LMFDB - Embedded newform 10000.2.a.bj.1.5

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			1.5
Root			$-2.30927$ 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)

$f(q)$	$=$	$q+0.474903 q^{3} +3.03582 q^{7} -2.77447 q^{9} -2.00000 q^{11} +1.42721 q^{13} -1.86025 q^{17} -0.903319 q^{19} +1.44172 q^{21} -3.32932 q^{23} -2.74231 q^{27} +3.96307 q^{29} +6.43997 q^{31} -0.949806 q^{33} -3.82022 q^{37} +0.677786 q^{39} -1.83422 q^{41} -3.59445 q^{43} -4.79995 q^{47} +2.21619 q^{49} -0.883436 q^{51} +9.50473 q^{53} -0.428989 q^{57} -10.6456 q^{59} +14.2742 q^{61} -8.42278 q^{63} -10.6902 q^{67} -1.58111 q^{69} -12.4598 q^{71} -0.267631 q^{73} -6.07163 q^{77} +8.57176 q^{79} +7.02107 q^{81} -12.6182 q^{83} +1.88207 q^{87} -4.76796 q^{89} +4.33275 q^{91} +3.05836 q^{93} +9.95805 q^{97} +5.54893 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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
$3$	0.474903	0.274185	0.137093	−	0.990558i	$-0.456224\pi$
$3$	0.474903	0.274185	0.137093	+	0.990558i	$0.456224\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.03582	1.14743	0.573716	−	0.819055i	$-0.305502\pi$
$7$	3.03582	1.14743	0.573716	+	0.819055i	$0.305502\pi$
$8$	0	0
$9$	−2.77447	−0.924822
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	1.42721	0.395837	0.197918	−	0.980218i	$-0.436582\pi$
$13$	1.42721	0.395837	0.197918	+	0.980218i	$0.436582\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.86025	−0.451176	−0.225588	−	0.974223i	$-0.572430\pi$
$17$	−1.86025	−0.451176	−0.225588	+	0.974223i	$0.572430\pi$
$18$	0	0
$19$	−0.903319	−0.207236	−0.103618	−	0.994617i	$-0.533042\pi$
$19$	−0.903319	−0.207236	−0.103618	+	0.994617i	$0.533042\pi$
$20$	0	0
$21$	1.44172	0.314609
$22$	0	0
$23$	−3.32932	−0.694212	−0.347106	−	0.937826i	$-0.612836\pi$
$23$	−3.32932	−0.694212	−0.347106	+	0.937826i	$0.612836\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.74231	−0.527758
$28$	0	0
$29$	3.96307	0.735924	0.367962	−	0.929841i	$-0.380056\pi$
$29$	3.96307	0.735924	0.367962	+	0.929841i	$0.380056\pi$
$30$	0	0
$31$	6.43997	1.15665	0.578326	−	0.815806i	$-0.303706\pi$
$31$	6.43997	1.15665	0.578326	+	0.815806i	$0.303706\pi$
$32$	0	0
$33$	−0.949806	−0.165340
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.82022	−0.628040	−0.314020	−	0.949416i	$-0.601676\pi$
$37$	−3.82022	−0.628040	−0.314020	+	0.949416i	$0.601676\pi$
$38$	0	0
$39$	0.677786	0.108533
$40$	0	0
$41$	−1.83422	−0.286457	−0.143228	−	0.989690i	$-0.545748\pi$
$41$	−1.83422	−0.286457	−0.143228	+	0.989690i	$0.545748\pi$
$42$	0	0
$43$	−3.59445	−0.548149	−0.274074	−	0.961708i	$-0.588371\pi$
$43$	−3.59445	−0.548149	−0.274074	+	0.961708i	$0.588371\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.79995	−0.700144	−0.350072	−	0.936723i	$-0.613843\pi$
$47$	−4.79995	−0.700144	−0.350072	+	0.936723i	$0.613843\pi$
$48$	0	0
$49$	2.21619	0.316598
$50$	0	0
$51$	−0.883436	−0.123706
$52$	0	0
$53$	9.50473	1.30558	0.652788	−	0.757541i	$-0.273599\pi$
$53$	9.50473	1.30558	0.652788	+	0.757541i	$0.273599\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.428989	−0.0568209
$58$	0	0
$59$	−10.6456	−1.38594	−0.692971	−	0.720966i	$-0.743698\pi$
$59$	−10.6456	−1.38594	−0.692971	+	0.720966i	$0.743698\pi$
$60$	0	0
$61$	14.2742	1.82762	0.913811	−	0.406140i	$-0.133125\pi$
$61$	14.2742	1.82762	0.913811	+	0.406140i	$0.133125\pi$
$62$	0	0
$63$	−8.42278	−1.06117
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.6902	−1.30601	−0.653007	−	0.757352i	$-0.726493\pi$
$67$	−10.6902	−1.30601	−0.653007	+	0.757352i	$0.726493\pi$
$68$	0	0
$69$	−1.58111	−0.190343
$70$	0	0
$71$	−12.4598	−1.47871	−0.739356	−	0.673315i	$-0.764870\pi$
$71$	−12.4598	−1.47871	−0.739356	+	0.673315i	$0.764870\pi$
$72$	0	0
$73$	−0.267631	−0.0313239	−0.0156619	−	0.999877i	$-0.504986\pi$
$73$	−0.267631	−0.0313239	−0.0156619	+	0.999877i	$0.504986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.07163	−0.691927
$78$	0	0
$79$	8.57176	0.964398	0.482199	−	0.876062i	$-0.339838\pi$
$79$	8.57176	0.964398	0.482199	+	0.876062i	$0.339838\pi$
$80$	0	0
$81$	7.02107	0.780119
$82$	0	0
$83$	−12.6182	−1.38502	−0.692512	−	0.721406i	$-0.743496\pi$
$83$	−12.6182	−1.38502	−0.692512	+	0.721406i	$0.743496\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.88207	0.201779
$88$	0	0
$89$	−4.76796	−0.505402	−0.252701	−	0.967544i	$-0.581319\pi$
$89$	−4.76796	−0.505402	−0.252701	+	0.967544i	$0.581319\pi$
$90$	0	0
$91$	4.33275	0.454196
$92$	0	0
$93$	3.05836	0.317137
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.95805	1.01109	0.505543	−	0.862801i	$-0.331292\pi$
$97$	9.95805	1.01109	0.505543	+	0.862801i	$0.331292\pi$
