LMFDB - Embedded newform 4830.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4830,2,Mod(1,4830)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4830, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4830.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4830.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:	$38.5677441763$
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	$\chi$	$=$	4830.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} -2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -8.00000 q^{29} -1.00000 q^{30} -10.0000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} +6.00000 q^{38} -2.00000 q^{39} -1.00000 q^{40} -10.0000 q^{41} -1.00000 q^{42} +2.00000 q^{43} +2.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} -2.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +2.00000 q^{55} -1.00000 q^{56} -6.00000 q^{57} +8.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} +10.0000 q^{61} +10.0000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -2.00000 q^{66} -14.0000 q^{67} +1.00000 q^{69} -1.00000 q^{70} +6.00000 q^{71} -1.00000 q^{72} +10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{75} -6.00000 q^{76} +2.00000 q^{77} +2.00000 q^{78} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -18.0000 q^{83} +1.00000 q^{84} -2.00000 q^{86} -8.00000 q^{87} -2.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} -2.00000 q^{91} +1.00000 q^{92} -10.0000 q^{93} +2.00000 q^{94} -6.00000 q^{95} -1.00000 q^{96} +14.0000 q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−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	$0.741620\pi$
$20$	1.00000	0.223607
$21$	1.00000	0.218218
$22$	−2.00000	−0.426401
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	−1.00000	−0.182574
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	0	0
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	6.00000	0.973329
$39$	−2.00000	−0.320256
$40$	−1.00000	−0.158114
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	−1.00000	−0.154303
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	2.00000	0.301511
$45$	1.00000	0.149071
$46$	−1.00000	−0.147442
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	2.00000	0.269680
$56$	−1.00000	−0.133631
$57$	−6.00000	−0.794719
$58$	8.00000	1.05045
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	1.00000	0.129099
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	10.0000	1.27000
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	−2.00000	−0.246183
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	−1.00000	−0.119523
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	−1.00000	−0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	10.0000	1.16248
$75$	1.00000	0.115470
$76$	−6.00000	−0.688247
$77$	2.00000	0.227921
$78$	2.00000	0.226455
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	−18.0000	−1.97576	−0.987878	−	0.155230i	$-0.950388\pi$
$83$	−18.0000	−1.97576	−0.987878	+	0.155230i	$0.950388\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	−2.00000	−0.215666
$87$	−8.00000	−0.857690
$88$	−2.00000	−0.213201
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	−1.00000	−0.105409
$91$	−2.00000	−0.209657
$92$	1.00000	0.104257
$93$	−10.0000	−1.03695
$94$	2.00000	0.206284
$95$	−6.00000	−0.615587
$96$	−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
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	2.00000	0.196116
$105$	1.00000	0.0975900
$106$	6.00000	0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	1.00000	0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	−2.00000	−0.190693
