LMFDB - Embedded newform 639.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [639,2,Mod(1,639)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(639, 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("639.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 639 = 3^{2} \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	639.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:	$5.10244068916$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 213)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	639.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{7} +3.00000 q^{8} +O(q^{10})$

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

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	2.00000	0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−2.00000	−0.534522
$15$	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		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	−2.00000	−0.377964
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	0	0
$40$	−6.00000	−0.948683
$41$	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	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	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
$55$	0	0
$56$	6.00000	0.801784
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	10.0000	1.27000
$63$	0	0
$64$	7.00000	0.875000
$65$	4.00000	0.496139
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	0	0
$70$	4.00000	0.478091
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	2.00000	0.223607
$81$	0	0
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	12.0000	1.23771
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−4.00000	−0.388514
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	−2.00000	−0.188982
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	0	0
$118$	12.0000	1.10469
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−10.0000	−0.905357
$123$	0	0
$124$	10.0000	0.898027
$125$	12.0000	1.07331
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	−4.00000	−0.350823
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	0	0
$134$	−2.00000	−0.172774
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−1.00000	−0.0839181
$143$	0	0
$144$	0	0
$145$	−4.00000	−0.332182
$146$	10.0000	0.827606
$147$	0	0
$148$	6.00000	0.493197
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	20.0000	1.60644
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	10.0000	0.790569
$161$	0	0
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	4.00000	0.304997
$173$	24.0000	1.82469	0.912343	−	0.409426i	$-0.134271\pi$
$173$	24.0000	1.82469	0.912343	+	0.409426i	$0.134271\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	−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
$181$	−26.0000	−1.93256	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000	−1.93256	−0.966282	+	0.257485i	$0.917106\pi$
$182$	4.00000	0.296500
$183$	0	0
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	0	0
$188$	12.0000	0.875190
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	3.00000	0.214286
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	10.0000	0.703598
$203$	4.00000	0.280745
$204$	0	0
$205$	0	0
$206$	−16.0000	−1.11477
$207$	0	0
$208$	2.00000	0.138675
$209$	0	0
$210$	0	0
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	12.0000	0.820303
$215$	8.00000	0.545595
$216$	0	0
$217$	−20.0000	−1.35769
$218$	−10.0000	−0.677285
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	−10.0000	−0.668153
$225$	0	0
$226$	8.00000	0.532152
$227$	−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
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	0	0
$232$	6.00000	0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	12.0000	0.781133
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	−10.0000	−0.640184
$245$	6.00000	0.383326
$246$	0	0
$247$	0	0
$248$	−30.0000	−1.90500
$249$	0	0
$250$	−12.0000	−0.758947
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	0	0
$254$	6.00000	0.376473
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−28.0000	−1.74659	−0.873296	−	0.487190i	$-0.838022\pi$
$257$	−28.0000	−1.74659	−0.873296	+	0.487190i	$0.838022\pi$
$258$	0	0
$259$	−12.0000	−0.745644
$260$	−4.00000	−0.248069
$261$	0	0
$262$	−4.00000	−0.247121
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	0	0
$267$	0	0
$268$	−2.00000	−0.122169
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−14.0000	−0.839664
$279$	0	0
$280$	−12.0000	−0.717137
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	−2.00000	−0.118888	−0.0594438	−	0.998232i	$-0.518933\pi$
$283$	−2.00000	−0.118888	−0.0594438	+	0.998232i	$0.518933\pi$
$284$	−1.00000	−0.0593391
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	4.00000	0.234888
