LMFDB - Embedded newform 285.2.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [285,2,Mod(1,285)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("285.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(285, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 285 = 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	285.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-1,-1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$2.27573645761$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	285.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -2.00000 q^{22} -4.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} +1.00000 q^{30} +5.00000 q^{32} +2.00000 q^{33} +2.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} -1.00000 q^{38} +4.00000 q^{39} +3.00000 q^{40} +2.00000 q^{42} -10.0000 q^{43} +2.00000 q^{44} -1.00000 q^{45} -4.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +4.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +2.00000 q^{55} +6.00000 q^{56} +1.00000 q^{57} +4.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -2.00000 q^{63} +7.00000 q^{64} +4.00000 q^{65} +2.00000 q^{66} -16.0000 q^{67} -2.00000 q^{68} +4.00000 q^{69} +2.00000 q^{70} -3.00000 q^{72} -2.00000 q^{73} -1.00000 q^{75} +1.00000 q^{76} +4.00000 q^{77} +4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -12.0000 q^{83} -2.00000 q^{84} -2.00000 q^{85} -10.0000 q^{86} -4.00000 q^{87} +6.00000 q^{88} -1.00000 q^{90} +8.00000 q^{91} +4.00000 q^{92} +12.0000 q^{94} +1.00000 q^{95} -5.00000 q^{96} -16.0000 q^{97} -3.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	1.00000	0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−2.00000	−0.534522
$15$	1.00000	0.258199
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.00000	0.223607
$21$	2.00000	0.436436
$22$	−2.00000	−0.426401
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	3.00000	0.612372
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	1.00000	0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	5.00000	0.883883
$33$	2.00000	0.348155
$34$	2.00000	0.342997
$35$	2.00000	0.338062
$36$	−1.00000	−0.166667
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−1.00000	−0.162221
$39$	4.00000	0.640513
$40$	3.00000	0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	2.00000	0.308607
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	2.00000	0.301511
$45$	−1.00000	−0.149071
$46$	−4.00000	−0.589768
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	−2.00000	−0.280056
$52$	4.00000	0.554700
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−1.00000	−0.136083
$55$	2.00000	0.269680
$56$	6.00000	0.801784
$57$	1.00000	0.132453
$58$	4.00000	0.525226
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−1.00000	−0.129099
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	7.00000	0.875000
$65$	4.00000	0.496139
$66$	2.00000	0.246183
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	−2.00000	−0.242536
$69$	4.00000	0.481543
$70$	2.00000	0.239046
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−3.00000	−0.353553
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	1.00000	0.114708
$77$	4.00000	0.455842
$78$	4.00000	0.452911
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−2.00000	−0.218218
$85$	−2.00000	−0.216930
$86$	−10.0000	−1.07833
$87$	−4.00000	−0.428845
$88$	6.00000	0.639602
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	−1.00000	−0.105409
$91$	8.00000	0.838628
$92$	4.00000	0.417029
$93$	0	0
$94$	12.0000	1.23771
$95$	1.00000	0.102598
$96$	−5.00000	−0.510310
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	−3.00000	−0.303046
