LMFDB - Embedded newform 6664.2.a.l.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6664,2,Mod(1,6664)] 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("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6664 = 2^{3} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6664.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$53.2123079070$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	6664.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.86081 q^{3} +0.462598 q^{5} +5.18421 q^{9} -2.00000 q^{11} -6.64681 q^{13} +1.32340 q^{15} +1.00000 q^{17} -4.64681 q^{19} -5.72161 q^{23} -4.78600 q^{25} +6.24860 q^{27} +0.796415 q^{29} -0.184210 q^{31} -5.72161 q^{33} -7.44322 q^{37} -19.0152 q^{39} +6.71120 q^{41} +5.50761 q^{43} +2.39821 q^{45} +5.72161 q^{47} +2.86081 q^{51} -4.33382 q^{53} -0.925197 q^{55} -13.2936 q^{57} -0.646809 q^{59} -4.86081 q^{61} -3.07480 q^{65} -15.7562 q^{67} -16.3684 q^{69} -0.556777 q^{71} +4.73202 q^{73} -13.6918 q^{75} -13.1648 q^{79} +2.32340 q^{81} +12.2188 q^{83} +0.462598 q^{85} +2.27839 q^{87} +6.36842 q^{89} -0.526989 q^{93} -2.14961 q^{95} -14.6814 q^{97} -10.3684 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - q^{5} + 2 q^{9} - 6 q^{11} - 4 q^{13} - 4 q^{15} + 3 q^{17} + 2 q^{19} - 6 q^{23} - 4 q^{25} + 6 q^{27} - 4 q^{29} + 13 q^{31} - 6 q^{33} - 14 q^{39} + 5 q^{41} - 5 q^{43} + 4 q^{45} + 6 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - q^5 + 2 * q^9 - 6 * q^11 - 4 * q^13 - 4 * q^15 + 3 * q^17 + 2 * q^19 - 6 * q^23 - 4 * q^25 + 6 * q^27 - 4 * q^29 + 13 * q^31 - 6 * q^33 - 14 * q^39 + 5 * q^41 - 5 * q^43 + 4 * q^45 + 6 * q^47 + 3 * q^51 - 9 * q^53 + 2 * q^55 - 8 * q^57 + 14 * q^59 - 9 * q^61 - 14 * q^65 - 13 * q^67 - 22 * q^69 - 24 * q^71 + 7 * q^73 - 6 * q^75 - 6 * q^79 - q^81 - q^85 + 18 * q^87 - 8 * q^89 - 16 * q^95 - 5 * q^97 - 4 * q^99

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
$3$	2.86081	1.65169	0.825844	−	0.563899i	$-0.190699\pi$
$3$	2.86081	1.65169	0.825844	+	0.563899i	$0.190699\pi$
$4$	0	0
$5$	0.462598	0.206880	0.103440	−	0.994636i	$-0.467015\pi$
$5$	0.462598	0.206880	0.103440	+	0.994636i	$0.467015\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.18421	1.72807
$10$	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
$13$	−6.64681	−1.84349	−0.921747	−	0.387793i	$-0.873238\pi$
$13$	−6.64681	−1.84349	−0.921747	+	0.387793i	$0.873238\pi$
$14$	0	0
$15$	1.32340	0.341702
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−4.64681	−1.06605	−0.533025	−	0.846099i	$-0.678945\pi$
$19$	−4.64681	−1.06605	−0.533025	+	0.846099i	$0.678945\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.72161	−1.19304	−0.596519	−	0.802599i	$-0.703450\pi$
$23$	−5.72161	−1.19304	−0.596519	+	0.802599i	$0.703450\pi$
$24$	0	0
$25$	−4.78600	−0.957201
$26$	0	0
$27$	6.24860	1.20254
$28$	0	0
$29$	0.796415	0.147891	0.0739453	−	0.997262i	$-0.476441\pi$
$29$	0.796415	0.147891	0.0739453	+	0.997262i	$0.476441\pi$
$30$	0	0
$31$	−0.184210	−0.0330851	−0.0165426	−	0.999863i	$-0.505266\pi$
$31$	−0.184210	−0.0330851	−0.0165426	+	0.999863i	$0.505266\pi$
$32$	0	0
$33$	−5.72161	−0.996005
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.44322	−1.22366	−0.611829	−	0.790990i	$-0.709566\pi$
$37$	−7.44322	−1.22366	−0.611829	+	0.790990i	$0.709566\pi$
$38$	0	0
$39$	−19.0152	−3.04487
$40$	0	0
$41$	6.71120	1.04811	0.524057	−	0.851683i	$-0.324418\pi$
$41$	6.71120	1.04811	0.524057	+	0.851683i	$0.324418\pi$
$42$	0	0
$43$	5.50761	0.839903	0.419952	−	0.907546i	$-0.362047\pi$
$43$	5.50761	0.839903	0.419952	+	0.907546i	$0.362047\pi$
$44$	0	0
$45$	2.39821	0.357504
$46$	0	0
$47$	5.72161	0.834583	0.417291	−	0.908773i	$-0.362979\pi$
$47$	5.72161	0.834583	0.417291	+	0.908773i	$0.362979\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.86081	0.400593
$52$	0	0
$53$	−4.33382	−0.595295	−0.297648	−	0.954676i	$-0.596202\pi$
$53$	−4.33382	−0.595295	−0.297648	+	0.954676i	$0.596202\pi$
$54$	0	0
$55$	−0.925197	−0.124754
$56$	0	0
$57$	−13.2936	−1.76078
$58$	0	0
$59$	−0.646809	−0.0842073	−0.0421037	−	0.999113i	$-0.513406\pi$
$59$	−0.646809	−0.0842073	−0.0421037	+	0.999113i	$0.513406\pi$
$60$	0	0
$61$	−4.86081	−0.622362	−0.311181	−	0.950351i	$-0.600725\pi$
$61$	−4.86081	−0.622362	−0.311181	+	0.950351i	$0.600725\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.07480	−0.381382
$66$	0	0
$67$	−15.7562	−1.92493	−0.962464	−	0.271409i	$-0.912510\pi$
$67$	−15.7562	−1.92493	−0.962464	+	0.271409i	$0.912510\pi$
$68$	0	0
$69$	−16.3684	−1.97053
$70$	0	0
$71$	−0.556777	−0.0660772	−0.0330386	−	0.999454i	$-0.510518\pi$
