LMFDB - Embedded newform 8379.2.a.bb.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 = [2,0,0,2,0,0,0,0,0,-6,0,0,8,0,0,-10,0] 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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1197)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} +1.73205 q^{5} +1.73205 q^{8} -3.00000 q^{10} +5.19615 q^{11} +4.00000 q^{13} -5.00000 q^{16} +6.92820 q^{17} -1.00000 q^{19} +1.73205 q^{20} -9.00000 q^{22} -8.66025 q^{23} -2.00000 q^{25} -6.92820 q^{26} -3.46410 q^{29} -2.00000 q^{31} +5.19615 q^{32} -12.0000 q^{34} +8.00000 q^{37} +1.73205 q^{38} +3.00000 q^{40} -3.46410 q^{41} -7.00000 q^{43} +5.19615 q^{44} +15.0000 q^{46} +8.66025 q^{47} +3.46410 q^{50} +4.00000 q^{52} -3.46410 q^{53} +9.00000 q^{55} +6.00000 q^{58} +6.92820 q^{59} +7.00000 q^{61} +3.46410 q^{62} +1.00000 q^{64} +6.92820 q^{65} +8.00000 q^{67} +6.92820 q^{68} +3.46410 q^{71} +7.00000 q^{73} -13.8564 q^{74} -1.00000 q^{76} +8.00000 q^{79} -8.66025 q^{80} +6.00000 q^{82} +5.19615 q^{83} +12.0000 q^{85} +12.1244 q^{86} +9.00000 q^{88} -6.92820 q^{89} -8.66025 q^{92} -15.0000 q^{94} -1.73205 q^{95} +10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 6 q^{10} + 8 q^{13} - 10 q^{16} - 2 q^{19} - 18 q^{22} - 4 q^{25} - 4 q^{31} - 24 q^{34} + 16 q^{37} + 6 q^{40} - 14 q^{43} + 30 q^{46} + 8 q^{52} + 18 q^{55} + 12 q^{58} + 14 q^{61} + 2 q^{64}+ \cdots + 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 - 6 * q^10 + 8 * q^13 - 10 * q^16 - 2 * q^19 - 18 * q^22 - 4 * q^25 - 4 * q^31 - 24 * q^34 + 16 * q^37 + 6 * q^40 - 14 * q^43 + 30 * q^46 + 8 * q^52 + 18 * q^55 + 12 * q^58 + 14 * q^61 + 2 * q^64 + 16 * q^67 + 14 * q^73 - 2 * q^76 + 16 * q^79 + 12 * q^82 + 24 * q^85 + 18 * q^88 - 30 * q^94 + 20 * 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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.73205	0.612372
$9$	0	0
$10$	−3.00000	−0.948683
$11$	5.19615	1.56670	0.783349	−	0.621582i	$-0.213510\pi$
$11$	5.19615	1.56670	0.783349	+	0.621582i	$0.213510\pi$
$12$	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
$16$	−5.00000	−1.25000
$17$	6.92820	1.68034	0.840168	−	0.542326i	$-0.182456\pi$
$17$	6.92820	1.68034	0.840168	+	0.542326i	$0.182456\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.73205	0.387298
$21$	0	0
$22$	−9.00000	−1.91881
$23$	−8.66025	−1.80579	−0.902894	−	0.429863i	$-0.858562\pi$
$23$	−8.66025	−1.80579	−0.902894	+	0.429863i	$0.858562\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	−6.92820	−1.35873
$27$	0	0
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	−12.0000	−2.05798
$35$	0	0
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	1.73205	0.280976
$39$	0	0
$40$	3.00000	0.474342
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	5.19615	0.783349
$45$	0	0
$46$	15.0000	2.21163
$47$	8.66025	1.26323	0.631614	−	0.775283i	$-0.282393\pi$
$47$	8.66025	1.26323	0.631614	+	0.775283i	$0.282393\pi$
$48$	0	0
$49$	0	0
$50$	3.46410	0.489898
$51$	0	0
$52$	4.00000	0.554700
$53$	−3.46410	−0.475831	−0.237915	−	0.971286i	$-0.576464\pi$
$53$	−3.46410	−0.475831	−0.237915	+	0.971286i	$0.576464\pi$
$54$	0	0
$55$	9.00000	1.21356
$56$	0	0
$57$	0	0
$58$	6.00000	0.787839
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	3.46410	0.439941
$63$	0	0
$64$	1.00000	0.125000
$65$	6.92820	0.859338
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	6.92820	0.840168
$69$	0	0
$70$	0	0
$71$	3.46410	0.411113	0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410	0.411113	0.205557	+	0.978645i	$0.434100\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	−13.8564	−1.61077
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−8.66025	−0.968246
$81$	0	0
$82$	6.00000	0.662589
$83$	5.19615	0.570352	0.285176	−	0.958475i	$-0.407948\pi$
$83$	5.19615	0.570352	0.285176	+	0.958475i	$0.407948\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	12.1244	1.30740
$87$	0	0
$88$	9.00000	0.959403
$89$	−6.92820	−0.734388	−0.367194	−	0.930144i	$-0.619682\pi$
$89$	−6.92820	−0.734388	−0.367194	+	0.930144i	$0.619682\pi$
$90$	0	0
