LMFDB - Embedded newform 5077.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5077,2,Mod(1,5077)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5077, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5077.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5077 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5077.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:	$40.5400491062$
Analytic rank:	$3$
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	$\chi$	$=$	5077.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -3.00000 q^{3} +2.00000 q^{4} -4.00000 q^{5} +6.00000 q^{6} -4.00000 q^{7} +6.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -3.00000 q^{3} +2.00000 q^{4} -4.00000 q^{5} +6.00000 q^{6} -4.00000 q^{7} +6.00000 q^{9} +8.00000 q^{10} -6.00000 q^{11} -6.00000 q^{12} -4.00000 q^{13} +8.00000 q^{14} +12.0000 q^{15} -4.00000 q^{16} -4.00000 q^{17} -12.0000 q^{18} -7.00000 q^{19} -8.00000 q^{20} +12.0000 q^{21} +12.0000 q^{22} -6.00000 q^{23} +11.0000 q^{25} +8.00000 q^{26} -9.00000 q^{27} -8.00000 q^{28} -6.00000 q^{29} -24.0000 q^{30} -2.00000 q^{31} +8.00000 q^{32} +18.0000 q^{33} +8.00000 q^{34} +16.0000 q^{35} +12.0000 q^{36} +14.0000 q^{38} +12.0000 q^{39} -24.0000 q^{42} -8.00000 q^{43} -12.0000 q^{44} -24.0000 q^{45} +12.0000 q^{46} -9.00000 q^{47} +12.0000 q^{48} +9.00000 q^{49} -22.0000 q^{50} +12.0000 q^{51} -8.00000 q^{52} -9.00000 q^{53} +18.0000 q^{54} +24.0000 q^{55} +21.0000 q^{57} +12.0000 q^{58} -11.0000 q^{59} +24.0000 q^{60} -2.00000 q^{61} +4.00000 q^{62} -24.0000 q^{63} -8.00000 q^{64} +16.0000 q^{65} -36.0000 q^{66} -12.0000 q^{67} -8.00000 q^{68} +18.0000 q^{69} -32.0000 q^{70} -8.00000 q^{71} -14.0000 q^{73} -33.0000 q^{75} -14.0000 q^{76} +24.0000 q^{77} -24.0000 q^{78} +9.00000 q^{79} +16.0000 q^{80} +9.00000 q^{81} -2.00000 q^{83} +24.0000 q^{84} +16.0000 q^{85} +16.0000 q^{86} +18.0000 q^{87} +11.0000 q^{89} +48.0000 q^{90} +16.0000 q^{91} -12.0000 q^{92} +6.00000 q^{93} +18.0000 q^{94} +28.0000 q^{95} -24.0000 q^{96} +6.00000 q^{97} -18.0000 q^{98} -36.0000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	2.00000	1.00000
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	6.00000	2.44949
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	8.00000	2.52982
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	−6.00000	−1.73205
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	8.00000	2.13809
$15$	12.0000	3.09839
$16$	−4.00000	−1.00000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−12.0000	−2.82843
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	−8.00000	−1.78885
$21$	12.0000	2.61861
$22$	12.0000	2.55841
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	8.00000	1.56893
$27$	−9.00000	−1.73205
$28$	−8.00000	−1.51186
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−24.0000	−4.38178
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	8.00000	1.41421
$33$	18.0000	3.13340
$34$	8.00000	1.37199
$35$	16.0000	2.70449
$36$	12.0000	2.00000
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	14.0000	2.27110
$39$	12.0000	1.92154
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−24.0000	−3.70328
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−12.0000	−1.80907
$45$	−24.0000	−3.57771
$46$	12.0000	1.76930
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	12.0000	1.73205
$49$	9.00000	1.28571
$50$	−22.0000	−3.11127
$51$	12.0000	1.68034
$52$	−8.00000	−1.10940
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	18.0000	2.44949
$55$	24.0000	3.23616
$56$	0	0
$57$	21.0000	2.78152
$58$	12.0000	1.57568
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	24.0000	3.09839
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.00000	0.508001
$63$	−24.0000	−3.02372
$64$	−8.00000	−1.00000
$65$	16.0000	1.98456
$66$	−36.0000	−4.43129
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−8.00000	−0.970143
$69$	18.0000	2.16695
$70$	−32.0000	−3.82473
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	−33.0000	−3.81051
$76$	−14.0000	−1.60591
