LMFDB - Embedded newform 8208.2.a.ca.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8208,2,Mod(1,8208)] 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("8208.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8208 = 2^{4} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8208.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,0,-3,0,-1,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$65.5412099791$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.7986588.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 16x^{3} + 23x^{2} + 24x - 36 $ x^5 - x^4 - 16x^3 + 23x^2 + 24*x - 36
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.37470$ of defining polynomial
Character	$\chi$	$=$	8208.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.59676 q^{5} +1.07564 q^{7} +0.137382 q^{11} -6.04711 q^{13} +5.68766 q^{17} +1.00000 q^{19} -3.37470 q^{23} -2.45034 q^{25} +0.161676 q^{29} -8.61202 q^{31} +1.71755 q^{35} +5.51208 q^{37} -6.86883 q^{41} -4.03185 q^{43} +7.19083 q^{47} -5.84299 q^{49} -2.17558 q^{53} +0.219367 q^{55} -11.9250 q^{59} +9.67376 q^{61} -9.65581 q^{65} -10.4503 q^{67} -8.48854 q^{71} +16.3136 q^{73} +0.147774 q^{77} -8.80420 q^{79} -0.617611 q^{83} +9.08186 q^{85} -3.11789 q^{89} -6.50452 q^{91} +1.59676 q^{95} +1.20681 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{5} - q^{7} - 4 q^{11} + 3 q^{13} - 5 q^{17} + 5 q^{19} - q^{23} + 10 q^{25} - 4 q^{29} - 16 q^{31} - 6 q^{35} + 7 q^{37} - 7 q^{41} - 3 q^{43} + 7 q^{47} + 18 q^{49} + 2 q^{53} - q^{55} - 15 q^{59}+ \cdots + 27 q^{97}+O(q^{100}) $ 5 * q - 3 * q^5 - q^7 - 4 * q^11 + 3 * q^13 - 5 * q^17 + 5 * q^19 - q^23 + 10 * q^25 - 4 * q^29 - 16 * q^31 - 6 * q^35 + 7 * q^37 - 7 * q^41 - 3 * q^43 + 7 * q^47 + 18 * q^49 + 2 * q^53 - q^55 - 15 * q^59 + 23 * q^61 - 32 * q^65 - 30 * q^67 - 12 * q^71 + 13 * q^73 + 4 * q^77 - 3 * q^79 + 5 * q^83 + 7 * q^85 - 29 * q^91 - 3 * q^95 + 27 * 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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.59676	0.714095	0.357047	−	0.934086i	$-0.383783\pi$
$5$	1.59676	0.714095	0.357047	+	0.934086i	$0.383783\pi$
$6$	0	0
$7$	1.07564	0.406555	0.203277	−	0.979121i	$-0.434841\pi$
$7$	1.07564	0.406555	0.203277	+	0.979121i	$0.434841\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.137382	0.0414223	0.0207111	−	0.999786i	$-0.493407\pi$
$11$	0.137382	0.0414223	0.0207111	+	0.999786i	$0.493407\pi$
$12$	0	0
$13$	−6.04711	−1.67717	−0.838583	−	0.544774i	$-0.816615\pi$
$13$	−6.04711	−1.67717	−0.838583	+	0.544774i	$0.816615\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.68766	1.37946	0.689730	−	0.724067i	$-0.257729\pi$
$17$	5.68766	1.37946	0.689730	+	0.724067i	$0.257729\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.37470	−0.703674	−0.351837	−	0.936061i	$-0.614443\pi$
$23$	−3.37470	−0.703674	−0.351837	+	0.936061i	$0.614443\pi$
$24$	0	0
$25$	−2.45034	−0.490068
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.161676	0.0300225	0.0150112	−	0.999887i	$-0.495222\pi$
$29$	0.161676	0.0300225	0.0150112	+	0.999887i	$0.495222\pi$
$30$	0	0
$31$	−8.61202	−1.54676	−0.773382	−	0.633940i	$-0.781437\pi$
$31$	−8.61202	−1.54676	−0.773382	+	0.633940i	$0.781437\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.71755	0.290318
$36$	0	0
$37$	5.51208	0.906181	0.453090	−	0.891465i	$-0.350321\pi$
$37$	5.51208	0.906181	0.453090	+	0.891465i	$0.350321\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.86883	−1.07273	−0.536366	−	0.843986i	$-0.680203\pi$
$41$	−6.86883	−1.07273	−0.536366	+	0.843986i	$0.680203\pi$
$42$	0	0
$43$	−4.03185	−0.614852	−0.307426	−	0.951572i	$-0.599468\pi$
$43$	−4.03185	−0.614852	−0.307426	+	0.951572i	$0.599468\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.19083	1.04889	0.524445	−	0.851444i	$-0.324273\pi$
$47$	7.19083	1.04889	0.524445	+	0.851444i	$0.324273\pi$
$48$	0	0
$49$	−5.84299	−0.834713
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.17558	−0.298839	−0.149419	−	0.988774i	$-0.547740\pi$
$53$	−2.17558	−0.298839	−0.149419	+	0.988774i	$0.547740\pi$
$54$	0	0
$55$	0.219367	0.0295794
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.9250	−1.55250	−0.776250	−	0.630425i	$-0.782881\pi$
$59$	−11.9250	−1.55250	−0.776250	+	0.630425i	$0.782881\pi$
$60$	0	0
$61$	9.67376	1.23860	0.619299	−	0.785155i	$-0.287417\pi$
$61$	9.67376	1.23860	0.619299	+	0.785155i	$0.287417\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−9.65581	−1.19766
$66$	0	0
$67$	−10.4503	−1.27671	−0.638356	−	0.769741i	$-0.720385\pi$