$91$	0	0
$92$	−8.66025	−0.902894
$93$	0	0
$94$	−15.0000	−1.54713
$95$	−1.73205	−0.177705
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	0	0
$100$	−2.00000	−0.200000
$101$	8.66025	0.861727	0.430864	−	0.902417i	$-0.358209\pi$
$101$	8.66025	0.861727	0.430864	+	0.902417i	$0.358209\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	6.92820	0.679366
$105$	0	0
$106$	6.00000	0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	−15.5885	−1.48630
$111$	0	0
$112$	0	0
$113$	−10.3923	−0.977626	−0.488813	−	0.872389i	$-0.662570\pi$
$113$	−10.3923	−0.977626	−0.488813	+	0.872389i	$0.662570\pi$
$114$	0	0
$115$	−15.0000	−1.39876
$116$	−3.46410	−0.321634
$117$	0	0
$118$	−12.0000	−1.10469
$119$	0	0
$120$	0	0
$121$	16.0000	1.45455
$122$	−12.1244	−1.09769
$123$	0	0
$124$	−2.00000	−0.179605
$125$	−12.1244	−1.08444
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	−12.0000	−1.05247
$131$	−3.46410	−0.302660	−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410	−0.302660	−0.151330	+	0.988483i	$0.548356\pi$
$132$	0	0
$133$	0	0
$134$	−13.8564	−1.19701
$135$	0	0
$136$	12.0000	1.02899
$137$	−8.66025	−0.739895	−0.369948	−	0.929053i	$-0.620624\pi$
$137$	−8.66025	−0.739895	−0.369948	+	0.929053i	$0.620624\pi$
$138$	0	0
$139$	−17.0000	−1.44192	−0.720961	−	0.692976i	$-0.756299\pi$
$139$	−17.0000	−1.44192	−0.720961	+	0.692976i	$0.756299\pi$
$140$	0	0
$141$	0	0
$142$	−6.00000	−0.503509
$143$	20.7846	1.73810
$144$	0	0
$145$	−6.00000	−0.498273
$146$	−12.1244	−1.00342
$147$	0	0
$148$	8.00000	0.657596
$149$	−1.73205	−0.141895	−0.0709476	−	0.997480i	$-0.522602\pi$
$149$	−1.73205	−0.141895	−0.0709476	+	0.997480i	$0.522602\pi$
$150$	0	0
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	−1.73205	−0.140488
$153$	0	0
$154$	0	0
$155$	−3.46410	−0.278243
$156$	0	0
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	−13.8564	−1.10236
$159$	0	0
$160$	9.00000	0.711512
$161$	0	0
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	−3.46410	−0.270501
$165$	0	0
$166$	−9.00000	−0.698535
$167$	−17.3205	−1.34030	−0.670151	−	0.742225i	$-0.733770\pi$
$167$	−17.3205	−1.34030	−0.670151	+	0.742225i	$0.733770\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−20.7846	−1.59411
$171$	0	0
$172$	−7.00000	−0.533745
$173$	6.92820	0.526742	0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820	0.526742	0.263371	+	0.964695i	$0.415166\pi$
$174$	0	0
$175$	0	0
$176$	−25.9808	−1.95837
$177$	0	0
$178$	12.0000	0.899438
$179$	−10.3923	−0.776757	−0.388379	−	0.921500i	$-0.626965\pi$
$179$	−10.3923	−0.776757	−0.388379	+	0.921500i	$0.626965\pi$
$180$	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
$184$	−15.0000	−1.10581
$185$	13.8564	1.01874
$186$	0	0
$187$	36.0000	2.63258
$188$	8.66025	0.631614
$189$	0	0
$190$	3.00000	0.217643
$191$	−5.19615	−0.375980	−0.187990	−	0.982171i	$-0.560197\pi$
$191$	−5.19615	−0.375980	−0.187990	+	0.982171i	$0.560197\pi$
$192$	0	0
$193$	8.00000	0.575853	0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000	0.575853	0.287926	+	0.957653i	$0.407034\pi$
$194$	−17.3205	−1.24354
$195$	0	0
$196$	0	0
$197$	−15.5885	−1.11063	−0.555316	−	0.831640i	$-0.687403\pi$
$197$	−15.5885	−1.11063	−0.555316	+	0.831640i	$0.687403\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	−3.46410	−0.244949
$201$	0	0
$202$	−15.0000	−1.05540
$203$	0	0
$204$	0	0
$205$	−6.00000	−0.419058
$206$	24.2487	1.68949
$207$	0	0
$208$	−20.0000	−1.38675
$209$	−5.19615	−0.359425
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−3.46410	−0.237915
$213$	0	0
$214$	0	0
$215$	−12.1244	−0.826874
$216$	0	0
$217$	0	0
$218$	−34.6410	−2.34619
$219$	0	0
$220$	9.00000	0.606780
$221$	27.7128	1.86417
$222$	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
$226$	18.0000	1.19734
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	25.9808	1.71312
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−6.92820	−0.453882	−0.226941	−	0.973909i	$-0.572872\pi$
$233$	−6.92820	−0.453882	−0.226941	+	0.973909i	$0.572872\pi$
$234$	0	0
$235$	15.0000	0.978492
$236$	6.92820	0.450988
$237$	0	0
$238$	0	0
