LMFDB - Embedded newform 2898.2.a.t.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:	$0$
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)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +3.00000 q^{10} -6.00000 q^{11} +5.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +2.00000 q^{19} +3.00000 q^{20} -6.00000 q^{22} +1.00000 q^{23} +4.00000 q^{25} +5.00000 q^{26} +1.00000 q^{28} -9.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +3.00000 q^{35} -1.00000 q^{37} +2.00000 q^{38} +3.00000 q^{40} +9.00000 q^{41} +11.0000 q^{43} -6.00000 q^{44} +1.00000 q^{46} +3.00000 q^{47} +1.00000 q^{49} +4.00000 q^{50} +5.00000 q^{52} -6.00000 q^{53} -18.0000 q^{55} +1.00000 q^{56} -9.00000 q^{58} +12.0000 q^{59} +2.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +15.0000 q^{65} -4.00000 q^{67} +6.00000 q^{68} +3.00000 q^{70} -6.00000 q^{71} +2.00000 q^{73} -1.00000 q^{74} +2.00000 q^{76} -6.00000 q^{77} -16.0000 q^{79} +3.00000 q^{80} +9.00000 q^{82} -12.0000 q^{83} +18.0000 q^{85} +11.0000 q^{86} -6.00000 q^{88} +12.0000 q^{89} +5.00000 q^{91} +1.00000 q^{92} +3.00000 q^{94} +6.00000 q^{95} +5.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	3.00000	0.948683
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	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
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−6.00000	−1.27920
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	4.00000	0.800000
$26$	5.00000	0.980581
$27$	0	0
$28$	1.00000	0.188982
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	3.00000	0.507093
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	3.00000	0.474342
$41$	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
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	1.00000	0.147442
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.00000	0.565685
$51$	0	0
$52$	5.00000	0.693375
$53$	−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
$55$	−18.0000	−2.42712
$56$	1.00000	0.133631
$57$	0	0
$58$	−9.00000	−1.18176
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	15.0000	1.86052
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	3.00000	0.358569
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	2.00000	0.229416
$77$	−6.00000	−0.683763
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	9.00000	0.993884
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	18.0000	1.95237
$86$	11.0000	1.18616
$87$	0	0
$88$	−6.00000	−0.639602
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	5.00000	0.524142
$92$	1.00000	0.104257
$93$	0	0
$94$	3.00000	0.309426
$95$	6.00000	0.615587
$96$	0	0
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	4.00000	0.400000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	−18.0000	−1.71623
$111$	0	0
$112$	1.00000	0.0944911
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	−9.00000	−0.835629
$117$	0	0
$118$	12.0000	1.10469
$119$	6.00000	0.550019
$120$	0	0
$121$	25.0000	2.27273
$122$	2.00000	0.181071
$123$	0	0
$124$	−4.00000	−0.359211
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−19.0000	−1.68598	−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−19.0000	−1.68598	−0.842989	+	0.537931i	$0.819206\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	15.0000	1.31559
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	0	0
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	3.00000	0.253546
$141$	0	0
$142$	−6.00000	−0.503509
$143$	−30.0000	−2.50873
$144$	0	0
$145$	−27.0000	−2.24223
$146$	2.00000	0.165521