$71$	−0.556777	−0.0660772	−0.0330386	+	0.999454i	$0.510518\pi$
$72$	0	0
$73$	4.73202	0.553842	0.276921	−	0.960893i	$-0.410686\pi$
$73$	4.73202	0.553842	0.276921	+	0.960893i	$0.410686\pi$
$74$	0	0
$75$	−13.6918	−1.58100
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.1648	−1.48116	−0.740580	−	0.671968i	$-0.765449\pi$
$79$	−13.1648	−1.48116	−0.740580	+	0.671968i	$0.765449\pi$
$80$	0	0
$81$	2.32340	0.258156
$82$	0	0
$83$	12.2188	1.34119	0.670595	−	0.741824i	$-0.266039\pi$
$83$	12.2188	1.34119	0.670595	+	0.741824i	$0.266039\pi$
$84$	0	0
$85$	0.462598	0.0501758
$86$	0	0
$87$	2.27839	0.244269
$88$	0	0
$89$	6.36842	0.675051	0.337526	−	0.941316i	$-0.390410\pi$
$89$	6.36842	0.675051	0.337526	+	0.941316i	$0.390410\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.526989	−0.0546463
$94$	0	0
$95$	−2.14961	−0.220545
$96$	0	0
$97$	−14.6814	−1.49067	−0.745336	−	0.666689i	$-0.767711\pi$
$97$	−14.6814	−1.49067	−0.745336	+	0.666689i	$0.767711\pi$
$98$	0	0
$99$	−10.3684	−1.04207
$100$	0	0
$101$	1.59283	0.158492	0.0792462	−	0.996855i	$-0.474749\pi$
$101$	1.59283	0.158492	0.0792462	+	0.996855i	$0.474749\pi$
$102$	0	0
$103$	3.57201	0.351960	0.175980	−	0.984394i	$-0.443691\pi$
$103$	3.57201	0.351960	0.175980	+	0.984394i	$0.443691\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.2936	1.09180	0.545898	−	0.837852i	$-0.316189\pi$
$107$	11.2936	1.09180	0.545898	+	0.837852i	$0.316189\pi$
$108$	0	0
$109$	10.6468	1.01978	0.509890	−	0.860240i	$-0.329686\pi$
$109$	10.6468	1.01978	0.509890	+	0.860240i	$0.329686\pi$
$110$	0	0
$111$	−21.2936	−2.02110
$112$	0	0
$113$	−10.9252	−1.02776	−0.513878	−	0.857863i	$-0.671792\pi$
$113$	−10.9252	−1.02776	−0.513878	+	0.857863i	$0.671792\pi$
$114$	0	0
$115$	−2.64681	−0.246816
$116$	0	0
$117$	−34.4585	−3.18569
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	19.1994	1.73116
$124$	0	0
$125$	−4.52699	−0.404906
$126$	0	0
$127$	−18.8310	−1.67098	−0.835491	−	0.549504i	$-0.814817\pi$
$127$	−18.8310	−1.67098	−0.835491	+	0.549504i	$0.814817\pi$
$128$	0	0
$129$	15.7562	1.38726
$130$	0	0
$131$	−17.0361	−1.48845	−0.744223	−	0.667931i	$-0.767180\pi$
$131$	−17.0361	−1.48845	−0.744223	+	0.667931i	$0.767180\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.89059	0.248783
$136$	0	0
$137$	3.80683	0.325239	0.162620	−	0.986689i	$-0.448006\pi$
$137$	3.80683	0.325239	0.162620	+	0.986689i	$0.448006\pi$
$138$	0	0
$139$	5.10941	0.433374	0.216687	−	0.976241i	$-0.430475\pi$
$139$	5.10941	0.433374	0.216687	+	0.976241i	$0.430475\pi$
$140$	0	0
$141$	16.3684	1.37847
$142$	0	0
$143$	13.2936	1.11167
$144$	0	0
$145$	0.368420	0.0305956
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.6724	−1.69355	−0.846777	−	0.531949i	$-0.821460\pi$
$149$	−20.6724	−1.69355	−0.846777	+	0.531949i	$0.821460\pi$
$150$	0	0
$151$	11.3580	0.924302	0.462151	−	0.886801i	$-0.347078\pi$
$151$	11.3580	0.924302	0.462151	+	0.886801i	$0.347078\pi$
$152$	0	0
$153$	5.18421	0.419119
$154$	0	0
$155$	−0.0852153	−0.00684466
$156$	0	0
$157$	16.7368	1.33575	0.667873	−	0.744276i	$-0.267205\pi$
$157$	16.7368	1.33575	0.667873	+	0.744276i	$0.267205\pi$
$158$	0	0
$159$	−12.3982	−0.983242
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.6468	0.990574	0.495287	−	0.868729i	$-0.335063\pi$
$163$	12.6468	0.990574	0.495287	+	0.868729i	$0.335063\pi$
$164$	0	0
$165$	−2.64681	−0.206054
$166$	0	0
$167$	−16.0256	−1.24010	−0.620051	−	0.784562i	$-0.712888\pi$
$167$	−16.0256	−1.24010	−0.620051	+	0.784562i	$0.712888\pi$
$168$	0	0
$169$	31.1801	2.39847
$170$	0	0
$171$	−24.0900	−1.84221
$172$	0	0
$173$	−1.41758	−0.107777	−0.0538884	−	0.998547i	$-0.517162\pi$
$173$	−1.41758	−0.107777	−0.0538884	+	0.998547i	$0.517162\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.85039	−0.139084
$178$	0	0
$179$	19.2292	1.43726	0.718630	−	0.695393i	$-0.244770\pi$
$179$	19.2292	1.43726	0.718630	+	0.695393i	$0.244770\pi$
$180$	0	0
$181$	16.5872	1.23292	0.616460	−	0.787386i	$-0.288566\pi$
$181$	16.5872	1.23292	0.616460	+	0.787386i	$0.288566\pi$
$182$	0	0
$183$	−13.9058	−1.02795
$184$	0	0
$185$	−3.44322	−0.253151
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.2742	1.61171	0.805854	−	0.592115i	$-0.201707\pi$
$191$	22.2742	1.61171	0.805854	+	0.592115i	$0.201707\pi$
$192$	0	0
$193$	−3.59283	−0.258618	−0.129309	−	0.991604i	$-0.541276\pi$
$193$	−3.59283	−0.258618	−0.129309	+	0.991604i	$0.541276\pi$
$194$	0	0
$195$	−8.79641	−0.629924
$196$	0	0
$197$	−4.55678	−0.324657	−0.162328	−	0.986737i	$-0.551900\pi$
$197$	−4.55678	−0.324657	−0.162328	+	0.986737i	$0.551900\pi$
$198$	0	0
