LMFDB - Embedded newform 3328.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] 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("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-1,0,-1,0,-1,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1664)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} -2.00000 q^{11} +1.00000 q^{13} +1.00000 q^{15} +1.00000 q^{17} +8.00000 q^{19} +1.00000 q^{21} +6.00000 q^{23} -4.00000 q^{25} +5.00000 q^{27} +6.00000 q^{29} -8.00000 q^{31} +2.00000 q^{33} +1.00000 q^{35} -1.00000 q^{37} -1.00000 q^{39} -4.00000 q^{41} +5.00000 q^{43} +2.00000 q^{45} +11.0000 q^{47} -6.00000 q^{49} -1.00000 q^{51} -6.00000 q^{53} +2.00000 q^{55} -8.00000 q^{57} -8.00000 q^{59} +4.00000 q^{61} +2.00000 q^{63} -1.00000 q^{65} -2.00000 q^{67} -6.00000 q^{69} -3.00000 q^{71} -8.00000 q^{73} +4.00000 q^{75} +2.00000 q^{77} -10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{83} -1.00000 q^{85} -6.00000 q^{87} -16.0000 q^{89} -1.00000 q^{91} +8.00000 q^{93} -8.00000 q^{95} -2.00000 q^{97} +4.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−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
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	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
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	11.0000	1.60451	0.802257	−	0.596978i	$-0.203632\pi$
$47$	11.0000	1.60451	0.802257	+	0.596978i	$0.203632\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	−1.00000	−0.124035
$66$	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
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	−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
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	4.00000	0.360668
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	22.0000	1.95218	0.976092	−	0.217357i	$-0.0697436\pi$
$127$	22.0000	1.95218	0.976092	+	0.217357i	$0.0697436\pi$
$128$	0	0
$129$	−5.00000	−0.440225
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	−11.0000	−0.926367
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	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
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	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
$169$	1.00000	0.0769231
$170$	0	0
$171$	−16.0000	−1.22355
$172$	0	0
$173$	−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
$175$	4.00000	0.302372
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	13.0000	0.971666	0.485833	−	0.874052i	$-0.338516\pi$
$179$	13.0000	0.971666	0.485833	+	0.874052i	$0.338516\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	−23.0000	−1.63868	−0.819341	−	0.573306i	$-0.805660\pi$
$197$	−23.0000	−1.63868	−0.819341	+	0.573306i	$0.805660\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	−12.0000	−0.834058
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−15.0000	−1.03264	−0.516321	−	0.856395i	$-0.672699\pi$
$211$	−15.0000	−1.03264	−0.516321	+	0.856395i	$0.672699\pi$
$212$	0	0
$213$	3.00000	0.205557
$214$	0	0
$215$	−5.00000	−0.340997
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	8.00000	0.540590
$220$	0	0
$221$	1.00000	0.0672673
$222$	0	0
$223$	−3.00000	−0.200895	−0.100447	−	0.994942i	$-0.532027\pi$
$223$	−3.00000	−0.200895	−0.100447	+	0.994942i	$0.532027\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	0	0
$229$	21.0000	1.38772	0.693860	−	0.720110i	$-0.255909\pi$
$229$	21.0000	1.38772	0.693860	+	0.720110i	$0.255909\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	−7.00000	−0.458585	−0.229293	−	0.973358i	$-0.573641\pi$
$233$	−7.00000	−0.458585	−0.229293	+	0.973358i	$0.573641\pi$
$234$	0	0
$235$	−11.0000	−0.717561
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	11.0000	0.711531	0.355765	−	0.934575i	$-0.384220\pi$
$239$	11.0000	0.711531	0.355765	+	0.934575i	$0.384220\pi$
$240$	0	0
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	1.00000	0.0626224
$256$	0	0
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	0	0
$261$	−12.0000	−0.742781
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	16.0000	0.979184
$268$	0	0
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	3.00000	0.182237	0.0911185	−	0.995840i	$-0.470956\pi$
