LMFDB - Embedded newform 3871.2.a.i.1.12

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			1.12
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)

$f(q)$	$=$	\(q-0.0847728 q^{2} +2.25088 q^{3} -1.99281 q^{4} +0.792214 q^{5} -0.190813 q^{6} +0.338482 q^{8} +2.06645 q^{9} -0.0671582 q^{10} +4.82174 q^{11} -4.48558 q^{12} +3.03783 q^{13} +1.78318 q^{15} +3.95693 q^{16} +5.93429 q^{17} -0.175179 q^{18} -4.99844 q^{19} -1.57874 q^{20} -0.408752 q^{22} +2.59183 q^{23} +0.761882 q^{24} -4.37240 q^{25} -0.257525 q^{26} -2.10130 q^{27} -5.54886 q^{29} -0.151165 q^{30} +1.01430 q^{31} -1.01240 q^{32} +10.8531 q^{33} -0.503066 q^{34} -4.11805 q^{36} +2.40651 q^{37} +0.423732 q^{38} +6.83778 q^{39} +0.268150 q^{40} +7.84236 q^{41} +4.98433 q^{43} -9.60882 q^{44} +1.63707 q^{45} -0.219717 q^{46} -2.19804 q^{47} +8.90657 q^{48} +0.370660 q^{50} +13.3574 q^{51} -6.05382 q^{52} +8.23376 q^{53} +0.178133 q^{54} +3.81985 q^{55} -11.2509 q^{57} +0.470393 q^{58} -5.31212 q^{59} -3.55354 q^{60} +14.5460 q^{61} -0.0859851 q^{62} -7.82804 q^{64} +2.40661 q^{65} -0.920051 q^{66} -13.0920 q^{67} -11.8259 q^{68} +5.83389 q^{69} -16.5753 q^{71} +0.699457 q^{72} +15.8536 q^{73} -0.204006 q^{74} -9.84173 q^{75} +9.96095 q^{76} -0.579658 q^{78} -1.00000 q^{79} +3.13474 q^{80} -10.9291 q^{81} -0.664819 q^{82} +2.63695 q^{83} +4.70123 q^{85} -0.422536 q^{86} -12.4898 q^{87} +1.63207 q^{88} -9.86298 q^{89} -0.138779 q^{90} -5.16503 q^{92} +2.28307 q^{93} +0.186334 q^{94} -3.95983 q^{95} -2.27880 q^{96} -9.69545 q^{97} +9.96389 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.0847728	−0.0599434	−0.0299717	−	0.999551i	$-0.509542\pi$
$2$	−0.0847728	−0.0599434	−0.0299717	+	0.999551i	$0.509542\pi$
$3$	2.25088	1.29954	0.649772	−	0.760129i	$-0.274864\pi$
$3$	2.25088	1.29954	0.649772	+	0.760129i	$0.274864\pi$
$4$	−1.99281	−0.996407
$5$	0.792214	0.354289	0.177144	−	0.984185i	$-0.443314\pi$
$5$	0.792214	0.354289	0.177144	+	0.984185i	$0.443314\pi$
$6$	−0.190813	−0.0778992
$7$	0	0
$7$	0	0
$8$	0.338482	0.119671
$9$	2.06645	0.688817
$10$	−0.0671582	−0.0212373
$11$	4.82174	1.45381	0.726904	−	0.686739i	$-0.240958\pi$
$11$	4.82174	1.45381	0.726904	+	0.686739i	$0.240958\pi$
$12$	−4.48558	−1.29488
$13$	3.03783	0.842542	0.421271	−	0.906935i	$-0.361584\pi$
$13$	3.03783	0.842542	0.421271	+	0.906935i	$0.361584\pi$
$14$	0	0
$15$	1.78318	0.460414
$16$	3.95693	0.989233
$17$	5.93429	1.43928	0.719638	−	0.694349i	$-0.244308\pi$
$17$	5.93429	1.43928	0.719638	+	0.694349i	$0.244308\pi$
$18$	−0.175179	−0.0412901
$19$	−4.99844	−1.14672	−0.573360	−	0.819303i	$-0.694360\pi$
$19$	−4.99844	−1.14672	−0.573360	+	0.819303i	$0.694360\pi$
$20$	−1.57874	−0.353016
$21$	0	0
$22$	−0.408752	−0.0871463
$23$	2.59183	0.540433	0.270217	−	0.962800i	$-0.412905\pi$
$23$	2.59183	0.540433	0.270217	+	0.962800i	$0.412905\pi$
$24$	0.761882	0.155518
$25$	−4.37240	−0.874479
$26$	−0.257525	−0.0505048
$27$	−2.10130	−0.404396
$28$	0	0
$29$	−5.54886	−1.03040	−0.515199	−	0.857071i	$-0.672282\pi$
$29$	−5.54886	−1.03040	−0.515199	+	0.857071i	$0.672282\pi$
$30$	−0.151165	−0.0275988
$31$	1.01430	0.182174	0.0910869	−	0.995843i	$-0.470966\pi$
$31$	1.01430	0.182174	0.0910869	+	0.995843i	$0.470966\pi$
$32$	−1.01240	−0.178970
$33$	10.8531	1.88929
$34$	−0.503066	−0.0862752
$35$	0	0
$36$	−4.11805	−0.686342
$37$	2.40651	0.395627	0.197814	−	0.980240i	$-0.436616\pi$
$37$	2.40651	0.395627	0.197814	+	0.980240i	$0.436616\pi$
$38$	0.423732	0.0687384
$39$	6.83778	1.09492
$40$	0.268150	0.0423983
$41$	7.84236	1.22477	0.612385	−	0.790560i	$-0.290210\pi$
$41$	7.84236	1.22477	0.612385	+	0.790560i	$0.290210\pi$
$42$	0	0
$43$	4.98433	0.760104	0.380052	−	0.924965i	$-0.375906\pi$
$43$	4.98433	0.760104	0.380052	+	0.924965i	$0.375906\pi$
$44$	−9.60882	−1.44858
$45$	1.63707	0.244040
$46$	−0.219717	−0.0323954
$47$	−2.19804	−0.320617	−0.160308	−	0.987067i	$-0.551249\pi$
$47$	−2.19804	−0.320617	−0.160308	+	0.987067i	$0.551249\pi$
$48$	8.90657	1.28555
$49$	0	0
$50$	0.370660	0.0524193
$51$	13.3574	1.87040
$52$	−6.05382	−0.839514
$53$	8.23376	1.13099	0.565497	−	0.824751i	$-0.308685\pi$
$53$	8.23376	1.13099	0.565497	+	0.824751i	$0.308685\pi$
$54$	0.178133	0.0242409
$55$	3.81985	0.515068
$56$	0	0
$57$	−11.2509	−1.49021
$58$	0.470393	0.0617656
$59$	−5.31212	−0.691579	−0.345790	−	0.938312i	$-0.612389\pi$
$59$	−5.31212	−0.691579	−0.345790	+	0.938312i	$0.612389\pi$
$60$	−3.55354	−0.458760
$61$	14.5460	1.86242	0.931210	−	0.364483i	$-0.118754\pi$
$61$	14.5460	1.86242	0.931210	+	0.364483i	$0.118754\pi$
$62$	−0.0859851	−0.0109201
$63$	0	0
$64$	−7.82804	−0.978505
$65$	2.40661	0.298503
$66$	−0.920051	−0.113251
$67$	−13.0920	−1.59944	−0.799720	−	0.600374i	$-0.795019\pi$
$67$	−13.0920	−1.59944	−0.799720	+	0.600374i	$0.795019\pi$
$68$	−11.8259	−1.43410
$69$	5.83389	0.702317
$70$	0	0
$71$	−16.5753	−1.96713	−0.983566	−	0.180549i	$-0.942213\pi$
$71$	−16.5753	−1.96713	−0.983566	+	0.180549i	$0.942213\pi$
$72$	0.699457	0.0824318
$73$	15.8536	1.85553	0.927763	−	0.373169i	$-0.121729\pi$
$73$	15.8536	1.85553	0.927763	+	0.373169i	$0.121729\pi$
$74$	−0.204006	−0.0237153
$75$	−9.84173	−1.13643
$76$	9.96095	1.14260
$77$	0	0
$78$	−0.579658	−0.0656333
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	3.13474	0.350474
$81$	−10.9291	−1.21435
$82$	−0.664819	−0.0734169