$199$	−14.1752	−1.00486	−0.502428	−	0.864619i	$-0.667560\pi$
$199$	−14.1752	−1.00486	−0.502428	+	0.864619i	$0.667560\pi$
$200$	0	0
$201$	−45.0755	−3.17938
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.10459	0.216834
$206$	0	0
$207$	−29.6620	−2.06165
$208$	0	0
$209$	9.29362	0.642853
$210$	0	0
$211$	−14.6260	−1.00689	−0.503447	−	0.864026i	$-0.667935\pi$
$211$	−14.6260	−1.00689	−0.503447	+	0.864026i	$0.667935\pi$
$212$	0	0
$213$	−1.59283	−0.109139
$214$	0	0
$215$	2.54781	0.173759
$216$	0	0
$217$	0	0
$218$	0	0
$219$	13.5374	0.914773
$220$	0	0
$221$	−6.64681	−0.447113
$222$	0	0
$223$	−2.40717	−0.161196	−0.0805980	−	0.996747i	$-0.525683\pi$
$223$	−2.40717	−0.161196	−0.0805980	+	0.996747i	$0.525683\pi$
$224$	0	0
$225$	−24.8116	−1.65411
$226$	0	0
$227$	29.2895	1.94401	0.972005	−	0.234959i	$-0.0754957\pi$
$227$	29.2895	1.94401	0.972005	+	0.234959i	$0.0754957\pi$
$228$	0	0
$229$	−3.50280	−0.231471	−0.115736	−	0.993280i	$-0.536923\pi$
$229$	−3.50280	−0.231471	−0.115736	+	0.993280i	$0.536923\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.1496	−0.795947	−0.397974	−	0.917397i	$-0.630286\pi$
$233$	−12.1496	−0.795947	−0.397974	+	0.917397i	$0.630286\pi$
$234$	0	0
$235$	2.64681	0.172659
$236$	0	0
$237$	−37.6620	−2.44641
$238$	0	0
$239$	−3.52844	−0.228236	−0.114118	−	0.993467i	$-0.536404\pi$
$239$	−3.52844	−0.228236	−0.114118	+	0.993467i	$0.536404\pi$
$240$	0	0
$241$	7.90582	0.509259	0.254629	−	0.967039i	$-0.418047\pi$
$241$	7.90582	0.509259	0.254629	+	0.967039i	$0.418047\pi$
$242$	0	0
$243$	−12.0990	−0.776151
$244$	0	0
$245$	0	0
$246$	0	0
$247$	30.8864	1.96526
$248$	0	0
$249$	34.9557	2.21522
$250$	0	0
$251$	10.2396	0.646320	0.323160	−	0.946344i	$-0.395255\pi$
$251$	10.2396	0.646320	0.323160	+	0.946344i	$0.395255\pi$
$252$	0	0
$253$	11.4432	0.719429
$254$	0	0
$255$	1.32340	0.0828748
$256$	0	0
$257$	23.1648	1.44498	0.722491	−	0.691380i	$-0.242997\pi$
$257$	23.1648	1.44498	0.722491	+	0.691380i	$0.242997\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	4.12878	0.255565
$262$	0	0
$263$	−29.2936	−1.80632	−0.903161	−	0.429302i	$-0.858760\pi$
$263$	−29.2936	−1.80632	−0.903161	+	0.429302i	$0.858760\pi$
$264$	0	0
$265$	−2.00482	−0.123155
$266$	0	0
$267$	18.2188	1.11497
$268$	0	0
$269$	15.2936	0.932468	0.466234	−	0.884661i	$-0.345610\pi$
$269$	15.2936	0.932468	0.466234	+	0.884661i	$0.345610\pi$
$270$	0	0
$271$	−21.1648	−1.28567	−0.642836	−	0.766004i	$-0.722242\pi$
$271$	−21.1648	−1.28567	−0.642836	+	0.766004i	$0.722242\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.57201	0.577214
$276$	0	0
$277$	19.5720	1.17597	0.587984	−	0.808873i	$-0.299922\pi$
$277$	19.5720	1.17597	0.587984	+	0.808873i	$0.299922\pi$
$278$	0	0
$279$	−0.954984	−0.0571734
$280$	0	0
$281$	−26.5526	−1.58400	−0.791999	−	0.610523i	$-0.790959\pi$
$281$	−26.5526	−1.58400	−0.791999	+	0.610523i	$0.790959\pi$
$282$	0	0
$283$	24.1842	1.43760	0.718801	−	0.695216i	$-0.244691\pi$
$283$	24.1842	1.43760	0.718801	+	0.695216i	$0.244691\pi$
$284$	0	0
$285$	−6.14961	−0.364271
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−42.0007	−2.46212
$292$	0	0
$293$	18.7756	1.09688	0.548441	−	0.836189i	$-0.315222\pi$
$293$	18.7756	1.09688	0.548441	+	0.836189i	$0.315222\pi$
$294$	0	0
$295$	−0.299213	−0.0174208
$296$	0	0
$297$	−12.4972	−0.725161
$298$	0	0
$299$	38.0305	2.19936
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4.55678	0.261780
$304$	0	0
$305$	−2.24860	−0.128755
$306$	0	0
$307$	−0.835165	−0.0476654	−0.0238327	−	0.999716i	$-0.507587\pi$
$307$	−0.835165	−0.0476654	−0.0238327	+	0.999716i	$0.507587\pi$
$308$	0	0
$309$	10.2188	0.581328
$310$	0	0
$311$	−8.55263	−0.484975	−0.242488	−	0.970155i	$-0.577963\pi$
$311$	−8.55263	−0.484975	−0.242488	+	0.970155i	$0.577963\pi$
$312$	0	0
$313$	10.0554	0.568366	0.284183	−	0.958770i	$-0.408278\pi$
$313$	10.0554	0.568366	0.284183	+	0.958770i	$0.408278\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.1053	0.960727	0.480364	−	0.877069i	$-0.340505\pi$
$317$	17.1053	0.960727	0.480364	+	0.877069i	$0.340505\pi$
$318$	0	0
$319$	−1.59283	−0.0891813
$320$	0	0
$321$	32.3088	1.80330
$322$	0	0
$323$	−4.64681	−0.258555
$324$	0	0
$325$	31.8116	1.76459
$326$	0	0
$327$	30.4585	1.68436
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6.46260	0.355217	0.177608	−	0.984101i	$-0.443164\pi$
$331$	6.46260	0.355217	0.177608	+	0.984101i	$0.443164\pi$
$332$	0	0
$333$	−38.5872	−2.11457
$334$	0	0
$335$	−7.28880	−0.398230
$336$	0	0
$337$	−1.50280	−0.0818626	−0.0409313	−	0.999162i	$-0.513032\pi$
$337$	−1.50280	−0.0818626	−0.0409313	+	0.999162i	$0.513032\pi$
