LMFDB - Embedded newform 1083.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1083,2,Mod(1,1083)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1083, 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("1083.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1083 = 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1083.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:	$8.64779853890$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1083.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} -5.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +1.00000 q^{21} +2.00000 q^{22} -4.00000 q^{23} +3.00000 q^{24} -5.00000 q^{25} +5.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +8.00000 q^{29} +3.00000 q^{31} -5.00000 q^{32} -2.00000 q^{33} +4.00000 q^{34} -1.00000 q^{36} -3.00000 q^{37} -5.00000 q^{39} +12.0000 q^{41} -1.00000 q^{42} -1.00000 q^{43} +2.00000 q^{44} +4.00000 q^{46} -6.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +5.00000 q^{50} -4.00000 q^{51} +5.00000 q^{52} -4.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} -8.00000 q^{58} -10.0000 q^{59} -13.0000 q^{61} -3.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} +2.00000 q^{66} -11.0000 q^{67} +4.00000 q^{68} -4.00000 q^{69} -6.00000 q^{71} +3.00000 q^{72} -11.0000 q^{73} +3.00000 q^{74} -5.00000 q^{75} -2.00000 q^{77} +5.00000 q^{78} -1.00000 q^{79} +1.00000 q^{81} -12.0000 q^{82} -1.00000 q^{84} +1.00000 q^{86} +8.00000 q^{87} -6.00000 q^{88} +6.00000 q^{89} -5.00000 q^{91} +4.00000 q^{92} +3.00000 q^{93} +6.00000 q^{94} -5.00000 q^{96} -2.00000 q^{97} +6.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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	$0.615027\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.00000	−0.288675
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−1.00000	−0.235702
$19$	0	0
$19$	0	0
$20$	0	0
$21$	1.00000	0.218218
$22$	2.00000	0.426401
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	3.00000	0.612372
$25$	−5.00000	−1.00000
$26$	5.00000	0.980581
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	−5.00000	−0.883883
$33$	−2.00000	−0.348155
$34$	4.00000	0.685994
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	−5.00000	−0.800641
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	−1.00000	−0.154303
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	4.00000	0.589768
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	5.00000	0.707107
$51$	−4.00000	−0.560112
$52$	5.00000	0.693375
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	0	0
$58$	−8.00000	−1.05045
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	−3.00000	−0.381000
$63$	1.00000	0.125988
$64$	7.00000	0.875000
$65$	0	0
$66$	2.00000	0.246183
$67$	−11.0000	−1.34386	−0.671932	−	0.740613i	$-0.734535\pi$
$67$	−11.0000	−1.34386	−0.671932	+	0.740613i	$0.734535\pi$
$68$	4.00000	0.485071
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	3.00000	0.353553
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	3.00000	0.348743
$75$	−5.00000	−0.577350
$76$	0	0
$77$	−2.00000	−0.227921
$78$	5.00000	0.566139
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−12.0000	−1.32518
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	1.00000	0.107833
$87$	8.00000	0.857690
$88$	−6.00000	−0.639602
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−5.00000	−0.524142
$92$	4.00000	0.417029
$93$	3.00000	0.311086
$94$	6.00000	0.618853
$95$	0	0
$96$	−5.00000	−0.510310
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	6.00000	0.606092
$99$	−2.00000	−0.201008
$100$	5.00000	0.500000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	4.00000	0.396059
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	−15.0000	−1.47087