$67$	−10.4503	−1.27671	−0.638356	+	0.769741i	$0.720385\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.48854	−1.00740	−0.503702	−	0.863877i	$-0.668029\pi$
$71$	−8.48854	−1.00740	−0.503702	+	0.863877i	$0.668029\pi$
$72$	0	0
$73$	16.3136	1.90936	0.954680	−	0.297634i	$-0.0961974\pi$
$73$	16.3136	1.90936	0.954680	+	0.297634i	$0.0961974\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.147774	0.0168404
$78$	0	0
$79$	−8.80420	−0.990549	−0.495275	−	0.868736i	$-0.664933\pi$
$79$	−8.80420	−0.990549	−0.495275	+	0.868736i	$0.664933\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.617611	−0.0677916	−0.0338958	−	0.999425i	$-0.510791\pi$
$83$	−0.617611	−0.0677916	−0.0338958	+	0.999425i	$0.510791\pi$
$84$	0	0
$85$	9.08186	0.985066
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.11789	−0.330495	−0.165248	−	0.986252i	$-0.552842\pi$
$89$	−3.11789	−0.330495	−0.165248	+	0.986252i	$0.552842\pi$
$90$	0	0
$91$	−6.50452	−0.681859
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.59676	0.163825
$96$	0	0
$97$	1.20681	0.122533	0.0612665	−	0.998121i	$-0.480486\pi$
$97$	1.20681	0.122533	0.0612665	+	0.998121i	$0.480486\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.80285	0.776413	0.388206	−	0.921572i	$-0.373095\pi$
$101$	7.80285	0.776413	0.388206	+	0.921572i	$0.373095\pi$
$102$	0	0
$103$	13.3149	1.31196	0.655980	−	0.754779i	$-0.272256\pi$
$103$	13.3149	1.31196	0.655980	+	0.754779i	$0.272256\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.9943	−1.73957	−0.869786	−	0.493429i	$-0.835743\pi$
$107$	−17.9943	−1.73957	−0.869786	+	0.493429i	$0.835743\pi$
$108$	0	0
$109$	−10.8208	−1.03644	−0.518222	−	0.855246i	$-0.673406\pi$
$109$	−10.8208	−1.03644	−0.518222	+	0.855246i	$0.673406\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.1622	1.33226	0.666132	−	0.745834i	$-0.267949\pi$
$113$	14.1622	1.33226	0.666132	+	0.745834i	$0.267949\pi$
$114$	0	0
$115$	−5.38860	−0.502490
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.11789	0.560826
$120$	0	0
$121$	−10.9811	−0.998284
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.8964	−1.06405
$126$	0	0
$127$	−5.48368	−0.486598	−0.243299	−	0.969951i	$-0.578230\pi$
$127$	−5.48368	−0.486598	−0.243299	+	0.969951i	$0.578230\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.82442	−0.159400	−0.0797002	−	0.996819i	$-0.525396\pi$
$131$	−1.82442	−0.159400	−0.0797002	+	0.996819i	$0.525396\pi$
$132$	0	0
$133$	1.07564	0.0932700
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−22.4869	−1.92118	−0.960591	−	0.277965i	$-0.910340\pi$
$137$	−22.4869	−1.92118	−0.960591	+	0.277965i	$0.910340\pi$
$138$	0	0
$139$	−5.23886	−0.444354	−0.222177	−	0.975006i	$-0.571316\pi$
$139$	−5.23886	−0.444354	−0.222177	+	0.975006i	$0.571316\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.830765	−0.0694720
$144$	0	0
$145$	0.258159	0.0214389
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.9193	−1.63185	−0.815925	−	0.578158i	$-0.803772\pi$
$149$	−19.9193	−1.63185	−0.815925	+	0.578158i	$0.803772\pi$
$150$	0	0
$151$	−8.80420	−0.716475	−0.358238	−	0.933630i	$-0.616622\pi$
$151$	−8.80420	−0.716475	−0.358238	+	0.933630i	$0.616622\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−13.7514	−1.10454
$156$	0	0
$157$	17.3948	1.38826	0.694129	−	0.719851i	$-0.255790\pi$
$157$	17.3948	1.38826	0.694129	+	0.719851i	$0.255790\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.62997	−0.286082
$162$	0	0
$163$	12.1942	0.955120	0.477560	−	0.878599i	$-0.341521\pi$
$163$	12.1942	0.955120	0.477560	+	0.878599i	$0.341521\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.9296	1.23267	0.616337	−	0.787483i	$-0.288616\pi$
$167$	15.9296	1.23267	0.616337	+	0.787483i	$0.288616\pi$
$168$	0	0
$169$	23.5675	1.81289
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.41787	−0.259856	−0.129928	−	0.991523i	$-0.541475\pi$
$173$	−3.41787	−0.259856	−0.129928	+	0.991523i	$0.541475\pi$
$174$	0	0
$175$	−2.63569	−0.199240
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−17.6543	−1.31954	−0.659771	−	0.751467i	$-0.729347\pi$
$179$	−17.6543	−1.31954	−0.659771	+	0.751467i	$0.729347\pi$
$180$	0	0
$181$	19.3919	1.44139	0.720695	−	0.693252i	$-0.243823\pi$
$181$	19.3919	1.44139	0.720695	+	0.693252i	$0.243823\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.80150	0.647099
$186$	0	0
$187$	0.781383	0.0571404
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.0381	1.08812	0.544058	−	0.839047i	$-0.316887\pi$
$191$	15.0381	1.08812	0.544058	+	0.839047i	$0.316887\pi$
$192$	0	0
$193$	−6.29698	−0.453266	−0.226633	−	0.973980i	$-0.572772\pi$