$338$	0	0
$339$	−31.2549	−1.69753
$340$	0	0
$341$	0.368420	0.0199511
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−7.57201	−0.407663
$346$	0	0
$347$	1.14401	0.0614137	0.0307069	−	0.999528i	$-0.490224\pi$
$347$	1.14401	0.0614137	0.0307069	+	0.999528i	$0.490224\pi$
$348$	0	0
$349$	−15.0748	−0.806936	−0.403468	−	0.914994i	$-0.632195\pi$
$349$	−15.0748	−0.806936	−0.403468	+	0.914994i	$0.632195\pi$
$350$	0	0
$351$	−41.5333	−2.21688
$352$	0	0
$353$	−2.73684	−0.145667	−0.0728337	−	0.997344i	$-0.523204\pi$
$353$	−2.73684	−0.145667	−0.0728337	+	0.997344i	$0.523204\pi$
$354$	0	0
$355$	−0.257564	−0.0136701
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−27.8254	−1.46857	−0.734285	−	0.678842i	$-0.762482\pi$
$359$	−27.8254	−1.46857	−0.734285	+	0.678842i	$0.762482\pi$
$360$	0	0
$361$	2.59283	0.136465
$362$	0	0
$363$	−20.0256	−1.05107
$364$	0	0
$365$	2.18903	0.114579
$366$	0	0
$367$	−8.02564	−0.418935	−0.209468	−	0.977816i	$-0.567173\pi$
$367$	−8.02564	−0.418935	−0.209468	+	0.977816i	$0.567173\pi$
$368$	0	0
$369$	34.7923	1.81121
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−26.0948	−1.35114	−0.675570	−	0.737296i	$-0.736102\pi$
$373$	−26.0948	−1.35114	−0.675570	+	0.737296i	$0.736102\pi$
$374$	0	0
$375$	−12.9508	−0.668778
$376$	0	0
$377$	−5.29362	−0.272635
$378$	0	0
$379$	−22.4376	−1.15254	−0.576272	−	0.817258i	$-0.695493\pi$
$379$	−22.4376	−1.15254	−0.576272	+	0.817258i	$0.695493\pi$
$380$	0	0
$381$	−53.8719	−2.75994
$382$	0	0
$383$	13.7908	0.704678	0.352339	−	0.935872i	$-0.385386\pi$
$383$	13.7908	0.704678	0.352339	+	0.935872i	$0.385386\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	28.5526	1.45141
$388$	0	0
$389$	33.6966	1.70849	0.854244	−	0.519873i	$-0.174021\pi$
$389$	33.6966	1.70849	0.854244	+	0.519873i	$0.174021\pi$
$390$	0	0
$391$	−5.72161	−0.289354
$392$	0	0
$393$	−48.7368	−2.45845
$394$	0	0
$395$	−6.09003	−0.306423
$396$	0	0
$397$	−18.9806	−0.952610	−0.476305	−	0.879280i	$-0.658024\pi$
$397$	−18.9806	−0.952610	−0.476305	+	0.879280i	$0.658024\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	0	0
$403$	1.22441	0.0609922
$404$	0	0
$405$	1.07480	0.0534074
$406$	0	0
$407$	14.8864	0.737894
$408$	0	0
$409$	−20.5180	−1.01455	−0.507276	−	0.861784i	$-0.669347\pi$
$409$	−20.5180	−1.01455	−0.507276	+	0.861784i	$0.669347\pi$
$410$	0	0
$411$	10.8906	0.537193
$412$	0	0
$413$	0	0
$414$	0	0
$415$	5.65240	0.277466
$416$	0	0
$417$	14.6170	0.715799
$418$	0	0
$419$	7.92665	0.387242	0.193621	−	0.981076i	$-0.437977\pi$
$419$	7.92665	0.387242	0.193621	+	0.981076i	$0.437977\pi$
$420$	0	0
$421$	23.3788	1.13941	0.569707	−	0.821848i	$-0.307057\pi$
$421$	23.3788	1.13941	0.569707	+	0.821848i	$0.307057\pi$
$422$	0	0
$423$	29.6620	1.44222
$424$	0	0
$425$	−4.78600	−0.232155
$426$	0	0
$427$	0	0
$428$	0	0
$429$	38.0305	1.83613
$430$	0	0
$431$	19.8116	0.954293	0.477147	−	0.878824i	$-0.341671\pi$
$431$	19.8116	0.954293	0.477147	+	0.878824i	$0.341671\pi$
$432$	0	0
$433$	−39.0457	−1.87642	−0.938208	−	0.346072i	$-0.887515\pi$
$433$	−39.0457	−1.87642	−0.938208	+	0.346072i	$0.887515\pi$
$434$	0	0
$435$	1.05398	0.0505344
$436$	0	0
$437$	26.5872	1.27184
$438$	0	0
$439$	−23.3401	−1.11396	−0.556981	−	0.830525i	$-0.688040\pi$
$439$	−23.3401	−1.11396	−0.556981	+	0.830525i	$0.688040\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.85039	0.468006	0.234003	−	0.972236i	$-0.424817\pi$
$443$	9.85039	0.468006	0.234003	+	0.972236i	$0.424817\pi$
$444$	0	0
$445$	2.94602	0.139655
$446$	0	0
$447$	−59.1399	−2.79722
$448$	0	0
$449$	−13.5028	−0.637236	−0.318618	−	0.947883i	$-0.603219\pi$
$449$	−13.5028	−0.637236	−0.318618	+	0.947883i	$0.603219\pi$
$450$	0	0
$451$	−13.4224	−0.632036
$452$	0	0
$453$	32.4931	1.52666
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9.21736	−0.431170	−0.215585	−	0.976485i	$-0.569166\pi$
$457$	−9.21736	−0.431170	−0.215585	+	0.976485i	$0.569166\pi$
$458$	0	0
$459$	6.24860	0.291660
$460$	0	0
$461$	−7.81164	−0.363825	−0.181912	−	0.983315i	$-0.558229\pi$
$461$	−7.81164	−0.363825	−0.181912	+	0.983315i	$0.558229\pi$
$462$	0	0
$463$	−27.7860	−1.29133	−0.645663	−	0.763623i	$-0.723419\pi$
$463$	−27.7860	−1.29133	−0.645663	+	0.763623i	$0.723419\pi$
$464$	0	0
$465$	−0.243784	−0.0113052
$466$	0	0
$467$	−8.45845	−0.391410	−0.195705	−	0.980663i	$-0.562700\pi$
$467$	−8.45845	−0.391410	−0.195705	+	0.980663i	$0.562700\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	47.8809	2.20623
$472$	0	0
$473$	−11.0152	−0.506481
$474$	0	0
$475$	22.2396	1.02042
$476$	0	0
$477$	−22.4674	−1.02871
$478$	0	0