$99$	−2.00000	−0.201008
$100$	−1.00000	−0.100000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	−2.00000	−0.198030
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	12.0000	1.17670
$105$	−2.00000	−0.195180
$106$	−2.00000	−0.194257
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	2.00000	0.188982
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	1.00000	0.0936586
$115$	4.00000	0.373002
$116$	−4.00000	−0.371391
$117$	−4.00000	−0.369800
$118$	4.00000	0.368230
$119$	−4.00000	−0.366679
$120$	−3.00000	−0.273861
$121$	−7.00000	−0.636364
$122$	2.00000	0.181071
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	−2.00000	−0.178174
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−3.00000	−0.265165
$129$	10.0000	0.880451
$130$	4.00000	0.350823
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	−2.00000	−0.174078
$133$	2.00000	0.173422
$134$	−16.0000	−1.38219
$135$	1.00000	0.0860663
$136$	−6.00000	−0.514496
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	4.00000	0.340503
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−2.00000	−0.169031
$141$	−12.0000	−1.01058
$142$	0	0
$143$	8.00000	0.668994
$144$	−1.00000	−0.0833333
$145$	−4.00000	−0.332182
$146$	−2.00000	−0.165521
$147$	3.00000	0.247436
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	−1.00000	−0.0816497
$151$	24.0000	1.95309	0.976546	−	0.215308i	$-0.0690756\pi$
$151$	24.0000	1.95309	0.976546	+	0.215308i	$0.0690756\pi$
$152$	3.00000	0.243332
$153$	2.00000	0.161690
$154$	4.00000	0.322329
$155$	0	0
$156$	−4.00000	−0.320256
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−8.00000	−0.636446
$159$	2.00000	0.158610
$160$	−5.00000	−0.395285
$161$	8.00000	0.630488
$162$	1.00000	0.0785674
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	−12.0000	−0.931381
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	−6.00000	−0.462910
$169$	3.00000	0.230769
$170$	−2.00000	−0.153393
$171$	−1.00000	−0.0764719
$172$	10.0000	0.762493
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	−4.00000	−0.303239
$175$	−2.00000	−0.151186
$176$	2.00000	0.150756
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	1.00000	0.0745356
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	8.00000	0.592999
$183$	−2.00000	−0.147844
$184$	12.0000	0.884652
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	−12.0000	−0.875190
$189$	2.00000	0.145479
$190$	1.00000	0.0725476
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	−7.00000	−0.505181
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−16.0000	−1.14873
$195$	−4.00000	−0.286446
$196$	3.00000	0.214286
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−2.00000	−0.142134
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	−3.00000	−0.212132
$201$	16.0000	1.12855
$202$	14.0000	0.985037
$203$	−8.00000	−0.561490
$204$	2.00000	0.140028
$205$	0	0
$206$	−8.00000	−0.557386
$207$	−4.00000	−0.278019
$208$	4.00000	0.277350
$209$	2.00000	0.138343
$210$	−2.00000	−0.138013
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−12.0000	−0.820303
$215$	10.0000	0.681994
$216$	3.00000	0.204124
$217$	0	0
$218$	10.0000	0.677285
$219$	2.00000	0.135147
$220$	−2.00000	−0.134840
$221$	−8.00000	−0.538138
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	−10.0000	−0.668153
$225$	1.00000	0.0666667
$226$	14.0000	0.931266
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−1.00000	−0.0662266
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	4.00000	0.263752
$231$	−4.00000	−0.263181
$232$	−12.0000	−0.787839
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−4.00000	−0.261488
$235$	−12.0000	−0.782794
