LMFDB - Embedded newform 442.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [442,2,Mod(1,442)] 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("442.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(442, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 442 = 2 \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	442.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,2,1,4] 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:	$3.52938776934$
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	$\chi$	$=$	442.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +2.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} -2.00000 q^{11} +2.00000 q^{12} -1.00000 q^{13} -4.00000 q^{14} +8.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +4.00000 q^{20} -8.00000 q^{21} -2.00000 q^{22} -4.00000 q^{23} +2.00000 q^{24} +11.0000 q^{25} -1.00000 q^{26} -4.00000 q^{27} -4.00000 q^{28} -8.00000 q^{29} +8.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} -1.00000 q^{34} -16.0000 q^{35} +1.00000 q^{36} +8.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +4.00000 q^{40} +10.0000 q^{41} -8.00000 q^{42} -2.00000 q^{44} +4.00000 q^{45} -4.00000 q^{46} +8.00000 q^{47} +2.00000 q^{48} +9.00000 q^{49} +11.0000 q^{50} -2.00000 q^{51} -1.00000 q^{52} +2.00000 q^{53} -4.00000 q^{54} -8.00000 q^{55} -4.00000 q^{56} -8.00000 q^{57} -8.00000 q^{58} +8.00000 q^{60} +12.0000 q^{61} +4.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{66} +8.00000 q^{67} -1.00000 q^{68} -8.00000 q^{69} -16.0000 q^{70} +1.00000 q^{72} -10.0000 q^{73} +8.00000 q^{74} +22.0000 q^{75} -4.00000 q^{76} +8.00000 q^{77} -2.00000 q^{78} -4.00000 q^{79} +4.00000 q^{80} -11.0000 q^{81} +10.0000 q^{82} -8.00000 q^{84} -4.00000 q^{85} -16.0000 q^{87} -2.00000 q^{88} -14.0000 q^{89} +4.00000 q^{90} +4.00000 q^{91} -4.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} -16.0000 q^{95} +2.00000 q^{96} -6.00000 q^{97} +9.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
$2$	1.00000	0.707107
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	1.00000	0.500000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	2.00000	0.816497
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	4.00000	1.26491
$11$	−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
$14$	−4.00000	−1.06904
$15$	8.00000	2.06559
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	4.00000	0.894427
$21$	−8.00000	−1.74574
$22$	−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	$0.636929\pi$
$24$	2.00000	0.408248
$25$	11.0000	2.20000
$26$	−1.00000	−0.196116
$27$	−4.00000	−0.769800
$28$	−4.00000	−0.755929
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	8.00000	1.46059
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	−4.00000	−0.696311
$34$	−1.00000	−0.171499
$35$	−16.0000	−2.70449
$36$	1.00000	0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−4.00000	−0.648886
$39$	−2.00000	−0.320256
$40$	4.00000	0.632456
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	−8.00000	−1.23443
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−2.00000	−0.301511
$45$	4.00000	0.596285
$46$	−4.00000	−0.589768
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	2.00000	0.288675
$49$	9.00000	1.28571
$50$	11.0000	1.55563
$51$	−2.00000	−0.280056
$52$	−1.00000	−0.138675
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	−4.00000	−0.544331
$55$	−8.00000	−1.07872
$56$	−4.00000	−0.534522
$57$	−8.00000	−1.05963
$58$	−8.00000	−1.05045
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	8.00000	1.03280
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	4.00000	0.508001
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−4.00000	−0.492366
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−1.00000	−0.121268
$69$	−8.00000	−0.963087
$70$	−16.0000	−1.91237
