LMFDB - Embedded newform 2898.2.a.j.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:	no (minimal twist has level 966)
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)

$f(q)$

$=$

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

$q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{10} +5.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +8.00000 q^{19} +3.00000 q^{20} +1.00000 q^{23} +4.00000 q^{25} -5.00000 q^{26} +1.00000 q^{28} -3.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} +3.00000 q^{35} -7.00000 q^{37} -8.00000 q^{38} -3.00000 q^{40} -9.00000 q^{41} -1.00000 q^{43} -1.00000 q^{46} +3.00000 q^{47} +1.00000 q^{49} -4.00000 q^{50} +5.00000 q^{52} +12.0000 q^{53} -1.00000 q^{56} +3.00000 q^{58} +6.00000 q^{59} +14.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +15.0000 q^{65} -4.00000 q^{67} -3.00000 q^{70} -6.00000 q^{71} -4.00000 q^{73} +7.00000 q^{74} +8.00000 q^{76} -16.0000 q^{79} +3.00000 q^{80} +9.00000 q^{82} +12.0000 q^{83} +1.00000 q^{86} -6.00000 q^{89} +5.00000 q^{91} +1.00000 q^{92} -3.00000 q^{94} +24.0000 q^{95} -1.00000 q^{97} -1.00000 q^{98} +O(q^{100})$

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
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	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
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	0	0
$23$	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
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−8.00000	−1.29777
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	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
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	3.00000	0.393919
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	−2.00000	−0.254000
$63$	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	$0.578571\pi$
$68$	0	0
$69$	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
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	8.00000	0.917663
$77$	0	0
$78$	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	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	0	0
$86$	1.00000	0.107833
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	5.00000	0.524142
$92$	1.00000	0.104257
$93$	0	0
$94$	−3.00000	−0.309426
$95$	24.0000	2.46235
$96$	0	0
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	4.00000	0.400000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	−5.00000	−0.490290
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	0	0
$111$	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
$116$	−3.00000	−0.278543
$117$	0	0
$118$	−6.00000	−0.552345
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−14.0000	−1.26750
$123$	0	0
$124$	2.00000	0.179605
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−15.0000	−1.31559
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	4.00000	0.345547
$135$	0	0
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	3.00000	0.253546
$141$	0	0
$142$	6.00000	0.503509
$143$	0	0
$144$	0	0
$145$	−9.00000	−0.747409
$146$	4.00000	0.331042
$147$	0	0
$148$	−7.00000	−0.575396
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	11.0000	0.895167	0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000	0.895167	0.447584	+	0.894242i	$0.352285\pi$
$152$	−8.00000	−0.648886
$153$	0	0
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	16.0000	1.27289
$159$	0	0
$160$	−3.00000	−0.237171
$161$	1.00000	0.0788110
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	−12.0000	−0.931381
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	−1.00000	−0.0762493
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	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
$185$	−21.0000	−1.54395
$186$	0	0
$187$	0	0
$188$	3.00000	0.218797
$189$	0	0
$190$	−24.0000	−1.74114
