LMFDB - Embedded newform 4002.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, 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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$0$
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	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +2.00000 q^{20} -4.00000 q^{21} +1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +1.00000 q^{29} -2.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} +8.00000 q^{35} +1.00000 q^{36} +10.0000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +2.00000 q^{40} +10.0000 q^{41} -4.00000 q^{42} +4.00000 q^{43} +2.00000 q^{45} +1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +4.00000 q^{56} -4.00000 q^{57} +1.00000 q^{58} +12.0000 q^{59} -2.00000 q^{60} -14.0000 q^{61} +8.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +16.0000 q^{67} -6.00000 q^{68} -1.00000 q^{69} +8.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} +10.0000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +2.00000 q^{78} -16.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -12.0000 q^{83} -4.00000 q^{84} -12.0000 q^{85} +4.00000 q^{86} -1.00000 q^{87} +10.0000 q^{89} +2.00000 q^{90} -8.00000 q^{91} +1.00000 q^{92} -8.00000 q^{93} -8.00000 q^{94} +8.00000 q^{95} -1.00000 q^{96} +6.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	−1.00000	−0.408248
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−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	$0.589456\pi$
$14$	4.00000	1.06904
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	2.00000	0.447214
$21$	−4.00000	−0.872872
$22$	0	0
$23$	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
$28$	4.00000	0.755929
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−2.00000	−0.365148
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.00000	−1.02899
$35$	8.00000	1.35225
$36$	1.00000	0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	4.00000	0.648886
$39$	2.00000	0.320256
$40$	2.00000	0.316228
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	−4.00000	−0.617213
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	1.00000	0.147442
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−1.00000	−0.144338
$49$	9.00000	1.28571
$50$	−1.00000	−0.141421
$51$	6.00000	0.840168
$52$	−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
$55$	0	0
$56$	4.00000	0.534522
$57$	−4.00000	−0.529813
$58$	1.00000	0.131306
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	−2.00000	−0.258199
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	8.00000	1.01600
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	16.0000	1.95471	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	16.0000	1.95471	0.977356	+	0.211604i	$0.0678686\pi$
$68$	−6.00000	−0.727607
$69$	−1.00000	−0.120386
$70$	8.00000	0.956183
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	10.0000	1.16248
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	2.00000	0.226455
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−4.00000	−0.436436
$85$	−12.0000	−1.30158
$86$	4.00000	0.431331
$87$	−1.00000	−0.107211
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	2.00000	0.210819
$91$	−8.00000	−0.838628
$92$	1.00000	0.104257
$93$	−8.00000	−0.829561
$94$	−8.00000	−0.825137
$95$	8.00000	0.820783
$96$	−1.00000	−0.102062
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	9.00000	0.909137
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	6.00000	0.594089
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−2.00000	−0.196116
$105$	−8.00000	−0.780720
$106$	−6.00000	−0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	4.00000	0.377964
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	−4.00000	−0.374634
$115$	2.00000	0.186501
$116$	1.00000	0.0928477
$117$	−2.00000	−0.184900
$118$	12.0000	1.10469
$119$	−24.0000	−2.20008
$120$	−2.00000	−0.182574
$121$	−11.0000	−1.00000
$122$	−14.0000	−1.26750
$123$	−10.0000	−0.901670
