LMFDB - Embedded newform 4655.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4655,2,Mod(1,4655)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4655, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4655.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4655 = 5 \cdot 7^{2} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4655.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:	$37.1703621409$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 665)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4655.1

$q$ -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^{4} -1.00000 q^{5} -3.00000 q^{9} +1.00000 q^{11} +2.00000 q^{13} +4.00000 q^{16} +1.00000 q^{17} -1.00000 q^{19} +2.00000 q^{20} -3.00000 q^{23} +1.00000 q^{25} -2.00000 q^{29} -2.00000 q^{31} +6.00000 q^{36} +12.0000 q^{37} +10.0000 q^{41} -4.00000 q^{43} -2.00000 q^{44} +3.00000 q^{45} +3.00000 q^{47} -4.00000 q^{52} -8.00000 q^{53} -1.00000 q^{55} +10.0000 q^{59} -13.0000 q^{61} -8.00000 q^{64} -2.00000 q^{65} +6.00000 q^{67} -2.00000 q^{68} -2.00000 q^{71} -11.0000 q^{73} +2.00000 q^{76} +12.0000 q^{79} -4.00000 q^{80} +9.00000 q^{81} -9.00000 q^{83} -1.00000 q^{85} -6.00000 q^{89} +6.00000 q^{92} +1.00000 q^{95} -2.00000 q^{97} -3.00000 q^{99} +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

)

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

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−2.00000	−1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.00000	0.447214
$21$	0	0
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	6.00000	1.00000
$37$	12.0000	1.97279	0.986394	−	0.164399i	$-0.0525685\pi$
$37$	12.0000	1.97279	0.986394	+	0.164399i	$0.0525685\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−2.00000	−0.301511
$45$	3.00000	0.447214
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	−4.00000	−0.447214
$81$	9.00000	1.00000
$82$	0	0
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	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
$91$	0	0
$92$	6.00000	0.625543
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	−2.00000	−0.200000
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.0000	1.35343	0.676716	−	0.736245i	$-0.263403\pi$
$107$	14.0000	1.35343	0.676716	+	0.736245i	$0.263403\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	4.00000	0.371391
$117$	−6.00000	−0.554700
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0	0
$124$	4.00000	0.359211
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.0000	−1.66004	−0.830019	−	0.557735i	$-0.811670\pi$
$131$	−19.0000	−1.66004	−0.830019	+	0.557735i	$0.811670\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.00000	0.427179	0.213589	−	0.976924i	$-0.431485\pi$
$137$	5.00000	0.427179	0.213589	+	0.976924i	$0.431485\pi$
$138$	0	0
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.00000	0.167248
$144$	−12.0000	−1.00000
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	−24.0000	−1.97279
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	−20.0000	−1.56174
$165$	0	0
$166$	0	0
$167$	−22.0000	−1.70241	−0.851206	−	0.524832i	$-0.824128\pi$
$167$	−22.0000	−1.70241	−0.851206	+	0.524832i	$0.824128\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	3.00000	0.229416
$172$	8.00000	0.609994
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	0	0
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	−6.00000	−0.447214
$181$	−24.0000	−1.78391	−0.891953	−	0.452128i	$-0.850665\pi$
$181$	−24.0000	−1.78391	−0.891953	+	0.452128i	$0.850665\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	1.00000	0.0731272