$105$	0	0
$106$	4.00000	0.388514
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	−1.00000	−0.0962250
$109$	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
$111$	−3.00000	−0.284747
$112$	−1.00000	−0.0944911
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	0	0
$116$	−8.00000	−0.742781
$117$	−5.00000	−0.462250
$118$	10.0000	0.920575
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	13.0000	1.17696
$123$	12.0000	1.08200
$124$	−3.00000	−0.269408
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	3.00000	0.265165
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	11.0000	0.950255
$135$	0	0
$136$	−12.0000	−1.02899
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	4.00000	0.340503
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	6.00000	0.503509
$143$	10.0000	0.836242
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	11.0000	0.910366
$147$	−6.00000	−0.494872
$148$	3.00000	0.246598
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	5.00000	0.408248
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	2.00000	0.161165
$155$	0	0
$156$	5.00000	0.400320
$157$	3.00000	0.239426	0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000	0.239426	0.119713	+	0.992809i	$0.461803\pi$
$158$	1.00000	0.0795557
$159$	−4.00000	−0.317221
$160$	0	0
$161$	−4.00000	−0.315244
$162$	−1.00000	−0.0785674
$163$	3.00000	0.234978	0.117489	−	0.993074i	$-0.462515\pi$
$163$	3.00000	0.234978	0.117489	+	0.993074i	$0.462515\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	0	0
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	3.00000	0.231455
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	1.00000	0.0762493
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	−8.00000	−0.606478
$175$	−5.00000	−0.377964
$176$	2.00000	0.150756
$177$	−10.0000	−0.751646
$178$	−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
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	5.00000	0.370625
$183$	−13.0000	−0.960988
$184$	−12.0000	−0.884652
$185$	0	0
$186$	−3.00000	−0.219971
$187$	8.00000	0.585018
$188$	6.00000	0.437595
$189$	1.00000	0.0727393
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	7.00000	0.505181
$193$	−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
$196$	6.00000	0.428571
$197$	−26.0000	−1.85242	−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000	−1.85242	−0.926212	+	0.377004i	$0.876954\pi$
$198$	2.00000	0.142134
$199$	21.0000	1.48865	0.744325	−	0.667817i	$-0.232771\pi$
$199$	21.0000	1.48865	0.744325	+	0.667817i	$0.232771\pi$
$200$	−15.0000	−1.06066
$201$	−11.0000	−0.775880
$202$	−14.0000	−0.985037
$203$	8.00000	0.561490
$204$	4.00000	0.280056
$205$	0	0
$206$	13.0000	0.905753
$207$	−4.00000	−0.278019
$208$	5.00000	0.346688
$209$	0	0
$210$	0	0
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	4.00000	0.274721
$213$	−6.00000	−0.411113
$214$	−18.0000	−1.23045
$215$	0	0
$216$	3.00000	0.204124
$217$	3.00000	0.203653
$218$	−2.00000	−0.135457
$219$	−11.0000	−0.743311
$220$	0	0
$221$	20.0000	1.34535
$222$	3.00000	0.201347
$223$	−9.00000	−0.602685	−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000	−0.602685	−0.301342	+	0.953516i	$0.597435\pi$
$224$	−5.00000	−0.334077
$225$	−5.00000	−0.333333
$226$	2.00000	0.133038
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	24.0000	1.57568
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	5.00000	0.326860
$235$	0	0
$236$	10.0000	0.650945
$237$	−1.00000	−0.0649570
$238$	4.00000	0.259281
$239$	−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
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	13.0000	0.832240
$245$	0	0
$246$	−12.0000	−0.765092
$247$	0	0
$248$	9.00000	0.571501
$249$	0	0
$250$	0	0
