LMFDB - Embedded newform 273.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(1,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, 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("273.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 273 = 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	273.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:	$2.17991597518$
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	$\chi$	$=$	273.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{10} -2.00000 q^{11} +2.00000 q^{12} -1.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} -4.00000 q^{16} +2.00000 q^{18} +1.00000 q^{19} +2.00000 q^{20} -1.00000 q^{21} -4.00000 q^{22} +3.00000 q^{23} -4.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} -5.00000 q^{29} +2.00000 q^{30} +9.00000 q^{31} -8.00000 q^{32} -2.00000 q^{33} -1.00000 q^{35} +2.00000 q^{36} +2.00000 q^{38} -1.00000 q^{39} +2.00000 q^{41} -2.00000 q^{42} -1.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} +6.00000 q^{46} +3.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} -8.00000 q^{50} -2.00000 q^{52} -9.00000 q^{53} +2.00000 q^{54} -2.00000 q^{55} +1.00000 q^{57} -10.0000 q^{58} +2.00000 q^{60} -2.00000 q^{61} +18.0000 q^{62} -1.00000 q^{63} -8.00000 q^{64} -1.00000 q^{65} -4.00000 q^{66} +10.0000 q^{67} +3.00000 q^{69} -2.00000 q^{70} -12.0000 q^{71} +15.0000 q^{73} -4.00000 q^{75} +2.00000 q^{76} +2.00000 q^{77} -2.00000 q^{78} +11.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} +3.00000 q^{83} -2.00000 q^{84} -2.00000 q^{86} -5.00000 q^{87} -17.0000 q^{89} +2.00000 q^{90} +1.00000 q^{91} +6.00000 q^{92} +9.00000 q^{93} +6.00000 q^{94} +1.00000 q^{95} -8.00000 q^{96} +3.00000 q^{97} +2.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.00000	1.00000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	2.00000	0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	2.00000	0.577350
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.00000	−0.534522
$15$	1.00000	0.258199
$16$	−4.00000	−1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	2.00000	0.471405
$19$	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
$21$	−1.00000	−0.218218
$22$	−4.00000	−0.852803
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	−2.00000	−0.377964
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	2.00000	0.365148
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	−8.00000	−1.41421
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−1.00000	−0.169031
$36$	2.00000	0.333333
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	2.00000	0.324443
$39$	−1.00000	−0.160128
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−2.00000	−0.308607
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−4.00000	−0.603023
$45$	1.00000	0.149071
$46$	6.00000	0.884652
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−4.00000	−0.577350
$49$	1.00000	0.142857
$50$	−8.00000	−1.13137
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	2.00000	0.272166
$55$	−2.00000	−0.269680
$56$	0	0
$57$	1.00000	0.132453
$58$	−10.0000	−1.31306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	2.00000	0.258199
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	18.0000	2.28600
$63$	−1.00000	−0.125988
$64$	−8.00000	−1.00000
$65$	−1.00000	−0.124035
$66$	−4.00000	−0.492366
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	−2.00000	−0.239046
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	15.0000	1.75562	0.877809	−	0.479012i	$-0.159005\pi$
$73$	15.0000	1.75562	0.877809	+	0.479012i	$0.159005\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	2.00000	0.229416
$77$	2.00000	0.227921
$78$	−2.00000	−0.226455
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−2.00000	−0.215666
$87$	−5.00000	−0.536056
$88$	0	0
$89$	−17.0000	−1.80200	−0.900998	−	0.433823i	$-0.857164\pi$
$89$	−17.0000	−1.80200	−0.900998	+	0.433823i	$0.857164\pi$
$90$	2.00000	0.210819
$91$	1.00000	0.104828
$92$	6.00000	0.625543
$93$	9.00000	0.933257