$188$	−6.00000	−0.437595
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.0000	−0.926212	−0.463106	−	0.886303i	$-0.653265\pi$
$197$	−13.0000	−0.926212	−0.463106	+	0.886303i	$0.653265\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−10.0000	−0.698430
$206$	0	0
$207$	9.00000	0.625543
$208$	8.00000	0.554700
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	16.0000	1.09888
$213$	0	0
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	2.00000	0.134840
$221$	2.00000	0.134535
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.0000	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$234$	0	0
$235$	−3.00000	−0.195698
$236$	−20.0000	−1.30189
$237$	0	0
$238$	0	0
$239$	−5.00000	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	0	0
$244$	26.0000	1.66448
$245$	0	0
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	8.00000	0.499026	0.249513	−	0.968371i	$-0.419729\pi$
$257$	8.00000	0.499026	0.249513	+	0.968371i	$0.419729\pi$
$258$	0	0
$259$	0	0
$260$	4.00000	0.248069
$261$	6.00000	0.371391
$262$	0	0
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	8.00000	0.491436
$266$	0	0
$267$	0	0
$268$	−12.0000	−0.733017
$269$	−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
$271$	−9.00000	−0.546711	−0.273356	−	0.961913i	$-0.588134\pi$
$271$	−9.00000	−0.546711	−0.273356	+	0.961913i	$0.588134\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	22.0000	1.28745
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	−10.0000	−0.582223
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.00000	−0.346989
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−4.00000	−0.229416
$305$	13.0000	0.744378
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	0	0
$313$	−9.00000	−0.508710	−0.254355	−	0.967111i	$-0.581863\pi$
$313$	−9.00000	−0.508710	−0.254355	+	0.967111i	$0.581863\pi$
$314$	0	0
$315$	0	0
$316$	−24.0000	−1.35011
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	8.00000	0.447214
$321$	0	0
$322$	0	0
$323$	−1.00000	−0.0556415
$324$	−18.0000	−1.00000
$325$	2.00000	0.110940
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	22.0000	1.20923	0.604615	−	0.796518i	$-0.293327\pi$
$331$	22.0000	1.20923	0.604615	+	0.796518i	$0.293327\pi$
$332$	18.0000	0.987878
$333$	−36.0000	−1.97279
$334$	0	0
$335$	−6.00000	−0.327815
$336$	0	0
$337$	28.0000	1.52526	0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000	1.52526	0.762629	+	0.646837i	$0.223908\pi$
$338$	0	0
$339$	0	0
$340$	2.00000	0.108465
$341$	−2.00000	−0.108306
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−22.0000	−1.17094	−0.585471	−	0.810693i	$-0.699090\pi$
$353$	−22.0000	−1.17094	−0.585471	+	0.810693i	$0.699090\pi$
$354$	0	0
$355$	2.00000	0.106149
$356$	12.0000	0.635999
$357$	0	0
$358$	0	0
$359$	29.0000	1.53056	0.765281	−	0.643697i	$-0.222600\pi$
$359$	29.0000	1.53056	0.765281	+	0.643697i	$0.222600\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11.0000	0.575766
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−12.0000	−0.625543
$369$	−30.0000	−1.56174
$370$	0	0
$371$	0	0
$372$	0	0
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−2.00000	−0.102598
$381$	0	0
$382$	0	0
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.0000	0.609994
$388$	4.00000	0.203069
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−12.0000	−0.603786
$396$	6.00000	0.301511
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	0	0
$399$	0	0
$400$	4.00000	0.200000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	−18.0000	−0.895533
$405$	−9.00000	−0.447214
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	0	0