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	−1.00000	−0.0629941
$253$	8.00000	0.502956
$254$	−8.00000	−0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	20.0000	1.24757	0.623783	−	0.781598i	$-0.285595\pi$
$257$	20.0000	1.24757	0.623783	+	0.781598i	$0.285595\pi$
$258$	1.00000	0.0622573
$259$	−3.00000	−0.186411
$260$	0	0
$261$	8.00000	0.495188
$262$	14.0000	0.864923
$263$	26.0000	1.60323	0.801614	−	0.597841i	$-0.203975\pi$
$263$	26.0000	1.60323	0.801614	+	0.597841i	$0.203975\pi$
$264$	−6.00000	−0.369274
$265$	0	0
$266$	0	0
$267$	6.00000	0.367194
$268$	11.0000	0.671932
$269$	−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
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	4.00000	0.242536
$273$	−5.00000	−0.302614
$274$	−18.0000	−1.08742
$275$	10.0000	0.603023
$276$	4.00000	0.240772
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−5.00000	−0.299880
$279$	3.00000	0.179605
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	6.00000	0.357295
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	−10.0000	−0.591312
$287$	12.0000	0.708338
$288$	−5.00000	−0.294628
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−2.00000	−0.117242
$292$	11.0000	0.643726
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	6.00000	0.349927
$295$	0	0
$296$	−9.00000	−0.523114
$297$	−2.00000	−0.116052
$298$	6.00000	0.347571
$299$	20.0000	1.15663
$300$	5.00000	0.288675
$301$	−1.00000	−0.0576390
$302$	12.0000	0.690522
$303$	14.0000	0.804279
$304$	0	0
$305$	0	0
$306$	4.00000	0.228665
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	2.00000	0.113961
$309$	−13.0000	−0.739544
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	−15.0000	−0.849208
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	−3.00000	−0.169300
$315$	0	0
$316$	1.00000	0.0562544
$317$	4.00000	0.224662	0.112331	−	0.993671i	$-0.464168\pi$
$317$	4.00000	0.224662	0.112331	+	0.993671i	$0.464168\pi$
$318$	4.00000	0.224309
$319$	−16.0000	−0.895828
$320$	0	0
$321$	18.0000	1.00466
$322$	4.00000	0.222911
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	25.0000	1.38675
$326$	−3.00000	−0.166155
$327$	2.00000	0.110600
$328$	36.0000	1.98777
$329$	−6.00000	−0.330791
$330$	0	0
$331$	25.0000	1.37412	0.687062	−	0.726599i	$-0.258900\pi$
$331$	25.0000	1.37412	0.687062	+	0.726599i	$0.258900\pi$
$332$	0	0
$333$	−3.00000	−0.164399
$334$	−2.00000	−0.109435
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−12.0000	−0.652714
$339$	−2.00000	−0.108625
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	−13.0000	−0.701934
$344$	−3.00000	−0.161749
$345$	0	0
$346$	0	0
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	−8.00000	−0.428845
$349$	−21.0000	−1.12410	−0.562052	−	0.827102i	$-0.689988\pi$
$349$	−21.0000	−1.12410	−0.562052	+	0.827102i	$0.689988\pi$
$350$	5.00000	0.267261
$351$	−5.00000	−0.266880
$352$	10.0000	0.533002
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	10.0000	0.531494
$355$	0	0
$356$	−6.00000	−0.317999
$357$	−4.00000	−0.211702
$358$	12.0000	0.634220
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	0	0
$362$	−2.00000	−0.105118
$363$	−7.00000	−0.367405
$364$	5.00000	0.262071
$365$	0	0
$366$	13.0000	0.679521
$367$	7.00000	0.365397	0.182699	−	0.983169i	$-0.441517\pi$
$367$	7.00000	0.365397	0.182699	+	0.983169i	$0.441517\pi$
$368$	4.00000	0.208514
$369$	12.0000	0.624695
$370$	0	0
$371$	−4.00000	−0.207670
$372$	−3.00000	−0.155543
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	−18.0000	−0.928279
$377$	−40.0000	−2.06010
$378$	−1.00000	−0.0514344
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	−24.0000	−1.22795
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	19.0000	0.967075
$387$	−1.00000	−0.0508329
$388$	2.00000	0.101535