$94$	6.00000	0.618853
$95$	1.00000	0.102598
$96$	−8.00000	−0.816497
$97$	3.00000	0.304604	0.152302	−	0.988334i	$-0.451331\pi$
$97$	3.00000	0.304604	0.152302	+	0.988334i	$0.451331\pi$
$98$	2.00000	0.202031
$99$	−2.00000	−0.201008
$100$	−8.00000	−0.800000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	−18.0000	−1.74831
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	2.00000	0.192450
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	4.00000	0.377964
$113$	−11.0000	−1.03479	−0.517396	−	0.855746i	$-0.673099\pi$
$113$	−11.0000	−1.03479	−0.517396	+	0.855746i	$0.673099\pi$
$114$	2.00000	0.187317
$115$	3.00000	0.279751
$116$	−10.0000	−0.928477
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−4.00000	−0.362143
$123$	2.00000	0.180334
$124$	18.0000	1.61645
$125$	−9.00000	−0.804984
$126$	−2.00000	−0.178174
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	−2.00000	−0.175412
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	−4.00000	−0.348155
$133$	−1.00000	−0.0867110
$134$	20.0000	1.72774
$135$	1.00000	0.0860663
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	6.00000	0.510754
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	−2.00000	−0.169031
$141$	3.00000	0.252646
$142$	−24.0000	−2.01404
$143$	2.00000	0.167248
$144$	−4.00000	−0.333333
$145$	−5.00000	−0.415227
$146$	30.0000	2.48282
$147$	1.00000	0.0824786
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−8.00000	−0.653197
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	0	0
$154$	4.00000	0.322329
$155$	9.00000	0.722897
$156$	−2.00000	−0.160128
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	22.0000	1.75023
$159$	−9.00000	−0.713746
$160$	−8.00000	−0.632456
$161$	−3.00000	−0.236433
$162$	2.00000	0.157135
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	4.00000	0.312348
$165$	−2.00000	−0.155700
$166$	6.00000	0.465690
$167$	25.0000	1.93456	0.967279	−	0.253715i	$-0.0816525\pi$
$167$	25.0000	1.93456	0.967279	+	0.253715i	$0.0816525\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.00000	0.0764719
$172$	−2.00000	−0.152499
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	−10.0000	−0.758098
$175$	4.00000	0.302372
$176$	8.00000	0.603023
$177$	0	0
$178$	−34.0000	−2.54841
$179$	−1.00000	−0.0747435	−0.0373718	−	0.999301i	$-0.511899\pi$
$179$	−1.00000	−0.0747435	−0.0373718	+	0.999301i	$0.511899\pi$
$180$	2.00000	0.149071
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	2.00000	0.148250
$183$	−2.00000	−0.147844
$184$	0	0
$185$	0	0
$186$	18.0000	1.31982
$187$	0	0
$188$	6.00000	0.437595
$189$	−1.00000	−0.0727393
$190$	2.00000	0.145095
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	−8.00000	−0.577350
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	6.00000	0.430775
$195$	−1.00000	−0.0716115
$196$	2.00000	0.142857
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	−4.00000	−0.284268
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	10.0000	0.705346
$202$	28.0000	1.97007
$203$	5.00000	0.350931
$204$	0	0
$205$	2.00000	0.139686
$206$	−8.00000	−0.557386
$207$	3.00000	0.208514
$208$	4.00000	0.277350
$209$	−2.00000	−0.138343
$210$	−2.00000	−0.138013
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	−18.0000	−1.23625
$213$	−12.0000	−0.822226
$214$	8.00000	0.546869
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	−9.00000	−0.610960
$218$	36.0000	2.43823
$219$	15.0000	1.01361
$220$	−4.00000	−0.269680
$221$	0	0
$222$	0	0
$223$	−29.0000	−1.94198	−0.970992	−	0.239113i	$-0.923143\pi$
$223$	−29.0000	−1.94198	−0.970992	+	0.239113i	$0.923143\pi$
$224$	8.00000	0.534522
$225$	−4.00000	−0.266667
$226$	−22.0000	−1.46342
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	2.00000	0.132453
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	6.00000	0.395628
$231$	2.00000	0.131590
$232$	0	0
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	−2.00000	−0.130744
$235$	3.00000	0.195698
$236$	0	0
$237$	11.0000	0.714527
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	−4.00000	−0.258199