$239$	10.3923	0.672222	0.336111	−	0.941822i	$-0.390888\pi$
$239$	10.3923	0.672222	0.336111	+	0.941822i	$0.390888\pi$
$240$	0	0
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	−27.7128	−1.78145
$243$	0	0
$244$	7.00000	0.448129
$245$	0	0
$246$	0	0
$247$	−4.00000	−0.254514
$248$	−3.46410	−0.219971
$249$	0	0
$250$	21.0000	1.32816
$251$	−1.73205	−0.109326	−0.0546630	−	0.998505i	$-0.517408\pi$
$251$	−1.73205	−0.109326	−0.0546630	+	0.998505i	$0.517408\pi$
$252$	0	0
$253$	−45.0000	−2.82913
$254$	−34.6410	−2.17357
$255$	0	0
$256$	19.0000	1.18750
$257$	24.2487	1.51259	0.756297	−	0.654229i	$-0.227007\pi$
$257$	24.2487	1.51259	0.756297	+	0.654229i	$0.227007\pi$
$258$	0	0
$259$	0	0
$260$	6.92820	0.429669
$261$	0	0
$262$	6.00000	0.370681
$263$	24.2487	1.49524	0.747620	−	0.664127i	$-0.231197\pi$
$263$	24.2487	1.49524	0.747620	+	0.664127i	$0.231197\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	8.00000	0.488678
$269$	6.92820	0.422420	0.211210	−	0.977441i	$-0.432260\pi$
$269$	6.92820	0.422420	0.211210	+	0.977441i	$0.432260\pi$
$270$	0	0
$271$	−11.0000	−0.668202	−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000	−0.668202	−0.334101	+	0.942537i	$0.608433\pi$
$272$	−34.6410	−2.10042
$273$	0	0
$274$	15.0000	0.906183
$275$	−10.3923	−0.626680
$276$	0	0
$277$	−13.0000	−0.781094	−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000	−0.781094	−0.390547	+	0.920583i	$0.627714\pi$
$278$	29.4449	1.76599
$279$	0	0
$280$	0	0
$281$	17.3205	1.03325	0.516627	−	0.856210i	$-0.327187\pi$
$281$	17.3205	1.03325	0.516627	+	0.856210i	$0.327187\pi$
$282$	0	0
$283$	31.0000	1.84276	0.921379	−	0.388664i	$-0.127063\pi$
$283$	31.0000	1.84276	0.921379	+	0.388664i	$0.127063\pi$
$284$	3.46410	0.205557
$285$	0	0
$286$	−36.0000	−2.12872
$287$	0	0
$288$	0	0
$289$	31.0000	1.82353
$290$	10.3923	0.610257
$291$	0	0
$292$	7.00000	0.409644
$293$	31.1769	1.82137	0.910687	−	0.413096i	$-0.135553\pi$
$293$	31.1769	1.82137	0.910687	+	0.413096i	$0.135553\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	13.8564	0.805387
$297$	0	0
$298$	3.00000	0.173785
$299$	−34.6410	−2.00334
$300$	0	0
$301$	0	0
$302$	38.1051	2.19270
$303$	0	0
$304$	5.00000	0.286770
$305$	12.1244	0.694239
$306$	0	0
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	0	0
$309$	0	0
$310$	6.00000	0.340777
$311$	−31.1769	−1.76788	−0.883940	−	0.467600i	$-0.845119\pi$
$311$	−31.1769	−1.76788	−0.883940	+	0.467600i	$0.845119\pi$
$312$	0	0
$313$	−29.0000	−1.63918	−0.819588	−	0.572953i	$-0.805798\pi$
$313$	−29.0000	−1.63918	−0.819588	+	0.572953i	$0.805798\pi$
$314$	−22.5167	−1.27069
$315$	0	0
$316$	8.00000	0.450035
$317$	−3.46410	−0.194563	−0.0972817	−	0.995257i	$-0.531015\pi$
$317$	−3.46410	−0.194563	−0.0972817	+	0.995257i	$0.531015\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	1.73205	0.0968246
$321$	0	0
$322$	0	0
$323$	−6.92820	−0.385496
$324$	0	0
$325$	−8.00000	−0.443760
$326$	−19.0526	−1.05522
$327$	0	0
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	5.19615	0.285176
$333$	0	0
$334$	30.0000	1.64153
$335$	13.8564	0.757056
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−5.19615	−0.282633
$339$	0	0
$340$	12.0000	0.650791
$341$	−10.3923	−0.562775
$342$	0	0
$343$	0	0
$344$	−12.1244	−0.653701
$345$	0	0
$346$	−12.0000	−0.645124
$347$	−1.73205	−0.0929814	−0.0464907	−	0.998919i	$-0.514804\pi$
$347$	−1.73205	−0.0929814	−0.0464907	+	0.998919i	$0.514804\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	0	0
$352$	27.0000	1.43910
$353$	20.7846	1.10625	0.553127	−	0.833097i	$-0.313435\pi$
$353$	20.7846	1.10625	0.553127	+	0.833097i	$0.313435\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	−6.92820	−0.367194
$357$	0	0
$358$	18.0000	0.951330
$359$	8.66025	0.457071	0.228535	−	0.973536i	$-0.426606\pi$
$359$	8.66025	0.457071	0.228535	+	0.973536i	$0.426606\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	24.2487	1.27448
$363$	0	0
$364$	0	0
$365$	12.1244	0.634618
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	43.3013	2.25723
$369$	0	0
$370$	−24.0000	−1.24770
$371$	0	0