$193$	−6.29698	−0.453266	−0.226633	+	0.973980i	$0.572772\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.26167	0.446126	0.223063	−	0.974804i	$-0.428394\pi$
$197$	6.26167	0.446126	0.223063	+	0.974804i	$0.428394\pi$
$198$	0	0
$199$	1.74806	0.123916	0.0619582	−	0.998079i	$-0.480265\pi$
$199$	1.74806	0.123916	0.0619582	+	0.998079i	$0.480265\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.173906	0.0122058
$204$	0	0
$205$	−10.9679	−0.766032
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.137382	0.00950292
$210$	0	0
$211$	−24.6321	−1.69575	−0.847873	−	0.530199i	$-0.822117\pi$
$211$	−24.6321	−1.69575	−0.847873	+	0.530199i	$0.822117\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.43792	−0.439063
$216$	0	0
$217$	−9.26345	−0.628844
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−34.3939	−2.31358
$222$	0	0
$223$	−5.39131	−0.361029	−0.180514	−	0.983572i	$-0.557776\pi$
$223$	−5.39131	−0.361029	−0.180514	+	0.983572i	$0.557776\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.07430	−0.469538	−0.234769	−	0.972051i	$-0.575433\pi$
$227$	−7.07430	−0.469538	−0.234769	+	0.972051i	$0.575433\pi$
$228$	0	0
$229$	−3.46228	−0.228794	−0.114397	−	0.993435i	$-0.536494\pi$
$229$	−3.46228	−0.228794	−0.114397	+	0.993435i	$0.536494\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.24912	0.212857	0.106429	−	0.994320i	$-0.466058\pi$
$233$	3.24912	0.212857	0.106429	+	0.994320i	$0.466058\pi$
$234$	0	0
$235$	11.4821	0.749007
$236$	0	0
$237$	0	0
$238$	0	0
$239$	18.3308	1.18572	0.592860	−	0.805305i	$-0.297999\pi$
$239$	18.3308	1.18572	0.592860	+	0.805305i	$0.297999\pi$
$240$	0	0
$241$	17.0983	1.10140	0.550698	−	0.834705i	$-0.314362\pi$
$241$	17.0983	1.10140	0.550698	+	0.834705i	$0.314362\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−9.32989	−0.596065
$246$	0	0
$247$	−6.04711	−0.384768
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−11.3504	−0.716431	−0.358216	−	0.933639i	$-0.616615\pi$
$251$	−11.3504	−0.716431	−0.358216	+	0.933639i	$0.616615\pi$
$252$	0	0
$253$	−0.463624	−0.0291478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.29906	−0.330546	−0.165273	−	0.986248i	$-0.552851\pi$
$257$	−5.29906	−0.330546	−0.165273	+	0.986248i	$0.552851\pi$
$258$	0	0
$259$	5.92903	0.368412
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.0533	−0.866565	−0.433282	−	0.901258i	$-0.642645\pi$
$263$	−14.0533	−0.866565	−0.433282	+	0.901258i	$0.642645\pi$
$264$	0	0
$265$	−3.47389	−0.213399
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.18935	−0.560285	−0.280142	−	0.959958i	$-0.590382\pi$
$269$	−9.18935	−0.560285	−0.280142	+	0.959958i	$0.590382\pi$
$270$	0	0
$271$	−30.1525	−1.83164	−0.915818	−	0.401594i	$-0.868456\pi$
$271$	−30.1525	−1.83164	−0.915818	+	0.401594i	$0.868456\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.336633	−0.0202998
$276$	0	0
$277$	−13.3620	−0.802847	−0.401424	−	0.915893i	$-0.631484\pi$
$277$	−13.3620	−0.802847	−0.401424	+	0.915893i	$0.631484\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.11371	0.126093	0.0630467	−	0.998011i	$-0.479918\pi$
$281$	2.11371	0.126093	0.0630467	+	0.998011i	$0.479918\pi$
$282$	0	0
$283$	22.8124	1.35606	0.678028	−	0.735036i	$-0.262835\pi$
$283$	22.8124	1.35606	0.678028	+	0.735036i	$0.262835\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.38840	−0.436124
$288$	0	0
$289$	15.3495	0.902911
$290$	0	0
$291$	0	0
$292$	0	0
$293$	30.0517	1.75564	0.877819	−	0.478993i	$-0.158998\pi$
$293$	30.0517	1.75564	0.877819	+	0.478993i	$0.158998\pi$
$294$	0	0
$295$	−19.0414	−1.10863
$296$	0	0
$297$	0	0
$298$	0	0
$299$	20.4072	1.18018
$300$	0	0
$301$	−4.33683	−0.249971
$302$	0	0
$303$	0	0
$304$	0	0
$305$	15.4467	0.884476
$306$	0	0
$307$	−7.93462	−0.452853	−0.226426	−	0.974028i	$-0.572704\pi$
$307$	−7.93462	−0.452853	−0.226426	+	0.974028i	$0.572704\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.17268	0.236611	0.118306	−	0.992977i	$-0.462254\pi$
$311$	4.17268	0.236611	0.118306	+	0.992977i	$0.462254\pi$
$312$	0	0
$313$	18.9734	1.07244	0.536221	−	0.844078i	$-0.319851\pi$
$313$	18.9734	1.07244	0.536221	+	0.844078i	$0.319851\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.2372	−1.13663	−0.568317	−	0.822810i	$-0.692405\pi$
$317$	−20.2372	−1.13663	−0.568317	+	0.822810i	$0.692405\pi$
$318$	0	0
$319$	0.0222114	0.00124360
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.68766	0.316470
$324$	0	0
$325$	14.8175	0.821926
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.73476	0.426431
$330$	0	0
$331$	−0.875174	−0.0481039	−0.0240520	−	0.999711i	$-0.507657\pi$
