$	6.34795	0.248415	0.124207	+	0.992256i	$0.460361\pi$
$654$	−20.3403	−0.795367
$655$	−6.43017	−0.251248
$656$	−18.7868	−0.733501
$657$	−9.69440	−0.378214
$658$	0	0
$659$	−37.9331	−1.47766	−0.738832	−	0.673890i	$-0.764622\pi$
$659$	−37.9331	−1.47766	−0.738832	+	0.673890i	$0.764622\pi$
$660$	14.8559	0.578263
$661$	−26.4586	−1.02912	−0.514561	−	0.857454i	$-0.672045\pi$
$661$	−26.4586	−1.02912	−0.514561	+	0.857454i	$0.672045\pi$
$662$	39.3552	1.52958
$663$	−3.34136	−0.129768
$664$	10.6686	0.414023
$665$	0	0
$666$	28.6236	1.10914
$667$	−1.49365	−0.0578343
$668$	15.7863	0.610790
$669$	−0.556323	−0.0215087
$670$	18.3312	0.708198
$671$	5.15371	0.198957
$672$	0	0
$673$	−7.42610	−0.286255	−0.143127	−	0.989704i	$-0.545716\pi$
$673$	−7.42610	−0.286255	−0.143127	+	0.989704i	$0.545716\pi$
$674$	−26.1740	−1.00818
$675$	−26.4079	−1.01644
$676$	−26.3706	−1.01425
$677$	−43.3125	−1.66463	−0.832317	−	0.554300i	$-0.812986\pi$
$677$	−43.3125	−1.66463	−0.832317	+	0.554300i	$0.812986\pi$
$678$	−56.6211	−2.17452
$679$	0	0
$680$	−3.53709	−0.135641
$681$	14.5190	0.556369
$682$	−91.6385	−3.50902
$683$	−13.4607	−0.515059	−0.257530	−	0.966270i	$-0.582909\pi$
$683$	−13.4607	−0.515059	−0.257530	+	0.966270i	$0.582909\pi$
$684$	38.1258	1.45778
$685$	−12.6278	−0.482485
$686$	0	0
$687$	−19.9999	−0.763043
$688$	−39.7262	−1.51455
$689$	−27.4172	−1.04451
$690$	−1.90733	−0.0726108
$691$	−32.7765	−1.24688	−0.623438	−	0.781873i	$-0.714265\pi$
$691$	−32.7765	−1.24688	−0.623438	+	0.781873i	$0.714265\pi$
$692$	22.5249	0.856270
$693$	0	0
$694$	18.9698	0.720083
$695$	6.61559	0.250944
$696$	14.0849	0.533886
$697$	2.04740	0.0775509
$698$	72.9476	2.76111
$699$	19.3890	0.733359
$700$	0	0
$701$	37.6433	1.42177	0.710884	−	0.703309i	$-0.248295\pi$
$701$	37.6433	1.42177	0.710884	+	0.703309i	$0.248295\pi$
$702$	−39.4288	−1.48814
$703$	63.6941	2.40227
$704$	−15.8851	−0.598691
$705$	7.69201	0.289698
$706$	86.3073	3.24822
$707$	0	0
$708$	36.4758	1.37084
$709$	26.4030	0.991585	0.495793	−	0.868441i	$-0.334878\pi$
$709$	26.4030	0.991585	0.495793	+	0.868441i	$0.334878\pi$
$710$	−0.248104	−0.00931116
$711$	−1.19311	−0.0447452
$712$	−84.9332	−3.18301
$713$	8.23374	0.308356
$714$	0	0
$715$	6.42338	0.240221
$716$	−22.1559	−0.828005
$717$	25.3800	0.947835
$718$	82.8012	3.09011
$719$	−15.7490	−0.587340	−0.293670	−	0.955907i	$-0.594877\pi$
$719$	−15.7490	−0.587340	−0.293670	+	0.955907i	$0.594877\pi$
$720$	−5.63382	−0.209960
$721$	0	0
$722$	72.1843	2.68642
$723$	33.4628	1.24450
$724$	−48.0491	−1.78573
$725$	7.14241	0.265262
$726$	−23.7582	−0.881750
$727$	28.3361	1.05093	0.525463	−	0.850816i	$-0.323892\pi$
