LMFDB - Embedded newform 930.2.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [930,2,Mod(1,930)] 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("930.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(930, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,-1,1,1,1,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.42608738798$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	930.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +1.00000 q^{20} +4.00000 q^{21} -2.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -8.00000 q^{29} +1.00000 q^{30} +1.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} -4.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{39} -1.00000 q^{40} +6.00000 q^{41} -4.00000 q^{42} +4.00000 q^{43} +2.00000 q^{44} +1.00000 q^{45} +6.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} +4.00000 q^{56} +8.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} -8.00000 q^{61} -1.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +2.00000 q^{66} -4.00000 q^{67} +6.00000 q^{69} +4.00000 q^{70} -8.00000 q^{71} -1.00000 q^{72} +4.00000 q^{73} +2.00000 q^{74} -1.00000 q^{75} -8.00000 q^{77} +2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -16.0000 q^{83} +4.00000 q^{84} -4.00000 q^{86} +8.00000 q^{87} -2.00000 q^{88} -2.00000 q^{89} -1.00000 q^{90} -8.00000 q^{91} -6.00000 q^{92} -1.00000 q^{93} +12.0000 q^{94} +1.00000 q^{96} -2.00000 q^{97} -9.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−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
$10$	−1.00000	−0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−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	$0.410544\pi$
$14$	4.00000	1.06904
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	1.00000	0.223607
$21$	4.00000	0.872872
$22$	−2.00000	−0.426401
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	−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	$0.766494\pi$
$30$	1.00000	0.182574
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−4.00000	−0.676123
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	−1.00000	−0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−4.00000	−0.617213
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	2.00000	0.301511
$45$	1.00000	0.149071
$46$	6.00000	0.884652
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	−1.00000	−0.144338
$49$	9.00000	1.28571
$50$	−1.00000	−0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	2.00000	0.269680
$56$	4.00000	0.534522
$57$	0	0
$58$	8.00000	1.05045
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	−1.00000	−0.129099
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	−1.00000	−0.127000
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	2.00000	0.246183
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	4.00000	0.478091
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−1.00000	−0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	2.00000	0.232495
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−8.00000	−0.911685
$78$	2.00000	0.226455
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	−4.00000	−0.431331
$87$	8.00000	0.857690
$88$	−2.00000	−0.213201
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	−1.00000	−0.105409
$91$	−8.00000	−0.838628
$92$	−6.00000	−0.625543
$93$	−1.00000	−0.103695
$94$	12.0000	1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−9.00000	−0.909137
$99$	2.00000	0.201008
$100$	1.00000	0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−2.00000	−0.196116
$105$	4.00000	0.390360
$106$	2.00000	0.194257
$107$	−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
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	−2.00000	−0.190693
$111$	2.00000	0.189832
$112$	−4.00000	−0.377964
$113$	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
$115$	−6.00000	−0.559503
$116$	−8.00000	−0.742781
$117$	2.00000	0.184900
$118$	12.0000	1.10469
$119$	0	0
$120$	1.00000	0.0912871
$121$	−7.00000	−0.636364
$122$	8.00000	0.724286
$123$	−6.00000	−0.541002
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	4.00000	0.356348