$241$	−13.0000	−0.837404	−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000	−0.837404	−0.418702	+	0.908124i	$0.637515\pi$
$242$	−14.0000	−0.899954
$243$	1.00000	0.0641500
$244$	−4.00000	−0.256074
$245$	1.00000	0.0638877
$246$	4.00000	0.255031
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	3.00000	0.190117
$250$	−18.0000	−1.13842
$251$	−14.0000	−0.883672	−0.441836	−	0.897096i	$-0.645673\pi$
$251$	−14.0000	−0.883672	−0.441836	+	0.897096i	$0.645673\pi$
$252$	−2.00000	−0.125988
$253$	−6.00000	−0.377217
$254$	24.0000	1.50589
$255$	0	0
$256$	16.0000	1.00000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−2.00000	−0.124515
$259$	0	0
$260$	−2.00000	−0.124035
$261$	−5.00000	−0.309492
$262$	−16.0000	−0.988483
$263$	−23.0000	−1.41824	−0.709120	−	0.705087i	$-0.750908\pi$
$263$	−23.0000	−1.41824	−0.709120	+	0.705087i	$0.750908\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	−2.00000	−0.122628
$267$	−17.0000	−1.04038
$268$	20.0000	1.22169
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	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	$0.578126\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	8.00000	0.482418
$276$	6.00000	0.361158
$277$	−23.0000	−1.38194	−0.690968	−	0.722885i	$-0.742815\pi$
$277$	−23.0000	−1.38194	−0.690968	+	0.722885i	$0.742815\pi$
$278$	−12.0000	−0.719712
$279$	9.00000	0.538816
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	6.00000	0.357295
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	−24.0000	−1.42414
$285$	1.00000	0.0592349
$286$	4.00000	0.236525
$287$	−2.00000	−0.118056
$288$	−8.00000	−0.471405
$289$	−17.0000	−1.00000
$290$	−10.0000	−0.587220
$291$	3.00000	0.175863
$292$	30.0000	1.75562
$293$	−31.0000	−1.81104	−0.905520	−	0.424304i	$-0.860519\pi$
$293$	−31.0000	−1.81104	−0.905520	+	0.424304i	$0.860519\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	0	0
$297$	−2.00000	−0.116052
$298$	−12.0000	−0.695141
$299$	−3.00000	−0.173494
$300$	−8.00000	−0.461880
$301$	1.00000	0.0576390
$302$	40.0000	2.30174
$303$	14.0000	0.804279
$304$	−4.00000	−0.229416
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−5.00000	−0.285365	−0.142683	−	0.989769i	$-0.545573\pi$
$307$	−5.00000	−0.285365	−0.142683	+	0.989769i	$0.545573\pi$
$308$	4.00000	0.227921
$309$	−4.00000	−0.227552
$310$	18.0000	1.02233
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	8.00000	0.451466
$315$	−1.00000	−0.0563436
$316$	22.0000	1.23760
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	−18.0000	−1.00939
$319$	10.0000	0.559893
$320$	−8.00000	−0.447214
$321$	4.00000	0.223258
$322$	−6.00000	−0.334367
$323$	0	0
$324$	2.00000	0.111111
$325$	4.00000	0.221880
$326$	−40.0000	−2.21540
$327$	18.0000	0.995402
$328$	0	0
$329$	−3.00000	−0.165395
$330$	−4.00000	−0.220193
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	50.0000	2.73588
$335$	10.0000	0.546358
$336$	4.00000	0.218218
$337$	1.00000	0.0544735	0.0272367	−	0.999629i	$-0.491329\pi$
$337$	1.00000	0.0544735	0.0272367	+	0.999629i	$0.491329\pi$
$338$	2.00000	0.108786
$339$	−11.0000	−0.597438
$340$	0	0
$341$	−18.0000	−0.974755
$342$	2.00000	0.108148
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.00000	0.161515
$346$	−8.00000	−0.430083
$347$	32.0000	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$348$	−10.0000	−0.536056
$349$	31.0000	1.65939	0.829696	−	0.558216i	$-0.188514\pi$
$349$	31.0000	1.65939	0.829696	+	0.558216i	$0.188514\pi$
$350$	8.00000	0.427618
$351$	−1.00000	−0.0533761
$352$	16.0000	0.852803
$353$	22.0000	1.17094	0.585471	−	0.810693i	$-0.300910\pi$
$353$	22.0000	1.17094	0.585471	+	0.810693i	$0.300910\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	−34.0000	−1.80200
$357$	0	0
$358$	−2.00000	−0.105703
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−28.0000	−1.47165
$363$	−7.00000	−0.367405
$364$	2.00000	0.104828
$365$	15.0000	0.785136
$366$	−4.00000	−0.209083
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	−12.0000	−0.625543
$369$	2.00000	0.104116
$370$	0	0
$371$	9.00000	0.467257