$727$	28.3361	1.05093	0.525463	+	0.850816i	$0.323892\pi$
$728$	0	0
$729$	27.4985	1.01846
$730$	11.7670	0.435515
$731$	4.32940	0.160129
$732$	7.64627	0.282614
$733$	−23.7282	−0.876421	−0.438210	−	0.898872i	$-0.644388\pi$
$733$	−23.7282	−0.876421	−0.438210	+	0.898872i	$0.644388\pi$
$734$	−16.5831	−0.612094
$735$	0	0
$736$	7.81669	0.288127
$737$	−53.4752	−1.96979
$738$	6.87445	0.253052
$739$	7.34954	0.270357	0.135179	−	0.990821i	$-0.456839\pi$
$739$	7.34954	0.270357	0.135179	+	0.990821i	$0.456839\pi$
$740$	−24.3141	−0.893805
$741$	−24.9651	−0.917117
$742$	0	0
$743$	43.1316	1.58234	0.791172	−	0.611594i	$-0.209471\pi$
$743$	43.1316	1.58234	0.791172	+	0.611594i	$0.209471\pi$
$744$	−77.6429	−2.84653
$745$	7.93695	0.290787
$746$	−12.3847	−0.453436
$747$	−1.85189	−0.0677571
$748$	18.0681	0.660636
$749$	0	0
$750$	18.8539	0.688446
$751$	17.1766	0.626785	0.313392	−	0.949624i	$-0.398535\pi$
$751$	17.1766	0.626785	0.313392	+	0.949624i	$0.398535\pi$
$752$	−85.8429	−3.13037
$753$	−31.5136	−1.14842
$754$	10.6641	0.388364
$755$	7.14570	0.260059
$756$	0	0
$757$	21.6086	0.785379	0.392689	−	0.919671i	$-0.371545\pi$
$757$	21.6086	0.785379	0.392689	+	0.919671i	$0.371545\pi$
$758$	52.6848	1.91360
$759$	5.56399	0.201960
$760$	−26.4275	−0.958627
$761$	−31.6890	−1.14873	−0.574363	−	0.818601i	$-0.694750\pi$
$761$	−31.6890	−1.14873	−0.574363	+	0.818601i	$0.694750\pi$
$762$	47.2646	1.71221
$763$	0	0
$764$	81.3762	2.94409
$765$	0.613979	0.0221985
$766$	−44.3401	−1.60207
$767$	15.7714	0.569473
$768$	31.7944	1.14728
$769$	4.99916	0.180274	0.0901371	−	0.995929i	$-0.471269\pi$
$769$	4.99916	0.180274	0.0901371	+	0.995929i	$0.471269\pi$
$770$	0	0
$771$	20.2048	0.727658
$772$	−72.3026	−2.60223
$773$	−1.86671	−0.0671410	−0.0335705	−	0.999436i	$-0.510688\pi$
$773$	−1.86671	−0.0671410	−0.0335705	+	0.999436i	$0.510688\pi$
$774$	14.5366	0.522507
$775$	−39.3725	−1.41430
$776$	46.9084	1.68391
$777$	0	0
$778$	−9.87233	−0.353940
$779$	15.2972	0.548081
$780$	9.53001	0.341229
$781$	0.723759	0.0258981
$782$	−2.31975	−0.0829541
$783$	−8.59242	−0.307068
$784$	0	0
$785$	−7.36897	−0.263010
$786$	39.7669	1.41844
$787$	−19.1046	−0.681005	−0.340502	−	0.940244i	$-0.610597\pi$
$787$	−19.1046	−0.681005	−0.340502	+	0.940244i	$0.610597\pi$
$788$	68.1319	2.42710
$789$	13.3986	0.477001
$790$	1.44819	0.0515242
$791$	0	0
$792$	34.6451	1.23106
$793$	3.30610	0.117403
$794$	−8.08670	−0.286986
$795$	−7.62962	−0.270595
$796$	−24.0192	−0.851336
$797$	−9.33694	−0.330731	−0.165366	−	0.986232i	$-0.552880\pi$
$797$	−9.33694	−0.330731	−0.165366	+	0.986232i	$0.552880\pi$
$798$	0	0
$799$	9.35524	0.330965
