LMFDB - Embedded newform 10000.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$79.8504020213$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.14884000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 $ x^8 - 11x^6 + 36x^4 - 31*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.13370$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.60278 q^{3} +0.407162 q^{7} +3.77447 q^{9} +O(q^{10})$	$q-2.60278 q^{3} +0.407162 q^{7} +3.77447 q^{9} -2.00000 q^{11} +0.700668 q^{13} -1.58273 q^{17} -4.95078 q^{19} -1.05975 q^{21} +1.20145 q^{23} -2.01577 q^{27} +5.50906 q^{29} -8.20390 q^{31} +5.20556 q^{33} +5.13532 q^{37} -1.82368 q^{39} +7.21619 q^{41} +9.16531 q^{43} -1.27323 q^{47} -6.83422 q^{49} +4.11950 q^{51} +5.07996 q^{53} +12.8858 q^{57} +6.49972 q^{59} -9.42008 q^{61} +1.53682 q^{63} -3.08173 q^{67} -3.12710 q^{69} -6.86639 q^{71} -0.545146 q^{73} -0.814323 q^{77} -5.48159 q^{79} -6.07680 q^{81} +0.974135 q^{83} -14.3389 q^{87} -2.26649 q^{89} +0.285285 q^{91} +21.3529 q^{93} -15.2185 q^{97} -7.54893 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^9	$ 8 q + 4 q^{9} - 16 q^{11} - 10 q^{19} + 6 q^{21} + 20 q^{29} - 16 q^{31} - 18 q^{39} + 26 q^{41} - 14 q^{49} + 4 q^{51} - 30 q^{59} + 6 q^{61} + 8 q^{69} - 46 q^{71} - 10 q^{79} - 32 q^{81} + 30 q^{89} + 14 q^{91} - 8 q^{99}+O(q^{100}) $ 8 * q + 4 * q^9 - 16 * q^11 - 10 * q^19 + 6 * q^21 + 20 * q^29 - 16 * q^31 - 18 * q^39 + 26 * q^41 - 14 * q^49 + 4 * q^51 - 30 * q^59 + 6 * q^61 + 8 * q^69 - 46 * q^71 - 10 * q^79 - 32 * q^81 + 30 * q^89 + 14 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.60278	−1.50272	−0.751358	−	0.659895i	$-0.770601\pi$
$3$	−2.60278	−1.50272	−0.751358	+	0.659895i	$0.770601\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.407162	0.153893	0.0769463	−	0.997035i	$-0.475483\pi$
$7$	0.407162	0.153893	0.0769463	+	0.997035i	$0.475483\pi$
$8$	0	0
$9$	3.77447	1.25816
$10$	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
$13$	0.700668	0.194330	0.0971651	−	0.995268i	$-0.469023\pi$
$13$	0.700668	0.194330	0.0971651	+	0.995268i	$0.469023\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.58273	−0.383869	−0.191934	−	0.981408i	$-0.561476\pi$
$17$	−1.58273	−0.383869	−0.191934	+	0.981408i	$0.561476\pi$
$18$	0	0
$19$	−4.95078	−1.13579	−0.567894	−	0.823102i	$-0.692242\pi$
$19$	−4.95078	−1.13579	−0.567894	+	0.823102i	$0.692242\pi$
$20$	0	0
$21$	−1.05975	−0.231257
$22$	0	0
$23$	1.20145	0.250519	0.125259	−	0.992124i	$-0.460024\pi$
$23$	1.20145	0.250519	0.125259	+	0.992124i	$0.460024\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.01577	−0.387935
$28$	0	0
$29$	5.50906	1.02301	0.511504	−	0.859281i	$-0.329089\pi$
$29$	5.50906	1.02301	0.511504	+	0.859281i	$0.329089\pi$
$30$	0	0
$31$	−8.20390	−1.47346	−0.736732	−	0.676185i	$-0.763632\pi$
$31$	−8.20390	−1.47346	−0.736732	+	0.676185i	$0.763632\pi$
$32$	0	0
$33$	5.20556	0.906172
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.13532	0.844241	0.422121	−	0.906540i	$-0.361286\pi$
$37$	5.13532	0.844241	0.422121	+	0.906540i	$0.361286\pi$
$38$	0	0
$39$	−1.82368	−0.292023
$40$	0	0
$41$	7.21619	1.12698	0.563489	−	0.826123i	$-0.309459\pi$
$41$	7.21619	1.12698	0.563489	+	0.826123i	$0.309459\pi$
$42$	0	0
$43$	9.16531	1.39770	0.698848	−	0.715270i	$-0.253696\pi$
$43$	9.16531	1.39770	0.698848	+	0.715270i	$0.253696\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.27323	−0.185720	−0.0928602	−	0.995679i	$-0.529601\pi$
$47$	−1.27323	−0.185720	−0.0928602	+	0.995679i	$0.529601\pi$
$48$	0	0
$49$	−6.83422	−0.976317
$50$	0	0
$51$	4.11950	0.576846
$52$	0	0
$53$	5.07996	0.697786	0.348893	−	0.937162i	$-0.386558\pi$
$53$	5.07996	0.697786	0.348893	+	0.937162i	$0.386558\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.8858	1.70677
$58$	0	0
$59$	6.49972	0.846191	0.423096	−	0.906085i	$-0.360943\pi$
$59$	6.49972	0.846191	0.423096	+	0.906085i	$0.360943\pi$
$60$	0	0
$61$	−9.42008	−1.20612	−0.603059	−	0.797697i	$-0.706052\pi$
$61$	−9.42008	−1.20612	−0.603059	+	0.797697i	$0.706052\pi$
$62$	0	0
$63$	1.53682	0.193621
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.08173	−0.376493	−0.188247	−	0.982122i	$-0.560280\pi$
$67$	−3.08173	−0.376493	−0.188247	+	0.982122i	$0.560280\pi$
$68$	0	0
$69$	−3.12710	−0.376458
$70$	0	0
$71$	−6.86639	−0.814891	−0.407445	−	0.913230i	$-0.633580\pi$
$71$	−6.86639	−0.814891	−0.407445	+	0.913230i	$0.633580\pi$
$72$	0	0
$73$	−0.545146	−0.0638045	−0.0319022	−	0.999491i	$-0.510157\pi$
$73$	−0.545146	−0.0638045	−0.0319022	+	0.999491i	$0.510157\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.814323	−0.0928007
$78$	0	0
$79$	−5.48159	−0.616727	−0.308363	−	0.951269i	$-0.599781\pi$
$79$	−5.48159	−0.616727	−0.308363	+	0.951269i	$0.599781\pi$
$80$	0	0
