LMFDB - Embedded newform 8379.2.a.bw.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8379,2,Mod(1,8379)] 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("8379.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8379, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,10,0,0,0,0,0,12,0,0,-8,0,0,22,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.9066518536$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13068.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} - x + 1 $ x^4 - x^3 - 6*x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 171)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.328543$ of defining polynomial
Character	$\chi$	$=$	8379.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.71519 q^{2} +5.37228 q^{4} +3.22060 q^{5} +9.15640 q^{8} +8.74456 q^{10} -2.20979 q^{11} -2.00000 q^{13} +14.1168 q^{16} +3.22060 q^{17} -1.00000 q^{19} +17.3020 q^{20} -6.00000 q^{22} +1.01082 q^{23} +5.37228 q^{25} -5.43039 q^{26} +1.01082 q^{29} -4.74456 q^{31} +20.0172 q^{32} +8.74456 q^{34} +10.7446 q^{37} -2.71519 q^{38} +29.4891 q^{40} +5.43039 q^{41} -11.1168 q^{43} -11.8716 q^{44} +2.74456 q^{46} +4.23142 q^{47} +14.5868 q^{50} -10.7446 q^{52} +9.84996 q^{53} -7.11684 q^{55} +2.74456 q^{58} -10.8608 q^{59} +5.11684 q^{61} -12.8824 q^{62} +26.1168 q^{64} -6.44121 q^{65} -4.00000 q^{67} +17.3020 q^{68} +2.02163 q^{71} +5.11684 q^{73} +29.1736 q^{74} -5.37228 q^{76} -4.00000 q^{79} +45.4647 q^{80} +14.7446 q^{82} +11.8716 q^{83} +10.3723 q^{85} -30.1844 q^{86} -20.2337 q^{88} -9.84996 q^{89} +5.43039 q^{92} +11.4891 q^{94} -3.22060 q^{95} -7.48913 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{4} + 12 q^{10} - 8 q^{13} + 22 q^{16} - 4 q^{19} - 24 q^{22} + 10 q^{25} + 4 q^{31} + 12 q^{34} + 20 q^{37} + 72 q^{40} - 10 q^{43} - 12 q^{46} - 20 q^{52} + 6 q^{55} - 12 q^{58} - 14 q^{61}+ \cdots + 16 q^{97}+O(q^{100}) $ 4 * q + 10 * q^4 + 12 * q^10 - 8 * q^13 + 22 * q^16 - 4 * q^19 - 24 * q^22 + 10 * q^25 + 4 * q^31 + 12 * q^34 + 20 * q^37 + 72 * q^40 - 10 * q^43 - 12 * q^46 - 20 * q^52 + 6 * q^55 - 12 * q^58 - 14 * q^61 + 70 * q^64 - 16 * q^67 - 14 * q^73 - 10 * q^76 - 16 * q^79 + 36 * q^82 + 30 * q^85 - 12 * q^88 + 16 * q^97

Coefficient data

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

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

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

$2$	2.71519	1.91993	0.959966	−	0.280116i	$-0.0903729\pi$
$2$	2.71519	1.91993	0.959966	+	0.280116i	$0.0903729\pi$
$3$	0	0
$3$	0	0
$4$	5.37228	2.68614
$5$	3.22060	1.44030	0.720149	−	0.693820i	$-0.244074\pi$
$5$	3.22060	1.44030	0.720149	+	0.693820i	$0.244074\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	9.15640	3.23728
$9$	0	0
$10$	8.74456	2.76527
$11$	−2.20979	−0.666276	−0.333138	−	0.942878i	$-0.608107\pi$
$11$	−2.20979	−0.666276	−0.333138	+	0.942878i	$0.608107\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	14.1168	3.52921
$17$	3.22060	0.781111	0.390555	−	0.920579i	$-0.372283\pi$
$17$	3.22060	0.781111	0.390555	+	0.920579i	$0.372283\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	17.3020	3.86884
$21$	0	0
$22$	−6.00000	−1.27920
$23$	1.01082	0.210770	0.105385	−	0.994432i	$-0.466393\pi$
$23$	1.01082	0.210770	0.105385	+	0.994432i	$0.466393\pi$
$24$	0	0
$25$	5.37228	1.07446
$26$	−5.43039	−1.06499
$27$	0	0
$28$	0	0
$29$	1.01082	0.187704	0.0938519	−	0.995586i	$-0.470082\pi$
$29$	1.01082	0.187704	0.0938519	+	0.995586i	$0.470082\pi$
$30$	0	0
$31$	−4.74456	−0.852149	−0.426074	−	0.904688i	$-0.640104\pi$
$31$	−4.74456	−0.852149	−0.426074	+	0.904688i	$0.640104\pi$
$32$	20.0172	3.53857
$33$	0	0
$34$	8.74456	1.49968
$35$	0	0
$36$	0	0
$37$	10.7446	1.76640	0.883198	−	0.469001i	$-0.155386\pi$
$37$	10.7446	1.76640	0.883198	+	0.469001i	$0.155386\pi$
$38$	−2.71519	−0.440463
$39$	0	0
$40$	29.4891	4.66264
$41$	5.43039	0.848084	0.424042	−	0.905642i	$-0.360611\pi$
$41$	5.43039	0.848084	0.424042	+	0.905642i	$0.360611\pi$
$42$	0	0
$43$	−11.1168	−1.69530	−0.847651	−	0.530554i	$-0.821984\pi$
$43$	−11.1168	−1.69530	−0.847651	+	0.530554i	$0.821984\pi$
$44$	−11.8716	−1.78971
$45$	0	0
$46$	2.74456	0.404664
$47$	4.23142	0.617216	0.308608	−	0.951189i	$-0.400137\pi$
$47$	4.23142	0.617216	0.308608	+	0.951189i	$0.400137\pi$
$48$	0	0
$49$	0	0
$50$	14.5868	2.06288
$51$	0	0
$52$	−10.7446	−1.49000
$53$	9.84996	1.35300	0.676498	−	0.736444i	$-0.263497\pi$
$53$	9.84996	1.35300	0.676498	+	0.736444i	$0.263497\pi$
$54$	0	0
$55$	−7.11684	−0.959635
$56$	0	0
$57$	0	0
$58$	2.74456	0.360379
$59$	−10.8608	−1.41395	−0.706976	−	0.707237i	$-0.749941\pi$
$59$	−10.8608	−1.41395	−0.706976	+	0.707237i	$0.749941\pi$
$60$	0	0
