LMFDB - Embedded newform 142.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [142,2,Mod(1,142)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(142, 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("142.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 142 = 2 \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	142.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:	$1.13387570870$
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	$\chi$	$=$	142.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -3.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -3.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} -2.00000 q^{10} -6.00000 q^{11} +3.00000 q^{12} -5.00000 q^{13} +3.00000 q^{14} +6.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -6.00000 q^{18} +1.00000 q^{19} +2.00000 q^{20} -9.00000 q^{21} +6.00000 q^{22} +5.00000 q^{23} -3.00000 q^{24} -1.00000 q^{25} +5.00000 q^{26} +9.00000 q^{27} -3.00000 q^{28} -2.00000 q^{29} -6.00000 q^{30} -5.00000 q^{31} -1.00000 q^{32} -18.0000 q^{33} -6.00000 q^{34} -6.00000 q^{35} +6.00000 q^{36} -2.00000 q^{37} -1.00000 q^{38} -15.0000 q^{39} -2.00000 q^{40} +10.0000 q^{41} +9.00000 q^{42} +1.00000 q^{43} -6.00000 q^{44} +12.0000 q^{45} -5.00000 q^{46} -1.00000 q^{47} +3.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} +18.0000 q^{51} -5.00000 q^{52} +6.00000 q^{53} -9.00000 q^{54} -12.0000 q^{55} +3.00000 q^{56} +3.00000 q^{57} +2.00000 q^{58} -2.00000 q^{59} +6.00000 q^{60} -2.00000 q^{61} +5.00000 q^{62} -18.0000 q^{63} +1.00000 q^{64} -10.0000 q^{65} +18.0000 q^{66} +2.00000 q^{67} +6.00000 q^{68} +15.0000 q^{69} +6.00000 q^{70} +1.00000 q^{71} -6.00000 q^{72} +7.00000 q^{73} +2.00000 q^{74} -3.00000 q^{75} +1.00000 q^{76} +18.0000 q^{77} +15.0000 q^{78} -6.00000 q^{79} +2.00000 q^{80} +9.00000 q^{81} -10.0000 q^{82} -4.00000 q^{83} -9.00000 q^{84} +12.0000 q^{85} -1.00000 q^{86} -6.00000 q^{87} +6.00000 q^{88} +9.00000 q^{89} -12.0000 q^{90} +15.0000 q^{91} +5.00000 q^{92} -15.0000 q^{93} +1.00000 q^{94} +2.00000 q^{95} -3.00000 q^{96} +2.00000 q^{97} -2.00000 q^{98} -36.0000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	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	$0.352416\pi$
$6$	−3.00000	−1.22474
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	6.00000	2.00000
$10$	−2.00000	−0.632456
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	3.00000	0.866025
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	3.00000	0.801784
$15$	6.00000	1.54919
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−6.00000	−1.41421
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	2.00000	0.447214
$21$	−9.00000	−1.96396
$22$	6.00000	1.27920
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	−3.00000	−0.612372
$25$	−1.00000	−0.200000
$26$	5.00000	0.980581
$27$	9.00000	1.73205
$28$	−3.00000	−0.566947
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	−6.00000	−1.09545
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	−1.00000	−0.176777
$33$	−18.0000	−3.13340
$34$	−6.00000	−1.02899
$35$	−6.00000	−1.01419
$36$	6.00000	1.00000
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−1.00000	−0.162221
$39$	−15.0000	−2.40192
$40$	−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	$0.214777\pi$
$42$	9.00000	1.38873
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	−6.00000	−0.904534
$45$	12.0000	1.78885
$46$	−5.00000	−0.737210
$47$	−1.00000	−0.145865	−0.0729325	−	0.997337i	$-0.523236\pi$
$47$	−1.00000	−0.145865	−0.0729325	+	0.997337i	$0.523236\pi$
$48$	3.00000	0.433013
$49$	2.00000	0.285714
$50$	1.00000	0.141421
$51$	18.0000	2.52050
$52$	−5.00000	−0.693375
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−9.00000	−1.22474
$55$	−12.0000	−1.61808
$56$	3.00000	0.400892
$57$	3.00000	0.397360
$58$	2.00000	0.262613
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	6.00000	0.774597
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	5.00000	0.635001
$63$	−18.0000	−2.26779
$64$	1.00000	0.125000
$65$	−10.0000	−1.24035
$66$	18.0000	2.21565
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	6.00000	0.727607
$69$	15.0000	1.80579
$70$	6.00000	0.717137
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	−6.00000	−0.707107
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	2.00000	0.232495
$75$	−3.00000	−0.346410
$76$	1.00000	0.114708
$77$	18.0000	2.05129