$479$	−25.1905	−1.15098	−0.575491	−	0.817808i	$-0.695189\pi$
$479$	−25.1905	−1.15098	−0.575491	+	0.817808i	$0.695189\pi$
$480$	0	0
$481$	49.4737	2.25581
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.79160	−0.308391
$486$	0	0
$487$	11.8116	0.535237	0.267618	−	0.963525i	$-0.413763\pi$
$487$	11.8116	0.535237	0.267618	+	0.963525i	$0.413763\pi$
$488$	0	0
$489$	36.1801	1.63612
$490$	0	0
$491$	−4.01378	−0.181139	−0.0905697	−	0.995890i	$-0.528869\pi$
$491$	−4.01378	−0.181139	−0.0905697	+	0.995890i	$0.528869\pi$
$492$	0	0
$493$	0.796415	0.0358687
$494$	0	0
$495$	−4.79641	−0.215583
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12.2784	0.549656	0.274828	−	0.961493i	$-0.411379\pi$
$499$	12.2784	0.549656	0.274828	+	0.961493i	$0.411379\pi$
$500$	0	0
$501$	−45.8462	−2.04826
$502$	0	0
$503$	22.7022	1.01224	0.506121	−	0.862462i	$-0.331079\pi$
$503$	22.7022	1.01224	0.506121	+	0.862462i	$0.331079\pi$
$504$	0	0
$505$	0.736841	0.0327890
$506$	0	0
$507$	89.2001	3.96152
$508$	0	0
$509$	−32.9944	−1.46245	−0.731226	−	0.682136i	$-0.761051\pi$
$509$	−32.9944	−1.46245	−0.731226	+	0.682136i	$0.761051\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−29.0361	−1.28197
$514$	0	0
$515$	1.65240	0.0728136
$516$	0	0
$517$	−11.4432	−0.503272
$518$	0	0
$519$	−4.05543	−0.178014
$520$	0	0
$521$	−21.5976	−0.946210	−0.473105	−	0.881006i	$-0.656867\pi$
$521$	−21.5976	−0.946210	−0.473105	+	0.881006i	$0.656867\pi$
$522$	0	0
$523$	36.6981	1.60470	0.802348	−	0.596857i	$-0.203584\pi$
$523$	36.6981	1.60470	0.802348	+	0.596857i	$0.203584\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.184210	−0.00802432
$528$	0	0
$529$	9.73684	0.423341
$530$	0	0
$531$	−3.35319	−0.145516
$532$	0	0
$533$	−44.6081	−1.93219
$534$	0	0
$535$	5.22441	0.225871
$536$	0	0
$537$	55.0111	2.37390
$538$	0	0
$539$	0	0
$540$	0	0
$541$	16.3088	0.701172	0.350586	−	0.936531i	$-0.385982\pi$
$541$	16.3088	0.701172	0.350586	+	0.936531i	$0.385982\pi$
$542$	0	0
$543$	47.4529	2.03640
$544$	0	0
$545$	4.92520	0.210972
$546$	0	0
$547$	29.3836	1.25635	0.628177	−	0.778070i	$-0.283801\pi$
$547$	29.3836	1.25635	0.628177	+	0.778070i	$0.283801\pi$
$548$	0	0
$549$	−25.1994	−1.07549
$550$	0	0
$551$	−3.70079	−0.157659
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−9.85039	−0.418126
$556$	0	0
$557$	11.3353	0.480291	0.240145	−	0.970737i	$-0.422805\pi$
$557$	11.3353	0.480291	0.240145	+	0.970737i	$0.422805\pi$
$558$	0	0
$559$	−36.6081	−1.54836
$560$	0	0
$561$	−5.72161	−0.241567
$562$	0	0
$563$	0.448819	0.0189155	0.00945773	−	0.999955i	$-0.496989\pi$
$563$	0.448819	0.0189155	0.00945773	+	0.999955i	$0.496989\pi$
$564$	0	0
$565$	−5.05398	−0.212622
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17.4868	−0.733084	−0.366542	−	0.930401i	$-0.619458\pi$
$569$	−17.4868	−0.733084	−0.366542	+	0.930401i	$0.619458\pi$
$570$	0	0
$571$	20.2701	0.848277	0.424139	−	0.905597i	$-0.360577\pi$
$571$	20.2701	0.848277	0.424139	+	0.905597i	$0.360577\pi$
$572$	0	0
$573$	63.7223	2.66204
$574$	0	0
$575$	27.3836	1.14198
$576$	0	0
$577$	32.5277	1.35414	0.677072	−	0.735917i	$-0.263249\pi$
$577$	32.5277	1.35414	0.677072	+	0.735917i	$0.263249\pi$
$578$	0	0
$579$	−10.2784	−0.427155
$580$	0	0
$581$	0	0
$582$	0	0
$583$	8.66763	0.358977
$584$	0	0
$585$	−15.9404	−0.659055
$586$	0	0
$587$	−24.7577	−1.02186	−0.510929	−	0.859623i	$-0.670699\pi$
$587$	−24.7577	−1.02186	−0.510929	+	0.859623i	$0.670699\pi$
$588$	0	0
$589$	0.855989	0.0352704
$590$	0	0
$591$	−13.0361	−0.536232
$592$	0	0
$593$	29.6829	1.21893	0.609465	−	0.792813i	$-0.291384\pi$
$593$	29.6829	1.21893	0.609465	+	0.792813i	$0.291384\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−40.5526	−1.65971
$598$	0	0
$599$	22.3732	0.914146	0.457073	−	0.889429i	$-0.348898\pi$
$599$	22.3732	0.914146	0.457073	+	0.889429i	$0.348898\pi$
$600$	0	0
$601$	18.1801	0.741580	0.370790	−	0.928717i	$-0.379087\pi$
$601$	18.1801	0.741580	0.370790	+	0.928717i	$0.379087\pi$
$602$	0	0
$603$	−81.6835	−3.32641
$604$	0	0
$605$	−3.23819	−0.131651
$606$	0	0
$607$	−0.0733538	−0.00297734	−0.00148867	−	0.999999i	$-0.500474\pi$
$607$	−0.0733538	−0.00297734	−0.00148867	+	0.999999i	$0.500474\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−38.0305	−1.53855
$612$	0	0
$613$	30.2534	1.22192	0.610962	−	0.791660i	$-0.290783\pi$
$613$	30.2534	1.22192	0.610962	+	0.791660i	$0.290783\pi$
$614$	0	0
$615$	8.88163	0.358142
$616$	0	0
$617$	26.9944	1.08675	0.543377	−	0.839489i	$-0.317145\pi$
$617$	26.9944	1.08675	0.543377	+	0.839489i	$0.317145\pi$
$618$	0	0
$619$	28.2217	1.13433	0.567163	−	0.823605i	$-0.308041\pi$