$61$	5.11684	0.655145	0.327572	−	0.944826i	$-0.393769\pi$
$61$	5.11684	0.655145	0.327572	+	0.944826i	$0.393769\pi$
$62$	−12.8824	−1.63607
$63$	0	0
$64$	26.1168	3.26461
$65$	−6.44121	−0.798933
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	17.3020	2.09817
$69$	0	0
$70$	0	0
$71$	2.02163	0.239924	0.119962	−	0.992779i	$-0.461723\pi$
$71$	2.02163	0.239924	0.119962	+	0.992779i	$0.461723\pi$
$72$	0	0
$73$	5.11684	0.598881	0.299441	−	0.954115i	$-0.403200\pi$
$73$	5.11684	0.598881	0.299441	+	0.954115i	$0.403200\pi$
$74$	29.1736	3.39136
$75$	0	0
$76$	−5.37228	−0.616243
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	45.4647	5.08311
$81$	0	0
$82$	14.7446	1.62826
$83$	11.8716	1.30308	0.651538	−	0.758616i	$-0.274124\pi$
$83$	11.8716	1.30308	0.651538	+	0.758616i	$0.274124\pi$
$84$	0	0
$85$	10.3723	1.12503
$86$	−30.1844	−3.25487
$87$	0	0
$88$	−20.2337	−2.15692
$89$	−9.84996	−1.04409	−0.522047	−	0.852917i	$-0.674831\pi$
$89$	−9.84996	−1.04409	−0.522047	+	0.852917i	$0.674831\pi$
$90$	0	0
$91$	0	0
$92$	5.43039	0.566157
$93$	0	0
$94$	11.4891	1.18501
$95$	−3.22060	−0.330427
$96$	0	0
$97$	−7.48913	−0.760405	−0.380203	−	0.924903i	$-0.624146\pi$
$97$	−7.48913	−0.760405	−0.380203	+	0.924903i	$0.624146\pi$
$98$	0	0
$99$	0	0
$100$	28.8614	2.88614
$101$	−12.8824	−1.28185	−0.640924	−	0.767604i	$-0.721449\pi$
$101$	−12.8824	−1.28185	−0.640924	+	0.767604i	$0.721449\pi$
$102$	0	0
$103$	7.25544	0.714899	0.357450	−	0.933932i	$-0.383646\pi$
$103$	7.25544	0.714899	0.357450	+	0.933932i	$0.383646\pi$
$104$	−18.3128	−1.79572
$105$	0	0
$106$	26.7446	2.59766
$107$	4.41957	0.427256	0.213628	−	0.976915i	$-0.431472\pi$
$107$	4.41957	0.427256	0.213628	+	0.976915i	$0.431472\pi$
$108$	0	0
$109$	5.25544	0.503380	0.251690	−	0.967808i	$-0.419014\pi$
$109$	5.25544	0.503380	0.251690	+	0.967808i	$0.419014\pi$
$110$	−19.3236	−1.84243
$111$	0	0
$112$	0	0
$113$	−11.8716	−1.11679	−0.558393	−	0.829577i	$-0.688582\pi$
$113$	−11.8716	−1.11679	−0.558393	+	0.829577i	$0.688582\pi$
$114$	0	0
$115$	3.25544	0.303571
$116$	5.43039	0.504199
$117$	0	0
$118$	−29.4891	−2.71469
$119$	0	0
$120$	0	0
$121$	−6.11684	−0.556077
$122$	13.8932	1.25783
$123$	0	0
$124$	−25.4891	−2.28899
$125$	1.19897	0.107239
$126$	0	0
$127$	−12.7446	−1.13090	−0.565449	−	0.824784i	$-0.691297\pi$
$127$	−12.7446	−1.13090	−0.565449	+	0.824784i	$0.691297\pi$
$128$	30.8780	2.72925
$129$	0	0
$130$	−17.4891	−1.53390
$131$	15.0922	1.31861	0.659306	−	0.751875i	$-0.270850\pi$
$131$	15.0922	1.31861	0.659306	+	0.751875i	$0.270850\pi$
$132$	0	0
$133$	0	0
$134$	−10.8608	−0.938228
$135$	0	0
$136$	29.4891	2.52867
$137$	12.0597	1.03033	0.515167	−	0.857090i	$-0.327730\pi$
$137$	12.0597	1.03033	0.515167	+	0.857090i	$0.327730\pi$
$138$	0	0
$139$	−3.11684	−0.264367	−0.132184	−	0.991225i	$-0.542199\pi$
$139$	−3.11684	−0.264367	−0.132184	+	0.991225i	$0.542199\pi$
$140$	0	0
$141$	0	0
$142$	5.48913	0.460637
$143$	4.41957	0.369583
$144$	0	0
$145$	3.25544	0.270349
$146$	13.8932	1.14981
$147$	0	0
$148$	57.7228	4.74479
$149$	20.5226	1.68128	0.840638	−	0.541598i	$-0.182180\pi$
$149$	20.5226	1.68128	0.840638	+	0.541598i	$0.182180\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	−9.15640	−0.742682
$153$	0	0
$154$	0	0
$155$	−15.2804	−1.22735
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−10.8608	−0.864037
$159$	0	0
$160$	64.4674	5.09659
$161$	0	0
$162$	0	0
$163$	−21.4891	−1.68316	−0.841579	−	0.540134i	$-0.818374\pi$
$163$	−21.4891	−1.68316	−0.841579	+	0.540134i	$0.818374\pi$
$164$	29.1736	2.27807
$165$	0	0
$166$	32.2337	2.50182
$167$	6.44121	0.498435	0.249218	−	0.968447i	$-0.419827\pi$
$167$	6.44121	0.498435	0.249218	+	0.968447i	$0.419827\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	28.1628	2.15999
$171$	0	0
$172$	−59.7228	−4.55382
$173$	1.01082	0.0768509	0.0384255	−	0.999261i	$-0.487766\pi$
$173$	1.01082	0.0768509	0.0384255	+	0.999261i	$0.487766\pi$
$174$	0	0
$175$	0	0
$176$	−31.1952	−2.35143
$177$	0	0
$178$	−26.7446	−2.00459
$179$	−4.41957	−0.330334	−0.165167	−	0.986266i	$-0.552816\pi$
$179$	−4.41957	−0.330334	−0.165167	+	0.986266i	$0.552816\pi$
$180$	0	0
$181$	−19.4891	−1.44862	−0.724308	−	0.689477i	$-0.757840\pi$
$181$	−19.4891	−1.44862	−0.724308	+	0.689477i	$0.757840\pi$
$182$	0	0
$183$	0	0
$184$	9.25544	0.682320
$185$	34.6040	2.54413
$186$	0	0
$187$	−7.11684	−0.520435
$188$	22.7324	1.65793
$189$	0	0
$190$	−8.74456	−0.634397
$191$	−21.5334	−1.55810	−0.779051	−	0.626960i	$-0.784299\pi$
$191$	−21.5334	−1.55810	−0.779051	+	0.626960i	$0.784299\pi$
$192$	0	0
$193$	16.2337	1.16853	0.584263	−	0.811564i	$-0.301384\pi$
$193$	16.2337	1.16853	0.584263	+	0.811564i	$0.301384\pi$
$194$	−20.3344	−1.45993
$195$	0	0
$196$	0	0
$197$	−23.7432	−1.69163	−0.845816	−	0.533475i	$-0.820886\pi$