$81$	−6.07680	−0.675200
$82$	0	0
$83$	0.974135	0.106925	0.0534626	−	0.998570i	$-0.482974\pi$
$83$	0.974135	0.106925	0.0534626	+	0.998570i	$0.482974\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−14.3389	−1.53729
$88$	0	0
$89$	−2.26649	−0.240247	−0.120124	−	0.992759i	$-0.538329\pi$
$89$	−2.26649	−0.240247	−0.120124	+	0.992759i	$0.538329\pi$
$90$	0	0
$91$	0.285285	0.0299060
$92$	0	0
$93$	21.3529	2.21420
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.2185	−1.54520	−0.772600	−	0.634893i	$-0.781044\pi$
$97$	−15.2185	−1.54520	−0.772600	+	0.634893i	$0.781044\pi$
$98$	0	0
$99$	−7.54893	−0.758696
$100$	0	0
$101$	18.3965	1.83052	0.915261	−	0.402861i	$-0.131984\pi$
$101$	18.3965	1.83052	0.915261	+	0.402861i	$0.131984\pi$
$102$	0	0
$103$	11.8842	1.17099	0.585495	−	0.810676i	$-0.300900\pi$
$103$	11.8842	1.17099	0.585495	+	0.810676i	$0.300900\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.754919	0.0729808	0.0364904	−	0.999334i	$-0.488382\pi$
$107$	0.754919	0.0729808	0.0364904	+	0.999334i	$0.488382\pi$
$108$	0	0
$109$	−9.15752	−0.877131	−0.438566	−	0.898699i	$-0.644513\pi$
$109$	−9.15752	−0.877131	−0.438566	+	0.898699i	$0.644513\pi$
$110$	0	0
$111$	−13.3661	−1.26865
$112$	0	0
$113$	12.8399	1.20788	0.603938	−	0.797032i	$-0.293598\pi$
$113$	12.8399	1.20788	0.603938	+	0.797032i	$0.293598\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.64465	0.244498
$118$	0	0
$119$	−0.644428	−0.0590746
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−18.7821	−1.69353
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.10517	0.630482	0.315241	−	0.949012i	$-0.397915\pi$
$127$	7.10517	0.630482	0.315241	+	0.949012i	$0.397915\pi$
$128$	0	0
$129$	−23.8553	−2.10034
$130$	0	0
$131$	−2.39425	−0.209187	−0.104593	−	0.994515i	$-0.533354\pi$
$131$	−2.39425	−0.209187	−0.104593	+	0.994515i	$0.533354\pi$
$132$	0	0
$133$	−2.01577	−0.174789
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.0193	−1.62493	−0.812466	−	0.583009i	$-0.801875\pi$
$137$	−19.0193	−1.62493	−0.812466	+	0.583009i	$0.801875\pi$
$138$	0	0
$139$	6.73470	0.571230	0.285615	−	0.958344i	$-0.407802\pi$
$139$	6.73470	0.571230	0.285615	+	0.958344i	$0.407802\pi$
$140$	0	0
$141$	3.31395	0.279085
$142$	0	0
$143$	−1.40134	−0.117186
$144$	0	0
$145$	0	0
$146$	0	0
$147$	17.7880	1.46713
$148$	0	0
$149$	−0.720492	−0.0590250	−0.0295125	−	0.999564i	$-0.509395\pi$
$149$	−0.720492	−0.0590250	−0.0295125	+	0.999564i	$0.509395\pi$
$150$	0	0
$151$	15.5178	1.26282	0.631412	−	0.775447i	$-0.282476\pi$
$151$	15.5178	1.26282	0.631412	+	0.775447i	$0.282476\pi$
$152$	0	0
$153$	−5.97397	−0.482967
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.78418	0.222202	0.111101	−	0.993809i	$-0.464562\pi$
$157$	2.78418	0.222202	0.111101	+	0.993809i	$0.464562\pi$
$158$	0	0
$159$	−13.2220	−1.04857
$160$	0	0
$161$	0.489182	0.0385530
$162$	0	0
$163$	24.2074	1.89607	0.948034	−	0.318170i	$-0.103068\pi$
$163$	24.2074	1.89607	0.948034	+	0.318170i	$0.103068\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.0462	1.47384	0.736919	−	0.675982i	$-0.236280\pi$
$167$	19.0462	1.47384	0.736919	+	0.675982i	$0.236280\pi$
$168$	0	0
$169$	−12.5091	−0.962236
$170$	0	0
$171$	−18.6866	−1.42900
$172$	0	0
$173$	−13.2470	−1.00715	−0.503577	−	0.863950i	$-0.667983\pi$
$173$	−13.2470	−1.00715	−0.503577	+	0.863950i	$0.667983\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−16.9173	−1.27159
$178$	0	0
$179$	12.2250	0.913737	0.456869	−	0.889534i	$-0.348971\pi$
$179$	12.2250	0.913737	0.456869	+	0.889534i	$0.348971\pi$
$180$	0	0
$181$	10.9168	0.811439	0.405719	−	0.913998i	$-0.367021\pi$
$181$	10.9168	0.811439	0.405719	+	0.913998i	$0.367021\pi$
$182$	0	0
$183$	24.5184	1.81245
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.16546	0.231482
$188$	0	0
$189$	−0.820743	−0.0597003
$190$	0	0
$191$	−1.56763	−0.113430	−0.0567148	−	0.998390i	$-0.518063\pi$
$191$	−1.56763	−0.113430	−0.0567148	+	0.998390i	$0.518063\pi$
$192$	0	0
$193$	1.65786	0.119335	0.0596675	−	0.998218i	$-0.480996\pi$
$193$	1.65786	0.119335	0.0596675	+	0.998218i	$0.480996\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.2831	0.946380	0.473190	−	0.880960i	$-0.343102\pi$
$197$	13.2831	0.946380	0.473190	+	0.880960i	$0.343102\pi$
$198$	0	0
$199$	−12.1025	−0.857921	−0.428960	−	0.903323i	$-0.641120\pi$
$199$	−12.1025	−0.857921	−0.428960	+	0.903323i	$0.641120\pi$
$200$	0	0
$201$	8.02107	0.565763
$202$	0	0
$203$	2.24308	0.157433
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.53482	0.315192
$208$	0	0
$209$	9.90157	0.684906
$210$	0	0
$211$	6.70061	0.461289	0.230644	−	0.973038i	$-0.425917\pi$
$211$	6.70061	0.461289	0.230644	+	0.973038i	$0.425917\pi$
$212$	0	0
$213$	17.8717	1.22455
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.34031	−0.226755
