LMFDB - Embedded newform 9920.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9920,2,Mod(1,9920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9920, 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("9920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9920 = 2^{6} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9920.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:	$79.2115988051$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 620)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	9920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -1.00000 q^{5} -2.00000 q^{7} +6.00000 q^{9} -2.00000 q^{11} -2.00000 q^{13} -3.00000 q^{15} -3.00000 q^{17} +3.00000 q^{19} -6.00000 q^{21} -4.00000 q^{23} +1.00000 q^{25} +9.00000 q^{27} +4.00000 q^{29} -1.00000 q^{31} -6.00000 q^{33} +2.00000 q^{35} +7.00000 q^{37} -6.00000 q^{39} -3.00000 q^{41} +5.00000 q^{43} -6.00000 q^{45} -6.00000 q^{47} -3.00000 q^{49} -9.00000 q^{51} -1.00000 q^{53} +2.00000 q^{55} +9.00000 q^{57} +11.0000 q^{59} -8.00000 q^{61} -12.0000 q^{63} +2.00000 q^{65} -4.00000 q^{67} -12.0000 q^{69} +15.0000 q^{71} -13.0000 q^{73} +3.00000 q^{75} +4.00000 q^{77} -12.0000 q^{79} +9.00000 q^{81} -7.00000 q^{83} +3.00000 q^{85} +12.0000 q^{87} -18.0000 q^{89} +4.00000 q^{91} -3.00000 q^{93} -3.00000 q^{95} +10.0000 q^{97} -12.0000 q^{99} +O(q^{100})$

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
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	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
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	−6.00000	−1.30931
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	9.00000	1.73205
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	0	0
$45$	−6.00000	−0.894427
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−9.00000	−1.26025
$52$	0	0
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	9.00000	1.19208
$58$	0	0
$59$	11.0000	1.43208	0.716039	−	0.698060i	$-0.245953\pi$
$59$	11.0000	1.43208	0.716039	+	0.698060i	$0.245953\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	−12.0000	−1.51186
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−12.0000	−1.44463
$70$	0	0
$71$	15.0000	1.78017	0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000	1.78017	0.890086	+	0.455792i	$0.150644\pi$
$72$	0	0
$73$	−13.0000	−1.52153	−0.760767	−	0.649025i	$-0.775177\pi$
$73$	−13.0000	−1.52153	−0.760767	+	0.649025i	$0.775177\pi$
$74$	0	0
$75$	3.00000	0.346410
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−7.00000	−0.768350	−0.384175	−	0.923260i	$-0.625514\pi$
$83$	−7.00000	−0.768350	−0.384175	+	0.923260i	$0.625514\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	12.0000	1.28654
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	−3.00000	−0.311086
$94$	0	0
$95$	−3.00000	−0.307794
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−12.0000	−1.20605
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	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
$105$	6.00000	0.585540
$106$	0	0
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	0	0
$111$	21.0000	1.99323
$112$	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
$115$	4.00000	0.373002
$116$	0	0
$117$	−12.0000	−1.10940
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−9.00000	−0.811503
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	15.0000	1.32068