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	24.0000	1.96616	0.983078	−	0.183186i	$-0.0586410\pi$
$149$	24.0000	1.96616	0.983078	+	0.183186i	$0.0586410\pi$
$150$	0	0
$151$	−19.0000	−1.54620	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−19.0000	−1.54620	−0.773099	+	0.634285i	$0.781294\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	−6.00000	−0.483494
$155$	−12.0000	−0.963863
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−16.0000	−1.27289
$159$	0	0
$160$	3.00000	0.237171
$161$	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
$166$	−12.0000	−0.931381
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	18.0000	1.38054
$171$	0	0
$172$	11.0000	0.838742
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	−6.00000	−0.452267
$177$	0	0
$178$	12.0000	0.899438
$179$	−3.00000	−0.224231	−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000	−0.224231	−0.112115	+	0.993695i	$0.535763\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	5.00000	0.370625
$183$	0	0
$184$	1.00000	0.0737210
$185$	−3.00000	−0.220564
$186$	0	0
$187$	−36.0000	−2.63258
$188$	3.00000	0.218797
$189$	0	0
$190$	6.00000	0.435286
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	5.00000	0.358979
$195$	0	0
$196$	1.00000	0.0714286
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	−6.00000	−0.422159
$203$	−9.00000	−0.631676
$204$	0	0
$205$	27.0000	1.88576
$206$	11.0000	0.766406
$207$	0	0
$208$	5.00000	0.346688
$209$	−12.0000	−0.830057
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−6.00000	−0.410152
$215$	33.0000	2.25058
$216$	0	0
$217$	−4.00000	−0.271538
$218$	11.0000	0.745014
$219$	0	0
$220$	−18.0000	−1.21356
$221$	30.0000	2.01802
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−15.0000	−0.997785
$227$	21.0000	1.39382	0.696909	−	0.717159i	$-0.254558\pi$
$227$	21.0000	1.39382	0.696909	+	0.717159i	$0.254558\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	3.00000	0.197814
$231$	0	0
$232$	−9.00000	−0.590879
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	12.0000	0.781133
$237$	0	0
$238$	6.00000	0.388922
$239$	−18.0000	−1.16432	−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000	−1.16432	−0.582162	+	0.813073i	$0.697793\pi$
$240$	0	0
$241$	11.0000	0.708572	0.354286	−	0.935137i	$-0.384724\pi$
$241$	11.0000	0.708572	0.354286	+	0.935137i	$0.384724\pi$
$242$	25.0000	1.60706
$243$	0	0
$244$	2.00000	0.128037
$245$	3.00000	0.191663
$246$	0	0
$247$	10.0000	0.636285
$248$	−4.00000	−0.254000
$249$	0	0
$250$	−3.00000	−0.189737
$251$	21.0000	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	−19.0000	−1.19217
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	15.0000	0.930261
$261$	0	0
$262$	−6.00000	−0.370681
$263$	3.00000	0.184988	0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000	0.184988	0.0924940	+	0.995713i	$0.470516\pi$
$264$	0	0
$265$	−18.0000	−1.10573
$266$	2.00000	0.122628
$267$	0	0
$268$	−4.00000	−0.244339
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	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
$274$	−15.0000	−0.906183
$275$	−24.0000	−1.44725
$276$	0	0
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	11.0000	0.659736
$279$	0	0
$280$	3.00000	0.179284
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	0	0
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−30.0000	−1.77394
$287$	9.00000	0.531253
$288$	0	0
$289$	19.0000	1.11765
$290$	−27.0000	−1.58549
$291$	0	0
$292$	2.00000	0.117041
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	36.0000	2.09600