$218$	0	0
$219$	1.41889	0.0958800
$220$	0	0
$221$	−1.10897	−0.0745973
$222$	0	0
$223$	10.5917	0.709272	0.354636	−	0.935004i	$-0.384605\pi$
$223$	10.5917	0.709272	0.354636	+	0.935004i	$0.384605\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−19.6112	−1.30164	−0.650822	−	0.759231i	$-0.725575\pi$
$227$	−19.6112	−1.30164	−0.650822	+	0.759231i	$0.725575\pi$
$228$	0	0
$229$	8.83898	0.584096	0.292048	−	0.956404i	$-0.405663\pi$
$229$	8.83898	0.584096	0.292048	+	0.956404i	$0.405663\pi$
$230$	0	0
$231$	2.11950	0.139453
$232$	0	0
$233$	11.0624	0.724723	0.362362	−	0.932038i	$-0.381971\pi$
$233$	11.0624	0.724723	0.362362	+	0.932038i	$0.381971\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	14.2674	0.926765
$238$	0	0
$239$	−16.1055	−1.04178	−0.520888	−	0.853625i	$-0.674399\pi$
$239$	−16.1055	−1.04178	−0.520888	+	0.853625i	$0.674399\pi$
$240$	0	0
$241$	5.98873	0.385768	0.192884	−	0.981222i	$-0.438216\pi$
$241$	5.98873	0.385768	0.192884	+	0.981222i	$0.438216\pi$
$242$	0	0
$243$	21.8639	1.40257
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.46885	−0.220718
$248$	0	0
$249$	−2.53546	−0.160678
$250$	0	0
$251$	−3.73176	−0.235547	−0.117773	−	0.993041i	$-0.537576\pi$
$251$	−3.73176	−0.235547	−0.117773	+	0.993041i	$0.537576\pi$
$252$	0	0
$253$	−2.40289	−0.151068
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−23.5935	−1.47172	−0.735862	−	0.677132i	$-0.763223\pi$
$257$	−23.5935	−1.47172	−0.735862	+	0.677132i	$0.763223\pi$
$258$	0	0
$259$	2.09090	0.129922
$260$	0	0
$261$	20.7938	1.28710
$262$	0	0
$263$	6.64126	0.409518	0.204759	−	0.978812i	$-0.434359\pi$
$263$	6.64126	0.409518	0.204759	+	0.978812i	$0.434359\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.89917	0.361023
$268$	0	0
$269$	−13.0439	−0.795300	−0.397650	−	0.917537i	$-0.630174\pi$
$269$	−13.0439	−0.795300	−0.397650	+	0.917537i	$0.630174\pi$
$270$	0	0
$271$	11.8488	0.719762	0.359881	−	0.932998i	$-0.382817\pi$
$271$	11.8488	0.719762	0.359881	+	0.932998i	$0.382817\pi$
$272$	0	0
$273$	−0.742534	−0.0449402
$274$	0	0
$275$	0	0
$276$	0	0
$277$	16.9559	1.01878	0.509390	−	0.860536i	$-0.329871\pi$
$277$	16.9559	1.01878	0.509390	+	0.860536i	$0.329871\pi$
$278$	0	0
$279$	−30.9653	−1.85385
$280$	0	0
$281$	−5.07498	−0.302748	−0.151374	−	0.988477i	$-0.548370\pi$
$281$	−5.07498	−0.302748	−0.151374	+	0.988477i	$0.548370\pi$
$282$	0	0
$283$	−11.8248	−0.702914	−0.351457	−	0.936204i	$-0.614314\pi$
$283$	−11.8248	−0.702914	−0.351457	+	0.936204i	$0.614314\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.93815	0.173434
$288$	0	0
$289$	−14.4950	−0.852645
$290$	0	0
$291$	39.6103	2.32200
$292$	0	0
$293$	−19.4348	−1.13540	−0.567698	−	0.823237i	$-0.692166\pi$
$293$	−19.4348	−1.13540	−0.567698	+	0.823237i	$0.692166\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.03154	0.233934
$298$	0	0
$299$	0.841814	0.0486834
$300$	0	0
$301$	3.73176	0.215095
$302$	0	0
$303$	−47.8821	−2.75076
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−25.4169	−1.45062	−0.725310	−	0.688423i	$-0.758303\pi$
$307$	−25.4169	−1.45062	−0.725310	+	0.688423i	$0.758303\pi$
$308$	0	0
$309$	−30.9321	−1.75967
$310$	0	0
$311$	−15.8578	−0.899212	−0.449606	−	0.893227i	$-0.648436\pi$
$311$	−15.8578	−0.899212	−0.449606	+	0.893227i	$0.648436\pi$
$312$	0	0
$313$	7.01644	0.396592	0.198296	−	0.980142i	$-0.436459\pi$
$313$	7.01644	0.396592	0.198296	+	0.980142i	$0.436459\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.0337	0.732044	0.366022	−	0.930606i	$-0.380720\pi$
$317$	13.0337	0.732044	0.366022	+	0.930606i	$0.380720\pi$
$318$	0	0
$319$	−11.0181	−0.616897
$320$	0	0
$321$	−1.96489	−0.109669
$322$	0	0
$323$	7.83576	0.435994
$324$	0	0
$325$	0	0
$326$	0	0
$327$	23.8350	1.31808
$328$	0	0
$329$	−0.518412	−0.0285810
$330$	0	0
$331$	17.6439	0.969794	0.484897	−	0.874571i	$-0.338857\pi$
$331$	17.6439	0.969794	0.484897	+	0.874571i	$0.338857\pi$
$332$	0	0
$333$	19.3831	1.06219
$334$	0	0
$335$	0	0
$336$	0	0
$337$	24.8495	1.35364	0.676819	−	0.736149i	$-0.263358\pi$
$337$	24.8495	1.35364	0.676819	+	0.736149i	$0.263358\pi$
$338$	0	0
$339$	−33.4194	−1.81509
$340$	0	0
$341$	16.4078	0.888532
$342$	0	0
$343$	−5.63276	−0.304141
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.2775	−0.712773	−0.356386	−	0.934339i	$-0.615991\pi$
$347$	−13.2775	−0.712773	−0.356386	+	0.934339i	$0.615991\pi$
$348$	0	0
$349$	−18.1283	−0.970385	−0.485192	−	0.874407i	$-0.661250\pi$
$349$	−18.1283	−0.970385	−0.485192	+	0.874407i	$0.661250\pi$
$350$	0	0
$351$	−1.41238	−0.0753875
$352$	0	0
$353$	−17.7042	−0.942302	−0.471151	−	0.882053i	$-0.656161\pi$
$353$	−17.7042	−0.942302	−0.471151	+	0.882053i	$0.656161\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.67730	0.0887723
$358$	0	0
