LMFDB - Embedded newform 930.2.a.e.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,-1,1,1,1,3] 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)

$f(q)$

$=$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +8.00000 q^{17} -1.00000 q^{18} -7.00000 q^{19} +1.00000 q^{20} -3.00000 q^{21} -3.00000 q^{22} +7.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} -8.00000 q^{29} +1.00000 q^{30} -1.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} -8.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} -4.00000 q^{37} +7.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} +3.00000 q^{42} +1.00000 q^{43} +3.00000 q^{44} +1.00000 q^{45} -7.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} -8.00000 q^{51} -2.00000 q^{52} +5.00000 q^{53} +1.00000 q^{54} +3.00000 q^{55} -3.00000 q^{56} +7.00000 q^{57} +8.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} +1.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +3.00000 q^{66} +10.0000 q^{67} +8.00000 q^{68} -7.00000 q^{69} -3.00000 q^{70} +9.00000 q^{71} -1.00000 q^{72} +1.00000 q^{73} +4.00000 q^{74} -1.00000 q^{75} -7.00000 q^{76} +9.00000 q^{77} -2.00000 q^{78} +13.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -16.0000 q^{83} -3.00000 q^{84} +8.00000 q^{85} -1.00000 q^{86} +8.00000 q^{87} -3.00000 q^{88} -3.00000 q^{89} -1.00000 q^{90} -6.00000 q^{91} +7.00000 q^{92} +1.00000 q^{93} -6.00000 q^{94} -7.00000 q^{95} +1.00000 q^{96} +6.00000 q^{97} -2.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−3.00000	−0.801784
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	8.00000	1.94029	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000	1.94029	0.970143	+	0.242536i	$0.0779791\pi$
$18$	−1.00000	−0.235702
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	1.00000	0.223607
$21$	−3.00000	−0.654654
$22$	−3.00000	−0.639602
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	3.00000	0.566947
$29$	−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
$33$	−3.00000	−0.522233
$34$	−8.00000	−1.37199
$35$	3.00000	0.507093
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	7.00000	1.13555
$39$	2.00000	0.320256
$40$	−1.00000	−0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	3.00000	0.462910
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	3.00000	0.452267
$45$	1.00000	0.149071
$46$	−7.00000	−1.03209
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	−8.00000	−1.12022
$52$	−2.00000	−0.277350
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	1.00000	0.136083
$55$	3.00000	0.404520
$56$	−3.00000	−0.400892
$57$	7.00000	0.927173
$58$	8.00000	1.05045
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	−1.00000	−0.129099
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	1.00000	0.127000
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	3.00000	0.369274
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	8.00000	0.970143
$69$	−7.00000	−0.842701
$70$	−3.00000	−0.358569
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	−1.00000	−0.117851
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	4.00000	0.464991
$75$	−1.00000	−0.115470
$76$	−7.00000	−0.802955
$77$	9.00000	1.02565
$78$	−2.00000	−0.226455
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	−3.00000	−0.327327
$85$	8.00000	0.867722
$86$	−1.00000	−0.107833
$87$	8.00000	0.857690
$88$	−3.00000	−0.319801
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	−1.00000	−0.105409
$91$	−6.00000	−0.628971
$92$	7.00000	0.729800
$93$	1.00000	0.103695
$94$	−6.00000	−0.618853
$95$	−7.00000	−0.718185
$96$	1.00000	0.102062