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	8.00000	0.929981
$75$	22.0000	2.54034
$76$	−4.00000	−0.458831
$77$	8.00000	0.911685
$78$	−2.00000	−0.226455
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	4.00000	0.447214
$81$	−11.0000	−1.22222
$82$	10.0000	1.10432
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−8.00000	−0.872872
$85$	−4.00000	−0.433861
$86$	0	0
$87$	−16.0000	−1.71538
$88$	−2.00000	−0.213201
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	4.00000	0.421637
$91$	4.00000	0.419314
$92$	−4.00000	−0.417029
$93$	8.00000	0.829561
$94$	8.00000	0.825137
$95$	−16.0000	−1.64157
$96$	2.00000	0.204124
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	9.00000	0.909137
$99$	−2.00000	−0.201008
$100$	11.0000	1.10000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	−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	$0.628955\pi$
$104$	−1.00000	−0.0980581
$105$	−32.0000	−3.12288
$106$	2.00000	0.194257
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	−4.00000	−0.384900
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	−8.00000	−0.762770
$111$	16.0000	1.51865
$112$	−4.00000	−0.377964
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−8.00000	−0.749269
$115$	−16.0000	−1.49201
$116$	−8.00000	−0.742781
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	4.00000	0.366679
$120$	8.00000	0.730297
$121$	−7.00000	−0.636364
$122$	12.0000	1.08643
$123$	20.0000	1.80334
$124$	4.00000	0.359211
$125$	24.0000	2.14663
$126$	−4.00000	−0.356348
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.00000	−0.350823
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	−4.00000	−0.348155
$133$	16.0000	1.38738
$134$	8.00000	0.691095
$135$	−16.0000	−1.37706
$136$	−1.00000	−0.0857493
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	−8.00000	−0.681005
$139$	−18.0000	−1.52674	−0.763370	−	0.645961i	$-0.776457\pi$
$139$	−18.0000	−1.52674	−0.763370	+	0.645961i	$0.776457\pi$
$140$	−16.0000	−1.35225
$141$	16.0000	1.34744
$142$	0	0
$143$	2.00000	0.167248
$144$	1.00000	0.0833333
$145$	−32.0000	−2.65746
$146$	−10.0000	−0.827606
$147$	18.0000	1.48461
$148$	8.00000	0.657596
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	22.0000	1.79629
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−4.00000	−0.324443
$153$	−1.00000	−0.0808452
$154$	8.00000	0.644658
$155$	16.0000	1.28515
$156$	−2.00000	−0.160128
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−4.00000	−0.318223
$159$	4.00000	0.317221
$160$	4.00000	0.316228
$161$	16.0000	1.26098
$162$	−11.0000	−0.864242
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	10.0000	0.780869
$165$	−16.0000	−1.24560
$166$	0	0
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	−8.00000	−0.617213
$169$	1.00000	0.0769231
$170$	−4.00000	−0.306786
$171$	−4.00000	−0.305888
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	−16.0000	−1.21296
$175$	−44.0000	−3.32609
$176$	−2.00000	−0.150756
$177$	0	0
$178$	−14.0000	−1.04934
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	4.00000	0.298142
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	4.00000	0.296500
$183$	24.0000	1.77413
$184$	−4.00000	−0.294884
$185$	32.0000	2.35269
$186$	8.00000	0.586588
$187$	2.00000	0.146254
$188$	8.00000	0.583460
$189$	16.0000	1.16383
$190$	−16.0000	−1.16076
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	2.00000	0.144338
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−6.00000	−0.430775
$195$	−8.00000	−0.572892
$196$	9.00000	0.642857
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	−2.00000	−0.142134
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	11.0000	0.777817
$201$	16.0000	1.12855
$202$	−6.00000	−0.422159
$203$	32.0000	2.24596
$204$	−2.00000	−0.140028
$205$	40.0000	2.79372
$206$	−8.00000	−0.557386
$207$	−4.00000	−0.278019
$208$	−1.00000	−0.0693375
$209$	8.00000	0.553372