$77$	24.0000	2.73505
$78$	−24.0000	−2.71746
$79$	9.00000	1.01258	0.506290	−	0.862364i	$-0.331017\pi$
$79$	9.00000	1.01258	0.506290	+	0.862364i	$0.331017\pi$
$80$	16.0000	1.78885
$81$	9.00000	1.00000
$82$	0	0
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	24.0000	2.61861
$85$	16.0000	1.73544
$86$	16.0000	1.72532
$87$	18.0000	1.92980
$88$	0	0
$89$	11.0000	1.16600	0.582999	−	0.812473i	$-0.301879\pi$
$89$	11.0000	1.16600	0.582999	+	0.812473i	$0.301879\pi$
$90$	48.0000	5.05964
$91$	16.0000	1.67726
$92$	−12.0000	−1.25109
$93$	6.00000	0.622171
$94$	18.0000	1.85656
$95$	28.0000	2.87274
$96$	−24.0000	−2.44949
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	−18.0000	−1.81827
$99$	−36.0000	−3.61814
$100$	22.0000	2.20000
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	−24.0000	−2.37635
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−48.0000	−4.68432
$106$	18.0000	1.74831
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−18.0000	−1.73205
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	−48.0000	−4.57662
$111$	0	0
$112$	16.0000	1.51186
$113$	−5.00000	−0.470360	−0.235180	−	0.971952i	$-0.575568\pi$
$113$	−5.00000	−0.470360	−0.235180	+	0.971952i	$0.575568\pi$
$114$	−42.0000	−3.93366
$115$	24.0000	2.23801
$116$	−12.0000	−1.11417
$117$	−24.0000	−2.21880
$118$	22.0000	2.02526
$119$	16.0000	1.46672
$120$	0	0
$121$	25.0000	2.27273
$122$	4.00000	0.362143
$123$	0	0
$124$	−4.00000	−0.359211
$125$	−24.0000	−2.14663
$126$	48.0000	4.27618
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	24.0000	2.11308
$130$	−32.0000	−2.80659
$131$	−7.00000	−0.611593	−0.305796	−	0.952097i	$-0.598923\pi$
$131$	−7.00000	−0.611593	−0.305796	+	0.952097i	$0.598923\pi$
$132$	36.0000	3.13340
$133$	28.0000	2.42791
$134$	24.0000	2.07328
$135$	36.0000	3.09839
$136$	0	0
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	−36.0000	−3.06452
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	32.0000	2.70449
$141$	27.0000	2.27381
$142$	16.0000	1.34269
$143$	24.0000	2.00698
$144$	−24.0000	−2.00000
$145$	24.0000	1.99309
$146$	28.0000	2.31730
$147$	−27.0000	−2.22692
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	66.0000	5.38888
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	0	0
$153$	−24.0000	−1.94029
$154$	−48.0000	−3.86795
$155$	8.00000	0.642575
$156$	24.0000	1.92154
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−18.0000	−1.43200
$159$	27.0000	2.14124
$160$	−32.0000	−2.52982
$161$	24.0000	1.89146
$162$	−18.0000	−1.41421
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	−72.0000	−5.60519
$166$	4.00000	0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−32.0000	−2.45429
$171$	−42.0000	−3.21182
$172$	−16.0000	−1.21999
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	−36.0000	−2.72915
$175$	−44.0000	−3.32609
$176$	24.0000	1.80907
$177$	33.0000	2.48043
$178$	−22.0000	−1.64897
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	−48.0000	−3.57771
$181$	19.0000	1.41226	0.706129	−	0.708083i	$-0.250440\pi$
$181$	19.0000	1.41226	0.706129	+	0.708083i	$0.250440\pi$
$182$	−32.0000	−2.37200
$183$	6.00000	0.443533
$184$	0	0
$185$	0	0
$186$	−12.0000	−0.879883
$187$	24.0000	1.75505
$188$	−18.0000	−1.31278
$189$	36.0000	2.61861
$190$	−56.0000	−4.06267
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	24.0000	1.73205
$193$	−13.0000	−0.935760	−0.467880	−	0.883792i	$-0.654982\pi$
$193$	−13.0000	−0.935760	−0.467880	+	0.883792i	$0.654982\pi$
$194$	−12.0000	−0.861550
$195$	−48.0000	−3.43735
$196$	18.0000	1.28571
$197$	−20.0000	−1.42494	−0.712470	−	0.701702i	$-0.752424\pi$
$197$	−20.0000	−1.42494	−0.712470	+	0.701702i	$0.752424\pi$
$198$	72.0000	5.11682
$199$	−13.0000	−0.921546	−0.460773	−	0.887518i	$-0.652428\pi$
$199$	−13.0000	−0.921546	−0.460773	+	0.887518i	$0.652428\pi$
$200$	0	0
$201$	36.0000	2.53924
$202$	−8.00000	−0.562878
$203$	24.0000	1.68447
$204$	24.0000	1.68034
$205$	0	0
$206$	0	0
$207$	−36.0000	−2.50217
$208$	16.0000	1.10940
$209$	42.0000	2.90520
$210$	96.0000	6.62463
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−18.0000	−1.23625