$78$	15.0000	1.69842
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	2.00000	0.223607
$81$	9.00000	1.00000
$82$	−10.0000	−1.10432
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−9.00000	−0.981981
$85$	12.0000	1.30158
$86$	−1.00000	−0.107833
$87$	−6.00000	−0.643268
$88$	6.00000	0.639602
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	−12.0000	−1.26491
$91$	15.0000	1.57243
$92$	5.00000	0.521286
$93$	−15.0000	−1.55543
$94$	1.00000	0.103142
$95$	2.00000	0.205196
$96$	−3.00000	−0.306186
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−2.00000	−0.202031
$99$	−36.0000	−3.61814
$100$	−1.00000	−0.100000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	−18.0000	−1.78227
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	5.00000	0.490290
$105$	−18.0000	−1.75662
$106$	−6.00000	−0.582772
$107$	11.0000	1.06341	0.531705	−	0.846930i	$-0.321551\pi$
$107$	11.0000	1.06341	0.531705	+	0.846930i	$0.321551\pi$
$108$	9.00000	0.866025
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	12.0000	1.14416
$111$	−6.00000	−0.569495
$112$	−3.00000	−0.283473
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−3.00000	−0.280976
$115$	10.0000	0.932505
$116$	−2.00000	−0.185695
$117$	−30.0000	−2.77350
$118$	2.00000	0.184115
$119$	−18.0000	−1.65006
$120$	−6.00000	−0.547723
$121$	25.0000	2.27273
$122$	2.00000	0.181071
$123$	30.0000	2.70501
$124$	−5.00000	−0.449013
$125$	−12.0000	−1.07331
$126$	18.0000	1.60357
$127$	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
$129$	3.00000	0.264135
$130$	10.0000	0.877058
$131$	−17.0000	−1.48530	−0.742648	−	0.669681i	$-0.766431\pi$
$131$	−17.0000	−1.48530	−0.742648	+	0.669681i	$0.766431\pi$
$132$	−18.0000	−1.56670
$133$	−3.00000	−0.260133
$134$	−2.00000	−0.172774
$135$	18.0000	1.54919
$136$	−6.00000	−0.514496
$137$	16.0000	1.36697	0.683486	−	0.729964i	$-0.260463\pi$
$137$	16.0000	1.36697	0.683486	+	0.729964i	$0.260463\pi$
$138$	−15.0000	−1.27688
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	−6.00000	−0.507093
$141$	−3.00000	−0.252646
$142$	−1.00000	−0.0839181
$143$	30.0000	2.50873
$144$	6.00000	0.500000
$145$	−4.00000	−0.332182
$146$	−7.00000	−0.579324
$147$	6.00000	0.494872
$148$	−2.00000	−0.164399
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	3.00000	0.244949
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−1.00000	−0.0811107
$153$	36.0000	2.91043
$154$	−18.0000	−1.45048
$155$	−10.0000	−0.803219
$156$	−15.0000	−1.20096
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	6.00000	0.477334
$159$	18.0000	1.42749
$160$	−2.00000	−0.158114
$161$	−15.0000	−1.18217
$162$	−9.00000	−0.707107
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	10.0000	0.780869
$165$	−36.0000	−2.80260
$166$	4.00000	0.310460
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	9.00000	0.694365
$169$	12.0000	0.923077
$170$	−12.0000	−0.920358
$171$	6.00000	0.458831
$172$	1.00000	0.0762493
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	6.00000	0.454859
$175$	3.00000	0.226779
$176$	−6.00000	−0.452267
$177$	−6.00000	−0.450988
$178$	−9.00000	−0.674579
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	12.0000	0.894427
$181$	−21.0000	−1.56092	−0.780459	−	0.625207i	$-0.785014\pi$
$181$	−21.0000	−1.56092	−0.780459	+	0.625207i	$0.785014\pi$
$182$	−15.0000	−1.11187
$183$	−6.00000	−0.443533
$184$	−5.00000	−0.368605
$185$	−4.00000	−0.294086
$186$	15.0000	1.09985
$187$	−36.0000	−2.63258
$188$	−1.00000	−0.0729325
$189$	−27.0000	−1.96396
$190$	−2.00000	−0.145095
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	3.00000	0.216506
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−2.00000	−0.143592
$195$	−30.0000	−2.14834
$196$	2.00000	0.142857
$197$	21.0000	1.49619	0.748094	−	0.663593i	$-0.230969\pi$
$197$	21.0000	1.49619	0.748094	+	0.663593i	$0.230969\pi$
$198$	36.0000	2.55841
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	1.00000	0.0707107
$201$	6.00000	0.423207
$202$	12.0000	0.844317
$203$	6.00000	0.421117
$204$	18.0000	1.26025
$205$	20.0000	1.39686
$206$	4.00000	0.278693
$207$	30.0000	2.08514
$208$	−5.00000	−0.346688
$209$	−6.00000	−0.415029
$210$	18.0000	1.24212
$211$	18.0000	1.23917	0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000	1.23917	0.619586	+	0.784929i	$0.287301\pi$
$212$	6.00000	0.412082
$213$	3.00000	0.205557
$214$	−11.0000	−0.751945
$215$	2.00000	0.136399
$216$	−9.00000	−0.612372
