LMFDB - Embedded newform 3871.2.a.e.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 = [8,3,-3,7,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

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

Embedding invariants

Embedding label			1.4
Root			$-0.174832$ of defining polynomial
Character	$\chi$	$=$	3871.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.174832 q^{2} +2.31835 q^{3} -1.96943 q^{4} +1.01700 q^{5} -0.405323 q^{6} +0.693986 q^{8} +2.37476 q^{9} -0.177804 q^{10} -4.96725 q^{11} -4.56584 q^{12} +5.62124 q^{13} +2.35776 q^{15} +3.81754 q^{16} -3.76485 q^{17} -0.415185 q^{18} -3.14061 q^{19} -2.00291 q^{20} +0.868437 q^{22} +4.19183 q^{23} +1.60890 q^{24} -3.96572 q^{25} -0.982775 q^{26} -1.44953 q^{27} -7.96005 q^{29} -0.412213 q^{30} -9.91356 q^{31} -2.05540 q^{32} -11.5158 q^{33} +0.658218 q^{34} -4.67693 q^{36} -0.106445 q^{37} +0.549081 q^{38} +13.0320 q^{39} +0.705782 q^{40} -8.41414 q^{41} +0.0223469 q^{43} +9.78267 q^{44} +2.41512 q^{45} -0.732868 q^{46} -5.74459 q^{47} +8.85039 q^{48} +0.693336 q^{50} -8.72825 q^{51} -11.0707 q^{52} +3.47513 q^{53} +0.253426 q^{54} -5.05168 q^{55} -7.28104 q^{57} +1.39168 q^{58} +10.8960 q^{59} -4.64345 q^{60} +10.6898 q^{61} +1.73321 q^{62} -7.27572 q^{64} +5.71679 q^{65} +2.01334 q^{66} +8.90858 q^{67} +7.41462 q^{68} +9.71814 q^{69} -4.03217 q^{71} +1.64805 q^{72} +12.9662 q^{73} +0.0186101 q^{74} -9.19393 q^{75} +6.18522 q^{76} -2.27842 q^{78} +1.00000 q^{79} +3.88243 q^{80} -10.4848 q^{81} +1.47106 q^{82} -5.20485 q^{83} -3.82884 q^{85} -0.00390697 q^{86} -18.4542 q^{87} -3.44720 q^{88} -9.29307 q^{89} -0.422242 q^{90} -8.25553 q^{92} -22.9831 q^{93} +1.00434 q^{94} -3.19399 q^{95} -4.76514 q^{96} +4.37822 q^{97} -11.7960 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{2} - 3 q^{3} + 7 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{8} - q^{9} - 10 q^{10} + 3 q^{11} - 10 q^{12} - 3 q^{13} + 11 q^{15} + 13 q^{16} - 14 q^{17} + 4 q^{18} - 12 q^{19} - 12 q^{20} - 15 q^{22}+ \cdots - 25 q^{99}+O(q^{100}) $ 8 * q + 3 * q^2 - 3 * q^3 + 7 * q^4 - 4 * q^5 - 6 * q^6 + 6 * q^8 - q^9 - 10 * q^10 + 3 * q^11 - 10 * q^12 - 3 * q^13 + 11 * q^15 + 13 * q^16 - 14 * q^17 + 4 * q^18 - 12 * q^19 - 12 * q^20 - 15 * q^22 + 17 * q^23 - 14 * q^24 - 6 * q^25 + 13 * q^26 + 6 * q^27 - 6 * q^29 + 17 * q^30 - 15 * q^31 + 5 * q^32 - 2 * q^33 + 2 * q^34 - 33 * q^36 - 2 * q^37 - 19 * q^38 + 6 * q^39 + 13 * q^40 - 8 * q^41 + 9 * q^43 + 7 * q^44 - 5 * q^45 - 24 * q^46 - 34 * q^47 + 9 * q^48 - 3 * q^50 - 11 * q^51 - 17 * q^52 + 2 * q^53 + 10 * q^54 - 16 * q^55 + 6 * q^57 - 14 * q^58 - 21 * q^59 - 14 * q^60 - 11 * q^61 + 15 * q^62 - 16 * q^64 - 5 * q^65 + 18 * q^66 + 14 * q^67 - 7 * q^68 + 8 * q^69 - 2 * q^71 - 18 * q^72 - 7 * q^73 - 8 * q^74 - 28 * q^75 + 23 * q^76 - 8 * q^78 + 8 * q^79 - 6 * q^80 - 28 * q^81 + 10 * q^82 - 32 * q^83 - 9 * q^85 - 6 * q^86 - 43 * q^87 - 48 * q^88 - 6 * q^89 - q^90 - 3 * q^92 - 12 * q^93 - 33 * q^94 - 15 * q^95 + 9 * q^96 + 39 * q^97 - 25 * q^99

Coefficient data

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

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

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

$2$	−0.174832	−0.123625	−0.0618126	−	0.998088i	$-0.519688\pi$
$2$	−0.174832	−0.123625	−0.0618126	+	0.998088i	$0.519688\pi$
$3$	2.31835	1.33850	0.669251	−	0.743037i	$-0.266615\pi$
$3$	2.31835	1.33850	0.669251	+	0.743037i	$0.266615\pi$
$4$	−1.96943	−0.984717
$5$	1.01700	0.454815	0.227408	−	0.973800i	$-0.426975\pi$
$5$	1.01700	0.454815	0.227408	+	0.973800i	$0.426975\pi$
$6$	−0.405323	−0.165473
$7$	0	0
$7$	0	0
$8$	0.693986	0.245361
$9$	2.37476	0.791586
$10$	−0.177804	−0.0562266
$11$	−4.96725	−1.49768	−0.748841	−	0.662749i	$-0.769389\pi$
$11$	−4.96725	−1.49768	−0.748841	+	0.662749i	$0.769389\pi$
$12$	−4.56584	−1.31804
$13$	5.62124	1.55905	0.779526	−	0.626370i	$-0.215460\pi$
$13$	5.62124	1.55905	0.779526	+	0.626370i	$0.215460\pi$
$14$	0	0
$15$	2.35776	0.608771
$16$	3.81754	0.954384
$17$	−3.76485	−0.913110	−0.456555	−	0.889695i	$-0.650917\pi$
$17$	−3.76485	−0.913110	−0.456555	+	0.889695i	$0.650917\pi$
$18$	−0.415185	−0.0978599
$19$	−3.14061	−0.720505	−0.360253	−	0.932855i	$-0.617310\pi$
$19$	−3.14061	−0.720505	−0.360253	+	0.932855i	$0.617310\pi$
$20$	−2.00291	−0.447864
$21$	0	0
$22$	0.868437	0.185151
$23$	4.19183	0.874057	0.437029	−	0.899448i	$-0.356031\pi$
$23$	4.19183	0.874057	0.437029	+	0.899448i	$0.356031\pi$
$24$	1.60890	0.328416
$25$	−3.96572	−0.793143
$26$	−0.982775	−0.192738
$27$	−1.44953	−0.278963
$28$	0	0
$29$	−7.96005	−1.47814	−0.739072	−	0.673626i	$-0.764736\pi$
$29$	−7.96005	−1.47814	−0.739072	+	0.673626i	$0.764736\pi$
$30$	−0.412213	−0.0752594
$31$	−9.91356	−1.78053	−0.890264	−	0.455445i	$-0.849480\pi$
$31$	−9.91356	−1.78053	−0.890264	+	0.455445i	$0.849480\pi$
$32$	−2.05540	−0.363347
$33$	−11.5158	−2.00465
$34$	0.658218	0.112883
$35$	0	0
$36$	−4.67693	−0.779488
$37$	−0.106445	−0.0174995	−0.00874976	−	0.999962i	$-0.502785\pi$
$37$	−0.106445	−0.0174995	−0.00874976	+	0.999962i	$0.502785\pi$
$38$	0.549081	0.0890726
$39$	13.0320	2.08679
$40$	0.705782	0.111594
$41$	−8.41414	−1.31407	−0.657034	−	0.753861i	$-0.728189\pi$
$41$	−8.41414	−1.31407	−0.657034	+	0.753861i	$0.728189\pi$
$42$	0	0
$43$	0.0223469	0.00340787	0.00170394	−	0.999999i	$-0.499458\pi$
$43$	0.0223469	0.00340787	0.00170394	+	0.999999i	$0.499458\pi$
$44$	9.78267	1.47479
$45$	2.41512	0.360025
$46$	−0.732868	−0.108055
$47$	−5.74459	−0.837934	−0.418967	−	0.908001i	$-0.637608\pi$
$47$	−5.74459	−0.837934	−0.418967	+	0.908001i	$0.637608\pi$
$48$	8.85039	1.27744
