LMFDB - Embedded newform 8379.2.a.o.1.1

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 = [1,1,0,-1,4,0,0,-3,0,4,2,0,-4,0,0,-1,0] 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:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{4} +4.00000 q^{5} -3.00000 q^{8} +4.00000 q^{10} +2.00000 q^{11} -4.00000 q^{13} -1.00000 q^{16} +1.00000 q^{19} -4.00000 q^{20} +2.00000 q^{22} +6.00000 q^{23} +11.0000 q^{25} -4.00000 q^{26} -10.0000 q^{29} +5.00000 q^{32} +6.00000 q^{37} +1.00000 q^{38} -12.0000 q^{40} -10.0000 q^{41} +8.00000 q^{43} -2.00000 q^{44} +6.00000 q^{46} +12.0000 q^{47} +11.0000 q^{50} +4.00000 q^{52} +6.00000 q^{53} +8.00000 q^{55} -10.0000 q^{58} -12.0000 q^{59} +2.00000 q^{61} +7.00000 q^{64} -16.0000 q^{65} -2.00000 q^{67} +12.0000 q^{71} +6.00000 q^{73} +6.00000 q^{74} -1.00000 q^{76} +2.00000 q^{79} -4.00000 q^{80} -10.0000 q^{82} +8.00000 q^{86} -6.00000 q^{88} -2.00000 q^{89} -6.00000 q^{92} +12.0000 q^{94} +4.00000 q^{95} +12.0000 q^{97} +O(q^{100})$

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−3.00000	−1.06066
$9$	0	0
$10$	4.00000	1.26491
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−4.00000	−0.894427
$21$	0	0
$22$	2.00000	0.426401
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	−4.00000	−0.784465
$27$	0	0
$28$	0	0
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−12.0000	−1.89737
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	6.00000	0.884652
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	0	0
$50$	11.0000	1.55563
$51$	0	0
$52$	4.00000	0.554700
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	0	0
$58$	−10.0000	−1.31306
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	7.00000	0.875000
$65$	−16.0000	−1.98456
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	−10.0000	−1.10432
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	−6.00000	−0.639602
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	0	0
$94$	12.0000	1.23771
$95$	4.00000	0.410391
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	0	0
$100$	−11.0000	−1.10000
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	12.0000	1.17670
$105$	0	0
$106$	6.00000	0.582772
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	8.00000	0.762770
$111$	0	0
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	24.0000	2.23801
$116$	10.0000	0.928477
$117$	0	0
$118$	−12.0000	−1.10469
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	2.00000	0.181071
$123$	0	0
$124$	0	0
$125$	24.0000	2.14663
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	−16.0000	−1.40329
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	0	0
$134$	−2.00000	−0.172774
$135$	0	0
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	12.0000	1.00702
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−40.0000	−3.32182
$146$	6.00000	0.496564
$147$	0	0
$148$	−6.00000	−0.493197
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−3.00000	−0.243332
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	2.00000	0.159111
$159$	0	0
$160$	20.0000	1.58114
$161$	0	0
$162$	0	0
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	−2.00000	−0.149906
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	0	0
$183$	0	0
$184$	−18.0000	−1.32698
$185$	24.0000	1.76452
$186$	0	0
$187$	0	0
$188$	−12.0000	−0.875190
$189$	0	0
$190$	4.00000	0.290191
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	0	0
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	−33.0000	−2.33345
$201$	0	0
$202$	−4.00000	−0.281439
$203$	0	0
$204$	0	0
$205$	−40.0000	−2.79372
$206$	16.0000	1.11477
$207$	0	0
$208$	4.00000	0.277350
$209$	2.00000	0.138343
$210$	0	0
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−4.00000	−0.273434
$215$	32.0000	2.18238
$216$	0	0
$217$	0	0
$218$	6.00000	0.406371