$98$	0	0
$99$	5.54893	0.557689
$100$	0	0
$101$	9.34612	0.929974	0.464987	−	0.885318i	$-0.346059\pi$
$101$	9.34612	0.929974	0.464987	+	0.885318i	$0.346059\pi$
$102$	0	0
$103$	9.08408	0.895081	0.447540	−	0.894264i	$-0.352300\pi$
$103$	9.08408	0.895081	0.447540	+	0.894264i	$0.352300\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.62871	0.544148	0.272074	−	0.962276i	$-0.412290\pi$
$107$	5.62871	0.544148	0.272074	+	0.962276i	$0.412290\pi$
$108$	0	0
$109$	−10.1130	−0.968649	−0.484325	−	0.874888i	$-0.660935\pi$
$109$	−10.1130	−0.968649	−0.484325	+	0.874888i	$0.660935\pi$
$110$	0	0
$111$	−1.81423	−0.172199
$112$	0	0
$113$	−10.7120	−1.00770	−0.503851	−	0.863791i	$-0.668084\pi$
$113$	−10.7120	−1.00770	−0.503851	+	0.863791i	$0.668084\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.95975	−0.366079
$118$	0	0
$119$	−5.64737	−0.517693
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−0.871076	−0.0785423
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.3609	−1.00812	−0.504060	−	0.863669i	$-0.668161\pi$
$127$	−11.3609	−1.00812	−0.504060	+	0.863669i	$0.668161\pi$
$128$	0	0
$129$	−1.70702	−0.150294
$130$	0	0
$131$	−7.98771	−0.697890	−0.348945	−	0.937143i	$-0.613460\pi$
$131$	−7.98771	−0.697890	−0.348945	+	0.937143i	$0.613460\pi$
$132$	0	0
$133$	−2.74231	−0.237789
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.33726	−0.797736	−0.398868	−	0.917008i	$-0.630597\pi$
$137$	−9.33726	−0.797736	−0.398868	+	0.917008i	$0.630597\pi$
$138$	0	0
$139$	−17.9150	−1.51953	−0.759767	−	0.650195i	$-0.774687\pi$
$139$	−17.9150	−1.51953	−0.759767	+	0.650195i	$0.774687\pi$
$140$	0	0
$141$	−2.27951	−0.191969
$142$	0	0
$143$	−2.85442	−0.238699
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.05247	0.0868065
$148$	0	0
$149$	−6.31395	−0.517259	−0.258629	−	0.965977i	$-0.583271\pi$
$149$	−6.31395	−0.517259	−0.258629	+	0.965977i	$0.583271\pi$
$150$	0	0
$151$	−4.71947	−0.384065	−0.192033	−	0.981389i	$-0.561508\pi$
$151$	−4.71947	−0.384065	−0.192033	+	0.981389i	$0.561508\pi$
$152$	0	0
$153$	5.16119	0.417258
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1.46908	−0.117245	−0.0586225	−	0.998280i	$-0.518671\pi$
$157$	−1.46908	−0.117245	−0.0586225	+	0.998280i	$0.518671\pi$
$158$	0	0
$159$	4.51382	0.357969
$160$	0	0
$161$	−10.1072	−0.796560
$162$	0	0
$163$	4.45969	0.349310	0.174655	−	0.984630i	$-0.444119\pi$
$163$	4.45969	0.349310	0.174655	+	0.984630i	$0.444119\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.4337	0.807381	0.403691	−	0.914896i	$-0.367727\pi$
$167$	10.4337	0.807381	0.403691	+	0.914896i	$0.367727\pi$
$168$	0	0
$169$	−10.9631	−0.843313
$170$	0	0
$171$	2.50623	0.191656
$172$	0	0
$173$	7.67619	0.583610	0.291805	−	0.956478i	$-0.405744\pi$
$173$	7.67619	0.583610	0.291805	+	0.956478i	$0.405744\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.05563	−0.380005
$178$	0	0
$179$	−15.5168	−1.15978	−0.579889	−	0.814696i	$-0.696904\pi$
$179$	−15.5168	−1.15978	−0.579889	+	0.814696i	$0.696904\pi$
$180$	0	0
$181$	−1.59056	−0.118225	−0.0591126	−	0.998251i	$-0.518827\pi$
$181$	−1.59056	−0.118225	−0.0591126	+	0.998251i	$0.518827\pi$
$182$	0	0
$183$	6.77885	0.501107
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.72049	0.272069
$188$	0	0
$189$	−8.32515	−0.605566
$190$	0	0
$191$	−19.6684	−1.42316	−0.711579	−	0.702606i	$-0.752020\pi$
$191$	−19.6684	−1.42316	−0.711579	+	0.702606i	$0.752020\pi$
$192$	0	0
$193$	−13.1100	−0.943680	−0.471840	−	0.881684i	$-0.656410\pi$
$193$	−13.1100	−0.943680	−0.471840	+	0.881684i	$0.656410\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.42949	0.244341	0.122170	−	0.992509i	$-0.461015\pi$
$197$	3.42949	0.244341	0.122170	+	0.992509i	$0.461015\pi$
$198$	0	0
$199$	−17.6959	−1.25443	−0.627215	−	0.778846i	$-0.715805\pi$
$199$	−17.6959	−1.25443	−0.627215	+	0.778846i	$0.715805\pi$
$200$	0	0
$201$	−5.07680	−0.358090
$202$	0	0
$203$	12.0312	0.844422
$204$	0	0
$205$	0	0
$206$	0	0
$207$	9.23710	0.642023
$208$	0	0
$209$	1.80664	0.124968
$210$	0	0
$211$	3.24366	0.223303	0.111651	−	0.993747i	$-0.464386\pi$
$211$	3.24366	0.223303	0.111651	+	0.993747i	$0.464386\pi$
$212$	0	0
$213$	−5.91722	−0.405441
$214$	0	0
$215$	0	0
$216$	0	0
$217$	19.5506	1.32718
$218$	0	0
$219$	−0.127099	−0.00858854
$220$	0	0
$221$	−2.65496	−0.178592
$222$	0	0
$223$	28.7148	1.92288	0.961441	−	0.275010i	$-0.0886812\pi$
$223$	28.7148	1.92288	0.961441	+	0.275010i	$0.0886812\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.7206	0.777926	0.388963	−	0.921253i	$-0.372833\pi$
$227$	11.7206	0.777926	0.388963	+	0.921253i	$0.372833\pi$
$228$	0	0
$229$	−16.4013	−1.08383	−0.541914	−	0.840434i	$-0.682300\pi$