$331$	−0.875174	−0.0481039	−0.0240520	+	0.999711i	$0.507657\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−16.6867	−0.911694
$336$	0	0
$337$	−2.44554	−0.133217	−0.0666086	−	0.997779i	$-0.521218\pi$
$337$	−2.44554	−0.133217	−0.0666086	+	0.997779i	$0.521218\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.18314	−0.0640705
$342$	0	0
$343$	−13.8145	−0.745911
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−35.3274	−1.89647	−0.948236	−	0.317567i	$-0.897134\pi$
$347$	−35.3274	−1.89647	−0.948236	+	0.317567i	$0.897134\pi$
$348$	0	0
$349$	4.06328	0.217503	0.108751	−	0.994069i	$-0.465315\pi$
$349$	4.06328	0.217503	0.108751	+	0.994069i	$0.465315\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−29.6410	−1.57763	−0.788815	−	0.614630i	$-0.789305\pi$
$353$	−29.6410	−1.57763	−0.788815	+	0.614630i	$0.789305\pi$
$354$	0	0
$355$	−13.5542	−0.719382
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.78023	0.0939569	0.0469785	−	0.998896i	$-0.485041\pi$
$359$	1.78023	0.0939569	0.0469785	+	0.998896i	$0.485041\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	26.0490	1.36346
$366$	0	0
$367$	30.0178	1.56692	0.783459	−	0.621443i	$-0.213453\pi$
$367$	30.0178	1.56692	0.783459	+	0.621443i	$0.213453\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.34014	−0.121494
$372$	0	0
$373$	19.9907	1.03508	0.517539	−	0.855660i	$-0.326848\pi$
$373$	19.9907	1.03508	0.517539	+	0.855660i	$0.326848\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.977673	−0.0503527
$378$	0	0
$379$	9.52457	0.489244	0.244622	−	0.969618i	$-0.421336\pi$
$379$	9.52457	0.489244	0.244622	+	0.969618i	$0.421336\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.5930	−1.00116	−0.500578	−	0.865691i	$-0.666879\pi$
$383$	−19.5930	−1.00116	−0.500578	+	0.865691i	$0.666879\pi$
$384$	0	0
$385$	0.235960	0.0120257
$386$	0	0
$387$	0	0
$388$	0	0
$389$	11.5492	0.585569	0.292785	−	0.956178i	$-0.405418\pi$
$389$	11.5492	0.585569	0.292785	+	0.956178i	$0.405418\pi$
$390$	0	0
$391$	−19.1942	−0.970690
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−14.0582	−0.707346
$396$	0	0
$397$	−34.2938	−1.72116	−0.860579	−	0.509317i	$-0.829898\pi$
$397$	−34.2938	−1.72116	−0.860579	+	0.509317i	$0.829898\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	29.2673	1.46154	0.730770	−	0.682623i	$-0.239161\pi$
$401$	29.2673	1.46154	0.730770	+	0.682623i	$0.239161\pi$
$402$	0	0
$403$	52.0778	2.59418
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.757262	0.0375361
$408$	0	0
$409$	29.1844	1.44307	0.721537	−	0.692376i	$-0.243436\pi$
$409$	29.1844	1.44307	0.721537	+	0.692376i	$0.243436\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−12.8270	−0.631176
$414$	0	0
$415$	−0.986180	−0.0484097
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.3440	1.04272	0.521360	−	0.853337i	$-0.325425\pi$
$419$	21.3440	1.04272	0.521360	+	0.853337i	$0.325425\pi$
$420$	0	0
$421$	12.6633	0.617172	0.308586	−	0.951197i	$-0.400144\pi$
$421$	12.6633	0.617172	0.308586	+	0.951197i	$0.400144\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−13.9367	−0.676030
$426$	0	0
$427$	10.4055	0.503558
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.8690	−0.523539	−0.261770	−	0.965130i	$-0.584306\pi$
$431$	−10.8690	−0.523539	−0.261770	+	0.965130i	$0.584306\pi$
$432$	0	0
$433$	6.80568	0.327060	0.163530	−	0.986538i	$-0.447712\pi$
$433$	6.80568	0.327060	0.163530	+	0.986538i	$0.447712\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.37470	−0.161434
$438$	0	0
$439$	20.2397	0.965989	0.482994	−	0.875623i	$-0.339549\pi$
$439$	20.2397	0.965989	0.482994	+	0.875623i	$0.339549\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	35.8867	1.70503	0.852514	−	0.522705i	$-0.175077\pi$
$443$	35.8867	1.70503	0.852514	+	0.522705i	$0.175077\pi$
$444$	0	0
$445$	−4.97853	−0.236005
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.7680	0.602558	0.301279	−	0.953536i	$-0.402586\pi$
$449$	12.7680	0.602558	0.301279	+	0.953536i	$0.402586\pi$
$450$	0	0
$451$	−0.943655	−0.0444350
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10.3862	−0.486912
$456$	0	0
$457$	−11.5626	−0.540877	−0.270439	−	0.962737i	$-0.587169\pi$
$457$	−11.5626	−0.540877	−0.270439	+	0.962737i	$0.587169\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−39.7582	−1.85172	−0.925862	−	0.377862i	$-0.876659\pi$
$461$	−39.7582	−1.85172	−0.925862	+	0.377862i	$0.876659\pi$
$462$	0	0
$463$	−5.16782	−0.240169	−0.120085	−	0.992764i	$-0.538317\pi$
$463$	−5.16782	−0.240169	−0.120085	+	0.992764i	$0.538317\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.0172	−1.48158	−0.740790	−	0.671736i	$-0.765549\pi$