$49$	0	0
$50$	0.693336	0.0980525
$51$	−8.72825	−1.22220
$52$	−11.0707	−1.53522
$53$	3.47513	0.477346	0.238673	−	0.971100i	$-0.423288\pi$
$53$	3.47513	0.477346	0.238673	+	0.971100i	$0.423288\pi$
$54$	0.253426	0.0344869
$55$	−5.05168	−0.681169
$56$	0	0
$57$	−7.28104	−0.964397
$58$	1.39168	0.182736
$59$	10.8960	1.41854	0.709272	−	0.704935i	$-0.249024\pi$
$59$	10.8960	1.41854	0.709272	+	0.704935i	$0.249024\pi$
$60$	−4.64345	−0.599467
$61$	10.6898	1.36869	0.684343	−	0.729160i	$-0.260089\pi$
$61$	10.6898	1.36869	0.684343	+	0.729160i	$0.260089\pi$
$62$	1.73321	0.220118
$63$	0	0
$64$	−7.27572	−0.909465
$65$	5.71679	0.709081
$66$	2.01334	0.247825
$67$	8.90858	1.08836	0.544178	−	0.838970i	$-0.316841\pi$
$67$	8.90858	1.08836	0.544178	+	0.838970i	$0.316841\pi$
$68$	7.41462	0.899155
$69$	9.71814	1.16993
$70$	0	0
$71$	−4.03217	−0.478530	−0.239265	−	0.970954i	$-0.576906\pi$
$71$	−4.03217	−0.478530	−0.239265	+	0.970954i	$0.576906\pi$
$72$	1.64805	0.194224
$73$	12.9662	1.51758	0.758790	−	0.651335i	$-0.225791\pi$
$73$	12.9662	1.51758	0.758790	+	0.651335i	$0.225791\pi$
$74$	0.0186101	0.00216338
$75$	−9.19393	−1.06162
$76$	6.18522	0.709494
$77$	0	0
$78$	−2.27842	−0.257980
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	3.88243	0.434068
$81$	−10.4848	−1.16498
$82$	1.47106	0.162452
$83$	−5.20485	−0.571306	−0.285653	−	0.958333i	$-0.592211\pi$
$83$	−5.20485	−0.571306	−0.285653	+	0.958333i	$0.592211\pi$
$84$	0	0
$85$	−3.82884	−0.415296
$86$	−0.00390697	−0.000421299 0
$87$	−18.4542	−1.97850
$88$	−3.44720	−0.367473
$89$	−9.29307	−0.985064	−0.492532	−	0.870294i	$-0.663929\pi$
$89$	−9.29307	−0.985064	−0.492532	+	0.870294i	$0.663929\pi$
$90$	−0.422242	−0.0445082
$91$	0	0
$92$	−8.25553	−0.860699
$93$	−22.9831	−2.38324
$94$	1.00434	0.103590
$95$	−3.19399	−0.327697
$96$	−4.76514	−0.486340
$97$	4.37822	0.444541	0.222270	−	0.974985i	$-0.428653\pi$
$97$	4.37822	0.444541	0.222270	+	0.974985i	$0.428653\pi$
$98$	0	0
$99$	−11.7960	−1.18554
$100$	7.81021	0.781021
$101$	−16.3293	−1.62483	−0.812414	−	0.583081i	$-0.801847\pi$
$101$	−16.3293	−1.62483	−0.812414	+	0.583081i	$0.801847\pi$
$102$	1.52598	0.151095
$103$	−5.66701	−0.558387	−0.279193	−	0.960235i	$-0.590067\pi$
$103$	−5.66701	−0.558387	−0.279193	+	0.960235i	$0.590067\pi$
$104$	3.90106	0.382531
$105$	0	0
$106$	−0.607566	−0.0590120
$107$	3.18070	0.307490	0.153745	−	0.988111i	$-0.450867\pi$
$107$	3.18070	0.307490	0.153745	+	0.988111i	$0.450867\pi$
$108$	2.85476	0.274699
$109$	−12.8742	−1.23312	−0.616560	−	0.787308i	$-0.711474\pi$
$109$	−12.8742	−1.23312	−0.616560	+	0.787308i	$0.711474\pi$
$110$	0.883198	0.0842096
$111$	−0.246778	−0.0234231
$112$	0	0
$113$	−7.92079	−0.745125	−0.372562	−	0.928007i	$-0.621521\pi$
$113$	−7.92079	−0.745125	−0.372562	+	0.928007i	$0.621521\pi$
$114$	1.27296	0.119224
$115$	4.26308	0.397534
$116$	15.6768	1.45555
$117$	13.3491	1.23412
$118$	−1.90498	−0.175368
$119$	0	0
$120$	1.63625	0.149369
$121$	13.6736	1.24305
$122$	−1.86892	−0.169204
$123$	−19.5069	−1.75888
$124$	19.5241	1.75332
$125$	−9.11811	−0.815549
$126$	0	0
$127$	16.9952	1.50808	0.754040	−	0.656828i	$-0.228102\pi$
$127$	16.9952	1.50808	0.754040	+	0.656828i	$0.228102\pi$
$128$	5.38283	0.475780
$129$	0.0518080	0.00456144
$130$	−0.999480	−0.0876602
$131$	3.84363	0.335820	0.167910	−	0.985802i	$-0.446298\pi$
$131$	3.84363	0.335820	0.167910	+	0.985802i	$0.446298\pi$
$132$	22.6797	1.97401
$133$	0	0
$134$	−1.55751	−0.134548
$135$	−1.47417	−0.126877
$136$	−2.61275	−0.224042
$137$	−11.3973	−0.973738	−0.486869	−	0.873475i	$-0.661861\pi$
$137$	−11.3973	−0.973738	−0.486869	+	0.873475i	$0.661861\pi$
$138$	−1.69905	−0.144632
$139$	−13.7334	−1.16485	−0.582425	−	0.812884i	$-0.697896\pi$
$139$	−13.7334	−1.16485	−0.582425	+	0.812884i	$0.697896\pi$
$140$	0	0
$141$	−13.3180	−1.12158
$142$	0.704954	0.0591584
$143$	−27.9221	−2.33496
$144$	9.06572	0.755477
$145$	−8.09536	−0.672283
$146$	−2.26691	−0.187611
$147$	0	0
$148$	0.209637	0.0172321
$149$	12.1713	0.997111	0.498556	−	0.866858i	$-0.333864\pi$
$149$	12.1713	0.997111	0.498556	+	0.866858i	$0.333864\pi$
$150$	1.60740	0.131243
$151$	−23.5784	−1.91879	−0.959393	−	0.282073i	$-0.908978\pi$
$151$	−23.5784	−1.91879	−0.959393	+	0.282073i	$0.908978\pi$
$152$	−2.17954	−0.176784
$153$	−8.94060	−0.722805
$154$	0	0
$155$	−10.0821	−0.809811
$156$	−25.6657	−2.05490
$157$	−5.23538	−0.417829	−0.208914	−	0.977934i	$-0.566993\pi$
$157$	−5.23538	−0.417829	−0.208914	+	0.977934i	$0.566993\pi$
$158$	−0.174832	−0.0139089
$159$	8.05658	0.638928
$160$	−2.09034	−0.165256
$161$	0	0
$162$	1.83308	0.144021
$163$	5.74363	0.449876	0.224938	−	0.974373i	$-0.427782\pi$
$163$	5.74363	0.449876	0.224938	+	0.974373i	$0.427782\pi$
$164$	16.5711	1.29398
$165$	−11.7116	−0.911745
$166$	0.909976	0.0706279
$167$	−15.7035	−1.21517	−0.607587	−	0.794253i	$-0.707862\pi$
$167$	−15.7035	−1.21517	−0.607587	+	0.794253i	$0.707862\pi$
$168$	0	0
$169$	18.5984	1.43064
$170$	0.669406	0.0513411
$171$	−7.45819	−0.570342
$172$	−0.0440108	−0.00335579
$173$	−14.5568	−1.10673	−0.553367	−	0.832937i	$-0.686657\pi$
$173$	−14.5568	−1.10673	−0.553367	+	0.832937i	$0.686657\pi$
$174$	3.22639	0.244592
$175$	0	0
$176$	−18.9627	−1.42936
$177$	25.2609	1.89872
$178$	1.62473	0.121779
$179$	13.5591	1.01345	0.506727	−	0.862106i	$-0.330855\pi$
$179$	13.5591	1.01345	0.506727	+	0.862106i	$0.330855\pi$
$180$	−4.75642	−0.354523
$181$	−14.5676	−1.08280	−0.541400	−	0.840765i	$-0.682106\pi$
$181$	−14.5676	−1.08280	−0.541400	+	0.840765i	$0.682106\pi$
$182$	0	0
$183$	24.7827	1.83199
$184$	2.90907	0.214460
$185$	−0.108255	−0.00795905
$186$	4.01820	0.294628
$187$	18.7010	1.36755