$229$	−16.4013	−1.08383	−0.541914	+	0.840434i	$0.682300\pi$
$230$	0	0
$231$	−2.88344	−0.189716
$232$	0	0
$233$	−22.5146	−1.47498	−0.737490	−	0.675358i	$-0.763989\pi$
$233$	−22.5146	−1.47498	−0.737490	+	0.675358i	$0.763989\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.07075	0.264424
$238$	0	0
$239$	6.63333	0.429074	0.214537	−	0.976716i	$-0.431176\pi$
$239$	6.63333	0.429074	0.214537	+	0.976716i	$0.431176\pi$
$240$	0	0
$241$	26.2261	1.68937	0.844684	−	0.535265i	$-0.179788\pi$
$241$	26.2261	1.68937	0.844684	+	0.535265i	$0.179788\pi$
$242$	0	0
$243$	11.5613	0.741655
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.28923	−0.0820315
$248$	0	0
$249$	−5.99241	−0.379753
$250$	0	0
$251$	10.9121	0.688766	0.344383	−	0.938829i	$-0.388088\pi$
$251$	10.9121	0.688766	0.344383	+	0.938829i	$0.388088\pi$
$252$	0	0
$253$	6.65865	0.418626
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.58051	−0.410481	−0.205240	−	0.978712i	$-0.565798\pi$
$257$	−6.58051	−0.410481	−0.205240	+	0.978712i	$0.565798\pi$
$258$	0	0
$259$	−11.5975	−0.720632
$260$	0	0
$261$	−10.9954	−0.680599
$262$	0	0
$263$	−27.1073	−1.67151	−0.835753	−	0.549106i	$-0.814968\pi$
$263$	−27.1073	−1.67151	−0.835753	+	0.549106i	$0.814968\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.26432	−0.138574
$268$	0	0
$269$	1.00945	0.0615474	0.0307737	−	0.999526i	$-0.490203\pi$
$269$	1.00945	0.0615474	0.0307737	+	0.999526i	$0.490203\pi$
$270$	0	0
$271$	−6.25203	−0.379784	−0.189892	−	0.981805i	$-0.560814\pi$
$271$	−6.25203	−0.379784	−0.189892	+	0.981805i	$0.560814\pi$
$272$	0	0
$273$	2.05763	0.124534
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.6703	1.48230	0.741148	−	0.671342i	$-0.234282\pi$
$277$	24.6703	1.48230	0.741148	+	0.671342i	$0.234282\pi$
$278$	0	0
$279$	−17.8675	−1.06970
$280$	0	0
$281$	1.83891	0.109700	0.0548502	−	0.998495i	$-0.482532\pi$
$281$	1.83891	0.109700	0.0548502	+	0.998495i	$0.482532\pi$
$282$	0	0
$283$	−8.64116	−0.513664	−0.256832	−	0.966456i	$-0.582679\pi$
$283$	−8.64116	−0.513664	−0.256832	+	0.966456i	$0.582679\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.56835	−0.328690
$288$	0	0
$289$	−13.5395	−0.796440
$290$	0	0
$291$	4.72910	0.277225
$292$	0	0
$293$	−6.29156	−0.367557	−0.183779	−	0.982968i	$-0.558833\pi$
$293$	−6.29156	−0.367557	−0.183779	+	0.982968i	$0.558833\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.48462	0.318250
$298$	0	0
$299$	−4.75164	−0.274795
$300$	0	0
$301$	−10.9121	−0.628963
$302$	0	0
$303$	4.43850	0.254985
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−28.6661	−1.63606	−0.818030	−	0.575175i	$-0.804934\pi$
$307$	−28.6661	−1.63606	−0.818030	+	0.575175i	$0.804934\pi$
$308$	0	0
$309$	4.31405	0.245418
$310$	0	0
$311$	7.83649	0.444367	0.222183	−	0.975005i	$-0.428682\pi$
$311$	7.83649	0.444367	0.222183	+	0.975005i	$0.428682\pi$
$312$	0	0
$313$	−21.4093	−1.21012	−0.605061	−	0.796179i	$-0.706851\pi$
$313$	−21.4093	−1.21012	−0.605061	+	0.796179i	$0.706851\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.01983	−0.225776	−0.112888	−	0.993608i	$-0.536010\pi$
$317$	−4.01983	−0.225776	−0.112888	+	0.993608i	$0.536010\pi$
$318$	0	0
$319$	−7.92614	−0.443779
$320$	0	0
$321$	2.67309	0.149197
$322$	0	0
$323$	1.68040	0.0934997
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−4.80269	−0.265589
$328$	0	0
$329$	−14.5718	−0.803367
$330$	0	0
$331$	−11.6439	−0.640005	−0.320002	−	0.947417i	$-0.603684\pi$
$331$	−11.6439	−0.640005	−0.320002	+	0.947417i	$0.603684\pi$
$332$	0	0
$333$	10.5991	0.580825
$334$	0	0
$335$	0	0
$336$	0	0
$337$	21.5348	1.17307	0.586537	−	0.809922i	$-0.300491\pi$
$337$	21.5348	1.17307	0.586537	+	0.809922i	$0.300491\pi$
$338$	0	0
$339$	−5.08716	−0.276297
$340$	0	0
$341$	−12.8799	−0.697487
$342$	0	0
$343$	−14.5228	−0.784157
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.5972	0.837303	0.418652	−	0.908147i	$-0.362503\pi$
$347$	15.5972	0.837303	0.418652	+	0.908147i	$0.362503\pi$
$348$	0	0
$349$	5.56598	0.297940	0.148970	−	0.988842i	$-0.452404\pi$
$349$	5.56598	0.297940	0.148970	+	0.988842i	$0.452404\pi$
$350$	0	0
$351$	−3.91385	−0.208906
$352$	0	0
$353$	−8.02216	−0.426977	−0.213488	−	0.976946i	$-0.568483\pi$
$353$	−8.02216	−0.426977	−0.213488	+	0.976946i	$0.568483\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.68195	−0.141944
$358$	0	0
$359$	12.3427	0.651424	0.325712	−	0.945469i	$-0.394396\pi$
$359$	12.3427	0.651424	0.325712	+	0.945469i	$0.394396\pi$
$360$	0	0
$361$	−18.1840	−0.957053
$362$	0	0
$363$	−3.32432	−0.174482
$364$	0	0
$365$	0	0
$366$	0	0
$367$	26.8749	1.40286	0.701430	−	0.712738i	$-0.252545\pi$
$367$	26.8749	1.40286	0.701430	+	0.712738i	$0.252545\pi$
