LMFDB - Embedded newform 4840.2.a.bd.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4840,2,Mod(1,4840)] 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("4840.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-2,0,6,0,4,0,4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.6475945783$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.22733568.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 $ x^6 - 8x^4 - 2x^3 + 16x^2 + 8x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.184585$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.96593 q^{3} +1.00000 q^{5} +1.02876 q^{7} +5.79673 q^{9} -0.504296 q^{13} -2.96593 q^{15} -3.93186 q^{17} +2.50430 q^{19} -3.05124 q^{21} -5.73921 q^{23} +1.00000 q^{25} -8.29490 q^{27} +6.92327 q^{29} +4.47755 q^{31} +1.02876 q^{35} +4.78329 q^{37} +1.49570 q^{39} +11.8404 q^{41} -7.10060 q^{43} +5.79673 q^{45} -0.182642 q^{47} -5.94165 q^{49} +11.6616 q^{51} +12.4669 q^{53} -7.42756 q^{57} -10.5494 q^{59} -10.9209 q^{61} +5.96346 q^{63} -0.504296 q^{65} +9.76386 q^{67} +17.0221 q^{69} -13.9992 q^{71} -7.32321 q^{73} -2.96593 q^{75} +10.2743 q^{79} +7.21190 q^{81} +4.33368 q^{83} -3.93186 q^{85} -20.5339 q^{87} +17.2789 q^{89} -0.518800 q^{91} -13.2801 q^{93} +2.50430 q^{95} -7.80167 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9} - 2 q^{15} + 8 q^{17} + 12 q^{19} - 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} + 16 q^{29} - 4 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} + 32 q^{41} - 4 q^{43}+ \cdots + 12 q^{95}+O(q^{100}) $ 6 * q - 2 * q^3 + 6 * q^5 + 4 * q^7 + 4 * q^9 - 2 * q^15 + 8 * q^17 + 12 * q^19 - 8 * q^21 - 8 * q^23 + 6 * q^25 - 14 * q^27 + 16 * q^29 - 4 * q^31 + 4 * q^35 + 8 * q^37 + 12 * q^39 + 32 * q^41 - 4 * q^43 + 4 * q^45 - 6 * q^47 + 16 * q^49 + 40 * q^51 + 8 * q^53 - 16 * q^57 + 4 * q^59 + 16 * q^61 + 28 * q^63 - 2 * q^67 + 8 * q^69 - 28 * q^71 + 16 * q^73 - 2 * q^75 - 10 * q^81 + 12 * q^83 + 8 * q^85 - 24 * q^87 + 18 * q^89 - 24 * q^91 + 20 * q^93 + 12 * q^95

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	0
$2$	0	0
$3$	−2.96593	−1.71238	−0.856190	−	0.516661i	$-0.827175\pi$
$3$	−2.96593	−1.71238	−0.856190	+	0.516661i	$0.827175\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.02876	0.388836	0.194418	−	0.980919i	$-0.437718\pi$
$7$	1.02876	0.388836	0.194418	+	0.980919i	$0.437718\pi$
$8$	0	0
$9$	5.79673	1.93224
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.504296	−0.139866	−0.0699332	−	0.997552i	$-0.522279\pi$
$13$	−0.504296	−0.139866	−0.0699332	+	0.997552i	$0.522279\pi$
$14$	0	0
$15$	−2.96593	−0.765799
$16$	0	0
$17$	−3.93186	−0.953615	−0.476808	−	0.879008i	$-0.658206\pi$
$17$	−3.93186	−0.953615	−0.476808	+	0.879008i	$0.658206\pi$
$18$	0	0
$19$	2.50430	0.574525	0.287262	−	0.957852i	$-0.407255\pi$
$19$	2.50430	0.574525	0.287262	+	0.957852i	$0.407255\pi$
$20$	0	0
$21$	−3.05124	−0.665834
$22$	0	0
$23$	−5.73921	−1.19671	−0.598354	−	0.801232i	$-0.704178\pi$
$23$	−5.73921	−1.19671	−0.598354	+	0.801232i	$0.704178\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−8.29490	−1.59636
$28$	0	0
$29$	6.92327	1.28562	0.642809	−	0.766026i	$-0.277769\pi$
$29$	6.92327	1.28562	0.642809	+	0.766026i	$0.277769\pi$
$30$	0	0
$31$	4.47755	0.804191	0.402096	−	0.915598i	$-0.368282\pi$
$31$	4.47755	0.804191	0.402096	+	0.915598i	$0.368282\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.02876	0.173893
$36$	0	0
$37$	4.78329	0.786367	0.393184	−	0.919460i	$-0.371374\pi$
$37$	4.78329	0.786367	0.393184	+	0.919460i	$0.371374\pi$
$38$	0	0
$39$	1.49570	0.239504
$40$	0	0
$41$	11.8404	1.84916	0.924582	−	0.380983i	$-0.124414\pi$
$41$	11.8404	1.84916	0.924582	+	0.380983i	$0.124414\pi$
$42$	0	0
$43$	−7.10060	−1.08283	−0.541416	−	0.840755i	$-0.682111\pi$
$43$	−7.10060	−1.08283	−0.541416	+	0.840755i	$0.682111\pi$
$44$	0	0
$45$	5.79673	0.864126
$46$	0	0
$47$	−0.182642	−0.0266410	−0.0133205	−	0.999911i	$-0.504240\pi$
$47$	−0.182642	−0.0266410	−0.0133205	+	0.999911i	$0.504240\pi$
$48$	0	0
$49$	−5.94165	−0.848807
$50$	0	0
$51$	11.6616	1.63295
$52$	0	0
$53$	12.4669	1.71246	0.856232	−	0.516591i	$-0.172799\pi$
$53$	12.4669	1.71246	0.856232	+	0.516591i	$0.172799\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−7.42756	−0.983805
$58$	0	0
$59$	−10.5494	−1.37341	−0.686706	−	0.726935i	$-0.740944\pi$
$59$	−10.5494	−1.37341	−0.686706	+	0.726935i	$0.740944\pi$
$60$	0	0
$61$	−10.9209	−1.39827	−0.699136	−	0.714989i	$-0.746432\pi$
$61$	−10.9209	−1.39827	−0.699136	+	0.714989i	$0.746432\pi$
$62$	0	0
$63$	5.96346	0.751325
$64$	0	0
$65$	−0.504296	−0.0625502
$66$	0	0
$67$	9.76386	1.19285	0.596423	−	0.802671i	$-0.296588\pi$