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	−2.00000	−0.175412
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	−2.00000	−0.174078
$133$	0	0
$134$	4.00000	0.345547
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	−6.00000	−0.510754
$139$	18.0000	1.52674	0.763370	−	0.645961i	$-0.223543\pi$
$139$	18.0000	1.52674	0.763370	+	0.645961i	$0.223543\pi$
$140$	−4.00000	−0.338062
$141$	12.0000	1.01058
$142$	8.00000	0.671345
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	−8.00000	−0.664364
$146$	−4.00000	−0.331042
$147$	−9.00000	−0.742307
$148$	−2.00000	−0.164399
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	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	$0.197401\pi$
$152$	0	0
$153$	0	0
$154$	8.00000	0.644658
$155$	1.00000	0.0803219
$156$	−2.00000	−0.160128
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	−4.00000	−0.318223
$159$	2.00000	0.158610
$160$	−1.00000	−0.0790569
$161$	24.0000	1.89146
$162$	−1.00000	−0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	6.00000	0.468521
$165$	−2.00000	−0.155700
$166$	16.0000	1.24184
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	−4.00000	−0.308607
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	4.00000	0.304997
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	−8.00000	−0.606478
$175$	−4.00000	−0.302372
$176$	2.00000	0.150756
$177$	12.0000	0.901975
$178$	2.00000	0.149906
$179$	14.0000	1.04641	0.523205	−	0.852207i	$-0.324736\pi$
$179$	14.0000	1.04641	0.523205	+	0.852207i	$0.324736\pi$
$180$	1.00000	0.0745356
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	8.00000	0.592999
$183$	8.00000	0.591377
$184$	6.00000	0.442326
$185$	−2.00000	−0.147043
$186$	1.00000	0.0733236
$187$	0	0
$188$	−12.0000	−0.875190
$189$	4.00000	0.290957
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	−1.00000	−0.0721688
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	2.00000	0.143592
$195$	−2.00000	−0.143223
$196$	9.00000	0.642857
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−2.00000	−0.142134
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	−1.00000	−0.0707107
$201$	4.00000	0.282138
$202$	10.0000	0.703598
$203$	32.0000	2.24596
$204$	0	0
$205$	6.00000	0.419058
$206$	4.00000	0.278693
$207$	−6.00000	−0.417029
$208$	2.00000	0.138675
$209$	0	0
$210$	−4.00000	−0.276026
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−2.00000	−0.137361
$213$	8.00000	0.548151
$214$	4.00000	0.273434
$215$	4.00000	0.272798
$216$	1.00000	0.0680414
$217$	−4.00000	−0.271538
$218$	18.0000	1.21911
$219$	−4.00000	−0.270295
$220$	2.00000	0.134840
$221$	0	0
$222$	−2.00000	−0.134231
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	4.00000	0.267261
$225$	1.00000	0.0666667
$226$	−18.0000	−1.19734
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	6.00000	0.395628
$231$	8.00000	0.526361
$232$	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	$0.562969\pi$
$234$	−2.00000	−0.130744
$235$	−12.0000	−0.782794
$236$	−12.0000	−0.781133
$237$	−4.00000	−0.259828
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	−1.00000	−0.0645497
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	7.00000	0.449977
$243$	−1.00000	−0.0641500
$244$	−8.00000	−0.512148
$245$	9.00000	0.574989
$246$	6.00000	0.382546
$247$	0	0
$248$	−1.00000	−0.0635001
$249$	16.0000	1.01396
$250$	−1.00000	−0.0632456
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	−4.00000	−0.251976
$253$	−12.0000	−0.754434
$254$	6.00000	0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	4.00000	0.249029
$259$	8.00000	0.497096
$260$	2.00000	0.124035
$261$	−8.00000	−0.495188
$262$	4.00000	0.247121
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	2.00000	0.123091
$265$	−2.00000	−0.122859
$266$	0	0
$267$	2.00000	0.122398
$268$	−4.00000	−0.244339
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	1.00000	0.0608581
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	12.0000	0.724947
$275$	2.00000	0.120605
$276$	6.00000	0.361158
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−18.0000	−1.07957