$219$	0	0
$220$	−8.00000	−0.539360
$221$	0	0
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	0	0
$226$	14.0000	0.931266
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	24.0000	1.58251
$231$	0	0
$232$	30.0000	1.96960
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	48.0000	3.13117
$236$	12.0000	0.781133
$237$	0	0
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	0	0
$250$	24.0000	1.51789
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	−18.0000	−1.12942
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	0	0
$260$	16.0000	0.992278
$261$	0	0
$262$	4.00000	0.247121
$263$	26.0000	1.60323	0.801614	−	0.597841i	$-0.203975\pi$
$263$	26.0000	1.60323	0.801614	+	0.597841i	$0.203975\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	0	0
$267$	0	0
$268$	2.00000	0.122169
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	0	0
$274$	18.0000	1.08742
$275$	22.0000	1.32665
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−12.0000	−0.719712
$279$	0	0
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−8.00000	−0.473050
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−40.0000	−2.34888
$291$	0	0
$292$	−6.00000	−0.351123
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	−48.0000	−2.79467
$296$	−18.0000	−1.04623
$297$	0	0
$298$	6.00000	0.347571
$299$	−24.0000	−1.38796
$300$	0	0
$301$	0	0
$302$	2.00000	0.115087
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	8.00000	0.458079
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−2.00000	−0.112509
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	−20.0000	−1.11979
$320$	28.0000	1.56525
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−44.0000	−2.44068
$326$	16.0000	0.886158
$327$	0	0
$328$	30.0000	1.65647
$329$	0	0
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	0	0
$333$	0	0
$334$	16.0000	0.875481
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	3.00000	0.163178
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−24.0000	−1.29399
$345$	0	0
$346$	2.00000	0.107521
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	0	0
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	0	0
$352$	10.0000	0.533002
$353$	16.0000	0.851594	0.425797	−	0.904819i	$-0.359994\pi$
$353$	16.0000	0.851594	0.425797	+	0.904819i	$0.359994\pi$
$354$	0	0
$355$	48.0000	2.54758
$356$	2.00000	0.106000
$357$	0	0
$358$	4.00000	0.211407
$359$	−14.0000	−0.738892	−0.369446	−	0.929252i	$-0.620452\pi$
$359$	−14.0000	−0.738892	−0.369446	+	0.929252i	$0.620452\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	8.00000	0.420471
$363$	0	0
$364$	0	0
$365$	24.0000	1.25622
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	24.0000	1.24770
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	0	0
$376$	−36.0000	−1.85656
$377$	40.0000	2.06010
$378$	0	0
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	−10.0000	−0.511645
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	0	0
$386$	2.00000	0.101797
$387$	0	0
$388$	−12.0000	−0.609208
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	14.0000	0.705310
$395$	8.00000	0.402524
$396$	0	0
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	−24.0000	−1.20301
$399$	0	0
$400$	−11.0000	−0.550000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	0	0
$404$	4.00000	0.199007
$405$	0	0
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	−40.0000	−1.97546
$411$	0	0
$412$	−16.0000	−0.788263
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−20.0000	−0.980581
$417$	0	0
$418$	2.00000	0.0978232
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	22.0000	1.07094
$423$	0	0
$424$	−18.0000	−0.874157
$425$	0	0
$426$	0	0
$427$	0	0
$428$	4.00000	0.193347
$429$	0	0
$430$	32.0000	1.54318
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	0	0
$435$	0	0
$436$	−6.00000	−0.287348