$83$	2.63695	0.289443	0.144721	−	0.989472i	$-0.453771\pi$
$83$	2.63695	0.289443	0.144721	+	0.989472i	$0.453771\pi$
$84$	0	0
$85$	4.70123	0.509920
$86$	−0.422536	−0.0455632
$87$	−12.4898	−1.33905
$88$	1.63207	0.173979
$89$	−9.86298	−1.04547	−0.522737	−	0.852494i	$-0.675089\pi$
$89$	−9.86298	−1.04547	−0.522737	+	0.852494i	$0.675089\pi$
$90$	−0.138779	−0.0146286
$91$	0	0
$92$	−5.16503	−0.538491
$93$	2.28307	0.236743
$94$	0.186334	0.0192189
$95$	−3.95983	−0.406270
$96$	−2.27880	−0.232579
$97$	−9.69545	−0.984424	−0.492212	−	0.870475i	$-0.663812\pi$
$97$	−9.69545	−0.984424	−0.492212	+	0.870475i	$0.663812\pi$
$98$	0	0
$99$	9.96389	1.00141
$100$	8.71337	0.871337
$101$	11.1292	1.10739	0.553697	−	0.832718i	$-0.313217\pi$
$101$	11.1292	1.10739	0.553697	+	0.832718i	$0.313217\pi$
$102$	−1.13234	−0.112118
$103$	6.22826	0.613689	0.306845	−	0.951760i	$-0.400727\pi$
$103$	6.22826	0.613689	0.306845	+	0.951760i	$0.400727\pi$
$104$	1.02825	0.100828
$105$	0	0
$106$	−0.697999	−0.0677956
$107$	13.4518	1.30044	0.650218	−	0.759748i	$-0.274678\pi$
$107$	13.4518	1.30044	0.650218	+	0.759748i	$0.274678\pi$
$108$	4.18751	0.402943
$109$	15.9791	1.53052	0.765259	−	0.643722i	$-0.222611\pi$
$109$	15.9791	1.53052	0.765259	+	0.643722i	$0.222611\pi$
$110$	−0.323819	−0.0308750
$111$	5.41675	0.514136
$112$	0	0
$113$	−8.94774	−0.841732	−0.420866	−	0.907123i	$-0.638274\pi$
$113$	−8.94774	−0.841732	−0.420866	+	0.907123i	$0.638274\pi$
$114$	0.953768	0.0893286
$115$	2.05328	0.191470
$116$	11.0578	1.02670
$117$	6.27752	0.580357
$118$	0.450324	0.0414556
$119$	0	0
$120$	0.603574	0.0550985
$121$	12.2491	1.11356
$122$	−1.23310	−0.111640
$123$	17.6522	1.59164
$124$	−2.02131	−0.181519
$125$	−7.42495	−0.664107
$126$	0	0
$127$	8.14554	0.722800	0.361400	−	0.932411i	$-0.382299\pi$
$127$	8.14554	0.722800	0.361400	+	0.932411i	$0.382299\pi$
$128$	2.68841	0.237625
$129$	11.2191	0.987789
$130$	−0.204015	−0.0178933
$131$	−12.7634	−1.11514	−0.557571	−	0.830129i	$-0.688267\pi$
$131$	−12.7634	−1.11514	−0.557571	+	0.830129i	$0.688267\pi$
$132$	−21.6283	−1.88250
$133$	0	0
$134$	1.10984	0.0958759
$135$	−1.66468	−0.143273
$136$	2.00865	0.172240
$137$	13.8404	1.18247	0.591234	−	0.806500i	$-0.298641\pi$
$137$	13.8404	1.18247	0.591234	+	0.806500i	$0.298641\pi$
$138$	−0.494555	−0.0420993
$139$	−10.1945	−0.864684	−0.432342	−	0.901710i	$-0.642313\pi$
$139$	−10.1945	−0.864684	−0.432342	+	0.901710i	$0.642313\pi$
$140$	0	0
$141$	−4.94751	−0.416656
$142$	1.40514	0.117917
$143$	14.6476	1.22489
$144$	8.17681	0.681401
$145$	−4.39589	−0.365059
$146$	−1.34396	−0.111227
$147$	0	0
$148$	−4.79572	−0.394206
$149$	0.0894068	0.00732449	0.00366224	−	0.999993i	$-0.498834\pi$
$149$	0.0894068	0.00732449	0.00366224	+	0.999993i	$0.498834\pi$
$150$	0.834311	0.0681212
$151$	16.3581	1.33120	0.665602	−	0.746307i	$-0.268175\pi$
$151$	16.3581	1.33120	0.665602	+	0.746307i	$0.268175\pi$
$152$	−1.69188	−0.137230
$153$	12.2629	0.991398
$154$	0	0
$155$	0.803543	0.0645421
$156$	−13.6264	−1.09099
$157$	13.4954	1.07705	0.538527	−	0.842608i	$-0.318981\pi$
$157$	13.4954	1.07705	0.538527	+	0.842608i	$0.318981\pi$
$158$	0.0847728	0.00674416
$159$	18.5332	1.46978
$160$	−0.802041	−0.0634069
$161$	0	0
$162$	0.926493	0.0727922
$163$	16.9182	1.32514	0.662569	−	0.749001i	$-0.269466\pi$
$163$	16.9182	1.32514	0.662569	+	0.749001i	$0.269466\pi$
$164$	−15.6284	−1.22037
$165$	8.59801	0.669354
$166$	−0.223542	−0.0173502
$167$	1.46920	0.113690	0.0568449	−	0.998383i	$-0.481896\pi$
$167$	1.46920	0.113690	0.0568449	+	0.998383i	$0.481896\pi$
$168$	0	0
$169$	−3.77161	−0.290124
$170$	−0.398536	−0.0305663
$171$	−10.3290	−0.789881
$172$	−9.93285	−0.757372
$173$	3.67774	0.279614	0.139807	−	0.990179i	$-0.455352\pi$
$173$	3.67774	0.279614	0.139807	+	0.990179i	$0.455352\pi$
$174$	1.05880	0.0802672
$175$	0	0
$176$	19.0793	1.43816
$177$	−11.9569	−0.898738
$178$	0.836113	0.0626693
$179$	9.68683	0.724028	0.362014	−	0.932173i	$-0.382089\pi$
$179$	9.68683	0.724028	0.362014	+	0.932173i	$0.382089\pi$
$180$	−3.26238	−0.243163
$181$	−7.11939	−0.529180	−0.264590	−	0.964361i	$-0.585237\pi$
$181$	−7.11939	−0.529180	−0.264590	+	0.964361i	$0.585237\pi$
$182$	0	0
$183$	32.7412	2.42030
$184$	0.877287	0.0646745
$185$	1.90647	0.140166
$186$	−0.193542	−0.0141912
$187$	28.6136	2.09243
$188$	4.38028	0.319465
$189$	0	0
$190$	0.335686	0.0243532
$191$	−4.18228	−0.302619	−0.151309	−	0.988486i	$-0.548349\pi$
$191$	−4.18228	−0.302619	−0.151309	+	0.988486i	$0.548349\pi$
$192$	−17.6200	−1.27161
$193$	−1.54825	−0.111445	−0.0557226	−	0.998446i	$-0.517746\pi$
$193$	−1.54825	−0.111445	−0.0557226	+	0.998446i	$0.517746\pi$
$194$	0.821911	0.0590098
$195$	5.41698	0.387918
$196$	0	0
$197$	−7.85465	−0.559621	−0.279810	−	0.960055i	$-0.590272\pi$
$197$	−7.85465	−0.559621	−0.279810	+	0.960055i	$0.590272\pi$
$198$	−0.844667	−0.0600279
$199$	−19.3246	−1.36989	−0.684943	−	0.728596i	$-0.740173\pi$
$199$	−19.3246	−1.36989	−0.684943	+	0.728596i	$0.740173\pi$
$200$	−1.47998	−0.104650
$201$	−29.4684	−2.07854
$202$	−0.943451	−0.0663810
$203$	0	0
$204$	−26.6187	−1.86368
$205$	6.21283	0.433923
$206$	−0.527988	−0.0367866
$207$	5.35588	0.372260
$208$	12.0205	0.833470
$209$	−24.1012	−1.66711
$210$	0	0
$211$	23.8096	1.63912	0.819561	−	0.572992i	$-0.194217\pi$
$211$	23.8096	1.63912	0.819561	+	0.572992i	$0.194217\pi$