$210$	−32.0000	−2.20821
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	6.00000	0.410152
$215$	0	0
$216$	−4.00000	−0.272166
$217$	−16.0000	−1.08615
$218$	4.00000	0.270914
$219$	−20.0000	−1.35147
$220$	−8.00000	−0.539360
$221$	1.00000	0.0672673
$222$	16.0000	1.07385
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	−4.00000	−0.267261
$225$	11.0000	0.733333
$226$	2.00000	0.133038
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	−8.00000	−0.529813
$229$	30.0000	1.98246	0.991228	−	0.132164i	$-0.0421925\pi$
$229$	30.0000	1.98246	0.991228	+	0.132164i	$0.0421925\pi$
$230$	−16.0000	−1.05501
$231$	16.0000	1.05272
$232$	−8.00000	−0.525226
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−1.00000	−0.0653720
$235$	32.0000	2.08745
$236$	0	0
$237$	−8.00000	−0.519656
$238$	4.00000	0.259281
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	8.00000	0.516398
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	−7.00000	−0.449977
$243$	−10.0000	−0.641500
$244$	12.0000	0.768221
$245$	36.0000	2.29996
$246$	20.0000	1.27515
$247$	4.00000	0.254514
$248$	4.00000	0.254000
$249$	0	0
$250$	24.0000	1.51789
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−4.00000	−0.251976
$253$	8.00000	0.502956
$254$	16.0000	1.00393
$255$	−8.00000	−0.500979
$256$	1.00000	0.0625000
$257$	−10.0000	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$258$	0	0
$259$	−32.0000	−1.98838
$260$	−4.00000	−0.248069
$261$	−8.00000	−0.495188
$262$	14.0000	0.864923
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	−4.00000	−0.246183
$265$	8.00000	0.491436
$266$	16.0000	0.981023
$267$	−28.0000	−1.71357
$268$	8.00000	0.488678
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	−16.0000	−0.973729
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−1.00000	−0.0606339
$273$	8.00000	0.484182
$274$	22.0000	1.32907
$275$	−22.0000	−1.32665
$276$	−8.00000	−0.481543
$277$	32.0000	1.92269	0.961347	−	0.275340i	$-0.0887905\pi$
$277$	32.0000	1.92269	0.961347	+	0.275340i	$0.0887905\pi$
$278$	−18.0000	−1.07957
$279$	4.00000	0.239474
$280$	−16.0000	−0.956183
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	16.0000	0.952786
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	0	0
$285$	−32.0000	−1.89552
$286$	2.00000	0.118262
$287$	−40.0000	−2.36113
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	−32.0000	−1.87910
$291$	−12.0000	−0.703452
$292$	−10.0000	−0.585206
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	18.0000	1.04978
$295$	0	0
$296$	8.00000	0.464991
$297$	8.00000	0.464207
$298$	−18.0000	−1.04271
$299$	4.00000	0.231326
$300$	22.0000	1.27017
$301$	0	0
$302$	16.0000	0.920697
$303$	−12.0000	−0.689382
$304$	−4.00000	−0.229416
$305$	48.0000	2.74847
$306$	−1.00000	−0.0571662
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	8.00000	0.455842
$309$	−16.0000	−0.910208
$310$	16.0000	0.908739
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	−2.00000	−0.113228
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	−10.0000	−0.564333
$315$	−16.0000	−0.901498
$316$	−4.00000	−0.225018
$317$	−8.00000	−0.449325	−0.224662	−	0.974437i	$-0.572128\pi$
$317$	−8.00000	−0.449325	−0.224662	+	0.974437i	$0.572128\pi$
$318$	4.00000	0.224309
$319$	16.0000	0.895828
$320$	4.00000	0.223607
$321$	12.0000	0.669775
$322$	16.0000	0.891645
$323$	4.00000	0.222566
$324$	−11.0000	−0.611111
$325$	−11.0000	−0.610170
$326$	−14.0000	−0.775388
$327$	8.00000	0.442401
$328$	10.0000	0.552158
$329$	−32.0000	−1.76422
$330$	−16.0000	−0.880771
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	0	0
$333$	8.00000	0.438397
$334$	−20.0000	−1.09435
$335$	32.0000	1.74835
$336$	−8.00000	−0.436436
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	1.00000	0.0543928
$339$	4.00000	0.217250
$340$	−4.00000	−0.216930
$341$	−8.00000	−0.433224
$342$	−4.00000	−0.216295