$271$	3.00000	0.182237	0.0911185	+	0.995840i	$0.470956\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	8.00000	0.482418
$276$	0	0
$277$	24.0000	1.44202	0.721010	−	0.692925i	$-0.243678\pi$
$277$	24.0000	1.44202	0.721010	+	0.692925i	$0.243678\pi$
$278$	0	0
$279$	16.0000	0.957895
$280$	0	0
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	0	0
$283$	−32.0000	−1.90220	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−32.0000	−1.90220	−0.951101	+	0.308879i	$0.900046\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	−10.0000	−0.580259
$298$	0	0
$299$	6.00000	0.346989
$300$	0	0
$301$	−5.00000	−0.288195
$302$	0	0
$303$	8.00000	0.459588
$304$	0	0
$305$	−4.00000	−0.229039
$306$	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
$309$	6.00000	0.341328
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	11.0000	0.621757	0.310878	−	0.950450i	$-0.399377\pi$
$313$	11.0000	0.621757	0.310878	+	0.950450i	$0.399377\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	1.00000	0.0553001
$328$	0	0
$329$	−11.0000	−0.606450
$330$	0	0
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	2.00000	0.109272
$336$	0	0
$337$	−25.0000	−1.36184	−0.680918	−	0.732359i	$-0.738419\pi$
$337$	−25.0000	−1.36184	−0.680918	+	0.732359i	$0.738419\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	16.0000	0.866449
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	6.00000	0.323029
$346$	0	0
$347$	−15.0000	−0.805242	−0.402621	−	0.915367i	$-0.631901\pi$
$347$	−15.0000	−0.805242	−0.402621	+	0.915367i	$0.631901\pi$
$348$	0	0
$349$	−25.0000	−1.33822	−0.669110	−	0.743164i	$-0.733324\pi$
$349$	−25.0000	−1.33822	−0.669110	+	0.743164i	$0.733324\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	3.00000	0.159223
$356$	0	0
$357$	1.00000	0.0529256
$358$	0	0
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	8.00000	0.418739
$366$	0	0
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	0	0
$369$	8.00000	0.416463
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	24.0000	1.24267	0.621336	−	0.783544i	$-0.286590\pi$
$373$	24.0000	1.24267	0.621336	+	0.783544i	$0.286590\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	0	0
$381$	−22.0000	−1.12709
$382$	0	0
$383$	−15.0000	−0.766464	−0.383232	−	0.923652i	$-0.625189\pi$
$383$	−15.0000	−0.766464	−0.383232	+	0.923652i	$0.625189\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	0	0
$387$	−10.0000	−0.508329
$388$	0	0
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	3.00000	0.151330
$394$	0	0
$395$	10.0000	0.503155
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	2.00000	0.0991363
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	14.0000	0.690569
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	−13.0000	−0.636613
$418$	0	0
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	0	0
$421$	15.0000	0.731055	0.365528	−	0.930800i	$-0.380889\pi$
$421$	15.0000	0.731055	0.365528	+	0.930800i	$0.380889\pi$
$422$	0	0
$423$	−22.0000	−1.06968
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	−4.00000	−0.193574
$428$	0	0
$429$	2.00000	0.0965609
$430$	0	0
$431$	15.0000	0.722525	0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000	0.722525	0.361262	+	0.932464i	$0.382346\pi$
$432$	0	0
$433$	31.0000	1.48976	0.744882	−	0.667196i	$-0.232506\pi$
$433$	31.0000	1.48976	0.744882	+	0.667196i	$0.232506\pi$
$434$	0	0
$435$	6.00000	0.287678
$436$	0	0
$437$	48.0000	2.29615
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	35.0000	1.66290	0.831450	−	0.555599i	$-0.187511\pi$
$443$	35.0000	1.66290	0.831450	+	0.555599i	$0.187511\pi$
$444$	0	0
$445$	16.0000	0.758473
$446$	0	0
$447$	−6.00000	−0.283790
$448$	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
$452$	0	0
$453$	−5.00000	−0.234920
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	32.0000	1.49690	0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000	1.49690	0.748448	+	0.663193i	$0.230799\pi$
$458$	0	0
$459$	5.00000	0.233380
$460$	0	0
$461$	−37.0000	−1.72326	−0.861631	−	0.507535i	$-0.830557\pi$
$461$	−37.0000	−1.72326	−0.861631	+	0.507535i	$0.830557\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	−10.0000	−0.459800