$236$	−4.00000	−0.260378
$237$	8.00000	0.519656
$238$	−4.00000	−0.259281
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	−1.00000	−0.0645497
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	−7.00000	−0.449977
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	3.00000	0.191663
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	12.0000	0.760469
$250$	−1.00000	−0.0632456
$251$	22.0000	1.38863	0.694314	−	0.719672i	$-0.255708\pi$
$251$	22.0000	1.38863	0.694314	+	0.719672i	$0.255708\pi$
$252$	2.00000	0.125988
$253$	8.00000	0.502956
$254$	4.00000	0.250982
$255$	2.00000	0.125245
$256$	−17.0000	−1.06250
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	10.0000	0.622573
$259$	0	0
$260$	−4.00000	−0.248069
$261$	4.00000	0.247594
$262$	−14.0000	−0.864923
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	−6.00000	−0.369274
$265$	2.00000	0.122859
$266$	2.00000	0.122628
$267$	0	0
$268$	16.0000	0.977356
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	1.00000	0.0608581
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	−2.00000	−0.121268
$273$	−8.00000	−0.484182
$274$	6.00000	0.362473
$275$	−2.00000	−0.120605
$276$	−4.00000	−0.240772
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	8.00000	0.479808
$279$	0	0
$280$	−6.00000	−0.358569
$281$	28.0000	1.67034	0.835170	−	0.549992i	$-0.185369\pi$
$281$	28.0000	1.67034	0.835170	+	0.549992i	$0.185369\pi$
$282$	−12.0000	−0.714590
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	0	0
$285$	−1.00000	−0.0592349
$286$	8.00000	0.473050
$287$	0	0
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	16.0000	0.937937
$292$	2.00000	0.117041
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	3.00000	0.174964
$295$	−4.00000	−0.232889
$296$	0	0
$297$	2.00000	0.116052
$298$	−18.0000	−1.04271
$299$	16.0000	0.925304
$300$	1.00000	0.0577350
$301$	20.0000	1.15278
$302$	24.0000	1.38104
$303$	−14.0000	−0.804279
$304$	1.00000	0.0573539
$305$	−2.00000	−0.114520
$306$	2.00000	0.114332
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	−4.00000	−0.227921
$309$	8.00000	0.455104
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	−12.0000	−0.679366
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−18.0000	−1.01580
$315$	2.00000	0.112687
$316$	8.00000	0.450035
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	2.00000	0.112154
$319$	−8.00000	−0.447914
$320$	−7.00000	−0.391312
$321$	12.0000	0.669775
$322$	8.00000	0.445823
$323$	−2.00000	−0.111283
$324$	−1.00000	−0.0555556
$325$	−4.00000	−0.221880
$326$	−6.00000	−0.332309
$327$	−10.0000	−0.553001
$328$	0	0
$329$	−24.0000	−1.32316
$330$	−2.00000	−0.110096
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	−8.00000	−0.437741
$335$	16.0000	0.874173
$336$	−2.00000	−0.109109
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	3.00000	0.163178
$339$	−14.0000	−0.760376
$340$	2.00000	0.108465
$341$	0	0
$342$	−1.00000	−0.0540738
$343$	20.0000	1.07990
$344$	30.0000	1.61749
$345$	−4.00000	−0.215353
$346$	18.0000	0.967686
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	4.00000	0.214423
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	−2.00000	−0.106904
$351$	4.00000	0.213504
$352$	−10.0000	−0.533002
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	0	0
$357$	4.00000	0.211702
$358$	−12.0000	−0.634220
$359$	−2.00000	−0.105556	−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000	−0.105556	−0.0527780	+	0.998606i	$0.516808\pi$
$360$	3.00000	0.158114
$361$	1.00000	0.0526316
$362$	10.0000	0.525588
$363$	7.00000	0.367405
$364$	−8.00000	−0.419314
$365$	2.00000	0.104685
$366$	−2.00000	−0.104542
