LMFDB - Embedded newform 930.2.a.g.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:	$0$
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	$\chi$	$=$	930.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{21} +4.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} -2.00000 q^{29} +1.00000 q^{30} -1.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} -4.00000 q^{42} -4.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} -4.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -4.00000 q^{56} +4.00000 q^{57} +2.00000 q^{58} +8.00000 q^{59} -1.00000 q^{60} +10.0000 q^{61} +1.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} -12.0000 q^{67} +2.00000 q^{68} +4.00000 q^{69} +4.00000 q^{70} -1.00000 q^{72} +14.0000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -16.0000 q^{77} -2.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +4.00000 q^{83} +4.00000 q^{84} -2.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} -6.00000 q^{89} +1.00000 q^{90} +8.00000 q^{91} +4.00000 q^{92} -1.00000 q^{93} -8.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} -6.00000 q^{97} -9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−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	$0.410544\pi$
$14$	−4.00000	−1.06904
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−1.00000	−0.223607
$21$	4.00000	0.872872
$22$	4.00000	0.852803
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	1.00000	0.182574
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	−2.00000	−0.342997
$35$	−4.00000	−0.676123
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−4.00000	−0.648886
$39$	2.00000	0.320256
$40$	1.00000	0.158114
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	−4.00000	−0.617213
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−4.00000	−0.603023
$45$	−1.00000	−0.149071
$46$	−4.00000	−0.589768
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	−1.00000	−0.141421
$51$	2.00000	0.280056
$52$	2.00000	0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	4.00000	0.539360
$56$	−4.00000	−0.534522
$57$	4.00000	0.529813
$58$	2.00000	0.262613
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	−1.00000	−0.129099
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	1.00000	0.127000
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	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	$0.761888\pi$
$68$	2.00000	0.242536
$69$	4.00000	0.481543
$70$	4.00000	0.478091
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	6.00000	0.697486
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	−16.0000	−1.82337
$78$	−2.00000	−0.226455
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	4.00000	0.436436
$85$	−2.00000	−0.216930
$86$	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	$0.603011\pi$
$90$	1.00000	0.105409
$91$	8.00000	0.838628
$92$	4.00000	0.417029
$93$	−1.00000	−0.103695
$94$	−8.00000	−0.825137
$95$	−4.00000	−0.410391
$96$	−1.00000	−0.102062
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	−9.00000	−0.909137
$99$	−4.00000	−0.402015
$100$	1.00000	0.100000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	−2.00000	−0.198030
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−2.00000	−0.196116
$105$	−4.00000	−0.390360
$106$	6.00000	0.582772
$107$	−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	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−4.00000	−0.381385
$111$	−6.00000	−0.569495
$112$	4.00000	0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−4.00000	−0.374634
$115$	−4.00000	−0.373002
$116$	−2.00000	−0.185695
$117$	2.00000	0.184900
$118$	−8.00000	−0.736460
$119$	8.00000	0.733359
$120$	1.00000	0.0912871
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	10.0000	0.901670
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	−4.00000	−0.356348
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	2.00000	0.175412
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	−4.00000	−0.348155
$133$	16.0000	1.38738
$134$	12.0000	1.03664
$135$	−1.00000	−0.0860663
$136$	−2.00000	−0.171499