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	−62.3538	−3.22424
$375$	0	0
$376$	15.0000	0.773566
$377$	−13.8564	−0.713641
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	−1.73205	−0.0888523
$381$	0	0
$382$	9.00000	0.460480
$383$	−34.6410	−1.77007	−0.885037	−	0.465521i	$-0.845867\pi$
$383$	−34.6410	−1.77007	−0.885037	+	0.465521i	$0.845867\pi$
$384$	0	0
$385$	0	0
$386$	−13.8564	−0.705273
$387$	0	0
$388$	10.0000	0.507673
$389$	27.7128	1.40510	0.702548	−	0.711637i	$-0.252046\pi$
$389$	27.7128	1.40510	0.702548	+	0.711637i	$0.252046\pi$
$390$	0	0
$391$	−60.0000	−3.03433
$392$	0	0
$393$	0	0
$394$	27.0000	1.36024
$395$	13.8564	0.697191
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−12.1244	−0.607739
$399$	0	0
$400$	10.0000	0.500000
$401$	−27.7128	−1.38391	−0.691956	−	0.721940i	$-0.743251\pi$
$401$	−27.7128	−1.38391	−0.691956	+	0.721940i	$0.743251\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	8.66025	0.430864
$405$	0	0
$406$	0	0
$407$	41.5692	2.06051
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	10.3923	0.513239
$411$	0	0
$412$	−14.0000	−0.689730
$413$	0	0
$414$	0	0
$415$	9.00000	0.441793
$416$	20.7846	1.01905
$417$	0	0
$418$	9.00000	0.440204
$419$	15.5885	0.761546	0.380773	−	0.924669i	$-0.375658\pi$
$419$	15.5885	0.761546	0.380773	+	0.924669i	$0.375658\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	6.92820	0.337260
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−13.8564	−0.672134
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	21.0000	1.01271
$431$	38.1051	1.83546	0.917729	−	0.397206i	$-0.130020\pi$
$431$	38.1051	1.83546	0.917729	+	0.397206i	$0.130020\pi$
$432$	0	0
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	0	0
$435$	0	0
$436$	20.0000	0.957826
$437$	8.66025	0.414276
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	15.5885	0.743151
$441$	0	0
$442$	−48.0000	−2.28313
$443$	−3.46410	−0.164584	−0.0822922	−	0.996608i	$-0.526224\pi$
$443$	−3.46410	−0.164584	−0.0822922	+	0.996608i	$0.526224\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	−17.3205	−0.820150
$447$	0	0
$448$	0	0
$449$	−20.7846	−0.980886	−0.490443	−	0.871473i	$-0.663165\pi$
$449$	−20.7846	−0.980886	−0.490443	+	0.871473i	$0.663165\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	−10.3923	−0.488813
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.0000	1.07589	0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000	1.07589	0.537947	+	0.842978i	$0.319200\pi$
$458$	3.46410	0.161867
$459$	0	0
$460$	−15.0000	−0.699379
$461$	−8.66025	−0.403348	−0.201674	−	0.979453i	$-0.564638\pi$
$461$	−8.66025	−0.403348	−0.201674	+	0.979453i	$0.564638\pi$
$462$	0	0
$463$	−7.00000	−0.325318	−0.162659	−	0.986682i	$-0.552007\pi$
$463$	−7.00000	−0.325318	−0.162659	+	0.986682i	$0.552007\pi$
$464$	17.3205	0.804084
$465$	0	0
$466$	12.0000	0.555889
$467$	−36.3731	−1.68314	−0.841572	−	0.540144i	$-0.818370\pi$
$467$	−36.3731	−1.68314	−0.841572	+	0.540144i	$0.818370\pi$
$468$	0	0
$469$	0	0
$470$	−25.9808	−1.19840
$471$	0	0
$472$	12.0000	0.552345
$473$	−36.3731	−1.67244
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	0	0
$478$	−18.0000	−0.823301
$479$	−22.5167	−1.02881	−0.514406	−	0.857547i	$-0.671988\pi$
$479$	−22.5167	−1.02881	−0.514406	+	0.857547i	$0.671988\pi$
$480$	0	0
$481$	32.0000	1.45907
$482$	−48.4974	−2.20900
$483$	0	0
$484$	16.0000	0.727273
$485$	17.3205	0.786484
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	12.1244	0.548844
$489$	0	0
$490$	0	0
$491$	−22.5167	−1.01616	−0.508081	−	0.861309i	$-0.669645\pi$
$491$	−22.5167	−1.01616	−0.508081	+	0.861309i	$0.669645\pi$
$492$	0	0
$493$	−24.0000	−1.08091
$494$	6.92820	0.311715
$495$	0	0
$496$	10.0000	0.449013
$497$	0	0
$498$	0	0
$499$	5.00000	0.223831	0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000	0.223831	0.111915	+	0.993718i	$0.464301\pi$
$500$	−12.1244	−0.542218
$501$	0	0
$502$	3.00000	0.133897
$503$	−1.73205	−0.0772283	−0.0386142	−	0.999254i	$-0.512294\pi$
$503$	−1.73205	−0.0772283	−0.0386142	+	0.999254i	$0.512294\pi$
$504$	0	0
$505$	15.0000	0.667491
$506$	77.9423	3.46496
$507$	0	0
