LMFDB - Embedded newform 2898.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2898,2,Mod(1,2898)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2898, 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("2898.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2898.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:	$23.1406465058$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2898.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +2.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -6.00000 q^{19} -1.00000 q^{20} -2.00000 q^{22} -1.00000 q^{23} -4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{28} +5.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} +1.00000 q^{35} +3.00000 q^{37} +6.00000 q^{38} +1.00000 q^{40} +9.00000 q^{41} +3.00000 q^{43} +2.00000 q^{44} +1.00000 q^{46} -9.00000 q^{47} +1.00000 q^{49} +4.00000 q^{50} -1.00000 q^{52} -6.00000 q^{53} -2.00000 q^{55} +1.00000 q^{56} -5.00000 q^{58} -8.00000 q^{59} +2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +4.00000 q^{67} +6.00000 q^{68} -1.00000 q^{70} +10.0000 q^{71} +2.00000 q^{73} -3.00000 q^{74} -6.00000 q^{76} -2.00000 q^{77} -4.00000 q^{79} -1.00000 q^{80} -9.00000 q^{82} +4.00000 q^{83} -6.00000 q^{85} -3.00000 q^{86} -2.00000 q^{88} +1.00000 q^{91} -1.00000 q^{92} +9.00000 q^{94} +6.00000 q^{95} -5.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.00000	−0.800000
$26$	1.00000	0.196116
$27$	0	0
$28$	−1.00000	−0.188982
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.00000	−1.02899
$35$	1.00000	0.169031
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	1.00000	0.158114
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	3.00000	0.457496	0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000	0.457496	0.228748	+	0.973486i	$0.426537\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	1.00000	0.147442
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.00000	0.565685
$51$	0	0
$52$	−1.00000	−0.138675
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	1.00000	0.133631
$57$	0	0
$58$	−5.00000	−0.656532
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	−1.00000	−0.119523
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−3.00000	−0.348743
$75$	0	0
$76$	−6.00000	−0.688247
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−9.00000	−0.993884
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	−3.00000	−0.323498
$87$	0	0
$88$	−2.00000	−0.213201
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−1.00000	−0.104257
$93$	0	0
$94$	9.00000	0.928279
$95$	6.00000	0.615587
$96$	0	0
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	−15.0000	−1.47799	−0.738997	−	0.673709i	$-0.764700\pi$
$103$	−15.0000	−1.47799	−0.738997	+	0.673709i	$0.764700\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	6.00000	0.582772
$107$	−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
$109$	−17.0000	−1.62830	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−17.0000	−1.62830	−0.814152	+	0.580651i	$0.802798\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	5.00000	0.464238
$117$	0	0
$118$	8.00000	0.736460
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−2.00000	−0.181071
$123$	0	0
$124$	−8.00000	−0.718421
$125$	9.00000	0.804984
$126$	0	0
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.00000	−0.0877058
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−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
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−10.0000	−0.839181
$143$	−2.00000	−0.167248
$144$	0	0
$145$	−5.00000	−0.415227
$146$	−2.00000	−0.165521
$147$	0	0
$148$	3.00000	0.246598
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	2.00000	0.161165