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	−2.00000	−0.202031
$99$	3.00000	0.301511
$100$	1.00000	0.100000
$101$	5.00000	0.497519	0.248759	−	0.968565i	$-0.419977\pi$
$101$	5.00000	0.497519	0.248759	+	0.968565i	$0.419977\pi$
$102$	8.00000	0.792118
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	2.00000	0.196116
$105$	−3.00000	−0.292770
$106$	−5.00000	−0.485643
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	−1.00000	−0.0962250
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	−3.00000	−0.286039
$111$	4.00000	0.379663
$112$	3.00000	0.283473
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	−7.00000	−0.655610
$115$	7.00000	0.652753
$116$	−8.00000	−0.742781
$117$	−2.00000	−0.184900
$118$	−6.00000	−0.552345
$119$	24.0000	2.20008
$120$	1.00000	0.0912871
$121$	−2.00000	−0.181818
$122$	−2.00000	−0.181071
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	1.00000	0.0894427
$126$	−3.00000	−0.267261
$127$	−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
$129$	−1.00000	−0.0880451
$130$	2.00000	0.175412
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	−3.00000	−0.261116
$133$	−21.0000	−1.82093
$134$	−10.0000	−0.863868
$135$	−1.00000	−0.0860663
$136$	−8.00000	−0.685994
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	7.00000	0.595880
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	3.00000	0.253546
$141$	−6.00000	−0.505291
$142$	−9.00000	−0.755263
$143$	−6.00000	−0.501745
$144$	1.00000	0.0833333
$145$	−8.00000	−0.664364
$146$	−1.00000	−0.0827606
$147$	−2.00000	−0.164957
$148$	−4.00000	−0.328798
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	1.00000	0.0816497
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	7.00000	0.567775
$153$	8.00000	0.646762
$154$	−9.00000	−0.725241
$155$	−1.00000	−0.0803219
$156$	2.00000	0.160128
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	−13.0000	−1.03422
$159$	−5.00000	−0.396526
$160$	−1.00000	−0.0790569
$161$	21.0000	1.65503
$162$	−1.00000	−0.0785674
$163$	18.0000	1.40987	0.704934	−	0.709273i	$-0.250976\pi$
$163$	18.0000	1.40987	0.704934	+	0.709273i	$0.250976\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	16.0000	1.24184
$167$	15.0000	1.16073	0.580367	−	0.814355i	$-0.302909\pi$
$167$	15.0000	1.16073	0.580367	+	0.814355i	$0.302909\pi$
$168$	3.00000	0.231455
$169$	−9.00000	−0.692308
$170$	−8.00000	−0.613572
$171$	−7.00000	−0.535303
$172$	1.00000	0.0762493
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−8.00000	−0.606478
$175$	3.00000	0.226779
$176$	3.00000	0.226134
$177$	−6.00000	−0.450988
$178$	3.00000	0.224860
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	1.00000	0.0745356
$181$	−21.0000	−1.56092	−0.780459	−	0.625207i	$-0.785014\pi$
$181$	−21.0000	−1.56092	−0.780459	+	0.625207i	$0.785014\pi$
$182$	6.00000	0.444750
$183$	−2.00000	−0.147844
$184$	−7.00000	−0.516047
$185$	−4.00000	−0.294086
$186$	−1.00000	−0.0733236
$187$	24.0000	1.75505
$188$	6.00000	0.437595
$189$	−3.00000	−0.218218
$190$	7.00000	0.507833
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	−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
$196$	2.00000	0.142857
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−3.00000	−0.213201
$199$	−3.00000	−0.212664	−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000	−0.212664	−0.106332	+	0.994331i	$0.533911\pi$
$200$	−1.00000	−0.0707107
$201$	−10.0000	−0.705346
$202$	−5.00000	−0.351799
$203$	−24.0000	−1.68447
$204$	−8.00000	−0.560112
$205$	0	0
$206$	−8.00000	−0.557386
$207$	7.00000	0.486534
$208$	−2.00000	−0.138675
$209$	−21.0000	−1.45260
$210$	3.00000	0.207020
$211$	17.0000	1.17033	0.585164	−	0.810915i	$-0.301030\pi$
