LMFDB - Embedded newform 3871.2.a.i.1.10

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 = [24,4,0,28,0] 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:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	\(q-0.380222 q^{2} +1.62042 q^{3} -1.85543 q^{4} -3.79610 q^{5} -0.616119 q^{6} +1.46592 q^{8} -0.374245 q^{9} +1.44336 q^{10} -0.391089 q^{11} -3.00657 q^{12} -7.07220 q^{13} -6.15126 q^{15} +3.15349 q^{16} -4.17204 q^{17} +0.142296 q^{18} -6.56066 q^{19} +7.04340 q^{20} +0.148701 q^{22} +1.55903 q^{23} +2.37540 q^{24} +9.41035 q^{25} +2.68901 q^{26} -5.46769 q^{27} -2.62613 q^{29} +2.33885 q^{30} +5.73522 q^{31} -4.13087 q^{32} -0.633728 q^{33} +1.58630 q^{34} +0.694386 q^{36} +4.14653 q^{37} +2.49451 q^{38} -11.4599 q^{39} -5.56478 q^{40} -1.34115 q^{41} -11.5513 q^{43} +0.725639 q^{44} +1.42067 q^{45} -0.592779 q^{46} -5.55629 q^{47} +5.10997 q^{48} -3.57802 q^{50} -6.76044 q^{51} +13.1220 q^{52} +7.60160 q^{53} +2.07894 q^{54} +1.48461 q^{55} -10.6310 q^{57} +0.998512 q^{58} -4.38862 q^{59} +11.4132 q^{60} -5.80738 q^{61} -2.18066 q^{62} -4.73633 q^{64} +26.8468 q^{65} +0.240957 q^{66} +2.49625 q^{67} +7.74093 q^{68} +2.52629 q^{69} +1.76933 q^{71} -0.548613 q^{72} +14.3344 q^{73} -1.57660 q^{74} +15.2487 q^{75} +12.1729 q^{76} +4.35732 q^{78} -1.00000 q^{79} -11.9709 q^{80} -7.73721 q^{81} +0.509935 q^{82} -2.87184 q^{83} +15.8375 q^{85} +4.39207 q^{86} -4.25542 q^{87} -0.573306 q^{88} +11.4266 q^{89} -0.540170 q^{90} -2.89268 q^{92} +9.29345 q^{93} +2.11262 q^{94} +24.9049 q^{95} -6.69373 q^{96} -4.69311 q^{97} +0.146363 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{2} + 28 q^{4} + 18 q^{8} + 50 q^{9} + 14 q^{11} + 18 q^{15} + 36 q^{16} + 10 q^{18} - 12 q^{22} + 22 q^{23} + 56 q^{25} + 36 q^{29} + 14 q^{30} + 30 q^{32} + 8 q^{36} + 24 q^{37} + 56 q^{39}+ \cdots + 6 q^{99}+O(q^{100}) $ 24 * q + 4 * q^2 + 28 * q^4 + 18 * q^8 + 50 * q^9 + 14 * q^11 + 18 * q^15 + 36 * q^16 + 10 * q^18 - 12 * q^22 + 22 * q^23 + 56 * q^25 + 36 * q^29 + 14 * q^30 + 30 * q^32 + 8 * q^36 + 24 * q^37 + 56 * q^39 - 2 * q^43 + 74 * q^44 + 48 * q^46 - 42 * q^50 + 22 * q^51 + 48 * q^53 - 16 * q^57 + 12 * q^58 + 120 * q^60 - 38 * q^64 + 62 * q^65 + 32 * q^67 + 44 * q^71 + 26 * q^72 + 48 * q^74 - 144 * q^78 - 24 * q^79 + 108 * q^81 - 42 * q^85 + 72 * q^86 + 36 * q^88 - 16 * q^92 + 48 * q^93 + 2 * q^95 + 6 * 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 $

$2$	−0.380222	−0.268858	−0.134429	−	0.990923i	$-0.542920\pi$
$2$	−0.380222	−0.268858	−0.134429	+	0.990923i	$0.542920\pi$
$3$	1.62042	0.935549	0.467774	−	0.883848i	$-0.345056\pi$
$3$	1.62042	0.935549	0.467774	+	0.883848i	$0.345056\pi$
$4$	−1.85543	−0.927716
$5$	−3.79610	−1.69767	−0.848833	−	0.528661i	$-0.822694\pi$
$5$	−3.79610	−1.69767	−0.848833	+	0.528661i	$0.822694\pi$
$6$	−0.616119	−0.251529
$7$	0	0
$7$	0	0
$8$	1.46592	0.518281
$9$	−0.374245	−0.124748
$10$	1.44336	0.456431
$11$	−0.391089	−0.117918	−0.0589589	−	0.998260i	$-0.518778\pi$
$11$	−0.391089	−0.117918	−0.0589589	+	0.998260i	$0.518778\pi$
$12$	−3.00657	−0.867923
$13$	−7.07220	−1.96148	−0.980738	−	0.195328i	$-0.937423\pi$
$13$	−7.07220	−1.96148	−0.980738	+	0.195328i	$0.937423\pi$
$14$	0	0
$15$	−6.15126	−1.58825
$16$	3.15349	0.788372
$17$	−4.17204	−1.01187	−0.505934	−	0.862572i	$-0.668852\pi$
$17$	−4.17204	−1.01187	−0.505934	+	0.862572i	$0.668852\pi$
$18$	0.142296	0.0335396
$19$	−6.56066	−1.50512	−0.752559	−	0.658525i	$-0.771181\pi$
$19$	−6.56066	−1.50512	−0.752559	+	0.658525i	$0.771181\pi$
$20$	7.04340	1.57495
$21$	0	0
$22$	0.148701	0.0317031
$23$	1.55903	0.325081	0.162541	−	0.986702i	$-0.448031\pi$
$23$	1.55903	0.325081	0.162541	+	0.986702i	$0.448031\pi$
$24$	2.37540	0.484877
$25$	9.41035	1.88207
$26$	2.68901	0.527358
$27$	−5.46769	−1.05226
$28$	0	0
$29$	−2.62613	−0.487660	−0.243830	−	0.969818i	$-0.578404\pi$
$29$	−2.62613	−0.487660	−0.243830	+	0.969818i	$0.578404\pi$
$30$	2.33885	0.427013
$31$	5.73522	1.03008	0.515038	−	0.857168i	$-0.327778\pi$
$31$	5.73522	1.03008	0.515038	+	0.857168i	$0.327778\pi$
$32$	−4.13087	−0.730241
$33$	−0.633728	−0.110318
$34$	1.58630	0.272048
$35$	0	0
$36$	0.694386	0.115731
$37$	4.14653	0.681686	0.340843	−	0.940120i	$-0.389288\pi$
$37$	4.14653	0.681686	0.340843	+	0.940120i	$0.389288\pi$
$38$	2.49451	0.404663
$39$	−11.4599	−1.83506
$40$	−5.56478	−0.879868
$41$	−1.34115	−0.209452	−0.104726	−	0.994501i	$-0.533397\pi$
$41$	−1.34115	−0.209452	−0.104726	+	0.994501i	$0.533397\pi$
$42$	0	0
$43$	−11.5513	−1.76156	−0.880780	−	0.473526i	$-0.842981\pi$
$43$	−11.5513	−1.76156	−0.880780	+	0.473526i	$0.842981\pi$
$44$	0.725639	0.109394
$45$	1.42067	0.211781
$46$	−0.592779	−0.0874005
$47$	−5.55629	−0.810468	−0.405234	−	0.914213i	$-0.632810\pi$
$47$	−5.55629	−0.810468	−0.405234	+	0.914213i	$0.632810\pi$
$48$	5.10997	0.737560
$49$	0	0
$50$	−3.57802	−0.506009
$51$	−6.76044	−0.946651
$52$	13.1220	1.81969
$53$	7.60160	1.04416	0.522080	−	0.852896i	$-0.325156\pi$
$53$	7.60160	1.04416	0.522080	+	0.852896i	$0.325156\pi$
$54$	2.07894	0.282907
$55$	1.48461	0.200185
$56$	0	0
$57$	−10.6310	−1.40811
$58$	0.998512	0.131111
$59$	−4.38862	−0.571349	−0.285675	−	0.958327i	$-0.592218\pi$
$59$	−4.38862	−0.571349	−0.285675	+	0.958327i	$0.592218\pi$
$60$	11.4132	1.47344
$61$	−5.80738	−0.743559	−0.371779	−	0.928321i	$-0.621252\pi$
$61$	−5.80738	−0.743559	−0.371779	+	0.928321i	$0.621252\pi$
$62$	−2.18066	−0.276944
$63$	0	0