$155$	8.00000	0.642575
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	1.00000	0.0790569
$161$	1.00000	0.0788110
$162$	0	0
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	6.00000	0.460179
$171$	0	0
$172$	3.00000	0.228748
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	2.00000	0.150756
$177$	0	0
$178$	0	0
$179$	11.0000	0.822179	0.411089	−	0.911595i	$-0.365148\pi$
$179$	11.0000	0.822179	0.411089	+	0.911595i	$0.365148\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	1.00000	0.0737210
$185$	−3.00000	−0.220564
$186$	0	0
$187$	12.0000	0.877527
$188$	−9.00000	−0.656392
$189$	0	0
$190$	−6.00000	−0.435286
$191$	−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
$193$	3.00000	0.215945	0.107972	−	0.994154i	$-0.465564\pi$
$193$	3.00000	0.215945	0.107972	+	0.994154i	$0.465564\pi$
$194$	5.00000	0.358979
$195$	0	0
$196$	1.00000	0.0714286
$197$	−9.00000	−0.641223	−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000	−0.641223	−0.320612	+	0.947211i	$0.603888\pi$
$198$	0	0
$199$	−21.0000	−1.48865	−0.744325	−	0.667817i	$-0.767229\pi$
$199$	−21.0000	−1.48865	−0.744325	+	0.667817i	$0.767229\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	10.0000	0.703598
$203$	−5.00000	−0.350931
$204$	0	0
$205$	−9.00000	−0.628587
$206$	15.0000	1.04510
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	18.0000	1.23045
$215$	−3.00000	−0.204598
$216$	0	0
$217$	8.00000	0.543075
$218$	17.0000	1.15139
$219$	0	0
$220$	−2.00000	−0.134840
$221$	−6.00000	−0.403604
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	9.00000	0.598671
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	0	0
$229$	−30.0000	−1.98246	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	−30.0000	−1.98246	−0.991228	+	0.132164i	$0.957808\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	−5.00000	−0.328266
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	−8.00000	−0.520756
$237$	0	0
$238$	6.00000	0.388922
$239$	−14.0000	−0.905585	−0.452792	−	0.891616i	$-0.649572\pi$
$239$	−14.0000	−0.905585	−0.452792	+	0.891616i	$0.649572\pi$
$240$	0	0
$241$	29.0000	1.86805	0.934027	−	0.357202i	$-0.116269\pi$
$241$	29.0000	1.86805	0.934027	+	0.357202i	$0.116269\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	2.00000	0.128037
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	6.00000	0.381771
$248$	8.00000	0.508001
$249$	0	0
$250$	−9.00000	−0.569210
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	−1.00000	−0.0627456
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	1.00000	0.0620174
$261$	0	0
$262$	10.0000	0.617802
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	−6.00000	−0.367884
$267$	0	0
$268$	4.00000	0.244339
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	1.00000	0.0604122
$275$	−8.00000	−0.482418
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−13.0000	−0.779688
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	13.0000	0.775515	0.387757	−	0.921761i	$-0.373250\pi$
$281$	13.0000	0.775515	0.387757	+	0.921761i	$0.373250\pi$
$282$	0	0
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	10.0000	0.593391
$285$	0	0
$286$	2.00000	0.118262
$287$	−9.00000	−0.531253
$288$	0	0
$289$	19.0000	1.11765
$290$	5.00000	0.293610
$291$	0	0
$292$	2.00000	0.117041
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	−3.00000	−0.174371
$297$	0	0
$298$	−12.0000	−0.695141
$299$	1.00000	0.0578315
$300$	0	0
$301$	−3.00000	−0.172917
$302$	−9.00000	−0.517892
$303$	0	0
$304$	−6.00000	−0.344124
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−17.0000	−0.970241	−0.485121	−	0.874447i	$-0.661224\pi$
$307$	−17.0000	−0.970241	−0.485121	+	0.874447i	$0.661224\pi$
$308$	−2.00000	−0.113961
$309$	0	0