$343$	−8.00000	−0.431959
$344$	0	0
$345$	−32.0000	−1.72282
$346$	0	0
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	−16.0000	−0.857690
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−44.0000	−2.35190
$351$	4.00000	0.213504
$352$	−2.00000	−0.106600
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	8.00000	0.423405
$358$	−4.00000	−0.211407
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	4.00000	0.210819
$361$	−3.00000	−0.157895
$362$	−20.0000	−1.05118
$363$	−14.0000	−0.734809
$364$	4.00000	0.209657
$365$	−40.0000	−2.09370
$366$	24.0000	1.25450
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	−4.00000	−0.208514
$369$	10.0000	0.520579
$370$	32.0000	1.66360
$371$	−8.00000	−0.415339
$372$	8.00000	0.414781
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	2.00000	0.103418
$375$	48.0000	2.47871
$376$	8.00000	0.412568
$377$	8.00000	0.412021
$378$	16.0000	0.822951
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	−16.0000	−0.820783
$381$	32.0000	1.63941
$382$	0	0
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	2.00000	0.102062
$385$	32.0000	1.63087
$386$	−6.00000	−0.305392
$387$	0	0
$388$	−6.00000	−0.304604
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	−8.00000	−0.405096
$391$	4.00000	0.202289
$392$	9.00000	0.454569
$393$	28.0000	1.41241
$394$	24.0000	1.20910
$395$	−16.0000	−0.805047
$396$	−2.00000	−0.100504
$397$	−32.0000	−1.60603	−0.803017	−	0.595956i	$-0.796773\pi$
$397$	−32.0000	−1.60603	−0.803017	+	0.595956i	$0.796773\pi$
$398$	−20.0000	−1.00251
$399$	32.0000	1.60200
$400$	11.0000	0.550000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	16.0000	0.798007
$403$	−4.00000	−0.199254
$404$	−6.00000	−0.298511
$405$	−44.0000	−2.18638
$406$	32.0000	1.58813
$407$	−16.0000	−0.793091
$408$	−2.00000	−0.0990148
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	40.0000	1.97546
$411$	44.0000	2.17036
$412$	−8.00000	−0.394132
$413$	0	0
$414$	−4.00000	−0.196589
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	−36.0000	−1.76293
$418$	8.00000	0.391293
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	−32.0000	−1.56144
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−22.0000	−1.07094
$423$	8.00000	0.388973
$424$	2.00000	0.0971286
$425$	−11.0000	−0.533578
$426$	0	0
$427$	−48.0000	−2.32288
$428$	6.00000	0.290021
$429$	4.00000	0.193122
$430$	0	0
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	−4.00000	−0.192450
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−16.0000	−0.768025
$435$	−64.0000	−3.06857
$436$	4.00000	0.191565
$437$	16.0000	0.765384
$438$	−20.0000	−0.955637
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	−8.00000	−0.381385
$441$	9.00000	0.428571
$442$	1.00000	0.0475651
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	16.0000	0.759326
$445$	−56.0000	−2.65465
$446$	24.0000	1.13643
$447$	−36.0000	−1.70274
$448$	−4.00000	−0.188982
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	11.0000	0.518545
$451$	−20.0000	−0.941763
$452$	2.00000	0.0940721
$453$	32.0000	1.50349
$454$	−14.0000	−0.657053
$455$	16.0000	0.750092
$456$	−8.00000	−0.374634
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	30.0000	1.40181
$459$	4.00000	0.186704
$460$	−16.0000	−0.746004
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	16.0000	0.744387
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	−8.00000	−0.371391
$465$	32.0000	1.48396
$466$	−6.00000	−0.277945
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	−1.00000	−0.0462250
$469$	−32.0000	−1.47762
$470$	32.0000	1.47605
$471$	−20.0000	−0.921551
$472$	0	0
$473$	0	0
$474$	−8.00000	−0.367452
$475$	−44.0000	−2.01886
$476$	4.00000	0.183340
$477$	2.00000	0.0915737
$478$	24.0000	1.09773
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	8.00000	0.365148
$481$	−8.00000	−0.364769
$482$	18.0000	0.819878
$483$	32.0000	1.45605