$67$	9.76386	1.19285	0.596423	+	0.802671i	$0.296588\pi$
$68$	0	0
$69$	17.0221	2.04922
$70$	0	0
$71$	−13.9992	−1.66140	−0.830698	−	0.556723i	$-0.812058\pi$
$71$	−13.9992	−1.66140	−0.830698	+	0.556723i	$0.812058\pi$
$72$	0	0
$73$	−7.32321	−0.857117	−0.428559	−	0.903514i	$-0.640978\pi$
$73$	−7.32321	−0.857117	−0.428559	+	0.903514i	$0.640978\pi$
$74$	0	0
$75$	−2.96593	−0.342476
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.2743	1.15595	0.577973	−	0.816056i	$-0.303844\pi$
$79$	10.2743	1.15595	0.577973	+	0.816056i	$0.303844\pi$
$80$	0	0
$81$	7.21190	0.801322
$82$	0	0
$83$	4.33368	0.475683	0.237842	−	0.971304i	$-0.423560\pi$
$83$	4.33368	0.475683	0.237842	+	0.971304i	$0.423560\pi$
$84$	0	0
$85$	−3.93186	−0.426470
$86$	0	0
$87$	−20.5339	−2.20147
$88$	0	0
$89$	17.2789	1.83156	0.915779	−	0.401683i	$-0.131575\pi$
$89$	17.2789	1.83156	0.915779	+	0.401683i	$0.131575\pi$
$90$	0	0
$91$	−0.518800	−0.0543851
$92$	0	0
$93$	−13.2801	−1.37708
$94$	0	0
$95$	2.50430	0.256935
$96$	0	0
$97$	−7.80167	−0.792139	−0.396070	−	0.918220i	$-0.629626\pi$
$97$	−7.80167	−0.792139	−0.396070	+	0.918220i	$0.629626\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.91035	−0.687605	−0.343803	−	0.939042i	$-0.611715\pi$
$101$	−6.91035	−0.687605	−0.343803	+	0.939042i	$0.611715\pi$
$102$	0	0
$103$	1.07395	0.105819	0.0529096	−	0.998599i	$-0.483150\pi$
$103$	1.07395	0.105819	0.0529096	+	0.998599i	$0.483150\pi$
$104$	0	0
$105$	−3.05124	−0.297770
$106$	0	0
$107$	−15.6135	−1.50942	−0.754708	−	0.656061i	$-0.772222\pi$
$107$	−15.6135	−1.50942	−0.754708	+	0.656061i	$0.772222\pi$
$108$	0	0
$109$	10.5632	1.01177	0.505886	−	0.862600i	$-0.331166\pi$
$109$	10.5632	1.01177	0.505886	+	0.862600i	$0.331166\pi$
$110$	0	0
$111$	−14.1869	−1.34656
$112$	0	0
$113$	1.72975	0.162721	0.0813605	−	0.996685i	$-0.474073\pi$
$113$	1.72975	0.162721	0.0813605	+	0.996685i	$0.474073\pi$
$114$	0	0
$115$	−5.73921	−0.535184
$116$	0	0
$117$	−2.92327	−0.270256
$118$	0	0
$119$	−4.04495	−0.370800
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−35.1179	−3.16647
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.80700	−0.870231	−0.435115	−	0.900375i	$-0.643292\pi$
$127$	−9.80700	−0.870231	−0.435115	+	0.900375i	$0.643292\pi$
$128$	0	0
$129$	21.0599	1.85422
$130$	0	0
$131$	−3.79284	−0.331382	−0.165691	−	0.986178i	$-0.552985\pi$
$131$	−3.79284	−0.331382	−0.165691	+	0.986178i	$0.552985\pi$
$132$	0	0
$133$	2.57633	0.223396
$134$	0	0
$135$	−8.29490	−0.713912
$136$	0	0
$137$	−4.58496	−0.391719	−0.195860	−	0.980632i	$-0.562750\pi$
$137$	−4.58496	−0.391719	−0.195860	+	0.980632i	$0.562750\pi$
$138$	0	0
$139$	9.11697	0.773291	0.386646	−	0.922228i	$-0.373634\pi$
$139$	9.11697	0.773291	0.386646	+	0.922228i	$0.373634\pi$
$140$	0	0
$141$	0.541702	0.0456196
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.92327	0.574946
$146$	0	0
$147$	17.6225	1.45348
$148$	0	0
$149$	18.1001	1.48281	0.741407	−	0.671055i	$-0.234159\pi$
$149$	18.1001	1.48281	0.741407	+	0.671055i	$0.234159\pi$
$150$	0	0
$151$	−10.6982	−0.870606	−0.435303	−	0.900284i	$-0.643359\pi$
$151$	−10.6982	−0.870606	−0.435303	+	0.900284i	$0.643359\pi$
$152$	0	0
$153$	−22.7919	−1.84262
$154$	0	0
$155$	4.47755	0.359645
$156$	0	0
$157$	−3.32321	−0.265221	−0.132611	−	0.991168i	$-0.542336\pi$
$157$	−3.32321	−0.265221	−0.132611	+	0.991168i	$0.542336\pi$
$158$	0	0
$159$	−36.9760	−2.93239
$160$	0	0
$161$	−5.90428	−0.465322
$162$	0	0
$163$	−10.0868	−0.790058	−0.395029	−	0.918669i	$-0.629265\pi$
$163$	−10.0868	−0.790058	−0.395029	+	0.918669i	$0.629265\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.39010	0.571863	0.285931	−	0.958250i	$-0.407697\pi$
$167$	7.39010	0.571863	0.285931	+	0.958250i	$0.407697\pi$
$168$	0	0
$169$	−12.7457	−0.980437
$170$	0	0
$171$	14.5167	1.11012
$172$	0	0
$173$	18.5390	1.40949	0.704745	−	0.709460i	$-0.251061\pi$
$173$	18.5390	1.40949	0.704745	+	0.709460i	$0.251061\pi$
$174$	0	0
$175$	1.02876	0.0777671
$176$	0	0
$177$	31.2887	2.35180
$178$	0	0
$179$	26.2551	1.96240	0.981199	−	0.192997i	$-0.0618206\pi$
$179$	26.2551	1.96240	0.981199	+	0.192997i	$0.0618206\pi$
$180$	0	0
$181$	11.5551	0.858887	0.429444	−	0.903094i	$-0.358710\pi$
$181$	11.5551	0.858887	0.429444	+	0.903094i	$0.358710\pi$
$182$	0	0
$183$	32.3905	2.39437
$184$	0	0
$185$	4.78329	0.351674
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−8.53349	−0.620720
$190$	0	0
$191$	10.4804	0.758334	0.379167	−	0.925328i	$-0.376211\pi$
$191$	10.4804	0.758334	0.379167	+	0.925328i	$0.376211\pi$
$192$	0	0
$193$	19.1446	1.37806	0.689030	−	0.724733i	$-0.258037\pi$
$193$	19.1446	1.37806	0.689030	+	0.724733i	$0.258037\pi$