$359$	−32.5443	−1.71762	−0.858812	−	0.512290i	$-0.828797\pi$
$359$	−32.5443	−1.71762	−0.858812	+	0.512290i	$0.828797\pi$
$360$	0	0
$361$	5.51025	0.290013
$362$	0	0
$363$	18.2195	0.956274
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−32.9481	−1.71988	−0.859939	−	0.510397i	$-0.829499\pi$
$367$	−32.9481	−1.71988	−0.859939	+	0.510397i	$0.829499\pi$
$368$	0	0
$369$	27.2373	1.41791
$370$	0	0
$371$	2.06837	0.107384
$372$	0	0
$373$	3.46648	0.179487	0.0897436	−	0.995965i	$-0.471395\pi$
$373$	3.46648	0.179487	0.0897436	+	0.995965i	$0.471395\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.86002	0.198801
$378$	0	0
$379$	1.92863	0.0990670	0.0495335	−	0.998772i	$-0.484227\pi$
$379$	1.92863	0.0990670	0.0495335	+	0.998772i	$0.484227\pi$
$380$	0	0
$381$	−18.4932	−0.947436
$382$	0	0
$383$	4.93003	0.251913	0.125956	−	0.992036i	$-0.459800\pi$
$383$	4.93003	0.251913	0.125956	+	0.992036i	$0.459800\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	34.5942	1.75852
$388$	0	0
$389$	13.3860	0.678697	0.339348	−	0.940661i	$-0.389793\pi$
$389$	13.3860	0.678697	0.339348	+	0.940661i	$0.389793\pi$
$390$	0	0
$391$	−1.90157	−0.0961663
$392$	0	0
$393$	6.23172	0.314349
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.13887	0.107347	0.0536733	−	0.998559i	$-0.482907\pi$
$397$	2.13887	0.107347	0.0536733	+	0.998559i	$0.482907\pi$
$398$	0	0
$399$	5.24660	0.262659
$400$	0	0
$401$	−26.8213	−1.33939	−0.669696	−	0.742635i	$-0.733576\pi$
$401$	−26.8213	−1.33939	−0.669696	+	0.742635i	$0.733576\pi$
$402$	0	0
$403$	−5.74821	−0.286339
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.2706	−0.509097
$408$	0	0
$409$	13.9982	0.692169	0.346084	−	0.938203i	$-0.387511\pi$
$409$	13.9982	0.692169	0.346084	+	0.938203i	$0.387511\pi$
$410$	0	0
$411$	49.5032	2.44181
$412$	0	0
$413$	2.64644	0.130223
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−17.5290	−0.858397
$418$	0	0
$419$	−30.9020	−1.50966	−0.754831	−	0.655919i	$-0.772281\pi$
$419$	−30.9020	−1.50966	−0.754831	+	0.655919i	$0.772281\pi$
$420$	0	0
$421$	−8.23974	−0.401581	−0.200790	−	0.979634i	$-0.564351\pi$
$421$	−8.23974	−0.401581	−0.200790	+	0.979634i	$0.564351\pi$
$422$	0	0
$423$	−4.80578	−0.233665
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−3.83550	−0.185613
$428$	0	0
$429$	3.64737	0.176097
$430$	0	0
$431$	−27.1482	−1.30768	−0.653841	−	0.756632i	$-0.726843\pi$
$431$	−27.1482	−1.30768	−0.653841	+	0.756632i	$0.726843\pi$
$432$	0	0
$433$	−21.4905	−1.03277	−0.516383	−	0.856357i	$-0.672722\pi$
$433$	−21.4905	−1.03277	−0.516383	+	0.856357i	$0.672722\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.94810	−0.284536
$438$	0	0
$439$	25.7956	1.23116	0.615578	−	0.788076i	$-0.288922\pi$
$439$	25.7956	1.23116	0.615578	+	0.788076i	$0.288922\pi$
$440$	0	0
$441$	−25.7955	−1.22836
$442$	0	0
$443$	3.18479	0.151314	0.0756570	−	0.997134i	$-0.475895\pi$
$443$	3.18479	0.151314	0.0756570	+	0.997134i	$0.475895\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.87528	0.0886978
$448$	0	0
$449$	36.0785	1.70265	0.851325	−	0.524639i	$-0.175800\pi$
$449$	36.0785	1.70265	0.851325	+	0.524639i	$0.175800\pi$
$450$	0	0
$451$	−14.4324	−0.679594
$452$	0	0
$453$	−40.3896	−1.89767
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−25.5245	−1.19399	−0.596994	−	0.802246i	$-0.703638\pi$
$457$	−25.5245	−1.19399	−0.596994	+	0.802246i	$0.703638\pi$
$458$	0	0
$459$	3.19042	0.148916
$460$	0	0
$461$	−16.6392	−0.774963	−0.387482	−	0.921877i	$-0.626655\pi$
$461$	−16.6392	−0.774963	−0.387482	+	0.921877i	$0.626655\pi$
$462$	0	0
$463$	17.8178	0.828062	0.414031	−	0.910263i	$-0.364120\pi$
$463$	17.8178	0.828062	0.414031	+	0.910263i	$0.364120\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14.2480	−0.659321	−0.329660	−	0.944100i	$-0.606934\pi$
$467$	−14.2480	−0.659321	−0.329660	+	0.944100i	$0.606934\pi$
$468$	0	0
$469$	−1.25476	−0.0579395
$470$	0	0
$471$	−7.24660	−0.333906
$472$	0	0
$473$	−18.3306	−0.842843
$474$	0	0
$475$	0	0
$476$	0	0
$477$	19.1742	0.877924
$478$	0	0
$479$	−2.90864	−0.132899	−0.0664496	−	0.997790i	$-0.521167\pi$
$479$	−2.90864	−0.132899	−0.0664496	+	0.997790i	$0.521167\pi$
$480$	0	0
$481$	3.59815	0.164062
$482$	0	0
$483$	−1.27323	−0.0579342
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−17.0652	−0.773298	−0.386649	−	0.922227i	$-0.626368\pi$
$487$	−17.0652	−0.773298	−0.386649	+	0.922227i	$0.626368\pi$
$488$	0	0
$489$	−63.0065	−2.84925
$490$	0	0
$491$	−30.7768	−1.38894	−0.694470	−	0.719522i	$-0.744361\pi$
$491$	−30.7768	−1.38894	−0.694470	+	0.719522i	$0.744361\pi$
$492$	0	0
$493$	−8.71937	−0.392701
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.79573	−0.125406
$498$	0	0
$499$	11.8824	0.531927	0.265964	−	0.963983i	$-0.414310\pi$
$499$	11.8824	0.531927	0.265964	+	0.963983i	$0.414310\pi$