$800$	−37.3783	−1.32152
$801$	14.7430	0.520917
$802$	−39.4215	−1.39202
$803$	−34.3262	−1.21134
$804$	−79.3382	−2.79804
$805$	0	0
$806$	−58.7859	−2.07065
$807$	−21.4683	−0.755721
$808$	60.6740	2.13450
$809$	−45.7286	−1.60773	−0.803865	−	0.594812i	$-0.797227\pi$
$809$	−45.7286	−1.60773	−0.803865	+	0.594812i	$0.797227\pi$
$810$	−5.78861	−0.203391
$811$	−12.0671	−0.423732	−0.211866	−	0.977299i	$-0.567954\pi$
$811$	−12.0671	−0.423732	−0.211866	+	0.977299i	$0.567954\pi$
$812$	0	0
$813$	−3.13911	−0.110093
$814$	101.351	3.55236
$815$	−6.34897	−0.222395
$816$	10.3769	0.363264
$817$	32.3473	1.13169
$818$	50.7344	1.77389
$819$	0	0
$820$	−5.83946	−0.203923
$821$	−46.2328	−1.61354	−0.806768	−	0.590869i	$-0.798785\pi$
$821$	−46.2328	−1.61354	−0.806768	+	0.590869i	$0.798785\pi$
$822$	78.0960	2.72391
$823$	3.51542	0.122540	0.0612700	−	0.998121i	$-0.480485\pi$
$823$	3.51542	0.122540	0.0612700	+	0.998121i	$0.480485\pi$
$824$	−54.6199	−1.90278
$825$	−26.6062	−0.926309
$826$	0	0
$827$	52.1132	1.81216	0.906078	−	0.423112i	$-0.139062\pi$
$827$	52.1132	1.81216	0.906078	+	0.423112i	$0.139062\pi$
$828$	−5.45089	−0.189431
$829$	−14.2537	−0.495051	−0.247526	−	0.968881i	$-0.579617\pi$
$829$	−14.2537	−0.495051	−0.247526	+	0.968881i	$0.579617\pi$
$830$	2.24781	0.0780225
$831$	24.8659	0.862589
$832$	−10.1902	−0.353283
$833$	0	0
$834$	−40.9137	−1.41672
$835$	1.89944	0.0657328
$836$	134.997	4.66896
$837$	47.3657	1.63720
$838$	−18.3570	−0.634132
$839$	−44.8944	−1.54993	−0.774964	−	0.632006i	$-0.782232\pi$
$839$	−44.8944	−1.54993	−0.774964	+	0.632006i	$0.782232\pi$
$840$	0	0
$841$	−26.6760	−0.919864
$842$	−85.4372	−2.94436
$843$	37.8272	1.30284
$844$	32.7400	1.12696
$845$	−3.17296	−0.109153
$846$	31.4116	1.07995
$847$	0	0
$848$	85.1467	2.92395
$849$	−38.5553	−1.32321
$850$	11.0927	0.380476
$851$	−9.10643	−0.312164
$852$	1.07380	0.0367878
$853$	−11.8835	−0.406885	−0.203442	−	0.979087i	$-0.565213\pi$
$853$	−11.8835	−0.406885	−0.203442	+	0.979087i	$0.565213\pi$
$854$	0	0
$855$	4.58737	0.156885
$856$	38.4959	1.31576
$857$	40.9638	1.39930	0.699649	−	0.714487i	$-0.253340\pi$
$857$	40.9638	1.39930	0.699649	+	0.714487i	$0.253340\pi$
$858$	−39.7249	−1.35619
$859$	48.4730	1.65388	0.826938	−	0.562292i	$-0.190080\pi$
$859$	48.4730	1.65388	0.826938	+	0.562292i	$0.190080\pi$
$860$	−12.3480	−0.421064
$861$	0	0
$862$	16.7781	0.571463
$863$	17.0772	0.581314	0.290657	−	0.956827i	$-0.406126\pi$
$863$	17.0772	0.581314	0.290657	+	0.956827i	$0.406126\pi$
$864$	44.9666	1.52979
$865$	2.71025	0.0921512
$866$	72.3304	2.45789
$867$	21.7206	0.737670
$868$	0	0
$869$	−4.22461	−0.143310
