LMFDB - Embedded newform 459.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [459,2,Mod(1,459)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(459, 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("459.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$459 = 3^{3} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	459.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:	$3.66513345278$
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	$\chi$	$=$	459.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-2.00000 q^{2} +2.00000 q^{4} +2.00000 q^{5} +4.00000 q^{7} -4.00000 q^{10} -3.00000 q^{11} +7.00000 q^{13} -8.00000 q^{14} -4.00000 q^{16} -1.00000 q^{17} -4.00000 q^{19} +4.00000 q^{20} +6.00000 q^{22} -1.00000 q^{23} -1.00000 q^{25} -14.0000 q^{26} +8.00000 q^{28} +9.00000 q^{29} -2.00000 q^{31} +8.00000 q^{32} +2.00000 q^{34} +8.00000 q^{35} -8.00000 q^{37} +8.00000 q^{38} +9.00000 q^{41} +7.00000 q^{43} -6.00000 q^{44} +2.00000 q^{46} +9.00000 q^{49} +2.00000 q^{50} +14.0000 q^{52} -6.00000 q^{53} -6.00000 q^{55} -18.0000 q^{58} +2.00000 q^{61} +4.00000 q^{62} -8.00000 q^{64} +14.0000 q^{65} +7.00000 q^{67} -2.00000 q^{68} -16.0000 q^{70} +7.00000 q^{71} +6.00000 q^{73} +16.0000 q^{74} -8.00000 q^{76} -12.0000 q^{77} -12.0000 q^{79} -8.00000 q^{80} -18.0000 q^{82} -14.0000 q^{83} -2.00000 q^{85} -14.0000 q^{86} +8.00000 q^{89} +28.0000 q^{91} -2.00000 q^{92} -8.00000 q^{95} -10.0000 q^{97} -18.0000 q^{98} +O(q^{100})

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	0	0
$10$	−4.00000	−1.26491
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	7.00000	1.94145	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	7.00000	1.94145	0.970725	+	0.240192i	$0.0772105\pi$
$14$	−8.00000	−2.13809
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	4.00000	0.894427
$21$	0	0
$22$	6.00000	1.27920
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	−14.0000	−2.74563
$27$	0	0
$28$	8.00000	1.51186
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	2.00000	0.342997
$35$	8.00000	1.35225
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	8.00000	1.29777
$39$	0	0
$40$	0	0
$41$	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
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	2.00000	0.294884
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	2.00000	0.282843
$51$	0	0
$52$	14.0000	1.94145
$53$	−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
$55$	−6.00000	−0.809040
$56$	0	0
$57$	0	0
$58$	−18.0000	−2.36352
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	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
$64$	−8.00000	−1.00000
$65$	14.0000	1.73649
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	−16.0000	−1.91237
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	16.0000	1.85996
$75$	0	0
$76$	−8.00000	−0.917663
$77$	−12.0000	−1.36753
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	−8.00000	−0.894427
$81$	0	0
$82$	−18.0000	−1.98777
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	−14.0000	−1.50966
$87$	0	0
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	28.0000	2.93520
$92$	−2.00000	−0.208514
$93$	0	0
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−18.0000	−1.81827
$99$	0	0
$100$	−2.00000	−0.200000
