LMFDB - Embedded newform 966.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(1,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, 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("966.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	966.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:	$7.71354883526$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	966.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} -1.00000 q^{21} +2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{32} -2.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +6.00000 q^{38} -6.00000 q^{39} +6.00000 q^{41} +1.00000 q^{42} -6.00000 q^{43} -2.00000 q^{44} -1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +5.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} -4.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} -6.00000 q^{57} +6.00000 q^{58} -8.00000 q^{61} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -2.00000 q^{67} +2.00000 q^{68} +1.00000 q^{69} -1.00000 q^{72} -2.00000 q^{73} -5.00000 q^{75} -6.00000 q^{76} +2.00000 q^{77} +6.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -2.00000 q^{83} -1.00000 q^{84} +6.00000 q^{86} -6.00000 q^{87} +2.00000 q^{88} -2.00000 q^{89} +6.00000 q^{91} +1.00000 q^{92} -8.00000 q^{94} -1.00000 q^{96} +14.0000 q^{97} -1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
$2$	−1.00000	−0.707107
$3$	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
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	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
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−1.00000	−0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	2.00000	0.426401
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	−5.00000	−1.00000
$26$	6.00000	1.17670
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	−2.00000	−0.348155
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	6.00000	0.973329
$39$	−6.00000	−0.960769
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	1.00000	0.154303
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−1.00000	−0.147442
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	5.00000	0.707107
$51$	2.00000	0.280056
$52$	−6.00000	−0.832050
$53$	−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
$56$	1.00000	0.133631
$57$	−6.00000	−0.794719
$58$	6.00000	0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	2.00000	0.242536
$69$	1.00000	0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	−6.00000	−0.688247
$77$	2.00000	0.227921
$78$	6.00000	0.679366
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	6.00000	0.646997
$87$	−6.00000	−0.643268
$88$	2.00000	0.213201
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	1.00000	0.104257
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−1.00000	−0.101015
$99$	−2.00000	−0.201008
$100$	−5.00000	−0.500000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	−2.00000	−0.198030
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	4.00000	0.388514
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	1.00000	0.0962250
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	6.00000	0.561951
$115$	0	0
$116$	−6.00000	−0.557086
$117$	−6.00000	−0.554700
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−7.00000	−0.636364
$122$	8.00000	0.724286
$123$	6.00000	0.541002
$124$	0	0
$125$	0	0
$126$	1.00000	0.0890871
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	−6.00000	−0.528271
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−2.00000	−0.174078
$133$	6.00000	0.520266
$134$	2.00000	0.172774
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−1.00000	−0.0851257
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	12.0000	1.00349
$144$	1.00000	0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	1.00000	0.0824786
$148$	0	0
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	5.00000	0.408248
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	6.00000	0.486664
$153$	2.00000	0.161690
$154$	−2.00000	−0.161165
$155$	0	0
$156$	−6.00000	−0.480384
$157$	−20.0000	−1.59617	−0.798087	−	0.602542i	$-0.794154\pi$
$157$	−20.0000	−1.59617	−0.798087	+	0.602542i	$0.794154\pi$
$158$	−8.00000	−0.636446
$159$	−4.00000	−0.317221