$111$	−10.0000	−0.949158
$112$	1.00000	0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	6.00000	0.561951
$115$	1.00000	0.0932505
$116$	−8.00000	−0.742781
$117$	−2.00000	−0.184900
$118$	4.00000	0.368230
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−7.00000	−0.636364
$122$	−10.0000	−0.905357
$123$	−10.0000	−0.901670
$124$	−10.0000	−0.898027
$125$	1.00000	0.0894427
$126$	−1.00000	−0.0890871
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−1.00000	−0.0883883
$129$	2.00000	0.176090
$130$	2.00000	0.175412
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	2.00000	0.174078
$133$	−6.00000	−0.520266
$134$	14.0000	1.20942
$135$	1.00000	0.0860663
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−1.00000	−0.0851257
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	1.00000	0.0845154
$141$	−2.00000	−0.168430
$142$	−6.00000	−0.503509
$143$	−4.00000	−0.334497
$144$	1.00000	0.0833333
$145$	−8.00000	−0.664364
$146$	−10.0000	−0.827606
$147$	1.00000	0.0824786
$148$	−10.0000	−0.821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−1.00000	−0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	−2.00000	−0.161165
$155$	−10.0000	−0.803219
$156$	−2.00000	−0.160128
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	1.00000	0.0788110
$162$	−1.00000	−0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−10.0000	−0.780869
$165$	2.00000	0.155700
$166$	18.0000	1.39707
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	2.00000	0.152499
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	8.00000	0.606478
$175$	1.00000	0.0755929
$176$	2.00000	0.150756
$177$	−4.00000	−0.300658
$178$	6.00000	0.449719
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	1.00000	0.0745356
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	2.00000	0.148250
$183$	10.0000	0.739221
$184$	−1.00000	−0.0737210
$185$	−10.0000	−0.735215
$186$	10.0000	0.733236
$187$	0	0
$188$	−2.00000	−0.145865
$189$	1.00000	0.0727393
$190$	6.00000	0.435286
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	1.00000	0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−14.0000	−1.00514
$195$	−2.00000	−0.143223
$196$	1.00000	0.0714286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	−2.00000	−0.142134
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	−1.00000	−0.0707107
$201$	−14.0000	−0.987484
$202$	6.00000	0.422159
$203$	−8.00000	−0.561490
$204$	0	0
$205$	−10.0000	−0.698430
$206$	0	0
$207$	1.00000	0.0695048
$208$	−2.00000	−0.138675
$209$	−12.0000	−0.830057
$210$	−1.00000	−0.0690066
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−6.00000	−0.412082
$213$	6.00000	0.411113
$214$	0	0
$215$	2.00000	0.136399
$216$	−1.00000	−0.0680414
$217$	−10.0000	−0.678844
$218$	−14.0000	−0.948200
$219$	10.0000	0.675737
$220$	2.00000	0.134840
$221$	0	0
$222$	10.0000	0.671156
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−1.00000	−0.0668153
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	−26.0000	−1.72568	−0.862840	−	0.505477i	$-0.831317\pi$
$227$	−26.0000	−1.72568	−0.862840	+	0.505477i	$0.831317\pi$
$228$	−6.00000	−0.397360
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	−1.00000	−0.0659380
$231$	2.00000	0.131590
$232$	8.00000	0.525226
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	2.00000	0.130744
$235$	−2.00000	−0.130466
$236$	−4.00000	−0.260378
$237$	0	0
$238$	0	0
$239$	−2.00000	−0.129369	−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000	−0.129369	−0.0646846	+	0.997906i	$0.520604\pi$
$240$	1.00000	0.0645497
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	10.0000	0.640184