$372$	18.0000	0.933257
$373$	38.0000	1.96757	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	38.0000	1.96757	0.983783	+	0.179364i	$0.0574041\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	5.00000	0.257513
$378$	−2.00000	−0.102869
$379$	−18.0000	−0.924598	−0.462299	−	0.886724i	$-0.652975\pi$
$379$	−18.0000	−0.924598	−0.462299	+	0.886724i	$0.652975\pi$
$380$	2.00000	0.102598
$381$	12.0000	0.614779
$382$	32.0000	1.63726
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	−20.0000	−1.01797
$387$	−1.00000	−0.0508329
$388$	6.00000	0.304604
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	−2.00000	−0.101274
$391$	0	0
$392$	0	0
$393$	−8.00000	−0.403547
$394$	20.0000	1.00759
$395$	11.0000	0.553470
$396$	−4.00000	−0.201008
$397$	15.0000	0.752828	0.376414	−	0.926451i	$-0.377157\pi$
$397$	15.0000	0.752828	0.376414	+	0.926451i	$0.377157\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	16.0000	0.800000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	20.0000	0.997509
$403$	−9.00000	−0.448322
$404$	28.0000	1.39305
$405$	1.00000	0.0496904
$406$	10.0000	0.496292
$407$	0	0
$408$	0	0
$409$	23.0000	1.13728	0.568638	−	0.822588i	$-0.307470\pi$
$409$	23.0000	1.13728	0.568638	+	0.822588i	$0.307470\pi$
$410$	4.00000	0.197546
$411$	0	0
$412$	−8.00000	−0.394132
$413$	0	0
$414$	6.00000	0.294884
$415$	3.00000	0.147264
$416$	8.00000	0.392232
$417$	−6.00000	−0.293821
$418$	−4.00000	−0.195646
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	−2.00000	−0.0975900
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	−10.0000	−0.486792
$423$	3.00000	0.145865
$424$	0	0
$425$	0	0
$426$	−24.0000	−1.16280
$427$	2.00000	0.0967868
$428$	8.00000	0.386695
$429$	2.00000	0.0965609
$430$	−2.00000	−0.0964486
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	−4.00000	−0.192450
$433$	−20.0000	−0.961139	−0.480569	−	0.876957i	$-0.659570\pi$
$433$	−20.0000	−0.961139	−0.480569	+	0.876957i	$0.659570\pi$
$434$	−18.0000	−0.864028
$435$	−5.00000	−0.239732
$436$	36.0000	1.72409
$437$	3.00000	0.143509
$438$	30.0000	1.43346
$439$	−6.00000	−0.286364	−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000	−0.286364	−0.143182	+	0.989696i	$0.545733\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	27.0000	1.28281	0.641404	−	0.767203i	$-0.278352\pi$
$443$	27.0000	1.28281	0.641404	+	0.767203i	$0.278352\pi$
$444$	0	0
$445$	−17.0000	−0.805877
$446$	−58.0000	−2.74638
$447$	−6.00000	−0.283790
$448$	8.00000	0.377964
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	−8.00000	−0.377124
$451$	−4.00000	−0.188353
$452$	−22.0000	−1.03479
$453$	20.0000	0.939682
$454$	24.0000	1.12638
$455$	1.00000	0.0468807
$456$	0	0
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	28.0000	1.30835
$459$	0	0
$460$	6.00000	0.279751
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	4.00000	0.186097
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	20.0000	0.928477
$465$	9.00000	0.417365
$466$	−18.0000	−0.833834
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	−2.00000	−0.0924500
$469$	−10.0000	−0.461757
$470$	6.00000	0.276759
$471$	4.00000	0.184310
$472$	0	0
$473$	2.00000	0.0919601
$474$	22.0000	1.01049
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−9.00000	−0.412082
$478$	−24.0000	−1.09773
$479$	17.0000	0.776750	0.388375	−	0.921501i	$-0.373037\pi$
$479$	17.0000	0.776750	0.388375	+	0.921501i	$0.373037\pi$
$480$	−8.00000	−0.365148
$481$	0	0
$482$	−26.0000	−1.18427
$483$	−3.00000	−0.136505
$484$	−14.0000	−0.636364
$485$	3.00000	0.136223
$486$	2.00000	0.0907218
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	2.00000	0.0903508
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	4.00000	0.180334
$493$	0	0
$494$	−2.00000	−0.0899843
$495$	−2.00000	−0.0898933
$496$	−36.0000	−1.61645
$497$	12.0000	0.538274
$498$	6.00000	0.268866
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	−18.0000	−0.804984
$501$	25.0000	1.11692
$502$	−28.0000	−1.24970
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	−12.0000	−0.533465