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	−1.00000	−0.0785674
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	2.00000	0.155230
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	1.00000	0.0771517
$169$	23.0000	1.76923
$170$	0	0
$171$	−6.00000	−0.458831
$172$	−6.00000	−0.457496
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	6.00000	0.454859
$175$	5.00000	0.377964
$176$	−2.00000	−0.150756
$177$	0	0
$178$	2.00000	0.149906
$179$	−8.00000	−0.597948	−0.298974	−	0.954261i	$-0.596644\pi$
$179$	−8.00000	−0.597948	−0.298974	+	0.954261i	$0.596644\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	−6.00000	−0.444750
$183$	−8.00000	−0.591377
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	8.00000	0.583460
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	1.00000	0.0721688
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	1.00000	0.0714286
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	2.00000	0.142134
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	5.00000	0.353553
$201$	−2.00000	−0.141069
$202$	14.0000	0.985037
$203$	6.00000	0.421117
$204$	2.00000	0.140028
$205$	0	0
$206$	4.00000	0.278693
$207$	1.00000	0.0695048
$208$	−6.00000	−0.416025
$209$	12.0000	0.830057
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	−2.00000	−0.136717
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	12.0000	0.812743
$219$	−2.00000	−0.135147
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	1.00000	0.0668153
$225$	−5.00000	−0.333333
$226$	−6.00000	−0.399114
$227$	26.0000	1.72568	0.862840	−	0.505477i	$-0.168683\pi$
$227$	26.0000	1.72568	0.862840	+	0.505477i	$0.168683\pi$
$228$	−6.00000	−0.397360
$229$	24.0000	1.58596	0.792982	−	0.609245i	$-0.208527\pi$
$229$	24.0000	1.58596	0.792982	+	0.609245i	$0.208527\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	6.00000	0.393919
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	0	0
$237$	8.00000	0.519656
$238$	2.00000	0.129641
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	−8.00000	−0.512148
$245$	0	0
$246$	−6.00000	−0.382546
$247$	36.0000	2.29063
$248$	0	0
$249$	−2.00000	−0.126745
$250$	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
$253$	−2.00000	−0.125739
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	6.00000	0.373544
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	−6.00000	−0.367884
$267$	−2.00000	−0.122398
$268$	−2.00000	−0.122169
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	2.00000	0.121268
$273$	6.00000	0.363137
$274$	6.00000	0.362473
$275$	10.0000	0.603023
$276$	1.00000	0.0601929
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	−8.00000	−0.476393
$283$	30.0000	1.78331	0.891657	−	0.452711i	$-0.149543\pi$
$283$	30.0000	1.78331	0.891657	+	0.452711i	$0.149543\pi$
$284$	0	0
$285$	0	0
$286$	−12.0000	−0.709575
$287$	−6.00000	−0.354169
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	14.0000	0.820695
$292$	−2.00000	−0.117041
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	−1.00000	−0.0583212
$295$	0	0
$296$	0	0
$297$	−2.00000	−0.116052
$298$	−4.00000	−0.231714
$299$	−6.00000	−0.346989
$300$	−5.00000	−0.288675
$301$	6.00000	0.345834
$302$	24.0000	1.38104
$303$	−14.0000	−0.804279
$304$	−6.00000	−0.344124
$305$	0	0
$306$	−2.00000	−0.114332
$307$	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
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	6.00000	0.339683
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	20.0000	1.12867
$315$	0	0
$316$	8.00000	0.450035
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	4.00000	0.224309
$319$	12.0000	0.671871
$320$	0	0
$321$	2.00000	0.111629
$322$	1.00000	0.0557278
$323$	−12.0000	−0.667698
$324$	1.00000	0.0555556
$325$	30.0000	1.66410
$326$	−16.0000	−0.886158
$327$	−12.0000	−0.663602
$328$	−6.00000	−0.331295
$329$	−8.00000	−0.441054
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−2.00000	−0.109764
$333$	0	0
$334$	−16.0000	−0.875481
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	−23.0000	−1.25104
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	6.00000	0.324443
$343$	−1.00000	−0.0539949
$344$	6.00000	0.323498
$345$	0	0
$346$	−6.00000	−0.322562
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	−6.00000	−0.321634
