LMFDB - Embedded newform 114.2.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [114,2,Mod(1,114)] 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("114.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(114, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 114 = 2 \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	114.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-1] 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:	$0.910294583043$
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	$\chi$	$=$	114.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +2.00000 q^{20} -4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} -2.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} -1.00000 q^{38} -2.00000 q^{39} +2.00000 q^{40} +10.0000 q^{41} +4.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} -4.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} -8.00000 q^{55} +1.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} -2.00000 q^{60} +14.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +4.00000 q^{66} -12.0000 q^{67} -6.00000 q^{68} +4.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} +10.0000 q^{74} +1.00000 q^{75} -1.00000 q^{76} -2.00000 q^{78} -4.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +12.0000 q^{83} -12.0000 q^{85} +4.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} -4.00000 q^{92} -4.00000 q^{93} -4.00000 q^{94} -2.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} -7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.00000	0.447214
$21$	0	0
$22$	−4.00000	−0.852803
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	−2.00000	−0.365148
$31$	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
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	−1.00000	−0.162221
$39$	−2.00000	−0.320256
$40$	2.00000	0.316228
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−4.00000	−0.603023
$45$	2.00000	0.298142
$46$	−4.00000	−0.589768
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	−1.00000	−0.141421
$51$	6.00000	0.840168
$52$	2.00000	0.277350
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	−1.00000	−0.136083
$55$	−8.00000	−1.07872
$56$	0	0
$57$	1.00000	0.132453
$58$	−2.00000	−0.262613
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	−2.00000	−0.258199
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	4.00000	0.492366
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−6.00000	−0.727607
$69$	4.00000	0.481543
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	1.00000	0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	10.0000	1.16248
$75$	1.00000	0.115470
$76$	−1.00000	−0.114708
$77$	0	0
$78$	−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	$0.572244\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	4.00000	0.431331
$87$	2.00000	0.214423
$88$	−4.00000	−0.426401
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	−4.00000	−0.417029
$93$	−4.00000	−0.414781
$94$	−4.00000	−0.412568
$95$	−2.00000	−0.205196
$96$	−1.00000	−0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−7.00000	−0.707107
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	6.00000	0.594089
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−10.0000	−0.971286
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	−1.00000	−0.0962250
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	−8.00000	−0.762770
$111$	−10.0000	−0.949158
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	1.00000	0.0936586
$115$	−8.00000	−0.746004
$116$	−2.00000	−0.185695
$117$	2.00000	0.184900
$118$	12.0000	1.10469
$119$	0	0
$120$	−2.00000	−0.182574
$121$	5.00000	0.454545
$122$	14.0000	1.26750
$123$	−10.0000	−0.901670
$124$	4.00000	0.359211
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	4.00000	0.350823
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	−12.0000	−1.03664