$279$	1.00000	0.0598684
$280$	4.00000	0.239046
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−12.0000	−0.714590
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	−4.00000	−0.236525
$287$	−24.0000	−1.41668
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	8.00000	0.469776
$291$	2.00000	0.117242
$292$	4.00000	0.234082
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	9.00000	0.524891
$295$	−12.0000	−0.698667
$296$	2.00000	0.116248
$297$	−2.00000	−0.116052
$298$	−14.0000	−0.810998
$299$	−12.0000	−0.693978
$300$	−1.00000	−0.0577350
$301$	−16.0000	−0.922225
$302$	−20.0000	−1.15087
$303$	10.0000	0.574485
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	−8.00000	−0.455842
$309$	4.00000	0.227552
$310$	−1.00000	−0.0567962
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	2.00000	0.113228
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	−18.0000	−1.01580
$315$	−4.00000	−0.225374
$316$	4.00000	0.225018
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	−2.00000	−0.112154
$319$	−16.0000	−0.895828
$320$	1.00000	0.0559017
$321$	4.00000	0.223258
$322$	−24.0000	−1.33747
$323$	0	0
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	−12.0000	−0.664619
$327$	18.0000	0.995402
$328$	−6.00000	−0.331295
$329$	48.0000	2.64633
$330$	2.00000	0.110096
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	−16.0000	−0.878114
$333$	−2.00000	−0.109599
$334$	18.0000	0.984916
$335$	−4.00000	−0.218543
$336$	4.00000	0.218218
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	9.00000	0.489535
$339$	−18.0000	−0.977626
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	−8.00000	−0.431959
$344$	−4.00000	−0.215666
$345$	6.00000	0.323029
$346$	−18.0000	−0.967686
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	8.00000	0.428845
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	4.00000	0.213809
$351$	−2.00000	−0.106752
$352$	−2.00000	−0.106600
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	−12.0000	−0.637793
$355$	−8.00000	−0.424596
$356$	−2.00000	−0.106000
$357$	0	0
$358$	−14.0000	−0.739923
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−1.00000	−0.0527046
$361$	−19.0000	−1.00000
$362$	12.0000	0.630706
$363$	7.00000	0.367405
$364$	−8.00000	−0.419314
$365$	4.00000	0.209370
$366$	−8.00000	−0.418167
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	−6.00000	−0.312772
$369$	6.00000	0.312348
$370$	2.00000	0.103975
$371$	8.00000	0.415339
$372$	−1.00000	−0.0518476
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	12.0000	0.618853
$377$	−16.0000	−0.824042
$378$	−4.00000	−0.205738
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	−8.00000	−0.409316
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	1.00000	0.0510310
$385$	−8.00000	−0.407718
$386$	−22.0000	−1.11977
$387$	4.00000	0.203331
$388$	−2.00000	−0.101535
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	2.00000	0.101274
$391$	0	0
$392$	−9.00000	−0.454569
$393$	4.00000	0.201773
$394$	−2.00000	−0.100759
$395$	4.00000	0.201262
$396$	2.00000	0.100504
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	12.0000	0.601506
$399$	0	0
$400$	1.00000	0.0500000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	−4.00000	−0.199502
$403$	2.00000	0.0996271
$404$	−10.0000	−0.497519
$405$	1.00000	0.0496904
$406$	−32.0000	−1.58813
$407$	−4.00000	−0.198273
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	−6.00000	−0.296319
$411$	12.0000	0.591916
$412$	−4.00000	−0.197066
$413$	48.0000	2.36193
$414$	6.00000	0.294884
$415$	−16.0000	−0.785409
$416$	−2.00000	−0.0980581
$417$	−18.0000	−0.881464
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	4.00000	0.195180
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	20.0000	0.973585
$423$	−12.0000	−0.583460
$424$	2.00000	0.0971286
$425$	0	0
$426$	−8.00000	−0.387601
$427$	32.0000	1.54859
$428$	−4.00000	−0.193347
$429$	−4.00000	−0.193122
$430$	−4.00000	−0.192897
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	−1.00000	−0.0481125
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	4.00000	0.192006