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	12.0000	1.16554
$107$	7.00000	0.676716	0.338358	−	0.941018i	$-0.390129\pi$
$107$	7.00000	0.676716	0.338358	+	0.941018i	$0.390129\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	12.0000	1.14416
$111$	0	0
$112$	−16.0000	−1.51186
$113$	−13.0000	−1.22294	−0.611469	−	0.791269i	$-0.709421\pi$
$113$	−13.0000	−1.22294	−0.611469	+	0.791269i	$0.709421\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	18.0000	1.67126
$117$	0	0
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−4.00000	−0.362143
$123$	0	0
$124$	−4.00000	−0.359211
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	0	0
$129$	0	0
$130$	−28.0000	−2.45576
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	−16.0000	−1.38738
$134$	−14.0000	−1.20942
$135$	0	0
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	16.0000	1.35225
$141$	0	0
$142$	−14.0000	−1.17485
$143$	−21.0000	−1.75611
$144$	0	0
$145$	18.0000	1.49482
$146$	−12.0000	−0.993127
$147$	0	0
$148$	−16.0000	−1.31519
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−13.0000	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$152$	0	0
$153$	0	0
$154$	24.0000	1.93398
$155$	−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	$0.409888\pi$
$158$	24.0000	1.90934
$159$	0	0
$160$	16.0000	1.26491
$161$	−4.00000	−0.315244
$162$	0	0
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	18.0000	1.40556
$165$	0	0
$166$	28.0000	2.17322
$167$	1.00000	0.0773823	0.0386912	−	0.999251i	$-0.487681\pi$
$167$	1.00000	0.0773823	0.0386912	+	0.999251i	$0.487681\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	4.00000	0.306786
$171$	0	0
$172$	14.0000	1.06749
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	12.0000	0.904534
$177$	0	0
$178$	−16.0000	−1.19925
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	4.00000	0.297318	0.148659	−	0.988889i	$-0.452504\pi$
$181$	4.00000	0.297318	0.148659	+	0.988889i	$0.452504\pi$
$182$	−56.0000	−4.15100
$183$	0	0
$184$	0	0
$185$	−16.0000	−1.17634
$186$	0	0
$187$	3.00000	0.219382
$188$	0	0
$189$	0	0
$190$	16.0000	1.16076
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	20.0000	1.43592
$195$	0	0
$196$	18.0000	1.28571
$197$	15.0000	1.06871	0.534353	−	0.845262i	$-0.320555\pi$
$197$	15.0000	1.06871	0.534353	+	0.845262i	$0.320555\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	0	0
$202$	16.0000	1.12576
$203$	36.0000	2.52670
$204$	0	0
$205$	18.0000	1.25717
$206$	−16.0000	−1.11477
$207$	0	0
$208$	−28.0000	−1.94145
$209$	12.0000	0.830057
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	−14.0000	−0.957020
$215$	14.0000	0.954792
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	0	0
$220$	−12.0000	−0.809040
$221$	−7.00000	−0.470871
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	32.0000	2.13809
$225$	0	0
$226$	26.0000	1.72949
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	21.0000	1.38772	0.693860	−	0.720110i	$-0.255909\pi$
$229$	21.0000	1.38772	0.693860	+	0.720110i	$0.255909\pi$
$230$	4.00000	0.263752
$231$	0	0
$232$	0	0
$233$	−3.00000	−0.196537	−0.0982683	−	0.995160i	$-0.531330\pi$
$233$	−3.00000	−0.196537	−0.0982683	+	0.995160i	$0.531330\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	8.00000	0.518563
$239$	−26.0000	−1.68180	−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000	−1.68180	−0.840900	+	0.541190i	$0.817974\pi$