$124$	8.00000	0.718421
$125$	−12.0000	−1.07331
$126$	4.00000	0.356348
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	−4.00000	−0.350823
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	16.0000	1.38219
$135$	−2.00000	−0.172133
$136$	−6.00000	−0.514496
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	−1.00000	−0.0851257
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	8.00000	0.676123
$141$	8.00000	0.673722
$142$	−8.00000	−0.671345
$143$	0	0
$144$	1.00000	0.0833333
$145$	2.00000	0.166091
$146$	2.00000	0.165521
$147$	−9.00000	−0.742307
$148$	10.0000	0.821995
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	1.00000	0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	4.00000	0.324443
$153$	−6.00000	−0.485071
$154$	0	0
$155$	16.0000	1.28515
$156$	2.00000	0.160128
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−16.0000	−1.27289
$159$	6.00000	0.475831
$160$	2.00000	0.158114
$161$	4.00000	0.315244
$162$	1.00000	0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−12.0000	−0.931381
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	−4.00000	−0.308607
$169$	−9.00000	−0.692308
$170$	−12.0000	−0.920358
$171$	4.00000	0.305888
$172$	4.00000	0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	−1.00000	−0.0758098
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−12.0000	−0.901975
$178$	10.0000	0.749532
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	2.00000	0.149071
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	−8.00000	−0.592999
$183$	14.0000	1.03491
$184$	1.00000	0.0737210
$185$	20.0000	1.47043
$186$	−8.00000	−0.586588
$187$	0	0
$188$	−8.00000	−0.583460
$189$	−4.00000	−0.290957
$190$	8.00000	0.580381
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.00000	−0.0721688
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	6.00000	0.430775
$195$	4.00000	0.286446
$196$	9.00000	0.642857
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	−1.00000	−0.0707107
$201$	−16.0000	−1.12855
$202$	−10.0000	−0.703598
$203$	4.00000	0.280745
$204$	6.00000	0.420084
$205$	20.0000	1.39686
$206$	4.00000	0.278693
$207$	1.00000	0.0695048
$208$	−2.00000	−0.138675
$209$	0	0
$210$	−8.00000	−0.552052
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−6.00000	−0.412082
$213$	8.00000	0.548151
$214$	−12.0000	−0.820303
$215$	8.00000	0.545595
$216$	−1.00000	−0.0680414
$217$	32.0000	2.17230
$218$	−2.00000	−0.135457
$219$	−2.00000	−0.135147
$220$	0	0
$221$	12.0000	0.807207
$222$	−10.0000	−0.671156
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	4.00000	0.267261
$225$	−1.00000	−0.0666667
$226$	10.0000	0.665190
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	−4.00000	−0.264906
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	1.00000	0.0656532
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−2.00000	−0.130744
$235$	−16.0000	−1.04372
$236$	12.0000	0.781133
$237$	16.0000	1.03931
$238$	−24.0000	−1.55569
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	−2.00000	−0.129099
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−11.0000	−0.707107
$243$	−1.00000	−0.0641500
$244$	−14.0000	−0.896258
$245$	18.0000	1.14998
$246$	−10.0000	−0.637577
$247$	−8.00000	−0.509028
$248$	8.00000	0.508001
$249$	12.0000	0.760469
$250$	−12.0000	−0.758947
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	8.00000	0.501965
$255$	12.0000	0.751469
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	−4.00000	−0.249029
$259$	40.0000	2.48548
$260$	−4.00000	−0.248069
$261$	1.00000	0.0618984
$262$	12.0000	0.741362
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	16.0000	0.981023
$267$	−10.0000	−0.611990
$268$	16.0000	0.977356
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	−2.00000	−0.121716
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−6.00000	−0.363803
$273$	8.00000	0.484182
$274$	10.0000	0.604122
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	4.00000	0.239904
$279$	8.00000	0.478947
$280$	8.00000	0.478091
