LMFDB - Embedded newform 459.2.a.a.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	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} -4.00000 q^{5} +1.00000 q^{7} +8.00000 q^{10} +6.00000 q^{11} +1.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} -1.00000 q^{17} -7.00000 q^{19} -8.00000 q^{20} -12.0000 q^{22} -4.00000 q^{23} +11.0000 q^{25} -2.00000 q^{26} +2.00000 q^{28} -6.00000 q^{29} -8.00000 q^{31} +8.00000 q^{32} +2.00000 q^{34} -4.00000 q^{35} +1.00000 q^{37} +14.0000 q^{38} +4.00000 q^{43} +12.0000 q^{44} +8.00000 q^{46} -6.00000 q^{47} -6.00000 q^{49} -22.0000 q^{50} +2.00000 q^{52} -24.0000 q^{55} +12.0000 q^{58} -6.00000 q^{59} -7.00000 q^{61} +16.0000 q^{62} -8.00000 q^{64} -4.00000 q^{65} +1.00000 q^{67} -2.00000 q^{68} +8.00000 q^{70} +4.00000 q^{71} +3.00000 q^{73} -2.00000 q^{74} -14.0000 q^{76} +6.00000 q^{77} -9.00000 q^{79} +16.0000 q^{80} -14.0000 q^{83} +4.00000 q^{85} -8.00000 q^{86} +14.0000 q^{89} +1.00000 q^{91} -8.00000 q^{92} +12.0000 q^{94} +28.0000 q^{95} -1.00000 q^{97} +12.0000 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
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	0	0
$10$	8.00000	2.52982
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	−8.00000	−1.78885
$21$	0	0
$22$	−12.0000	−2.55841
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	2.00000	0.377964
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	2.00000	0.342997
$35$	−4.00000	−0.676123
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	14.0000	2.27110
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	12.0000	1.80907
$45$	0	0
$46$	8.00000	1.17954
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	−22.0000	−3.11127
$51$	0	0
$52$	2.00000	0.277350
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−24.0000	−3.23616
$56$	0	0
$57$	0	0
$58$	12.0000	1.57568
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	16.0000	2.03200
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	1.00000	0.122169	0.0610847	−	0.998133i	$-0.480544\pi$
$67$	1.00000	0.122169	0.0610847	+	0.998133i	$0.480544\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	8.00000	0.956183
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	3.00000	0.351123	0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000	0.351123	0.175562	+	0.984468i	$0.443826\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−14.0000	−1.60591
$77$	6.00000	0.683763
$78$	0	0
$79$	−9.00000	−1.01258	−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000	−1.01258	−0.506290	+	0.862364i	$0.668983\pi$
$80$	16.0000	1.78885
$81$	0	0
$82$	0	0
$83$	−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
$85$	4.00000	0.433861
$86$	−8.00000	−0.862662
$87$	0	0
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−8.00000	−0.834058
$93$	0	0
$94$	12.0000	1.23771
$95$	28.0000	2.87274
$96$	0	0
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	12.0000	1.21218
$99$	0	0
$100$	22.0000	2.20000
$101$	−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
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−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
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	48.0000	4.57662
$111$	0	0
$112$	−4.00000	−0.377964
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	16.0000	1.49201
