LMFDB - Embedded newform 6498.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6498,2,Mod(1,6498)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6498, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6498.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6498 = 2 \cdot 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6498.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:	$51.8867912334$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6498.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -3.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -3.00000 q^{7} -1.00000 q^{8} -4.00000 q^{10} -2.00000 q^{11} -7.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} +4.00000 q^{20} +2.00000 q^{22} +4.00000 q^{23} +11.0000 q^{25} +7.00000 q^{26} -3.00000 q^{28} -4.00000 q^{29} +1.00000 q^{31} -1.00000 q^{32} -12.0000 q^{35} +7.00000 q^{37} -4.00000 q^{40} -4.00000 q^{41} +7.00000 q^{43} -2.00000 q^{44} -4.00000 q^{46} -2.00000 q^{47} +2.00000 q^{49} -11.0000 q^{50} -7.00000 q^{52} +4.00000 q^{53} -8.00000 q^{55} +3.00000 q^{56} +4.00000 q^{58} +6.00000 q^{59} -1.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -28.0000 q^{65} +3.00000 q^{67} +12.0000 q^{70} -2.00000 q^{71} -3.00000 q^{73} -7.00000 q^{74} +6.00000 q^{77} +5.00000 q^{79} +4.00000 q^{80} +4.00000 q^{82} +12.0000 q^{83} -7.00000 q^{86} +2.00000 q^{88} -18.0000 q^{89} +21.0000 q^{91} +4.00000 q^{92} +2.00000 q^{94} +10.0000 q^{97} -2.00000 q^{98} +O(q^{100})$

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
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−4.00000	−1.26491
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−7.00000	−1.94145	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−7.00000	−1.94145	−0.970725	+	0.240192i	$0.922790\pi$
$14$	3.00000	0.801784
$15$	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
$19$	0	0
$19$	0	0
$20$	4.00000	0.894427
$21$	0	0
$22$	2.00000	0.426401
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	7.00000	1.37281
$27$	0	0
$28$	−3.00000	−0.566947
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	−12.0000	−2.02837
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	0	0
$39$	0	0
$40$	−4.00000	−0.632456
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−4.00000	−0.589768
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−11.0000	−1.55563
$51$	0	0
$52$	−7.00000	−0.970725
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	3.00000	0.400892
$57$	0	0
$58$	4.00000	0.525226
$59$	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
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	−28.0000	−3.47297
$66$	0	0
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	0	0
$69$	0	0
$70$	12.0000	1.43427
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	−3.00000	−0.351123	−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000	−0.351123	−0.175562	+	0.984468i	$0.556174\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	4.00000	0.441726
$83$	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
$86$	−7.00000	−0.754829
$87$	0	0
$88$	2.00000	0.213201
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	21.0000	2.20140
$92$	4.00000	0.417029
$93$	0	0
$94$	2.00000	0.206284
$95$	0	0
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	11.0000	1.10000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	9.00000	0.886796	0.443398	−	0.896325i	$-0.353773\pi$
