LMFDB - Embedded newform 8379.2.a.b.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,-2,0,2,-1,0,0,0,0,2,-4,0,4,0,0,-4,-3] 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)

$f(q)$

$=$

$q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{10} -4.00000 q^{11} +4.00000 q^{13} -4.00000 q^{16} -3.00000 q^{17} -1.00000 q^{19} -2.00000 q^{20} +8.00000 q^{22} +3.00000 q^{23} -4.00000 q^{25} -8.00000 q^{26} -10.0000 q^{29} +8.00000 q^{32} +6.00000 q^{34} -6.00000 q^{37} +2.00000 q^{38} -2.00000 q^{41} -7.00000 q^{43} -8.00000 q^{44} -6.00000 q^{46} +8.00000 q^{50} +8.00000 q^{52} +12.0000 q^{53} +4.00000 q^{55} +20.0000 q^{58} -12.0000 q^{59} +10.0000 q^{61} -8.00000 q^{64} -4.00000 q^{65} +10.0000 q^{67} -6.00000 q^{68} -6.00000 q^{71} +6.00000 q^{73} +12.0000 q^{74} -2.00000 q^{76} -10.0000 q^{79} +4.00000 q^{80} +4.00000 q^{82} -3.00000 q^{83} +3.00000 q^{85} +14.0000 q^{86} +14.0000 q^{89} +6.00000 q^{92} +1.00000 q^{95} -12.0000 q^{97} +O(q^{100})$

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	2.00000	0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.00000	−0.447214
$21$	0	0
$22$	8.00000	1.70561
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−8.00000	−1.56893
$27$	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	$0.5\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	−8.00000	−1.20605
$45$	0	0
$46$	−6.00000	−0.884652
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	8.00000	1.13137
$51$	0	0
$52$	8.00000	1.10940
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	0	0
$58$	20.0000	2.62613
$59$	−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
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	12.0000	1.39497
$75$	0	0
$76$	−2.00000	−0.229416
$77$	0	0
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	4.00000	0.441726
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	14.0000	1.50966
$87$	0	0
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	0	0
$92$	6.00000	0.625543
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	0	0
$100$	−8.00000	−0.800000
$101$	−11.0000	−1.09454	−0.547270	−	0.836956i	$-0.684333\pi$
$101$	−11.0000	−1.09454	−0.547270	+	0.836956i	$0.684333\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	−24.0000	−2.33109
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	−8.00000	−0.762770
$111$	0	0
$112$	0	0
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	−20.0000	−1.85695
$117$	0	0
$118$	24.0000	2.20938
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	−20.0000	−1.81071
$123$	0	0
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	0	0
$130$	8.00000	0.701646
$131$	5.00000	0.436852	0.218426	−	0.975854i	$-0.429908\pi$
$131$	5.00000	0.436852	0.218426	+	0.975854i	$0.429908\pi$
$132$	0	0
$133$	0	0
$134$	−20.0000	−1.72774
$135$	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
$143$	−16.0000	−1.33799
$144$	0	0
$145$	10.0000	0.830455
$146$	−12.0000	−0.993127
$147$	0	0
$148$	−12.0000	−0.986394
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.00000	0.0798087	0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000	0.0798087	0.0399043	+	0.999204i	$0.487295\pi$
$158$	20.0000	1.59111
$159$	0	0
$160$	−8.00000	−0.632456
$161$	0	0
$162$	0	0
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	6.00000	0.465690
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−6.00000	−0.460179
$171$	0	0
$172$	−14.0000	−1.06749
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	0	0
