LMFDB - Embedded newform 8512.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.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:	$67.9686622005$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 532)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61803 q^{3} +1.00000 q^{5} +1.00000 q^{7} +3.85410 q^{9} -6.09017 q^{11} +2.23607 q^{13} +2.61803 q^{15} -7.61803 q^{17} +1.00000 q^{19} +2.61803 q^{21} -8.70820 q^{23} -4.00000 q^{25} +2.23607 q^{27} -0.854102 q^{29} -2.38197 q^{31} -15.9443 q^{33} +1.00000 q^{35} +6.70820 q^{37} +5.85410 q^{39} +0.0901699 q^{41} -8.47214 q^{43} +3.85410 q^{45} +4.70820 q^{47} +1.00000 q^{49} -19.9443 q^{51} +14.3262 q^{53} -6.09017 q^{55} +2.61803 q^{57} +10.7082 q^{59} -5.47214 q^{61} +3.85410 q^{63} +2.23607 q^{65} +2.09017 q^{67} -22.7984 q^{69} -11.1803 q^{71} -0.909830 q^{73} -10.4721 q^{75} -6.09017 q^{77} -6.94427 q^{79} -5.70820 q^{81} -14.6180 q^{83} -7.61803 q^{85} -2.23607 q^{87} -12.1803 q^{89} +2.23607 q^{91} -6.23607 q^{93} +1.00000 q^{95} +8.70820 q^{97} -23.4721 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 2 q^{5} + 2 q^{7} + q^{9} - q^{11} + 3 q^{15} - 13 q^{17} + 2 q^{19} + 3 q^{21} - 4 q^{23} - 8 q^{25} + 5 q^{29} - 7 q^{31} - 14 q^{33} + 2 q^{35} + 5 q^{39} - 11 q^{41} - 8 q^{43} + q^{45}+ \cdots - 38 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + 2 * q^5 + 2 * q^7 + q^9 - q^11 + 3 * q^15 - 13 * q^17 + 2 * q^19 + 3 * q^21 - 4 * q^23 - 8 * q^25 + 5 * q^29 - 7 * q^31 - 14 * q^33 + 2 * q^35 + 5 * q^39 - 11 * q^41 - 8 * q^43 + q^45 - 4 * q^47 + 2 * q^49 - 22 * q^51 + 13 * q^53 - q^55 + 3 * q^57 + 8 * q^59 - 2 * q^61 + q^63 - 7 * q^67 - 21 * q^69 - 13 * q^73 - 12 * q^75 - q^77 + 4 * q^79 + 2 * q^81 - 27 * q^83 - 13 * q^85 - 2 * q^89 - 8 * q^93 + 2 * q^95 + 4 * q^97 - 38 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.61803	1.51152	0.755761	−	0.654847i	$-0.227267\pi$
$3$	2.61803	1.51152	0.755761	+	0.654847i	$0.227267\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.85410	1.28470
$10$	0	0
$11$	−6.09017	−1.83626	−0.918128	−	0.396285i	$-0.870299\pi$
$11$	−6.09017	−1.83626	−0.918128	+	0.396285i	$0.870299\pi$
$12$	0	0
$13$	2.23607	0.620174	0.310087	−	0.950708i	$-0.399642\pi$
$13$	2.23607	0.620174	0.310087	+	0.950708i	$0.399642\pi$
$14$	0	0
$15$	2.61803	0.675973
$16$	0	0
$17$	−7.61803	−1.84764	−0.923822	−	0.382822i	$-0.874952\pi$
$17$	−7.61803	−1.84764	−0.923822	+	0.382822i	$0.874952\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.61803	0.571302
$22$	0	0
$23$	−8.70820	−1.81579	−0.907893	−	0.419202i	$-0.862310\pi$
$23$	−8.70820	−1.81579	−0.907893	+	0.419202i	$0.862310\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	−0.854102	−0.158603	−0.0793014	−	0.996851i	$-0.525269\pi$
$29$	−0.854102	−0.158603	−0.0793014	+	0.996851i	$0.525269\pi$
$30$	0	0
$31$	−2.38197	−0.427814	−0.213907	−	0.976854i	$-0.568619\pi$
$31$	−2.38197	−0.427814	−0.213907	+	0.976854i	$0.568619\pi$
$32$	0	0
$33$	−15.9443	−2.77554
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	6.70820	1.10282	0.551411	−	0.834234i	$-0.314090\pi$
$37$	6.70820	1.10282	0.551411	+	0.834234i	$0.314090\pi$
$38$	0	0
$39$	5.85410	0.937407
$40$	0	0
$41$	0.0901699	0.0140822	0.00704109	−	0.999975i	$-0.497759\pi$
$41$	0.0901699	0.0140822	0.00704109	+	0.999975i	$0.497759\pi$
$42$	0	0
$43$	−8.47214	−1.29199	−0.645994	−	0.763342i	$-0.723557\pi$
$43$	−8.47214	−1.29199	−0.645994	+	0.763342i	$0.723557\pi$
$44$	0	0
$45$	3.85410	0.574536
$46$	0	0
$47$	4.70820	0.686762	0.343381	−	0.939196i	$-0.388428\pi$
$47$	4.70820	0.686762	0.343381	+	0.939196i	$0.388428\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−19.9443	−2.79276
$52$	0	0
$53$	14.3262	1.96786	0.983930	−	0.178554i	$-0.0571420\pi$
$53$	14.3262	1.96786	0.983930	+	0.178554i	$0.0571420\pi$
$54$	0	0
$55$	−6.09017	−0.821198
$56$	0	0
$57$	2.61803	0.346767
$58$	0	0
$59$	10.7082	1.39409	0.697045	−	0.717028i	$-0.254498\pi$
$59$	10.7082	1.39409	0.697045	+	0.717028i	$0.254498\pi$
$60$	0	0
$61$	−5.47214	−0.700635	−0.350318	−	0.936631i	$-0.613926\pi$
$61$	−5.47214	−0.700635	−0.350318	+	0.936631i	$0.613926\pi$
$62$	0	0
$63$	3.85410	0.485571
$64$	0	0
$65$	2.23607	0.277350
$66$	0	0
$67$	2.09017	0.255355	0.127677	−	0.991816i	$-0.459248\pi$
$67$	2.09017	0.255355	0.127677	+	0.991816i	$0.459248\pi$
$68$	0	0
$69$	−22.7984	−2.74460
$70$	0	0
$71$	−11.1803	−1.32686	−0.663431	−	0.748237i	$-0.730900\pi$
$71$	−11.1803	−1.32686	−0.663431	+	0.748237i	$0.730900\pi$
$72$	0	0
$73$	−0.909830	−0.106488	−0.0532438	−	0.998582i	$-0.516956\pi$
$73$	−0.909830	−0.106488	−0.0532438	+	0.998582i	$0.516956\pi$
$74$	0	0
$75$	−10.4721	−1.20922