$213$	24.0000	1.64445
$214$	−24.0000	−1.64061
$215$	32.0000	2.18238
$216$	0	0
$217$	8.00000	0.543075
$218$	−14.0000	−0.948200
$219$	42.0000	2.83810
$220$	48.0000	3.23616
$221$	16.0000	1.07628
$222$	0	0
$223$	21.0000	1.40626	0.703132	−	0.711059i	$-0.251784\pi$
$223$	21.0000	1.40626	0.703132	+	0.711059i	$0.251784\pi$
$224$	−32.0000	−2.13809
$225$	66.0000	4.40000
$226$	10.0000	0.665190
$227$	−26.0000	−1.72568	−0.862840	−	0.505477i	$-0.831317\pi$
$227$	−26.0000	−1.72568	−0.862840	+	0.505477i	$0.831317\pi$
$228$	42.0000	2.78152
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−48.0000	−3.16503
$231$	−72.0000	−4.73725
$232$	0	0
$233$	15.0000	0.982683	0.491341	−	0.870967i	$-0.336507\pi$
$233$	15.0000	0.982683	0.491341	+	0.870967i	$0.336507\pi$
$234$	48.0000	3.13786
$235$	36.0000	2.34838
$236$	−22.0000	−1.43208
$237$	−27.0000	−1.75384
$238$	−32.0000	−2.07425
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	−48.0000	−3.09839
$241$	−29.0000	−1.86805	−0.934027	−	0.357202i	$-0.883731\pi$
$241$	−29.0000	−1.86805	−0.934027	+	0.357202i	$0.883731\pi$
$242$	−50.0000	−3.21412
$243$	0	0
$244$	−4.00000	−0.256074
$245$	−36.0000	−2.29996
$246$	0	0
$247$	28.0000	1.78160
$248$	0	0
$249$	6.00000	0.380235
$250$	48.0000	3.03579
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	−48.0000	−3.02372
$253$	36.0000	2.26330
$254$	−14.0000	−0.878438
$255$	−48.0000	−3.00588
$256$	16.0000	1.00000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	−48.0000	−2.98835
$259$	0	0
$260$	32.0000	1.98456
$261$	−36.0000	−2.22834
$262$	14.0000	0.864923
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	36.0000	2.21146
$266$	−56.0000	−3.43358
$267$	−33.0000	−2.01957
$268$	−24.0000	−1.46603
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	−72.0000	−4.38178
$271$	18.0000	1.09342	0.546711	−	0.837321i	$-0.315880\pi$
$271$	18.0000	1.09342	0.546711	+	0.837321i	$0.315880\pi$
$272$	16.0000	0.970143
$273$	−48.0000	−2.90509
$274$	6.00000	0.362473
$275$	−66.0000	−3.97995
$276$	36.0000	2.16695
$277$	−13.0000	−0.781094	−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000	−0.781094	−0.390547	+	0.920583i	$0.627714\pi$
$278$	−8.00000	−0.479808
$279$	−12.0000	−0.718421
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	−54.0000	−3.21565
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	−16.0000	−0.949425
$285$	−84.0000	−4.97573
$286$	−48.0000	−2.83830
$287$	0	0
$288$	48.0000	2.82843
$289$	−1.00000	−0.0588235
$290$	−48.0000	−2.81866
$291$	−18.0000	−1.05518
$292$	−28.0000	−1.63858
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	54.0000	3.14934
$295$	44.0000	2.56178
$296$	0	0
$297$	54.0000	3.13340
$298$	−12.0000	−0.695141
$299$	24.0000	1.38796
$300$	−66.0000	−3.81051
$301$	32.0000	1.84445
$302$	−18.0000	−1.03578
$303$	−12.0000	−0.689382
$304$	28.0000	1.60591
$305$	8.00000	0.458079
$306$	48.0000	2.74398
$307$	−17.0000	−0.970241	−0.485121	−	0.874447i	$-0.661224\pi$
$307$	−17.0000	−0.970241	−0.485121	+	0.874447i	$0.661224\pi$
$308$	48.0000	2.73505
$309$	0	0
$310$	−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	$0.861843\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−8.00000	−0.451466
$315$	96.0000	5.40899
$316$	18.0000	1.01258
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	−54.0000	−3.02817
$319$	36.0000	2.01561
$320$	32.0000	1.78885
$321$	−36.0000	−2.00932
$322$	−48.0000	−2.67494
$323$	28.0000	1.55796
$324$	18.0000	1.00000
$325$	−44.0000	−2.44068
$326$	8.00000	0.443079
$327$	−21.0000	−1.16130
$328$	0	0
$329$	36.0000	1.98474
$330$	144.000	7.92694
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	16.0000	0.875481
$335$	48.0000	2.62252
$336$	−48.0000	−2.61861
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	−6.00000	−0.326357
$339$	15.0000	0.814688
$340$	32.0000	1.73544
$341$	12.0000	0.649836
$342$	84.0000	4.54220
$343$	−8.00000	−0.431959
$344$	0	0
$345$	−72.0000	−3.87635
$346$	18.0000	0.967686
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	36.0000	1.92980
$349$	−35.0000	−1.87351	−0.936754	−	0.349990i	$-0.886185\pi$