$211$	17.0000	1.17033	0.585164	+	0.810915i	$0.301030\pi$
$212$	5.00000	0.343401
$213$	−9.00000	−0.616670
$214$	3.00000	0.205076
$215$	1.00000	0.0681994
$216$	1.00000	0.0680414
$217$	−3.00000	−0.203653
$218$	−4.00000	−0.270914
$219$	−1.00000	−0.0675737
$220$	3.00000	0.202260
$221$	−16.0000	−1.07628
$222$	−4.00000	−0.268462
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	−3.00000	−0.200446
$225$	1.00000	0.0666667
$226$	−1.00000	−0.0665190
$227$	25.0000	1.65931	0.829654	−	0.558278i	$-0.188538\pi$
$227$	25.0000	1.65931	0.829654	+	0.558278i	$0.188538\pi$
$228$	7.00000	0.463586
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	−7.00000	−0.461566
$231$	−9.00000	−0.592157
$232$	8.00000	0.525226
$233$	−25.0000	−1.63780	−0.818902	−	0.573933i	$-0.805417\pi$
$233$	−25.0000	−1.63780	−0.818902	+	0.573933i	$0.805417\pi$
$234$	2.00000	0.130744
$235$	6.00000	0.391397
$236$	6.00000	0.390567
$237$	−13.0000	−0.844441
$238$	−24.0000	−1.55569
$239$	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
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	2.00000	0.127775
$246$	0	0
$247$	14.0000	0.890799
$248$	1.00000	0.0635001
$249$	16.0000	1.01396
$250$	−1.00000	−0.0632456
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	3.00000	0.188982
$253$	21.0000	1.32026
$254$	12.0000	0.752947
$255$	−8.00000	−0.500979
$256$	1.00000	0.0625000
$257$	−23.0000	−1.43470	−0.717350	−	0.696713i	$-0.754645\pi$
$257$	−23.0000	−1.43470	−0.717350	+	0.696713i	$0.754645\pi$
$258$	1.00000	0.0622573
$259$	−12.0000	−0.745644
$260$	−2.00000	−0.124035
$261$	−8.00000	−0.495188
$262$	−6.00000	−0.370681
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	3.00000	0.184637
$265$	5.00000	0.307148
$266$	21.0000	1.28759
$267$	3.00000	0.183597
$268$	10.0000	0.610847
$269$	−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
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	8.00000	0.485071
$273$	6.00000	0.363137
$274$	−10.0000	−0.604122
$275$	3.00000	0.180907
$276$	−7.00000	−0.421350
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	10.0000	0.599760
$279$	−1.00000	−0.0598684
$280$	−3.00000	−0.179284
$281$	−28.0000	−1.67034	−0.835170	−	0.549992i	$-0.814631\pi$
$281$	−28.0000	−1.67034	−0.835170	+	0.549992i	$0.814631\pi$
$282$	6.00000	0.357295
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	9.00000	0.534052
$285$	7.00000	0.414644
$286$	6.00000	0.354787
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	47.0000	2.76471
$290$	8.00000	0.469776
$291$	−6.00000	−0.351726
$292$	1.00000	0.0585206
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	2.00000	0.116642
$295$	6.00000	0.349334
$296$	4.00000	0.232495
$297$	−3.00000	−0.174078
$298$	9.00000	0.521356
$299$	−14.0000	−0.809641
$300$	−1.00000	−0.0577350
$301$	3.00000	0.172917
$302$	8.00000	0.460348
$303$	−5.00000	−0.287242
$304$	−7.00000	−0.401478
$305$	2.00000	0.114520
$306$	−8.00000	−0.457330
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	9.00000	0.512823
$309$	−8.00000	−0.455104
$310$	1.00000	0.0567962
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	−2.00000	−0.113228
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	7.00000	0.395033
$315$	3.00000	0.169031
$316$	13.0000	0.731307
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	5.00000	0.280386
$319$	−24.0000	−1.34374
$320$	1.00000	0.0559017
$321$	3.00000	0.167444
$322$	−21.0000	−1.17028
$323$	−56.0000	−3.11592
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	−18.0000	−0.996928
$327$	−4.00000	−0.221201
$328$	0	0
$329$	18.0000	0.992372
$330$	3.00000	0.165145
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	−16.0000	−0.878114