$412$	−8.00000	−0.394132
$413$	0	0
$414$	0	0
$415$	9.00000	0.441793
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	0	0
$423$	−9.00000	−0.437595
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	0	0
$428$	−28.0000	−1.35343
$429$	0	0
$430$	0	0
$431$	−34.0000	−1.63772	−0.818861	−	0.573992i	$-0.805394\pi$
$431$	−34.0000	−1.63772	−0.818861	+	0.573992i	$0.805394\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	0	0
$436$	−4.00000	−0.191565
$437$	3.00000	0.143509
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	37.0000	1.75792	0.878962	−	0.476893i	$-0.158237\pi$
$443$	37.0000	1.75792	0.878962	+	0.476893i	$0.158237\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	24.0000	1.12887
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.00000	−0.233890	−0.116945	−	0.993138i	$-0.537310\pi$
$457$	−5.00000	−0.233890	−0.116945	+	0.993138i	$0.537310\pi$
$458$	0	0
$459$	0	0
$460$	−6.00000	−0.279751
$461$	−11.0000	−0.512321	−0.256161	−	0.966634i	$-0.582458\pi$
$461$	−11.0000	−0.512321	−0.256161	+	0.966634i	$0.582458\pi$
$462$	0	0
$463$	−5.00000	−0.232370	−0.116185	−	0.993228i	$-0.537067\pi$
$463$	−5.00000	−0.232370	−0.116185	+	0.993228i	$0.537067\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	0	0
$467$	−21.0000	−0.971764	−0.485882	−	0.874024i	$-0.661502\pi$
$467$	−21.0000	−0.971764	−0.485882	+	0.874024i	$0.661502\pi$
$468$	12.0000	0.554700
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	24.0000	1.09888
$478$	0	0
$479$	11.0000	0.502603	0.251301	−	0.967909i	$-0.419141\pi$
$479$	11.0000	0.502603	0.251301	+	0.967909i	$0.419141\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	0	0
$483$	0	0
$484$	20.0000	0.909091
$485$	2.00000	0.0908153
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−25.0000	−1.12823	−0.564117	−	0.825695i	$-0.690783\pi$
$491$	−25.0000	−1.12823	−0.564117	+	0.825695i	$0.690783\pi$
$492$	0	0
$493$	−2.00000	−0.0900755
$494$	0	0
$495$	3.00000	0.134840
$496$	−8.00000	−0.359211
$497$	0	0
$498$	0	0
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	2.00000	0.0894427
$501$	0	0
$502$	0	0
$503$	13.0000	0.579641	0.289821	−	0.957081i	$-0.406404\pi$
$503$	13.0000	0.579641	0.289821	+	0.957081i	$0.406404\pi$
$504$	0	0
$505$	−9.00000	−0.400495
$506$	0	0
$507$	0	0
$508$	16.0000	0.709885
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.00000	−0.176261
$516$	0	0
$517$	3.00000	0.131940
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.0000	1.40195	0.700973	−	0.713188i	$-0.252749\pi$
$521$	32.0000	1.40195	0.700973	+	0.713188i	$0.252749\pi$
$522$	0	0
$523$	26.0000	1.13690	0.568450	−	0.822718i	$-0.307543\pi$
$523$	26.0000	1.13690	0.568450	+	0.822718i	$0.307543\pi$
$524$	38.0000	1.66004
$525$	0	0
$526$	0	0
$527$	−2.00000	−0.0871214
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−30.0000	−1.30189
$532$	0	0
$533$	20.0000	0.866296
$534$	0	0
$535$	−14.0000	−0.605273
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.00000	−0.257960	−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000	−0.257960	−0.128980	+	0.991647i	$0.541170\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−14.0000	−0.598597	−0.299298	−	0.954160i	$-0.596753\pi$
$547$	−14.0000	−0.598597	−0.299298	+	0.954160i	$0.596753\pi$
$548$	−10.0000	−0.427179
$549$	39.0000	1.66448
$550$	0	0
$551$	2.00000	0.0852029
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	22.0000	0.933008
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	42.0000	1.77009	0.885044	−	0.465506i	$-0.154128\pi$
$563$	42.0000	1.77009	0.885044	+	0.465506i	$0.154128\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−33.0000	−1.38101	−0.690504	−	0.723329i	$-0.742611\pi$