$217$	15.0000	1.01827
$218$	−12.0000	−0.812743
$219$	21.0000	1.41905
$220$	−12.0000	−0.809040
$221$	−30.0000	−2.01802
$222$	6.00000	0.402694
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	3.00000	0.200446
$225$	−6.00000	−0.400000
$226$	14.0000	0.931266
$227$	10.0000	0.663723	0.331862	−	0.943328i	$-0.392323\pi$
$227$	10.0000	0.663723	0.331862	+	0.943328i	$0.392323\pi$
$228$	3.00000	0.198680
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−10.0000	−0.659380
$231$	54.0000	3.55294
$232$	2.00000	0.131306
$233$	−25.0000	−1.63780	−0.818902	−	0.573933i	$-0.805417\pi$
$233$	−25.0000	−1.63780	−0.818902	+	0.573933i	$0.805417\pi$
$234$	30.0000	1.96116
$235$	−2.00000	−0.130466
$236$	−2.00000	−0.130189
$237$	−18.0000	−1.16923
$238$	18.0000	1.16677
$239$	11.0000	0.711531	0.355765	−	0.934575i	$-0.384220\pi$
$239$	11.0000	0.711531	0.355765	+	0.934575i	$0.384220\pi$
$240$	6.00000	0.387298
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	−25.0000	−1.60706
$243$	0	0
$244$	−2.00000	−0.128037
$245$	4.00000	0.255551
$246$	−30.0000	−1.91273
$247$	−5.00000	−0.318142
$248$	5.00000	0.317500
$249$	−12.0000	−0.760469
$250$	12.0000	0.758947
$251$	−17.0000	−1.07303	−0.536515	−	0.843891i	$-0.680260\pi$
$251$	−17.0000	−1.07303	−0.536515	+	0.843891i	$0.680260\pi$
$252$	−18.0000	−1.13389
$253$	−30.0000	−1.88608
$254$	−8.00000	−0.501965
$255$	36.0000	2.25441
$256$	1.00000	0.0625000
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	−3.00000	−0.186772
$259$	6.00000	0.372822
$260$	−10.0000	−0.620174
$261$	−12.0000	−0.742781
$262$	17.0000	1.05026
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	18.0000	1.10782
$265$	12.0000	0.737154
$266$	3.00000	0.183942
$267$	27.0000	1.65237
$268$	2.00000	0.122169
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	−18.0000	−1.09545
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	6.00000	0.363803
$273$	45.0000	2.72352
$274$	−16.0000	−0.966595
$275$	6.00000	0.361814
$276$	15.0000	0.902894
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	16.0000	0.959616
$279$	−30.0000	−1.79605
$280$	6.00000	0.358569
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	3.00000	0.178647
$283$	18.0000	1.06999	0.534994	−	0.844856i	$-0.320314\pi$
$283$	18.0000	1.06999	0.534994	+	0.844856i	$0.320314\pi$
$284$	1.00000	0.0593391
$285$	6.00000	0.355409
$286$	−30.0000	−1.77394
$287$	−30.0000	−1.77084
$288$	−6.00000	−0.353553
$289$	19.0000	1.11765
$290$	4.00000	0.234888
$291$	6.00000	0.351726
$292$	7.00000	0.409644
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	−6.00000	−0.349927
$295$	−4.00000	−0.232889
$296$	2.00000	0.116248
$297$	−54.0000	−3.13340
$298$	11.0000	0.637213
$299$	−25.0000	−1.44579
$300$	−3.00000	−0.173205
$301$	−3.00000	−0.172917
$302$	−4.00000	−0.230174
$303$	−36.0000	−2.06815
$304$	1.00000	0.0573539
$305$	−4.00000	−0.229039
$306$	−36.0000	−2.05798
$307$	−34.0000	−1.94048	−0.970241	−	0.242140i	$-0.922151\pi$
$307$	−34.0000	−1.94048	−0.970241	+	0.242140i	$0.922151\pi$
$308$	18.0000	1.02565
$309$	−12.0000	−0.682656
$310$	10.0000	0.567962
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	15.0000	0.849208
$313$	27.0000	1.52613	0.763065	−	0.646322i	$-0.223694\pi$
$313$	27.0000	1.52613	0.763065	+	0.646322i	$0.223694\pi$
$314$	−22.0000	−1.24153
$315$	−36.0000	−2.02837
$316$	−6.00000	−0.337526
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−18.0000	−1.00939
$319$	12.0000	0.671871
$320$	2.00000	0.111803
$321$	33.0000	1.84188
$322$	15.0000	0.835917
$323$	6.00000	0.333849
$324$	9.00000	0.500000
$325$	5.00000	0.277350
$326$	−10.0000	−0.553849
$327$	36.0000	1.99080
$328$	−10.0000	−0.552158
$329$	3.00000	0.165395
$330$	36.0000	1.98173
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−4.00000	−0.219529
$333$	−12.0000	−0.657596
$334$	−16.0000	−0.875481
$335$	4.00000	0.218543
$336$	−9.00000	−0.490990
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−12.0000	−0.652714
$339$	−42.0000	−2.28113
$340$	12.0000	0.650791
$341$	30.0000	1.62459
$342$	−6.00000	−0.324443
$343$	15.0000	0.809924
$344$	−1.00000	−0.0539164
$345$	30.0000	1.61515
$346$	−9.00000	−0.483843
$347$	−22.0000	−1.18102	−0.590511	−	0.807030i	$-0.701074\pi$
$347$	−22.0000	−1.18102	−0.590511	+	0.807030i	$0.701074\pi$
$348$	−6.00000	−0.321634
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	−3.00000	−0.160357
$351$	−45.0000	−2.40192
$352$	6.00000	0.319801