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	−5.00000	−0.267261
$351$	−6.00000	−0.320256
$352$	2.00000	0.106600
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	0	0
$356$	−2.00000	−0.106000
$357$	−2.00000	−0.105851
$358$	8.00000	0.422813
$359$	−28.0000	−1.47778	−0.738892	−	0.673824i	$-0.764651\pi$
$359$	−28.0000	−1.47778	−0.738892	+	0.673824i	$0.764651\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	8.00000	0.420471
$363$	−7.00000	−0.367405
$364$	6.00000	0.314485
$365$	0	0
$366$	8.00000	0.418167
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	1.00000	0.0521286
$369$	6.00000	0.312348
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	−8.00000	−0.414224	−0.207112	−	0.978317i	$-0.566407\pi$
$373$	−8.00000	−0.414224	−0.207112	+	0.978317i	$0.566407\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	−8.00000	−0.412568
$377$	36.0000	1.85409
$378$	1.00000	0.0514344
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	12.0000	0.613973
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−10.0000	−0.508987
$387$	−6.00000	−0.304997
$388$	14.0000	0.710742
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−10.0000	−0.503793
$395$	0	0
$396$	−2.00000	−0.100504
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−16.0000	−0.802008
$399$	6.00000	0.300376
$400$	−5.00000	−0.250000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	2.00000	0.0997509
$403$	0	0
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−6.00000	−0.297775
$407$	0	0
$408$	−2.00000	−0.0990148
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	−4.00000	−0.197066
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	6.00000	0.294174
$417$	0	0
$418$	−12.0000	−0.586939
$419$	38.0000	1.85642	0.928211	−	0.372055i	$-0.121347\pi$
$419$	38.0000	1.85642	0.928211	+	0.372055i	$0.121347\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	−12.0000	−0.584151
$423$	8.00000	0.388973
$424$	4.00000	0.194257
$425$	−10.0000	−0.485071
$426$	0	0
$427$	8.00000	0.387147
$428$	2.00000	0.0966736
$429$	12.0000	0.579365
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	1.00000	0.0481125
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	0	0
$436$	−12.0000	−0.574696
$437$	−6.00000	−0.287019
$438$	2.00000	0.0955637
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	12.0000	0.570782
$443$	−32.0000	−1.52037	−0.760183	−	0.649709i	$-0.774891\pi$
$443$	−32.0000	−1.52037	−0.760183	+	0.649709i	$0.774891\pi$
$444$	0	0
$445$	0	0
$446$	−16.0000	−0.757622
$447$	4.00000	0.189194
$448$	−1.00000	−0.0472456
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	5.00000	0.235702
$451$	−12.0000	−0.565058
$452$	6.00000	0.282216
$453$	−24.0000	−1.12762
$454$	−26.0000	−1.22024
$455$	0	0
$456$	6.00000	0.280976
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	−24.0000	−1.12145
$459$	2.00000	0.0933520
$460$	0	0
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	−2.00000	−0.0930484
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−26.0000	−1.20443
$467$	22.0000	1.01804	0.509019	−	0.860755i	$-0.330008\pi$
$467$	22.0000	1.01804	0.509019	+	0.860755i	$0.330008\pi$
$468$	−6.00000	−0.277350
$469$	2.00000	0.0923514
$470$	0	0
$471$	−20.0000	−0.921551
$472$	0	0
$473$	12.0000	0.551761
$474$	−8.00000	−0.367452
$475$	30.0000	1.37649
$476$	−2.00000	−0.0916698
$477$	−4.00000	−0.183147
$478$	−24.0000	−1.09773
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	0	0
$482$	2.00000	0.0910975
$483$	−1.00000	−0.0455016
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	8.00000	0.362143
$489$	16.0000	0.723545
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	6.00000	0.270501
$493$	−12.0000	−0.540453
$494$	−36.0000	−1.61972
$495$	0	0
$496$	0	0
$497$	0	0
$498$	2.00000	0.0896221
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	2.00000	0.0892644
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	2.00000	0.0889108
$507$	23.0000	1.02147
$508$	−8.00000	−0.354943
$509$	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
$511$	2.00000	0.0884748
$512$	−1.00000	−0.0441942
$513$	−6.00000	−0.264906
$514$	2.00000	0.0882162
$515$	0	0
$516$	−6.00000	−0.264135
$517$	−16.0000	−0.703679
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	6.00000	0.262613