$870$	2.96759	0.100611
$871$	−34.3043	−1.16236
$872$	40.2936	1.36451
$873$	−8.14250	−0.275582
$874$	−17.3321	−0.586267
$875$	0	0
$876$	−50.9278	−1.72069
$877$	−38.9785	−1.31621	−0.658106	−	0.752926i	$-0.728642\pi$
$877$	−38.9785	−1.31621	−0.658106	+	0.752926i	$0.728642\pi$
$878$	12.0021	0.405051
$879$	−11.0249	−0.371862
$880$	−19.9484	−0.672459
$881$	−8.10231	−0.272974	−0.136487	−	0.990642i	$-0.543581\pi$
$881$	−8.10231	−0.272974	−0.136487	+	0.990642i	$0.543581\pi$
$882$	0	0
$883$	−9.89363	−0.332947	−0.166474	−	0.986046i	$-0.553238\pi$
$883$	−9.89363	−0.332947	−0.166474	+	0.986046i	$0.553238\pi$
$884$	11.5907	0.389836
$885$	4.38884	0.147529
$886$	−12.4474	−0.418177
$887$	−43.3446	−1.45537	−0.727684	−	0.685913i	$-0.759403\pi$
$887$	−43.3446	−1.45537	−0.727684	+	0.685913i	$0.759403\pi$
$888$	85.8722	2.88168
$889$	0	0
$890$	−17.8949	−0.599837
$891$	16.8863	0.565714
$892$	1.92980	0.0646144
$893$	69.8981	2.33905
$894$	−49.0855	−1.64166
$895$	−2.66584	−0.0891094
$896$	0	0
$897$	3.56929	0.119175
$898$	21.7158	0.724667
$899$	−12.8108	−0.427263
$900$	26.0653	0.868844
$901$	−9.27937	−0.309141
$902$	24.3412	0.810474
$903$	0	0
$904$	112.165	3.73056
$905$	−5.78137	−0.192179
$906$	−44.1921	−1.46818
$907$	3.09162	0.102655	0.0513277	−	0.998682i	$-0.483655\pi$
$907$	3.09162	0.102655	0.0513277	+	0.998682i	$0.483655\pi$
$908$	−50.3642	−1.67139
$909$	−10.5320	−0.349323
$910$	0	0
$911$	−51.3384	−1.70092	−0.850458	−	0.526043i	$-0.823675\pi$
$911$	−51.3384	−1.70092	−0.850458	+	0.526043i	$0.823675\pi$
$912$	77.5314	2.56732
$913$	−6.55722	−0.217012
$914$	−30.8248	−1.01959
$915$	0.920014	0.0304147
$916$	69.3765	2.29226
$917$	0	0
$918$	−13.3447	−0.440440
$919$	2.36061	0.0778692	0.0389346	−	0.999242i	$-0.487604\pi$
$919$	2.36061	0.0778692	0.0389346	+	0.999242i	$0.487604\pi$
$920$	3.77838	0.124569
$921$	16.9594	0.558833
$922$	−95.5807	−3.14778
$923$	0.464290	0.0152823
$924$	0	0
$925$	43.5456	1.43177
$926$	74.5036	2.44834
$927$	9.48109	0.311400
$928$	−12.1619	−0.399234
$929$	−32.1347	−1.05431	−0.527153	−	0.849770i	$-0.676740\pi$
$929$	−32.1347	−1.05431	−0.527153	+	0.849770i	$0.676740\pi$
$930$	−16.3588	−0.536428
$931$	0	0
$932$	−67.2575	−2.20309
$933$	18.9381	0.620007
$934$	53.7531	1.75885
$935$	2.17399	0.0710972
$936$	22.2248	0.726441
$937$	7.75488	0.253341	0.126670	−	0.991945i	$-0.459571\pi$
$937$	7.75488	0.253341	0.126670	+	0.991945i	$0.459571\pi$
$938$	0	0
$939$	39.2009	1.27927
$940$	−26.6824	−0.870283
$941$	48.1023	1.56809	0.784045	−	0.620703i	$-0.213153\pi$
$941$	48.1023	1.56809	0.784045	+	0.620703i	$0.213153\pi$
$942$	45.5729	1.48484
$943$	−2.18707	−0.0712206