$130$	0	0
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	−9.00000	−0.774597
$136$	0	0
$137$	−1.00000	−0.0854358	−0.0427179	−	0.999087i	$-0.513602\pi$
$137$	−1.00000	−0.0854358	−0.0427179	+	0.999087i	$0.513602\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−18.0000	−1.51587
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	−9.00000	−0.742307
$148$	0	0
$149$	−21.0000	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	−18.0000	−1.45521
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	0	0
$159$	−3.00000	−0.237915
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	0	0
$165$	6.00000	0.467099
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	18.0000	1.37649
$172$	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
$175$	−2.00000	−0.151186
$176$	0	0
$177$	33.0000	2.48043
$178$	0	0
$179$	−14.0000	−1.04641	−0.523205	−	0.852207i	$-0.675264\pi$
$179$	−14.0000	−1.04641	−0.523205	+	0.852207i	$0.675264\pi$
$180$	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
$183$	−24.0000	−1.77413
$184$	0	0
$185$	−7.00000	−0.514650
$186$	0	0
$187$	6.00000	0.438763
$188$	0	0
$189$	−18.0000	−1.30931
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	6.00000	0.429669
$196$	0	0
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	−8.00000	−0.561490
$204$	0	0
$205$	3.00000	0.209529
$206$	0	0
$207$	−24.0000	−1.66812
$208$	0	0
$209$	−6.00000	−0.415029
$210$	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
$213$	45.0000	3.08335
$214$	0	0
$215$	−5.00000	−0.340997
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	−39.0000	−2.63538
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	21.0000	1.40626	0.703132	−	0.711059i	$-0.251784\pi$
$223$	21.0000	1.40626	0.703132	+	0.711059i	$0.251784\pi$
$224$	0	0
$225$	6.00000	0.400000
$226$	0	0
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	0	0
$237$	−36.0000	−2.33845
$238$	0	0
$239$	−2.00000	−0.129369	−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000	−0.129369	−0.0646846	+	0.997906i	$0.520604\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	−6.00000	−0.381771
$248$	0	0
$249$	−21.0000	−1.33082
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	0	0
$255$	9.00000	0.563602
$256$	0	0
$257$	16.0000	0.998053	0.499026	−	0.866587i	$-0.333691\pi$
$257$	16.0000	0.998053	0.499026	+	0.866587i	$0.333691\pi$
$258$	0	0
$259$	−14.0000	−0.869918
$260$	0	0
$261$	24.0000	1.48556
$262$	0	0
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	1.00000	0.0614295
$266$	0	0
$267$	−54.0000	−3.30475
$268$	0	0
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	12.0000	0.726273
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−17.0000	−1.02143	−0.510716	−	0.859750i	$-0.670619\pi$
$277$	−17.0000	−1.02143	−0.510716	+	0.859750i	$0.670619\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	0	0
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	0	0
$285$	−9.00000	−0.533114
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	30.0000	1.75863
$292$	0	0
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	0	0
$295$	−11.0000	−0.640445
$296$	0	0
$297$	−18.0000	−1.04447
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	−10.0000	−0.576390
$302$	0	0
$303$	9.00000	0.517036
$304$	0	0
$305$	8.00000	0.458079
$306$	0	0
$307$	−6.00000	−0.342438	−0.171219	−	0.985233i	$-0.554771\pi$