$212$	−16.4083	−1.12693
$213$	−37.3091	−2.55638
$214$	−1.14035	−0.0779526
$215$	3.94866	0.269296
$216$	−0.711254	−0.0483947
$217$	0	0
$218$	−1.35459	−0.0917446
$219$	35.6846	2.41134
$220$	−7.61225	−0.513217
$221$	18.0273	1.21265
$222$	−0.459194	−0.0308191
$223$	−7.55781	−0.506108	−0.253054	−	0.967452i	$-0.581435\pi$
$223$	−7.55781	−0.506108	−0.253054	+	0.967452i	$0.581435\pi$
$224$	0	0
$225$	−9.03535	−0.602356
$226$	0.758525	0.0504563
$227$	11.9790	0.795071	0.397535	−	0.917587i	$-0.369866\pi$
$227$	11.9790	0.795071	0.397535	+	0.917587i	$0.369866\pi$
$228$	22.4209	1.48486
$229$	−6.16032	−0.407086	−0.203543	−	0.979066i	$-0.565246\pi$
$229$	−6.16032	−0.407086	−0.203543	+	0.979066i	$0.565246\pi$
$230$	−0.174063	−0.0114773
$231$	0	0
$232$	−1.87819	−0.123309
$233$	13.5069	0.884868	0.442434	−	0.896801i	$-0.354115\pi$
$233$	13.5069	0.884868	0.442434	+	0.896801i	$0.354115\pi$
$234$	−0.532163	−0.0347886
$235$	−1.74132	−0.113591
$236$	10.5861	0.689094
$237$	−2.25088	−0.146210
$238$	0	0
$239$	10.2744	0.664595	0.332298	−	0.943175i	$-0.392176\pi$
$239$	10.2744	0.664595	0.332298	+	0.943175i	$0.392176\pi$
$240$	7.05591	0.455457
$241$	−2.33555	−0.150446	−0.0752230	−	0.997167i	$-0.523967\pi$
$241$	−2.33555	−0.150446	−0.0752230	+	0.997167i	$0.523967\pi$
$242$	−1.03840	−0.0667506
$243$	−18.2962	−1.17370
$244$	−28.9874	−1.85573
$245$	0	0
$246$	−1.49643	−0.0954086
$247$	−15.1844	−0.966159
$248$	0.343323	0.0218010
$249$	5.93545	0.376144
$250$	0.629434	0.0398089
$251$	−15.6121	−0.985424	−0.492712	−	0.870192i	$-0.663994\pi$
$251$	−15.6121	−0.985424	−0.492712	+	0.870192i	$0.663994\pi$
$252$	0	0
$253$	12.4971	0.785687
$254$	−0.690520	−0.0433271
$255$	10.5819	0.662664
$256$	15.4282	0.964261
$257$	31.8247	1.98517	0.992585	−	0.121552i	$-0.0387871\pi$
$257$	31.8247	1.98517	0.992585	+	0.121552i	$0.0387871\pi$
$258$	−0.951077	−0.0592115
$259$	0	0
$260$	−4.79592	−0.297431
$261$	−11.4665	−0.709756
$262$	1.08199	0.0668454
$263$	−29.2379	−1.80289	−0.901444	−	0.432896i	$-0.857492\pi$
$263$	−29.2379	−1.80289	−0.901444	+	0.432896i	$0.857492\pi$
$264$	3.67359	0.226094
$265$	6.52290	0.400698
$266$	0	0
$267$	−22.2004	−1.35864
$268$	26.0899	1.59369
$269$	15.6038	0.951380	0.475690	−	0.879613i	$-0.342198\pi$
$269$	15.6038	0.951380	0.475690	+	0.879613i	$0.342198\pi$
$270$	0.141120	0.00858828
$271$	−14.6440	−0.889562	−0.444781	−	0.895639i	$-0.646718\pi$
$271$	−14.6440	−0.889562	−0.444781	+	0.895639i	$0.646718\pi$
$272$	23.4816	1.42378
$273$	0	0
$274$	−1.17329	−0.0708812
$275$	−21.0825	−1.27133
$276$	−11.6258	−0.699794
$277$	−2.14604	−0.128943	−0.0644715	−	0.997920i	$-0.520536\pi$
$277$	−2.14604	−0.128943	−0.0644715	+	0.997920i	$0.520536\pi$
$278$	0.864215	0.0518322
$279$	2.09600	0.125484
$280$	0	0
$281$	−8.56660	−0.511041	−0.255520	−	0.966804i	$-0.582247\pi$
$281$	−8.56660	−0.511041	−0.255520	+	0.966804i	$0.582247\pi$
$282$	0.419415	0.0249758
$283$	6.56059	0.389987	0.194993	−	0.980805i	$-0.437531\pi$
$283$	6.56059	0.389987	0.194993	+	0.980805i	$0.437531\pi$
$284$	33.0316	1.96006
$285$	−8.91310	−0.527966
$286$	−1.24172	−0.0734244
$287$	0	0
$288$	−2.09208	−0.123277
$289$	18.2158	1.07152
$290$	0.372652	0.0218829
$291$	−21.8233	−1.27930
$292$	−31.5933	−1.84886
$293$	−14.5262	−0.848630	−0.424315	−	0.905515i	$-0.639485\pi$
$293$	−14.5262	−0.848630	−0.424315	+	0.905515i	$0.639485\pi$
$294$	0	0
$295$	−4.20834	−0.245019
$296$	0.814560	0.0473453
$297$	−10.1319	−0.587915
$298$	−0.00757927	−0.000439055 0
$299$	7.87352	0.455338
$300$	19.6127	1.13234
$301$	0	0
$302$	−1.38672	−0.0797969
$303$	25.0504	1.43911
$304$	−19.7785	−1.13437
$305$	11.5235	0.659835
$306$	−1.03956	−0.0594278
$307$	−6.01140	−0.343089	−0.171544	−	0.985176i	$-0.554876\pi$
$307$	−6.01140	−0.343089	−0.171544	+	0.985176i	$0.554876\pi$
$308$	0	0
$309$	14.0191	0.797517
$310$	−0.0681186	−0.00386888
$311$	−29.3317	−1.66325	−0.831623	−	0.555340i	$-0.812588\pi$
$311$	−29.3317	−1.66325	−0.831623	+	0.555340i	$0.812588\pi$
$312$	2.31446	0.131031
$313$	11.4089	0.644870	0.322435	−	0.946592i	$-0.395499\pi$
$313$	11.4089	0.644870	0.322435	+	0.946592i	$0.395499\pi$
$314$	−1.14405	−0.0645623
$315$	0	0
$316$	1.99281	0.112105
$317$	−13.3876	−0.751921	−0.375961	−	0.926636i	$-0.622687\pi$
$317$	−13.3876	−0.751921	−0.375961	+	0.926636i	$0.622687\pi$
$318$	−1.57111	−0.0881035
$319$	−26.7552	−1.49800
$320$	−6.20149	−0.346674
$321$	30.2784	1.68998
$322$	0	0
$323$	−29.6622	−1.65045
$324$	21.7797	1.20998
$325$	−13.2826	−0.736785
$326$	−1.43421	−0.0794333
$327$	35.9670	1.98898
$328$	2.65450	0.146570
$329$	0	0
$330$	−0.728878	−0.0401234
$331$	−23.2790	−1.27953	−0.639765	−	0.768571i	$-0.720968\pi$
$331$	−23.2790	−1.27953	−0.639765	+	0.768571i	$0.720968\pi$
$332$	−5.25495	−0.288403
$333$	4.97293	0.272515
$334$	−0.124548	−0.00681496
$335$	−10.3716	−0.566664
$336$	0	0
$337$	32.4217	1.76612	0.883060	−	0.469260i	$-0.155479\pi$
$337$	32.4217	1.76612	0.883060	+	0.469260i	$0.155479\pi$
$338$	0.319730	0.0173910
$339$	−20.1403	−1.09387
$340$	−9.36867	−0.508088
$341$	4.89069	0.264846
$342$	0.875621	0.0473482
$343$	0	0
$344$	1.68711	0.0909627
$345$	4.62169	0.248823
$346$	−0.311773	−0.0167610
$347$	−6.94452	−0.372802	−0.186401	−	0.982474i	$-0.559682\pi$
$347$	−6.94452	−0.372802	−0.186401	+	0.982474i	$0.559682\pi$
$348$	24.8899	1.33424