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−4.00000	−0.340503
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	−4.00000	−0.338062
$141$	8.00000	0.673722
$142$	0	0
$143$	−8.00000	−0.668994
$144$	1.00000	0.0833333
$145$	2.00000	0.166091
$146$	−14.0000	−1.15865
$147$	9.00000	0.742307
$148$	−6.00000	−0.493197
$149$	−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
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−4.00000	−0.324443
$153$	2.00000	0.161690
$154$	16.0000	1.28932
$155$	1.00000	0.0803219
$156$	2.00000	0.160128
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	8.00000	0.636446
$159$	−6.00000	−0.475831
$160$	1.00000	0.0790569
$161$	16.0000	1.26098
$162$	−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	$0.344270\pi$
$164$	10.0000	0.780869
$165$	4.00000	0.311400
$166$	−4.00000	−0.310460
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	−4.00000	−0.308607
$169$	−9.00000	−0.692308
$170$	2.00000	0.153393
$171$	4.00000	0.305888
$172$	−4.00000	−0.304997
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	2.00000	0.151620
$175$	4.00000	0.302372
$176$	−4.00000	−0.301511
$177$	8.00000	0.601317
$178$	6.00000	0.449719
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	−1.00000	−0.0745356
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	−8.00000	−0.592999
$183$	10.0000	0.739221
$184$	−4.00000	−0.294884
$185$	6.00000	0.441129
$186$	1.00000	0.0733236
$187$	−8.00000	−0.585018
$188$	8.00000	0.583460
$189$	4.00000	0.290957
$190$	4.00000	0.290191
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	1.00000	0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	6.00000	0.430775
$195$	−2.00000	−0.143223
$196$	9.00000	0.642857
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	4.00000	0.284268
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	−1.00000	−0.0707107
$201$	−12.0000	−0.846415
$202$	−18.0000	−1.26648
$203$	−8.00000	−0.561490
$204$	2.00000	0.140028
$205$	−10.0000	−0.698430
$206$	−4.00000	−0.278693
$207$	4.00000	0.278019
$208$	2.00000	0.138675
$209$	−16.0000	−1.10674
$210$	4.00000	0.276026
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	4.00000	0.273434
$215$	4.00000	0.272798
$216$	−1.00000	−0.0680414
$217$	−4.00000	−0.271538
$218$	−6.00000	−0.406371
$219$	14.0000	0.946032
$220$	4.00000	0.269680
$221$	4.00000	0.269069
$222$	6.00000	0.402694
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−4.00000	−0.267261
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	4.00000	0.264906
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	4.00000	0.263752
$231$	−16.0000	−1.05272
$232$	2.00000	0.131306
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	−2.00000	−0.130744
$235$	−8.00000	−0.521862
$236$	8.00000	0.520756
$237$	−8.00000	−0.519656
$238$	−8.00000	−0.518563
$239$	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
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	10.0000	0.640184
$245$	−9.00000	−0.574989
$246$	−10.0000	−0.637577
$247$	8.00000	0.509028
$248$	1.00000	0.0635001
$249$	4.00000	0.253490
$250$	1.00000	0.0632456
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	4.00000	0.251976
$253$	−16.0000	−1.00591
$254$	8.00000	0.501965
$255$	−2.00000	−0.125245
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	4.00000	0.249029
$259$	−24.0000	−1.49129
$260$	−2.00000	−0.124035
$261$	−2.00000	−0.123797
$262$	8.00000	0.494242
$263$	−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
$265$	6.00000	0.368577
$266$	−16.0000	−0.981023
$267$	−6.00000	−0.367194
$268$	−12.0000	−0.733017
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	1.00000	0.0608581
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	2.00000	0.121268
$273$	8.00000	0.484182
$274$	6.00000	0.362473
$275$	−4.00000	−0.241209
$276$	4.00000	0.240772
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−16.0000	−0.959616
$279$	−1.00000	−0.0598684
$280$	4.00000	0.239046
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	−8.00000	−0.476393
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	8.00000	0.473050
$287$	40.0000	2.36113