$76$	0	0
$77$	−6.09017	−0.694039
$78$	0	0
$79$	−6.94427	−0.781292	−0.390646	−	0.920541i	$-0.627748\pi$
$79$	−6.94427	−0.781292	−0.390646	+	0.920541i	$0.627748\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	0	0
$83$	−14.6180	−1.60454	−0.802269	−	0.596963i	$-0.796374\pi$
$83$	−14.6180	−1.60454	−0.802269	+	0.596963i	$0.796374\pi$
$84$	0	0
$85$	−7.61803	−0.826292
$86$	0	0
$87$	−2.23607	−0.239732
$88$	0	0
$89$	−12.1803	−1.29111	−0.645557	−	0.763712i	$-0.723375\pi$
$89$	−12.1803	−1.29111	−0.645557	+	0.763712i	$0.723375\pi$
$90$	0	0
$91$	2.23607	0.234404
$92$	0	0
$93$	−6.23607	−0.646650
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	8.70820	0.884184	0.442092	−	0.896970i	$-0.354236\pi$
$97$	8.70820	0.884184	0.442092	+	0.896970i	$0.354236\pi$
$98$	0	0
$99$	−23.4721	−2.35904
$100$	0	0
$101$	8.23607	0.819519	0.409760	−	0.912194i	$-0.365613\pi$
$101$	8.23607	0.819519	0.409760	+	0.912194i	$0.365613\pi$
$102$	0	0
$103$	−7.76393	−0.765003	−0.382501	−	0.923955i	$-0.624937\pi$
$103$	−7.76393	−0.765003	−0.382501	+	0.923955i	$0.624937\pi$
$104$	0	0
$105$	2.61803	0.255494
$106$	0	0
$107$	−3.76393	−0.363873	−0.181937	−	0.983310i	$-0.558237\pi$
$107$	−3.76393	−0.363873	−0.181937	+	0.983310i	$0.558237\pi$
$108$	0	0
$109$	−18.8885	−1.80919	−0.904597	−	0.426267i	$-0.859828\pi$
$109$	−18.8885	−1.80919	−0.904597	+	0.426267i	$0.859828\pi$
$110$	0	0
$111$	17.5623	1.66694
$112$	0	0
$113$	−14.6180	−1.37515	−0.687574	−	0.726114i	$-0.741325\pi$
$113$	−14.6180	−1.37515	−0.687574	+	0.726114i	$0.741325\pi$
$114$	0	0
$115$	−8.70820	−0.812044
$116$	0	0
$117$	8.61803	0.796738
$118$	0	0
$119$	−7.61803	−0.698344
$120$	0	0
$121$	26.0902	2.37183
$122$	0	0
$123$	0.236068	0.0212855
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	15.2361	1.35198	0.675991	−	0.736910i	$-0.263716\pi$
$127$	15.2361	1.35198	0.675991	+	0.736910i	$0.263716\pi$
$128$	0	0
$129$	−22.1803	−1.95287
$130$	0	0
$131$	4.38197	0.382854	0.191427	−	0.981507i	$-0.438688\pi$
$131$	4.38197	0.382854	0.191427	+	0.981507i	$0.438688\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	2.23607	0.192450
$136$	0	0
$137$	3.52786	0.301406	0.150703	−	0.988579i	$-0.451846\pi$
$137$	3.52786	0.301406	0.150703	+	0.988579i	$0.451846\pi$
$138$	0	0
$139$	5.18034	0.439391	0.219695	−	0.975569i	$-0.429494\pi$
$139$	5.18034	0.439391	0.219695	+	0.975569i	$0.429494\pi$
$140$	0	0
$141$	12.3262	1.03806
$142$	0	0
$143$	−13.6180	−1.13880
$144$	0	0
$145$	−0.854102	−0.0709293
$146$	0	0
$147$	2.61803	0.215932
$148$	0	0
$149$	−7.00000	−0.573462	−0.286731	−	0.958011i	$-0.592569\pi$
$149$	−7.00000	−0.573462	−0.286731	+	0.958011i	$0.592569\pi$
$150$	0	0
$151$	−4.38197	−0.356599	−0.178300	−	0.983976i	$-0.557060\pi$
$151$	−4.38197	−0.356599	−0.178300	+	0.983976i	$0.557060\pi$
$152$	0	0
$153$	−29.3607	−2.37367
$154$	0	0
$155$	−2.38197	−0.191324
$156$	0	0
$157$	−13.6180	−1.08684	−0.543419	−	0.839462i	$-0.682870\pi$
$157$	−13.6180	−1.08684	−0.543419	+	0.839462i	$0.682870\pi$
$158$	0	0
$159$	37.5066	2.97447
$160$	0	0
$161$	−8.70820	−0.686303
$162$	0	0
$163$	10.7984	0.845794	0.422897	−	0.906178i	$-0.361013\pi$
$163$	10.7984	0.845794	0.422897	+	0.906178i	$0.361013\pi$
$164$	0	0
$165$	−15.9443	−1.24126
$166$	0	0
$167$	1.47214	0.113917	0.0569587	−	0.998377i	$-0.481860\pi$
$167$	1.47214	0.113917	0.0569587	+	0.998377i	$0.481860\pi$
$168$	0	0
$169$	−8.00000	−0.615385
$170$	0	0
$171$	3.85410	0.294731
$172$	0	0
$173$	10.2918	0.782471	0.391235	−	0.920291i	$-0.372048\pi$
$173$	10.2918	0.782471	0.391235	+	0.920291i	$0.372048\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	28.0344	2.10720
$178$	0	0
$179$	10.3820	0.775985	0.387992	−	0.921663i	$-0.373169\pi$
$179$	10.3820	0.775985	0.387992	+	0.921663i	$0.373169\pi$
$180$	0	0
$181$	13.5066	1.00394	0.501968	−	0.864886i	$-0.332610\pi$
$181$	13.5066	1.00394	0.501968	+	0.864886i	$0.332610\pi$
$182$	0	0
$183$	−14.3262	−1.05903
$184$	0	0
$185$	6.70820	0.493197
$186$	0	0
$187$	46.3951	3.39275
$188$	0	0
$189$	2.23607	0.162650
$190$	0	0
$191$	8.03444	0.581352	0.290676	−	0.956822i	$-0.406120\pi$
$191$	8.03444	0.581352	0.290676	+	0.956822i	$0.406120\pi$
$192$	0	0
$193$	−1.43769	−0.103487	−0.0517437	−	0.998660i	$-0.516478\pi$
$193$	−1.43769	−0.103487	−0.0517437	+	0.998660i	$0.516478\pi$
$194$	0	0
$195$	5.85410	0.419221
$196$	0	0
$197$	−8.79837	−0.626858	−0.313429	−	0.949612i	$-0.601478\pi$
$197$	−8.79837	−0.626858	−0.313429	+	0.949612i	$0.601478\pi$
$198$	0	0
$199$	2.23607	0.158511	0.0792553	−	0.996854i	$-0.474746\pi$
$199$	2.23607	0.158511	0.0792553	+	0.996854i	$0.474746\pi$
$200$	0	0
$201$	5.47214	0.385975
$202$	0	0
$203$	−0.854102	−0.0599462