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	6.00000	0.318896
$355$	2.00000	0.106149
$356$	9.00000	0.476999
$357$	−54.0000	−2.85798
$358$	−9.00000	−0.475665
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	−12.0000	−0.632456
$361$	−18.0000	−0.947368
$362$	21.0000	1.10374
$363$	75.0000	3.93648
$364$	15.0000	0.786214
$365$	14.0000	0.732793
$366$	6.00000	0.313625
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	5.00000	0.260643
$369$	60.0000	3.12348
$370$	4.00000	0.207950
$371$	−18.0000	−0.934513
$372$	−15.0000	−0.777714
$373$	−28.0000	−1.44979	−0.724893	−	0.688862i	$-0.758111\pi$
$373$	−28.0000	−1.44979	−0.724893	+	0.688862i	$0.758111\pi$
$374$	36.0000	1.86152
$375$	−36.0000	−1.85903
$376$	1.00000	0.0515711
$377$	10.0000	0.515026
$378$	27.0000	1.38873
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	2.00000	0.102598
$381$	24.0000	1.22956
$382$	8.00000	0.409316
$383$	−1.00000	−0.0510976	−0.0255488	−	0.999674i	$-0.508133\pi$
$383$	−1.00000	−0.0510976	−0.0255488	+	0.999674i	$0.508133\pi$
$384$	−3.00000	−0.153093
$385$	36.0000	1.83473
$386$	14.0000	0.712581
$387$	6.00000	0.304997
$388$	2.00000	0.101535
$389$	13.0000	0.659126	0.329563	−	0.944134i	$-0.393099\pi$
$389$	13.0000	0.659126	0.329563	+	0.944134i	$0.393099\pi$
$390$	30.0000	1.51911
$391$	30.0000	1.51717
$392$	−2.00000	−0.101015
$393$	−51.0000	−2.57261
$394$	−21.0000	−1.05796
$395$	−12.0000	−0.603786
$396$	−36.0000	−1.80907
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	14.0000	0.701757
$399$	−9.00000	−0.450564
$400$	−1.00000	−0.0500000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	−6.00000	−0.299253
$403$	25.0000	1.24534
$404$	−12.0000	−0.597022
$405$	18.0000	0.894427
$406$	−6.00000	−0.297775
$407$	12.0000	0.594818
$408$	−18.0000	−0.891133
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	−20.0000	−0.987730
$411$	48.0000	2.36767
$412$	−4.00000	−0.197066
$413$	6.00000	0.295241
$414$	−30.0000	−1.47442
$415$	−8.00000	−0.392705
$416$	5.00000	0.245145
$417$	−48.0000	−2.35057
$418$	6.00000	0.293470
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	−18.0000	−0.878310
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	−18.0000	−0.876226
$423$	−6.00000	−0.291730
$424$	−6.00000	−0.291386
$425$	−6.00000	−0.291043
$426$	−3.00000	−0.145350
$427$	6.00000	0.290360
$428$	11.0000	0.531705
$429$	90.0000	4.34524
$430$	−2.00000	−0.0964486
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	9.00000	0.433013
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−15.0000	−0.720023
$435$	−12.0000	−0.575356
$436$	12.0000	0.574696
$437$	5.00000	0.239182
$438$	−21.0000	−1.00342
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	12.0000	0.572078
$441$	12.0000	0.571429
$442$	30.0000	1.42695
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−6.00000	−0.284747
$445$	18.0000	0.853282
$446$	14.0000	0.662919
$447$	−33.0000	−1.56085
$448$	−3.00000	−0.141737
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	6.00000	0.282843
$451$	−60.0000	−2.82529
$452$	−14.0000	−0.658505
$453$	12.0000	0.563809
$454$	−10.0000	−0.469323
$455$	30.0000	1.40642
$456$	−3.00000	−0.140488
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	10.0000	0.467269
$459$	54.0000	2.52050
$460$	10.0000	0.466252
$461$	−13.0000	−0.605470	−0.302735	−	0.953075i	$-0.597900\pi$
$461$	−13.0000	−0.605470	−0.302735	+	0.953075i	$0.597900\pi$
$462$	−54.0000	−2.51231
$463$	−6.00000	−0.278844	−0.139422	−	0.990233i	$-0.544524\pi$
$463$	−6.00000	−0.278844	−0.139422	+	0.990233i	$0.544524\pi$
$464$	−2.00000	−0.0928477
$465$	−30.0000	−1.39122
$466$	25.0000	1.15810
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	−30.0000	−1.38675
$469$	−6.00000	−0.277054
$470$	2.00000	0.0922531
$471$	66.0000	3.04112
$472$	2.00000	0.0920575
$473$	−6.00000	−0.275880
$474$	18.0000	0.826767
$475$	−1.00000	−0.0458831
$476$	−18.0000	−0.825029
$477$	36.0000	1.64833
$478$	−11.0000	−0.503128
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	−6.00000	−0.273861
$481$	10.0000	0.455961
$482$	−14.0000	−0.637683
$483$	−45.0000	−2.04757
$484$	25.0000	1.13636
$485$	4.00000	0.181631
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	2.00000	0.0905357
$489$	30.0000	1.35665
$490$	−4.00000	−0.180702
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	30.0000	1.35250
$493$	−12.0000	−0.540453
$494$	5.00000	0.224961
$495$	−72.0000	−3.23616
$496$	−5.00000	−0.224507