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	−4.00000	−0.206835
$375$	1.00000	0.0516398
$376$	−36.0000	−1.85656
$377$	−16.0000	−0.824042
$378$	2.00000	0.102869
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−1.00000	−0.0512989
$381$	−4.00000	−0.204926
$382$	−18.0000	−0.920960
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	3.00000	0.153093
$385$	−4.00000	−0.203859
$386$	4.00000	0.203595
$387$	−10.0000	−0.508329
$388$	16.0000	0.812277
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	−4.00000	−0.202548
$391$	−8.00000	−0.404577
$392$	9.00000	0.454569
$393$	14.0000	0.706207
$394$	−2.00000	−0.100759
$395$	8.00000	0.402524
$396$	2.00000	0.100504
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	12.0000	0.601506
$399$	−2.00000	−0.100125
$400$	−1.00000	−0.0500000
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	16.0000	0.798007
$403$	0	0
$404$	−14.0000	−0.696526
$405$	−1.00000	−0.0496904
$406$	−8.00000	−0.397033
$407$	0	0
$408$	6.00000	0.297044
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	8.00000	0.394132
$413$	−8.00000	−0.393654
$414$	−4.00000	−0.196589
$415$	12.0000	0.589057
$416$	−20.0000	−0.980581
$417$	−8.00000	−0.391762
$418$	2.00000	0.0978232
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	2.00000	0.0975900
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	−12.0000	−0.584151
$423$	12.0000	0.583460
$424$	6.00000	0.291386
$425$	2.00000	0.0970143
$426$	0	0
$427$	−4.00000	−0.193574
$428$	12.0000	0.580042
$429$	−8.00000	−0.386244
$430$	10.0000	0.482243
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	1.00000	0.0481125
$433$	−36.0000	−1.73005	−0.865025	−	0.501729i	$-0.832697\pi$
$433$	−36.0000	−1.73005	−0.865025	+	0.501729i	$0.832697\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	−10.0000	−0.478913
$437$	4.00000	0.191346
$438$	2.00000	0.0955637
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	−6.00000	−0.286039
$441$	−3.00000	−0.142857
$442$	−8.00000	−0.380521
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	0	0
$445$	0	0
$446$	24.0000	1.13643
$447$	18.0000	0.851371
$448$	−14.0000	−0.661438
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−14.0000	−0.658505
$453$	−24.0000	−1.12762
$454$	12.0000	0.563188
$455$	−8.00000	−0.375046
$456$	−3.00000	−0.140488
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	−14.0000	−0.654177
$459$	−2.00000	−0.0933520
$460$	−4.00000	−0.186501
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	−4.00000	−0.186097
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	−18.0000	−0.833834
$467$	16.0000	0.740392	0.370196	−	0.928954i	$-0.379291\pi$
$467$	16.0000	0.740392	0.370196	+	0.928954i	$0.379291\pi$
$468$	4.00000	0.184900
$469$	32.0000	1.47762
$470$	−12.0000	−0.553519
$471$	18.0000	0.829396
$472$	−12.0000	−0.552345
$473$	20.0000	0.919601
$474$	8.00000	0.367452
$475$	−1.00000	−0.0458831
$476$	4.00000	0.183340
$477$	−2.00000	−0.0915737
$478$	−6.00000	−0.274434
$479$	38.0000	1.73626	0.868132	−	0.496333i	$-0.165321\pi$
$479$	38.0000	1.73626	0.868132	+	0.496333i	$0.165321\pi$
$480$	5.00000	0.228218
$481$	0	0
$482$	26.0000	1.18427
$483$	−8.00000	−0.364013
$484$	7.00000	0.318182
$485$	16.0000	0.726523
$486$	−1.00000	−0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−6.00000	−0.271607
$489$	6.00000	0.271329
$490$	3.00000	0.135526
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	8.00000	0.360302
$494$	4.00000	0.179969
$495$	2.00000	0.0898933
$496$	0	0
$497$	0	0
$498$	12.0000	0.537733
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	1.00000	0.0447214
$501$	8.00000	0.357414
$502$	22.0000	0.981908