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	24.0000	1.39028
$299$	5.00000	0.289157
$300$	0	0
$301$	11.0000	0.634029
$302$	−19.0000	−1.09333
$303$	0	0
$304$	2.00000	0.114708
$305$	6.00000	0.343559
$306$	0	0
$307$	17.0000	0.970241	0.485121	−	0.874447i	$-0.338776\pi$
$307$	17.0000	0.970241	0.485121	+	0.874447i	$0.338776\pi$
$308$	−6.00000	−0.341882
$309$	0	0
$310$	−12.0000	−0.681554
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−16.0000	−0.900070
$317$	−3.00000	−0.168497	−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000	−0.168497	−0.0842484	+	0.996445i	$0.526849\pi$
$318$	0	0
$319$	54.0000	3.02342
$320$	3.00000	0.167705
$321$	0	0
$322$	1.00000	0.0557278
$323$	12.0000	0.667698
$324$	0	0
$325$	20.0000	1.10940
$326$	14.0000	0.775388
$327$	0	0
$328$	9.00000	0.496942
$329$	3.00000	0.165395
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	−24.0000	−1.31322
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	12.0000	0.652714
$339$	0	0
$340$	18.0000	0.976187
$341$	24.0000	1.29967
$342$	0	0
$343$	1.00000	0.0539949
$344$	11.0000	0.593080
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−27.0000	−1.44944	−0.724718	−	0.689046i	$-0.758030\pi$
$347$	−27.0000	−1.44944	−0.724718	+	0.689046i	$0.758030\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	−6.00000	−0.319801
$353$	−21.0000	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$354$	0	0
$355$	−18.0000	−0.955341
$356$	12.0000	0.635999
$357$	0	0
$358$	−3.00000	−0.158555
$359$	−3.00000	−0.158334	−0.0791670	−	0.996861i	$-0.525226\pi$
$359$	−3.00000	−0.158334	−0.0791670	+	0.996861i	$0.525226\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	2.00000	0.105118
$363$	0	0
$364$	5.00000	0.262071
$365$	6.00000	0.314054
$366$	0	0
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−3.00000	−0.155963
$371$	−6.00000	−0.311504
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−36.0000	−1.86152
$375$	0	0
$376$	3.00000	0.154713
$377$	−45.0000	−2.31762
$378$	0	0
$379$	−19.0000	−0.975964	−0.487982	−	0.872854i	$-0.662267\pi$
$379$	−19.0000	−0.975964	−0.487982	+	0.872854i	$0.662267\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	0	0
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	−18.0000	−0.917365
$386$	11.0000	0.559885
$387$	0	0
$388$	5.00000	0.253837
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	1.00000	0.0505076
$393$	0	0
$394$	−3.00000	−0.151138
$395$	−48.0000	−2.41514
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	−7.00000	−0.350878
$399$	0	0
$400$	4.00000	0.200000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	−20.0000	−0.996271
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−9.00000	−0.446663
$407$	6.00000	0.297409
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	27.0000	1.33343
$411$	0	0
$412$	11.0000	0.541931
$413$	12.0000	0.590481
$414$	0	0
$415$	−36.0000	−1.76717
$416$	5.00000	0.245145
$417$	0	0
$418$	−12.0000	−0.586939
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−37.0000	−1.80327	−0.901635	−	0.432498i	$-0.857632\pi$
$421$	−37.0000	−1.80327	−0.901635	+	0.432498i	$0.857632\pi$
$422$	8.00000	0.389434
$423$	0	0
$424$	−6.00000	−0.291386
$425$	24.0000	1.16417
$426$	0	0
$427$	2.00000	0.0967868
$428$	−6.00000	−0.290021
$429$	0	0
$430$	33.0000	1.59140
$431$	−33.0000	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$432$	0	0
$433$	−37.0000	−1.77811	−0.889053	−	0.457804i	$-0.848636\pi$
$433$	−37.0000	−1.77811	−0.889053	+	0.457804i	$0.848636\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	11.0000	0.526804
$437$	2.00000	0.0956730