$204$	0	0
$205$	0.0901699	0.00629774
$206$	0	0
$207$	−33.5623	−2.33274
$208$	0	0
$209$	−6.09017	−0.421266
$210$	0	0
$211$	−26.9787	−1.85729	−0.928646	−	0.370968i	$-0.879026\pi$
$211$	−26.9787	−1.85729	−0.928646	+	0.370968i	$0.879026\pi$
$212$	0	0
$213$	−29.2705	−2.00558
$214$	0	0
$215$	−8.47214	−0.577795
$216$	0	0
$217$	−2.38197	−0.161698
$218$	0	0
$219$	−2.38197	−0.160958
$220$	0	0
$221$	−17.0344	−1.14586
$222$	0	0
$223$	5.94427	0.398058	0.199029	−	0.979994i	$-0.436221\pi$
$223$	5.94427	0.398058	0.199029	+	0.979994i	$0.436221\pi$
$224$	0	0
$225$	−15.4164	−1.02776
$226$	0	0
$227$	−16.0902	−1.06794	−0.533971	−	0.845503i	$-0.679301\pi$
$227$	−16.0902	−1.06794	−0.533971	+	0.845503i	$0.679301\pi$
$228$	0	0
$229$	12.4721	0.824182	0.412091	−	0.911143i	$-0.364799\pi$
$229$	12.4721	0.824182	0.412091	+	0.911143i	$0.364799\pi$
$230$	0	0
$231$	−15.9443	−1.04906
$232$	0	0
$233$	−7.32624	−0.479958	−0.239979	−	0.970778i	$-0.577141\pi$
$233$	−7.32624	−0.479958	−0.239979	+	0.970778i	$0.577141\pi$
$234$	0	0
$235$	4.70820	0.307129
$236$	0	0
$237$	−18.1803	−1.18094
$238$	0	0
$239$	−18.1246	−1.17238	−0.586192	−	0.810172i	$-0.699374\pi$
$239$	−18.1246	−1.17238	−0.586192	+	0.810172i	$0.699374\pi$
$240$	0	0
$241$	23.7082	1.52718	0.763590	−	0.645702i	$-0.223435\pi$
$241$	23.7082	1.52718	0.763590	+	0.645702i	$0.223435\pi$
$242$	0	0
$243$	−21.6525	−1.38901
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	2.23607	0.142278
$248$	0	0
$249$	−38.2705	−2.42530
$250$	0	0
$251$	0.673762	0.0425275	0.0212637	−	0.999774i	$-0.493231\pi$
$251$	0.673762	0.0425275	0.0212637	+	0.999774i	$0.493231\pi$
$252$	0	0
$253$	53.0344	3.33425
$254$	0	0
$255$	−19.9443	−1.24896
$256$	0	0
$257$	13.2705	0.827792	0.413896	−	0.910324i	$-0.364168\pi$
$257$	13.2705	0.827792	0.413896	+	0.910324i	$0.364168\pi$
$258$	0	0
$259$	6.70820	0.416828
$260$	0	0
$261$	−3.29180	−0.203757
$262$	0	0
$263$	18.7984	1.15916	0.579579	−	0.814916i	$-0.303217\pi$
$263$	18.7984	1.15916	0.579579	+	0.814916i	$0.303217\pi$
$264$	0	0
$265$	14.3262	0.880054
$266$	0	0
$267$	−31.8885	−1.95155
$268$	0	0
$269$	−4.79837	−0.292562	−0.146281	−	0.989243i	$-0.546730\pi$
$269$	−4.79837	−0.292562	−0.146281	+	0.989243i	$0.546730\pi$
$270$	0	0
$271$	0.562306	0.0341577	0.0170788	−	0.999854i	$-0.494563\pi$
$271$	0.562306	0.0341577	0.0170788	+	0.999854i	$0.494563\pi$
$272$	0	0
$273$	5.85410	0.354306
$274$	0	0
$275$	24.3607	1.46900
$276$	0	0
$277$	8.65248	0.519877	0.259938	−	0.965625i	$-0.416298\pi$
$277$	8.65248	0.519877	0.259938	+	0.965625i	$0.416298\pi$
$278$	0	0
$279$	−9.18034	−0.549613
$280$	0	0
$281$	−11.0000	−0.656205	−0.328102	−	0.944642i	$-0.606409\pi$
$281$	−11.0000	−0.656205	−0.328102	+	0.944642i	$0.606409\pi$
$282$	0	0
$283$	18.6180	1.10673	0.553364	−	0.832940i	$-0.313344\pi$
$283$	18.6180	1.10673	0.553364	+	0.832940i	$0.313344\pi$
$284$	0	0
$285$	2.61803	0.155079
$286$	0	0
$287$	0.0901699	0.00532256
$288$	0	0
$289$	41.0344	2.41379
$290$	0	0
$291$	22.7984	1.33646
$292$	0	0
$293$	19.4721	1.13757	0.568787	−	0.822485i	$-0.307413\pi$
$293$	19.4721	1.13757	0.568787	+	0.822485i	$0.307413\pi$
$294$	0	0
$295$	10.7082	0.623456
$296$	0	0
$297$	−13.6180	−0.790198
$298$	0	0
$299$	−19.4721	−1.12610
$300$	0	0
$301$	−8.47214	−0.488326
$302$	0	0
$303$	21.5623	1.23872
$304$	0	0
$305$	−5.47214	−0.313334
$306$	0	0
$307$	−22.7426	−1.29799	−0.648996	−	0.760792i	$-0.724811\pi$
$307$	−22.7426	−1.29799	−0.648996	+	0.760792i	$0.724811\pi$
$308$	0	0
$309$	−20.3262	−1.15632
$310$	0	0
$311$	−3.32624	−0.188614	−0.0943068	−	0.995543i	$-0.530063\pi$
$311$	−3.32624	−0.188614	−0.0943068	+	0.995543i	$0.530063\pi$
$312$	0	0
$313$	14.1803	0.801520	0.400760	−	0.916183i	$-0.368746\pi$
$313$	14.1803	0.801520	0.400760	+	0.916183i	$0.368746\pi$
$314$	0	0
$315$	3.85410	0.217154
$316$	0	0
$317$	−4.47214	−0.251180	−0.125590	−	0.992082i	$-0.540082\pi$
$317$	−4.47214	−0.251180	−0.125590	+	0.992082i	$0.540082\pi$
$318$	0	0
$319$	5.20163	0.291235
$320$	0	0
$321$	−9.85410	−0.550002
$322$	0	0
$323$	−7.61803	−0.423879
$324$	0	0
$325$	−8.94427	−0.496139
$326$	0	0
$327$	−49.4508	−2.73464
$328$	0	0
$329$	4.70820	0.259572
$330$	0	0
$331$	0.729490	0.0400964	0.0200482	−	0.999799i	$-0.493618\pi$
$331$	0.729490	0.0400964	0.0200482	+	0.999799i	$0.493618\pi$
$332$	0	0
$333$	25.8541	1.41680
$334$	0	0
$335$	2.09017	0.114198
$336$	0	0
$337$	−9.32624	−0.508033	−0.254016	−	0.967200i	$-0.581752\pi$
$337$	−9.32624	−0.508033	−0.254016	+	0.967200i	$0.581752\pi$
$338$	0	0
$339$	−38.2705	−2.07857
$340$	0	0
$341$	14.5066	0.785575
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−22.7984	−1.22742
$346$	0	0