$503$	−8.00000	−0.356702	−0.178351	−	0.983967i	$-0.557076\pi$
$503$	−8.00000	−0.356702	−0.178351	+	0.983967i	$0.557076\pi$
$504$	6.00000	0.267261
$505$	−14.0000	−0.622992
$506$	8.00000	0.355643
$507$	−3.00000	−0.133235
$508$	−4.00000	−0.177471
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	2.00000	0.0885615
$511$	4.00000	0.176950
$512$	−11.0000	−0.486136
$513$	1.00000	0.0441511
$514$	22.0000	0.970378
$515$	8.00000	0.352522
$516$	−10.0000	−0.440225
$517$	−24.0000	−1.05552
$518$	0	0
$519$	−18.0000	−0.790112
$520$	−12.0000	−0.526235
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	4.00000	0.175075
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	14.0000	0.611593
$525$	2.00000	0.0872872
$526$	−16.0000	−0.697633
$527$	0	0
$528$	−2.00000	−0.0870388
$529$	−7.00000	−0.304348
$530$	2.00000	0.0868744
$531$	4.00000	0.173585
$532$	−2.00000	−0.0867110
$533$	0	0
$534$	0	0
$535$	12.0000	0.518805
$536$	48.0000	2.07328
$537$	12.0000	0.517838
$538$	4.00000	0.172452
$539$	6.00000	0.258438
$540$	−1.00000	−0.0430331
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	12.0000	0.515444
$543$	−10.0000	−0.429141
$544$	10.0000	0.428746
$545$	−10.0000	−0.428353
$546$	−8.00000	−0.342368
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	−6.00000	−0.256307
$549$	2.00000	0.0853579
$550$	−2.00000	−0.0852803
$551$	−4.00000	−0.170406
$552$	−12.0000	−0.510754
$553$	16.0000	0.680389
$554$	22.0000	0.934690
$555$	0	0
$556$	−8.00000	−0.339276
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	0	0
$559$	40.0000	1.69182
$560$	−2.00000	−0.0845154
$561$	4.00000	0.168880
$562$	28.0000	1.18111
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	12.0000	0.505291
$565$	−14.0000	−0.588984
$566$	26.0000	1.09286
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	−1.00000	−0.0418854
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	−8.00000	−0.334497
$573$	18.0000	0.751961
$574$	0	0
$575$	−4.00000	−0.166812
$576$	7.00000	0.291667
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	−13.0000	−0.540729
$579$	−4.00000	−0.166234
$580$	4.00000	0.166091
$581$	24.0000	0.995688
$582$	16.0000	0.663221
$583$	4.00000	0.165663
$584$	6.00000	0.248282
$585$	4.00000	0.165380
$586$	26.0000	1.07405
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	−3.00000	−0.123718
$589$	0	0
$590$	−4.00000	−0.164677
$591$	2.00000	0.0822690
$592$	0	0
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	2.00000	0.0820610
$595$	4.00000	0.163984
$596$	18.0000	0.737309
$597$	−12.0000	−0.491127
$598$	16.0000	0.654289
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	3.00000	0.122474
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	20.0000	0.815139
$603$	−16.0000	−0.651570
$604$	−24.0000	−0.976546
$605$	7.00000	0.284590
$606$	−14.0000	−0.568711
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	−5.00000	−0.202777
$609$	8.00000	0.324176
$610$	−2.00000	−0.0809776
$611$	−48.0000	−1.94187
$612$	−2.00000	−0.0808452
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	0	0
$615$	0	0
$616$	−12.0000	−0.483494
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	8.00000	0.321807
$619$	16.0000	0.643094	0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000	0.643094	0.321547	+	0.946894i	$0.395797\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	−18.0000	−0.721734
$623$	0	0
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−14.0000	−0.559553
$627$	−2.00000	−0.0798723
$628$	18.0000	0.718278
$629$	0	0
$630$	2.00000	0.0796819
$631$	48.0000	1.91085	0.955425	−	0.295234i	$-0.0953977\pi$
$631$	48.0000	1.91085	0.955425	+	0.295234i	$0.0953977\pi$
$632$	24.0000	0.954669
$633$	12.0000	0.476957