$116$	−12.0000	−1.11417
$117$	0	0
$118$	12.0000	1.10469
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	25.0000	2.27273
$122$	14.0000	1.26750
$123$	0	0
$124$	−16.0000	−1.43684
$125$	−24.0000	−2.14663
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	0	0
$130$	8.00000	0.701646
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	−7.00000	−0.606977
$134$	−2.00000	−0.172774
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	1.00000	0.0848189	0.0424094	−	0.999100i	$-0.486497\pi$
$139$	1.00000	0.0848189	0.0424094	+	0.999100i	$0.486497\pi$
$140$	−8.00000	−0.676123
$141$	0	0
$142$	−8.00000	−0.671345
$143$	6.00000	0.501745
$144$	0	0
$145$	24.0000	1.99309
$146$	−6.00000	−0.496564
$147$	0	0
$148$	2.00000	0.164399
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	0	0
$153$	0	0
$154$	−12.0000	−0.966988
$155$	32.0000	2.57030
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	18.0000	1.43200
$159$	0	0
$160$	−32.0000	−2.52982
$161$	−4.00000	−0.315244
$162$	0	0
$163$	21.0000	1.64485	0.822423	−	0.568876i	$-0.192621\pi$
$163$	21.0000	1.64485	0.822423	+	0.568876i	$0.192621\pi$
$164$	0	0
$165$	0	0
$166$	28.0000	2.17322
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−8.00000	−0.613572
$171$	0	0
$172$	8.00000	0.609994
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	11.0000	0.831522
$176$	−24.0000	−1.80907
$177$	0	0
$178$	−28.0000	−2.09869
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−6.00000	−0.438763
$188$	−12.0000	−0.875190
$189$	0	0
$190$	−56.0000	−4.06267
$191$	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
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−12.0000	−0.857143
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	0	0
$202$	4.00000	0.281439
$203$	−6.00000	−0.421117
$204$	0	0
$205$	0	0
$206$	2.00000	0.139347
$207$	0	0
$208$	−4.00000	−0.277350
$209$	−42.0000	−2.90520
$210$	0	0
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	0	0
$213$	0	0
$214$	4.00000	0.273434
$215$	−16.0000	−1.09119
$216$	0	0
$217$	−8.00000	−0.543075
$218$	36.0000	2.43823
$219$	0	0
$220$	−48.0000	−3.23616
$221$	−1.00000	−0.0672673
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	8.00000	0.534522
$225$	0	0
$226$	−4.00000	−0.266076
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	−32.0000	−2.11002
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	−12.0000	−0.781133
$237$	0	0
$238$	2.00000	0.129641
$239$	10.0000	0.646846	0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000	0.646846	0.323423	+	0.946254i	$0.395166\pi$
$240$	0	0
$241$	15.0000	0.966235	0.483117	−	0.875556i	$-0.339504\pi$
$241$	15.0000	0.966235	0.483117	+	0.875556i	$0.339504\pi$
$242$	−50.0000	−3.21412
$243$	0	0
$244$	−14.0000	−0.896258
$245$	24.0000	1.53330
$246$	0	0
$247$	−7.00000	−0.445399
$248$	0	0
$249$	0	0
$250$	48.0000	3.03579
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−24.0000	−1.50887
$254$	−16.0000	−1.00393
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	−8.00000	−0.496139
$261$	0	0
$262$	20.0000	1.23560
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	0	0
$266$	14.0000	0.858395
$267$	0	0
$268$	2.00000	0.122169
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−7.00000	−0.425220	−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000	−0.425220	−0.212610	+	0.977137i	$0.568196\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	24.0000	1.44989