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	−18.0000	−0.858116
$441$	0	0
$442$	30.0000	1.42695
$443$	−9.00000	−0.427603	−0.213801	−	0.976877i	$-0.568585\pi$
$443$	−9.00000	−0.427603	−0.213801	+	0.976877i	$0.568585\pi$
$444$	0	0
$445$	36.0000	1.70656
$446$	8.00000	0.378811
$447$	0	0
$448$	1.00000	0.0472456
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	−54.0000	−2.54276
$452$	−15.0000	−0.705541
$453$	0	0
$454$	21.0000	0.985579
$455$	15.0000	0.703211
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	3.00000	0.139876
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	23.0000	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	18.0000	0.833834
$467$	−21.0000	−0.971764	−0.485882	−	0.874024i	$-0.661502\pi$
$467$	−21.0000	−0.971764	−0.485882	+	0.874024i	$0.661502\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	9.00000	0.415139
$471$	0	0
$472$	12.0000	0.552345
$473$	−66.0000	−3.03468
$474$	0	0
$475$	8.00000	0.367065
$476$	6.00000	0.275010
$477$	0	0
$478$	−18.0000	−0.823301
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−5.00000	−0.227980
$482$	11.0000	0.501036
$483$	0	0
$484$	25.0000	1.13636
$485$	15.0000	0.681115
$486$	0	0
$487$	−25.0000	−1.13286	−0.566429	−	0.824110i	$-0.691675\pi$
$487$	−25.0000	−1.13286	−0.566429	+	0.824110i	$0.691675\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	3.00000	0.135526
$491$	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
$493$	−54.0000	−2.43204
$494$	10.0000	0.449921
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−6.00000	−0.269137
$498$	0	0
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	21.0000	0.937276
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	−6.00000	−0.266733
$507$	0	0
$508$	−19.0000	−0.842989
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	1.00000	0.0441942
$513$	0	0
$514$	−18.0000	−0.793946
$515$	33.0000	1.45415
$516$	0	0
$517$	−18.0000	−0.791639
$518$	−1.00000	−0.0439375
$519$	0	0
$520$	15.0000	0.657794
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	3.00000	0.130806
$527$	−24.0000	−1.04546
$528$	0	0
$529$	1.00000	0.0434783
$530$	−18.0000	−0.781870
$531$	0	0
$532$	2.00000	0.0867110
$533$	45.0000	1.94917
$534$	0	0
$535$	−18.0000	−0.778208
$536$	−4.00000	−0.172774
$537$	0	0
$538$	−30.0000	−1.29339
$539$	−6.00000	−0.258438
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	14.0000	0.601351
$543$	0	0
$544$	6.00000	0.257248
$545$	33.0000	1.41356
$546$	0	0
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	−15.0000	−0.640768
$549$	0	0
$550$	−24.0000	−1.02336
$551$	−18.0000	−0.766826
$552$	0	0
$553$	−16.0000	−0.680389
$554$	−4.00000	−0.169944
$555$	0	0
$556$	11.0000	0.466504
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	55.0000	2.32625
$560$	3.00000	0.126773
$561$	0	0
$562$	3.00000	0.126547
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	0	0
$565$	−45.0000	−1.89316
$566$	−28.0000	−1.17693
$567$	0	0
$568$	−6.00000	−0.251754
$569$	−9.00000	−0.377300	−0.188650	−	0.982044i	$-0.560411\pi$
$569$	−9.00000	−0.377300	−0.188650	+	0.982044i	$0.560411\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−30.0000	−1.25436
$573$	0	0
$574$	9.00000	0.375653
$575$	4.00000	0.166812
$576$	0	0
$577$	−4.00000	−0.166522	−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000	−0.166522	−0.0832611	+	0.996528i	$0.526534\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	−27.0000	−1.12111
$581$	−12.0000	−0.497844
$582$	0	0
$583$	36.0000	1.49097
$584$	2.00000	0.0827606
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	36.0000	1.48210
