LMFDB - Embedded newform 3871.2.a.i.1.4

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			1.4
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)

$f(q)$	$=$	\(q-2.35106 q^{2} +0.333714 q^{3} +3.52747 q^{4} +1.89309 q^{5} -0.784581 q^{6} -3.59117 q^{8} -2.88863 q^{9} -4.45076 q^{10} -0.703021 q^{11} +1.17717 q^{12} +1.83326 q^{13} +0.631750 q^{15} +1.38810 q^{16} +3.07498 q^{17} +6.79134 q^{18} +7.10178 q^{19} +6.67781 q^{20} +1.65284 q^{22} -4.90913 q^{23} -1.19842 q^{24} -1.41622 q^{25} -4.31009 q^{26} -1.96512 q^{27} +5.01477 q^{29} -1.48528 q^{30} +6.47747 q^{31} +3.91883 q^{32} -0.234608 q^{33} -7.22946 q^{34} -10.1896 q^{36} -7.47724 q^{37} -16.6967 q^{38} +0.611784 q^{39} -6.79839 q^{40} +8.80318 q^{41} -6.42517 q^{43} -2.47988 q^{44} -5.46844 q^{45} +11.5416 q^{46} +0.334805 q^{47} +0.463228 q^{48} +3.32962 q^{50} +1.02617 q^{51} +6.46676 q^{52} +6.88460 q^{53} +4.62011 q^{54} -1.33088 q^{55} +2.36996 q^{57} -11.7900 q^{58} -0.651763 q^{59} +2.22848 q^{60} -13.6365 q^{61} -15.2289 q^{62} -11.9896 q^{64} +3.47052 q^{65} +0.551577 q^{66} -4.86177 q^{67} +10.8469 q^{68} -1.63825 q^{69} -6.10525 q^{71} +10.3736 q^{72} +16.2784 q^{73} +17.5794 q^{74} -0.472613 q^{75} +25.0513 q^{76} -1.43834 q^{78} -1.00000 q^{79} +2.62779 q^{80} +8.01012 q^{81} -20.6968 q^{82} -4.82664 q^{83} +5.82121 q^{85} +15.1059 q^{86} +1.67350 q^{87} +2.52466 q^{88} +4.21687 q^{89} +12.8566 q^{90} -17.3168 q^{92} +2.16162 q^{93} -0.787145 q^{94} +13.4443 q^{95} +1.30777 q^{96} +12.3793 q^{97} +2.03077 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.35106	−1.66245	−0.831224	−	0.555937i	$-0.812359\pi$
$2$	−2.35106	−1.66245	−0.831224	+	0.555937i	$0.812359\pi$
$3$	0.333714	0.192670	0.0963350	−	0.995349i	$-0.469288\pi$
$3$	0.333714	0.192670	0.0963350	+	0.995349i	$0.469288\pi$
$4$	3.52747	1.76373
$5$	1.89309	0.846614	0.423307	−	0.905986i	$-0.360869\pi$
$5$	1.89309	0.846614	0.423307	+	0.905986i	$0.360869\pi$
$6$	−0.784581	−0.320304
$7$	0	0
$7$	0	0
$8$	−3.59117	−1.26967
$9$	−2.88863	−0.962878
$10$	−4.45076	−1.40745
$11$	−0.703021	−0.211969	−0.105984	−	0.994368i	$-0.533799\pi$
$11$	−0.703021	−0.211969	−0.105984	+	0.994368i	$0.533799\pi$
$12$	1.17717	0.339819
$13$	1.83326	0.508454	0.254227	−	0.967145i	$-0.418179\pi$
$13$	1.83326	0.508454	0.254227	+	0.967145i	$0.418179\pi$
$14$	0	0
$15$	0.631750	0.163117
$16$	1.38810	0.347025
$17$	3.07498	0.745793	0.372896	−	0.927873i	$-0.378365\pi$
$17$	3.07498	0.745793	0.372896	+	0.927873i	$0.378365\pi$
$18$	6.79134	1.60074
$19$	7.10178	1.62926	0.814630	−	0.579981i	$-0.196940\pi$
$19$	7.10178	1.62926	0.814630	+	0.579981i	$0.196940\pi$
$20$	6.67781	1.49320
$21$	0	0
$22$	1.65284	0.352387
$23$	−4.90913	−1.02362	−0.511812	−	0.859097i	$-0.671026\pi$
$23$	−4.90913	−1.02362	−0.511812	+	0.859097i	$0.671026\pi$
$24$	−1.19842	−0.244627
$25$	−1.41622	−0.283244
$26$	−4.31009	−0.845279
$27$	−1.96512	−0.378188
$28$	0	0
$29$	5.01477	0.931220	0.465610	−	0.884990i	$-0.345835\pi$
$29$	5.01477	0.931220	0.465610	+	0.884990i	$0.345835\pi$
$30$	−1.48528	−0.271174
$31$	6.47747	1.16339	0.581694	−	0.813408i	$-0.302390\pi$
$31$	6.47747	1.16339	0.581694	+	0.813408i	$0.302390\pi$
$32$	3.91883	0.692759
$33$	−0.234608	−0.0408400
$34$	−7.22946	−1.23984
$35$	0	0
$36$	−10.1896	−1.69826
$37$	−7.47724	−1.22925	−0.614625	−	0.788819i	$-0.710693\pi$
$37$	−7.47724	−1.22925	−0.614625	+	0.788819i	$0.710693\pi$
$38$	−16.6967	−2.70856
$39$	0.611784	0.0979639
$40$	−6.79839	−1.07492
$41$	8.80318	1.37483	0.687413	−	0.726267i	$-0.258746\pi$
$41$	8.80318	1.37483	0.687413	+	0.726267i	$0.258746\pi$
$42$	0	0
$43$	−6.42517	−0.979829	−0.489915	−	0.871770i	$-0.662972\pi$
$43$	−6.42517	−0.979829	−0.489915	+	0.871770i	$0.662972\pi$
$44$	−2.47988	−0.373857
$45$	−5.46844	−0.815187
$46$	11.5416	1.70172
$47$	0.334805	0.0488363	0.0244181	−	0.999702i	$-0.492227\pi$
$47$	0.334805	0.0488363	0.0244181	+	0.999702i	$0.492227\pi$
$48$	0.463228	0.0668612
$49$	0	0
$50$	3.32962	0.470879
$51$	1.02617	0.143692
$52$	6.46676	0.896778
$53$	6.88460	0.945673	0.472836	−	0.881150i	$-0.343230\pi$
$53$	6.88460	0.945673	0.472836	+	0.881150i	$0.343230\pi$
$54$	4.62011	0.628718
$55$	−1.33088	−0.179456
$56$	0	0
$57$	2.36996	0.313910
$58$	−11.7900	−1.54810
$59$	−0.651763	−0.0848523	−0.0424261	−	0.999100i	$-0.513509\pi$
$59$	−0.651763	−0.0848523	−0.0424261	+	0.999100i	$0.513509\pi$
$60$	2.22848	0.287695
$61$	−13.6365	−1.74597	−0.872986	−	0.487744i	$-0.837820\pi$
$61$	−13.6365	−1.74597	−0.872986	+	0.487744i	$0.837820\pi$
$62$	−15.2289	−1.93407
$63$	0	0
$64$	−11.9896	−1.49870
$65$	3.47052	0.430465
$66$	0.551577	0.0678944
$67$	−4.86177	−0.593959	−0.296980	−	0.954884i	$-0.595979\pi$
$67$	−4.86177	−0.593959	−0.296980	+	0.954884i	$0.595979\pi$
$68$	10.8469	1.31538
$69$	−1.63825	−0.197222
$70$	0	0
$71$	−6.10525	−0.724560	−0.362280	−	0.932069i	$-0.618002\pi$
$71$	−6.10525	−0.724560	−0.362280	+	0.932069i	$0.618002\pi$
$72$	10.3736	1.22254
$73$	16.2784	1.90524	0.952619	−	0.304165i	$-0.0983775\pi$
$73$	16.2784	1.90524	0.952619	+	0.304165i	$0.0983775\pi$
$74$	17.5794	2.04357
$75$	−0.472613	−0.0545727
$76$	25.0513	2.87358
$77$	0	0
$78$	−1.43834	−0.162860
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	2.62779	0.293796
$81$	8.01012	0.890013