$103$	9.00000	0.886796	0.443398	+	0.896325i	$0.353773\pi$
$104$	7.00000	0.686406
$105$	0	0
$106$	−4.00000	−0.388514
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	8.00000	0.762770
$111$	0	0
$112$	−3.00000	−0.283473
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	16.0000	1.49201
$116$	−4.00000	−0.371391
$117$	0	0
$118$	−6.00000	−0.552345
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	1.00000	0.0905357
$123$	0	0
$124$	1.00000	0.0898027
$125$	24.0000	2.14663
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	28.0000	2.45576
$131$	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
$133$	0	0
$134$	−3.00000	−0.259161
$135$	0	0
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	21.0000	1.78120	0.890598	−	0.454791i	$-0.150286\pi$
$139$	21.0000	1.78120	0.890598	+	0.454791i	$0.150286\pi$
$140$	−12.0000	−1.01419
$141$	0	0
$142$	2.00000	0.167836
$143$	14.0000	1.17074
$144$	0	0
$145$	−16.0000	−1.32873
$146$	3.00000	0.248282
$147$	0	0
$148$	7.00000	0.575396
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	0	0
$154$	−6.00000	−0.483494
$155$	4.00000	0.321288
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	−5.00000	−0.397779
$159$	0	0
$160$	−4.00000	−0.316228
$161$	−12.0000	−0.945732
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	−12.0000	−0.931381
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	0	0
$171$	0	0
$172$	7.00000	0.533745
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	0	0
$175$	−33.0000	−2.49457
$176$	−2.00000	−0.150756
$177$	0	0
$178$	18.0000	1.34916
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	−21.0000	−1.55662
$183$	0	0
$184$	−4.00000	−0.294884
$185$	28.0000	2.05860
$186$	0	0
$187$	0	0
$188$	−2.00000	−0.145865
$189$	0	0
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	−21.0000	−1.51161	−0.755807	−	0.654795i	$-0.772755\pi$
$193$	−21.0000	−1.51161	−0.755807	+	0.654795i	$0.772755\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	2.00000	0.142857
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	−11.0000	−0.777817
$201$	0	0
$202$	2.00000	0.140720
$203$	12.0000	0.842235
$204$	0	0
$205$	−16.0000	−1.11749
$206$	−9.00000	−0.627060
$207$	0	0
$208$	−7.00000	−0.485363
$209$	0	0
$210$	0	0
$211$	9.00000	0.619586	0.309793	−	0.950804i	$-0.399740\pi$
$211$	9.00000	0.619586	0.309793	+	0.950804i	$0.399740\pi$
$212$	4.00000	0.274721
$213$	0	0
$214$	2.00000	0.136717
$215$	28.0000	1.90958
$216$	0	0
$217$	−3.00000	−0.203653
$218$	−6.00000	−0.406371
$219$	0	0
$220$	−8.00000	−0.539360
$221$	0	0
$222$	0	0
$223$	5.00000	0.334825	0.167412	−	0.985887i	$-0.446459\pi$
$223$	5.00000	0.334825	0.167412	+	0.985887i	$0.446459\pi$
$224$	3.00000	0.200446
$225$	0	0
$226$	2.00000	0.133038
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	−16.0000	−1.05501
$231$	0	0
$232$	4.00000	0.262613
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−1.00000	−0.0640184
$245$	8.00000	0.511101
$246$	0	0
$247$	0	0
$248$	−1.00000	−0.0635001
$249$	0	0
$250$	−24.0000	−1.51789
$251$	−14.0000	−0.883672	−0.441836	−	0.897096i	$-0.645673\pi$
$251$	−14.0000	−0.883672	−0.441836	+	0.897096i	$0.645673\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	28.0000	1.74659	0.873296	−	0.487190i	$-0.161978\pi$
$257$	28.0000	1.74659	0.873296	+	0.487190i	$0.161978\pi$
$258$	0	0
$259$	−21.0000	−1.30488
$260$	−28.0000	−1.73649
$261$	0	0
$262$	−18.0000	−1.11204