$197$	−23.7432	−1.69163	−0.845816	+	0.533475i	$0.820886\pi$
$198$	0	0
$199$	−0.883156	−0.0626053	−0.0313026	−	0.999510i	$-0.509966\pi$
$199$	−0.883156	−0.0626053	−0.0313026	+	0.999510i	$0.509966\pi$
$200$	49.1908	3.47831
$201$	0	0
$202$	−34.9783	−2.46106
$203$	0	0
$204$	0	0
$205$	17.4891	1.22149
$206$	19.6999	1.37256
$207$	0	0
$208$	−28.2337	−1.95765
$209$	2.20979	0.152854
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	52.9168	3.63434
$213$	0	0
$214$	12.0000	0.820303
$215$	−35.8029	−2.44174
$216$	0	0
$217$	0	0
$218$	14.2695	0.966455
$219$	0	0
$220$	−38.2337	−2.57771
$221$	−6.44121	−0.433282
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	0	0
$226$	−32.2337	−2.14415
$227$	−19.3236	−1.28255	−0.641277	−	0.767310i	$-0.721595\pi$
$227$	−19.3236	−1.28255	−0.641277	+	0.767310i	$0.721595\pi$
$228$	0	0
$229$	−15.6277	−1.03271	−0.516354	−	0.856375i	$-0.672711\pi$
$229$	−15.6277	−1.03271	−0.516354	+	0.856375i	$0.672711\pi$
$230$	8.83915	0.582836
$231$	0	0
$232$	9.25544	0.607649
$233$	−7.64018	−0.500525	−0.250262	−	0.968178i	$-0.580517\pi$
$233$	−7.64018	−0.500525	−0.250262	+	0.968178i	$0.580517\pi$
$234$	0	0
$235$	13.6277	0.888974
$236$	−58.3472	−3.79808
$237$	0	0
$238$	0	0
$239$	−12.6943	−0.821123	−0.410562	−	0.911833i	$-0.634667\pi$
$239$	−12.6943	−0.821123	−0.410562	+	0.911833i	$0.634667\pi$
$240$	0	0
$241$	24.2337	1.56103	0.780515	−	0.625138i	$-0.214957\pi$
$241$	24.2337	1.56103	0.780515	+	0.625138i	$0.214957\pi$
$242$	−16.6084	−1.06763
$243$	0	0
$244$	27.4891	1.75981
$245$	0	0
$246$	0	0
$247$	2.00000	0.127257
$248$	−43.4431	−2.75864
$249$	0	0
$250$	3.25544	0.205892
$251$	−23.9313	−1.51053	−0.755266	−	0.655418i	$-0.772493\pi$
$251$	−23.9313	−1.51053	−0.755266	+	0.655418i	$0.772493\pi$
$252$	0	0
$253$	−2.23369	−0.140431
$254$	−34.6040	−2.17125
$255$	0	0
$256$	31.6060	1.97537
$257$	−5.43039	−0.338738	−0.169369	−	0.985553i	$-0.554173\pi$
$257$	−5.43039	−0.338738	−0.169369	+	0.985553i	$0.554173\pi$
$258$	0	0
$259$	0	0
$260$	−34.6040	−2.14605
$261$	0	0
$262$	40.9783	2.53164
$263$	13.0706	0.805966	0.402983	−	0.915208i	$-0.367973\pi$
$263$	13.0706	0.805966	0.402983	+	0.915208i	$0.367973\pi$
$264$	0	0
$265$	31.7228	1.94872
$266$	0	0
$267$	0	0
$268$	−21.4891	−1.31266
$269$	−7.82833	−0.477302	−0.238651	−	0.971105i	$-0.576705\pi$
$269$	−7.82833	−0.477302	−0.238651	+	0.971105i	$0.576705\pi$
$270$	0	0
$271$	−25.4891	−1.54835	−0.774177	−	0.632969i	$-0.781836\pi$
$271$	−25.4891	−1.54835	−0.774177	+	0.632969i	$0.781836\pi$
$272$	45.4647	2.75671
$273$	0	0
$274$	32.7446	1.97817
$275$	−11.8716	−0.715884
$276$	0	0
$277$	9.11684	0.547778	0.273889	−	0.961761i	$-0.411690\pi$
$277$	9.11684	0.547778	0.273889	+	0.961761i	$0.411690\pi$
$278$	−8.46284	−0.507567
$279$	0	0
$280$	0	0
$281$	24.7540	1.47670	0.738350	−	0.674418i	$-0.235605\pi$
$281$	24.7540	1.47670	0.738350	+	0.674418i	$0.235605\pi$
$282$	0	0
$283$	−6.37228	−0.378793	−0.189396	−	0.981901i	$-0.560653\pi$
$283$	−6.37228	−0.378793	−0.189396	+	0.981901i	$0.560653\pi$
$284$	10.8608	0.644469
$285$	0	0
$286$	12.0000	0.709575
$287$	0	0
$288$	0	0
$289$	−6.62772	−0.389866
$290$	8.83915	0.519053
$291$	0	0
$292$	27.4891	1.60868
$293$	25.1303	1.46813	0.734064	−	0.679080i	$-0.237621\pi$
$293$	25.1303	1.46813	0.734064	+	0.679080i	$0.237621\pi$
$294$	0	0
$295$	−34.9783	−2.03651
$296$	98.3815	5.71831
$297$	0	0
$298$	55.7228	3.22794
$299$	−2.02163	−0.116914
$300$	0	0
$301$	0	0
$302$	−10.8608	−0.624968
$303$	0	0
$304$	−14.1168	−0.809657
$305$	16.4793	0.943603
$306$	0	0
$307$	−25.4891	−1.45474	−0.727371	−	0.686245i	$-0.759258\pi$
$307$	−25.4891	−1.45474	−0.727371	+	0.686245i	$0.759258\pi$
$308$	0	0
$309$	0	0
$310$	−41.4891	−2.35642
$311$	−12.6943	−0.719825	−0.359913	−	0.932986i	$-0.617194\pi$
$311$	−12.6943	−0.719825	−0.359913	+	0.932986i	$0.617194\pi$
$312$	0	0
$313$	−7.48913	−0.423310	−0.211655	−	0.977344i	$-0.567885\pi$
$313$	−7.48913	−0.423310	−0.211655	+	0.977344i	$0.567885\pi$
$314$	−5.43039	−0.306455
$315$	0	0
$316$	−21.4891	−1.20886
$317$	−12.2479	−0.687911	−0.343955	−	0.938986i	$-0.611767\pi$
$317$	−12.2479	−0.687911	−0.343955	+	0.938986i	$0.611767\pi$
$318$	0	0
$319$	−2.23369	−0.125063
$320$	84.1120	4.70200
$321$	0	0
$322$	0	0
$323$	−3.22060	−0.179199
$324$	0	0
$325$	−10.7446	−0.596001
$326$	−58.3472	−3.23155
$327$	0	0
$328$	49.7228	2.74548
$329$	0	0
$330$	0	0
$331$	18.9783	1.04314	0.521569	−	0.853209i	$-0.325347\pi$
$331$	18.9783	1.04314	0.521569	+	0.853209i	$0.325347\pi$
$332$	63.7775	3.50025
$333$	0	0
$334$	17.4891	0.956962
$335$	−12.8824	−0.703841
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−24.4368	−1.32918
$339$	0	0
$340$	55.7228	3.02199
$341$	10.4845	0.567766