$347$	11.5623	0.620697	0.310349	−	0.950623i	$-0.399554\pi$
$347$	11.5623	0.620697	0.310349	+	0.950623i	$0.399554\pi$
$348$	0	0
$349$	3.27051	0.175066	0.0875332	−	0.996162i	$-0.472102\pi$
$349$	3.27051	0.175066	0.0875332	+	0.996162i	$0.472102\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	−21.6180	−1.15061	−0.575306	−	0.817938i	$-0.695117\pi$
$353$	−21.6180	−1.15061	−0.575306	+	0.817938i	$0.695117\pi$
$354$	0	0
$355$	−11.1803	−0.593391
$356$	0	0
$357$	−19.9443	−1.05556
$358$	0	0
$359$	−3.43769	−0.181435	−0.0907173	−	0.995877i	$-0.528916\pi$
$359$	−3.43769	−0.181435	−0.0907173	+	0.995877i	$0.528916\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	68.3050	3.58508
$364$	0	0
$365$	−0.909830	−0.0476227
$366$	0	0
$367$	12.8885	0.672777	0.336388	−	0.941723i	$-0.390795\pi$
$367$	12.8885	0.672777	0.336388	+	0.941723i	$0.390795\pi$
$368$	0	0
$369$	0.347524	0.0180914
$370$	0	0
$371$	14.3262	0.743781
$372$	0	0
$373$	16.5066	0.854678	0.427339	−	0.904091i	$-0.359451\pi$
$373$	16.5066	0.854678	0.427339	+	0.904091i	$0.359451\pi$
$374$	0	0
$375$	−23.5623	−1.21675
$376$	0	0
$377$	−1.90983	−0.0983613
$378$	0	0
$379$	−6.52786	−0.335314	−0.167657	−	0.985845i	$-0.553620\pi$
$379$	−6.52786	−0.335314	−0.167657	+	0.985845i	$0.553620\pi$
$380$	0	0
$381$	39.8885	2.04355
$382$	0	0
$383$	−16.2361	−0.829624	−0.414812	−	0.909907i	$-0.636153\pi$
$383$	−16.2361	−0.829624	−0.414812	+	0.909907i	$0.636153\pi$
$384$	0	0
$385$	−6.09017	−0.310384
$386$	0	0
$387$	−32.6525	−1.65982
$388$	0	0
$389$	−5.61803	−0.284846	−0.142423	−	0.989806i	$-0.545489\pi$
$389$	−5.61803	−0.284846	−0.142423	+	0.989806i	$0.545489\pi$
$390$	0	0
$391$	66.3394	3.35493
$392$	0	0
$393$	11.4721	0.578693
$394$	0	0
$395$	−6.94427	−0.349404
$396$	0	0
$397$	15.2361	0.764676	0.382338	−	0.924022i	$-0.375119\pi$
$397$	15.2361	0.764676	0.382338	+	0.924022i	$0.375119\pi$
$398$	0	0
$399$	2.61803	0.131066
$400$	0	0
$401$	−26.7426	−1.33546	−0.667732	−	0.744402i	$-0.732735\pi$
$401$	−26.7426	−1.33546	−0.667732	+	0.744402i	$0.732735\pi$
$402$	0	0
$403$	−5.32624	−0.265319
$404$	0	0
$405$	−5.70820	−0.283643
$406$	0	0
$407$	−40.8541	−2.02506
$408$	0	0
$409$	−1.27051	−0.0628227	−0.0314113	−	0.999507i	$-0.510000\pi$
$409$	−1.27051	−0.0628227	−0.0314113	+	0.999507i	$0.510000\pi$
$410$	0	0
$411$	9.23607	0.455582
$412$	0	0
$413$	10.7082	0.526916
$414$	0	0
$415$	−14.6180	−0.717571
$416$	0	0
$417$	13.5623	0.664149
$418$	0	0
$419$	8.70820	0.425424	0.212712	−	0.977115i	$-0.431770\pi$
$419$	8.70820	0.425424	0.212712	+	0.977115i	$0.431770\pi$
$420$	0	0
$421$	9.18034	0.447422	0.223711	−	0.974655i	$-0.428183\pi$
$421$	9.18034	0.447422	0.223711	+	0.974655i	$0.428183\pi$
$422$	0	0
$423$	18.1459	0.882284
$424$	0	0
$425$	30.4721	1.47812
$426$	0	0
$427$	−5.47214	−0.264815
$428$	0	0
$429$	−35.6525	−1.72132
$430$	0	0
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	0	0
$433$	31.4721	1.51245	0.756227	−	0.654309i	$-0.227040\pi$
$433$	31.4721	1.51245	0.756227	+	0.654309i	$0.227040\pi$
$434$	0	0
$435$	−2.23607	−0.107211
$436$	0	0
$437$	−8.70820	−0.416570
$438$	0	0
$439$	28.3607	1.35358	0.676791	−	0.736175i	$-0.263370\pi$
$439$	28.3607	1.35358	0.676791	+	0.736175i	$0.263370\pi$
$440$	0	0
$441$	3.85410	0.183529
$442$	0	0
$443$	−34.7984	−1.65332	−0.826660	−	0.562701i	$-0.809762\pi$
$443$	−34.7984	−1.65332	−0.826660	+	0.562701i	$0.809762\pi$
$444$	0	0
$445$	−12.1803	−0.577403
$446$	0	0
$447$	−18.3262	−0.866801
$448$	0	0
$449$	−1.38197	−0.0652190	−0.0326095	−	0.999468i	$-0.510382\pi$
$449$	−1.38197	−0.0652190	−0.0326095	+	0.999468i	$0.510382\pi$
$450$	0	0
$451$	−0.549150	−0.0258585
$452$	0	0
$453$	−11.4721	−0.539008
$454$	0	0
$455$	2.23607	0.104828
$456$	0	0
$457$	16.2148	0.758495	0.379248	−	0.925295i	$-0.376183\pi$
$457$	16.2148	0.758495	0.379248	+	0.925295i	$0.376183\pi$
$458$	0	0
$459$	−17.0344	−0.795100
$460$	0	0
$461$	24.0344	1.11940	0.559698	−	0.828697i	$-0.310917\pi$
$461$	24.0344	1.11940	0.559698	+	0.828697i	$0.310917\pi$
$462$	0	0
$463$	−40.7771	−1.89507	−0.947536	−	0.319649i	$-0.896435\pi$
$463$	−40.7771	−1.89507	−0.947536	+	0.319649i	$0.896435\pi$
$464$	0	0
$465$	−6.23607	−0.289191
$466$	0	0
$467$	−19.8541	−0.918738	−0.459369	−	0.888245i	$-0.651924\pi$
$467$	−19.8541	−0.918738	−0.459369	+	0.888245i	$0.651924\pi$
$468$	0	0
$469$	2.09017	0.0965151
$470$	0	0
$471$	−35.6525	−1.64278
$472$	0	0
$473$	51.5967	2.37242
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	55.2148	2.52811
$478$	0	0
$479$	−15.9787	−0.730086	−0.365043	−	0.930991i	$-0.618946\pi$
$479$	−15.9787	−0.730086	−0.365043	+	0.930991i	$0.618946\pi$
$480$	0	0
$481$	15.0000	0.683941
$482$	0	0
$483$	−22.7984	−1.03736