$634$	14.0000	0.556011
$635$	−4.00000	−0.158735
$636$	−2.00000	−0.0793052
$637$	12.0000	0.475457
$638$	−8.00000	−0.316723
$639$	0	0
$640$	3.00000	0.118585
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	12.0000	0.473602
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	−8.00000	−0.315244
$645$	−10.0000	−0.393750
$646$	−2.00000	−0.0786889
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	−3.00000	−0.117851
$649$	−8.00000	−0.314027
$650$	−4.00000	−0.156893
$651$	0	0
$652$	6.00000	0.234978
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−10.0000	−0.391031
$655$	14.0000	0.547025
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	−24.0000	−0.935617
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	2.00000	0.0778499
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	−28.0000	−1.08825
$663$	8.00000	0.310694
$664$	36.0000	1.39707
$665$	−2.00000	−0.0775567
$666$	0	0
$667$	−16.0000	−0.619522
$668$	8.00000	0.309529
$669$	−24.0000	−0.927894
$670$	16.0000	0.618134
$671$	−4.00000	−0.154418
$672$	10.0000	0.385758
$673$	24.0000	0.925132	0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000	0.925132	0.462566	+	0.886585i	$0.346929\pi$
$674$	16.0000	0.616297
$675$	−1.00000	−0.0384900
$676$	−3.00000	−0.115385
$677$	−50.0000	−1.92166	−0.960828	−	0.277145i	$-0.910612\pi$
$677$	−50.0000	−1.92166	−0.960828	+	0.277145i	$0.910612\pi$
$678$	−14.0000	−0.537667
$679$	32.0000	1.22805
$680$	6.00000	0.230089
$681$	−12.0000	−0.459841
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	1.00000	0.0382360
$685$	−6.00000	−0.229248
$686$	20.0000	0.763604
$687$	14.0000	0.534133
$688$	10.0000	0.381246
$689$	8.00000	0.304776
$690$	−4.00000	−0.152277
$691$	−16.0000	−0.608669	−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000	−0.608669	−0.304334	+	0.952565i	$0.598434\pi$
$692$	−18.0000	−0.684257
$693$	4.00000	0.151947
$694$	−28.0000	−1.06287
$695$	−8.00000	−0.303457
$696$	12.0000	0.454859
$697$	0	0
$698$	−18.0000	−0.681310
$699$	18.0000	0.680823
$700$	2.00000	0.0755929
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	4.00000	0.150970
$703$	0	0
$704$	−14.0000	−0.527645
$705$	12.0000	0.451946
$706$	−2.00000	−0.0752710
$707$	−28.0000	−1.05305
$708$	4.00000	0.150329
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	0	0
$714$	4.00000	0.149696
$715$	−8.00000	−0.299183
$716$	12.0000	0.448461
$717$	6.00000	0.224074
$718$	−2.00000	−0.0746393
$719$	42.0000	1.56634	0.783168	−	0.621810i	$-0.213603\pi$
$719$	42.0000	1.56634	0.783168	+	0.621810i	$0.213603\pi$
$720$	1.00000	0.0372678
$721$	16.0000	0.595871
$722$	1.00000	0.0372161
$723$	−26.0000	−0.966950
$724$	−10.0000	−0.371647
$725$	4.00000	0.148556
$726$	7.00000	0.259794
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	−24.0000	−0.889499
$729$	1.00000	0.0370370
$730$	2.00000	0.0740233
$731$	−20.0000	−0.739727
$732$	2.00000	0.0739221
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	10.0000	0.369107
$735$	−3.00000	−0.110657
$736$	−20.0000	−0.737210
$737$	32.0000	1.17874
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	4.00000	0.146845
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	4.00000	0.146450
$747$	−12.0000	−0.439057
$748$	4.00000	0.146254
$749$	24.0000	0.876941
$750$	1.00000	0.0365148
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	−12.0000	−0.437595
$753$	−22.0000	−0.801725
$754$	−16.0000	−0.582686
$755$	−24.0000	−0.873449
$756$	−2.00000	−0.0727393
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	−20.0000	−0.726433
$759$	−8.00000	−0.290382
$760$	−3.00000	−0.108821
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−4.00000	−0.144905