$507$	1.00000	0.0444116
$508$	24.0000	1.06483
$509$	41.0000	1.81729	0.908647	−	0.417566i	$-0.137117\pi$
$509$	41.0000	1.81729	0.908647	+	0.417566i	$0.137117\pi$
$510$	0	0
$511$	−15.0000	−0.663561
$512$	32.0000	1.41421
$513$	1.00000	0.0441511
$514$	4.00000	0.176432
$515$	−4.00000	−0.176261
$516$	−2.00000	−0.0880451
$517$	−6.00000	−0.263880
$518$	0	0
$519$	−4.00000	−0.175581
$520$	0	0
$521$	24.0000	1.05146	0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000	1.05146	0.525730	+	0.850652i	$0.323792\pi$
$522$	−10.0000	−0.437688
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	−16.0000	−0.698963
$525$	4.00000	0.174574
$526$	−46.0000	−2.00570
$527$	0	0
$528$	8.00000	0.348155
$529$	−14.0000	−0.608696
$530$	−18.0000	−0.781870
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	−2.00000	−0.0866296
$534$	−34.0000	−1.47132
$535$	4.00000	0.172935
$536$	0	0
$537$	−1.00000	−0.0431532
$538$	−48.0000	−2.06943
$539$	−2.00000	−0.0861461
$540$	2.00000	0.0860663
$541$	−4.00000	−0.171973	−0.0859867	−	0.996296i	$-0.527404\pi$
$541$	−4.00000	−0.171973	−0.0859867	+	0.996296i	$0.527404\pi$
$542$	−16.0000	−0.687259
$543$	−14.0000	−0.600798
$544$	0	0
$545$	18.0000	0.771035
$546$	2.00000	0.0855921
$547$	1.00000	0.0427569	0.0213785	−	0.999771i	$-0.493195\pi$
$547$	1.00000	0.0427569	0.0213785	+	0.999771i	$0.493195\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	16.0000	0.682242
$551$	−5.00000	−0.213007
$552$	0	0
$553$	−11.0000	−0.467768
$554$	−46.0000	−1.95435
$555$	0	0
$556$	−12.0000	−0.508913
$557$	−44.0000	−1.86434	−0.932170	−	0.362021i	$-0.882087\pi$
$557$	−44.0000	−1.86434	−0.932170	+	0.362021i	$0.882087\pi$
$558$	18.0000	0.762001
$559$	1.00000	0.0422955
$560$	4.00000	0.169031
$561$	0	0
$562$	−20.0000	−0.843649
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	6.00000	0.252646
$565$	−11.0000	−0.462773
$566$	−16.0000	−0.672530
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−41.0000	−1.71881	−0.859405	−	0.511296i	$-0.829166\pi$
$569$	−41.0000	−1.71881	−0.859405	+	0.511296i	$0.829166\pi$
$570$	2.00000	0.0837708
$571$	−25.0000	−1.04622	−0.523109	−	0.852266i	$-0.675228\pi$
$571$	−25.0000	−1.04622	−0.523109	+	0.852266i	$0.675228\pi$
$572$	4.00000	0.167248
$573$	16.0000	0.668410
$574$	−4.00000	−0.166957
$575$	−12.0000	−0.500435
$576$	−8.00000	−0.333333
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−34.0000	−1.41421
$579$	−10.0000	−0.415586
$580$	−10.0000	−0.415227
$581$	−3.00000	−0.124461
$582$	6.00000	0.248708
$583$	18.0000	0.745484
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	−62.0000	−2.56120
$587$	19.0000	0.784214	0.392107	−	0.919920i	$-0.371746\pi$
$587$	19.0000	0.784214	0.392107	+	0.919920i	$0.371746\pi$
$588$	2.00000	0.0824786
$589$	9.00000	0.370839
$590$	0	0
$591$	10.0000	0.411345
$592$	0	0
$593$	33.0000	1.35515	0.677574	−	0.735455i	$-0.263031\pi$
$593$	33.0000	1.35515	0.677574	+	0.735455i	$0.263031\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−12.0000	−0.491539
$597$	0	0
$598$	−6.00000	−0.245358
$599$	27.0000	1.10319	0.551595	−	0.834112i	$-0.314019\pi$
$599$	27.0000	1.10319	0.551595	+	0.834112i	$0.314019\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	2.00000	0.0815139
$603$	10.0000	0.407231
$604$	40.0000	1.62758
$605$	−7.00000	−0.284590
$606$	28.0000	1.13742
$607$	38.0000	1.54237	0.771186	−	0.636610i	$-0.219664\pi$
$607$	38.0000	1.54237	0.771186	+	0.636610i	$0.219664\pi$
$608$	−8.00000	−0.324443
$609$	5.00000	0.202610
$610$	−4.00000	−0.161955
$611$	−3.00000	−0.121367
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	−10.0000	−0.403567
$615$	2.00000	0.0806478
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	−8.00000	−0.321807
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	18.0000	0.722897
$621$	3.00000	0.120386
$622$	12.0000	0.481156
$623$	17.0000	0.681091
$624$	4.00000	0.160128
$625$	11.0000	0.440000
$626$	−44.0000	−1.75859
$627$	−2.00000	−0.0798723
$628$	8.00000	0.319235
$629$	0	0
$630$	−2.00000	−0.0796819