$64$	−4.73633	−0.592041
$65$	26.8468	3.32993
$66$	0.240957	0.0296598
$67$	2.49625	0.304966	0.152483	−	0.988306i	$-0.451273\pi$
$67$	2.49625	0.304966	0.152483	+	0.988306i	$0.451273\pi$
$68$	7.74093	0.938725
$69$	2.52629	0.304129
$70$	0	0
$71$	1.76933	0.209981	0.104990	−	0.994473i	$-0.466519\pi$
$71$	1.76933	0.209981	0.104990	+	0.994473i	$0.466519\pi$
$72$	−0.548613	−0.0646547
$73$	14.3344	1.67772	0.838858	−	0.544351i	$-0.183224\pi$
$73$	14.3344	1.67772	0.838858	+	0.544351i	$0.183224\pi$
$74$	−1.57660	−0.183276
$75$	15.2487	1.76077
$76$	12.1729	1.39632
$77$	0	0
$78$	4.35732	0.493369
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	−11.9709	−1.33839
$81$	−7.73721	−0.859689
$82$	0.509935	0.0563129
$83$	−2.87184	−0.315226	−0.157613	−	0.987501i	$-0.550380\pi$
$83$	−2.87184	−0.315226	−0.157613	+	0.987501i	$0.550380\pi$
$84$	0	0
$85$	15.8375	1.71781
$86$	4.39207	0.473609
$87$	−4.25542	−0.456229
$88$	−0.573306	−0.0611146
$89$	11.4266	1.21122	0.605609	−	0.795763i	$-0.292930\pi$
$89$	11.4266	1.21122	0.605609	+	0.795763i	$0.292930\pi$
$90$	−0.540170	−0.0569390
$91$	0	0
$92$	−2.89268	−0.301583
$93$	9.29345	0.963686
$94$	2.11262	0.217900
$95$	24.9049	2.55519
$96$	−6.69373	−0.683176
$97$	−4.69311	−0.476514	−0.238257	−	0.971202i	$-0.576576\pi$
$97$	−4.69311	−0.476514	−0.238257	+	0.971202i	$0.576576\pi$
$98$	0	0
$99$	0.146363	0.0147101
$100$	−17.4603	−1.74603
$101$	7.43342	0.739653	0.369827	−	0.929101i	$-0.379417\pi$
$101$	7.43342	0.739653	0.369827	+	0.929101i	$0.379417\pi$
$102$	2.57047	0.254514
$103$	−5.20633	−0.512995	−0.256498	−	0.966545i	$-0.582569\pi$
$103$	−5.20633	−0.512995	−0.256498	+	0.966545i	$0.582569\pi$
$104$	−10.3673	−1.01660
$105$	0	0
$106$	−2.89030	−0.280730
$107$	−17.5404	−1.69570	−0.847848	−	0.530239i	$-0.822102\pi$
$107$	−17.5404	−1.69570	−0.847848	+	0.530239i	$0.822102\pi$
$108$	10.1449	0.976195
$109$	5.46934	0.523868	0.261934	−	0.965086i	$-0.415640\pi$
$109$	5.46934	0.523868	0.261934	+	0.965086i	$0.415640\pi$
$110$	−0.564483	−0.0538213
$111$	6.71912	0.637750
$112$	0	0
$113$	13.8208	1.30015	0.650075	−	0.759870i	$-0.274737\pi$
$113$	13.8208	1.30015	0.650075	+	0.759870i	$0.274737\pi$
$114$	4.04215	0.378582
$115$	−5.91825	−0.551879
$116$	4.87260	0.452409
$117$	2.64674	0.244691
$118$	1.66865	0.153612
$119$	0	0
$120$	−9.01726	−0.823160
$121$	−10.8470	−0.986095
$122$	2.20809	0.199911
$123$	−2.17322	−0.195953
$124$	−10.6413	−0.955617
$125$	−16.7421	−1.49746
$126$	0	0
$127$	−5.30396	−0.470650	−0.235325	−	0.971917i	$-0.575616\pi$
$127$	−5.30396	−0.470650	−0.235325	+	0.971917i	$0.575616\pi$
$128$	10.0626	0.889416
$129$	−18.7180	−1.64802
$130$	−10.2077	−0.895278
$131$	14.6059	1.27612	0.638060	−	0.769987i	$-0.279737\pi$
$131$	14.6059	1.27612	0.638060	+	0.769987i	$0.279737\pi$
$132$	1.17584	0.102344
$133$	0	0
$134$	−0.949131	−0.0819924
$135$	20.7559	1.78638
$136$	−6.11587	−0.524432
$137$	−11.4271	−0.976279	−0.488140	−	0.872765i	$-0.662324\pi$
$137$	−11.4271	−0.976279	−0.488140	+	0.872765i	$0.662324\pi$
$138$	−0.960550	−0.0817675
$139$	15.8047	1.34053	0.670267	−	0.742120i	$-0.266180\pi$
$139$	15.8047	1.34053	0.670267	+	0.742120i	$0.266180\pi$
$140$	0	0
$141$	−9.00351	−0.758232
$142$	−0.672738	−0.0564549
$143$	2.76586	0.231293
$144$	−1.18018	−0.0983481
$145$	9.96903	0.827883
$146$	−5.45026	−0.451067
$147$	0	0
$148$	−7.69361	−0.632411
$149$	−10.4829	−0.858796	−0.429398	−	0.903115i	$-0.641274\pi$
$149$	−10.4829	−0.858796	−0.429398	+	0.903115i	$0.641274\pi$
$150$	−5.79790	−0.473396
$151$	−15.6069	−1.27007	−0.635037	−	0.772482i	$-0.719015\pi$
$151$	−15.6069	−1.27007	−0.635037	+	0.772482i	$0.719015\pi$
$152$	−9.61740	−0.780074
$153$	1.56136	0.126229
$154$	0	0
$155$	−21.7714	−1.74872
$156$	21.2631	1.70241
$157$	−13.8941	−1.10887	−0.554437	−	0.832226i	$-0.687066\pi$
$157$	−13.8941	−1.10887	−0.554437	+	0.832226i	$0.687066\pi$
$158$	0.380222	0.0302488
$159$	12.3178	0.976863
$160$	15.6812	1.23971
$161$	0	0
$162$	2.94186	0.231134
$163$	25.1091	1.96669	0.983347	−	0.181735i	$-0.0581714\pi$
$163$	25.1091	1.96669	0.983347	+	0.181735i	$0.0581714\pi$
$164$	2.48841	0.194312
$165$	2.40569	0.187283
$166$	1.09194	0.0847509
$167$	8.47630	0.655916	0.327958	−	0.944692i	$-0.393640\pi$
$167$	8.47630	0.655916	0.327958	+	0.944692i	$0.393640\pi$
$168$	0	0
$169$	37.0160	2.84739
$170$	−6.02175	−0.461847
$171$	2.45529	0.187761
$172$	21.4327	1.63423
$173$	−26.1289	−1.98655	−0.993273	−	0.115795i	$-0.963059\pi$
$173$	−26.1289	−1.98655	−0.993273	+	0.115795i	$0.963059\pi$
$174$	1.61801	0.122661
$175$	0	0
$176$	−1.23330	−0.0929631
$177$	−7.11139	−0.534525
$178$	−4.34465	−0.325645
$179$	17.2217	1.28721	0.643607	−	0.765356i	$-0.277437\pi$
$179$	17.2217	1.28721	0.643607	+	0.765356i	$0.277437\pi$
$180$	−2.63596	−0.196473
$181$	−4.80860	−0.357421	−0.178710	−	0.983902i	$-0.557192\pi$
$181$	−4.80860	−0.357421	−0.178710	+	0.983902i	$0.557192\pi$
$182$	0	0
$183$	−9.41038	−0.695636
$184$	2.28542	0.168483
$185$	−15.7406	−1.15728
$186$	−3.53357	−0.259094
$187$	1.63164	0.119317
$188$	10.3093	0.751884
$189$	0	0
$190$	−9.46939	−0.686982
$191$	−7.88049	−0.570212	−0.285106	−	0.958496i	$-0.592029\pi$
$191$	−7.88049	−0.570212	−0.285106	+	0.958496i	$0.592029\pi$
$192$	−7.67483	−0.553883
$193$	14.1925	1.02160	0.510801	−	0.859699i	$-0.329349\pi$
$193$	14.1925	1.02160	0.510801	+	0.859699i	$0.329349\pi$
$194$	1.78443	0.128114
$195$	43.5030	3.11531
$196$	0	0
$197$	16.5450	1.17878	0.589390	−	0.807848i	$-0.299368\pi$