$291$	0	0
$292$	10.0000	0.585206
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	−18.0000	−1.04623
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−8.00000	−0.461112
$302$	−8.00000	−0.460348
$303$	0	0
$304$	0	0
$305$	−20.0000	−1.14520
$306$	0	0
$307$	26.0000	1.48390	0.741949	−	0.670456i	$-0.233902\pi$
$307$	26.0000	1.48390	0.741949	+	0.670456i	$0.233902\pi$
$308$	0	0
$309$	0	0
$310$	−20.0000	−1.13592
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−16.0000	−0.898650	−0.449325	−	0.893368i	$-0.648335\pi$
$317$	−16.0000	−0.898650	−0.449325	+	0.893368i	$0.648335\pi$
$318$	0	0
$319$	0	0
$320$	−14.0000	−0.782624
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	2.00000	0.110940
$326$	−6.00000	−0.332309
$327$	0	0
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−2.00000	−0.109930	−0.0549650	−	0.998488i	$-0.517505\pi$
$331$	−2.00000	−0.109930	−0.0549650	+	0.998488i	$0.517505\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	8.00000	0.437741
$335$	−4.00000	−0.218543
$336$	0	0
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−24.0000	−1.29025
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	0	0
$353$	12.0000	0.638696	0.319348	−	0.947638i	$-0.396536\pi$
$353$	12.0000	0.638696	0.319348	+	0.947638i	$0.396536\pi$
$354$	0	0
$355$	−2.00000	−0.106149
$356$	6.00000	0.317999
$357$	0	0
$358$	12.0000	0.634220
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	26.0000	1.36653
$363$	0	0
$364$	4.00000	0.209657
$365$	20.0000	1.04685
$366$	0	0
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	0	0
$369$	0	0
$370$	−12.0000	−0.623850
$371$	8.00000	0.415339
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	0	0
$376$	−36.0000	−1.85656
$377$	−4.00000	−0.206010
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	0	0
$382$	−24.0000	−1.22795
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	0	0
$386$	−18.0000	−0.916176
$387$	0	0
$388$	2.00000	0.101535
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	0	0
$392$	−9.00000	−0.454569
$393$	0	0
$394$	−12.0000	−0.604551
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	20.0000	1.00251
$399$	0	0
$400$	1.00000	0.0500000
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	20.0000	0.996271
$404$	10.0000	0.497519
$405$	0	0
$406$	−4.00000	−0.198517
$407$	0	0
$408$	0	0
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	0	0
$411$	0	0
$412$	−16.0000	−0.788263
$413$	−24.0000	−1.18096
$414$	0	0
$415$	−8.00000	−0.392705
$416$	10.0000	0.490290
$417$	0	0
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	−22.0000	−1.07094
$423$	0	0
$424$	12.0000	0.582772
$425$	0	0
$426$	0	0
$427$	20.0000	0.967868
$428$	12.0000	0.580042
$429$	0	0
$430$	−8.00000	−0.385794
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	20.0000	0.960031
$435$	0	0
$436$	−10.0000	−0.478913
$437$	0	0
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	12.0000	0.568216
$447$	0	0
$448$	14.0000	0.661438
$449$	−8.00000	−0.377543	−0.188772	−	0.982021i	$-0.560451\pi$
$449$	−8.00000	−0.377543	−0.188772	+	0.982021i	$0.560451\pi$
$450$	0	0
$451$	0	0
$452$	8.00000	0.376288
$453$	0	0
$454$	4.00000	0.187729
$455$	8.00000	0.375046
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	26.0000	1.21490
$459$	0	0
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−6.00000	−0.277945
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	−24.0000	−1.10704
$471$	0	0
$472$	−36.0000	−1.65703
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	12.0000	0.548867
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	10.0000	0.455488
$483$	0	0
$484$	11.0000	0.500000
$485$	4.00000	0.181631
$486$	0	0
$487$	30.0000	1.35943	0.679715	−	0.733476i	$-0.262104\pi$
$487$	30.0000	1.35943	0.679715	+	0.733476i	$0.262104\pi$
$488$	30.0000	1.35804
$489$	0	0
$490$	−6.00000	−0.271052
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	10.0000	0.449013
$497$	2.00000	0.0897123
$498$	0	0
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	−20.0000	−0.892644
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	0	0
$505$	20.0000	0.889988
$506$	0	0
$507$	0	0
$508$	6.00000	0.266207
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	11.0000	0.486136
$513$	0	0
$514$	28.0000	1.23503
$515$	−32.0000	−1.41009
$516$	0	0
$517$	0	0
$518$	12.0000	0.527250
$519$	0	0
$520$	12.0000	0.526235
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	38.0000	1.66162	0.830812	−	0.556553i	$-0.187876\pi$
$523$	38.0000	1.66162	0.830812	+	0.556553i	$0.187876\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	8.00000	0.347498
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	24.0000	1.03761
$536$	6.00000	0.259161
$537$	0	0
$538$	4.00000	0.172452
$539$	0	0