$484$	−7.00000	−0.318182
$485$	−24.0000	−1.08978
$486$	−10.0000	−0.453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	12.0000	0.543214
$489$	−28.0000	−1.26620
$490$	36.0000	1.62631
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	20.0000	0.901670
$493$	8.00000	0.360302
$494$	4.00000	0.179969
$495$	−8.00000	−0.359573
$496$	4.00000	0.179605
$497$	0	0
$498$	0	0
$499$	−2.00000	−0.0895323	−0.0447661	−	0.998997i	$-0.514254\pi$
$499$	−2.00000	−0.0895323	−0.0447661	+	0.998997i	$0.514254\pi$
$500$	24.0000	1.07331
$501$	−40.0000	−1.78707
$502$	0	0
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	−4.00000	−0.178174
$505$	−24.0000	−1.06799
$506$	8.00000	0.355643
$507$	2.00000	0.0888231
$508$	16.0000	0.709885
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	−8.00000	−0.354246
$511$	40.0000	1.76950
$512$	1.00000	0.0441942
$513$	16.0000	0.706417
$514$	−10.0000	−0.441081
$515$	−32.0000	−1.41009
$516$	0	0
$517$	−16.0000	−0.703679
$518$	−32.0000	−1.40600
$519$	0	0
$520$	−4.00000	−0.175412
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	−8.00000	−0.350150
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	14.0000	0.611593
$525$	−88.0000	−3.84063
$526$	−8.00000	−0.348817
$527$	−4.00000	−0.174243
$528$	−4.00000	−0.174078
$529$	−7.00000	−0.304348
$530$	8.00000	0.347498
$531$	0	0
$532$	16.0000	0.693688
$533$	−10.0000	−0.433148
$534$	−28.0000	−1.21168
$535$	24.0000	1.03761
$536$	8.00000	0.345547
$537$	−8.00000	−0.345225
$538$	16.0000	0.689809
$539$	−18.0000	−0.775315
$540$	−16.0000	−0.688530
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	8.00000	0.343629
$543$	−40.0000	−1.71656
$544$	−1.00000	−0.0428746
$545$	16.0000	0.685365
$546$	8.00000	0.342368
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	22.0000	0.939793
$549$	12.0000	0.512148
$550$	−22.0000	−0.938083
$551$	32.0000	1.36325
$552$	−8.00000	−0.340503
$553$	16.0000	0.680389
$554$	32.0000	1.35955
$555$	64.0000	2.71665
$556$	−18.0000	−0.763370
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	4.00000	0.169334
$559$	0	0
$560$	−16.0000	−0.676123
$561$	4.00000	0.168880
$562$	−26.0000	−1.09674
$563$	16.0000	0.674320	0.337160	−	0.941447i	$-0.390534\pi$
$563$	16.0000	0.674320	0.337160	+	0.941447i	$0.390534\pi$
$564$	16.0000	0.673722
$565$	8.00000	0.336563
$566$	−6.00000	−0.252199
$567$	44.0000	1.84783
$568$	0	0
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	−32.0000	−1.34033
$571$	−2.00000	−0.0836974	−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000	−0.0836974	−0.0418487	+	0.999124i	$0.513325\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	−40.0000	−1.66957
$575$	−44.0000	−1.83493
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	1.00000	0.0415945
$579$	−12.0000	−0.498703
$580$	−32.0000	−1.32873
$581$	0	0
$582$	−12.0000	−0.497416
$583$	−4.00000	−0.165663
$584$	−10.0000	−0.413803
$585$	−4.00000	−0.165380
$586$	22.0000	0.908812
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	18.0000	0.742307
$589$	−16.0000	−0.659269
$590$	0	0
$591$	48.0000	1.97446
$592$	8.00000	0.328798
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	8.00000	0.328244
$595$	16.0000	0.655936
$596$	−18.0000	−0.737309
$597$	−40.0000	−1.63709
$598$	4.00000	0.163572
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	22.0000	0.898146
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	16.0000	0.651031
$605$	−28.0000	−1.13836
$606$	−12.0000	−0.487467
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	−4.00000	−0.162221
$609$	64.0000	2.59341
$610$	48.0000	1.94346
$611$	−8.00000	−0.323645
$612$	−1.00000	−0.0404226
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	4.00000	0.161427
$615$	80.0000	3.22591
$616$	8.00000	0.322329
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−16.0000	−0.643614
$619$	−46.0000	−1.84890	−0.924448	−	0.381308i	$-0.875474\pi$