$240$	0	0
$241$	−24.0000	−1.54598	−0.772988	−	0.634421i	$-0.781239\pi$
$241$	−24.0000	−1.54598	−0.772988	+	0.634421i	$0.781239\pi$
$242$	4.00000	0.257130
$243$	0	0
$244$	4.00000	0.256074
$245$	18.0000	1.14998
$246$	0	0
$247$	−28.0000	−1.78160
$248$	0	0
$249$	0	0
$250$	24.0000	1.51789
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	3.00000	0.188608
$254$	26.0000	1.63139
$255$	0	0
$256$	16.0000	1.00000
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	0	0
$259$	−32.0000	−1.98838
$260$	28.0000	1.73649
$261$	0	0
$262$	8.00000	0.494242
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	32.0000	1.96205
$267$	0	0
$268$	14.0000	0.855186
$269$	−7.00000	−0.426798	−0.213399	−	0.976965i	$-0.568453\pi$
$269$	−7.00000	−0.426798	−0.213399	+	0.976965i	$0.568453\pi$
$270$	0	0
$271$	−1.00000	−0.0607457	−0.0303728	−	0.999539i	$-0.509669\pi$
$271$	−1.00000	−0.0607457	−0.0303728	+	0.999539i	$0.509669\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−24.0000	−1.44989
$275$	3.00000	0.180907
$276$	0	0
$277$	32.0000	1.92269	0.961347	−	0.275340i	$-0.0887905\pi$
$277$	32.0000	1.92269	0.961347	+	0.275340i	$0.0887905\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	14.0000	0.830747
$285$	0	0
$286$	42.0000	2.48351
$287$	36.0000	2.12501
$288$	0	0
$289$	1.00000	0.0588235
$290$	−36.0000	−2.11399
$291$	0	0
$292$	12.0000	0.702247
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−20.0000	−1.15857
$299$	−7.00000	−0.404820
$300$	0	0
$301$	28.0000	1.61389
$302$	26.0000	1.49613
$303$	0	0
$304$	16.0000	0.917663
$305$	4.00000	0.229039
$306$	0	0
$307$	−9.00000	−0.513657	−0.256829	−	0.966457i	$-0.582678\pi$
$307$	−9.00000	−0.513657	−0.256829	+	0.966457i	$0.582678\pi$
$308$	−24.0000	−1.36753
$309$	0	0
$310$	8.00000	0.454369
$311$	−28.0000	−1.58773	−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000	−1.58773	−0.793867	+	0.608091i	$0.791935\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	−24.0000	−1.35011
$317$	−17.0000	−0.954815	−0.477408	−	0.878682i	$-0.658423\pi$
$317$	−17.0000	−0.954815	−0.477408	+	0.878682i	$0.658423\pi$
$318$	0	0
$319$	−27.0000	−1.51171
$320$	−16.0000	−0.894427
$321$	0	0
$322$	8.00000	0.445823
$323$	4.00000	0.222566
$324$	0	0
$325$	−7.00000	−0.388290
$326$	12.0000	0.664619
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−11.0000	−0.604615	−0.302307	−	0.953211i	$-0.597757\pi$
$331$	−11.0000	−0.604615	−0.302307	+	0.953211i	$0.597757\pi$
$332$	−28.0000	−1.53670
$333$	0	0
$334$	−2.00000	−0.109435
$335$	14.0000	0.764902
$336$	0	0
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	−72.0000	−3.91628
$339$	0	0
$340$	−4.00000	−0.216930
$341$	6.00000	0.324918
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	26.0000	1.39777
$347$	1.00000	0.0536828	0.0268414	−	0.999640i	$-0.491455\pi$
$347$	1.00000	0.0536828	0.0268414	+	0.999640i	$0.491455\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	8.00000	0.427618
$351$	0	0
$352$	−24.0000	−1.27920
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	14.0000	0.743043
$356$	16.0000	0.847998
$357$	0	0
$358$	−48.0000	−2.53688
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−8.00000	−0.420471
$363$	0	0
$364$	56.0000	2.93520
$365$	12.0000	0.628109
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	32.0000	1.66360