$349$	−22.3618	−1.19700	−0.598500	−	0.801123i	$-0.704236\pi$
$349$	−22.3618	−1.19700	−0.598500	+	0.801123i	$0.704236\pi$
$350$	0	0
$351$	−6.38340	−0.340721
$352$	−4.88155	−0.260187
$353$	4.89439	0.260502	0.130251	−	0.991481i	$-0.458422\pi$
$353$	4.89439	0.260502	0.130251	+	0.991481i	$0.458422\pi$
$354$	1.01362	0.0538735
$355$	−13.1312	−0.696933
$356$	19.6551	1.04172
$357$	0	0
$358$	−0.821180	−0.0434007
$359$	8.00023	0.422236	0.211118	−	0.977461i	$-0.432290\pi$
$359$	8.00023	0.422236	0.211118	+	0.977461i	$0.432290\pi$
$360$	0.554120	0.0292047
$361$	5.98438	0.314967
$362$	0.603530	0.0317209
$363$	27.5713	1.44712
$364$	0	0
$365$	12.5595	0.657393
$366$	−2.77556	−0.145081
$367$	29.0958	1.51879	0.759393	−	0.650632i	$-0.225496\pi$
$367$	29.0958	1.51879	0.759393	+	0.650632i	$0.225496\pi$
$368$	10.2557	0.534615
$369$	16.2058	0.843643
$370$	−0.161617	−0.00840206
$371$	0	0
$372$	−4.54973	−0.235892
$373$	−0.876666	−0.0453920	−0.0226960	−	0.999742i	$-0.507225\pi$
$373$	−0.876666	−0.0453920	−0.0226960	+	0.999742i	$0.507225\pi$
$374$	−2.42565	−0.125428
$375$	−16.7126	−0.863037
$376$	−0.743996	−0.0383687
$377$	−16.8565	−0.868153
$378$	0	0
$379$	7.05888	0.362590	0.181295	−	0.983429i	$-0.441971\pi$
$379$	7.05888	0.362590	0.181295	+	0.983429i	$0.441971\pi$
$380$	7.89121	0.404810
$381$	18.3346	0.939311
$382$	0.354543	0.0181400
$383$	−6.87591	−0.351342	−0.175671	−	0.984449i	$-0.556210\pi$
$383$	−6.87591	−0.351342	−0.175671	+	0.984449i	$0.556210\pi$
$384$	6.05129	0.308804
$385$	0	0
$386$	0.131249	0.00668041
$387$	10.2999	0.523572
$388$	19.3212	0.980887
$389$	−2.30306	−0.116770	−0.0583848	−	0.998294i	$-0.518595\pi$
$389$	−2.30306	−0.116770	−0.0583848	+	0.998294i	$0.518595\pi$
$390$	−0.459213	−0.0232532
$391$	15.3807	0.777833
$392$	0	0
$393$	−28.7288	−1.44918
$394$	0.665861	0.0335456
$395$	−0.792214	−0.0398606
$396$	−19.8562	−0.997810
$397$	−16.2023	−0.813171	−0.406585	−	0.913613i	$-0.633281\pi$
$397$	−16.2023	−0.813171	−0.406585	+	0.913613i	$0.633281\pi$
$398$	1.63820	0.0821157
$399$	0	0
$400$	−17.3013	−0.865064
$401$	−0.847274	−0.0423109	−0.0211554	−	0.999776i	$-0.506734\pi$
$401$	−0.847274	−0.0423109	−0.0211554	+	0.999776i	$0.506734\pi$
$402$	2.49812	0.124595
$403$	3.08127	0.153489
$404$	−22.1784	−1.10341
$405$	−8.65821	−0.430230
$406$	0	0
$407$	11.6035	0.575166
$408$	4.52123	0.223834
$409$	−4.77171	−0.235946	−0.117973	−	0.993017i	$-0.537640\pi$
$409$	−4.77171	−0.235946	−0.117973	+	0.993017i	$0.537640\pi$
$410$	−0.526679	−0.0260108
$411$	31.1531	1.53667
$412$	−12.4118	−0.611484
$413$	0	0
$414$	−0.454033	−0.0223145
$415$	2.08903	0.102546
$416$	−3.07551	−0.150789
$417$	−22.9465	−1.12370
$418$	2.04312	0.0999324
$419$	−19.6367	−0.959318	−0.479659	−	0.877455i	$-0.659240\pi$
$419$	−19.6367	−0.959318	−0.479659	+	0.877455i	$0.659240\pi$
$420$	0	0
$421$	−21.6191	−1.05365	−0.526826	−	0.849973i	$-0.676618\pi$
$421$	−21.6191	−1.05365	−0.526826	+	0.849973i	$0.676618\pi$
$422$	−2.01841	−0.0982546
$423$	−4.54214	−0.220846
$424$	2.78698	0.135348
$425$	−25.9471	−1.25862
$426$	3.16280	0.153238
$427$	0	0
$428$	−26.8070	−1.29576
$429$	32.9700	1.59180
$430$	−0.334739	−0.0161425
$431$	−2.67736	−0.128964	−0.0644819	−	0.997919i	$-0.520539\pi$
$431$	−2.67736	−0.128964	−0.0644819	+	0.997919i	$0.520539\pi$
$432$	−8.31472	−0.400042
$433$	−11.0838	−0.532653	−0.266326	−	0.963883i	$-0.585810\pi$
$433$	−11.0838	−0.532653	−0.266326	+	0.963883i	$0.585810\pi$
$434$	0	0
$435$	−9.89461	−0.474410
$436$	−31.8433	−1.52502
$437$	−12.9551	−0.619726
$438$	−3.02508	−0.144544
$439$	−5.90223	−0.281698	−0.140849	−	0.990031i	$-0.544983\pi$
$439$	−5.90223	−0.281698	−0.140849	+	0.990031i	$0.544983\pi$
$440$	1.29295	0.0616390
$441$	0	0
$442$	−1.52823	−0.0726904
$443$	−33.2952	−1.58190	−0.790951	−	0.611880i	$-0.790414\pi$
$443$	−33.2952	−1.58190	−0.790951	+	0.611880i	$0.790414\pi$
$444$	−10.7946	−0.512288
$445$	−7.81359	−0.370400
$446$	0.640697	0.0303379
$447$	0.201244	0.00951850
$448$	0	0
$449$	23.9754	1.13147	0.565736	−	0.824587i	$-0.308592\pi$
$449$	23.9754	1.13147	0.565736	+	0.824587i	$0.308592\pi$
$450$	0.765952	0.0361073
$451$	37.8138	1.78058
$452$	17.8312	0.838708
$453$	36.8201	1.72996
$454$	−1.01549	−0.0476593
$455$	0	0
$456$	−3.80822	−0.178336
$457$	−4.24088	−0.198380	−0.0991900	−	0.995069i	$-0.531625\pi$
$457$	−4.24088	−0.198380	−0.0991900	+	0.995069i	$0.531625\pi$
$458$	0.522228	0.0244021
$459$	−12.4697	−0.582038
$460$	−4.09181	−0.190782
$461$	−34.0610	−1.58638	−0.793190	−	0.608974i	$-0.791581\pi$
$461$	−34.0610	−1.58638	−0.793190	+	0.608974i	$0.791581\pi$
$462$	0	0
$463$	−32.8739	−1.52778	−0.763891	−	0.645346i	$-0.776713\pi$
$463$	−32.8739	−1.52778	−0.763891	+	0.645346i	$0.776713\pi$
$464$	−21.9565	−1.01930
$465$	1.80868	0.0838754
$466$	−1.14502	−0.0530420
$467$	36.3645	1.68275	0.841374	−	0.540453i	$-0.181747\pi$
$467$	36.3645	1.68275	0.841374	+	0.540453i	$0.181747\pi$
$468$	−12.5099	−0.578272
$469$	0	0
$470$	0.147616	0.00680903
$471$	30.3766	1.39968
$472$	−1.79806	−0.0827623
$473$	24.0331	1.10505
$474$	0.190813	0.00876434
$475$	21.8552	1.00278
$476$	0	0
$477$	17.0147	0.779048
$478$	−0.870989	−0.0398381
$479$	−3.48304	−0.159144	−0.0795720	−	0.996829i	$-0.525355\pi$
$479$	−3.48304	−0.159144	−0.0795720	+	0.996829i	$0.525355\pi$
$480$	−1.80530	−0.0824001
$481$	7.31055	0.333333