$342$	0	0
$343$	0	0
$344$	−101.790	−5.48816
$345$	0	0
$346$	2.74456	0.147549
$347$	32.3942	1.73901	0.869505	−	0.493924i	$-0.164438\pi$
$347$	32.3942	1.73901	0.869505	+	0.493924i	$0.164438\pi$
$348$	0	0
$349$	−17.8614	−0.956099	−0.478050	−	0.878333i	$-0.658656\pi$
$349$	−17.8614	−0.956099	−0.478050	+	0.878333i	$0.658656\pi$
$350$	0	0
$351$	0	0
$352$	−44.2337	−2.35766
$353$	−14.9040	−0.793262	−0.396631	−	0.917978i	$-0.629821\pi$
$353$	−14.9040	−0.793262	−0.396631	+	0.917978i	$0.629821\pi$
$354$	0	0
$355$	6.51087	0.345561
$356$	−52.9168	−2.80458
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−4.60773	−0.243187	−0.121593	−	0.992580i	$-0.538800\pi$
$359$	−4.60773	−0.243187	−0.121593	+	0.992580i	$0.538800\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−52.9168	−2.78124
$363$	0	0
$364$	0	0
$365$	16.4793	0.862567
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	14.2695	0.743851
$369$	0	0
$370$	93.9565	4.88457
$371$	0	0
$372$	0	0
$373$	−24.2337	−1.25477	−0.627386	−	0.778708i	$-0.715875\pi$
$373$	−24.2337	−1.25477	−0.627386	+	0.778708i	$0.715875\pi$
$374$	−19.3236	−0.999200
$375$	0	0
$376$	38.7446	1.99810
$377$	−2.02163	−0.104119
$378$	0	0
$379$	28.7446	1.47651	0.738255	−	0.674522i	$-0.235650\pi$
$379$	28.7446	1.47651	0.738255	+	0.674522i	$0.235650\pi$
$380$	−17.3020	−0.887573
$381$	0	0
$382$	−58.4674	−2.99145
$383$	17.3020	0.884090	0.442045	−	0.896993i	$-0.354253\pi$
$383$	17.3020	0.884090	0.442045	+	0.896993i	$0.354253\pi$
$384$	0	0
$385$	0	0
$386$	44.0776	2.24349
$387$	0	0
$388$	−40.2337	−2.04256
$389$	12.0597	0.611454	0.305727	−	0.952119i	$-0.401101\pi$
$389$	12.0597	0.611454	0.305727	+	0.952119i	$0.401101\pi$
$390$	0	0
$391$	3.25544	0.164635
$392$	0	0
$393$	0	0
$394$	−64.4674	−3.24782
$395$	−12.8824	−0.648184
$396$	0	0
$397$	17.1168	0.859070	0.429535	−	0.903050i	$-0.358678\pi$
$397$	17.1168	0.859070	0.429535	+	0.903050i	$0.358678\pi$
$398$	−2.39794	−0.120198
$399$	0	0
$400$	75.8397	3.79198
$401$	3.40876	0.170225	0.0851126	−	0.996371i	$-0.472875\pi$
$401$	3.40876	0.170225	0.0851126	+	0.996371i	$0.472875\pi$
$402$	0	0
$403$	9.48913	0.472687
$404$	−69.2079	−3.44322
$405$	0	0
$406$	0	0
$407$	−23.7432	−1.17691
$408$	0	0
$409$	−7.48913	−0.370313	−0.185157	−	0.982709i	$-0.559279\pi$
$409$	−7.48913	−0.370313	−0.185157	+	0.982709i	$0.559279\pi$
$410$	47.4864	2.34519
$411$	0	0
$412$	38.9783	1.92032
$413$	0	0
$414$	0	0
$415$	38.2337	1.87682
$416$	−40.0344	−1.96285
$417$	0	0
$418$	6.00000	0.293470
$419$	−11.8716	−0.579965	−0.289983	−	0.957032i	$-0.593650\pi$
$419$	−11.8716	−0.579965	−0.289983	+	0.957032i	$0.593650\pi$
$420$	0	0
$421$	−8.97825	−0.437573	−0.218787	−	0.975773i	$-0.570210\pi$
$421$	−8.97825	−0.437573	−0.218787	+	0.975773i	$0.570210\pi$
$422$	−10.8608	−0.528694
$423$	0	0
$424$	90.1902	4.38002
$425$	17.3020	0.839269
$426$	0	0
$427$	0	0
$428$	23.7432	1.14767
$429$	0	0
$430$	−97.2119	−4.68798
$431$	−30.1844	−1.45393	−0.726966	−	0.686674i	$-0.759070\pi$
$431$	−30.1844	−1.45393	−0.726966	+	0.686674i	$0.759070\pi$
$432$	0	0
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	0	0
$435$	0	0
$436$	28.2337	1.35215
$437$	−1.01082	−0.0483539
$438$	0	0
$439$	18.2337	0.870246	0.435123	−	0.900371i	$-0.356705\pi$
$439$	18.2337	0.870246	0.435123	+	0.900371i	$0.356705\pi$
$440$	−65.1647	−3.10660
$441$	0	0
$442$	−17.4891	−0.831873
$443$	−27.9746	−1.32911	−0.664557	−	0.747238i	$-0.731380\pi$
$443$	−27.9746	−1.32911	−0.664557	+	0.747238i	$0.731380\pi$
$444$	0	0
$445$	−31.7228	−1.50381
$446$	10.8608	0.514273
$447$	0	0
$448$	0	0
$449$	22.3561	1.05505	0.527524	−	0.849540i	$-0.323120\pi$
$449$	22.3561	1.05505	0.527524	+	0.849540i	$0.323120\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	−63.7775	−2.99984
$453$	0	0
$454$	−52.4674	−2.46242
$455$	0	0
$456$	0	0
$457$	−8.37228	−0.391639	−0.195819	−	0.980640i	$-0.562737\pi$
$457$	−8.37228	−0.391639	−0.195819	+	0.980640i	$0.562737\pi$
$458$	−42.4323	−1.98273
$459$	0	0
$460$	17.4891	0.815435
$461$	26.9638	1.25583	0.627914	−	0.778282i	$-0.283909\pi$
$461$	26.9638	1.25583	0.627914	+	0.778282i	$0.283909\pi$
$462$	0	0
$463$	−2.37228	−0.110249	−0.0551246	−	0.998479i	$-0.517556\pi$
$463$	−2.37228	−0.110249	−0.0551246	+	0.998479i	$0.517556\pi$
$464$	14.2695	0.662447
$465$	0	0
$466$	−20.7446	−0.960973
$467$	36.4374	1.68612	0.843062	−	0.537816i	$-0.180751\pi$
$467$	36.4374	1.68612	0.843062	+	0.537816i	$0.180751\pi$
$468$	0	0
$469$	0	0
$470$	37.0019	1.70677
$471$	0	0
$472$	−99.4456	−4.57736
$473$	24.5659	1.12954
$474$	0	0
$475$	−5.37228	−0.246497
$476$	0	0
$477$	0	0
$478$	−34.4674	−1.57650
$479$	−33.5932	−1.53491	−0.767455	−	0.641103i	$-0.778477\pi$
$479$	−33.5932	−1.53491	−0.767455	+	0.641103i	$0.778477\pi$