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	−18.0000	−0.909137
$393$	−14.0000	−0.706207
$394$	26.0000	1.30986
$395$	0	0
$396$	2.00000	0.100504
$397$	7.00000	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$398$	−21.0000	−1.05263
$399$	0	0
$400$	5.00000	0.250000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	11.0000	0.548630
$403$	−15.0000	−0.747203
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−8.00000	−0.397033
$407$	6.00000	0.297409
$408$	−12.0000	−0.594089
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	13.0000	0.640464
$413$	−10.0000	−0.492068
$414$	4.00000	0.196589
$415$	0	0
$416$	25.0000	1.22573
$417$	5.00000	0.244851
$418$	0	0
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−15.0000	−0.730189
$423$	−6.00000	−0.291730
$424$	−12.0000	−0.582772
$425$	20.0000	0.970143
$426$	6.00000	0.290701
$427$	−13.0000	−0.629114
$428$	−18.0000	−0.870063
$429$	10.0000	0.482805
$430$	0	0
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	−1.00000	−0.0481125
$433$	9.00000	0.432512	0.216256	−	0.976337i	$-0.430615\pi$
$433$	9.00000	0.432512	0.216256	+	0.976337i	$0.430615\pi$
$434$	−3.00000	−0.144005
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	11.0000	0.525600
$439$	−35.0000	−1.67046	−0.835229	−	0.549902i	$-0.814665\pi$
$439$	−35.0000	−1.67046	−0.835229	+	0.549902i	$0.814665\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	−20.0000	−0.951303
$443$	−32.0000	−1.52037	−0.760183	−	0.649709i	$-0.774891\pi$
$443$	−32.0000	−1.52037	−0.760183	+	0.649709i	$0.774891\pi$
$444$	3.00000	0.142374
$445$	0	0
$446$	9.00000	0.426162
$447$	−6.00000	−0.283790
$448$	7.00000	0.330719
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	5.00000	0.235702
$451$	−24.0000	−1.13012
$452$	2.00000	0.0940721
$453$	−12.0000	−0.563809
$454$	−24.0000	−1.12638
$455$	0	0
$456$	0	0
$457$	−11.0000	−0.514558	−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000	−0.514558	−0.257279	+	0.966337i	$0.582826\pi$
$458$	−13.0000	−0.607450
$459$	−4.00000	−0.186704
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	2.00000	0.0930484
$463$	23.0000	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	6.00000	0.277945
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	5.00000	0.231125
$469$	−11.0000	−0.507933
$470$	0	0
$471$	3.00000	0.138233
$472$	−30.0000	−1.38086
$473$	2.00000	0.0919601
$474$	1.00000	0.0459315
$475$	0	0
$476$	4.00000	0.183340
$477$	−4.00000	−0.183147
$478$	12.0000	0.548867
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	0	0
$481$	15.0000	0.683941
$482$	−7.00000	−0.318841
$483$	−4.00000	−0.182006
$484$	7.00000	0.318182
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	−39.0000	−1.76545
$489$	3.00000	0.135665
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	−12.0000	−0.541002
$493$	−32.0000	−1.44121
$494$	0	0
$495$	0	0
$496$	−3.00000	−0.134704
$497$	−6.00000	−0.269137
$498$	0	0
$499$	29.0000	1.29822	0.649109	−	0.760695i	$-0.275142\pi$
$499$	29.0000	1.29822	0.649109	+	0.760695i	$0.275142\pi$
$500$	0	0
$501$	2.00000	0.0893534
$502$	2.00000	0.0892644
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	−8.00000	−0.355643
$507$	12.0000	0.532939
$508$	−8.00000	−0.354943
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−11.0000	−0.486611
$512$	11.0000	0.486136
$513$	0	0
$514$	−20.0000	−0.882162
$515$	0	0
$516$	1.00000	0.0440225
$517$	12.0000	0.527759
$518$	3.00000	0.131812
$519$	0	0
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−8.00000	−0.350150
$523$	−29.0000	−1.26808	−0.634041	−	0.773300i	$-0.718605\pi$
$523$	−29.0000	−1.26808	−0.634041	+	0.773300i	$0.718605\pi$
$524$	14.0000	0.611593
$525$	−5.00000	−0.218218
$526$	−26.0000	−1.13365
$527$	−12.0000	−0.522728