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−4.00000	−0.171815
$543$	0	0
$544$	0	0
$545$	−20.0000	−0.856706
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	8.00000	0.340195
$554$	10.0000	0.424859
$555$	0	0
$556$	−14.0000	−0.593732
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	4.00000	0.169031
$561$	0	0
$562$	16.0000	0.674919
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	16.0000	0.673125
$566$	2.00000	0.0840663
$567$	0	0
$568$	3.00000	0.125877
$569$	−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
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	17.0000	0.707107
$579$	0	0
$580$	4.00000	0.166091
$581$	8.00000	0.331896
$582$	0	0
$583$	0	0
$584$	−30.0000	−1.24141
$585$	0	0
$586$	−26.0000	−1.07405
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	0	0
$590$	−24.0000	−0.988064
$591$	0	0
$592$	6.00000	0.246598
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	8.00000	0.326056
$603$	0	0
$604$	−8.00000	−0.325515
$605$	22.0000	0.894427
$606$	0	0
$607$	−38.0000	−1.54237	−0.771186	−	0.636610i	$-0.780336\pi$
$607$	−38.0000	−1.54237	−0.771186	+	0.636610i	$0.780336\pi$
$608$	0	0
$609$	0	0
$610$	20.0000	0.809776
$611$	24.0000	0.970936
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	−26.0000	−1.04927
$615$	0	0
$616$	0	0
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	−20.0000	−0.803219
$621$	0	0
$622$	32.0000	1.28308
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	0	0
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	12.0000	0.477334
$633$	0	0
$634$	16.0000	0.635441
$635$	12.0000	0.476205
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	0	0
$640$	−6.00000	−0.237171
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	0	0
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−6.00000	−0.234978
$653$	−20.0000	−0.782660	−0.391330	−	0.920250i	$-0.627985\pi$
$653$	−20.0000	−0.782660	−0.391330	+	0.920250i	$0.627985\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	0	0
$657$	0	0
$658$	24.0000	0.935617
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	2.00000	0.0777322
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	8.00000	0.309529
$669$	0	0
$670$	4.00000	0.154533
$671$	0	0
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−30.0000	−1.15556
$675$	0	0
$676$	9.00000	0.346154
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	0	0
$688$	4.00000	0.152499
$689$	−8.00000	−0.304776
$690$	0	0
$691$	−38.0000	−1.44559	−0.722794	−	0.691063i	$-0.757142\pi$
$691$	−38.0000	−1.44559	−0.722794	+	0.691063i	$0.757142\pi$
$692$	−24.0000	−0.912343
$693$	0	0
$694$	24.0000	0.911028
$695$	−28.0000	−1.06210
$696$	0	0
$697$	0	0
$698$	−10.0000	−0.378506
$699$	0	0
$700$	2.00000	0.0755929
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−12.0000	−0.451626
$707$	−20.0000	−0.752177
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	2.00000	0.0750587
$711$	0	0
$712$	−18.0000	−0.674579
$713$	0	0
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	−32.0000	−1.19423
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	19.0000	0.707107
$723$	0	0
$724$	26.0000	0.966282
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	−12.0000	−0.444750
$729$	0	0
$730$	−20.0000	−0.740233
$731$	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	$0.313086\pi$
$734$	−20.0000	−0.738213
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	−8.00000	−0.293689
$743$	44.0000	1.61420	0.807102	−	0.590412i	$-0.201035\pi$
$743$	44.0000	1.61420	0.807102	+	0.590412i	$0.201035\pi$
$744$	0	0
$745$	0	0
$746$	−26.0000	−0.951928
$747$	0	0
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	4.00000	0.145671
$755$	−16.0000	−0.582300
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−16.0000	−0.581146
$759$	0	0
$760$	0	0
$761$	8.00000	0.290000	0.145000	−	0.989432i	$-0.453682\pi$
$761$	8.00000	0.290000	0.145000	+	0.989432i	$0.453682\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	−24.0000	−0.868290
$765$	0	0
$766$	−4.00000	−0.144526
$767$	24.0000	0.866590
$768$	0	0
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	0	0
$772$	−18.0000	−0.647834
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	0	0
$775$	10.0000	0.359211
$776$	−6.00000	−0.215387
$777$	0	0
$778$	16.0000	0.573628
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	3.00000	0.107143
$785$	−4.00000	−0.142766
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	−12.0000	−0.427482
$789$	0	0
$790$	8.00000	0.284627
$791$	−16.0000	−0.568895
$792$	0	0
$793$	−20.0000	−0.710221
$794$	14.0000	0.496841
$795$	0	0
$796$	20.0000	0.708881
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	0	0
$800$	5.00000	0.176777
$801$	0	0
$802$	−24.0000	−0.847469
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−20.0000	−0.704470