$82$	−20.6968	−2.28558
$83$	−4.82664	−0.529793	−0.264896	−	0.964277i	$-0.585338\pi$
$83$	−4.82664	−0.529793	−0.264896	+	0.964277i	$0.585338\pi$
$84$	0	0
$85$	5.82121	0.631399
$86$	15.1059	1.62892
$87$	1.67350	0.179418
$88$	2.52466	0.269130
$89$	4.21687	0.446987	0.223494	−	0.974705i	$-0.428254\pi$
$89$	4.21687	0.446987	0.223494	+	0.974705i	$0.428254\pi$
$90$	12.8566	1.35521
$91$	0	0
$92$	−17.3168	−1.80540
$93$	2.16162	0.224150
$94$	−0.787145	−0.0811878
$95$	13.4443	1.37935
$96$	1.30777	0.133474
$97$	12.3793	1.25693	0.628463	−	0.777839i	$-0.283684\pi$
$97$	12.3793	1.25693	0.628463	+	0.777839i	$0.283684\pi$
$98$	0	0
$99$	2.03077	0.204100
$100$	−4.99568	−0.499568
$101$	11.5203	1.14632	0.573158	−	0.819445i	$-0.305718\pi$
$101$	11.5203	1.14632	0.573158	+	0.819445i	$0.305718\pi$
$102$	−2.41257	−0.238880
$103$	10.3962	1.02437	0.512184	−	0.858876i	$-0.328837\pi$
$103$	10.3962	1.02437	0.512184	+	0.858876i	$0.328837\pi$
$104$	−6.58353	−0.645569
$105$	0	0
$106$	−16.1861	−1.57213
$107$	14.4452	1.39647	0.698235	−	0.715868i	$-0.253969\pi$
$107$	14.4452	1.39647	0.698235	+	0.715868i	$0.253969\pi$
$108$	−6.93190	−0.667023
$109$	11.3519	1.08731	0.543657	−	0.839307i	$-0.317039\pi$
$109$	11.3519	1.08731	0.543657	+	0.839307i	$0.317039\pi$
$110$	3.12897	0.298336
$111$	−2.49526	−0.236840
$112$	0	0
$113$	1.27051	0.119520	0.0597600	−	0.998213i	$-0.480966\pi$
$113$	1.27051	0.119520	0.0597600	+	0.998213i	$0.480966\pi$
$114$	−5.57192	−0.521858
$115$	−9.29341	−0.866615
$116$	17.6894	1.64242
$117$	−5.29561	−0.489580
$118$	1.53233	0.141063
$119$	0	0
$120$	−2.26872	−0.207105
$121$	−10.5058	−0.955069
$122$	32.0602	2.90259
$123$	2.93775	0.264888
$124$	22.8491	2.05191
$125$	−12.1465	−1.08641
$126$	0	0
$127$	−0.277865	−0.0246565	−0.0123283	−	0.999924i	$-0.503924\pi$
$127$	−0.277865	−0.0246565	−0.0123283	+	0.999924i	$0.503924\pi$
$128$	20.3506	1.79875
$129$	−2.14417	−0.188784
$130$	−8.15938	−0.715625
$131$	2.96088	0.258693	0.129347	−	0.991599i	$-0.458712\pi$
$131$	2.96088	0.258693	0.129347	+	0.991599i	$0.458712\pi$
$132$	−0.827573	−0.0720310
$133$	0	0
$134$	11.4303	0.987427
$135$	−3.72015	−0.320179
$136$	−11.0428	−0.946910
$137$	−3.31852	−0.283520	−0.141760	−	0.989901i	$-0.545276\pi$
$137$	−3.31852	−0.283520	−0.141760	+	0.989901i	$0.545276\pi$
$138$	3.85161	0.327871
$139$	6.75969	0.573349	0.286675	−	0.958028i	$-0.407450\pi$
$139$	6.75969	0.573349	0.286675	+	0.958028i	$0.407450\pi$
$140$	0	0
$141$	0.111729	0.00940928
$142$	14.3538	1.20454
$143$	−1.28882	−0.107776
$144$	−4.00971	−0.334142
$145$	9.49340	0.788384
$146$	−38.2714	−3.16736
$147$	0	0
$148$	−26.3757	−2.16807
$149$	1.52867	0.125234	0.0626169	−	0.998038i	$-0.480055\pi$
$149$	1.52867	0.125234	0.0626169	+	0.998038i	$0.480055\pi$
$150$	1.11114	0.0907242
$151$	1.76178	0.143372	0.0716858	−	0.997427i	$-0.477162\pi$
$151$	1.76178	0.143372	0.0716858	+	0.997427i	$0.477162\pi$
$152$	−25.5037	−2.06862
$153$	−8.88250	−0.718108
$154$	0	0
$155$	12.2624	0.984941
$156$	2.15805	0.172782
$157$	−15.9448	−1.27253	−0.636265	−	0.771470i	$-0.719522\pi$
$157$	−15.9448	−1.27253	−0.636265	+	0.771470i	$0.719522\pi$
$158$	2.35106	0.187040
$159$	2.29749	0.182203
$160$	7.41869	0.586499
$161$	0	0
$162$	−18.8322	−1.47960
$163$	−3.74486	−0.293320	−0.146660	−	0.989187i	$-0.546852\pi$
$163$	−3.74486	−0.293320	−0.146660	+	0.989187i	$0.546852\pi$
$164$	31.0529	2.42483
$165$	−0.444134	−0.0345758
$166$	11.3477	0.880753
$167$	15.9785	1.23646	0.618228	−	0.785999i	$-0.287851\pi$
$167$	15.9785	1.23646	0.618228	+	0.785999i	$0.287851\pi$
$168$	0	0
$169$	−9.63917	−0.741474
$170$	−13.6860	−1.04967
$171$	−20.5144	−1.56878
$172$	−22.6646	−1.72816
$173$	−7.25400	−0.551512	−0.275756	−	0.961228i	$-0.588928\pi$
$173$	−7.25400	−0.551512	−0.275756	+	0.961228i	$0.588928\pi$
$174$	−3.93449	−0.298273
$175$	0	0
$176$	−0.975862	−0.0735584
$177$	−0.217502	−0.0163485
$178$	−9.91410	−0.743093
$179$	−11.7908	−0.881287	−0.440643	−	0.897682i	$-0.645250\pi$
$179$	−11.7908	−0.881287	−0.440643	+	0.897682i	$0.645250\pi$
$180$	−19.2897	−1.43777
$181$	−4.92956	−0.366412	−0.183206	−	0.983075i	$-0.558647\pi$
$181$	−4.92956	−0.366412	−0.183206	+	0.983075i	$0.558647\pi$
$182$	0	0
$183$	−4.55069	−0.336397
$184$	17.6295	1.29966
$185$	−14.1551	−1.04070
$186$	−5.08210	−0.372638
$187$	−2.16178	−0.158085
$188$	1.18101	0.0861342
$189$	0	0
$190$	−31.6083	−2.29311
$191$	−0.278538	−0.0201543	−0.0100771	−	0.999949i	$-0.503208\pi$
$191$	−0.278538	−0.0201543	−0.0100771	+	0.999949i	$0.503208\pi$
$192$	−4.00110	−0.288755
$193$	−5.76474	−0.414955	−0.207477	−	0.978240i	$-0.566525\pi$
$193$	−5.76474	−0.414955	−0.207477	+	0.978240i	$0.566525\pi$
$194$	−29.1044	−2.08958
$195$	1.15816	0.0829376
$196$	0	0
$197$	−9.61916	−0.685337	−0.342668	−	0.939456i	$-0.611331\pi$
$197$	−9.61916	−0.685337	−0.342668	+	0.939456i	$0.611331\pi$
$198$	−4.77446	−0.339306
$199$	9.34293	0.662303	0.331151	−	0.943578i	$-0.392563\pi$
$199$	9.34293	0.662303	0.331151	+	0.943578i	$0.392563\pi$
$200$	5.08588	0.359626
$201$	−1.62244	−0.114438
$202$	−27.0850	−1.90569
$203$	0	0
$204$	3.61977	0.253434
$205$	16.6652	1.16395
$206$	−24.4420	−1.70296
$207$	14.1807	0.985626
$208$	2.54474	0.176446
$209$	−4.99270	−0.345352
$210$	0	0
$211$	−2.50092	−0.172170	−0.0860851	−	0.996288i	$-0.527436\pi$