$528$	2.00000	0.0870388
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−10.0000	−0.433963
$532$	0	0
$533$	−60.0000	−2.59889
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−33.0000	−1.42538
$537$	−12.0000	−0.517838
$538$	10.0000	0.431131
$539$	12.0000	0.516877
$540$	0	0
$541$	−39.0000	−1.67674	−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000	−1.67674	−0.838370	+	0.545101i	$0.816491\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	20.0000	0.857493
$545$	0	0
$546$	5.00000	0.213980
$547$	29.0000	1.23995	0.619975	−	0.784621i	$-0.287143\pi$
$547$	29.0000	1.23995	0.619975	+	0.784621i	$0.287143\pi$
$548$	−18.0000	−0.768922
$549$	−13.0000	−0.554826
$550$	−10.0000	−0.426401
$551$	0	0
$552$	−12.0000	−0.510754
$553$	−1.00000	−0.0425243
$554$	2.00000	0.0849719
$555$	0	0
$556$	−5.00000	−0.212047
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	−3.00000	−0.127000
$559$	5.00000	0.211477
$560$	0	0
$561$	8.00000	0.337760
$562$	−8.00000	−0.337460
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	4.00000	0.168133
$567$	1.00000	0.0419961
$568$	−18.0000	−0.755263
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	33.0000	1.38101	0.690504	−	0.723329i	$-0.257389\pi$
$571$	33.0000	1.38101	0.690504	+	0.723329i	$0.257389\pi$
$572$	−10.0000	−0.418121
$573$	24.0000	1.00261
$574$	−12.0000	−0.500870
$575$	20.0000	0.834058
$576$	7.00000	0.291667
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	1.00000	0.0415945
$579$	−19.0000	−0.789613
$580$	0	0
$581$	0	0
$582$	2.00000	0.0829027
$583$	8.00000	0.331326
$584$	−33.0000	−1.36555
$585$	0	0
$586$	14.0000	0.578335
$587$	−26.0000	−1.07313	−0.536567	−	0.843857i	$-0.680279\pi$
$587$	−26.0000	−1.07313	−0.536567	+	0.843857i	$0.680279\pi$
$588$	6.00000	0.247436
$589$	0	0
$590$	0	0
$591$	−26.0000	−1.06950
$592$	3.00000	0.123299
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	6.00000	0.245770
$597$	21.0000	0.859473
$598$	−20.0000	−0.817861
$599$	−18.0000	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$599$	−18.0000	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$600$	−15.0000	−0.612372
$601$	21.0000	0.856608	0.428304	−	0.903635i	$-0.359111\pi$
$601$	21.0000	0.856608	0.428304	+	0.903635i	$0.359111\pi$
$602$	1.00000	0.0407570
$603$	−11.0000	−0.447955
$604$	12.0000	0.488273
$605$	0	0
$606$	−14.0000	−0.568711
$607$	43.0000	1.74532	0.872658	−	0.488332i	$-0.162394\pi$
$607$	43.0000	1.74532	0.872658	+	0.488332i	$0.162394\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	30.0000	1.21367
$612$	4.00000	0.161690
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	−6.00000	−0.241747
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	13.0000	0.522937
$619$	25.0000	1.00483	0.502417	−	0.864625i	$-0.332444\pi$
$619$	25.0000	1.00483	0.502417	+	0.864625i	$0.332444\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	−18.0000	−0.721734
$623$	6.00000	0.240385
$624$	5.00000	0.200160
$625$	25.0000	1.00000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	−3.00000	−0.119713
$629$	12.0000	0.478471
$630$	0	0
$631$	−15.0000	−0.597141	−0.298570	−	0.954388i	$-0.596510\pi$
$631$	−15.0000	−0.597141	−0.298570	+	0.954388i	$0.596510\pi$
$632$	−3.00000	−0.119334
$633$	15.0000	0.596196
$634$	−4.00000	−0.158860
$635$	0	0
$636$	4.00000	0.158610
$637$	30.0000	1.18864
$638$	16.0000	0.633446
$639$	−6.00000	−0.237356
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	−18.0000	−0.710403
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	3.00000	0.117851
$649$	20.0000	0.785069
$650$	−25.0000	−0.980581
$651$	3.00000	0.117579
$652$	−3.00000	−0.117489
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	−12.0000	−0.468521
$657$	−11.0000	−0.429151
$658$	6.00000	0.233904