$571$	−33.0000	−1.38101	−0.690504	+	0.723329i	$0.742611\pi$
$572$	−4.00000	−0.167248
$573$	0	0
$574$	0	0
$575$	−3.00000	−0.125109
$576$	24.0000	1.00000
$577$	−33.0000	−1.37381	−0.686904	−	0.726748i	$-0.741031\pi$
$577$	−33.0000	−1.37381	−0.686904	+	0.726748i	$0.741031\pi$
$578$	0	0
$579$	0	0
$580$	−4.00000	−0.166091
$581$	0	0
$582$	0	0
$583$	−8.00000	−0.331326
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	−39.0000	−1.60970	−0.804851	−	0.593477i	$-0.797755\pi$
$587$	−39.0000	−1.60970	−0.804851	+	0.593477i	$0.797755\pi$
$588$	0	0
$589$	2.00000	0.0824086
$590$	0	0
$591$	0	0
$592$	48.0000	1.97279
$593$	−33.0000	−1.35515	−0.677574	−	0.735455i	$-0.736969\pi$
$593$	−33.0000	−1.35515	−0.677574	+	0.735455i	$0.736969\pi$
$594$	0	0
$595$	0	0
$596$	−28.0000	−1.14692
$597$	0	0
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	−24.0000	−0.978980	−0.489490	−	0.872009i	$-0.662817\pi$
$601$	−24.0000	−0.978980	−0.489490	+	0.872009i	$0.662817\pi$
$602$	0	0
$603$	−18.0000	−0.733017
$604$	8.00000	0.325515
$605$	10.0000	0.406558
$606$	0	0
$607$	18.0000	0.730597	0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000	0.730597	0.365299	+	0.930890i	$0.380967\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.00000	0.242734
$612$	6.00000	0.242536
$613$	−15.0000	−0.605844	−0.302922	−	0.953015i	$-0.597962\pi$
$613$	−15.0000	−0.605844	−0.302922	+	0.953015i	$0.597962\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	−1.00000	−0.0401934	−0.0200967	−	0.999798i	$-0.506397\pi$
$619$	−1.00000	−0.0401934	−0.0200967	+	0.999798i	$0.506397\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	14.0000	0.558661
$629$	12.0000	0.478471
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.00000	0.317470
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.00000	0.237356
$640$	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
$643$	−13.0000	−0.512670	−0.256335	−	0.966588i	$-0.582515\pi$
$643$	−13.0000	−0.512670	−0.256335	+	0.966588i	$0.582515\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	10.0000	0.392534
$650$	0	0
$651$	0	0
$652$	22.0000	0.861586
$653$	33.0000	1.29139	0.645695	−	0.763596i	$-0.276568\pi$
$653$	33.0000	1.29139	0.645695	+	0.763596i	$0.276568\pi$
$654$	0	0
$655$	19.0000	0.742391
$656$	40.0000	1.56174
$657$	33.0000	1.28745
$658$	0	0
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	−20.0000	−0.777910	−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000	−0.777910	−0.388955	+	0.921257i	$0.627164\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.00000	0.232321
$668$	44.0000	1.70241
$669$	0	0
$670$	0	0
$671$	−13.0000	−0.501859
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	0	0
$675$	0	0
$676$	18.0000	0.692308
$677$	8.00000	0.307465	0.153732	−	0.988113i	$-0.450871\pi$
$677$	8.00000	0.307465	0.153732	+	0.988113i	$0.450871\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.0000	1.22445	0.612223	−	0.790685i	$-0.290275\pi$
$683$	32.0000	1.22445	0.612223	+	0.790685i	$0.290275\pi$
$684$	−6.00000	−0.229416
$685$	−5.00000	−0.191040
$686$	0	0
$687$	0	0
$688$	−16.0000	−0.609994
$689$	−16.0000	−0.609551
$690$	0	0
$691$	29.0000	1.10321	0.551606	−	0.834105i	$-0.314015\pi$
$691$	29.0000	1.10321	0.551606	+	0.834105i	$0.314015\pi$
$692$	24.0000	0.912343
$693$	0	0
$694$	0	0
$695$	11.0000	0.417254
$696$	0	0
$697$	10.0000	0.378777
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13.0000	0.491003	0.245502	−	0.969396i	$-0.421047\pi$
$701$	13.0000	0.491003	0.245502	+	0.969396i	$0.421047\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	−8.00000	−0.301511