$194$	0	0
$195$	1.49570	0.107110
$196$	0	0
$197$	0.935844	0.0666761	0.0333381	−	0.999444i	$-0.489386\pi$
$197$	0.935844	0.0666761	0.0333381	+	0.999444i	$0.489386\pi$
$198$	0	0
$199$	−26.4159	−1.87257	−0.936286	−	0.351238i	$-0.885761\pi$
$199$	−26.4159	−1.87257	−0.936286	+	0.351238i	$0.885761\pi$
$200$	0	0
$201$	−28.9589	−2.04260
$202$	0	0
$203$	7.12240	0.499894
$204$	0	0
$205$	11.8404	0.826971
$206$	0	0
$207$	−33.2686	−2.31233
$208$	0	0
$209$	0	0
$210$	0	0
$211$	10.1888	0.701423	0.350712	−	0.936483i	$-0.385940\pi$
$211$	10.1888	0.701423	0.350712	+	0.936483i	$0.385940\pi$
$212$	0	0
$213$	41.5205	2.84494
$214$	0	0
$215$	−7.10060	−0.484257
$216$	0	0
$217$	4.60633	0.312698
$218$	0	0
$219$	21.7201	1.46771
$220$	0	0
$221$	1.98282	0.133379
$222$	0	0
$223$	−3.62471	−0.242728	−0.121364	−	0.992608i	$-0.538727\pi$
$223$	−3.62471	−0.242728	−0.121364	+	0.992608i	$0.538727\pi$
$224$	0	0
$225$	5.79673	0.386449
$226$	0	0
$227$	10.6472	0.706676	0.353338	−	0.935496i	$-0.385046\pi$
$227$	10.6472	0.706676	0.353338	+	0.935496i	$0.385046\pi$
$228$	0	0
$229$	−2.43755	−0.161078	−0.0805388	−	0.996751i	$-0.525664\pi$
$229$	−2.43755	−0.161078	−0.0805388	+	0.996751i	$0.525664\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.6962	1.55239	0.776193	−	0.630495i	$-0.217148\pi$
$233$	23.6962	1.55239	0.776193	+	0.630495i	$0.217148\pi$
$234$	0	0
$235$	−0.182642	−0.0119142
$236$	0	0
$237$	−30.4728	−1.97942
$238$	0	0
$239$	3.50301	0.226591	0.113295	−	0.993561i	$-0.463859\pi$
$239$	3.50301	0.226591	0.113295	+	0.993561i	$0.463859\pi$
$240$	0	0
$241$	−7.01997	−0.452196	−0.226098	−	0.974105i	$-0.572597\pi$
$241$	−7.01997	−0.452196	−0.226098	+	0.974105i	$0.572597\pi$
$242$	0	0
$243$	3.49474	0.224188
$244$	0	0
$245$	−5.94165	−0.379598
$246$	0	0
$247$	−1.26291	−0.0803567
$248$	0	0
$249$	−12.8534	−0.814550
$250$	0	0
$251$	21.8188	1.37719	0.688596	−	0.725145i	$-0.258228\pi$
$251$	21.8188	1.37719	0.688596	+	0.725145i	$0.258228\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	11.6616	0.730278
$256$	0	0
$257$	1.66770	0.104029	0.0520143	−	0.998646i	$-0.483436\pi$
$257$	1.66770	0.104029	0.0520143	+	0.998646i	$0.483436\pi$
$258$	0	0
$259$	4.92087	0.305768
$260$	0	0
$261$	40.1323	2.48413
$262$	0	0
$263$	23.9489	1.47675	0.738376	−	0.674389i	$-0.235593\pi$
$263$	23.9489	1.47675	0.738376	+	0.674389i	$0.235593\pi$
$264$	0	0
$265$	12.4669	0.765837
$266$	0	0
$267$	−51.2479	−3.13632
$268$	0	0
$269$	12.1425	0.740340	0.370170	−	0.928964i	$-0.379300\pi$
$269$	12.1425	0.740340	0.370170	+	0.928964i	$0.379300\pi$
$270$	0	0
$271$	18.3731	1.11608	0.558042	−	0.829813i	$-0.311553\pi$
$271$	18.3731	1.11608	0.558042	+	0.829813i	$0.311553\pi$
$272$	0	0
$273$	1.53872	0.0931279
$274$	0	0
$275$	0	0
$276$	0	0
$277$	19.3747	1.16411	0.582057	−	0.813148i	$-0.302248\pi$
$277$	19.3747	1.16411	0.582057	+	0.813148i	$0.302248\pi$
$278$	0	0
$279$	25.9551	1.55389
$280$	0	0
$281$	5.28855	0.315488	0.157744	−	0.987480i	$-0.449578\pi$
$281$	5.28855	0.315488	0.157744	+	0.987480i	$0.449578\pi$
$282$	0	0
$283$	13.3392	0.792936	0.396468	−	0.918049i	$-0.370236\pi$
$283$	13.3392	0.792936	0.396468	+	0.918049i	$0.370236\pi$
$284$	0	0
$285$	−7.42756	−0.439971
$286$	0	0
$287$	12.1810	0.719021
$288$	0	0
$289$	−1.54050	−0.0906178
$290$	0	0
$291$	23.1392	1.35644
$292$	0	0
$293$	−9.70482	−0.566961	−0.283481	−	0.958978i	$-0.591489\pi$
$293$	−9.70482	−0.566961	−0.283481	+	0.958978i	$0.591489\pi$
$294$	0	0
$295$	−10.5494	−0.614209
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.89426	0.167379
$300$	0	0
$301$	−7.30483	−0.421043
$302$	0	0
$303$	20.4956	1.17744
$304$	0	0
$305$	−10.9209	−0.625326
$306$	0	0
$307$	13.7220	0.783156	0.391578	−	0.920145i	$-0.371929\pi$
$307$	13.7220	0.783156	0.391578	+	0.920145i	$0.371929\pi$
$308$	0	0
$309$	−3.18525	−0.181203
$310$	0	0
$311$	−0.293496	−0.0166426	−0.00832131	−	0.999965i	$-0.502649\pi$
$311$	−0.293496	−0.0166426	−0.00832131	+	0.999965i	$0.502649\pi$
$312$	0	0
$313$	1.45181	0.0820611	0.0410305	−	0.999158i	$-0.486936\pi$
$313$	1.45181	0.0820611	0.0410305	+	0.999158i	$0.486936\pi$
$314$	0	0
$315$	5.96346	0.336003
$316$	0	0
$317$	−27.4596	−1.54228	−0.771142	−	0.636664i	$-0.780314\pi$
$317$	−27.4596	−1.54228	−0.771142	+	0.636664i	$0.780314\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	46.3086	2.58469
$322$	0	0
$323$	−9.84653	−0.547876
$324$	0	0
$325$	−0.504296	−0.0279733
$326$	0	0
$327$	−31.3297	−1.73254
$328$	0	0
$329$	−0.187895	−0.0103590
$330$	0	0
$331$	16.6968	0.917741	0.458871	−	0.888503i	$-0.348254\pi$
$331$	16.6968	0.917741	0.458871	+	0.888503i	$0.348254\pi$
$332$	0	0
$333$	27.7274	1.51945
$334$	0	0
$335$	9.76386	0.533457
