LMFDB - Embedded newform 2028.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2028,2,Mod(1,2028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, 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("2028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2028.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:	$16.1936615299$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +4.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} +4.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} -4.00000 q^{15} +2.00000 q^{17} +2.00000 q^{19} -2.00000 q^{21} +11.0000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +10.0000 q^{31} -4.00000 q^{33} +8.00000 q^{35} -10.0000 q^{37} -8.00000 q^{41} +4.00000 q^{43} +4.00000 q^{45} +4.00000 q^{47} -3.00000 q^{49} -2.00000 q^{51} -10.0000 q^{53} +16.0000 q^{55} -2.00000 q^{57} +8.00000 q^{59} -14.0000 q^{61} +2.00000 q^{63} -2.00000 q^{67} -16.0000 q^{71} +10.0000 q^{73} -11.0000 q^{75} +8.00000 q^{77} -16.0000 q^{79} +1.00000 q^{81} +8.00000 q^{85} +6.00000 q^{87} +4.00000 q^{89} -10.0000 q^{93} +8.00000 q^{95} +2.00000 q^{97} +4.00000 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−4.00000	−1.03280
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	8.00000	1.35225
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	16.0000	2.15744
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	−11.0000	−1.27017
$76$	0	0
$77$	8.00000	0.911685
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−10.0000	−1.03695
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	8.00000	0.721336
$124$	0	0
$125$	24.0000	2.14663
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−24.0000	−1.99309
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	40.0000	3.21288
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	0	0
$165$	−16.0000	−1.24560
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	22.0000	1.66304
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	0	0
$185$	−40.0000	−2.94086
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	0	0
$203$	−12.0000	−0.842235
$204$	0	0
$205$	−32.0000	−2.23498
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	16.0000	1.09630
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	20.0000	1.35769
$218$	0	0
$219$	−10.0000	−0.675737
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	0	0
$237$	16.0000	1.03931
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−12.0000	−0.766652
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−8.00000	−0.500979
$256$	0	0
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	0	0
$259$	−20.0000	−1.24274
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	−40.0000	−2.45718
$266$	0	0
$267$	−4.00000	−0.244796
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	10.0000	0.607457	0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000	0.607457	0.303728	+	0.952759i	$0.401768\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	44.0000	2.65330
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	0	0
$287$	−16.0000	−0.944450
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	−8.00000	−0.467365	−0.233682	−	0.972313i	$-0.575078\pi$
$293$	−8.00000	−0.467365	−0.233682	+	0.972313i	$0.575078\pi$
$294$	0	0
$295$	32.0000	1.86311
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	−10.0000	−0.574485
$304$	0	0
$305$	−56.0000	−3.20655
$306$	0	0
$307$	−22.0000	−1.25561	−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000	−1.25561	−0.627803	+	0.778372i	$0.716046\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	8.00000	0.450749
$316$	0	0
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−2.00000	−0.109930	−0.0549650	−	0.998488i	$-0.517505\pi$
$331$	−2.00000	−0.109930	−0.0549650	+	0.998488i	$0.517505\pi$
$332$	0	0
$333$	−10.0000	−0.547997
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	40.0000	2.16612
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	16.0000	0.851594	0.425797	−	0.904819i	$-0.359994\pi$
$353$	16.0000	0.851594	0.425797	+	0.904819i	$0.359994\pi$
$354$	0	0
$355$	−64.0000	−3.39677
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−5.00000	−0.262432
$364$	0	0
$365$	40.0000	2.09370
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	−8.00000	−0.416463
$370$	0	0
$371$	−20.0000	−1.03835
$372$	0	0
$373$	−18.0000	−0.932005	−0.466002	−	0.884783i	$-0.654306\pi$
$373$	−18.0000	−0.932005	−0.466002	+	0.884783i	$0.654306\pi$
$374$	0	0
$375$	−24.0000	−1.23935
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	32.0000	1.63087
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	−64.0000	−3.22019
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	4.00000	0.198762
$406$	0	0
$407$	−40.0000	−1.98273
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	0	0
$413$	16.0000	0.787309
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	22.0000	1.06716
$426$	0	0
$427$	−28.0000	−1.35501
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	0	0
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	0	0
$435$	24.0000	1.15071
$436$	0	0
$437$	0	0
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	16.0000	0.758473
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	−32.0000	−1.50682