$368$	0	0
$369$	5.08898	0.264922
$370$	0	0
$371$	28.8546	1.49806
$372$	0	0
$373$	27.6389	1.43109	0.715544	−	0.698567i	$-0.246179\pi$
$373$	27.6389	1.43109	0.715544	+	0.698567i	$0.246179\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.65614	0.291306
$378$	0	0
$379$	3.47462	0.178479	0.0892397	−	0.996010i	$-0.471556\pi$
$379$	3.47462	0.178479	0.0892397	+	0.996010i	$0.471556\pi$
$380$	0	0
$381$	−5.39534	−0.276412
$382$	0	0
$383$	27.3719	1.39864	0.699319	−	0.714810i	$-0.253487\pi$
$383$	27.3719	1.39864	0.699319	+	0.714810i	$0.253487\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.97269	0.506940
$388$	0	0
$389$	10.8845	0.551867	0.275934	−	0.961177i	$-0.411013\pi$
$389$	10.8845	0.551867	0.275934	+	0.961177i	$0.411013\pi$
$390$	0	0
$391$	6.19336	0.313212
$392$	0	0
$393$	−3.79339	−0.191351
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−16.2212	−0.814120	−0.407060	−	0.913401i	$-0.633446\pi$
$397$	−16.2212	−0.814120	−0.407060	+	0.913401i	$0.633446\pi$
$398$	0	0
$399$	−1.30233	−0.0651981
$400$	0	0
$401$	3.78686	0.189107	0.0945534	−	0.995520i	$-0.469858\pi$
$401$	3.78686	0.189107	0.0945534	+	0.995520i	$0.469858\pi$
$402$	0	0
$403$	9.19118	0.457845
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.64044	0.378722
$408$	0	0
$409$	1.85585	0.0917661	0.0458831	−	0.998947i	$-0.485390\pi$
$409$	1.85585	0.0917661	0.0458831	+	0.998947i	$0.485390\pi$
$410$	0	0
$411$	−4.43429	−0.218728
$412$	0	0
$413$	−32.3181	−1.59027
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−8.50790	−0.416634
$418$	0	0
$419$	−14.3472	−0.700907	−0.350453	−	0.936580i	$-0.613972\pi$
$419$	−14.3472	−0.700907	−0.350453	+	0.936580i	$0.613972\pi$
$420$	0	0
$421$	15.4545	0.753207	0.376604	−	0.926374i	$-0.377092\pi$
$421$	15.4545	0.753207	0.376604	+	0.926374i	$0.377092\pi$
$422$	0	0
$423$	13.3173	0.647509
$424$	0	0
$425$	0	0
$426$	0	0
$427$	43.3338	2.09707
$428$	0	0
$429$	−1.35557	−0.0654477
$430$	0	0
$431$	−12.5043	−0.602311	−0.301156	−	0.953575i	$-0.597372\pi$
$431$	−12.5043	−0.602311	−0.301156	+	0.953575i	$0.597372\pi$
$432$	0	0
$433$	22.4951	1.08105	0.540524	−	0.841329i	$-0.318226\pi$
$433$	22.4951	1.08105	0.540524	+	0.841329i	$0.318226\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.00744	0.143865
$438$	0	0
$439$	−12.1776	−0.581204	−0.290602	−	0.956844i	$-0.593855\pi$
$439$	−12.1776	−0.581204	−0.290602	+	0.956844i	$0.593855\pi$
$440$	0	0
$441$	−6.14873	−0.292797
$442$	0	0
$443$	−20.7101	−0.983968	−0.491984	−	0.870604i	$-0.663728\pi$
$443$	−20.7101	−0.983968	−0.491984	+	0.870604i	$0.663728\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.99851	−0.141825
$448$	0	0
$449$	−25.9539	−1.22484	−0.612420	−	0.790533i	$-0.709804\pi$
$449$	−25.9539	−1.22484	−0.612420	+	0.790533i	$0.709804\pi$
$450$	0	0
$451$	3.66844	0.172740
$452$	0	0
$453$	−2.24129	−0.105305
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.50150	0.397684	0.198842	−	0.980032i	$-0.436282\pi$
$457$	8.50150	0.397684	0.198842	+	0.980032i	$0.436282\pi$
$458$	0	0
$459$	5.10137	0.238112
$460$	0	0
$461$	14.7851	0.688609	0.344305	−	0.938858i	$-0.388115\pi$
$461$	14.7851	0.688609	0.344305	+	0.938858i	$0.388115\pi$
$462$	0	0
$463$	−22.1921	−1.03135	−0.515677	−	0.856783i	$-0.672460\pi$
$463$	−22.1921	−1.03135	−0.515677	+	0.856783i	$0.672460\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−28.5014	−1.31889	−0.659443	−	0.751754i	$-0.729208\pi$
$467$	−28.5014	−1.31889	−0.659443	+	0.751754i	$0.729208\pi$
$468$	0	0
$469$	−32.4534	−1.49856
$470$	0	0
$471$	−0.697669	−0.0321469
$472$	0	0
$473$	7.18891	0.330546
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−26.3706	−1.20743
$478$	0	0
$479$	−25.0569	−1.14488	−0.572440	−	0.819947i	$-0.694003\pi$
$479$	−25.0569	−1.14488	−0.572440	+	0.819947i	$0.694003\pi$
$480$	0	0
$481$	−5.45225	−0.248601
$482$	0	0
$483$	−4.79995	−0.218405
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.46479	−0.0663758	−0.0331879	−	0.999449i	$-0.510566\pi$
$487$	−1.46479	−0.0663758	−0.0331879	+	0.999449i	$0.510566\pi$
$488$	0	0
$489$	2.11792	0.0957757
$490$	0	0
$491$	20.0686	0.905685	0.452843	−	0.891591i	$-0.350410\pi$
$491$	20.0686	0.905685	0.452843	+	0.891591i	$0.350410\pi$
$492$	0	0
$493$	−7.37229	−0.332031
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−37.8258	−1.69672
$498$	0	0
$499$	−0.624999	−0.0279788	−0.0139894	−	0.999902i	$-0.504453\pi$
$499$	−0.624999	−0.0279788	−0.0139894	+	0.999902i	$0.504453\pi$
$500$	0	0
$501$	4.95498	0.221372
$502$	0	0
$503$	−19.3052	−0.860776	−0.430388	−	0.902644i	$-0.641623\pi$
$503$	−19.3052	−0.860776	−0.430388	+	0.902644i	$0.641623\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−5.20639	−0.231224