$500$	0	0
$501$	−49.5730	−2.21476
$502$	0	0
$503$	−2.97823	−0.132793	−0.0663964	−	0.997793i	$-0.521150\pi$
$503$	−2.97823	−0.132793	−0.0663964	+	0.997793i	$0.521150\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	32.5584	1.44597
$508$	0	0
$509$	−36.7159	−1.62740	−0.813702	−	0.581282i	$-0.802551\pi$
$509$	−36.7159	−1.62740	−0.813702	+	0.581282i	$0.802551\pi$
$510$	0	0
$511$	−0.221962	−0.00981904
$512$	0	0
$513$	9.97963	0.440612
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.54647	0.111994
$518$	0	0
$519$	34.4792	1.51347
$520$	0	0
$521$	2.50039	0.109544	0.0547720	−	0.998499i	$-0.482557\pi$
$521$	2.50039	0.109544	0.0547720	+	0.998499i	$0.482557\pi$
$522$	0	0
$523$	−42.1258	−1.84203	−0.921017	−	0.389523i	$-0.872640\pi$
$523$	−42.1258	−1.84203	−0.921017	+	0.389523i	$0.872640\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.9846	0.565617
$528$	0	0
$529$	−21.5565	−0.937240
$530$	0	0
$531$	24.5330	1.06464
$532$	0	0
$533$	5.05615	0.219006
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−31.8189	−1.37309
$538$	0	0
$539$	13.6684	0.588741
$540$	0	0
$541$	−27.3492	−1.17583	−0.587916	−	0.808922i	$-0.700052\pi$
$541$	−27.3492	−1.17583	−0.587916	+	0.808922i	$0.700052\pi$
$542$	0	0
$543$	−28.4140	−1.21936
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−20.3392	−0.869640	−0.434820	−	0.900517i	$-0.643188\pi$
$547$	−20.3392	−0.869640	−0.434820	+	0.900517i	$0.643188\pi$
$548$	0	0
$549$	−35.5558	−1.51748
$550$	0	0
$551$	−27.2742	−1.16192
$552$	0	0
$553$	−2.23189	−0.0949097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.2605	1.19744	0.598718	−	0.800960i	$-0.295677\pi$
$557$	28.2605	1.19744	0.598718	+	0.800960i	$0.295677\pi$
$558$	0	0
$559$	6.42184	0.271615
$560$	0	0
$561$	−8.23901	−0.347851
$562$	0	0
$563$	−12.9254	−0.544742	−0.272371	−	0.962192i	$-0.587808\pi$
$563$	−12.9254	−0.544742	−0.272371	+	0.962192i	$0.587808\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.47424	−0.103908
$568$	0	0
$569$	−3.08443	−0.129306	−0.0646531	−	0.997908i	$-0.520594\pi$
$569$	−3.08443	−0.129306	−0.0646531	+	0.997908i	$0.520594\pi$
$570$	0	0
$571$	4.12766	0.172737	0.0863687	−	0.996263i	$-0.472474\pi$
$571$	4.12766	0.172737	0.0863687	+	0.996263i	$0.472474\pi$
$572$	0	0
$573$	4.08019	0.170453
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.35570	0.264591	0.132296	−	0.991210i	$-0.457765\pi$
$577$	6.35570	0.264591	0.132296	+	0.991210i	$0.457765\pi$
$578$	0	0
$579$	−4.31503	−0.179327
$580$	0	0
$581$	0.396630	0.0164550
$582$	0	0
$583$	−10.1599	−0.420781
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.1040	0.871054	0.435527	−	0.900176i	$-0.356562\pi$
$587$	21.1040	0.871054	0.435527	+	0.900176i	$0.356562\pi$
$588$	0	0
$589$	40.6157	1.67354
$590$	0	0
$591$	−34.5729	−1.42214
$592$	0	0
$593$	21.6529	0.889177	0.444589	−	0.895735i	$-0.353350\pi$
$593$	21.6529	0.889177	0.444589	+	0.895735i	$0.353350\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	31.5000	1.28921
$598$	0	0
$599$	3.38501	0.138308	0.0691539	−	0.997606i	$-0.477970\pi$
$599$	3.38501	0.138308	0.0691539	+	0.997606i	$0.477970\pi$
$600$	0	0
$601$	28.8265	1.17586	0.587928	−	0.808913i	$-0.299944\pi$
$601$	28.8265	1.17586	0.587928	+	0.808913i	$0.299944\pi$
$602$	0	0
$603$	−11.6319	−0.473687
$604$	0	0
$605$	0	0
$606$	0	0
$607$	15.6708	0.636059	0.318029	−	0.948081i	$-0.396979\pi$
$607$	15.6708	0.636059	0.318029	+	0.948081i	$0.396979\pi$
$608$	0	0
$609$	−5.83824	−0.236578
$610$	0	0
$611$	−0.892114	−0.0360911
$612$	0	0
$613$	38.2005	1.54290	0.771451	−	0.636288i	$-0.219531\pi$
$613$	38.2005	1.54290	0.771451	+	0.636288i	$0.219531\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.2318	−0.532691	−0.266346	−	0.963878i	$-0.585816\pi$
$617$	−13.2318	−0.532691	−0.266346	+	0.963878i	$0.585816\pi$
$618$	0	0
$619$	−6.04463	−0.242954	−0.121477	−	0.992594i	$-0.538763\pi$
$619$	−6.04463	−0.242954	−0.121477	+	0.992594i	$0.538763\pi$
$620$	0	0
$621$	−2.42184	−0.0971849
$622$	0	0
$623$	−0.922826	−0.0369722
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−25.7716	−1.02922
$628$	0	0
$629$	−8.12783	−0.324078
$630$	0	0
$631$	37.7855	1.50422	0.752108	−	0.659040i	$-0.229037\pi$
$631$	37.7855	1.50422	0.752108	+	0.659040i	$0.229037\pi$
$632$	0	0
$633$	−17.4402	−0.693186
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.78852	−0.189728
$638$	0	0
$639$	−25.9170	−1.02526
$640$	0	0
$641$	−20.8570	−0.823802	−0.411901	−	0.911229i	$-0.635135\pi$
$641$	−20.8570	−0.823802	−0.411901	+	0.911229i	$0.635135\pi$
$642$	0	0
$643$	−37.5552	−1.48103	−0.740516	−	0.672039i	$-0.765419\pi$
$643$	−37.5552	−1.48103	−0.740516	+	0.672039i	$0.765419\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27.4003	−1.07722	−0.538608	−	0.842557i	$-0.681049\pi$