$176$	16.0000	1.20605
$177$	0	0
$178$	−28.0000	−2.09869
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	12.0000	0.877527
$188$	0	0
$189$	0	0
$190$	−2.00000	−0.145095
$191$	17.0000	1.23008	0.615038	−	0.788497i	$-0.289140\pi$
$191$	17.0000	1.23008	0.615038	+	0.788497i	$0.289140\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	24.0000	1.72310
$195$	0	0
$196$	0	0
$197$	11.0000	0.783718	0.391859	−	0.920025i	$-0.371832\pi$
$197$	11.0000	0.783718	0.391859	+	0.920025i	$0.371832\pi$
$198$	0	0
$199$	−27.0000	−1.91398	−0.956990	−	0.290122i	$-0.906304\pi$
$199$	−27.0000	−1.91398	−0.956990	+	0.290122i	$0.906304\pi$
$200$	0	0
$201$	0	0
$202$	22.0000	1.54791
$203$	0	0
$204$	0	0
$205$	2.00000	0.139686
$206$	−16.0000	−1.11477
$207$	0	0
$208$	−16.0000	−1.10940
$209$	4.00000	0.276686
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	24.0000	1.64833
$213$	0	0
$214$	20.0000	1.36717
$215$	7.00000	0.477396
$216$	0	0
$217$	0	0
$218$	12.0000	0.812743
$219$	0	0
$220$	8.00000	0.539360
$221$	−12.0000	−0.807207
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	0	0
$225$	0	0
$226$	−16.0000	−1.06430
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	0	0
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	0	0
$235$	0	0
$236$	−24.0000	−1.56227
$237$	0	0
$238$	0	0
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	−10.0000	−0.642824
$243$	0	0
$244$	20.0000	1.28037
$245$	0	0
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	0	0
$250$	−18.0000	−1.13842
$251$	23.0000	1.45175	0.725874	−	0.687828i	$-0.241436\pi$
$251$	23.0000	1.45175	0.725874	+	0.687828i	$0.241436\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	−24.0000	−1.50589
$255$	0	0
$256$	16.0000	1.00000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	0	0
$260$	−8.00000	−0.496139
$261$	0	0
$262$	−10.0000	−0.617802
$263$	17.0000	1.04826	0.524132	−	0.851637i	$-0.324390\pi$
$263$	17.0000	1.04826	0.524132	+	0.851637i	$0.324390\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	0	0
$268$	20.0000	1.22169
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−23.0000	−1.39715	−0.698575	−	0.715537i	$-0.746182\pi$
$271$	−23.0000	−1.39715	−0.698575	+	0.715537i	$0.746182\pi$
$272$	12.0000	0.727607
$273$	0	0
$274$	−36.0000	−2.17484
$275$	16.0000	0.964836
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	24.0000	1.43942
$279$	0	0
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	0	0
$283$	1.00000	0.0594438	0.0297219	−	0.999558i	$-0.490538\pi$
$283$	1.00000	0.0594438	0.0297219	+	0.999558i	$0.490538\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	32.0000	1.89220
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−20.0000	−1.17444
$291$	0	0
$292$	12.0000	0.702247
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	0	0
$298$	−18.0000	−1.04271
$299$	12.0000	0.693978
$300$	0	0
$301$	0	0
$302$	20.0000	1.15087
$303$	0	0
$304$	4.00000	0.229416
$305$	−10.0000	−0.572598
$306$	0	0
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	0	0
$313$	11.0000	0.621757	0.310878	−	0.950450i	$-0.399377\pi$
$313$	11.0000	0.621757	0.310878	+	0.950450i	$0.399377\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−20.0000	−1.12509
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	40.0000	2.23957
$320$	8.00000	0.447214
$321$	0	0
$322$	0	0
$323$	3.00000	0.166924
$324$	0	0
$325$	−16.0000	−0.887520
$326$	−38.0000	−2.10463