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	16.0000	0.982872
$266$	0	0
$267$	0	0
$268$	3.00000	0.183254
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	0	0
$274$	−2.00000	−0.120824
$275$	−22.0000	−1.32665
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	−21.0000	−1.25950
$279$	0	0
$280$	12.0000	0.717137
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	−14.0000	−0.827837
$287$	12.0000	0.708338
$288$	0	0
$289$	−17.0000	−1.00000
$290$	16.0000	0.939552
$291$	0	0
$292$	−3.00000	−0.175562
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	−7.00000	−0.406867
$297$	0	0
$298$	−18.0000	−1.04271
$299$	−28.0000	−1.61928
$300$	0	0
$301$	−21.0000	−1.21042
$302$	−20.0000	−1.15087
$303$	0	0
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	6.00000	0.341882
$309$	0	0
$310$	−4.00000	−0.227185
$311$	−34.0000	−1.92796	−0.963982	−	0.265969i	$-0.914308\pi$
$311$	−34.0000	−1.92796	−0.963982	+	0.265969i	$0.914308\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−7.00000	−0.395033
$315$	0	0
$316$	5.00000	0.281272
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	4.00000	0.223607
$321$	0	0
$322$	12.0000	0.668734
$323$	0	0
$324$	0	0
$325$	−77.0000	−4.27119
$326$	−11.0000	−0.609234
$327$	0	0
$328$	4.00000	0.220863
$329$	6.00000	0.330791
$330$	0	0
$331$	23.0000	1.26419	0.632097	−	0.774889i	$-0.282194\pi$
$331$	23.0000	1.26419	0.632097	+	0.774889i	$0.282194\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	6.00000	0.328305
$335$	12.0000	0.655630
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	−36.0000	−1.95814
$339$	0	0
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	15.0000	0.809924
$344$	−7.00000	−0.377415
$345$	0	0
$346$	12.0000	0.645124
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	0	0
$349$	31.0000	1.65939	0.829696	−	0.558216i	$-0.188514\pi$
$349$	31.0000	1.65939	0.829696	+	0.558216i	$0.188514\pi$
$350$	33.0000	1.76392
$351$	0	0
$352$	2.00000	0.106600
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	−18.0000	−0.953998
$357$	0	0
$358$	−12.0000	−0.634220
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	0	0
$362$	2.00000	0.105118
$363$	0	0
$364$	21.0000	1.10070
$365$	−12.0000	−0.628109
$366$	0	0
$367$	19.0000	0.991792	0.495896	−	0.868382i	$-0.334840\pi$
$367$	19.0000	0.991792	0.495896	+	0.868382i	$0.334840\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	−28.0000	−1.45565
$371$	−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
$376$	2.00000	0.103142
$377$	28.0000	1.44207
$378$	0	0
$379$	−21.0000	−1.07870	−0.539349	−	0.842082i	$-0.681330\pi$
$379$	−21.0000	−1.07870	−0.539349	+	0.842082i	$0.681330\pi$
$380$	0	0
$381$	0	0
$382$	−20.0000	−1.02329
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	0	0
$385$	24.0000	1.22315
$386$	21.0000	1.06887
$387$	0	0
$388$	10.0000	0.507673
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	0	0
$392$	−2.00000	−0.101015
$393$	0	0
$394$	−10.0000	−0.503793
$395$	20.0000	1.00631
$396$	0	0
$397$	−13.0000	−0.652451	−0.326226	−	0.945292i	$-0.605777\pi$
$397$	−13.0000	−0.652451	−0.326226	+	0.945292i	$0.605777\pi$
$398$	7.00000	0.350878
$399$	0	0
$400$	11.0000	0.550000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	−7.00000	−0.348695
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	−12.0000	−0.595550
$407$	−14.0000	−0.693954
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	16.0000	0.790184
$411$	0	0