$647$	−27.4003	−1.07722	−0.538608	+	0.842557i	$0.681049\pi$
$648$	0	0
$649$	−12.9994	−0.510272
$650$	0	0
$651$	8.69410	0.340749
$652$	0	0
$653$	−23.0461	−0.901864	−0.450932	−	0.892558i	$-0.648908\pi$
$653$	−23.0461	−0.901864	−0.450932	+	0.892558i	$0.648908\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.05763	−0.0802760
$658$	0	0
$659$	−20.4837	−0.797931	−0.398966	−	0.916966i	$-0.630631\pi$
$659$	−20.4837	−0.797931	−0.398966	+	0.916966i	$0.630631\pi$
$660$	0	0
$661$	13.9900	0.544147	0.272074	−	0.962276i	$-0.412291\pi$
$661$	13.9900	0.544147	0.272074	+	0.962276i	$0.412291\pi$
$662$	0	0
$663$	2.88640	0.112099
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.61884	0.256283
$668$	0	0
$669$	−27.5678	−1.06583
$670$	0	0
$671$	18.8402	0.727317
$672$	0	0
$673$	−7.84572	−0.302430	−0.151215	−	0.988501i	$-0.548319\pi$
$673$	−7.84572	−0.302430	−0.151215	+	0.988501i	$0.548319\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14.3983	−0.553371	−0.276686	−	0.960960i	$-0.589236\pi$
$677$	−14.3983	−0.553371	−0.276686	+	0.960960i	$0.589236\pi$
$678$	0	0
$679$	−6.19637	−0.237795
$680$	0	0
$681$	51.0437	1.95600
$682$	0	0
$683$	−12.6740	−0.484956	−0.242478	−	0.970157i	$-0.577960\pi$
$683$	−12.6740	−0.484956	−0.242478	+	0.970157i	$0.577960\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−23.0059	−0.877731
$688$	0	0
$689$	3.55937	0.135601
$690$	0	0
$691$	−5.74530	−0.218562	−0.109281	−	0.994011i	$-0.534855\pi$
$691$	−5.74530	−0.218562	−0.109281	+	0.994011i	$0.534855\pi$
$692$	0	0
$693$	−3.07364	−0.116758
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−11.4213	−0.432612
$698$	0	0
$699$	−28.7931	−1.08905
$700$	0	0
$701$	−30.5834	−1.15512	−0.577560	−	0.816348i	$-0.695995\pi$
$701$	−30.5834	−1.15512	−0.577560	+	0.816348i	$0.695995\pi$
$702$	0	0
$703$	−25.4238	−0.958879
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.49036	0.281704
$708$	0	0
$709$	−27.0696	−1.01662	−0.508310	−	0.861174i	$-0.669730\pi$
$709$	−27.0696	−1.01662	−0.508310	+	0.861174i	$0.669730\pi$
$710$	0	0
$711$	−20.6901	−0.775938
$712$	0	0
$713$	−9.85653	−0.369130
$714$	0	0
$715$	0	0
$716$	0	0
$717$	41.9190	1.56549
$718$	0	0
$719$	16.7066	0.623052	0.311526	−	0.950238i	$-0.399160\pi$
$719$	16.7066	0.623052	0.311526	+	0.950238i	$0.399160\pi$
$720$	0	0
$721$	4.83881	0.180207
$722$	0	0
$723$	−15.5874	−0.579700
$724$	0	0
$725$	0	0
$726$	0	0
$727$	11.7006	0.433951	0.216975	−	0.976177i	$-0.430381\pi$
$727$	11.7006	0.433951	0.216975	+	0.976177i	$0.430381\pi$
$728$	0	0
$729$	−38.6765	−1.43246
$730$	0	0
$731$	−14.5062	−0.536532
$732$	0	0
$733$	−34.1078	−1.25980	−0.629901	−	0.776676i	$-0.716904\pi$
$733$	−34.1078	−1.25980	−0.629901	+	0.776676i	$0.716904\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.16346	0.227034
$738$	0	0
$739$	−5.74517	−0.211340	−0.105670	−	0.994401i	$-0.533699\pi$
$739$	−5.74517	−0.211340	−0.105670	+	0.994401i	$0.533699\pi$
$740$	0	0
$741$	9.02866	0.331676
$742$	0	0
$743$	36.4348	1.33666	0.668332	−	0.743863i	$-0.267009\pi$
$743$	36.4348	1.33666	0.668332	+	0.743863i	$0.267009\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.67684	0.134528
$748$	0	0
$749$	0.307374	0.0112312
$750$	0	0
$751$	−1.48912	−0.0543387	−0.0271693	−	0.999631i	$-0.508649\pi$
$751$	−1.48912	−0.0543387	−0.0271693	+	0.999631i	$0.508649\pi$
$752$	0	0
$753$	9.71296	0.353960
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.53316	0.201106	0.100553	−	0.994932i	$-0.467939\pi$
$757$	5.53316	0.201106	0.100553	+	0.994932i	$0.467939\pi$
$758$	0	0
$759$	6.25420	0.227013
$760$	0	0
$761$	−18.6697	−0.676776	−0.338388	−	0.941007i	$-0.609882\pi$
$761$	−18.6697	−0.676776	−0.338388	+	0.941007i	$0.609882\pi$
$762$	0	0
$763$	−3.72859	−0.134984
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.55414	0.164441
$768$	0	0
$769$	−13.1433	−0.473960	−0.236980	−	0.971515i	$-0.576158\pi$
$769$	−13.1433	−0.473960	−0.236980	+	0.971515i	$0.576158\pi$
$770$	0	0
$771$	61.4088	2.21158
$772$	0	0
$773$	−27.9736	−1.00614	−0.503070	−	0.864246i	$-0.667796\pi$
$773$	−27.9736	−1.00614	−0.503070	+	0.864246i	$0.667796\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−5.44217	−0.195237
$778$	0	0
$779$	−35.7258	−1.28001
$780$	0	0
$781$	13.7328	0.491398
$782$	0	0
$783$	−11.1050	−0.396860
$784$	0	0
$785$	0	0
$786$	0	0
$787$	33.3670	1.18940	0.594702	−	0.803946i	$-0.297270\pi$
$787$	33.3670	1.18940	0.594702	+	0.803946i	$0.297270\pi$
$788$	0	0
$789$	−17.2857	−0.615389
$790$	0	0
$791$	5.22791	0.185883
$792$	0	0
$793$	−6.60035	−0.234385
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−37.3986	−1.32473	−0.662364	−	0.749183i	$-0.730447\pi$
$797$	−37.3986	−1.32473	−0.662364	+	0.749183i	$0.730447\pi$
$798$	0	0
$799$	2.01519	0.0712923
$800$	0	0
$801$	−8.55478	−0.302268
$802$	0	0
$803$	1.09029	0.0384756