$484$	0	0
$485$	8.70820	0.395419
$486$	0	0
$487$	−20.7082	−0.938378	−0.469189	−	0.883098i	$-0.655454\pi$
$487$	−20.7082	−0.938378	−0.469189	+	0.883098i	$0.655454\pi$
$488$	0	0
$489$	28.2705	1.27844
$490$	0	0
$491$	−23.0557	−1.04049	−0.520245	−	0.854017i	$-0.674159\pi$
$491$	−23.0557	−1.04049	−0.520245	+	0.854017i	$0.674159\pi$
$492$	0	0
$493$	6.50658	0.293042
$494$	0	0
$495$	−23.4721	−1.05499
$496$	0	0
$497$	−11.1803	−0.501507
$498$	0	0
$499$	−6.90983	−0.309326	−0.154663	−	0.987967i	$-0.549429\pi$
$499$	−6.90983	−0.309326	−0.154663	+	0.987967i	$0.549429\pi$
$500$	0	0
$501$	3.85410	0.172189
$502$	0	0
$503$	0.944272	0.0421030	0.0210515	−	0.999778i	$-0.493299\pi$
$503$	0.944272	0.0421030	0.0210515	+	0.999778i	$0.493299\pi$
$504$	0	0
$505$	8.23607	0.366500
$506$	0	0
$507$	−20.9443	−0.930168
$508$	0	0
$509$	−33.4164	−1.48116	−0.740578	−	0.671970i	$-0.765448\pi$
$509$	−33.4164	−1.48116	−0.740578	+	0.671970i	$0.765448\pi$
$510$	0	0
$511$	−0.909830	−0.0402485
$512$	0	0
$513$	2.23607	0.0987248
$514$	0	0
$515$	−7.76393	−0.342120
$516$	0	0
$517$	−28.6738	−1.26107
$518$	0	0
$519$	26.9443	1.18272
$520$	0	0
$521$	6.88854	0.301793	0.150896	−	0.988550i	$-0.451784\pi$
$521$	6.88854	0.301793	0.150896	+	0.988550i	$0.451784\pi$
$522$	0	0
$523$	−4.41641	−0.193116	−0.0965580	−	0.995327i	$-0.530783\pi$
$523$	−4.41641	−0.193116	−0.0965580	+	0.995327i	$0.530783\pi$
$524$	0	0
$525$	−10.4721	−0.457041
$526$	0	0
$527$	18.1459	0.790448
$528$	0	0
$529$	52.8328	2.29708
$530$	0	0
$531$	41.2705	1.79099
$532$	0	0
$533$	0.201626	0.00873340
$534$	0	0
$535$	−3.76393	−0.162729
$536$	0	0
$537$	27.1803	1.17292
$538$	0	0
$539$	−6.09017	−0.262322
$540$	0	0
$541$	10.5836	0.455024	0.227512	−	0.973775i	$-0.426941\pi$
$541$	10.5836	0.455024	0.227512	+	0.973775i	$0.426941\pi$
$542$	0	0
$543$	35.3607	1.51747
$544$	0	0
$545$	−18.8885	−0.809096
$546$	0	0
$547$	−10.6180	−0.453994	−0.226997	−	0.973895i	$-0.572891\pi$
$547$	−10.6180	−0.453994	−0.226997	+	0.973895i	$0.572891\pi$
$548$	0	0
$549$	−21.0902	−0.900107
$550$	0	0
$551$	−0.854102	−0.0363860
$552$	0	0
$553$	−6.94427	−0.295300
$554$	0	0
$555$	17.5623	0.745478
$556$	0	0
$557$	15.0344	0.637030	0.318515	−	0.947918i	$-0.396816\pi$
$557$	15.0344	0.637030	0.318515	+	0.947918i	$0.396816\pi$
$558$	0	0
$559$	−18.9443	−0.801257
$560$	0	0
$561$	121.464	5.12821
$562$	0	0
$563$	−37.5410	−1.58217	−0.791083	−	0.611709i	$-0.790482\pi$
$563$	−37.5410	−1.58217	−0.791083	+	0.611709i	$0.790482\pi$
$564$	0	0
$565$	−14.6180	−0.614985
$566$	0	0
$567$	−5.70820	−0.239722
$568$	0	0
$569$	−13.2918	−0.557221	−0.278611	−	0.960404i	$-0.589874\pi$
$569$	−13.2918	−0.557221	−0.278611	+	0.960404i	$0.589874\pi$
$570$	0	0
$571$	5.94427	0.248760	0.124380	−	0.992235i	$-0.460306\pi$
$571$	5.94427	0.248760	0.124380	+	0.992235i	$0.460306\pi$
$572$	0	0
$573$	21.0344	0.878726
$574$	0	0
$575$	34.8328	1.45263
$576$	0	0
$577$	9.67376	0.402724	0.201362	−	0.979517i	$-0.435463\pi$
$577$	9.67376	0.402724	0.201362	+	0.979517i	$0.435463\pi$
$578$	0	0
$579$	−3.76393	−0.156424
$580$	0	0
$581$	−14.6180	−0.606458
$582$	0	0
$583$	−87.2492	−3.61349
$584$	0	0
$585$	8.61803	0.356312
$586$	0	0
$587$	34.3607	1.41822	0.709109	−	0.705099i	$-0.249098\pi$
$587$	34.3607	1.41822	0.709109	+	0.705099i	$0.249098\pi$
$588$	0	0
$589$	−2.38197	−0.0981472
$590$	0	0
$591$	−23.0344	−0.947510
$592$	0	0
$593$	27.2918	1.12074	0.560370	−	0.828242i	$-0.310659\pi$
$593$	27.2918	1.12074	0.560370	+	0.828242i	$0.310659\pi$
$594$	0	0
$595$	−7.61803	−0.312309
$596$	0	0
$597$	5.85410	0.239592
$598$	0	0
$599$	40.1459	1.64032	0.820158	−	0.572136i	$-0.193885\pi$
$599$	40.1459	1.64032	0.820158	+	0.572136i	$0.193885\pi$
$600$	0	0
$601$	18.9787	0.774158	0.387079	−	0.922047i	$-0.373484\pi$
$601$	18.9787	0.774158	0.387079	+	0.922047i	$0.373484\pi$
$602$	0	0
$603$	8.05573	0.328055
$604$	0	0
$605$	26.0902	1.06072
$606$	0	0
$607$	−36.5967	−1.48542	−0.742708	−	0.669615i	$-0.766459\pi$
$607$	−36.5967	−1.48542	−0.742708	+	0.669615i	$0.766459\pi$
$608$	0	0
$609$	−2.23607	−0.0906100
$610$	0	0
$611$	10.5279	0.425912
$612$	0	0
$613$	−2.03444	−0.0821703	−0.0410852	−	0.999156i	$-0.513081\pi$
$613$	−2.03444	−0.0821703	−0.0410852	+	0.999156i	$0.513081\pi$
$614$	0	0
$615$	0.236068	0.00951918
$616$	0	0
$617$	−26.8541	−1.08111	−0.540553	−	0.841310i	$-0.681785\pi$
$617$	−26.8541	−1.08111	−0.540553	+	0.841310i	$0.681785\pi$
$618$	0	0
$619$	46.7984	1.88099	0.940493	−	0.339814i	$-0.110364\pi$
$619$	46.7984	1.88099	0.940493	+	0.339814i	$0.110364\pi$
$620$	0	0
$621$	−19.4721	−0.781390
$622$	0	0
$623$	−12.1803	−0.487995
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−15.9443	−0.636753