$275$	66.0000	3.97995
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−2.00000	−0.119952
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−12.0000	−0.709575
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	−48.0000	−2.81866
$291$	0	0
$292$	6.00000	0.351123
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	0	0
$297$	0	0
$298$	28.0000	1.62200
$299$	−4.00000	−0.231326
$300$	0	0
$301$	4.00000	0.230556
$302$	−34.0000	−1.95648
$303$	0	0
$304$	28.0000	1.60591
$305$	28.0000	1.60328
$306$	0	0
$307$	24.0000	1.36975	0.684876	−	0.728659i	$-0.259856\pi$
$307$	24.0000	1.36975	0.684876	+	0.728659i	$0.259856\pi$
$308$	12.0000	0.683763
$309$	0	0
$310$	−64.0000	−3.63496
$311$	−22.0000	−1.24751	−0.623753	−	0.781622i	$-0.714393\pi$
$311$	−22.0000	−1.24751	−0.623753	+	0.781622i	$0.714393\pi$
$312$	0	0
$313$	29.0000	1.63918	0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000	1.63918	0.819588	+	0.572953i	$0.194202\pi$
$314$	−20.0000	−1.12867
$315$	0	0
$316$	−18.0000	−1.01258
$317$	−8.00000	−0.449325	−0.224662	−	0.974437i	$-0.572128\pi$
$317$	−8.00000	−0.449325	−0.224662	+	0.974437i	$0.572128\pi$
$318$	0	0
$319$	−36.0000	−2.01561
$320$	32.0000	1.78885
$321$	0	0
$322$	8.00000	0.445823
$323$	7.00000	0.389490
$324$	0	0
$325$	11.0000	0.610170
$326$	−42.0000	−2.32616
$327$	0	0
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	7.00000	0.384755	0.192377	−	0.981321i	$-0.438380\pi$
$331$	7.00000	0.384755	0.192377	+	0.981321i	$0.438380\pi$
$332$	−28.0000	−1.53670
$333$	0	0
$334$	−32.0000	−1.75096
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	24.0000	1.30543
$339$	0	0
$340$	8.00000	0.433861
$341$	−48.0000	−2.59935
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	44.0000	2.36545
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	0	0
$349$	−33.0000	−1.76645	−0.883225	−	0.468950i	$-0.844632\pi$
$349$	−33.0000	−1.76645	−0.883225	+	0.468950i	$0.844632\pi$
$350$	−22.0000	−1.17595
$351$	0	0
$352$	48.0000	2.55841
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−16.0000	−0.849192
$356$	28.0000	1.48400
$357$	0	0
$358$	24.0000	1.26844
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	10.0000	0.525588
$363$	0	0
$364$	2.00000	0.104828
$365$	−12.0000	−0.628109
$366$	0	0
$367$	1.00000	0.0521996	0.0260998	−	0.999659i	$-0.491691\pi$
$367$	1.00000	0.0521996	0.0260998	+	0.999659i	$0.491691\pi$
$368$	16.0000	0.834058
$369$	0	0
$370$	8.00000	0.415900
$371$	0	0
$372$	0	0
$373$	27.0000	1.39801	0.699004	−	0.715118i	$-0.253627\pi$
$373$	27.0000	1.39801	0.699004	+	0.715118i	$0.253627\pi$
$374$	12.0000	0.620505
$375$	0	0
$376$	0	0
$377$	−6.00000	−0.309016
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	56.0000	2.87274
$381$	0	0
$382$	−8.00000	−0.409316
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	38.0000	1.93415
$387$	0	0
$388$	−2.00000	−0.101535
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	0	0
$394$	−36.0000	−1.81365
$395$	36.0000	1.81136
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	22.0000	1.10276
$399$	0	0
$400$	−44.0000	−2.20000
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	−4.00000	−0.199007
$405$	0	0
$406$	12.0000	0.595550
$407$	6.00000	0.297409
$408$	0	0
