LMFDB - Embedded newform 238.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [238,2,Mod(1,238)] 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("238.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(238, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 238 = 2 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	238.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$1.90043956811$
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	$\chi$	$=$	238.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -2.00000 q^{21} -2.00000 q^{22} +4.00000 q^{23} +2.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} -4.00000 q^{27} -1.00000 q^{28} +4.00000 q^{29} +1.00000 q^{32} -4.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} -4.00000 q^{39} -2.00000 q^{41} -2.00000 q^{42} -2.00000 q^{44} +4.00000 q^{46} +2.00000 q^{48} +1.00000 q^{49} -5.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} +2.00000 q^{53} -4.00000 q^{54} -1.00000 q^{56} +4.00000 q^{58} +4.00000 q^{59} -12.0000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} -8.00000 q^{67} -1.00000 q^{68} +8.00000 q^{69} +12.0000 q^{71} +1.00000 q^{72} -14.0000 q^{73} +8.00000 q^{74} -10.0000 q^{75} +2.00000 q^{77} -4.00000 q^{78} +12.0000 q^{79} -11.0000 q^{81} -2.00000 q^{82} +4.00000 q^{83} -2.00000 q^{84} +8.00000 q^{87} -2.00000 q^{88} -6.00000 q^{89} +2.00000 q^{91} +4.00000 q^{92} +2.00000 q^{96} +6.00000 q^{97} +1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	2.00000	0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−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
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	−2.00000	−0.426401
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	2.00000	0.408248
$25$	−5.00000	−1.00000
$26$	−2.00000	−0.392232
$27$	−4.00000	−0.769800
$28$	−1.00000	−0.188982
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	−4.00000	−0.696311
$34$	−1.00000	−0.171499
$35$	0	0
$36$	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	$0.271573\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−2.00000	−0.308607
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	4.00000	0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	2.00000	0.288675
$49$	1.00000	0.142857
$50$	−5.00000	−0.707107
$51$	−2.00000	−0.280056
$52$	−2.00000	−0.277350
$53$	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
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	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	$0.416153\pi$
$60$	0	0
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−1.00000	−0.121268
$69$	8.00000	0.963087
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	1.00000	0.117851
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	8.00000	0.929981
$75$	−10.0000	−1.15470
$76$	0	0
$77$	2.00000	0.227921
$78$	−4.00000	−0.452911
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	−2.00000	−0.220863
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	0	0
$87$	8.00000	0.857690
$88$	−2.00000	−0.213201
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	4.00000	0.417029
$93$	0	0
$94$	0	0
$95$	0	0
$96$	2.00000	0.204124
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	1.00000	0.101015
$99$	−2.00000	−0.201008
$100$	−5.00000	−0.500000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−2.00000	−0.198030
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	2.00000	0.194257
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	−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	$0.438644\pi$
$110$	0	0
$111$	16.0000	1.51865
$112$	−1.00000	−0.0944911
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	0	0
$116$	4.00000	0.371391
$117$	−2.00000	−0.184900
$118$	4.00000	0.368230
$119$	1.00000	0.0916698
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−12.0000	−1.08643
$123$	−4.00000	−0.360668
$124$	0	0
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	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
$133$	0	0
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	8.00000	0.681005
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	0	0
$142$	12.0000	1.00702
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	2.00000	0.164957
$148$	8.00000	0.657596