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	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
$595$	18.0000	0.737928
$596$	24.0000	0.983078
$597$	0	0
$598$	5.00000	0.204465
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	11.0000	0.448327
$603$	0	0
$604$	−19.0000	−0.773099
$605$	75.0000	3.04918
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	6.00000	0.242933
$611$	15.0000	0.606835
$612$	0	0
$613$	5.00000	0.201948	0.100974	−	0.994889i	$-0.467804\pi$
$613$	5.00000	0.201948	0.100974	+	0.994889i	$0.467804\pi$
$614$	17.0000	0.686064
$615$	0	0
$616$	−6.00000	−0.241747
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	−12.0000	−0.481932
$621$	0	0
$622$	0	0
$623$	12.0000	0.480770
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	14.0000	0.558661
$629$	−6.00000	−0.239236
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	−16.0000	−0.636446
$633$	0	0
$634$	−3.00000	−0.119145
$635$	−57.0000	−2.26198
$636$	0	0
$637$	5.00000	0.198107
$638$	54.0000	2.13788
$639$	0	0
$640$	3.00000	0.118585
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	0	0
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	12.0000	0.472134
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	−72.0000	−2.82625
$650$	20.0000	0.784465
$651$	0	0
$652$	14.0000	0.548282
$653$	−21.0000	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$653$	−21.0000	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$654$	0	0
$655$	−18.0000	−0.703318
$656$	9.00000	0.351391
$657$	0	0
$658$	3.00000	0.116952
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	26.0000	1.01052
$663$	0	0
$664$	−12.0000	−0.465690
$665$	6.00000	0.232670
$666$	0	0
$667$	−9.00000	−0.348481
$668$	−24.0000	−0.928588
$669$	0	0
$670$	−12.0000	−0.463600
$671$	−12.0000	−0.463255
$672$	0	0
$673$	−37.0000	−1.42625	−0.713123	−	0.701039i	$-0.752720\pi$
$673$	−37.0000	−1.42625	−0.713123	+	0.701039i	$0.752720\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	12.0000	0.461538
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	5.00000	0.191882
$680$	18.0000	0.690268
$681$	0	0
$682$	24.0000	0.919007
$683$	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
$685$	−45.0000	−1.71936
$686$	1.00000	0.0381802
$687$	0	0
$688$	11.0000	0.419371
$689$	−30.0000	−1.14291
$690$	0	0
$691$	−31.0000	−1.17930	−0.589648	−	0.807661i	$-0.700733\pi$
$691$	−31.0000	−1.17930	−0.589648	+	0.807661i	$0.700733\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	−27.0000	−1.02491
$695$	33.0000	1.25176
$696$	0	0
$697$	54.0000	2.04540
$698$	−10.0000	−0.378506
$699$	0	0
$700$	4.00000	0.151186
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	−6.00000	−0.226134
$705$	0	0
$706$	−21.0000	−0.790345
$707$	−6.00000	−0.225653
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	−18.0000	−0.675528
$711$	0	0
$712$	12.0000	0.449719
$713$	−4.00000	−0.149801
$714$	0	0
$715$	−90.0000	−3.36581
$716$	−3.00000	−0.112115
$717$	0	0
$718$	−3.00000	−0.111959
$719$	−21.0000	−0.783168	−0.391584	−	0.920142i	$-0.628073\pi$
$719$	−21.0000	−0.783168	−0.391584	+	0.920142i	$0.628073\pi$
$720$	0	0
$721$	11.0000	0.409661
$722$	−15.0000	−0.558242
$723$	0	0
$724$	2.00000	0.0743294
$725$	−36.0000	−1.33701
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	5.00000	0.185312
$729$	0	0
$730$	6.00000	0.222070
$731$	66.0000	2.44110
$732$	0	0
$733$	20.0000	0.738717	0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000	0.738717	0.369358	+	0.929287i	$0.379577\pi$
$734$	17.0000	0.627481
$735$	0	0