$135$	−2.00000	−0.172133
$136$	−6.00000	−0.514496
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	4.00000	0.340503
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	8.00000	0.671345
$143$	−8.00000	−0.668994
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	−6.00000	−0.496564
$147$	7.00000	0.577350
$148$	10.0000	0.821995
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	1.00000	0.0816497
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	−1.00000	−0.0811107
$153$	−6.00000	−0.485071
$154$	0	0
$155$	8.00000	0.642575
$156$	−2.00000	−0.160128
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	−4.00000	−0.318223
$159$	10.0000	0.793052
$160$	2.00000	0.158114
$161$	0	0
$162$	1.00000	0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	10.0000	0.780869
$165$	8.00000	0.622799
$166$	12.0000	0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−12.0000	−0.920358
$171$	−1.00000	−0.0764719
$172$	4.00000	0.304997
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	−4.00000	−0.301511
$177$	−12.0000	−0.901975
$178$	−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	$0.351972\pi$
$180$	2.00000	0.149071
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	−14.0000	−1.03491
$184$	−4.00000	−0.294884
$185$	20.0000	1.47043
$186$	−4.00000	−0.293294
$187$	24.0000	1.75505
$188$	−4.00000	−0.291730
$189$	0	0
$190$	−2.00000	−0.145095
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	−1.00000	−0.0721688
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	10.0000	0.717958
$195$	−4.00000	−0.286446
$196$	−7.00000	−0.500000
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	−4.00000	−0.284268
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−1.00000	−0.0707107
$201$	12.0000	0.846415
$202$	2.00000	0.140720
$203$	0	0
$204$	6.00000	0.420084
$205$	20.0000	1.39686
$206$	−12.0000	−0.836080
$207$	−4.00000	−0.278019
$208$	2.00000	0.138675
$209$	4.00000	0.276686
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−10.0000	−0.686803
$213$	−8.00000	−0.548151
$214$	−4.00000	−0.273434
$215$	8.00000	0.545595
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−6.00000	−0.406371
$219$	6.00000	0.405442
$220$	−8.00000	−0.539360
$221$	−12.0000	−0.807207
$222$	−10.0000	−0.671156
$223$	−28.0000	−1.87502	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−28.0000	−1.87502	−0.937509	+	0.347960i	$0.886874\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	2.00000	0.133038
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	1.00000	0.0662266
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	−2.00000	−0.131306
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	2.00000	0.130744
$235$	−8.00000	−0.521862
$236$	12.0000	0.781133
$237$	4.00000	0.259828
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	−2.00000	−0.129099
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	14.0000	0.896258
$245$	−14.0000	−0.894427
$246$	−10.0000	−0.637577
$247$	−2.00000	−0.127257
$248$	4.00000	0.254000
$249$	−12.0000	−0.760469
$250$	−12.0000	−0.758947
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	−12.0000	−0.752947
$255$	12.0000	0.751469
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	4.00000	0.248069
$261$	−2.00000	−0.123797
$262$	12.0000	0.741362
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	4.00000	0.246183
$265$	−20.0000	−1.22859
$266$	0	0
$267$	6.00000	0.367194
$268$	−12.0000	−0.733017
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	−2.00000	−0.121716
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−14.0000	−0.845771
$275$	4.00000	0.241209
$276$	4.00000	0.240772
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	12.0000	0.719712
$279$	4.00000	0.239474
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	4.00000	0.238197
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	8.00000	0.474713
$285$	2.00000	0.118470
$286$	−8.00000	−0.473050