$804$	0	0
$805$	0	0
$806$	0	0
$807$	33.9504	1.19511
$808$	0	0
$809$	33.6017	1.18137	0.590687	−	0.806901i	$-0.298857\pi$
$809$	33.6017	1.18137	0.590687	+	0.806901i	$0.298857\pi$
$810$	0	0
$811$	38.0056	1.33456	0.667278	−	0.744809i	$-0.267459\pi$
$811$	38.0056	1.33456	0.667278	+	0.744809i	$0.267459\pi$
$812$	0	0
$813$	−30.8398	−1.08160
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−45.3755	−1.58749
$818$	0	0
$819$	1.07680	0.0376264
$820$	0	0
$821$	24.9332	0.870176	0.435088	−	0.900388i	$-0.356717\pi$
$821$	24.9332	0.870176	0.435088	+	0.900388i	$0.356717\pi$
$822$	0	0
$823$	−25.4606	−0.887499	−0.443750	−	0.896151i	$-0.646352\pi$
$823$	−25.4606	−0.887499	−0.443750	+	0.896151i	$0.646352\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.3387	−1.12453	−0.562263	−	0.826958i	$-0.690069\pi$
$827$	−32.3387	−1.12453	−0.562263	+	0.826958i	$0.690069\pi$
$828$	0	0
$829$	24.9812	0.867633	0.433816	−	0.901001i	$-0.357167\pi$
$829$	24.9812	0.867633	0.433816	+	0.901001i	$0.357167\pi$
$830$	0	0
$831$	−44.1324	−1.53094
$832$	0	0
$833$	10.8167	0.374778
$834$	0	0
$835$	0	0
$836$	0	0
$837$	16.5372	0.571608
$838$	0	0
$839$	−19.3526	−0.668125	−0.334062	−	0.942551i	$-0.608420\pi$
$839$	−19.3526	−0.668125	−0.334062	+	0.942551i	$0.608420\pi$
$840$	0	0
$841$	1.34980	0.0465447
$842$	0	0
$843$	13.2091	0.454944
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.85013	−0.0979317
$848$	0	0
$849$	30.7775	1.05628
$850$	0	0
$851$	6.16980	0.211498
$852$	0	0
$853$	26.7876	0.917190	0.458595	−	0.888645i	$-0.348353\pi$
$853$	26.7876	0.917190	0.458595	+	0.888645i	$0.348353\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−47.5186	−1.62320	−0.811602	−	0.584210i	$-0.801404\pi$
$857$	−47.5186	−1.62320	−0.811602	+	0.584210i	$0.801404\pi$
$858$	0	0
$859$	−11.5290	−0.393365	−0.196682	−	0.980467i	$-0.563017\pi$
$859$	−11.5290	−0.393365	−0.196682	+	0.980467i	$0.563017\pi$
$860$	0	0
$861$	−7.64737	−0.260622
$862$	0	0
$863$	−18.0057	−0.612922	−0.306461	−	0.951883i	$-0.599145\pi$
$863$	−18.0057	−0.612922	−0.306461	+	0.951883i	$0.599145\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	37.7272	1.28128
$868$	0	0
$869$	10.9632	0.371900
$870$	0	0
$871$	−2.15927	−0.0731641
$872$	0	0
$873$	−57.4415	−1.94410
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−17.0973	−0.577335	−0.288668	−	0.957429i	$-0.593212\pi$
$877$	−17.0973	−0.577335	−0.288668	+	0.957429i	$0.593212\pi$
$878$	0	0
$879$	50.5846	1.70618
$880$	0	0
$881$	−38.0291	−1.28123	−0.640617	−	0.767861i	$-0.721321\pi$
$881$	−38.0291	−1.28123	−0.640617	+	0.767861i	$0.721321\pi$
$882$	0	0
$883$	36.8494	1.24008	0.620040	−	0.784570i	$-0.287116\pi$
$883$	36.8494	1.24008	0.620040	+	0.784570i	$0.287116\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−29.8738	−1.00306	−0.501532	−	0.865139i	$-0.667230\pi$
$887$	−29.8738	−1.00306	−0.501532	+	0.865139i	$0.667230\pi$
$888$	0	0
$889$	2.89295	0.0970265
$890$	0	0
$891$	12.1536	0.407161
$892$	0	0
$893$	6.30351	0.210939
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.19106	−0.0731573
$898$	0	0
$899$	−45.1958	−1.50736
$900$	0	0
$901$	−8.04022	−0.267859
$902$	0	0
$903$	−9.71296	−0.323227
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−43.4897	−1.44405	−0.722026	−	0.691866i	$-0.756789\pi$
$907$	−43.4897	−1.44405	−0.722026	+	0.691866i	$0.756789\pi$
$908$	0	0
$909$	69.4371	2.30308
$910$	0	0
$911$	24.3690	0.807383	0.403691	−	0.914895i	$-0.367727\pi$
$911$	24.3690	0.807383	0.403691	+	0.914895i	$0.367727\pi$
$912$	0	0
$913$	−1.94827	−0.0644783
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.974848	−0.0321923
$918$	0	0
$919$	−45.5143	−1.50138	−0.750690	−	0.660655i	$-0.770279\pi$
$919$	−45.5143	−1.50138	−0.750690	+	0.660655i	$0.770279\pi$
$920$	0	0
$921$	66.1546	2.17987
$922$	0	0
$923$	−4.81106	−0.158358
$924$	0	0
$925$	0	0
$926$	0	0
$927$	44.8567	1.47329
$928$	0	0
$929$	9.03795	0.296525	0.148263	−	0.988948i	$-0.452632\pi$
$929$	9.03795	0.296525	0.148263	+	0.988948i	$0.452632\pi$
$930$	0	0
$931$	33.8347	1.10889
$932$	0	0
$933$	41.2743	1.35126
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−33.3272	−1.08875	−0.544376	−	0.838841i	$-0.683234\pi$
$937$	−33.3272	−1.08875	−0.544376	+	0.838841i	$0.683234\pi$
$938$	0	0
$939$	−18.2622	−0.595966
$940$	0	0
$941$	−29.3010	−0.955186	−0.477593	−	0.878581i	$-0.658491\pi$
$941$	−29.3010	−0.955186	−0.477593	+	0.878581i	$0.658491\pi$
$942$	0	0
$943$	8.66985	0.282329
$944$	0	0
$945$	0	0
$946$	0	0
$947$	56.8456	1.84723	0.923617	−	0.383317i	$-0.125218\pi$
$947$	56.8456	1.84723	0.923617	+	0.383317i	$0.125218\pi$
$948$	0	0
$949$	−0.381966	−0.0123991
$950$	0	0
$951$	−33.9238	−1.10005
$952$	0	0
$953$	−19.0368	−0.616663	−0.308331	−	0.951279i	$-0.599771\pi$