$508$	20.0000	0.887357
$509$	−38.1051	−1.68898	−0.844490	−	0.535572i	$-0.820096\pi$
$509$	−38.1051	−1.68898	−0.844490	+	0.535572i	$0.820096\pi$
$510$	0	0
$511$	0	0
$512$	−8.66025	−0.382733
$513$	0	0
$514$	−42.0000	−1.85254
$515$	−24.2487	−1.06853
$516$	0	0
$517$	45.0000	1.97910
$518$	0	0
$519$	0	0
$520$	12.0000	0.526235
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−2.00000	−0.0874539	−0.0437269	−	0.999044i	$-0.513923\pi$
$523$	−2.00000	−0.0874539	−0.0437269	+	0.999044i	$0.513923\pi$
$524$	−3.46410	−0.151330
$525$	0	0
$526$	−42.0000	−1.83129
$527$	−13.8564	−0.603595
$528$	0	0
$529$	52.0000	2.26087
$530$	10.3923	0.451413
$531$	0	0
$532$	0	0
$533$	−13.8564	−0.600188
$534$	0	0
$535$	0	0
$536$	13.8564	0.598506
$537$	0	0
$538$	−12.0000	−0.517357
$539$	0	0
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	19.0526	0.818377
$543$	0	0
$544$	36.0000	1.54349
$545$	34.6410	1.48386
$546$	0	0
$547$	38.0000	1.62476	0.812381	−	0.583127i	$-0.198171\pi$
$547$	38.0000	1.62476	0.812381	+	0.583127i	$0.198171\pi$
$548$	−8.66025	−0.369948
$549$	0	0
$550$	18.0000	0.767523
$551$	3.46410	0.147576
$552$	0	0
$553$	0	0
$554$	22.5167	0.956641
$555$	0	0
$556$	−17.0000	−0.720961
$557$	−19.0526	−0.807283	−0.403641	−	0.914917i	$-0.632256\pi$
$557$	−19.0526	−0.807283	−0.403641	+	0.914917i	$0.632256\pi$
$558$	0	0
$559$	−28.0000	−1.18427
$560$	0	0
$561$	0	0
$562$	−30.0000	−1.26547
$563$	−20.7846	−0.875967	−0.437983	−	0.898983i	$-0.644307\pi$
$563$	−20.7846	−0.875967	−0.437983	+	0.898983i	$0.644307\pi$
$564$	0	0
$565$	−18.0000	−0.757266
$566$	−53.6936	−2.25691
$567$	0	0
$568$	6.00000	0.251754
$569$	31.1769	1.30700	0.653502	−	0.756925i	$-0.273299\pi$
$569$	31.1769	1.30700	0.653502	+	0.756925i	$0.273299\pi$
$570$	0	0
$571$	−25.0000	−1.04622	−0.523109	−	0.852266i	$-0.675228\pi$
$571$	−25.0000	−1.04622	−0.523109	+	0.852266i	$0.675228\pi$
$572$	20.7846	0.869048
$573$	0	0
$574$	0	0
$575$	17.3205	0.722315
$576$	0	0
$577$	−17.0000	−0.707719	−0.353860	−	0.935299i	$-0.615131\pi$
$577$	−17.0000	−0.707719	−0.353860	+	0.935299i	$0.615131\pi$
$578$	−53.6936	−2.23336
$579$	0	0
$580$	−6.00000	−0.249136
$581$	0	0
$582$	0	0
$583$	−18.0000	−0.745484
$584$	12.1244	0.501709
$585$	0	0
$586$	−54.0000	−2.23072
$587$	−3.46410	−0.142979	−0.0714894	−	0.997441i	$-0.522775\pi$
$587$	−3.46410	−0.142979	−0.0714894	+	0.997441i	$0.522775\pi$
$588$	0	0
$589$	2.00000	0.0824086
$590$	−20.7846	−0.855689
$591$	0	0
$592$	−40.0000	−1.64399
$593$	8.66025	0.355634	0.177817	−	0.984064i	$-0.443096\pi$
$593$	8.66025	0.355634	0.177817	+	0.984064i	$0.443096\pi$
$594$	0	0
$595$	0	0
$596$	−1.73205	−0.0709476
$597$	0	0
$598$	60.0000	2.45358
$599$	17.3205	0.707697	0.353848	−	0.935303i	$-0.384873\pi$
$599$	17.3205	0.707697	0.353848	+	0.935303i	$0.384873\pi$
$600$	0	0
$601$	40.0000	1.63163	0.815817	−	0.578310i	$-0.196288\pi$
$601$	40.0000	1.63163	0.815817	+	0.578310i	$0.196288\pi$
$602$	0	0
$603$	0	0
$604$	−22.0000	−0.895167
$605$	27.7128	1.12669
$606$	0	0
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	−5.19615	−0.210732
$609$	0	0
$610$	−21.0000	−0.850265
$611$	34.6410	1.40143
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	3.46410	0.139800
$615$	0	0
$616$	0	0
$617$	−19.0526	−0.767027	−0.383514	−	0.923535i	$-0.625286\pi$
$617$	−19.0526	−0.767027	−0.383514	+	0.923535i	$0.625286\pi$
$618$	0	0
$619$	−41.0000	−1.64793	−0.823965	−	0.566641i	$-0.808243\pi$
$619$	−41.0000	−1.64793	−0.823965	+	0.566641i	$0.808243\pi$
$620$	−3.46410	−0.139122
$621$	0	0
$622$	54.0000	2.16520
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	50.2295	2.00757
$627$	0	0
$628$	13.0000	0.518756
$629$	55.4256	2.20996
$630$	0	0
$631$	−13.0000	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$632$	13.8564	0.551178
$633$	0	0
$634$	6.00000	0.238290
$635$	34.6410	1.37469
$636$	0	0
$637$	0	0
$638$	31.1769	1.23431
$639$	0	0
$640$	−21.0000	−0.830098
$641$	13.8564	0.547295	0.273648	−	0.961830i	$-0.411770\pi$
$641$	13.8564	0.547295	0.273648	+	0.961830i	$0.411770\pi$
$642$	0	0
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	−25.9808	−1.02141	−0.510705	−	0.859756i	$-0.670615\pi$