$523$	18.0000	0.787085	0.393543	−	0.919306i	$-0.371249\pi$
$523$	18.0000	0.787085	0.393543	+	0.919306i	$0.371249\pi$
$524$	0	0
$525$	5.00000	0.218218
$526$	24.0000	1.04645
$527$	0	0
$528$	−2.00000	−0.0870388
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	6.00000	0.260133
$533$	−36.0000	−1.55933
$534$	2.00000	0.0865485
$535$	0	0
$536$	2.00000	0.0863868
$537$	−8.00000	−0.345225
$538$	−6.00000	−0.258678
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	−8.00000	−0.343629
$543$	−8.00000	−0.343313
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	−6.00000	−0.256776
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	−6.00000	−0.256307
$549$	−8.00000	−0.341432
$550$	−10.0000	−0.426401
$551$	36.0000	1.53365
$552$	−1.00000	−0.0425628
$553$	−8.00000	−0.340195
$554$	22.0000	0.934690
$555$	0	0
$556$	0	0
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	0	0
$559$	36.0000	1.52264
$560$	0	0
$561$	−4.00000	−0.168880
$562$	10.0000	0.421825
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	8.00000	0.336861
$565$	0	0
$566$	−30.0000	−1.26099
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−46.0000	−1.92842	−0.964210	−	0.265139i	$-0.914582\pi$
$569$	−46.0000	−1.92842	−0.964210	+	0.265139i	$0.914582\pi$
$570$	0	0
$571$	−2.00000	−0.0836974	−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000	−0.0836974	−0.0418487	+	0.999124i	$0.513325\pi$
$572$	12.0000	0.501745
$573$	−12.0000	−0.501307
$574$	6.00000	0.250435
$575$	−5.00000	−0.208514
$576$	1.00000	0.0416667
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	13.0000	0.540729
$579$	10.0000	0.415586
$580$	0	0
$581$	2.00000	0.0829740
$582$	−14.0000	−0.580319
$583$	8.00000	0.331326
$584$	2.00000	0.0827606
$585$	0	0
$586$	24.0000	0.991431
$587$	48.0000	1.98117	0.990586	−	0.136892i	$-0.0437113\pi$
$587$	48.0000	1.98117	0.990586	+	0.136892i	$0.0437113\pi$
$588$	1.00000	0.0412393
$589$	0	0
$590$	0	0
$591$	10.0000	0.411345
$592$	0	0
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	4.00000	0.163846
$597$	16.0000	0.654836
$598$	6.00000	0.245358
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	5.00000	0.204124
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	−6.00000	−0.244542
$603$	−2.00000	−0.0814463
$604$	−24.0000	−0.976546
$605$	0	0
$606$	14.0000	0.568711
$607$	48.0000	1.94826	0.974130	−	0.225989i	$-0.0725612\pi$
$607$	48.0000	1.94826	0.974130	+	0.225989i	$0.0725612\pi$
$608$	6.00000	0.243332
$609$	6.00000	0.243132
$610$	0	0
$611$	−48.0000	−1.94187
$612$	2.00000	0.0808452
$613$	−20.0000	−0.807792	−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000	−0.807792	−0.403896	+	0.914805i	$0.632344\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	4.00000	0.160904
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	16.0000	0.641542
$623$	2.00000	0.0801283
$624$	−6.00000	−0.240192
$625$	25.0000	1.00000
$626$	26.0000	1.03917
$627$	12.0000	0.479234
$628$	−20.0000	−0.798087
$629$	0	0
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	−8.00000	−0.318223
$633$	12.0000	0.476957
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	−4.00000	−0.158610
$637$	−6.00000	−0.237729
$638$	−12.0000	−0.475085
$639$	0	0
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−2.00000	−0.0789337
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	12.0000	0.472134
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−30.0000	−1.17670
$651$	0	0
$652$	16.0000	0.626608
$653$	22.0000	0.860927	0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000	0.860927	0.430463	+	0.902608i	$0.358350\pi$
$654$	12.0000	0.469237
$655$	0	0
$656$	6.00000	0.234261
$657$	−2.00000	−0.0780274
$658$	8.00000	0.311872
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	−20.0000	−0.777322
$663$	−12.0000	−0.466041
$664$	2.00000	0.0776151
$665$	0	0
$666$	0	0
$667$	−6.00000	−0.232321
$668$	16.0000	0.619059
$669$	16.0000	0.618596
$670$	0	0
$671$	16.0000	0.617673
$672$	1.00000	0.0385758
$673$	−50.0000	−1.92736	−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000	−1.92736	−0.963679	+	0.267063i	$0.913947\pi$
$674$	6.00000	0.231111
$675$	−5.00000	−0.192450
$676$	23.0000	0.884615
$677$	−8.00000	−0.307465	−0.153732	−	0.988113i	$-0.549129\pi$