$497$	−3.00000	−0.134568
$498$	12.0000	0.537733
$499$	−29.0000	−1.29822	−0.649109	−	0.760695i	$-0.724858\pi$
$499$	−29.0000	−1.29822	−0.649109	+	0.760695i	$0.724858\pi$
$500$	−12.0000	−0.536656
$501$	48.0000	2.14448
$502$	17.0000	0.758747
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	18.0000	0.801784
$505$	−24.0000	−1.06799
$506$	30.0000	1.33366
$507$	36.0000	1.59882
$508$	8.00000	0.354943
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	−36.0000	−1.59411
$511$	−21.0000	−0.928985
$512$	−1.00000	−0.0441942
$513$	9.00000	0.397360
$514$	8.00000	0.352865
$515$	−8.00000	−0.352522
$516$	3.00000	0.132068
$517$	6.00000	0.263880
$518$	−6.00000	−0.263625
$519$	27.0000	1.18517
$520$	10.0000	0.438529
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	12.0000	0.525226
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	−17.0000	−0.742648
$525$	9.00000	0.392792
$526$	−6.00000	−0.261612
$527$	−30.0000	−1.30682
$528$	−18.0000	−0.783349
$529$	2.00000	0.0869565
$530$	−12.0000	−0.521247
$531$	−12.0000	−0.520756
$532$	−3.00000	−0.130066
$533$	−50.0000	−2.16574
$534$	−27.0000	−1.16840
$535$	22.0000	0.951143
$536$	−2.00000	−0.0863868
$537$	27.0000	1.16514
$538$	15.0000	0.646696
$539$	−12.0000	−0.516877
$540$	18.0000	0.774597
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	16.0000	0.687259
$543$	−63.0000	−2.70359
$544$	−6.00000	−0.257248
$545$	24.0000	1.02805
$546$	−45.0000	−1.92582
$547$	23.0000	0.983409	0.491704	−	0.870762i	$-0.336374\pi$
$547$	23.0000	0.983409	0.491704	+	0.870762i	$0.336374\pi$
$548$	16.0000	0.683486
$549$	−12.0000	−0.512148
$550$	−6.00000	−0.255841
$551$	−2.00000	−0.0852029
$552$	−15.0000	−0.638442
$553$	18.0000	0.765438
$554$	10.0000	0.424859
$555$	−12.0000	−0.509372
$556$	−16.0000	−0.678551
$557$	−26.0000	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$558$	30.0000	1.27000
$559$	−5.00000	−0.211477
$560$	−6.00000	−0.253546
$561$	−108.000	−4.55976
$562$	4.00000	0.168730
$563$	26.0000	1.09577	0.547885	−	0.836554i	$-0.315433\pi$
$563$	26.0000	1.09577	0.547885	+	0.836554i	$0.315433\pi$
$564$	−3.00000	−0.126323
$565$	−28.0000	−1.17797
$566$	−18.0000	−0.756596
$567$	−27.0000	−1.13389
$568$	−1.00000	−0.0419591
$569$	27.0000	1.13190	0.565949	−	0.824440i	$-0.308510\pi$
$569$	27.0000	1.13190	0.565949	+	0.824440i	$0.308510\pi$
$570$	−6.00000	−0.251312
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	30.0000	1.25436
$573$	−24.0000	−1.00261
$574$	30.0000	1.25218
$575$	−5.00000	−0.208514
$576$	6.00000	0.250000
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	−19.0000	−0.790296
$579$	−42.0000	−1.74546
$580$	−4.00000	−0.166091
$581$	12.0000	0.497844
$582$	−6.00000	−0.248708
$583$	−36.0000	−1.49097
$584$	−7.00000	−0.289662
$585$	−60.0000	−2.48069
$586$	−12.0000	−0.495715
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	6.00000	0.247436
$589$	−5.00000	−0.206021
$590$	4.00000	0.164677
$591$	63.0000	2.59147
$592$	−2.00000	−0.0821995
$593$	−39.0000	−1.60154	−0.800769	−	0.598973i	$-0.795576\pi$
$593$	−39.0000	−1.60154	−0.800769	+	0.598973i	$0.795576\pi$
$594$	54.0000	2.21565
$595$	−36.0000	−1.47586
$596$	−11.0000	−0.450578
$597$	−42.0000	−1.71895
$598$	25.0000	1.02233
$599$	35.0000	1.43006	0.715031	−	0.699093i	$-0.246413\pi$
$599$	35.0000	1.43006	0.715031	+	0.699093i	$0.246413\pi$
$600$	3.00000	0.122474
$601$	4.00000	0.163163	0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000	0.163163	0.0815817	+	0.996667i	$0.474003\pi$
$602$	3.00000	0.122271
$603$	12.0000	0.488678
$604$	4.00000	0.162758
$605$	50.0000	2.03279
$606$	36.0000	1.46240
$607$	20.0000	0.811775	0.405887	−	0.913923i	$-0.366962\pi$
$607$	20.0000	0.811775	0.405887	+	0.913923i	$0.366962\pi$
$608$	−1.00000	−0.0405554
$609$	18.0000	0.729397
$610$	4.00000	0.161955
$611$	5.00000	0.202278
$612$	36.0000	1.45521
$613$	40.0000	1.61558	0.807792	−	0.589467i	$-0.200662\pi$
$613$	40.0000	1.61558	0.807792	+	0.589467i	$0.200662\pi$
$614$	34.0000	1.37213
$615$	60.0000	2.41943
$616$	−18.0000	−0.725241
$617$	3.00000	0.120775	0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000	0.120775	0.0603877	+	0.998175i	$0.480766\pi$
$618$	12.0000	0.482711
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	−10.0000	−0.401610
$621$	45.0000	1.80579
$622$	−8.00000	−0.320771
$623$	−27.0000	−1.08173
$624$	−15.0000	−0.600481
$625$	−19.0000	−0.760000
$626$	−27.0000	−1.07914
$627$	−18.0000	−0.718851