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−1.00000	−0.0719816	−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000	−0.0719816	−0.0359908	+	0.999352i	$0.511459\pi$
$194$	1.00000	0.0717958
$195$	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
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	−6.00000	−0.422159
$203$	−3.00000	−0.210559
$204$	0	0
$205$	−27.0000	−1.88576
$206$	1.00000	0.0696733
$207$	0	0
$208$	5.00000	0.346688
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	12.0000	0.824163
$213$	0	0
$214$	12.0000	0.820303
$215$	−3.00000	−0.204598
$216$	0	0
$217$	2.00000	0.135769
$218$	19.0000	1.28684
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	15.0000	0.997785
$227$	15.0000	0.995585	0.497792	−	0.867296i	$-0.334144\pi$
$227$	15.0000	0.995585	0.497792	+	0.867296i	$0.334144\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	−3.00000	−0.197814
$231$	0	0
$232$	3.00000	0.196960
$233$	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
$235$	9.00000	0.587095
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	14.0000	0.896258
$245$	3.00000	0.191663
$246$	0	0
$247$	40.0000	2.54514
$248$	−2.00000	−0.127000
$249$	0	0
$250$	3.00000	0.189737
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	0	0
$253$	0	0
$254$	13.0000	0.815693
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	15.0000	0.930261
$261$	0	0
$262$	−18.0000	−1.11204
$263$	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
$265$	36.0000	2.21146
$266$	−8.00000	−0.490511
$267$	0	0
$268$	−4.00000	−0.244339
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	0	0
$274$	−9.00000	−0.543710
$275$	0	0
$276$	0	0
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	−5.00000	−0.299880
$279$	0	0
$280$	−3.00000	−0.179284
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	0	0
$287$	−9.00000	−0.531253
$288$	0	0
$289$	−17.0000	−1.00000
$290$	9.00000	0.528498
$291$	0	0
$292$	−4.00000	−0.234082
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	18.0000	1.04800
$296$	7.00000	0.406867
$297$	0	0
$298$	0	0
$299$	5.00000	0.289157
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	−11.0000	−0.632979
$303$	0	0
$304$	8.00000	0.458831
$305$	42.0000	2.40491
$306$	0	0
$307$	11.0000	0.627803	0.313902	−	0.949456i	$-0.398364\pi$
$307$	11.0000	0.627803	0.313902	+	0.949456i	$0.398364\pi$
$308$	0	0
$309$	0	0
$310$	−6.00000	−0.340777
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−16.0000	−0.900070
$317$	27.0000	1.51647	0.758236	−	0.651981i	$-0.226062\pi$
$317$	27.0000	1.51647	0.758236	+	0.651981i	$0.226062\pi$
$318$	0	0
$319$	0	0
$320$	3.00000	0.167705
$321$	0	0
$322$	−1.00000	−0.0557278
$323$	0	0
$324$	0	0
$325$	20.0000	1.10940
$326$	−20.0000	−1.10770
$327$	0	0
$328$	9.00000	0.496942
$329$	3.00000	0.165395
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	−12.0000	−0.656611
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	1.00000	0.0539164
$345$	0	0
$346$	6.00000	0.322562
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	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
$352$	0	0
$353$	−27.0000	−1.43706	−0.718532	−	0.695493i	$-0.755186\pi$
$353$	−27.0000	−1.43706	−0.718532	+	0.695493i	$0.755186\pi$
$354$	0	0
$355$	−18.0000	−0.955341
$356$	−6.00000	−0.317999
$357$	0	0
$358$	15.0000	0.792775
$359$	−21.0000	−1.10834	−0.554169	−	0.832404i	$-0.686964\pi$
$359$	−21.0000	−1.10834	−0.554169	+	0.832404i	$0.686964\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	−2.00000	−0.105118
$363$	0	0
$364$	5.00000	0.262071
$365$	−12.0000	−0.628109
$366$	0	0
$367$	29.0000	1.51379	0.756894	−	0.653538i	$-0.226716\pi$
$367$	29.0000	1.51379	0.756894	+	0.653538i	$0.226716\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	21.0000	1.09174
$371$	12.0000	0.623009
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	−3.00000	−0.154713
$377$	−15.0000	−0.772539
$378$	0	0
$379$	29.0000	1.48963	0.744815	−	0.667271i	$-0.232538\pi$