$953$	−19.0368	−0.616663	−0.308331	+	0.951279i	$0.599771\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	28.6778	0.927021
$958$	0	0
$959$	−7.74394	−0.250065
$960$	0	0
$961$	36.3039	1.17109
$962$	0	0
$963$	2.84942	0.0918212
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−14.7105	−0.473059	−0.236529	−	0.971624i	$-0.576010\pi$
$967$	−14.7105	−0.473059	−0.236529	+	0.971624i	$0.576010\pi$
$968$	0	0
$969$	−20.3948	−0.655174
$970$	0	0
$971$	38.8453	1.24661	0.623303	−	0.781980i	$-0.285790\pi$
$971$	38.8453	1.24661	0.623303	+	0.781980i	$0.285790\pi$
$972$	0	0
$973$	2.74211	0.0879081
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−58.4544	−1.87012	−0.935060	−	0.354489i	$-0.884655\pi$
$977$	−58.4544	−1.87012	−0.935060	+	0.354489i	$0.884655\pi$
$978$	0	0
$979$	4.53297	0.144874
$980$	0	0
$981$	−34.5647	−1.10357
$982$	0	0
$983$	−28.1595	−0.898147	−0.449074	−	0.893495i	$-0.648246\pi$
$983$	−28.1595	−0.898147	−0.449074	+	0.893495i	$0.648246\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.34931	0.0429491
$988$	0	0
$989$	11.0116	0.350149
$990$	0	0
$991$	−39.3437	−1.24979	−0.624897	−	0.780707i	$-0.714859\pi$
$991$	−39.3437	−1.24979	−0.624897	+	0.780707i	$0.714859\pi$
$992$	0	0
$993$	−45.9231	−1.45733
$994$	0	0
$995$	0	0
$996$	0	0
$997$	43.1178	1.36555	0.682777	−	0.730627i	$-0.260772\pi$
$997$	43.1178	1.36555	0.682777	+	0.730627i	$0.260772\pi$
$998$	0	0
$999$	−10.3516	−0.327511

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bj.1.1		8
4.3	odd	2		625.2.a.f.1.3		8
5.4	even	2	inner	10000.2.a.bj.1.8		8
12.11	even	2		5625.2.a.x.1.6		8
20.3	even	4		625.2.b.c.624.6		8
20.7	even	4		625.2.b.c.624.3		8
20.19	odd	2		625.2.a.f.1.6		8
25.3	odd	20		400.2.y.c.209.2		8
25.17	odd	20		400.2.y.c.289.2		8
60.59	even	2		5625.2.a.x.1.3		8
100.3	even	20		25.2.e.a.9.2	✓	8
100.11	odd	10		625.2.d.o.501.2		16
100.19	odd	10		125.2.d.b.51.2		16
100.23	even	20		625.2.e.i.124.2		8
100.27	even	20		625.2.e.a.124.1		8
100.31	odd	10		125.2.d.b.51.3		16
100.39	odd	10		625.2.d.o.501.3		16
100.47	even	20		125.2.e.b.49.1		8
100.59	odd	10		625.2.d.o.126.3		16
100.63	even	20		625.2.e.a.499.1		8
100.67	even	20		25.2.e.a.14.2	yes	8
100.71	odd	10		125.2.d.b.76.3		16
100.79	odd	10		125.2.d.b.76.2		16
100.83	even	20		125.2.e.b.74.1		8
100.87	even	20		625.2.e.i.499.2		8
100.91	odd	10		625.2.d.o.126.2		16
300.167	odd	20		225.2.m.a.64.1		8
300.203	odd	20		225.2.m.a.109.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.e.a.9.2	✓	8	100.3	even	20
25.2.e.a.14.2	yes	8	100.67	even	20
125.2.d.b.51.2		16	100.19	odd	10
125.2.d.b.51.3		16	100.31	odd	10
125.2.d.b.76.2		16	100.79	odd	10
125.2.d.b.76.3		16	100.71	odd	10
125.2.e.b.49.1		8	100.47	even	20
125.2.e.b.74.1		8	100.83	even	20
225.2.m.a.64.1		8	300.167	odd	20
225.2.m.a.109.1		8	300.203	odd	20
400.2.y.c.209.2		8	25.3	odd	20
400.2.y.c.289.2		8	25.17	odd	20
625.2.a.f.1.3		8	4.3	odd	2
625.2.a.f.1.6		8	20.19	odd	2
625.2.b.c.624.3		8	20.7	even	4
625.2.b.c.624.6		8	20.3	even	4
625.2.d.o.126.2		16	100.91	odd	10
625.2.d.o.126.3		16	100.59	odd	10
625.2.d.o.501.2		16	100.11	odd	10
625.2.d.o.501.3		16	100.39	odd	10
625.2.e.a.124.1		8	100.27	even	20
625.2.e.a.499.1		8	100.63	even	20
625.2.e.i.124.2		8	100.23	even	20
625.2.e.i.499.2		8	100.87	even	20
5625.2.a.x.1.3		8	60.59	even	2
5625.2.a.x.1.6		8	12.11	even	2
10000.2.a.bj.1.1		8	1.1	even	1	trivial
10000.2.a.bj.1.8		8	5.4	even	2	inner

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q-2.60278 q^{3} +0.407162 q^{7} +3.77447 q^{9} +O(q^{10})\)	\(q-2.60278 q^{3} +0.407162 q^{7} +3.77447 q^{9} -2.00000 q^{11} +0.700668 q^{13} -1.58273 q^{17} -4.95078 q^{19} -1.05975 q^{21} +1.20145 q^{23} -2.01577 q^{27} +5.50906 q^{29} -8.20390 q^{31} +5.20556 q^{33} +5.13532 q^{37} -1.82368 q^{39} +7.21619 q^{41} +9.16531 q^{43} -1.27323 q^{47} -6.83422 q^{49} +4.11950 q^{51} +5.07996 q^{53} +12.8858 q^{57} +6.49972 q^{59} -9.42008 q^{61} +1.53682 q^{63} -3.08173 q^{67} -3.12710 q^{69} -6.86639 q^{71} -0.545146 q^{73} -0.814323 q^{77} -5.48159 q^{79} -6.07680 q^{81} +0.974135 q^{83} -14.3389 q^{87} -2.26649 q^{89} +0.285285 q^{91} +21.3529 q^{93} -15.2185 q^{97} -7.54893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^9	\( 8 q + 4 q^{9} - 16 q^{11} - 10 q^{19} + 6 q^{21} + 20 q^{29} - 16 q^{31} - 18 q^{39} + 26 q^{41} - 14 q^{49} + 4 q^{51} - 30 q^{59} + 6 q^{61} + 8 q^{69} - 46 q^{71} - 10 q^{79} - 32 q^{81} + 30 q^{89} + 14 q^{91} - 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^9 - 16 * q^11 - 10 * q^19 + 6 * q^21 + 20 * q^29 - 16 * q^31 - 18 * q^39 + 26 * q^41 - 14 * q^49 + 4 * q^51 - 30 * q^59 + 6 * q^61 + 8 * q^69 - 46 * q^71 - 10 * q^79 - 32 * q^81 + 30 * q^89 + 14 * q^91 - 8 * q^99

Introduction

L-functions

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