$467$	−32.0172	−1.48158	−0.740790	+	0.671736i	$0.765549\pi$
$468$	0	0
$469$	−11.2408	−0.519053
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.553905	−0.0254686
$474$	0	0
$475$	−2.45034	−0.112429
$476$	0	0
$477$	0	0
$478$	0	0
$479$	27.6819	1.26482	0.632409	−	0.774635i	$-0.282066\pi$
$479$	27.6819	1.26482	0.632409	+	0.774635i	$0.282066\pi$
$480$	0	0
$481$	−33.3322	−1.51982
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.92699	0.0875002
$486$	0	0
$487$	−2.16249	−0.0979917	−0.0489959	−	0.998799i	$-0.515602\pi$
$487$	−2.16249	−0.0979917	−0.0489959	+	0.998799i	$0.515602\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−26.2401	−1.18420	−0.592100	−	0.805865i	$-0.701701\pi$
$491$	−26.2401	−1.18420	−0.592100	+	0.805865i	$0.701701\pi$
$492$	0	0
$493$	0.919559	0.0414148
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.13063	−0.409565
$498$	0	0
$499$	−3.89878	−0.174533	−0.0872666	−	0.996185i	$-0.527813\pi$
$499$	−3.89878	−0.174533	−0.0872666	+	0.996185i	$0.527813\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.97915	−0.222009	−0.111005	−	0.993820i	$-0.535407\pi$
$503$	−4.97915	−0.222009	−0.111005	+	0.993820i	$0.535407\pi$
$504$	0	0
$505$	12.4593	0.554432
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.9115	−0.926886	−0.463443	−	0.886127i	$-0.653386\pi$
$509$	−20.9115	−0.926886	−0.463443	+	0.886127i	$0.653386\pi$
$510$	0	0
$511$	17.5476	0.776259
$512$	0	0
$513$	0	0
$514$	0	0
$515$	21.2608	0.936863
$516$	0	0
$517$	0.987892	0.0434474
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.01174	0.0443250	0.0221625	−	0.999754i	$-0.492945\pi$
$521$	1.01174	0.0443250	0.0221625	+	0.999754i	$0.492945\pi$
$522$	0	0
$523$	13.0132	0.569027	0.284513	−	0.958672i	$-0.408168\pi$
$523$	13.0132	0.569027	0.284513	+	0.958672i	$0.408168\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−48.9822	−2.13370
$528$	0	0
$529$	−11.6114	−0.504843
$530$	0	0
$531$	0	0
$532$	0	0
$533$	41.5366	1.79915
$534$	0	0
$535$	−28.7326	−1.24222
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.802723	−0.0345757
$540$	0	0
$541$	−6.08511	−0.261619	−0.130810	−	0.991407i	$-0.541758\pi$
$541$	−6.08511	−0.261619	−0.130810	+	0.991407i	$0.541758\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−17.2783	−0.740120
$546$	0	0
$547$	36.8872	1.57718	0.788592	−	0.614917i	$-0.210810\pi$
$547$	36.8872	1.57718	0.788592	+	0.614917i	$0.210810\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.161676	0.00688763
$552$	0	0
$553$	−9.47016	−0.402712
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.312954	0.0132603	0.00663015	−	0.999978i	$-0.497890\pi$
$557$	0.312954	0.0132603	0.00663015	+	0.999978i	$0.497890\pi$
$558$	0	0
$559$	24.3810	1.03121
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.42672	−0.270854	−0.135427	−	0.990787i	$-0.543241\pi$
$563$	−6.42672	−0.270854	−0.135427	+	0.990787i	$0.543241\pi$
$564$	0	0
$565$	22.6137	0.951363
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−28.1809	−1.18140	−0.590702	−	0.806890i	$-0.701149\pi$
$569$	−28.1809	−1.18140	−0.590702	+	0.806890i	$0.701149\pi$
$570$	0	0
$571$	−2.54616	−0.106553	−0.0532767	−	0.998580i	$-0.516967\pi$
$571$	−2.54616	−0.106553	−0.0532767	+	0.998580i	$0.516967\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.26917	0.344848
$576$	0	0
$577$	−5.47255	−0.227825	−0.113913	−	0.993491i	$-0.536338\pi$
$577$	−5.47255	−0.227825	−0.113913	+	0.993491i	$0.536338\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.664329	−0.0275610
$582$	0	0
$583$	−0.298886	−0.0123786
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−39.2274	−1.61909	−0.809544	−	0.587060i	$-0.800285\pi$
$587$	−39.2274	−1.61909	−0.809544	+	0.587060i	$0.800285\pi$
$588$	0	0
$589$	−8.61202	−0.354852
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−37.2124	−1.52813	−0.764065	−	0.645139i	$-0.776800\pi$
$593$	−37.2124	−1.52813	−0.764065	+	0.645139i	$0.776800\pi$
$594$	0	0
$595$	9.76883	0.400483
$596$	0	0
$597$	0	0
$598$	0	0
$599$	23.9261	0.977596	0.488798	−	0.872397i	$-0.337436\pi$
$599$	23.9261	0.977596	0.488798	+	0.872397i	$0.337436\pi$
$600$	0	0
$601$	−11.5113	−0.469555	−0.234778	−	0.972049i	$-0.575436\pi$
$601$	−11.5113	−0.469555	−0.234778	+	0.972049i	$0.575436\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−17.5343	−0.712870
$606$	0	0
$607$	−25.2247	−1.02384	−0.511919	−	0.859034i	$-0.671065\pi$
$607$	−25.2247	−1.02384	−0.511919	+	0.859034i	$0.671065\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−43.4837	−1.75916
$612$	0	0
$613$	9.16426	0.370141	0.185071	−	0.982725i	$-0.440749\pi$