$409$	17.0000	0.840596	0.420298	−	0.907386i	$-0.361926\pi$
$409$	17.0000	0.840596	0.420298	+	0.907386i	$0.361926\pi$
$410$	0	0
$411$	0	0
$412$	−2.00000	−0.0985329
$413$	−6.00000	−0.295241
$414$	0	0
$415$	56.0000	2.74893
$416$	8.00000	0.392232
$417$	0	0
$418$	84.0000	4.10857
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	−11.0000	−0.536107	−0.268054	−	0.963404i	$-0.586380\pi$
$421$	−11.0000	−0.536107	−0.268054	+	0.963404i	$0.586380\pi$
$422$	38.0000	1.84981
$423$	0	0
$424$	0	0
$425$	−11.0000	−0.533578
$426$	0	0
$427$	−7.00000	−0.338754
$428$	−4.00000	−0.193347
$429$	0	0
$430$	32.0000	1.54318
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	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
$436$	−36.0000	−1.72409
$437$	28.0000	1.33942
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	0	0
$442$	2.00000	0.0951303
$443$	26.0000	1.23530	0.617649	−	0.786454i	$-0.288085\pi$
$443$	26.0000	1.23530	0.617649	+	0.786454i	$0.288085\pi$
$444$	0	0
$445$	−56.0000	−2.65465
$446$	−48.0000	−2.27287
$447$	0	0
$448$	−8.00000	−0.377964
$449$	38.0000	1.79333	0.896665	−	0.442709i	$-0.145982\pi$
$449$	38.0000	1.79333	0.896665	+	0.442709i	$0.145982\pi$
$450$	0	0
$451$	0	0
$452$	4.00000	0.188144
$453$	0	0
$454$	−12.0000	−0.563188
$455$	−4.00000	−0.187523
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−12.0000	−0.560723
$459$	0	0
$460$	32.0000	1.49201
$461$	−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
$463$	11.0000	0.511213	0.255607	−	0.966781i	$-0.417725\pi$
$463$	11.0000	0.511213	0.255607	+	0.966781i	$0.417725\pi$
$464$	24.0000	1.11417
$465$	0	0
$466$	0	0
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	1.00000	0.0461757
$470$	−48.0000	−2.21407
$471$	0	0
$472$	0	0
$473$	24.0000	1.10352
$474$	0	0
$475$	−77.0000	−3.53300
$476$	−2.00000	−0.0916698
$477$	0	0
$478$	−20.0000	−0.914779
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	1.00000	0.0455961
$482$	−30.0000	−1.36646
$483$	0	0
$484$	50.0000	2.27273
$485$	4.00000	0.181631
$486$	0	0
$487$	−19.0000	−0.860972	−0.430486	−	0.902597i	$-0.641658\pi$
$487$	−19.0000	−0.860972	−0.430486	+	0.902597i	$0.641658\pi$
$488$	0	0
$489$	0	0
$490$	−48.0000	−2.16842
$491$	10.0000	0.451294	0.225647	−	0.974209i	$-0.427550\pi$
$491$	10.0000	0.451294	0.225647	+	0.974209i	$0.427550\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	14.0000	0.629890
$495$	0	0
$496$	32.0000	1.43684
$497$	4.00000	0.179425
$498$	0	0
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	−48.0000	−2.14663
$501$	0	0
$502$	0	0
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	48.0000	2.13386
$507$	0	0
$508$	16.0000	0.709885
$509$	−4.00000	−0.177297	−0.0886484	−	0.996063i	$-0.528255\pi$
$509$	−4.00000	−0.177297	−0.0886484	+	0.996063i	$0.528255\pi$
$510$	0	0
$511$	3.00000	0.132712
$512$	−32.0000	−1.41421
$513$	0	0
$514$	0	0
$515$	4.00000	0.176261
$516$	0	0
$517$	−36.0000	−1.58328
$518$	−2.00000	−0.0878750
$519$	0	0
$520$	0	0
$521$	−32.0000	−1.40195	−0.700973	−	0.713188i	$-0.747251\pi$
$521$	−32.0000	−1.40195	−0.700973	+	0.713188i	$0.747251\pi$
$522$	0	0
$523$	9.00000	0.393543	0.196771	−	0.980449i	$-0.436954\pi$
$523$	9.00000	0.393543	0.196771	+	0.980449i	$0.436954\pi$
$524$	−20.0000	−0.873704
$525$	0	0
$526$	−8.00000	−0.348817
$527$	8.00000	0.348485