$245$	1.00000	0.0638877
$246$	10.0000	0.637577
$247$	12.0000	0.763542
$248$	10.0000	0.635001
$249$	−18.0000	−1.14070
$250$	−1.00000	−0.0632456
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	1.00000	0.0629941
$253$	2.00000	0.125739
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	−2.00000	−0.124515
$259$	−10.0000	−0.621370
$260$	−2.00000	−0.124035
$261$	−8.00000	−0.495188
$262$	−20.0000	−1.23560
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	−2.00000	−0.123091
$265$	−6.00000	−0.368577
$266$	6.00000	0.367884
$267$	−6.00000	−0.367194
$268$	−14.0000	−0.855186
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	−1.00000	−0.0608581
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	18.0000	1.08742
$275$	2.00000	0.120605
$276$	1.00000	0.0601929
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	−16.0000	−0.959616
$279$	−10.0000	−0.598684
$280$	−1.00000	−0.0597614
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	2.00000	0.119098
$283$	−32.0000	−1.90220	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−32.0000	−1.90220	−0.951101	+	0.308879i	$0.900046\pi$
$284$	6.00000	0.356034
$285$	−6.00000	−0.355409
$286$	4.00000	0.236525
$287$	−10.0000	−0.590281
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	8.00000	0.469776
$291$	14.0000	0.820695
$292$	10.0000	0.585206
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−1.00000	−0.0583212
$295$	−4.00000	−0.232889
$296$	10.0000	0.581238
$297$	2.00000	0.116052
$298$	−6.00000	−0.347571
$299$	−2.00000	−0.115663
$300$	1.00000	0.0577350
$301$	2.00000	0.115278
$302$	0	0
$303$	−6.00000	−0.344691
$304$	−6.00000	−0.344124
$305$	10.0000	0.572598
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	2.00000	0.113961
$309$	0	0
$310$	10.0000	0.567962
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	2.00000	0.113228
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−18.0000	−1.01580
$315$	1.00000	0.0563436
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	6.00000	0.336463
$319$	−16.0000	−0.895828
$320$	1.00000	0.0559017
$321$	0	0
$322$	−1.00000	−0.0557278
$323$	0	0
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	−4.00000	−0.221540
$327$	14.0000	0.774202
$328$	10.0000	0.552158
$329$	−2.00000	−0.110264
$330$	−2.00000	−0.110096
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−18.0000	−0.987878
$333$	−10.0000	−0.547997
$334$	2.00000	0.109435
$335$	−14.0000	−0.764902
$336$	1.00000	0.0545545
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	9.00000	0.489535
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−20.0000	−1.08306
$342$	6.00000	0.324443
$343$	1.00000	0.0539949
$344$	−2.00000	−0.107833
$345$	1.00000	0.0538382
$346$	−12.0000	−0.645124
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−8.00000	−0.428845
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	−1.00000	−0.0534522
$351$	−2.00000	−0.106752
$352$	−2.00000	−0.106600
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	4.00000	0.212598
$355$	6.00000	0.318447
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−20.0000	−1.05703
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	−1.00000	−0.0527046
$361$	17.0000	0.894737
$362$	−26.0000	−1.36653
$363$	−7.00000	−0.367405
$364$	−2.00000	−0.104828
$365$	10.0000	0.523424
$366$	−10.0000	−0.522708
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	1.00000	0.0521286
$369$	−10.0000	−0.520579
$370$	10.0000	0.519875
$371$	−6.00000	−0.311504
$372$	−10.0000	−0.518476
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	2.00000	0.103142
$377$	16.0000	0.824042