$371$	−24.0000	−1.24602
$372$	0	0
$373$	−3.00000	−0.155334	−0.0776671	−	0.996979i	$-0.524747\pi$
$373$	−3.00000	−0.155334	−0.0776671	+	0.996979i	$0.524747\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	0	0
$377$	63.0000	3.24467
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	−16.0000	−0.820783
$381$	0	0
$382$	−8.00000	−0.409316
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	−40.0000	−2.03595
$387$	0	0
$388$	−20.0000	−1.01535
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	1.00000	0.0505722
$392$	0	0
$393$	0	0
$394$	−30.0000	−1.51138
$395$	−24.0000	−1.20757
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	4.00000	0.200000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	−14.0000	−0.697390
$404$	−16.0000	−0.796030
$405$	0	0
$406$	−72.0000	−3.57330
$407$	24.0000	1.18964
$408$	0	0
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	−36.0000	−1.77791
$411$	0	0
$412$	16.0000	0.788263
$413$	0	0
$414$	0	0
$415$	−28.0000	−1.37447
$416$	56.0000	2.74563
$417$	0	0
$418$	−24.0000	−1.17388
$419$	25.0000	1.22133	0.610665	−	0.791889i	$-0.290902\pi$
$419$	25.0000	1.22133	0.610665	+	0.791889i	$0.290902\pi$
$420$	0	0
$421$	−17.0000	−0.828529	−0.414265	−	0.910156i	$-0.635961\pi$
$421$	−17.0000	−0.828529	−0.414265	+	0.910156i	$0.635961\pi$
$422$	44.0000	2.14189
$423$	0	0
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	8.00000	0.387147
$428$	14.0000	0.676716
$429$	0	0
$430$	−28.0000	−1.35028
$431$	−27.0000	−1.30054	−0.650272	−	0.759701i	$-0.725345\pi$
$431$	−27.0000	−1.30054	−0.650272	+	0.759701i	$0.725345\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	0	0
$442$	14.0000	0.665912
$443$	−40.0000	−1.90046	−0.950229	−	0.311553i	$-0.899151\pi$
$443$	−40.0000	−1.90046	−0.950229	+	0.311553i	$0.899151\pi$
$444$	0	0
$445$	16.0000	0.758473
$446$	24.0000	1.13643
$447$	0	0
$448$	−32.0000	−1.51186
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	−27.0000	−1.27138
$452$	−26.0000	−1.22294
$453$	0	0
$454$	24.0000	1.12638
$455$	56.0000	2.62532
$456$	0	0
$457$	33.0000	1.54367	0.771837	−	0.635820i	$-0.219338\pi$
$457$	33.0000	1.54367	0.771837	+	0.635820i	$0.219338\pi$
$458$	−42.0000	−1.96253
$459$	0	0
$460$	−4.00000	−0.186501
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−36.0000	−1.67126
$465$	0	0
$466$	6.00000	0.277945
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	28.0000	1.29292
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−21.0000	−0.965581
$474$	0	0
$475$	4.00000	0.183533
$476$	−8.00000	−0.366679
$477$	0	0
$478$	52.0000	2.37842
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−56.0000	−2.55338
$482$	48.0000	2.18634
$483$	0	0
$484$	−4.00000	−0.181818
$485$	−20.0000	−0.908153
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	0	0
$490$	−36.0000	−1.62631
$491$	34.0000	1.53440	0.767199	−	0.641409i	$-0.221650\pi$
$491$	34.0000	1.53440	0.767199	+	0.641409i	$0.221650\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	56.0000	2.51956
$495$	0	0
$496$	8.00000	0.359211
$497$	28.0000	1.25597
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	−24.0000	−1.07331
$501$	0	0
$502$	36.0000	1.60676
$503$	−3.00000	−0.133763	−0.0668817	−	0.997761i	$-0.521305\pi$
$503$	−3.00000	−0.133763	−0.0668817	+	0.997761i	$0.521305\pi$
$504$	0	0
$505$	−16.0000	−0.711991
$506$	−6.00000	−0.266733