$310$	−8.00000	−0.454369
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−17.0000	−0.954815	−0.477408	−	0.878682i	$-0.658423\pi$
$317$	−17.0000	−0.954815	−0.477408	+	0.878682i	$0.658423\pi$
$318$	0	0
$319$	10.0000	0.559893
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−1.00000	−0.0557278
$323$	−36.0000	−2.00309
$324$	0	0
$325$	4.00000	0.221880
$326$	−14.0000	−0.775388
$327$	0	0
$328$	−9.00000	−0.496942
$329$	9.00000	0.496186
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	4.00000	0.219529
$333$	0	0
$334$	8.00000	0.437741
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	12.0000	0.652714
$339$	0	0
$340$	−6.00000	−0.325396
$341$	−16.0000	−0.866449
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−3.00000	−0.161749
$345$	0	0
$346$	−6.00000	−0.322562
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	−2.00000	−0.106600
$353$	11.0000	0.585471	0.292735	−	0.956193i	$-0.405434\pi$
$353$	11.0000	0.585471	0.292735	+	0.956193i	$0.405434\pi$
$354$	0	0
$355$	−10.0000	−0.530745
$356$	0	0
$357$	0	0
$358$	−11.0000	−0.581368
$359$	7.00000	0.369446	0.184723	−	0.982791i	$-0.440861\pi$
$359$	7.00000	0.369446	0.184723	+	0.982791i	$0.440861\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	2.00000	0.105118
$363$	0	0
$364$	1.00000	0.0524142
$365$	−2.00000	−0.104685
$366$	0	0
$367$	3.00000	0.156599	0.0782994	−	0.996930i	$-0.475051\pi$
$367$	3.00000	0.156599	0.0782994	+	0.996930i	$0.475051\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	3.00000	0.155963
$371$	6.00000	0.311504
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	9.00000	0.464140
$377$	−5.00000	−0.257513
$378$	0	0
$379$	29.0000	1.48963	0.744815	−	0.667271i	$-0.232538\pi$
$379$	29.0000	1.48963	0.744815	+	0.667271i	$0.232538\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	8.00000	0.409316
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	−3.00000	−0.152696
$387$	0	0
$388$	−5.00000	−0.253837
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	9.00000	0.453413
$395$	4.00000	0.201262
$396$	0	0
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	21.0000	1.05263
$399$	0	0
$400$	−4.00000	−0.200000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	−10.0000	−0.497519
$405$	0	0
$406$	5.00000	0.248146
$407$	6.00000	0.297409
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	9.00000	0.444478
$411$	0	0
$412$	−15.0000	−0.738997
$413$	8.00000	0.393654
$414$	0	0
$415$	−4.00000	−0.196352
$416$	1.00000	0.0490290
$417$	0	0
$418$	12.0000	0.586939
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	31.0000	1.51085	0.755424	−	0.655237i	$-0.227431\pi$
$421$	31.0000	1.51085	0.755424	+	0.655237i	$0.227431\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	6.00000	0.291386
$425$	−24.0000	−1.16417
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	−18.0000	−0.870063
$429$	0	0
$430$	3.00000	0.144673
$431$	−19.0000	−0.915198	−0.457599	−	0.889159i	$-0.651290\pi$
$431$	−19.0000	−0.915198	−0.457599	+	0.889159i	$0.651290\pi$
$432$	0	0
$433$	13.0000	0.624740	0.312370	−	0.949960i	$-0.398877\pi$
$433$	13.0000	0.624740	0.312370	+	0.949960i	$0.398877\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−17.0000	−0.814152
$437$	6.00000	0.287019
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	2.00000	0.0953463
$441$	0	0
$442$	6.00000	0.285391
$443$	−23.0000	−1.09276	−0.546381	−	0.837536i	$-0.683995\pi$
$443$	−23.0000	−1.09276	−0.546381	+	0.837536i	$0.683995\pi$
$444$	0	0
$445$	0	0
$446$	16.0000	0.757622
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	18.0000	0.847587
$452$	−9.00000	−0.423324
$453$	0	0
$454$	3.00000	0.140797
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−30.0000	−1.40334	−0.701670	−	0.712502i	$-0.747562\pi$