$437$	6.00000	0.287019
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	−24.0000	−1.14416
$441$	0	0
$442$	0	0
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	−4.00000	−0.189405
$447$	0	0
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	−20.0000	−0.941763
$452$	−14.0000	−0.658505
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−14.0000	−0.654177
$459$	0	0
$460$	−24.0000	−1.11901
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	10.0000	0.464238
$465$	0	0
$466$	18.0000	0.833834
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	0	0
$469$	0	0
$470$	48.0000	2.21407
$471$	0	0
$472$	36.0000	1.65703
$473$	16.0000	0.735681
$474$	0	0
$475$	11.0000	0.504715
$476$	0	0
$477$	0	0
$478$	6.00000	0.274434
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	−8.00000	−0.364390
$483$	0	0
$484$	7.00000	0.318182
$485$	48.0000	2.17957
$486$	0	0
$487$	−26.0000	−1.17817	−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000	−1.17817	−0.589086	+	0.808070i	$0.700512\pi$
$488$	−6.00000	−0.271607
$489$	0	0
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	0	0
$494$	−4.00000	−0.179969
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	−24.0000	−1.07331
$501$	0	0
$502$	4.00000	0.178529
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	−16.0000	−0.711991
$506$	12.0000	0.533465
$507$	0	0
$508$	18.0000	0.798621
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	0	0
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−22.0000	−0.970378
$515$	64.0000	2.82018
$516$	0	0
$517$	24.0000	1.05552
$518$	0	0
$519$	0	0
$520$	48.0000	2.10494
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	26.0000	1.13365
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	24.0000	1.04249
$531$	0	0
$532$	0	0
$533$	40.0000	1.73259
$534$	0	0
$535$	−16.0000	−0.691740
$536$	6.00000	0.259161
$537$	0	0
$538$	−6.00000	−0.258678
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	0	0
$545$	24.0000	1.02805
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	−18.0000	−0.768922
$549$	0	0
$550$	22.0000	0.938083
$551$	−10.0000	−0.426014
$552$	0	0
$553$	0	0
$554$	22.0000	0.934690
$555$	0	0
$556$	12.0000	0.508913
$557$	−34.0000	−1.44063	−0.720313	−	0.693649i	$-0.756002\pi$
$557$	−34.0000	−1.44063	−0.720313	+	0.693649i	$0.756002\pi$
$558$	0	0
$559$	−32.0000	−1.35346
$560$	0	0
$561$	0	0
$562$	−26.0000	−1.09674
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	56.0000	2.35594
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−36.0000	−1.51053
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	8.00000	0.334497
$573$	0	0
$574$	0	0
$575$	66.0000	2.75239
$576$	0	0
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	40.0000	1.66091
$581$	0	0
$582$	0	0
$583$	12.0000	0.496989
$584$	−18.0000	−0.744845
$585$	0	0
$586$	2.00000	0.0826192
$587$	−8.00000	−0.330195	−0.165098	−	0.986277i	$-0.552794\pi$
$587$	−8.00000	−0.330195	−0.165098	+	0.986277i	$0.552794\pi$
$588$	0	0
$589$	0	0
$590$	−48.0000	−1.97613
$591$	0	0
$592$	−6.00000	−0.246598
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	−24.0000	−0.981433
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	12.0000	0.489490	0.244745	−	0.969587i	$-0.421296\pi$
$601$	12.0000	0.489490	0.244745	+	0.969587i	$0.421296\pi$
$602$	0	0
$603$	0	0
$604$	−2.00000	−0.0813788
$605$	−28.0000	−1.13836
$606$	0	0
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	8.00000	0.323911
$611$	−48.0000	−1.94187
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	−24.0000	−0.962312
$623$	0	0
$624$	0	0
$625$	41.0000	1.64000
$626$	22.0000	0.879297
$627$	0	0
$628$	10.0000	0.399043
$629$	0	0
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	−6.00000	−0.238667
$633$	0	0
$634$	−6.00000	−0.238290
$635$	−72.0000	−2.85723
$636$	0	0
$637$	0	0