$197$	16.5450	1.17878	0.589390	+	0.807848i	$0.299368\pi$
$198$	−0.0556506	−0.00395491
$199$	7.55864	0.535817	0.267909	−	0.963444i	$-0.413667\pi$
$199$	7.55864	0.535817	0.267909	+	0.963444i	$0.413667\pi$
$200$	13.7948	0.975442
$201$	4.04498	0.285311
$202$	−2.82635	−0.198861
$203$	0	0
$204$	12.5435	0.878223
$205$	5.09113	0.355580
$206$	1.97956	0.137923
$207$	−0.583461	−0.0405533
$208$	−22.3021	−1.54637
$209$	2.56580	0.177480
$210$	0	0
$211$	−5.81483	−0.400309	−0.200155	−	0.979764i	$-0.564144\pi$
$211$	−5.81483	−0.400309	−0.200155	+	0.979764i	$0.564144\pi$
$212$	−14.1042	−0.968684
$213$	2.86705	0.196447
$214$	6.66925	0.455901
$215$	43.8499	2.99054
$216$	−8.01519	−0.545365
$217$	0	0
$218$	−2.07957	−0.140846
$219$	23.2277	1.56958
$220$	−2.75460	−0.185715
$221$	29.5055	1.98475
$222$	−2.55476	−0.171464
$223$	−12.9648	−0.868190	−0.434095	−	0.900867i	$-0.642932\pi$
$223$	−12.9648	−0.868190	−0.434095	+	0.900867i	$0.642932\pi$
$224$	0	0
$225$	−3.52178	−0.234785
$226$	−5.25497	−0.349555
$227$	−25.0355	−1.66166	−0.830832	−	0.556523i	$-0.812135\pi$
$227$	−25.0355	−1.66166	−0.830832	+	0.556523i	$0.812135\pi$
$228$	19.7251	1.30633
$229$	−8.40522	−0.555433	−0.277716	−	0.960663i	$-0.589578\pi$
$229$	−8.40522	−0.555433	−0.277716	+	0.960663i	$0.589578\pi$
$230$	2.25025	0.148377
$231$	0	0
$232$	−3.84969	−0.252745
$233$	−0.0651189	−0.00426608	−0.00213304	−	0.999998i	$-0.500679\pi$
$233$	−0.0651189	−0.00426608	−0.00213304	+	0.999998i	$0.500679\pi$
$234$	−1.00635	−0.0657870
$235$	21.0922	1.37590
$236$	8.14277	0.530049
$237$	−1.62042	−0.105257
$238$	0	0
$239$	−7.03630	−0.455140	−0.227570	−	0.973762i	$-0.573078\pi$
$239$	−7.03630	−0.455140	−0.227570	+	0.973762i	$0.573078\pi$
$240$	−19.3979	−1.25213
$241$	5.09495	0.328195	0.164097	−	0.986444i	$-0.447529\pi$
$241$	5.09495	0.328195	0.164097	+	0.986444i	$0.447529\pi$
$242$	4.12429	0.265119
$243$	3.86556	0.247976
$244$	10.7752	0.689811
$245$	0	0
$246$	0.826307	0.0526834
$247$	46.3983	2.95225
$248$	8.40737	0.533868
$249$	−4.65359	−0.294909
$250$	6.36573	0.402604
$251$	0.625014	0.0394505	0.0197253	−	0.999805i	$-0.493721\pi$
$251$	0.625014	0.0394505	0.0197253	+	0.999805i	$0.493721\pi$
$252$	0	0
$253$	−0.609722	−0.0383329
$254$	2.01668	0.126538
$255$	25.6633	1.60710
$256$	5.64664	0.352915
$257$	25.9276	1.61732	0.808659	−	0.588277i	$-0.200194\pi$
$257$	25.9276	1.61732	0.808659	+	0.588277i	$0.200194\pi$
$258$	7.11698	0.443084
$259$	0	0
$260$	−49.8123	−3.08923
$261$	0.982815	0.0608347
$262$	−5.55347	−0.343094
$263$	11.0204	0.679547	0.339773	−	0.940507i	$-0.389650\pi$
$263$	11.0204	0.679547	0.339773	+	0.940507i	$0.389650\pi$
$264$	−0.928995	−0.0571757
$265$	−28.8564	−1.77264
$266$	0	0
$267$	18.5159	1.13315
$268$	−4.63163	−0.282922
$269$	21.4108	1.30544	0.652721	−	0.757599i	$-0.273627\pi$
$269$	21.4108	1.30544	0.652721	+	0.757599i	$0.273627\pi$
$270$	−7.89184	−0.480282
$271$	8.91945	0.541818	0.270909	−	0.962605i	$-0.412676\pi$
$271$	8.91945	0.541818	0.270909	+	0.962605i	$0.412676\pi$
$272$	−13.1565	−0.797728
$273$	0	0
$274$	4.34482	0.262480
$275$	−3.68029	−0.221930
$276$	−4.68735	−0.282145
$277$	−16.5864	−0.996579	−0.498290	−	0.867011i	$-0.666038\pi$
$277$	−16.5864	−0.996579	−0.498290	+	0.867011i	$0.666038\pi$
$278$	−6.00928	−0.360413
$279$	−2.14638	−0.128500
$280$	0	0
$281$	14.6728	0.875308	0.437654	−	0.899144i	$-0.355810\pi$
$281$	14.6728	0.875308	0.437654	+	0.899144i	$0.355810\pi$
$282$	3.42333	0.203857
$283$	2.19960	0.130753	0.0653763	−	0.997861i	$-0.479175\pi$
$283$	2.19960	0.130753	0.0653763	+	0.997861i	$0.479175\pi$
$284$	−3.28287	−0.194802
$285$	40.3564	2.39050
$286$	−1.05164	−0.0621849
$287$	0	0
$288$	1.54596	0.0910964
$289$	0.405885	0.0238756
$290$	−3.79045	−0.222583
$291$	−7.60481	−0.445802
$292$	−26.5965	−1.55644
$293$	−27.2921	−1.59442	−0.797211	−	0.603700i	$-0.793692\pi$
$293$	−27.2921	−1.59442	−0.797211	+	0.603700i	$0.793692\pi$
$294$	0	0
$295$	16.6596	0.969960
$296$	6.07849	0.353305
$297$	2.13835	0.124080
$298$	3.98585	0.230894
$299$	−11.0258	−0.637639
$300$	−28.2929	−1.63349
$301$	0	0
$302$	5.93410	0.341469
$303$	12.0453	0.691982
$304$	−20.6890	−1.18659
$305$	22.0454	1.26231
$306$	−0.593665	−0.0339376
$307$	−14.8235	−0.846024	−0.423012	−	0.906124i	$-0.639027\pi$
$307$	−14.8235	−0.846024	−0.423012	+	0.906124i	$0.639027\pi$
$308$	0	0
$309$	−8.43644	−0.479932
$310$	8.27798	0.470158
$311$	19.7831	1.12180	0.560900	−	0.827884i	$-0.310455\pi$
$311$	19.7831	1.12180	0.560900	+	0.827884i	$0.310455\pi$
$312$	−16.7993	−0.951075
$313$	−16.2823	−0.920330	−0.460165	−	0.887833i	$-0.652210\pi$
$313$	−16.2823	−0.920330	−0.460165	+	0.887833i	$0.652210\pi$
$314$	5.28286	0.298129
$315$	0	0
$316$	1.85543	0.104376
$317$	20.7179	1.16363	0.581815	−	0.813321i	$-0.302343\pi$
$317$	20.7179	1.16363	0.581815	+	0.813321i	$0.302343\pi$
$318$	−4.68349	−0.262637
$319$	1.02705	0.0575038
$320$	17.9796	1.00509
$321$	−28.4228	−1.58641
$322$	0	0
$323$	27.3713	1.52298
$324$	14.3559	0.797547
$325$	−66.5519	−3.69164
$326$	−9.54703	−0.528761
$327$	8.86263	0.490104
$328$	−1.96602	−0.108555
$329$	0	0
$330$	−0.914698	−0.0503525
$331$	13.7123	0.753695	0.376848	−	0.926275i	$-0.377008\pi$
$331$	13.7123	0.753695	0.376848	+	0.926275i	$0.377008\pi$
$332$	5.32851	0.292440
$333$	−1.55182	−0.0850392
$334$	−3.22288	−0.176348
$335$	−9.47602	−0.517731
$336$	0	0
$337$	−18.7359	−1.02061	−0.510305	−	0.859993i	$-0.670468\pi$
$337$	−18.7359	−1.02061	−0.510305	+	0.859993i	$0.670468\pi$