$307$	−6.00000	−0.342438	−0.171219	+	0.985233i	$0.554771\pi$
$308$	0	0
$309$	−24.0000	−1.36531
$310$	0	0
$311$	31.0000	1.75785	0.878924	−	0.476961i	$-0.158262\pi$
$311$	31.0000	1.75785	0.878924	+	0.476961i	$0.158262\pi$
$312$	0	0
$313$	9.00000	0.508710	0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000	0.508710	0.254355	+	0.967111i	$0.418137\pi$
$314$	0	0
$315$	12.0000	0.676123
$316$	0	0
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	−54.0000	−3.01399
$322$	0	0
$323$	−9.00000	−0.500773
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	−45.0000	−2.48851
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	42.0000	2.30159
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	25.0000	1.36184	0.680918	−	0.732359i	$-0.261581\pi$
$337$	25.0000	1.36184	0.680918	+	0.732359i	$0.261581\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	12.0000	0.646058
$346$	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
$349$	17.0000	0.909989	0.454995	−	0.890494i	$-0.349641\pi$
$349$	17.0000	0.909989	0.454995	+	0.890494i	$0.349641\pi$
$350$	0	0
$351$	−18.0000	−0.960769
$352$	0	0
$353$	−22.0000	−1.17094	−0.585471	−	0.810693i	$-0.699090\pi$
$353$	−22.0000	−1.17094	−0.585471	+	0.810693i	$0.699090\pi$
$354$	0	0
$355$	−15.0000	−0.796117
$356$	0	0
$357$	18.0000	0.952661
$358$	0	0
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	−21.0000	−1.10221
$364$	0	0
$365$	13.0000	0.680451
$366$	0	0
$367$	27.0000	1.40939	0.704694	−	0.709511i	$-0.251084\pi$
$367$	27.0000	1.40939	0.704694	+	0.709511i	$0.251084\pi$
$368$	0	0
$369$	−18.0000	−0.937043
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	−12.0000	−0.621336	−0.310668	−	0.950518i	$-0.600553\pi$
$373$	−12.0000	−0.621336	−0.310668	+	0.950518i	$0.600553\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	−8.00000	−0.412021
$378$	0	0
$379$	27.0000	1.38690	0.693448	−	0.720506i	$-0.256091\pi$
$379$	27.0000	1.38690	0.693448	+	0.720506i	$0.256091\pi$
$380$	0	0
$381$	−24.0000	−1.22956
$382$	0	0
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	30.0000	1.52499
$388$	0	0
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	−27.0000	−1.36197
$394$	0	0
$395$	12.0000	0.603786
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	0	0
$399$	−18.0000	−0.901127
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	0	0
$405$	−9.00000	−0.447214
$406$	0	0
$407$	−14.0000	−0.693954
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	−3.00000	−0.147979
$412$	0	0
$413$	−22.0000	−1.08255
$414$	0	0
$415$	7.00000	0.343616
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	−19.0000	−0.928211	−0.464105	−	0.885780i	$-0.653624\pi$
$419$	−19.0000	−0.928211	−0.464105	+	0.885780i	$0.653624\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	0	0
$423$	−36.0000	−1.75038
$424$	0	0
$425$	−3.00000	−0.145521
$426$	0	0
$427$	16.0000	0.774294
$428$	0	0
$429$	12.0000	0.579365
$430$	0	0
$431$	33.0000	1.58955	0.794777	−	0.606902i	$-0.207588\pi$
$431$	33.0000	1.58955	0.794777	+	0.606902i	$0.207588\pi$
$432$	0	0
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	0	0
$435$	−12.0000	−0.575356
$436$	0	0
$437$	−12.0000	−0.574038
$438$	0	0
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	0	0
$443$	−2.00000	−0.0950229	−0.0475114	−	0.998871i	$-0.515129\pi$
$443$	−2.00000	−0.0950229	−0.0475114	+	0.998871i	$0.515129\pi$
$444$	0	0
$445$	18.0000	0.853282
$446$	0	0
$447$	−63.0000	−2.97980