$287$	0	0
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	−4.00000	−0.234888
$291$	−10.0000	−0.586210
$292$	−6.00000	−0.351123
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	7.00000	0.408248
$295$	24.0000	1.39733
$296$	10.0000	0.581238
$297$	4.00000	0.232104
$298$	−6.00000	−0.347571
$299$	−8.00000	−0.462652
$300$	1.00000	0.0577350
$301$	0	0
$302$	20.0000	1.15087
$303$	−2.00000	−0.114897
$304$	−1.00000	−0.0573539
$305$	28.0000	1.60328
$306$	−6.00000	−0.342997
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	8.00000	0.454369
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	−2.00000	−0.113228
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	−4.00000	−0.225018
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	10.0000	0.560772
$319$	8.00000	0.447914
$320$	2.00000	0.111803
$321$	4.00000	0.223258
$322$	0	0
$323$	6.00000	0.333849
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	20.0000	1.10770
$327$	6.00000	0.331801
$328$	10.0000	0.552158
$329$	0	0
$330$	8.00000	0.440386
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	12.0000	0.658586
$333$	10.0000	0.547997
$334$	0	0
$335$	−24.0000	−1.31126
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	−9.00000	−0.489535
$339$	−2.00000	−0.108625
$340$	−12.0000	−0.650791
$341$	−16.0000	−0.866449
$342$	−1.00000	−0.0540738
$343$	0	0
$344$	4.00000	0.215666
$345$	8.00000	0.430706
$346$	6.00000	0.322562
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	2.00000	0.107211
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	−4.00000	−0.213201
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	−12.0000	−0.637793
$355$	16.0000	0.849192
$356$	−6.00000	−0.317999
$357$	0	0
$358$	12.0000	0.634220
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	2.00000	0.105409
$361$	1.00000	0.0526316
$362$	−14.0000	−0.735824
$363$	−5.00000	−0.262432
$364$	0	0
$365$	−12.0000	−0.628109
$366$	−14.0000	−0.731792
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−4.00000	−0.208514
$369$	10.0000	0.520579
$370$	20.0000	1.03975
$371$	0	0
$372$	−4.00000	−0.207390
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	24.0000	1.24101
$375$	12.0000	0.619677
$376$	−4.00000	−0.206284
$377$	−4.00000	−0.206010
$378$	0	0
$379$	−36.0000	−1.84920	−0.924598	−	0.380945i	$-0.875599\pi$
$379$	−36.0000	−1.84920	−0.924598	+	0.380945i	$0.875599\pi$
$380$	−2.00000	−0.102598
$381$	12.0000	0.614779
$382$	4.00000	0.204658
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	4.00000	0.203331
$388$	10.0000	0.507673
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	−4.00000	−0.202548
$391$	24.0000	1.21373
$392$	−7.00000	−0.353553
$393$	−12.0000	−0.605320
$394$	−22.0000	−1.10834
$395$	−8.00000	−0.402524
$396$	−4.00000	−0.201008
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	12.0000	0.598506
$403$	8.00000	0.398508
$404$	2.00000	0.0995037
$405$	2.00000	0.0993808
$406$	0	0
$407$	−40.0000	−1.98273
$408$	6.00000	0.297044
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	20.0000	0.987730
$411$	14.0000	0.690569
$412$	−12.0000	−0.591198
$413$	0	0
$414$	−4.00000	−0.196589
$415$	24.0000	1.17811
$416$	2.00000	0.0980581
$417$	−12.0000	−0.587643
$418$	4.00000	0.195646
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	12.0000	0.584151
$423$	−4.00000	−0.194487
$424$	−10.0000	−0.485643
$425$	6.00000	0.291043
$426$	−8.00000	−0.387601
$427$	0	0
$428$	−4.00000	−0.193347
$429$	8.00000	0.386244
$430$	8.00000	0.385794
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−1.00000	−0.0481125
$433$	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
$435$	4.00000	0.191785
$436$	−6.00000	−0.287348
$437$	4.00000	0.191346
$438$	6.00000	0.286691
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	−8.00000	−0.381385
$441$	−7.00000	−0.333333
$442$	−12.0000	−0.570782
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	−10.0000	−0.474579