$480$	0	0
$481$	−21.4891	−0.979820
$482$	65.7992	2.99707
$483$	0	0
$484$	−32.8614	−1.49370
$485$	−24.1195	−1.09521
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	46.8519	2.12088
$489$	0	0
$490$	0	0
$491$	−1.01082	−0.0456175	−0.0228087	−	0.999740i	$-0.507261\pi$
$491$	−1.01082	−0.0456175	−0.0228087	+	0.999740i	$0.507261\pi$
$492$	0	0
$493$	3.25544	0.146618
$494$	5.43039	0.244325
$495$	0	0
$496$	−66.9783	−3.00741
$497$	0	0
$498$	0	0
$499$	−11.1168	−0.497658	−0.248829	−	0.968547i	$-0.580046\pi$
$499$	−11.1168	−0.497658	−0.248829	+	0.968547i	$0.580046\pi$
$500$	6.44121	0.288059
$501$	0	0
$502$	−64.9783	−2.90012
$503$	31.5715	1.40770	0.703852	−	0.710346i	$-0.251462\pi$
$503$	31.5715	1.40770	0.703852	+	0.710346i	$0.251462\pi$
$504$	0	0
$505$	−41.4891	−1.84624
$506$	−6.06490	−0.269618
$507$	0	0
$508$	−68.4674	−3.03775
$509$	−24.7540	−1.09720	−0.548601	−	0.836084i	$-0.684839\pi$
$509$	−24.7540	−1.09720	−0.548601	+	0.836084i	$0.684839\pi$
$510$	0	0
$511$	0	0
$512$	24.0604	1.06333
$513$	0	0
$514$	−14.7446	−0.650355
$515$	23.3669	1.02967
$516$	0	0
$517$	−9.35053	−0.411236
$518$	0	0
$519$	0	0
$520$	−58.9783	−2.58637
$521$	−16.2912	−0.713729	−0.356864	−	0.934156i	$-0.616154\pi$
$521$	−16.2912	−0.713729	−0.356864	+	0.934156i	$0.616154\pi$
$522$	0	0
$523$	18.2337	0.797304	0.398652	−	0.917102i	$-0.369478\pi$
$523$	18.2337	0.797304	0.398652	+	0.917102i	$0.369478\pi$
$524$	81.0795	3.54198
$525$	0	0
$526$	35.4891	1.54740
$527$	−15.2804	−0.665623
$528$	0	0
$529$	−21.9783	−0.955576
$530$	86.1336	3.74140
$531$	0	0
$532$	0	0
$533$	−10.8608	−0.470433
$534$	0	0
$535$	14.2337	0.615376
$536$	−36.6256	−1.58198
$537$	0	0
$538$	−21.2554	−0.916387
$539$	0	0
$540$	0	0
$541$	21.1168	0.907884	0.453942	−	0.891031i	$-0.350017\pi$
$541$	21.1168	0.907884	0.453942	+	0.891031i	$0.350017\pi$
$542$	−69.2079	−2.97274
$543$	0	0
$544$	64.4674	2.76402
$545$	16.9257	0.725016
$546$	0	0
$547$	22.2337	0.950644	0.475322	−	0.879812i	$-0.342332\pi$
$547$	22.2337	0.950644	0.475322	+	0.879812i	$0.342332\pi$
$548$	64.7884	2.76762
$549$	0	0
$550$	−32.2337	−1.37445
$551$	−1.01082	−0.0430622
$552$	0	0
$553$	0	0
$554$	24.7540	1.05170
$555$	0	0
$556$	−16.7446	−0.710128
$557$	−20.5226	−0.869570	−0.434785	−	0.900534i	$-0.643176\pi$
$557$	−20.5226	−0.869570	−0.434785	+	0.900534i	$0.643176\pi$
$558$	0	0
$559$	22.2337	0.940385
$560$	0	0
$561$	0	0
$562$	67.2119	2.83516
$563$	38.6472	1.62879	0.814393	−	0.580313i	$-0.197070\pi$
$563$	38.6472	1.62879	0.814393	+	0.580313i	$0.197070\pi$
$564$	0	0
$565$	−38.2337	−1.60850
$566$	−17.3020	−0.727257
$567$	0	0
$568$	18.5109	0.776699
$569$	3.03245	0.127127	0.0635634	−	0.997978i	$-0.479753\pi$
$569$	3.03245	0.127127	0.0635634	+	0.997978i	$0.479753\pi$
$570$	0	0
$571$	2.51087	0.105077	0.0525384	−	0.998619i	$-0.483269\pi$
$571$	2.51087	0.105077	0.0525384	+	0.998619i	$0.483269\pi$
$572$	23.7432	0.992753
$573$	0	0
$574$	0	0
$575$	5.43039	0.226463
$576$	0	0
$577$	41.1168	1.71172	0.855858	−	0.517210i	$-0.173030\pi$
$577$	41.1168	1.71172	0.855858	+	0.517210i	$0.173030\pi$
$578$	−17.9955	−0.748516
$579$	0	0
$580$	17.4891	0.726196
$581$	0	0
$582$	0	0
$583$	−21.7663	−0.901469
$584$	46.8519	1.93874
$585$	0	0
$586$	68.2337	2.81871
$587$	1.83348	0.0756758	0.0378379	−	0.999284i	$-0.487953\pi$
$587$	1.83348	0.0756758	0.0378379	+	0.999284i	$0.487953\pi$
$588$	0	0
$589$	4.74456	0.195496
$590$	−94.9728	−3.90997
$591$	0	0
$592$	151.679	6.23398
$593$	−4.04326	−0.166037	−0.0830185	−	0.996548i	$-0.526456\pi$
$593$	−4.04326	−0.166037	−0.0830185	+	0.996548i	$0.526456\pi$
$594$	0	0
$595$	0	0
$596$	110.253	4.51614
$597$	0	0
$598$	−5.48913	−0.224467
$599$	12.8824	0.526361	0.263181	−	0.964747i	$-0.415229\pi$
$599$	12.8824	0.526361	0.263181	+	0.964747i	$0.415229\pi$
$600$	0	0
$601$	25.2554	1.03019	0.515095	−	0.857133i	$-0.327756\pi$
$601$	25.2554	1.03019	0.515095	+	0.857133i	$0.327756\pi$
$602$	0	0
$603$	0	0
$604$	−21.4891	−0.874380
$605$	−19.6999	−0.800916
$606$	0	0
$607$	−28.7446	−1.16671	−0.583353	−	0.812219i	$-0.698260\pi$
$607$	−28.7446	−1.16671	−0.583353	+	0.812219i	$0.698260\pi$
$608$	−20.0172	−0.811804
$609$	0	0
$610$	44.7446	1.81165
$611$	−8.46284	−0.342370
$612$	0	0
$613$	2.60597	0.105254	0.0526271	−	0.998614i	$-0.483241\pi$
$613$	2.60597	0.105254	0.0526271	+	0.998614i	$0.483241\pi$
$614$	−69.2079	−2.79300
$615$	0	0
$616$	0	0
$617$	3.22060	0.129657	0.0648283	−	0.997896i	$-0.479350\pi$
$617$	3.22060	0.129657	0.0648283	+	0.997896i	$0.479350\pi$
$618$	0	0
$619$	−6.97825	−0.280480	−0.140240	−	0.990118i	$-0.544787\pi$
$619$	−6.97825	−0.280480	−0.140240	+	0.990118i	$0.544787\pi$
$620$	−82.0903	−3.29683
$621$	0	0
$622$	−34.4674	−1.38202
$623$	0	0
$624$	0	0
$625$	−23.0000	−0.920000