$211$	−2.50092	−0.172170	−0.0860851	+	0.996288i	$0.527436\pi$
$212$	24.2852	1.66792
$213$	−2.03741	−0.139601
$214$	−33.9615	−2.32156
$215$	−12.1634	−0.829537
$216$	7.05708	0.480173
$217$	0	0
$218$	−26.6889	−1.80760
$219$	5.43232	0.367082
$220$	−4.69464	−0.316512
$221$	5.63723	0.379201
$222$	5.86650	0.393734
$223$	0.717213	0.0480281	0.0240141	−	0.999712i	$-0.492355\pi$
$223$	0.717213	0.0480281	0.0240141	+	0.999712i	$0.492355\pi$
$224$	0	0
$225$	4.09095	0.272730
$226$	−2.98705	−0.198696
$227$	27.4469	1.82171	0.910857	−	0.412722i	$-0.135422\pi$
$227$	27.4469	1.82171	0.910857	+	0.412722i	$0.135422\pi$
$228$	8.35998	0.553653
$229$	12.4869	0.825155	0.412577	−	0.910923i	$-0.364629\pi$
$229$	12.4869	0.825155	0.412577	+	0.910923i	$0.364629\pi$
$230$	21.8493	1.44070
$231$	0	0
$232$	−18.0089	−1.18234
$233$	5.97118	0.391185	0.195592	−	0.980685i	$-0.437337\pi$
$233$	5.97118	0.391185	0.195592	+	0.980685i	$0.437337\pi$
$234$	12.4503	0.813901
$235$	0.633814	0.0413455
$236$	−2.29907	−0.149657
$237$	−0.333714	−0.0216771
$238$	0	0
$239$	18.5375	1.19909	0.599544	−	0.800342i	$-0.295349\pi$
$239$	18.5375	1.19909	0.599544	+	0.800342i	$0.295349\pi$
$240$	0.876931	0.0566057
$241$	26.7118	1.72066	0.860330	−	0.509738i	$-0.170258\pi$
$241$	26.7118	1.72066	0.860330	+	0.509738i	$0.170258\pi$
$242$	24.6996	1.58775
$243$	8.56845	0.549667
$244$	−48.1023	−3.07943
$245$	0	0
$246$	−6.90681	−0.440362
$247$	13.0194	0.828404
$248$	−23.2617	−1.47712
$249$	−1.61072	−0.102075
$250$	28.5570	1.80611
$251$	−5.62714	−0.355182	−0.177591	−	0.984104i	$-0.556830\pi$
$251$	−5.62714	−0.355182	−0.177591	+	0.984104i	$0.556830\pi$
$252$	0	0
$253$	3.45122	0.216976
$254$	0.653276	0.0409902
$255$	1.94262	0.121652
$256$	−23.8661	−1.49163
$257$	11.6657	0.727688	0.363844	−	0.931460i	$-0.381464\pi$
$257$	11.6657	0.727688	0.363844	+	0.931460i	$0.381464\pi$
$258$	5.04107	0.313843
$259$	0	0
$260$	12.2421	0.759225
$261$	−14.4858	−0.896651
$262$	−6.96120	−0.430064
$263$	−3.16523	−0.195177	−0.0975883	−	0.995227i	$-0.531113\pi$
$263$	−3.16523	−0.195177	−0.0975883	+	0.995227i	$0.531113\pi$
$264$	0.842517	0.0518533
$265$	13.0332	0.800620
$266$	0	0
$267$	1.40723	0.0861211
$268$	−17.1497	−1.04759
$269$	−6.95721	−0.424189	−0.212094	−	0.977249i	$-0.568028\pi$
$269$	−6.95721	−0.424189	−0.212094	+	0.977249i	$0.568028\pi$
$270$	8.74628	0.532281
$271$	−18.5226	−1.12517	−0.562583	−	0.826741i	$-0.690192\pi$
$271$	−18.5226	−1.12517	−0.562583	+	0.826741i	$0.690192\pi$
$272$	4.26838	0.258808
$273$	0	0
$274$	7.80203	0.471338
$275$	0.995633	0.0600389
$276$	−5.77887	−0.347847
$277$	−20.7817	−1.24865	−0.624325	−	0.781164i	$-0.714626\pi$
$277$	−20.7817	−1.24865	−0.624325	+	0.781164i	$0.714626\pi$
$278$	−15.8924	−0.953164
$279$	−18.7110	−1.12020
$280$	0	0
$281$	32.5498	1.94176	0.970880	−	0.239565i	$-0.0770047\pi$
$281$	32.5498	1.94176	0.970880	+	0.239565i	$0.0770047\pi$
$282$	−0.262681	−0.0156424
$283$	−15.1676	−0.901620	−0.450810	−	0.892620i	$-0.648865\pi$
$283$	−15.1676	−0.901620	−0.450810	+	0.892620i	$0.648865\pi$
$284$	−21.5361	−1.27793
$285$	4.48655	0.265760
$286$	3.03009	0.179173
$287$	0	0
$288$	−11.3201	−0.667042
$289$	−7.54449	−0.443793
$290$	−22.3195	−1.31065
$291$	4.13115	0.242172
$292$	57.4214	3.36033
$293$	9.08429	0.530710	0.265355	−	0.964151i	$-0.414511\pi$
$293$	9.08429	0.530710	0.265355	+	0.964151i	$0.414511\pi$
$294$	0	0
$295$	−1.23384	−0.0718371
$296$	26.8520	1.56074
$297$	1.38152	0.0801640
$298$	−3.59400	−0.208195
$299$	−8.99970	−0.520466
$300$	−1.66713	−0.0962517
$301$	0	0
$302$	−4.14204	−0.238348
$303$	3.84450	0.220861
$304$	9.85797	0.565393
$305$	−25.8151	−1.47817
$306$	20.8833	1.19382
$307$	4.20375	0.239921	0.119960	−	0.992779i	$-0.461723\pi$
$307$	4.20375	0.239921	0.119960	+	0.992779i	$0.461723\pi$
$308$	0	0
$309$	3.46936	0.197365
$310$	−28.8296	−1.63741
$311$	17.1938	0.974969	0.487484	−	0.873132i	$-0.337915\pi$
$311$	17.1938	0.974969	0.487484	+	0.873132i	$0.337915\pi$
$312$	−2.19702	−0.124382
$313$	−4.79410	−0.270979	−0.135489	−	0.990779i	$-0.543261\pi$
$313$	−4.79410	−0.270979	−0.135489	+	0.990779i	$0.543261\pi$
$314$	37.4870	2.11552
$315$	0	0
$316$	−3.52747	−0.198436
$317$	−7.16341	−0.402337	−0.201169	−	0.979557i	$-0.564474\pi$
$317$	−7.16341	−0.402337	−0.201169	+	0.979557i	$0.564474\pi$
$318$	−5.40153	−0.302903
$319$	−3.52549	−0.197389
$320$	−22.6974	−1.26882
$321$	4.82057	0.269058
$322$	0	0
$323$	21.8378	1.21509
$324$	28.2554	1.56975
$325$	−2.59630	−0.144017
$326$	8.80439	0.487630
$327$	3.78829	0.209493
$328$	−31.6137	−1.74557
$329$	0	0
$330$	1.04418	0.0574804
$331$	3.67598	0.202050	0.101025	−	0.994884i	$-0.467788\pi$
$331$	3.67598	0.202050	0.101025	+	0.994884i	$0.467788\pi$
$332$	−17.0258	−0.934414
$333$	21.5990	1.18362
$334$	−37.5665	−2.05554
$335$	−9.20375	−0.502854
$336$	0	0
$337$	13.1255	0.714994	0.357497	−	0.933914i	$-0.383630\pi$
$337$	13.1255	0.714994	0.357497	+	0.933914i	$0.383630\pi$
$338$	22.6622	1.23266
$339$	0.423989	0.0230279
$340$	20.5341	1.11362
$341$	−4.55380	−0.246602
$342$	48.2306	2.60801
$343$	0	0
$344$	23.0738	1.24406
$345$	−3.10134	−0.166971
$346$	17.0546	0.916859
$347$	6.37663	0.342315	0.171158	−	0.985244i	$-0.445249\pi$
$347$	6.37663	0.342315	0.171158	+	0.985244i	$0.445249\pi$
$348$	5.90322	0.316446