$763$	−20.0000	−0.724049
$764$	18.0000	0.651217
$765$	−2.00000	−0.0723102
$766$	8.00000	0.289052
$767$	−16.0000	−0.577727
$768$	17.0000	0.613435
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	−4.00000	−0.144150
$771$	−22.0000	−0.792311
$772$	−4.00000	−0.143963
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	−10.0000	−0.359443
$775$	0	0
$776$	48.0000	1.72310
$777$	0	0
$778$	−30.0000	−1.07555
$779$	0	0
$780$	4.00000	0.143223
$781$	0	0
$782$	−8.00000	−0.286079
$783$	−4.00000	−0.142948
$784$	3.00000	0.107143
$785$	18.0000	0.642448
$786$	14.0000	0.499363
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	2.00000	0.0712470
$789$	16.0000	0.569615
$790$	8.00000	0.284627
$791$	−28.0000	−0.995565
$792$	6.00000	0.213201
$793$	−8.00000	−0.284088
$794$	−2.00000	−0.0709773
$795$	−2.00000	−0.0709327
$796$	−12.0000	−0.425329
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	−2.00000	−0.0707992
$799$	24.0000	0.849059
$800$	5.00000	0.176777
$801$	0	0
$802$	12.0000	0.423735
$803$	4.00000	0.141157
$804$	−16.0000	−0.564276
$805$	−8.00000	−0.281963
$806$	0	0
$807$	−4.00000	−0.140807
$808$	−42.0000	−1.47755
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	−1.00000	−0.0351364
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	8.00000	0.280745
$813$	−12.0000	−0.420858
$814$	0	0
$815$	6.00000	0.210171
$816$	2.00000	0.0700140
$817$	10.0000	0.349856
$818$	−26.0000	−0.909069
$819$	8.00000	0.279543
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	−6.00000	−0.209274
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	24.0000	0.836080
$825$	2.00000	0.0696311
$826$	−8.00000	−0.278356
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	4.00000	0.139010
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	12.0000	0.416526
$831$	−22.0000	−0.763172
$832$	−28.0000	−0.970725
$833$	−6.00000	−0.207888
$834$	−8.00000	−0.277017
$835$	8.00000	0.276851
$836$	−2.00000	−0.0691714
$837$	0	0
$838$	−26.0000	−0.898155
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	6.00000	0.207020
$841$	−13.0000	−0.448276
$842$	−34.0000	−1.17172
$843$	−28.0000	−0.964371
$844$	12.0000	0.413057
$845$	−3.00000	−0.103203
$846$	12.0000	0.412568
$847$	14.0000	0.481046
$848$	2.00000	0.0686803
$849$	−26.0000	−0.892318
$850$	2.00000	0.0685994
$851$	0	0
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	−4.00000	−0.136877
$855$	1.00000	0.0341993
$856$	36.0000	1.23045
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	−8.00000	−0.273115
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	−10.0000	−0.340997
$861$	0	0
$862$	−4.00000	−0.136241
$863$	40.0000	1.36162	0.680808	−	0.732462i	$-0.261629\pi$
$863$	40.0000	1.36162	0.680808	+	0.732462i	$0.261629\pi$
$864$	−5.00000	−0.170103
$865$	−18.0000	−0.612018
$866$	−36.0000	−1.22333
$867$	13.0000	0.441503
$868$	0	0
$869$	16.0000	0.542763
$870$	4.00000	0.135613
$871$	64.0000	2.16856
$872$	−30.0000	−1.01593
$873$	−16.0000	−0.541518
$874$	4.00000	0.135302
$875$	2.00000	0.0676123
$876$	−2.00000	−0.0675737
$877$	12.0000	0.405211	0.202606	−	0.979260i	$-0.435059\pi$
$877$	12.0000	0.405211	0.202606	+	0.979260i	$0.435059\pi$
$878$	−40.0000	−1.34993
$879$	−26.0000	−0.876958
$880$	−2.00000	−0.0674200
$881$	−10.0000	−0.336909	−0.168454	−	0.985709i	$-0.553878\pi$
$881$	−10.0000	−0.336909	−0.168454	+	0.985709i	$0.553878\pi$
$882$	−3.00000	−0.101015
$883$	−46.0000	−1.54802	−0.774012	−	0.633171i	$-0.781753\pi$
$883$	−46.0000	−1.54802	−0.774012	+	0.633171i	$0.781753\pi$
$884$	8.00000	0.269069
$885$	4.00000	0.134459
$886$	16.0000	0.537531
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	−24.0000	−0.803579