$445$	−12.0000	−0.568855
$446$	−28.0000	−1.32584
$447$	6.00000	0.283790
$448$	0	0
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	−1.00000	−0.0471405
$451$	−40.0000	−1.88353
$452$	2.00000	0.0940721
$453$	−20.0000	−0.939682
$454$	28.0000	1.31411
$455$	0	0
$456$	1.00000	0.0468293
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	−10.0000	−0.467269
$459$	6.00000	0.280056
$460$	−8.00000	−0.373002
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−2.00000	−0.0928477
$465$	−8.00000	−0.370991
$466$	−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	$0.653135\pi$
$468$	2.00000	0.0924500
$469$	0	0
$470$	−8.00000	−0.369012
$471$	−22.0000	−1.01371
$472$	12.0000	0.552345
$473$	−16.0000	−0.735681
$474$	4.00000	0.183726
$475$	1.00000	0.0458831
$476$	0	0
$477$	−10.0000	−0.457869
$478$	12.0000	0.548867
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	−2.00000	−0.0912871
$481$	20.0000	0.911922
$482$	10.0000	0.455488
$483$	0	0
$484$	5.00000	0.227273
$485$	20.0000	0.908153
$486$	−1.00000	−0.0453609
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	14.0000	0.633750
$489$	−20.0000	−0.904431
$490$	−14.0000	−0.632456
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−10.0000	−0.450835
$493$	12.0000	0.540453
$494$	−2.00000	−0.0899843
$495$	−8.00000	−0.359573
$496$	4.00000	0.179605
$497$	0	0
$498$	−12.0000	−0.537733
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	−28.0000	−1.24970
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	16.0000	0.711287
$507$	9.00000	0.399704
$508$	−12.0000	−0.532414
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	12.0000	0.531369
$511$	0	0
$512$	1.00000	0.0441942
$513$	1.00000	0.0441511
$514$	2.00000	0.0882162
$515$	−24.0000	−1.05757
$516$	−4.00000	−0.176090
$517$	16.0000	0.703679
$518$	0	0
$519$	−6.00000	−0.263371
$520$	4.00000	0.175412
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	−2.00000	−0.0875376
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−12.0000	−0.523225
$527$	−24.0000	−1.04546
$528$	4.00000	0.174078
$529$	−7.00000	−0.304348
$530$	−20.0000	−0.868744
$531$	12.0000	0.520756
$532$	0	0
$533$	20.0000	0.866296
$534$	6.00000	0.259645
$535$	−8.00000	−0.345870
$536$	−12.0000	−0.518321
$537$	−12.0000	−0.517838
$538$	6.00000	0.258678
$539$	28.0000	1.20605
$540$	−2.00000	−0.0860663
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	−8.00000	−0.343629
$543$	14.0000	0.600798
$544$	−6.00000	−0.257248
$545$	−12.0000	−0.514024
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	−14.0000	−0.598050
$549$	14.0000	0.597505
$550$	4.00000	0.170561
$551$	2.00000	0.0852029
$552$	4.00000	0.170251
$553$	0	0
$554$	−26.0000	−1.10463
$555$	−20.0000	−0.848953
$556$	12.0000	0.508913
$557$	34.0000	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$558$	4.00000	0.169334
$559$	8.00000	0.338364
$560$	0	0
$561$	−24.0000	−1.01328
$562$	10.0000	0.421825
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	4.00000	0.168430
$565$	4.00000	0.168281
$566$	12.0000	0.504398
$567$	0	0
$568$	8.00000	0.335673
$569$	−38.0000	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$569$	−38.0000	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$570$	2.00000	0.0837708
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	−8.00000	−0.334497
$573$	−4.00000	−0.167102
$574$	0	0
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	19.0000	0.790296
$579$	6.00000	0.249351
$580$	−4.00000	−0.166091
$581$	0	0
$582$	−10.0000	−0.414513
$583$	40.0000	1.65663
$584$	−6.00000	−0.248282
$585$	4.00000	0.165380
$586$	−18.0000	−0.743573
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	7.00000	0.288675
$589$	−4.00000	−0.164817
$590$	24.0000	0.988064
$591$	22.0000	0.904959
$592$	10.0000	0.410997
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−6.00000	−0.245770
$597$	16.0000	0.654836
$598$	−8.00000	−0.327144