$508$	0	0
$509$	−10.5202	−0.466298	−0.233149	−	0.972441i	$-0.574903\pi$
$509$	−10.5202	−0.466298	−0.233149	+	0.972441i	$0.574903\pi$
$510$	0	0
$511$	−0.812479	−0.0359420
$512$	0	0
$513$	2.47718	0.109370
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9.59989	0.422203
$518$	0	0
$519$	3.64545	0.160017
$520$	0	0
$521$	−10.0070	−0.438413	−0.219207	−	0.975678i	$-0.570347\pi$
$521$	−10.0070	−0.438413	−0.219207	+	0.975678i	$0.570347\pi$
$522$	0	0
$523$	22.7830	0.996233	0.498117	−	0.867110i	$-0.334025\pi$
$523$	22.7830	0.996233	0.498117	+	0.867110i	$0.334025\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.9799	−0.521854
$528$	0	0
$529$	−11.9156	−0.518070
$530$	0	0
$531$	29.5359	1.28175
$532$	0	0
$533$	−2.61782	−0.113390
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7.36896	−0.317994
$538$	0	0
$539$	−4.43237	−0.190916
$540$	0	0
$541$	3.25900	0.140115	0.0700576	−	0.997543i	$-0.477682\pi$
$541$	3.25900	0.140115	0.0700576	+	0.997543i	$0.477682\pi$
$542$	0	0
$543$	−0.755360	−0.0324156
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−13.5883	−0.580993	−0.290497	−	0.956876i	$-0.593821\pi$
$547$	−13.5883	−0.580993	−0.290497	+	0.956876i	$0.593821\pi$
$548$	0	0
$549$	−39.6033	−1.69023
$550$	0	0
$551$	−3.57992	−0.152510
$552$	0	0
$553$	26.0223	1.10658
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.6399	1.17114	0.585571	−	0.810621i	$-0.300870\pi$
$557$	27.6399	1.17114	0.585571	+	0.810621i	$0.300870\pi$
$558$	0	0
$559$	−5.13004	−0.216978
$560$	0	0
$561$	1.76687	0.0745974
$562$	0	0
$563$	−1.65925	−0.0699291	−0.0349646	−	0.999389i	$-0.511132\pi$
$563$	−1.65925	−0.0699291	−0.0349646	+	0.999389i	$0.511132\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	21.3147	0.895133
$568$	0	0
$569$	17.8828	0.749686	0.374843	−	0.927088i	$-0.377697\pi$
$569$	17.8828	0.749686	0.374843	+	0.927088i	$0.377697\pi$
$570$	0	0
$571$	36.8723	1.54306	0.771530	−	0.636193i	$-0.219492\pi$
$571$	36.8723	1.54306	0.771530	+	0.636193i	$0.219492\pi$
$572$	0	0
$573$	−9.34060	−0.390209
$574$	0	0
$575$	0	0
$576$	0	0
$577$	22.8137	0.949746	0.474873	−	0.880054i	$-0.342494\pi$
$577$	22.8137	0.949746	0.474873	+	0.880054i	$0.342494\pi$
$578$	0	0
$579$	−6.22599	−0.258743
$580$	0	0
$581$	−38.3065	−1.58922
$582$	0	0
$583$	−19.0095	−0.787291
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−11.0855	−0.457546	−0.228773	−	0.973480i	$-0.573471\pi$
$587$	−11.0855	−0.457546	−0.228773	+	0.973480i	$0.573471\pi$
$588$	0	0
$589$	−5.81734	−0.239699
$590$	0	0
$591$	1.62867	0.0669947
$592$	0	0
$593$	−11.1321	−0.457139	−0.228570	−	0.973528i	$-0.573405\pi$
$593$	−11.1321	−0.457139	−0.228570	+	0.973528i	$0.573405\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.40384	−0.343946
$598$	0	0
$599$	−36.2736	−1.48210	−0.741049	−	0.671451i	$-0.765671\pi$
$599$	−36.2736	−1.48210	−0.741049	+	0.671451i	$0.765671\pi$
$600$	0	0
$601$	−15.1051	−0.616150	−0.308075	−	0.951362i	$-0.599685\pi$
$601$	−15.1051	−0.616150	−0.308075	+	0.951362i	$0.599685\pi$
$602$	0	0
$603$	29.6596	1.20783
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−33.5066	−1.35999	−0.679996	−	0.733216i	$-0.738018\pi$
$607$	−33.5066	−1.35999	−0.679996	+	0.733216i	$0.738018\pi$
$608$	0	0
$609$	5.71363	0.231528
$610$	0	0
$611$	−6.85053	−0.277143
$612$	0	0
$613$	28.0289	1.13208	0.566038	−	0.824379i	$-0.308476\pi$
$613$	28.0289	1.13208	0.566038	+	0.824379i	$0.308476\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.5223	−1.22878	−0.614391	−	0.789002i	$-0.710598\pi$
$617$	−30.5223	−1.22878	−0.614391	+	0.789002i	$0.710598\pi$
$618$	0	0
$619$	21.6971	0.872080	0.436040	−	0.899927i	$-0.356381\pi$
$619$	21.6971	0.872080	0.436040	+	0.899927i	$0.356381\pi$
$620$	0	0
$621$	9.13004	0.366376
$622$	0	0
$623$	−14.4746	−0.579914
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.857977	0.0342643
$628$	0	0
$629$	7.10655	0.283357
$630$	0	0
$631$	16.2277	0.646015	0.323007	−	0.946396i	$-0.395306\pi$
$631$	16.2277	0.646015	0.323007	+	0.946396i	$0.395306\pi$
$632$	0	0
$633$	1.54042	0.0612264
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.16296	0.125321
$638$	0	0
$639$	34.5694	1.36755
$640$	0	0
$641$	−22.1774	−0.875956	−0.437978	−	0.898986i	$-0.644305\pi$
$641$	−22.1774	−0.875956	−0.437978	+	0.898986i	$0.644305\pi$
$642$	0	0
$643$	−13.2767	−0.523583	−0.261792	−	0.965124i	$-0.584313\pi$
$643$	−13.2767	−0.523583	−0.261792	+	0.965124i	$0.584313\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11.2853	−0.443671	−0.221835	−	0.975084i	$-0.571205\pi$
$647$	−11.2853	−0.443671	−0.221835	+	0.975084i	$0.571205\pi$
$648$	0	0
$649$	21.2912	0.835754
$650$	0	0
$651$	9.28462	0.363893