$736$	1.00000	0.0368605
$737$	24.0000	0.884051
$738$	0	0
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	−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
$745$	72.0000	2.63788
$746$	14.0000	0.512576
$747$	0	0
$748$	−36.0000	−1.31629
$749$	−6.00000	−0.219235
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	3.00000	0.109399
$753$	0	0
$754$	−45.0000	−1.63880
$755$	−57.0000	−2.07444
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	−19.0000	−0.690111
$759$	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
$763$	11.0000	0.398227
$764$	0	0
$765$	0	0
$766$	6.00000	0.216789
$767$	60.0000	2.16647
$768$	0	0
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	−18.0000	−0.648675
$771$	0	0
$772$	11.0000	0.395899
$773$	27.0000	0.971123	0.485561	−	0.874203i	$-0.338615\pi$
$773$	27.0000	0.971123	0.485561	+	0.874203i	$0.338615\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	5.00000	0.179490
$777$	0	0
$778$	−24.0000	−0.860442
$779$	18.0000	0.644917
$780$	0	0
$781$	36.0000	1.28818
$782$	6.00000	0.214560
$783$	0	0
$784$	1.00000	0.0357143
$785$	42.0000	1.49904
$786$	0	0
$787$	−16.0000	−0.570338	−0.285169	−	0.958477i	$-0.592050\pi$
$787$	−16.0000	−0.570338	−0.285169	+	0.958477i	$0.592050\pi$
$788$	−3.00000	−0.106871
$789$	0	0
$790$	−48.0000	−1.70776
$791$	−15.0000	−0.533339
$792$	0	0
$793$	10.0000	0.355110
$794$	26.0000	0.922705
$795$	0	0
$796$	−7.00000	−0.248108
$797$	15.0000	0.531327	0.265664	−	0.964066i	$-0.414409\pi$
$797$	15.0000	0.531327	0.265664	+	0.964066i	$0.414409\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	4.00000	0.141421
$801$	0	0
$802$	6.00000	0.211867
$803$	−12.0000	−0.423471
$804$	0	0
$805$	3.00000	0.105736
$806$	−20.0000	−0.704470
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	11.0000	0.386262	0.193131	−	0.981173i	$-0.438136\pi$
$811$	11.0000	0.386262	0.193131	+	0.981173i	$0.438136\pi$
$812$	−9.00000	−0.315838
$813$	0	0
$814$	6.00000	0.210300
$815$	42.0000	1.47120
$816$	0	0
$817$	22.0000	0.769683
$818$	−22.0000	−0.769212
$819$	0	0
$820$	27.0000	0.942881
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	29.0000	1.01088	0.505438	−	0.862863i	$-0.331331\pi$
$823$	29.0000	1.01088	0.505438	+	0.862863i	$0.331331\pi$
$824$	11.0000	0.383203
$825$	0	0
$826$	12.0000	0.417533
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	−36.0000	−1.24958
$831$	0	0
$832$	5.00000	0.173344
$833$	6.00000	0.207888
$834$	0	0
$835$	−72.0000	−2.49166
$836$	−12.0000	−0.415029
$837$	0	0
$838$	0	0
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	−37.0000	−1.27510
$843$	0	0
$844$	8.00000	0.275371
$845$	36.0000	1.23844
$846$	0	0
$847$	25.0000	0.859010
$848$	−6.00000	−0.206041
$849$	0	0
$850$	24.0000	0.823193
$851$	−1.00000	−0.0342796
$852$	0	0
$853$	−37.0000	−1.26686	−0.633428	−	0.773802i	$-0.718353\pi$
$853$	−37.0000	−1.26686	−0.633428	+	0.773802i	$0.718353\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	−6.00000	−0.205076
$857$	33.0000	1.12726	0.563629	−	0.826028i	$-0.309405\pi$
$857$	33.0000	1.12726	0.563629	+	0.826028i	$0.309405\pi$
$858$	0	0
$859$	23.0000	0.784750	0.392375	−	0.919805i	$-0.371654\pi$
$859$	23.0000	0.784750	0.392375	+	0.919805i	$0.371654\pi$
$860$	33.0000	1.12529
$861$	0	0
$862$	−33.0000	−1.12398
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−54.0000	−1.83606
$866$	−37.0000	−1.25731
$867$	0	0
$868$	−4.00000	−0.135769
$869$	96.0000	3.25658
$870$	0	0
$871$	−20.0000	−0.677674
$872$	11.0000	0.372507
$873$	0	0
$874$	2.00000	0.0676510