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	−2.00000	−0.117444
$291$	−6.00000	−0.351726
$292$	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	$0.556077\pi$
$294$	−9.00000	−0.524891
$295$	−8.00000	−0.465778
$296$	6.00000	0.348743
$297$	−4.00000	−0.232104
$298$	6.00000	0.347571
$299$	8.00000	0.462652
$300$	1.00000	0.0577350
$301$	−16.0000	−0.922225
$302$	−8.00000	−0.460348
$303$	18.0000	1.03407
$304$	4.00000	0.229416
$305$	−10.0000	−0.572598
$306$	−2.00000	−0.114332
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	−16.0000	−0.911685
$309$	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
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	22.0000	1.24153
$315$	−4.00000	−0.225374
$316$	−8.00000	−0.450035
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	6.00000	0.336463
$319$	8.00000	0.447914
$320$	−1.00000	−0.0559017
$321$	−4.00000	−0.223258
$322$	−16.0000	−0.891645
$323$	8.00000	0.445132
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	−12.0000	−0.664619
$327$	6.00000	0.331801
$328$	−10.0000	−0.552158
$329$	32.0000	1.76422
$330$	−4.00000	−0.220193
$331$	24.0000	1.31916	0.659580	−	0.751635i	$-0.270734\pi$
$331$	24.0000	1.31916	0.659580	+	0.751635i	$0.270734\pi$
$332$	4.00000	0.219529
$333$	−6.00000	−0.328798
$334$	20.0000	1.09435
$335$	12.0000	0.655630
$336$	4.00000	0.218218
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	9.00000	0.489535
$339$	6.00000	0.325875
$340$	−2.00000	−0.108465
$341$	4.00000	0.216612
$342$	−4.00000	−0.216295
$343$	8.00000	0.431959
$344$	4.00000	0.215666
$345$	−4.00000	−0.215353
$346$	14.0000	0.752645
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	−2.00000	−0.107211
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	−4.00000	−0.213809
$351$	2.00000	0.106752
$352$	4.00000	0.213201
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	−8.00000	−0.425195
$355$	0	0
$356$	−6.00000	−0.317999
$357$	8.00000	0.423405
$358$	20.0000	1.05703
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	6.00000	0.315353
$363$	5.00000	0.262432
$364$	8.00000	0.419314
$365$	−14.0000	−0.732793
$366$	−10.0000	−0.522708
$367$	−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
$370$	−6.00000	−0.311925
$371$	−24.0000	−1.24602
$372$	−1.00000	−0.0518476
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	8.00000	0.413670
$375$	−1.00000	−0.0516398
$376$	−8.00000	−0.412568
$377$	−4.00000	−0.206010
$378$	−4.00000	−0.205738
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−4.00000	−0.205196
$381$	−8.00000	−0.409852
$382$	−24.0000	−1.22795
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	−1.00000	−0.0510310
$385$	16.0000	0.815436
$386$	14.0000	0.712581
$387$	−4.00000	−0.203331
$388$	−6.00000	−0.304604
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	2.00000	0.101274
$391$	8.00000	0.404577
$392$	−9.00000	−0.454569
$393$	−8.00000	−0.403547
$394$	14.0000	0.705310
$395$	8.00000	0.402524
$396$	−4.00000	−0.201008
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	8.00000	0.401004
$399$	16.0000	0.801002
$400$	1.00000	0.0500000
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	12.0000	0.598506
$403$	−2.00000	−0.0996271
$404$	18.0000	0.895533
$405$	−1.00000	−0.0496904
$406$	8.00000	0.397033
$407$	24.0000	1.18964
$408$	−2.00000	−0.0990148
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	10.0000	0.493865
$411$	−6.00000	−0.295958
$412$	4.00000	0.197066
$413$	32.0000	1.57462
$414$	−4.00000	−0.196589
$415$	−4.00000	−0.196352
$416$	−2.00000	−0.0980581
$417$	16.0000	0.783523
$418$	16.0000	0.782586
$419$	8.00000	0.390826	0.195413	−	0.980721i	$-0.437395\pi$
$419$	8.00000	0.390826	0.195413	+	0.980721i	$0.437395\pi$
$420$	−4.00000	−0.195180
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	12.0000	0.584151
$423$	8.00000	0.388973
$424$	6.00000	0.291386
$425$	2.00000	0.0970143
$426$	0	0
$427$	40.0000	1.93574
$428$	−4.00000	−0.193347
$429$	−8.00000	−0.386244
$430$	−4.00000	−0.192897
$431$	−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	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	4.00000	0.192006