$619$	−46.0000	−1.84890	−0.924448	+	0.381308i	$0.875474\pi$
$620$	16.0000	0.642575
$621$	16.0000	0.642058
$622$	8.00000	0.320771
$623$	56.0000	2.24359
$624$	−2.00000	−0.0800641
$625$	41.0000	1.64000
$626$	−26.0000	−1.03917
$627$	16.0000	0.638978
$628$	−10.0000	−0.399043
$629$	−8.00000	−0.318981
$630$	−16.0000	−0.637455
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−4.00000	−0.159111
$633$	−44.0000	−1.74884
$634$	−8.00000	−0.317721
$635$	64.0000	2.53976
$636$	4.00000	0.158610
$637$	−9.00000	−0.356593
$638$	16.0000	0.633446
$639$	0	0
$640$	4.00000	0.158114
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	12.0000	0.473602
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	16.0000	0.630488
$645$	0	0
$646$	4.00000	0.157378
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	−11.0000	−0.432121
$649$	0	0
$650$	−11.0000	−0.431455
$651$	−32.0000	−1.25418
$652$	−14.0000	−0.548282
$653$	−16.0000	−0.626128	−0.313064	−	0.949732i	$-0.601356\pi$
$653$	−16.0000	−0.626128	−0.313064	+	0.949732i	$0.601356\pi$
$654$	8.00000	0.312825
$655$	56.0000	2.18810
$656$	10.0000	0.390434
$657$	−10.0000	−0.390137
$658$	−32.0000	−1.24749
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	−16.0000	−0.622799
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	16.0000	0.621858
$663$	2.00000	0.0776736
$664$	0	0
$665$	64.0000	2.48181
$666$	8.00000	0.309994
$667$	32.0000	1.23904
$668$	−20.0000	−0.773823
$669$	48.0000	1.85579
$670$	32.0000	1.23627
$671$	−24.0000	−0.926510
$672$	−8.00000	−0.308607
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	−22.0000	−0.847408
$675$	−44.0000	−1.69356
$676$	1.00000	0.0384615
$677$	4.00000	0.153732	0.0768662	−	0.997041i	$-0.475509\pi$
$677$	4.00000	0.153732	0.0768662	+	0.997041i	$0.475509\pi$
$678$	4.00000	0.153619
$679$	24.0000	0.921035
$680$	−4.00000	−0.153393
$681$	−28.0000	−1.07296
$682$	−8.00000	−0.306336
$683$	46.0000	1.76014	0.880071	−	0.474843i	$-0.157495\pi$
$683$	46.0000	1.76014	0.880071	+	0.474843i	$0.157495\pi$
$684$	−4.00000	−0.152944
$685$	88.0000	3.36231
$686$	−8.00000	−0.305441
$687$	60.0000	2.28914
$688$	0	0
$689$	−2.00000	−0.0761939
$690$	−32.0000	−1.21822
$691$	−18.0000	−0.684752	−0.342376	−	0.939563i	$-0.611232\pi$
$691$	−18.0000	−0.684752	−0.342376	+	0.939563i	$0.611232\pi$
$692$	0	0
$693$	8.00000	0.303895
$694$	6.00000	0.227757
$695$	−72.0000	−2.73112
$696$	−16.0000	−0.606478
$697$	−10.0000	−0.378777
$698$	2.00000	0.0757011
$699$	−12.0000	−0.453882
$700$	−44.0000	−1.66304
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	4.00000	0.150970
$703$	−32.0000	−1.20690
$704$	−2.00000	−0.0753778
$705$	64.0000	2.41038
$706$	14.0000	0.526897
$707$	24.0000	0.902613
$708$	0	0
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	−14.0000	−0.524672
$713$	−16.0000	−0.599205
$714$	8.00000	0.299392
$715$	8.00000	0.299183
$716$	−4.00000	−0.149487
$717$	48.0000	1.79259
$718$	−16.0000	−0.597115
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	4.00000	0.149071
$721$	32.0000	1.19174
$722$	−3.00000	−0.111648
$723$	36.0000	1.33885
$724$	−20.0000	−0.743294
$725$	−88.0000	−3.26824
$726$	−14.0000	−0.519589
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	4.00000	0.148250
$729$	13.0000	0.481481
$730$	−40.0000	−1.48047
$731$	0	0
$732$	24.0000	0.887066
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−20.0000	−0.738213
$735$	72.0000	2.65576
$736$	−4.00000	−0.147442
$737$	−16.0000	−0.589368
$738$	10.0000	0.368105
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	32.0000	1.17634
$741$	8.00000	0.293887
$742$	−8.00000	−0.293689
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	8.00000	0.293294
$745$	−72.0000	−2.63788
$746$	−14.0000	−0.512576
$747$	0	0
$748$	2.00000	0.0731272
$749$	−24.0000	−0.876941
$750$	48.0000	1.75271
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	8.00000	0.291343