$349$	−35.0000	−1.87351	−0.936754	+	0.349990i	$0.886185\pi$
$350$	88.0000	4.70380
$351$	36.0000	1.92154
$352$	−48.0000	−2.55841
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	−66.0000	−3.50786
$355$	32.0000	1.69838
$356$	22.0000	1.16600
$357$	−48.0000	−2.54043
$358$	−18.0000	−0.951330
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	−38.0000	−1.99724
$363$	−75.0000	−3.93648
$364$	32.0000	1.67726
$365$	56.0000	2.93117
$366$	−12.0000	−0.627250
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	24.0000	1.25109
$369$	0	0
$370$	0	0
$371$	36.0000	1.86903
$372$	12.0000	0.622171
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	−48.0000	−2.48202
$375$	72.0000	3.71806
$376$	0	0
$377$	24.0000	1.23606
$378$	−72.0000	−3.70328
$379$	15.0000	0.770498	0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000	0.770498	0.385249	+	0.922813i	$0.374116\pi$
$380$	56.0000	2.87274
$381$	−21.0000	−1.07586
$382$	32.0000	1.63726
$383$	−27.0000	−1.37964	−0.689818	−	0.723983i	$-0.742309\pi$
$383$	−27.0000	−1.37964	−0.689818	+	0.723983i	$0.742309\pi$
$384$	0	0
$385$	−96.0000	−4.89261
$386$	26.0000	1.32337
$387$	−48.0000	−2.43998
$388$	12.0000	0.609208
$389$	−3.00000	−0.152106	−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000	−0.152106	−0.0760530	+	0.997104i	$0.524232\pi$
$390$	96.0000	4.86115
$391$	24.0000	1.21373
$392$	0	0
$393$	21.0000	1.05931
$394$	40.0000	2.01517
$395$	−36.0000	−1.81136
$396$	−72.0000	−3.61814
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	26.0000	1.30326
$399$	−84.0000	−4.20526
$400$	−44.0000	−2.20000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−72.0000	−3.59103
$403$	8.00000	0.398508
$404$	8.00000	0.398015
$405$	−36.0000	−1.78885
$406$	−48.0000	−2.38220
$407$	0	0
$408$	0	0
$409$	−31.0000	−1.53285	−0.766426	−	0.642333i	$-0.777967\pi$
$409$	−31.0000	−1.53285	−0.766426	+	0.642333i	$0.777967\pi$
$410$	0	0
$411$	9.00000	0.443937
$412$	0	0
$413$	44.0000	2.16510
$414$	72.0000	3.53861
$415$	8.00000	0.392705
$416$	−32.0000	−1.56893
$417$	−12.0000	−0.587643
$418$	−84.0000	−4.10857
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	−96.0000	−4.68432
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	24.0000	1.16830
$423$	−54.0000	−2.62557
$424$	0	0
$425$	−44.0000	−2.13431
$426$	−48.0000	−2.32561
$427$	8.00000	0.387147
$428$	24.0000	1.16008
$429$	−72.0000	−3.47619
$430$	−64.0000	−3.08635
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	36.0000	1.73205
$433$	12.0000	0.576683	0.288342	−	0.957528i	$-0.406896\pi$
$433$	12.0000	0.576683	0.288342	+	0.957528i	$0.406896\pi$
$434$	−16.0000	−0.768025
$435$	−72.0000	−3.45214
$436$	14.0000	0.670478
$437$	42.0000	2.00913
$438$	−84.0000	−4.01368
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	54.0000	2.57143
$442$	−32.0000	−1.52208
$443$	−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
$445$	−44.0000	−2.08580
$446$	−42.0000	−1.98876
$447$	−18.0000	−0.851371
$448$	32.0000	1.51186
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	−132.000	−6.22254
$451$	0	0
$452$	−10.0000	−0.470360
$453$	−27.0000	−1.26857
$454$	52.0000	2.44048
$455$	−64.0000	−3.00037
$456$	0	0
$457$	−21.0000	−0.982339	−0.491169	−	0.871064i	$-0.663430\pi$
$457$	−21.0000	−0.982339	−0.491169	+	0.871064i	$0.663430\pi$
$458$	20.0000	0.934539
$459$	36.0000	1.68034
$460$	48.0000	2.23801
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	144.000	6.69949
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	24.0000	1.11417
$465$	−24.0000	−1.11297
$466$	−30.0000	−1.38972
$467$	−11.0000	−0.509019	−0.254510	−	0.967070i	$-0.581914\pi$
$467$	−11.0000	−0.509019	−0.254510	+	0.967070i	$0.581914\pi$
$468$	−48.0000	−2.21880
$469$	48.0000	2.21643
$470$	−72.0000	−3.32111
$471$	−12.0000	−0.552931
$472$	0	0
$473$	48.0000	2.20704
$474$	54.0000	2.48030
$475$	−77.0000	−3.53300
$476$	32.0000	1.46672
$477$	−54.0000	−2.47249
$478$	30.0000	1.37217
$479$	−26.0000	−1.18797	−0.593985	−	0.804476i	$-0.702446\pi$
$479$	−26.0000	−1.18797	−0.593985	+	0.804476i	$0.702446\pi$
$480$	96.0000	4.38178
$481$	0	0