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	0	0
$633$	−5.00000	−0.198732
$634$	−48.0000	−1.90632
$635$	12.0000	0.476205
$636$	−18.0000	−0.713746
$637$	−1.00000	−0.0396214
$638$	20.0000	0.791808
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−15.0000	−0.592464	−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000	−0.592464	−0.296232	+	0.955116i	$0.595730\pi$
$642$	8.00000	0.315735
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	−6.00000	−0.236433
$645$	−1.00000	−0.0393750
$646$	0	0
$647$	38.0000	1.49393	0.746967	−	0.664861i	$-0.231509\pi$
$647$	38.0000	1.49393	0.746967	+	0.664861i	$0.231509\pi$
$648$	0	0
$649$	0	0
$650$	8.00000	0.313786
$651$	−9.00000	−0.352738
$652$	−40.0000	−1.56652
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	36.0000	1.40771
$655$	−8.00000	−0.312586
$656$	−8.00000	−0.312348
$657$	15.0000	0.585206
$658$	−6.00000	−0.233904
$659$	−39.0000	−1.51922	−0.759612	−	0.650376i	$-0.774611\pi$
$659$	−39.0000	−1.51922	−0.759612	+	0.650376i	$0.774611\pi$
$660$	−4.00000	−0.155700
$661$	−5.00000	−0.194477	−0.0972387	−	0.995261i	$-0.531001\pi$
$661$	−5.00000	−0.194477	−0.0972387	+	0.995261i	$0.531001\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	0	0
$665$	−1.00000	−0.0387783
$666$	0	0
$667$	−15.0000	−0.580802
$668$	50.0000	1.93456
$669$	−29.0000	−1.12120
$670$	20.0000	0.772667
$671$	4.00000	0.154418
$672$	8.00000	0.308607
$673$	49.0000	1.88881	0.944406	−	0.328783i	$-0.106638\pi$
$673$	49.0000	1.88881	0.944406	+	0.328783i	$0.106638\pi$
$674$	2.00000	0.0770371
$675$	−4.00000	−0.153960
$676$	2.00000	0.0769231
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	−22.0000	−0.844905
$679$	−3.00000	−0.115129
$680$	0	0
$681$	12.0000	0.459841
$682$	−36.0000	−1.37851
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	−2.00000	−0.0763604
$687$	14.0000	0.534133
$688$	4.00000	0.152499
$689$	9.00000	0.342873
$690$	6.00000	0.228416
$691$	−25.0000	−0.951045	−0.475522	−	0.879704i	$-0.657741\pi$
$691$	−25.0000	−0.951045	−0.475522	+	0.879704i	$0.657741\pi$
$692$	−8.00000	−0.304114
$693$	2.00000	0.0759737
$694$	64.0000	2.42941
$695$	−6.00000	−0.227593
$696$	0	0
$697$	0	0
$698$	62.0000	2.34673
$699$	−9.00000	−0.340411
$700$	8.00000	0.302372
$701$	21.0000	0.793159	0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000	0.793159	0.396580	+	0.918000i	$0.370197\pi$
$702$	−2.00000	−0.0754851
$703$	0	0
$704$	16.0000	0.603023
$705$	3.00000	0.112987
$706$	44.0000	1.65596
$707$	−14.0000	−0.526524
$708$	0	0
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	−24.0000	−0.900704
$711$	11.0000	0.412532
$712$	0	0
$713$	27.0000	1.01116
$714$	0	0
$715$	2.00000	0.0747958
$716$	−2.00000	−0.0747435
$717$	−12.0000	−0.448148
$718$	−32.0000	−1.19423
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	−4.00000	−0.149071
$721$	4.00000	0.148968
$722$	−36.0000	−1.33978
$723$	−13.0000	−0.483475
$724$	−28.0000	−1.04061
$725$	20.0000	0.742781
$726$	−14.0000	−0.519589
$727$	38.0000	1.40934	0.704671	−	0.709534i	$-0.251095\pi$
$727$	38.0000	1.40934	0.704671	+	0.709534i	$0.251095\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	30.0000	1.11035
$731$	0	0
$732$	−4.00000	−0.147844
$733$	15.0000	0.554038	0.277019	−	0.960864i	$-0.410654\pi$
$733$	15.0000	0.554038	0.277019	+	0.960864i	$0.410654\pi$
$734$	36.0000	1.32878
$735$	1.00000	0.0368856
$736$	−24.0000	−0.884652
$737$	−20.0000	−0.736709
$738$	4.00000	0.147242
$739$	−6.00000	−0.220714	−0.110357	−	0.993892i	$-0.535199\pi$
$739$	−6.00000	−0.220714	−0.110357	+	0.993892i	$0.535199\pi$
$740$	0	0
$741$	−1.00000	−0.0367359
$742$	18.0000	0.660801
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	76.0000	2.78256
$747$	3.00000	0.109764
$748$	0	0
$749$	−4.00000	−0.146157
$750$	−18.0000	−0.657267
$751$	−41.0000	−1.49611	−0.748056	−	0.663636i	$-0.769012\pi$
$751$	−41.0000	−1.49611	−0.748056	+	0.663636i	$0.769012\pi$
$752$	−12.0000	−0.437595
$753$	−14.0000	−0.510188
$754$	10.0000	0.364179
$755$	20.0000	0.727875
$756$	−2.00000	−0.0727393