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−10.0000	−0.816497
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	2.00000	0.161165
$155$	0	0
$156$	−4.00000	−0.320256
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	12.0000	0.954669
$159$	4.00000	0.317221
$160$	0	0
$161$	−4.00000	−0.315244
$162$	−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	$0.684718\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	4.00000	0.310460
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−2.00000	−0.154303
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	8.00000	0.606478
$175$	5.00000	0.377964
$176$	−2.00000	−0.150756
$177$	8.00000	0.601317
$178$	−6.00000	−0.449719
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	2.00000	0.148250
$183$	−24.0000	−1.77413
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	2.00000	0.146254
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	2.00000	0.144338
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	1.00000	0.0714286
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	−2.00000	−0.142134
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−5.00000	−0.353553
$201$	−16.0000	−1.12855
$202$	−10.0000	−0.703598
$203$	−4.00000	−0.280745
$204$	−2.00000	−0.140028
$205$	0	0
$206$	8.00000	0.557386
$207$	4.00000	0.278019
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	2.00000	0.137361
$213$	24.0000	1.64445
$214$	10.0000	0.683586
$215$	0	0
$216$	−4.00000	−0.272166
$217$	0	0
$218$	4.00000	0.270914
$219$	−28.0000	−1.89206
$220$	0	0
$221$	2.00000	0.134535
$222$	16.0000	1.07385
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	−1.00000	−0.0668153
$225$	−5.00000	−0.333333
$226$	18.0000	1.19734
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	4.00000	0.262613
$233$	−30.0000	−1.96537	−0.982683	−	0.185296i	$-0.940675\pi$
$233$	−30.0000	−1.96537	−0.982683	+	0.185296i	$0.940675\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	4.00000	0.260378
$237$	24.0000	1.55897
$238$	1.00000	0.0648204
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	−7.00000	−0.449977
$243$	−10.0000	−0.641500
$244$	−12.0000	−0.768221
$245$	0	0
$246$	−4.00000	−0.255031
$247$	0	0
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	−1.00000	−0.0629941
$253$	−8.00000	−0.502956
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	4.00000	0.247594
$262$	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
$265$	0	0
$266$	0	0
$267$	−12.0000	−0.734388
$268$	−8.00000	−0.488678
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	−1.00000	−0.0606339
$273$	4.00000	0.242091
$274$	6.00000	0.362473
$275$	10.0000	0.603023
$276$	8.00000	0.481543
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	14.0000	0.839664
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	18.0000	1.06999	0.534994	−	0.844856i	$-0.320314\pi$
$283$	18.0000	1.06999	0.534994	+	0.844856i	$0.320314\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	4.00000	0.236525
$287$	2.00000	0.118056
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	12.0000	0.703452
$292$	−14.0000	−0.819288
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	8.00000	0.464991
$297$	8.00000	0.464207
$298$	6.00000	0.347571
$299$	−8.00000	−0.462652
$300$	−10.0000	−0.577350
$301$	0	0
$302$	−8.00000	−0.460348
$303$	−20.0000	−1.14897
$304$	0	0
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	2.00000	0.113961
$309$	16.0000	0.910208
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−4.00000	−0.226455
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	12.0000	0.675053
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	4.00000	0.224309
$319$	−8.00000	−0.447914
$320$	0	0
$321$	20.0000	1.11629
$322$	−4.00000	−0.222911
$323$	0	0
$324$	−11.0000	−0.611111
$325$	10.0000	0.554700
$326$	−14.0000	−0.775388
$327$	8.00000	0.442401
$328$	−2.00000	−0.110432
$329$	0	0
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	4.00000	0.219529
$333$	8.00000	0.438397
$334$	0	0
$335$	0	0
$336$	−2.00000	−0.109109
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−9.00000	−0.489535
$339$	36.0000	1.95525
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	8.00000	0.428845
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	5.00000	0.267261