$613$	9.16426	0.370141	0.185071	+	0.982725i	$0.440749\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−44.8623	−1.80609	−0.903043	−	0.429549i	$-0.858672\pi$
$617$	−44.8623	−1.80609	−0.903043	+	0.429549i	$0.858672\pi$
$618$	0	0
$619$	−38.0677	−1.53007	−0.765036	−	0.643988i	$-0.777279\pi$
$619$	−38.0677	−1.53007	−0.765036	+	0.643988i	$0.777279\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.35373	−0.134364
$624$	0	0
$625$	−6.74411	−0.269764
$626$	0	0
$627$	0	0
$628$	0	0
$629$	31.3509	1.25004
$630$	0	0
$631$	4.06667	0.161892	0.0809458	−	0.996719i	$-0.474206\pi$
$631$	4.06667	0.161892	0.0809458	+	0.996719i	$0.474206\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8.75614	−0.347477
$636$	0	0
$637$	35.3332	1.39995
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−42.4358	−1.67611	−0.838057	−	0.545583i	$-0.816308\pi$
$641$	−42.4358	−1.67611	−0.838057	+	0.545583i	$0.816308\pi$
$642$	0	0
$643$	36.7103	1.44771	0.723856	−	0.689951i	$-0.242368\pi$
$643$	36.7103	1.44771	0.723856	+	0.689951i	$0.242368\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19.5585	−0.768925	−0.384462	−	0.923141i	$-0.625613\pi$
$647$	−19.5585	−0.768925	−0.384462	+	0.923141i	$0.625613\pi$
$648$	0	0
$649$	−1.63828	−0.0643081
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.3587	−0.405365	−0.202683	−	0.979244i	$-0.564966\pi$
$653$	−10.3587	−0.405365	−0.202683	+	0.979244i	$0.564966\pi$
$654$	0	0
$655$	−2.91317	−0.113827
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.61527	0.101877	0.0509383	−	0.998702i	$-0.483779\pi$
$659$	2.61527	0.101877	0.0509383	+	0.998702i	$0.483779\pi$
$660$	0	0
$661$	26.7546	1.04063	0.520317	−	0.853973i	$-0.325814\pi$
$661$	26.7546	1.04063	0.520317	+	0.853973i	$0.325814\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.71755	0.0666036
$666$	0	0
$667$	−0.545608	−0.0211260
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.32900	0.0513055
$672$	0	0
$673$	−0.601674	−0.0231928	−0.0115964	−	0.999933i	$-0.503691\pi$
$673$	−0.601674	−0.0231928	−0.0115964	+	0.999933i	$0.503691\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−31.4529	−1.20883	−0.604417	−	0.796668i	$-0.706594\pi$
$677$	−31.4529	−1.20883	−0.604417	+	0.796668i	$0.706594\pi$
$678$	0	0
$679$	1.29810	0.0498164
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20.0877	0.768634	0.384317	−	0.923201i	$-0.374437\pi$
$683$	20.0877	0.768634	0.384317	+	0.923201i	$0.374437\pi$
$684$	0	0
$685$	−35.9062	−1.37191
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13.1560	0.501202
$690$	0	0
$691$	−17.5735	−0.668528	−0.334264	−	0.942479i	$-0.608488\pi$
$691$	−17.5735	−0.668528	−0.334264	+	0.942479i	$0.608488\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.36523	−0.317311
$696$	0	0
$697$	−39.0676	−1.47979
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.2468	0.689172	0.344586	−	0.938755i	$-0.388019\pi$
$701$	18.2468	0.689172	0.344586	+	0.938755i	$0.388019\pi$
$702$	0	0
$703$	5.51208	0.207892
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.39307	0.315654
$708$	0	0
$709$	10.6778	0.401014	0.200507	−	0.979692i	$-0.435741\pi$
$709$	10.6778	0.401014	0.200507	+	0.979692i	$0.435741\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	29.0630	1.08842
$714$	0	0
$715$	−1.32654	−0.0496096
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.24925	−0.158470	−0.0792352	−	0.996856i	$-0.525248\pi$
$719$	−4.24925	−0.158470	−0.0792352	+	0.996856i	$0.525248\pi$
$720$	0	0
$721$	14.3221	0.533383
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.396162	−0.0147131
$726$	0	0
$727$	−9.79017	−0.363097	−0.181549	−	0.983382i	$-0.558111\pi$
$727$	−9.79017	−0.363097	−0.181549	+	0.983382i	$0.558111\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−22.9318	−0.848164
$732$	0	0
$733$	−40.4117	−1.49264	−0.746320	−	0.665587i	$-0.768181\pi$
$733$	−40.4117	−1.49264	−0.746320	+	0.665587i	$0.768181\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.43569	−0.0528843
$738$	0	0
$739$	−48.5858	−1.78726	−0.893628	−	0.448808i	$-0.851849\pi$
$739$	−48.5858	−1.78726	−0.893628	+	0.448808i	$0.851849\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29.7109	−1.08999	−0.544994	−	0.838440i	$-0.683468\pi$
$743$	−29.7109	−1.08999	−0.544994	+	0.838440i	$0.683468\pi$
$744$	0	0
$745$	−31.8064	−1.16530
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−19.3554	−0.707231
$750$	0	0
$751$	−1.77879	−0.0649089	−0.0324545	−	0.999473i	$-0.510332\pi$
$751$	−1.77879	−0.0649089	−0.0324545	+	0.999473i	$0.510332\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−14.0582	−0.511631
$756$	0	0