$482$	58.0000	2.64183
$483$	−72.0000	−3.27611
$484$	50.0000	2.27273
$485$	−24.0000	−1.08978
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	72.0000	3.25263
$491$	40.0000	1.80517	0.902587	−	0.430507i	$-0.141665\pi$
$491$	40.0000	1.80517	0.902587	+	0.430507i	$0.141665\pi$
$492$	0	0
$493$	24.0000	1.08091
$494$	−56.0000	−2.51956
$495$	144.000	6.47232
$496$	8.00000	0.359211
$497$	32.0000	1.43540
$498$	−12.0000	−0.537733
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	−48.0000	−2.14663
$501$	24.0000	1.07224
$502$	−16.0000	−0.714115
$503$	−41.0000	−1.82810	−0.914050	−	0.405602i	$-0.867062\pi$
$503$	−41.0000	−1.82810	−0.914050	+	0.405602i	$0.867062\pi$
$504$	0	0
$505$	−16.0000	−0.711991
$506$	−72.0000	−3.20079
$507$	−9.00000	−0.399704
$508$	14.0000	0.621150
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	96.0000	4.25095
$511$	56.0000	2.47729
$512$	−32.0000	−1.41421
$513$	63.0000	2.78152
$514$	36.0000	1.58789
$515$	0	0
$516$	48.0000	2.11308
$517$	54.0000	2.37492
$518$	0	0
$519$	27.0000	1.18517
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	72.0000	3.15135
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	−14.0000	−0.611593
$525$	132.000	5.76095
$526$	−4.00000	−0.174408
$527$	8.00000	0.348485
$528$	−72.0000	−3.13340
$529$	13.0000	0.565217
$530$	−72.0000	−3.12748
$531$	−66.0000	−2.86416
$532$	56.0000	2.42791
$533$	0	0
$534$	66.0000	2.85610
$535$	−48.0000	−2.07522
$536$	0	0
$537$	−27.0000	−1.16514
$538$	28.0000	1.20717
$539$	−54.0000	−2.32594
$540$	72.0000	3.09839
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	−36.0000	−1.54633
$543$	−57.0000	−2.44610
$544$	−32.0000	−1.37199
$545$	−28.0000	−1.19939
$546$	96.0000	4.10842
$547$	9.00000	0.384812	0.192406	−	0.981315i	$-0.438371\pi$
$547$	9.00000	0.384812	0.192406	+	0.981315i	$0.438371\pi$
$548$	−6.00000	−0.256307
$549$	−12.0000	−0.512148
$550$	132.000	5.62850
$551$	42.0000	1.78926
$552$	0	0
$553$	−36.0000	−1.53088
$554$	26.0000	1.10463
$555$	0	0
$556$	8.00000	0.339276
$557$	17.0000	0.720313	0.360157	−	0.932892i	$-0.382723\pi$
$557$	17.0000	0.720313	0.360157	+	0.932892i	$0.382723\pi$
$558$	24.0000	1.01600
$559$	32.0000	1.35346
$560$	−64.0000	−2.70449
$561$	−72.0000	−3.03984
$562$	−48.0000	−2.02476
$563$	−33.0000	−1.39078	−0.695392	−	0.718631i	$-0.744769\pi$
$563$	−33.0000	−1.39078	−0.695392	+	0.718631i	$0.744769\pi$
$564$	54.0000	2.27381
$565$	20.0000	0.841406
$566$	32.0000	1.34506
$567$	−36.0000	−1.51186
$568$	0	0
$569$	2.00000	0.0838444	0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000	0.0838444	0.0419222	+	0.999121i	$0.486652\pi$
$570$	168.000	7.03675
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	48.0000	2.00698
$573$	48.0000	2.00523
$574$	0	0
$575$	−66.0000	−2.75239
$576$	−48.0000	−2.00000
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	2.00000	0.0831890
$579$	39.0000	1.62078
$580$	48.0000	1.99309
$581$	8.00000	0.331896
$582$	36.0000	1.49225
$583$	54.0000	2.23645
$584$	0	0
$585$	96.0000	3.96911
$586$	−18.0000	−0.743573
$587$	−26.0000	−1.07313	−0.536567	−	0.843857i	$-0.680279\pi$
$587$	−26.0000	−1.07313	−0.536567	+	0.843857i	$0.680279\pi$
$588$	−54.0000	−2.22692
$589$	14.0000	0.576860
$590$	−88.0000	−3.62290
$591$	60.0000	2.46807
$592$	0	0
$593$	5.00000	0.205325	0.102663	−	0.994716i	$-0.467264\pi$
$593$	5.00000	0.205325	0.102663	+	0.994716i	$0.467264\pi$
$594$	−108.000	−4.43129
$595$	−64.0000	−2.62374
$596$	12.0000	0.491539
$597$	39.0000	1.59616
$598$	−48.0000	−1.96287
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	−64.0000	−2.60845
$603$	−72.0000	−2.93207
$604$	18.0000	0.732410
$605$	−100.000	−4.06558
$606$	24.0000	0.974933
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	−56.0000	−2.27110
$609$	−72.0000	−2.91759
$610$	−16.0000	−0.647821
$611$	36.0000	1.45640
$612$	−48.0000	−1.94029
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	34.0000	1.37213
$615$	0	0
$616$	0	0
$617$	−4.00000	−0.161034	−0.0805170	−	0.996753i	$-0.525657\pi$
$617$	−4.00000	−0.161034	−0.0805170	+	0.996753i	$0.525657\pi$
$618$	0	0