$351$	8.00000	0.427008
$352$	−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	$0.378476\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	−6.00000	−0.317999
$357$	2.00000	0.105851
$358$	12.0000	0.634220
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	12.0000	0.630706
$363$	−14.0000	−0.734809
$364$	2.00000	0.104828
$365$	0	0
$366$	−24.0000	−1.25450
$367$	12.0000	0.626395	0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000	0.626395	0.313197	+	0.949688i	$0.398600\pi$
$368$	4.00000	0.208514
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	0	0
$377$	−8.00000	−0.412021
$378$	4.00000	0.205738
$379$	14.0000	0.719132	0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000	0.719132	0.359566	+	0.933120i	$0.382925\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	0	0
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	−26.0000	−1.32337
$387$	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	$0.224939\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	1.00000	0.0505076
$393$	28.0000	1.41241
$394$	8.00000	0.403034
$395$	0	0
$396$	−2.00000	−0.100504
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	20.0000	1.00251
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−16.0000	−0.798007
$403$	0	0
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−4.00000	−0.198517
$407$	−16.0000	−0.793091
$408$	−2.00000	−0.0990148
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	8.00000	0.394132
$413$	−4.00000	−0.196827
$414$	4.00000	0.196589
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	28.0000	1.37117
$418$	0	0
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	−10.0000	−0.486792
$423$	0	0
$424$	2.00000	0.0971286
$425$	5.00000	0.242536
$426$	24.0000	1.16280
$427$	12.0000	0.580721
$428$	10.0000	0.483368
$429$	8.00000	0.386244
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	−4.00000	−0.192450
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	0	0
$436$	4.00000	0.191565
$437$	0	0
$438$	−28.0000	−1.33789
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	2.00000	0.0951303
$443$	−40.0000	−1.90046	−0.950229	−	0.311553i	$-0.899151\pi$
$443$	−40.0000	−1.90046	−0.950229	+	0.311553i	$0.899151\pi$
$444$	16.0000	0.759326
$445$	0	0
$446$	−24.0000	−1.13643
$447$	12.0000	0.567581
$448$	−1.00000	−0.0472456
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	−5.00000	−0.235702
$451$	4.00000	0.188353
$452$	18.0000	0.846649
$453$	−16.0000	−0.751746
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−14.0000	−0.654177
$459$	4.00000	0.186704
$460$	0	0
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	4.00000	0.186097
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	−30.0000	−1.38972
$467$	16.0000	0.740392	0.370196	−	0.928954i	$-0.379291\pi$
$467$	16.0000	0.740392	0.370196	+	0.928954i	$0.379291\pi$
$468$	−2.00000	−0.0924500
$469$	8.00000	0.369406
$470$	0	0
$471$	−44.0000	−2.02741
$472$	4.00000	0.184115
$473$	0	0
$474$	24.0000	1.10236
$475$	0	0
$476$	1.00000	0.0458349
$477$	2.00000	0.0915737
$478$	−24.0000	−1.09773
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	−26.0000	−1.18427
$483$	−8.00000	−0.364013
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−10.0000	−0.453609
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−12.0000	−0.543214
$489$	−28.0000	−1.26620
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−4.00000	−0.180334
$493$	−4.00000	−0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.0000	−0.538274
$498$	8.00000	0.358489
$499$	30.0000	1.34298	0.671492	−	0.741012i	$-0.265654\pi$
$499$	30.0000	1.34298	0.671492	+	0.741012i	$0.265654\pi$
$500$	0	0
$501$	0	0
$502$	−28.0000	−1.24970
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	−8.00000	−0.355643
$507$	−18.0000	−0.799408
$508$	8.00000	0.354943
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	1.00000	0.0441942
$513$	0	0
$514$	30.0000	1.32324
$515$	0	0
$516$	0	0
$517$	0	0
$518$	−8.00000	−0.351500
$519$	0	0
$520$	0	0
$521$	−34.0000	−1.48957	−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000	−1.48957	−0.744784	+	0.667306i	$0.767447\pi$
$522$	4.00000	0.175075
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	14.0000	0.611593
$525$	10.0000	0.436436
$526$	−8.00000	−0.348817