$435$	8.00000	0.383571
$436$	−18.0000	−0.862044
$437$	0	0
$438$	4.00000	0.191127
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	−2.00000	−0.0953463
$441$	9.00000	0.428571
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	2.00000	0.0949158
$445$	−2.00000	−0.0948091
$446$	2.00000	0.0947027
$447$	−14.0000	−0.662177
$448$	−4.00000	−0.188982
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	−1.00000	−0.0471405
$451$	12.0000	0.565058
$452$	18.0000	0.846649
$453$	−20.0000	−0.939682
$454$	4.00000	0.187729
$455$	−8.00000	−0.375046
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	−4.00000	−0.186908
$459$	0	0
$460$	−6.00000	−0.279751
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	−8.00000	−0.372194
$463$	−42.0000	−1.95191	−0.975953	−	0.217982i	$-0.930053\pi$
$463$	−42.0000	−1.95191	−0.975953	+	0.217982i	$0.930053\pi$
$464$	−8.00000	−0.371391
$465$	−1.00000	−0.0463739
$466$	6.00000	0.277945
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	2.00000	0.0924500
$469$	16.0000	0.738811
$470$	12.0000	0.553519
$471$	−18.0000	−0.829396
$472$	12.0000	0.552345
$473$	8.00000	0.367840
$474$	4.00000	0.183726
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	−8.00000	−0.365911
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	1.00000	0.0456435
$481$	−4.00000	−0.182384
$482$	−10.0000	−0.455488
$483$	−24.0000	−1.09204
$484$	−7.00000	−0.318182
$485$	−2.00000	−0.0908153
$486$	1.00000	0.0453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	8.00000	0.362143
$489$	−12.0000	−0.542659
$490$	−9.00000	−0.406579
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	−6.00000	−0.270501
$493$	0	0
$494$	0	0
$495$	2.00000	0.0898933
$496$	1.00000	0.0449013
$497$	32.0000	1.43540
$498$	−16.0000	−0.716977
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	1.00000	0.0447214
$501$	18.0000	0.804181
$502$	−2.00000	−0.0892644
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	4.00000	0.178174
$505$	−10.0000	−0.444994
$506$	12.0000	0.533465
$507$	9.00000	0.399704
$508$	−6.00000	−0.266207
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	14.0000	0.617514
$515$	−4.00000	−0.176261
$516$	−4.00000	−0.176090
$517$	−24.0000	−1.05552
$518$	−8.00000	−0.351500
$519$	−18.0000	−0.790112
$520$	−2.00000	−0.0877058
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	8.00000	0.350150
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	−4.00000	−0.174741
$525$	4.00000	0.174574
$526$	6.00000	0.261612
$527$	0	0
$528$	−2.00000	−0.0870388
$529$	13.0000	0.565217
$530$	2.00000	0.0868744
$531$	−12.0000	−0.520756
$532$	0	0
$533$	12.0000	0.519778
$534$	−2.00000	−0.0865485
$535$	−4.00000	−0.172935
$536$	4.00000	0.172774
$537$	−14.0000	−0.604145
$538$	0	0
$539$	18.0000	0.775315
$540$	−1.00000	−0.0430331
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	0	0
$543$	12.0000	0.514969
$544$	0	0
$545$	−18.0000	−0.771035
$546$	−8.00000	−0.342368
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−12.0000	−0.512615
$549$	−8.00000	−0.341432
$550$	−2.00000	−0.0852803
$551$	0	0
$552$	−6.00000	−0.255377
$553$	−16.0000	−0.680389
$554$	2.00000	0.0849719
$555$	2.00000	0.0848953
$556$	18.0000	0.763370
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	−1.00000	−0.0423334
$559$	8.00000	0.338364
$560$	−4.00000	−0.169031
$561$	0	0
$562$	−6.00000	−0.253095
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	12.0000	0.505291
$565$	18.0000	0.757266
$566$	−20.0000	−0.840663
$567$	−4.00000	−0.167984
$568$	8.00000	0.335673
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	4.00000	0.167248
$573$	−8.00000	−0.334205
$574$	24.0000	1.00174
$575$	−6.00000	−0.250217
$576$	1.00000	0.0416667
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	17.0000	0.707107
$579$	−22.0000	−0.914289
$580$	−8.00000	−0.332182
$581$	64.0000	2.65517
$582$	−2.00000	−0.0829027
$583$	−4.00000	−0.165663
$584$	−4.00000	−0.165521
$585$	2.00000	0.0826898
$586$	−14.0000	−0.578335
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	−9.00000	−0.371154
$589$	0	0
$590$	12.0000	0.494032
$591$	−2.00000	−0.0822690