$378$	−1.00000	−0.0514344
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	−6.00000	−0.307794
$381$	−2.00000	−0.102463
$382$	8.00000	0.409316
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	−1.00000	−0.0510310
$385$	2.00000	0.101929
$386$	14.0000	0.712581
$387$	2.00000	0.101666
$388$	14.0000	0.710742
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	2.00000	0.101274
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	20.0000	1.00887
$394$	18.0000	0.906827
$395$	0	0
$396$	2.00000	0.100504
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	4.00000	0.200502
$399$	−6.00000	−0.300376
$400$	1.00000	0.0500000
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	14.0000	0.698257
$403$	20.0000	0.996271
$404$	−6.00000	−0.298511
$405$	1.00000	0.0496904
$406$	8.00000	0.397033
$407$	−20.0000	−0.991363
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	10.0000	0.493865
$411$	−18.0000	−0.887875
$412$	0	0
$413$	−4.00000	−0.196827
$414$	−1.00000	−0.0491473
$415$	−18.0000	−0.883585
$416$	2.00000	0.0980581
$417$	16.0000	0.783523
$418$	12.0000	0.586939
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	1.00000	0.0487950
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	4.00000	0.194717
$423$	−2.00000	−0.0972433
$424$	6.00000	0.291386
$425$	0	0
$426$	−6.00000	−0.290701
$427$	10.0000	0.483934
$428$	0	0
$429$	−4.00000	−0.193122
$430$	−2.00000	−0.0964486
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	1.00000	0.0481125
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	10.0000	0.480015
$435$	−8.00000	−0.383571
$436$	14.0000	0.670478
$437$	−6.00000	−0.287019
$438$	−10.0000	−0.477818
$439$	38.0000	1.81364	0.906821	−	0.421517i	$-0.138502\pi$
$439$	38.0000	1.81364	0.906821	+	0.421517i	$0.138502\pi$
$440$	−2.00000	−0.0953463
$441$	1.00000	0.0476190
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−10.0000	−0.474579
$445$	−6.00000	−0.284427
$446$	−8.00000	−0.378811
$447$	6.00000	0.283790
$448$	1.00000	0.0472456
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	−1.00000	−0.0471405
$451$	−20.0000	−0.941763
$452$	−6.00000	−0.282216
$453$	0	0
$454$	26.0000	1.22024
$455$	−2.00000	−0.0937614
$456$	6.00000	0.280976
$457$	−24.0000	−1.12267	−0.561336	−	0.827588i	$-0.689713\pi$
$457$	−24.0000	−1.12267	−0.561336	+	0.827588i	$0.689713\pi$
$458$	−2.00000	−0.0934539
$459$	0	0
$460$	1.00000	0.0466252
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	−2.00000	−0.0930484
$463$	6.00000	0.278844	0.139422	−	0.990233i	$-0.455476\pi$
$463$	6.00000	0.278844	0.139422	+	0.990233i	$0.455476\pi$
$464$	−8.00000	−0.371391
$465$	−10.0000	−0.463739
$466$	−10.0000	−0.463241
$467$	−30.0000	−1.38823	−0.694117	−	0.719862i	$-0.744205\pi$
$467$	−30.0000	−1.38823	−0.694117	+	0.719862i	$0.744205\pi$
$468$	−2.00000	−0.0924500
$469$	−14.0000	−0.646460
$470$	2.00000	0.0922531
$471$	18.0000	0.829396
$472$	4.00000	0.184115
$473$	4.00000	0.183920
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	−6.00000	−0.274721
$478$	2.00000	0.0914779
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	−1.00000	−0.0456435
$481$	20.0000	0.911922
$482$	−20.0000	−0.910975
$483$	1.00000	0.0455016
$484$	−7.00000	−0.318182
$485$	14.0000	0.635707
$486$	−1.00000	−0.0453609
$487$	−18.0000	−0.815658	−0.407829	−	0.913058i	$-0.633714\pi$
$487$	−18.0000	−0.815658	−0.407829	+	0.913058i	$0.633714\pi$
$488$	−10.0000	−0.452679
$489$	4.00000	0.180886
$490$	−1.00000	−0.0451754
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	−10.0000	−0.450835
$493$	0	0
$494$	−12.0000	−0.539906
$495$	2.00000	0.0898933