$944$	−48.9795	−1.59415
$945$	0	0
$946$	51.4715	1.67348
$947$	7.10963	0.231032	0.115516	−	0.993306i	$-0.463148\pi$
$947$	7.10963	0.231032	0.115516	+	0.993306i	$0.463148\pi$
$948$	−6.26781	−0.203569
$949$	−22.0202	−0.714805
$950$	82.8795	2.68897
$951$	7.77566	0.252143
$952$	0	0
$953$	−33.3566	−1.08053	−0.540264	−	0.841496i	$-0.681675\pi$
$953$	−33.3566	−1.08053	−0.540264	+	0.841496i	$0.681675\pi$
$954$	−31.1568	−1.00874
$955$	9.79134	0.316840
$956$	−88.0395	−2.84740
$957$	−8.65695	−0.279840
$958$	−28.4558	−0.919365
$959$	0	0
$960$	−2.83572	−0.0915225
$961$	39.6194	1.27805
$962$	65.0166	2.09622
$963$	−6.68223	−0.215332
$964$	−116.077	−3.73860
$965$	−8.69960	−0.280050
$966$	0	0
$967$	40.4562	1.30098	0.650492	−	0.759513i	$-0.274563\pi$
$967$	40.4562	1.30098	0.650492	+	0.759513i	$0.274563\pi$
$968$	47.0645	1.51271
$969$	−8.44945	−0.271435
$970$	9.88328	0.317333
$971$	−22.7548	−0.730236	−0.365118	−	0.930961i	$-0.618971\pi$
$971$	−22.7548	−0.730236	−0.365118	+	0.930961i	$0.618971\pi$
$972$	−53.7916	−1.72537
$973$	0	0
$974$	26.7972	0.858639
$975$	−17.0678	−0.546608
$976$	−10.2674	−0.328651
$977$	6.38128	0.204155	0.102078	−	0.994776i	$-0.467451\pi$
$977$	6.38128	0.204155	0.102078	+	0.994776i	$0.467451\pi$
$978$	39.2647	1.25555
$979$	52.2022	1.66839
$980$	0	0
$981$	−6.99428	−0.223310
$982$	94.9808	3.03096
$983$	−53.6034	−1.70968	−0.854841	−	0.518889i	$-0.826346\pi$
$983$	−53.6034	−1.70968	−0.854841	+	0.518889i	$0.826346\pi$
$984$	20.6237	0.657460
$985$	8.19777	0.261203
$986$	3.60927	0.114943
$987$	0	0
$988$	86.6002	2.75512
$989$	−4.62473	−0.147058
$990$	7.29950	0.231993
$991$	4.52460	0.143729	0.0718644	−	0.997414i	$-0.477105\pi$
$991$	4.52460	0.143729	0.0718644	+	0.997414i	$0.477105\pi$
$992$	67.0425	2.12860
$993$	−20.4945	−0.650374
$994$	0	0
$995$	−2.89003	−0.0916202
$996$	−9.72858	−0.308262
$997$	−42.8267	−1.35634	−0.678168	−	0.734907i	$-0.737226\pi$
$997$	−42.8267	−1.35634	−0.678168	+	0.734907i	$0.737226\pi$
$998$	103.086	3.26312
$999$	−52.3860	−1.65742

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3871.2.a.e.1.8		8
7.6	odd	2		553.2.a.b.1.8	✓	8
21.20	even	2		4977.2.a.j.1.1		8
28.27	even	2		8848.2.a.s.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
553.2.a.b.1.8	✓	8	7.6	odd	2
3871.2.a.e.1.8		8	1.1	even	1	trivial
4977.2.a.j.1.1		8	21.20	even	2
8848.2.a.s.1.3		8	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3871 = 7^{2} \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3871.a (trivial)