$448$	0	0
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	0	0
$453$	−48.0000	−2.25524
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	0	0
$459$	−27.0000	−1.26025
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	3.00000	0.139122
$466$	0	0
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	54.0000	2.48819
$472$	0	0
$473$	−10.0000	−0.459800
$474$	0	0
$475$	3.00000	0.137649
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	0	0
$483$	24.0000	1.09204
$484$	0	0
$485$	−10.0000	−0.454077
$486$	0	0
$487$	15.0000	0.679715	0.339857	−	0.940477i	$-0.389621\pi$
$487$	15.0000	0.679715	0.339857	+	0.940477i	$0.389621\pi$
$488$	0	0
$489$	−48.0000	−2.17064
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	−30.0000	−1.34568
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	9.00000	0.402090
$502$	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
$505$	−3.00000	−0.133498
$506$	0	0
$507$	−27.0000	−1.19911
$508$	0	0
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	26.0000	1.15017
$512$	0	0
$513$	27.0000	1.19208
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−29.0000	−1.27051	−0.635257	−	0.772301i	$-0.719106\pi$
$521$	−29.0000	−1.27051	−0.635257	+	0.772301i	$0.719106\pi$
$522$	0	0
$523$	23.0000	1.00572	0.502860	−	0.864368i	$-0.332281\pi$
$523$	23.0000	1.00572	0.502860	+	0.864368i	$0.332281\pi$
$524$	0	0
$525$	−6.00000	−0.261861
$526$	0	0
$527$	3.00000	0.130682
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	66.0000	2.86416
$532$	0	0
$533$	6.00000	0.259889
$534$	0	0
$535$	18.0000	0.778208
$536$	0	0
$537$	−42.0000	−1.81243
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	6.00000	0.257485
$544$	0	0
$545$	15.0000	0.642529
$546$	0	0
$547$	34.0000	1.45374	0.726868	−	0.686778i	$-0.240975\pi$
$547$	34.0000	1.45374	0.726868	+	0.686778i	$0.240975\pi$
$548$	0	0
$549$	−48.0000	−2.04859
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	24.0000	1.02058
$554$	0	0
$555$	−21.0000	−0.891400
$556$	0	0
$557$	34.0000	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	0	0
$561$	18.0000	0.759961
$562$	0	0
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	−18.0000	−0.755929
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	−48.0000	−2.00523
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	28.0000	1.16566	0.582828	−	0.812596i	$-0.301946\pi$
$577$	28.0000	1.16566	0.582828	+	0.812596i	$0.301946\pi$
$578$	0	0
$579$	−66.0000	−2.74287
$580$	0	0
$581$	14.0000	0.580818
$582$	0	0
$583$	2.00000	0.0828315
$584$	0	0
$585$	12.0000	0.496139
$586$	0	0
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	0	0
$589$	−3.00000	−0.123613
$590$	0	0
$591$	42.0000	1.72765
$592$	0	0
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	0	0
$597$	−42.0000	−1.71895
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	0	0
$603$	−24.0000	−0.977356
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	13.0000	0.525065	0.262533	−	0.964923i	$-0.415442\pi$
$613$	13.0000	0.525065	0.262533	+	0.964923i	$0.415442\pi$
$614$	0	0
$615$	9.00000	0.362915
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	0	0
$621$	−36.0000	−1.44463
$622$	0	0
$623$	36.0000	1.44231
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−18.0000	−0.718851
$628$	0	0
$629$	−21.0000	−0.837325
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	−36.0000	−1.43087
$634$	0	0
$635$	8.00000	0.317470