$592$	−2.00000	−0.0821995
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	14.0000	0.573462
$597$	12.0000	0.491127
$598$	12.0000	0.490716
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	1.00000	0.0408248
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	16.0000	0.652111
$603$	−4.00000	−0.162893
$604$	20.0000	0.813788
$605$	−7.00000	−0.284590
$606$	−10.0000	−0.406222
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	−32.0000	−1.29671
$610$	8.00000	0.323911
$611$	−24.0000	−0.970936
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	4.00000	0.161427
$615$	−6.00000	−0.241943
$616$	8.00000	0.322329
$617$	−34.0000	−1.36879	−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000	−1.36879	−0.684394	+	0.729112i	$0.739933\pi$
$618$	−4.00000	−0.160904
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	1.00000	0.0401610
$621$	6.00000	0.240772
$622$	−24.0000	−0.962312
$623$	8.00000	0.320513
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	4.00000	0.159872
$627$	0	0
$628$	18.0000	0.718278
$629$	0	0
$630$	4.00000	0.159364
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	−4.00000	−0.159111
$633$	20.0000	0.794929
$634$	22.0000	0.873732
$635$	−6.00000	−0.238103
$636$	2.00000	0.0793052
$637$	18.0000	0.713186
$638$	16.0000	0.633446
$639$	−8.00000	−0.316475
$640$	−1.00000	−0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−4.00000	−0.157867
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	24.0000	0.945732
$645$	−4.00000	−0.157500
$646$	0	0
$647$	14.0000	0.550397	0.275198	−	0.961387i	$-0.411256\pi$
$647$	14.0000	0.550397	0.275198	+	0.961387i	$0.411256\pi$
$648$	−1.00000	−0.0392837
$649$	−24.0000	−0.942082
$650$	−2.00000	−0.0784465
$651$	4.00000	0.156772
$652$	12.0000	0.469956
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	−18.0000	−0.703856
$655$	−4.00000	−0.156293
$656$	6.00000	0.234261
$657$	4.00000	0.156055
$658$	−48.0000	−1.87123
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	−2.00000	−0.0778499
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	−26.0000	−1.01052
$663$	0	0
$664$	16.0000	0.620920
$665$	0	0
$666$	2.00000	0.0774984
$667$	48.0000	1.85857
$668$	−18.0000	−0.696441
$669$	2.00000	0.0773245
$670$	4.00000	0.154533
$671$	−16.0000	−0.617673
$672$	−4.00000	−0.154303
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	16.0000	0.616297
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	50.0000	1.92166	0.960828	−	0.277145i	$-0.0893883\pi$
$677$	50.0000	1.92166	0.960828	+	0.277145i	$0.0893883\pi$
$678$	18.0000	0.691286
$679$	8.00000	0.307012
$680$	0	0
$681$	4.00000	0.153280
$682$	−2.00000	−0.0765840
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	8.00000	0.305441
$687$	−4.00000	−0.152610
$688$	4.00000	0.152499
$689$	−4.00000	−0.152388
$690$	−6.00000	−0.228416
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	18.0000	0.684257
$693$	−8.00000	−0.303895
$694$	−16.0000	−0.607352
$695$	18.0000	0.682779
$696$	−8.00000	−0.303239
$697$	0	0
$698$	−2.00000	−0.0757011
$699$	6.00000	0.226941
$700$	−4.00000	−0.151186
$701$	26.0000	0.982006	0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000	0.982006	0.491003	+	0.871158i	$0.336630\pi$
$702$	2.00000	0.0754851
$703$	0	0
$704$	2.00000	0.0753778
$705$	12.0000	0.451946
$706$	−4.00000	−0.150542
$707$	40.0000	1.50435
$708$	12.0000	0.450988
$709$	−40.0000	−1.50223	−0.751116	−	0.660171i	$-0.770484\pi$
$709$	−40.0000	−1.50223	−0.751116	+	0.660171i	$0.770484\pi$
$710$	8.00000	0.300235
$711$	4.00000	0.150012
$712$	2.00000	0.0749532
$713$	−6.00000	−0.224702
$714$	0	0
$715$	4.00000	0.149592
$716$	14.0000	0.523205
$717$	−8.00000	−0.298765
$718$	0	0
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	1.00000	0.0372678
$721$	16.0000	0.595871
$722$	19.0000	0.707107
$723$	−10.0000	−0.371904
$724$	−12.0000	−0.445976
$725$	−8.00000	−0.297113
$726$	−7.00000	−0.259794
$727$	24.0000	0.890111	0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000	0.890111	0.445055	+	0.895503i	$0.353184\pi$
$728$	8.00000	0.296500
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	0	0