$457$	−30.0000	−1.40334	−0.701670	+	0.712502i	$0.747562\pi$
$458$	30.0000	1.40181
$459$	0	0
$460$	1.00000	0.0466252
$461$	40.0000	1.86299	0.931493	−	0.363760i	$-0.118507\pi$
$461$	40.0000	1.86299	0.931493	+	0.363760i	$0.118507\pi$
$462$	0	0
$463$	−29.0000	−1.34774	−0.673872	−	0.738848i	$-0.735370\pi$
$463$	−29.0000	−1.34774	−0.673872	+	0.738848i	$0.735370\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	22.0000	1.01913
$467$	19.0000	0.879215	0.439608	−	0.898190i	$-0.355118\pi$
$467$	19.0000	0.879215	0.439608	+	0.898190i	$0.355118\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	−9.00000	−0.415139
$471$	0	0
$472$	8.00000	0.368230
$473$	6.00000	0.275880
$474$	0	0
$475$	24.0000	1.10120
$476$	−6.00000	−0.275010
$477$	0	0
$478$	14.0000	0.640345
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	−29.0000	−1.32091
$483$	0	0
$484$	−7.00000	−0.318182
$485$	5.00000	0.227038
$486$	0	0
$487$	35.0000	1.58600	0.793001	−	0.609221i	$-0.208518\pi$
$487$	35.0000	1.58600	0.793001	+	0.609221i	$0.208518\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	1.00000	0.0451754
$491$	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
$493$	30.0000	1.35113
$494$	−6.00000	−0.269953
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−10.0000	−0.448561
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	3.00000	0.133897
$503$	14.0000	0.624229	0.312115	−	0.950044i	$-0.398963\pi$
$503$	14.0000	0.624229	0.312115	+	0.950044i	$0.398963\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	2.00000	0.0889108
$507$	0	0
$508$	1.00000	0.0443678
$509$	−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
$511$	−2.00000	−0.0884748
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−6.00000	−0.264649
$515$	15.0000	0.660979
$516$	0	0
$517$	−18.0000	−0.791639
$518$	3.00000	0.131812
$519$	0	0
$520$	−1.00000	−0.0438529
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	−10.0000	−0.436852
$525$	0	0
$526$	−9.00000	−0.392419
$527$	−48.0000	−2.09091
$528$	0	0
$529$	1.00000	0.0434783
$530$	−6.00000	−0.260623
$531$	0	0
$532$	6.00000	0.260133
$533$	−9.00000	−0.389833
$534$	0	0
$535$	18.0000	0.778208
$536$	−4.00000	−0.172774
$537$	0	0
$538$	−6.00000	−0.258678
$539$	2.00000	0.0861461
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	−14.0000	−0.601351
$543$	0	0
$544$	−6.00000	−0.257248
$545$	17.0000	0.728200
$546$	0	0
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	−1.00000	−0.0427179
$549$	0	0
$550$	8.00000	0.341121
$551$	−30.0000	−1.27804
$552$	0	0
$553$	4.00000	0.170097
$554$	−8.00000	−0.339887
$555$	0	0
$556$	13.0000	0.551323
$557$	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
$559$	−3.00000	−0.126886
$560$	1.00000	0.0422577
$561$	0	0
$562$	−13.0000	−0.548372
$563$	−3.00000	−0.126435	−0.0632175	−	0.998000i	$-0.520136\pi$
$563$	−3.00000	−0.126435	−0.0632175	+	0.998000i	$0.520136\pi$
$564$	0	0
$565$	9.00000	0.378633
$566$	24.0000	1.00880
$567$	0	0
$568$	−10.0000	−0.419591
$569$	33.0000	1.38343	0.691716	−	0.722170i	$-0.256855\pi$
$569$	33.0000	1.38343	0.691716	+	0.722170i	$0.256855\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	9.00000	0.375653
$575$	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	$0.301946\pi$
$578$	−19.0000	−0.790296
$579$	0	0
$580$	−5.00000	−0.207614
$581$	−4.00000	−0.165948
$582$	0	0
$583$	−12.0000	−0.496989
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−10.0000	−0.413096
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	48.0000	1.97781
$590$	−8.00000	−0.329355
$591$	0	0
$592$	3.00000	0.123299
$593$	21.0000	0.862367	0.431183	−	0.902264i	$-0.358096\pi$
$593$	21.0000	0.862367	0.431183	+	0.902264i	$0.358096\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	12.0000	0.491539