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−39.0000	−1.46468	−0.732338	−	0.680941i	$-0.761571\pi$
$709$	−39.0000	−1.46468	−0.732338	+	0.680941i	$0.761571\pi$
$710$	0	0
$711$	−36.0000	−1.35011
$712$	0	0
$713$	6.00000	0.224702
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	−4.00000	−0.149487
$717$	0	0
$718$	0	0
$719$	7.00000	0.261056	0.130528	−	0.991445i	$-0.458333\pi$
$719$	7.00000	0.261056	0.130528	+	0.991445i	$0.458333\pi$
$720$	12.0000	0.447214
$721$	0	0
$722$	0	0
$723$	0	0
$724$	48.0000	1.78391
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	52.0000	1.92857	0.964287	−	0.264861i	$-0.0853260\pi$
$727$	52.0000	1.92857	0.964287	+	0.264861i	$0.0853260\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−4.00000	−0.147945
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.00000	0.221013
$738$	0	0
$739$	−9.00000	−0.331070	−0.165535	−	0.986204i	$-0.552935\pi$
$739$	−9.00000	−0.331070	−0.165535	+	0.986204i	$0.552935\pi$
$740$	24.0000	0.882258
$741$	0	0
$742$	0	0
$743$	52.0000	1.90769	0.953847	−	0.300291i	$-0.0970839\pi$
$743$	52.0000	1.90769	0.953847	+	0.300291i	$0.0970839\pi$
$744$	0	0
$745$	−14.0000	−0.512920
$746$	0	0
$747$	27.0000	0.987878
$748$	−2.00000	−0.0731272
$749$	0	0
$750$	0	0
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	0	0
$755$	4.00000	0.145575
$756$	0	0
$757$	35.0000	1.27210	0.636048	−	0.771649i	$-0.280568\pi$
$757$	35.0000	1.27210	0.636048	+	0.771649i	$0.280568\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3.00000	0.108465
$766$	0	0
$767$	20.0000	0.722158
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	−8.00000	−0.287926
$773$	−4.00000	−0.143870	−0.0719350	−	0.997409i	$-0.522917\pi$
$773$	−4.00000	−0.143870	−0.0719350	+	0.997409i	$0.522917\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10.0000	−0.358287
$780$	0	0
$781$	−2.00000	−0.0715656
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.00000	0.249841
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	26.0000	0.926212
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−26.0000	−0.923287
$794$	0	0
$795$	0	0
$796$	16.0000	0.567105
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	3.00000	0.106132
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	−11.0000	−0.388182
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	22.0000	0.772524	0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000	0.772524	0.386262	+	0.922389i	$0.373766\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11.0000	0.385313
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	0	0
$820$	20.0000	0.698430
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	0	0
$823$	−35.0000	−1.22002	−0.610012	−	0.792392i	$-0.708835\pi$
$823$	−35.0000	−1.22002	−0.610012	+	0.792392i	$0.708835\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.0000	0.486828	0.243414	−	0.969923i	$-0.421733\pi$
$827$	14.0000	0.486828	0.243414	+	0.969923i	$0.421733\pi$
$828$	−18.0000	−0.625543
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	0	0
$832$	−16.0000	−0.554700
$833$	0	0
$834$	0	0
$835$	22.0000	0.761341
$836$	2.00000	0.0691714
$837$	0	0
$838$	0	0
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	−16.0000	−0.550743
$845$	9.00000	0.309609
$846$	0	0
$847$	0	0
$848$	−32.0000	−1.09888
$849$	0	0
$850$	0	0
$851$	−36.0000	−1.23406
$852$	0	0
$853$	−1.00000	−0.0342393	−0.0171197	−	0.999853i	$-0.505450\pi$
$853$	−1.00000	−0.0342393	−0.0171197	+	0.999853i	$0.505450\pi$
$854$	0	0
$855$	−3.00000	−0.102598
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	0	0
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12.0000	0.407072