$659$	−34.0000	−1.32445	−0.662226	−	0.749304i	$-0.730388\pi$
$659$	−34.0000	−1.32445	−0.662226	+	0.749304i	$0.730388\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	−25.0000	−0.971653
$663$	20.0000	0.776736
$664$	0	0
$665$	0	0
$666$	3.00000	0.116248
$667$	−32.0000	−1.23904
$668$	−2.00000	−0.0773823
$669$	−9.00000	−0.347960
$670$	0	0
$671$	26.0000	1.00372
$672$	−5.00000	−0.192879
$673$	33.0000	1.27206	0.636028	−	0.771666i	$-0.280576\pi$
$673$	33.0000	1.27206	0.636028	+	0.771666i	$0.280576\pi$
$674$	13.0000	0.500741
$675$	−5.00000	−0.192450
$676$	−12.0000	−0.461538
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	2.00000	0.0768095
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	24.0000	0.919682
$682$	6.00000	0.229752
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	13.0000	0.495981
$688$	1.00000	0.0381246
$689$	20.0000	0.761939
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	16.0000	0.607352
$695$	0	0
$696$	24.0000	0.909718
$697$	−48.0000	−1.81813
$698$	21.0000	0.794862
$699$	−6.00000	−0.226941
$700$	5.00000	0.188982
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	5.00000	0.188713
$703$	0	0
$704$	−14.0000	−0.527645
$705$	0	0
$706$	−4.00000	−0.150542
$707$	14.0000	0.526524
$708$	10.0000	0.375823
$709$	−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
$711$	−1.00000	−0.0375029
$712$	18.0000	0.674579
$713$	−12.0000	−0.449404
$714$	4.00000	0.149696
$715$	0	0
$716$	12.0000	0.448461
$717$	−12.0000	−0.448148
$718$	20.0000	0.746393
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−13.0000	−0.484145
$722$	0	0
$723$	7.00000	0.260333
$724$	−2.00000	−0.0743294
$725$	−40.0000	−1.48556
$726$	7.00000	0.259794
$727$	−47.0000	−1.74313	−0.871567	−	0.490277i	$-0.836896\pi$
$727$	−47.0000	−1.74313	−0.871567	+	0.490277i	$0.836896\pi$
$728$	−15.0000	−0.555937
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.00000	0.147945
$732$	13.0000	0.480494
$733$	−50.0000	−1.84679	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000	−1.84679	−0.923396	+	0.383849i	$0.874598\pi$
$734$	−7.00000	−0.258375
$735$	0	0
$736$	20.0000	0.737210
$737$	22.0000	0.810380
$738$	−12.0000	−0.441726
$739$	19.0000	0.698926	0.349463	−	0.936950i	$-0.386364\pi$
$739$	19.0000	0.698926	0.349463	+	0.936950i	$0.386364\pi$
$740$	0	0
$741$	0	0
$742$	4.00000	0.146845
$743$	50.0000	1.83432	0.917161	−	0.398517i	$-0.130475\pi$
$743$	50.0000	1.83432	0.917161	+	0.398517i	$0.130475\pi$
$744$	9.00000	0.329956
$745$	0	0
$746$	10.0000	0.366126
$747$	0	0
$748$	−8.00000	−0.292509
$749$	18.0000	0.657706
$750$	0	0
$751$	−45.0000	−1.64207	−0.821037	−	0.570875i	$-0.806604\pi$
$751$	−45.0000	−1.64207	−0.821037	+	0.570875i	$0.806604\pi$
$752$	6.00000	0.218797
$753$	−2.00000	−0.0728841
$754$	40.0000	1.45671
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	13.0000	0.472493	0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000	0.472493	0.236247	+	0.971693i	$0.424083\pi$
$758$	−5.00000	−0.181608
$759$	8.00000	0.290382
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−8.00000	−0.289809
$763$	2.00000	0.0724049
$764$	−24.0000	−0.868290
$765$	0	0
$766$	14.0000	0.505841
$767$	50.0000	1.80540
$768$	−17.0000	−0.613435
$769$	−37.0000	−1.33425	−0.667127	−	0.744944i	$-0.732476\pi$
$769$	−37.0000	−1.33425	−0.667127	+	0.744944i	$0.732476\pi$
$770$	0	0
$771$	20.0000	0.720282
$772$	19.0000	0.683825
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	1.00000	0.0359443
$775$	−15.0000	−0.538816
$776$	−6.00000	−0.215387
$777$	−3.00000	−0.107624
$778$	24.0000	0.860442
$779$	0	0
$780$	0	0
$781$	12.0000	0.429394
$782$	−16.0000	−0.572159
$783$	8.00000	0.285897
$784$	6.00000	0.214286
$785$	0	0
$786$	14.0000	0.499363