$619$	−3.00000	−0.120580	−0.0602901	−	0.998181i	$-0.519203\pi$
$619$	−3.00000	−0.120580	−0.0602901	+	0.998181i	$0.519203\pi$
$620$	16.0000	0.642575
$621$	54.0000	2.16695
$622$	64.0000	2.56617
$623$	−44.0000	−1.76282
$624$	−48.0000	−1.92154
$625$	41.0000	1.64000
$626$	28.0000	1.11911
$627$	−126.000	−5.03196
$628$	8.00000	0.319235
$629$	0	0
$630$	−192.000	−7.64946
$631$	47.0000	1.87104	0.935520	−	0.353273i	$-0.114931\pi$
$631$	47.0000	1.87104	0.935520	+	0.353273i	$0.114931\pi$
$632$	0	0
$633$	36.0000	1.43087
$634$	44.0000	1.74746
$635$	−28.0000	−1.11115
$636$	54.0000	2.14124
$637$	−36.0000	−1.42637
$638$	−72.0000	−2.85051
$639$	−48.0000	−1.89885
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	72.0000	2.84161
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	48.0000	1.89146
$645$	−96.0000	−3.78000
$646$	−56.0000	−2.20329
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	0	0
$649$	66.0000	2.59073
$650$	88.0000	3.45164
$651$	−24.0000	−0.940634
$652$	−8.00000	−0.313304
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	42.0000	1.64233
$655$	28.0000	1.09405
$656$	0	0
$657$	−84.0000	−3.27715
$658$	−72.0000	−2.80685
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	−144.000	−5.60519
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	8.00000	0.310929
$663$	−48.0000	−1.86417
$664$	0	0
$665$	−112.000	−4.34317
$666$	0	0
$667$	36.0000	1.39393
$668$	−16.0000	−0.619059
$669$	−63.0000	−2.43572
$670$	−96.0000	−3.70880
$671$	12.0000	0.463255
$672$	96.0000	3.70328
$673$	16.0000	0.616755	0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000	0.616755	0.308377	+	0.951264i	$0.400214\pi$
$674$	−16.0000	−0.616297
$675$	−99.0000	−3.81051
$676$	6.00000	0.230769
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	−30.0000	−1.15214
$679$	−24.0000	−0.921035
$680$	0	0
$681$	78.0000	2.98897
$682$	−24.0000	−0.919007
$683$	−45.0000	−1.72188	−0.860939	−	0.508709i	$-0.830123\pi$
$683$	−45.0000	−1.72188	−0.860939	+	0.508709i	$0.830123\pi$
$684$	−84.0000	−3.21182
$685$	12.0000	0.458496
$686$	16.0000	0.610883
$687$	30.0000	1.14457
$688$	32.0000	1.21999
$689$	36.0000	1.37149
$690$	144.000	5.48199
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	−18.0000	−0.684257
$693$	144.000	5.47011
$694$	−24.0000	−0.911028
$695$	−16.0000	−0.606915
$696$	0	0
$697$	0	0
$698$	70.0000	2.64954
$699$	−45.0000	−1.70206
$700$	−88.0000	−3.32609
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	−72.0000	−2.71746
$703$	0	0
$704$	48.0000	1.80907
$705$	−108.000	−4.06752
$706$	68.0000	2.55921
$707$	−16.0000	−0.601742
$708$	66.0000	2.48043
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	−64.0000	−2.40188
$711$	54.0000	2.02516
$712$	0	0
$713$	12.0000	0.449404
$714$	96.0000	3.59271
$715$	−96.0000	−3.59020
$716$	18.0000	0.672692
$717$	45.0000	1.68056
$718$	30.0000	1.11959
$719$	−10.0000	−0.372937	−0.186469	−	0.982461i	$-0.559704\pi$
$719$	−10.0000	−0.372937	−0.186469	+	0.982461i	$0.559704\pi$
$720$	96.0000	3.57771
$721$	0	0
$722$	−60.0000	−2.23297
$723$	87.0000	3.23556
$724$	38.0000	1.41226
$725$	−66.0000	−2.45118
$726$	150.000	5.56702
$727$	−33.0000	−1.22390	−0.611951	−	0.790896i	$-0.709615\pi$
$727$	−33.0000	−1.22390	−0.611951	+	0.790896i	$0.709615\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	−112.000	−4.14531
$731$	32.0000	1.18356
$732$	12.0000	0.443533
$733$	−18.0000	−0.664845	−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000	−0.664845	−0.332423	+	0.943131i	$0.607866\pi$
$734$	−64.0000	−2.36228
$735$	108.000	3.98364
$736$	−48.0000	−1.76930
$737$	72.0000	2.65215
$738$	0	0
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	0	0
$741$	−84.0000	−3.08582
$742$	−72.0000	−2.64320
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	−24.0000	−0.879292
$746$	0	0
$747$	−12.0000	−0.439057
$748$	48.0000	1.75505
$749$	−48.0000	−1.75388
$750$	−144.000	−5.25814
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	36.0000	1.31278
$753$	−24.0000	−0.874609
$754$	−48.0000	−1.74806
$755$	−36.0000	−1.31017
$756$	72.0000	2.61861
$757$	−33.0000	−1.19941	−0.599703	−	0.800223i	$-0.704714\pi$