$349$	15.3035	0.819176	0.409588	−	0.912271i	$-0.365672\pi$
$349$	15.3035	0.819176	0.409588	+	0.912271i	$0.365672\pi$
$350$	0	0
$351$	−3.60257	−0.192291
$352$	−2.75502	−0.146843
$353$	18.8437	1.00295	0.501475	−	0.865172i	$-0.332791\pi$
$353$	18.8437	1.00295	0.501475	+	0.865172i	$0.332791\pi$
$354$	0.511361	0.0271785
$355$	−11.5578	−0.613423
$356$	14.8749	0.788367
$357$	0	0
$358$	27.7209	1.46509
$359$	13.1014	0.691467	0.345734	−	0.938333i	$-0.387630\pi$
$359$	13.1014	0.691467	0.345734	+	0.938333i	$0.387630\pi$
$360$	19.6381	1.03502
$361$	31.4353	1.65449
$362$	11.5897	0.609140
$363$	−3.50592	−0.184013
$364$	0	0
$365$	30.8164	1.61300
$366$	10.6989	0.559242
$367$	−29.1113	−1.51960	−0.759799	−	0.650157i	$-0.774703\pi$
$367$	−29.1113	−1.51960	−0.759799	+	0.650157i	$0.774703\pi$
$368$	−6.81436	−0.355223
$369$	−25.4292	−1.32379
$370$	33.2794	1.73011
$371$	0	0
$372$	7.62506	0.395341
$373$	−13.5929	−0.703815	−0.351908	−	0.936035i	$-0.614467\pi$
$373$	−13.5929	−0.703815	−0.351908	+	0.936035i	$0.614467\pi$
$374$	5.08246	0.262808
$375$	−4.05345	−0.209319
$376$	−1.20234	−0.0620059
$377$	9.19337	0.473482
$378$	0	0
$379$	38.0652	1.95528	0.977638	−	0.210295i	$-0.0674425\pi$
$379$	38.0652	1.95528	0.977638	+	0.210295i	$0.0674425\pi$
$380$	47.4243	2.43282
$381$	−0.0927274	−0.00475057
$382$	0.654858	0.0335055
$383$	−27.0353	−1.38144	−0.690720	−	0.723122i	$-0.742706\pi$
$383$	−27.0353	−1.38144	−0.690720	+	0.723122i	$0.742706\pi$
$384$	6.79127	0.346566
$385$	0	0
$386$	13.5532	0.689841
$387$	18.5600	0.943456
$388$	43.6676	2.21688
$389$	11.7462	0.595556	0.297778	−	0.954635i	$-0.403754\pi$
$389$	11.7462	0.595556	0.297778	+	0.954635i	$0.403754\pi$
$390$	−2.72290	−0.137880
$391$	−15.0955	−0.763412
$392$	0	0
$393$	0.988088	0.0498425
$394$	22.6152	1.13934
$395$	−1.89309	−0.0952516
$396$	7.16348	0.359978
$397$	36.4690	1.83032	0.915162	−	0.403086i	$-0.132062\pi$
$397$	36.4690	1.83032	0.915162	+	0.403086i	$0.132062\pi$
$398$	−21.9658	−1.10104
$399$	0	0
$400$	−1.96585	−0.0982927
$401$	33.8240	1.68909	0.844544	−	0.535486i	$-0.179871\pi$
$401$	33.8240	1.68909	0.844544	+	0.535486i	$0.179871\pi$
$402$	3.81445	0.190248
$403$	11.8749	0.591530
$404$	40.6376	2.02180
$405$	15.1638	0.753498
$406$	0	0
$407$	5.25666	0.260563
$408$	−3.68513	−0.182441
$409$	−1.16517	−0.0576137	−0.0288069	−	0.999585i	$-0.509171\pi$
$409$	−1.16517	−0.0576137	−0.0288069	+	0.999585i	$0.509171\pi$
$410$	−39.1808	−1.93500
$411$	−1.10744	−0.0546258
$412$	36.6722	1.80671
$413$	0	0
$414$	−33.3396	−1.63855
$415$	−9.13726	−0.448530
$416$	7.18423	0.352236
$417$	2.25580	0.110467
$418$	11.7381	0.574130
$419$	40.3909	1.97322	0.986612	−	0.163083i	$-0.0521440\pi$
$419$	40.3909	1.97322	0.986612	+	0.163083i	$0.0521440\pi$
$420$	0	0
$421$	28.5030	1.38915	0.694576	−	0.719419i	$-0.255592\pi$
$421$	28.5030	1.38915	0.694576	+	0.719419i	$0.255592\pi$
$422$	5.87980	0.286224
$423$	−0.967128	−0.0470234
$424$	−24.7237	−1.20069
$425$	−4.35485	−0.211241
$426$	4.79006	0.232079
$427$	0	0
$428$	50.9550	2.46300
$429$	−0.430097	−0.0207653
$430$	28.5969	1.37906
$431$	36.0228	1.73516	0.867578	−	0.497301i	$-0.165675\pi$
$431$	36.0228	1.73516	0.867578	+	0.497301i	$0.165675\pi$
$432$	−2.72778	−0.131240
$433$	−19.0901	−0.917410	−0.458705	−	0.888589i	$-0.651687\pi$
$433$	−19.0901	−0.917410	−0.458705	+	0.888589i	$0.651687\pi$
$434$	0	0
$435$	3.16808	0.151898
$436$	40.0434	1.91773
$437$	−34.8636	−1.66775
$438$	−12.7717	−0.610255
$439$	−37.8067	−1.80442	−0.902208	−	0.431302i	$-0.858054\pi$
$439$	−37.8067	−1.80442	−0.902208	+	0.431302i	$0.858054\pi$
$440$	4.77941	0.227849
$441$	0	0
$442$	−13.2535	−0.630403
$443$	−34.3769	−1.63330	−0.816649	−	0.577135i	$-0.804171\pi$
$443$	−34.3769	−1.63330	−0.816649	+	0.577135i	$0.804171\pi$
$444$	−8.80196	−0.417722
$445$	7.98290	0.378426
$446$	−1.68621	−0.0798443
$447$	0.510140	0.0241288
$448$	0	0
$449$	28.5444	1.34709	0.673547	−	0.739144i	$-0.264770\pi$
$449$	28.5444	1.34709	0.673547	+	0.739144i	$0.264770\pi$
$450$	−9.61805	−0.453399
$451$	−6.18882	−0.291420
$452$	4.48170	0.210801
$453$	0.587931	0.0276234
$454$	−64.5292	−3.02850
$455$	0	0
$456$	−8.51094	−0.398561
$457$	16.8828	0.789746	0.394873	−	0.918736i	$-0.370789\pi$
$457$	16.8828	0.789746	0.394873	+	0.918736i	$0.370789\pi$
$458$	−29.3573	−1.37178
$459$	−6.04271	−0.282050
$460$	−32.7822	−1.52848
$461$	−38.7527	−1.80489	−0.902446	−	0.430802i	$-0.858231\pi$
$461$	−38.7527	−1.80489	−0.902446	+	0.430802i	$0.858231\pi$
$462$	0	0
$463$	14.2136	0.660564	0.330282	−	0.943882i	$-0.392856\pi$
$463$	14.2136	0.660564	0.330282	+	0.943882i	$0.392856\pi$
$464$	6.96099	0.323156
$465$	4.09214	0.189769
$466$	−14.0386	−0.650324
$467$	−6.06817	−0.280802	−0.140401	−	0.990095i	$-0.544839\pi$
$467$	−6.06817	−0.280802	−0.140401	+	0.990095i	$0.544839\pi$
$468$	−18.6801	−0.863488
$469$	0	0
$470$	−1.49013	−0.0687347
$471$	−5.32099	−0.245178
$472$	2.34059	0.107734
$473$	4.51703	0.207693
$474$	0.784581	0.0360370
$475$	−10.0577	−0.461478
$476$	0	0
$477$	−19.8871	−0.910568
$478$	−43.5826	−1.99342
$479$	−3.63220	−0.165960	−0.0829798	−	0.996551i	$-0.526444\pi$
$479$	−3.63220	−0.165960	−0.0829798	+	0.996551i	$0.526444\pi$
$480$	2.47572	0.113001
$481$	−13.7077	−0.625018
$482$	−62.8010	−2.86051
$483$	0	0
$484$	−37.0587	−1.68449