$452$	0	0
$453$	18.0000	0.845714
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	0	0
$465$	−40.0000	−1.85496
$466$	0	0
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	16.0000	0.735681
$474$	0	0
$475$	22.0000	1.00943
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.00000	0.363261
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	0	0
$495$	16.0000	0.719147
$496$	0	0
$497$	−32.0000	−1.43540
$498$	0	0
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	40.0000	1.77998
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	20.0000	0.884748
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	−32.0000	−1.41009
$516$	0	0
$517$	16.0000	0.703679
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	−2.00000	−0.0876216	−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000	−0.0876216	−0.0438108	+	0.999040i	$0.513950\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	0	0
$525$	−22.0000	−0.960159
$526$	0	0
$527$	20.0000	0.871214
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	0	0
$534$	0	0
$535$	48.0000	2.07522
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−12.0000	−0.516877
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	8.00000	0.342682
$546$	0	0
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	0	0
$549$	−14.0000	−0.597505
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	−32.0000	−1.36078
$554$	0	0
$555$	40.0000	1.69791
$556$	0	0
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	0	0
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−40.0000	−1.65663
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	0	0
$589$	20.0000	0.824086
$590$	0	0
$591$	−12.0000	−0.493614
$592$	0	0
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	0	0
$597$	−4.00000	−0.163709
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	0	0
$605$	20.0000	0.813116
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	32.0000	1.29036
$616$	0	0
$617$	48.0000	1.93241	0.966204	−	0.257780i	$-0.0829910\pi$
$617$	48.0000	1.93241	0.966204	+	0.257780i	$0.0829910\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.00000	0.320513
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	−8.00000	−0.319489
$628$	0	0
$629$	−20.0000	−0.797452
$630$	0	0
$631$	30.0000	1.19428	0.597141	−	0.802137i	$-0.296303\pi$
$631$	30.0000	1.19428	0.597141	+	0.802137i	$0.296303\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	0	0
$635$	48.0000	1.90482
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−16.0000	−0.632950
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	0	0
$645$	−16.0000	−0.629999
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	−20.0000	−0.783862
$652$	0	0
$653$	−46.0000	−1.80012	−0.900060	−	0.435767i	$-0.856477\pi$
$653$	−46.0000	−1.80012	−0.900060	+	0.435767i	$0.856477\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	0	0
$668$	0	0
$669$	14.0000	0.541271
$670$	0	0
$671$	−56.0000	−2.16186
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	0	0
$675$	−11.0000	−0.423390
$676$	0	0
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	−32.0000	−1.22266
$686$	0	0
$687$	−2.00000	−0.0763048
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	0	0
$693$	8.00000	0.303895
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−16.0000	−0.606043
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	−20.0000	−0.754314
$704$	0	0
$705$	−16.0000	−0.602595
$706$	0	0
$707$	20.0000	0.752177
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	−66.0000	−2.45118
$726$	0	0
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	0	0
$735$	12.0000	0.442627
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−22.0000	−0.809283	−0.404642	−	0.914475i	$-0.632604\pi$
$739$	−22.0000	−0.809283	−0.404642	+	0.914475i	$0.632604\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	24.0000	0.876941
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	−72.0000	−2.62035
$756$	0	0
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−40.0000	−1.45000	−0.724999	−	0.688749i	$-0.758160\pi$
$761$	−40.0000	−1.45000	−0.724999	+	0.688749i	$0.758160\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	0	0
$765$	8.00000	0.289241
$766$	0	0
$767$	0	0
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	0	0
$773$	48.0000	1.72644	0.863220	−	0.504828i	$-0.168444\pi$
$773$	48.0000	1.72644	0.863220	+	0.504828i	$0.168444\pi$
$774$	0	0
$775$	110.000	3.95132
$776$	0	0
$777$	20.0000	0.717496
$778$	0	0
$779$	−16.0000	−0.573259
$780$	0	0
$781$	−64.0000	−2.29010
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	8.00000	0.285532
$786$	0	0
$787$	50.0000	1.78231	0.891154	−	0.453701i	$-0.149897\pi$
$787$	50.0000	1.78231	0.891154	+	0.453701i	$0.149897\pi$
$788$	0	0
$789$	16.0000	0.569615
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	0	0
$794$	0	0
$795$	40.0000	1.41865
$796$	0	0
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	4.00000	0.141333
$802$	0	0
$803$	40.0000	1.41157
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.0000	0.492823
$808$	0	0
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	0	0
$811$	−6.00000	−0.210688	−0.105344	−	0.994436i	$-0.533594\pi$
$811$	−6.00000	−0.210688	−0.105344	+	0.994436i	$0.533594\pi$
$812$	0	0