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	90.0000	3.56034
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	7.00000	0.276053	0.138027	−	0.990429i	$-0.455924\pi$
$643$	7.00000	0.276053	0.138027	+	0.990429i	$0.455924\pi$
$644$	0	0
$645$	−15.0000	−0.590624
$646$	0	0
$647$	−27.0000	−1.06148	−0.530740	−	0.847535i	$-0.678086\pi$
$647$	−27.0000	−1.06148	−0.530740	+	0.847535i	$0.678086\pi$
$648$	0	0
$649$	−22.0000	−0.863576
$650$	0	0
$651$	6.00000	0.235159
$652$	0	0
$653$	−20.0000	−0.782660	−0.391330	−	0.920250i	$-0.627985\pi$
$653$	−20.0000	−0.782660	−0.391330	+	0.920250i	$0.627985\pi$
$654$	0	0
$655$	9.00000	0.351659
$656$	0	0
$657$	−78.0000	−3.04307
$658$	0	0
$659$	7.00000	0.272681	0.136341	−	0.990662i	$-0.456466\pi$
$659$	7.00000	0.272681	0.136341	+	0.990662i	$0.456466\pi$
$660$	0	0
$661$	11.0000	0.427850	0.213925	−	0.976850i	$-0.431375\pi$
$661$	11.0000	0.427850	0.213925	+	0.976850i	$0.431375\pi$
$662$	0	0
$663$	18.0000	0.699062
$664$	0	0
$665$	6.00000	0.232670
$666$	0	0
$667$	−16.0000	−0.619522
$668$	0	0
$669$	63.0000	2.43572
$670$	0	0
$671$	16.0000	0.617673
$672$	0	0
$673$	15.0000	0.578208	0.289104	−	0.957298i	$-0.406643\pi$
$673$	15.0000	0.578208	0.289104	+	0.957298i	$0.406643\pi$
$674$	0	0
$675$	9.00000	0.346410
$676$	0	0
$677$	21.0000	0.807096	0.403548	−	0.914959i	$-0.367777\pi$
$677$	21.0000	0.807096	0.403548	+	0.914959i	$0.367777\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	72.0000	2.75905
$682$	0	0
$683$	2.00000	0.0765279	0.0382639	−	0.999268i	$-0.487817\pi$
$683$	2.00000	0.0765279	0.0382639	+	0.999268i	$0.487817\pi$
$684$	0	0
$685$	1.00000	0.0382080
$686$	0	0
$687$	60.0000	2.28914
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	51.0000	1.94013	0.970066	−	0.242843i	$-0.0780798\pi$
$691$	51.0000	1.94013	0.970066	+	0.242843i	$0.0780798\pi$
$692$	0	0
$693$	24.0000	0.911685
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	9.00000	0.340899
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	0	0
$703$	21.0000	0.792030
$704$	0	0
$705$	18.0000	0.677919
$706$	0	0
$707$	−6.00000	−0.225653
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	−72.0000	−2.70021
$712$	0	0
$713$	4.00000	0.149801
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	−6.00000	−0.224074
$718$	0	0
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	−30.0000	−1.11571
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−15.0000	−0.554795
$732$	0	0
$733$	20.0000	0.738717	0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000	0.738717	0.369358	+	0.929287i	$0.379577\pi$
$734$	0	0
$735$	9.00000	0.331970
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	38.0000	1.39785	0.698926	−	0.715194i	$-0.253662\pi$
$739$	38.0000	1.39785	0.698926	+	0.715194i	$0.253662\pi$
$740$	0	0
$741$	−18.0000	−0.661247
$742$	0	0
$743$	−33.0000	−1.21065	−0.605326	−	0.795977i	$-0.706957\pi$
$743$	−33.0000	−1.21065	−0.605326	+	0.795977i	$0.706957\pi$
$744$	0	0
$745$	21.0000	0.769380
$746$	0	0
$747$	−42.0000	−1.53670
$748$	0	0
$749$	36.0000	1.31541
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	6.00000	0.218652
$754$	0	0
$755$	16.0000	0.582300
$756$	0	0
$757$	−35.0000	−1.27210	−0.636048	−	0.771649i	$-0.719432\pi$
$757$	−35.0000	−1.27210	−0.636048	+	0.771649i	$0.719432\pi$
$758$	0	0
$759$	24.0000	0.871145
$760$	0	0