$597$	0	0
$598$	−1.00000	−0.0408930
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	3.00000	0.122271
$603$	0	0
$604$	9.00000	0.366205
$605$	7.00000	0.284590
$606$	0	0
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	2.00000	0.0809776
$611$	9.00000	0.364101
$612$	0	0
$613$	−7.00000	−0.282727	−0.141364	−	0.989958i	$-0.545149\pi$
$613$	−7.00000	−0.282727	−0.141364	+	0.989958i	$0.545149\pi$
$614$	17.0000	0.686064
$615$	0	0
$616$	2.00000	0.0805823
$617$	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
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	24.0000	0.962312
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	22.0000	0.879297
$627$	0	0
$628$	−18.0000	−0.718278
$629$	18.0000	0.717707
$630$	0	0
$631$	34.0000	1.35352	0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000	1.35352	0.676759	+	0.736204i	$0.263384\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	17.0000	0.675156
$635$	−1.00000	−0.0396838
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	−10.0000	−0.395904
$639$	0	0
$640$	1.00000	0.0395285
$641$	21.0000	0.829450	0.414725	−	0.909947i	$-0.363878\pi$
$641$	21.0000	0.829450	0.414725	+	0.909947i	$0.363878\pi$
$642$	0	0
$643$	30.0000	1.18308	0.591542	−	0.806274i	$-0.298519\pi$
$643$	30.0000	1.18308	0.591542	+	0.806274i	$0.298519\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	36.0000	1.41640
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	−4.00000	−0.156893
$651$	0	0
$652$	14.0000	0.548282
$653$	−47.0000	−1.83925	−0.919626	−	0.392795i	$-0.871508\pi$
$653$	−47.0000	−1.83925	−0.919626	+	0.392795i	$0.871508\pi$
$654$	0	0
$655$	10.0000	0.390732
$656$	9.00000	0.351391
$657$	0	0
$658$	−9.00000	−0.350857
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	0	0
$661$	24.0000	0.933492	0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000	0.933492	0.466746	+	0.884391i	$0.345426\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	−4.00000	−0.155230
$665$	−6.00000	−0.232670
$666$	0	0
$667$	−5.00000	−0.193601
$668$	−8.00000	−0.309529
$669$	0	0
$670$	4.00000	0.154533
$671$	4.00000	0.154418
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	34.0000	1.30963
$675$	0	0
$676$	−12.0000	−0.461538
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	0	0
$679$	5.00000	0.191882
$680$	6.00000	0.230089
$681$	0	0
$682$	16.0000	0.612672
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	1.00000	0.0382080
$686$	1.00000	0.0381802
$687$	0	0
$688$	3.00000	0.114374
$689$	6.00000	0.228582
$690$	0	0
$691$	−17.0000	−0.646710	−0.323355	−	0.946278i	$-0.604811\pi$
$691$	−17.0000	−0.646710	−0.323355	+	0.946278i	$0.604811\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−3.00000	−0.113878
$695$	−13.0000	−0.493118
$696$	0	0
$697$	54.0000	2.04540
$698$	6.00000	0.227103
$699$	0	0
$700$	4.00000	0.151186
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	−18.0000	−0.678883
$704$	2.00000	0.0753778
$705$	0	0
$706$	−11.0000	−0.413990
$707$	10.0000	0.376089
$708$	0	0
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	10.0000	0.375293
$711$	0	0
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	2.00000	0.0747958
$716$	11.0000	0.411089
$717$	0	0
$718$	−7.00000	−0.261238
$719$	−49.0000	−1.82739	−0.913696	−	0.406399i	$-0.866784\pi$
$719$	−49.0000	−1.82739	−0.913696	+	0.406399i	$0.866784\pi$
$720$	0	0
$721$	15.0000	0.558629
$722$	−17.0000	−0.632674
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	−20.0000	−0.742781
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	2.00000	0.0740233
$731$	18.0000	0.665754
$732$	0	0
$733$	−16.0000	−0.590973	−0.295487	−	0.955347i	$-0.595482\pi$