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	1.00000	0.0408248
$601$	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
$603$	−12.0000	−0.488678
$604$	20.0000	0.813788
$605$	10.0000	0.406558
$606$	−2.00000	−0.0812444
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	28.0000	1.13369
$611$	−8.00000	−0.323645
$612$	−6.00000	−0.242536
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−12.0000	−0.484281
$615$	−20.0000	−0.806478
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	12.0000	0.482711
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	8.00000	0.321288
$621$	4.00000	0.160514
$622$	4.00000	0.160385
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	−19.0000	−0.760000
$626$	−22.0000	−0.879297
$627$	−4.00000	−0.159745
$628$	22.0000	0.877896
$629$	−60.0000	−2.39236
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	−4.00000	−0.159111
$633$	−12.0000	−0.476957
$634$	6.00000	0.238290
$635$	−24.0000	−0.952411
$636$	10.0000	0.396526
$637$	−14.0000	−0.554700
$638$	8.00000	0.316723
$639$	8.00000	0.316475
$640$	2.00000	0.0790569
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	4.00000	0.157867
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	6.00000	0.236067
$647$	−4.00000	−0.157256	−0.0786281	−	0.996904i	$-0.525054\pi$
$647$	−4.00000	−0.157256	−0.0786281	+	0.996904i	$0.525054\pi$
$648$	1.00000	0.0392837
$649$	−48.0000	−1.88416
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	20.0000	0.783260
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	6.00000	0.234619
$655$	24.0000	0.937758
$656$	10.0000	0.390434
$657$	−6.00000	−0.234082
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	8.00000	0.311400
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	−4.00000	−0.155464
$663$	12.0000	0.466041
$664$	12.0000	0.465690
$665$	0	0
$666$	10.0000	0.387492
$667$	8.00000	0.309761
$668$	0	0
$669$	28.0000	1.08254
$670$	−24.0000	−0.927201
$671$	−56.0000	−2.16186
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−14.0000	−0.539260
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	−2.00000	−0.0768095
$679$	0	0
$680$	−12.0000	−0.460179
$681$	−28.0000	−1.07296
$682$	−16.0000	−0.612672
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	−1.00000	−0.0382360
$685$	−28.0000	−1.06983
$686$	0	0
$687$	10.0000	0.381524
$688$	4.00000	0.152499
$689$	−20.0000	−0.761939
$690$	8.00000	0.304555
$691$	−36.0000	−1.36950	−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000	−1.36950	−0.684752	+	0.728776i	$0.740090\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−20.0000	−0.759190
$695$	24.0000	0.910372
$696$	2.00000	0.0758098
$697$	−60.0000	−2.27266
$698$	−26.0000	−0.984115
$699$	6.00000	0.226941
$700$	0	0
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	−2.00000	−0.0754851
$703$	−10.0000	−0.377157
$704$	−4.00000	−0.150756
$705$	8.00000	0.301297
$706$	18.0000	0.677439
$707$	0	0
$708$	−12.0000	−0.450988
$709$	−42.0000	−1.57734	−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000	−1.57734	−0.788672	+	0.614815i	$0.789231\pi$
$710$	16.0000	0.600469
$711$	−4.00000	−0.150012
$712$	−6.00000	−0.224860
$713$	−16.0000	−0.599205
$714$	0	0
$715$	−16.0000	−0.598366
$716$	12.0000	0.448461
$717$	−12.0000	−0.448148
$718$	12.0000	0.447836
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	1.00000	0.0372161
$723$	−10.0000	−0.371904
$724$	−14.0000	−0.520306
$725$	2.00000	0.0742781
$726$	−5.00000	−0.185567
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−12.0000	−0.444140
$731$	−24.0000	−0.887672
$732$	−14.0000	−0.517455
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	−8.00000	−0.295285
$735$	14.0000	0.516398
$736$	−4.00000	−0.147442
$737$	48.0000	1.76810
$738$	10.0000	0.368105
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	20.0000	0.735215
$741$	2.00000	0.0734718
$742$	0	0
$743$	32.0000	1.17397	0.586983	−	0.809599i	$-0.300316\pi$