$496$	−10.0000	−0.449013
$497$	6.00000	0.269137
$498$	18.0000	0.806599
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	1.00000	0.0447214
$501$	−2.00000	−0.0893534
$502$	−24.0000	−1.07117
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	−1.00000	−0.0445435
$505$	−6.00000	−0.266996
$506$	−2.00000	−0.0889108
$507$	−9.00000	−0.399704
$508$	−2.00000	−0.0887357
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	−1.00000	−0.0441942
$513$	−6.00000	−0.264906
$514$	6.00000	0.264649
$515$	0	0
$516$	2.00000	0.0880451
$517$	−4.00000	−0.175920
$518$	10.0000	0.439375
$519$	12.0000	0.526742
$520$	2.00000	0.0877058
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	8.00000	0.350150
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	20.0000	0.873704
$525$	1.00000	0.0436436
$526$	−16.0000	−0.697633
$527$	0	0
$528$	2.00000	0.0870388
$529$	1.00000	0.0434783
$530$	6.00000	0.260623
$531$	−4.00000	−0.173585
$532$	−6.00000	−0.260133
$533$	20.0000	0.866296
$534$	6.00000	0.259645
$535$	0	0
$536$	14.0000	0.604708
$537$	20.0000	0.863064
$538$	18.0000	0.776035
$539$	2.00000	0.0861461
$540$	1.00000	0.0430331
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	−22.0000	−0.944981
$543$	26.0000	1.11577
$544$	0	0
$545$	14.0000	0.599694
$546$	2.00000	0.0855921
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	−18.0000	−0.768922
$549$	10.0000	0.426790
$550$	−2.00000	−0.0852803
$551$	48.0000	2.04487
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	−4.00000	−0.169944
$555$	−10.0000	−0.424476
$556$	16.0000	0.678551
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	10.0000	0.423334
$559$	−4.00000	−0.169182
$560$	1.00000	0.0422577
$561$	0	0
$562$	16.0000	0.674919
$563$	−34.0000	−1.43293	−0.716465	−	0.697623i	$-0.754241\pi$
$563$	−34.0000	−1.43293	−0.716465	+	0.697623i	$0.754241\pi$
$564$	−2.00000	−0.0842152
$565$	−6.00000	−0.252422
$566$	32.0000	1.34506
$567$	1.00000	0.0419961
$568$	−6.00000	−0.251754
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	6.00000	0.251312
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	−4.00000	−0.167248
$573$	−8.00000	−0.334205
$574$	10.0000	0.417392
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	17.0000	0.707107
$579$	−14.0000	−0.581820
$580$	−8.00000	−0.332182
$581$	−18.0000	−0.746766
$582$	−14.0000	−0.580319
$583$	−12.0000	−0.496989
$584$	−10.0000	−0.413803
$585$	−2.00000	−0.0826898
$586$	−6.00000	−0.247858
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	1.00000	0.0412393
$589$	60.0000	2.47226
$590$	4.00000	0.164677
$591$	−18.0000	−0.740421
$592$	−10.0000	−0.410997
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	6.00000	0.245770
$597$	−4.00000	−0.163709
$598$	2.00000	0.0817861
$599$	38.0000	1.55264	0.776319	−	0.630340i	$-0.217085\pi$
$599$	38.0000	1.55264	0.776319	+	0.630340i	$0.217085\pi$
$600$	−1.00000	−0.0408248
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	−2.00000	−0.0815139
$603$	−14.0000	−0.570124
$604$	0	0
$605$	−7.00000	−0.284590
$606$	6.00000	0.243733
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	6.00000	0.243332
$609$	−8.00000	−0.324176
$610$	−10.0000	−0.404888
$611$	4.00000	0.161823
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	−4.00000	−0.161427
$615$	−10.0000	−0.403239
$616$	−2.00000	−0.0805823
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	−10.0000	−0.401610
$621$	1.00000	0.0401286
$622$	−12.0000	−0.481156
$623$	−6.00000	−0.240385
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	−6.00000	−0.239808
$627$	−12.0000	−0.479234
$628$	18.0000	0.718278
$629$	0	0
$630$	−1.00000	−0.0398410