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	8.00000	0.476393
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	−8.00000	−0.474713
$285$	−8.00000	−0.473879
$286$	0	0
$287$	40.0000	2.36113
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	2.00000	0.117444
$291$	−6.00000	−0.351726
$292$	2.00000	0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−9.00000	−0.524891
$295$	24.0000	1.39733
$296$	10.0000	0.581238
$297$	0	0
$298$	10.0000	0.579284
$299$	−2.00000	−0.115663
$300$	1.00000	0.0577350
$301$	16.0000	0.922225
$302$	0	0
$303$	10.0000	0.574485
$304$	4.00000	0.229416
$305$	−28.0000	−1.60328
$306$	−6.00000	−0.342997
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	16.0000	0.908739
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	2.00000	0.113228
$313$	−30.0000	−1.69570	−0.847850	−	0.530236i	$-0.822103\pi$
$313$	−30.0000	−1.69570	−0.847850	+	0.530236i	$0.822103\pi$
$314$	2.00000	0.112867
$315$	8.00000	0.450749
$316$	−16.0000	−0.900070
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	6.00000	0.336463
$319$	0	0
$320$	2.00000	0.111803
$321$	12.0000	0.669775
$322$	4.00000	0.222911
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	12.0000	0.664619
$327$	2.00000	0.110600
$328$	10.0000	0.552158
$329$	−32.0000	−1.76422
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−12.0000	−0.658586
$333$	10.0000	0.547997
$334$	16.0000	0.875481
$335$	32.0000	1.74835
$336$	−4.00000	−0.218218
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−9.00000	−0.489535
$339$	−10.0000	−0.543125
$340$	−12.0000	−0.650791
$341$	0	0
$342$	4.00000	0.216295
$343$	8.00000	0.431959
$344$	4.00000	0.215666
$345$	−2.00000	−0.107676
$346$	−18.0000	−0.967686
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	−1.00000	−0.0536056
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	−4.00000	−0.213809
$351$	2.00000	0.106752
$352$	0	0
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	−12.0000	−0.637793
$355$	−16.0000	−0.849192
$356$	10.0000	0.529999
$357$	24.0000	1.27021
$358$	−20.0000	−1.05703
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	2.00000	0.105409
$361$	−3.00000	−0.157895
$362$	−2.00000	−0.105118
$363$	11.0000	0.577350
$364$	−8.00000	−0.419314
$365$	4.00000	0.209370
$366$	14.0000	0.731792
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	1.00000	0.0521286
$369$	10.0000	0.520579
$370$	20.0000	1.03975
$371$	−24.0000	−1.24602
$372$	−8.00000	−0.414781
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	−8.00000	−0.412568
$377$	−2.00000	−0.103005
$378$	−4.00000	−0.205738
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	8.00000	0.410391
$381$	−8.00000	−0.409852
$382$	12.0000	0.613973
$383$	−32.0000	−1.63512	−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000	−1.63512	−0.817562	+	0.575841i	$0.804675\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	26.0000	1.32337
$387$	4.00000	0.203331
$388$	6.00000	0.304604
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	4.00000	0.202548
$391$	−6.00000	−0.303433
$392$	9.00000	0.454569
$393$	−12.0000	−0.605320
$394$	22.0000	1.10834
$395$	−32.0000	−1.61009
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	4.00000	0.200502
$399$	−16.0000	−0.801002
$400$	−1.00000	−0.0500000
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	−16.0000	−0.798007
$403$	−16.0000	−0.797017
$404$	−10.0000	−0.497519
$405$	2.00000	0.0993808
$406$	4.00000	0.198517
$407$	0	0
$408$	6.00000	0.297044
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	20.0000	0.987730
$411$	−10.0000	−0.493264
$412$	4.00000	0.197066
$413$	48.0000	2.36193
$414$	1.00000	0.0491473
$415$	−24.0000	−1.17811
$416$	−2.00000	−0.0980581
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	−8.00000	−0.390360
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−20.0000	−0.973585
$423$	−8.00000	−0.388973
$424$	−6.00000	−0.291386
$425$	6.00000	0.291043
$426$	8.00000	0.387601
$427$	−56.0000	−2.71003
$428$	−12.0000	−0.580042