$628$	0	0
$629$	−51.1033	−2.03762
$630$	0	0
$631$	0.888544	0.0353724	0.0176862	−	0.999844i	$-0.494370\pi$
$631$	0.888544	0.0353724	0.0176862	+	0.999844i	$0.494370\pi$
$632$	0	0
$633$	−70.6312	−2.80734
$634$	0	0
$635$	15.2361	0.604625
$636$	0	0
$637$	2.23607	0.0885962
$638$	0	0
$639$	−43.0902	−1.70462
$640$	0	0
$641$	6.38197	0.252073	0.126036	−	0.992026i	$-0.459774\pi$
$641$	6.38197	0.252073	0.126036	+	0.992026i	$0.459774\pi$
$642$	0	0
$643$	7.47214	0.294672	0.147336	−	0.989086i	$-0.452930\pi$
$643$	7.47214	0.294672	0.147336	+	0.989086i	$0.452930\pi$
$644$	0	0
$645$	−22.1803	−0.873350
$646$	0	0
$647$	−27.6525	−1.08713	−0.543566	−	0.839367i	$-0.682926\pi$
$647$	−27.6525	−1.08713	−0.543566	+	0.839367i	$0.682926\pi$
$648$	0	0
$649$	−65.2148	−2.55990
$650$	0	0
$651$	−6.23607	−0.244411
$652$	0	0
$653$	22.1246	0.865803	0.432901	−	0.901441i	$-0.357490\pi$
$653$	22.1246	0.865803	0.432901	+	0.901441i	$0.357490\pi$
$654$	0	0
$655$	4.38197	0.171218
$656$	0	0
$657$	−3.50658	−0.136805
$658$	0	0
$659$	−22.0902	−0.860511	−0.430255	−	0.902707i	$-0.641577\pi$
$659$	−22.0902	−0.860511	−0.430255	+	0.902707i	$0.641577\pi$
$660$	0	0
$661$	−12.4721	−0.485110	−0.242555	−	0.970138i	$-0.577985\pi$
$661$	−12.4721	−0.485110	−0.242555	+	0.970138i	$0.577985\pi$
$662$	0	0
$663$	−44.5967	−1.73199
$664$	0	0
$665$	1.00000	0.0387783
$666$	0	0
$667$	7.43769	0.287989
$668$	0	0
$669$	15.5623	0.601674
$670$	0	0
$671$	33.3262	1.28655
$672$	0	0
$673$	−44.5066	−1.71560	−0.857801	−	0.513982i	$-0.828170\pi$
$673$	−44.5066	−1.71560	−0.857801	+	0.513982i	$0.828170\pi$
$674$	0	0
$675$	−8.94427	−0.344265
$676$	0	0
$677$	47.9230	1.84183	0.920915	−	0.389764i	$-0.127443\pi$
$677$	47.9230	1.84183	0.920915	+	0.389764i	$0.127443\pi$
$678$	0	0
$679$	8.70820	0.334190
$680$	0	0
$681$	−42.1246	−1.61422
$682$	0	0
$683$	−7.29180	−0.279013	−0.139506	−	0.990221i	$-0.544552\pi$
$683$	−7.29180	−0.279013	−0.139506	+	0.990221i	$0.544552\pi$
$684$	0	0
$685$	3.52786	0.134793
$686$	0	0
$687$	32.6525	1.24577
$688$	0	0
$689$	32.0344	1.22042
$690$	0	0
$691$	39.1246	1.48837	0.744185	−	0.667973i	$-0.232838\pi$
$691$	39.1246	1.48837	0.744185	+	0.667973i	$0.232838\pi$
$692$	0	0
$693$	−23.4721	−0.891633
$694$	0	0
$695$	5.18034	0.196501
$696$	0	0
$697$	−0.686918	−0.0260189
$698$	0	0
$699$	−19.1803	−0.725467
$700$	0	0
$701$	6.47214	0.244449	0.122225	−	0.992502i	$-0.460997\pi$
$701$	6.47214	0.244449	0.122225	+	0.992502i	$0.460997\pi$
$702$	0	0
$703$	6.70820	0.253005
$704$	0	0
$705$	12.3262	0.464233
$706$	0	0
$707$	8.23607	0.309749
$708$	0	0
$709$	22.7082	0.852824	0.426412	−	0.904529i	$-0.359777\pi$
$709$	22.7082	0.852824	0.426412	+	0.904529i	$0.359777\pi$
$710$	0	0
$711$	−26.7639	−1.00373
$712$	0	0
$713$	20.7426	0.776818
$714$	0	0
$715$	−13.6180	−0.509286
$716$	0	0
$717$	−47.4508	−1.77208
$718$	0	0
$719$	28.3607	1.05767	0.528837	−	0.848723i	$-0.322628\pi$
$719$	28.3607	1.05767	0.528837	+	0.848723i	$0.322628\pi$
$720$	0	0
$721$	−7.76393	−0.289144
$722$	0	0
$723$	62.0689	2.30837
$724$	0	0
$725$	3.41641	0.126882
$726$	0	0
$727$	−18.5279	−0.687160	−0.343580	−	0.939123i	$-0.611640\pi$
$727$	−18.5279	−0.687160	−0.343580	+	0.939123i	$0.611640\pi$
$728$	0	0
$729$	−39.5623	−1.46527
$730$	0	0
$731$	64.5410	2.38714
$732$	0	0
$733$	−38.8328	−1.43432	−0.717161	−	0.696907i	$-0.754559\pi$
$733$	−38.8328	−1.43432	−0.717161	+	0.696907i	$0.754559\pi$
$734$	0	0
$735$	2.61803	0.0965676
$736$	0	0
$737$	−12.7295	−0.468897
$738$	0	0
$739$	−3.18034	−0.116991	−0.0584953	−	0.998288i	$-0.518630\pi$
$739$	−3.18034	−0.116991	−0.0584953	+	0.998288i	$0.518630\pi$
$740$	0	0
$741$	5.85410	0.215056
$742$	0	0
$743$	−8.34752	−0.306241	−0.153120	−	0.988208i	$-0.548932\pi$
$743$	−8.34752	−0.306241	−0.153120	+	0.988208i	$0.548932\pi$
$744$	0	0
$745$	−7.00000	−0.256460
$746$	0	0
$747$	−56.3394	−2.06135
$748$	0	0
$749$	−3.76393	−0.137531
$750$	0	0
$751$	−33.3820	−1.21813	−0.609063	−	0.793122i	$-0.708454\pi$
$751$	−33.3820	−1.21813	−0.609063	+	0.793122i	$0.708454\pi$
$752$	0	0
$753$	1.76393	0.0642813
$754$	0	0
$755$	−4.38197	−0.159476
$756$	0	0
$757$	−28.8328	−1.04795	−0.523973	−	0.851735i	$-0.675551\pi$
$757$	−28.8328	−1.04795	−0.523973	+	0.851735i	$0.675551\pi$
$758$	0	0
$759$	138.846	5.03979
$760$	0	0
$761$	−5.94427	−0.215480	−0.107740	−	0.994179i	$-0.534361\pi$
$761$	−5.94427	−0.215480	−0.107740	+	0.994179i	$0.534361\pi$
$762$	0	0
$763$	−18.8885	−0.683811
$764$	0	0
$765$	−29.3607	−1.06154
$766$	0	0
$767$	23.9443	0.864578
$768$	0	0
$769$	−28.3607	−1.02271	−0.511356	−	0.859369i	$-0.670857\pi$
$769$	−28.3607	−1.02271	−0.511356	+	0.859369i	$0.670857\pi$
$770$	0	0