$757$	41.0000	1.49017	0.745085	−	0.666969i	$-0.232409\pi$
$757$	41.0000	1.49017	0.745085	+	0.666969i	$0.232409\pi$
$758$	−36.0000	−1.30758
$759$	−6.00000	−0.217786
$760$	0	0
$761$	21.0000	0.761249	0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000	0.761249	0.380625	+	0.924730i	$0.375709\pi$
$762$	24.0000	0.869428
$763$	−18.0000	−0.651644
$764$	32.0000	1.15772
$765$	0	0
$766$	72.0000	2.60147
$767$	0	0
$768$	16.0000	0.577350
$769$	−7.00000	−0.252426	−0.126213	−	0.992003i	$-0.540282\pi$
$769$	−7.00000	−0.252426	−0.126213	+	0.992003i	$0.540282\pi$
$770$	4.00000	0.144150
$771$	2.00000	0.0720282
$772$	−20.0000	−0.719816
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	−2.00000	−0.0718885
$775$	−36.0000	−1.29316
$776$	0	0
$777$	0	0
$778$	−36.0000	−1.29066
$779$	2.00000	0.0716574
$780$	−2.00000	−0.0716115
$781$	24.0000	0.858788
$782$	0	0
$783$	−5.00000	−0.178685
$784$	−4.00000	−0.142857
$785$	4.00000	0.142766
$786$	−16.0000	−0.570701
$787$	−47.0000	−1.67537	−0.837685	−	0.546154i	$-0.816091\pi$
$787$	−47.0000	−1.67537	−0.837685	+	0.546154i	$0.816091\pi$
$788$	20.0000	0.712470
$789$	−23.0000	−0.818822
$790$	22.0000	0.782725
$791$	11.0000	0.391115
$792$	0	0
$793$	2.00000	0.0710221
$794$	30.0000	1.06466
$795$	−9.00000	−0.319197
$796$	0	0
$797$	−38.0000	−1.34603	−0.673015	−	0.739629i	$-0.735001\pi$
$797$	−38.0000	−1.34603	−0.673015	+	0.739629i	$0.735001\pi$
$798$	−2.00000	−0.0707992
$799$	0	0
$800$	32.0000	1.13137
$801$	−17.0000	−0.600665
$802$	0	0
$803$	−30.0000	−1.05868
$804$	20.0000	0.705346
$805$	−3.00000	−0.105736
$806$	−18.0000	−0.634023
$807$	−24.0000	−0.844840
$808$	0	0
$809$	−15.0000	−0.527372	−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000	−0.527372	−0.263686	+	0.964609i	$0.584938\pi$
$810$	2.00000	0.0702728
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	10.0000	0.350931
$813$	−8.00000	−0.280572
$814$	0	0
$815$	−20.0000	−0.700569
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	46.0000	1.60835
$819$	1.00000	0.0349428
$820$	4.00000	0.139686
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	0	0
$825$	8.00000	0.278524
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	6.00000	0.208514
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	6.00000	0.208263
$831$	−23.0000	−0.797861
$832$	8.00000	0.277350
$833$	0	0
$834$	−12.0000	−0.415526
$835$	25.0000	0.865161
$836$	−4.00000	−0.138343
$837$	9.00000	0.311086
$838$	−60.0000	−2.07267
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	16.0000	0.551396
$843$	−10.0000	−0.344418
$844$	−10.0000	−0.344214
$845$	1.00000	0.0344010
$846$	6.00000	0.206284
$847$	7.00000	0.240523
$848$	36.0000	1.23625
$849$	−8.00000	−0.274559
$850$	0	0
$851$	0	0
$852$	−24.0000	−0.822226
$853$	17.0000	0.582069	0.291034	−	0.956713i	$-0.406001\pi$
$853$	17.0000	0.582069	0.291034	+	0.956713i	$0.406001\pi$
$854$	4.00000	0.136877
$855$	1.00000	0.0341993
$856$	0	0
$857$	−38.0000	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$858$	4.00000	0.136558
$859$	−58.0000	−1.97893	−0.989467	−	0.144757i	$-0.953760\pi$
$859$	−58.0000	−1.97893	−0.989467	+	0.144757i	$0.953760\pi$
$860$	−2.00000	−0.0681994
$861$	−2.00000	−0.0681598
$862$	60.0000	2.04361
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	−8.00000	−0.272166
$865$	−4.00000	−0.136004
$866$	−40.0000	−1.35926
$867$	−17.0000	−0.577350
$868$	−18.0000	−0.610960
$869$	−22.0000	−0.746299
$870$	−10.0000	−0.339032
$871$	−10.0000	−0.338837
$872$	0	0
$873$	3.00000	0.101535
$874$	6.00000	0.202953
$875$	9.00000	0.304256
$876$	30.0000	1.01361
$877$	4.00000	0.135070	0.0675352	−	0.997717i	$-0.478487\pi$
$877$	4.00000	0.135070	0.0675352	+	0.997717i	$0.478487\pi$
$878$	−12.0000	−0.404980
$879$	−31.0000	−1.04560
$880$	8.00000	0.269680
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	2.00000	0.0673435
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	0	0
$886$	54.0000	1.81417
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	−34.0000	−1.13968
$891$	−2.00000	−0.0670025
$892$	−58.0000	−1.94198
$893$	3.00000	0.100391