$647$	−25.9808	−1.02141	−0.510705	+	0.859756i	$0.670615\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	13.8564	0.543493
$651$	0	0
$652$	11.0000	0.430793
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	17.3205	0.676252
$657$	0	0
$658$	0	0
$659$	17.3205	0.674711	0.337356	−	0.941377i	$-0.390468\pi$
$659$	17.3205	0.674711	0.337356	+	0.941377i	$0.390468\pi$
$660$	0	0
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	−24.2487	−0.942453
$663$	0	0
$664$	9.00000	0.349268
$665$	0	0
$666$	0	0
$667$	30.0000	1.16160
$668$	−17.3205	−0.670151
$669$	0	0
$670$	−24.0000	−0.927201
$671$	36.3731	1.40417
$672$	0	0
$673$	−4.00000	−0.154189	−0.0770943	−	0.997024i	$-0.524564\pi$
$673$	−4.00000	−0.154189	−0.0770943	+	0.997024i	$0.524564\pi$
$674$	−24.2487	−0.934025
$675$	0	0
$676$	3.00000	0.115385
$677$	20.7846	0.798817	0.399409	−	0.916773i	$-0.369215\pi$
$677$	20.7846	0.798817	0.399409	+	0.916773i	$0.369215\pi$
$678$	0	0
$679$	0	0
$680$	20.7846	0.797053
$681$	0	0
$682$	18.0000	0.689256
$683$	10.3923	0.397650	0.198825	−	0.980035i	$-0.436287\pi$
$683$	10.3923	0.397650	0.198825	+	0.980035i	$0.436287\pi$
$684$	0	0
$685$	−15.0000	−0.573121
$686$	0	0
$687$	0	0
$688$	35.0000	1.33436
$689$	−13.8564	−0.527887
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	6.92820	0.263371
$693$	0	0
$694$	3.00000	0.113878
$695$	−29.4449	−1.11691
$696$	0	0
$697$	−24.0000	−0.909065
$698$	−17.3205	−0.655591
$699$	0	0
$700$	0	0
$701$	−1.73205	−0.0654187	−0.0327093	−	0.999465i	$-0.510414\pi$
$701$	−1.73205	−0.0654187	−0.0327093	+	0.999465i	$0.510414\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	5.19615	0.195837
$705$	0	0
$706$	−36.0000	−1.35488
$707$	0	0
$708$	0	0
$709$	17.0000	0.638448	0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000	0.638448	0.319224	+	0.947679i	$0.396578\pi$
$710$	−10.3923	−0.390016
$711$	0	0
$712$	−12.0000	−0.449719
$713$	17.3205	0.648658
$714$	0	0
$715$	36.0000	1.34632
$716$	−10.3923	−0.388379
$717$	0	0
$718$	−15.0000	−0.559795
$719$	−3.46410	−0.129189	−0.0645946	−	0.997912i	$-0.520575\pi$
$719$	−3.46410	−0.129189	−0.0645946	+	0.997912i	$0.520575\pi$
$720$	0	0
$721$	0	0
$722$	−1.73205	−0.0644603
$723$	0	0
$724$	−14.0000	−0.520306
$725$	6.92820	0.257307
$726$	0	0
$727$	−11.0000	−0.407967	−0.203984	−	0.978974i	$-0.565389\pi$
$727$	−11.0000	−0.407967	−0.203984	+	0.978974i	$0.565389\pi$
$728$	0	0
$729$	0	0
$730$	−21.0000	−0.777245
$731$	−48.4974	−1.79374
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	13.8564	0.511449
$735$	0	0
$736$	−45.0000	−1.65872
$737$	41.5692	1.53122
$738$	0	0
$739$	32.0000	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$740$	13.8564	0.509372
$741$	0	0
$742$	0	0
$743$	34.6410	1.27086	0.635428	−	0.772160i	$-0.280824\pi$
$743$	34.6410	1.27086	0.635428	+	0.772160i	$0.280824\pi$
$744$	0	0
$745$	−3.00000	−0.109911
$746$	6.92820	0.253660
$747$	0	0
$748$	36.0000	1.31629
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	−43.3013	−1.57903
$753$	0	0
$754$	24.0000	0.874028
$755$	−38.1051	−1.38679
$756$	0	0
$757$	41.0000	1.49017	0.745085	−	0.666969i	$-0.232409\pi$
$757$	41.0000	1.49017	0.745085	+	0.666969i	$0.232409\pi$
$758$	27.7128	1.00657
$759$	0	0
$760$	−3.00000	−0.108821
$761$	15.5885	0.565081	0.282541	−	0.959255i	$-0.408823\pi$
$761$	15.5885	0.565081	0.282541	+	0.959255i	$0.408823\pi$
$762$	0	0
$763$	0	0
$764$	−5.19615	−0.187990
$765$	0	0
$766$	60.0000	2.16789
$767$	27.7128	1.00065
$768$	0	0
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	0	0
$772$	8.00000	0.287926
$773$	34.6410	1.24595	0.622975	−	0.782241i	$-0.285924\pi$
$773$	34.6410	1.24595	0.622975	+	0.782241i	$0.285924\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	17.3205	0.621770
$777$	0	0
$778$	−48.0000	−1.72088
$779$	3.46410	0.124114
$780$	0	0
$781$	18.0000	0.644091
$782$	103.923	3.71628
$783$	0	0
$784$	0	0
$785$	22.5167	0.803654
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	−15.5885	−0.555316
$789$	0	0
$790$	−24.0000	−0.853882
$791$	0	0
$792$	0	0
$793$	28.0000	0.994309