$743$	32.0000	1.17397	0.586983	+	0.809599i	$0.300316\pi$
$744$	−4.00000	−0.146647
$745$	−12.0000	−0.439646
$746$	26.0000	0.951928
$747$	12.0000	0.439057
$748$	24.0000	0.877527
$749$	0	0
$750$	12.0000	0.438178
$751$	44.0000	1.60558	0.802791	−	0.596260i	$-0.203347\pi$
$751$	44.0000	1.60558	0.802791	+	0.596260i	$0.203347\pi$
$752$	−4.00000	−0.145865
$753$	28.0000	1.02038
$754$	−4.00000	−0.145671
$755$	40.0000	1.45575
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−36.0000	−1.30758
$759$	−16.0000	−0.580763
$760$	−2.00000	−0.0725476
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	12.0000	0.434714
$763$	0	0
$764$	4.00000	0.144715
$765$	−12.0000	−0.433861
$766$	16.0000	0.578103
$767$	24.0000	0.866590
$768$	−1.00000	−0.0360844
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	−6.00000	−0.215945
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	4.00000	0.143777
$775$	−4.00000	−0.143684
$776$	10.0000	0.358979
$777$	0	0
$778$	18.0000	0.645331
$779$	−10.0000	−0.358287
$780$	−4.00000	−0.143223
$781$	−32.0000	−1.14505
$782$	24.0000	0.858238
$783$	2.00000	0.0714742
$784$	−7.00000	−0.250000
$785$	44.0000	1.57043
$786$	−12.0000	−0.428026
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	−22.0000	−0.783718
$789$	12.0000	0.427211
$790$	−8.00000	−0.284627
$791$	0	0
$792$	−4.00000	−0.142134
$793$	28.0000	0.994309
$794$	−10.0000	−0.354887
$795$	20.0000	0.709327
$796$	−16.0000	−0.567105
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	−1.00000	−0.0353553
$801$	−6.00000	−0.212000
$802$	−14.0000	−0.494357
$803$	24.0000	0.846942
$804$	12.0000	0.423207
$805$	0	0
$806$	8.00000	0.281788
$807$	−6.00000	−0.211210
$808$	2.00000	0.0703598
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	2.00000	0.0702728
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	−40.0000	−1.40200
$815$	40.0000	1.40114
$816$	6.00000	0.210042
$817$	−4.00000	−0.139942
$818$	−14.0000	−0.489499
$819$	0	0
$820$	20.0000	0.698430
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	14.0000	0.488306
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	−12.0000	−0.418040
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	−4.00000	−0.139010
$829$	42.0000	1.45872	0.729360	−	0.684130i	$-0.239818\pi$
$829$	42.0000	1.45872	0.729360	+	0.684130i	$0.239818\pi$
$830$	24.0000	0.833052
$831$	26.0000	0.901930
$832$	2.00000	0.0693375
$833$	42.0000	1.45521
$834$	−12.0000	−0.415526
$835$	0	0
$836$	4.00000	0.138343
$837$	−4.00000	−0.138260
$838$	−12.0000	−0.414533
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	26.0000	0.896019
$843$	−10.0000	−0.344418
$844$	12.0000	0.413057
$845$	−18.0000	−0.619219
$846$	−4.00000	−0.137523
$847$	0	0
$848$	−10.0000	−0.343401
$849$	−12.0000	−0.411839
$850$	6.00000	0.205798
$851$	−40.0000	−1.37118
$852$	−8.00000	−0.274075
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	−4.00000	−0.136717
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	8.00000	0.273115
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−24.0000	−0.817443
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−1.00000	−0.0340207
$865$	12.0000	0.408012
$866$	26.0000	0.883516
$867$	−19.0000	−0.645274
$868$	0	0
$869$	16.0000	0.542763
$870$	4.00000	0.135613
$871$	−24.0000	−0.813209
$872$	−6.00000	−0.203186
$873$	10.0000	0.338449
$874$	4.00000	0.135302
$875$	0	0
$876$	6.00000	0.202721
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	−4.00000	−0.134993
$879$	18.0000	0.607125
$880$	−8.00000	−0.269680
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	−7.00000	−0.235702
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	−12.0000	−0.403604
$885$	−24.0000	−0.806751
$886$	−20.0000	−0.671913
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	−10.0000	−0.335578
$889$	0	0
$890$	−12.0000	−0.402241
$891$	−4.00000	−0.134005
$892$	−28.0000	−0.937509
$893$	4.00000	0.133855