$771$	34.7426	1.25123
$772$	0	0
$773$	43.1246	1.55108	0.775542	−	0.631296i	$-0.217477\pi$
$773$	43.1246	1.55108	0.775542	+	0.631296i	$0.217477\pi$
$774$	0	0
$775$	9.52786	0.342251
$776$	0	0
$777$	17.5623	0.630044
$778$	0	0
$779$	0.0901699	0.00323067
$780$	0	0
$781$	68.0902	2.43646
$782$	0	0
$783$	−1.90983	−0.0682518
$784$	0	0
$785$	−13.6180	−0.486048
$786$	0	0
$787$	0.360680	0.0128568	0.00642842	−	0.999979i	$-0.497954\pi$
$787$	0.360680	0.0128568	0.00642842	+	0.999979i	$0.497954\pi$
$788$	0	0
$789$	49.2148	1.75209
$790$	0	0
$791$	−14.6180	−0.519757
$792$	0	0
$793$	−12.2361	−0.434516
$794$	0	0
$795$	37.5066	1.33022
$796$	0	0
$797$	19.7295	0.698854	0.349427	−	0.936964i	$-0.386376\pi$
$797$	19.7295	0.698854	0.349427	+	0.936964i	$0.386376\pi$
$798$	0	0
$799$	−35.8673	−1.26889
$800$	0	0
$801$	−46.9443	−1.65869
$802$	0	0
$803$	5.54102	0.195538
$804$	0	0
$805$	−8.70820	−0.306924
$806$	0	0
$807$	−12.5623	−0.442214
$808$	0	0
$809$	−15.0689	−0.529794	−0.264897	−	0.964277i	$-0.585338\pi$
$809$	−15.0689	−0.529794	−0.264897	+	0.964277i	$0.585338\pi$
$810$	0	0
$811$	−7.88854	−0.277004	−0.138502	−	0.990362i	$-0.544229\pi$
$811$	−7.88854	−0.277004	−0.138502	+	0.990362i	$0.544229\pi$
$812$	0	0
$813$	1.47214	0.0516301
$814$	0	0
$815$	10.7984	0.378251
$816$	0	0
$817$	−8.47214	−0.296403
$818$	0	0
$819$	8.61803	0.301138
$820$	0	0
$821$	−23.4164	−0.817238	−0.408619	−	0.912705i	$-0.633990\pi$
$821$	−23.4164	−0.817238	−0.408619	+	0.912705i	$0.633990\pi$
$822$	0	0
$823$	−33.2361	−1.15854	−0.579268	−	0.815137i	$-0.696662\pi$
$823$	−33.2361	−1.15854	−0.579268	+	0.815137i	$0.696662\pi$
$824$	0	0
$825$	63.7771	2.22043
$826$	0	0
$827$	−24.8197	−0.863064	−0.431532	−	0.902098i	$-0.642027\pi$
$827$	−24.8197	−0.863064	−0.431532	+	0.902098i	$0.642027\pi$
$828$	0	0
$829$	38.2492	1.32845	0.664225	−	0.747533i	$-0.268762\pi$
$829$	38.2492	1.32845	0.664225	+	0.747533i	$0.268762\pi$
$830$	0	0
$831$	22.6525	0.785806
$832$	0	0
$833$	−7.61803	−0.263949
$834$	0	0
$835$	1.47214	0.0509454
$836$	0	0
$837$	−5.32624	−0.184102
$838$	0	0
$839$	−36.3607	−1.25531	−0.627655	−	0.778492i	$-0.715985\pi$
$839$	−36.3607	−1.25531	−0.627655	+	0.778492i	$0.715985\pi$
$840$	0	0
$841$	−28.2705	−0.974845
$842$	0	0
$843$	−28.7984	−0.991869
$844$	0	0
$845$	−8.00000	−0.275208
$846$	0	0
$847$	26.0902	0.896469
$848$	0	0
$849$	48.7426	1.67284
$850$	0	0
$851$	−58.4164	−2.00249
$852$	0	0
$853$	−38.2148	−1.30845	−0.654225	−	0.756300i	$-0.727005\pi$
$853$	−38.2148	−1.30845	−0.654225	+	0.756300i	$0.727005\pi$
$854$	0	0
$855$	3.85410	0.131808
$856$	0	0
$857$	24.3820	0.832872	0.416436	−	0.909165i	$-0.363279\pi$
$857$	24.3820	0.832872	0.416436	+	0.909165i	$0.363279\pi$
$858$	0	0
$859$	−31.9787	−1.09110	−0.545550	−	0.838078i	$-0.683679\pi$
$859$	−31.9787	−1.09110	−0.545550	+	0.838078i	$0.683679\pi$
$860$	0	0
$861$	0.236068	0.00804518
$862$	0	0
$863$	22.0344	0.750061	0.375031	−	0.927012i	$-0.377632\pi$
$863$	22.0344	0.750061	0.375031	+	0.927012i	$0.377632\pi$
$864$	0	0
$865$	10.2918	0.349932
$866$	0	0
$867$	107.430	3.64850
$868$	0	0
$869$	42.2918	1.43465
$870$	0	0
$871$	4.67376	0.158364
$872$	0	0
$873$	33.5623	1.13591
$874$	0	0
$875$	−9.00000	−0.304256
$876$	0	0
$877$	37.7082	1.27332	0.636658	−	0.771146i	$-0.280316\pi$
$877$	37.7082	1.27332	0.636658	+	0.771146i	$0.280316\pi$
$878$	0	0
$879$	50.9787	1.71947
$880$	0	0
$881$	16.5066	0.556121	0.278060	−	0.960564i	$-0.410308\pi$
$881$	16.5066	0.556121	0.278060	+	0.960564i	$0.410308\pi$
$882$	0	0
$883$	38.1246	1.28300	0.641498	−	0.767125i	$-0.278313\pi$
$883$	38.1246	1.28300	0.641498	+	0.767125i	$0.278313\pi$
$884$	0	0
$885$	28.0344	0.942367
$886$	0	0
$887$	−55.8328	−1.87468	−0.937341	−	0.348413i	$-0.886721\pi$
$887$	−55.8328	−1.87468	−0.937341	+	0.348413i	$0.886721\pi$
$888$	0	0
$889$	15.2361	0.511001
$890$	0	0
$891$	34.7639	1.16464
$892$	0	0
$893$	4.70820	0.157554
$894$	0	0
$895$	10.3820	0.347031
$896$	0	0
$897$	−50.9787	−1.70213
$898$	0	0
$899$	2.03444	0.0678524
$900$	0	0
$901$	−109.138	−3.63591
$902$	0	0
$903$	−22.1803	−0.738115
$904$	0	0
$905$	13.5066	0.448974
$906$	0	0
$907$	−34.3050	−1.13908	−0.569539	−	0.821965i	$-0.692878\pi$
$907$	−34.3050	−1.13908	−0.569539	+	0.821965i	$0.692878\pi$
$908$	0	0
$909$	31.7426	1.05284
$910$	0	0
$911$	0.472136	0.0156426	0.00782128	−	0.999969i	$-0.497510\pi$
$911$	0.472136	0.0156426	0.00782128	+	0.999969i	$0.497510\pi$
$912$	0	0
$913$	89.0263	2.94634
$914$	0	0
$915$	−14.3262	−0.473611
$916$	0	0
$917$	4.38197	0.144705
$918$	0	0
$919$	31.3607	1.03449	0.517247	−	0.855836i	$-0.326957\pi$
$919$	31.3607	1.03449	0.517247	+	0.855836i	$0.326957\pi$
$920$	0	0
$921$	−59.5410	−1.96194
$922$	0	0