$794$	3.46410	0.122936
$795$	0	0
$796$	7.00000	0.248108
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	60.0000	2.12265
$800$	−10.3923	−0.367423
$801$	0	0
$802$	48.0000	1.69494
$803$	36.3731	1.28358
$804$	0	0
$805$	0	0
$806$	13.8564	0.488071
$807$	0	0
$808$	15.0000	0.527698
$809$	−5.19615	−0.182687	−0.0913435	−	0.995819i	$-0.529116\pi$
$809$	−5.19615	−0.182687	−0.0913435	+	0.995819i	$0.529116\pi$
$810$	0	0
$811$	22.0000	0.772524	0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000	0.772524	0.386262	+	0.922389i	$0.373766\pi$
$812$	0	0
$813$	0	0
$814$	−72.0000	−2.52360
$815$	19.0526	0.667382
$816$	0	0
$817$	7.00000	0.244899
$818$	24.2487	0.847836
$819$	0	0
$820$	−6.00000	−0.209529
$821$	−50.2295	−1.75302	−0.876510	−	0.481383i	$-0.840135\pi$
$821$	−50.2295	−1.75302	−0.876510	+	0.481383i	$0.840135\pi$
$822$	0	0
$823$	41.0000	1.42917	0.714585	−	0.699549i	$-0.246616\pi$
$823$	41.0000	1.42917	0.714585	+	0.699549i	$0.246616\pi$
$824$	−24.2487	−0.844744
$825$	0	0
$826$	0	0
$827$	27.7128	0.963669	0.481834	−	0.876262i	$-0.339971\pi$
$827$	27.7128	0.963669	0.481834	+	0.876262i	$0.339971\pi$
$828$	0	0
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	−15.5885	−0.541083
$831$	0	0
$832$	4.00000	0.138675
$833$	0	0
$834$	0	0
$835$	−30.0000	−1.03819
$836$	−5.19615	−0.179713
$837$	0	0
$838$	−27.0000	−0.932700
$839$	27.7128	0.956753	0.478376	−	0.878155i	$-0.341226\pi$
$839$	27.7128	0.956753	0.478376	+	0.878155i	$0.341226\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	−3.46410	−0.119381
$843$	0	0
$844$	−4.00000	−0.137686
$845$	5.19615	0.178753
$846$	0	0
$847$	0	0
$848$	17.3205	0.594789
$849$	0	0
$850$	24.0000	0.823193
$851$	−69.2820	−2.37496
$852$	0	0
$853$	−35.0000	−1.19838	−0.599189	−	0.800608i	$-0.704510\pi$
$853$	−35.0000	−1.19838	−0.599189	+	0.800608i	$0.704510\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34.6410	−1.18331	−0.591657	−	0.806190i	$-0.701526\pi$
$857$	−34.6410	−1.18331	−0.591657	+	0.806190i	$0.701526\pi$
$858$	0	0
$859$	37.0000	1.26242	0.631212	−	0.775610i	$-0.282558\pi$
$859$	37.0000	1.26242	0.631212	+	0.775610i	$0.282558\pi$
$860$	−12.1244	−0.413437
$861$	0	0
$862$	−66.0000	−2.24797
$863$	10.3923	0.353758	0.176879	−	0.984233i	$-0.443400\pi$
$863$	10.3923	0.353758	0.176879	+	0.984233i	$0.443400\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	−6.92820	−0.235430
$867$	0	0
$868$	0	0
$869$	41.5692	1.41014
$870$	0	0
$871$	32.0000	1.08428
$872$	34.6410	1.17309
$873$	0	0
$874$	−15.0000	−0.507383
$875$	0	0
$876$	0	0
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	−48.4974	−1.63671
$879$	0	0
$880$	−45.0000	−1.51695
$881$	−20.7846	−0.700251	−0.350126	−	0.936703i	$-0.613861\pi$
$881$	−20.7846	−0.700251	−0.350126	+	0.936703i	$0.613861\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	27.7128	0.932083
$885$	0	0
$886$	6.00000	0.201574
$887$	3.46410	0.116313	0.0581566	−	0.998307i	$-0.481478\pi$
$887$	3.46410	0.116313	0.0581566	+	0.998307i	$0.481478\pi$
$888$	0	0
$889$	0	0
$890$	20.7846	0.696702
$891$	0	0
$892$	10.0000	0.334825
$893$	−8.66025	−0.289804
$894$	0	0
$895$	−18.0000	−0.601674
$896$	0	0
$897$	0	0
$898$	36.0000	1.20134
$899$	6.92820	0.231069
$900$	0	0
$901$	−24.0000	−0.799556
$902$	31.1769	1.03808
$903$	0	0
$904$	−18.0000	−0.598671
$905$	−24.2487	−0.806054
$906$	0	0
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31.1769	1.03294	0.516469	−	0.856306i	$-0.327246\pi$
$911$	31.1769	1.03294	0.516469	+	0.856306i	$0.327246\pi$
$912$	0	0
$913$	27.0000	0.893570
$914$	−39.8372	−1.31770
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	0	0
$918$	0	0
$919$	17.0000	0.560778	0.280389	−	0.959886i	$-0.409536\pi$
$919$	17.0000	0.560778	0.280389	+	0.959886i	$0.409536\pi$
$920$	−25.9808	−0.856560
$921$	0	0
$922$	15.0000	0.493999
$923$	13.8564	0.456089
$924$	0	0
$925$	−16.0000	−0.526077
$926$	12.1244	0.398431
$927$	0	0
$928$	−18.0000	−0.590879
$929$	43.3013	1.42067	0.710334	−	0.703864i	$-0.248544\pi$
$929$	43.3013	1.42067	0.710334	+	0.703864i	$0.248544\pi$
$930$	0	0
$931$	0	0
$932$	−6.92820	−0.226941
$933$	0	0