$188$	11.3136	0.825128
$189$	0	0
$190$	0.558414	0.0405116
$191$	−11.6478	−0.842803	−0.421401	−	0.906874i	$-0.638462\pi$
$191$	−11.6478	−0.842803	−0.421401	+	0.906874i	$0.638462\pi$
$192$	−16.8677	−1.21732
$193$	0.432888	0.0311599	0.0155800	−	0.999879i	$-0.495041\pi$
$193$	0.432888	0.0311599	0.0155800	+	0.999879i	$0.495041\pi$
$194$	−0.765454	−0.0549564
$195$	13.2535	0.949105
$196$	0	0
$197$	22.7867	1.62349	0.811744	−	0.584014i	$-0.198519\pi$
$197$	22.7867	1.62349	0.811744	+	0.584014i	$0.198519\pi$
$198$	2.06233	0.146563
$199$	6.74403	0.478071	0.239036	−	0.971011i	$-0.423169\pi$
$199$	6.74403	0.478071	0.239036	+	0.971011i	$0.423169\pi$
$200$	−2.75215	−0.194606
$201$	20.6532	1.45677
$202$	2.85489	0.200870
$203$	0	0
$204$	17.1897	1.20352
$205$	−8.55716	−0.597658
$206$	0.990777	0.0690307
$207$	9.95458	0.691891
$208$	21.4593	1.48793
$209$	15.6002	1.07909
$210$	0	0
$211$	−17.8692	−1.23017	−0.615085	−	0.788461i	$-0.710878\pi$
$211$	−17.8692	−1.23017	−0.615085	+	0.788461i	$0.710878\pi$
$212$	−6.84404	−0.470051
$213$	−9.34798	−0.640513
$214$	−0.556090	−0.0380135
$215$	0.0227268	0.00154995
$216$	−1.00596	−0.0684466
$217$	0	0
$218$	2.25082	0.152445
$219$	30.0603	2.03128
$220$	9.94895	0.670758
$221$	−21.1631	−1.42359
$222$	0.0431448	0.00289569
$223$	20.8494	1.39618	0.698091	−	0.716009i	$-0.254033\pi$
$223$	20.8494	1.39618	0.698091	+	0.716009i	$0.254033\pi$
$224$	0	0
$225$	−9.41761	−0.627841
$226$	1.38481	0.0921162
$227$	−28.1258	−1.86677	−0.933387	−	0.358872i	$-0.883162\pi$
$227$	−28.1258	−1.86677	−0.933387	+	0.358872i	$0.883162\pi$
$228$	14.3395	0.949658
$229$	−13.4191	−0.886756	−0.443378	−	0.896335i	$-0.646220\pi$
$229$	−13.4191	−0.886756	−0.443378	+	0.896335i	$0.646220\pi$
$230$	−0.745325	−0.0491453
$231$	0	0
$232$	−5.52416	−0.362679
$233$	0.839988	0.0550295	0.0275147	−	0.999621i	$-0.491241\pi$
$233$	0.839988	0.0550295	0.0275147	+	0.999621i	$0.491241\pi$
$234$	−2.33385	−0.152569
$235$	−5.84223	−0.381105
$236$	−21.4590	−1.39686
$237$	2.31835	0.150593
$238$	0	0
$239$	17.2643	1.11673	0.558367	−	0.829594i	$-0.311428\pi$
$239$	17.2643	1.11673	0.558367	+	0.829594i	$0.311428\pi$
$240$	9.00083	0.581001
$241$	14.3630	0.925199	0.462600	−	0.886567i	$-0.346917\pi$
$241$	14.3630	0.925199	0.462600	+	0.886567i	$0.346917\pi$
$242$	−2.39058	−0.153673
$243$	−19.9589	−1.28036
$244$	−21.0528	−1.34777
$245$	0	0
$246$	3.41045	0.217442
$247$	−17.6541	−1.12331
$248$	−6.87987	−0.436872
$249$	−12.0667	−0.764694
$250$	1.59414	0.100822
$251$	18.2123	1.14955	0.574774	−	0.818312i	$-0.305090\pi$
$251$	18.2123	1.14955	0.574774	+	0.818312i	$0.305090\pi$
$252$	0	0
$253$	−20.8219	−1.30906
$254$	−2.97131	−0.186437
$255$	−8.87661	−0.555875
$256$	13.6103	0.850647
$257$	−13.7295	−0.856423	−0.428211	−	0.903679i	$-0.640856\pi$
$257$	−13.7295	−0.856423	−0.428211	+	0.903679i	$0.640856\pi$
$258$	−0.00905773	−0.000563909 0
$259$	0	0
$260$	−11.2588	−0.698244
$261$	−18.9032	−1.17008
$262$	−0.671992	−0.0415158
$263$	20.6349	1.27240	0.636201	−	0.771523i	$-0.280505\pi$
$263$	20.6349	1.27240	0.636201	+	0.771523i	$0.280505\pi$
$264$	−7.99183	−0.491863
$265$	3.53420	0.217104
$266$	0	0
$267$	−21.5446	−1.31851
$268$	−17.5449	−1.07172
$269$	2.50791	0.152910	0.0764549	−	0.997073i	$-0.475640\pi$
$269$	2.50791	0.152910	0.0764549	+	0.997073i	$0.475640\pi$
$270$	0.257733	0.0156851
$271$	15.3358	0.931584	0.465792	−	0.884894i	$-0.345769\pi$
$271$	15.3358	0.931584	0.465792	+	0.884894i	$0.345769\pi$
$272$	−14.3724	−0.871458
$273$	0	0
$274$	1.99262	0.120379
$275$	19.6987	1.18788
$276$	−19.1392	−1.15205
$277$	−7.92951	−0.476438	−0.238219	−	0.971211i	$-0.576564\pi$
$277$	−7.92951	−0.476438	−0.238219	+	0.971211i	$0.576564\pi$
$278$	2.40104	0.144005
$279$	−23.5423	−1.40944
$280$	0	0
$281$	14.6334	0.872958	0.436479	−	0.899715i	$-0.356225\pi$
$281$	14.6334	0.872958	0.436479	+	0.899715i	$0.356225\pi$
$282$	2.32841	0.138655
$283$	−9.05580	−0.538311	−0.269156	−	0.963097i	$-0.586745\pi$
$283$	−9.05580	−0.538311	−0.269156	+	0.963097i	$0.586745\pi$
$284$	7.94108	0.471217
$285$	−7.40480	−0.438623
$286$	4.88169	0.288660
$287$	0	0
$288$	−4.88108	−0.287620
$289$	−2.82591	−0.166230
$290$	1.41533	0.0831111
$291$	10.1502	0.595018
$292$	−25.5361	−1.49439
$293$	−32.4375	−1.89502	−0.947510	−	0.319726i	$-0.896409\pi$
$293$	−32.4375	−1.89502	−0.947510	+	0.319726i	$0.896409\pi$
$294$	0	0
$295$	11.0812	0.645175
$296$	−0.0738716	−0.00429370
$297$	7.20020	0.417798
$298$	−2.12794	−0.123268
$299$	23.5633	1.36270
$300$	18.1068	1.04540
$301$	0	0
$302$	4.12228	0.237210
$303$	−37.8571	−2.17483
$304$	−11.9894	−0.687639
$305$	10.8715	0.622499
$306$	1.56311	0.0893569
$307$	18.9900	1.08382	0.541908	−	0.840438i	$-0.317702\pi$
$307$	18.9900	1.08382	0.541908	+	0.840438i	$0.317702\pi$
$308$	0	0
$309$	−13.1381	−0.747401
$310$	1.76267	0.100113
$311$	−2.39628	−0.135881	−0.0679403	−	0.997689i	$-0.521643\pi$
$311$	−2.39628	−0.135881	−0.0679403	+	0.997689i	$0.521643\pi$
$312$	9.04404	0.512018
$313$	−6.57897	−0.371865	−0.185933	−	0.982562i	$-0.559531\pi$
$313$	−6.57897	−0.371865	−0.185933	+	0.982562i	$0.559531\pi$
$314$	0.915314	0.0516541
$315$	0	0
$316$	−1.96943	−0.110789
$317$	−10.5945	−0.595048	−0.297524	−	0.954714i	$-0.596161\pi$
$317$	−10.5945	−0.595048	−0.297524	+	0.954714i	$0.596161\pi$
$318$	−1.40855	−0.0789876
$319$	39.5396	2.21379
$320$	−7.39939	−0.413639
$321$	7.37399	0.411576
$322$	0	0
$323$	11.8239	0.657901
$324$	20.6491	1.14717
$325$	−22.2922	−1.23655
$326$	−1.00417	−0.0556160
$327$	−29.8468	−1.65053
$328$	−5.83929	−0.322421
$329$	0	0
$330$	2.04756	0.112715
$331$	−12.1120	−0.665736	−0.332868	−	0.942973i	$-0.608016\pi$