$435$	2.00000	0.0958927
$436$	6.00000	0.287348
$437$	16.0000	0.765384
$438$	−14.0000	−0.668946
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	−4.00000	−0.190693
$441$	9.00000	0.428571
$442$	−4.00000	−0.190261
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	−6.00000	−0.284747
$445$	6.00000	0.284427
$446$	−16.0000	−0.757622
$447$	−6.00000	−0.283790
$448$	4.00000	0.188982
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	−1.00000	−0.0471405
$451$	−40.0000	−1.88353
$452$	6.00000	0.282216
$453$	8.00000	0.375873
$454$	−4.00000	−0.187729
$455$	−8.00000	−0.375046
$456$	−4.00000	−0.187317
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	14.0000	0.654177
$459$	2.00000	0.0933520
$460$	−4.00000	−0.186501
$461$	38.0000	1.76984	0.884918	−	0.465746i	$-0.154214\pi$
$461$	38.0000	1.76984	0.884918	+	0.465746i	$0.154214\pi$
$462$	16.0000	0.744387
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	−2.00000	−0.0928477
$465$	1.00000	0.0463739
$466$	10.0000	0.463241
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	2.00000	0.0924500
$469$	−48.0000	−2.21643
$470$	8.00000	0.369012
$471$	−22.0000	−1.01371
$472$	−8.00000	−0.368230
$473$	16.0000	0.735681
$474$	8.00000	0.367452
$475$	4.00000	0.183533
$476$	8.00000	0.366679
$477$	−6.00000	−0.274721
$478$	−8.00000	−0.365911
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	1.00000	0.0456435
$481$	−12.0000	−0.547153
$482$	14.0000	0.637683
$483$	16.0000	0.728025
$484$	5.00000	0.227273
$485$	6.00000	0.272446
$486$	−1.00000	−0.0453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	−10.0000	−0.452679
$489$	12.0000	0.542659
$490$	9.00000	0.406579
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	10.0000	0.450835
$493$	−4.00000	−0.180151
$494$	−8.00000	−0.359937
$495$	4.00000	0.179787
$496$	−1.00000	−0.0449013
$497$	0	0
$498$	−4.00000	−0.179244
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	−1.00000	−0.0447214
$501$	−20.0000	−0.893534
$502$	28.0000	1.24970
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	−4.00000	−0.178174
$505$	−18.0000	−0.800989
$506$	16.0000	0.711287
$507$	−9.00000	−0.399704
$508$	−8.00000	−0.354943
$509$	22.0000	0.975133	0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000	0.975133	0.487566	+	0.873086i	$0.337885\pi$
$510$	2.00000	0.0885615
$511$	56.0000	2.47729
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	−6.00000	−0.264649
$515$	−4.00000	−0.176261
$516$	−4.00000	−0.176090
$517$	−32.0000	−1.40736
$518$	24.0000	1.05450
$519$	−14.0000	−0.614532
$520$	2.00000	0.0877058
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	2.00000	0.0875376
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	−8.00000	−0.349482
$525$	4.00000	0.174574
$526$	12.0000	0.523225
$527$	−2.00000	−0.0871214
$528$	−4.00000	−0.174078
$529$	−7.00000	−0.304348
$530$	−6.00000	−0.260623
$531$	8.00000	0.347170
$532$	16.0000	0.693688
$533$	20.0000	0.866296
$534$	6.00000	0.259645
$535$	4.00000	0.172935
$536$	12.0000	0.518321
$537$	−20.0000	−0.863064
$538$	18.0000	0.776035
$539$	−36.0000	−1.55063
$540$	−1.00000	−0.0430331
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−8.00000	−0.343629
$543$	−6.00000	−0.257485
$544$	−2.00000	−0.0857493
$545$	−6.00000	−0.257012
$546$	−8.00000	−0.342368
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−6.00000	−0.256307
$549$	10.0000	0.426790
$550$	4.00000	0.170561
$551$	−8.00000	−0.340811
$552$	−4.00000	−0.170251
$553$	−32.0000	−1.36078
$554$	22.0000	0.934690
$555$	6.00000	0.254686
$556$	16.0000	0.678551
$557$	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
$561$	−8.00000	−0.337760
$562$	30.0000	1.26547
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	8.00000	0.336861
$565$	−6.00000	−0.252422
$566$	−20.0000	−0.840663
$567$	4.00000	0.167984
$568$	0	0
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	4.00000	0.167542
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	−8.00000	−0.334497
$573$	24.0000	1.00261
$574$	−40.0000	−1.66957
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	13.0000	0.540729
$579$	−14.0000	−0.581820