$755$	64.0000	2.32920
$756$	16.0000	0.581914
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−34.0000	−1.23494
$759$	16.0000	0.580763
$760$	−16.0000	−0.580381
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	32.0000	1.15924
$763$	−16.0000	−0.579239
$764$	0	0
$765$	−4.00000	−0.144620
$766$	32.0000	1.15621
$767$	0	0
$768$	2.00000	0.0721688
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	32.0000	1.15320
$771$	−20.0000	−0.720282
$772$	−6.00000	−0.215945
$773$	46.0000	1.65451	0.827253	−	0.561830i	$-0.189903\pi$
$773$	46.0000	1.65451	0.827253	+	0.561830i	$0.189903\pi$
$774$	0	0
$775$	44.0000	1.58053
$776$	−6.00000	−0.215387
$777$	−64.0000	−2.29599
$778$	30.0000	1.07555
$779$	−40.0000	−1.43315
$780$	−8.00000	−0.286446
$781$	0	0
$782$	4.00000	0.143040
$783$	32.0000	1.14359
$784$	9.00000	0.321429
$785$	−40.0000	−1.42766
$786$	28.0000	0.998727
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	24.0000	0.854965
$789$	−16.0000	−0.569615
$790$	−16.0000	−0.569254
$791$	−8.00000	−0.284447
$792$	−2.00000	−0.0710669
$793$	−12.0000	−0.426132
$794$	−32.0000	−1.13564
$795$	16.0000	0.567462
$796$	−20.0000	−0.708881
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	32.0000	1.13279
$799$	−8.00000	−0.283020
$800$	11.0000	0.388909
$801$	−14.0000	−0.494666
$802$	−6.00000	−0.211867
$803$	20.0000	0.705785
$804$	16.0000	0.564276
$805$	64.0000	2.25570
$806$	−4.00000	−0.140894
$807$	32.0000	1.12645
$808$	−6.00000	−0.211079
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	−44.0000	−1.54600
$811$	−18.0000	−0.632065	−0.316033	−	0.948748i	$-0.602351\pi$
$811$	−18.0000	−0.632065	−0.316033	+	0.948748i	$0.602351\pi$
$812$	32.0000	1.12298
$813$	16.0000	0.561144
$814$	−16.0000	−0.560800
$815$	−56.0000	−1.96159
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	14.0000	0.489499
$819$	4.00000	0.139771
$820$	40.0000	1.39686
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	44.0000	1.53468
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	−8.00000	−0.278693
$825$	−44.0000	−1.53188
$826$	0	0
$827$	30.0000	1.04320	0.521601	−	0.853189i	$-0.325335\pi$
$827$	30.0000	1.04320	0.521601	+	0.853189i	$0.325335\pi$
$828$	−4.00000	−0.139010
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	0	0
$831$	64.0000	2.22014
$832$	−1.00000	−0.0346688
$833$	−9.00000	−0.311832
$834$	−36.0000	−1.24658
$835$	−80.0000	−2.76851
$836$	8.00000	0.276686
$837$	−16.0000	−0.553041
$838$	6.00000	0.207267
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	−32.0000	−1.10410
$841$	35.0000	1.20690
$842$	10.0000	0.344623
$843$	−52.0000	−1.79098
$844$	−22.0000	−0.757271
$845$	4.00000	0.137604
$846$	8.00000	0.275046
$847$	28.0000	0.962091
$848$	2.00000	0.0686803
$849$	−12.0000	−0.411839
$850$	−11.0000	−0.377297
$851$	−32.0000	−1.09695
$852$	0	0
$853$	4.00000	0.136957	0.0684787	−	0.997653i	$-0.478185\pi$
$853$	4.00000	0.136957	0.0684787	+	0.997653i	$0.478185\pi$
$854$	−48.0000	−1.64253
$855$	−16.0000	−0.547188
$856$	6.00000	0.205076
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	4.00000	0.136558
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	−80.0000	−2.72639
$862$	−32.0000	−1.08992
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	2.00000	0.0679628
$867$	2.00000	0.0679236
$868$	−16.0000	−0.543075
$869$	8.00000	0.271381
$870$	−64.0000	−2.16980
$871$	−8.00000	−0.271070
$872$	4.00000	0.135457
$873$	−6.00000	−0.203069
$874$	16.0000	0.541208
$875$	−96.0000	−3.24539
$876$	−20.0000	−0.675737
$877$	−4.00000	−0.135070	−0.0675352	−	0.997717i	$-0.521513\pi$
$877$	−4.00000	−0.135070	−0.0675352	+	0.997717i	$0.521513\pi$
$878$	28.0000	0.944954
$879$	44.0000	1.48408
$880$	−8.00000	−0.269680
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	9.00000	0.303046
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	1.00000	0.0336336
$885$	0	0