$331$	−12.1120	−0.665736	−0.332868	+	0.942973i	$0.608016\pi$
$332$	10.2506	0.562575
$333$	−0.252782	−0.0138524
$334$	2.74548	0.150226
$335$	9.06001	0.495001
$336$	0	0
$337$	7.11902	0.387798	0.193899	−	0.981022i	$-0.437887\pi$
$337$	7.11902	0.387798	0.193899	+	0.981022i	$0.437887\pi$
$338$	−3.25160	−0.176864
$339$	−18.3632	−0.997351
$340$	7.54065	0.408949
$341$	49.2431	2.66667
$342$	1.30393	0.0705086
$343$	0	0
$344$	0.0155084	0.000836159 0
$345$	9.88333	0.532100
$346$	2.54500	0.136820
$347$	10.3178	0.553886	0.276943	−	0.960886i	$-0.410679\pi$
$347$	10.3178	0.553886	0.276943	+	0.960886i	$0.410679\pi$
$348$	36.3443	1.94826
$349$	−2.74123	−0.146735	−0.0733673	−	0.997305i	$-0.523375\pi$
$349$	−2.74123	−0.146735	−0.0733673	+	0.997305i	$0.523375\pi$
$350$	0	0
$351$	−8.14818	−0.434918
$352$	10.2097	0.544178
$353$	−7.72828	−0.411335	−0.205667	−	0.978622i	$-0.565936\pi$
$353$	−7.72828	−0.411335	−0.205667	+	0.978622i	$0.565936\pi$
$354$	−4.41642	−0.234730
$355$	−4.10070	−0.217643
$356$	18.3021	0.970009
$357$	0	0
$358$	−2.37057	−0.125289
$359$	9.05392	0.477848	0.238924	−	0.971038i	$-0.423205\pi$
$359$	9.05392	0.477848	0.238924	+	0.971038i	$0.423205\pi$
$360$	1.67606	0.0883362
$361$	−9.13657	−0.480872
$362$	2.54689	0.133861
$363$	31.7002	1.66383
$364$	0	0
$365$	13.1866	0.690219
$366$	−4.33282	−0.226480
$367$	−23.4483	−1.22399	−0.611995	−	0.790861i	$-0.709633\pi$
$367$	−23.4483	−1.22399	−0.611995	+	0.790861i	$0.709633\pi$
$368$	16.0025	0.834186
$369$	−19.9815	−1.04020
$370$	0.0189264	0.000983939 0
$371$	0	0
$372$	45.2637	2.34682
$373$	−3.12851	−0.161988	−0.0809941	−	0.996715i	$-0.525810\pi$
$373$	−3.12851	−0.161988	−0.0809941	+	0.996715i	$0.525810\pi$
$374$	−3.26953	−0.169064
$375$	−21.1390	−1.09161
$376$	−3.98666	−0.205596
$377$	−44.7454	−2.30450
$378$	0	0
$379$	32.4090	1.66474	0.832369	−	0.554222i	$-0.186984\pi$
$379$	32.4090	1.66474	0.832369	+	0.554222i	$0.186984\pi$
$380$	6.29036	0.322689
$381$	39.4009	2.01857
$382$	2.03641	0.104192
$383$	21.9878	1.12352	0.561761	−	0.827300i	$-0.310124\pi$
$383$	21.9878	1.12352	0.561761	+	0.827300i	$0.310124\pi$
$384$	12.4793	0.636832
$385$	0	0
$386$	−0.0756828	−0.00385215
$387$	0.0530685	0.00269762
$388$	−8.62261	−0.437747
$389$	−27.9765	−1.41847	−0.709233	−	0.704974i	$-0.750959\pi$
$389$	−27.9765	−1.41847	−0.709233	+	0.704974i	$0.750959\pi$
$390$	−2.31715	−0.117333
$391$	−15.7816	−0.798110
$392$	0	0
$393$	8.91089	0.449495
$394$	−3.98386	−0.200704
$395$	1.01700	0.0511707
$396$	23.2315	1.16742
$397$	29.2626	1.46865	0.734323	−	0.678800i	$-0.237500\pi$
$397$	29.2626	1.46865	0.734323	+	0.678800i	$0.237500\pi$
$398$	−1.17907	−0.0591017
$399$	0	0
$400$	−15.1393	−0.756963
$401$	9.69085	0.483938	0.241969	−	0.970284i	$-0.422207\pi$
$401$	9.69085	0.483938	0.241969	+	0.970284i	$0.422207\pi$
$402$	−3.61086	−0.180093
$403$	−55.7265	−2.77594
$404$	32.1595	1.60000
$405$	−10.6630	−0.529850
$406$	0	0
$407$	0.528741	0.0262087
$408$	−6.05728	−0.299880
$409$	13.6651	0.675698	0.337849	−	0.941200i	$-0.390301\pi$
$409$	13.6651	0.675698	0.337849	+	0.941200i	$0.390301\pi$
$410$	1.49607	0.0738856
$411$	−26.4230	−1.30335
$412$	11.1608	0.549853
$413$	0	0
$414$	−1.74038	−0.0855352
$415$	−5.29332	−0.259839
$416$	−11.5539	−0.566477
$417$	−31.8388	−1.55915
$418$	−2.72742	−0.133403
$419$	13.7918	0.673772	0.336886	−	0.941545i	$-0.390626\pi$
$419$	13.7918	0.673772	0.336886	+	0.941545i	$0.390626\pi$
$420$	0	0
$421$	−16.3813	−0.798375	−0.399188	−	0.916869i	$-0.630708\pi$
$421$	−16.3813	−0.798375	−0.399188	+	0.916869i	$0.630708\pi$
$422$	3.12412	0.152080
$423$	−13.6420	−0.663297
$424$	2.41169	0.117122
$425$	14.9303	0.724227
$426$	1.63433	0.0791836
$427$	0	0
$428$	−6.26418	−0.302791
$429$	−64.7333	−3.12535
$430$	−0.00397338	−0.000191613 0
$431$	16.7256	0.805644	0.402822	−	0.915278i	$-0.368029\pi$
$431$	16.7256	0.805644	0.402822	+	0.915278i	$0.368029\pi$
$432$	−5.53365	−0.266238
$433$	13.9398	0.669905	0.334953	−	0.942235i	$-0.391280\pi$
$433$	13.9398	0.669905	0.334953	+	0.942235i	$0.391280\pi$
$434$	0	0
$435$	−18.7679	−0.899851
$436$	25.3548	1.21427
$437$	−13.1649	−0.629763
$438$	−5.25551	−0.251118
$439$	−24.5976	−1.17398	−0.586990	−	0.809594i	$-0.699687\pi$
$439$	−24.5976	−1.17398	−0.586990	+	0.809594i	$0.699687\pi$
$440$	−3.50580	−0.167132
$441$	0	0
$442$	3.70000	0.175991
$443$	25.4642	1.20984	0.604920	−	0.796286i	$-0.293205\pi$
$443$	25.4642	1.20984	0.604920	+	0.796286i	$0.293205\pi$
$444$	0.486013	0.0230652
$445$	−9.45104	−0.448022
$446$	−3.64516	−0.172603
$447$	28.2173	1.33463
$448$	0	0
$449$	−6.07943	−0.286906	−0.143453	−	0.989657i	$-0.545821\pi$
$449$	−6.07943	−0.286906	−0.143453	+	0.989657i	$0.545821\pi$
$450$	1.64650	0.0776169
$451$	41.7951	1.96806
$452$	15.5995	0.733737
$453$	−54.6631	−2.56830
$454$	4.91730	0.230780
$455$	0	0
$456$	−5.05294	−0.236626
$457$	33.8987	1.58571	0.792857	−	0.609408i	$-0.208593\pi$
$457$	33.8987	1.58571	0.792857	+	0.609408i	$0.208593\pi$
$458$	2.34609	0.109625
$459$	5.45728	0.254724
$460$	−8.39586	−0.391459
$461$	−22.0227	−1.02570	−0.512850	−	0.858478i	$-0.671410\pi$
$461$	−22.0227	−1.02570	−0.512850	+	0.858478i	$0.671410\pi$
$462$	0	0
$463$	16.7218	0.777128	0.388564	−	0.921422i	$-0.372971\pi$
$463$	16.7218	0.777128	0.388564	+	0.921422i	$0.372971\pi$
$464$	−30.3878	−1.41072
$465$	−23.3738	−1.08393
$466$	−0.146857	−0.00680303
$467$	−5.20269	−0.240752	−0.120376	−	0.992728i	$-0.538410\pi$
$467$	−5.20269	−0.240752	−0.120376	+	0.992728i	$0.538410\pi$
$468$	−26.2901	−1.21526
$469$	0	0
$470$	1.02141	0.0471142
$471$	−12.1374	−0.559264
$472$	7.56170	0.348055
$473$	−0.111003	−0.00510391
$474$	−0.405323	−0.0186171