$626$	−20.3344	−0.812727
$627$	0	0
$628$	−10.7446	−0.428755
$629$	34.6040	1.37975
$630$	0	0
$631$	15.1168	0.601792	0.300896	−	0.953657i	$-0.402714\pi$
$631$	15.1168	0.601792	0.300896	+	0.953657i	$0.402714\pi$
$632$	−36.6256	−1.45689
$633$	0	0
$634$	−33.2554	−1.32074
$635$	−41.0452	−1.62883
$636$	0	0
$637$	0	0
$638$	−6.06490	−0.240112
$639$	0	0
$640$	99.4456	3.93093
$641$	−24.7540	−0.977724	−0.488862	−	0.872361i	$-0.662588\pi$
$641$	−24.7540	−0.977724	−0.488862	+	0.872361i	$0.662588\pi$
$642$	0	0
$643$	35.1168	1.38487	0.692437	−	0.721479i	$-0.256537\pi$
$643$	35.1168	1.38487	0.692437	+	0.721479i	$0.256537\pi$
$644$	0	0
$645$	0	0
$646$	−8.74456	−0.344050
$647$	17.1138	0.672814	0.336407	−	0.941717i	$-0.390788\pi$
$647$	17.1138	0.672814	0.336407	+	0.941717i	$0.390788\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	−29.1736	−1.14428
$651$	0	0
$652$	−115.446	−4.52120
$653$	14.0814	0.551047	0.275524	−	0.961294i	$-0.411149\pi$
$653$	14.0814	0.551047	0.275524	+	0.961294i	$0.411149\pi$
$654$	0	0
$655$	48.6060	1.89919
$656$	76.6600	2.99307
$657$	0	0
$658$	0	0
$659$	11.2371	0.437735	0.218867	−	0.975755i	$-0.429764\pi$
$659$	11.2371	0.437735	0.218867	+	0.975755i	$0.429764\pi$
$660$	0	0
$661$	6.74456	0.262333	0.131167	−	0.991360i	$-0.458128\pi$
$661$	6.74456	0.262333	0.131167	+	0.991360i	$0.458128\pi$
$662$	51.5296	2.00276
$663$	0	0
$664$	108.701	4.21842
$665$	0	0
$666$	0	0
$667$	1.02175	0.0395623
$668$	34.6040	1.33887
$669$	0	0
$670$	−34.9783	−1.35133
$671$	−11.3071	−0.436507
$672$	0	0
$673$	−25.2554	−0.973526	−0.486763	−	0.873534i	$-0.661822\pi$
$673$	−25.2554	−0.973526	−0.486763	+	0.873534i	$0.661822\pi$
$674$	38.0127	1.46420
$675$	0	0
$676$	−48.3505	−1.85964
$677$	−44.4539	−1.70850	−0.854252	−	0.519860i	$-0.825984\pi$
$677$	−44.4539	−1.70850	−0.854252	+	0.519860i	$0.825984\pi$
$678$	0	0
$679$	0	0
$680$	94.9728	3.64204
$681$	0	0
$682$	28.4674	1.09007
$683$	34.2277	1.30968	0.654842	−	0.755765i	$-0.272735\pi$
$683$	34.2277	1.30968	0.654842	+	0.755765i	$0.272735\pi$
$684$	0	0
$685$	38.8397	1.48399
$686$	0	0
$687$	0	0
$688$	−156.935	−5.98308
$689$	−19.6999	−0.750507
$690$	0	0
$691$	−27.1168	−1.03157	−0.515787	−	0.856717i	$-0.672500\pi$
$691$	−27.1168	−1.03157	−0.515787	+	0.856717i	$0.672500\pi$
$692$	5.43039	0.206432
$693$	0	0
$694$	87.9565	3.33878
$695$	−10.0381	−0.380767
$696$	0	0
$697$	17.4891	0.662448
$698$	−48.4972	−1.83565
$699$	0	0
$700$	0	0
$701$	−30.5607	−1.15426	−0.577131	−	0.816652i	$-0.695828\pi$
$701$	−30.5607	−1.15426	−0.577131	+	0.816652i	$0.695828\pi$
$702$	0	0
$703$	−10.7446	−0.405239
$704$	−57.7126	−2.17513
$705$	0	0
$706$	−40.4674	−1.52301
$707$	0	0
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	17.6783	0.663454
$711$	0	0
$712$	−90.1902	−3.38002
$713$	−4.79588	−0.179607
$714$	0	0
$715$	14.2337	0.532310
$716$	−23.7432	−0.887325
$717$	0	0
$718$	−12.5109	−0.466902
$719$	38.8354	1.44832	0.724158	−	0.689634i	$-0.242229\pi$
$719$	38.8354	1.44832	0.724158	+	0.689634i	$0.242229\pi$
$720$	0	0
$721$	0	0
$722$	2.71519	0.101049
$723$	0	0
$724$	−104.701	−3.89118
$725$	5.43039	0.201680
$726$	0	0
$727$	37.3505	1.38525	0.692627	−	0.721296i	$-0.256453\pi$
$727$	37.3505	1.38525	0.692627	+	0.721296i	$0.256453\pi$
$728$	0	0
$729$	0	0
$730$	44.7446	1.65607
$731$	−35.8029	−1.32422
$732$	0	0
$733$	−18.4674	−0.682108	−0.341054	−	0.940044i	$-0.610784\pi$
$733$	−18.4674	−0.682108	−0.341054	+	0.940044i	$0.610784\pi$
$734$	−21.7216	−0.801757
$735$	0	0
$736$	20.2337	0.745824
$737$	8.83915	0.325594
$738$	0	0
$739$	27.1168	0.997509	0.498755	−	0.866743i	$-0.333791\pi$
$739$	27.1168	0.997509	0.498755	+	0.866743i	$0.333791\pi$
$740$	185.902	6.83390
$741$	0	0
$742$	0	0
$743$	28.5391	1.04700	0.523498	−	0.852027i	$-0.324627\pi$
$743$	28.5391	1.04700	0.523498	+	0.852027i	$0.324627\pi$
$744$	0	0
$745$	66.0951	2.42154
$746$	−65.7992	−2.40908
$747$	0	0
$748$	−38.2337	−1.39796
$749$	0	0
$750$	0	0
$751$	24.4674	0.892827	0.446414	−	0.894827i	$-0.352701\pi$
$751$	24.4674	0.892827	0.446414	+	0.894827i	$0.352701\pi$
$752$	59.7343	2.17829
$753$	0	0
$754$	−5.48913	−0.199902
$755$	−12.8824	−0.468839
$756$	0	0
$757$	−5.11684	−0.185975	−0.0929874	−	0.995667i	$-0.529642\pi$
$757$	−5.11684	−0.185975	−0.0929874	+	0.995667i	$0.529642\pi$
$758$	78.0471	2.83480
$759$	0	0
$760$	−29.4891	−1.06968
$761$	−0.822662	−0.0298215	−0.0149107	−	0.999889i	$-0.504746\pi$
$761$	−0.822662	−0.0298215	−0.0149107	+	0.999889i	$0.504746\pi$
$762$	0	0
$763$	0	0
$764$	−115.683	−4.18528
$765$	0	0
$766$	46.9783	1.69739
$767$	21.7216	0.784320
$768$	0	0
$769$	−6.88316	−0.248213	−0.124106	−	0.992269i	$-0.539606\pi$
$769$	−6.88316	−0.248213	−0.124106	+	0.992269i	$0.539606\pi$
$770$	0	0
$771$	0	0