$485$	23.4351	1.06413
$486$	−20.1449	−0.913792
$487$	−15.2264	−0.689976	−0.344988	−	0.938607i	$-0.612117\pi$
$487$	−15.2264	−0.689976	−0.344988	+	0.938607i	$0.612117\pi$
$488$	48.9709	2.21681
$489$	−1.24971	−0.0565140
$490$	0	0
$491$	−3.72789	−0.168237	−0.0841186	−	0.996456i	$-0.526807\pi$
$491$	−3.72789	−0.168237	−0.0841186	+	0.996456i	$0.526807\pi$
$492$	10.3628	0.467191
$493$	15.4203	0.694497
$494$	−30.6093	−1.37718
$495$	3.84443	0.172794
$496$	8.99137	0.403724
$497$	0	0
$498$	3.78689	0.169695
$499$	6.68732	0.299366	0.149683	−	0.988734i	$-0.452175\pi$
$499$	6.68732	0.299366	0.149683	+	0.988734i	$0.452175\pi$
$500$	−42.8463	−1.91614
$501$	5.33227	0.238228
$502$	13.2297	0.590472
$503$	2.65119	0.118211	0.0591054	−	0.998252i	$-0.481175\pi$
$503$	2.65119	0.118211	0.0591054	+	0.998252i	$0.481175\pi$
$504$	0	0
$505$	21.8090	0.970488
$506$	−8.11402	−0.360712
$507$	−3.21673	−0.142860
$508$	−0.980159	−0.0434875
$509$	33.9270	1.50379	0.751894	−	0.659285i	$-0.229141\pi$
$509$	33.9270	1.50379	0.751894	+	0.659285i	$0.229141\pi$
$510$	−4.56721	−0.202240
$511$	0	0
$512$	15.4095	0.681010
$513$	−13.9559	−0.616166
$514$	−27.4268	−1.20974
$515$	19.6809	0.867244
$516$	−7.56349	−0.332964
$517$	−0.235375	−0.0103518
$518$	0	0
$519$	−2.42076	−0.106260
$520$	−12.4632	−0.546548
$521$	−16.4363	−0.720089	−0.360045	−	0.932935i	$-0.617239\pi$
$521$	−16.4363	−0.720089	−0.360045	+	0.932935i	$0.617239\pi$
$522$	34.0570	1.49064
$523$	27.7912	1.21522	0.607611	−	0.794234i	$-0.292128\pi$
$523$	27.7912	1.21522	0.607611	+	0.794234i	$0.292128\pi$
$524$	10.4444	0.456266
$525$	0	0
$526$	7.44164	0.324471
$527$	19.9181	0.867646
$528$	−0.325659	−0.0141725
$529$	1.09957	0.0478075
$530$	−30.6417	−1.33099
$531$	1.88270	0.0817024
$532$	0	0
$533$	16.1385	0.699036
$534$	−3.30848	−0.143172
$535$	27.3460	1.18227
$536$	17.4594	0.754132
$537$	−3.93476	−0.169798
$538$	16.3568	0.705192
$539$	0	0
$540$	−13.1227	−0.564711
$541$	−3.89180	−0.167321	−0.0836607	−	0.996494i	$-0.526661\pi$
$541$	−3.89180	−0.167321	−0.0836607	+	0.996494i	$0.526661\pi$
$542$	43.5476	1.87053
$543$	−1.64507	−0.0705965
$544$	12.0503	0.516654
$545$	21.4901	0.920536
$546$	0	0
$547$	−27.9363	−1.19447	−0.597235	−	0.802066i	$-0.703734\pi$
$547$	−27.9363	−1.19447	−0.597235	+	0.802066i	$0.703734\pi$
$548$	−11.7060	−0.500054
$549$	39.3908	1.68116
$550$	−2.34079	−0.0998116
$551$	35.6138	1.51720
$552$	5.88322	0.250406
$553$	0	0
$554$	48.8589	2.07582
$555$	−4.72375	−0.200512
$556$	23.8446	1.01124
$557$	7.59365	0.321753	0.160877	−	0.986975i	$-0.448568\pi$
$557$	7.59365	0.321753	0.160877	+	0.986975i	$0.448568\pi$
$558$	43.9907	1.86228
$559$	−11.7790	−0.498198
$560$	0	0
$561$	−0.721416	−0.0304582
$562$	−76.5265	−3.22808
$563$	−5.65920	−0.238507	−0.119254	−	0.992864i	$-0.538050\pi$
$563$	−5.65920	−0.238507	−0.119254	+	0.992864i	$0.538050\pi$
$564$	0.394121	0.0165955
$565$	2.40520	0.101187
$566$	35.6599	1.49890
$567$	0	0
$568$	21.9250	0.919951
$569$	−32.5994	−1.36664	−0.683319	−	0.730120i	$-0.739464\pi$
$569$	−32.5994	−1.36664	−0.683319	+	0.730120i	$0.739464\pi$
$570$	−10.5481	−0.441813
$571$	−16.9712	−0.710223	−0.355111	−	0.934824i	$-0.615557\pi$
$571$	−16.9712	−0.710223	−0.355111	+	0.934824i	$0.615557\pi$
$572$	−4.54627	−0.190089
$573$	−0.0929521	−0.00388313
$574$	0	0
$575$	6.95242	0.289936
$576$	34.6336	1.44307
$577$	5.39249	0.224492	0.112246	−	0.993680i	$-0.464196\pi$
$577$	5.39249	0.224492	0.112246	+	0.993680i	$0.464196\pi$
$578$	17.7375	0.737783
$579$	−1.92377	−0.0799494
$580$	33.4877	1.39050
$581$	0	0
$582$	−9.71256	−0.402598
$583$	−4.84002	−0.200453
$584$	−58.4583	−2.41902
$585$	−10.0251	−0.414485
$586$	−21.3577	−0.882277
$587$	26.7043	1.10220	0.551102	−	0.834438i	$-0.314207\pi$
$587$	26.7043	1.10220	0.551102	+	0.834438i	$0.314207\pi$
$588$	0	0
$589$	46.0016	1.89546
$590$	2.90084	0.119426
$591$	−3.21005	−0.132044
$592$	−10.3791	−0.426580
$593$	25.2557	1.03713	0.518564	−	0.855039i	$-0.326467\pi$
$593$	25.2557	1.03713	0.518564	+	0.855039i	$0.326467\pi$
$594$	−3.24804	−0.133269
$595$	0	0
$596$	5.39235	0.220879
$597$	3.11787	0.127606
$598$	21.1588	0.865248
$599$	12.0680	0.493086	0.246543	−	0.969132i	$-0.420705\pi$
$599$	12.0680	0.493086	0.246543	+	0.969132i	$0.420705\pi$
$600$	1.69723	0.0692892
$601$	−15.8026	−0.644603	−0.322302	−	0.946637i	$-0.604457\pi$
$601$	−15.8026	−0.644603	−0.322302	+	0.946637i	$0.604457\pi$
$602$	0	0
$603$	14.0439	0.571911
$604$	6.21462	0.252869
$605$	−19.8883	−0.808575
$606$	−9.03864	−0.367170
$607$	−16.1703	−0.656334	−0.328167	−	0.944620i	$-0.606431\pi$
$607$	−16.1703	−0.656334	−0.328167	+	0.944620i	$0.606431\pi$
$608$	27.8307	1.12868
$609$	0	0
$610$	60.6927	2.45737
$611$	0.613783	0.0248310
$612$	−31.3327	−1.26655
$613$	−44.3319	−1.79055	−0.895275	−	0.445515i	$-0.853021\pi$
$613$	−44.3319	−1.79055	−0.895275	+	0.445515i	$0.853021\pi$
$614$	−9.88327	−0.398856
$615$	5.56141	0.224258
$616$	0	0
$617$	−24.9956	−1.00629	−0.503143	−	0.864203i	$-0.667823\pi$
$617$	−24.9956	−1.00629	−0.503143	+	0.864203i	$0.667823\pi$
$618$	−8.15666	−0.328109
$619$	−38.1139	−1.53193	−0.765964	−	0.642884i	$-0.777738\pi$
$619$	−38.1139	−1.53193	−0.765964	+	0.642884i	$0.777738\pi$
$620$	43.2553	1.73717
$621$	9.64704	0.387122
$622$	−40.4235	−1.62084
$623$	0	0