$338$	−14.0743	−0.765542
$339$	22.3955	1.21635
$340$	−29.3853	−1.59364
$341$	−2.24298	−0.121464
$342$	−0.933557	−0.0504810
$343$	0	0
$344$	−16.9333	−0.912983
$345$	−9.59003	−0.516310
$346$	9.93480	0.534098
$347$	−13.7814	−0.739824	−0.369912	−	0.929067i	$-0.620612\pi$
$347$	−13.7814	−0.739824	−0.369912	+	0.929067i	$0.620612\pi$
$348$	7.89565	0.423251
$349$	6.94330	0.371666	0.185833	−	0.982581i	$-0.440502\pi$
$349$	6.94330	0.371666	0.185833	+	0.982581i	$0.440502\pi$
$350$	0	0
$351$	38.6686	2.06398
$352$	1.61554	0.0861084
$353$	26.3457	1.40224	0.701119	−	0.713045i	$-0.252684\pi$
$353$	26.3457	1.40224	0.701119	+	0.713045i	$0.252684\pi$
$354$	2.70391	0.143711
$355$	−6.71654	−0.356477
$356$	−21.2013	−1.12366
$357$	0	0
$358$	−6.54809	−0.346077
$359$	−4.26829	−0.225272	−0.112636	−	0.993636i	$-0.535929\pi$
$359$	−4.26829	−0.225272	−0.112636	+	0.993636i	$0.535929\pi$
$360$	2.08259	0.109762
$361$	24.0423	1.26538
$362$	1.82834	0.0960953
$363$	−17.5768	−0.922540
$364$	0	0
$365$	−54.4148	−2.84820
$366$	3.57804	0.187027
$367$	−31.3359	−1.63572	−0.817860	−	0.575417i	$-0.804840\pi$
$367$	−31.3359	−1.63572	−0.817860	+	0.575417i	$0.804840\pi$
$368$	4.91639	0.256285
$369$	0.501919	0.0261288
$370$	5.98494	0.311142
$371$	0	0
$372$	−17.2434	−0.894026
$373$	−11.7044	−0.606029	−0.303014	−	0.952986i	$-0.597993\pi$
$373$	−11.7044	−0.606029	−0.303014	+	0.952986i	$0.597993\pi$
$374$	−0.620385	−0.0320794
$375$	−27.1293	−1.40095
$376$	−8.14507	−0.420050
$377$	18.5725	0.956533
$378$	0	0
$379$	−8.15168	−0.418724	−0.209362	−	0.977838i	$-0.567139\pi$
$379$	−8.15168	−0.418724	−0.209362	+	0.977838i	$0.567139\pi$
$380$	−46.2093	−2.37049
$381$	−8.59463	−0.440316
$382$	2.99634	0.153306
$383$	−4.84769	−0.247705	−0.123853	−	0.992301i	$-0.539525\pi$
$383$	−4.84769	−0.247705	−0.123853	+	0.992301i	$0.539525\pi$
$384$	16.3056	0.832092
$385$	0	0
$386$	−5.39632	−0.274665
$387$	4.32302	0.219752
$388$	8.70775	0.442069
$389$	8.27923	0.419773	0.209887	−	0.977726i	$-0.432690\pi$
$389$	8.27923	0.419773	0.209887	+	0.977726i	$0.432690\pi$
$390$	−16.5408	−0.837576
$391$	−6.50435	−0.328939
$392$	0	0
$393$	23.6676	1.19387
$394$	−6.29077	−0.316924
$395$	3.79610	0.191002
$396$	−0.271567	−0.0136468
$397$	−18.1149	−0.909161	−0.454581	−	0.890706i	$-0.650211\pi$
$397$	−18.1149	−0.909161	−0.454581	+	0.890706i	$0.650211\pi$
$398$	−2.87396	−0.144059
$399$	0	0
$400$	29.6754	1.48377
$401$	3.69872	0.184705	0.0923526	−	0.995726i	$-0.470561\pi$
$401$	3.69872	0.184705	0.0923526	+	0.995726i	$0.470561\pi$
$402$	−1.53799	−0.0767079
$403$	−40.5606	−2.02047
$404$	−13.7922	−0.686188
$405$	29.3712	1.45947
$406$	0	0
$407$	−1.62166	−0.0803829
$408$	−9.91027	−0.490631
$409$	−36.0056	−1.78036	−0.890181	−	0.455608i	$-0.849422\pi$
$409$	−36.0056	−1.78036	−0.890181	+	0.455608i	$0.849422\pi$
$410$	−1.93576	−0.0956004
$411$	−18.5166	−0.913357
$412$	9.65999	0.475914
$413$	0	0
$414$	0.221845	0.0109031
$415$	10.9018	0.535148
$416$	29.2143	1.43235
$417$	25.6102	1.25413
$418$	−0.975575	−0.0477170
$419$	−12.6697	−0.618955	−0.309478	−	0.950907i	$-0.600154\pi$
$419$	−12.6697	−0.618955	−0.309478	+	0.950907i	$0.600154\pi$
$420$	0	0
$421$	28.7588	1.40162	0.700809	−	0.713349i	$-0.252823\pi$
$421$	28.7588	1.40162	0.700809	+	0.713349i	$0.252823\pi$
$422$	2.21093	0.107626
$423$	2.07941	0.101105
$424$	11.1433	0.541168
$425$	−39.2603	−1.90441
$426$	−1.09012	−0.0528163
$427$	0	0
$428$	32.5450	1.57312
$429$	4.48185	0.216386
$430$	−16.6727	−0.804030
$431$	−25.7829	−1.24192	−0.620958	−	0.783844i	$-0.713256\pi$
$431$	−25.7829	−1.24192	−0.620958	+	0.783844i	$0.713256\pi$
$432$	−17.2423	−0.829570
$433$	4.70806	0.226255	0.113127	−	0.993580i	$-0.463913\pi$
$433$	4.70806	0.226255	0.113127	+	0.993580i	$0.463913\pi$
$434$	0	0
$435$	16.1540	0.774525
$436$	−10.1480	−0.486001
$437$	−10.2283	−0.489286
$438$	−8.83170	−0.421995
$439$	14.0326	0.669739	0.334870	−	0.942264i	$-0.391308\pi$
$439$	14.0326	0.669739	0.334870	+	0.942264i	$0.391308\pi$
$440$	2.17632	0.103752
$441$	0	0
$442$	−11.2186	−0.533616
$443$	21.7044	1.03121	0.515603	−	0.856827i	$-0.327568\pi$
$443$	21.7044	1.03121	0.515603	+	0.856827i	$0.327568\pi$
$444$	−12.4669	−0.591651
$445$	−43.3765	−2.05624
$446$	4.92952	0.233420
$447$	−16.9868	−0.803446
$448$	0	0
$449$	11.9975	0.566198	0.283099	−	0.959091i	$-0.408637\pi$
$449$	11.9975	0.566198	0.283099	+	0.959091i	$0.408637\pi$
$450$	1.33906	0.0631238
$451$	0.524509	0.0246982
$452$	−25.6435	−1.20617
$453$	−25.2898	−1.18822
$454$	9.51905	0.446751
$455$	0	0
$456$	−15.5842	−0.729798
$457$	−19.1804	−0.897222	−0.448611	−	0.893727i	$-0.648081\pi$
$457$	−19.1804	−0.897222	−0.448611	+	0.893727i	$0.648081\pi$
$458$	3.19585	0.149332
$459$	22.8114	1.06474
$460$	10.9809	0.511987
$461$	1.24178	0.0578355	0.0289177	−	0.999582i	$-0.490794\pi$
$461$	1.24178	0.0578355	0.0289177	+	0.999582i	$0.490794\pi$
$462$	0	0
$463$	12.7019	0.590307	0.295153	−	0.955450i	$-0.404629\pi$
$463$	12.7019	0.590307	0.295153	+	0.955450i	$0.404629\pi$
$464$	−8.28146	−0.384457
$465$	−35.2788	−1.63602
$466$	0.0247597	0.00114697
$467$	−12.1596	−0.562681	−0.281340	−	0.959608i	$-0.590779\pi$
$467$	−12.1596	−0.562681	−0.281340	+	0.959608i	$0.590779\pi$
$468$	−4.91084	−0.227004
$469$	0	0
$470$	−8.01972	−0.369922
$471$	−22.5143	−1.03740
$472$	−6.43336	−0.296119
$473$	4.51760	0.207719
$474$	0.616119	0.0282993
$475$	−61.7381	−2.83274
$476$	0	0
$477$	−2.84486	−0.130257
$478$	2.67536	0.122368