$474$	0	0
$475$	−32.0000	−1.46826
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	35.0000	1.59919	0.799595	−	0.600539i	$-0.205047\pi$
$479$	35.0000	1.59919	0.799595	+	0.600539i	$0.205047\pi$
$480$	0	0
$481$	−1.00000	−0.0455961
$482$	0	0
$483$	6.00000	0.273009
$484$	0	0
$485$	2.00000	0.0908153
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	−6.00000	−0.271329
$490$	0	0
$491$	13.0000	0.586682	0.293341	−	0.956008i	$-0.405233\pi$
$491$	13.0000	0.586682	0.293341	+	0.956008i	$0.405233\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	3.00000	0.134568
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	0	0
$513$	40.0000	1.76604
$514$	0	0
$515$	6.00000	0.264392
$516$	0	0
$517$	−22.0000	−0.967559
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	11.0000	0.481919	0.240959	−	0.970535i	$-0.422538\pi$
$521$	11.0000	0.481919	0.240959	+	0.970535i	$0.422538\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	16.0000	0.694341
$532$	0	0
$533$	−4.00000	−0.173259
$534$	0	0
$535$	4.00000	0.172935
$536$	0	0
$537$	−13.0000	−0.560991
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	1.00000	0.0429934	0.0214967	−	0.999769i	$-0.493157\pi$
$541$	1.00000	0.0429934	0.0214967	+	0.999769i	$0.493157\pi$
$542$	0	0
$543$	−22.0000	−0.944110
$544$	0	0
$545$	1.00000	0.0428353
$546$	0	0
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	0	0
$549$	−8.00000	−0.341432
$550$	0	0
$551$	48.0000	2.04487
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	−1.00000	−0.0424476
$556$	0	0
$557$	−15.0000	−0.635570	−0.317785	−	0.948163i	$-0.602939\pi$
$557$	−15.0000	−0.635570	−0.317785	+	0.948163i	$0.602939\pi$
$558$	0	0
$559$	5.00000	0.211477
$560$	0	0
$561$	2.00000	0.0844401
$562$	0	0
$563$	−31.0000	−1.30649	−0.653247	−	0.757145i	$-0.726594\pi$
$563$	−31.0000	−1.30649	−0.653247	+	0.757145i	$0.726594\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−1.00000	−0.0419222	−0.0209611	−	0.999780i	$-0.506673\pi$
$569$	−1.00000	−0.0419222	−0.0209611	+	0.999780i	$0.506673\pi$
$570$	0	0
$571$	7.00000	0.292941	0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000	0.292941	0.146470	+	0.989215i	$0.453209\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	0	0
$575$	−24.0000	−1.00087
$576$	0	0
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	0	0
$579$	−6.00000	−0.249351
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	−64.0000	−2.63707
$590$	0	0
$591$	23.0000	0.946094
$592$	0	0
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	1.00000	0.0409960
$596$	0	0
$597$	2.00000	0.0818546
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	45.0000	1.83559	0.917794	−	0.397057i	$-0.129968\pi$
$601$	45.0000	1.83559	0.917794	+	0.397057i	$0.129968\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	11.0000	0.445012
$612$	0	0
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	0	0
$615$	−4.00000	−0.161296
$616$	0	0
$617$	−20.0000	−0.805170	−0.402585	−	0.915383i	$-0.631888\pi$
$617$	−20.0000	−0.805170	−0.402585	+	0.915383i	$0.631888\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	30.0000	1.20386
$622$	0	0
$623$	16.0000	0.641026
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	16.0000	0.638978
$628$	0	0
$629$	−1.00000	−0.0398726
$630$	0	0
$631$	−47.0000	−1.87104	−0.935520	−	0.353273i	$-0.885069\pi$
$631$	−47.0000	−1.87104	−0.935520	+	0.353273i	$0.885069\pi$
$632$	0	0
$633$	15.0000	0.596196
$634$	0	0
$635$	−22.0000	−0.873043
$636$	0	0
$637$	−6.00000	−0.237729
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	0	0
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	0	0
$645$	5.00000	0.196875
$646$	0	0
$647$	4.00000	0.157256	0.0786281	−	0.996904i	$-0.474946\pi$
$647$	4.00000	0.157256	0.0786281	+	0.996904i	$0.474946\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	0	0
$655$	3.00000	0.117220
$656$	0	0
$657$	16.0000	0.624219
$658$	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
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	0	0
$663$	−1.00000	−0.0388368
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	36.0000	1.39393
$668$	0	0
$669$	3.00000	0.115987
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	0	0