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−22.0000	−1.20923	−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000	−1.20923	−0.604615	+	0.796518i	$0.706673\pi$
$332$	−6.00000	−0.329293
$333$	0	0
$334$	−4.00000	−0.218870
$335$	−10.0000	−0.546358
$336$	0	0
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	−6.00000	−0.326357
$339$	0	0
$340$	6.00000	0.325396
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	28.0000	1.50529
$347$	−29.0000	−1.55680	−0.778401	−	0.627768i	$-0.783969\pi$
$347$	−29.0000	−1.55680	−0.778401	+	0.627768i	$0.783969\pi$
$348$	0	0
$349$	13.0000	0.695874	0.347937	−	0.937518i	$-0.386882\pi$
$349$	13.0000	0.695874	0.347937	+	0.937518i	$0.386882\pi$
$350$	0	0
$351$	0	0
$352$	−32.0000	−1.70561
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	28.0000	1.48400
$357$	0	0
$358$	−32.0000	−1.69125
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	28.0000	1.47165
$363$	0	0
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−12.0000	−0.625543
$369$	0	0
$370$	−12.0000	−0.623850
$371$	0	0
$372$	0	0
$373$	32.0000	1.65690	0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000	1.65690	0.828449	+	0.560065i	$0.189224\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	0	0
$377$	−40.0000	−2.06010
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	−34.0000	−1.73959
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	0	0
$386$	32.0000	1.62876
$387$	0	0
$388$	−24.0000	−1.21842
$389$	3.00000	0.152106	0.0760530	−	0.997104i	$-0.475768\pi$
$389$	3.00000	0.152106	0.0760530	+	0.997104i	$0.475768\pi$
$390$	0	0
$391$	−9.00000	−0.455150
$392$	0	0
$393$	0	0
$394$	−22.0000	−1.10834
$395$	10.0000	0.503155
$396$	0	0
$397$	−15.0000	−0.752828	−0.376414	−	0.926451i	$-0.622843\pi$
$397$	−15.0000	−0.752828	−0.376414	+	0.926451i	$0.622843\pi$
$398$	54.0000	2.70678
$399$	0	0
$400$	16.0000	0.800000
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	0	0
$404$	−22.0000	−1.09454
$405$	0	0
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	16.0000	0.788263
$413$	0	0
$414$	0	0
$415$	3.00000	0.147264
$416$	32.0000	1.56893
$417$	0	0
$418$	−8.00000	−0.391293
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	0	0
$428$	−20.0000	−0.966736
$429$	0	0
$430$	−14.0000	−0.675140
$431$	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
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	−12.0000	−0.574696
$437$	−3.00000	−0.143509
$438$	0	0
$439$	30.0000	1.43182	0.715911	−	0.698192i	$-0.246012\pi$
$439$	30.0000	1.43182	0.715911	+	0.698192i	$0.246012\pi$
$440$	0	0
$441$	0	0
$442$	24.0000	1.14156
$443$	21.0000	0.997740	0.498870	−	0.866677i	$-0.333748\pi$
$443$	21.0000	0.997740	0.498870	+	0.866677i	$0.333748\pi$
$444$	0	0
$445$	−14.0000	−0.663664
$446$	−20.0000	−0.947027
$447$	0	0
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	16.0000	0.752577
$453$	0	0
$454$	0	0
$455$	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
$460$	−6.00000	−0.279751
$461$	−26.0000	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	40.0000	1.85695
$465$	0	0
$466$	18.0000	0.833834
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	28.0000	1.28744
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	0	0
$478$	18.0000	0.823301
$479$	−11.0000	−0.502603	−0.251301	−	0.967909i	$-0.580859\pi$
$479$	−11.0000	−0.502603	−0.251301	+	0.967909i	$0.580859\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	56.0000	2.55073
$483$	0	0
$484$	10.0000	0.454545
$485$	12.0000	0.544892