$757$	−33.0000	−1.19941	−0.599703	+	0.800223i	$0.704714\pi$
$758$	−30.0000	−1.08965
$759$	−108.000	−3.92015
$760$	0	0
$761$	−38.0000	−1.37750	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000	−1.37750	−0.688749	+	0.724999i	$0.741840\pi$
$762$	42.0000	1.52150
$763$	−28.0000	−1.01367
$764$	−32.0000	−1.15772
$765$	96.0000	3.47089
$766$	54.0000	1.95110
$767$	44.0000	1.58875
$768$	−48.0000	−1.73205
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	192.000	6.91920
$771$	54.0000	1.94476
$772$	−26.0000	−0.935760
$773$	3.00000	0.107903	0.0539513	−	0.998544i	$-0.482818\pi$
$773$	3.00000	0.107903	0.0539513	+	0.998544i	$0.482818\pi$
$774$	96.0000	3.45065
$775$	−22.0000	−0.790263
$776$	0	0
$777$	0	0
$778$	6.00000	0.215110
$779$	0	0
$780$	−96.0000	−3.43735
$781$	48.0000	1.71758
$782$	−48.0000	−1.71648
$783$	54.0000	1.92980
$784$	−36.0000	−1.28571
$785$	−16.0000	−0.571064
$786$	−42.0000	−1.49809
$787$	−1.00000	−0.0356462	−0.0178231	−	0.999841i	$-0.505674\pi$
$787$	−1.00000	−0.0356462	−0.0178231	+	0.999841i	$0.505674\pi$
$788$	−40.0000	−1.42494
$789$	−6.00000	−0.213606
$790$	72.0000	2.56165
$791$	20.0000	0.711118
$792$	0	0
$793$	8.00000	0.284088
$794$	−20.0000	−0.709773
$795$	−108.000	−3.83037
$796$	−26.0000	−0.921546
$797$	39.0000	1.38145	0.690725	−	0.723117i	$-0.257291\pi$
$797$	39.0000	1.38145	0.690725	+	0.723117i	$0.257291\pi$
$798$	168.000	5.94714
$799$	36.0000	1.27359
$800$	88.0000	3.11127
$801$	66.0000	2.33200
$802$	36.0000	1.27120
$803$	84.0000	2.96430
$804$	72.0000	2.53924
$805$	−96.0000	−3.38356
$806$	−16.0000	−0.563576
$807$	42.0000	1.47847
$808$	0	0
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	72.0000	2.52982
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	48.0000	1.68447
$813$	−54.0000	−1.89386
$814$	0	0
$815$	16.0000	0.560456
$816$	−48.0000	−1.68034
$817$	56.0000	1.95919
$818$	62.0000	2.16778
$819$	96.0000	3.35451
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	−18.0000	−0.627822
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	0	0
$825$	198.000	6.89348
$826$	−88.0000	−3.06191
$827$	24.0000	0.834562	0.417281	−	0.908778i	$-0.362983\pi$
$827$	24.0000	0.834562	0.417281	+	0.908778i	$0.362983\pi$
$828$	−72.0000	−2.50217
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	−16.0000	−0.555368
$831$	39.0000	1.35290
$832$	32.0000	1.10940
$833$	−36.0000	−1.24733
$834$	24.0000	0.831052
$835$	32.0000	1.10741
$836$	84.0000	2.90520
$837$	18.0000	0.622171
$838$	40.0000	1.38178
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−20.0000	−0.689246
$843$	−72.0000	−2.47981
$844$	−24.0000	−0.826114
$845$	−12.0000	−0.412813
$846$	108.000	3.71312
$847$	−100.000	−3.43604
$848$	36.0000	1.23625
$849$	48.0000	1.64736
$850$	88.0000	3.01838
$851$	0	0
$852$	48.0000	1.64445
$853$	15.0000	0.513590	0.256795	−	0.966466i	$-0.417333\pi$
$853$	15.0000	0.513590	0.256795	+	0.966466i	$0.417333\pi$
$854$	−16.0000	−0.547509
$855$	168.000	5.74548
$856$	0	0
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	144.000	4.91608
$859$	−41.0000	−1.39890	−0.699451	−	0.714681i	$-0.746572\pi$
$859$	−41.0000	−1.39890	−0.699451	+	0.714681i	$0.746572\pi$
$860$	64.0000	2.18238
$861$	0	0
$862$	60.0000	2.04361
$863$	21.0000	0.714848	0.357424	−	0.933942i	$-0.383655\pi$
$863$	21.0000	0.714848	0.357424	+	0.933942i	$0.383655\pi$
$864$	−72.0000	−2.44949
$865$	36.0000	1.22404
$866$	−24.0000	−0.815553
$867$	3.00000	0.101885
$868$	16.0000	0.543075
$869$	−54.0000	−1.83182
$870$	144.000	4.88206
$871$	48.0000	1.62642
$872$	0	0
$873$	36.0000	1.21842
$874$	−84.0000	−2.84134
$875$	96.0000	3.24539
$876$	84.0000	2.83810
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	−20.0000	−0.674967
$879$	−27.0000	−0.910687
$880$	−96.0000	−3.23616
$881$	−33.0000	−1.11180	−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000	−1.11180	−0.555899	+	0.831250i	$0.687626\pi$
$882$	−108.000	−3.63655
$883$	−22.0000	−0.740359	−0.370179	−	0.928960i	$-0.620704\pi$
$883$	−22.0000	−0.740359	−0.370179	+	0.928960i	$0.620704\pi$
$884$	32.0000	1.07628
$885$	−132.000	−4.43713