$412$	9.00000	0.443398
$413$	−18.0000	−0.885722
$414$	0	0
$415$	48.0000	2.35623
$416$	7.00000	0.343203
$417$	0	0
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	−9.00000	−0.438113
$423$	0	0
$424$	−4.00000	−0.194257
$425$	0	0
$426$	0	0
$427$	3.00000	0.145180
$428$	−2.00000	−0.0966736
$429$	0	0
$430$	−28.0000	−1.35028
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	0	0
$433$	23.0000	1.10531	0.552655	−	0.833410i	$-0.313615\pi$
$433$	23.0000	1.10531	0.552655	+	0.833410i	$0.313615\pi$
$434$	3.00000	0.144005
$435$	0	0
$436$	6.00000	0.287348
$437$	0	0
$438$	0	0
$439$	23.0000	1.09773	0.548865	−	0.835911i	$-0.315060\pi$
$439$	23.0000	1.09773	0.548865	+	0.835911i	$0.315060\pi$
$440$	8.00000	0.381385
$441$	0	0
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	−72.0000	−3.41313
$446$	−5.00000	−0.236757
$447$	0	0
$448$	−3.00000	−0.141737
$449$	−16.0000	−0.755087	−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000	−0.755087	−0.377543	+	0.925992i	$0.623231\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	−2.00000	−0.0940721
$453$	0	0
$454$	4.00000	0.187729
$455$	84.0000	3.93798
$456$	0	0
$457$	29.0000	1.35656	0.678281	−	0.734802i	$-0.262725\pi$
$457$	29.0000	1.35656	0.678281	+	0.734802i	$0.262725\pi$
$458$	7.00000	0.327089
$459$	0	0
$460$	16.0000	0.746004
$461$	34.0000	1.58354	0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000	1.58354	0.791769	+	0.610821i	$0.209160\pi$
$462$	0	0
$463$	19.0000	0.883005	0.441502	−	0.897260i	$-0.354446\pi$
$463$	19.0000	0.883005	0.441502	+	0.897260i	$0.354446\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	6.00000	0.277945
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	8.00000	0.369012
$471$	0	0
$472$	−6.00000	−0.276172
$473$	−14.0000	−0.643721
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	24.0000	1.09773
$479$	−34.0000	−1.55350	−0.776750	−	0.629809i	$-0.783133\pi$
$479$	−34.0000	−1.55350	−0.776750	+	0.629809i	$0.783133\pi$
$480$	0	0
$481$	−49.0000	−2.23421
$482$	−1.00000	−0.0455488
$483$	0	0
$484$	−7.00000	−0.318182
$485$	40.0000	1.81631
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	1.00000	0.0452679
$489$	0	0
$490$	−8.00000	−0.361403
$491$	−26.0000	−1.17336	−0.586682	−	0.809818i	$-0.699566\pi$
$491$	−26.0000	−1.17336	−0.586682	+	0.809818i	$0.699566\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	1.00000	0.0449013
$497$	6.00000	0.269137
$498$	0	0
$499$	−11.0000	−0.492428	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−11.0000	−0.492428	−0.246214	+	0.969216i	$0.579187\pi$
$500$	24.0000	1.07331
$501$	0	0
$502$	14.0000	0.624851
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	8.00000	0.355643
$507$	0	0
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	9.00000	0.398137
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−28.0000	−1.23503
$515$	36.0000	1.58635
$516$	0	0
$517$	4.00000	0.175920
$518$	21.0000	0.922687
$519$	0	0
$520$	28.0000	1.22788
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	0	0
$523$	29.0000	1.26808	0.634041	−	0.773300i	$-0.281395\pi$
$523$	29.0000	1.26808	0.634041	+	0.773300i	$0.281395\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	−18.0000	−0.784837
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−16.0000	−0.694996
$531$	0	0
$532$	0	0
$533$	28.0000	1.21281
$534$	0	0
$535$	−8.00000	−0.345870
$536$	−3.00000	−0.129580
$537$	0	0
$538$	18.0000	0.776035
$539$	−4.00000	−0.172292