$733$	−16.0000	−0.590973	−0.295487	+	0.955347i	$0.595482\pi$
$734$	−3.00000	−0.110732
$735$	0	0
$736$	1.00000	0.0368605
$737$	8.00000	0.294684
$738$	0	0
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	−3.00000	−0.110282
$741$	0	0
$742$	−6.00000	−0.220267
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	−22.0000	−0.805477
$747$	0	0
$748$	12.0000	0.438763
$749$	18.0000	0.657706
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	−9.00000	−0.328196
$753$	0	0
$754$	5.00000	0.182089
$755$	−9.00000	−0.327544
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	−29.0000	−1.05333
$759$	0	0
$760$	−6.00000	−0.217643
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	17.0000	0.615441
$764$	−8.00000	−0.289430
$765$	0	0
$766$	6.00000	0.216789
$767$	8.00000	0.288863
$768$	0	0
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	−2.00000	−0.0720750
$771$	0	0
$772$	3.00000	0.107972
$773$	−25.0000	−0.899188	−0.449594	−	0.893233i	$-0.648431\pi$
$773$	−25.0000	−0.899188	−0.449594	+	0.893233i	$0.648431\pi$
$774$	0	0
$775$	32.0000	1.14947
$776$	5.00000	0.179490
$777$	0	0
$778$	−16.0000	−0.573628
$779$	−54.0000	−1.93475
$780$	0	0
$781$	20.0000	0.715656
$782$	6.00000	0.214560
$783$	0	0
$784$	1.00000	0.0357143
$785$	18.0000	0.642448
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−9.00000	−0.320612
$789$	0	0
$790$	−4.00000	−0.142314
$791$	9.00000	0.320003
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	26.0000	0.922705
$795$	0	0
$796$	−21.0000	−0.744325
$797$	−5.00000	−0.177109	−0.0885545	−	0.996071i	$-0.528225\pi$
$797$	−5.00000	−0.177109	−0.0885545	+	0.996071i	$0.528225\pi$
$798$	0	0
$799$	−54.0000	−1.91038
$800$	4.00000	0.141421
$801$	0	0
$802$	−18.0000	−0.635602
$803$	4.00000	0.141157
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	−8.00000	−0.281788
$807$	0	0
$808$	10.0000	0.351799
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	13.0000	0.456492	0.228246	−	0.973604i	$-0.426701\pi$
$811$	13.0000	0.456492	0.228246	+	0.973604i	$0.426701\pi$
$812$	−5.00000	−0.175466
$813$	0	0
$814$	−6.00000	−0.210300
$815$	−14.0000	−0.490399
$816$	0	0
$817$	−18.0000	−0.629740
$818$	−2.00000	−0.0699284
$819$	0	0
$820$	−9.00000	−0.314294
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	0	0
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	15.0000	0.522550
$825$	0	0
$826$	−8.00000	−0.278356
$827$	−2.00000	−0.0695468	−0.0347734	−	0.999395i	$-0.511071\pi$
$827$	−2.00000	−0.0695468	−0.0347734	+	0.999395i	$0.511071\pi$
$828$	0	0
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	4.00000	0.138842
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	6.00000	0.207888
$834$	0	0
$835$	8.00000	0.276851
$836$	−12.0000	−0.415029
$837$	0	0
$838$	16.0000	0.552711
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−31.0000	−1.06833
$843$	0	0
$844$	−20.0000	−0.688428
$845$	12.0000	0.412813
$846$	0	0
$847$	7.00000	0.240523
$848$	−6.00000	−0.206041
$849$	0	0
$850$	24.0000	0.823193
$851$	−3.00000	−0.102839
$852$	0	0
$853$	−23.0000	−0.787505	−0.393753	−	0.919216i	$-0.628823\pi$
$853$	−23.0000	−0.787505	−0.393753	+	0.919216i	$0.628823\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	18.0000	0.615227
$857$	1.00000	0.0341593	0.0170797	−	0.999854i	$-0.494563\pi$
$857$	1.00000	0.0341593	0.0170797	+	0.999854i	$0.494563\pi$
$858$	0	0
$859$	−39.0000	−1.33066	−0.665331	−	0.746548i	$-0.731710\pi$
$859$	−39.0000	−1.33066	−0.665331	+	0.746548i	$0.731710\pi$
$860$	−3.00000	−0.102299
$861$	0	0
$862$	19.0000	0.647143
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	−13.0000	−0.441758
$867$	0	0
$868$	8.00000	0.271538
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−4.00000	−0.135535
$872$	17.0000	0.575693