$507$	0	0
$508$	−26.0000	−1.15356
$509$	8.00000	0.354594	0.177297	−	0.984157i	$-0.443265\pi$
$509$	8.00000	0.354594	0.177297	+	0.984157i	$0.443265\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	−32.0000	−1.41421
$513$	0	0
$514$	60.0000	2.64649
$515$	16.0000	0.705044
$516$	0	0
$517$	0	0
$518$	64.0000	2.81200
$519$	0	0
$520$	0	0
$521$	25.0000	1.09527	0.547635	−	0.836717i	$-0.315528\pi$
$521$	25.0000	1.09527	0.547635	+	0.836717i	$0.315528\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	16.0000	0.697633
$527$	2.00000	0.0871214
$528$	0	0
$529$	−22.0000	−0.956522
$530$	24.0000	1.04249
$531$	0	0
$532$	−32.0000	−1.38738
$533$	63.0000	2.72883
$534$	0	0
$535$	14.0000	0.605273
$536$	0	0
$537$	0	0
$538$	14.0000	0.603583
$539$	−27.0000	−1.16297
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	2.00000	0.0859074
$543$	0	0
$544$	−8.00000	−0.342997
$545$	0	0
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	24.0000	1.02523
$549$	0	0
$550$	−6.00000	−0.255841
$551$	−36.0000	−1.53365
$552$	0	0
$553$	−48.0000	−2.04117
$554$	−64.0000	−2.71910
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−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
$559$	49.0000	2.07248
$560$	−32.0000	−1.35225
$561$	0	0
$562$	12.0000	0.506189
$563$	22.0000	0.927189	0.463595	−	0.886047i	$-0.346559\pi$
$563$	22.0000	0.927189	0.463595	+	0.886047i	$0.346559\pi$
$564$	0	0
$565$	−26.0000	−1.09383
$566$	−16.0000	−0.672530
$567$	0	0
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	46.0000	1.92504	0.962520	−	0.271211i	$-0.0874240\pi$
$571$	46.0000	1.92504	0.962520	+	0.271211i	$0.0874240\pi$
$572$	−42.0000	−1.75611
$573$	0	0
$574$	−72.0000	−3.00522
$575$	1.00000	0.0417029
$576$	0	0
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	−2.00000	−0.0831890
$579$	0	0
$580$	36.0000	1.49482
$581$	−56.0000	−2.32327
$582$	0	0
$583$	18.0000	0.745484
$584$	0	0
$585$	0	0
$586$	36.0000	1.48715
$587$	2.00000	0.0825488	0.0412744	−	0.999148i	$-0.486858\pi$
$587$	2.00000	0.0825488	0.0412744	+	0.999148i	$0.486858\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	0	0
$592$	32.0000	1.31519
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	0	0
$595$	−8.00000	−0.327968
$596$	20.0000	0.819232
$597$	0	0
$598$	14.0000	0.572503
$599$	18.0000	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	−56.0000	−2.28239
$603$	0	0
$604$	−26.0000	−1.05792
$605$	−4.00000	−0.162623
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	−32.0000	−1.29777
$609$	0	0
$610$	−8.00000	−0.323911
$611$	0	0
$612$	0	0
$613$	−29.0000	−1.17130	−0.585649	−	0.810564i	$-0.699160\pi$
$613$	−29.0000	−1.17130	−0.585649	+	0.810564i	$0.699160\pi$
$614$	18.0000	0.726421
$615$	0	0
$616$	0	0
$617$	−3.00000	−0.120775	−0.0603877	−	0.998175i	$-0.519234\pi$
$617$	−3.00000	−0.120775	−0.0603877	+	0.998175i	$0.519234\pi$
$618$	0	0
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	56.0000	2.24540
$623$	32.0000	1.28205
$624$	0	0
$625$	−19.0000	−0.760000
$626$	32.0000	1.27898
$627$	0	0
$628$	14.0000	0.558661
$629$	8.00000	0.318981
$630$	0	0
$631$	13.0000	0.517522	0.258761	−	0.965941i	$-0.416686\pi$
$631$	13.0000	0.517522	0.258761	+	0.965941i	$0.416686\pi$
$632$	0	0
$633$	0	0
$634$	34.0000	1.35031
$635$	−26.0000	−1.03178
$636$	0	0
$637$	63.0000	2.49615
$638$	54.0000	2.13788