$652$	0	0
$653$	35.8134	1.40149	0.700743	−	0.713414i	$-0.252852\pi$
$653$	35.8134	1.40149	0.700743	+	0.713414i	$0.252852\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.742534	0.0289690
$658$	0	0
$659$	−39.7655	−1.54905	−0.774523	−	0.632546i	$-0.782010\pi$
$659$	−39.7655	−1.54905	−0.774523	+	0.632546i	$0.782010\pi$
$660$	0	0
$661$	−6.24734	−0.242993	−0.121497	−	0.992592i	$-0.538769\pi$
$661$	−6.24734	−0.242993	−0.121497	+	0.992592i	$0.538769\pi$
$662$	0	0
$663$	−1.26085	−0.0489673
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−13.1943	−0.510887
$668$	0	0
$669$	13.6367	0.527226
$670$	0	0
$671$	−28.5484	−1.10210
$672$	0	0
$673$	41.4627	1.59827	0.799135	−	0.601151i	$-0.205291\pi$
$673$	41.4627	1.59827	0.799135	+	0.601151i	$0.205291\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.43915	0.0553112	0.0276556	−	0.999618i	$-0.491196\pi$
$677$	1.43915	0.0553112	0.0276556	+	0.999618i	$0.491196\pi$
$678$	0	0
$679$	30.2308	1.16015
$680$	0	0
$681$	5.56616	0.213296
$682$	0	0
$683$	−8.48623	−0.324716	−0.162358	−	0.986732i	$-0.551910\pi$
$683$	−8.48623	−0.324716	−0.162358	+	0.986732i	$0.551910\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−7.78902	−0.297170
$688$	0	0
$689$	13.5652	0.516795
$690$	0	0
$691$	43.7797	1.66546	0.832730	−	0.553679i	$-0.186777\pi$
$691$	43.7797	1.66546	0.832730	+	0.553679i	$0.186777\pi$
$692$	0	0
$693$	16.8456	0.639910
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.41210	0.129243
$698$	0	0
$699$	−10.6922	−0.404418
$700$	0	0
$701$	0.840795	0.0317564	0.0158782	−	0.999874i	$-0.494946\pi$
$701$	0.840795	0.0317564	0.0158782	+	0.999874i	$0.494946\pi$
$702$	0	0
$703$	3.45087	0.130152
$704$	0	0
$705$	0	0
$706$	0	0
$707$	28.3731	1.06708
$708$	0	0
$709$	−13.3812	−0.502543	−0.251271	−	0.967917i	$-0.580849\pi$
$709$	−13.3812	−0.502543	−0.251271	+	0.967917i	$0.580849\pi$
$710$	0	0
$711$	−23.7821	−0.891897
$712$	0	0
$713$	−21.4407	−0.802962
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.15019	0.117646
$718$	0	0
$719$	−43.4148	−1.61910	−0.809550	−	0.587051i	$-0.800289\pi$
$719$	−43.4148	−1.61910	−0.809550	+	0.587051i	$0.800289\pi$
$720$	0	0
$721$	27.5776	1.02704
$722$	0	0
$723$	12.4548	0.463200
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−32.0480	−1.18859	−0.594297	−	0.804245i	$-0.702570\pi$
$727$	−32.0480	−1.18859	−0.594297	+	0.804245i	$0.702570\pi$
$728$	0	0
$729$	−15.5727	−0.576768
$730$	0	0
$731$	6.68657	0.247312
$732$	0	0
$733$	−8.13928	−0.300631	−0.150316	−	0.988638i	$-0.548029\pi$
$733$	−8.13928	−0.300631	−0.150316	+	0.988638i	$0.548029\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.3804	0.787556
$738$	0	0
$739$	7.12714	0.262176	0.131088	−	0.991371i	$-0.458153\pi$
$739$	7.12714	0.262176	0.131088	+	0.991371i	$0.458153\pi$
$740$	0	0
$741$	−0.612257	−0.0224918
$742$	0	0
$743$	21.9040	0.803578	0.401789	−	0.915732i	$-0.368388\pi$
$743$	21.9040	0.803578	0.401789	+	0.915732i	$0.368388\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	35.0087	1.28090
$748$	0	0
$749$	17.0877	0.624373
$750$	0	0
$751$	−9.21909	−0.336409	−0.168205	−	0.985752i	$-0.553797\pi$
$751$	−9.21909	−0.336409	−0.168205	+	0.985752i	$0.553797\pi$
$752$	0	0
$753$	5.18219	0.188849
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−45.6524	−1.65926	−0.829632	−	0.558311i	$-0.811450\pi$
$757$	−45.6524	−1.65926	−0.829632	+	0.558311i	$0.811450\pi$
$758$	0	0
$759$	3.16221	0.114781
$760$	0	0
$761$	39.9058	1.44658	0.723291	−	0.690543i	$-0.242628\pi$
$761$	39.9058	1.44658	0.723291	+	0.690543i	$0.242628\pi$
$762$	0	0
$763$	−30.7012	−1.11146
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−15.1935	−0.548607
$768$	0	0
$769$	−44.3420	−1.59901	−0.799506	−	0.600658i	$-0.794906\pi$
$769$	−44.3420	−1.59901	−0.799506	+	0.600658i	$0.794906\pi$
$770$	0	0
$771$	−3.12510	−0.112548
$772$	0	0
$773$	38.4944	1.38455	0.692273	−	0.721635i	$-0.256609\pi$
$773$	38.4944	1.38455	0.692273	+	0.721635i	$0.256609\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−5.50768	−0.197587
$778$	0	0
$779$	1.65689	0.0593641
$780$	0	0
$781$	24.9197	0.891697
$782$	0	0
$783$	−10.8680	−0.388390
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−53.0202	−1.88997	−0.944983	−	0.327120i	$-0.893922\pi$
$787$	−53.0202	−1.88997	−0.944983	+	0.327120i	$0.893922\pi$
$788$	0	0
$789$	−12.8733	−0.458302
$790$	0	0
$791$	−32.5197	−1.15627
$792$	0	0
$793$	20.3723	0.723440
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.6769	0.449037	0.224519	−	0.974470i	$-0.427919\pi$
$797$	12.6769	0.449037	0.224519	+	0.974470i	$0.427919\pi$
$798$	0	0
$799$	8.92908	0.315888
$800$	0	0
$801$	13.2285	0.467407
$802$	0	0