$527$	0	0
$528$	−4.00000	−0.174078
$529$	−7.00000	−0.304348
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	4.00000	0.173259
$534$	−12.0000	−0.519291
$535$	0	0
$536$	−8.00000	−0.345547
$537$	24.0000	1.03568
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	0	0
$543$	24.0000	1.02994
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	4.00000	0.171184
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	6.00000	0.256307
$549$	−12.0000	−0.512148
$550$	10.0000	0.426401
$551$	0	0
$552$	8.00000	0.340503
$553$	−12.0000	−0.510292
$554$	−12.0000	−0.509831
$555$	0	0
$556$	14.0000	0.593732
$557$	34.0000	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	4.00000	0.168880
$562$	−10.0000	−0.421825
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	0	0
$566$	18.0000	0.756596
$567$	11.0000	0.461957
$568$	12.0000	0.503509
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	2.00000	0.0834784
$575$	−20.0000	−0.834058
$576$	1.00000	0.0416667
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	1.00000	0.0415945
$579$	−52.0000	−2.16105
$580$	0	0
$581$	−4.00000	−0.165948
$582$	12.0000	0.497416
$583$	−4.00000	−0.165663
$584$	−14.0000	−0.579324
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−8.00000	−0.330195	−0.165098	−	0.986277i	$-0.552794\pi$
$587$	−8.00000	−0.330195	−0.165098	+	0.986277i	$0.552794\pi$
$588$	2.00000	0.0824786
$589$	0	0
$590$	0	0
$591$	16.0000	0.658152
$592$	8.00000	0.328798
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	8.00000	0.328244
$595$	0	0
$596$	6.00000	0.245770
$597$	40.0000	1.63709
$598$	−8.00000	−0.327144
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	−10.0000	−0.408248
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−20.0000	−0.812444
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	0	0
$612$	−1.00000	−0.0404226
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	2.00000	0.0805823
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	16.0000	0.643614
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	−16.0000	−0.642058
$622$	0	0
$623$	6.00000	0.240385
$624$	−4.00000	−0.160128
$625$	25.0000	1.00000
$626$	6.00000	0.239808
$627$	0	0
$628$	−22.0000	−0.877896
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	12.0000	0.477334
$633$	−20.0000	−0.794929
$634$	24.0000	0.953162
$635$	0	0
$636$	4.00000	0.158610
$637$	−2.00000	−0.0792429
$638$	−8.00000	−0.316723
$639$	12.0000	0.474713
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	20.0000	0.789337
$643$	50.0000	1.97181	0.985904	−	0.167313i	$-0.0535092\pi$
$643$	50.0000	1.97181	0.985904	+	0.167313i	$0.0535092\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	−11.0000	−0.432121
$649$	−8.00000	−0.314027
$650$	10.0000	0.392232
$651$	0	0
$652$	−14.0000	−0.548282
$653$	12.0000	0.469596	0.234798	−	0.972044i	$-0.424557\pi$
$653$	12.0000	0.469596	0.234798	+	0.972044i	$0.424557\pi$
$654$	8.00000	0.312825
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	−14.0000	−0.546192
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−8.00000	−0.310929
$663$	4.00000	0.155347
$664$	4.00000	0.155230
$665$	0	0
$666$	8.00000	0.309994
$667$	16.0000	0.619522
$668$	0	0
$669$	−48.0000	−1.85579
$670$	0	0
$671$	24.0000	0.926510
$672$	−2.00000	−0.0771517
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	2.00000	0.0770371
$675$	20.0000	0.769800
$676$	−9.00000	−0.346154
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	36.0000	1.38257
$679$	−6.00000	−0.230259
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	22.0000	0.841807	0.420903	−	0.907106i	$-0.361713\pi$
$683$	22.0000	0.841807	0.420903	+	0.907106i	$0.361713\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	−28.0000	−1.06827
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−6.00000	−0.228251	−0.114125	−	0.993466i	$-0.536407\pi$
$691$	−6.00000	−0.228251	−0.114125	+	0.993466i	$0.536407\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	18.0000	0.683271
$695$	0	0
$696$	8.00000	0.303239
$697$	2.00000	0.0757554
$698$	14.0000	0.529908
$699$	−60.0000	−2.26941
$700$	5.00000	0.188982
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	8.00000	0.301941
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	14.0000	0.526897