$628$	22.0000	0.877896
$629$	−12.0000	−0.478471
$630$	36.0000	1.43427
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	6.00000	0.238667
$633$	54.0000	2.14631
$634$	−18.0000	−0.714871
$635$	16.0000	0.634941
$636$	18.0000	0.713746
$637$	−10.0000	−0.396214
$638$	−12.0000	−0.475085
$639$	6.00000	0.237356
$640$	−2.00000	−0.0790569
$641$	37.0000	1.46141	0.730706	−	0.682692i	$-0.239191\pi$
$641$	37.0000	1.46141	0.730706	+	0.682692i	$0.239191\pi$
$642$	−33.0000	−1.30241
$643$	−21.0000	−0.828159	−0.414080	−	0.910241i	$-0.635896\pi$
$643$	−21.0000	−0.828159	−0.414080	+	0.910241i	$0.635896\pi$
$644$	−15.0000	−0.591083
$645$	6.00000	0.236250
$646$	−6.00000	−0.236067
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	−9.00000	−0.353553
$649$	12.0000	0.471041
$650$	−5.00000	−0.196116
$651$	45.0000	1.76369
$652$	10.0000	0.391630
$653$	−46.0000	−1.80012	−0.900060	−	0.435767i	$-0.856477\pi$
$653$	−46.0000	−1.80012	−0.900060	+	0.435767i	$0.856477\pi$
$654$	−36.0000	−1.40771
$655$	−34.0000	−1.32849
$656$	10.0000	0.390434
$657$	42.0000	1.63858
$658$	−3.00000	−0.116952
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	−36.0000	−1.40130
$661$	45.0000	1.75030	0.875149	−	0.483854i	$-0.160764\pi$
$661$	45.0000	1.75030	0.875149	+	0.483854i	$0.160764\pi$
$662$	8.00000	0.310929
$663$	−90.0000	−3.49531
$664$	4.00000	0.155230
$665$	−6.00000	−0.232670
$666$	12.0000	0.464991
$667$	−10.0000	−0.387202
$668$	16.0000	0.619059
$669$	−42.0000	−1.62381
$670$	−4.00000	−0.154533
$671$	12.0000	0.463255
$672$	9.00000	0.347183
$673$	−12.0000	−0.462566	−0.231283	−	0.972887i	$-0.574292\pi$
$673$	−12.0000	−0.462566	−0.231283	+	0.972887i	$0.574292\pi$
$674$	22.0000	0.847408
$675$	−9.00000	−0.346410
$676$	12.0000	0.461538
$677$	8.00000	0.307465	0.153732	−	0.988113i	$-0.450871\pi$
$677$	8.00000	0.307465	0.153732	+	0.988113i	$0.450871\pi$
$678$	42.0000	1.61300
$679$	−6.00000	−0.230259
$680$	−12.0000	−0.460179
$681$	30.0000	1.14960
$682$	−30.0000	−1.14876
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	6.00000	0.229416
$685$	32.0000	1.22266
$686$	−15.0000	−0.572703
$687$	−30.0000	−1.14457
$688$	1.00000	0.0381246
$689$	−30.0000	−1.14291
$690$	−30.0000	−1.14208
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	9.00000	0.342129
$693$	108.000	4.10258
$694$	22.0000	0.835109
$695$	−32.0000	−1.21383
$696$	6.00000	0.227429
$697$	60.0000	2.27266
$698$	−19.0000	−0.719161
$699$	−75.0000	−2.83676
$700$	3.00000	0.113389
$701$	−3.00000	−0.113308	−0.0566542	−	0.998394i	$-0.518043\pi$
$701$	−3.00000	−0.113308	−0.0566542	+	0.998394i	$0.518043\pi$
$702$	45.0000	1.69842
$703$	−2.00000	−0.0754314
$704$	−6.00000	−0.226134
$705$	−6.00000	−0.225973
$706$	18.0000	0.677439
$707$	36.0000	1.35392
$708$	−6.00000	−0.225494
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	−2.00000	−0.0750587
$711$	−36.0000	−1.35011
$712$	−9.00000	−0.337289
$713$	−25.0000	−0.936257
$714$	54.0000	2.02090
$715$	60.0000	2.24387
$716$	9.00000	0.336346
$717$	33.0000	1.23241
$718$	12.0000	0.447836
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	12.0000	0.447214
$721$	12.0000	0.446903
$722$	18.0000	0.669891
$723$	42.0000	1.56200
$724$	−21.0000	−0.780459
$725$	2.00000	0.0742781
$726$	−75.0000	−2.78351
$727$	33.0000	1.22390	0.611951	−	0.790896i	$-0.290385\pi$
$727$	33.0000	1.22390	0.611951	+	0.790896i	$0.290385\pi$
$728$	−15.0000	−0.555937
$729$	−27.0000	−1.00000
$730$	−14.0000	−0.518163
$731$	6.00000	0.221918
$732$	−6.00000	−0.221766
$733$	−5.00000	−0.184679	−0.0923396	−	0.995728i	$-0.529435\pi$
$733$	−5.00000	−0.184679	−0.0923396	+	0.995728i	$0.529435\pi$
$734$	−14.0000	−0.516749
$735$	12.0000	0.442627
$736$	−5.00000	−0.184302
$737$	−12.0000	−0.442026
$738$	−60.0000	−2.20863
$739$	41.0000	1.50821	0.754105	−	0.656754i	$-0.228071\pi$
$739$	41.0000	1.50821	0.754105	+	0.656754i	$0.228071\pi$
$740$	−4.00000	−0.147043
$741$	−15.0000	−0.551039
$742$	18.0000	0.660801
$743$	−17.0000	−0.623670	−0.311835	−	0.950136i	$-0.600944\pi$
$743$	−17.0000	−0.623670	−0.311835	+	0.950136i	$0.600944\pi$
$744$	15.0000	0.549927
$745$	−22.0000	−0.806018
$746$	28.0000	1.02515
$747$	−24.0000	−0.878114
$748$	−36.0000	−1.31629
$749$	−33.0000	−1.20579
$750$	36.0000	1.31453
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	−1.00000	−0.0364662
$753$	−51.0000	−1.85854
$754$	−10.0000	−0.364179
$755$	8.00000	0.291150
$756$	−27.0000	−0.981981