$761$	−20.0000	−0.724999	−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000	−0.724999	−0.362500	+	0.931984i	$0.618077\pi$
$762$	0	0
$763$	30.0000	1.08607
$764$	0	0
$765$	18.0000	0.650791
$766$	0	0
$767$	−22.0000	−0.794374
$768$	0	0
$769$	−29.0000	−1.04577	−0.522883	−	0.852404i	$-0.675144\pi$
$769$	−29.0000	−1.04577	−0.522883	+	0.852404i	$0.675144\pi$
$770$	0	0
$771$	48.0000	1.72868
$772$	0	0
$773$	22.0000	0.791285	0.395643	−	0.918405i	$-0.370522\pi$
$773$	22.0000	0.791285	0.395643	+	0.918405i	$0.370522\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−42.0000	−1.50674
$778$	0	0
$779$	−9.00000	−0.322458
$780$	0	0
$781$	−30.0000	−1.07348
$782$	0	0
$783$	36.0000	1.28654
$784$	0	0
$785$	−18.0000	−0.642448
$786$	0	0
$787$	−36.0000	−1.28326	−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000	−1.28326	−0.641631	+	0.767014i	$0.721742\pi$
$788$	0	0
$789$	63.0000	2.24286
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	16.0000	0.568177
$794$	0	0
$795$	3.00000	0.106399
$796$	0	0
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	0	0
$801$	−108.000	−3.81599
$802$	0	0
$803$	26.0000	0.917520
$804$	0	0
$805$	−8.00000	−0.281963
$806$	0	0
$807$	−36.0000	−1.26726
$808$	0	0
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	33.0000	1.15879	0.579393	−	0.815048i	$-0.303290\pi$
$811$	33.0000	1.15879	0.579393	+	0.815048i	$0.303290\pi$
$812$	0	0
$813$	−48.0000	−1.68343
$814$	0	0
$815$	16.0000	0.560456
$816$	0	0
$817$	15.0000	0.524784
$818$	0	0
$819$	24.0000	0.838628
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	−37.0000	−1.28974	−0.644869	−	0.764293i	$-0.723088\pi$
$823$	−37.0000	−1.28974	−0.644869	+	0.764293i	$0.723088\pi$
$824$	0	0
$825$	−6.00000	−0.208893
$826$	0	0
$827$	1.00000	0.0347734	0.0173867	−	0.999849i	$-0.494465\pi$
$827$	1.00000	0.0347734	0.0173867	+	0.999849i	$0.494465\pi$
$828$	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
$831$	−51.0000	−1.76917
$832$	0	0
$833$	9.00000	0.311832
$834$	0	0
$835$	−3.00000	−0.103819
$836$	0	0
$837$	−9.00000	−0.311086
$838$	0	0
$839$	11.0000	0.379762	0.189881	−	0.981807i	$-0.439190\pi$
$839$	11.0000	0.379762	0.189881	+	0.981807i	$0.439190\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	81.0000	2.78979
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	14.0000	0.481046
$848$	0	0
$849$	66.0000	2.26511
$850$	0	0
$851$	−28.0000	−0.959828
$852$	0	0
$853$	12.0000	0.410872	0.205436	−	0.978671i	$-0.434139\pi$
$853$	12.0000	0.410872	0.205436	+	0.978671i	$0.434139\pi$
$854$	0	0
$855$	−18.0000	−0.615587
$856$	0	0
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0	0
$861$	18.0000	0.613438
$862$	0	0
$863$	1.00000	0.0340404	0.0170202	−	0.999855i	$-0.494582\pi$
$863$	1.00000	0.0340404	0.0170202	+	0.999855i	$0.494582\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	0	0
$867$	−24.0000	−0.815083
$868$	0	0
$869$	24.0000	0.814144
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	60.0000	2.03069
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	0	0
$879$	36.0000	1.21425
$880$	0	0
$881$	−44.0000	−1.48240	−0.741199	−	0.671286i	$-0.765742\pi$
$881$	−44.0000	−1.48240	−0.741199	+	0.671286i	$0.765742\pi$
$882$	0	0
$883$	33.0000	1.11054	0.555269	−	0.831671i	$-0.312615\pi$
$883$	33.0000	1.11054	0.555269	+	0.831671i	$0.312615\pi$
$884$	0	0
$885$	−33.0000	−1.10928
$886$	0	0
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	−18.0000	−0.603023
$892$	0	0