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.00000	0.0451294	0.0225647	−	0.999745i	$-0.492817\pi$
$491$	1.00000	0.0451294	0.0225647	+	0.999745i	$0.492817\pi$
$492$	0	0
$493$	30.0000	1.35113
$494$	8.00000	0.359937
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	7.00000	0.313363	0.156682	−	0.987649i	$-0.449920\pi$
$499$	7.00000	0.313363	0.156682	+	0.987649i	$0.449920\pi$
$500$	18.0000	0.804984
$501$	0	0
$502$	−46.0000	−2.05308
$503$	3.00000	0.133763	0.0668817	−	0.997761i	$-0.478695\pi$
$503$	3.00000	0.133763	0.0668817	+	0.997761i	$0.478695\pi$
$504$	0	0
$505$	11.0000	0.489494
$506$	24.0000	1.06693
$507$	0	0
$508$	24.0000	1.06483
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	0	0
$511$	0	0
$512$	−32.0000	−1.41421
$513$	0	0
$514$	4.00000	0.176432
$515$	−8.00000	−0.352522
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	40.0000	1.74908	0.874539	−	0.484955i	$-0.161164\pi$
$523$	40.0000	1.74908	0.874539	+	0.484955i	$0.161164\pi$
$524$	10.0000	0.436852
$525$	0	0
$526$	−34.0000	−1.48247
$527$	0	0
$528$	0	0
$529$	−14.0000	−0.608696
$530$	24.0000	1.04249
$531$	0	0
$532$	0	0
$533$	−8.00000	−0.346518
$534$	0	0
$535$	10.0000	0.432338
$536$	0	0
$537$	0	0
$538$	−36.0000	−1.55207
$539$	0	0
$540$	0	0
$541$	35.0000	1.50477	0.752384	−	0.658725i	$-0.228904\pi$
$541$	35.0000	1.50477	0.752384	+	0.658725i	$0.228904\pi$
$542$	46.0000	1.97587
$543$	0	0
$544$	−24.0000	−1.02899
$545$	6.00000	0.257012
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	36.0000	1.53784
$549$	0	0
$550$	−32.0000	−1.36448
$551$	10.0000	0.426014
$552$	0	0
$553$	0	0
$554$	−44.0000	−1.86938
$555$	0	0
$556$	−24.0000	−1.01783
$557$	−7.00000	−0.296600	−0.148300	−	0.988942i	$-0.547380\pi$
$557$	−7.00000	−0.296600	−0.148300	+	0.988942i	$0.547380\pi$
$558$	0	0
$559$	−28.0000	−1.18427
$560$	0	0
$561$	0	0
$562$	40.0000	1.68730
$563$	−10.0000	−0.421450	−0.210725	−	0.977545i	$-0.567582\pi$
$563$	−10.0000	−0.421450	−0.210725	+	0.977545i	$0.567582\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	−2.00000	−0.0840663
$567$	0	0
$568$	0	0
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	−32.0000	−1.33799
$573$	0	0
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	16.0000	0.665512
$579$	0	0
$580$	20.0000	0.830455
$581$	0	0
$582$	0	0
$583$	−48.0000	−1.98796
$584$	0	0
$585$	0	0
$586$	−32.0000	−1.32191
$587$	−7.00000	−0.288921	−0.144460	−	0.989511i	$-0.546145\pi$
$587$	−7.00000	−0.288921	−0.144460	+	0.989511i	$0.546145\pi$
$588$	0	0
$589$	0	0
$590$	−24.0000	−0.988064
$591$	0	0
$592$	24.0000	0.986394
$593$	39.0000	1.60154	0.800769	−	0.598973i	$-0.204424\pi$
$593$	39.0000	1.60154	0.800769	+	0.598973i	$0.204424\pi$
$594$	0	0
$595$	0	0
$596$	18.0000	0.737309
$597$	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
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	0	0
$604$	−20.0000	−0.813788
$605$	−5.00000	−0.203279
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−8.00000	−0.324443
$609$	0	0
$610$	20.0000	0.809776
$611$	0	0
$612$	0	0
$613$	25.0000	1.00974	0.504870	−	0.863195i	$-0.331540\pi$
$613$	25.0000	1.00974	0.504870	+	0.863195i	$0.331540\pi$
$614$	−44.0000	−1.77570
$615$	0	0
$616$	0	0
$617$	−27.0000	−1.08698	−0.543490	−	0.839416i	$-0.682897\pi$
$617$	−27.0000	−1.08698	−0.543490	+	0.839416i	$0.682897\pi$
$618$	0	0
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	0	0
$621$	0	0
$622$	6.00000	0.240578
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	2.00000	0.0798087