$757$	−35.0000	−1.27210	−0.636048	−	0.771649i	$-0.719432\pi$
$757$	−35.0000	−1.27210	−0.636048	+	0.771649i	$0.719432\pi$
$758$	8.00000	0.290573
$759$	−90.0000	−3.26679
$760$	−2.00000	−0.0725476
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	−24.0000	−0.869428
$763$	−36.0000	−1.30329
$764$	−8.00000	−0.289430
$765$	72.0000	2.60317
$766$	1.00000	0.0361315
$767$	10.0000	0.361079
$768$	3.00000	0.108253
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	−36.0000	−1.29735
$771$	−24.0000	−0.864339
$772$	−14.0000	−0.503871
$773$	−1.00000	−0.0359675	−0.0179838	−	0.999838i	$-0.505725\pi$
$773$	−1.00000	−0.0359675	−0.0179838	+	0.999838i	$0.505725\pi$
$774$	−6.00000	−0.215666
$775$	5.00000	0.179605
$776$	−2.00000	−0.0717958
$777$	18.0000	0.645746
$778$	−13.0000	−0.466073
$779$	10.0000	0.358287
$780$	−30.0000	−1.07417
$781$	−6.00000	−0.214697
$782$	−30.0000	−1.07280
$783$	−18.0000	−0.643268
$784$	2.00000	0.0714286
$785$	44.0000	1.57043
$786$	51.0000	1.81911
$787$	−5.00000	−0.178231	−0.0891154	−	0.996021i	$-0.528404\pi$
$787$	−5.00000	−0.178231	−0.0891154	+	0.996021i	$0.528404\pi$
$788$	21.0000	0.748094
$789$	18.0000	0.640817
$790$	12.0000	0.426941
$791$	42.0000	1.49335
$792$	36.0000	1.27920
$793$	10.0000	0.355110
$794$	18.0000	0.638796
$795$	36.0000	1.27679
$796$	−14.0000	−0.496217
$797$	26.0000	0.920967	0.460484	−	0.887668i	$-0.347676\pi$
$797$	26.0000	0.920967	0.460484	+	0.887668i	$0.347676\pi$
$798$	9.00000	0.318597
$799$	−6.00000	−0.212265
$800$	1.00000	0.0353553
$801$	54.0000	1.90800
$802$	0	0
$803$	−42.0000	−1.48215
$804$	6.00000	0.211604
$805$	−30.0000	−1.05736
$806$	−25.0000	−0.880587
$807$	−45.0000	−1.58408
$808$	12.0000	0.422159
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	−18.0000	−0.632456
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	6.00000	0.210559
$813$	−48.0000	−1.68343
$814$	−12.0000	−0.420600
$815$	20.0000	0.700569
$816$	18.0000	0.630126
$817$	1.00000	0.0349856
$818$	19.0000	0.664319
$819$	90.0000	3.14485
$820$	20.0000	0.698430
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	−48.0000	−1.67419
$823$	−5.00000	−0.174289	−0.0871445	−	0.996196i	$-0.527774\pi$
$823$	−5.00000	−0.174289	−0.0871445	+	0.996196i	$0.527774\pi$
$824$	4.00000	0.139347
$825$	18.0000	0.626680
$826$	−6.00000	−0.208767
$827$	30.0000	1.04320	0.521601	−	0.853189i	$-0.325335\pi$
$827$	30.0000	1.04320	0.521601	+	0.853189i	$0.325335\pi$
$828$	30.0000	1.04257
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	8.00000	0.277684
$831$	−30.0000	−1.04069
$832$	−5.00000	−0.173344
$833$	12.0000	0.415775
$834$	48.0000	1.66210
$835$	32.0000	1.10741
$836$	−6.00000	−0.207514
$837$	−45.0000	−1.55543
$838$	−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	$0.713448\pi$
$840$	18.0000	0.621059
$841$	−25.0000	−0.862069
$842$	1.00000	0.0344623
$843$	−12.0000	−0.413302
$844$	18.0000	0.619586
$845$	24.0000	0.825625
$846$	6.00000	0.206284
$847$	−75.0000	−2.57703
$848$	6.00000	0.206041
$849$	54.0000	1.85328
$850$	6.00000	0.205798
$851$	−10.0000	−0.342796
$852$	3.00000	0.102778
$853$	−28.0000	−0.958702	−0.479351	−	0.877623i	$-0.659128\pi$
$853$	−28.0000	−0.958702	−0.479351	+	0.877623i	$0.659128\pi$
$854$	−6.00000	−0.205316
$855$	12.0000	0.410391
$856$	−11.0000	−0.375972
$857$	29.0000	0.990621	0.495311	−	0.868716i	$-0.335054\pi$
$857$	29.0000	0.990621	0.495311	+	0.868716i	$0.335054\pi$
$858$	−90.0000	−3.07255
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	2.00000	0.0681994
$861$	−90.0000	−3.06719
$862$	−20.0000	−0.681203
$863$	−1.00000	−0.0340404	−0.0170202	−	0.999855i	$-0.505418\pi$
$863$	−1.00000	−0.0340404	−0.0170202	+	0.999855i	$0.505418\pi$
$864$	−9.00000	−0.306186
$865$	18.0000	0.612018
$866$	−2.00000	−0.0679628
$867$	57.0000	1.93582
$868$	15.0000	0.509133
$869$	36.0000	1.22122
$870$	12.0000	0.406838
$871$	−10.0000	−0.338837
$872$	−12.0000	−0.406371
$873$	12.0000	0.406138
$874$	−5.00000	−0.169128
$875$	36.0000	1.21702
$876$	21.0000	0.709524
$877$	40.0000	1.35070	0.675352	−	0.737496i	$-0.263992\pi$
$877$	40.0000	1.35070	0.675352	+	0.737496i	$0.263992\pi$
$878$	−28.0000	−0.944954
$879$	36.0000	1.21425
$880$	−12.0000	−0.404520
$881$	1.00000	0.0336909	0.0168454	−	0.999858i	$-0.494638\pi$
$881$	1.00000	0.0336909	0.0168454	+	0.999858i	$0.494638\pi$
$882$	−12.0000	−0.404061
$883$	−42.0000	−1.41341	−0.706706	−	0.707507i	$-0.749820\pi$
$883$	−42.0000	−1.41341	−0.706706	+	0.707507i	$0.749820\pi$