$479$	−21.1681	−0.967197	−0.483598	−	0.875290i	$-0.660670\pi$
$479$	−21.1681	−0.967197	−0.483598	+	0.875290i	$0.660670\pi$
$480$	25.4100	1.15980
$481$	−29.3251	−1.33711
$482$	−1.93721	−0.0882376
$483$	0	0
$484$	20.1260	0.914816
$485$	17.8155	0.808961
$486$	−1.46977	−0.0666701
$487$	−31.3358	−1.41996	−0.709981	−	0.704221i	$-0.751296\pi$
$487$	−31.3358	−1.41996	−0.709981	+	0.704221i	$0.751296\pi$
$488$	−8.51315	−0.385372
$489$	40.6872	1.83994
$490$	0	0
$491$	−14.7254	−0.664548	−0.332274	−	0.943183i	$-0.607816\pi$
$491$	−14.7254	−0.664548	−0.332274	+	0.943183i	$0.607816\pi$
$492$	4.03226	0.181789
$493$	10.9563	0.493447
$494$	−17.6417	−0.793736
$495$	−0.555609	−0.0249728
$496$	18.0859	0.812082
$497$	0	0
$498$	1.76940	0.0792886
$499$	−20.4261	−0.914396	−0.457198	−	0.889365i	$-0.651147\pi$
$499$	−20.4261	−0.914396	−0.457198	+	0.889365i	$0.651147\pi$
$500$	31.0639	1.38922
$501$	13.7352	0.613641
$502$	−0.237644	−0.0106066
$503$	30.9877	1.38168	0.690838	−	0.723010i	$-0.257242\pi$
$503$	30.9877	1.38168	0.690838	+	0.723010i	$0.257242\pi$
$504$	0	0
$505$	−28.2180	−1.25568
$506$	0.231830	0.0103061
$507$	59.9815	2.66387
$508$	9.84113	0.436630
$509$	−39.2483	−1.73965	−0.869824	−	0.493361i	$-0.835768\pi$
$509$	−39.2483	−1.73965	−0.869824	+	0.493361i	$0.835768\pi$
$510$	−9.75775	−0.432081
$511$	0	0
$512$	−22.2722	−0.984299
$513$	35.8716	1.58377
$514$	−9.85824	−0.434829
$515$	19.7637	0.870895
$516$	34.7299	1.52890
$517$	2.17300	0.0955686
$518$	0	0
$519$	−42.3398	−1.85851
$520$	39.3552	1.72584
$521$	18.9383	0.829703	0.414851	−	0.909889i	$-0.363834\pi$
$521$	18.9383	0.829703	0.414851	+	0.909889i	$0.363834\pi$
$522$	−0.373688	−0.0163559
$523$	13.0537	0.570797	0.285399	−	0.958409i	$-0.407874\pi$
$523$	13.0537	0.570797	0.285399	+	0.958409i	$0.407874\pi$
$524$	−27.1001	−1.18388
$525$	0	0
$526$	−4.19020	−0.182701
$527$	−23.9275	−1.04230
$528$	−1.99845	−0.0869715
$529$	−20.5694	−0.894322
$530$	10.9718	0.476587
$531$	1.64242	0.0712749
$532$	0	0
$533$	9.48488	0.410836
$534$	−7.04014	−0.304657
$535$	66.5851	2.87873
$536$	3.65931	0.158058
$537$	27.9064	1.20425
$538$	−8.14087	−0.350978
$539$	0	0
$540$	−38.5111	−1.65725
$541$	−14.7251	−0.633083	−0.316541	−	0.948579i	$-0.602522\pi$
$541$	−14.7251	−0.633083	−0.316541	+	0.948579i	$0.602522\pi$
$542$	−3.39137	−0.145672
$543$	−7.79195	−0.334384
$544$	17.2341	0.738907
$545$	−20.7622	−0.889353
$546$	0	0
$547$	19.0904	0.816245	0.408123	−	0.912927i	$-0.366184\pi$
$547$	19.0904	0.816245	0.408123	+	0.912927i	$0.366184\pi$
$548$	21.2021	0.905710
$549$	2.17338	0.0927577
$550$	1.39933	0.0596675
$551$	17.2291	0.733985
$552$	3.70334	0.157624
$553$	0	0
$554$	6.30651	0.267938
$555$	−25.5064	−1.08269
$556$	−29.3245	−1.24363
$557$	−8.88689	−0.376550	−0.188275	−	0.982116i	$-0.560290\pi$
$557$	−8.88689	−0.376550	−0.188275	+	0.982116i	$0.560290\pi$
$558$	0.816100	0.0345483
$559$	81.6933	3.45526
$560$	0	0
$561$	2.64394	0.111627
$562$	−5.57893	−0.235333
$563$	42.8327	1.80518	0.902591	−	0.430500i	$-0.141663\pi$
$563$	42.8327	1.80518	0.902591	+	0.430500i	$0.141663\pi$
$564$	16.7054	0.703424
$565$	−52.4651	−2.20722
$566$	−0.836336	−0.0351538
$567$	0	0
$568$	2.59369	0.108829
$569$	−30.4631	−1.27708	−0.638539	−	0.769589i	$-0.720461\pi$
$569$	−30.4631	−1.27708	−0.638539	+	0.769589i	$0.720461\pi$
$570$	−15.3444	−0.642705
$571$	−21.0691	−0.881712	−0.440856	−	0.897578i	$-0.645325\pi$
$571$	−21.0691	−0.881712	−0.440856	+	0.897578i	$0.645325\pi$
$572$	−5.13187	−0.214574
$573$	−12.7697	−0.533461
$574$	0	0
$575$	14.6711	0.611826
$576$	1.77255	0.0738561
$577$	36.6032	1.52381	0.761905	−	0.647689i	$-0.224264\pi$
$577$	36.6032	1.52381	0.761905	+	0.647689i	$0.224264\pi$
$578$	−0.154327	−0.00641914
$579$	22.9979	0.955758
$580$	−18.4969	−0.768040
$581$	0	0
$582$	2.89152	0.119857
$583$	−2.97290	−0.123125
$584$	21.0131	0.869528
$585$	−10.0473	−0.415404
$586$	10.3771	0.428673
$587$	0.730785	0.0301627	0.0150814	−	0.999886i	$-0.495199\pi$
$587$	0.730785	0.0301627	0.0150814	+	0.999886i	$0.495199\pi$
$588$	0	0
$589$	−37.6268	−1.55039
$590$	−6.33435	−0.260781
$591$	26.8098	1.10281
$592$	13.0760	0.537422
$593$	22.6253	0.929110	0.464555	−	0.885544i	$-0.346214\pi$
$593$	22.6253	0.929110	0.464555	+	0.885544i	$0.346214\pi$
$594$	−0.813050	−0.0333598
$595$	0	0
$596$	19.4504	0.796719
$597$	12.2481	0.501283
$598$	4.19226	0.171434
$599$	−20.3335	−0.830803	−0.415402	−	0.909638i	$-0.636359\pi$
$599$	−20.3335	−0.830803	−0.415402	+	0.909638i	$0.636359\pi$
$600$	22.3534	0.912573
$601$	−0.591631	−0.0241331	−0.0120666	−	0.999927i	$-0.503841\pi$
$601$	−0.591631	−0.0241331	−0.0120666	+	0.999927i	$0.503841\pi$
$602$	0	0
$603$	−0.934211	−0.0380440
$604$	28.9576	1.17827
$605$	41.1765	1.67406
$606$	−4.57987	−0.186045
$607$	−2.76989	−0.112427	−0.0562133	−	0.998419i	$-0.517903\pi$
$607$	−2.76989	−0.112427	−0.0562133	+	0.998419i	$0.517903\pi$
$608$	27.1012	1.09910
$609$	0	0
$610$	−8.38214	−0.339383
$611$	39.2952	1.58971
$612$	−2.89700	−0.117104
$613$	−44.4642	−1.79589	−0.897945	−	0.440107i	$-0.854941\pi$
$613$	−44.4642	−1.79589	−0.897945	+	0.440107i	$0.854941\pi$
$614$	5.63624	0.227460
$615$	8.24976	0.332663
$616$	0	0
$617$	25.7855	1.03809	0.519043	−	0.854748i	$-0.326288\pi$
$617$	25.7855	1.03809	0.519043	+	0.854748i	$0.326288\pi$
$618$	3.20772	0.129033
$619$	−23.0573	−0.926753	−0.463376	−	0.886162i	$-0.653362\pi$
$619$	−23.0573	−0.926753	−0.463376	+	0.886162i	$0.653362\pi$
$620$	40.3954	1.62232