$619$	28.2217	1.13433	0.567163	+	0.823605i	$0.308041\pi$
$620$	0	0
$621$	−35.7521	−1.43468
$622$	0	0
$623$	0	0
$624$	0	0
$625$	21.8358	0.873433
$626$	0	0
$627$	26.5872	1.06179
$628$	0	0
$629$	−7.44322	−0.296781
$630$	0	0
$631$	20.4238	0.813061	0.406530	−	0.913637i	$-0.366739\pi$
$631$	20.4238	0.813061	0.406530	+	0.913637i	$0.366739\pi$
$632$	0	0
$633$	−41.8421	−1.66307
$634$	0	0
$635$	−8.71120	−0.345693
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.88645	−0.114186
$640$	0	0
$641$	−12.2784	−0.484967	−0.242484	−	0.970156i	$-0.577962\pi$
$641$	−12.2784	−0.484967	−0.242484	+	0.970156i	$0.577962\pi$
$642$	0	0
$643$	34.4328	1.35790	0.678949	−	0.734186i	$-0.262436\pi$
$643$	34.4328	1.35790	0.678949	+	0.734186i	$0.262436\pi$
$644$	0	0
$645$	7.28880	0.286996
$646$	0	0
$647$	−17.0540	−0.670461	−0.335231	−	0.942136i	$-0.608814\pi$
$647$	−17.0540	−0.670461	−0.335231	+	0.942136i	$0.608814\pi$
$648$	0	0
$649$	1.29362	0.0507789
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−28.2396	−1.10510	−0.552551	−	0.833479i	$-0.686346\pi$
$653$	−28.2396	−1.10510	−0.552551	+	0.833479i	$0.686346\pi$
$654$	0	0
$655$	−7.88085	−0.307930
$656$	0	0
$657$	24.5318	0.957077
$658$	0	0
$659$	27.6870	1.07853	0.539266	−	0.842135i	$-0.318702\pi$
$659$	27.6870	1.07853	0.539266	+	0.842135i	$0.318702\pi$
$660$	0	0
$661$	−27.6412	−1.07512	−0.537559	−	0.843226i	$-0.680654\pi$
$661$	−27.6412	−1.07512	−0.537559	+	0.843226i	$0.680654\pi$
$662$	0	0
$663$	−19.0152	−0.738490
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.55678	−0.176439
$668$	0	0
$669$	−6.88645	−0.266245
$670$	0	0
$671$	9.72161	0.375299
$672$	0	0
$673$	−38.8656	−1.49816	−0.749080	−	0.662480i	$-0.769504\pi$
$673$	−38.8656	−1.49816	−0.749080	+	0.662480i	$0.769504\pi$
$674$	0	0
$675$	−29.9058	−1.15108
$676$	0	0
$677$	−43.2161	−1.66093	−0.830465	−	0.557071i	$-0.811925\pi$
$677$	−43.2161	−1.66093	−0.830465	+	0.557071i	$0.811925\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	83.7915	3.21090
$682$	0	0
$683$	−29.9404	−1.14564	−0.572819	−	0.819682i	$-0.694150\pi$
$683$	−29.9404	−1.14564	−0.572819	+	0.819682i	$0.694150\pi$
$684$	0	0
$685$	1.76103	0.0672856
$686$	0	0
$687$	−10.0208	−0.382318
$688$	0	0
$689$	28.8060	1.09742
$690$	0	0
$691$	17.8248	0.678086	0.339043	−	0.940771i	$-0.389897\pi$
$691$	17.8248	0.678086	0.339043	+	0.940771i	$0.389897\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.36360	0.0896566
$696$	0	0
$697$	6.71120	0.254205
$698$	0	0
$699$	−34.7577	−1.31466
$700$	0	0
$701$	−37.0665	−1.39998	−0.699991	−	0.714151i	$-0.746813\pi$
$701$	−37.0665	−1.39998	−0.699991	+	0.714151i	$0.746813\pi$
$702$	0	0
$703$	34.5872	1.30448
$704$	0	0
$705$	7.57201	0.285178
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−21.8325	−0.819936	−0.409968	−	0.912100i	$-0.634460\pi$
$709$	−21.8325	−0.819936	−0.409968	+	0.912100i	$0.634460\pi$
$710$	0	0
$711$	−68.2493	−2.55955
$712$	0	0
$713$	1.05398	0.0394718
$714$	0	0
$715$	6.14961	0.229982
$716$	0	0
$717$	−10.0942	−0.376974
$718$	0	0
$719$	−3.84558	−0.143416	−0.0717079	−	0.997426i	$-0.522845\pi$
$719$	−3.84558	−0.143416	−0.0717079	+	0.997426i	$0.522845\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	22.6170	0.841136
$724$	0	0
$725$	−3.81164	−0.141561
$726$	0	0
$727$	19.8116	0.734773	0.367387	−	0.930068i	$-0.380253\pi$
$727$	19.8116	0.734773	0.367387	+	0.930068i	$0.380253\pi$
$728$	0	0
$729$	−41.5831	−1.54011
$730$	0	0
$731$	5.50761	0.203706
$732$	0	0
$733$	−37.2853	−1.37716	−0.688582	−	0.725158i	$-0.741767\pi$
$733$	−37.2853	−1.37716	−0.688582	+	0.725158i	$0.741767\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	31.5124	1.16078
$738$	0	0
$739$	−11.1004	−0.408336	−0.204168	−	0.978936i	$-0.565449\pi$
$739$	−11.1004	−0.408336	−0.204168	+	0.978936i	$0.565449\pi$
$740$	0	0
$741$	88.3601	3.24599
$742$	0	0
$743$	25.1469	0.922551	0.461275	−	0.887257i	$-0.347392\pi$
$743$	25.1469	0.922551	0.461275	+	0.887257i	$0.347392\pi$
$744$	0	0
$745$	−9.56304	−0.350363
$746$	0	0
$747$	63.3449	2.31767
$748$	0	0
$749$	0	0
$750$	0	0
$751$	5.89204	0.215004	0.107502	−	0.994205i	$-0.465715\pi$
$751$	5.89204	0.215004	0.107502	+	0.994205i	$0.465715\pi$
$752$	0	0
$753$	29.2936	1.06752
$754$	0	0
$755$	5.25420	0.191220
$756$	0	0
$757$	−9.83728	−0.357542	−0.178771	−	0.983891i	$-0.557212\pi$
$757$	−9.83728	−0.357542	−0.178771	+	0.983891i	$0.557212\pi$
$758$	0	0
$759$	32.7368	1.18827
$760$	0	0
$761$	26.0513	0.944358	0.472179	−	0.881503i	$-0.343468\pi$
$761$	26.0513	0.944358	0.472179	+	0.881503i	$0.343468\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.39821	0.0867074