$787$	−25.0000	−0.891154	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	−25.0000	−0.891154	−0.445577	+	0.895244i	$0.647001\pi$
$788$	26.0000	0.926212
$789$	26.0000	0.925625
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	−6.00000	−0.213201
$793$	65.0000	2.30822
$794$	−7.00000	−0.248421
$795$	0	0
$796$	−21.0000	−0.744325
$797$	14.0000	0.495905	0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000	0.495905	0.247953	+	0.968772i	$0.420242\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	25.0000	0.883883
$801$	6.00000	0.212000
$802$	−6.00000	−0.211867
$803$	22.0000	0.776363
$804$	11.0000	0.387940
$805$	0	0
$806$	15.0000	0.528352
$807$	−10.0000	−0.352017
$808$	42.0000	1.47755
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	−8.00000	−0.280745
$813$	0	0
$814$	−6.00000	−0.210300
$815$	0	0
$816$	4.00000	0.140028
$817$	0	0
$818$	−14.0000	−0.489499
$819$	−5.00000	−0.174714
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	−18.0000	−0.627822
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	−39.0000	−1.35863
$825$	10.0000	0.348155
$826$	10.0000	0.347945
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	4.00000	0.139010
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	−35.0000	−1.21341
$833$	24.0000	0.831551
$834$	−5.00000	−0.173136
$835$	0	0
$836$	0	0
$837$	3.00000	0.103695
$838$	14.0000	0.483622
$839$	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
$841$	35.0000	1.20690
$842$	−22.0000	−0.758170
$843$	8.00000	0.275535
$844$	−15.0000	−0.516321
$845$	0	0
$846$	6.00000	0.206284
$847$	−7.00000	−0.240523
$848$	4.00000	0.137361
$849$	−4.00000	−0.137280
$850$	−20.0000	−0.685994
$851$	12.0000	0.411355
$852$	6.00000	0.205557
$853$	−5.00000	−0.171197	−0.0855984	−	0.996330i	$-0.527280\pi$
$853$	−5.00000	−0.171197	−0.0855984	+	0.996330i	$0.527280\pi$
$854$	13.0000	0.444851
$855$	0	0
$856$	54.0000	1.84568
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	−10.0000	−0.341394
$859$	15.0000	0.511793	0.255897	−	0.966704i	$-0.417629\pi$
$859$	15.0000	0.511793	0.255897	+	0.966704i	$0.417629\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	−10.0000	−0.340601
$863$	−10.0000	−0.340404	−0.170202	−	0.985409i	$-0.554442\pi$
$863$	−10.0000	−0.340404	−0.170202	+	0.985409i	$0.554442\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	−9.00000	−0.305832
$867$	−1.00000	−0.0339618
$868$	−3.00000	−0.101827
$869$	2.00000	0.0678454
$870$	0	0
$871$	55.0000	1.86360
$872$	6.00000	0.203186
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	0	0
$876$	11.0000	0.371656
$877$	−15.0000	−0.506514	−0.253257	−	0.967399i	$-0.581502\pi$
$877$	−15.0000	−0.506514	−0.253257	+	0.967399i	$0.581502\pi$
$878$	35.0000	1.18119
$879$	−14.0000	−0.472208
$880$	0	0
$881$	−28.0000	−0.943344	−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000	−0.943344	−0.471672	+	0.881774i	$0.656349\pi$
$882$	6.00000	0.202031
$883$	−1.00000	−0.0336527	−0.0168263	−	0.999858i	$-0.505356\pi$
$883$	−1.00000	−0.0336527	−0.0168263	+	0.999858i	$0.505356\pi$
$884$	−20.0000	−0.672673
$885$	0	0
$886$	32.0000	1.07506
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	−9.00000	−0.302020
$889$	8.00000	0.268311
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	9.00000	0.301342
$893$	0	0
$894$	6.00000	0.200670
$895$	0	0
$896$	3.00000	0.100223
$897$	20.0000	0.667781
$898$	−12.0000	−0.400445
$899$	24.0000	0.800445
$900$	5.00000	0.166667
$901$	16.0000	0.533037
$902$	24.0000	0.799113
$903$	−1.00000	−0.0332779
$904$	−6.00000	−0.199557
$905$	0	0
$906$	12.0000	0.398673
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	−24.0000	−0.796468
$909$	14.0000	0.464351
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	0	0
$914$	11.0000	0.363848
$915$	0	0
$916$	−13.0000	−0.429532
$917$	−14.0000	−0.462321