$429$	0	0
$430$	8.00000	0.385794
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	−1.00000	−0.0481125
$433$	−10.0000	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$434$	32.0000	1.53605
$435$	−2.00000	−0.0958927
$436$	−2.00000	−0.0957826
$437$	4.00000	0.191346
$438$	−2.00000	−0.0955637
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	12.0000	0.570782
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	−10.0000	−0.474579
$445$	20.0000	0.948091
$446$	−16.0000	−0.757622
$447$	−10.0000	−0.472984
$448$	4.00000	0.188982
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	−1.00000	−0.0471405
$451$	0	0
$452$	10.0000	0.470360
$453$	0	0
$454$	−4.00000	−0.187729
$455$	−16.0000	−0.750092
$456$	−4.00000	−0.187317
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	−6.00000	−0.280362
$459$	6.00000	0.280056
$460$	2.00000	0.0932505
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	1.00000	0.0464238
$465$	−16.0000	−0.741982
$466$	−6.00000	−0.277945
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	−2.00000	−0.0924500
$469$	64.0000	2.95525
$470$	−16.0000	−0.738025
$471$	−2.00000	−0.0921551
$472$	12.0000	0.552345
$473$	0	0
$474$	16.0000	0.734904
$475$	−4.00000	−0.183533
$476$	−24.0000	−1.10004
$477$	−6.00000	−0.274721
$478$	−24.0000	−1.09773
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	−2.00000	−0.0912871
$481$	−20.0000	−0.911922
$482$	2.00000	0.0910975
$483$	−4.00000	−0.182006
$484$	−11.0000	−0.500000
$485$	12.0000	0.544892
$486$	−1.00000	−0.0453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−14.0000	−0.633750
$489$	−12.0000	−0.542659
$490$	18.0000	0.813157
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	−10.0000	−0.450835
$493$	−6.00000	−0.270226
$494$	−8.00000	−0.359937
$495$	0	0
$496$	8.00000	0.359211
$497$	−32.0000	−1.43540
$498$	12.0000	0.537733
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−12.0000	−0.536656
$501$	−16.0000	−0.714827
$502$	−8.00000	−0.357057
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	4.00000	0.178174
$505$	−20.0000	−0.889988
$506$	0	0
$507$	9.00000	0.399704
$508$	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	$0.399581\pi$
$510$	12.0000	0.531369
$511$	8.00000	0.353899
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	−14.0000	−0.617514
$515$	8.00000	0.352522
$516$	−4.00000	−0.176090
$517$	0	0
$518$	40.0000	1.75750
$519$	18.0000	0.790112
$520$	−4.00000	−0.175412
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	1.00000	0.0437688
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	12.0000	0.524222
$525$	4.00000	0.174574
$526$	4.00000	0.174408
$527$	−48.0000	−2.09091
$528$	0	0
$529$	1.00000	0.0434783
$530$	−12.0000	−0.521247
$531$	12.0000	0.520756
$532$	16.0000	0.693688
$533$	−20.0000	−0.866296
$534$	−10.0000	−0.432742
$535$	−24.0000	−1.03761
$536$	16.0000	0.691095
$537$	20.0000	0.863064
$538$	−18.0000	−0.776035
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	−8.00000	−0.343629
$543$	2.00000	0.0858282
$544$	−6.00000	−0.257248
$545$	−4.00000	−0.171341
$546$	8.00000	0.342368
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	10.0000	0.427179
$549$	−14.0000	−0.597505
$550$	0	0
$551$	4.00000	0.170406
$552$	−1.00000	−0.0425628
$553$	−64.0000	−2.72156
$554$	−10.0000	−0.424859
$555$	−20.0000	−0.848953
$556$	4.00000	0.169638
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	8.00000	0.338667
$559$	−8.00000	−0.338364
$560$	8.00000	0.338062
$561$	0	0
$562$	6.00000	0.253095
$563$	−8.00000	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$563$	−8.00000	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$564$	8.00000	0.336861
$565$	20.0000	0.841406
$566$	16.0000	0.672530
$567$	4.00000	0.167984
$568$	−8.00000	−0.335673
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	−8.00000	−0.335083
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	40.0000	1.66957
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	19.0000	0.790296
$579$	−26.0000	−1.08052