$803$	0.535262	0.0188890
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.479392	0.0168754
$808$	0	0
$809$	−41.8935	−1.47290	−0.736449	−	0.676493i	$-0.763499\pi$
$809$	−41.8935	−1.47290	−0.736449	+	0.676493i	$0.763499\pi$
$810$	0	0
$811$	34.5486	1.21317	0.606583	−	0.795020i	$-0.292540\pi$
$811$	34.5486	1.21317	0.606583	+	0.795020i	$0.292540\pi$
$812$	0	0
$813$	−2.96911	−0.104131
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.24694	0.113596
$818$	0	0
$819$	−12.0211	−0.420050
$820$	0	0
$821$	−21.1349	−0.737612	−0.368806	−	0.929506i	$-0.620233\pi$
$821$	−21.1349	−0.737612	−0.368806	+	0.929506i	$0.620233\pi$
$822$	0	0
$823$	3.17714	0.110748	0.0553740	−	0.998466i	$-0.482365\pi$
$823$	3.17714	0.110748	0.0553740	+	0.998466i	$0.482365\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.74482	0.338860	0.169430	−	0.985542i	$-0.445807\pi$
$827$	9.74482	0.338860	0.169430	+	0.985542i	$0.445807\pi$
$828$	0	0
$829$	23.4352	0.813938	0.406969	−	0.913442i	$-0.366586\pi$
$829$	23.4352	0.813938	0.406969	+	0.913442i	$0.366586\pi$
$830$	0	0
$831$	11.7160	0.406424
$832$	0	0
$833$	−4.12265	−0.142841
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−17.6604	−0.610432
$838$	0	0
$839$	−42.6819	−1.47354	−0.736771	−	0.676142i	$-0.763650\pi$
$839$	−42.6819	−1.47354	−0.736771	+	0.676142i	$0.763650\pi$
$840$	0	0
$841$	−13.2941	−0.458416
$842$	0	0
$843$	0.873305	0.0300782
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−21.2507	−0.730183
$848$	0	0
$849$	−4.10371	−0.140839
$850$	0	0
$851$	12.7187	0.435993
$852$	0	0
$853$	−16.9610	−0.580733	−0.290367	−	0.956915i	$-0.593777\pi$
$853$	−16.9610	−0.580733	−0.290367	+	0.956915i	$0.593777\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.3176	1.34306	0.671531	−	0.740976i	$-0.265637\pi$
$857$	39.3176	1.34306	0.671531	+	0.740976i	$0.265637\pi$
$858$	0	0
$859$	−0.707056	−0.0241244	−0.0120622	−	0.999927i	$-0.503840\pi$
$859$	−0.707056	−0.0241244	−0.0120622	+	0.999927i	$0.503840\pi$
$860$	0	0
$861$	−2.64443	−0.0901219
$862$	0	0
$863$	0.909409	0.0309567	0.0154783	−	0.999880i	$-0.495073\pi$
$863$	0.909409	0.0309567	0.0154783	+	0.999880i	$0.495073\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−6.42994	−0.218372
$868$	0	0
$869$	−17.1435	−0.581554
$870$	0	0
$871$	−15.2571	−0.516968
$872$	0	0
$873$	−27.6283	−0.935075
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−33.6613	−1.13666	−0.568331	−	0.822800i	$-0.692411\pi$
$877$	−33.6613	−1.13666	−0.568331	+	0.822800i	$0.692411\pi$
$878$	0	0
$879$	−2.98788	−0.100779
$880$	0	0
$881$	−19.9283	−0.671402	−0.335701	−	0.941969i	$-0.608973\pi$
$881$	−19.9283	−0.671402	−0.335701	+	0.941969i	$0.608973\pi$
$882$	0	0
$883$	15.6081	0.525255	0.262627	−	0.964897i	$-0.415411\pi$
$883$	15.6081	0.525255	0.262627	+	0.964897i	$0.415411\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	46.8968	1.57464	0.787320	−	0.616544i	$-0.211468\pi$
$887$	46.8968	1.57464	0.787320	+	0.616544i	$0.211468\pi$
$888$	0	0
$889$	−34.4897	−1.15675
$890$	0	0
$891$	−14.0421	−0.470429
$892$	0	0
$893$	4.33588	0.145095
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.25657	−0.0753447
$898$	0	0
$899$	25.5220	0.851208
$900$	0	0
$901$	−17.6811	−0.589044
$902$	0	0
$903$	−5.18219	−0.172452
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.43447	0.0476308	0.0238154	−	0.999716i	$-0.492419\pi$
$907$	1.43447	0.0476308	0.0238154	+	0.999716i	$0.492419\pi$
$908$	0	0
$909$	−25.9305	−0.860061
$910$	0	0
$911$	2.81129	0.0931422	0.0465711	−	0.998915i	$-0.485171\pi$
$911$	2.81129	0.0931422	0.0465711	+	0.998915i	$0.485171\pi$
$912$	0	0
$913$	25.2363	0.835201
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−24.2492	−0.800780
$918$	0	0
$919$	−0.992236	−0.0327308	−0.0163654	−	0.999866i	$-0.505210\pi$
$919$	−0.992236	−0.0327308	−0.0163654	+	0.999866i	$0.505210\pi$
$920$	0	0
$921$	−13.6136	−0.448584
$922$	0	0
$923$	−17.7828	−0.585329
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−25.2035	−0.827791
$928$	0	0
$929$	33.3227	1.09328	0.546642	−	0.837367i	$-0.315906\pi$
$929$	33.3227	1.09328	0.546642	+	0.837367i	$0.315906\pi$
$930$	0	0
$931$	−2.00192	−0.0656104
$932$	0	0
$933$	3.72157	0.121839
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−13.3675	−0.436698	−0.218349	−	0.975871i	$-0.570067\pi$
$937$	−13.3675	−0.436698	−0.218349	+	0.975871i	$0.570067\pi$
$938$	0	0
$939$	−10.1673	−0.331798
$940$	0	0
$941$	−2.14982	−0.0700820	−0.0350410	−	0.999386i	$-0.511156\pi$
$941$	−2.14982	−0.0700820	−0.0350410	+	0.999386i	$0.511156\pi$
$942$	0	0
$943$	6.10671	0.198862
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.9183	1.13469	0.567346	−	0.823479i	$-0.307970\pi$