$886$	64.0000	2.15012
$887$	1.00000	0.0335767	0.0167884	−	0.999859i	$-0.494656\pi$
$887$	1.00000	0.0335767	0.0167884	+	0.999859i	$0.494656\pi$
$888$	0	0
$889$	−28.0000	−0.939090
$890$	88.0000	2.94977
$891$	−54.0000	−1.80907
$892$	42.0000	1.40626
$893$	63.0000	2.10821
$894$	36.0000	1.20402
$895$	−36.0000	−1.20335
$896$	0	0
$897$	−72.0000	−2.40401
$898$	−28.0000	−0.934372
$899$	12.0000	0.400222
$900$	132.000	4.40000
$901$	36.0000	1.19933
$902$	0	0
$903$	−96.0000	−3.19468
$904$	0	0
$905$	−76.0000	−2.52633
$906$	54.0000	1.79403
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−52.0000	−1.72568
$909$	24.0000	0.796030
$910$	128.000	4.24316
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	−84.0000	−2.78152
$913$	12.0000	0.397142
$914$	42.0000	1.38924
$915$	−24.0000	−0.793416
$916$	−20.0000	−0.660819
$917$	28.0000	0.924641
$918$	−72.0000	−2.37635
$919$	−25.0000	−0.824674	−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000	−0.824674	−0.412337	+	0.911031i	$0.635287\pi$
$920$	0	0
$921$	51.0000	1.68051
$922$	12.0000	0.395199
$923$	32.0000	1.05329
$924$	−144.000	−4.73725
$925$	0	0
$926$	−32.0000	−1.05159
$927$	0	0
$928$	−48.0000	−1.57568
$929$	−28.0000	−0.918650	−0.459325	−	0.888268i	$-0.651909\pi$
$929$	−28.0000	−0.918650	−0.459325	+	0.888268i	$0.651909\pi$
$930$	48.0000	1.57398
$931$	−63.0000	−2.06474
$932$	30.0000	0.982683
$933$	96.0000	3.14290
$934$	22.0000	0.719862
$935$	−96.0000	−3.13954
$936$	0	0
$937$	37.0000	1.20874	0.604369	−	0.796705i	$-0.293425\pi$
$937$	37.0000	1.20874	0.604369	+	0.796705i	$0.293425\pi$
$938$	−96.0000	−3.13451
$939$	42.0000	1.37062
$940$	72.0000	2.34838
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	24.0000	0.781962
$943$	0	0
$944$	44.0000	1.43208
$945$	−144.000	−4.68432
$946$	−96.0000	−3.12123
$947$	−17.0000	−0.552426	−0.276213	−	0.961096i	$-0.589079\pi$
$947$	−17.0000	−0.552426	−0.276213	+	0.961096i	$0.589079\pi$
$948$	−54.0000	−1.75384
$949$	56.0000	1.81784
$950$	154.000	4.99642
$951$	66.0000	2.14020
$952$	0	0
$953$	48.0000	1.55487	0.777436	−	0.628962i	$-0.216520\pi$
$953$	48.0000	1.55487	0.777436	+	0.628962i	$0.216520\pi$
$954$	108.000	3.49663
$955$	64.0000	2.07099
$956$	−30.0000	−0.970269
$957$	−108.000	−3.49114
$958$	52.0000	1.68004
$959$	12.0000	0.387500
$960$	−96.0000	−3.09839
$961$	−27.0000	−0.870968
$962$	0	0
$963$	72.0000	2.32017
$964$	−58.0000	−1.86805
$965$	52.0000	1.67394
$966$	144.000	4.63312
$967$	−23.0000	−0.739630	−0.369815	−	0.929105i	$-0.620579\pi$
$967$	−23.0000	−0.739630	−0.369815	+	0.929105i	$0.620579\pi$
$968$	0	0
$969$	−84.0000	−2.69847
$970$	48.0000	1.54119
$971$	−2.00000	−0.0641831	−0.0320915	−	0.999485i	$-0.510217\pi$
$971$	−2.00000	−0.0641831	−0.0320915	+	0.999485i	$0.510217\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	48.0000	1.53802
$975$	132.000	4.22738
$976$	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	$0.448862\pi$
$978$	−24.0000	−0.767435
$979$	−66.0000	−2.10937
$980$	−72.0000	−2.29996
$981$	42.0000	1.34096
$982$	−80.0000	−2.55290
$983$	17.0000	0.542216	0.271108	−	0.962549i	$-0.412610\pi$
$983$	17.0000	0.542216	0.271108	+	0.962549i	$0.412610\pi$
$984$	0	0
$985$	80.0000	2.54901
$986$	−48.0000	−1.52863
$987$	−108.000	−3.43768
$988$	56.0000	1.78160
$989$	48.0000	1.52631
$990$	−288.000	−9.15324
$991$	−13.0000	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$991$	−13.0000	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$992$	−16.0000	−0.508001
$993$	12.0000	0.380808
$994$	−64.0000	−2.02996
$995$	52.0000	1.64851
$996$	12.0000	0.380235
$997$	49.0000	1.55185	0.775923	−	0.630828i	$-0.217285\pi$
$997$	49.0000	1.55185	0.775923	+	0.630828i	$0.217285\pi$
$998$	−16.0000	−0.506471
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5077.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5077.2.a.a.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:	\( N \)	\(=\)	\( 5077 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5077.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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