$580$	2.00000	0.0830455
$581$	16.0000	0.663792
$582$	6.00000	0.248708
$583$	24.0000	0.993978
$584$	−14.0000	−0.579324
$585$	−2.00000	−0.0826898
$586$	6.00000	0.247858
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	9.00000	0.371154
$589$	−4.00000	−0.164817
$590$	8.00000	0.329355
$591$	−14.0000	−0.575883
$592$	−6.00000	−0.246598
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	4.00000	0.164122
$595$	−8.00000	−0.327968
$596$	−6.00000	−0.245770
$597$	−8.00000	−0.327418
$598$	−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
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	16.0000	0.652111
$603$	−12.0000	−0.488678
$604$	8.00000	0.325515
$605$	−5.00000	−0.203279
$606$	−18.0000	−0.731200
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	−4.00000	−0.162221
$609$	−8.00000	−0.324176
$610$	10.0000	0.404888
$611$	16.0000	0.647291
$612$	2.00000	0.0808452
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	20.0000	0.807134
$615$	−10.0000	−0.403239
$616$	16.0000	0.644658
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−4.00000	−0.160904
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	1.00000	0.0401610
$621$	4.00000	0.160514
$622$	−24.0000	−0.962312
$623$	−24.0000	−0.961540
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	34.0000	1.35891
$627$	−16.0000	−0.638978
$628$	−22.0000	−0.877896
$629$	−12.0000	−0.478471
$630$	4.00000	0.159364
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	8.00000	0.318223
$633$	−12.0000	−0.476957
$634$	−2.00000	−0.0794301
$635$	8.00000	0.317470
$636$	−6.00000	−0.237915
$637$	18.0000	0.713186
$638$	−8.00000	−0.316723
$639$	0	0
$640$	1.00000	0.0395285
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	4.00000	0.157867
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	16.0000	0.630488
$645$	4.00000	0.157500
$646$	−8.00000	−0.314756
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	−1.00000	−0.0392837
$649$	−32.0000	−1.25611
$650$	−2.00000	−0.0784465
$651$	−4.00000	−0.156772
$652$	12.0000	0.469956
$653$	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
$655$	8.00000	0.312586
$656$	10.0000	0.390434
$657$	14.0000	0.546192
$658$	−32.0000	−1.24749
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	4.00000	0.155700
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	−24.0000	−0.932786
$663$	4.00000	0.155347
$664$	−4.00000	−0.155230
$665$	−16.0000	−0.620453
$666$	6.00000	0.232495
$667$	−8.00000	−0.309761
$668$	−20.0000	−0.773823
$669$	16.0000	0.618596
$670$	−12.0000	−0.463600
$671$	−40.0000	−1.54418
$672$	−4.00000	−0.154303
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	2.00000	0.0770371
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	−6.00000	−0.230429
$679$	−24.0000	−0.921035
$680$	2.00000	0.0766965
$681$	4.00000	0.153280
$682$	−4.00000	−0.153168
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	4.00000	0.152944
$685$	6.00000	0.229248
$686$	−8.00000	−0.305441
$687$	−14.0000	−0.534133
$688$	−4.00000	−0.152499
$689$	−12.0000	−0.457164
$690$	4.00000	0.152277
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−14.0000	−0.532200
$693$	−16.0000	−0.607790
$694$	−20.0000	−0.759190
$695$	−16.0000	−0.606915
$696$	2.00000	0.0758098
$697$	20.0000	0.757554
$698$	−30.0000	−1.13552
$699$	−10.0000	−0.378235
$700$	4.00000	0.151186
$701$	−46.0000	−1.73740	−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−46.0000	−1.73740	−0.868698	+	0.495342i	$0.835043\pi$
$702$	−2.00000	−0.0754851
$703$	−24.0000	−0.905177
$704$	−4.00000	−0.150756
$705$	−8.00000	−0.301297
$706$	−10.0000	−0.376355
$707$	72.0000	2.70784
$708$	8.00000	0.300658
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	6.00000	0.224860
$713$	−4.00000	−0.149801
$714$	−8.00000	−0.299392
$715$	8.00000	0.299183
$716$	−20.0000	−0.747435
$717$	8.00000	0.298765
$718$	−16.0000	−0.597115
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	−1.00000	−0.0372678
$721$	16.0000	0.595871
$722$	3.00000	0.111648
$723$	−14.0000	−0.520666
$724$	−6.00000	−0.222988
$725$	−2.00000	−0.0742781
$726$	−5.00000	−0.185567