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	−18.0000	−0.714871
$635$	−2.00000	−0.0793676
$636$	−6.00000	−0.237915
$637$	−2.00000	−0.0792429
$638$	16.0000	0.633446
$639$	6.00000	0.237356
$640$	−1.00000	−0.0395285
$641$	−8.00000	−0.315981	−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000	−0.315981	−0.157991	+	0.987441i	$0.550502\pi$
$642$	0	0
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	1.00000	0.0394055
$645$	2.00000	0.0787499
$646$	0	0
$647$	−38.0000	−1.49393	−0.746967	−	0.664861i	$-0.768491\pi$
$647$	−38.0000	−1.49393	−0.746967	+	0.664861i	$0.768491\pi$
$648$	−1.00000	−0.0392837
$649$	−8.00000	−0.314027
$650$	2.00000	0.0784465
$651$	−10.0000	−0.391931
$652$	4.00000	0.156652
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	−14.0000	−0.547443
$655$	20.0000	0.781465
$656$	−10.0000	−0.390434
$657$	10.0000	0.390137
$658$	2.00000	0.0779681
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	2.00000	0.0778499
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	18.0000	0.698535
$665$	−6.00000	−0.232670
$666$	10.0000	0.387492
$667$	−8.00000	−0.309761
$668$	−2.00000	−0.0773823
$669$	8.00000	0.309298
$670$	14.0000	0.540867
$671$	20.0000	0.772091
$672$	−1.00000	−0.0385758
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	8.00000	0.308148
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	6.00000	0.230429
$679$	14.0000	0.537271
$680$	0	0
$681$	−26.0000	−0.996322
$682$	20.0000	0.765840
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	−6.00000	−0.229416
$685$	−18.0000	−0.687745
$686$	−1.00000	−0.0381802
$687$	2.00000	0.0763048
$688$	2.00000	0.0762493
$689$	12.0000	0.457164
$690$	−1.00000	−0.0380693
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	12.0000	0.456172
$693$	2.00000	0.0759737
$694$	−12.0000	−0.455514
$695$	16.0000	0.606915
$696$	8.00000	0.303239
$697$	0	0
$698$	8.00000	0.302804
$699$	10.0000	0.378235
$700$	1.00000	0.0377964
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	2.00000	0.0754851
$703$	60.0000	2.26294
$704$	2.00000	0.0753778
$705$	−2.00000	−0.0753244
$706$	−6.00000	−0.225813
$707$	−6.00000	−0.225653
$708$	−4.00000	−0.150329
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	−6.00000	−0.225176
$711$	0	0
$712$	6.00000	0.224860
$713$	−10.0000	−0.374503
$714$	0	0
$715$	−4.00000	−0.149592
$716$	20.0000	0.747435
$717$	−2.00000	−0.0746914
$718$	24.0000	0.895672
$719$	4.00000	0.149175	0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000	0.149175	0.0745874	+	0.997214i	$0.476236\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	−17.0000	−0.632674
$723$	20.0000	0.743808
$724$	26.0000	0.966282
$725$	−8.00000	−0.297113
$726$	7.00000	0.259794
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	2.00000	0.0741249
$729$	1.00000	0.0370370
$730$	−10.0000	−0.370117
$731$	0	0
$732$	10.0000	0.369611
$733$	−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
$735$	1.00000	0.0368856
$736$	−1.00000	−0.0368605
$737$	−28.0000	−1.03139
$738$	10.0000	0.368105
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	−10.0000	−0.367607
$741$	12.0000	0.440831
$742$	6.00000	0.220267
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	10.0000	0.366618
$745$	6.00000	0.219823
$746$	2.00000	0.0732252
$747$	−18.0000	−0.658586
$748$	0	0
$749$	0	0
$750$	−1.00000	−0.0365148
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	−2.00000	−0.0729325
$753$	24.0000	0.874609
$754$	−16.0000	−0.582686
$755$	0	0
$756$	1.00000	0.0363696
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	8.00000	0.290573
$759$	2.00000	0.0725954
$760$	6.00000	0.217643