$540$	0	0
$541$	−19.0000	−0.816874	−0.408437	−	0.912787i	$-0.633926\pi$
$541$	−19.0000	−0.816874	−0.408437	+	0.912787i	$0.633926\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	0	0
$545$	24.0000	1.02805
$546$	0	0
$547$	−29.0000	−1.23995	−0.619975	−	0.784621i	$-0.712857\pi$
$547$	−29.0000	−1.23995	−0.619975	+	0.784621i	$0.712857\pi$
$548$	2.00000	0.0854358
$549$	0	0
$550$	22.0000	0.938083
$551$	0	0
$552$	0	0
$553$	−15.0000	−0.637865
$554$	26.0000	1.10463
$555$	0	0
$556$	21.0000	0.890598
$557$	−22.0000	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$558$	0	0
$559$	−49.0000	−2.07248
$560$	−12.0000	−0.507093
$561$	0	0
$562$	4.00000	0.168730
$563$	42.0000	1.77009	0.885044	−	0.465506i	$-0.154128\pi$
$563$	42.0000	1.77009	0.885044	+	0.465506i	$0.154128\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	12.0000	0.504398
$567$	0	0
$568$	2.00000	0.0839181
$569$	−16.0000	−0.670755	−0.335377	−	0.942084i	$-0.608864\pi$
$569$	−16.0000	−0.670755	−0.335377	+	0.942084i	$0.608864\pi$
$570$	0	0
$571$	−7.00000	−0.292941	−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000	−0.292941	−0.146470	+	0.989215i	$0.546791\pi$
$572$	14.0000	0.585369
$573$	0	0
$574$	−12.0000	−0.500870
$575$	44.0000	1.83493
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	17.0000	0.707107
$579$	0	0
$580$	−16.0000	−0.664364
$581$	−36.0000	−1.49353
$582$	0	0
$583$	−8.00000	−0.331326
$584$	3.00000	0.124141
$585$	0	0
$586$	−18.0000	−0.743573
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	0	0
$589$	0	0
$590$	−24.0000	−0.988064
$591$	0	0
$592$	7.00000	0.287698
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	0	0
$596$	18.0000	0.737309
$597$	0	0
$598$	28.0000	1.14501
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	21.0000	0.855896
$603$	0	0
$604$	20.0000	0.813788
$605$	−28.0000	−1.13836
$606$	0	0
$607$	41.0000	1.66414	0.832069	−	0.554672i	$-0.187156\pi$
$607$	41.0000	1.66414	0.832069	+	0.554672i	$0.187156\pi$
$608$	0	0
$609$	0	0
$610$	4.00000	0.161955
$611$	14.0000	0.566379
$612$	0	0
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	−12.0000	−0.484281
$615$	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
$619$	−23.0000	−0.924448	−0.462224	−	0.886763i	$-0.652948\pi$
$619$	−23.0000	−0.924448	−0.462224	+	0.886763i	$0.652948\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	34.0000	1.36328
$623$	54.0000	2.16346
$624$	0	0
$625$	41.0000	1.64000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	7.00000	0.279330
$629$	0	0
$630$	0	0
$631$	−19.0000	−0.756378	−0.378189	−	0.925728i	$-0.623453\pi$
$631$	−19.0000	−0.756378	−0.378189	+	0.925728i	$0.623453\pi$
$632$	−5.00000	−0.198889
$633$	0	0
$634$	−24.0000	−0.953162
$635$	0	0
$636$	0	0
$637$	−14.0000	−0.554700
$638$	−8.00000	−0.316723
$639$	0	0
$640$	−4.00000	−0.158114
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	13.0000	0.512670	0.256335	−	0.966588i	$-0.417485\pi$
$643$	13.0000	0.512670	0.256335	+	0.966588i	$0.417485\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	0	0
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	77.0000	3.02019
$651$	0	0
$652$	11.0000	0.430793
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	0	0
$655$	72.0000	2.81327
$656$	−4.00000	−0.156174
$657$	0	0
$658$	−6.00000	−0.233904
$659$	10.0000	0.389545	0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000	0.389545	0.194772	+	0.980848i	$0.437603\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	−23.0000	−0.893920