$482$	0.197991	0.00901825
$483$	0	0
$484$	−24.4103	−1.10956
$485$	−7.68087	−0.348771
$486$	1.55102	0.0703558
$487$	1.73690	0.0787066	0.0393533	−	0.999225i	$-0.487470\pi$
$487$	1.73690	0.0787066	0.0393533	+	0.999225i	$0.487470\pi$
$488$	4.92355	0.222879
$489$	38.0809	1.72208
$490$	0	0
$491$	38.7407	1.74834	0.874172	−	0.485617i	$-0.161405\pi$
$491$	38.7407	1.74834	0.874172	+	0.485617i	$0.161405\pi$
$492$	−35.1775	−1.58592
$493$	−32.9286	−1.48303
$494$	1.28722	0.0579149
$495$	7.89353	0.354788
$496$	4.01352	0.180212
$497$	0	0
$498$	−0.503165	−0.0225474
$499$	10.0411	0.449501	0.224750	−	0.974416i	$-0.427843\pi$
$499$	10.0411	0.449501	0.224750	+	0.974416i	$0.427843\pi$
$500$	14.7965	0.661721
$501$	3.30698	0.147745
$502$	1.32348	0.0590697
$503$	−39.8106	−1.77507	−0.887533	−	0.460745i	$-0.847582\pi$
$503$	−39.8106	−1.77507	−0.887533	+	0.460745i	$0.847582\pi$
$504$	0	0
$505$	8.81669	0.392337
$506$	−1.05942	−0.0470968
$507$	−8.48943	−0.377029
$508$	−16.2325	−0.720203
$509$	−41.2887	−1.83009	−0.915045	−	0.403351i	$-0.867845\pi$
$509$	−41.2887	−1.83009	−0.915045	+	0.403351i	$0.867845\pi$
$510$	−0.897057	−0.0397223
$511$	0	0
$512$	−6.68472	−0.295426
$513$	10.5032	0.463729
$514$	−2.69787	−0.118998
$515$	4.93412	0.217423
$516$	−22.3576	−0.984239
$517$	−10.5984	−0.466115
$518$	0	0
$519$	8.27815	0.363371
$520$	0.814594	0.0357223
$521$	−24.1019	−1.05592	−0.527962	−	0.849268i	$-0.677044\pi$
$521$	−24.1019	−1.05592	−0.527962	+	0.849268i	$0.677044\pi$
$522$	0.972044	0.0425452
$523$	−32.7731	−1.43307	−0.716534	−	0.697552i	$-0.754273\pi$
$523$	−32.7731	−1.43307	−0.716534	+	0.697552i	$0.754273\pi$
$524$	25.4350	1.11114
$525$	0	0
$526$	2.47858	0.108071
$527$	6.01915	0.262198
$528$	42.9452	1.86895
$529$	−16.2824	−0.707932
$530$	−0.552964	−0.0240192
$531$	−10.9772	−0.476372
$532$	0	0
$533$	23.8237	1.03192
$534$	1.88199	0.0814416
$535$	10.6567	0.460730
$536$	−4.43140	−0.191407
$537$	21.8039	0.940907
$538$	−1.32278	−0.0570290
$539$	0	0
$540$	3.31740	0.142758
$541$	−28.4384	−1.22266	−0.611332	−	0.791374i	$-0.709366\pi$
$541$	−28.4384	−1.22266	−0.611332	+	0.791374i	$0.709366\pi$
$542$	1.24142	0.0533234
$543$	−16.0249	−0.687693
$544$	−6.00790	−0.257587
$545$	12.6589	0.542246
$546$	0	0
$547$	8.67034	0.370717	0.185358	−	0.982671i	$-0.440655\pi$
$547$	8.67034	0.370717	0.185358	+	0.982671i	$0.440655\pi$
$548$	−27.5814	−1.17822
$549$	30.0585	1.28287
$550$	1.78723	0.0762076
$551$	27.7356	1.18158
$552$	1.97467	0.0840474
$553$	0	0
$554$	0.181926	0.00772929
$555$	4.29123	0.182153
$556$	20.3157	0.861577
$557$	−12.7601	−0.540664	−0.270332	−	0.962767i	$-0.587133\pi$
$557$	−12.7601	−0.540664	−0.270332	+	0.962767i	$0.587133\pi$
$558$	−0.177684	−0.00752197
$559$	15.1415	0.640419
$560$	0	0
$561$	64.4057	2.71921
$562$	0.726215	0.0306335
$563$	−9.72295	−0.409774	−0.204887	−	0.978786i	$-0.565683\pi$
$563$	−9.72295	−0.409774	−0.204887	+	0.978786i	$0.565683\pi$
$564$	9.85947	0.415159
$565$	−7.08853	−0.298217
$566$	−0.556160	−0.0233772
$567$	0	0
$568$	−5.61046	−0.235410
$569$	23.4306	0.982263	0.491131	−	0.871085i	$-0.336583\pi$
$569$	23.4306	0.982263	0.491131	+	0.871085i	$0.336583\pi$
$570$	0.755589	0.0316481
$571$	−18.6552	−0.780698	−0.390349	−	0.920667i	$-0.627646\pi$
$571$	−18.6552	−0.780698	−0.390349	+	0.920667i	$0.627646\pi$
$572$	−29.1899	−1.22049
$573$	−9.41379	−0.393267
$574$	0	0
$575$	−11.3325	−0.472598
$576$	−16.1763	−0.674011
$577$	14.3500	0.597399	0.298700	−	0.954347i	$-0.403447\pi$
$577$	14.3500	0.597399	0.298700	+	0.954347i	$0.403447\pi$
$578$	−1.54420	−0.0642304
$579$	−3.48491	−0.144828
$580$	8.76018	0.363747
$581$	0	0
$582$	1.85002	0.0766858
$583$	39.7010	1.64425
$584$	5.36617	0.222054
$585$	4.97314	0.205614
$586$	1.23143	0.0508698
$587$	19.2156	0.793112	0.396556	−	0.918011i	$-0.370205\pi$
$587$	19.2156	0.793112	0.396556	+	0.918011i	$0.370205\pi$
$588$	0	0
$589$	−5.06992	−0.208902
$590$	0.356753	0.0146873
$591$	−17.6799	−0.727252
$592$	9.52239	0.391368
$593$	−24.3195	−0.998682	−0.499341	−	0.866406i	$-0.666424\pi$
$593$	−24.3195	−0.998682	−0.499341	+	0.866406i	$0.666424\pi$
$594$	0.858913	0.0352416
$595$	0	0
$596$	−0.178171	−0.00729817
$597$	−43.4974	−1.78023
$598$	−0.667461	−0.0272945
$599$	10.2639	0.419370	0.209685	−	0.977769i	$-0.432756\pi$
$599$	10.2639	0.419370	0.209685	+	0.977769i	$0.432756\pi$
$600$	−3.33125	−0.135998
$601$	−23.1772	−0.945419	−0.472709	−	0.881218i	$-0.656724\pi$
$601$	−23.1772	−0.945419	−0.472709	+	0.881218i	$0.656724\pi$
$602$	0	0
$603$	−27.0539	−1.10172
$604$	−32.5987	−1.32642
$605$	9.70395	0.394522
$606$	−2.12359	−0.0862651
$607$	4.30242	0.174630	0.0873150	−	0.996181i	$-0.472171\pi$
$607$	4.30242	0.174630	0.0873150	+	0.996181i	$0.472171\pi$
$608$	5.06044	0.205228
$609$	0	0
$610$	−0.976881	−0.0395528
$611$	−6.67725	−0.270133
$612$	−24.4377	−0.987836
$613$	−24.2401	−0.979050	−0.489525	−	0.871989i	$-0.662830\pi$
$613$	−24.2401	−0.979050	−0.489525	+	0.871989i	$0.662830\pi$
$614$	0.509604	0.0205659
$615$	13.9843	0.563902
$616$	0	0
$617$	−0.987912	−0.0397718	−0.0198859	−	0.999802i	$-0.506330\pi$
$617$	−0.987912	−0.0397718	−0.0198859	+	0.999802i	$0.506330\pi$
$618$	−1.18844	−0.0478059
$619$	0.109181	0.00438834	0.00219417	−	0.999998i	$-0.499302\pi$
$619$	0.109181	0.00438834	0.00219417	+	0.999998i	$0.499302\pi$
$620$	−1.60131	−0.0643102