$475$	12.4548	0.571464
$476$	0	0
$477$	8.25259	0.377860
$478$	−3.01836	−0.138057
$479$	22.3035	1.01907	0.509536	−	0.860449i	$-0.329817\pi$
$479$	22.3035	1.01907	0.509536	+	0.860449i	$0.329817\pi$
$480$	−4.84614	−0.221195
$481$	−0.598355	−0.0272827
$482$	−2.51111	−0.114378
$483$	0	0
$484$	−26.9292	−1.22405
$485$	4.45264	0.202184
$486$	3.48946	0.158285
$487$	37.2799	1.68931	0.844657	−	0.535308i	$-0.179804\pi$
$487$	37.2799	1.68931	0.844657	+	0.535308i	$0.179804\pi$
$488$	7.41856	0.335822
$489$	13.3158	0.602160
$490$	0	0
$491$	31.7335	1.43211	0.716056	−	0.698043i	$-0.245946\pi$
$491$	31.7335	1.43211	0.716056	+	0.698043i	$0.245946\pi$
$492$	38.4176	1.73200
$493$	29.9684	1.34971
$494$	3.08651	0.138869
$495$	−11.9965	−0.539203
$496$	−37.8454	−1.69931
$497$	0	0
$498$	2.10965	0.0945355
$499$	−38.2680	−1.71311	−0.856556	−	0.516054i	$-0.827400\pi$
$499$	−38.2680	−1.71311	−0.856556	+	0.516054i	$0.827400\pi$
$500$	17.9575	0.803085
$501$	−36.4063	−1.62651
$502$	−3.18409	−0.142113
$503$	30.0417	1.33949	0.669747	−	0.742589i	$-0.266402\pi$
$503$	30.0417	1.33949	0.669747	+	0.742589i	$0.266402\pi$
$504$	0	0
$505$	−16.6069	−0.738996
$506$	3.64034	0.161833
$507$	43.1175	1.91492
$508$	−33.4709	−1.48503
$509$	2.58556	0.114603	0.0573015	−	0.998357i	$-0.481750\pi$
$509$	2.58556	0.114603	0.0573015	+	0.998357i	$0.481750\pi$
$510$	1.55192	0.0687201
$511$	0	0
$512$	−13.1452	−0.580941
$513$	4.55242	0.200994
$514$	2.40036	0.105875
$515$	−5.76333	−0.253963
$516$	−0.102032	−0.00449173
$517$	28.5348	1.25496
$518$	0	0
$519$	−33.7478	−1.48137
$520$	3.96737	0.173981
$521$	15.2994	0.670277	0.335138	−	0.942169i	$-0.391217\pi$
$521$	15.2994	0.670277	0.335138	+	0.942169i	$0.391217\pi$
$522$	3.30489	0.144651
$523$	−10.4292	−0.456036	−0.228018	−	0.973657i	$-0.573224\pi$
$523$	−10.4292	−0.456036	−0.228018	+	0.973657i	$0.573224\pi$
$524$	−7.56978	−0.330687
$525$	0	0
$526$	−3.60765	−0.157301
$527$	37.3231	1.62582
$528$	−43.9621	−1.91321
$529$	−5.42856	−0.236024
$530$	−0.617893	−0.0268396
$531$	25.8754	1.12290
$532$	0	0
$533$	−47.2979	−2.04870
$534$	3.76670	0.163001
$535$	3.23477	0.139851
$536$	6.18243	0.267040
$537$	31.4348	1.35651
$538$	−0.438464	−0.0189035
$539$	0	0
$540$	2.90329	0.124938
$541$	23.2498	0.999588	0.499794	−	0.866144i	$-0.333409\pi$
$541$	23.2498	0.999588	0.499794	+	0.866144i	$0.333409\pi$
$542$	−2.68120	−0.115167
$543$	−33.7728	−1.44933
$544$	7.73827	0.331776
$545$	−13.0930	−0.560842
$546$	0	0
$547$	32.5034	1.38974	0.694872	−	0.719134i	$-0.255461\pi$
$547$	32.5034	1.38974	0.694872	+	0.719134i	$0.255461\pi$
$548$	22.4463	0.958857
$549$	25.3856	1.08343
$550$	−3.44397	−0.146851
$551$	24.9994	1.06501
$552$	6.74425	0.287054
$553$	0	0
$554$	1.38634	0.0588998
$555$	−0.250973	−0.0106532
$556$	27.0470	1.14705
$557$	−24.7618	−1.04919	−0.524595	−	0.851352i	$-0.675783\pi$
$557$	−24.7618	−1.04919	−0.524595	+	0.851352i	$0.675783\pi$
$558$	4.11596	0.174242
$559$	0.125617	0.00531305
$560$	0	0
$561$	43.3554	1.83047
$562$	−2.55840	−0.107920
$563$	6.42852	0.270930	0.135465	−	0.990782i	$-0.456747\pi$
$563$	6.42852	0.270930	0.135465	+	0.990782i	$0.456747\pi$
$564$	26.2289	1.10443
$565$	−8.05542	−0.338894
$566$	1.58325	0.0665489
$567$	0	0
$568$	−2.79827	−0.117413
$569$	−34.4527	−1.44433	−0.722166	−	0.691720i	$-0.756853\pi$
$569$	−34.4527	−1.44433	−0.722166	+	0.691720i	$0.756853\pi$
$570$	1.29460	0.0542248
$571$	−4.62310	−0.193471	−0.0967354	−	0.995310i	$-0.530840\pi$
$571$	−4.62310	−0.193471	−0.0967354	+	0.995310i	$0.530840\pi$
$572$	54.9908	2.29928
$573$	−27.0036	−1.12809
$574$	0	0
$575$	−16.6236	−0.693252
$576$	−17.2781	−0.719920
$577$	−15.8151	−0.658392	−0.329196	−	0.944262i	$-0.606778\pi$
$577$	−15.8151	−0.658392	−0.329196	+	0.944262i	$0.606778\pi$
$578$	0.494061	0.0205502
$579$	1.00359	0.0417076
$580$	15.9433	0.662008
$581$	0	0
$582$	−1.77459	−0.0735592
$583$	−17.2618	−0.714913
$584$	8.99837	0.372355
$585$	13.5760	0.561298
$586$	5.67113	0.234272
$587$	−5.30042	−0.218772	−0.109386	−	0.993999i	$-0.534888\pi$
$587$	−5.30042	−0.218772	−0.109386	+	0.993999i	$0.534888\pi$
$588$	0	0
$589$	31.1346	1.28288
$590$	−1.93736	−0.0797599
$591$	52.8277	2.17304
$592$	−0.406359	−0.0167013
$593$	14.7983	0.607695	0.303847	−	0.952721i	$-0.401729\pi$
$593$	14.7983	0.607695	0.303847	+	0.952721i	$0.401729\pi$
$594$	−1.25883	−0.0516503
$595$	0	0
$596$	−23.9706	−0.981872
$597$	15.6350	0.639899
$598$	−4.11963	−0.168464
$599$	3.82550	0.156306	0.0781528	−	0.996941i	$-0.475098\pi$
$599$	3.82550	0.156306	0.0781528	+	0.996941i	$0.475098\pi$
$600$	−6.38045	−0.260481
$601$	23.3831	0.953816	0.476908	−	0.878953i	$-0.341757\pi$
$601$	23.3831	0.953816	0.476908	+	0.878953i	$0.341757\pi$
$602$	0	0
$603$	21.1557	0.861527
$604$	46.4362	1.88946
$605$	13.9060	0.565359
$606$	6.61865	0.268864
$607$	−41.8519	−1.69872	−0.849358	−	0.527817i	$-0.823011\pi$
$607$	−41.8519	−1.69872	−0.849358	+	0.527817i	$0.823011\pi$
$608$	6.45521	0.261793
$609$	0	0
$610$	−1.90069	−0.0769566
$611$	−32.2917	−1.30638
$612$	17.6079	0.711758
$613$	−45.1455	−1.82341	−0.911705	−	0.410844i	$-0.865234\pi$
$613$	−45.1455	−1.82341	−0.911705	+	0.410844i	$0.865234\pi$
$614$	−3.32007	−0.133987
$615$	−19.8385	−0.799966
$616$	0	0
$617$	−31.2423	−1.25777	−0.628884	−	0.777499i	$-0.716488\pi$
$617$	−31.2423	−1.25777	−0.628884	+	0.777499i	$0.716488\pi$
$618$	2.29697	0.0923977
$619$	21.7576	0.874513	0.437256	−	0.899337i	$-0.355950\pi$
$619$	21.7576	0.874513	0.437256	+	0.899337i	$0.355950\pi$
$620$	19.8560	0.797435
$621$	−6.07620	−0.243830
$622$	0.418948	0.0167983
$623$	0	0
$624$	49.7502	1.99160
$625$	10.5555	0.422219
$626$	1.15022	0.0459719