$624$	0.849217	0.0339959
$625$	−15.9132	−0.636529
$626$	11.2712	0.450488
$627$	−1.66613	−0.0665390
$628$	−56.2446	−2.24441
$629$	−22.9924	−0.916766
$630$	0	0
$631$	−14.7258	−0.586224	−0.293112	−	0.956078i	$-0.594691\pi$
$631$	−14.7258	−0.586224	−0.293112	+	0.956078i	$0.594691\pi$
$632$	3.59117	0.142849
$633$	−0.834592	−0.0331720
$634$	16.8416	0.668865
$635$	−0.526022	−0.0208746
$636$	8.10432	0.321357
$637$	0	0
$638$	8.28862	0.328150
$639$	17.6358	0.697663
$640$	38.5254	1.52285
$641$	31.0164	1.22507	0.612537	−	0.790442i	$-0.290149\pi$
$641$	31.0164	1.22507	0.612537	+	0.790442i	$0.290149\pi$
$642$	−11.3334	−0.447295
$643$	3.76583	0.148510	0.0742550	−	0.997239i	$-0.476342\pi$
$643$	3.76583	0.148510	0.0742550	+	0.997239i	$0.476342\pi$
$644$	0	0
$645$	−4.05910	−0.159827
$646$	−51.3420	−2.02002
$647$	42.1849	1.65846	0.829231	−	0.558907i	$-0.188779\pi$
$647$	42.1849	1.65846	0.829231	+	0.558907i	$0.188779\pi$
$648$	−28.7657	−1.13002
$649$	0.458203	0.0179860
$650$	6.10404	0.239420
$651$	0	0
$652$	−13.2099	−0.517339
$653$	−4.68498	−0.183337	−0.0916686	−	0.995790i	$-0.529220\pi$
$653$	−4.68498	−0.183337	−0.0916686	+	0.995790i	$0.529220\pi$
$654$	−8.90648	−0.348271
$655$	5.60521	0.219014
$656$	12.2197	0.477098
$657$	−47.0223	−1.83451
$658$	0	0
$659$	25.4657	0.992001	0.496001	−	0.868322i	$-0.334801\pi$
$659$	25.4657	0.992001	0.496001	+	0.868322i	$0.334801\pi$
$660$	−1.56667	−0.0609824
$661$	−1.42535	−0.0554396	−0.0277198	−	0.999616i	$-0.508825\pi$
$661$	−1.42535	−0.0554396	−0.0277198	+	0.999616i	$0.508825\pi$
$662$	−8.64243	−0.335898
$663$	1.88123	0.0730607
$664$	17.3333	0.672662
$665$	0	0
$666$	−50.7805	−1.96771
$667$	−24.6182	−0.953219
$668$	56.3638	2.18078
$669$	0.239344	0.00925358
$670$	21.6385	0.835970
$671$	9.58673	0.370092
$672$	0	0
$673$	44.0238	1.69699	0.848497	−	0.529201i	$-0.177508\pi$
$673$	44.0238	1.69699	0.848497	+	0.529201i	$0.177508\pi$
$674$	−30.8589	−1.18864
$675$	2.78305	0.107119
$676$	−34.0019	−1.30776
$677$	−1.58863	−0.0610559	−0.0305280	−	0.999534i	$-0.509719\pi$
$677$	−1.58863	−0.0610559	−0.0305280	+	0.999534i	$0.509719\pi$
$678$	−0.996822	−0.0382827
$679$	0	0
$680$	−20.9049	−0.801667
$681$	9.15942	0.350990
$682$	10.7062	0.409963
$683$	−35.3892	−1.35413	−0.677066	−	0.735922i	$-0.736749\pi$
$683$	−35.3892	−1.35413	−0.677066	+	0.735922i	$0.736749\pi$
$684$	−72.3641	−2.76691
$685$	−6.28225	−0.240032
$686$	0	0
$687$	4.16704	0.158983
$688$	−8.91877	−0.340025
$689$	12.6213	0.480831
$690$	7.29144	0.277580
$691$	16.9553	0.645012	0.322506	−	0.946567i	$-0.395475\pi$
$691$	16.9553	0.645012	0.322506	+	0.946567i	$0.395475\pi$
$692$	−25.5883	−0.972720
$693$	0	0
$694$	−14.9918	−0.569081
$695$	12.7967	0.485406
$696$	−6.00982	−0.227802
$697$	27.0696	1.02533
$698$	−35.9793	−1.36184
$699$	1.99267	0.0753696
$700$	0	0
$701$	39.9698	1.50964	0.754819	−	0.655933i	$-0.227725\pi$
$701$	39.9698	1.50964	0.754819	+	0.655933i	$0.227725\pi$
$702$	8.46986	0.319674
$703$	−53.1017	−2.00277
$704$	8.42894	0.317678
$705$	0.211513	0.00796603
$706$	−44.3027	−1.66735
$707$	0	0
$708$	−0.767233	−0.0288344
$709$	−4.67881	−0.175716	−0.0878581	−	0.996133i	$-0.528002\pi$
$709$	−4.67881	−0.175716	−0.0878581	+	0.996133i	$0.528002\pi$
$710$	27.1730	1.01978
$711$	2.88863	0.108332
$712$	−15.1435	−0.567526
$713$	−31.7988	−1.19087
$714$	0	0
$715$	−2.43985	−0.0912451
$716$	−41.5917	−1.55436
$717$	6.18621	0.231028
$718$	−30.8022	−1.14953
$719$	5.78026	0.215567	0.107784	−	0.994174i	$-0.465625\pi$
$719$	5.78026	0.215567	0.107784	+	0.994174i	$0.465625\pi$
$720$	−7.59073	−0.282890
$721$	0	0
$722$	−73.9061	−2.75050
$723$	8.91411	0.331519
$724$	−17.3889	−0.646253
$725$	−7.10202	−0.263763
$726$	8.24262	0.305912
$727$	−4.01519	−0.148915	−0.0744576	−	0.997224i	$-0.523723\pi$
$727$	−4.01519	−0.148915	−0.0744576	+	0.997224i	$0.523723\pi$
$728$	0	0
$729$	−21.1709	−0.784109
$730$	−72.4510	−2.68153
$731$	−19.7573	−0.730749
$732$	−16.0524	−0.593314
$733$	44.7733	1.65374	0.826870	−	0.562393i	$-0.190119\pi$
$733$	44.7733	1.65374	0.826870	+	0.562393i	$0.190119\pi$
$734$	68.4424	2.52625
$735$	0	0
$736$	−19.2381	−0.709125
$737$	3.41792	0.125901
$738$	59.7854	2.20073
$739$	−11.0827	−0.407684	−0.203842	−	0.979004i	$-0.565343\pi$
$739$	−11.0827	−0.407684	−0.203842	+	0.979004i	$0.565343\pi$
$740$	−49.9316	−1.83552
$741$	4.34476	0.159609
$742$	0	0
$743$	6.02279	0.220955	0.110477	−	0.993879i	$-0.464762\pi$
$743$	6.02279	0.220955	0.110477	+	0.993879i	$0.464762\pi$
$744$	−7.76275	−0.284596
$745$	2.89391	0.106025
$746$	31.9578	1.17006
$747$	13.9424	0.510126
$748$	−7.62560	−0.278820
$749$	0	0
$750$	9.52989	0.347982
$751$	35.5872	1.29859	0.649297	−	0.760535i	$-0.275063\pi$
$751$	35.5872	1.29859	0.649297	+	0.760535i	$0.275063\pi$
$752$	0.464742	0.0169474
$753$	−1.87786	−0.0684329
$754$	−21.6141	−0.787140
$755$	3.33520	0.121380
$756$	0	0
$757$	28.4248	1.03312	0.516558	−	0.856252i	$-0.327213\pi$
$757$	28.4248	1.03312	0.516558	+	0.856252i	$0.327213\pi$
$758$	−89.4934	−3.25055
$759$	1.15172	0.0418049
$760$	−48.2807	−1.75132
$761$	29.0380	1.05263	0.526313	−	0.850291i	$-0.323574\pi$
$761$	29.0380	1.05263	0.526313	+	0.850291i	$0.323574\pi$
$762$	0.218007	0.00789758
$763$	0	0
$764$	−0.982534	−0.0355468
$765$	−16.8153	−0.607960
$766$	63.5616	2.29657