$580$	2.00000	0.0830455
$581$	−48.0000	−1.99138
$582$	−6.00000	−0.248708
$583$	0	0
$584$	2.00000	0.0827606
$585$	−4.00000	−0.165380
$586$	6.00000	0.247858
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	−9.00000	−0.371154
$589$	32.0000	1.31854
$590$	24.0000	0.988064
$591$	−22.0000	−0.904959
$592$	10.0000	0.410997
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	−48.0000	−1.96781
$596$	10.0000	0.409616
$597$	−4.00000	−0.163709
$598$	−2.00000	−0.0817861
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	1.00000	0.0408248
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	16.0000	0.652111
$603$	16.0000	0.651570
$604$	0	0
$605$	−22.0000	−0.894427
$606$	10.0000	0.406222
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	4.00000	0.162221
$609$	−4.00000	−0.162088
$610$	−28.0000	−1.13369
$611$	16.0000	0.647291
$612$	−6.00000	−0.242536
$613$	30.0000	1.21169	0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000	1.21169	0.605844	+	0.795583i	$0.292835\pi$
$614$	28.0000	1.12999
$615$	−20.0000	−0.806478
$616$	0	0
$617$	34.0000	1.36879	0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000	1.36879	0.684394	+	0.729112i	$0.260067\pi$
$618$	−4.00000	−0.160904
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	16.0000	0.642575
$621$	−1.00000	−0.0401286
$622$	−32.0000	−1.28308
$623$	40.0000	1.60257
$624$	2.00000	0.0800641
$625$	−19.0000	−0.760000
$626$	−30.0000	−1.19904
$627$	0	0
$628$	2.00000	0.0798087
$629$	−60.0000	−2.39236
$630$	8.00000	0.318728
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	−16.0000	−0.636446
$633$	20.0000	0.794929
$634$	30.0000	1.19145
$635$	16.0000	0.634941
$636$	6.00000	0.237915
$637$	−18.0000	−0.713186
$638$	0	0
$639$	−8.00000	−0.316475
$640$	2.00000	0.0790569
$641$	−46.0000	−1.81689	−0.908445	−	0.418004i	$-0.862730\pi$
$641$	−46.0000	−1.81689	−0.908445	+	0.418004i	$0.862730\pi$
$642$	12.0000	0.473602
$643$	40.0000	1.57745	0.788723	−	0.614749i	$-0.210743\pi$
$643$	40.0000	1.57745	0.788723	+	0.614749i	$0.210743\pi$
$644$	4.00000	0.157622
$645$	−8.00000	−0.315000
$646$	−24.0000	−0.944267
$647$	−48.0000	−1.88707	−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000	−1.88707	−0.943537	+	0.331266i	$0.892524\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	2.00000	0.0784465
$651$	−32.0000	−1.25418
$652$	12.0000	0.469956
$653$	−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
$655$	24.0000	0.937758
$656$	10.0000	0.390434
$657$	2.00000	0.0780274
$658$	−32.0000	−1.24749
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	−20.0000	−0.777322
$663$	−12.0000	−0.466041
$664$	−12.0000	−0.465690
$665$	32.0000	1.24091
$666$	10.0000	0.387492
$667$	1.00000	0.0387202
$668$	16.0000	0.619059
$669$	16.0000	0.618596
$670$	32.0000	1.23627
$671$	0	0
$672$	−4.00000	−0.154303
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	22.0000	0.847408
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	−10.0000	−0.384048
$679$	24.0000	0.921035
$680$	−12.0000	−0.460179
$681$	4.00000	0.153280
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	4.00000	0.152944
$685$	20.0000	0.764161
$686$	8.00000	0.305441
$687$	6.00000	0.228914
$688$	4.00000	0.152499
$689$	12.0000	0.457164
$690$	−2.00000	−0.0761387
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	−4.00000	−0.151838
$695$	8.00000	0.303457
$696$	−1.00000	−0.0379049
$697$	−60.0000	−2.27266
$698$	−2.00000	−0.0757011
$699$	6.00000	0.226941
$700$	−4.00000	−0.151186
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	2.00000	0.0754851
$703$	40.0000	1.50863
$704$	0	0
$705$	16.0000	0.602595
$706$	10.0000	0.376355
$707$	−40.0000	−1.50435
$708$	−12.0000	−0.450988
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	−16.0000	−0.600469
$711$	−16.0000	−0.600047
$712$	10.0000	0.374766
$713$	8.00000	0.299602
$714$	24.0000	0.898177
$715$	0	0
$716$	−20.0000	−0.747435
$717$	24.0000	0.896296
$718$	−4.00000	−0.149279