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	−16.0000	−0.619522
$668$	−6.00000	−0.232147
$669$	0	0
$670$	−12.0000	−0.463600
$671$	2.00000	0.0772091
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	36.0000	1.38462
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	0	0
$679$	−30.0000	−1.15129
$680$	0	0
$681$	0	0
$682$	2.00000	0.0765840
$683$	−40.0000	−1.53056	−0.765279	−	0.643699i	$-0.777399\pi$
$683$	−40.0000	−1.53056	−0.765279	+	0.643699i	$0.777399\pi$
$684$	0	0
$685$	8.00000	0.305664
$686$	−15.0000	−0.572703
$687$	0	0
$688$	7.00000	0.266872
$689$	−28.0000	−1.06672
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	28.0000	1.06287
$695$	84.0000	3.18630
$696$	0	0
$697$	0	0
$698$	−31.0000	−1.17337
$699$	0	0
$700$	−33.0000	−1.24728
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	0	0
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	−3.00000	−0.112667	−0.0563337	−	0.998412i	$-0.517941\pi$
$709$	−3.00000	−0.112667	−0.0563337	+	0.998412i	$0.517941\pi$
$710$	8.00000	0.300235
$711$	0	0
$712$	18.0000	0.674579
$713$	4.00000	0.149801
$714$	0	0
$715$	56.0000	2.09428
$716$	12.0000	0.448461
$717$	0	0
$718$	−12.0000	−0.447836
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	−27.0000	−1.00553
$722$	0	0
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	−44.0000	−1.63412
$726$	0	0
$727$	37.0000	1.37225	0.686127	−	0.727482i	$-0.259309\pi$
$727$	37.0000	1.37225	0.686127	+	0.727482i	$0.259309\pi$
$728$	−21.0000	−0.778312
$729$	0	0
$730$	12.0000	0.444140
$731$	0	0
$732$	0	0
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	−19.0000	−0.701303
$735$	0	0
$736$	−4.00000	−0.147442
$737$	−6.00000	−0.221013
$738$	0	0
$739$	35.0000	1.28750	0.643748	−	0.765238i	$-0.277379\pi$
$739$	35.0000	1.28750	0.643748	+	0.765238i	$0.277379\pi$
$740$	28.0000	1.02930
$741$	0	0
$742$	12.0000	0.440534
$743$	−26.0000	−0.953847	−0.476924	−	0.878945i	$-0.658248\pi$
$743$	−26.0000	−0.953847	−0.476924	+	0.878945i	$0.658248\pi$
$744$	0	0
$745$	72.0000	2.63788
$746$	22.0000	0.805477
$747$	0	0
$748$	0	0
$749$	6.00000	0.219235
$750$	0	0
$751$	−31.0000	−1.13121	−0.565603	−	0.824678i	$-0.691357\pi$
$751$	−31.0000	−1.13121	−0.565603	+	0.824678i	$0.691357\pi$
$752$	−2.00000	−0.0729325
$753$	0	0
$754$	−28.0000	−1.01970
$755$	80.0000	2.91150
$756$	0	0
$757$	−23.0000	−0.835949	−0.417975	−	0.908459i	$-0.637260\pi$
$757$	−23.0000	−0.835949	−0.417975	+	0.908459i	$0.637260\pi$
$758$	21.0000	0.762754
$759$	0	0
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	−18.0000	−0.651644
$764$	20.0000	0.723575
$765$	0	0
$766$	−14.0000	−0.505841
$767$	−42.0000	−1.51653
$768$	0	0
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	−24.0000	−0.864900
$771$	0	0
$772$	−21.0000	−0.755807
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	11.0000	0.395132
$776$	−10.0000	−0.358979
$777$	0	0
$778$	−24.0000	−0.860442
$779$	0	0
$780$	0	0
$781$	4.00000	0.143131
$782$	0	0
$783$	0	0
$784$	2.00000	0.0714286
$785$	28.0000	0.999363
$786$	0	0
$787$	49.0000	1.74666	0.873331	−	0.487128i	$-0.161955\pi$
$787$	49.0000	1.74666	0.873331	+	0.487128i	$0.161955\pi$
$788$	10.0000	0.356235
$789$	0	0
$790$	−20.0000	−0.711568
$791$	6.00000	0.213335
$792$	0	0
$793$	7.00000	0.248577
$794$	13.0000	0.461353
$795$	0	0
$796$	−7.00000	−0.248108
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	0	0