$621$	−5.44622	−0.218549
$622$	2.48653	0.0997007
$623$	0	0
$624$	27.0566	1.08313
$625$	15.9798	0.639193
$626$	−0.967165	−0.0386557
$627$	−54.2488	−2.16649
$628$	−26.8939	−1.07318
$629$	14.2809	0.569417
$630$	0	0
$631$	−24.2703	−0.966186	−0.483093	−	0.875569i	$-0.660487\pi$
$631$	−24.2703	−0.966186	−0.483093	+	0.875569i	$0.660487\pi$
$632$	−0.338482	−0.0134641
$633$	53.5926	2.13011
$634$	1.13490	0.0450727
$635$	6.45301	0.256080
$636$	−36.9332	−1.46450
$637$	0	0
$638$	2.26811	0.0897953
$639$	−34.2522	−1.35499
$640$	2.12980	0.0841877
$641$	−0.0426367	−0.00168405	−0.000842025	−	1.00000i	$-0.500268\pi$
$641$	−0.0426367	−0.00168405	−0.000842025		1.00000i	$0.500268\pi$
$642$	−2.56678	−0.101303
$643$	29.1391	1.14913	0.574566	−	0.818458i	$-0.305171\pi$
$643$	29.1391	1.14913	0.574566	+	0.818458i	$0.305171\pi$
$644$	0	0
$645$	8.88795	0.349963
$646$	2.51455	0.0989335
$647$	−35.9839	−1.41467	−0.707337	−	0.706877i	$-0.750103\pi$
$647$	−35.9839	−1.41467	−0.707337	+	0.706877i	$0.750103\pi$
$648$	−3.69932	−0.145323
$649$	−25.6137	−1.00542
$650$	1.12600	0.0441654
$651$	0	0
$652$	−33.7149	−1.32038
$653$	33.5789	1.31404	0.657022	−	0.753871i	$-0.271816\pi$
$653$	33.5789	1.31404	0.657022	+	0.753871i	$0.271816\pi$
$654$	−3.04902	−0.119226
$655$	−10.1113	−0.395082
$656$	31.0317	1.21158
$657$	32.7607	1.27812
$658$	0	0
$659$	18.5502	0.722614	0.361307	−	0.932447i	$-0.382331\pi$
$659$	18.5502	0.722614	0.361307	+	0.932447i	$0.382331\pi$
$660$	−17.1342	−0.666949
$661$	47.0598	1.83041	0.915207	−	0.402984i	$-0.132027\pi$
$661$	47.0598	1.83041	0.915207	+	0.402984i	$0.132027\pi$
$662$	1.97343	0.0766994
$663$	40.5773	1.57589
$664$	0.892560	0.0346381
$665$	0	0
$666$	−0.421569	−0.0163355
$667$	−14.3817	−0.556861
$668$	−2.92783	−0.113281
$669$	−17.0117	−0.657710
$670$	0.879234	0.0339678
$671$	70.1368	2.70760
$672$	0	0
$673$	−30.6004	−1.17956	−0.589779	−	0.807565i	$-0.700785\pi$
$673$	−30.6004	−1.17956	−0.589779	+	0.807565i	$0.700785\pi$
$674$	−2.74847	−0.105867
$675$	9.18773	0.353636
$676$	7.51611	0.289081
$677$	−24.9894	−0.960421	−0.480211	−	0.877153i	$-0.659440\pi$
$677$	−24.9894	−0.960421	−0.480211	+	0.877153i	$0.659440\pi$
$678$	1.70735	0.0655703
$679$	0	0
$680$	1.59128	0.0610229
$681$	26.9632	1.03323
$682$	−0.414598	−0.0158758
$683$	15.4855	0.592535	0.296267	−	0.955105i	$-0.404258\pi$
$683$	15.4855	0.592535	0.296267	+	0.955105i	$0.404258\pi$
$684$	20.5838	0.787042
$685$	10.9646	0.418935
$686$	0	0
$687$	−13.8661	−0.529026
$688$	19.7227	0.751920
$689$	25.0127	0.952909
$690$	−0.391793	−0.0149153
$691$	−46.0971	−1.75362	−0.876809	−	0.480839i	$-0.840332\pi$
$691$	−46.0971	−1.75362	−0.876809	+	0.480839i	$0.840332\pi$
$692$	−7.32906	−0.278609
$693$	0	0
$694$	0.588707	0.0223470
$695$	−8.07621	−0.306348
$696$	−4.22758	−0.160246
$697$	46.5388	1.76278
$698$	1.89567	0.0717523
$699$	30.4024	1.14993
$700$	0	0
$701$	18.5437	0.700387	0.350193	−	0.936677i	$-0.386116\pi$
$701$	18.5437	0.700387	0.350193	+	0.936677i	$0.386116\pi$
$702$	0.541139	0.0204240
$703$	−12.0288	−0.453674
$704$	−37.7448	−1.42256
$705$	−3.91949	−0.147616
$706$	−0.414911	−0.0156154
$707$	0	0
$708$	23.8279	0.895509
$709$	5.73344	0.215324	0.107662	−	0.994188i	$-0.465664\pi$
$709$	5.73344	0.215324	0.107662	+	0.994188i	$0.465664\pi$
$710$	1.11317	0.0417766
$711$	−2.06645	−0.0774980
$712$	−3.33844	−0.125113
$713$	2.62889	0.0984528
$714$	0	0
$715$	11.6040	0.433966
$716$	−19.3041	−0.721426
$717$	23.1264	0.863671
$718$	−0.678202	−0.0253103
$719$	−53.0755	−1.97938	−0.989692	−	0.143215i	$-0.954256\pi$
$719$	−53.0755	−1.97938	−0.989692	+	0.143215i	$0.954256\pi$
$720$	6.47778	0.241413
$721$	0	0
$722$	−0.507313	−0.0188802
$723$	−5.25703	−0.195511
$724$	14.1876	0.527278
$725$	24.2618	0.901062
$726$	−2.33730	−0.0867454
$727$	9.88620	0.366659	0.183329	−	0.983052i	$-0.441312\pi$
$727$	9.88620	0.366659	0.183329	+	0.983052i	$0.441312\pi$
$728$	0	0
$729$	−8.39519	−0.310933
$730$	−1.06470	−0.0394064
$731$	29.5785	1.09400
$732$	−65.2471	−2.41160
$733$	45.1755	1.66860	0.834298	−	0.551314i	$-0.185873\pi$
$733$	45.1755	1.66860	0.834298	+	0.551314i	$0.185873\pi$
$734$	−2.46653	−0.0910413
$735$	0	0
$736$	−2.62398	−0.0967211
$737$	−63.1261	−2.32528
$738$	−1.37382	−0.0505708
$739$	11.5593	0.425216	0.212608	−	0.977138i	$-0.431804\pi$
$739$	11.5593	0.425216	0.212608	+	0.977138i	$0.431804\pi$
$740$	−3.79924	−0.139663
$741$	−34.1782	−1.25557
$742$	0	0
$743$	19.1615	0.702967	0.351484	−	0.936194i	$-0.385677\pi$
$743$	19.1615	0.702967	0.351484	+	0.936194i	$0.385677\pi$
$744$	0.772777	0.0283314
$745$	0.0708293	0.00259499
$746$	0.0743174	0.00272095
$747$	5.44913	0.199373
$748$	−57.0215	−2.08491
$749$	0	0
$750$	1.41678	0.0517334
$751$	5.77892	0.210876	0.105438	−	0.994426i	$-0.466376\pi$
$751$	5.77892	0.210876	0.105438	+	0.994426i	$0.466376\pi$
$752$	−8.69748	−0.317165
$753$	−35.1408	−1.28060
$754$	1.42897	0.0520401
$755$	12.9591	0.471631
$756$	0	0
$757$	46.7327	1.69853	0.849264	−	0.527968i	$-0.177046\pi$
$757$	46.7327	1.69853	0.849264	+	0.527968i	$0.177046\pi$
$758$	−0.598401	−0.0217349
$759$	28.1295	1.02103
$760$	−1.34033	−0.0486190
$761$	−28.2213	−1.02302	−0.511510	−	0.859277i	$-0.670914\pi$
$761$	−28.2213	−1.02302	−0.511510	+	0.859277i	$0.670914\pi$
$762$	−1.55428	−0.0563055
$763$	0	0
$764$	8.33450	0.301531
$765$	9.71486	0.351241