$379$	29.0000	1.48963	0.744815	+	0.667271i	$0.232538\pi$
$380$	24.0000	1.23117
$381$	0	0
$382$	−12.0000	−0.613973
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	0	0
$385$	0	0
$386$	1.00000	0.0508987
$387$	0	0
$388$	−1.00000	−0.0507673
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	9.00000	0.453413
$395$	−48.0000	−2.41514
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−17.0000	−0.852133
$399$	0	0
$400$	4.00000	0.200000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	6.00000	0.298511
$405$	0	0
$406$	3.00000	0.148888
$407$	0	0
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	27.0000	1.33343
$411$	0	0
$412$	−1.00000	−0.0492665
$413$	6.00000	0.295241
$414$	0	0
$415$	36.0000	1.76717
$416$	−5.00000	−0.245145
$417$	0	0
$418$	0	0
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	−12.0000	−0.582772
$425$	0	0
$426$	0	0
$427$	14.0000	0.677507
$428$	−12.0000	−0.580042
$429$	0	0
$430$	3.00000	0.144673
$431$	−15.0000	−0.722525	−0.361262	−	0.932464i	$-0.617654\pi$
$431$	−15.0000	−0.722525	−0.361262	+	0.932464i	$0.617654\pi$
$432$	0	0
$433$	17.0000	0.816968	0.408484	−	0.912766i	$-0.366058\pi$
$433$	17.0000	0.816968	0.408484	+	0.912766i	$0.366058\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	−19.0000	−0.909935
$437$	8.00000	0.382692
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	27.0000	1.28281	0.641404	−	0.767203i	$-0.278352\pi$
$443$	27.0000	1.28281	0.641404	+	0.767203i	$0.278352\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	−14.0000	−0.662919
$447$	0	0
$448$	1.00000	0.0472456
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	0	0
$452$	−15.0000	−0.705541
$453$	0	0
$454$	−15.0000	−0.703985
$455$	15.0000	0.703211
$456$	0	0
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	4.00000	0.186908
$459$	0	0
$460$	3.00000	0.139876
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	−31.0000	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−6.00000	−0.277945
$467$	33.0000	1.52706	0.763529	−	0.645774i	$-0.223465\pi$
$467$	33.0000	1.52706	0.763529	+	0.645774i	$0.223465\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	−9.00000	−0.415139
$471$	0	0
$472$	−6.00000	−0.276172
$473$	0	0
$474$	0	0
$475$	32.0000	1.46826
$476$	0	0
$477$	0	0
$478$	12.0000	0.548867
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−35.0000	−1.59586
$482$	19.0000	0.865426
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−3.00000	−0.136223
$486$	0	0
$487$	−31.0000	−1.40474	−0.702372	−	0.711810i	$-0.747876\pi$
$487$	−31.0000	−1.40474	−0.702372	+	0.711810i	$0.747876\pi$
$488$	−14.0000	−0.633750
$489$	0	0
$490$	−3.00000	−0.135526
$491$	−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
$493$	0	0
$494$	−40.0000	−1.79969
$495$	0	0
$496$	2.00000	0.0898027
$497$	−6.00000	−0.269137
$498$	0	0
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	−15.0000	−0.669483
$503$	−18.0000	−0.802580	−0.401290	−	0.915951i	$-0.631438\pi$
$503$	−18.0000	−0.802580	−0.401290	+	0.915951i	$0.631438\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	0	0
$507$	0	0
$508$	−13.0000	−0.576782
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	−3.00000	−0.132196
$516$	0	0
$517$	0	0
$518$	7.00000	0.307562
$519$	0	0
$520$	−15.0000	−0.657794
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	26.0000	1.13690	0.568450	−	0.822718i	$-0.307543\pi$
$523$	26.0000	1.13690	0.568450	+	0.822718i	$0.307543\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	−9.00000	−0.392419
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	−36.0000	−1.56374
$531$	0	0
$532$	8.00000	0.346844
$533$	−45.0000	−1.94917
$534$	0	0
$535$	−36.0000	−1.55642
$536$	4.00000	0.172774
$537$	0	0
$538$	−18.0000	−0.776035
$539$	0	0
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	28.0000	1.20270
$543$	0	0
$544$	0	0
$545$	−57.0000	−2.44161
$546$	0	0
$547$	−10.0000	−0.427569	−0.213785	−	0.976881i	$-0.568579\pi$