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	−10.0000	−0.394362	−0.197181	−	0.980367i	$-0.563179\pi$
$643$	−10.0000	−0.394362	−0.197181	+	0.980367i	$0.563179\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	−8.00000	−0.314756
$647$	−38.0000	−1.49393	−0.746967	−	0.664861i	$-0.768491\pi$
$647$	−38.0000	−1.49393	−0.746967	+	0.664861i	$0.768491\pi$
$648$	0	0
$649$	0	0
$650$	14.0000	0.549125
$651$	0	0
$652$	−12.0000	−0.469956
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	−36.0000	−1.40556
$657$	0	0
$658$	0	0
$659$	26.0000	1.01282	0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000	1.01282	0.506408	+	0.862294i	$0.330973\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	22.0000	0.855054
$663$	0	0
$664$	0	0
$665$	−32.0000	−1.24091
$666$	0	0
$667$	−9.00000	−0.348481
$668$	2.00000	0.0773823
$669$	0	0
$670$	−28.0000	−1.08173
$671$	−6.00000	−0.231627
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−8.00000	−0.308148
$675$	0	0
$676$	72.0000	2.76923
$677$	−3.00000	−0.115299	−0.0576497	−	0.998337i	$-0.518361\pi$
$677$	−3.00000	−0.115299	−0.0576497	+	0.998337i	$0.518361\pi$
$678$	0	0
$679$	−40.0000	−1.53506
$680$	0	0
$681$	0	0
$682$	−12.0000	−0.459504
$683$	31.0000	1.18618	0.593091	−	0.805135i	$-0.297907\pi$
$683$	31.0000	1.18618	0.593091	+	0.805135i	$0.297907\pi$
$684$	0	0
$685$	24.0000	0.916993
$686$	−16.0000	−0.610883
$687$	0	0
$688$	−28.0000	−1.06749
$689$	−42.0000	−1.60007
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	−26.0000	−0.988372
$693$	0	0
$694$	−2.00000	−0.0759190
$695$	−4.00000	−0.151729
$696$	0	0
$697$	−9.00000	−0.340899
$698$	−60.0000	−2.27103
$699$	0	0
$700$	−8.00000	−0.302372
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	24.0000	0.904534
$705$	0	0
$706$	48.0000	1.80650
$707$	−32.0000	−1.20348
$708$	0	0
$709$	44.0000	1.65245	0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000	1.65245	0.826227	+	0.563337i	$0.190483\pi$
$710$	−28.0000	−1.05082
$711$	0	0
$712$	0	0
$713$	2.00000	0.0749006
$714$	0	0
$715$	−42.0000	−1.57071
$716$	48.0000	1.79384
$717$	0	0
$718$	12.0000	0.447836
$719$	44.0000	1.64092	0.820462	−	0.571702i	$-0.193717\pi$
$719$	44.0000	1.64092	0.820462	+	0.571702i	$0.193717\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	6.00000	0.223297
$723$	0	0
$724$	8.00000	0.297318
$725$	−9.00000	−0.334252
$726$	0	0
$727$	−23.0000	−0.853023	−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000	−0.853023	−0.426511	+	0.904482i	$0.640258\pi$
$728$	0	0
$729$	0	0
$730$	−24.0000	−0.888280
$731$	−7.00000	−0.258904
$732$	0	0
$733$	7.00000	0.258551	0.129275	−	0.991609i	$-0.458735\pi$
$733$	7.00000	0.258551	0.129275	+	0.991609i	$0.458735\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−8.00000	−0.294884
$737$	−21.0000	−0.773545
$738$	0	0
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	−32.0000	−1.17634
$741$	0	0
$742$	48.0000	1.76214
$743$	−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
$745$	20.0000	0.732743
$746$	6.00000	0.219676
$747$	0	0
$748$	6.00000	0.219382
$749$	28.0000	1.02310
$750$	0	0
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	0	0
$753$	0	0
$754$	−126.000	−4.58865
$755$	−26.0000	−0.946237
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	0	0
$764$	8.00000	0.289430
$765$	0	0
$766$	−36.0000	−1.30073
$767$	0	0