$766$	0.582890	0.0210607
$767$	−16.1373	−0.582684
$768$	34.7269	1.25310
$769$	−2.16739	−0.0781581	−0.0390791	−	0.999236i	$-0.512442\pi$
$769$	−2.16739	−0.0781581	−0.0390791	+	0.999236i	$0.512442\pi$
$770$	0	0
$771$	71.6335	2.57982
$772$	3.08536	0.111045
$773$	1.63826	0.0589243	0.0294621	−	0.999566i	$-0.490621\pi$
$773$	1.63826	0.0589243	0.0294621	+	0.999566i	$0.490621\pi$
$774$	−0.873150	−0.0313847
$775$	−4.43492	−0.159307
$776$	−3.28174	−0.117807
$777$	0	0
$778$	0.195237	0.00699957
$779$	−39.1995	−1.40447
$780$	−10.7950	−0.386524
$781$	−79.9220	−2.85983
$782$	−1.30386	−0.0466260
$783$	11.6598	0.416689
$784$	0	0
$785$	10.6913	0.381588
$786$	2.43542	0.0868687
$787$	−27.0166	−0.963037	−0.481518	−	0.876436i	$-0.659915\pi$
$787$	−27.0166	−0.963037	−0.481518	+	0.876436i	$0.659915\pi$
$788$	15.6528	0.557610
$789$	−65.8110	−2.34293
$790$	0.0671582	0.00238938
$791$	0	0
$792$	3.37260	0.119840
$793$	44.1881	1.56917
$794$	1.37352	0.0487443
$795$	14.6822	0.520726
$796$	38.5104	1.36496
$797$	−30.3636	−1.07553	−0.537766	−	0.843094i	$-0.680732\pi$
$797$	−30.3636	−1.07553	−0.537766	+	0.843094i	$0.680732\pi$
$798$	0	0
$799$	−13.0438	−0.461456
$800$	4.42663	0.156505
$801$	−20.3814	−0.720140
$802$	0.0718258	0.00253626
$803$	76.4420	2.69758
$804$	58.7251	2.07107
$805$	0	0
$806$	−0.261208	−0.00920066
$807$	35.1222	1.23636
$808$	3.76702	0.132523
$809$	20.7559	0.729740	0.364870	−	0.931058i	$-0.381113\pi$
$809$	20.7559	0.729740	0.364870	+	0.931058i	$0.381113\pi$
$810$	0.733981	0.0257895
$811$	45.8919	1.61148	0.805742	−	0.592267i	$-0.201767\pi$
$811$	45.8919	1.61148	0.805742	+	0.592267i	$0.201767\pi$
$812$	0	0
$813$	−32.9619	−1.15603
$814$	−0.983665	−0.0344775
$815$	13.4029	0.469482
$816$	52.8542	1.85027
$817$	−24.9139	−0.871626
$818$	0.404512	0.0141434
$819$	0	0
$820$	−12.3810	−0.432363
$821$	−27.5321	−0.960879	−0.480439	−	0.877028i	$-0.659523\pi$
$821$	−27.5321	−0.960879	−0.480439	+	0.877028i	$0.659523\pi$
$822$	−2.64094	−0.0921133
$823$	−4.57527	−0.159484	−0.0797419	−	0.996816i	$-0.525410\pi$
$823$	−4.57527	−0.159484	−0.0797419	+	0.996816i	$0.525410\pi$
$824$	2.10816	0.0734411
$825$	−47.4542	−1.65214
$826$	0	0
$827$	−45.0134	−1.56527	−0.782635	−	0.622481i	$-0.786125\pi$
$827$	−45.0134	−1.56527	−0.782635	+	0.622481i	$0.786125\pi$
$828$	−10.6733	−0.370922
$829$	−48.8234	−1.69571	−0.847853	−	0.530232i	$-0.822105\pi$
$829$	−48.8234	−1.69571	−0.847853	+	0.530232i	$0.822105\pi$
$830$	−0.177093	−0.00614698
$831$	−4.83047	−0.167567
$832$	−23.7802	−0.824431
$833$	0	0
$834$	1.94524	0.0673582
$835$	1.16392	0.0402790
$836$	48.0291	1.66112
$837$	−2.13135	−0.0736704
$838$	1.66466	0.0575048
$839$	17.3617	0.599391	0.299695	−	0.954035i	$-0.403115\pi$
$839$	17.3617	0.599391	0.299695	+	0.954035i	$0.403115\pi$
$840$	0	0
$841$	1.78988	0.0617200
$842$	1.83271	0.0631595
$843$	−19.2824	−0.664120
$844$	−47.4482	−1.63323
$845$	−2.98792	−0.102788
$846$	0.385050	0.0132383
$847$	0	0
$848$	32.5804	1.11882
$849$	14.7671	0.506805
$850$	2.19961	0.0754459
$851$	6.23725	0.213810
$852$	74.3501	2.54719
$853$	−16.2893	−0.557735	−0.278867	−	0.960330i	$-0.589959\pi$
$853$	−16.2893	−0.557735	−0.278867	+	0.960330i	$0.589959\pi$
$854$	0	0
$855$	−8.18280	−0.279846
$856$	4.55320	0.155625
$857$	19.5205	0.666809	0.333405	−	0.942784i	$-0.391802\pi$
$857$	19.5205	0.666809	0.333405	+	0.942784i	$0.391802\pi$
$858$	−2.79496	−0.0954183
$859$	−43.0960	−1.47042	−0.735208	−	0.677841i	$-0.762916\pi$
$859$	−43.0960	−1.47042	−0.735208	+	0.677841i	$0.762916\pi$
$860$	−7.86894	−0.268329
$861$	0	0
$862$	0.226967	0.00773054
$863$	39.6169	1.34857	0.674287	−	0.738469i	$-0.264451\pi$
$863$	39.6169	1.34857	0.674287	+	0.738469i	$0.264451\pi$
$864$	2.12737	0.0723746
$865$	2.91356	0.0990640
$866$	0.939604	0.0319290
$867$	41.0015	1.39248
$868$	0	0
$869$	−4.82174	−0.163566
$870$	0.838794	0.0284378
$871$	−39.7712	−1.34759
$872$	5.40863	0.183159
$873$	−20.0352	−0.678088
$874$	1.09824	0.0371485
$875$	0	0
$876$	−71.1127	−2.40268
$877$	−53.8932	−1.81984	−0.909922	−	0.414780i	$-0.863859\pi$
$877$	−53.8932	−1.81984	−0.909922	+	0.414780i	$0.863859\pi$
$878$	0.500349	0.0168859
$879$	−32.6967	−1.10283
$880$	15.1149	0.509523
$881$	−5.58577	−0.188190	−0.0940948	−	0.995563i	$-0.529996\pi$
$881$	−5.58577	−0.188190	−0.0940948	+	0.995563i	$0.529996\pi$
$882$	0	0
$883$	24.3533	0.819553	0.409776	−	0.912186i	$-0.365607\pi$
$883$	24.3533	0.819553	0.409776	+	0.912186i	$0.365607\pi$
$884$	−35.9251	−1.20829
$885$	−9.47245	−0.318413
$886$	2.82253	0.0948246
$887$	16.4606	0.552693	0.276346	−	0.961058i	$-0.410876\pi$
$887$	16.4606	0.552693	0.276346	+	0.961058i	$0.410876\pi$
$888$	1.83347	0.0615274
$889$	0	0
$890$	0.662380	0.0222030
$891$	−52.6974	−1.76543
$892$	15.0613	0.504290
$893$	10.9867	0.367657
$894$	−0.0170600	−0.000570572 0
$895$	7.67405	0.256515
$896$	0	0
$897$	17.7223	0.591732
$898$	−2.03247	−0.0678243
$899$	−5.62821	−0.187711
$900$	18.0058	0.600192
$901$	48.8615	1.62781
$902$	−3.20558	−0.106734
$903$	0	0
$904$	−3.02865	−0.100731
$905$	−5.64008	−0.187483
$906$	−3.12134	−0.103700
$907$	30.0872	0.999028	0.499514	−	0.866306i	$-0.333512\pi$
$907$	30.0872	0.999028	0.499514	+	0.866306i	$0.333512\pi$
$908$	−23.8718	−0.792214
$909$	22.9979	0.762792
$910$	0	0
$911$	19.1588	0.634758	0.317379	−	0.948299i	$-0.397197\pi$