$800$	−11.0000	−0.388909
$801$	0	0
$802$	10.0000	0.353112
$803$	6.00000	0.211735
$804$	0	0
$805$	−48.0000	−1.69178
$806$	7.00000	0.246564
$807$	0	0
$808$	2.00000	0.0703598
$809$	−36.0000	−1.26569	−0.632846	−	0.774277i	$-0.718114\pi$
$809$	−36.0000	−1.26569	−0.632846	+	0.774277i	$0.718114\pi$
$810$	0	0
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	12.0000	0.421117
$813$	0	0
$814$	14.0000	0.490700
$815$	44.0000	1.54125
$816$	0	0
$817$	0	0
$818$	−10.0000	−0.349642
$819$	0	0
$820$	−16.0000	−0.558744
$821$	−12.0000	−0.418803	−0.209401	−	0.977830i	$-0.567152\pi$
$821$	−12.0000	−0.418803	−0.209401	+	0.977830i	$0.567152\pi$
$822$	0	0
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	−9.00000	−0.313530
$825$	0	0
$826$	18.0000	0.626300
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	0	0
$829$	45.0000	1.56291	0.781457	−	0.623959i	$-0.214477\pi$
$829$	45.0000	1.56291	0.781457	+	0.623959i	$0.214477\pi$
$830$	−48.0000	−1.66610
$831$	0	0
$832$	−7.00000	−0.242681
$833$	0	0
$834$	0	0
$835$	−24.0000	−0.830554
$836$	0	0
$837$	0	0
$838$	30.0000	1.03633
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−26.0000	−0.896019
$843$	0	0
$844$	9.00000	0.309793
$845$	144.000	4.95375
$846$	0	0
$847$	21.0000	0.721569
$848$	4.00000	0.137361
$849$	0	0
$850$	0	0
$851$	28.0000	0.959828
$852$	0	0
$853$	−9.00000	−0.308154	−0.154077	−	0.988059i	$-0.549240\pi$
$853$	−9.00000	−0.308154	−0.154077	+	0.988059i	$0.549240\pi$
$854$	−3.00000	−0.102658
$855$	0	0
$856$	2.00000	0.0683586
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	7.00000	0.238837	0.119418	−	0.992844i	$-0.461897\pi$
$859$	7.00000	0.238837	0.119418	+	0.992844i	$0.461897\pi$
$860$	28.0000	0.954792
$861$	0	0
$862$	6.00000	0.204361
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	0	0
$865$	−48.0000	−1.63205
$866$	−23.0000	−0.781572
$867$	0	0
$868$	−3.00000	−0.101827
$869$	−10.0000	−0.339227
$870$	0	0
$871$	−21.0000	−0.711558
$872$	−6.00000	−0.203186
$873$	0	0
$874$	0	0
$875$	−72.0000	−2.43404
$876$	0	0
$877$	19.0000	0.641584	0.320792	−	0.947150i	$-0.396051\pi$
$877$	19.0000	0.641584	0.320792	+	0.947150i	$0.396051\pi$
$878$	−23.0000	−0.776212
$879$	0	0
$880$	−8.00000	−0.269680
$881$	24.0000	0.808581	0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000	0.808581	0.404290	+	0.914631i	$0.367519\pi$
$882$	0	0
$883$	−41.0000	−1.37976	−0.689880	−	0.723924i	$-0.742337\pi$
$883$	−41.0000	−1.37976	−0.689880	+	0.723924i	$0.742337\pi$
$884$	0	0
$885$	0	0
$886$	4.00000	0.134383
$887$	−10.0000	−0.335767	−0.167884	−	0.985807i	$-0.553693\pi$
$887$	−10.0000	−0.335767	−0.167884	+	0.985807i	$0.553693\pi$
$888$	0	0
$889$	0	0
$890$	72.0000	2.41345
$891$	0	0
$892$	5.00000	0.167412
$893$	0	0
$894$	0	0
$895$	48.0000	1.60446
$896$	3.00000	0.100223
$897$	0	0
$898$	16.0000	0.533927
$899$	−4.00000	−0.133407
$900$	0	0
$901$	0	0
$902$	−8.00000	−0.266371
$903$	0	0
$904$	2.00000	0.0665190
$905$	−8.00000	−0.265929
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	−4.00000	−0.132745
$909$	0	0
$910$	−84.0000	−2.78457
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	−29.0000	−0.959235
$915$	0	0
$916$	−7.00000	−0.231287
$917$	−54.0000	−1.78324
$918$	0	0
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	−16.0000	−0.527504
$921$	0	0
$922$	−34.0000	−1.11973
$923$	14.0000	0.460816
$924$	0	0
$925$	77.0000	2.53174
$926$	−19.0000	−0.624379
$927$	0	0