$757$	−31.9188	−1.16011	−0.580055	−	0.814577i	$-0.696969\pi$
$757$	−31.9188	−1.16011	−0.580055	+	0.814577i	$0.696969\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.36782	−0.0495834	−0.0247917	−	0.999693i	$-0.507892\pi$
$761$	−1.36782	−0.0495834	−0.0247917	+	0.999693i	$0.507892\pi$
$762$	0	0
$763$	−11.6393	−0.421371
$764$	0	0
$765$	0	0
$766$	0	0
$767$	72.1116	2.60380
$768$	0	0
$769$	7.67093	0.276621	0.138310	−	0.990389i	$-0.455833\pi$
$769$	7.67093	0.276621	0.138310	+	0.990389i	$0.455833\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.39561	0.0501966	0.0250983	−	0.999685i	$-0.492010\pi$
$773$	1.39561	0.0501966	0.0250983	+	0.999685i	$0.492010\pi$
$774$	0	0
$775$	21.1024	0.758020
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.86883	−0.246101
$780$	0	0
$781$	−1.16617	−0.0417290
$782$	0	0
$783$	0	0
$784$	0	0
$785$	27.7754	0.991348
$786$	0	0
$787$	−37.2778	−1.32881	−0.664405	−	0.747373i	$-0.731315\pi$
$787$	−37.2778	−1.32881	−0.664405	+	0.747373i	$0.731315\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15.2334	0.541638
$792$	0	0
$793$	−58.4983	−2.07733
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.8925	−0.988003	−0.494001	−	0.869461i	$-0.664466\pi$
$797$	−27.8925	−0.988003	−0.494001	+	0.869461i	$0.664466\pi$
$798$	0	0
$799$	40.8990	1.44690
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.24119	0.0790900
$804$	0	0
$805$	−5.79621	−0.204289
$806$	0	0
$807$	0	0
$808$	0	0
$809$	45.3933	1.59594	0.797972	−	0.602694i	$-0.205906\pi$
$809$	45.3933	1.59594	0.797972	+	0.602694i	$0.205906\pi$
$810$	0	0
$811$	28.9081	1.01510	0.507551	−	0.861622i	$-0.330551\pi$
$811$	28.9081	1.01510	0.507551	+	0.861622i	$0.330551\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	19.4712	0.682046
$816$	0	0
$817$	−4.03185	−0.141057
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.3202	1.05818	0.529091	−	0.848565i	$-0.322533\pi$
$821$	30.3202	1.05818	0.529091	+	0.848565i	$0.322533\pi$
$822$	0	0
$823$	15.7062	0.547482	0.273741	−	0.961803i	$-0.411739\pi$
$823$	15.7062	0.547482	0.273741	+	0.961803i	$0.411739\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.3271	1.19367	0.596835	−	0.802364i	$-0.296425\pi$
$827$	34.3271	1.19367	0.596835	+	0.802364i	$0.296425\pi$
$828$	0	0
$829$	8.53916	0.296577	0.148289	−	0.988944i	$-0.452624\pi$
$829$	8.53916	0.296577	0.148289	+	0.988944i	$0.452624\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−33.2330	−1.15145
$834$	0	0
$835$	25.4359	0.880246
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8.34634	0.288148	0.144074	−	0.989567i	$-0.453980\pi$
$839$	8.34634	0.288148	0.144074	+	0.989567i	$0.453980\pi$
$840$	0	0
$841$	−28.9739	−0.999099
$842$	0	0
$843$	0	0
$844$	0	0
$845$	37.6318	1.29457
$846$	0	0
$847$	−11.8118	−0.405857
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−18.6016	−0.637656
$852$	0	0
$853$	−20.2084	−0.691922	−0.345961	−	0.938249i	$-0.612447\pi$
$853$	−20.2084	−0.691922	−0.345961	+	0.938249i	$0.612447\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38.9708	−1.33122	−0.665608	−	0.746302i	$-0.731828\pi$
$857$	−38.9708	−1.33122	−0.665608	+	0.746302i	$0.731828\pi$
$858$	0	0
$859$	6.42450	0.219201	0.109601	−	0.993976i	$-0.465043\pi$
$859$	6.42450	0.219201	0.109601	+	0.993976i	$0.465043\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22.2874	0.758672	0.379336	−	0.925259i	$-0.376152\pi$
$863$	22.2874	0.758672	0.379336	+	0.925259i	$0.376152\pi$
$864$	0	0
$865$	−5.45753	−0.185562
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.20954	−0.0410308
$870$	0	0
$871$	63.1943	2.14126
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12.7963	−0.432594
$876$	0	0
$877$	−49.5580	−1.67345	−0.836727	−	0.547621i	$-0.815534\pi$
$877$	−49.5580	−1.67345	−0.836727	+	0.547621i	$0.815534\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	17.5988	0.592918	0.296459	−	0.955046i	$-0.404194\pi$
$881$	17.5988	0.592918	0.296459	+	0.955046i	$0.404194\pi$
$882$	0	0
$883$	−25.9531	−0.873393	−0.436697	−	0.899609i	$-0.643852\pi$
$883$	−25.9531	−0.873393	−0.436697	+	0.899609i	$0.643852\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.6966	0.661347	0.330673	−	0.943745i	$-0.392724\pi$
$887$	19.6966	0.661347	0.330673	+	0.943745i	$0.392724\pi$
$888$	0	0
$889$	−5.89847	−0.197828
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.19083	0.240632
$894$	0	0
$895$	−28.1897	−0.942278
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.39236	−0.0464377
$900$	0	0
$901$	−12.3740	−0.412236
$902$	0	0
$903$	0	0
$904$	0	0
$905$	30.9643	1.02929
$906$	0	0
$907$	32.8390	1.09040	0.545201	−	0.838305i	$-0.316453\pi$