$629$	18.0000	0.717707
$630$	0	0
$631$	5.00000	0.199047	0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000	0.199047	0.0995234	+	0.995035i	$0.468268\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12.0000	−0.476205
$636$	0	0
$637$	0	0
$638$	−80.0000	−3.16723
$639$	0	0
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	11.0000	0.433798	0.216899	−	0.976194i	$-0.430406\pi$
$643$	11.0000	0.433798	0.216899	+	0.976194i	$0.430406\pi$
$644$	0	0
$645$	0	0
$646$	−6.00000	−0.236067
$647$	−23.0000	−0.904223	−0.452112	−	0.891961i	$-0.649329\pi$
$647$	−23.0000	−0.904223	−0.452112	+	0.891961i	$0.649329\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	32.0000	1.25514
$651$	0	0
$652$	38.0000	1.48819
$653$	21.0000	0.821794	0.410897	−	0.911682i	$-0.365216\pi$
$653$	21.0000	0.821794	0.410897	+	0.911682i	$0.365216\pi$
$654$	0	0
$655$	−5.00000	−0.195366
$656$	8.00000	0.312348
$657$	0	0
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	40.0000	1.55582	0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	44.0000	1.71011
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−30.0000	−1.16160
$668$	4.00000	0.154765
$669$	0	0
$670$	20.0000	0.772667
$671$	−40.0000	−1.54418
$672$	0	0
$673$	16.0000	0.616755	0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000	0.616755	0.308377	+	0.951264i	$0.400214\pi$
$674$	16.0000	0.616297
$675$	0	0
$676$	6.00000	0.230769
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	0	0
$688$	28.0000	1.06749
$689$	48.0000	1.82865
$690$	0	0
$691$	41.0000	1.55971	0.779857	−	0.625958i	$-0.215292\pi$
$691$	41.0000	1.55971	0.779857	+	0.625958i	$0.215292\pi$
$692$	−28.0000	−1.06440
$693$	0	0
$694$	58.0000	2.20165
$695$	12.0000	0.455186
$696$	0	0
$697$	6.00000	0.227266
$698$	−26.0000	−0.984115
$699$	0	0
$700$	0	0
$701$	21.0000	0.793159	0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000	0.793159	0.396580	+	0.918000i	$0.370197\pi$
$702$	0	0
$703$	6.00000	0.226294
$704$	32.0000	1.20605
$705$	0	0
$706$	−52.0000	−1.95705
$707$	0	0
$708$	0	0
$709$	21.0000	0.788672	0.394336	−	0.918966i	$-0.370975\pi$
$709$	21.0000	0.788672	0.394336	+	0.918966i	$0.370975\pi$
$710$	−12.0000	−0.450352
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	16.0000	0.598366
$716$	32.0000	1.19590
$717$	0	0
$718$	40.0000	1.49279
$719$	−21.0000	−0.783168	−0.391584	−	0.920142i	$-0.628073\pi$
$719$	−21.0000	−0.783168	−0.391584	+	0.920142i	$0.628073\pi$
$720$	0	0
$721$	0	0
$722$	−2.00000	−0.0744323
$723$	0	0
$724$	−28.0000	−1.04061
$725$	40.0000	1.48556
$726$	0	0
$727$	47.0000	1.74313	0.871567	−	0.490277i	$-0.163104\pi$
$727$	47.0000	1.74313	0.871567	+	0.490277i	$0.163104\pi$
$728$	0	0
$729$	0	0
$730$	12.0000	0.444140
$731$	21.0000	0.776713
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	0	0
$735$	0	0
$736$	24.0000	0.884652
$737$	−40.0000	−1.47342
$738$	0	0
$739$	51.0000	1.87607	0.938033	−	0.346547i	$-0.112646\pi$
$739$	51.0000	1.87607	0.938033	+	0.346547i	$0.112646\pi$
$740$	12.0000	0.441129
$741$	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
$745$	−9.00000	−0.329734
$746$	−64.0000	−2.34321
$747$	0	0
$748$	24.0000	0.877527
$749$	0	0
$750$	0	0
$751$	−14.0000	−0.510867	−0.255434	−	0.966827i	$-0.582218\pi$
$751$	−14.0000	−0.510867	−0.255434	+	0.966827i	$0.582218\pi$
$752$	0	0
$753$	0	0
$754$	80.0000	2.91343
$755$	10.0000	0.363937
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−24.0000	−0.871719
$759$	0	0
$760$	0	0