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	−14.0000	−0.606977
$533$	0	0
$534$	0	0
$535$	8.00000	0.345870
$536$	0	0
$537$	0	0
$538$	20.0000	0.862261
$539$	−36.0000	−1.55063
$540$	0	0
$541$	−9.00000	−0.386940	−0.193470	−	0.981106i	$-0.561974\pi$
$541$	−9.00000	−0.386940	−0.193470	+	0.981106i	$0.561974\pi$
$542$	14.0000	0.601351
$543$	0	0
$544$	−8.00000	−0.342997
$545$	72.0000	3.08414
$546$	0	0
$547$	−35.0000	−1.49649	−0.748246	−	0.663421i	$-0.769104\pi$
$547$	−35.0000	−1.49649	−0.748246	+	0.663421i	$0.769104\pi$
$548$	−24.0000	−1.02523
$549$	0	0
$550$	−132.000	−5.62850
$551$	42.0000	1.78926
$552$	0	0
$553$	−9.00000	−0.382719
$554$	44.0000	1.86938
$555$	0	0
$556$	2.00000	0.0848189
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	16.0000	0.676123
$561$	0	0
$562$	−36.0000	−1.51857
$563$	16.0000	0.674320	0.337160	−	0.941447i	$-0.390534\pi$
$563$	16.0000	0.674320	0.337160	+	0.941447i	$0.390534\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	8.00000	0.336265
$567$	0	0
$568$	0	0
$569$	8.00000	0.335377	0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000	0.335377	0.167689	+	0.985840i	$0.446370\pi$
$570$	0	0
$571$	−5.00000	−0.209243	−0.104622	−	0.994512i	$-0.533363\pi$
$571$	−5.00000	−0.209243	−0.104622	+	0.994512i	$0.533363\pi$
$572$	12.0000	0.501745
$573$	0	0
$574$	0	0
$575$	−44.0000	−1.83493
$576$	0	0
$577$	−5.00000	−0.208153	−0.104076	−	0.994569i	$-0.533189\pi$
$577$	−5.00000	−0.208153	−0.104076	+	0.994569i	$0.533189\pi$
$578$	−2.00000	−0.0831890
$579$	0	0
$580$	48.0000	1.99309
$581$	−14.0000	−0.580818
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−48.0000	−1.98286
$587$	−10.0000	−0.412744	−0.206372	−	0.978474i	$-0.566166\pi$
$587$	−10.0000	−0.412744	−0.206372	+	0.978474i	$0.566166\pi$
$588$	0	0
$589$	56.0000	2.30744
$590$	−48.0000	−1.97613
$591$	0	0
$592$	−4.00000	−0.164399
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	−28.0000	−1.14692
$597$	0	0
$598$	8.00000	0.327144
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	−8.00000	−0.326056
$603$	0	0
$604$	34.0000	1.38344
$605$	−100.000	−4.06558
$606$	0	0
$607$	17.0000	0.690009	0.345004	−	0.938601i	$-0.387877\pi$
$607$	17.0000	0.690009	0.345004	+	0.938601i	$0.387877\pi$
$608$	−56.0000	−2.27110
$609$	0	0
$610$	−56.0000	−2.26737
$611$	−6.00000	−0.242734
$612$	0	0
$613$	−5.00000	−0.201948	−0.100974	−	0.994889i	$-0.532196\pi$
$613$	−5.00000	−0.201948	−0.100974	+	0.994889i	$0.532196\pi$
$614$	−48.0000	−1.93712
$615$	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
$619$	−21.0000	−0.844061	−0.422031	−	0.906582i	$-0.638683\pi$
$619$	−21.0000	−0.844061	−0.422031	+	0.906582i	$0.638683\pi$
$620$	64.0000	2.57030
$621$	0	0
$622$	44.0000	1.76424
$623$	14.0000	0.560898
$624$	0	0
$625$	41.0000	1.64000
$626$	−58.0000	−2.31815
$627$	0	0
$628$	20.0000	0.798087
$629$	−1.00000	−0.0398726
$630$	0	0
$631$	1.00000	0.0398094	0.0199047	−	0.999802i	$-0.493664\pi$
$631$	1.00000	0.0398094	0.0199047	+	0.999802i	$0.493664\pi$
$632$	0	0
$633$	0	0
$634$	16.0000	0.635441
$635$	−32.0000	−1.26988
$636$	0	0
$637$	−6.00000	−0.237729
$638$	72.0000	2.85051
$639$	0	0
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	−14.0000	−0.550823
$647$	40.0000	1.57256	0.786281	−	0.617869i	$-0.212004\pi$
$647$	40.0000	1.57256	0.786281	+	0.617869i	$0.212004\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	−22.0000	−0.862911