$638$	−20.0000	−0.791808
$639$	0	0
$640$	−12.0000	−0.474342
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	−44.0000	−1.72582
$651$	0	0
$652$	−16.0000	−0.626608
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	10.0000	0.390434
$657$	0	0
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	2.00000	0.0777322
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−60.0000	−2.32321
$668$	−16.0000	−0.619059
$669$	0	0
$670$	−8.00000	−0.309067
$671$	4.00000	0.154418
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	−3.00000	−0.115385
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	72.0000	2.75098
$686$	0	0
$687$	0	0
$688$	−8.00000	−0.304997
$689$	−24.0000	−0.914327
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	−2.00000	−0.0759190
$695$	−48.0000	−1.82074
$696$	0	0
$697$	0	0
$698$	−34.0000	−1.28692
$699$	0	0
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	6.00000	0.226294
$704$	14.0000	0.527645
$705$	0	0
$706$	16.0000	0.602168
$707$	0	0
$708$	0	0
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	48.0000	1.80141
$711$	0	0
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	−32.0000	−1.19673
$716$	−4.00000	−0.149487
$717$	0	0
$718$	−14.0000	−0.522475
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	0	0
$722$	1.00000	0.0372161
$723$	0	0
$724$	−8.00000	−0.297318
$725$	−110.000	−4.08530
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	0	0
$730$	24.0000	0.888280
$731$	0	0
$732$	0	0
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	0	0
$735$	0	0
$736$	30.0000	1.10581
$737$	−4.00000	−0.147342
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	−24.0000	−0.882258
$741$	0	0
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	24.0000	0.879292
$746$	26.0000	0.951928
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−50.0000	−1.82453	−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000	−1.82453	−0.912263	+	0.409605i	$0.865667\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	40.0000	1.45671
$755$	8.00000	0.291150
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−6.00000	−0.217930
$759$	0	0
$760$	−12.0000	−0.435286
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	0	0
$763$	0	0
$764$	10.0000	0.361787
$765$	0	0
$766$	−24.0000	−0.867155
$767$	48.0000	1.73318
$768$	0	0
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	0	0
$775$	0	0
$776$	−36.0000	−1.29232
$777$	0	0
$778$	−30.0000	−1.07555
$779$	−10.0000	−0.358287
$780$	0	0
$781$	24.0000	0.858788
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−40.0000	−1.42766
$786$	0	0
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	−14.0000	−0.498729
$789$	0	0
$790$	8.00000	0.284627
$791$	0	0
$792$	0	0
$793$	−8.00000	−0.284088
$794$	30.0000	1.06466
$795$	0	0
$796$	24.0000	0.850657
$797$	−38.0000	−1.34603	−0.673015	−	0.739629i	$-0.735001\pi$
$797$	−38.0000	−1.34603	−0.673015	+	0.739629i	$0.735001\pi$
$798$	0	0
$799$	0	0
$800$	55.0000	1.94454
$801$	0	0
$802$	18.0000	0.635602
$803$	12.0000	0.423471
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	12.0000	0.422159
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	−8.00000	−0.280918	−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000	−0.280918	−0.140459	+	0.990086i	$0.544858\pi$
$812$	0	0
$813$	0	0
$814$	12.0000	0.420600
$815$	64.0000	2.24182
$816$	0	0
$817$	8.00000	0.279885
$818$	4.00000	0.139857
$819$	0	0
$820$	40.0000	1.39686
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	−48.0000	−1.67216
$825$	0	0
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	0	0
$832$	−28.0000	−0.970725
$833$	0	0
$834$	0	0
$835$	64.0000	2.21481
$836$	−2.00000	−0.0691714
$837$	0	0
$838$	36.0000	1.24360
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	−22.0000	−0.758170
$843$	0	0
$844$	−22.0000	−0.757271
$845$	12.0000	0.412813