$918$	4.00000	0.132020
$919$	−25.0000	−0.824674	−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000	−0.824674	−0.412337	+	0.911031i	$0.635287\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	−30.0000	−0.987997
$923$	30.0000	0.987462
$924$	2.00000	0.0657952
$925$	15.0000	0.493197
$926$	−23.0000	−0.755827
$927$	−13.0000	−0.426976
$928$	−40.0000	−1.31306
$929$	28.0000	0.918650	0.459325	−	0.888268i	$-0.348091\pi$
$929$	28.0000	0.918650	0.459325	+	0.888268i	$0.348091\pi$
$930$	0	0
$931$	0	0
$932$	6.00000	0.196537
$933$	18.0000	0.589294
$934$	8.00000	0.261768
$935$	0	0
$936$	−15.0000	−0.490290
$937$	−9.00000	−0.294017	−0.147009	−	0.989135i	$-0.546964\pi$
$937$	−9.00000	−0.294017	−0.147009	+	0.989135i	$0.546964\pi$
$938$	11.0000	0.359163
$939$	22.0000	0.717943
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	−3.00000	−0.0977453
$943$	−48.0000	−1.56310
$944$	10.0000	0.325472
$945$	0	0
$946$	−2.00000	−0.0650256
$947$	50.0000	1.62478	0.812391	−	0.583113i	$-0.198166\pi$
$947$	50.0000	1.62478	0.812391	+	0.583113i	$0.198166\pi$
$948$	1.00000	0.0324785
$949$	55.0000	1.78538
$950$	0	0
$951$	4.00000	0.129709
$952$	−12.0000	−0.388922
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	4.00000	0.129505
$955$	0	0
$956$	12.0000	0.388108
$957$	−16.0000	−0.517207
$958$	−14.0000	−0.452319
$959$	18.0000	0.581250
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−15.0000	−0.483619
$963$	18.0000	0.580042
$964$	−7.00000	−0.225455
$965$	0	0
$966$	4.00000	0.128698
$967$	−29.0000	−0.932577	−0.466289	−	0.884633i	$-0.654409\pi$
$967$	−29.0000	−0.932577	−0.466289	+	0.884633i	$0.654409\pi$
$968$	−21.0000	−0.674966
$969$	0	0
$970$	0	0
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	−1.00000	−0.0320750
$973$	5.00000	0.160293
$974$	−16.0000	−0.512673
$975$	25.0000	0.800641
$976$	13.0000	0.416120
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	−3.00000	−0.0959294
$979$	−12.0000	−0.383522
$980$	0	0
$981$	2.00000	0.0638551
$982$	−6.00000	−0.191468
$983$	−10.0000	−0.318950	−0.159475	−	0.987202i	$-0.550980\pi$
$983$	−10.0000	−0.318950	−0.159475	+	0.987202i	$0.550980\pi$
$984$	36.0000	1.14764
$985$	0	0
$986$	32.0000	1.01909
$987$	−6.00000	−0.190982
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	−47.0000	−1.49300	−0.746502	−	0.665383i	$-0.768268\pi$
$991$	−47.0000	−1.49300	−0.746502	+	0.665383i	$0.768268\pi$
$992$	−15.0000	−0.476250
$993$	25.0000	0.793351
$994$	6.00000	0.190308
$995$	0	0
$996$	0	0
$997$	7.00000	0.221692	0.110846	−	0.993838i	$-0.464644\pi$
$997$	7.00000	0.221692	0.110846	+	0.993838i	$0.464644\pi$
$998$	−29.0000	−0.917979
$999$	−3.00000	−0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1083.2.a.b.1.1		1
3.2	odd	2		3249.2.a.f.1.1		1
19.8	odd	6		57.2.e.a.7.1	✓	2
19.12	odd	6		57.2.e.a.49.1	yes	2
19.18	odd	2		1083.2.a.c.1.1		1
57.8	even	6		171.2.f.a.64.1		2
57.50	even	6		171.2.f.a.163.1		2
57.56	even	2		3249.2.a.c.1.1		1
76.27	even	6		912.2.q.a.577.1		2
76.31	even	6		912.2.q.a.49.1		2
228.107	odd	6		2736.2.s.j.1873.1		2
228.179	odd	6		2736.2.s.j.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.e.a.7.1	✓	2	19.8	odd	6
57.2.e.a.49.1	yes	2	19.12	odd	6
171.2.f.a.64.1		2	57.8	even	6
171.2.f.a.163.1		2	57.50	even	6
912.2.q.a.49.1		2	76.31	even	6
912.2.q.a.577.1		2	76.27	even	6
1083.2.a.b.1.1		1	1.1	even	1	trivial
1083.2.a.c.1.1		1	19.18	odd	2
2736.2.s.j.577.1		2	228.179	odd	6
2736.2.s.j.1873.1		2	228.107	odd	6
3249.2.a.c.1.1		1	57.56	even	2
3249.2.a.f.1.1		1	3.2	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:	\( N \)	\(=\)	\( 1083 = 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1083.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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