$547$	−10.0000	−0.427569	−0.213785	+	0.976881i	$0.568579\pi$
$548$	9.00000	0.384461
$549$	0	0
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	−16.0000	−0.680389
$554$	4.00000	0.169944
$555$	0	0
$556$	5.00000	0.212047
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	3.00000	0.126773
$561$	0	0
$562$	−15.0000	−0.632737
$563$	3.00000	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$563$	3.00000	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$564$	0	0
$565$	−45.0000	−1.89316
$566$	−14.0000	−0.588464
$567$	0	0
$568$	6.00000	0.251754
$569$	−45.0000	−1.88650	−0.943249	−	0.332086i	$-0.892248\pi$
$569$	−45.0000	−1.88650	−0.943249	+	0.332086i	$0.892248\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	0	0
$574$	9.00000	0.375653
$575$	4.00000	0.166812
$576$	0	0
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	17.0000	0.707107
$579$	0	0
$580$	−9.00000	−0.373705
$581$	12.0000	0.497844
$582$	0	0
$583$	0	0
$584$	4.00000	0.165521
$585$	0	0
$586$	−30.0000	−1.23929
$587$	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
$589$	16.0000	0.659269
$590$	−18.0000	−0.741048
$591$	0	0
$592$	−7.00000	−0.287698
$593$	27.0000	1.10876	0.554379	−	0.832265i	$-0.312956\pi$
$593$	27.0000	1.10876	0.554379	+	0.832265i	$0.312956\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	−5.00000	−0.204465
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−16.0000	−0.652654	−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000	−0.652654	−0.326327	+	0.945257i	$0.605811\pi$
$602$	1.00000	0.0407570
$603$	0	0
$604$	11.0000	0.447584
$605$	−33.0000	−1.34164
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	−8.00000	−0.324443
$609$	0	0
$610$	−42.0000	−1.70053
$611$	15.0000	0.606835
$612$	0	0
$613$	−1.00000	−0.0403896	−0.0201948	−	0.999796i	$-0.506429\pi$
$613$	−1.00000	−0.0403896	−0.0201948	+	0.999796i	$0.506429\pi$
$614$	−11.0000	−0.443924
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	6.00000	0.240966
$621$	0	0
$622$	12.0000	0.481156
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−29.0000	−1.16000
$626$	10.0000	0.399680
$627$	0	0
$628$	−4.00000	−0.159617
$629$	0	0
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	16.0000	0.636446
$633$	0	0
$634$	−27.0000	−1.07231
$635$	−39.0000	−1.54767
$636$	0	0
$637$	5.00000	0.198107
$638$	0	0
$639$	0	0
$640$	−3.00000	−0.118585
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	0	0
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	−20.0000	−0.784465
$651$	0	0
$652$	20.0000	0.783260
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	0	0
$655$	54.0000	2.10995
$656$	−9.00000	−0.351391
$657$	0	0
$658$	−3.00000	−0.116952
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	16.0000	0.621858
$663$	0	0
$664$	−12.0000	−0.465690
$665$	24.0000	0.930680
$666$	0	0
$667$	−3.00000	−0.116160
$668$	12.0000	0.464294
$669$	0	0
$670$	12.0000	0.463600
$671$	0	0
$672$	0	0
$673$	35.0000	1.34915	0.674575	−	0.738206i	$-0.264327\pi$
$673$	35.0000	1.34915	0.674575	+	0.738206i	$0.264327\pi$
$674$	4.00000	0.154074
$675$	0	0
$676$	12.0000	0.461538
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−1.00000	−0.0383765
$680$	0	0
$681$	0	0
$682$	0	0
$683$	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
$685$	27.0000	1.03162
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−1.00000	−0.0381246
$689$	60.0000	2.28582
$690$	0	0
$691$	−49.0000	−1.86405	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−49.0000	−1.86405	−0.932024	+	0.362397i	$0.881959\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	3.00000	0.113878
$695$	15.0000	0.568982
$696$	0	0
$697$	0	0
$698$	10.0000	0.378506
$699$	0	0
$700$	4.00000	0.151186
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	0	0
$703$	−56.0000	−2.11208
$704$	0	0
$705$	0	0
$706$	27.0000	1.01616
$707$	6.00000	0.225653