$336$	0	0
$337$	−9.17166	−0.499612	−0.249806	−	0.968296i	$-0.580367\pi$
$337$	−9.17166	−0.499612	−0.249806	+	0.968296i	$0.580367\pi$
$338$	0	0
$339$	−5.13031	−0.278640
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−13.3139	−0.718882
$344$	0	0
$345$	17.0221	0.916438
$346$	0	0
$347$	23.7060	1.27260	0.636301	−	0.771441i	$-0.280463\pi$
$347$	23.7060	1.27260	0.636301	+	0.771441i	$0.280463\pi$
$348$	0	0
$349$	13.3834	0.716396	0.358198	−	0.933646i	$-0.383391\pi$
$349$	13.3834	0.716396	0.358198	+	0.933646i	$0.383391\pi$
$350$	0	0
$351$	4.18308	0.223276
$352$	0	0
$353$	−22.7286	−1.20972	−0.604861	−	0.796331i	$-0.706771\pi$
$353$	−22.7286	−1.20972	−0.604861	+	0.796331i	$0.706771\pi$
$354$	0	0
$355$	−13.9992	−0.742999
$356$	0	0
$357$	11.9970	0.634950
$358$	0	0
$359$	5.39337	0.284651	0.142325	−	0.989820i	$-0.454542\pi$
$359$	5.39337	0.284651	0.142325	+	0.989820i	$0.454542\pi$
$360$	0	0
$361$	−12.7285	−0.669921
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.32321	−0.383314
$366$	0	0
$367$	8.58151	0.447951	0.223976	−	0.974595i	$-0.428096\pi$
$367$	8.58151	0.447951	0.223976	+	0.974595i	$0.428096\pi$
$368$	0	0
$369$	68.6358	3.57304
$370$	0	0
$371$	12.8255	0.665867
$372$	0	0
$373$	33.0945	1.71357	0.856784	−	0.515676i	$-0.172459\pi$
$373$	33.0945	1.71357	0.856784	+	0.515676i	$0.172459\pi$
$374$	0	0
$375$	−2.96593	−0.153160
$376$	0	0
$377$	−3.49137	−0.179815
$378$	0	0
$379$	−7.28378	−0.374143	−0.187071	−	0.982346i	$-0.559900\pi$
$379$	−7.28378	−0.374143	−0.187071	+	0.982346i	$0.559900\pi$
$380$	0	0
$381$	29.0869	1.49017
$382$	0	0
$383$	−8.90242	−0.454892	−0.227446	−	0.973791i	$-0.573038\pi$
$383$	−8.90242	−0.454892	−0.227446	+	0.973791i	$0.573038\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−41.1603	−2.09229
$388$	0	0
$389$	−3.12005	−0.158193	−0.0790963	−	0.996867i	$-0.525203\pi$
$389$	−3.12005	−0.158193	−0.0790963	+	0.996867i	$0.525203\pi$
$390$	0	0
$391$	22.5657	1.14120
$392$	0	0
$393$	11.2493	0.567452
$394$	0	0
$395$	10.2743	0.516955
$396$	0	0
$397$	28.9534	1.45313	0.726566	−	0.687097i	$-0.241115\pi$
$397$	28.9534	1.45313	0.726566	+	0.687097i	$0.241115\pi$
$398$	0	0
$399$	−7.64120	−0.382538
$400$	0	0
$401$	−22.7102	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$401$	−22.7102	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$402$	0	0
$403$	−2.25801	−0.112479
$404$	0	0
$405$	7.21190	0.358362
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−27.8432	−1.37676	−0.688379	−	0.725351i	$-0.741677\pi$
$409$	−27.8432	−1.37676	−0.688379	+	0.725351i	$0.741677\pi$
$410$	0	0
$411$	13.5986	0.670772
$412$	0	0
$413$	−10.8528	−0.534032
$414$	0	0
$415$	4.33368	0.212732
$416$	0	0
$417$	−27.0403	−1.32417
$418$	0	0
$419$	−34.6969	−1.69506	−0.847528	−	0.530751i	$-0.821910\pi$
$419$	−34.6969	−1.69506	−0.847528	+	0.530751i	$0.821910\pi$
$420$	0	0
$421$	6.16073	0.300256	0.150128	−	0.988667i	$-0.452031\pi$
$421$	6.16073	0.300256	0.150128	+	0.988667i	$0.452031\pi$
$422$	0	0
$423$	−1.05872	−0.0514770
$424$	0	0
$425$	−3.93186	−0.190723
$426$	0	0
$427$	−11.2350	−0.543698
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.9212	1.24858	0.624291	−	0.781192i	$-0.285388\pi$
$431$	25.9212	1.24858	0.624291	+	0.781192i	$0.285388\pi$
$432$	0	0
$433$	−16.6393	−0.799634	−0.399817	−	0.916595i	$-0.630926\pi$
$433$	−16.6393	−0.799634	−0.399817	+	0.916595i	$0.630926\pi$
$434$	0	0
$435$	−20.5339	−0.984526
$436$	0	0
$437$	−14.3727	−0.687538
$438$	0	0
$439$	22.3385	1.06616	0.533079	−	0.846066i	$-0.321035\pi$
$439$	22.3385	1.06616	0.533079	+	0.846066i	$0.321035\pi$
$440$	0	0
$441$	−34.4421	−1.64010
$442$	0	0
$443$	18.3803	0.873275	0.436637	−	0.899638i	$-0.356169\pi$
$443$	18.3803	0.873275	0.436637	+	0.899638i	$0.356169\pi$
$444$	0	0
$445$	17.2789	0.819098
$446$	0	0
$447$	−53.6835	−2.53914
$448$	0	0
$449$	13.8228	0.652338	0.326169	−	0.945311i	$-0.394242\pi$
$449$	13.8228	0.652338	0.326169	+	0.945311i	$0.394242\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	31.7301	1.49081
$454$	0	0
$455$	−0.518800	−0.0243217
$456$	0	0
$457$	−15.7689	−0.737638	−0.368819	−	0.929501i	$-0.620238\pi$
$457$	−15.7689	−0.737638	−0.368819	+	0.929501i	$0.620238\pi$
$458$	0	0
$459$	32.6144	1.52231
$460$	0	0
$461$	30.8758	1.43803	0.719014	−	0.694996i	$-0.244594\pi$
$461$	30.8758	1.43803	0.719014	+	0.694996i	$0.244594\pi$
$462$	0	0
$463$	19.1958	0.892106	0.446053	−	0.895007i	$-0.352829\pi$
$463$	19.1958	0.892106	0.446053	+	0.895007i	$0.352829\pi$
$464$	0	0
$465$	−13.2801	−0.615849
$466$	0	0
$467$	−38.8289	−1.79679	−0.898393	−	0.439193i	$-0.855264\pi$
$467$	−38.8289	−1.79679	−0.898393	+	0.439193i	$0.855264\pi$