$651$	0	0
$652$	42.0000	1.64485
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	0	0
$655$	40.0000	1.56293
$656$	0	0
$657$	0	0
$658$	12.0000	0.467809
$659$	38.0000	1.48027	0.740135	−	0.672458i	$-0.234762\pi$
$659$	38.0000	1.48027	0.740135	+	0.672458i	$0.234762\pi$
$660$	0	0
$661$	23.0000	0.894596	0.447298	−	0.894385i	$-0.352386\pi$
$661$	23.0000	0.894596	0.447298	+	0.894385i	$0.352386\pi$
$662$	−14.0000	−0.544125
$663$	0	0
$664$	0	0
$665$	28.0000	1.08579
$666$	0	0
$667$	24.0000	0.929284
$668$	32.0000	1.23812
$669$	0	0
$670$	8.00000	0.309067
$671$	−42.0000	−1.62139
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	−24.0000	−0.923077
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	0	0
$679$	−1.00000	−0.0383765
$680$	0	0
$681$	0	0
$682$	96.0000	3.67603
$683$	10.0000	0.382639	0.191320	−	0.981528i	$-0.438723\pi$
$683$	10.0000	0.382639	0.191320	+	0.981528i	$0.438723\pi$
$684$	0	0
$685$	48.0000	1.83399
$686$	26.0000	0.992685
$687$	0	0
$688$	−16.0000	−0.609994
$689$	0	0
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	−44.0000	−1.67263
$693$	0	0
$694$	40.0000	1.51838
$695$	−4.00000	−0.151729
$696$	0	0
$697$	0	0
$698$	66.0000	2.49814
$699$	0	0
$700$	22.0000	0.831522
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	−7.00000	−0.264010
$704$	−48.0000	−1.80907
$705$	0	0
$706$	−12.0000	−0.451626
$707$	−2.00000	−0.0752177
$708$	0	0
$709$	−13.0000	−0.488225	−0.244113	−	0.969747i	$-0.578497\pi$
$709$	−13.0000	−0.488225	−0.244113	+	0.969747i	$0.578497\pi$
$710$	32.0000	1.20094
$711$	0	0
$712$	0	0
$713$	32.0000	1.19841
$714$	0	0
$715$	−24.0000	−0.897549
$716$	−24.0000	−0.896922
$717$	0	0
$718$	−36.0000	−1.34351
$719$	38.0000	1.41716	0.708580	−	0.705630i	$-0.249336\pi$
$719$	38.0000	1.41716	0.708580	+	0.705630i	$0.249336\pi$
$720$	0	0
$721$	−1.00000	−0.0372419
$722$	−60.0000	−2.23297
$723$	0	0
$724$	−10.0000	−0.371647
$725$	−66.0000	−2.45118
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	0	0
$730$	24.0000	0.888280
$731$	−4.00000	−0.147945
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	−2.00000	−0.0738213
$735$	0	0
$736$	−32.0000	−1.17954
$737$	6.00000	0.221013
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	−8.00000	−0.294086
$741$	0	0
$742$	0	0
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	0	0
$745$	56.0000	2.05168
$746$	−54.0000	−1.97708
$747$	0	0
$748$	−12.0000	−0.438763
$749$	−2.00000	−0.0730784
$750$	0	0
$751$	11.0000	0.401396	0.200698	−	0.979653i	$-0.435679\pi$
$751$	11.0000	0.401396	0.200698	+	0.979653i	$0.435679\pi$
$752$	24.0000	0.875190
$753$	0	0
$754$	12.0000	0.437014
$755$	−68.0000	−2.47477
$756$	0	0
$757$	37.0000	1.34479	0.672394	−	0.740193i	$-0.265266\pi$
$757$	37.0000	1.34479	0.672394	+	0.740193i	$0.265266\pi$
$758$	58.0000	2.10665
$759$	0	0
$760$	0	0
$761$	40.0000	1.45000	0.724999	−	0.688749i	$-0.241840\pi$
$761$	40.0000	1.45000	0.724999	+	0.688749i	$0.241840\pi$
$762$	0	0
$763$	−18.0000	−0.651644
$764$	8.00000	0.289430
$765$	0	0
$766$	12.0000	0.433578
$767$	−6.00000	−0.216647
$768$	0	0
$769$	−41.0000	−1.47850	−0.739249	−	0.673432i	$-0.764819\pi$
$769$	−41.0000	−1.47850	−0.739249	+	0.673432i	$0.764819\pi$
$770$	48.0000	1.72980
$771$	0	0
$772$	−38.0000	−1.36765
$773$	32.0000	1.15096	0.575480	−	0.817816i	$-0.304815\pi$