$621$	−8.52431	−0.342069
$622$	−7.52199	−0.301604
$623$	0	0
$624$	−36.1387	−1.44671
$625$	16.5030	0.660120
$626$	6.19089	0.247438
$627$	4.15768	0.166042
$628$	25.7796	1.02872
$629$	−17.2995	−0.689776
$630$	0	0
$631$	2.22752	0.0886762	0.0443381	−	0.999017i	$-0.485882\pi$
$631$	2.22752	0.0886762	0.0443381	+	0.999017i	$0.485882\pi$
$632$	−1.46592	−0.0583112
$633$	−9.42245	−0.374509
$634$	−7.87739	−0.312851
$635$	20.1344	0.799007
$636$	−22.8548	−0.906251
$637$	0	0
$638$	−0.390507	−0.0154603
$639$	−0.662162	−0.0261947
$640$	−38.1986	−1.50993
$641$	31.7248	1.25305	0.626527	−	0.779400i	$-0.284476\pi$
$641$	31.7248	1.25305	0.626527	+	0.779400i	$0.284476\pi$
$642$	10.8070	0.426518
$643$	−26.5663	−1.04767	−0.523836	−	0.851819i	$-0.675499\pi$
$643$	−26.5663	−1.04767	−0.523836	+	0.851819i	$0.675499\pi$
$644$	0	0
$645$	71.0552	2.79780
$646$	−10.4072	−0.409465
$647$	27.0631	1.06396	0.531981	−	0.846756i	$-0.321448\pi$
$647$	27.0631	1.06396	0.531981	+	0.846756i	$0.321448\pi$
$648$	−11.3421	−0.445561
$649$	1.71634	0.0673723
$650$	25.3045	0.992525
$651$	0	0
$652$	−46.5882	−1.82453
$653$	−13.2000	−0.516557	−0.258279	−	0.966070i	$-0.583155\pi$
$653$	−13.2000	−0.516557	−0.258279	+	0.966070i	$0.583155\pi$
$654$	−3.36977	−0.131768
$655$	−55.4452	−2.16642
$656$	−4.22930	−0.165126
$657$	−5.36458	−0.209292
$658$	0	0
$659$	45.3253	1.76562	0.882811	−	0.469728i	$-0.155648\pi$
$659$	45.3253	1.76562	0.882811	+	0.469728i	$0.155648\pi$
$660$	−4.46360	−0.173745
$661$	34.6598	1.34811	0.674055	−	0.738682i	$-0.264551\pi$
$661$	34.6598	1.34811	0.674055	+	0.738682i	$0.264551\pi$
$662$	−5.21371	−0.202637
$663$	47.8112	1.85683
$664$	−4.20989	−0.163376
$665$	0	0
$666$	0.590036	0.0228634
$667$	−4.09422	−0.158529
$668$	−15.7272	−0.608503
$669$	−21.0085	−0.812234
$670$	3.60299	0.139196
$671$	2.27120	0.0876789
$672$	0	0
$673$	−40.7052	−1.56907	−0.784535	−	0.620084i	$-0.787098\pi$
$673$	−40.7052	−1.56907	−0.784535	+	0.620084i	$0.787098\pi$
$674$	7.12381	0.274399
$675$	−51.4529	−1.98042
$676$	−68.6807	−2.64157
$677$	−42.1098	−1.61841	−0.809206	−	0.587525i	$-0.800102\pi$
$677$	−42.1098	−1.61841	−0.809206	+	0.587525i	$0.800102\pi$
$678$	−8.51525	−0.327026
$679$	0	0
$680$	23.2164	0.890310
$681$	−40.5680	−1.55457
$682$	0.852831	0.0326566
$683$	24.4829	0.936813	0.468407	−	0.883513i	$-0.344828\pi$
$683$	24.4829	0.936813	0.468407	+	0.883513i	$0.344828\pi$
$684$	−4.55563	−0.174189
$685$	43.3782	1.65740
$686$	0	0
$687$	−13.6200	−0.519634
$688$	−36.4269	−1.38876
$689$	−53.7601	−2.04809
$690$	3.64634	0.138814
$691$	−20.6837	−0.786845	−0.393422	−	0.919358i	$-0.628709\pi$
$691$	−20.6837	−0.786845	−0.393422	+	0.919358i	$0.628709\pi$
$692$	48.4804	1.84295
$693$	0	0
$694$	5.23999	0.198907
$695$	−59.9960	−2.27578
$696$	−6.23811	−0.236455
$697$	5.59532	0.211938
$698$	−2.64000	−0.0999253
$699$	−0.105520	−0.00399113
$700$	0	0
$701$	38.4860	1.45360	0.726799	−	0.686850i	$-0.241007\pi$
$701$	38.4860	1.45360	0.726799	+	0.686850i	$0.241007\pi$
$702$	−14.7027	−0.554916
$703$	−27.2040	−1.02602
$704$	1.85233	0.0698122
$705$	34.1782	1.28723
$706$	−10.0172	−0.377002
$707$	0	0
$708$	13.1947	0.495887
$709$	−38.0519	−1.42907	−0.714535	−	0.699600i	$-0.753362\pi$
$709$	−38.0519	−1.42907	−0.714535	+	0.699600i	$0.753362\pi$
$710$	2.55378	0.0958416
$711$	0.374245	0.0140353
$712$	16.7505	0.627751
$713$	8.94140	0.334858
$714$	0	0
$715$	−10.4995	−0.392658
$716$	−31.9538	−1.19417
$717$	−11.4017	−0.425806
$718$	1.62290	0.0605661
$719$	−22.7535	−0.848562	−0.424281	−	0.905531i	$-0.639473\pi$
$719$	−22.7535	−0.848562	−0.424281	+	0.905531i	$0.639473\pi$
$720$	4.48007	0.166962
$721$	0	0
$722$	−9.14140	−0.340208
$723$	8.25595	0.307042
$724$	8.92203	0.331585
$725$	−24.7128	−0.917810
$726$	6.68307	0.248032
$727$	14.9856	0.555783	0.277892	−	0.960612i	$-0.410364\pi$
$727$	14.9856	0.555783	0.277892	+	0.960612i	$0.410364\pi$
$728$	0	0
$729$	29.4754	1.09168
$730$	20.6897	0.765761
$731$	48.1925	1.78246
$732$	17.4603	0.645352
$733$	−18.8867	−0.697595	−0.348797	−	0.937198i	$-0.613410\pi$
$733$	−18.8867	−0.697595	−0.348797	+	0.937198i	$0.613410\pi$
$734$	11.9146	0.439776
$735$	0	0
$736$	−6.44016	−0.237388
$737$	−0.976258	−0.0359609
$738$	−0.190841	−0.00702494
$739$	50.5809	1.86065	0.930325	−	0.366736i	$-0.119525\pi$
$739$	50.5809	1.86065	0.930325	+	0.366736i	$0.119525\pi$
$740$	29.2057	1.07362
$741$	75.1847	2.76198
$742$	0	0
$743$	−34.9627	−1.28266	−0.641329	−	0.767266i	$-0.721617\pi$
$743$	−34.9627	−1.28266	−0.641329	+	0.767266i	$0.721617\pi$
$744$	13.6235	0.499460
$745$	39.7943	1.45795
$746$	4.45026	0.162935
$747$	1.07477	0.0393239
$748$	−3.02739	−0.110692
$749$	0	0
$750$	10.3151	0.376656
$751$	−2.98091	−0.108775	−0.0543875	−	0.998520i	$-0.517321\pi$
$751$	−2.98091	−0.108775	−0.0543875	+	0.998520i	$0.517321\pi$
$752$	−17.5217	−0.638950
$753$	1.01278	0.0369079
$754$	−7.06168	−0.257171
$755$	59.2454	2.15616
$756$	0	0
$757$	6.83336	0.248363	0.124181	−	0.992260i	$-0.460370\pi$
$757$	6.83336	0.248363	0.124181	+	0.992260i	$0.460370\pi$
$758$	3.09945	0.112577
$759$	−0.988004	−0.0358623
$760$	36.5086	1.32431
$761$	−3.91753	−0.142010	−0.0710052	−	0.997476i	$-0.522621\pi$
$761$	−3.91753	−0.142010	−0.0710052	+	0.997476i	$0.522621\pi$
$762$	3.26787	0.118382
$763$	0	0
$764$	14.6217	0.528995
$765$	−5.92709	−0.214294
$766$	1.84320	0.0665975
$767$	31.0372	1.12069
$768$	9.14991	0.330169
$769$	13.9269	0.502218	0.251109	−	0.967959i	$-0.419205\pi$