$767$	−1.19485	−0.0431435
$768$	−7.96447	−0.287393
$769$	−28.6495	−1.03313	−0.516563	−	0.856249i	$-0.672789\pi$
$769$	−28.6495	−1.03313	−0.516563	+	0.856249i	$0.672789\pi$
$770$	0	0
$771$	3.89302	0.140204
$772$	−20.3349	−0.731870
$773$	24.5726	0.883813	0.441907	−	0.897061i	$-0.354302\pi$
$773$	24.5726	0.883813	0.441907	+	0.897061i	$0.354302\pi$
$774$	−43.6355	−1.56845
$775$	−9.17353	−0.329523
$776$	−44.4561	−1.59588
$777$	0	0
$778$	−27.6160	−0.990081
$779$	62.5182	2.23995
$780$	4.08538	0.146280
$781$	4.29212	0.153584
$782$	35.4904	1.26913
$783$	−9.85463	−0.352176
$784$	0	0
$785$	−30.1848	−1.07734
$786$	−2.32305	−0.0828605
$787$	−50.5704	−1.80264	−0.901319	−	0.433155i	$-0.857400\pi$
$787$	−50.5704	−1.80264	−0.901319	+	0.433155i	$0.857400\pi$
$788$	−33.9313	−1.20875
$789$	−1.05628	−0.0376047
$790$	4.45076	0.158351
$791$	0	0
$792$	−7.29283	−0.259140
$793$	−24.9992	−0.887747
$794$	−85.7406	−3.04282
$795$	4.34935	0.154255
$796$	32.9569	1.16813
$797$	18.3867	0.651289	0.325645	−	0.945492i	$-0.394419\pi$
$797$	18.3867	0.651289	0.325645	+	0.945492i	$0.394419\pi$
$798$	0	0
$799$	1.02952	0.0364217
$800$	−5.54994	−0.196220
$801$	−12.1810	−0.430394
$802$	−79.5221	−2.80802
$803$	−11.4440	−0.403851
$804$	−5.72311	−0.201839
$805$	0	0
$806$	−27.9185	−0.983387
$807$	−2.32172	−0.0817285
$808$	−41.3714	−1.45544
$809$	−44.9845	−1.58157	−0.790785	−	0.612094i	$-0.790327\pi$
$809$	−44.9845	−1.58157	−0.790785	+	0.612094i	$0.790327\pi$
$810$	−35.6511	−1.25265
$811$	39.7104	1.39442	0.697210	−	0.716867i	$-0.254424\pi$
$811$	39.7104	1.39442	0.697210	+	0.716867i	$0.254424\pi$
$812$	0	0
$813$	−6.18125	−0.216786
$814$	−12.3587	−0.433172
$815$	−7.08935	−0.248329
$816$	1.42442	0.0498646
$817$	−45.6301	−1.59640
$818$	2.73937	0.0957798
$819$	0	0
$820$	58.7859	2.05289
$821$	30.4242	1.06181	0.530906	−	0.847431i	$-0.321852\pi$
$821$	30.4242	1.06181	0.530906	+	0.847431i	$0.321852\pi$
$822$	2.60365	0.0908126
$823$	−48.4546	−1.68902	−0.844510	−	0.535540i	$-0.820108\pi$
$823$	−48.4546	−1.68902	−0.844510	+	0.535540i	$0.820108\pi$
$824$	−37.3345	−1.30061
$825$	0.332257	0.0115677
$826$	0	0
$827$	−41.0783	−1.42843	−0.714216	−	0.699926i	$-0.753217\pi$
$827$	−41.0783	−1.42843	−0.714216	+	0.699926i	$0.753217\pi$
$828$	50.0219	1.73838
$829$	17.6480	0.612941	0.306471	−	0.951880i	$-0.400852\pi$
$829$	17.6480	0.612941	0.306471	+	0.951880i	$0.400852\pi$
$830$	21.4822	0.745658
$831$	−6.93515	−0.240578
$832$	−21.9800	−0.762020
$833$	0	0
$834$	−5.30353	−0.183646
$835$	30.2488	1.04680
$836$	−17.6116	−0.609110
$837$	−12.7290	−0.439979
$838$	−94.9613	−3.28038
$839$	−0.952084	−0.0328696	−0.0164348	−	0.999865i	$-0.505232\pi$
$839$	−0.952084	−0.0328696	−0.0164348	+	0.999865i	$0.505232\pi$
$840$	0	0
$841$	−3.85208	−0.132830
$842$	−67.0122	−2.30939
$843$	10.8623	0.374119
$844$	−8.82191	−0.303662
$845$	−18.2478	−0.627743
$846$	2.27377	0.0781739
$847$	0	0
$848$	9.55650	0.328172
$849$	−5.06164	−0.173715
$850$	10.2385	0.351178
$851$	36.7068	1.25829
$852$	−7.18689	−0.246219
$853$	36.9306	1.26448	0.632240	−	0.774773i	$-0.282136\pi$
$853$	36.9306	1.26448	0.632240	+	0.774773i	$0.282136\pi$
$854$	0	0
$855$	−38.8356	−1.32815
$856$	−51.8751	−1.77306
$857$	22.7125	0.775843	0.387922	−	0.921692i	$-0.373193\pi$
$857$	22.7125	0.775843	0.387922	+	0.921692i	$0.373193\pi$
$858$	1.01118	0.0345212
$859$	−3.98918	−0.136109	−0.0680545	−	0.997682i	$-0.521679\pi$
$859$	−3.98918	−0.136109	−0.0680545	+	0.997682i	$0.521679\pi$
$860$	−42.9060	−1.46308
$861$	0	0
$862$	−84.6916	−2.88461
$863$	−12.7373	−0.433582	−0.216791	−	0.976218i	$-0.569559\pi$
$863$	−12.7373	−0.433582	−0.216791	+	0.976218i	$0.569559\pi$
$864$	−7.70098	−0.261993
$865$	−13.7325	−0.466918
$866$	44.8818	1.52515
$867$	−2.51770	−0.0855057
$868$	0	0
$869$	0.703021	0.0238484
$870$	−7.44834	−0.252522
$871$	−8.91287	−0.302001
$872$	−40.7665	−1.38053
$873$	−35.7592	−1.21027
$874$	81.9662	2.77255
$875$	0	0
$876$	19.1623	0.647436
$877$	−16.8880	−0.570267	−0.285134	−	0.958488i	$-0.592038\pi$
$877$	−16.8880	−0.570267	−0.285134	+	0.958488i	$0.592038\pi$
$878$	88.8857	2.99975
$879$	3.03156	0.102252
$880$	−1.84739	−0.0622756
$881$	−1.39323	−0.0469390	−0.0234695	−	0.999725i	$-0.507471\pi$
$881$	−1.39323	−0.0469390	−0.0234695	+	0.999725i	$0.507471\pi$
$882$	0	0
$883$	−29.2134	−0.983110	−0.491555	−	0.870846i	$-0.663571\pi$
$883$	−29.2134	−0.983110	−0.491555	+	0.870846i	$0.663571\pi$
$884$	19.8852	0.668811
$885$	−0.411751	−0.0138409
$886$	80.8221	2.71527
$887$	18.7513	0.629606	0.314803	−	0.949157i	$-0.398062\pi$
$887$	18.7513	0.629606	0.314803	+	0.949157i	$0.398062\pi$
$888$	8.96090	0.300708
$889$	0	0
$890$	−18.7683	−0.629114
$891$	−5.63128	−0.188655
$892$	2.52995	0.0847088
$893$	2.37771	0.0795670
$894$	−1.19937	−0.0401129
$895$	−22.3210	−0.746110
$896$	0	0
$897$	−3.00333	−0.100278
$898$	−67.1096	−2.23948
$899$	32.4830	1.08337
$900$	14.4307	0.481023
$901$	21.1700	0.705276
$902$	14.5503	0.484471
$903$	0	0
$904$	−4.56263	−0.151751
$905$	−9.33209	−0.310209
$906$	−1.38226	−0.0459225
$907$	−27.3052	−0.906655	−0.453327	−	0.891344i	$-0.649763\pi$
$907$	−27.3052	−0.906655	−0.453327	+	0.891344i	$0.649763\pi$
$908$	96.8180	3.21302
$909$	−33.2780	−1.10376
$910$	0	0
$911$	−37.1703	−1.23151	−0.615753	−	0.787939i	$-0.711148\pi$