$947$	34.9183	1.13469	0.567346	+	0.823479i	$0.307970\pi$
$948$	0	0
$949$	−0.381966	−0.0123991
$950$	0	0
$951$	−1.90903	−0.0619046
$952$	0	0
$953$	8.27883	0.268178	0.134089	−	0.990969i	$-0.457189\pi$
$953$	8.27883	0.268178	0.134089	+	0.990969i	$0.457189\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3.76415	−0.121678
$958$	0	0
$959$	−28.3462	−0.915347
$960$	0	0
$961$	10.4732	0.337844
$962$	0	0
$963$	−15.6167	−0.503241
$964$	0	0
$965$	0	0
$966$	0	0
$967$	29.4871	0.948241	0.474121	−	0.880460i	$-0.342766\pi$
$967$	29.4871	0.948241	0.474121	+	0.880460i	$0.342766\pi$
$968$	0	0
$969$	0.798025	0.0256363
$970$	0	0
$971$	−21.8666	−0.701733	−0.350867	−	0.936425i	$-0.614113\pi$
$971$	−21.8666	−0.701733	−0.350867	+	0.936425i	$0.614113\pi$
$972$	0	0
$973$	−54.3868	−1.74356
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.8482	−0.443043	−0.221521	−	0.975155i	$-0.571102\pi$
$977$	−13.8482	−0.443043	−0.221521	+	0.975155i	$0.571102\pi$
$978$	0	0
$979$	9.53591	0.304769
$980$	0	0
$981$	28.0582	0.895828
$982$	0	0
$983$	2.93538	0.0936241	0.0468120	−	0.998904i	$-0.485094\pi$
$983$	2.93538	0.0936241	0.0468120	+	0.998904i	$0.485094\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−6.92017	−0.220271
$988$	0	0
$989$	11.9671	0.380531
$990$	0	0
$991$	−31.6137	−1.00424	−0.502122	−	0.864797i	$-0.667447\pi$
$991$	−31.6137	−1.00424	−0.502122	+	0.864797i	$0.667447\pi$
$992$	0	0
$993$	−5.52970	−0.175480
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.0885	0.382845	0.191423	−	0.981508i	$-0.438690\pi$
$997$	12.0885	0.382845	0.191423	+	0.981508i	$0.438690\pi$
$998$	0	0
$999$	10.4762	0.331453

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bj.1.5		8
4.3	odd	2		625.2.a.f.1.1		8
5.4	even	2	inner	10000.2.a.bj.1.4		8
12.11	even	2		5625.2.a.x.1.8		8
20.3	even	4		625.2.b.c.624.8		8
20.7	even	4		625.2.b.c.624.1		8
20.19	odd	2		625.2.a.f.1.8		8
25.8	odd	20		400.2.y.c.289.1		8
25.22	odd	20		400.2.y.c.209.1		8
60.59	even	2		5625.2.a.x.1.1		8
100.3	even	20		125.2.e.b.49.2		8
100.11	odd	10		625.2.d.o.501.1		16
100.19	odd	10		125.2.d.b.51.1		16
100.23	even	20		625.2.e.a.124.2		8
100.27	even	20		625.2.e.i.124.1		8
100.31	odd	10		125.2.d.b.51.4		16
100.39	odd	10		625.2.d.o.501.4		16
100.47	even	20		25.2.e.a.9.1	✓	8
100.59	odd	10		625.2.d.o.126.4		16
100.63	even	20		625.2.e.i.499.1		8
100.67	even	20		125.2.e.b.74.2		8
100.71	odd	10		125.2.d.b.76.4		16
100.79	odd	10		125.2.d.b.76.1		16
100.83	even	20		25.2.e.a.14.1	yes	8
100.87	even	20		625.2.e.a.499.2		8
100.91	odd	10		625.2.d.o.126.1		16
300.47	odd	20		225.2.m.a.109.2		8
300.83	odd	20		225.2.m.a.64.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.e.a.9.1	✓	8	100.47	even	20
25.2.e.a.14.1	yes	8	100.83	even	20
125.2.d.b.51.1		16	100.19	odd	10
125.2.d.b.51.4		16	100.31	odd	10
125.2.d.b.76.1		16	100.79	odd	10
125.2.d.b.76.4		16	100.71	odd	10
125.2.e.b.49.2		8	100.3	even	20
125.2.e.b.74.2		8	100.67	even	20
225.2.m.a.64.2		8	300.83	odd	20
225.2.m.a.109.2		8	300.47	odd	20
400.2.y.c.209.1		8	25.22	odd	20
400.2.y.c.289.1		8	25.8	odd	20
625.2.a.f.1.1		8	4.3	odd	2
625.2.a.f.1.8		8	20.19	odd	2
625.2.b.c.624.1		8	20.7	even	4
625.2.b.c.624.8		8	20.3	even	4
625.2.d.o.126.1		16	100.91	odd	10
625.2.d.o.126.4		16	100.59	odd	10
625.2.d.o.501.1		16	100.11	odd	10
625.2.d.o.501.4		16	100.39	odd	10
625.2.e.a.124.2		8	100.23	even	20
625.2.e.a.499.2		8	100.87	even	20
625.2.e.i.124.1		8	100.27	even	20
625.2.e.i.499.1		8	100.63	even	20
5625.2.a.x.1.1		8	60.59	even	2
5625.2.a.x.1.8		8	12.11	even	2
10000.2.a.bj.1.4		8	5.4	even	2	inner
10000.2.a.bj.1.5		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.474903 q^{3} +3.03582 q^{7} -2.77447 q^{9} -2.00000 q^{11} +1.42721 q^{13} -1.86025 q^{17} -0.903319 q^{19} +1.44172 q^{21} -3.32932 q^{23} -2.74231 q^{27} +3.96307 q^{29} +6.43997 q^{31} -0.949806 q^{33} -3.82022 q^{37} +0.677786 q^{39} -1.83422 q^{41} -3.59445 q^{43} -4.79995 q^{47} +2.21619 q^{49} -0.883436 q^{51} +9.50473 q^{53} -0.428989 q^{57} -10.6456 q^{59} +14.2742 q^{61} -8.42278 q^{63} -10.6902 q^{67} -1.58111 q^{69} -12.4598 q^{71} -0.267631 q^{73} -6.07163 q^{77} +8.57176 q^{79} +7.02107 q^{81} -12.6182 q^{83} +1.88207 q^{87} -4.76796 q^{89} +4.33275 q^{91} +3.05836 q^{93} +9.95805 q^{97} +5.54893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9} - 16 q^{11} - 10 q^{19} + 6 q^{21} + 20 q^{29} - 16 q^{31} - 18 q^{39} + 26 q^{41} - 14 q^{49} + 4 q^{51} - 30 q^{59} + 6 q^{61} + 8 q^{69} - 46 q^{71} - 10 q^{79} - 32 q^{81} + 30 q^{89}+ \cdots - 8 q^{99}+O(q^{100}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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