$769$	13.9269	0.502218	0.251109	+	0.967959i	$0.419205\pi$
$770$	0	0
$771$	42.0135	1.51308
$772$	−26.3333	−0.947756
$773$	−7.79151	−0.280241	−0.140121	−	0.990134i	$-0.544749\pi$
$773$	−7.79151	−0.280241	−0.140121	+	0.990134i	$0.544749\pi$
$774$	−1.64371	−0.0590819
$775$	53.9704	1.93867
$776$	−6.87973	−0.246968
$777$	0	0
$778$	−3.14794	−0.112859
$779$	8.79882	0.315251
$780$	−80.7168	−2.89013
$781$	−0.691965	−0.0247605
$782$	2.47310	0.0884378
$783$	14.3588	0.513143
$784$	0	0
$785$	52.7435	1.88250
$786$	−8.99894	−0.320982
$787$	46.9568	1.67383	0.836914	−	0.547334i	$-0.184357\pi$
$787$	46.9568	1.67383	0.836914	+	0.547334i	$0.184357\pi$
$788$	−30.6981	−1.09357
$789$	17.8577	0.635749
$790$	−1.44336	−0.0513524
$791$	0	0
$792$	0.214557	0.00762395
$793$	41.0710	1.45847
$794$	6.88769	0.244435
$795$	−46.7595	−1.65839
$796$	−14.0245	−0.497086
$797$	39.3199	1.39278	0.696391	−	0.717663i	$-0.254788\pi$
$797$	39.3199	1.39278	0.696391	+	0.717663i	$0.254788\pi$
$798$	0	0
$799$	23.1810	0.820086
$800$	−38.8729	−1.37437
$801$	−4.27635	−0.151097
$802$	−1.40633	−0.0496594
$803$	−5.60603	−0.197833
$804$	−7.50517	−0.264687
$805$	0	0
$806$	15.4220	0.543218
$807$	34.6945	1.22130
$808$	10.8968	0.383348
$809$	−28.9552	−1.01801	−0.509005	−	0.860764i	$-0.669986\pi$
$809$	−28.9552	−1.01801	−0.509005	+	0.860764i	$0.669986\pi$
$810$	−11.1676	−0.392389
$811$	9.62380	0.337937	0.168969	−	0.985621i	$-0.445956\pi$
$811$	9.62380	0.337937	0.168969	+	0.985621i	$0.445956\pi$
$812$	0	0
$813$	14.4532	0.506897
$814$	0.616593	0.0216116
$815$	−95.3165	−3.33879
$816$	−21.3190	−0.746313
$817$	75.7843	2.65136
$818$	13.6901	0.478664
$819$	0	0
$820$	−9.44625	−0.329877
$821$	16.5062	0.576071	0.288035	−	0.957620i	$-0.406998\pi$
$821$	16.5062	0.576071	0.288035	+	0.957620i	$0.406998\pi$
$822$	7.04043	0.245563
$823$	37.0563	1.29170	0.645850	−	0.763464i	$-0.276503\pi$
$823$	37.0563	1.29170	0.645850	+	0.763464i	$0.276503\pi$
$824$	−7.63207	−0.265876
$825$	−5.96361	−0.207626
$826$	0	0
$827$	−48.0080	−1.66940	−0.834702	−	0.550703i	$-0.814360\pi$
$827$	−48.0080	−1.66940	−0.834702	+	0.550703i	$0.814360\pi$
$828$	1.08257	0.0376220
$829$	−26.6163	−0.924421	−0.462210	−	0.886770i	$-0.652944\pi$
$829$	−26.6163	−0.924421	−0.462210	+	0.886770i	$0.652944\pi$
$830$	−4.14511	−0.143879
$831$	−26.8769	−0.932349
$832$	33.4963	1.16127
$833$	0	0
$834$	−9.73755	−0.337184
$835$	−32.1769	−1.11353
$836$	−4.76067	−0.164651
$837$	−31.3584	−1.08390
$838$	4.81730	0.166411
$839$	10.2678	0.354485	0.177242	−	0.984167i	$-0.443282\pi$
$839$	10.2678	0.354485	0.177242	+	0.984167i	$0.443282\pi$
$840$	0	0
$841$	−22.1035	−0.762188
$842$	−10.9347	−0.376836
$843$	23.7761	0.818893
$844$	10.7890	0.371373
$845$	−140.517	−4.83391
$846$	−0.790639	−0.0271827
$847$	0	0
$848$	23.9715	0.823186
$849$	3.56427	0.122325
$850$	14.9276	0.512014
$851$	6.46459	0.221603
$852$	−5.31962	−0.182247
$853$	−50.0631	−1.71413	−0.857064	−	0.515211i	$-0.827714\pi$
$853$	−50.0631	−1.71413	−0.857064	+	0.515211i	$0.827714\pi$
$854$	0	0
$855$	−9.32054	−0.318756
$856$	−25.7129	−0.878847
$857$	0.635163	0.0216967	0.0108484	−	0.999941i	$-0.496547\pi$
$857$	0.635163	0.0216967	0.0108484	+	0.999941i	$0.496547\pi$
$858$	−1.70410	−0.0581770
$859$	−52.1857	−1.78055	−0.890276	−	0.455422i	$-0.849488\pi$
$859$	−52.1857	−1.78055	−0.890276	+	0.455422i	$0.849488\pi$
$860$	−81.3605	−2.77437
$861$	0	0
$862$	9.80321	0.333899
$863$	−44.1620	−1.50329	−0.751646	−	0.659567i	$-0.770740\pi$
$863$	−44.1620	−1.50329	−0.751646	+	0.659567i	$0.770740\pi$
$864$	22.5863	0.768401
$865$	99.1880	3.37249
$866$	−1.79011	−0.0608303
$867$	0.657704	0.0223368
$868$	0	0
$869$	0.391089	0.0132668
$870$	−6.14211	−0.208237
$871$	−17.6540	−0.598184
$872$	8.01762	0.271511
$873$	1.75638	0.0594443
$874$	3.88902	0.131548
$875$	0	0
$876$	−43.0975	−1.45613
$877$	−27.3287	−0.922825	−0.461413	−	0.887186i	$-0.652657\pi$
$877$	−27.3287	−0.922825	−0.461413	+	0.887186i	$0.652657\pi$
$878$	−5.33550	−0.180065
$879$	−44.2246	−1.49166
$880$	4.68171	0.157820
$881$	23.8567	0.803752	0.401876	−	0.915694i	$-0.368358\pi$
$881$	23.8567	0.803752	0.401876	+	0.915694i	$0.368358\pi$
$882$	0	0
$883$	28.9826	0.975342	0.487671	−	0.873028i	$-0.337847\pi$
$883$	28.9826	0.975342	0.487671	+	0.873028i	$0.337847\pi$
$884$	−54.7454	−1.84129
$885$	26.9955	0.907445
$886$	−8.25249	−0.277248
$887$	54.4751	1.82910	0.914548	−	0.404478i	$-0.132547\pi$
$887$	54.4751	1.82910	0.914548	+	0.404478i	$0.132547\pi$
$888$	9.84969	0.330534
$889$	0	0
$890$	16.4927	0.552836
$891$	3.02594	0.101373
$892$	24.0554	0.805434
$893$	36.4529	1.21985
$894$	6.45874	0.216013
$895$	−65.3754	−2.18526
$896$	0	0
$897$	−17.8664	−0.596542
$898$	−4.56172	−0.152227
$899$	−15.0614	−0.502326
$900$	6.53442	0.217814
$901$	−31.7142	−1.05655
$902$	−0.199430	−0.00664029
$903$	0	0
$904$	20.2602	0.673843
$905$	18.2539	0.606781
$906$	9.61572	0.319461
$907$	−38.1878	−1.26801	−0.634003	−	0.773330i	$-0.718589\pi$
$907$	−38.1878	−1.26801	−0.634003	+	0.773330i	$0.718589\pi$
$908$	46.4516	1.54155
$909$	−2.78192	−0.0922705
$910$	0	0
$911$	46.8766	1.55309	0.776546	−	0.630061i	$-0.216970\pi$
$911$	46.8766	1.55309	0.776546	+	0.630061i	$0.216970\pi$
$912$	−33.5248	−1.11012
$913$	1.12315	0.0371708
$914$	7.29282	0.241225
$915$	35.7227	1.18096
$916$	15.5953	0.515283
$917$	0	0
$918$	−8.67340	−0.286265
$919$	51.8106	1.70907	0.854537	−	0.519390i	$-0.173841\pi$
$919$	51.8106	1.70907	0.854537	+	0.519390i	$0.173841\pi$
$920$	−8.67568	−0.286029