$468$	0	0
$469$	10.0447	0.463821
$470$	0	0
$471$	9.85641	0.454159
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.50430	0.114905
$476$	0	0
$477$	72.2674	3.30890
$478$	0	0
$479$	9.12135	0.416765	0.208383	−	0.978047i	$-0.433180\pi$
$479$	9.12135	0.416765	0.208383	+	0.978047i	$0.433180\pi$
$480$	0	0
$481$	−2.41219	−0.109986
$482$	0	0
$483$	17.5117	0.796809
$484$	0	0
$485$	−7.80167	−0.354256
$486$	0	0
$487$	15.8103	0.716431	0.358216	−	0.933639i	$-0.383385\pi$
$487$	15.8103	0.716431	0.358216	+	0.933639i	$0.383385\pi$
$488$	0	0
$489$	29.9167	1.35288
$490$	0	0
$491$	8.09594	0.365365	0.182682	−	0.983172i	$-0.441522\pi$
$491$	8.09594	0.365365	0.182682	+	0.983172i	$0.441522\pi$
$492$	0	0
$493$	−27.2213	−1.22599
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14.4018	−0.646010
$498$	0	0
$499$	38.2856	1.71390	0.856948	−	0.515402i	$-0.172358\pi$
$499$	38.2856	1.71390	0.856948	+	0.515402i	$0.172358\pi$
$500$	0	0
$501$	−21.9185	−0.979246
$502$	0	0
$503$	−26.1242	−1.16482	−0.582410	−	0.812895i	$-0.697890\pi$
$503$	−26.1242	−1.16482	−0.582410	+	0.812895i	$0.697890\pi$
$504$	0	0
$505$	−6.91035	−0.307506
$506$	0	0
$507$	37.8028	1.67888
$508$	0	0
$509$	38.3357	1.69920	0.849600	−	0.527427i	$-0.176843\pi$
$509$	38.3357	1.69920	0.849600	+	0.527427i	$0.176843\pi$
$510$	0	0
$511$	−7.53385	−0.333278
$512$	0	0
$513$	−20.7729	−0.917146
$514$	0	0
$515$	1.07395	0.0473238
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−54.9852	−2.41358
$520$	0	0
$521$	−22.0519	−0.966111	−0.483055	−	0.875590i	$-0.660473\pi$
$521$	−22.0519	−0.966111	−0.483055	+	0.875590i	$0.660473\pi$
$522$	0	0
$523$	11.3086	0.494492	0.247246	−	0.968953i	$-0.420474\pi$
$523$	11.3086	0.494492	0.247246	+	0.968953i	$0.420474\pi$
$524$	0	0
$525$	−3.05124	−0.133167
$526$	0	0
$527$	−17.6051	−0.766889
$528$	0	0
$529$	9.93849	0.432108
$530$	0	0
$531$	−61.1519	−2.65377
$532$	0	0
$533$	−5.97107	−0.258636
$534$	0	0
$535$	−15.6135	−0.675032
$536$	0	0
$537$	−77.8708	−3.36037
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−13.9019	−0.597688	−0.298844	−	0.954302i	$-0.596601\pi$
$541$	−13.9019	−0.597688	−0.298844	+	0.954302i	$0.596601\pi$
$542$	0	0
$543$	−34.2717	−1.47074
$544$	0	0
$545$	10.5632	0.452478
$546$	0	0
$547$	−24.7035	−1.05625	−0.528123	−	0.849168i	$-0.677104\pi$
$547$	−24.7035	−1.05625	−0.528123	+	0.849168i	$0.677104\pi$
$548$	0	0
$549$	−63.3053	−2.70180
$550$	0	0
$551$	17.3379	0.738620
$552$	0	0
$553$	10.5698	0.449473
$554$	0	0
$555$	−14.1869	−0.602200
$556$	0	0
$557$	17.4258	0.738355	0.369177	−	0.929359i	$-0.379639\pi$
$557$	17.4258	0.738355	0.369177	+	0.929359i	$0.379639\pi$
$558$	0	0
$559$	3.58080	0.151452
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.29128	0.265146	0.132573	−	0.991173i	$-0.457676\pi$
$563$	6.29128	0.265146	0.132573	+	0.991173i	$0.457676\pi$
$564$	0	0
$565$	1.72975	0.0727711
$566$	0	0
$567$	7.41933	0.311583
$568$	0	0
$569$	23.8241	0.998759	0.499380	−	0.866383i	$-0.333561\pi$
$569$	23.8241	0.998759	0.499380	+	0.866383i	$0.333561\pi$
$570$	0	0
$571$	32.9400	1.37850	0.689248	−	0.724525i	$-0.257941\pi$
$571$	32.9400	1.37850	0.689248	+	0.724525i	$0.257941\pi$
$572$	0	0
$573$	−31.0841	−1.29856
$574$	0	0
$575$	−5.73921	−0.239341
$576$	0	0
$577$	6.98162	0.290649	0.145324	−	0.989384i	$-0.453577\pi$
$577$	6.98162	0.290649	0.145324	+	0.989384i	$0.453577\pi$
$578$	0	0
$579$	−56.7816	−2.35976
$580$	0	0
$581$	4.45833	0.184963
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−2.92327	−0.120862
$586$	0	0
$587$	−19.4035	−0.800867	−0.400433	−	0.916326i	$-0.631140\pi$
$587$	−19.4035	−0.800867	−0.400433	+	0.916326i	$0.631140\pi$
$588$	0	0
$589$	11.2131	0.462028
$590$	0	0
$591$	−2.77565	−0.114175
$592$	0	0
$593$	1.22305	0.0502248	0.0251124	−	0.999685i	$-0.492006\pi$
$593$	1.22305	0.0502248	0.0251124	+	0.999685i	$0.492006\pi$
$594$	0	0
$595$	−4.04495	−0.165827
$596$	0	0
$597$	78.3476	3.20655
$598$	0	0
$599$	−18.5791	−0.759122	−0.379561	−	0.925167i	$-0.623925\pi$
$599$	−18.5791	−0.759122	−0.379561	+	0.925167i	$0.623925\pi$
$600$	0	0
$601$	10.7275	0.437585	0.218793	−	0.975771i	$-0.429788\pi$
$601$	10.7275	0.437585	0.218793	+	0.975771i	$0.429788\pi$
$602$	0	0
$603$	56.5985	2.30487
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.8219	0.845133	0.422567	−	0.906332i	$-0.361129\pi$
$607$	20.8219	0.845133	0.422567	+	0.906332i	$0.361129\pi$
$608$	0	0
$609$	−21.1245	−0.856009
$610$	0	0
$611$	0.0921054	0.00372619
$612$	0	0
$613$	−15.4592	−0.624390	−0.312195	−	0.950018i	$-0.601064\pi$
$613$	−15.4592	−0.624390	−0.312195	+	0.950018i	$0.601064\pi$
$614$	0	0
$615$	−35.1179	−1.41609
$616$	0	0
$617$	7.32523	0.294903	0.147451	−	0.989069i	$-0.452893\pi$