$675$	−20.0000	−0.769800
$676$	0	0
$677$	−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
$679$	2.00000	0.0767530
$680$	0	0
$681$	24.0000	0.919682
$682$	0	0
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	0	0
$685$	14.0000	0.534913
$686$	0	0
$687$	−21.0000	−0.801200
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	−13.0000	−0.493118
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	7.00000	0.264764
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	11.0000	0.414284
$706$	0	0
$707$	8.00000	0.300871
$708$	0	0
$709$	−42.0000	−1.57734	−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000	−1.57734	−0.788672	+	0.614815i	$0.789231\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	0	0
$713$	−48.0000	−1.79761
$714$	0	0
$715$	2.00000	0.0747958
$716$	0	0
$717$	−11.0000	−0.410803
$718$	0	0
$719$	−42.0000	−1.56634	−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000	−1.56634	−0.783168	+	0.621810i	$0.786397\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	0	0
$723$	20.0000	0.743808
$724$	0	0
$725$	−24.0000	−0.891338
$726$	0	0
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	5.00000	0.184932
$732$	0	0
$733$	1.00000	0.0369358	0.0184679	−	0.999829i	$-0.494121\pi$
$733$	1.00000	0.0369358	0.0184679	+	0.999829i	$0.494121\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	3.00000	0.110059	0.0550297	−	0.998485i	$-0.482475\pi$
$743$	3.00000	0.110059	0.0550297	+	0.998485i	$0.482475\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	−5.00000	−0.181969
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	0	0
$759$	12.0000	0.435572
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	1.00000	0.0362024
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	15.0000	0.540212
$772$	0	0
$773$	−33.0000	−1.18693	−0.593464	−	0.804861i	$-0.702240\pi$
$773$	−33.0000	−1.18693	−0.593464	+	0.804861i	$0.702240\pi$
$774$	0	0
$775$	32.0000	1.14947
$776$	0	0
$777$	−1.00000	−0.0358748
$778$	0	0
$779$	−32.0000	−1.14652
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	30.0000	1.07211
$784$	0	0
$785$	4.00000	0.142766
$786$	0	0
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	0	0
$789$	16.0000	0.569615
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	−6.00000	−0.212798
$796$	0	0
$797$	−36.0000	−1.27519	−0.637593	−	0.770374i	$-0.720070\pi$
$797$	−36.0000	−1.27519	−0.637593	+	0.770374i	$0.720070\pi$
$798$	0	0
$799$	11.0000	0.389152
$800$	0	0
$801$	32.0000	1.13066
$802$	0	0
$803$	16.0000	0.564628
$804$	0	0
$805$	6.00000	0.211472
$806$	0	0
$807$	4.00000	0.140807
$808$	0	0
$809$	3.00000	0.105474	0.0527372	−	0.998608i	$-0.483205\pi$
$809$	3.00000	0.105474	0.0527372	+	0.998608i	$0.483205\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	−3.00000	−0.105215
$814$	0	0
$815$	−6.00000	−0.210171
$816$	0	0
$817$	40.0000	1.39942
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	−45.0000	−1.57051	−0.785255	−	0.619172i	$-0.787468\pi$
$821$	−45.0000	−1.57051	−0.785255	+	0.619172i	$0.787468\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	0	0
$825$	−8.00000	−0.278524
$826$	0	0
$827$	−52.0000	−1.80822	−0.904109	−	0.427303i	$-0.859464\pi$
$827$	−52.0000	−1.80822	−0.904109	+	0.427303i	$0.859464\pi$
$828$	0	0
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	−24.0000	−0.832551
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−16.0000	−0.553703
$836$	0	0
$837$	−40.0000	−1.38260
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	8.00000	0.275535
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	32.0000	1.09824
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	33.0000	1.12990	0.564949	−	0.825126i	$-0.308896\pi$
$853$	33.0000	1.12990	0.564949	+	0.825126i	$0.308896\pi$
$854$	0	0
$855$	16.0000	0.547188
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	0	0
$863$	39.0000	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	16.0000	0.543388
$868$	0	0
$869$	20.0000	0.678454
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	0	0
$873$	4.00000	0.135379
$874$	0	0
$875$	−9.00000	−0.304256
$876$	0	0
$877$	−7.00000	−0.236373	−0.118187	−	0.992991i	$-0.537708\pi$
$877$	−7.00000	−0.236373	−0.118187	+	0.992991i	$0.537708\pi$