$813$	−10.0000	−0.350715
$814$	0	0
$815$	−8.00000	−0.280228
$816$	0	0
$817$	8.00000	0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	−44.0000	−1.53188
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	48.0000	1.66111
$836$	0	0
$837$	−10.0000	−0.345651
$838$	0	0
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−8.00000	−0.275535
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	16.0000	0.549119
$850$	0	0
$851$	0	0
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	0	0
$861$	16.0000	0.545279
$862$	0	0
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	8.00000	0.272008
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	48.0000	1.62270
$876$	0	0
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	0	0
$879$	8.00000	0.269833
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	−32.0000	−1.07689	−0.538443	−	0.842662i	$-0.680987\pi$
$883$	−32.0000	−1.07689	−0.538443	+	0.842662i	$0.680987\pi$
$884$	0	0
$885$	−32.0000	−1.07567
$886$	0	0
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	48.0000	1.60446
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−60.0000	−2.00111
$900$	0	0
$901$	−20.0000	−0.666297
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	8.00000	0.265929
$906$	0	0
$907$	−56.0000	−1.85945	−0.929725	−	0.368255i	$-0.879955\pi$
$907$	−56.0000	−1.85945	−0.929725	+	0.368255i	$0.879955\pi$
$908$	0	0
$909$	10.0000	0.331679
$910$	0	0
$911$	−56.0000	−1.85536	−0.927681	−	0.373373i	$-0.878201\pi$
$911$	−56.0000	−1.85536	−0.927681	+	0.373373i	$0.878201\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	56.0000	1.85130
$916$	0	0
$917$	8.00000	0.264183
$918$	0	0
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	22.0000	0.724925
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−110.000	−3.61678
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	−16.0000	−0.524943	−0.262471	−	0.964940i	$-0.584538\pi$
$929$	−16.0000	−0.524943	−0.262471	+	0.964940i	$0.584538\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	32.0000	1.04651
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	0	0
$941$	−8.00000	−0.260793	−0.130396	−	0.991462i	$-0.541625\pi$
$941$	−8.00000	−0.260793	−0.130396	+	0.991462i	$0.541625\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−8.00000	−0.260240
$946$	0	0
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	0	0
$955$	−32.0000	−1.03550
$956$	0	0
$957$	24.0000	0.775810
$958$	0	0
$959$	−16.0000	−0.516667
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	56.0000	1.80270
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	0	0
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	0	0
$979$	16.0000	0.511362
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	48.0000	1.52941
$986$	0	0
$987$	−8.00000	−0.254643
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−36.0000	−1.14358	−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000	−1.14358	−0.571789	+	0.820401i	$0.693750\pi$
$992$	0	0
$993$	2.00000	0.0634681
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.2.a.c.1.1		1
3.2	odd	2		6084.2.a.b.1.1		1
4.3	odd	2		8112.2.a.bi.1.1		1
13.2	odd	12		2028.2.q.h.1837.1		4
13.3	even	3		2028.2.i.g.529.1		2
13.4	even	6		2028.2.i.e.2005.1		2
13.5	odd	4		2028.2.b.a.337.1		2
13.6	odd	12		2028.2.q.h.361.1		4
13.7	odd	12		2028.2.q.h.361.2		4
13.8	odd	4		2028.2.b.a.337.2		2
13.9	even	3		2028.2.i.g.2005.1		2
13.10	even	6		2028.2.i.e.529.1		2
13.11	odd	12		2028.2.q.h.1837.2		4
13.12	even	2		156.2.a.a.1.1	✓	1
39.5	even	4		6084.2.b.j.4393.2		2
39.8	even	4		6084.2.b.j.4393.1		2
39.38	odd	2		468.2.a.d.1.1		1
52.51	odd	2		624.2.a.e.1.1		1
65.12	odd	4		3900.2.h.b.1249.2		2
65.38	odd	4		3900.2.h.b.1249.1		2
65.64	even	2		3900.2.a.m.1.1		1
91.90	odd	2		7644.2.a.k.1.1		1
104.51	odd	2		2496.2.a.o.1.1		1
104.77	even	2		2496.2.a.bc.1.1		1
117.25	even	6		4212.2.i.l.1405.1		2
117.38	odd	6		4212.2.i.b.1405.1		2
117.77	odd	6		4212.2.i.b.2809.1		2
117.103	even	6		4212.2.i.l.2809.1		2
156.155	even	2		1872.2.a.s.1.1		1
312.77	odd	2		7488.2.a.c.1.1		1
312.155	even	2		7488.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.a.a.1.1	✓	1	13.12	even	2
468.2.a.d.1.1		1	39.38	odd	2
624.2.a.e.1.1		1	52.51	odd	2
1872.2.a.s.1.1		1	156.155	even	2
2028.2.a.c.1.1		1	1.1	even	1	trivial
2028.2.b.a.337.1		2	13.5	odd	4
2028.2.b.a.337.2		2	13.8	odd	4
2028.2.i.e.529.1		2	13.10	even	6
2028.2.i.e.2005.1		2	13.4	even	6
2028.2.i.g.529.1		2	13.3	even	3
2028.2.i.g.2005.1		2	13.9	even	3
2028.2.q.h.361.1		4	13.6	odd	12
2028.2.q.h.361.2		4	13.7	odd	12
2028.2.q.h.1837.1		4	13.2	odd	12
2028.2.q.h.1837.2		4	13.11	odd	12
2496.2.a.o.1.1		1	104.51	odd	2
2496.2.a.bc.1.1		1	104.77	even	2
3900.2.a.m.1.1		1	65.64	even	2
3900.2.h.b.1249.1		2	65.38	odd	4
3900.2.h.b.1249.2		2	65.12	odd	4
4212.2.i.b.1405.1		2	117.38	odd	6
4212.2.i.b.2809.1		2	117.77	odd	6
4212.2.i.l.1405.1		2	117.25	even	6
4212.2.i.l.2809.1		2	117.103	even	6
6084.2.a.b.1.1		1	3.2	odd	2
6084.2.b.j.4393.1		2	39.8	even	4
6084.2.b.j.4393.2		2	39.5	even	4
7488.2.a.c.1.1		1	312.77	odd	2
7488.2.a.d.1.1		1	312.155	even	2
7644.2.a.k.1.1		1	91.90	odd	2
8112.2.a.bi.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2028.a (trivial)

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