$766$	0	0
$767$	4.29921	0.155236
$768$	0	0
$769$	−21.9404	−0.791192	−0.395596	−	0.918425i	$-0.629462\pi$
$769$	−21.9404	−0.791192	−0.395596	+	0.918425i	$0.629462\pi$
$770$	0	0
$771$	66.2701	2.38666
$772$	0	0
$773$	20.9073	0.751982	0.375991	−	0.926623i	$-0.377302\pi$
$773$	20.9073	0.751982	0.375991	+	0.926623i	$0.377302\pi$
$774$	0	0
$775$	0.881630	0.0316691
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−31.1857	−1.11734
$780$	0	0
$781$	1.11355	0.0398461
$782$	0	0
$783$	4.97648	0.177845
$784$	0	0
$785$	7.74244	0.276339
$786$	0	0
$787$	−15.3241	−0.546244	−0.273122	−	0.961979i	$-0.588056\pi$
$787$	−15.3241	−0.546244	−0.273122	+	0.961979i	$0.588056\pi$
$788$	0	0
$789$	−83.8034	−2.98348
$790$	0	0
$791$	0	0
$792$	0	0
$793$	32.3088	1.14732
$794$	0	0
$795$	−5.73539	−0.203413
$796$	0	0
$797$	−31.7521	−1.12472	−0.562358	−	0.826894i	$-0.690106\pi$
$797$	−31.7521	−1.12472	−0.562358	+	0.826894i	$0.690106\pi$
$798$	0	0
$799$	5.72161	0.202416
$800$	0	0
$801$	33.0152	1.16654
$802$	0	0
$803$	−9.46405	−0.333979
$804$	0	0
$805$	0	0
$806$	0	0
$807$	43.7521	1.54015
$808$	0	0
$809$	32.8685	1.15560	0.577798	−	0.816180i	$-0.303912\pi$
$809$	32.8685	1.15560	0.577798	+	0.816180i	$0.303912\pi$
$810$	0	0
$811$	−2.81916	−0.0989940	−0.0494970	−	0.998774i	$-0.515762\pi$
$811$	−2.81916	−0.0989940	−0.0494970	+	0.998774i	$0.515762\pi$
$812$	0	0
$813$	−60.5485	−2.12353
$814$	0	0
$815$	5.85039	0.204930
$816$	0	0
$817$	−25.5928	−0.895380
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−47.0249	−1.64118	−0.820590	−	0.571518i	$-0.806355\pi$
$821$	−47.0249	−1.64118	−0.820590	+	0.571518i	$0.806355\pi$
$822$	0	0
$823$	−8.18836	−0.285428	−0.142714	−	0.989764i	$-0.545583\pi$
$823$	−8.18836	−0.285428	−0.142714	+	0.989764i	$0.545583\pi$
$824$	0	0
$825$	27.3836	0.953376
$826$	0	0
$827$	13.0152	0.452584	0.226292	−	0.974060i	$-0.427340\pi$
$827$	13.0152	0.452584	0.226292	+	0.974060i	$0.427340\pi$
$828$	0	0
$829$	−23.5028	−0.816286	−0.408143	−	0.912918i	$-0.633824\pi$
$829$	−23.5028	−0.816286	−0.408143	+	0.912918i	$0.633824\pi$
$830$	0	0
$831$	55.9917	1.94233
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−7.41344	−0.256553
$836$	0	0
$837$	−1.15106	−0.0397863
$838$	0	0
$839$	−2.44882	−0.0845426	−0.0422713	−	0.999106i	$-0.513459\pi$
$839$	−2.44882	−0.0845426	−0.0422713	+	0.999106i	$0.513459\pi$
$840$	0	0
$841$	−28.3657	−0.978128
$842$	0	0
$843$	−75.9619	−2.61627
$844$	0	0
$845$	14.4238	0.496195
$846$	0	0
$847$	0	0
$848$	0	0
$849$	69.1863	2.37447
$850$	0	0
$851$	42.5872	1.45987
$852$	0	0
$853$	−38.9944	−1.33514	−0.667571	−	0.744546i	$-0.732666\pi$
$853$	−38.9944	−1.33514	−0.667571	+	0.744546i	$0.732666\pi$
$854$	0	0
$855$	−11.1440	−0.381117
$856$	0	0
$857$	−36.4626	−1.24554	−0.622769	−	0.782406i	$-0.713992\pi$
$857$	−36.4626	−1.24554	−0.622769	+	0.782406i	$0.713992\pi$
$858$	0	0
$859$	30.6260	1.04495	0.522473	−	0.852656i	$-0.325010\pi$
$859$	30.6260	1.04495	0.522473	+	0.852656i	$0.325010\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.42095	0.218572	0.109286	−	0.994010i	$-0.465144\pi$
$863$	6.42095	0.218572	0.109286	+	0.994010i	$0.465144\pi$
$864$	0	0
$865$	−0.655771	−0.0222969
$866$	0	0
$867$	2.86081	0.0971581
$868$	0	0
$869$	26.3297	0.893173
$870$	0	0
$871$	104.729	3.54859
$872$	0	0
$873$	−76.1115	−2.57598
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.2880	1.02275	0.511377	−	0.859357i	$-0.329136\pi$
$877$	30.2880	1.02275	0.511377	+	0.859357i	$0.329136\pi$
$878$	0	0
$879$	53.7133	1.81171
$880$	0	0
$881$	−32.1634	−1.08361	−0.541806	−	0.840504i	$-0.682259\pi$
$881$	−32.1634	−1.08361	−0.541806	+	0.840504i	$0.682259\pi$
$882$	0	0
$883$	53.0498	1.78527	0.892635	−	0.450781i	$-0.148855\pi$
$883$	53.0498	1.78527	0.892635	+	0.450781i	$0.148855\pi$
$884$	0	0
$885$	−0.855989	−0.0287738
$886$	0	0
$887$	29.1094	0.977398	0.488699	−	0.872452i	$-0.337472\pi$
$887$	29.1094	0.977398	0.488699	+	0.872452i	$0.337472\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.64681	−0.155674
$892$	0	0
$893$	−26.5872	−0.889708
$894$	0	0
$895$	8.89541	0.297341
$896$	0	0
$897$	108.798	3.63265
$898$	0	0
$899$	−0.146708	−0.00489298
$900$	0	0
$901$	−4.33382	−0.144380
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.67323	0.255067
$906$	0	0
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	0	0
$909$	8.25756	0.273886
$910$	0	0
$911$	38.0125	1.25941	0.629706	−	0.776834i	$-0.283175\pi$
$911$	38.0125	1.25941	0.629706	+	0.776834i	$0.283175\pi$
$912$	0	0
$913$	−24.4376	−0.808767
$914$	0	0
$915$	−6.43281	−0.212662
$916$	0	0
$917$	0	0
$918$	0	0