$627$	36.1668	1.44436
$628$	10.3107	0.411443
$629$	0.400751	0.0159790
$630$	0	0
$631$	41.1685	1.63889	0.819447	−	0.573156i	$-0.194281\pi$
$631$	41.1685	1.63889	0.819447	+	0.573156i	$0.194281\pi$
$632$	0.693986	0.0276053
$633$	−41.4272	−1.64658
$634$	1.85227	0.0735630
$635$	17.2841	0.685898
$636$	−15.8669	−0.629163
$637$	0	0
$638$	−6.91280	−0.273680
$639$	−9.57541	−0.378798
$640$	5.47433	0.216392
$641$	−1.09084	−0.0430854	−0.0215427	−	0.999768i	$-0.506858\pi$
$641$	−1.09084	−0.0430854	−0.0215427	+	0.999768i	$0.506858\pi$
$642$	−1.28921	−0.0508812
$643$	−1.59788	−0.0630142	−0.0315071	−	0.999504i	$-0.510031\pi$
$643$	−1.59788	−0.0630142	−0.0315071	+	0.999504i	$0.510031\pi$
$644$	0	0
$645$	0.0526887	0.00207461
$646$	−2.06721	−0.0813331
$647$	−4.74488	−0.186541	−0.0932703	−	0.995641i	$-0.529732\pi$
$647$	−4.74488	−0.186541	−0.0932703	+	0.995641i	$0.529732\pi$
$648$	−7.27630	−0.285840
$649$	−54.1234	−2.12453
$650$	3.89741	0.152869
$651$	0	0
$652$	−11.3117	−0.443001
$653$	17.0928	0.668892	0.334446	−	0.942415i	$-0.391451\pi$
$653$	17.0928	0.668892	0.334446	+	0.942415i	$0.391451\pi$
$654$	5.21819	0.204048
$655$	3.90897	0.152736
$656$	−32.1213	−1.25413
$657$	30.7916	1.20129
$658$	0	0
$659$	12.6754	0.493764	0.246882	−	0.969046i	$-0.420594\pi$
$659$	12.6754	0.493764	0.246882	+	0.969046i	$0.420594\pi$
$660$	23.0652	0.897811
$661$	1.17036	0.0455218	0.0227609	−	0.999741i	$-0.492754\pi$
$661$	1.17036	0.0455218	0.0227609	+	0.999741i	$0.492754\pi$
$662$	2.11757	0.0823018
$663$	−49.0636	−1.90547
$664$	−3.61209	−0.140176
$665$	0	0
$666$	0.0441945	0.00171250
$667$	−33.3672	−1.29198
$668$	30.9270	1.19660
$669$	48.3363	1.86879
$670$	−1.58398	−0.0611946
$671$	−53.0988	−2.04986
$672$	0	0
$673$	−5.80037	−0.223588	−0.111794	−	0.993731i	$-0.535660\pi$
$673$	−5.80037	−0.223588	−0.111794	+	0.993731i	$0.535660\pi$
$674$	−1.24464	−0.0479416
$675$	5.74844	0.221258
$676$	−36.6282	−1.40878
$677$	−15.9403	−0.612637	−0.306319	−	0.951929i	$-0.599097\pi$
$677$	−15.9403	−0.612637	−0.306319	+	0.951929i	$0.599097\pi$
$678$	3.21048	0.123298
$679$	0	0
$680$	−2.65716	−0.101898
$681$	−65.2055	−2.49868
$682$	−8.60930	−0.329667
$683$	9.94514	0.380540	0.190270	−	0.981732i	$-0.439064\pi$
$683$	9.94514	0.380540	0.190270	+	0.981732i	$0.439064\pi$
$684$	14.6884	0.561625
$685$	−11.5910	−0.442871
$686$	0	0
$687$	−31.1101	−1.18692
$688$	0.0853102	0.00325242
$689$	19.5346	0.744207
$690$	−1.72793	−0.0657810
$691$	−22.5353	−0.857284	−0.428642	−	0.903474i	$-0.641008\pi$
$691$	−22.5353	−0.857284	−0.428642	+	0.903474i	$0.641008\pi$
$692$	28.6687	1.08982
$693$	0	0
$694$	−1.80388	−0.0684743
$695$	−13.9668	−0.529792
$696$	−12.8070	−0.485446
$697$	31.6780	1.19989
$698$	0.479256	0.0181401
$699$	1.94739	0.0736570
$700$	0	0
$701$	40.1100	1.51493	0.757466	−	0.652874i	$-0.226437\pi$
$701$	40.1100	1.51493	0.757466	+	0.652874i	$0.226437\pi$
$702$	1.42457	0.0537668
$703$	0.334304	0.0126085
$704$	36.1403	1.36209
$705$	−13.5444	−0.510110
$706$	1.35115	0.0508513
$707$	0	0
$708$	−49.7496	−1.86970
$709$	36.3061	1.36350	0.681752	−	0.731584i	$-0.261218\pi$
$709$	36.3061	1.36350	0.681752	+	0.731584i	$0.261218\pi$
$710$	0.716936	0.0269061
$711$	2.37476	0.0890603
$712$	−6.44926	−0.241696
$713$	−41.5560	−1.55628
$714$	0	0
$715$	−28.3967	−1.06198
$716$	−26.7038	−0.997966
$717$	40.0247	1.49475
$718$	−1.58292	−0.0590740
$719$	4.17943	0.155867	0.0779333	−	0.996959i	$-0.475168\pi$
$719$	4.17943	0.155867	0.0779333	+	0.996959i	$0.475168\pi$
$720$	9.21982	0.343602
$721$	0	0
$722$	1.59737	0.0594479
$723$	33.2984	1.23838
$724$	28.6899	1.06625
$725$	31.5673	1.17238
$726$	−5.54222	−0.205691
$727$	−14.6673	−0.543979	−0.271989	−	0.962300i	$-0.587682\pi$
$727$	−14.6673	−0.543979	−0.271989	+	0.962300i	$0.587682\pi$
$728$	0	0
$729$	−14.8173	−0.548787
$730$	−2.30545	−0.0853284
$731$	−0.0841328	−0.00311176
$732$	−48.8078	−1.80399
$733$	−3.29672	−0.121767	−0.0608836	−	0.998145i	$-0.519392\pi$
$733$	−3.29672	−0.121767	−0.0608836	+	0.998145i	$0.519392\pi$
$734$	4.09952	0.151316
$735$	0	0
$736$	−8.61589	−0.317586
$737$	−44.2512	−1.63001
$738$	3.49342	0.128595
$739$	16.1740	0.594971	0.297485	−	0.954726i	$-0.403852\pi$
$739$	16.1740	0.594971	0.297485	+	0.954726i	$0.403852\pi$
$740$	0.213201	0.00783741
$741$	−40.9285	−1.50355
$742$	0	0
$743$	−25.4657	−0.934247	−0.467124	−	0.884192i	$-0.654710\pi$
$743$	−25.4657	−0.934247	−0.467124	+	0.884192i	$0.654710\pi$
$744$	−15.9500	−0.584754
$745$	12.3782	0.453501
$746$	0.546966	0.0200258
$747$	−12.3602	−0.452238
$748$	−36.8303	−1.34665
$749$	0	0
$750$	3.69578	0.134951
$751$	−40.1129	−1.46374	−0.731870	−	0.681445i	$-0.761352\pi$
$751$	−40.1129	−1.46374	−0.731870	+	0.681445i	$0.761352\pi$
$752$	−21.9302	−0.799711
$753$	42.2224	1.53867
$754$	7.82294	0.284895
$755$	−23.9792	−0.872693
$756$	0	0
$757$	−14.0698	−0.511377	−0.255689	−	0.966759i	$-0.582302\pi$
$757$	−14.0698	−0.511377	−0.255689	+	0.966759i	$0.582302\pi$
$758$	−5.66614	−0.205804
$759$	−48.2724	−1.75218
$760$	−2.21659	−0.0804040
$761$	19.4976	0.706787	0.353393	−	0.935475i	$-0.385028\pi$
$761$	19.4976	0.706787	0.353393	+	0.935475i	$0.385028\pi$
$762$	−6.88855	−0.249546
$763$	0	0
$764$	22.9395	0.829922
$765$	−9.09257	−0.328743
$766$	−3.84417	−0.138896
$767$	61.2493	2.21158
$768$	31.5536	1.13859
$769$	−12.5705	−0.453303	−0.226651	−	0.973976i	$-0.572778\pi$
$769$	−12.5705	−0.453303	−0.226651	+	0.973976i	$0.572778\pi$
$770$	0	0
$771$	−31.8298	−1.14632
$772$	−0.852543	−0.0306837
$773$	−21.7460	−0.782148	−0.391074	−	0.920359i	$-0.627896\pi$
$773$	−21.7460	−0.782148	−0.391074	+	0.920359i	$0.627896\pi$
$774$	−0.00927810	−0.000333494 0