$907$	32.8390	1.09040	0.545201	+	0.838305i	$0.316453\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	55.6644	1.84425	0.922123	−	0.386898i	$-0.126453\pi$
$911$	55.6644	1.84425	0.922123	+	0.386898i	$0.126453\pi$
$912$	0	0
$913$	−0.0848488	−0.00280808
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.96242	−0.0648050
$918$	0	0
$919$	2.36714	0.0780848	0.0390424	−	0.999238i	$-0.487569\pi$
$919$	2.36714	0.0780848	0.0390424	+	0.999238i	$0.487569\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	51.3311	1.68958
$924$	0	0
$925$	−13.5065	−0.444091
$926$	0	0
$927$	0	0
$928$	0	0
$929$	25.7194	0.843826	0.421913	−	0.906636i	$-0.361359\pi$
$929$	25.7194	0.843826	0.421913	+	0.906636i	$0.361359\pi$
$930$	0	0
$931$	−5.84299	−0.191496
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.24768	0.0408037
$936$	0	0
$937$	45.8335	1.49731	0.748657	−	0.662957i	$-0.230699\pi$
$937$	45.8335	1.49731	0.748657	+	0.662957i	$0.230699\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−24.5255	−0.799508	−0.399754	−	0.916622i	$-0.630904\pi$
$941$	−24.5255	−0.799508	−0.399754	+	0.916622i	$0.630904\pi$
$942$	0	0
$943$	23.1802	0.754853
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.41372	0.240913	0.120457	−	0.992719i	$-0.461564\pi$
$947$	7.41372	0.240913	0.120457	+	0.992719i	$0.461564\pi$
$948$	0	0
$949$	−98.6500	−3.20231
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5.75290	−0.186355	−0.0931773	−	0.995650i	$-0.529702\pi$
$953$	−5.75290	−0.186355	−0.0931773	+	0.995650i	$0.529702\pi$
$954$	0	0
$955$	24.0123	0.777018
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−24.1878	−0.781065
$960$	0	0
$961$	43.1669	1.39248
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10.0548	−0.323675
$966$	0	0
$967$	15.8014	0.508140	0.254070	−	0.967186i	$-0.418231\pi$
$967$	15.8014	0.508140	0.254070	+	0.967186i	$0.418231\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.6731	−0.406700	−0.203350	−	0.979106i	$-0.565183\pi$
$971$	−12.6731	−0.406700	−0.203350	+	0.979106i	$0.565183\pi$
$972$	0	0
$973$	−5.63514	−0.180654
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.3577	0.939236	0.469618	−	0.882870i	$-0.344392\pi$
$977$	29.3577	0.939236	0.469618	+	0.882870i	$0.344392\pi$
$978$	0	0
$979$	−0.428342	−0.0136899
$980$	0	0
$981$	0	0
$982$	0	0
$983$	33.3620	1.06408	0.532042	−	0.846718i	$-0.321425\pi$
$983$	33.3620	1.06408	0.532042	+	0.846718i	$0.321425\pi$
$984$	0	0
$985$	9.99842	0.318576
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13.6063	0.432655
$990$	0	0
$991$	61.1987	1.94404	0.972020	−	0.234900i	$-0.0754762\pi$
$991$	61.1987	1.94404	0.972020	+	0.234900i	$0.0754762\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.79123	0.0884880
$996$	0	0
$997$	30.4016	0.962827	0.481413	−	0.876494i	$-0.340124\pi$
$997$	30.4016	0.962827	0.481413	+	0.876494i	$0.340124\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8208.2.a.ca.1.4		5
3.2	odd	2		8208.2.a.cd.1.2		5
4.3	odd	2		4104.2.a.p.1.4	✓	5
12.11	even	2		4104.2.a.q.1.2	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4104.2.a.p.1.4	✓	5	4.3	odd	2
4104.2.a.q.1.2	yes	5	12.11	even	2
8208.2.a.ca.1.4		5	1.1	even	1	trivial
8208.2.a.cd.1.2		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8208 = 2^{4} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8208.a (trivial)

\(f(q)\)	\(=\)	\(q+1.59676 q^{5} +1.07564 q^{7} +0.137382 q^{11} -6.04711 q^{13} +5.68766 q^{17} +1.00000 q^{19} -3.37470 q^{23} -2.45034 q^{25} +0.161676 q^{29} -8.61202 q^{31} +1.71755 q^{35} +5.51208 q^{37} -6.86883 q^{41} -4.03185 q^{43} +7.19083 q^{47} -5.84299 q^{49} -2.17558 q^{53} +0.219367 q^{55} -11.9250 q^{59} +9.67376 q^{61} -9.65581 q^{65} -10.4503 q^{67} -8.48854 q^{71} +16.3136 q^{73} +0.147774 q^{77} -8.80420 q^{79} -0.617611 q^{83} +9.08186 q^{85} -3.11789 q^{89} -6.50452 q^{91} +1.59676 q^{95} +1.20681 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{5} - q^{7} - 4 q^{11} + 3 q^{13} - 5 q^{17} + 5 q^{19} - q^{23} + 10 q^{25} - 4 q^{29} - 16 q^{31} - 6 q^{35} + 7 q^{37} - 7 q^{41} - 3 q^{43} + 7 q^{47} + 18 q^{49} + 2 q^{53} - q^{55} - 15 q^{59}+ \cdots + 27 q^{97}+O(q^{100}) \) 5 * q - 3 * q^5 - q^7 - 4 * q^11 + 3 * q^13 - 5 * q^17 + 5 * q^19 - q^23 + 10 * q^25 - 4 * q^29 - 16 * q^31 - 6 * q^35 + 7 * q^37 - 7 * q^41 - 3 * q^43 + 7 * q^47 + 18 * q^49 + 2 * q^53 - q^55 - 15 * q^59 + 23 * q^61 - 32 * q^65 - 30 * q^67 - 12 * q^71 + 13 * q^73 + 4 * q^77 - 3 * q^79 + 5 * q^83 + 7 * q^85 - 29 * q^91 - 3 * q^95 + 27 * q^97

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