$719$	−32.0000	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$720$	2.00000	0.0745356
$721$	16.0000	0.595871
$722$	−3.00000	−0.111648
$723$	−2.00000	−0.0743808
$724$	−2.00000	−0.0743294
$725$	−1.00000	−0.0371391
$726$	11.0000	0.408248
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	−24.0000	−0.887672
$732$	14.0000	0.517455
$733$	50.0000	1.84679	0.923396	−	0.383849i	$-0.125402\pi$
$733$	50.0000	1.84679	0.923396	+	0.383849i	$0.125402\pi$
$734$	−32.0000	−1.18114
$735$	−18.0000	−0.663940
$736$	1.00000	0.0368605
$737$	0	0
$738$	10.0000	0.368105
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	20.0000	0.735215
$741$	8.00000	0.293887
$742$	−24.0000	−0.881068
$743$	−28.0000	−1.02722	−0.513610	−	0.858024i	$-0.671692\pi$
$743$	−28.0000	−1.02722	−0.513610	+	0.858024i	$0.671692\pi$
$744$	−8.00000	−0.293294
$745$	20.0000	0.732743
$746$	−10.0000	−0.366126
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−48.0000	−1.75388
$750$	12.0000	0.438178
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	−8.00000	−0.291730
$753$	8.00000	0.291536
$754$	−2.00000	−0.0728357
$755$	0	0
$756$	−4.00000	−0.145479
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	8.00000	0.290191
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	−8.00000	−0.289809
$763$	−8.00000	−0.289619
$764$	12.0000	0.434145
$765$	−12.0000	−0.433861
$766$	−32.0000	−1.15621
$767$	−24.0000	−0.866590
$768$	−1.00000	−0.0360844
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	26.0000	0.935760
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	4.00000	0.143777
$775$	−8.00000	−0.287368
$776$	6.00000	0.215387
$777$	−40.0000	−1.43499
$778$	6.00000	0.215110
$779$	40.0000	1.43315
$780$	4.00000	0.143223
$781$	0	0
$782$	−6.00000	−0.214560
$783$	−1.00000	−0.0357371
$784$	9.00000	0.321429
$785$	4.00000	0.142766
$786$	−12.0000	−0.428026
$787$	−24.0000	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$787$	−24.0000	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$788$	22.0000	0.783718
$789$	−4.00000	−0.142404
$790$	−32.0000	−1.13851
$791$	40.0000	1.42224
$792$	0	0
$793$	28.0000	0.994309
$794$	−18.0000	−0.638796
$795$	12.0000	0.425596
$796$	4.00000	0.141776
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	−16.0000	−0.566394
$799$	48.0000	1.69812
$800$	−1.00000	−0.0353553
$801$	10.0000	0.353333
$802$	38.0000	1.34183
$803$	0	0
$804$	−16.0000	−0.564276
$805$	8.00000	0.281963
$806$	−16.0000	−0.563576
$807$	18.0000	0.633630
$808$	−10.0000	−0.351799
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	2.00000	0.0702728
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	4.00000	0.140372
$813$	8.00000	0.280572
$814$	0	0
$815$	24.0000	0.840683
$816$	6.00000	0.210042
$817$	16.0000	0.559769
$818$	2.00000	0.0699284
$819$	−8.00000	−0.279543
$820$	20.0000	0.698430
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	−10.0000	−0.348790
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	48.0000	1.67013
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	1.00000	0.0347524
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	−24.0000	−0.833052
$831$	10.0000	0.346896
$832$	−2.00000	−0.0693375
$833$	−54.0000	−1.87099
$834$	−4.00000	−0.138509
$835$	32.0000	1.10741
$836$	0	0
$837$	−8.00000	−0.276520
$838$	−12.0000	−0.414533
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	−8.00000	−0.276026
$841$	1.00000	0.0344828
$842$	2.00000	0.0689246
$843$	−6.00000	−0.206651
$844$	−20.0000	−0.688428
$845$	−18.0000	−0.619219
$846$	−8.00000	−0.275046
$847$	−44.0000	−1.51186
$848$	−6.00000	−0.206041
$849$	−16.0000	−0.549119
$850$	6.00000	0.205798
$851$	10.0000	0.342796
$852$	8.00000	0.274075
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	−56.0000	−1.91628
$855$	8.00000	0.273594
$856$	−12.0000	−0.410152
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	8.00000	0.272798
$861$	−40.0000	−1.36320
$862$	−16.0000	−0.544962
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	−1.00000	−0.0340207
$865$	−36.0000	−1.22404