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	2.00000	0.0724524
$763$	14.0000	0.506834
$764$	−8.00000	−0.289430
$765$	0	0
$766$	4.00000	0.144526
$767$	8.00000	0.288863
$768$	1.00000	0.0360844
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	−2.00000	−0.0720750
$771$	−6.00000	−0.216085
$772$	−14.0000	−0.503871
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	−2.00000	−0.0718885
$775$	−10.0000	−0.359211
$776$	−14.0000	−0.502571
$777$	−10.0000	−0.358748
$778$	−10.0000	−0.358517
$779$	60.0000	2.14972
$780$	−2.00000	−0.0716115
$781$	12.0000	0.429394
$782$	0	0
$783$	−8.00000	−0.285897
$784$	1.00000	0.0357143
$785$	18.0000	0.642448
$786$	−20.0000	−0.713376
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	−18.0000	−0.641223
$789$	16.0000	0.569615
$790$	0	0
$791$	−6.00000	−0.213335
$792$	−2.00000	−0.0710669
$793$	−20.0000	−0.710221
$794$	2.00000	0.0709773
$795$	−6.00000	−0.212798
$796$	−4.00000	−0.141776
$797$	26.0000	0.920967	0.460484	−	0.887668i	$-0.347676\pi$
$797$	26.0000	0.920967	0.460484	+	0.887668i	$0.347676\pi$
$798$	6.00000	0.212398
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−6.00000	−0.212000
$802$	12.0000	0.423735
$803$	20.0000	0.705785
$804$	−14.0000	−0.493742
$805$	1.00000	0.0352454
$806$	−20.0000	−0.704470
$807$	−18.0000	−0.633630
$808$	6.00000	0.211079
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	−1.00000	−0.0351364
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	−8.00000	−0.280745
$813$	22.0000	0.771574
$814$	20.0000	0.701000
$815$	4.00000	0.140114
$816$	0	0
$817$	−12.0000	−0.419827
$818$	14.0000	0.489499
$819$	−2.00000	−0.0698857
$820$	−10.0000	−0.349215
$821$	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
$823$	−18.0000	−0.627441	−0.313720	−	0.949515i	$-0.601575\pi$
$823$	−18.0000	−0.627441	−0.313720	+	0.949515i	$0.601575\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	4.00000	0.139178
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	1.00000	0.0347524
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	18.0000	0.624789
$831$	4.00000	0.138758
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	−16.0000	−0.554035
$835$	−2.00000	−0.0692129
$836$	−12.0000	−0.415029
$837$	−10.0000	−0.345651
$838$	−12.0000	−0.414533
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	−1.00000	−0.0345033
$841$	35.0000	1.20690
$842$	−34.0000	−1.17172
$843$	−16.0000	−0.551069
$844$	−4.00000	−0.137686
$845$	−9.00000	−0.309609
$846$	2.00000	0.0687614
$847$	−7.00000	−0.240523
$848$	−6.00000	−0.206041
$849$	−32.0000	−1.09824
$850$	0	0
$851$	−10.0000	−0.342796
$852$	6.00000	0.205557
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	−10.0000	−0.342193
$855$	−6.00000	−0.205196
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	4.00000	0.136558
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	2.00000	0.0681994
$861$	−10.0000	−0.340799
$862$	20.0000	0.681203
$863$	−40.0000	−1.36162	−0.680808	−	0.732462i	$-0.738371\pi$
$863$	−40.0000	−1.36162	−0.680808	+	0.732462i	$0.738371\pi$
$864$	−1.00000	−0.0340207
$865$	12.0000	0.408012
$866$	−30.0000	−1.01944
$867$	−17.0000	−0.577350
$868$	−10.0000	−0.339422
$869$	0	0
$870$	8.00000	0.271225
$871$	28.0000	0.948744
$872$	−14.0000	−0.474100
$873$	14.0000	0.473828
$874$	6.00000	0.202953
$875$	1.00000	0.0338062
$876$	10.0000	0.337869
$877$	−48.0000	−1.62084	−0.810422	−	0.585846i	$-0.800762\pi$
$877$	−48.0000	−1.62084	−0.810422	+	0.585846i	$0.800762\pi$
$878$	−38.0000	−1.28244
$879$	6.00000	0.202375
$880$	2.00000	0.0674200
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	−1.00000	−0.0336718