$773$	32.0000	1.15096	0.575480	+	0.817816i	$0.304815\pi$
$774$	0	0
$775$	−88.0000	−3.16105
$776$	0	0
$777$	0	0
$778$	36.0000	1.29066
$779$	0	0
$780$	0	0
$781$	24.0000	0.858788
$782$	−8.00000	−0.286079
$783$	0	0
$784$	24.0000	0.857143
$785$	−40.0000	−1.42766
$786$	0	0
$787$	−17.0000	−0.605985	−0.302992	−	0.952993i	$-0.597986\pi$
$787$	−17.0000	−0.605985	−0.302992	+	0.952993i	$0.597986\pi$
$788$	36.0000	1.28245
$789$	0	0
$790$	−72.0000	−2.56165
$791$	2.00000	0.0711118
$792$	0	0
$793$	−7.00000	−0.248577
$794$	36.0000	1.27759
$795$	0	0
$796$	−22.0000	−0.779769
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	6.00000	0.212265
$800$	88.0000	3.11127
$801$	0	0
$802$	32.0000	1.12996
$803$	18.0000	0.635206
$804$	0	0
$805$	16.0000	0.563926
$806$	16.0000	0.563576
$807$	0	0
$808$	0	0
$809$	−20.0000	−0.703163	−0.351581	−	0.936157i	$-0.614356\pi$
$809$	−20.0000	−0.703163	−0.351581	+	0.936157i	$0.614356\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	−12.0000	−0.421117
$813$	0	0
$814$	−12.0000	−0.420600
$815$	−84.0000	−2.94239
$816$	0	0
$817$	−28.0000	−0.979596
$818$	−34.0000	−1.18878
$819$	0	0
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	0	0
$823$	−37.0000	−1.28974	−0.644869	−	0.764293i	$-0.723088\pi$
$823$	−37.0000	−1.28974	−0.644869	+	0.764293i	$0.723088\pi$
$824$	0	0
$825$	0	0
$826$	12.0000	0.417533
$827$	−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
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	−112.000	−3.88758
$831$	0	0
$832$	−8.00000	−0.277350
$833$	6.00000	0.207888
$834$	0	0
$835$	−64.0000	−2.21481
$836$	−84.0000	−2.90520
$837$	0	0
$838$	−56.0000	−1.93449
$839$	−8.00000	−0.276191	−0.138095	−	0.990419i	$-0.544098\pi$
$839$	−8.00000	−0.276191	−0.138095	+	0.990419i	$0.544098\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	22.0000	0.758170
$843$	0	0
$844$	−38.0000	−1.30801
$845$	48.0000	1.65125
$846$	0	0
$847$	25.0000	0.859010
$848$	0	0
$849$	0	0
$850$	22.0000	0.754594
$851$	−4.00000	−0.137118
$852$	0	0
$853$	29.0000	0.992941	0.496471	−	0.868054i	$-0.334629\pi$
$853$	29.0000	0.992941	0.496471	+	0.868054i	$0.334629\pi$
$854$	14.0000	0.479070
$855$	0	0
$856$	0	0
$857$	32.0000	1.09310	0.546550	−	0.837427i	$-0.315941\pi$
$857$	32.0000	1.09310	0.546550	+	0.837427i	$0.315941\pi$
$858$	0	0
$859$	7.00000	0.238837	0.119418	−	0.992844i	$-0.461897\pi$
$859$	7.00000	0.238837	0.119418	+	0.992844i	$0.461897\pi$
$860$	−32.0000	−1.09119
$861$	0	0
$862$	−36.0000	−1.22616
$863$	−40.0000	−1.36162	−0.680808	−	0.732462i	$-0.738371\pi$
$863$	−40.0000	−1.36162	−0.680808	+	0.732462i	$0.738371\pi$
$864$	0	0
$865$	88.0000	2.99209
$866$	68.0000	2.31073
$867$	0	0
$868$	−16.0000	−0.543075
$869$	−54.0000	−1.83182
$870$	0	0
$871$	1.00000	0.0338837
$872$	0	0
$873$	0	0
$874$	−56.0000	−1.89423
$875$	−24.0000	−0.811348
$876$	0	0
$877$	−27.0000	−0.911725	−0.455863	−	0.890050i	$-0.650669\pi$
$877$	−27.0000	−0.911725	−0.455863	+	0.890050i	$0.650669\pi$
$878$	−8.00000	−0.269987
$879$	0	0
$880$	96.0000	3.23616
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	0	0
$883$	15.0000	0.504790	0.252395	−	0.967624i	$-0.418782\pi$
$883$	15.0000	0.504790	0.252395	+	0.967624i	$0.418782\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	−52.0000	−1.74697
$887$	54.0000	1.81314	0.906571	−	0.422053i	$-0.138690\pi$