$732$	8.00000	0.295689
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	−18.0000	−0.664392
$735$	−9.00000	−0.331970
$736$	6.00000	0.221163
$737$	−8.00000	−0.294684
$738$	−6.00000	−0.220863
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	−2.00000	−0.0735215
$741$	0	0
$742$	−8.00000	−0.293689
$743$	26.0000	0.953847	0.476924	−	0.878945i	$-0.341752\pi$
$743$	26.0000	0.953847	0.476924	+	0.878945i	$0.341752\pi$
$744$	1.00000	0.0366618
$745$	14.0000	0.512920
$746$	−6.00000	−0.219676
$747$	−16.0000	−0.585409
$748$	0	0
$749$	16.0000	0.584627
$750$	1.00000	0.0365148
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−12.0000	−0.437595
$753$	−2.00000	−0.0728841
$754$	16.0000	0.582686
$755$	20.0000	0.727875
$756$	4.00000	0.145479
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	−16.0000	−0.581146
$759$	12.0000	0.435572
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	−6.00000	−0.217357
$763$	72.0000	2.60658
$764$	8.00000	0.289430
$765$	0	0
$766$	−6.00000	−0.216789
$767$	−24.0000	−0.866590
$768$	−1.00000	−0.0360844
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	8.00000	0.288300
$771$	14.0000	0.504198
$772$	22.0000	0.791797
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	−4.00000	−0.143777
$775$	1.00000	0.0359211
$776$	2.00000	0.0717958
$777$	−8.00000	−0.286998
$778$	36.0000	1.29066
$779$	0	0
$780$	−2.00000	−0.0716115
$781$	−16.0000	−0.572525
$782$	0	0
$783$	8.00000	0.285897
$784$	9.00000	0.321429
$785$	18.0000	0.642448
$786$	−4.00000	−0.142675
$787$	−16.0000	−0.570338	−0.285169	−	0.958477i	$-0.592050\pi$
$787$	−16.0000	−0.570338	−0.285169	+	0.958477i	$0.592050\pi$
$788$	2.00000	0.0712470
$789$	6.00000	0.213606
$790$	−4.00000	−0.142314
$791$	−72.0000	−2.56003
$792$	−2.00000	−0.0710669
$793$	−16.0000	−0.568177
$794$	−18.0000	−0.638796
$795$	2.00000	0.0709327
$796$	−12.0000	−0.425329
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−2.00000	−0.0706665
$802$	10.0000	0.353112
$803$	8.00000	0.282314
$804$	4.00000	0.141069
$805$	24.0000	0.845889
$806$	−2.00000	−0.0704470
$807$	0	0
$808$	10.0000	0.351799
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	−1.00000	−0.0351364
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	32.0000	1.12298
$813$	0	0
$814$	4.00000	0.140200
$815$	12.0000	0.420342
$816$	0	0
$817$	0	0
$818$	−30.0000	−1.04893
$819$	−8.00000	−0.279543
$820$	6.00000	0.209529
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	−12.0000	−0.418548
$823$	−22.0000	−0.766872	−0.383436	−	0.923567i	$-0.625259\pi$
$823$	−22.0000	−0.766872	−0.383436	+	0.923567i	$0.625259\pi$
$824$	4.00000	0.139347
$825$	−2.00000	−0.0696311
$826$	−48.0000	−1.67013
$827$	−16.0000	−0.556375	−0.278187	−	0.960527i	$-0.589734\pi$
$827$	−16.0000	−0.556375	−0.278187	+	0.960527i	$0.589734\pi$
$828$	−6.00000	−0.208514
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	16.0000	0.555368
$831$	2.00000	0.0693792
$832$	2.00000	0.0693375
$833$	0	0
$834$	18.0000	0.623289
$835$	−18.0000	−0.622916
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	−12.0000	−0.414533
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	−4.00000	−0.138013
$841$	35.0000	1.20690
$842$	−18.0000	−0.620321
$843$	−6.00000	−0.206651
$844$	−20.0000	−0.688428
$845$	−9.00000	−0.309609
$846$	12.0000	0.412568
$847$	28.0000	0.962091
$848$	−2.00000	−0.0686803
$849$	−20.0000	−0.686398
$850$	0	0
$851$	12.0000	0.411355
$852$	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	$0.467246\pi$
$854$	−32.0000	−1.09502
$855$	0	0
$856$	4.00000	0.136717
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	4.00000	0.136558
$859$	22.0000	0.750630	0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000	0.750630	0.375315	+	0.926897i	$0.377534\pi$
$860$	4.00000	0.136399
$861$	24.0000	0.817918
$862$	16.0000	0.544962
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	1.00000	0.0340207
$865$	18.0000	0.612018
$866$	4.00000	0.135926
$867$	17.0000	0.577350
$868$	−4.00000	−0.135769
$869$	8.00000	0.271381
$870$	−8.00000	−0.271225
$871$	−8.00000	−0.271070
$872$	18.0000	0.609557