$617$	7.32523	0.294903	0.147451	+	0.989069i	$0.452893\pi$
$618$	0	0
$619$	18.8171	0.756322	0.378161	−	0.925740i	$-0.376557\pi$
$619$	18.8171	0.756322	0.378161	+	0.925740i	$0.376557\pi$
$620$	0	0
$621$	47.6062	1.91037
$622$	0	0
$623$	17.7759	0.712175
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−18.8072	−0.749892
$630$	0	0
$631$	−40.5028	−1.61239	−0.806196	−	0.591648i	$-0.798477\pi$
$631$	−40.5028	−1.61239	−0.806196	+	0.591648i	$0.798477\pi$
$632$	0	0
$633$	−30.2191	−1.20110
$634$	0	0
$635$	−9.80700	−0.389179
$636$	0	0
$637$	2.99635	0.118720
$638$	0	0
$639$	−81.1494	−3.21022
$640$	0	0
$641$	21.5110	0.849632	0.424816	−	0.905280i	$-0.360339\pi$
$641$	21.5110	0.849632	0.424816	+	0.905280i	$0.360339\pi$
$642$	0	0
$643$	−7.91518	−0.312144	−0.156072	−	0.987746i	$-0.549883\pi$
$643$	−7.91518	−0.312144	−0.156072	+	0.987746i	$0.549883\pi$
$644$	0	0
$645$	21.0599	0.829231
$646$	0	0
$647$	19.1958	0.754666	0.377333	−	0.926078i	$-0.376841\pi$
$647$	19.1958	0.754666	0.377333	+	0.926078i	$0.376841\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−13.6620	−0.535458
$652$	0	0
$653$	−38.0894	−1.49055	−0.745277	−	0.666755i	$-0.767683\pi$
$653$	−38.0894	−1.49055	−0.745277	+	0.666755i	$0.767683\pi$
$654$	0	0
$655$	−3.79284	−0.148199
$656$	0	0
$657$	−42.4507	−1.65616
$658$	0	0
$659$	2.64662	0.103098	0.0515488	−	0.998670i	$-0.483584\pi$
$659$	2.64662	0.103098	0.0515488	+	0.998670i	$0.483584\pi$
$660$	0	0
$661$	−7.55682	−0.293926	−0.146963	−	0.989142i	$-0.546950\pi$
$661$	−7.55682	−0.293926	−0.146963	+	0.989142i	$0.546950\pi$
$662$	0	0
$663$	−5.88090	−0.228395
$664$	0	0
$665$	2.57633	0.0999056
$666$	0	0
$667$	−39.7340	−1.53851
$668$	0	0
$669$	10.7506	0.415643
$670$	0	0
$671$	0	0
$672$	0	0
$673$	37.5542	1.44761	0.723803	−	0.690006i	$-0.242392\pi$
$673$	37.5542	1.44761	0.723803	+	0.690006i	$0.242392\pi$
$674$	0	0
$675$	−8.29490	−0.319271
$676$	0	0
$677$	−27.3287	−1.05033	−0.525163	−	0.851002i	$-0.675996\pi$
$677$	−27.3287	−1.05033	−0.525163	+	0.851002i	$0.675996\pi$
$678$	0	0
$679$	−8.02606	−0.308012
$680$	0	0
$681$	−31.5787	−1.21010
$682$	0	0
$683$	29.6633	1.13503	0.567516	−	0.823362i	$-0.307904\pi$
$683$	29.6633	1.13503	0.567516	+	0.823362i	$0.307904\pi$
$684$	0	0
$685$	−4.58496	−0.175182
$686$	0	0
$687$	7.22959	0.275826
$688$	0	0
$689$	−6.28702	−0.239516
$690$	0	0
$691$	38.5532	1.46663	0.733317	−	0.679887i	$-0.237971\pi$
$691$	38.5532	1.46663	0.733317	+	0.679887i	$0.237971\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.11697	0.345826
$696$	0	0
$697$	−46.5549	−1.76339
$698$	0	0
$699$	−70.2811	−2.65828
$700$	0	0
$701$	27.6081	1.04274	0.521372	−	0.853330i	$-0.325420\pi$
$701$	27.6081	1.04274	0.521372	+	0.853330i	$0.325420\pi$
$702$	0	0
$703$	11.9788	0.451788
$704$	0	0
$705$	0.541702	0.0204017
$706$	0	0
$707$	−7.10910	−0.267365
$708$	0	0
$709$	7.98510	0.299887	0.149943	−	0.988695i	$-0.452091\pi$
$709$	7.98510	0.299887	0.149943	+	0.988695i	$0.452091\pi$
$710$	0	0
$711$	59.5572	2.23357
$712$	0	0
$713$	−25.6976	−0.962381
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−10.3897	−0.388010
$718$	0	0
$719$	7.46162	0.278271	0.139136	−	0.990273i	$-0.455568\pi$
$719$	7.46162	0.278271	0.139136	+	0.990273i	$0.455568\pi$
$720$	0	0
$721$	1.10484	0.0411463
$722$	0	0
$723$	20.8207	0.774331
$724$	0	0
$725$	6.92327	0.257124
$726$	0	0
$727$	30.2439	1.12168	0.560842	−	0.827923i	$-0.310478\pi$
$727$	30.2439	1.12168	0.560842	+	0.827923i	$0.310478\pi$
$728$	0	0
$729$	−32.0008	−1.18522
$730$	0	0
$731$	27.9185	1.03260
$732$	0	0
$733$	16.0298	0.592074	0.296037	−	0.955176i	$-0.404335\pi$
$733$	16.0298	0.592074	0.296037	+	0.955176i	$0.404335\pi$
$734$	0	0
$735$	17.6225	0.650016
$736$	0	0
$737$	0	0
$738$	0	0
$739$	52.7498	1.94043	0.970217	−	0.242238i	$-0.0778814\pi$
$739$	52.7498	1.94043	0.970217	+	0.242238i	$0.0778814\pi$
$740$	0	0
$741$	3.74569	0.137601
$742$	0	0
$743$	52.3073	1.91897	0.959484	−	0.281762i	$-0.0909190\pi$
$743$	52.3073	1.91897	0.959484	+	0.281762i	$0.0909190\pi$
$744$	0	0
$745$	18.1001	0.663135
$746$	0	0
$747$	25.1212	0.919136
$748$	0	0
$749$	−16.0626	−0.586915
$750$	0	0
$751$	−11.3042	−0.412497	−0.206249	−	0.978500i	$-0.566126\pi$
$751$	−11.3042	−0.412497	−0.206249	+	0.978500i	$0.566126\pi$
$752$	0	0
$753$	−64.7130	−2.35827
$754$	0	0
$755$	−10.6982	−0.389347
$756$	0	0
$757$	26.3105	0.956271	0.478135	−	0.878286i	$-0.341313\pi$
$757$	26.3105	0.956271	0.478135	+	0.878286i	$0.341313\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	45.5746	1.65208	0.826039	−	0.563612i	$-0.190589\pi$
$761$	45.5746	1.65208	0.826039	+	0.563612i	$0.190589\pi$
$762$	0	0
$763$	10.8670	0.393413