$923$	−25.0000	−0.822885
$924$	0	0
$925$	−26.8328	−0.882258
$926$	0	0
$927$	−29.9230	−0.982800
$928$	0	0
$929$	−11.2705	−0.369773	−0.184887	−	0.982760i	$-0.559192\pi$
$929$	−11.2705	−0.369773	−0.184887	+	0.982760i	$0.559192\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−8.70820	−0.285094
$934$	0	0
$935$	46.3951	1.51728
$936$	0	0
$937$	−17.1459	−0.560132	−0.280066	−	0.959981i	$-0.590356\pi$
$937$	−17.1459	−0.560132	−0.280066	+	0.959981i	$0.590356\pi$
$938$	0	0
$939$	37.1246	1.21152
$940$	0	0
$941$	−47.8885	−1.56112	−0.780561	−	0.625080i	$-0.785066\pi$
$941$	−47.8885	−1.56112	−0.780561	+	0.625080i	$0.785066\pi$
$942$	0	0
$943$	−0.785218	−0.0255702
$944$	0	0
$945$	2.23607	0.0727393
$946$	0	0
$947$	−26.0344	−0.846006	−0.423003	−	0.906128i	$-0.639024\pi$
$947$	−26.0344	−0.846006	−0.423003	+	0.906128i	$0.639024\pi$
$948$	0	0
$949$	−2.03444	−0.0660408
$950$	0	0
$951$	−11.7082	−0.379665
$952$	0	0
$953$	13.2148	0.428069	0.214034	−	0.976826i	$-0.431340\pi$
$953$	13.2148	0.428069	0.214034	+	0.976826i	$0.431340\pi$
$954$	0	0
$955$	8.03444	0.259988
$956$	0	0
$957$	13.6180	0.440209
$958$	0	0
$959$	3.52786	0.113921
$960$	0	0
$961$	−25.3262	−0.816975
$962$	0	0
$963$	−14.5066	−0.467468
$964$	0	0
$965$	−1.43769	−0.0462810
$966$	0	0
$967$	35.9098	1.15478	0.577391	−	0.816468i	$-0.304071\pi$
$967$	35.9098	1.15478	0.577391	+	0.816468i	$0.304071\pi$
$968$	0	0
$969$	−19.9443	−0.640702
$970$	0	0
$971$	−21.7639	−0.698438	−0.349219	−	0.937041i	$-0.613553\pi$
$971$	−21.7639	−0.698438	−0.349219	+	0.937041i	$0.613553\pi$
$972$	0	0
$973$	5.18034	0.166074
$974$	0	0
$975$	−23.4164	−0.749925
$976$	0	0
$977$	21.2361	0.679402	0.339701	−	0.940533i	$-0.389674\pi$
$977$	21.2361	0.679402	0.339701	+	0.940533i	$0.389674\pi$
$978$	0	0
$979$	74.1803	2.37081
$980$	0	0
$981$	−72.7984	−2.32427
$982$	0	0
$983$	45.1803	1.44103	0.720515	−	0.693440i	$-0.243906\pi$
$983$	45.1803	1.44103	0.720515	+	0.693440i	$0.243906\pi$
$984$	0	0
$985$	−8.79837	−0.280340
$986$	0	0
$987$	12.3262	0.392348
$988$	0	0
$989$	73.7771	2.34597
$990$	0	0
$991$	−11.1246	−0.353385	−0.176692	−	0.984266i	$-0.556540\pi$
$991$	−11.1246	−0.353385	−0.176692	+	0.984266i	$0.556540\pi$
$992$	0	0
$993$	1.90983	0.0606066
$994$	0	0
$995$	2.23607	0.0708881
$996$	0	0
$997$	33.2016	1.05151	0.525753	−	0.850637i	$-0.323784\pi$
$997$	33.2016	1.05151	0.525753	+	0.850637i	$0.323784\pi$
$998$	0	0
$999$	15.0000	0.474579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.bg.1.2		2
4.3	odd	2		8512.2.a.k.1.1		2
8.3	odd	2		2128.2.a.m.1.2		2
8.5	even	2		532.2.a.b.1.1	✓	2
24.5	odd	2		4788.2.a.l.1.1		2
56.13	odd	2		3724.2.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.a.b.1.1	✓	2	8.5	even	2
2128.2.a.m.1.2		2	8.3	odd	2
3724.2.a.g.1.2		2	56.13	odd	2
4788.2.a.l.1.1		2	24.5	odd	2
8512.2.a.k.1.1		2	4.3	odd	2
8512.2.a.bg.1.2		2	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:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61803 q^{3} +1.00000 q^{5} +1.00000 q^{7} +3.85410 q^{9} -6.09017 q^{11} +2.23607 q^{13} +2.61803 q^{15} -7.61803 q^{17} +1.00000 q^{19} +2.61803 q^{21} -8.70820 q^{23} -4.00000 q^{25} +2.23607 q^{27} -0.854102 q^{29} -2.38197 q^{31} -15.9443 q^{33} +1.00000 q^{35} +6.70820 q^{37} +5.85410 q^{39} +0.0901699 q^{41} -8.47214 q^{43} +3.85410 q^{45} +4.70820 q^{47} +1.00000 q^{49} -19.9443 q^{51} +14.3262 q^{53} -6.09017 q^{55} +2.61803 q^{57} +10.7082 q^{59} -5.47214 q^{61} +3.85410 q^{63} +2.23607 q^{65} +2.09017 q^{67} -22.7984 q^{69} -11.1803 q^{71} -0.909830 q^{73} -10.4721 q^{75} -6.09017 q^{77} -6.94427 q^{79} -5.70820 q^{81} -14.6180 q^{83} -7.61803 q^{85} -2.23607 q^{87} -12.1803 q^{89} +2.23607 q^{91} -6.23607 q^{93} +1.00000 q^{95} +8.70820 q^{97} -23.4721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 2 q^{5} + 2 q^{7} + q^{9} - q^{11} + 3 q^{15} - 13 q^{17} + 2 q^{19} + 3 q^{21} - 4 q^{23} - 8 q^{25} + 5 q^{29} - 7 q^{31} - 14 q^{33} + 2 q^{35} + 5 q^{39} - 11 q^{41} - 8 q^{43} + q^{45}+ \cdots - 38 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + 2 * q^5 + 2 * q^7 + q^9 - q^11 + 3 * q^15 - 13 * q^17 + 2 * q^19 + 3 * q^21 - 4 * q^23 - 8 * q^25 + 5 * q^29 - 7 * q^31 - 14 * q^33 + 2 * q^35 + 5 * q^39 - 11 * q^41 - 8 * q^43 + q^45 - 4 * q^47 + 2 * q^49 - 22 * q^51 + 13 * q^53 - q^55 + 3 * q^57 + 8 * q^59 - 2 * q^61 + q^63 - 7 * q^67 - 21 * q^69 - 13 * q^73 - 12 * q^75 - q^77 + 4 * q^79 + 2 * q^81 - 27 * q^83 - 13 * q^85 - 2 * q^89 - 8 * q^93 + 2 * q^95 + 4 * q^97 - 38 * q^99

Introduction

L-functions

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Database

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