$928$	4.00000	0.131306
$929$	−8.00000	−0.262471	−0.131236	−	0.991351i	$-0.541894\pi$
$929$	−8.00000	−0.262471	−0.131236	+	0.991351i	$0.541894\pi$
$930$	0	0
$931$	0	0
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−8.00000	−0.261768
$935$	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	$0.633898\pi$
$938$	9.00000	0.293860
$939$	0	0
$940$	−8.00000	−0.260931
$941$	−16.0000	−0.521585	−0.260793	−	0.965395i	$-0.583984\pi$
$941$	−16.0000	−0.521585	−0.260793	+	0.965395i	$0.583984\pi$
$942$	0	0
$943$	−16.0000	−0.521032
$944$	6.00000	0.195283
$945$	0	0
$946$	14.0000	0.455179
$947$	−54.0000	−1.75476	−0.877382	−	0.479792i	$-0.840712\pi$
$947$	−54.0000	−1.75476	−0.877382	+	0.479792i	$0.840712\pi$
$948$	0	0
$949$	21.0000	0.681689
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.0000	0.323932	0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000	0.323932	0.161966	+	0.986796i	$0.448217\pi$
$954$	0	0
$955$	80.0000	2.58874
$956$	−24.0000	−0.776215
$957$	0	0
$958$	34.0000	1.09849
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−30.0000	−0.967742
$962$	49.0000	1.57982
$963$	0	0
$964$	1.00000	0.0322078
$965$	−84.0000	−2.70406
$966$	0	0
$967$	−9.00000	−0.289420	−0.144710	−	0.989474i	$-0.546225\pi$
$967$	−9.00000	−0.289420	−0.144710	+	0.989474i	$0.546225\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	−40.0000	−1.28432
$971$	−14.0000	−0.449281	−0.224641	−	0.974442i	$-0.572121\pi$
$971$	−14.0000	−0.449281	−0.224641	+	0.974442i	$0.572121\pi$
$972$	0	0
$973$	−63.0000	−2.01969
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	−14.0000	−0.447900	−0.223950	−	0.974601i	$-0.571895\pi$
$977$	−14.0000	−0.447900	−0.223950	+	0.974601i	$0.571895\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	8.00000	0.255551
$981$	0	0
$982$	26.0000	0.829693
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	0	0
$985$	40.0000	1.27451
$986$	0	0
$987$	0	0
$988$	0	0
$989$	28.0000	0.890348
$990$	0	0
$991$	−29.0000	−0.921215	−0.460608	−	0.887604i	$-0.652368\pi$
$991$	−29.0000	−0.921215	−0.460608	+	0.887604i	$0.652368\pi$
$992$	−1.00000	−0.0317500
$993$	0	0
$994$	−6.00000	−0.190308
$995$	−28.0000	−0.887660
$996$	0	0
$997$	−53.0000	−1.67853	−0.839263	−	0.543725i	$-0.817013\pi$
$997$	−53.0000	−1.67853	−0.839263	+	0.543725i	$0.817013\pi$
$998$	11.0000	0.348199
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.l.1.1		1
3.2	odd	2		2166.2.a.f.1.1		1
19.7	even	3		342.2.g.d.163.1		2
19.11	even	3		342.2.g.d.235.1		2
19.18	odd	2		6498.2.a.x.1.1		1
57.11	odd	6		114.2.e.a.7.1	✓	2
57.26	odd	6		114.2.e.a.49.1	yes	2
57.56	even	2		2166.2.a.c.1.1		1
76.7	odd	6		2736.2.s.c.1873.1		2
76.11	odd	6		2736.2.s.c.577.1		2
228.11	even	6		912.2.q.d.577.1		2
228.83	even	6		912.2.q.d.49.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.e.a.7.1	✓	2	57.11	odd	6
114.2.e.a.49.1	yes	2	57.26	odd	6
342.2.g.d.163.1		2	19.7	even	3
342.2.g.d.235.1		2	19.11	even	3
912.2.q.d.49.1		2	228.83	even	6
912.2.q.d.577.1		2	228.11	even	6
2166.2.a.c.1.1		1	57.56	even	2
2166.2.a.f.1.1		1	3.2	odd	2
2736.2.s.c.577.1		2	76.11	odd	6
2736.2.s.c.1873.1		2	76.7	odd	6
6498.2.a.l.1.1		1	1.1	even	1	trivial
6498.2.a.x.1.1		1	19.18	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:	\( N \)	\(=\)	\( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6498.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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