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	−4.00000	−0.134383
$887$	−46.0000	−1.54453	−0.772264	−	0.635301i	$-0.780876\pi$
$887$	−46.0000	−1.54453	−0.772264	+	0.635301i	$0.780876\pi$
$888$	10.0000	0.335578
$889$	−2.00000	−0.0670778
$890$	6.00000	0.201120
$891$	2.00000	0.0670025
$892$	8.00000	0.267860
$893$	12.0000	0.401565
$894$	−6.00000	−0.200670
$895$	20.0000	0.668526
$896$	−1.00000	−0.0334077
$897$	−2.00000	−0.0667781
$898$	2.00000	0.0667409
$899$	80.0000	2.66815
$900$	1.00000	0.0333333
$901$	0	0
$902$	20.0000	0.665927
$903$	2.00000	0.0665558
$904$	6.00000	0.199557
$905$	26.0000	0.864269
$906$	0	0
$907$	38.0000	1.26177	0.630885	−	0.775877i	$-0.282692\pi$
$907$	38.0000	1.26177	0.630885	+	0.775877i	$0.282692\pi$
$908$	−26.0000	−0.862840
$909$	−6.00000	−0.199007
$910$	2.00000	0.0662994
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	−6.00000	−0.198680
$913$	−36.0000	−1.19143
$914$	24.0000	0.793849
$915$	10.0000	0.330590
$916$	2.00000	0.0660819
$917$	20.0000	0.660458
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	−1.00000	−0.0329690
$921$	4.00000	0.131804
$922$	−10.0000	−0.329332
$923$	−12.0000	−0.394985
$924$	2.00000	0.0657952
$925$	−10.0000	−0.328798
$926$	−6.00000	−0.197172
$927$	0	0
$928$	8.00000	0.262613
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	10.0000	0.327913
$931$	−6.00000	−0.196642
$932$	10.0000	0.327561
$933$	12.0000	0.392862
$934$	30.0000	0.981630
$935$	0	0
$936$	2.00000	0.0653720
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	14.0000	0.457116
$939$	6.00000	0.195803
$940$	−2.00000	−0.0652328
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	−18.0000	−0.586472
$943$	−10.0000	−0.325645
$944$	−4.00000	−0.130189
$945$	1.00000	0.0325300
$946$	−4.00000	−0.130051
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	6.00000	0.194666
$951$	18.0000	0.583690
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	6.00000	0.194257
$955$	−8.00000	−0.258874
$956$	−2.00000	−0.0646846
$957$	−16.0000	−0.517207
$958$	32.0000	1.03387
$959$	−18.0000	−0.581250
$960$	1.00000	0.0322749
$961$	69.0000	2.22581
$962$	−20.0000	−0.644826
$963$	0	0
$964$	20.0000	0.644157
$965$	−14.0000	−0.450676
$966$	−1.00000	−0.0321745
$967$	6.00000	0.192947	0.0964735	−	0.995336i	$-0.469244\pi$
$967$	6.00000	0.192947	0.0964735	+	0.995336i	$0.469244\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	−14.0000	−0.449513
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	1.00000	0.0320750
$973$	16.0000	0.512936
$974$	18.0000	0.576757
$975$	−2.00000	−0.0640513
$976$	10.0000	0.320092
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	−4.00000	−0.127906
$979$	−12.0000	−0.383522
$980$	1.00000	0.0319438
$981$	14.0000	0.446986
$982$	4.00000	0.127645
$983$	60.0000	1.91370	0.956851	−	0.290578i	$-0.0938475\pi$
$983$	60.0000	1.91370	0.956851	+	0.290578i	$0.0938475\pi$
$984$	10.0000	0.318788
$985$	−18.0000	−0.573528
$986$	0	0
$987$	−2.00000	−0.0636607
$988$	12.0000	0.381771
$989$	2.00000	0.0635963
$990$	−2.00000	−0.0635642
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	10.0000	0.317500
$993$	20.0000	0.634681
$994$	−6.00000	−0.190308
$995$	−4.00000	−0.126809
$996$	−18.0000	−0.570352
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	12.0000	0.379853
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4830.2.a.q.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4830.2.a.q.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4830.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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