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	48.0000	1.72980
$771$	0	0
$772$	40.0000	1.43963
$773$	−52.0000	−1.87031	−0.935155	−	0.354239i	$-0.884740\pi$
$773$	−52.0000	−1.87031	−0.935155	+	0.354239i	$0.884740\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	0	0
$777$	0	0
$778$	12.0000	0.430221
$779$	−36.0000	−1.28983
$780$	0	0
$781$	−21.0000	−0.751439
$782$	−2.00000	−0.0715199
$783$	0	0
$784$	−36.0000	−1.28571
$785$	14.0000	0.499681
$786$	0	0
$787$	46.0000	1.63972	0.819861	−	0.572562i	$-0.194050\pi$
$787$	46.0000	1.63972	0.819861	+	0.572562i	$0.194050\pi$
$788$	30.0000	1.06871
$789$	0	0
$790$	48.0000	1.70776
$791$	−52.0000	−1.84891
$792$	0	0
$793$	14.0000	0.497155
$794$	−12.0000	−0.425864
$795$	0	0
$796$	−4.00000	−0.141776
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	0	0
$799$	0	0
$800$	−8.00000	−0.282843
$801$	0	0
$802$	−4.00000	−0.141245
$803$	−18.0000	−0.635206
$804$	0	0
$805$	−8.00000	−0.281963
$806$	28.0000	0.986258
$807$	0	0
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	72.0000	2.52670
$813$	0	0
$814$	−48.0000	−1.68240
$815$	−12.0000	−0.420342
$816$	0	0
$817$	−28.0000	−0.979596
$818$	14.0000	0.489499
$819$	0	0
$820$	36.0000	1.25717
$821$	−21.0000	−0.732905	−0.366453	−	0.930437i	$-0.619428\pi$
$821$	−21.0000	−0.732905	−0.366453	+	0.930437i	$0.619428\pi$
$822$	0	0
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−24.0000	−0.834562	−0.417281	−	0.908778i	$-0.637017\pi$
$827$	−24.0000	−0.834562	−0.417281	+	0.908778i	$0.637017\pi$
$828$	0	0
$829$	53.0000	1.84077	0.920383	−	0.391018i	$-0.127877\pi$
$829$	53.0000	1.84077	0.920383	+	0.391018i	$0.127877\pi$
$830$	56.0000	1.94379
$831$	0	0
$832$	−56.0000	−1.94145
$833$	−9.00000	−0.311832
$834$	0	0
$835$	2.00000	0.0692129
$836$	24.0000	0.830057
$837$	0	0
$838$	−50.0000	−1.72722
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	34.0000	1.17172
$843$	0	0
$844$	−44.0000	−1.51454
$845$	72.0000	2.47688
$846$	0	0
$847$	−8.00000	−0.274883
$848$	24.0000	0.824163
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	8.00000	0.274236
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	−16.0000	−0.547509
$855$	0	0
$856$	0	0
$857$	14.0000	0.478231	0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000	0.478231	0.239115	+	0.970991i	$0.423143\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	28.0000	0.954792
$861$	0	0
$862$	54.0000	1.83925
$863$	−10.0000	−0.340404	−0.170202	−	0.985409i	$-0.554442\pi$
$863$	−10.0000	−0.340404	−0.170202	+	0.985409i	$0.554442\pi$
$864$	0	0
$865$	−26.0000	−0.884027
$866$	68.0000	2.31073
$867$	0	0
$868$	−16.0000	−0.543075
$869$	36.0000	1.22122
$870$	0	0
$871$	49.0000	1.66030
$872$	0	0
$873$	0	0
$874$	−8.00000	−0.270604
$875$	−48.0000	−1.62270
$876$	0	0
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	−32.0000	−1.07995
$879$	0	0
$880$	24.0000	0.809040
$881$	1.00000	0.0336909	0.0168454	−	0.999858i	$-0.494638\pi$
$881$	1.00000	0.0336909	0.0168454	+	0.999858i	$0.494638\pi$
$882$	0	0
$883$	−39.0000	−1.31245	−0.656227	−	0.754563i	$-0.727849\pi$
$883$	−39.0000	−1.31245	−0.656227	+	0.754563i	$0.727849\pi$
$884$	−14.0000	−0.470871
$885$	0	0
$886$	80.0000	2.68765
$887$	−45.0000	−1.51095	−0.755476	−	0.655176i	$-0.772594\pi$
$887$	−45.0000	−1.51095	−0.755476	+	0.655176i	$0.772594\pi$
$888$	0	0