$677$	−8.00000	−0.307465	−0.153732	+	0.988113i	$0.549129\pi$
$678$	−6.00000	−0.230429
$679$	−14.0000	−0.537271
$680$	0	0
$681$	26.0000	0.996322
$682$	0	0
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	1.00000	0.0381802
$687$	24.0000	0.915657
$688$	−6.00000	−0.228748
$689$	24.0000	0.914327
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	6.00000	0.228086
$693$	2.00000	0.0759737
$694$	−8.00000	−0.303676
$695$	0	0
$696$	6.00000	0.227429
$697$	12.0000	0.454532
$698$	2.00000	0.0757011
$699$	26.0000	0.983410
$700$	5.00000	0.188982
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	6.00000	0.226455
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	14.0000	0.526897
$707$	14.0000	0.526524
$708$	0	0
$709$	32.0000	1.20179	0.600893	−	0.799330i	$-0.294812\pi$
$709$	32.0000	1.20179	0.600893	+	0.799330i	$0.294812\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	2.00000	0.0749532
$713$	0	0
$714$	2.00000	0.0748481
$715$	0	0
$716$	−8.00000	−0.298974
$717$	24.0000	0.896296
$718$	28.0000	1.04495
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	−17.0000	−0.632674
$723$	−2.00000	−0.0743808
$724$	−8.00000	−0.297318
$725$	30.0000	1.11417
$726$	7.00000	0.259794
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.0000	−0.443836
$732$	−8.00000	−0.295689
$733$	20.0000	0.738717	0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000	0.738717	0.369358	+	0.929287i	$0.379577\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	4.00000	0.147342
$738$	−6.00000	−0.220863
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	0	0
$741$	36.0000	1.32249
$742$	−4.00000	−0.146845
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	0	0
$745$	0	0
$746$	8.00000	0.292901
$747$	−2.00000	−0.0731762
$748$	−4.00000	−0.146254
$749$	−2.00000	−0.0730784
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	8.00000	0.291730
$753$	−2.00000	−0.0728841
$754$	−36.0000	−1.31104
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	34.0000	1.23494
$759$	−2.00000	−0.0725954
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	8.00000	0.289809
$763$	12.0000	0.434429
$764$	−12.0000	−0.434145
$765$	0	0
$766$	0	0
$767$	0	0
$768$	1.00000	0.0360844
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	10.0000	0.359908
$773$	28.0000	1.00709	0.503545	−	0.863969i	$-0.332029\pi$
$773$	28.0000	1.00709	0.503545	+	0.863969i	$0.332029\pi$
$774$	6.00000	0.215666
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	0	0
$779$	−36.0000	−1.28983
$780$	0	0
$781$	0	0
$782$	−2.00000	−0.0715199
$783$	−6.00000	−0.214423
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	−10.0000	−0.356462	−0.178231	−	0.983989i	$-0.557037\pi$
$787$	−10.0000	−0.356462	−0.178231	+	0.983989i	$0.557037\pi$
$788$	10.0000	0.356235
$789$	−24.0000	−0.854423
$790$	0	0
$791$	−6.00000	−0.213335
$792$	2.00000	0.0710669
$793$	48.0000	1.70453
$794$	14.0000	0.496841
$795$	0	0
$796$	16.0000	0.567105
$797$	−52.0000	−1.84193	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	−52.0000	−1.84193	−0.920967	+	0.389640i	$0.872599\pi$
$798$	−6.00000	−0.212398
$799$	16.0000	0.566039
$800$	5.00000	0.176777
$801$	−2.00000	−0.0706665
$802$	−2.00000	−0.0706225
$803$	4.00000	0.141157
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	0	0
$807$	6.00000	0.211210
$808$	14.0000	0.492518
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	−48.0000	−1.68551	−0.842754	−	0.538299i	$-0.819067\pi$
$811$	−48.0000	−1.68551	−0.842754	+	0.538299i	$0.819067\pi$
$812$	6.00000	0.210559
$813$	8.00000	0.280572
$814$	0	0
$815$	0	0
$816$	2.00000	0.0700140
$817$	36.0000	1.25948
$818$	−10.0000	−0.349642
$819$	6.00000	0.209657
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	6.00000	0.209274
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	4.00000	0.139347
$825$	10.0000	0.348155
$826$	0	0
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	1.00000	0.0347524
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	−6.00000	−0.208013
$833$	2.00000	0.0692959
$834$	0	0
$835$	0	0
$836$	12.0000	0.415029
$837$	0	0
$838$	−38.0000	−1.31269
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−10.0000	−0.344418
$844$	12.0000	0.413057
$845$	0	0
$846$	−8.00000	−0.275046
$847$	7.00000	0.240523