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	14.0000	0.518163
$731$	−8.00000	−0.295891
$732$	10.0000	0.369611
$733$	18.0000	0.664845	0.332423	−	0.943131i	$-0.392134\pi$
$733$	18.0000	0.664845	0.332423	+	0.943131i	$0.392134\pi$
$734$	8.00000	0.295285
$735$	−9.00000	−0.331970
$736$	−4.00000	−0.147442
$737$	48.0000	1.76810
$738$	−10.0000	−0.368105
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	6.00000	0.220564
$741$	8.00000	0.293887
$742$	24.0000	0.881068
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	1.00000	0.0366618
$745$	6.00000	0.219823
$746$	14.0000	0.512576
$747$	4.00000	0.146352
$748$	−8.00000	−0.292509
$749$	−16.0000	−0.584627
$750$	1.00000	0.0365148
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	8.00000	0.291730
$753$	−28.0000	−1.02038
$754$	4.00000	0.145671
$755$	−8.00000	−0.291150
$756$	4.00000	0.145479
$757$	−54.0000	−1.96266	−0.981332	−	0.192323i	$-0.938398\pi$
$757$	−54.0000	−1.96266	−0.981332	+	0.192323i	$0.938398\pi$
$758$	−4.00000	−0.145287
$759$	−16.0000	−0.580763
$760$	4.00000	0.145095
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	8.00000	0.289809
$763$	24.0000	0.868858
$764$	24.0000	0.868290
$765$	−2.00000	−0.0723102
$766$	4.00000	0.144526
$767$	16.0000	0.577727
$768$	1.00000	0.0360844
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	−16.0000	−0.576600
$771$	6.00000	0.216085
$772$	−14.0000	−0.503871
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	4.00000	0.143777
$775$	−1.00000	−0.0359211
$776$	6.00000	0.215387
$777$	−24.0000	−0.860995
$778$	10.0000	0.358517
$779$	40.0000	1.43315
$780$	−2.00000	−0.0716115
$781$	0	0
$782$	−8.00000	−0.286079
$783$	−2.00000	−0.0714742
$784$	9.00000	0.321429
$785$	22.0000	0.785214
$786$	8.00000	0.285351
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−14.0000	−0.498729
$789$	−12.0000	−0.427211
$790$	−8.00000	−0.284627
$791$	24.0000	0.853342
$792$	4.00000	0.142134
$793$	20.0000	0.710221
$794$	−2.00000	−0.0709773
$795$	6.00000	0.212798
$796$	−8.00000	−0.283552
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	−16.0000	−0.566394
$799$	16.0000	0.566039
$800$	−1.00000	−0.0353553
$801$	−6.00000	−0.212000
$802$	−34.0000	−1.20058
$803$	−56.0000	−1.97620
$804$	−12.0000	−0.423207
$805$	−16.0000	−0.563926
$806$	2.00000	0.0704470
$807$	−18.0000	−0.633630
$808$	−18.0000	−0.633238
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	1.00000	0.0351364
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	−8.00000	−0.280745
$813$	8.00000	0.280572
$814$	−24.0000	−0.841200
$815$	−12.0000	−0.420342
$816$	2.00000	0.0700140
$817$	−16.0000	−0.559769
$818$	−18.0000	−0.629355
$819$	8.00000	0.279543
$820$	−10.0000	−0.349215
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	6.00000	0.209274
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	−4.00000	−0.139347
$825$	−4.00000	−0.139262
$826$	−32.0000	−1.11342
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	4.00000	0.139010
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	4.00000	0.138842
$831$	−22.0000	−0.763172
$832$	2.00000	0.0693375
$833$	18.0000	0.623663
$834$	−16.0000	−0.554035
$835$	20.0000	0.692129
$836$	−16.0000	−0.553372
$837$	−1.00000	−0.0345651
$838$	−8.00000	−0.276355
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	4.00000	0.138013
$841$	−25.0000	−0.862069
$842$	−14.0000	−0.482472
$843$	−30.0000	−1.03325
$844$	−12.0000	−0.413057
$845$	9.00000	0.309609
$846$	−8.00000	−0.275046
$847$	20.0000	0.687208
$848$	−6.00000	−0.206041
$849$	20.0000	0.686398
$850$	−2.00000	−0.0685994
$851$	−24.0000	−0.822709
$852$	0	0
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	−40.0000	−1.36877
$855$	−4.00000	−0.136797
$856$	4.00000	0.136717
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	8.00000	0.273115
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	4.00000	0.136399
$861$	40.0000	1.36320
$862$	24.0000	0.817443
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	−1.00000	−0.0340207
$865$	14.0000	0.476014
$866$	26.0000	0.883516
$867$	−13.0000	−0.441503
$868$	−4.00000	−0.135769
$869$	32.0000	1.08553
$870$	−2.00000	−0.0678064