$886$	16.0000	0.537531
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	16.0000	0.536925
$889$	−64.0000	−2.14649
$890$	−56.0000	−1.87712
$891$	22.0000	0.737028
$892$	24.0000	0.803579
$893$	−32.0000	−1.07084
$894$	−36.0000	−1.20402
$895$	−16.0000	−0.534821
$896$	−4.00000	−0.133631
$897$	8.00000	0.267112
$898$	−6.00000	−0.200223
$899$	−32.0000	−1.06726
$900$	11.0000	0.366667
$901$	−2.00000	−0.0666297
$902$	−20.0000	−0.665927
$903$	0	0
$904$	2.00000	0.0665190
$905$	−80.0000	−2.65929
$906$	32.0000	1.06313
$907$	−14.0000	−0.464862	−0.232431	−	0.972613i	$-0.574668\pi$
$907$	−14.0000	−0.464862	−0.232431	+	0.972613i	$0.574668\pi$
$908$	−14.0000	−0.464606
$909$	−6.00000	−0.199007
$910$	16.0000	0.530395
$911$	56.0000	1.85536	0.927681	−	0.373373i	$-0.121799\pi$
$911$	56.0000	1.85536	0.927681	+	0.373373i	$0.121799\pi$
$912$	−8.00000	−0.264906
$913$	0	0
$914$	10.0000	0.330771
$915$	96.0000	3.17366
$916$	30.0000	0.991228
$917$	−56.0000	−1.84928
$918$	4.00000	0.132020
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−16.0000	−0.527504
$921$	8.00000	0.263609
$922$	−6.00000	−0.197599
$923$	0	0
$924$	16.0000	0.526361
$925$	88.0000	2.89342
$926$	−24.0000	−0.788689
$927$	−8.00000	−0.262754
$928$	−8.00000	−0.262613
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	32.0000	1.04932
$931$	−36.0000	−1.17985
$932$	−6.00000	−0.196537
$933$	16.0000	0.523816
$934$	−20.0000	−0.654420
$935$	8.00000	0.261628
$936$	−1.00000	−0.0326860
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	−32.0000	−1.04484
$939$	−52.0000	−1.69696
$940$	32.0000	1.04372
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	−20.0000	−0.651635
$943$	−40.0000	−1.30258
$944$	0	0
$945$	64.0000	2.08192
$946$	0	0
$947$	42.0000	1.36482	0.682408	−	0.730971i	$-0.260933\pi$
$947$	42.0000	1.36482	0.682408	+	0.730971i	$0.260933\pi$
$948$	−8.00000	−0.259828
$949$	10.0000	0.324614
$950$	−44.0000	−1.42755
$951$	−16.0000	−0.518836
$952$	4.00000	0.129641
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	24.0000	0.776215
$957$	32.0000	1.03441
$958$	−8.00000	−0.258468
$959$	−88.0000	−2.84167
$960$	8.00000	0.258199
$961$	−15.0000	−0.483871
$962$	−8.00000	−0.257930
$963$	6.00000	0.193347
$964$	18.0000	0.579741
$965$	−24.0000	−0.772587
$966$	32.0000	1.02958
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	−7.00000	−0.224989
$969$	8.00000	0.256997
$970$	−24.0000	−0.770594
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	−10.0000	−0.320750
$973$	72.0000	2.30821
$974$	−16.0000	−0.512673
$975$	−22.0000	−0.704564
$976$	12.0000	0.384111
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−28.0000	−0.895341
$979$	28.0000	0.894884
$980$	36.0000	1.14998
$981$	4.00000	0.127710
$982$	20.0000	0.638226
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	20.0000	0.637577
$985$	96.0000	3.05881
$986$	8.00000	0.254772
$987$	−64.0000	−2.03714
$988$	4.00000	0.127257
$989$	0	0
$990$	−8.00000	−0.254257
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	4.00000	0.127000
$993$	32.0000	1.01549
$994$	0	0
$995$	−80.0000	−2.53617
$996$	0	0
$997$	20.0000	0.633406	0.316703	−	0.948525i	$-0.397424\pi$
$997$	20.0000	0.633406	0.316703	+	0.948525i	$0.397424\pi$
$998$	−2.00000	−0.0633089
$999$	−32.0000	−1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	442.2.a.e.1.1	✓	1
3.2	odd	2		3978.2.a.a.1.1		1
4.3	odd	2		3536.2.a.d.1.1		1
13.12	even	2		5746.2.a.f.1.1		1
17.16	even	2		7514.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
442.2.a.e.1.1	✓	1	1.1	even	1	trivial
3536.2.a.d.1.1		1	4.3	odd	2
3978.2.a.a.1.1		1	3.2	odd	2
5746.2.a.f.1.1		1	13.12	even	2
7514.2.a.e.1.1		1	17.16	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 442 = 2 \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	442.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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