$764$	0	0
$765$	−22.7919	−0.824043
$766$	0	0
$767$	5.32001	0.192094
$768$	0	0
$769$	−52.2177	−1.88302	−0.941509	−	0.336987i	$-0.890592\pi$
$769$	−52.2177	−1.88302	−0.941509	+	0.336987i	$0.890592\pi$
$770$	0	0
$771$	−4.94629	−0.178136
$772$	0	0
$773$	19.5400	0.702807	0.351403	−	0.936224i	$-0.385705\pi$
$773$	19.5400	0.702807	0.351403	+	0.936224i	$0.385705\pi$
$774$	0	0
$775$	4.47755	0.160838
$776$	0	0
$777$	−14.5949	−0.523590
$778$	0	0
$779$	29.6519	1.06239
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−57.4278	−2.05230
$784$	0	0
$785$	−3.32321	−0.118610
$786$	0	0
$787$	−30.8849	−1.10093	−0.550464	−	0.834859i	$-0.685549\pi$
$787$	−30.8849	−1.10093	−0.550464	+	0.834859i	$0.685549\pi$
$788$	0	0
$789$	−71.0307	−2.52876
$790$	0	0
$791$	1.77950	0.0632718
$792$	0	0
$793$	5.50734	0.195571
$794$	0	0
$795$	−36.9760	−1.31140
$796$	0	0
$797$	−23.6837	−0.838920	−0.419460	−	0.907774i	$-0.637781\pi$
$797$	−23.6837	−0.838920	−0.419460	+	0.907774i	$0.637781\pi$
$798$	0	0
$799$	0.718121	0.0254053
$800$	0	0
$801$	100.161	3.53902
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−5.90428	−0.208099
$806$	0	0
$807$	−36.0137	−1.26774
$808$	0	0
$809$	3.57949	0.125848	0.0629240	−	0.998018i	$-0.479957\pi$
$809$	3.57949	0.125848	0.0629240	+	0.998018i	$0.479957\pi$
$810$	0	0
$811$	−22.1256	−0.776933	−0.388467	−	0.921463i	$-0.626995\pi$
$811$	−22.1256	−0.776933	−0.388467	+	0.921463i	$0.626995\pi$
$812$	0	0
$813$	−54.4932	−1.91116
$814$	0	0
$815$	−10.0868	−0.353325
$816$	0	0
$817$	−17.7820	−0.622113
$818$	0	0
$819$	−3.00735	−0.105085
$820$	0	0
$821$	40.1154	1.40004	0.700019	−	0.714124i	$-0.253175\pi$
$821$	40.1154	1.40004	0.700019	+	0.714124i	$0.253175\pi$
$822$	0	0
$823$	25.9856	0.905801	0.452900	−	0.891561i	$-0.350389\pi$
$823$	25.9856	0.905801	0.452900	+	0.891561i	$0.350389\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.5217	−1.13089	−0.565445	−	0.824786i	$-0.691296\pi$
$827$	−32.5217	−1.13089	−0.565445	+	0.824786i	$0.691296\pi$
$828$	0	0
$829$	−4.05382	−0.140795	−0.0703975	−	0.997519i	$-0.522427\pi$
$829$	−4.05382	−0.140795	−0.0703975	+	0.997519i	$0.522427\pi$
$830$	0	0
$831$	−57.4640	−1.99341
$832$	0	0
$833$	23.3617	0.809435
$834$	0	0
$835$	7.39010	0.255745
$836$	0	0
$837$	−37.1408	−1.28377
$838$	0	0
$839$	14.1939	0.490029	0.245015	−	0.969519i	$-0.421207\pi$
$839$	14.1939	0.490029	0.245015	+	0.969519i	$0.421207\pi$
$840$	0	0
$841$	18.9316	0.652814
$842$	0	0
$843$	−15.6855	−0.540236
$844$	0	0
$845$	−12.7457	−0.438465
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−39.5633	−1.35781
$850$	0	0
$851$	−27.4523	−0.941052
$852$	0	0
$853$	−12.0363	−0.412115	−0.206057	−	0.978540i	$-0.566063\pi$
$853$	−12.0363	−0.412115	−0.206057	+	0.978540i	$0.566063\pi$
$854$	0	0
$855$	14.5167	0.496462
$856$	0	0
$857$	12.4433	0.425054	0.212527	−	0.977155i	$-0.431831\pi$
$857$	12.4433	0.425054	0.212527	+	0.977155i	$0.431831\pi$
$858$	0	0
$859$	46.4462	1.58473	0.792363	−	0.610050i	$-0.208851\pi$
$859$	46.4462	1.58473	0.792363	+	0.610050i	$0.208851\pi$
$860$	0	0
$861$	−36.1279	−1.23124
$862$	0	0
$863$	11.1358	0.379066	0.189533	−	0.981874i	$-0.439303\pi$
$863$	11.1358	0.379066	0.189533	+	0.981874i	$0.439303\pi$
$864$	0	0
$865$	18.5390	0.630343
$866$	0	0
$867$	4.56902	0.155172
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−4.92387	−0.166839
$872$	0	0
$873$	−45.2242	−1.53061
$874$	0	0
$875$	1.02876	0.0347785
$876$	0	0
$877$	−57.3312	−1.93594	−0.967969	−	0.251071i	$-0.919217\pi$
$877$	−57.3312	−1.93594	−0.967969	+	0.251071i	$0.919217\pi$
$878$	0	0
$879$	28.7838	0.970853
$880$	0	0
$881$	−22.0068	−0.741428	−0.370714	−	0.928747i	$-0.620887\pi$
$881$	−22.0068	−0.741428	−0.370714	+	0.928747i	$0.620887\pi$
$882$	0	0
$883$	−4.31647	−0.145261	−0.0726304	−	0.997359i	$-0.523139\pi$
$883$	−4.31647	−0.145261	−0.0726304	+	0.997359i	$0.523139\pi$
$884$	0	0
$885$	31.2887	1.05176
$886$	0	0
$887$	19.2472	0.646258	0.323129	−	0.946355i	$-0.395265\pi$
$887$	19.2472	0.646258	0.323129	+	0.946355i	$0.395265\pi$
$888$	0	0
$889$	−10.0891	−0.338377
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.457389	−0.0153059
$894$	0	0
$895$	26.2551	0.877611
$896$	0	0
$897$	−8.58416	−0.286617
$898$	0	0
$899$	30.9992	1.03388
$900$	0	0
$901$	−49.0182	−1.63303
$902$	0	0
$903$	21.6656	0.720986
$904$	0	0
$905$	11.5551	0.384106
$906$	0	0
$907$	−20.5740	−0.683148	−0.341574	−	0.939855i	$-0.610960\pi$
$907$	−20.5740	−0.683148	−0.341574	+	0.939855i	$0.610960\pi$
$908$	0	0
$909$	−40.0574	−1.32862
$910$	0	0
$911$	26.3198	0.872014	0.436007	−	0.899943i	$-0.356392\pi$
$911$	26.3198	0.872014	0.436007	+	0.899943i	$0.356392\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	32.3905	1.07080