$911$	19.1588	0.634758	0.317379	+	0.948299i	$0.397197\pi$
$912$	−44.5189	−1.47417
$913$	12.7147	0.420794
$914$	0.359511	0.0118916
$915$	25.9380	0.857485
$916$	12.2764	0.405623
$917$	0	0
$918$	1.05710	0.0348894
$919$	−18.7329	−0.617940	−0.308970	−	0.951072i	$-0.599984\pi$
$919$	−18.7329	−0.617940	−0.308970	+	0.951072i	$0.599984\pi$
$920$	0.694999	0.0229134
$921$	−13.5309	−0.445859
$922$	2.88745	0.0950931
$923$	−50.3530	−1.65739
$924$	0	0
$925$	−10.5222	−0.345968
$926$	2.78682	0.0915805
$927$	12.8704	0.422720
$928$	5.61769	0.184410
$929$	−38.4994	−1.26312	−0.631562	−	0.775325i	$-0.717586\pi$
$929$	−38.4994	−1.26312	−0.631562	+	0.775325i	$0.717586\pi$
$930$	−0.153327	−0.00502778
$931$	0	0
$932$	−26.9168	−0.881689
$933$	−66.0220	−2.16146
$934$	−3.08272	−0.100870
$935$	22.6681	0.741326
$936$	2.12483	0.0694522
$937$	−31.7402	−1.03691	−0.518454	−	0.855106i	$-0.673492\pi$
$937$	−31.7402	−1.03691	−0.518454	+	0.855106i	$0.673492\pi$
$938$	0	0
$939$	25.6801	0.838037
$940$	3.47012	0.113183
$941$	−20.7177	−0.675378	−0.337689	−	0.941258i	$-0.609645\pi$
$941$	−20.7177	−0.675378	−0.337689	+	0.941258i	$0.609645\pi$
$942$	−2.57511	−0.0839016
$943$	20.3260	0.661907
$944$	−21.0197	−0.684133
$945$	0	0
$946$	−2.03736	−0.0662402
$947$	−37.9316	−1.23261	−0.616305	−	0.787508i	$-0.711371\pi$
$947$	−37.9316	−1.23261	−0.616305	+	0.787508i	$0.711371\pi$
$948$	4.48558	0.145685
$949$	48.1606	1.56336
$950$	−1.85272	−0.0601103
$951$	−30.1338	−0.977156
$952$	0	0
$953$	21.6747	0.702113	0.351057	−	0.936354i	$-0.385822\pi$
$953$	21.6747	0.702113	0.351057	+	0.936354i	$0.385822\pi$
$954$	−1.44238	−0.0466988
$955$	−3.31326	−0.107214
$956$	−20.4749	−0.662207
$957$	−60.2226	−1.94672
$958$	0.295267	0.00953964
$959$	0	0
$960$	−13.9588	−0.450518
$961$	−29.9712	−0.966813
$962$	−0.619736	−0.0199811
$963$	27.7975	0.895763
$964$	4.65431	0.149905
$965$	−1.22654	−0.0394838
$966$	0	0
$967$	−20.9033	−0.672205	−0.336103	−	0.941825i	$-0.609109\pi$
$967$	−20.9033	−0.672205	−0.336103	+	0.941825i	$0.609109\pi$
$968$	4.14612	0.133261
$969$	−66.7659	−2.14483
$970$	0.651129	0.0209065
$971$	−24.2669	−0.778762	−0.389381	−	0.921077i	$-0.627311\pi$
$971$	−24.2669	−0.778762	−0.389381	+	0.921077i	$0.627311\pi$
$972$	36.4610	1.16949
$973$	0	0
$974$	−0.147242	−0.00471794
$975$	−29.8975	−0.957485
$976$	57.5574	1.84237
$977$	40.7081	1.30237	0.651184	−	0.758920i	$-0.274273\pi$
$977$	40.7081	1.30237	0.651184	+	0.758920i	$0.274273\pi$
$978$	−3.22822	−0.103227
$979$	−47.5567	−1.51992
$980$	0	0
$981$	33.0200	1.05425
$982$	−3.28416	−0.104802
$983$	5.66733	0.180760	0.0903798	−	0.995907i	$-0.471192\pi$
$983$	5.66733	0.180760	0.0903798	+	0.995907i	$0.471192\pi$
$984$	5.97495	0.190474
$985$	−6.22256	−0.198267
$986$	2.79145	0.0888978
$987$	0	0
$988$	30.2597	0.962688
$989$	12.9185	0.410785
$990$	−0.669157	−0.0212672
$991$	4.61955	0.146745	0.0733724	−	0.997305i	$-0.476624\pi$
$991$	4.61955	0.146745	0.0733724	+	0.997305i	$0.476624\pi$
$992$	−1.02688	−0.0326036
$993$	−52.3982	−1.66281
$994$	0	0
$995$	−15.3092	−0.485336
$996$	−11.8282	−0.374792
$997$	−4.09577	−0.129714	−0.0648571	−	0.997895i	$-0.520659\pi$
$997$	−4.09577	−0.129714	−0.0648571	+	0.997895i	$0.520659\pi$
$998$	−0.851211	−0.0269446
$999$	−5.05680	−0.159990

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3871.2.a.i.1.12	yes	24
7.6	odd	2	inner	3871.2.a.i.1.11	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3871.2.a.i.1.11	✓	24	7.6	odd	2	inner
3871.2.a.i.1.12	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)

\(f(q)\)	\(=\)	\(q-0.0847728 q^{2} +2.25088 q^{3} -1.99281 q^{4} +0.792214 q^{5} -0.190813 q^{6} +0.338482 q^{8} +2.06645 q^{9} -0.0671582 q^{10} +4.82174 q^{11} -4.48558 q^{12} +3.03783 q^{13} +1.78318 q^{15} +3.95693 q^{16} +5.93429 q^{17} -0.175179 q^{18} -4.99844 q^{19} -1.57874 q^{20} -0.408752 q^{22} +2.59183 q^{23} +0.761882 q^{24} -4.37240 q^{25} -0.257525 q^{26} -2.10130 q^{27} -5.54886 q^{29} -0.151165 q^{30} +1.01430 q^{31} -1.01240 q^{32} +10.8531 q^{33} -0.503066 q^{34} -4.11805 q^{36} +2.40651 q^{37} +0.423732 q^{38} +6.83778 q^{39} +0.268150 q^{40} +7.84236 q^{41} +4.98433 q^{43} -9.60882 q^{44} +1.63707 q^{45} -0.219717 q^{46} -2.19804 q^{47} +8.90657 q^{48} +0.370660 q^{50} +13.3574 q^{51} -6.05382 q^{52} +8.23376 q^{53} +0.178133 q^{54} +3.81985 q^{55} -11.2509 q^{57} +0.470393 q^{58} -5.31212 q^{59} -3.55354 q^{60} +14.5460 q^{61} -0.0859851 q^{62} -7.82804 q^{64} +2.40661 q^{65} -0.920051 q^{66} -13.0920 q^{67} -11.8259 q^{68} +5.83389 q^{69} -16.5753 q^{71} +0.699457 q^{72} +15.8536 q^{73} -0.204006 q^{74} -9.84173 q^{75} +9.96095 q^{76} -0.579658 q^{78} -1.00000 q^{79} +3.13474 q^{80} -10.9291 q^{81} -0.664819 q^{82} +2.63695 q^{83} +4.70123 q^{85} -0.422536 q^{86} -12.4898 q^{87} +1.63207 q^{88} -9.86298 q^{89} -0.138779 q^{90} -5.16503 q^{92} +2.28307 q^{93} +0.186334 q^{94} -3.95983 q^{95} -2.27880 q^{96} -9.69545 q^{97} +9.96389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{2} + 28 q^{4} + 18 q^{8} + 50 q^{9} + 14 q^{11} + 18 q^{15} + 36 q^{16} + 10 q^{18} - 12 q^{22} + 22 q^{23} + 56 q^{25} + 36 q^{29} + 14 q^{30} + 30 q^{32} + 8 q^{36} + 24 q^{37} + 56 q^{39}+ \cdots + 6 q^{99}+O(q^{100}) \) 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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