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	18.0000	0.675528
$711$	0	0
$712$	6.00000	0.224860
$713$	2.00000	0.0749006
$714$	0	0
$715$	0	0
$716$	−15.0000	−0.560576
$717$	0	0
$718$	21.0000	0.783713
$719$	−45.0000	−1.67822	−0.839108	−	0.543964i	$-0.816923\pi$
$719$	−45.0000	−1.67822	−0.839108	+	0.543964i	$0.816923\pi$
$720$	0	0
$721$	−1.00000	−0.0372419
$722$	−45.0000	−1.67473
$723$	0	0
$724$	2.00000	0.0743294
$725$	−12.0000	−0.445669
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	−5.00000	−0.185312
$729$	0	0
$730$	12.0000	0.444140
$731$	0	0
$732$	0	0
$733$	8.00000	0.295487	0.147743	−	0.989026i	$-0.452799\pi$
$733$	8.00000	0.295487	0.147743	+	0.989026i	$0.452799\pi$
$734$	−29.0000	−1.07041
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	0	0
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	−21.0000	−0.771975
$741$	0	0
$742$	−12.0000	−0.440534
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	0	0
$746$	22.0000	0.805477
$747$	0	0
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	3.00000	0.109399
$753$	0	0
$754$	15.0000	0.546268
$755$	33.0000	1.20099
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	−29.0000	−1.05333
$759$	0	0
$760$	−24.0000	−0.870572
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	−19.0000	−0.687846
$764$	12.0000	0.434145
$765$	0	0
$766$	36.0000	1.30073
$767$	30.0000	1.08324
$768$	0	0
$769$	−37.0000	−1.33425	−0.667127	−	0.744944i	$-0.732476\pi$
$769$	−37.0000	−1.33425	−0.667127	+	0.744944i	$0.732476\pi$
$770$	0	0
$771$	0	0
$772$	−1.00000	−0.0359908
$773$	51.0000	1.83434	0.917171	−	0.398493i	$-0.130467\pi$
$773$	51.0000	1.83434	0.917171	+	0.398493i	$0.130467\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	1.00000	0.0358979
$777$	0	0
$778$	−30.0000	−1.07555
$779$	−72.0000	−2.57967
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	−12.0000	−0.428298
$786$	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
$790$	48.0000	1.70776
$791$	−15.0000	−0.533339
$792$	0	0
$793$	70.0000	2.48577
$794$	22.0000	0.780751
$795$	0	0
$796$	17.0000	0.602549
$797$	−33.0000	−1.16892	−0.584460	−	0.811423i	$-0.698694\pi$
$797$	−33.0000	−1.16892	−0.584460	+	0.811423i	$0.698694\pi$
$798$	0	0
$799$	0	0
$800$	−4.00000	−0.141421
$801$	0	0
$802$	18.0000	0.635602
$803$	0	0
$804$	0	0
$805$	3.00000	0.105736
$806$	−10.0000	−0.352235
$807$	0	0
$808$	−6.00000	−0.211079
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	0	0
$811$	5.00000	0.175574	0.0877869	−	0.996139i	$-0.472021\pi$
$811$	5.00000	0.175574	0.0877869	+	0.996139i	$0.472021\pi$
$812$	−3.00000	−0.105279
$813$	0	0
$814$	0	0
$815$	60.0000	2.10171
$816$	0	0
$817$	−8.00000	−0.279885
$818$	−32.0000	−1.11885
$819$	0	0
$820$	−27.0000	−0.942881
$821$	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
$823$	−13.0000	−0.453152	−0.226576	−	0.973994i	$-0.572753\pi$
$823$	−13.0000	−0.453152	−0.226576	+	0.973994i	$0.572753\pi$
$824$	1.00000	0.0348367
$825$	0	0
$826$	−6.00000	−0.208767
$827$	−6.00000	−0.208640	−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000	−0.208640	−0.104320	+	0.994544i	$0.533267\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	−36.0000	−1.24958
$831$	0	0
$832$	5.00000	0.173344
$833$	0	0
$834$	0	0
$835$	36.0000	1.24583
$836$	0	0
$837$	0	0
$838$	36.0000	1.24360
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	19.0000	0.654783
$843$	0	0
$844$	8.00000	0.275371
$845$	36.0000	1.23844
$846$	0	0
$847$	−11.0000	−0.377964
$848$	12.0000	0.412082
$849$	0	0
$850$	0	0
$851$	−7.00000	−0.239957
$852$	0	0
$853$	−49.0000	−1.67773	−0.838864	−	0.544341i	$-0.816780\pi$
$853$	−49.0000	−1.67773	−0.838864	+	0.544341i	$0.816780\pi$
$854$	−14.0000	−0.479070
$855$	0	0
$856$	12.0000	0.410152
$857$	15.0000	0.512390	0.256195	−	0.966625i	$-0.417531\pi$
$857$	15.0000	0.512390	0.256195	+	0.966625i	$0.417531\pi$
$858$	0	0
$859$	−31.0000	−1.05771	−0.528853	−	0.848713i	$-0.677378\pi$