$884$	−30.0000	−1.00901
$885$	−12.0000	−0.403376
$886$	36.0000	1.20944
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	6.00000	0.201347
$889$	−24.0000	−0.804934
$890$	−18.0000	−0.603361
$891$	−54.0000	−1.80907
$892$	−14.0000	−0.468755
$893$	−1.00000	−0.0334637
$894$	33.0000	1.10369
$895$	18.0000	0.601674
$896$	3.00000	0.100223
$897$	−75.0000	−2.50418
$898$	6.00000	0.200223
$899$	10.0000	0.333519
$900$	−6.00000	−0.200000
$901$	36.0000	1.19933
$902$	60.0000	1.99778
$903$	−9.00000	−0.299501
$904$	14.0000	0.465633
$905$	−42.0000	−1.39613
$906$	−12.0000	−0.398673
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	10.0000	0.331862
$909$	−72.0000	−2.38809
$910$	−30.0000	−0.994490
$911$	51.0000	1.68971	0.844853	−	0.534999i	$-0.179688\pi$
$911$	51.0000	1.68971	0.844853	+	0.534999i	$0.179688\pi$
$912$	3.00000	0.0993399
$913$	24.0000	0.794284
$914$	−8.00000	−0.264616
$915$	−12.0000	−0.396708
$916$	−10.0000	−0.330409
$917$	51.0000	1.68417
$918$	−54.0000	−1.78227
$919$	−39.0000	−1.28649	−0.643246	−	0.765660i	$-0.722413\pi$
$919$	−39.0000	−1.28649	−0.643246	+	0.765660i	$0.722413\pi$
$920$	−10.0000	−0.329690
$921$	−102.000	−3.36101
$922$	13.0000	0.428132
$923$	−5.00000	−0.164577
$924$	54.0000	1.77647
$925$	2.00000	0.0657596
$926$	6.00000	0.197172
$927$	−24.0000	−0.788263
$928$	2.00000	0.0656532
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	30.0000	0.983739
$931$	2.00000	0.0655474
$932$	−25.0000	−0.818902
$933$	24.0000	0.785725
$934$	−28.0000	−0.916188
$935$	−72.0000	−2.35465
$936$	30.0000	0.980581
$937$	−16.0000	−0.522697	−0.261349	−	0.965244i	$-0.584167\pi$
$937$	−16.0000	−0.522697	−0.261349	+	0.965244i	$0.584167\pi$
$938$	6.00000	0.195907
$939$	81.0000	2.64334
$940$	−2.00000	−0.0652328
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	−66.0000	−2.15040
$943$	50.0000	1.62822
$944$	−2.00000	−0.0650945
$945$	−54.0000	−1.75662
$946$	6.00000	0.195077
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	−18.0000	−0.584613
$949$	−35.0000	−1.13615
$950$	1.00000	0.0324443
$951$	54.0000	1.75107
$952$	18.0000	0.583383
$953$	−10.0000	−0.323932	−0.161966	−	0.986796i	$-0.551783\pi$
$953$	−10.0000	−0.323932	−0.161966	+	0.986796i	$0.551783\pi$
$954$	−36.0000	−1.16554
$955$	−16.0000	−0.517748
$956$	11.0000	0.355765
$957$	36.0000	1.16371
$958$	−21.0000	−0.678479
$959$	−48.0000	−1.55000
$960$	6.00000	0.193649
$961$	−6.00000	−0.193548
$962$	−10.0000	−0.322413
$963$	66.0000	2.12682
$964$	14.0000	0.450910
$965$	−28.0000	−0.901352
$966$	45.0000	1.44785
$967$	9.00000	0.289420	0.144710	−	0.989474i	$-0.453775\pi$
$967$	9.00000	0.289420	0.144710	+	0.989474i	$0.453775\pi$
$968$	−25.0000	−0.803530
$969$	18.0000	0.578243
$970$	−4.00000	−0.128432
$971$	13.0000	0.417190	0.208595	−	0.978002i	$-0.433111\pi$
$971$	13.0000	0.417190	0.208595	+	0.978002i	$0.433111\pi$
$972$	0	0
$973$	48.0000	1.53881
$974$	16.0000	0.512673
$975$	15.0000	0.480384
$976$	−2.00000	−0.0640184
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−30.0000	−0.959294
$979$	−54.0000	−1.72585
$980$	4.00000	0.127775
$981$	72.0000	2.29878
$982$	−12.0000	−0.382935
$983$	18.0000	0.574111	0.287055	−	0.957914i	$-0.407324\pi$
$983$	18.0000	0.574111	0.287055	+	0.957914i	$0.407324\pi$
$984$	−30.0000	−0.956365
$985$	42.0000	1.33823
$986$	12.0000	0.382158
$987$	9.00000	0.286473
$988$	−5.00000	−0.159071
$989$	5.00000	0.158991
$990$	72.0000	2.28831
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	5.00000	0.158750
$993$	−24.0000	−0.761617
$994$	3.00000	0.0951542
$995$	−28.0000	−0.887660
$996$	−12.0000	−0.380235
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	29.0000	0.917979
$999$	−18.0000	−0.569495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	142.2.a.c.1.1	✓	1
3.2	odd	2		1278.2.a.g.1.1		1
4.3	odd	2		1136.2.a.a.1.1		1
5.4	even	2		3550.2.a.j.1.1		1
7.6	odd	2		6958.2.a.a.1.1		1
8.3	odd	2		4544.2.a.n.1.1		1
8.5	even	2		4544.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
142.2.a.c.1.1	✓	1	1.1	even	1	trivial
1136.2.a.a.1.1		1	4.3	odd	2
1278.2.a.g.1.1		1	3.2	odd	2
3550.2.a.j.1.1		1	5.4	even	2
4544.2.a.a.1.1		1	8.5	even	2
4544.2.a.n.1.1		1	8.3	odd	2
6958.2.a.a.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 142 = 2 \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	142.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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