$873$	0	0
$874$	−6.00000	−0.202953
$875$	−9.00000	−0.304256
$876$	0	0
$877$	28.0000	0.945493	0.472746	−	0.881199i	$-0.343263\pi$
$877$	28.0000	0.945493	0.472746	+	0.881199i	$0.343263\pi$
$878$	10.0000	0.337484
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	0	0
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	23.0000	0.772700
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	0	0
$891$	0	0
$892$	−16.0000	−0.535720
$893$	54.0000	1.80704
$894$	0	0
$895$	−11.0000	−0.367689
$896$	1.00000	0.0334077
$897$	0	0
$898$	36.0000	1.20134
$899$	−40.0000	−1.33407
$900$	0	0
$901$	−36.0000	−1.19933
$902$	−18.0000	−0.599334
$903$	0	0
$904$	9.00000	0.299336
$905$	2.00000	0.0664822
$906$	0	0
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	−3.00000	−0.0995585
$909$	0	0
$910$	1.00000	0.0331497
$911$	−9.00000	−0.298183	−0.149092	−	0.988823i	$-0.547635\pi$
$911$	−9.00000	−0.298183	−0.149092	+	0.988823i	$0.547635\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	30.0000	0.992312
$915$	0	0
$916$	−30.0000	−0.991228
$917$	10.0000	0.330229
$918$	0	0
$919$	26.0000	0.857661	0.428830	−	0.903385i	$-0.358926\pi$
$919$	26.0000	0.857661	0.428830	+	0.903385i	$0.358926\pi$
$920$	−1.00000	−0.0329690
$921$	0	0
$922$	−40.0000	−1.31733
$923$	−10.0000	−0.329154
$924$	0	0
$925$	−12.0000	−0.394558
$926$	29.0000	0.952999
$927$	0	0
$928$	−5.00000	−0.164133
$929$	25.0000	0.820223	0.410112	−	0.912035i	$-0.365490\pi$
$929$	25.0000	0.820223	0.410112	+	0.912035i	$0.365490\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	−22.0000	−0.720634
$933$	0	0
$934$	−19.0000	−0.621699
$935$	−12.0000	−0.392442
$936$	0	0
$937$	3.00000	0.0980057	0.0490029	−	0.998799i	$-0.484396\pi$
$937$	3.00000	0.0980057	0.0490029	+	0.998799i	$0.484396\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	9.00000	0.293548
$941$	19.0000	0.619382	0.309691	−	0.950837i	$-0.399774\pi$
$941$	19.0000	0.619382	0.309691	+	0.950837i	$0.399774\pi$
$942$	0	0
$943$	−9.00000	−0.293080
$944$	−8.00000	−0.260378
$945$	0	0
$946$	−6.00000	−0.195077
$947$	45.0000	1.46230	0.731152	−	0.682215i	$-0.238983\pi$
$947$	45.0000	1.46230	0.731152	+	0.682215i	$0.238983\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	−24.0000	−0.778663
$951$	0	0
$952$	6.00000	0.194461
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	−14.0000	−0.452792
$957$	0	0
$958$	16.0000	0.516937
$959$	1.00000	0.0322917
$960$	0	0
$961$	33.0000	1.06452
$962$	3.00000	0.0967239
$963$	0	0
$964$	29.0000	0.934027
$965$	−3.00000	−0.0965734
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	−5.00000	−0.160540
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	−13.0000	−0.416761
$974$	−35.0000	−1.12147
$975$	0	0
$976$	2.00000	0.0640184
$977$	39.0000	1.24772	0.623860	−	0.781536i	$-0.285563\pi$
$977$	39.0000	1.24772	0.623860	+	0.781536i	$0.285563\pi$
$978$	0	0
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−12.0000	−0.382935
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	0	0
$985$	9.00000	0.286764
$986$	−30.0000	−0.955395
$987$	0	0
$988$	6.00000	0.190885
$989$	−3.00000	−0.0953945
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	10.0000	0.317181
$995$	21.0000	0.665745
$996$	0	0
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	−24.0000	−0.759707
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.c.1.1	✓	1
3.2	odd	2		2898.2.a.q.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2898.2.a.c.1.1	✓	1	1.1	even	1	trivial
2898.2.a.q.1.1	yes	1	3.2	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:	\( N \)	\(=\)	\( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2898.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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