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	−4.00000	−0.135225
$876$	−4.00000	−0.135147
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	24.0000	0.809961
$879$	−14.0000	−0.472208
$880$	2.00000	0.0674200
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	−9.00000	−0.303046
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	−12.0000	−0.403148
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	−2.00000	−0.0671156
$889$	24.0000	0.804934
$890$	2.00000	0.0670402
$891$	2.00000	0.0670025
$892$	−2.00000	−0.0669650
$893$	0	0
$894$	14.0000	0.468230
$895$	14.0000	0.467968
$896$	4.00000	0.133631
$897$	12.0000	0.400668
$898$	34.0000	1.13459
$899$	−8.00000	−0.266815
$900$	1.00000	0.0333333
$901$	0	0
$902$	−12.0000	−0.399556
$903$	16.0000	0.532447
$904$	−18.0000	−0.598671
$905$	−12.0000	−0.398893
$906$	20.0000	0.664455
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	−4.00000	−0.132745
$909$	−10.0000	−0.331679
$910$	8.00000	0.265197
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	0	0
$913$	−32.0000	−1.05905
$914$	−8.00000	−0.264616
$915$	8.00000	0.264472
$916$	4.00000	0.132164
$917$	16.0000	0.528367
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	6.00000	0.197814
$921$	4.00000	0.131804
$922$	−24.0000	−0.790398
$923$	−16.0000	−0.526646
$924$	8.00000	0.263181
$925$	−2.00000	−0.0657596
$926$	42.0000	1.38021
$927$	−4.00000	−0.131377
$928$	8.00000	0.262613
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	1.00000	0.0327913
$931$	0	0
$932$	−6.00000	−0.196537
$933$	−24.0000	−0.785725
$934$	36.0000	1.17796
$935$	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	$0.510401\pi$
$938$	−16.0000	−0.522419
$939$	4.00000	0.130535
$940$	−12.0000	−0.391397
$941$	36.0000	1.17357	0.586783	−	0.809744i	$-0.300394\pi$
$941$	36.0000	1.17357	0.586783	+	0.809744i	$0.300394\pi$
$942$	18.0000	0.586472
$943$	−36.0000	−1.17232
$944$	−12.0000	−0.390567
$945$	4.00000	0.130120
$946$	−8.00000	−0.260102
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	−4.00000	−0.129914
$949$	8.00000	0.259691
$950$	0	0
$951$	22.0000	0.713399
$952$	0	0
$953$	4.00000	0.129573	0.0647864	−	0.997899i	$-0.479363\pi$
$953$	4.00000	0.129573	0.0647864	+	0.997899i	$0.479363\pi$
$954$	2.00000	0.0647524
$955$	8.00000	0.258874
$956$	8.00000	0.258738
$957$	16.0000	0.517207
$958$	8.00000	0.258468
$959$	48.0000	1.55000
$960$	−1.00000	−0.0322749
$961$	1.00000	0.0322581
$962$	4.00000	0.128965
$963$	−4.00000	−0.128898
$964$	10.0000	0.322078
$965$	22.0000	0.708205
$966$	24.0000	0.772187
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	2.00000	0.0642161
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	−1.00000	−0.0320750
$973$	−72.0000	−2.30821
$974$	−2.00000	−0.0640841
$975$	−2.00000	−0.0640513
$976$	−8.00000	−0.256074
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	12.0000	0.383718
$979$	−4.00000	−0.127841
$980$	9.00000	0.287494
$981$	−18.0000	−0.574696
$982$	42.0000	1.34027
$983$	−54.0000	−1.72233	−0.861166	−	0.508323i	$-0.830265\pi$
$983$	−54.0000	−1.72233	−0.861166	+	0.508323i	$0.830265\pi$
$984$	6.00000	0.191273
$985$	2.00000	0.0637253
$986$	0	0
$987$	−48.0000	−1.52786
$988$	0	0
$989$	−24.0000	−0.763156
$990$	−2.00000	−0.0635642
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	−1.00000	−0.0317500
$993$	−26.0000	−0.825085
$994$	−32.0000	−1.01498
$995$	−12.0000	−0.380426
$996$	16.0000	0.506979
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	34.0000	1.07625
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.a.c.1.1	✓	1
3.2	odd	2		2790.2.a.o.1.1		1
4.3	odd	2		7440.2.a.bb.1.1		1
5.2	odd	4		4650.2.d.v.3349.1		2
5.3	odd	4		4650.2.d.v.3349.2		2
5.4	even	2		4650.2.a.bw.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.c.1.1	✓	1	1.1	even	1	trivial
2790.2.a.o.1.1		1	3.2	odd	2
4650.2.a.bw.1.1		1	5.4	even	2
4650.2.d.v.3349.1		2	5.2	odd	4
4650.2.d.v.3349.2		2	5.3	odd	4
7440.2.a.bb.1.1		1	4.3	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:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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