$871$	−24.0000	−0.813209
$872$	−6.00000	−0.203186
$873$	−6.00000	−0.203069
$874$	−16.0000	−0.541208
$875$	−4.00000	−0.135225
$876$	14.0000	0.473016
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	−32.0000	−1.07995
$879$	−6.00000	−0.202375
$880$	4.00000	0.134840
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	−9.00000	−0.303046
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	4.00000	0.134535
$885$	−8.00000	−0.268917
$886$	−28.0000	−0.940678
$887$	40.0000	1.34307	0.671534	−	0.740973i	$-0.265636\pi$
$887$	40.0000	1.34307	0.671534	+	0.740973i	$0.265636\pi$
$888$	6.00000	0.201347
$889$	−32.0000	−1.07325
$890$	−6.00000	−0.201120
$891$	−4.00000	−0.134005
$892$	16.0000	0.535720
$893$	32.0000	1.07084
$894$	6.00000	0.200670
$895$	20.0000	0.668526
$896$	−4.00000	−0.133631
$897$	8.00000	0.267112
$898$	14.0000	0.467186
$899$	2.00000	0.0667037
$900$	1.00000	0.0333333
$901$	−12.0000	−0.399778
$902$	40.0000	1.33185
$903$	−16.0000	−0.532447
$904$	−6.00000	−0.199557
$905$	6.00000	0.199447
$906$	−8.00000	−0.265782
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	4.00000	0.132745
$909$	18.0000	0.597022
$910$	8.00000	0.265197
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	4.00000	0.132453
$913$	−16.0000	−0.529523
$914$	18.0000	0.595387
$915$	−10.0000	−0.330590
$916$	−14.0000	−0.462573
$917$	−32.0000	−1.05673
$918$	−2.00000	−0.0660098
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	4.00000	0.131876
$921$	−20.0000	−0.659022
$922$	−38.0000	−1.25146
$923$	0	0
$924$	−16.0000	−0.526361
$925$	−6.00000	−0.197279
$926$	24.0000	0.788689
$927$	4.00000	0.131377
$928$	2.00000	0.0656532
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	−1.00000	−0.0327913
$931$	36.0000	1.17985
$932$	−10.0000	−0.327561
$933$	24.0000	0.785725
$934$	12.0000	0.392652
$935$	8.00000	0.261628
$936$	−2.00000	−0.0653720
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	48.0000	1.56726
$939$	−34.0000	−1.10955
$940$	−8.00000	−0.260931
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	22.0000	0.716799
$943$	40.0000	1.30258
$944$	8.00000	0.260378
$945$	−4.00000	−0.130120
$946$	−16.0000	−0.520205
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	−8.00000	−0.259828
$949$	28.0000	0.908918
$950$	−4.00000	−0.129777
$951$	2.00000	0.0648544
$952$	−8.00000	−0.259281
$953$	−38.0000	−1.23094	−0.615470	−	0.788160i	$-0.711034\pi$
$953$	−38.0000	−1.23094	−0.615470	+	0.788160i	$0.711034\pi$
$954$	6.00000	0.194257
$955$	−24.0000	−0.776622
$956$	8.00000	0.258738
$957$	8.00000	0.258603
$958$	16.0000	0.516937
$959$	−24.0000	−0.775000
$960$	−1.00000	−0.0322749
$961$	1.00000	0.0322581
$962$	12.0000	0.386896
$963$	−4.00000	−0.128898
$964$	−14.0000	−0.450910
$965$	14.0000	0.450676
$966$	−16.0000	−0.514792
$967$	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
$969$	8.00000	0.256997
$970$	−6.00000	−0.192648
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	1.00000	0.0320750
$973$	64.0000	2.05175
$974$	−24.0000	−0.769010
$975$	2.00000	0.0640513
$976$	10.0000	0.320092
$977$	−10.0000	−0.319928	−0.159964	−	0.987123i	$-0.551138\pi$
$977$	−10.0000	−0.319928	−0.159964	+	0.987123i	$0.551138\pi$
$978$	−12.0000	−0.383718
$979$	24.0000	0.767043
$980$	−9.00000	−0.287494
$981$	6.00000	0.191565
$982$	20.0000	0.638226
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	−10.0000	−0.318788
$985$	14.0000	0.446077
$986$	4.00000	0.127386
$987$	32.0000	1.01857
$988$	8.00000	0.254514
$989$	−16.0000	−0.508770
$990$	−4.00000	−0.127128
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	1.00000	0.0317500
$993$	24.0000	0.761617
$994$	0	0
$995$	8.00000	0.253617
$996$	4.00000	0.126745
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	24.0000	0.759707
$999$	−6.00000	−0.189832

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.g.1.1	✓	1	1.1	even	1	trivial
2790.2.a.bc.1.1		1	3.2	odd	2
4650.2.a.w.1.1		1	5.4	even	2
4650.2.d.c.3349.1		2	5.2	odd	4
4650.2.d.c.3349.2		2	5.3	odd	4
7440.2.a.a.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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