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−52.0000	−1.75592	−0.877958	−	0.478738i	$-0.841094\pi$
$877$	−52.0000	−1.75592	−0.877958	+	0.478738i	$0.841094\pi$
$878$	−10.0000	−0.337484
$879$	0	0
$880$	−18.0000	−0.606780
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	30.0000	1.00901
$885$	0	0
$886$	−9.00000	−0.302361
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	−19.0000	−0.637240
$890$	36.0000	1.20672
$891$	0	0
$892$	8.00000	0.267860
$893$	6.00000	0.200782
$894$	0	0
$895$	−9.00000	−0.300837
$896$	1.00000	0.0334077
$897$	0	0
$898$	12.0000	0.400445
$899$	36.0000	1.20067
$900$	0	0
$901$	−36.0000	−1.19933
$902$	−54.0000	−1.79800
$903$	0	0
$904$	−15.0000	−0.498893
$905$	6.00000	0.199447
$906$	0	0
$907$	41.0000	1.36138	0.680691	−	0.732570i	$-0.261680\pi$
$907$	41.0000	1.36138	0.680691	+	0.732570i	$0.261680\pi$
$908$	21.0000	0.696909
$909$	0	0
$910$	15.0000	0.497245
$911$	45.0000	1.49092	0.745458	−	0.666552i	$-0.232231\pi$
$911$	45.0000	1.49092	0.745458	+	0.666552i	$0.232231\pi$
$912$	0	0
$913$	72.0000	2.38285
$914$	2.00000	0.0661541
$915$	0	0
$916$	−10.0000	−0.330409
$917$	−6.00000	−0.198137
$918$	0	0
$919$	26.0000	0.857661	0.428830	−	0.903385i	$-0.358926\pi$
$919$	26.0000	0.857661	0.428830	+	0.903385i	$0.358926\pi$
$920$	3.00000	0.0989071
$921$	0	0
$922$	12.0000	0.395199
$923$	−30.0000	−0.987462
$924$	0	0
$925$	−4.00000	−0.131519
$926$	23.0000	0.755827
$927$	0	0
$928$	−9.00000	−0.295439
$929$	57.0000	1.87011	0.935055	−	0.354504i	$-0.115350\pi$
$929$	57.0000	1.87011	0.935055	+	0.354504i	$0.115350\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	18.0000	0.589610
$933$	0	0
$934$	−21.0000	−0.687141
$935$	−108.000	−3.53198
$936$	0	0
$937$	53.0000	1.73143	0.865717	−	0.500533i	$-0.166863\pi$
$937$	53.0000	1.73143	0.865717	+	0.500533i	$0.166863\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	9.00000	0.293548
$941$	−33.0000	−1.07577	−0.537885	−	0.843018i	$-0.680776\pi$
$941$	−33.0000	−1.07577	−0.537885	+	0.843018i	$0.680776\pi$
$942$	0	0
$943$	9.00000	0.293080
$944$	12.0000	0.390567
$945$	0	0
$946$	−66.0000	−2.14585
$947$	27.0000	0.877382	0.438691	−	0.898638i	$-0.355442\pi$
$947$	27.0000	0.877382	0.438691	+	0.898638i	$0.355442\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	8.00000	0.259554
$951$	0	0
$952$	6.00000	0.194461
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	0	0
$955$	0	0
$956$	−18.0000	−0.582162
$957$	0	0
$958$	0	0
$959$	−15.0000	−0.484375
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−5.00000	−0.161206
$963$	0	0
$964$	11.0000	0.354286
$965$	33.0000	1.06231
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	25.0000	0.803530
$969$	0	0
$970$	15.0000	0.481621
$971$	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
$973$	11.0000	0.352644
$974$	−25.0000	−0.801052
$975$	0	0
$976$	2.00000	0.0640184
$977$	57.0000	1.82359	0.911796	−	0.410644i	$-0.134696\pi$
$977$	57.0000	1.82359	0.911796	+	0.410644i	$0.134696\pi$
$978$	0	0
$979$	−72.0000	−2.30113
$980$	3.00000	0.0958315
$981$	0	0
$982$	12.0000	0.382935
$983$	−12.0000	−0.382741	−0.191370	−	0.981518i	$-0.561293\pi$
$983$	−12.0000	−0.382741	−0.191370	+	0.981518i	$0.561293\pi$
$984$	0	0
$985$	−9.00000	−0.286764
$986$	−54.0000	−1.71971
$987$	0	0
$988$	10.0000	0.318142
$989$	11.0000	0.349780
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	−6.00000	−0.190308
$995$	−21.0000	−0.665745
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	8.00000	0.253236
$999$	0	0

Twists

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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