$934$	63.0000	2.06142
$935$	62.3538	2.03919
$936$	0	0
$937$	43.0000	1.40475	0.702374	−	0.711808i	$-0.252123\pi$
$937$	43.0000	1.40475	0.702374	+	0.711808i	$0.252123\pi$
$938$	0	0
$939$	0	0
$940$	15.0000	0.489246
$941$	3.46410	0.112926	0.0564632	−	0.998405i	$-0.482018\pi$
$941$	3.46410	0.112926	0.0564632	+	0.998405i	$0.482018\pi$
$942$	0	0
$943$	30.0000	0.976934
$944$	−34.6410	−1.12747
$945$	0	0
$946$	63.0000	2.04831
$947$	−17.3205	−0.562841	−0.281420	−	0.959585i	$-0.590806\pi$
$947$	−17.3205	−0.562841	−0.281420	+	0.959585i	$0.590806\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	−3.46410	−0.112390
$951$	0	0
$952$	0	0
$953$	−48.4974	−1.57099	−0.785493	−	0.618871i	$-0.787590\pi$
$953$	−48.4974	−1.57099	−0.785493	+	0.618871i	$0.787590\pi$
$954$	0	0
$955$	−9.00000	−0.291233
$956$	10.3923	0.336111
$957$	0	0
$958$	39.0000	1.26003
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−55.4256	−1.78699
$963$	0	0
$964$	28.0000	0.901819
$965$	13.8564	0.446054
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	27.7128	0.890724
$969$	0	0
$970$	−30.0000	−0.963242
$971$	−13.8564	−0.444673	−0.222337	−	0.974970i	$-0.571368\pi$
$971$	−13.8564	−0.444673	−0.222337	+	0.974970i	$0.571368\pi$
$972$	0	0
$973$	0	0
$974$	−3.46410	−0.110997
$975$	0	0
$976$	−35.0000	−1.12032
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	0	0
$981$	0	0
$982$	39.0000	1.24454
$983$	−45.0333	−1.43634	−0.718170	−	0.695868i	$-0.755020\pi$
$983$	−45.0333	−1.43634	−0.718170	+	0.695868i	$0.755020\pi$
$984$	0	0
$985$	−27.0000	−0.860292
$986$	41.5692	1.32383
$987$	0	0
$988$	−4.00000	−0.127257
$989$	60.6218	1.92766
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	−10.3923	−0.329956
$993$	0	0
$994$	0	0
$995$	12.1244	0.384368
$996$	0	0
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	−8.66025	−0.274136
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.bb.1.1		2
3.2	odd	2	inner	8379.2.a.bb.1.2		2
7.3	odd	6		1197.2.j.f.856.2	yes	4
7.5	odd	6		1197.2.j.f.172.2	yes	4
7.6	odd	2		8379.2.a.be.1.1		2
21.5	even	6		1197.2.j.f.172.1	✓	4
21.17	even	6		1197.2.j.f.856.1	yes	4
21.20	even	2		8379.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1197.2.j.f.172.1	✓	4	21.5	even	6
1197.2.j.f.172.2	yes	4	7.5	odd	6
1197.2.j.f.856.1	yes	4	21.17	even	6
1197.2.j.f.856.2	yes	4	7.3	odd	6
8379.2.a.bb.1.1		2	1.1	even	1	trivial
8379.2.a.bb.1.2		2	3.2	odd	2	inner
8379.2.a.be.1.1		2	7.6	odd	2
8379.2.a.be.1.2		2	21.20	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)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} +1.73205 q^{5} +1.73205 q^{8} -3.00000 q^{10} +5.19615 q^{11} +4.00000 q^{13} -5.00000 q^{16} +6.92820 q^{17} -1.00000 q^{19} +1.73205 q^{20} -9.00000 q^{22} -8.66025 q^{23} -2.00000 q^{25} -6.92820 q^{26} -3.46410 q^{29} -2.00000 q^{31} +5.19615 q^{32} -12.0000 q^{34} +8.00000 q^{37} +1.73205 q^{38} +3.00000 q^{40} -3.46410 q^{41} -7.00000 q^{43} +5.19615 q^{44} +15.0000 q^{46} +8.66025 q^{47} +3.46410 q^{50} +4.00000 q^{52} -3.46410 q^{53} +9.00000 q^{55} +6.00000 q^{58} +6.92820 q^{59} +7.00000 q^{61} +3.46410 q^{62} +1.00000 q^{64} +6.92820 q^{65} +8.00000 q^{67} +6.92820 q^{68} +3.46410 q^{71} +7.00000 q^{73} -13.8564 q^{74} -1.00000 q^{76} +8.00000 q^{79} -8.66025 q^{80} +6.00000 q^{82} +5.19615 q^{83} +12.0000 q^{85} +12.1244 q^{86} +9.00000 q^{88} -6.92820 q^{89} -8.66025 q^{92} -15.0000 q^{94} -1.73205 q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 6 q^{10} + 8 q^{13} - 10 q^{16} - 2 q^{19} - 18 q^{22} - 4 q^{25} - 4 q^{31} - 24 q^{34} + 16 q^{37} + 6 q^{40} - 14 q^{43} + 30 q^{46} + 8 q^{52} + 18 q^{55} + 12 q^{58} + 14 q^{61} + 2 q^{64}+ \cdots + 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 - 6 * q^10 + 8 * q^13 - 10 * q^16 - 2 * q^19 - 18 * q^22 - 4 * q^25 - 4 * q^31 - 24 * q^34 + 16 * q^37 + 6 * q^40 - 14 * q^43 + 30 * q^46 + 8 * q^52 + 18 * q^55 + 12 * q^58 + 14 * q^61 + 2 * q^64 + 16 * q^67 + 14 * q^73 - 2 * q^76 + 16 * q^79 + 12 * q^82 + 24 * q^85 + 18 * q^88 - 30 * q^94 + 20 * q^97

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