$772$	87.2119	3.13883
$773$	−5.43039	−0.195318	−0.0976588	−	0.995220i	$-0.531135\pi$
$773$	−5.43039	−0.195318	−0.0976588	+	0.995220i	$0.531135\pi$
$774$	0	0
$775$	−25.4891	−0.915596
$776$	−68.5734	−2.46164
$777$	0	0
$778$	32.7446	1.17395
$779$	−5.43039	−0.194564
$780$	0	0
$781$	−4.46738	−0.159855
$782$	8.83915	0.316087
$783$	0	0
$784$	0	0
$785$	−6.44121	−0.229896
$786$	0	0
$787$	49.9565	1.78076	0.890378	−	0.455221i	$-0.150440\pi$
$787$	49.9565	1.78076	0.890378	+	0.455221i	$0.150440\pi$
$788$	−127.555	−4.54396
$789$	0	0
$790$	−34.9783	−1.24447
$791$	0	0
$792$	0	0
$793$	−10.2337	−0.363409
$794$	46.4756	1.64936
$795$	0	0
$796$	−4.74456	−0.168167
$797$	7.45202	0.263964	0.131982	−	0.991252i	$-0.457866\pi$
$797$	7.45202	0.263964	0.131982	+	0.991252i	$0.457866\pi$
$798$	0	0
$799$	13.6277	0.482114
$800$	107.538	3.80204
$801$	0	0
$802$	9.25544	0.326821
$803$	−11.3071	−0.399020
$804$	0	0
$805$	0	0
$806$	25.7648	0.907527
$807$	0	0
$808$	−117.957	−4.14970
$809$	−3.59691	−0.126461	−0.0632303	−	0.997999i	$-0.520140\pi$
$809$	−3.59691	−0.126461	−0.0632303	+	0.997999i	$0.520140\pi$
$810$	0	0
$811$	36.7446	1.29028	0.645138	−	0.764066i	$-0.276800\pi$
$811$	36.7446	1.29028	0.645138	+	0.764066i	$0.276800\pi$
$812$	0	0
$813$	0	0
$814$	−64.4674	−2.25958
$815$	−69.2079	−2.42425
$816$	0	0
$817$	11.1168	0.388929
$818$	−20.3344	−0.710977
$819$	0	0
$820$	93.9565	3.28110
$821$	−29.3617	−1.02473	−0.512366	−	0.858767i	$-0.671231\pi$
$821$	−29.3617	−1.02473	−0.512366	+	0.858767i	$0.671231\pi$
$822$	0	0
$823$	8.60597	0.299985	0.149993	−	0.988687i	$-0.452075\pi$
$823$	8.60597	0.299985	0.149993	+	0.988687i	$0.452075\pi$
$824$	66.4337	2.31433
$825$	0	0
$826$	0	0
$827$	−28.5391	−0.992401	−0.496200	−	0.868208i	$-0.665272\pi$
$827$	−28.5391	−0.992401	−0.496200	+	0.868208i	$0.665272\pi$
$828$	0	0
$829$	1.25544	0.0436031	0.0218016	−	0.999762i	$-0.493060\pi$
$829$	1.25544	0.0436031	0.0218016	+	0.999762i	$0.493060\pi$
$830$	103.812	3.60336
$831$	0	0
$832$	−52.2337	−1.81088
$833$	0	0
$834$	0	0
$835$	20.7446	0.717895
$836$	11.8716	0.410588
$837$	0	0
$838$	−32.2337	−1.11349
$839$	−30.1844	−1.04208	−0.521041	−	0.853532i	$-0.674456\pi$
$839$	−30.1844	−1.04208	−0.521041	+	0.853532i	$0.674456\pi$
$840$	0	0
$841$	−27.9783	−0.964767
$842$	−24.3777	−0.840111
$843$	0	0
$844$	−21.4891	−0.739686
$845$	−28.9854	−0.997129
$846$	0	0
$847$	0	0
$848$	139.050	4.77501
$849$	0	0
$850$	46.9783	1.61134
$851$	10.8608	0.372303
$852$	0	0
$853$	38.4674	1.31710	0.658549	−	0.752538i	$-0.271171\pi$
$853$	38.4674	1.31710	0.658549	+	0.752538i	$0.271171\pi$
$854$	0	0
$855$	0	0
$856$	40.4674	1.38315
$857$	35.2385	1.20372	0.601862	−	0.798600i	$-0.294426\pi$
$857$	35.2385	1.20372	0.601862	+	0.798600i	$0.294426\pi$
$858$	0	0
$859$	−3.11684	−0.106345	−0.0531727	−	0.998585i	$-0.516933\pi$
$859$	−3.11684	−0.106345	−0.0531727	+	0.998585i	$0.516933\pi$
$860$	−192.343	−6.55886
$861$	0	0
$862$	−81.9565	−2.79145
$863$	−14.9040	−0.507340	−0.253670	−	0.967291i	$-0.581638\pi$
$863$	−14.9040	−0.507340	−0.253670	+	0.967291i	$0.581638\pi$
$864$	0	0
$865$	3.25544	0.110688
$866$	59.7343	2.02985
$867$	0	0
$868$	0	0
$869$	8.83915	0.299847
$870$	0	0
$871$	8.00000	0.271070
$872$	48.1209	1.62958
$873$	0	0
$874$	−2.74456	−0.0928362
$875$	0	0
$876$	0	0
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	49.5080	1.67081
$879$	0	0
$880$	−100.467	−3.38675
$881$	0.822662	0.0277162	0.0138581	−	0.999904i	$-0.495589\pi$
$881$	0.822662	0.0277162	0.0138581	+	0.999904i	$0.495589\pi$
$882$	0	0
$883$	3.11684	0.104890	0.0524451	−	0.998624i	$-0.483299\pi$
$883$	3.11684	0.104890	0.0524451	+	0.998624i	$0.483299\pi$
$884$	−34.6040	−1.16386
$885$	0	0
$886$	−75.9565	−2.55181
$887$	−21.7216	−0.729338	−0.364669	−	0.931137i	$-0.618818\pi$
$887$	−21.7216	−0.729338	−0.364669	+	0.931137i	$0.618818\pi$
$888$	0	0
$889$	0	0
$890$	−86.1336	−2.88721
$891$	0	0
$892$	21.4891	0.719509
$893$	−4.23142	−0.141599
$894$	0	0
$895$	−14.2337	−0.475780
$896$	0	0
$897$	0	0
$898$	60.7011	2.02562
$899$	−4.79588	−0.159952
$900$	0	0
$901$	31.7228	1.05684
$902$	−32.5823	−1.08487
$903$	0	0
$904$	−108.701	−3.61534
$905$	−62.7667	−2.08644
$906$	0	0
$907$	23.2554	0.772184	0.386092	−	0.922460i	$-0.373825\pi$
$907$	23.2554	0.772184	0.386092	+	0.922460i	$0.373825\pi$
$908$	−103.812	−3.44512
$909$	0	0
$910$	0	0
$911$	−37.0019	−1.22593	−0.612964	−	0.790111i	$-0.710023\pi$
$911$	−37.0019	−1.22593	−0.612964	+	0.790111i	$0.710023\pi$
$912$	0	0
$913$	−26.2337	−0.868208
$914$	−22.7324	−0.751920
$915$	0	0
$916$	−83.9565	−2.77400
$917$	0	0
$918$	0	0
$919$	18.9783	0.626035	0.313017	−	0.949747i	$-0.398660\pi$
$919$	18.9783	0.626035	0.313017	+	0.949747i	$0.398660\pi$
$920$	29.8081	0.982743
$921$	0	0