$887$	54.0000	1.81314	0.906571	+	0.422053i	$0.138690\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	112.000	3.75425
$891$	0	0
$892$	48.0000	1.60716
$893$	42.0000	1.40548
$894$	0	0
$895$	48.0000	1.60446
$896$	0	0
$897$	0	0
$898$	−76.0000	−2.53615
$899$	48.0000	1.60089
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.0000	0.664822
$906$	0	0
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	8.00000	0.265197
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	−84.0000	−2.77999
$914$	−36.0000	−1.19077
$915$	0	0
$916$	12.0000	0.396491
$917$	−10.0000	−0.330229
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	20.0000	0.658665
$923$	4.00000	0.131662
$924$	0	0
$925$	11.0000	0.361678
$926$	−22.0000	−0.722965
$927$	0	0
$928$	−48.0000	−1.57568
$929$	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
$931$	42.0000	1.37649
$932$	0	0
$933$	0	0
$934$	48.0000	1.57061
$935$	24.0000	0.784884
$936$	0	0
$937$	−41.0000	−1.33941	−0.669706	−	0.742627i	$-0.733580\pi$
$937$	−41.0000	−1.33941	−0.669706	+	0.742627i	$0.733580\pi$
$938$	−2.00000	−0.0653023
$939$	0	0
$940$	48.0000	1.56559
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	0	0
$944$	24.0000	0.781133
$945$	0	0
$946$	−48.0000	−1.56061
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	0	0
$949$	3.00000	0.0973841
$950$	154.000	4.99642
$951$	0	0
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	20.0000	0.646846
$957$	0	0
$958$	−36.0000	−1.16311
$959$	−12.0000	−0.387500
$960$	0	0
$961$	33.0000	1.06452
$962$	−2.00000	−0.0644826
$963$	0	0
$964$	30.0000	0.966235
$965$	76.0000	2.44653
$966$	0	0
$967$	−51.0000	−1.64005	−0.820025	−	0.572328i	$-0.806040\pi$
$967$	−51.0000	−1.64005	−0.820025	+	0.572328i	$0.806040\pi$
$968$	0	0
$969$	0	0
$970$	−8.00000	−0.256865
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	1.00000	0.0320585
$974$	38.0000	1.21760
$975$	0	0
$976$	28.0000	0.896258
$977$	−36.0000	−1.15174	−0.575871	−	0.817541i	$-0.695337\pi$
$977$	−36.0000	−1.15174	−0.575871	+	0.817541i	$0.695337\pi$
$978$	0	0
$979$	84.0000	2.68465
$980$	48.0000	1.53330
$981$	0	0
$982$	−20.0000	−0.638226
$983$	−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
$985$	−72.0000	−2.29411
$986$	−12.0000	−0.382158
$987$	0	0
$988$	−14.0000	−0.445399
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−27.0000	−0.857683	−0.428842	−	0.903380i	$-0.641078\pi$
$991$	−27.0000	−0.857683	−0.428842	+	0.903380i	$0.641078\pi$
$992$	−64.0000	−2.03200
$993$	0	0
$994$	−8.00000	−0.253745
$995$	44.0000	1.39489
$996$	0	0
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	−88.0000	−2.78559
$999$	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.a.1.1	✓	1
3.2	odd	2		459.2.a.h.1.1	yes	1
4.3	odd	2		7344.2.a.a.1.1		1
12.11	even	2		7344.2.a.bl.1.1		1
17.16	even	2		7803.2.a.d.1.1		1
51.50	odd	2		7803.2.a.r.1.1		1

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

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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Newspace parameters

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Embedding invariants

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Coefficient data

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.a.1.1

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