\(f(q)\)	\(=\)	\(q+2.58125 q^{2} -1.34420 q^{3} +4.66284 q^{4} +0.561042 q^{5} -3.46973 q^{6} +6.87345 q^{8} -1.19311 q^{9} +1.44819 q^{10} -4.22461 q^{11} -6.26781 q^{12} -2.71008 q^{13} -0.754156 q^{15} +8.41639 q^{16} -0.917226 q^{17} -3.07972 q^{18} -6.85309 q^{19} +2.61605 q^{20} -10.9048 q^{22} +0.979795 q^{23} -9.23932 q^{24} -4.68523 q^{25} -6.99538 q^{26} +5.63640 q^{27} -1.52445 q^{29} -1.94666 q^{30} +8.40354 q^{31} +7.97789 q^{32} +5.67874 q^{33} -2.36759 q^{34} -5.56330 q^{36} -9.29422 q^{37} -17.6895 q^{38} +3.64290 q^{39} +3.85629 q^{40} -2.23217 q^{41} -4.72010 q^{43} -19.6987 q^{44} -0.669387 q^{45} +2.52909 q^{46} -10.1995 q^{47} -11.3133 q^{48} -12.0937 q^{50} +1.23294 q^{51} -12.6367 q^{52} +10.1168 q^{53} +14.5490 q^{54} -2.37018 q^{55} +9.21196 q^{57} -3.93499 q^{58} -5.81954 q^{59} -3.51651 q^{60} -1.21993 q^{61} +21.6916 q^{62} +3.76013 q^{64} -1.52047 q^{65} +14.6582 q^{66} +12.6580 q^{67} -4.27688 q^{68} -1.31704 q^{69} -0.171320 q^{71} -8.20080 q^{72} +8.12529 q^{73} -23.9907 q^{74} +6.29791 q^{75} -31.9549 q^{76} +9.40323 q^{78} +1.00000 q^{79} +4.72195 q^{80} -3.99714 q^{81} -5.76178 q^{82} +1.55215 q^{83} -0.514603 q^{85} -12.1837 q^{86} +2.04917 q^{87} -29.0376 q^{88} -12.3567 q^{89} -1.72785 q^{90} +4.56862 q^{92} -11.2961 q^{93} -26.3274 q^{94} -3.84487 q^{95} -10.7239 q^{96} +6.82458 q^{97} +5.04043 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} - 3 q^{3} + 7 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{8} - q^{9} - 10 q^{10} + 3 q^{11} - 10 q^{12} - 3 q^{13} + 11 q^{15} + 13 q^{16} - 14 q^{17} + 4 q^{18} - 12 q^{19} - 12 q^{20} - 15 q^{22}+ \cdots - 25 q^{99}+O(q^{100}) \) 8 * q + 3 * q^2 - 3 * q^3 + 7 * q^4 - 4 * q^5 - 6 * q^6 + 6 * q^8 - q^9 - 10 * q^10 + 3 * q^11 - 10 * q^12 - 3 * q^13 + 11 * q^15 + 13 * q^16 - 14 * q^17 + 4 * q^18 - 12 * q^19 - 12 * q^20 - 15 * q^22 + 17 * q^23 - 14 * q^24 - 6 * q^25 + 13 * q^26 + 6 * q^27 - 6 * q^29 + 17 * q^30 - 15 * q^31 + 5 * q^32 - 2 * q^33 + 2 * q^34 - 33 * q^36 - 2 * q^37 - 19 * q^38 + 6 * q^39 + 13 * q^40 - 8 * q^41 + 9 * q^43 + 7 * q^44 - 5 * q^45 - 24 * q^46 - 34 * q^47 + 9 * q^48 - 3 * q^50 - 11 * q^51 - 17 * q^52 + 2 * q^53 + 10 * q^54 - 16 * q^55 + 6 * q^57 - 14 * q^58 - 21 * q^59 - 14 * q^60 - 11 * q^61 + 15 * q^62 - 16 * q^64 - 5 * q^65 + 18 * q^66 + 14 * q^67 - 7 * q^68 + 8 * q^69 - 2 * q^71 - 18 * q^72 - 7 * q^73 - 8 * q^74 - 28 * q^75 + 23 * q^76 - 8 * q^78 + 8 * q^79 - 6 * q^80 - 28 * q^81 + 10 * q^82 - 32 * q^83 - 9 * q^85 - 6 * q^86 - 43 * q^87 - 48 * q^88 - 6 * q^89 - q^90 - 3 * q^92 - 12 * q^93 - 33 * q^94 - 15 * q^95 + 9 * q^96 + 39 * q^97 - 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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