$894$	6.00000	0.200670
$895$	24.0000	0.802232
$896$	0	0
$897$	8.00000	0.267112
$898$	34.0000	1.13459
$899$	−8.00000	−0.266815
$900$	−1.00000	−0.0333333
$901$	60.0000	1.99889
$902$	−40.0000	−1.33185
$903$	0	0
$904$	2.00000	0.0665190
$905$	−28.0000	−0.930751
$906$	−20.0000	−0.664455
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	28.0000	0.929213
$909$	2.00000	0.0663358
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	1.00000	0.0331133
$913$	−48.0000	−1.58857
$914$	26.0000	0.860004
$915$	−28.0000	−0.925651
$916$	−10.0000	−0.330409
$917$	0	0
$918$	6.00000	0.198030
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	−8.00000	−0.263752
$921$	12.0000	0.395413
$922$	2.00000	0.0658665
$923$	16.0000	0.526646
$924$	0	0
$925$	−10.0000	−0.328798
$926$	−8.00000	−0.262896
$927$	−12.0000	−0.394132
$928$	−2.00000	−0.0656532
$929$	−22.0000	−0.721797	−0.360898	−	0.932605i	$-0.617530\pi$
$929$	−22.0000	−0.721797	−0.360898	+	0.932605i	$0.617530\pi$
$930$	−8.00000	−0.262330
$931$	7.00000	0.229416
$932$	−6.00000	−0.196537
$933$	−4.00000	−0.130954
$934$	−20.0000	−0.654420
$935$	48.0000	1.56977
$936$	2.00000	0.0653720
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	22.0000	0.717943
$940$	−8.00000	−0.260931
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	−22.0000	−0.716799
$943$	−40.0000	−1.30258
$944$	12.0000	0.390567
$945$	0	0
$946$	−16.0000	−0.520205
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	4.00000	0.129914
$949$	−12.0000	−0.389536
$950$	1.00000	0.0324443
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	−10.0000	−0.323762
$955$	8.00000	0.258874
$956$	12.0000	0.388108
$957$	−8.00000	−0.258603
$958$	4.00000	0.129234
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	−15.0000	−0.483871
$962$	20.0000	0.644826
$963$	−4.00000	−0.128898
$964$	10.0000	0.322078
$965$	−12.0000	−0.386294
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	5.00000	0.160706
$969$	−6.00000	−0.192748
$970$	20.0000	0.642161
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	4.00000	0.128168
$975$	2.00000	0.0640513
$976$	14.0000	0.448129
$977$	−46.0000	−1.47167	−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000	−1.47167	−0.735835	+	0.677161i	$0.763210\pi$
$978$	−20.0000	−0.639529
$979$	24.0000	0.767043
$980$	−14.0000	−0.447214
$981$	−6.00000	−0.191565
$982$	20.0000	0.638226
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−10.0000	−0.318788
$985$	−44.0000	−1.40196
$986$	12.0000	0.382158
$987$	0	0
$988$	−2.00000	−0.0636285
$989$	−16.0000	−0.508770
$990$	−8.00000	−0.254257
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	4.00000	0.127000
$993$	4.00000	0.126936
$994$	0	0
$995$	−32.0000	−1.01447
$996$	−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	$0.510083\pi$
$998$	−20.0000	−0.633089
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.2.a.b.1.1	✓	1
3.2	odd	2		342.2.a.b.1.1		1
4.3	odd	2		912.2.a.k.1.1		1
5.2	odd	4		2850.2.d.b.799.2		2
5.3	odd	4		2850.2.d.b.799.1		2
5.4	even	2		2850.2.a.j.1.1		1
7.6	odd	2		5586.2.a.y.1.1		1
8.3	odd	2		3648.2.a.c.1.1		1
8.5	even	2		3648.2.a.x.1.1		1
12.11	even	2		2736.2.a.d.1.1		1
15.14	odd	2		8550.2.a.ba.1.1		1
19.18	odd	2		2166.2.a.d.1.1		1
57.56	even	2		6498.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.a.b.1.1	✓	1	1.1	even	1	trivial
342.2.a.b.1.1		1	3.2	odd	2
912.2.a.k.1.1		1	4.3	odd	2
2166.2.a.d.1.1		1	19.18	odd	2
2736.2.a.d.1.1		1	12.11	even	2
2850.2.a.j.1.1		1	5.4	even	2
2850.2.d.b.799.1		2	5.3	odd	4
2850.2.d.b.799.2		2	5.2	odd	4
3648.2.a.c.1.1		1	8.3	odd	2
3648.2.a.x.1.1		1	8.5	even	2
5586.2.a.y.1.1		1	7.6	odd	2
6498.2.a.p.1.1		1	57.56	even	2
8550.2.a.ba.1.1		1	15.14	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:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	114.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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