$761$	21.0000	0.761249	0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000	0.761249	0.380625	+	0.924730i	$0.375709\pi$
$762$	0	0
$763$	0	0
$764$	34.0000	1.23008
$765$	0	0
$766$	24.0000	0.867155
$767$	−48.0000	−1.73318
$768$	0	0
$769$	−37.0000	−1.33425	−0.667127	−	0.744944i	$-0.732476\pi$
$769$	−37.0000	−1.33425	−0.667127	+	0.744944i	$0.732476\pi$
$770$	0	0
$771$	0	0
$772$	−32.0000	−1.15171
$773$	28.0000	1.00709	0.503545	−	0.863969i	$-0.332029\pi$
$773$	28.0000	1.00709	0.503545	+	0.863969i	$0.332029\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−6.00000	−0.215110
$779$	2.00000	0.0716574
$780$	0	0
$781$	24.0000	0.858788
$782$	18.0000	0.643679
$783$	0	0
$784$	0	0
$785$	−1.00000	−0.0356915
$786$	0	0
$787$	46.0000	1.63972	0.819861	−	0.572562i	$-0.194050\pi$
$787$	46.0000	1.63972	0.819861	+	0.572562i	$0.194050\pi$
$788$	22.0000	0.783718
$789$	0	0
$790$	−20.0000	−0.711568
$791$	0	0
$792$	0	0
$793$	40.0000	1.42044
$794$	30.0000	1.06466
$795$	0	0
$796$	−54.0000	−1.91398
$797$	−4.00000	−0.141687	−0.0708436	−	0.997487i	$-0.522569\pi$
$797$	−4.00000	−0.141687	−0.0708436	+	0.997487i	$0.522569\pi$
$798$	0	0
$799$	0	0
$800$	−32.0000	−1.13137
$801$	0	0
$802$	−24.0000	−0.847469
$803$	−24.0000	−0.846942
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	27.0000	0.949269	0.474635	−	0.880183i	$-0.342580\pi$
$809$	27.0000	0.949269	0.474635	+	0.880183i	$0.342580\pi$
$810$	0	0
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	0	0
$813$	0	0
$814$	−48.0000	−1.68240
$815$	−19.0000	−0.665541
$816$	0	0
$817$	7.00000	0.244899
$818$	−64.0000	−2.23771
$819$	0	0
$820$	4.00000	0.139686
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.00000	0.208640	0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000	0.208640	0.104320	+	0.994544i	$0.466733\pi$
$828$	0	0
$829$	−52.0000	−1.80603	−0.903017	−	0.429604i	$-0.858653\pi$
$829$	−52.0000	−1.80603	−0.903017	+	0.429604i	$0.858653\pi$
$830$	−6.00000	−0.208263
$831$	0	0
$832$	−32.0000	−1.10940
$833$	0	0
$834$	0	0
$835$	−2.00000	−0.0692129
$836$	8.00000	0.276686
$837$	0	0
$838$	−18.0000	−0.621800
$839$	56.0000	1.93333	0.966667	−	0.256036i	$-0.0824164\pi$
$839$	56.0000	1.93333	0.966667	+	0.256036i	$0.0824164\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	−16.0000	−0.551396
$843$	0	0
$844$	8.00000	0.275371
$845$	−3.00000	−0.103203
$846$	0	0
$847$	0	0
$848$	−48.0000	−1.64833
$849$	0	0
$850$	−24.0000	−0.823193
$851$	−18.0000	−0.617032
$852$	0	0
$853$	−37.0000	−1.26686	−0.633428	−	0.773802i	$-0.718353\pi$
$853$	−37.0000	−1.26686	−0.633428	+	0.773802i	$0.718353\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−40.0000	−1.36637	−0.683187	−	0.730243i	$-0.739407\pi$
$857$	−40.0000	−1.36637	−0.683187	+	0.730243i	$0.739407\pi$
$858$	0	0
$859$	17.0000	0.580033	0.290016	−	0.957022i	$-0.406339\pi$
$859$	17.0000	0.580033	0.290016	+	0.957022i	$0.406339\pi$
$860$	14.0000	0.477396
$861$	0	0
$862$	−24.0000	−0.817443
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	28.0000	0.951479
$867$	0	0
$868$	0	0
$869$	40.0000	1.35691
$870$	0	0
$871$	40.0000	1.35535
$872$	0	0
$873$	0	0
$874$	6.00000	0.202953
$875$	0	0
$876$	0	0
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	−60.0000	−2.02490
$879$	0	0
$880$	−16.0000	−0.539360
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	0	0
$883$	47.0000	1.58168	0.790838	−	0.612026i	$-0.209645\pi$
$883$	47.0000	1.58168	0.790838	+	0.612026i	$0.209645\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	−42.0000	−1.41102