$922$	73.2119	2.41111
$923$	−4.04326	−0.133086
$924$	0	0
$925$	57.7228	1.89791
$926$	−6.44121	−0.211671
$927$	0	0
$928$	20.2337	0.664203
$929$	−4.79588	−0.157348	−0.0786739	−	0.996900i	$-0.525069\pi$
$929$	−4.79588	−0.157348	−0.0786739	+	0.996900i	$0.525069\pi$
$930$	0	0
$931$	0	0
$932$	−41.0452	−1.34448
$933$	0	0
$934$	98.9348	3.23724
$935$	−22.9205	−0.749581
$936$	0	0
$937$	5.11684	0.167160	0.0835800	−	0.996501i	$-0.473365\pi$
$937$	5.11684	0.167160	0.0835800	+	0.996501i	$0.473365\pi$
$938$	0	0
$939$	0	0
$940$	73.2119	2.38791
$941$	−24.7540	−0.806957	−0.403479	−	0.914989i	$-0.632199\pi$
$941$	−24.7540	−0.806957	−0.403479	+	0.914989i	$0.632199\pi$
$942$	0	0
$943$	5.48913	0.178751
$944$	−153.320	−4.99014
$945$	0	0
$946$	66.7011	2.16864
$947$	−9.84996	−0.320081	−0.160040	−	0.987110i	$-0.551162\pi$
$947$	−9.84996	−0.320081	−0.160040	+	0.987110i	$0.551162\pi$
$948$	0	0
$949$	−10.2337	−0.332200
$950$	−14.5868	−0.473258
$951$	0	0
$952$	0	0
$953$	9.47365	0.306882	0.153441	−	0.988158i	$-0.450965\pi$
$953$	9.47365	0.306882	0.153441	+	0.988158i	$0.450965\pi$
$954$	0	0
$955$	−69.3505	−2.24413
$956$	−68.1971	−2.20565
$957$	0	0
$958$	−91.2119	−2.94692
$959$	0	0
$960$	0	0
$961$	−8.48913	−0.273843
$962$	−58.3472	−1.88119
$963$	0	0
$964$	130.190	4.19314
$965$	52.2823	1.68303
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−56.0083	−1.80017
$969$	0	0
$970$	−65.4891	−2.10273
$971$	51.5296	1.65366	0.826832	−	0.562448i	$-0.190140\pi$
$971$	51.5296	1.65366	0.826832	+	0.562448i	$0.190140\pi$
$972$	0	0
$973$	0	0
$974$	21.7216	0.696004
$975$	0	0
$976$	72.2337	2.31214
$977$	−20.7107	−0.662595	−0.331298	−	0.943526i	$-0.607486\pi$
$977$	−20.7107	−0.662595	−0.331298	+	0.943526i	$0.607486\pi$
$978$	0	0
$979$	21.7663	0.695654
$980$	0	0
$981$	0	0
$982$	−2.74456	−0.0875825
$983$	−52.2823	−1.66755	−0.833773	−	0.552108i	$-0.813824\pi$
$983$	−52.2823	−1.66755	−0.833773	+	0.552108i	$0.813824\pi$
$984$	0	0
$985$	−76.4674	−2.43645
$986$	8.83915	0.281496
$987$	0	0
$988$	10.7446	0.341830
$989$	−11.2371	−0.357319
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−94.9728	−3.01539
$993$	0	0
$994$	0	0
$995$	−2.84429	−0.0901702
$996$	0	0
$997$	19.3505	0.612837	0.306419	−	0.951897i	$-0.400869\pi$
$997$	19.3505	0.612837	0.306419	+	0.951897i	$0.400869\pi$
$998$	−30.1844	−0.955470
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.bw.1.4		4
3.2	odd	2	inner	8379.2.a.bw.1.1		4
7.6	odd	2		171.2.a.e.1.4	yes	4
21.20	even	2		171.2.a.e.1.1	✓	4
28.27	even	2		2736.2.a.bf.1.1		4
35.34	odd	2		4275.2.a.bp.1.1		4
84.83	odd	2		2736.2.a.bf.1.4		4
105.104	even	2		4275.2.a.bp.1.4		4
133.132	even	2		3249.2.a.bf.1.1		4
399.398	odd	2		3249.2.a.bf.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.a.e.1.1	✓	4	21.20	even	2
171.2.a.e.1.4	yes	4	7.6	odd	2
2736.2.a.bf.1.1		4	28.27	even	2
2736.2.a.bf.1.4		4	84.83	odd	2
3249.2.a.bf.1.1		4	133.132	even	2
3249.2.a.bf.1.4		4	399.398	odd	2
4275.2.a.bp.1.1		4	35.34	odd	2
4275.2.a.bp.1.4		4	105.104	even	2
8379.2.a.bw.1.1		4	3.2	odd	2	inner
8379.2.a.bw.1.4		4	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:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q+2.71519 q^{2} +5.37228 q^{4} +3.22060 q^{5} +9.15640 q^{8} +8.74456 q^{10} -2.20979 q^{11} -2.00000 q^{13} +14.1168 q^{16} +3.22060 q^{17} -1.00000 q^{19} +17.3020 q^{20} -6.00000 q^{22} +1.01082 q^{23} +5.37228 q^{25} -5.43039 q^{26} +1.01082 q^{29} -4.74456 q^{31} +20.0172 q^{32} +8.74456 q^{34} +10.7446 q^{37} -2.71519 q^{38} +29.4891 q^{40} +5.43039 q^{41} -11.1168 q^{43} -11.8716 q^{44} +2.74456 q^{46} +4.23142 q^{47} +14.5868 q^{50} -10.7446 q^{52} +9.84996 q^{53} -7.11684 q^{55} +2.74456 q^{58} -10.8608 q^{59} +5.11684 q^{61} -12.8824 q^{62} +26.1168 q^{64} -6.44121 q^{65} -4.00000 q^{67} +17.3020 q^{68} +2.02163 q^{71} +5.11684 q^{73} +29.1736 q^{74} -5.37228 q^{76} -4.00000 q^{79} +45.4647 q^{80} +14.7446 q^{82} +11.8716 q^{83} +10.3723 q^{85} -30.1844 q^{86} -20.2337 q^{88} -9.84996 q^{89} +5.43039 q^{92} +11.4891 q^{94} -3.22060 q^{95} -7.48913 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{4} + 12 q^{10} - 8 q^{13} + 22 q^{16} - 4 q^{19} - 24 q^{22} + 10 q^{25} + 4 q^{31} + 12 q^{34} + 20 q^{37} + 72 q^{40} - 10 q^{43} - 12 q^{46} - 20 q^{52} + 6 q^{55} - 12 q^{58} - 14 q^{61}+ \cdots + 16 q^{97}+O(q^{100}) \) 4 * q + 10 * q^4 + 12 * q^10 - 8 * q^13 + 22 * q^16 - 4 * q^19 - 24 * q^22 + 10 * q^25 + 4 * q^31 + 12 * q^34 + 20 * q^37 + 72 * q^40 - 10 * q^43 - 12 * q^46 - 20 * q^52 + 6 * q^55 - 12 * q^58 - 14 * q^61 + 70 * q^64 - 16 * q^67 - 14 * q^73 - 10 * q^76 - 16 * q^79 + 36 * q^82 + 30 * q^85 - 12 * q^88 + 16 * q^97

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