$848$	−4.00000	−0.137361
$849$	30.0000	1.02960
$850$	10.0000	0.342997
$851$	0	0
$852$	0	0
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	−2.00000	−0.0683586
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	−12.0000	−0.409673
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	36.0000	1.22616
$863$	−56.0000	−1.90626	−0.953131	−	0.302558i	$-0.902160\pi$
$863$	−56.0000	−1.90626	−0.953131	+	0.302558i	$0.902160\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	34.0000	1.15537
$867$	−13.0000	−0.441503
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	12.0000	0.406604
$872$	12.0000	0.406371
$873$	14.0000	0.473828
$874$	6.00000	0.202953
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	16.0000	0.539974
$879$	−24.0000	−0.809500
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	−1.00000	−0.0336718
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	32.0000	1.07506
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	16.0000	0.535720
$893$	−48.0000	−1.60626
$894$	−4.00000	−0.133780
$895$	0	0
$896$	1.00000	0.0334077
$897$	−6.00000	−0.200334
$898$	−6.00000	−0.200223
$899$	0	0
$900$	−5.00000	−0.166667
$901$	−8.00000	−0.266519
$902$	12.0000	0.399556
$903$	6.00000	0.199667
$904$	−6.00000	−0.199557
$905$	0	0
$906$	24.0000	0.797347
$907$	−38.0000	−1.26177	−0.630885	−	0.775877i	$-0.717308\pi$
$907$	−38.0000	−1.26177	−0.630885	+	0.775877i	$0.717308\pi$
$908$	26.0000	0.862840
$909$	−14.0000	−0.464351
$910$	0	0
$911$	44.0000	1.45779	0.728893	−	0.684628i	$-0.240035\pi$
$911$	44.0000	1.45779	0.728893	+	0.684628i	$0.240035\pi$
$912$	−6.00000	−0.198680
$913$	4.00000	0.132381
$914$	18.0000	0.595387
$915$	0	0
$916$	24.0000	0.792982
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	10.0000	0.329332
$923$	0	0
$924$	2.00000	0.0657952
$925$	0	0
$926$	−16.0000	−0.525793
$927$	−4.00000	−0.131377
$928$	6.00000	0.196960
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	26.0000	0.851658
$933$	−16.0000	−0.523816
$934$	−22.0000	−0.719862
$935$	0	0
$936$	6.00000	0.196116
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	−2.00000	−0.0653023
$939$	−26.0000	−0.848478
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	20.0000	0.651635
$943$	6.00000	0.195387
$944$	0	0
$945$	0	0
$946$	−12.0000	−0.390154
$947$	32.0000	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$947$	32.0000	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$948$	8.00000	0.259828
$949$	12.0000	0.389536
$950$	−30.0000	−0.973329
$951$	2.00000	0.0648544
$952$	2.00000	0.0648204
$953$	58.0000	1.87880	0.939402	−	0.342817i	$-0.111381\pi$
$953$	58.0000	1.87880	0.939402	+	0.342817i	$0.111381\pi$
$954$	4.00000	0.129505
$955$	0	0
$956$	24.0000	0.776215
$957$	12.0000	0.387905
$958$	24.0000	0.775405
$959$	6.00000	0.193750
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	2.00000	0.0644491
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	1.00000	0.0321745
$967$	−56.0000	−1.80084	−0.900419	−	0.435023i	$-0.856740\pi$
$967$	−56.0000	−1.80084	−0.900419	+	0.435023i	$0.856740\pi$
$968$	7.00000	0.224989
$969$	−12.0000	−0.385496
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	8.00000	0.256337
$975$	30.0000	0.960769
$976$	−8.00000	−0.256074
$977$	10.0000	0.319928	0.159964	−	0.987123i	$-0.448862\pi$
$977$	10.0000	0.319928	0.159964	+	0.987123i	$0.448862\pi$
$978$	−16.0000	−0.511624
$979$	4.00000	0.127841
$980$	0	0
$981$	−12.0000	−0.383131
$982$	0	0
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	12.0000	0.382158
$987$	−8.00000	−0.254643
$988$	36.0000	1.14531
$989$	−6.00000	−0.190789
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	0	0
$995$	0	0
$996$	−2.00000	−0.0633724
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	−16.0000	−0.506471
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.2.a.e.1.1	✓	1
3.2	odd	2		2898.2.a.n.1.1		1
4.3	odd	2		7728.2.a.g.1.1		1
7.6	odd	2		6762.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.e.1.1	✓	1	1.1	even	1	trivial
2898.2.a.n.1.1		1	3.2	odd	2
6762.2.a.f.1.1		1	7.6	odd	2
7728.2.a.g.1.1		1	4.3	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:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	966.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more