$775$	39.3144	1.41221
$776$	3.03842	0.109073
$777$	0	0
$778$	4.89121	0.175358
$779$	26.4255	0.946793
$780$	−26.1020	−0.934600
$781$	20.0288	0.716686
$782$	2.75914	0.0986666
$783$	11.5384	0.412348
$784$	0	0
$785$	−5.32437	−0.190035
$786$	−1.55791	−0.0555689
$787$	18.1082	0.645488	0.322744	−	0.946486i	$-0.395395\pi$
$787$	18.1082	0.645488	0.322744	+	0.946486i	$0.395395\pi$
$788$	−44.8770	−1.59868
$789$	47.8390	1.70311
$790$	−0.177804	−0.00632599
$791$	0	0
$792$	−8.18626	−0.290886
$793$	60.0898	2.13385
$794$	−5.11605	−0.181562
$795$	8.19352	0.290594
$796$	−13.2819	−0.470765
$797$	−33.5694	−1.18909	−0.594544	−	0.804063i	$-0.702667\pi$
$797$	−33.5694	−1.18909	−0.594544	+	0.804063i	$0.702667\pi$
$798$	0	0
$799$	21.6275	0.765126
$800$	8.15113	0.288186
$801$	−22.0688	−0.779762
$802$	−1.69428	−0.0598269
$803$	−64.4064	−2.27285
$804$	−40.6752	−1.43450
$805$	0	0
$806$	9.74280	0.343176
$807$	5.81421	0.204670
$808$	−11.3323	−0.398669
$809$	−21.3975	−0.752297	−0.376149	−	0.926559i	$-0.622752\pi$
$809$	−21.3975	−0.752297	−0.376149	+	0.926559i	$0.622752\pi$
$810$	1.86424	0.0655028
$811$	−42.6906	−1.49907	−0.749535	−	0.661965i	$-0.769723\pi$
$811$	−42.6906	−1.49907	−0.749535	+	0.661965i	$0.769723\pi$
$812$	0	0
$813$	35.5538	1.24693
$814$	−0.0924411	−0.00324006
$815$	5.84126	0.204611
$816$	−33.3204	−1.16645
$817$	−0.0701830	−0.00245539
$818$	−2.38911	−0.0835333
$819$	0	0
$820$	16.8528	0.588524
$821$	−9.68152	−0.337887	−0.168944	−	0.985626i	$-0.554036\pi$
$821$	−9.68152	−0.337887	−0.168944	+	0.985626i	$0.554036\pi$
$822$	4.61960	0.161127
$823$	−47.7906	−1.66588	−0.832938	−	0.553367i	$-0.813343\pi$
$823$	−47.7906	−1.66588	−0.832938	+	0.553367i	$0.813343\pi$
$824$	−3.93282	−0.137006
$825$	45.6685	1.58997
$826$	0	0
$827$	−28.2883	−0.983682	−0.491841	−	0.870685i	$-0.663676\pi$
$827$	−28.2883	−0.983682	−0.491841	+	0.870685i	$0.663676\pi$
$828$	−19.6049	−0.681317
$829$	4.46950	0.155232	0.0776160	−	0.996983i	$-0.475269\pi$
$829$	4.46950	0.155232	0.0776160	+	0.996983i	$0.475269\pi$
$830$	0.925444	0.0321226
$831$	−18.3834	−0.637713
$832$	−40.8986	−1.41790
$833$	0	0
$834$	5.56646	0.192751
$835$	−15.9704	−0.552679
$836$	−30.7236	−1.06260
$837$	14.3700	0.496701
$838$	−2.41125	−0.0832953
$839$	12.9646	0.447586	0.223793	−	0.974637i	$-0.428156\pi$
$839$	12.9646	0.447586	0.223793	+	0.974637i	$0.428156\pi$
$840$	0	0
$841$	34.3624	1.18491
$842$	2.86398	0.0986993
$843$	33.9255	1.16845
$844$	35.1923	1.21137
$845$	18.9145	0.650678
$846$	2.38506	0.0820002
$847$	0	0
$848$	13.2664	0.455571
$849$	−20.9945	−0.720531
$850$	−2.61030	−0.0895327
$851$	−0.446201	−0.0152956
$852$	18.4102	0.630724
$853$	2.98429	0.102180	0.0510901	−	0.998694i	$-0.483730\pi$
$853$	2.98429	0.102180	0.0510901	+	0.998694i	$0.483730\pi$
$854$	0	0
$855$	−7.58496	−0.259400
$856$	2.20736	0.0754461
$857$	−29.4626	−1.00642	−0.503212	−	0.864163i	$-0.667848\pi$
$857$	−29.4626	−1.00642	−0.503212	+	0.864163i	$0.667848\pi$
$858$	11.3175	0.386372
$859$	−16.6382	−0.567688	−0.283844	−	0.958871i	$-0.591610\pi$
$859$	−16.6382	−0.567688	−0.283844	+	0.958871i	$0.591610\pi$
$860$	−0.0447589	−0.00152626
$861$	0	0
$862$	−2.92418	−0.0995979
$863$	9.74068	0.331576	0.165788	−	0.986161i	$-0.446983\pi$
$863$	9.74068	0.331576	0.165788	+	0.986161i	$0.446983\pi$
$864$	2.97937	0.101360
$865$	−14.8043	−0.503360
$866$	−2.43713	−0.0828172
$867$	−6.55145	−0.222499
$868$	0	0
$869$	−4.96725	−0.168502
$870$	3.28124	0.111244
$871$	50.0773	1.69680
$872$	−8.93448	−0.302560
$873$	10.3972	0.351892
$874$	2.30165	0.0778546
$875$	0	0
$876$	−59.2017	−2.00024
$877$	−1.10333	−0.0372567	−0.0186284	−	0.999826i	$-0.505930\pi$
$877$	−1.10333	−0.0372567	−0.0186284	+	0.999826i	$0.505930\pi$
$878$	4.30046	0.145134
$879$	−75.2016	−2.53649
$880$	−19.2850	−0.650097
$881$	−36.7920	−1.23955	−0.619777	−	0.784778i	$-0.712777\pi$
$881$	−36.7920	−1.23955	−0.619777	+	0.784778i	$0.712777\pi$
$882$	0	0
$883$	32.6949	1.10027	0.550135	−	0.835076i	$-0.314576\pi$
$883$	32.6949	1.10027	0.550135	+	0.835076i	$0.314576\pi$
$884$	41.6794	1.40183
$885$	25.6902	0.863568
$886$	−4.45197	−0.149567
$887$	39.4131	1.32336	0.661680	−	0.749786i	$-0.269844\pi$
$887$	39.4131	1.32336	0.661680	+	0.749786i	$0.269844\pi$
$888$	−0.171260	−0.00574712
$889$	0	0
$890$	1.65235	0.0553868
$891$	52.0806	1.74477
$892$	−41.0616	−1.37484
$893$	18.0415	0.603736
$894$	−4.93331	−0.164995
$895$	13.7896	0.460935
$896$	0	0
$897$	54.6280	1.82398
$898$	1.06288	0.0354688
$899$	78.9125	2.63188
$900$	18.5474	0.618245
$901$	−13.0833	−0.435870
$902$	−7.30715	−0.243301
$903$	0	0
$904$	−5.49691	−0.182825
$905$	−14.8152	−0.492474
$906$	9.55689	0.317506
$907$	−45.4742	−1.50995	−0.754974	−	0.655755i	$-0.772350\pi$
$907$	−45.4742	−1.50995	−0.754974	+	0.655755i	$0.772350\pi$
$908$	55.3919	1.83824
$909$	−38.7782	−1.28619
$910$	0	0
$911$	−13.9176	−0.461112	−0.230556	−	0.973059i	$-0.574054\pi$
$911$	−13.9176	−0.461112	−0.230556	+	0.973059i	$0.574054\pi$
$912$	−27.7956	−0.920405
$913$	25.8538	0.855635
$914$	−5.92659	−0.196034
$915$	25.2039	0.833216
$916$	26.4279	0.873204
$917$	0	0
$918$	−0.954109	−0.0314903
$919$	−0.293717	−0.00968884	−0.00484442	−	0.999988i	$-0.501542\pi$
$919$	−0.293717	−0.00968884	−0.00484442	+	0.999988i	$0.501542\pi$
$920$	2.95852	0.0975395
$921$	44.0255	1.45069
$922$	3.85029	0.126802
$923$	−22.6658	−0.746053
$924$	0	0
$925$	0.422132	0.0138796
$926$	−2.92351	−0.0960726
$927$	−13.4578	−0.442011
$928$	16.3611	0.537079
$929$	12.9353	0.424392	0.212196	−	0.977227i	$-0.431938\pi$
$929$	12.9353	0.424392	0.212196	+	0.977227i	$0.431938\pi$
$930$	4.08650	0.134001
$931$	0	0
$932$	−1.65430	−0.0541884