$893$	−18.0000	−0.602347
$894$	0	0
$895$	14.0000	0.467968
$896$	0	0
$897$	24.0000	0.801337
$898$	0	0
$899$	−4.00000	−0.133407
$900$	0	0
$901$	3.00000	0.0999445
$902$	0	0
$903$	−30.0000	−0.998337
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	36.0000	1.19536	0.597680	−	0.801735i	$-0.296089\pi$
$907$	36.0000	1.19536	0.597680	+	0.801735i	$0.296089\pi$
$908$	0	0
$909$	18.0000	0.597022
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	14.0000	0.463332
$914$	0	0
$915$	24.0000	0.793416
$916$	0	0
$917$	18.0000	0.594412
$918$	0	0
$919$	5.00000	0.164935	0.0824674	−	0.996594i	$-0.473720\pi$
$919$	5.00000	0.164935	0.0824674	+	0.996594i	$0.473720\pi$
$920$	0	0
$921$	−18.0000	−0.593120
$922$	0	0
$923$	−30.0000	−0.987462
$924$	0	0
$925$	7.00000	0.230159
$926$	0	0
$927$	−48.0000	−1.57653
$928$	0	0
$929$	40.0000	1.31236	0.656179	−	0.754606i	$-0.272172\pi$
$929$	40.0000	1.31236	0.656179	+	0.754606i	$0.272172\pi$
$930$	0	0
$931$	−9.00000	−0.294963
$932$	0	0
$933$	93.0000	3.04468
$934$	0	0
$935$	−6.00000	−0.196221
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	27.0000	0.881112
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	12.0000	0.390774
$944$	0	0
$945$	18.0000	0.585540
$946$	0	0
$947$	31.0000	1.00736	0.503682	−	0.863889i	$-0.331978\pi$
$947$	31.0000	1.00736	0.503682	+	0.863889i	$0.331978\pi$
$948$	0	0
$949$	26.0000	0.843996
$950$	0	0
$951$	−36.0000	−1.16738
$952$	0	0
$953$	47.0000	1.52248	0.761240	−	0.648471i	$-0.224591\pi$
$953$	47.0000	1.52248	0.761240	+	0.648471i	$0.224591\pi$
$954$	0	0
$955$	16.0000	0.517748
$956$	0	0
$957$	−24.0000	−0.775810
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−108.000	−3.48025
$964$	0	0
$965$	22.0000	0.708205
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	0	0
$969$	−27.0000	−0.867365
$970$	0	0
$971$	1.00000	0.0320915	0.0160458	−	0.999871i	$-0.494892\pi$
$971$	1.00000	0.0320915	0.0160458	+	0.999871i	$0.494892\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	0	0
$975$	−6.00000	−0.192154
$976$	0	0
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	−90.0000	−2.87348
$982$	0	0
$983$	−20.0000	−0.637901	−0.318950	−	0.947771i	$-0.603330\pi$
$983$	−20.0000	−0.637901	−0.318950	+	0.947771i	$0.603330\pi$
$984$	0	0
$985$	−14.0000	−0.446077
$986$	0	0
$987$	36.0000	1.14589
$988$	0	0
$989$	−20.0000	−0.635963
$990$	0	0
$991$	34.0000	1.08005	0.540023	−	0.841650i	$-0.318416\pi$
$991$	34.0000	1.08005	0.540023	+	0.841650i	$0.318416\pi$
$992$	0	0
$993$	24.0000	0.761617
$994$	0	0
$995$	14.0000	0.443830
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	0	0
$999$	63.0000	1.99323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9920.2.a.bi.1.1		1
4.3	odd	2		9920.2.a.a.1.1		1
8.3	odd	2		2480.2.a.o.1.1		1
8.5	even	2		620.2.a.a.1.1	✓	1
24.5	odd	2		5580.2.a.a.1.1		1
40.13	odd	4		3100.2.c.a.249.1		2
40.29	even	2		3100.2.a.g.1.1		1
40.37	odd	4		3100.2.c.a.249.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
620.2.a.a.1.1	✓	1	8.5	even	2
2480.2.a.o.1.1		1	8.3	odd	2
3100.2.a.g.1.1		1	40.29	even	2
3100.2.c.a.249.1		2	40.13	odd	4
3100.2.c.a.249.2		2	40.37	odd	4
5580.2.a.a.1.1		1	24.5	odd	2
9920.2.a.a.1.1		1	4.3	odd	2
9920.2.a.bi.1.1		1	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:	\( N \)	\(=\)	\( 9920 = 2^{6} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9920.a (trivial)

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