$887$	−14.0000	−0.470074	−0.235037	−	0.971986i	$-0.575521\pi$
$887$	−14.0000	−0.470074	−0.235037	+	0.971986i	$0.575521\pi$
$888$	0	0
$889$	0	0
$890$	28.0000	0.938562
$891$	0	0
$892$	20.0000	0.669650
$893$	0	0
$894$	0	0
$895$	−16.0000	−0.534821
$896$	0	0
$897$	0	0
$898$	12.0000	0.400445
$899$	0	0
$900$	0	0
$901$	−36.0000	−1.19933
$902$	−16.0000	−0.532742
$903$	0	0
$904$	0	0
$905$	14.0000	0.465376
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	42.0000	1.39152	0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000	1.39152	0.695761	+	0.718273i	$0.255067\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	−44.0000	−1.45539
$915$	0	0
$916$	−14.0000	−0.462573
$917$	0	0
$918$	0	0
$919$	−7.00000	−0.230909	−0.115454	−	0.993313i	$-0.536832\pi$
$919$	−7.00000	−0.230909	−0.115454	+	0.993313i	$0.536832\pi$
$920$	0	0
$921$	0	0
$922$	52.0000	1.71253
$923$	−24.0000	−0.789970
$924$	0	0
$925$	24.0000	0.789115
$926$	−8.00000	−0.262896
$927$	0	0
$928$	−80.0000	−2.62613
$929$	55.0000	1.80449	0.902246	−	0.431222i	$-0.141918\pi$
$929$	55.0000	1.80449	0.902246	+	0.431222i	$0.141918\pi$
$930$	0	0
$931$	0	0
$932$	−18.0000	−0.589610
$933$	0	0
$934$	72.0000	2.35591
$935$	−12.0000	−0.392442
$936$	0	0
$937$	57.0000	1.86211	0.931054	−	0.364880i	$-0.118890\pi$
$937$	57.0000	1.86211	0.931054	+	0.364880i	$0.118890\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−20.0000	−0.651981	−0.325991	−	0.945373i	$-0.605698\pi$
$941$	−20.0000	−0.651981	−0.325991	+	0.945373i	$0.605698\pi$
$942$	0	0
$943$	−6.00000	−0.195387
$944$	48.0000	1.56227
$945$	0	0
$946$	−56.0000	−1.82072
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	−8.00000	−0.259554
$951$	0	0
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	0	0
$955$	−17.0000	−0.550107
$956$	−18.0000	−0.582162
$957$	0	0
$958$	22.0000	0.710788
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	48.0000	1.54758
$963$	0	0
$964$	−56.0000	−1.80364
$965$	16.0000	0.515058
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	0	0
$970$	−24.0000	−0.770594
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	0	0
$974$	64.0000	2.05069
$975$	0	0
$976$	−40.0000	−1.28037
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	−56.0000	−1.78977
$980$	0	0
$981$	0	0
$982$	−2.00000	−0.0638226
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	−11.0000	−0.350489
$986$	−60.0000	−1.91079
$987$	0	0
$988$	−8.00000	−0.254514
$989$	−21.0000	−0.667761
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	27.0000	0.855958
$996$	0	0
$997$	25.0000	0.791758	0.395879	−	0.918303i	$-0.370440\pi$
$997$	25.0000	0.791758	0.395879	+	0.918303i	$0.370440\pi$
$998$	−14.0000	−0.443162
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.b.1.1		1
3.2	odd	2		2793.2.a.l.1.1		1
7.2	even	3		1197.2.j.b.172.1		2
7.4	even	3		1197.2.j.b.856.1		2
7.6	odd	2		8379.2.a.c.1.1		1
21.2	odd	6		399.2.j.a.172.1	yes	2
21.11	odd	6		399.2.j.a.58.1	✓	2
21.20	even	2		2793.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.j.a.58.1	✓	2	21.11	odd	6
399.2.j.a.172.1	yes	2	21.2	odd	6
1197.2.j.b.172.1		2	7.2	even	3
1197.2.j.b.856.1		2	7.4	even	3
2793.2.a.k.1.1		1	21.20	even	2
2793.2.a.l.1.1		1	3.2	odd	2
8379.2.a.b.1.1		1	1.1	even	1	trivial
8379.2.a.c.1.1		1	7.6	odd	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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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