$807$	0	0
$808$	−30.0000	−1.05540
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−4.00000	−0.140372
$813$	0	0
$814$	0	0
$815$	−12.0000	−0.420342
$816$	0	0
$817$	0	0
$818$	38.0000	1.32864
$819$	0	0
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	0	0
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	48.0000	1.67216
$825$	0	0
$826$	24.0000	0.835067
$827$	−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
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	8.00000	0.277684
$831$	0	0
$832$	−14.0000	−0.485363
$833$	0	0
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−6.00000	−0.206774
$843$	0	0
$844$	−22.0000	−0.757271
$845$	18.0000	0.619219
$846$	0	0
$847$	−22.0000	−0.755929
$848$	−4.00000	−0.137361
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	−20.0000	−0.684386
$855$	0	0
$856$	−36.0000	−1.23045
$857$	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
$859$	−38.0000	−1.29654	−0.648272	−	0.761409i	$-0.724508\pi$
$859$	−38.0000	−1.29654	−0.648272	+	0.761409i	$0.724508\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−32.0000	−1.08992
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	0	0
$865$	−48.0000	−1.63205
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	20.0000	0.678844
$869$	0	0
$870$	0	0
$871$	−4.00000	−0.135535
$872$	30.0000	1.01593
$873$	0	0
$874$	0	0
$875$	24.0000	0.811348
$876$	0	0
$877$	−26.0000	−0.877958	−0.438979	−	0.898497i	$-0.644660\pi$
$877$	−26.0000	−0.877958	−0.438979	+	0.898497i	$0.644660\pi$
$878$	26.0000	0.877457
$879$	0	0
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	0	0
$885$	0	0
$886$	20.0000	0.671913
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	−12.0000	−0.402241
$891$	0	0
$892$	12.0000	0.401790
$893$	0	0
$894$	0	0
$895$	24.0000	0.802232
$896$	6.00000	0.200446
$897$	0	0
$898$	8.00000	0.266963
$899$	−20.0000	−0.667037
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−24.0000	−0.798228
$905$	52.0000	1.72854
$906$	0	0
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	4.00000	0.132745
$909$	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
$913$	0	0
$914$	18.0000	0.595387
$915$	0	0
$916$	26.0000	0.859064
$917$	8.00000	0.264183
$918$	0	0
$919$	42.0000	1.38545	0.692726	−	0.721201i	$-0.256409\pi$
$919$	42.0000	1.38545	0.692726	+	0.721201i	$0.256409\pi$
$920$	0	0
$921$	0	0
$922$	12.0000	0.395199
$923$	−2.00000	−0.0658308
$924$	0	0
$925$	6.00000	0.197279
$926$	−4.00000	−0.131448
$927$	0	0
$928$	−10.0000	−0.328266
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	0	0
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−20.0000	−0.654420
$935$	0	0
$936$	0	0
$937$	−46.0000	−1.50275	−0.751377	−	0.659873i	$-0.770610\pi$
$937$	−46.0000	−1.50275	−0.751377	+	0.659873i	$0.770610\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	−24.0000	−0.782794
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	20.0000	0.649227
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	−48.0000	−1.55324
$956$	12.0000	0.388108
$957$	0	0
$958$	−16.0000	−0.516937
$959$	0	0
$960$	0	0
$961$	69.0000	2.22581
$962$	−12.0000	−0.386896
$963$	0	0
$964$	10.0000	0.322078
$965$	−36.0000	−1.15888
$966$	0	0
$967$	6.00000	0.192947	0.0964735	−	0.995336i	$-0.469244\pi$
$967$	6.00000	0.192947	0.0964735	+	0.995336i	$0.469244\pi$
$968$	−33.0000	−1.06066
$969$	0	0
$970$	−4.00000	−0.128432
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	28.0000	0.897639
$974$	−30.0000	−0.961262
$975$	0	0
$976$	−10.0000	−0.320092
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	0	0
$979$	0	0
$980$	−6.00000	−0.191663
$981$	0	0
$982$	16.0000	0.510581
$983$	−56.0000	−1.78612	−0.893061	−	0.449935i	$-0.851447\pi$
$983$	−56.0000	−1.78612	−0.893061	+	0.449935i	$0.851447\pi$
$984$	0	0
$985$	−24.0000	−0.764704
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	6.00000	0.190596	0.0952981	−	0.995449i	$-0.469620\pi$
$991$	6.00000	0.190596	0.0952981	+	0.995449i	$0.469620\pi$
$992$	50.0000	1.58750
$993$	0	0
$994$	−2.00000	−0.0634361
$995$	40.0000	1.26809
$996$	0	0
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	8.00000	0.253236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	639.2.a.a.1.1		1
3.2	odd	2		213.2.a.a.1.1	✓	1
12.11	even	2		3408.2.a.c.1.1		1
15.14	odd	2		5325.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
213.2.a.a.1.1	✓	1	3.2	odd	2
639.2.a.a.1.1		1	1.1	even	1	trivial
3408.2.a.c.1.1		1	12.11	even	2
5325.2.a.c.1.1		1	15.14	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:	\( N \)	\(=\)	\( 639 = 3^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	639.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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