$919$	21.8940	0.722215	0.361108	−	0.932524i	$-0.382399\pi$
$919$	21.8940	0.722215	0.361108	+	0.932524i	$0.382399\pi$
$920$	0	0
$921$	−2.38924	−0.0787283
$922$	0	0
$923$	3.70079	0.121813
$924$	0	0
$925$	35.6233	1.17129
$926$	0	0
$927$	18.5180	0.608212
$928$	0	0
$929$	−38.7506	−1.27137	−0.635683	−	0.771950i	$-0.719282\pi$
$929$	−38.7506	−1.27137	−0.635683	+	0.771950i	$0.719282\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−24.4674	−0.801027
$934$	0	0
$935$	−0.925197	−0.0302572
$936$	0	0
$937$	−52.8685	−1.72714	−0.863570	−	0.504230i	$-0.831777\pi$
$937$	−52.8685	−1.72714	−0.863570	+	0.504230i	$0.831777\pi$
$938$	0	0
$939$	28.7666	0.938763
$940$	0	0
$941$	−49.8969	−1.62659	−0.813296	−	0.581851i	$-0.802329\pi$
$941$	−49.8969	−1.62659	−0.813296	+	0.581851i	$0.802329\pi$
$942$	0	0
$943$	−38.3989	−1.25044
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−60.5097	−1.96630	−0.983151	−	0.182795i	$-0.941486\pi$
$947$	−60.5097	−1.96630	−0.983151	+	0.182795i	$0.941486\pi$
$948$	0	0
$949$	−31.4529	−1.02100
$950$	0	0
$951$	48.9348	1.58682
$952$	0	0
$953$	52.2839	1.69364	0.846820	−	0.531879i	$-0.178514\pi$
$953$	52.2839	1.69364	0.846820	+	0.531879i	$0.178514\pi$
$954$	0	0
$955$	10.3040	0.333431
$956$	0	0
$957$	−4.55678	−0.147300
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.9661	−0.998905
$962$	0	0
$963$	58.5485	1.88670
$964$	0	0
$965$	−1.66204	−0.0535029
$966$	0	0
$967$	32.6516	1.05001	0.525003	−	0.851101i	$-0.324064\pi$
$967$	32.6516	1.05001	0.525003	+	0.851101i	$0.324064\pi$
$968$	0	0
$969$	−13.2936	−0.427053
$970$	0	0
$971$	−19.0361	−0.610896	−0.305448	−	0.952209i	$-0.598806\pi$
$971$	−19.0361	−0.610896	−0.305448	+	0.952209i	$0.598806\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	91.0069	2.91455
$976$	0	0
$977$	12.2922	0.393261	0.196631	−	0.980478i	$-0.437000\pi$
$977$	12.2922	0.393261	0.196631	+	0.980478i	$0.437000\pi$
$978$	0	0
$979$	−12.7368	−0.407071
$980$	0	0
$981$	55.1953	1.76225
$982$	0	0
$983$	27.2590	0.869427	0.434714	−	0.900569i	$-0.356850\pi$
$983$	27.2590	0.869427	0.434714	+	0.900569i	$0.356850\pi$
$984$	0	0
$985$	−2.10796	−0.0671651
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−31.5124	−1.00204
$990$	0	0
$991$	−17.0956	−0.543061	−0.271530	−	0.962430i	$-0.587530\pi$
$991$	−17.0956	−0.543061	−0.271530	+	0.962430i	$0.587530\pi$
$992$	0	0
$993$	18.4882	0.586707
$994$	0	0
$995$	−6.55745	−0.207885
$996$	0	0
$997$	−27.2321	−0.862450	−0.431225	−	0.902244i	$-0.641918\pi$
$997$	−27.2321	−0.862450	−0.431225	+	0.902244i	$0.641918\pi$
$998$	0	0
$999$	−46.5097	−1.47150

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.l.1.3		3
7.6	odd	2		952.2.a.d.1.1	✓	3
21.20	even	2		8568.2.a.z.1.2		3
28.27	even	2		1904.2.a.p.1.3		3
56.13	odd	2		7616.2.a.bg.1.3		3
56.27	even	2		7616.2.a.ba.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.d.1.1	✓	3	7.6	odd	2
1904.2.a.p.1.3		3	28.27	even	2
6664.2.a.l.1.3		3	1.1	even	1	trivial
7616.2.a.ba.1.1		3	56.27	even	2
7616.2.a.bg.1.3		3	56.13	odd	2
8568.2.a.z.1.2		3	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:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\(q+2.86081 q^{3} +0.462598 q^{5} +5.18421 q^{9} -2.00000 q^{11} -6.64681 q^{13} +1.32340 q^{15} +1.00000 q^{17} -4.64681 q^{19} -5.72161 q^{23} -4.78600 q^{25} +6.24860 q^{27} +0.796415 q^{29} -0.184210 q^{31} -5.72161 q^{33} -7.44322 q^{37} -19.0152 q^{39} +6.71120 q^{41} +5.50761 q^{43} +2.39821 q^{45} +5.72161 q^{47} +2.86081 q^{51} -4.33382 q^{53} -0.925197 q^{55} -13.2936 q^{57} -0.646809 q^{59} -4.86081 q^{61} -3.07480 q^{65} -15.7562 q^{67} -16.3684 q^{69} -0.556777 q^{71} +4.73202 q^{73} -13.6918 q^{75} -13.1648 q^{79} +2.32340 q^{81} +12.2188 q^{83} +0.462598 q^{85} +2.27839 q^{87} +6.36842 q^{89} -0.526989 q^{93} -2.14961 q^{95} -14.6814 q^{97} -10.3684 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - q^{5} + 2 q^{9} - 6 q^{11} - 4 q^{13} - 4 q^{15} + 3 q^{17} + 2 q^{19} - 6 q^{23} - 4 q^{25} + 6 q^{27} - 4 q^{29} + 13 q^{31} - 6 q^{33} - 14 q^{39} + 5 q^{41} - 5 q^{43} + 4 q^{45} + 6 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - q^5 + 2 * q^9 - 6 * q^11 - 4 * q^13 - 4 * q^15 + 3 * q^17 + 2 * q^19 - 6 * q^23 - 4 * q^25 + 6 * q^27 - 4 * q^29 + 13 * q^31 - 6 * q^33 - 14 * q^39 + 5 * q^41 - 5 * q^43 + 4 * q^45 + 6 * q^47 + 3 * q^51 - 9 * q^53 + 2 * q^55 - 8 * q^57 + 14 * q^59 - 9 * q^61 - 14 * q^65 - 13 * q^67 - 22 * q^69 - 24 * q^71 + 7 * q^73 - 6 * q^75 - 6 * q^79 - q^81 - q^85 + 18 * q^87 - 8 * q^89 - 16 * q^95 - 5 * q^97 - 4 * q^99

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