$866$	−10.0000	−0.339814
$867$	−19.0000	−0.645274
$868$	32.0000	1.08615
$869$	0	0
$870$	−2.00000	−0.0678064
$871$	−32.0000	−1.08428
$872$	−2.00000	−0.0677285
$873$	6.00000	0.203069
$874$	4.00000	0.135302
$875$	−48.0000	−1.62270
$876$	−2.00000	−0.0675737
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	40.0000	1.34993
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	9.00000	0.303046
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	12.0000	0.403604
$885$	−24.0000	−0.806751
$886$	−20.0000	−0.671913
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	−10.0000	−0.335578
$889$	32.0000	1.07325
$890$	20.0000	0.670402
$891$	0	0
$892$	−16.0000	−0.535720
$893$	−32.0000	−1.07084
$894$	−10.0000	−0.334450
$895$	−40.0000	−1.33705
$896$	4.00000	0.133631
$897$	2.00000	0.0667781
$898$	2.00000	0.0667409
$899$	8.00000	0.266815
$900$	−1.00000	−0.0333333
$901$	36.0000	1.19933
$902$	0	0
$903$	−16.0000	−0.532447
$904$	10.0000	0.332595
$905$	−4.00000	−0.132964
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	−4.00000	−0.132745
$909$	−10.0000	−0.331679
$910$	−16.0000	−0.530395
$911$	−28.0000	−0.927681	−0.463841	−	0.885919i	$-0.653529\pi$
$911$	−28.0000	−0.927681	−0.463841	+	0.885919i	$0.653529\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	2.00000	0.0661541
$915$	28.0000	0.925651
$916$	−6.00000	−0.198246
$917$	48.0000	1.58510
$918$	6.00000	0.198030
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	2.00000	0.0659380
$921$	−28.0000	−0.922631
$922$	14.0000	0.461065
$923$	16.0000	0.526646
$924$	0	0
$925$	−10.0000	−0.328798
$926$	8.00000	0.262896
$927$	4.00000	0.131377
$928$	1.00000	0.0328266
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	−16.0000	−0.524661
$931$	36.0000	1.17985
$932$	−6.00000	−0.196537
$933$	32.0000	1.04763
$934$	−24.0000	−0.785304
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	−54.0000	−1.76410	−0.882052	−	0.471153i	$-0.843838\pi$
$937$	−54.0000	−1.76410	−0.882052	+	0.471153i	$0.843838\pi$
$938$	64.0000	2.08967
$939$	30.0000	0.979013
$940$	−16.0000	−0.521862
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	−2.00000	−0.0651635
$943$	10.0000	0.325645
$944$	12.0000	0.390567
$945$	−8.00000	−0.260240
$946$	0	0
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	16.0000	0.519656
$949$	−4.00000	−0.129845
$950$	−4.00000	−0.129777
$951$	−30.0000	−0.972817
$952$	−24.0000	−0.777844
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	−6.00000	−0.194257
$955$	24.0000	0.776622
$956$	−24.0000	−0.776215
$957$	0	0
$958$	12.0000	0.387702
$959$	40.0000	1.29167
$960$	−2.00000	−0.0645497
$961$	33.0000	1.06452
$962$	−20.0000	−0.644826
$963$	−12.0000	−0.386695
$964$	2.00000	0.0644157
$965$	52.0000	1.67394
$966$	−4.00000	−0.128698
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	−11.0000	−0.353553
$969$	24.0000	0.770991
$970$	12.0000	0.385297
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	−1.00000	−0.0320750
$973$	16.0000	0.512936
$974$	−32.0000	−1.02535
$975$	−2.00000	−0.0640513
$976$	−14.0000	−0.448129
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	−12.0000	−0.383718
$979$	0	0
$980$	18.0000	0.574989
$981$	−2.00000	−0.0638551
$982$	−36.0000	−1.14881
$983$	−60.0000	−1.91370	−0.956851	−	0.290578i	$-0.906153\pi$
$983$	−60.0000	−1.91370	−0.956851	+	0.290578i	$0.906153\pi$
$984$	−10.0000	−0.318788
$985$	44.0000	1.40196
$986$	−6.00000	−0.191079
$987$	32.0000	1.01857
$988$	−8.00000	−0.254514
$989$	4.00000	0.127193
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	8.00000	0.254000
$993$	20.0000	0.634681
$994$	−32.0000	−1.01498
$995$	8.00000	0.253617
$996$	12.0000	0.380235
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	4.00000	0.126618
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.l.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.l.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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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