$893$	−12.0000	−0.401565
$894$	18.0000	0.602010
$895$	12.0000	0.401116
$896$	6.00000	0.200446
$897$	−16.0000	−0.534224
$898$	0	0
$899$	0	0
$900$	−1.00000	−0.0333333
$901$	−4.00000	−0.133259
$902$	0	0
$903$	−20.0000	−0.665558
$904$	−42.0000	−1.39690
$905$	−10.0000	−0.332411
$906$	−24.0000	−0.797347
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	−12.0000	−0.398234
$909$	14.0000	0.464351
$910$	−8.00000	−0.265197
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	−1.00000	−0.0331133
$913$	24.0000	0.794284
$914$	−14.0000	−0.463079
$915$	2.00000	0.0661180
$916$	14.0000	0.462573
$917$	28.0000	0.924641
$918$	−2.00000	−0.0660098
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	−12.0000	−0.395628
$921$	0	0
$922$	6.00000	0.197599
$923$	0	0
$924$	4.00000	0.131590
$925$	0	0
$926$	−22.0000	−0.722965
$927$	−8.00000	−0.262754
$928$	20.0000	0.656532
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	18.0000	0.589610
$933$	18.0000	0.589294
$934$	16.0000	0.523536
$935$	4.00000	0.130814
$936$	12.0000	0.392232
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	32.0000	1.04484
$939$	14.0000	0.456873
$940$	12.0000	0.391397
$941$	−60.0000	−1.95594	−0.977972	−	0.208736i	$-0.933065\pi$
$941$	−60.0000	−1.95594	−0.977972	+	0.208736i	$0.933065\pi$
$942$	18.0000	0.586472
$943$	0	0
$944$	−4.00000	−0.130189
$945$	−2.00000	−0.0650600
$946$	20.0000	0.650256
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	−8.00000	−0.259828
$949$	8.00000	0.259691
$950$	−1.00000	−0.0324443
$951$	−14.0000	−0.453981
$952$	12.0000	0.388922
$953$	46.0000	1.49009	0.745043	−	0.667016i	$-0.232429\pi$
$953$	46.0000	1.49009	0.745043	+	0.667016i	$0.232429\pi$
$954$	−2.00000	−0.0647524
$955$	18.0000	0.582466
$956$	6.00000	0.194054
$957$	8.00000	0.258603
$958$	38.0000	1.22772
$959$	−12.0000	−0.387500
$960$	7.00000	0.225924
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−12.0000	−0.386695
$964$	−26.0000	−0.837404
$965$	−4.00000	−0.128765
$966$	−8.00000	−0.257396
$967$	34.0000	1.09337	0.546683	−	0.837340i	$-0.315890\pi$
$967$	34.0000	1.09337	0.546683	+	0.837340i	$0.315890\pi$
$968$	21.0000	0.674966
$969$	2.00000	0.0642493
$970$	16.0000	0.513729
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	1.00000	0.0320750
$973$	−16.0000	−0.512936
$974$	8.00000	0.256337
$975$	4.00000	0.128103
$976$	−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	$0.407031\pi$
$978$	6.00000	0.191859
$979$	0	0
$980$	−3.00000	−0.0958315
$981$	10.0000	0.319275
$982$	−6.00000	−0.191468
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	8.00000	0.254772
$987$	24.0000	0.763928
$988$	−4.00000	−0.127257
$989$	40.0000	1.27193
$990$	2.00000	0.0635642
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	−12.0000	−0.380426
$996$	−12.0000	−0.380235
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	32.0000	1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	285.2.a.b.1.1	✓	1
3.2	odd	2		855.2.a.b.1.1		1
4.3	odd	2		4560.2.a.v.1.1		1
5.2	odd	4		1425.2.c.d.799.2		2
5.3	odd	4		1425.2.c.d.799.1		2
5.4	even	2		1425.2.a.d.1.1		1
15.14	odd	2		4275.2.a.o.1.1		1
19.18	odd	2		5415.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.b.1.1	✓	1	1.1	even	1	trivial
855.2.a.b.1.1		1	3.2	odd	2
1425.2.a.d.1.1		1	5.4	even	2
1425.2.c.d.799.1		2	5.3	odd	4
1425.2.c.d.799.2		2	5.2	odd	4
4275.2.a.o.1.1		1	15.14	odd	2
4560.2.a.v.1.1		1	4.3	odd	2
5415.2.a.c.1.1		1	19.18	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 285 = 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	285.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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