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	0	0
$879$	0	0
$880$	−4.00000	−0.134840
$881$	21.0000	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$882$	0	0
$883$	51.0000	1.71629	0.858143	−	0.513410i	$-0.171618\pi$
$883$	51.0000	1.71629	0.858143	+	0.513410i	$0.171618\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	0	0
$887$	30.0000	1.00730	0.503651	−	0.863907i	$-0.331990\pi$
$887$	30.0000	1.00730	0.503651	+	0.863907i	$0.331990\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.00000	0.301511
$892$	−12.0000	−0.401790
$893$	−3.00000	−0.100391
$894$	0	0
$895$	−2.00000	−0.0668526
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.00000	0.133407
$900$	6.00000	0.200000
$901$	−8.00000	−0.266519
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.0000	0.797787
$906$	0	0
$907$	38.0000	1.26177	0.630885	−	0.775877i	$-0.282692\pi$
$907$	38.0000	1.26177	0.630885	+	0.775877i	$0.282692\pi$
$908$	36.0000	1.19470
$909$	−27.0000	−0.895533
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	−9.00000	−0.297857
$914$	0	0
$915$	0	0
$916$	30.0000	0.991228
$917$	0	0
$918$	0	0
$919$	−13.0000	−0.428830	−0.214415	−	0.976743i	$-0.568785\pi$
$919$	−13.0000	−0.428830	−0.214415	+	0.976743i	$0.568785\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4.00000	−0.131662
$924$	0	0
$925$	12.0000	0.394558
$926$	0	0
$927$	−12.0000	−0.394132
$928$	0	0
$929$	−33.0000	−1.08269	−0.541347	−	0.840799i	$-0.682086\pi$
$929$	−33.0000	−1.08269	−0.541347	+	0.840799i	$0.682086\pi$
$930$	0	0
$931$	0	0
$932$	42.0000	1.37576
$933$	0	0
$934$	0	0
$935$	−1.00000	−0.0327035
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	0	0
$940$	6.00000	0.195698
$941$	−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
$943$	−30.0000	−0.976934
$944$	40.0000	1.30189
$945$	0	0
$946$	0	0
$947$	−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
$949$	−22.0000	−0.714150
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	0	0
$955$	0	0
$956$	10.0000	0.323423
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−42.0000	−1.35343
$964$	8.00000	0.257663
$965$	−4.00000	−0.128765
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−52.0000	−1.66448
$977$	−28.0000	−0.895799	−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000	−0.895799	−0.447900	+	0.894084i	$0.647828\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	0	0
$981$	−6.00000	−0.191565
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	13.0000	0.414214
$986$	0	0
$987$	0	0
$988$	4.00000	0.127257
$989$	12.0000	0.381578
$990$	0	0
$991$	14.0000	0.444725	0.222362	−	0.974964i	$-0.428623\pi$
$991$	14.0000	0.444725	0.222362	+	0.974964i	$0.428623\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	−51.0000	−1.61519	−0.807593	−	0.589740i	$-0.799230\pi$
$997$	−51.0000	−1.61519	−0.807593	+	0.589740i	$0.799230\pi$
$998$	0	0
$999$	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	4655.2.a.l.1.1		1
7.2	even	3		665.2.j.b.571.1	yes	2
7.4	even	3		665.2.j.b.191.1	✓	2
7.6	odd	2		4655.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
665.2.j.b.191.1	✓	2	7.4	even	3
665.2.j.b.571.1	yes	2	7.2	even	3
4655.2.a.l.1.1		1	1.1	even	1	trivial
4655.2.a.m.1.1		1	7.6	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 $\F_{q}$

Introduction

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Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

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	4655.2.a.l.1.1

Level	$4655$
Weight	$2$
Character	4655.1
Self dual	yes
Analytic conductor	$37.170$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$