$921$	−24.0203	−0.791497
$922$	−0.472152	−0.0155495
$923$	−12.5130	−0.411872
$924$	0	0
$925$	39.0203	1.28298
$926$	−4.82954	−0.158708
$927$	1.94845	0.0639953
$928$	10.8482	0.356109
$929$	−18.9454	−0.621578	−0.310789	−	0.950479i	$-0.600593\pi$
$929$	−18.9454	−0.621578	−0.310789	+	0.950479i	$0.600593\pi$
$930$	13.4138	0.439856
$931$	0	0
$932$	0.120824	0.00395771
$933$	32.0570	1.04950
$934$	4.62336	0.151281
$935$	−6.19386	−0.202561
$936$	3.87991	0.126819
$937$	2.37869	0.0777084	0.0388542	−	0.999245i	$-0.487629\pi$
$937$	2.37869	0.0777084	0.0388542	+	0.999245i	$0.487629\pi$
$938$	0	0
$939$	−26.3841	−0.861014
$940$	−39.1351	−1.27645
$941$	−3.05212	−0.0994961	−0.0497481	−	0.998762i	$-0.515842\pi$
$941$	−3.05212	−0.0994961	−0.0497481	+	0.998762i	$0.515842\pi$
$942$	8.56044	0.278914
$943$	−2.09090	−0.0680890
$944$	−13.8394	−0.450435
$945$	0	0
$946$	−1.71769	−0.0558469
$947$	15.0764	0.489918	0.244959	−	0.969533i	$-0.421226\pi$
$947$	15.0764	0.489918	0.244959	+	0.969533i	$0.421226\pi$
$948$	3.00657	0.0976490
$949$	−101.376	−3.29080
$950$	23.4742	0.761604
$951$	33.5716	1.08863
$952$	0	0
$953$	−38.2968	−1.24056	−0.620278	−	0.784382i	$-0.712980\pi$
$953$	−38.2968	−1.24056	−0.620278	+	0.784382i	$0.712980\pi$
$954$	1.08168	0.0350207
$955$	29.9151	0.968030
$956$	13.0554	0.422241
$957$	1.66425	0.0537976
$958$	8.04859	0.260038
$959$	0	0
$960$	29.1344	0.940309
$961$	1.89270	0.0610548
$962$	11.1501	0.359492
$963$	6.56442	0.211535
$964$	−9.45333	−0.304471
$965$	−53.8763	−1.73434
$966$	0	0
$967$	−0.529044	−0.0170129	−0.00850645	−	0.999964i	$-0.502708\pi$
$967$	−0.529044	−0.0170129	−0.00850645	+	0.999964i	$0.502708\pi$
$968$	−15.9009	−0.511075
$969$	44.3530	1.42482
$970$	−6.77385	−0.217495
$971$	−21.9726	−0.705133	−0.352567	−	0.935787i	$-0.614691\pi$
$971$	−21.9726	−0.705133	−0.352567	+	0.935787i	$0.614691\pi$
$972$	−7.17227	−0.230051
$973$	0	0
$974$	11.9146	0.381767
$975$	−107.842	−3.45371
$976$	−18.3135	−0.586201
$977$	41.7059	1.33429	0.667144	−	0.744928i	$-0.267516\pi$
$977$	41.7059	1.33429	0.667144	+	0.744928i	$0.267516\pi$
$978$	−15.4702	−0.494682
$979$	−4.46882	−0.142824
$980$	0	0
$981$	−2.04688	−0.0653517
$982$	5.59892	0.178669
$983$	−47.3252	−1.50944	−0.754719	−	0.656048i	$-0.772227\pi$
$983$	−47.3252	−1.50944	−0.754719	+	0.656048i	$0.772227\pi$
$984$	−3.18577	−0.101559
$985$	−62.8063	−2.00118
$986$	−4.16583	−0.132667
$987$	0	0
$988$	−86.0889	−2.73885
$989$	−18.0089	−0.572650
$990$	0.211255	0.00671412
$991$	−52.1478	−1.65653	−0.828265	−	0.560337i	$-0.810672\pi$
$991$	−52.1478	−1.65653	−0.828265	+	0.560337i	$0.810672\pi$
$992$	−23.6914	−0.752203
$993$	22.2196	0.705119
$994$	0	0
$995$	−28.6933	−0.909639
$996$	8.63441	0.273592
$997$	−6.73292	−0.213234	−0.106617	−	0.994300i	$-0.534002\pi$
$997$	−6.73292	−0.213234	−0.106617	+	0.994300i	$0.534002\pi$
$998$	7.76644	0.245842
$999$	−22.6719	−0.717309

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3871.2.a.i.1.10	yes	24
7.6	odd	2	inner	3871.2.a.i.1.9	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3871.2.a.i.1.9	✓	24	7.6	odd	2	inner
3871.2.a.i.1.10	yes	24	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-0.380222 q^{2} +1.62042 q^{3} -1.85543 q^{4} -3.79610 q^{5} -0.616119 q^{6} +1.46592 q^{8} -0.374245 q^{9} +1.44336 q^{10} -0.391089 q^{11} -3.00657 q^{12} -7.07220 q^{13} -6.15126 q^{15} +3.15349 q^{16} -4.17204 q^{17} +0.142296 q^{18} -6.56066 q^{19} +7.04340 q^{20} +0.148701 q^{22} +1.55903 q^{23} +2.37540 q^{24} +9.41035 q^{25} +2.68901 q^{26} -5.46769 q^{27} -2.62613 q^{29} +2.33885 q^{30} +5.73522 q^{31} -4.13087 q^{32} -0.633728 q^{33} +1.58630 q^{34} +0.694386 q^{36} +4.14653 q^{37} +2.49451 q^{38} -11.4599 q^{39} -5.56478 q^{40} -1.34115 q^{41} -11.5513 q^{43} +0.725639 q^{44} +1.42067 q^{45} -0.592779 q^{46} -5.55629 q^{47} +5.10997 q^{48} -3.57802 q^{50} -6.76044 q^{51} +13.1220 q^{52} +7.60160 q^{53} +2.07894 q^{54} +1.48461 q^{55} -10.6310 q^{57} +0.998512 q^{58} -4.38862 q^{59} +11.4132 q^{60} -5.80738 q^{61} -2.18066 q^{62} -4.73633 q^{64} +26.8468 q^{65} +0.240957 q^{66} +2.49625 q^{67} +7.74093 q^{68} +2.52629 q^{69} +1.76933 q^{71} -0.548613 q^{72} +14.3344 q^{73} -1.57660 q^{74} +15.2487 q^{75} +12.1729 q^{76} +4.35732 q^{78} -1.00000 q^{79} -11.9709 q^{80} -7.73721 q^{81} +0.509935 q^{82} -2.87184 q^{83} +15.8375 q^{85} +4.39207 q^{86} -4.25542 q^{87} -0.573306 q^{88} +11.4266 q^{89} -0.540170 q^{90} -2.89268 q^{92} +9.29345 q^{93} +2.11262 q^{94} +24.9049 q^{95} -6.69373 q^{96} -4.69311 q^{97} +0.146363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{2} + 28 q^{4} + 18 q^{8} + 50 q^{9} + 14 q^{11} + 18 q^{15} + 36 q^{16} + 10 q^{18} - 12 q^{22} + 22 q^{23} + 56 q^{25} + 36 q^{29} + 14 q^{30} + 30 q^{32} + 8 q^{36} + 24 q^{37} + 56 q^{39}+ \cdots + 6 q^{99}+O(q^{100}) \) 24 * q + 4 * q^2 + 28 * q^4 + 18 * q^8 + 50 * q^9 + 14 * q^11 + 18 * q^15 + 36 * q^16 + 10 * q^18 - 12 * q^22 + 22 * q^23 + 56 * q^25 + 36 * q^29 + 14 * q^30 + 30 * q^32 + 8 * q^36 + 24 * q^37 + 56 * q^39 - 2 * q^43 + 74 * q^44 + 48 * q^46 - 42 * q^50 + 22 * q^51 + 48 * q^53 - 16 * q^57 + 12 * q^58 + 120 * q^60 - 38 * q^64 + 62 * q^65 + 32 * q^67 + 44 * q^71 + 26 * q^72 + 48 * q^74 - 144 * q^78 - 24 * q^79 + 108 * q^81 - 42 * q^85 + 72 * q^86 + 36 * q^88 - 16 * q^92 + 48 * q^93 + 2 * q^95 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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