$859$	−31.0000	−1.05771	−0.528853	+	0.848713i	$0.677378\pi$
$860$	−3.00000	−0.102299
$861$	0	0
$862$	15.0000	0.510902
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	−17.0000	−0.577684
$867$	0	0
$868$	2.00000	0.0678844
$869$	0	0
$870$	0	0
$871$	−20.0000	−0.677674
$872$	19.0000	0.643421
$873$	0	0
$874$	−8.00000	−0.270604
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−40.0000	−1.35070	−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000	−1.35070	−0.675352	+	0.737496i	$0.736008\pi$
$878$	−26.0000	−0.877457
$879$	0	0
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	0	0
$885$	0	0
$886$	−27.0000	−0.907083
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−13.0000	−0.436006
$890$	18.0000	0.603361
$891$	0	0
$892$	14.0000	0.468755
$893$	24.0000	0.803129
$894$	0	0
$895$	−45.0000	−1.50418
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	18.0000	0.600668
$899$	−6.00000	−0.200111
$900$	0	0
$901$	0	0
$902$	0	0
$903$	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	$0.261680\pi$
$908$	15.0000	0.497792
$909$	0	0
$910$	−15.0000	−0.497245
$911$	−21.0000	−0.695761	−0.347881	−	0.937539i	$-0.613099\pi$
$911$	−21.0000	−0.695761	−0.347881	+	0.937539i	$0.613099\pi$
$912$	0	0
$913$	0	0
$914$	28.0000	0.926158
$915$	0	0
$916$	−4.00000	−0.132164
$917$	18.0000	0.594412
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−3.00000	−0.0989071
$921$	0	0
$922$	−30.0000	−0.987997
$923$	−30.0000	−0.987462
$924$	0	0
$925$	−28.0000	−0.920634
$926$	31.0000	1.01872
$927$	0	0
$928$	3.00000	0.0984798
$929$	−33.0000	−1.08269	−0.541347	−	0.840799i	$-0.682086\pi$
$929$	−33.0000	−1.08269	−0.541347	+	0.840799i	$0.682086\pi$
$930$	0	0
$931$	8.00000	0.262189
$932$	6.00000	0.196537
$933$	0	0
$934$	−33.0000	−1.07979
$935$	0	0
$936$	0	0
$937$	−25.0000	−0.816714	−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000	−0.816714	−0.408357	+	0.912822i	$0.633898\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	9.00000	0.293548
$941$	51.0000	1.66255	0.831276	−	0.555860i	$-0.187611\pi$
$941$	51.0000	1.66255	0.831276	+	0.555860i	$0.187611\pi$
$942$	0	0
$943$	−9.00000	−0.293080
$944$	6.00000	0.195283
$945$	0	0
$946$	0	0
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	−32.0000	−1.03822
$951$	0	0
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	0	0
$955$	36.0000	1.16493
$956$	−12.0000	−0.388108
$957$	0	0
$958$	−6.00000	−0.193851
$959$	9.00000	0.290625
$960$	0	0
$961$	−27.0000	−0.870968
$962$	35.0000	1.12845
$963$	0	0
$964$	−19.0000	−0.611949
$965$	−3.00000	−0.0965734
$966$	0	0
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	3.00000	0.0963242
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	5.00000	0.160293
$974$	31.0000	0.993304
$975$	0	0
$976$	14.0000	0.448129
$977$	21.0000	0.671850	0.335925	−	0.941889i	$-0.390951\pi$
$977$	21.0000	0.671850	0.335925	+	0.941889i	$0.390951\pi$
$978$	0	0
$979$	0	0
$980$	3.00000	0.0958315
$981$	0	0
$982$	12.0000	0.382935
$983$	−30.0000	−0.956851	−0.478426	−	0.878128i	$-0.658792\pi$
$983$	−30.0000	−0.956851	−0.478426	+	0.878128i	$0.658792\pi$
$984$	0	0
$985$	−27.0000	−0.860292
$986$	0	0
$987$	0	0
$988$	40.0000	1.27257
$989$	−1.00000	−0.0317982
$990$	0	0
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	6.00000	0.190308
$995$	51.0000	1.61681
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	−14.0000	−0.443162
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.j.1.1		1
3.2	odd	2		966.2.a.i.1.1	✓	1
12.11	even	2		7728.2.a.a.1.1		1
21.20	even	2		6762.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.i.1.1	✓	1	3.2	odd	2
2898.2.a.j.1.1		1	1.1	even	1	trivial
6762.2.a.bd.1.1		1	21.20	even	2
7728.2.a.a.1.1		1	12.11	even	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

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Representations

Groups

Database

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