$846$	0	0
$847$	0	0
$848$	−6.00000	−0.206041
$849$	0	0
$850$	0	0
$851$	36.0000	1.23406
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	−32.0000	−1.09119
$861$	0	0
$862$	12.0000	0.408722
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	8.00000	0.272008
$866$	−4.00000	−0.135926
$867$	0	0
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	8.00000	0.271070
$872$	−18.0000	−0.609557
$873$	0	0
$874$	6.00000	0.202953
$875$	0	0
$876$	0	0
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	0	0
$879$	0	0
$880$	−8.00000	−0.269680
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	0	0
$883$	56.0000	1.88455	0.942275	−	0.334840i	$-0.108682\pi$
$883$	56.0000	1.88455	0.942275	+	0.334840i	$0.108682\pi$
$884$	0	0
$885$	0	0
$886$	−6.00000	−0.201574
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	0	0
$890$	−8.00000	−0.268161
$891$	0	0
$892$	4.00000	0.133930
$893$	12.0000	0.401565
$894$	0	0
$895$	16.0000	0.534821
$896$	0	0
$897$	0	0
$898$	−18.0000	−0.600668
$899$	0	0
$900$	0	0
$901$	0	0
$902$	−20.0000	−0.665927
$903$	0	0
$904$	−42.0000	−1.39690
$905$	32.0000	1.06372
$906$	0	0
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	0	0
$914$	22.0000	0.727695
$915$	0	0
$916$	14.0000	0.462573
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−72.0000	−2.37377
$921$	0	0
$922$	20.0000	0.658665
$923$	−48.0000	−1.57994
$924$	0	0
$925$	66.0000	2.17007
$926$	−32.0000	−1.05159
$927$	0	0
$928$	−50.0000	−1.64133
$929$	32.0000	1.04989	0.524943	−	0.851137i	$-0.324087\pi$
$929$	32.0000	1.04989	0.524943	+	0.851137i	$0.324087\pi$
$930$	0	0
$931$	0	0
$932$	−18.0000	−0.589610
$933$	0	0
$934$	24.0000	0.785304
$935$	0	0
$936$	0	0
$937$	−54.0000	−1.76410	−0.882052	−	0.471153i	$-0.843838\pi$
$937$	−54.0000	−1.76410	−0.882052	+	0.471153i	$0.843838\pi$
$938$	0	0
$939$	0	0
$940$	−48.0000	−1.56559
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	0	0
$943$	−60.0000	−1.95387
$944$	12.0000	0.390567
$945$	0	0
$946$	16.0000	0.520205
$947$	46.0000	1.49480	0.747400	−	0.664375i	$-0.231302\pi$
$947$	46.0000	1.49480	0.747400	+	0.664375i	$0.231302\pi$
$948$	0	0
$949$	−24.0000	−0.779073
$950$	11.0000	0.356887
$951$	0	0
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	0	0
$955$	−40.0000	−1.29437
$956$	−6.00000	−0.194054
$957$	0	0
$958$	8.00000	0.258468
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−24.0000	−0.773791
$963$	0	0
$964$	8.00000	0.257663
$965$	8.00000	0.257529
$966$	0	0
$967$	−44.0000	−1.41494	−0.707472	−	0.706741i	$-0.750165\pi$
$967$	−44.0000	−1.41494	−0.707472	+	0.706741i	$0.750165\pi$
$968$	21.0000	0.674966
$969$	0	0
$970$	48.0000	1.54119
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	0	0
$974$	−26.0000	−0.833094
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	−4.00000	−0.127841
$980$	0	0
$981$	0	0
$982$	−2.00000	−0.0638226
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	56.0000	1.78431
$986$	0	0
$987$	0	0
$988$	4.00000	0.127257
$989$	48.0000	1.52631
$990$	0	0
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−96.0000	−3.04340
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	16.0000	0.506471
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.o.1.1		1
3.2	odd	2		2793.2.a.c.1.1		1
7.6	odd	2		1197.2.a.c.1.1		1
21.20	even	2		399.2.a.b.1.1	✓	1
84.83	odd	2		6384.2.a.q.1.1		1
105.104	even	2		9975.2.a.j.1.1		1
399.398	odd	2		7581.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.b.1.1	✓	1	21.20	even	2
1197.2.a.c.1.1		1	7.6	odd	2
2793.2.a.c.1.1		1	3.2	odd	2
6384.2.a.q.1.1		1	84.83	odd	2
7581.2.a.e.1.1		1	399.398	odd	2
8379.2.a.o.1.1		1	1.1	even	1	trivial
9975.2.a.j.1.1		1	105.104	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:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

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