$878$	0	0
$879$	9.00000	0.303562
$880$	0	0
$881$	−19.0000	−0.640126	−0.320063	−	0.947396i	$-0.603704\pi$
$881$	−19.0000	−0.640126	−0.320063	+	0.947396i	$0.603704\pi$
$882$	0	0
$883$	31.0000	1.04323	0.521617	−	0.853180i	$-0.325329\pi$
$883$	31.0000	1.04323	0.521617	+	0.853180i	$0.325329\pi$
$884$	0	0
$885$	−8.00000	−0.268917
$886$	0	0
$887$	−40.0000	−1.34307	−0.671534	−	0.740973i	$-0.734364\pi$
$887$	−40.0000	−1.34307	−0.671534	+	0.740973i	$0.734364\pi$
$888$	0	0
$889$	−22.0000	−0.737856
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	88.0000	2.94481
$894$	0	0
$895$	−13.0000	−0.434542
$896$	0	0
$897$	−6.00000	−0.200334
$898$	0	0
$899$	−48.0000	−1.60089
$900$	0	0
$901$	−6.00000	−0.199889
$902$	0	0
$903$	5.00000	0.166390
$904$	0	0
$905$	−22.0000	−0.731305
$906$	0	0
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	0	0
$909$	16.0000	0.530687
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	4.00000	0.132236
$916$	0	0
$917$	3.00000	0.0990687
$918$	0	0
$919$	36.0000	1.18753	0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000	1.18753	0.593765	+	0.804638i	$0.297641\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	0	0
$923$	−3.00000	−0.0987462
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	12.0000	0.394132
$928$	0	0
$929$	48.0000	1.57483	0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000	1.57483	0.787414	+	0.616424i	$0.211419\pi$
$930$	0	0
$931$	−48.0000	−1.57314
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	2.00000	0.0654070
$936$	0	0
$937$	46.0000	1.50275	0.751377	−	0.659873i	$-0.229390\pi$
$937$	46.0000	1.50275	0.751377	+	0.659873i	$0.229390\pi$
$938$	0	0
$939$	−11.0000	−0.358971
$940$	0	0
$941$	−27.0000	−0.880175	−0.440087	−	0.897955i	$-0.645053\pi$
$941$	−27.0000	−0.880175	−0.440087	+	0.897955i	$0.645053\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	5.00000	0.162650
$946$	0	0
$947$	−42.0000	−1.36482	−0.682408	−	0.730971i	$-0.739067\pi$
$947$	−42.0000	−1.36482	−0.682408	+	0.730971i	$0.739067\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−57.0000	−1.84641	−0.923206	−	0.384307i	$-0.874441\pi$
$953$	−57.0000	−1.84641	−0.923206	+	0.384307i	$0.874441\pi$
$954$	0	0
$955$	18.0000	0.582466
$956$	0	0
$957$	12.0000	0.387905
$958$	0	0
$959$	14.0000	0.452084
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	8.00000	0.257796
$964$	0	0
$965$	−6.00000	−0.193147
$966$	0	0
$967$	−59.0000	−1.89731	−0.948656	−	0.316310i	$-0.897556\pi$
$967$	−59.0000	−1.89731	−0.948656	+	0.316310i	$0.897556\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	0	0
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	0	0
$973$	−13.0000	−0.416761
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−10.0000	−0.319928	−0.159964	−	0.987123i	$-0.551138\pi$
$977$	−10.0000	−0.319928	−0.159964	+	0.987123i	$0.551138\pi$
$978$	0	0
$979$	32.0000	1.02272
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	23.0000	0.733586	0.366793	−	0.930303i	$-0.380456\pi$
$983$	23.0000	0.733586	0.366793	+	0.930303i	$0.380456\pi$
$984$	0	0
$985$	23.0000	0.732841
$986$	0	0
$987$	11.0000	0.350134
$988$	0	0
$989$	30.0000	0.953945
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	26.0000	0.825085
$994$	0	0
$995$	2.00000	0.0634043
$996$	0	0
$997$	50.0000	1.58352	0.791758	−	0.610835i	$-0.209166\pi$
$997$	50.0000	1.58352	0.791758	+	0.610835i	$0.209166\pi$
$998$	0	0
$999$	−5.00000	−0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.d.1.1		1
4.3	odd	2		3328.2.a.h.1.1		1
8.3	odd	2		3328.2.a.e.1.1		1
8.5	even	2		3328.2.a.i.1.1		1
16.3	odd	4		1664.2.b.b.833.2	yes	2
16.5	even	4		1664.2.b.c.833.2	yes	2
16.11	odd	4		1664.2.b.b.833.1	✓	2
16.13	even	4		1664.2.b.c.833.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.b.833.1	✓	2	16.11	odd	4
1664.2.b.b.833.2	yes	2	16.3	odd	4
1664.2.b.c.833.1	yes	2	16.13	even	4
1664.2.b.c.833.2	yes	2	16.5	even	4
3328.2.a.d.1.1		1	1.1	even	1	trivial
3328.2.a.e.1.1		1	8.3	odd	2
3328.2.a.h.1.1		1	4.3	odd	2
3328.2.a.i.1.1		1	8.5	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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

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