$894$	−12.0000	−0.401340
$895$	−1.00000	−0.0334263
$896$	0	0
$897$	−3.00000	−0.100167
$898$	−24.0000	−0.800890
$899$	−45.0000	−1.50083
$900$	−8.00000	−0.266667
$901$	0	0
$902$	−8.00000	−0.266371
$903$	1.00000	0.0332779
$904$	0	0
$905$	−14.0000	−0.465376
$906$	40.0000	1.32891
$907$	−25.0000	−0.830111	−0.415056	−	0.909796i	$-0.636238\pi$
$907$	−25.0000	−0.830111	−0.415056	+	0.909796i	$0.636238\pi$
$908$	24.0000	0.796468
$909$	14.0000	0.464351
$910$	2.00000	0.0662994
$911$	9.00000	0.298183	0.149092	−	0.988823i	$-0.452365\pi$
$911$	9.00000	0.298183	0.149092	+	0.988823i	$0.452365\pi$
$912$	−4.00000	−0.132453
$913$	−6.00000	−0.198571
$914$	56.0000	1.85232
$915$	−2.00000	−0.0661180
$916$	28.0000	0.925146
$917$	8.00000	0.264183
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−5.00000	−0.164756
$922$	4.00000	0.131733
$923$	12.0000	0.394985
$924$	4.00000	0.131590
$925$	0	0
$926$	44.0000	1.44593
$927$	−4.00000	−0.131377
$928$	40.0000	1.31306
$929$	41.0000	1.34517	0.672583	−	0.740022i	$-0.265185\pi$
$929$	41.0000	1.34517	0.672583	+	0.740022i	$0.265185\pi$
$930$	18.0000	0.590243
$931$	1.00000	0.0327737
$932$	−18.0000	−0.589610
$933$	6.00000	0.196431
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	36.0000	1.17607	0.588034	−	0.808836i	$-0.299902\pi$
$937$	36.0000	1.17607	0.588034	+	0.808836i	$0.299902\pi$
$938$	−20.0000	−0.653023
$939$	−22.0000	−0.717943
$940$	6.00000	0.195698
$941$	−13.0000	−0.423788	−0.211894	−	0.977293i	$-0.567963\pi$
$941$	−13.0000	−0.423788	−0.211894	+	0.977293i	$0.567963\pi$
$942$	8.00000	0.260654
$943$	6.00000	0.195387
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	4.00000	0.130051
$947$	−58.0000	−1.88475	−0.942373	−	0.334563i	$-0.891411\pi$
$947$	−58.0000	−1.88475	−0.942373	+	0.334563i	$0.891411\pi$
$948$	22.0000	0.714527
$949$	−15.0000	−0.486921
$950$	−8.00000	−0.259554
$951$	−24.0000	−0.778253
$952$	0	0
$953$	41.0000	1.32812	0.664060	−	0.747679i	$-0.268832\pi$
$953$	41.0000	1.32812	0.664060	+	0.747679i	$0.268832\pi$
$954$	−18.0000	−0.582772
$955$	16.0000	0.517748
$956$	−24.0000	−0.776215
$957$	10.0000	0.323254
$958$	34.0000	1.09849
$959$	0	0
$960$	−8.00000	−0.258199
$961$	50.0000	1.61290
$962$	0	0
$963$	4.00000	0.128898
$964$	−26.0000	−0.837404
$965$	−10.0000	−0.321911
$966$	−6.00000	−0.193047
$967$	34.0000	1.09337	0.546683	−	0.837340i	$-0.315890\pi$
$967$	34.0000	1.09337	0.546683	+	0.837340i	$0.315890\pi$
$968$	0	0
$969$	0	0
$970$	6.00000	0.192648
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	2.00000	0.0641500
$973$	6.00000	0.192351
$974$	−4.00000	−0.128168
$975$	4.00000	0.128103
$976$	8.00000	0.256074
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	−40.0000	−1.27906
$979$	34.0000	1.08664
$980$	2.00000	0.0638877
$981$	18.0000	0.574696
$982$	−56.0000	−1.78703
$983$	45.0000	1.43528	0.717639	−	0.696416i	$-0.245223\pi$
$983$	45.0000	1.43528	0.717639	+	0.696416i	$0.245223\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	−3.00000	−0.0954911
$988$	−2.00000	−0.0636285
$989$	−3.00000	−0.0953945
$990$	−4.00000	−0.127128
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	−72.0000	−2.28600
$993$	14.0000	0.444277
$994$	24.0000	0.761234
$995$	0	0
$996$	6.00000	0.190117
$997$	−20.0000	−0.633406	−0.316703	−	0.948525i	$-0.602576\pi$
$997$	−20.0000	−0.633406	−0.316703	+	0.948525i	$0.602576\pi$
$998$	56.0000	1.77265
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.a.b.1.1	✓	1
3.2	odd	2		819.2.a.a.1.1		1
4.3	odd	2		4368.2.a.i.1.1		1
5.4	even	2		6825.2.a.a.1.1		1
7.6	odd	2		1911.2.a.g.1.1		1
13.12	even	2		3549.2.a.b.1.1		1
21.20	even	2		5733.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.b.1.1	✓	1	1.1	even	1	trivial
819.2.a.a.1.1		1	3.2	odd	2
1911.2.a.g.1.1		1	7.6	odd	2
3549.2.a.b.1.1		1	13.12	even	2
4368.2.a.i.1.1		1	4.3	odd	2
5733.2.a.a.1.1		1	21.20	even	2
6825.2.a.a.1.1		1	5.4	even	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:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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