$916$	0	0
$917$	−3.90194	−0.128853
$918$	0	0
$919$	−38.9505	−1.28486	−0.642429	−	0.766345i	$-0.722073\pi$
$919$	−38.9505	−1.28486	−0.642429	+	0.766345i	$0.722073\pi$
$920$	0	0
$921$	−40.6985	−1.34106
$922$	0	0
$923$	7.05972	0.232374
$924$	0	0
$925$	4.78329	0.157273
$926$	0	0
$927$	6.22538	0.204468
$928$	0	0
$929$	−3.42694	−0.112434	−0.0562171	−	0.998419i	$-0.517904\pi$
$929$	−3.42694	−0.112434	−0.0562171	+	0.998419i	$0.517904\pi$
$930$	0	0
$931$	−14.8796	−0.487661
$932$	0	0
$933$	0.870488	0.0284985
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−48.2956	−1.57775	−0.788875	−	0.614554i	$-0.789336\pi$
$937$	−48.2956	−1.57775	−0.788875	+	0.614554i	$0.789336\pi$
$938$	0	0
$939$	−4.30596	−0.140520
$940$	0	0
$941$	−33.1280	−1.07994	−0.539971	−	0.841684i	$-0.681565\pi$
$941$	−33.1280	−1.07994	−0.539971	+	0.841684i	$0.681565\pi$
$942$	0	0
$943$	−67.9547	−2.21291
$944$	0	0
$945$	−8.53349	−0.277594
$946$	0	0
$947$	−38.8503	−1.26247	−0.631233	−	0.775593i	$-0.717451\pi$
$947$	−38.8503	−1.26247	−0.631233	+	0.775593i	$0.717451\pi$
$948$	0	0
$949$	3.69306	0.119882
$950$	0	0
$951$	81.4431	2.64097
$952$	0	0
$953$	−38.7874	−1.25645	−0.628224	−	0.778033i	$-0.716218\pi$
$953$	−38.7874	−1.25645	−0.628224	+	0.778033i	$0.716218\pi$
$954$	0	0
$955$	10.4804	0.339137
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.71683	−0.152314
$960$	0	0
$961$	−10.9516	−0.353277
$962$	0	0
$963$	−90.5074	−2.91656
$964$	0	0
$965$	19.1446	0.616287
$966$	0	0
$967$	−60.9782	−1.96093	−0.980464	−	0.196698i	$-0.936978\pi$
$967$	−60.9782	−1.96093	−0.980464	+	0.196698i	$0.936978\pi$
$968$	0	0
$969$	29.2041	0.938171
$970$	0	0
$971$	29.6498	0.951509	0.475754	−	0.879578i	$-0.342175\pi$
$971$	29.6498	0.951509	0.475754	+	0.879578i	$0.342175\pi$
$972$	0	0
$973$	9.37919	0.300683
$974$	0	0
$975$	1.49570	0.0479009
$976$	0	0
$977$	21.0925	0.674810	0.337405	−	0.941360i	$-0.390451\pi$
$977$	21.0925	0.674810	0.337405	+	0.941360i	$0.390451\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	61.2321	1.95499
$982$	0	0
$983$	33.0051	1.05270	0.526350	−	0.850268i	$-0.323560\pi$
$983$	33.0051	1.05270	0.526350	+	0.850268i	$0.323560\pi$
$984$	0	0
$985$	0.935844	0.0298185
$986$	0	0
$987$	0.557283	0.0177385
$988$	0	0
$989$	40.7518	1.29583
$990$	0	0
$991$	−19.6935	−0.625586	−0.312793	−	0.949821i	$-0.601265\pi$
$991$	−19.6935	−0.625586	−0.312793	+	0.949821i	$0.601265\pi$
$992$	0	0
$993$	−49.5216	−1.57152
$994$	0	0
$995$	−26.4159	−0.837440
$996$	0	0
$997$	9.30145	0.294580	0.147290	−	0.989093i	$-0.452945\pi$
$997$	9.30145	0.294580	0.147290	+	0.989093i	$0.452945\pi$
$998$	0	0
$999$	−39.6769	−1.25532

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.bd.1.1	yes	6
4.3	odd	2		9680.2.a.da.1.6		6
11.10	odd	2		4840.2.a.bc.1.1	✓	6
44.43	even	2		9680.2.a.db.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.bc.1.1	✓	6	11.10	odd	2
4840.2.a.bd.1.1	yes	6	1.1	even	1	trivial
9680.2.a.da.1.6		6	4.3	odd	2
9680.2.a.db.1.6		6	44.43	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:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q-2.96593 q^{3} +1.00000 q^{5} +1.02876 q^{7} +5.79673 q^{9} -0.504296 q^{13} -2.96593 q^{15} -3.93186 q^{17} +2.50430 q^{19} -3.05124 q^{21} -5.73921 q^{23} +1.00000 q^{25} -8.29490 q^{27} +6.92327 q^{29} +4.47755 q^{31} +1.02876 q^{35} +4.78329 q^{37} +1.49570 q^{39} +11.8404 q^{41} -7.10060 q^{43} +5.79673 q^{45} -0.182642 q^{47} -5.94165 q^{49} +11.6616 q^{51} +12.4669 q^{53} -7.42756 q^{57} -10.5494 q^{59} -10.9209 q^{61} +5.96346 q^{63} -0.504296 q^{65} +9.76386 q^{67} +17.0221 q^{69} -13.9992 q^{71} -7.32321 q^{73} -2.96593 q^{75} +10.2743 q^{79} +7.21190 q^{81} +4.33368 q^{83} -3.93186 q^{85} -20.5339 q^{87} +17.2789 q^{89} -0.518800 q^{91} -13.2801 q^{93} +2.50430 q^{95} -7.80167 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9} - 2 q^{15} + 8 q^{17} + 12 q^{19} - 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} + 16 q^{29} - 4 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} + 32 q^{41} - 4 q^{43}+ \cdots + 12 q^{95}+O(q^{100}) \) 6 * q - 2 * q^3 + 6 * q^5 + 4 * q^7 + 4 * q^9 - 2 * q^15 + 8 * q^17 + 12 * q^19 - 8 * q^21 - 8 * q^23 + 6 * q^25 - 14 * q^27 + 16 * q^29 - 4 * q^31 + 4 * q^35 + 8 * q^37 + 12 * q^39 + 32 * q^41 - 4 * q^43 + 4 * q^45 - 6 * q^47 + 16 * q^49 + 40 * q^51 + 8 * q^53 - 16 * q^57 + 4 * q^59 + 16 * q^61 + 28 * q^63 - 2 * q^67 + 8 * q^69 - 28 * q^71 + 16 * q^73 - 2 * q^75 - 10 * q^81 + 12 * q^83 + 8 * q^85 - 24 * q^87 + 18 * q^89 - 24 * q^91 + 20 * q^93 + 12 * q^95

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