$911$	−37.1703	−1.23151	−0.615753	+	0.787939i	$0.711148\pi$
$912$	3.28974	0.108934
$913$	3.39323	0.112300
$914$	−39.6925	−1.31291
$915$	−8.61485	−0.284798
$916$	44.0470	1.45535
$917$	0	0
$918$	14.2068	0.468893
$919$	−19.1946	−0.633171	−0.316586	−	0.948564i	$-0.602536\pi$
$919$	−19.1946	−0.633171	−0.316586	+	0.948564i	$0.602536\pi$
$920$	33.3742	1.10031
$921$	1.40285	0.0462256
$922$	91.1098	3.00054
$923$	−11.1925	−0.368405
$924$	0	0
$925$	10.5894	0.348178
$926$	−33.4171	−1.09815
$927$	−30.0308	−0.986341
$928$	19.6521	0.645110
$929$	−31.0110	−1.01744	−0.508718	−	0.860933i	$-0.669880\pi$
$929$	−31.0110	−1.01744	−0.508718	+	0.860933i	$0.669880\pi$
$930$	−9.62086	−0.315480
$931$	0	0
$932$	21.0631	0.689946
$933$	5.73780	0.187847
$934$	14.2666	0.466818
$935$	−4.09243	−0.133837
$936$	19.0174	0.621604
$937$	−8.44101	−0.275756	−0.137878	−	0.990449i	$-0.544028\pi$
$937$	−8.44101	−0.275756	−0.137878	+	0.990449i	$0.544028\pi$
$938$	0	0
$939$	−1.59986	−0.0522095
$940$	2.23576	0.0729224
$941$	29.9788	0.977282	0.488641	−	0.872485i	$-0.337493\pi$
$941$	29.9788	0.977282	0.488641	+	0.872485i	$0.337493\pi$
$942$	12.5100	0.407596
$943$	−43.2160	−1.40730
$944$	−0.904711	−0.0294458
$945$	0	0
$946$	−10.6198	−0.345279
$947$	35.4147	1.15082	0.575411	−	0.817864i	$-0.304842\pi$
$947$	35.4147	1.15082	0.575411	+	0.817864i	$0.304842\pi$
$948$	−1.17717	−0.0382326
$949$	29.8424	0.968727
$950$	23.6462	0.767184
$951$	−2.39053	−0.0775183
$952$	0	0
$953$	−33.3026	−1.07878	−0.539389	−	0.842057i	$-0.681345\pi$
$953$	−33.3026	−1.07878	−0.539389	+	0.842057i	$0.681345\pi$
$954$	46.7557	1.51377
$955$	−0.527296	−0.0170629
$956$	65.3903	2.11487
$957$	−1.17651	−0.0380310
$958$	8.53951	0.275899
$959$	0	0
$960$	−7.57443	−0.244464
$961$	10.9576	0.353472
$962$	32.2276	1.03906
$963$	−41.7269	−1.34463
$964$	94.2251	3.03479
$965$	−10.9131	−0.351307
$966$	0	0
$967$	−24.6945	−0.794122	−0.397061	−	0.917792i	$-0.629970\pi$
$967$	−24.6945	−0.794122	−0.397061	+	0.917792i	$0.629970\pi$
$968$	37.7279	1.21262
$969$	7.28760	0.234111
$970$	−55.0972	−1.76906
$971$	2.23403	0.0716935	0.0358467	−	0.999357i	$-0.488587\pi$
$971$	2.23403	0.0716935	0.0358467	+	0.999357i	$0.488587\pi$
$972$	30.2250	0.969466
$973$	0	0
$974$	35.7982	1.14705
$975$	−0.866422	−0.0277477
$976$	−18.9288	−0.605895
$977$	44.1258	1.41171	0.705855	−	0.708357i	$-0.250563\pi$
$977$	44.1258	1.41171	0.705855	+	0.708357i	$0.250563\pi$
$978$	2.93815	0.0939517
$979$	−2.96455	−0.0947474
$980$	0	0
$981$	−32.7915	−1.04695
$982$	8.76448	0.279686
$983$	−44.6809	−1.42510	−0.712550	−	0.701621i	$-0.752460\pi$
$983$	−44.6809	−1.42510	−0.712550	+	0.701621i	$0.752460\pi$
$984$	−10.5499	−0.336320
$985$	−18.2099	−0.580216
$986$	−36.2541	−1.15456
$987$	0	0
$988$	45.9255	1.46108
$989$	31.5420	1.00298
$990$	−9.03846	−0.287261
$991$	−55.6872	−1.76896	−0.884481	−	0.466577i	$-0.845487\pi$
$991$	−55.6872	−1.76896	−0.884481	+	0.466577i	$0.845487\pi$
$992$	25.3841	0.805947
$993$	1.22673	0.0389290
$994$	0	0
$995$	17.6870	0.560715
$996$	−5.68176	−0.180034
$997$	−3.60002	−0.114014	−0.0570069	−	0.998374i	$-0.518156\pi$
$997$	−3.60002	−0.114014	−0.0570069	+	0.998374i	$0.518156\pi$
$998$	−15.7223	−0.497680
$999$	14.6937	0.464888

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3871.2.a.i.1.4	yes	24
7.6	odd	2	inner	3871.2.a.i.1.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3871.2.a.i.1.3	✓	24	7.6	odd	2	inner
3871.2.a.i.1.4	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)

\(f(q)\)	\(=\)	\(q-2.35106 q^{2} +0.333714 q^{3} +3.52747 q^{4} +1.89309 q^{5} -0.784581 q^{6} -3.59117 q^{8} -2.88863 q^{9} -4.45076 q^{10} -0.703021 q^{11} +1.17717 q^{12} +1.83326 q^{13} +0.631750 q^{15} +1.38810 q^{16} +3.07498 q^{17} +6.79134 q^{18} +7.10178 q^{19} +6.67781 q^{20} +1.65284 q^{22} -4.90913 q^{23} -1.19842 q^{24} -1.41622 q^{25} -4.31009 q^{26} -1.96512 q^{27} +5.01477 q^{29} -1.48528 q^{30} +6.47747 q^{31} +3.91883 q^{32} -0.234608 q^{33} -7.22946 q^{34} -10.1896 q^{36} -7.47724 q^{37} -16.6967 q^{38} +0.611784 q^{39} -6.79839 q^{40} +8.80318 q^{41} -6.42517 q^{43} -2.47988 q^{44} -5.46844 q^{45} +11.5416 q^{46} +0.334805 q^{47} +0.463228 q^{48} +3.32962 q^{50} +1.02617 q^{51} +6.46676 q^{52} +6.88460 q^{53} +4.62011 q^{54} -1.33088 q^{55} +2.36996 q^{57} -11.7900 q^{58} -0.651763 q^{59} +2.22848 q^{60} -13.6365 q^{61} -15.2289 q^{62} -11.9896 q^{64} +3.47052 q^{65} +0.551577 q^{66} -4.86177 q^{67} +10.8469 q^{68} -1.63825 q^{69} -6.10525 q^{71} +10.3736 q^{72} +16.2784 q^{73} +17.5794 q^{74} -0.472613 q^{75} +25.0513 q^{76} -1.43834 q^{78} -1.00000 q^{79} +2.62779 q^{80} +8.01012 q^{81} -20.6968 q^{82} -4.82664 q^{83} +5.82121 q^{85} +15.1059 q^{86} +1.67350 q^{87} +2.52466 q^{88} +4.21687 q^{89} +12.8566 q^{90} -17.3168 q^{92} +2.16162 q^{93} -0.787145 q^{94} +13.4443 q^{95} +1.30777 q^{96} +12.3793 q^{97} +2.03077 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{2} + 28 q^{4} + 18 q^{8} + 50 q^{9} + 14 q^{11} + 18 q^{15} + 36 q^{16} + 10 q^{18} - 12 q^{22} + 22 q^{23} + 56 q^{25} + 36 q^{29} + 14 q^{30} + 30 q^{32} + 8 q^{36} + 24 q^{37} + 56 q^{39}+ \cdots + 6 q^{99}+O(q^{100}) \) 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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