$707$	10.0000	0.376089
$708$	8.00000	0.300658
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	−6.00000	−0.224860
$713$	0	0
$714$	2.00000	0.0748481
$715$	0	0
$716$	12.0000	0.448461
$717$	−48.0000	−1.79259
$718$	−32.0000	−1.19423
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	−19.0000	−0.707107
$723$	−52.0000	−1.93390
$724$	12.0000	0.445976
$725$	−20.0000	−0.742781
$726$	−14.0000	−0.519589
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	2.00000	0.0741249
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	−24.0000	−0.887066
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	12.0000	0.442928
$735$	0	0
$736$	4.00000	0.147442
$737$	16.0000	0.589368
$738$	−2.00000	−0.0736210
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	0	0
$745$	0	0
$746$	26.0000	0.951928
$747$	4.00000	0.146352
$748$	2.00000	0.0731272
$749$	−10.0000	−0.365392
$750$	0	0
$751$	36.0000	1.31366	0.656829	−	0.754039i	$-0.271897\pi$
$751$	36.0000	1.31366	0.656829	+	0.754039i	$0.271897\pi$
$752$	0	0
$753$	−56.0000	−2.04075
$754$	−8.00000	−0.291343
$755$	0	0
$756$	4.00000	0.145479
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	14.0000	0.508503
$759$	−16.0000	−0.580763
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	16.0000	0.579619
$763$	−4.00000	−0.144810
$764$	0	0
$765$	0	0
$766$	24.0000	0.867155
$767$	−8.00000	−0.288863
$768$	2.00000	0.0721688
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	60.0000	2.16085
$772$	−26.0000	−0.935760
$773$	34.0000	1.22290	0.611448	−	0.791285i	$-0.290588\pi$
$773$	34.0000	1.22290	0.611448	+	0.791285i	$0.290588\pi$
$774$	0	0
$775$	0	0
$776$	6.00000	0.215387
$777$	−16.0000	−0.573997
$778$	30.0000	1.07555
$779$	0	0
$780$	0	0
$781$	−24.0000	−0.858788
$782$	−4.00000	−0.143040
$783$	−16.0000	−0.571793
$784$	1.00000	0.0357143
$785$	0	0
$786$	28.0000	0.998727
$787$	−26.0000	−0.926800	−0.463400	−	0.886149i	$-0.653371\pi$
$787$	−26.0000	−0.926800	−0.463400	+	0.886149i	$0.653371\pi$
$788$	8.00000	0.284988
$789$	−16.0000	−0.569615
$790$	0	0
$791$	−18.0000	−0.640006
$792$	−2.00000	−0.0710669
$793$	24.0000	0.852265
$794$	20.0000	0.709773
$795$	0	0
$796$	20.0000	0.708881
$797$	46.0000	1.62940	0.814702	−	0.579880i	$-0.196901\pi$
$797$	46.0000	1.62940	0.814702	+	0.579880i	$0.196901\pi$
$798$	0	0
$799$	0	0
$800$	−5.00000	−0.176777
$801$	−6.00000	−0.212000
$802$	−18.0000	−0.635602
$803$	28.0000	0.988099
$804$	−16.0000	−0.564276
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−10.0000	−0.351799
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	34.0000	1.19390	0.596951	−	0.802278i	$-0.296379\pi$
$811$	34.0000	1.19390	0.596951	+	0.802278i	$0.296379\pi$
$812$	−4.00000	−0.140372
$813$	0	0
$814$	−16.0000	−0.560800
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	−10.0000	−0.349642
$819$	2.00000	0.0698857
$820$	0	0
$821$	−8.00000	−0.279202	−0.139601	−	0.990208i	$-0.544582\pi$
$821$	−8.00000	−0.279202	−0.139601	+	0.990208i	$0.544582\pi$
$822$	12.0000	0.418548
$823$	−12.0000	−0.418294	−0.209147	−	0.977884i	$-0.567069\pi$
$823$	−12.0000	−0.418294	−0.209147	+	0.977884i	$0.567069\pi$
$824$	8.00000	0.278693
$825$	20.0000	0.696311
$826$	−4.00000	−0.139178
$827$	−26.0000	−0.904109	−0.452054	−	0.891990i	$-0.649309\pi$
$827$	−26.0000	−0.904109	−0.452054	+	0.891990i	$0.649309\pi$
$828$	4.00000	0.139010
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	−24.0000	−0.832551
$832$	−2.00000	−0.0693375
$833$	−1.00000	−0.0346479
$834$	28.0000	0.969561
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−26.0000	−0.898155
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	18.0000	0.620321
$843$	−20.0000	−0.688837
$844$	−10.0000	−0.344214
$845$	0	0
$846$	0	0
$847$	7.00000	0.240523
$848$	2.00000	0.0686803
$849$	36.0000	1.23552
$850$	5.00000	0.171499
$851$	32.0000	1.09695
$852$	24.0000	0.822226
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	12.0000	0.410632
$855$	0	0
$856$	10.0000	0.341793
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	8.00000	0.273115
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	−20.0000	−0.681203
$863$	56.0000	1.90626	0.953131	−	0.302558i	$-0.0978405\pi$
$863$	56.0000	1.90626	0.953131	+	0.302558i	$0.0978405\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	26.0000	0.883516