$889$	−52.0000	−1.74402
$890$	−32.0000	−1.07264
$891$	0	0
$892$	−24.0000	−0.803579
$893$	0	0
$894$	0	0
$895$	48.0000	1.60446
$896$	0	0
$897$	0	0
$898$	−52.0000	−1.73526
$899$	−18.0000	−0.600334
$900$	0	0
$901$	6.00000	0.199889
$902$	54.0000	1.79800
$903$	0	0
$904$	0	0
$905$	8.00000	0.265929
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−24.0000	−0.796468
$909$	0	0
$910$	−112.000	−3.71276
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	42.0000	1.39000
$914$	−66.0000	−2.18309
$915$	0	0
$916$	42.0000	1.38772
$917$	−16.0000	−0.528367
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	20.0000	0.658665
$923$	49.0000	1.61285
$924$	0	0
$925$	8.00000	0.263038
$926$	32.0000	1.05159
$927$	0	0
$928$	72.0000	2.36352
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−24.0000	−0.785304
$935$	6.00000	0.196221
$936$	0	0
$937$	25.0000	0.816714	0.408357	−	0.912822i	$-0.366102\pi$
$937$	25.0000	0.816714	0.408357	+	0.912822i	$0.366102\pi$
$938$	−56.0000	−1.82846
$939$	0	0
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	−9.00000	−0.293080
$944$	0	0
$945$	0	0
$946$	42.0000	1.36554
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	0	0
$949$	42.0000	1.36338
$950$	−8.00000	−0.259554
$951$	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
$955$	8.00000	0.258874
$956$	−52.0000	−1.68180
$957$	0	0
$958$	−48.0000	−1.55081
$959$	48.0000	1.55000
$960$	0	0
$961$	−27.0000	−0.870968
$962$	112.000	3.61102
$963$	0	0
$964$	−48.0000	−1.54598
$965$	40.0000	1.28765
$966$	0	0
$967$	−36.0000	−1.15768	−0.578841	−	0.815440i	$-0.696495\pi$
$967$	−36.0000	−1.15768	−0.578841	+	0.815440i	$0.696495\pi$
$968$	0	0
$969$	0	0
$970$	40.0000	1.28432
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	−4.00000	−0.128168
$975$	0	0
$976$	−8.00000	−0.256074
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	36.0000	1.14998
$981$	0	0
$982$	−68.0000	−2.16997
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	30.0000	0.955879
$986$	18.0000	0.573237
$987$	0	0
$988$	−56.0000	−1.78160
$989$	−7.00000	−0.222587
$990$	0	0
$991$	30.0000	0.952981	0.476491	−	0.879180i	$-0.341909\pi$
$991$	30.0000	0.952981	0.476491	+	0.879180i	$0.341909\pi$
$992$	−16.0000	−0.508001
$993$	0	0
$994$	−56.0000	−1.77621
$995$	−4.00000	−0.126809
$996$	0	0
$997$	4.00000	0.126681	0.0633406	−	0.997992i	$-0.479825\pi$
$997$	4.00000	0.126681	0.0633406	+	0.997992i	$0.479825\pi$
$998$	44.0000	1.39280
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	459.2.a.b.1.1	✓	1
3.2	odd	2		459.2.a.g.1.1	yes	1
4.3	odd	2		7344.2.a.bc.1.1		1
12.11	even	2		7344.2.a.h.1.1		1
17.16	even	2		7803.2.a.a.1.1		1
51.50	odd	2		7803.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.2.a.b.1.1	✓	1	1.1	even	1	trivial
459.2.a.g.1.1	yes	1	3.2	odd	2
7344.2.a.h.1.1		1	12.11	even	2
7344.2.a.bc.1.1		1	4.3	odd	2
7803.2.a.a.1.1		1	17.16	even	2
7803.2.a.t.1.1		1	51.50	odd	2

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
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Higher genus families
Abelian varieties over $\F_{q}$

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Newspace parameters

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	459.2.a.b.1.1

Level	$459$
Weight	$2$
Character	459.1
Self dual	yes
Analytic conductor	$3.665$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$