$933$	−5.55542	−0.181876
$934$	0.909599	0.0297630
$935$	19.0188	0.621982
$936$	9.26407	0.302806
$937$	−26.8258	−0.876362	−0.438181	−	0.898887i	$-0.644377\pi$
$937$	−26.8258	−0.876362	−0.438181	+	0.898887i	$0.644377\pi$
$938$	0	0
$939$	−15.2524	−0.497742
$940$	11.5059	0.375281
$941$	34.9829	1.14041	0.570205	−	0.821503i	$-0.306864\pi$
$941$	34.9829	1.14041	0.570205	+	0.821503i	$0.306864\pi$
$942$	2.12202	0.0691391
$943$	−35.2706	−1.14857
$944$	41.5960	1.35384
$945$	0	0
$946$	0.0194069	0.000630972 0
$947$	−18.4196	−0.598555	−0.299278	−	0.954166i	$-0.596746\pi$
$947$	−18.4196	−0.598555	−0.299278	+	0.954166i	$0.596746\pi$
$948$	−4.56584	−0.148292
$949$	72.8862	2.36599
$950$	−2.17750	−0.0706473
$951$	−24.5619	−0.796473
$952$	0	0
$953$	20.5996	0.667288	0.333644	−	0.942699i	$-0.391722\pi$
$953$	20.5996	0.667288	0.333644	+	0.942699i	$0.391722\pi$
$954$	−1.44282	−0.0467131
$955$	−11.8458	−0.383319
$956$	−34.0009	−1.09967
$957$	91.6667	2.96316
$958$	−3.89937	−0.125983
$959$	0	0
$960$	−17.1544	−0.553656
$961$	67.2787	2.17028
$962$	0.104612	0.00337282
$963$	7.55339	0.243405
$964$	−28.2869	−0.911059
$965$	0.440246	0.0141720
$966$	0	0
$967$	−16.8801	−0.542827	−0.271414	−	0.962463i	$-0.587491\pi$
$967$	−16.8801	−0.542827	−0.271414	+	0.962463i	$0.587491\pi$
$968$	9.48927	0.304997
$969$	27.4120	0.880601
$970$	−0.778465	−0.0249950
$971$	14.0403	0.450574	0.225287	−	0.974292i	$-0.427668\pi$
$971$	14.0403	0.450574	0.225287	+	0.974292i	$0.427668\pi$
$972$	39.3076	1.26079
$973$	0	0
$974$	−6.51774	−0.208842
$975$	−51.6813	−1.65513
$976$	40.8086	1.30625
$977$	57.3649	1.83526	0.917632	−	0.397431i	$-0.130098\pi$
$977$	57.3649	1.83526	0.917632	+	0.397431i	$0.130098\pi$
$978$	−2.32803	−0.0744421
$979$	46.1610	1.47531
$980$	0	0
$981$	−30.5730	−0.976120
$982$	−5.54804	−0.177045
$983$	−50.5584	−1.61256	−0.806282	−	0.591532i	$-0.798523\pi$
$983$	−50.5584	−1.61256	−0.806282	+	0.591532i	$0.798523\pi$
$984$	−13.5375	−0.431561
$985$	23.1741	0.738387
$986$	−5.23945	−0.166858
$987$	0	0
$988$	34.7686	1.10614
$989$	0.0936745	0.00297868
$990$	2.09738	0.0666591
$991$	4.81875	0.153073	0.0765364	−	0.997067i	$-0.475614\pi$
$991$	4.81875	0.153073	0.0765364	+	0.997067i	$0.475614\pi$
$992$	20.3763	0.646949
$993$	−28.0799	−0.891089
$994$	0	0
$995$	6.85866	0.217434
$996$	23.7645	0.753007
$997$	4.79199	0.151764	0.0758819	−	0.997117i	$-0.475823\pi$
$997$	4.79199	0.151764	0.0758819	+	0.997117i	$0.475823\pi$
$998$	6.69049	0.211784
$999$	0.154296	0.00488172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3871.2.a.e.1.4		8
7.6	odd	2		553.2.a.b.1.4	✓	8
21.20	even	2		4977.2.a.j.1.5		8
28.27	even	2		8848.2.a.s.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
553.2.a.b.1.4	✓	8	7.6	odd	2
3871.2.a.e.1.4		8	1.1	even	1	trivial
4977.2.a.j.1.5		8	21.20	even	2
8848.2.a.s.1.8		8	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.174832 q^{2} +2.31835 q^{3} -1.96943 q^{4} +1.01700 q^{5} -0.405323 q^{6} +0.693986 q^{8} +2.37476 q^{9} -0.177804 q^{10} -4.96725 q^{11} -4.56584 q^{12} +5.62124 q^{13} +2.35776 q^{15} +3.81754 q^{16} -3.76485 q^{17} -0.415185 q^{18} -3.14061 q^{19} -2.00291 q^{20} +0.868437 q^{22} +4.19183 q^{23} +1.60890 q^{24} -3.96572 q^{25} -0.982775 q^{26} -1.44953 q^{27} -7.96005 q^{29} -0.412213 q^{30} -9.91356 q^{31} -2.05540 q^{32} -11.5158 q^{33} +0.658218 q^{34} -4.67693 q^{36} -0.106445 q^{37} +0.549081 q^{38} +13.0320 q^{39} +0.705782 q^{40} -8.41414 q^{41} +0.0223469 q^{43} +9.78267 q^{44} +2.41512 q^{45} -0.732868 q^{46} -5.74459 q^{47} +8.85039 q^{48} +0.693336 q^{50} -8.72825 q^{51} -11.0707 q^{52} +3.47513 q^{53} +0.253426 q^{54} -5.05168 q^{55} -7.28104 q^{57} +1.39168 q^{58} +10.8960 q^{59} -4.64345 q^{60} +10.6898 q^{61} +1.73321 q^{62} -7.27572 q^{64} +5.71679 q^{65} +2.01334 q^{66} +8.90858 q^{67} +7.41462 q^{68} +9.71814 q^{69} -4.03217 q^{71} +1.64805 q^{72} +12.9662 q^{73} +0.0186101 q^{74} -9.19393 q^{75} +6.18522 q^{76} -2.27842 q^{78} +1.00000 q^{79} +3.88243 q^{80} -10.4848 q^{81} +1.47106 q^{82} -5.20485 q^{83} -3.82884 q^{85} -0.00390697 q^{86} -18.4542 q^{87} -3.44720 q^{88} -9.29307 q^{89} -0.422242 q^{90} -8.25553 q^{92} -22.9831 q^{93} +1.00434 q^{94} -3.19399 q^{95} -4.76514 q^{96} +4.37822 q^{97} -11.7960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} - 3 q^{3} + 7 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{8} - q^{9} - 10 q^{10} + 3 q^{11} - 10 q^{12} - 3 q^{13} + 11 q^{15} + 13 q^{16} - 14 q^{17} + 4 q^{18} - 12 q^{19} - 12 q^{20} - 15 q^{22}+ \cdots - 25 q^{99}+O(q^{100}) \) 8 * q + 3 * q^2 - 3 * q^3 + 7 * q^4 - 4 * q^5 - 6 * q^6 + 6 * q^8 - q^9 - 10 * q^10 + 3 * q^11 - 10 * q^12 - 3 * q^13 + 11 * q^15 + 13 * q^16 - 14 * q^17 + 4 * q^18 - 12 * q^19 - 12 * q^20 - 15 * q^22 + 17 * q^23 - 14 * q^24 - 6 * q^25 + 13 * q^26 + 6 * q^27 - 6 * q^29 + 17 * q^30 - 15 * q^31 + 5 * q^32 - 2 * q^33 + 2 * q^34 - 33 * q^36 - 2 * q^37 - 19 * q^38 + 6 * q^39 + 13 * q^40 - 8 * q^41 + 9 * q^43 + 7 * q^44 - 5 * q^45 - 24 * q^46 - 34 * q^47 + 9 * q^48 - 3 * q^50 - 11 * q^51 - 17 * q^52 + 2 * q^53 + 10 * q^54 - 16 * q^55 + 6 * q^57 - 14 * q^58 - 21 * q^59 - 14 * q^60 - 11 * q^61 + 15 * q^62 - 16 * q^64 - 5 * q^65 + 18 * q^66 + 14 * q^67 - 7 * q^68 + 8 * q^69 - 2 * q^71 - 18 * q^72 - 7 * q^73 - 8 * q^74 - 28 * q^75 + 23 * q^76 - 8 * q^78 + 8 * q^79 - 6 * q^80 - 28 * q^81 + 10 * q^82 - 32 * q^83 - 9 * q^85 - 6 * q^86 - 43 * q^87 - 48 * q^88 - 6 * q^89 - q^90 - 3 * q^92 - 12 * q^93 - 33 * q^94 - 15 * q^95 + 9 * q^96 + 39 * q^97 - 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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