$867$	2.00000	0.0679236
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	16.0000	0.542139
$872$	4.00000	0.135457
$873$	6.00000	0.203069
$874$	0	0
$875$	0	0
$876$	−28.0000	−0.946032
$877$	20.0000	0.675352	0.337676	−	0.941262i	$-0.390359\pi$
$877$	20.0000	0.675352	0.337676	+	0.941262i	$0.390359\pi$
$878$	−20.0000	−0.674967
$879$	−12.0000	−0.404750
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	1.00000	0.0336718
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	−40.0000	−1.34383
$887$	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
$889$	−8.00000	−0.268311
$890$	0	0
$891$	22.0000	0.737028
$892$	−24.0000	−0.803579
$893$	0	0
$894$	12.0000	0.401340
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	−16.0000	−0.534224
$898$	−10.0000	−0.333704
$899$	0	0
$900$	−5.00000	−0.166667
$901$	−2.00000	−0.0666297
$902$	4.00000	0.133185
$903$	0	0
$904$	18.0000	0.598671
$905$	0	0
$906$	−16.0000	−0.531564
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	6.00000	0.199117
$909$	−10.0000	−0.331679
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	−22.0000	−0.727695
$915$	0	0
$916$	−14.0000	−0.462573
$917$	−14.0000	−0.462321
$918$	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	$0.585007\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	−34.0000	−1.11973
$923$	−24.0000	−0.789970
$924$	4.00000	0.131590
$925$	−40.0000	−1.31519
$926$	8.00000	0.262896
$927$	8.00000	0.262754
$928$	4.00000	0.131306
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	0	0
$932$	−30.0000	−0.982683
$933$	0	0
$934$	16.0000	0.523536
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	8.00000	0.261209
$939$	12.0000	0.391605
$940$	0	0
$941$	4.00000	0.130396	0.0651981	−	0.997872i	$-0.479232\pi$
$941$	4.00000	0.130396	0.0651981	+	0.997872i	$0.479232\pi$
$942$	−44.0000	−1.43360
$943$	−8.00000	−0.260516
$944$	4.00000	0.130189
$945$	0	0
$946$	0	0
$947$	−38.0000	−1.23483	−0.617417	−	0.786636i	$-0.711821\pi$
$947$	−38.0000	−1.23483	−0.617417	+	0.786636i	$0.711821\pi$
$948$	24.0000	0.779484
$949$	28.0000	0.908918
$950$	0	0
$951$	48.0000	1.55651
$952$	1.00000	0.0324102
$953$	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
$957$	−16.0000	−0.517207
$958$	−36.0000	−1.16311
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−16.0000	−0.515861
$963$	10.0000	0.322245
$964$	−26.0000	−0.837404
$965$	0	0
$966$	−8.00000	−0.257396
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	0	0
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	−10.0000	−0.320750
$973$	−14.0000	−0.448819
$974$	−4.00000	−0.128168
$975$	20.0000	0.640513
$976$	−12.0000	−0.384111
$977$	−22.0000	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$977$	−22.0000	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$978$	−28.0000	−0.895341
$979$	12.0000	0.383522
$980$	0	0
$981$	4.00000	0.127710
$982$	20.0000	0.638226
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	−4.00000	−0.127515
$985$	0	0
$986$	−4.00000	−0.127386
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	12.0000	0.381193	0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000	0.381193	0.190596	+	0.981669i	$0.438958\pi$
$992$	0	0
$993$	−16.0000	−0.507745
$994$	−12.0000	−0.380617
$995$	0	0
$996$	8.00000	0.253490
$997$	−20.0000	−0.633406	−0.316703	−	0.948525i	$-0.602576\pi$
$997$	−20.0000	−0.633406	−0.316703	+	0.948525i	$0.602576\pi$
$998$	30.0000	0.949633
$999$	−32.0000	−1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	238.2.a.e.1.1	✓	1
3.2	odd	2		2142.2.a.d.1.1		1
4.3	odd	2		1904.2.a.a.1.1		1
5.4	even	2		5950.2.a.b.1.1		1
7.6	odd	2		1666.2.a.i.1.1		1
8.3	odd	2		7616.2.a.j.1.1		1
8.5	even	2		7616.2.a.c.1.1		1
17.16	even	2		4046.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
238.2.a.e.1.1	✓	1	1.1	even	1	trivial
1666.2.a.i.1.1		1	7.6	odd	2
1904.2.a.a.1.1		1	4.3	odd	2
2142.2.a.d.1.1		1	3.2	odd	2
4046.2.a.i.1.1		1	17.16	even	2
5950.2.a.b.1.1		1	5.4	even	2
7616.2.a.c.1.1		1	8.5	even	2
7616.2.a.j.1.1		1	8.3	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:	\( N \)	\(=\)	\( 238 = 2 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	238.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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