$333$	−4.00000	−0.219199
$334$	−15.0000	−0.820763
$335$	10.0000	0.546358
$336$	−3.00000	−0.163663
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	9.00000	0.489535
$339$	−1.00000	−0.0543125
$340$	8.00000	0.433861
$341$	−3.00000	−0.162459
$342$	7.00000	0.378517
$343$	−15.0000	−0.809924
$344$	−1.00000	−0.0539164
$345$	−7.00000	−0.376867
$346$	6.00000	0.322562
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	8.00000	0.428845
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	−3.00000	−0.160357
$351$	2.00000	0.106752
$352$	−3.00000	−0.159901
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	6.00000	0.318896
$355$	9.00000	0.477670
$356$	−3.00000	−0.159000
$357$	−24.0000	−1.27021
$358$	−4.00000	−0.211407
$359$	1.00000	0.0527780	0.0263890	−	0.999652i	$-0.491599\pi$
$359$	1.00000	0.0527780	0.0263890	+	0.999652i	$0.491599\pi$
$360$	−1.00000	−0.0527046
$361$	30.0000	1.57895
$362$	21.0000	1.10374
$363$	2.00000	0.104973
$364$	−6.00000	−0.314485
$365$	1.00000	0.0523424
$366$	2.00000	0.104542
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	7.00000	0.364900
$369$	0	0
$370$	4.00000	0.207950
$371$	15.0000	0.778761
$372$	1.00000	0.0518476
$373$	13.0000	0.673114	0.336557	−	0.941663i	$-0.390737\pi$
$373$	13.0000	0.673114	0.336557	+	0.941663i	$0.390737\pi$
$374$	−24.0000	−1.24101
$375$	−1.00000	−0.0516398
$376$	−6.00000	−0.309426
$377$	16.0000	0.824042
$378$	3.00000	0.154303
$379$	19.0000	0.975964	0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000	0.975964	0.487982	+	0.872854i	$0.337733\pi$
$380$	−7.00000	−0.359092
$381$	12.0000	0.614779
$382$	24.0000	1.22795
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	1.00000	0.0510310
$385$	9.00000	0.458682
$386$	14.0000	0.712581
$387$	1.00000	0.0508329
$388$	6.00000	0.304604
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	−2.00000	−0.101274
$391$	56.0000	2.83204
$392$	−2.00000	−0.101015
$393$	−6.00000	−0.302660
$394$	−6.00000	−0.302276
$395$	13.0000	0.654101
$396$	3.00000	0.150756
$397$	25.0000	1.25471	0.627357	−	0.778732i	$-0.284137\pi$
$397$	25.0000	1.25471	0.627357	+	0.778732i	$0.284137\pi$
$398$	3.00000	0.150376
$399$	21.0000	1.05131
$400$	1.00000	0.0500000
$401$	−35.0000	−1.74782	−0.873908	−	0.486091i	$-0.838422\pi$
$401$	−35.0000	−1.74782	−0.873908	+	0.486091i	$0.838422\pi$
$402$	10.0000	0.498755
$403$	2.00000	0.0996271
$404$	5.00000	0.248759
$405$	1.00000	0.0496904
$406$	24.0000	1.19110
$407$	−12.0000	−0.594818
$408$	8.00000	0.396059
$409$	12.0000	0.593362	0.296681	−	0.954977i	$-0.404120\pi$
$409$	12.0000	0.593362	0.296681	+	0.954977i	$0.404120\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	8.00000	0.394132
$413$	18.0000	0.885722
$414$	−7.00000	−0.344031
$415$	−16.0000	−0.785409
$416$	2.00000	0.0980581
$417$	10.0000	0.489702
$418$	21.0000	1.02714
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	−3.00000	−0.146385
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	−17.0000	−0.827547
$423$	6.00000	0.291730
$424$	−5.00000	−0.242821
$425$	8.00000	0.388057
$426$	9.00000	0.436051
$427$	6.00000	0.290360
$428$	−3.00000	−0.145010
$429$	6.00000	0.289683
$430$	−1.00000	−0.0482243
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	−1.00000	−0.0481125
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	3.00000	0.144005
$435$	8.00000	0.383571
$436$	4.00000	0.191565
$437$	−49.0000	−2.34399
$438$	1.00000	0.0477818
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	−3.00000	−0.143019
$441$	2.00000	0.0952381
$442$	16.0000	0.761042
$443$	19.0000	0.902717	0.451359	−	0.892343i	$-0.350940\pi$
$443$	19.0000	0.902717	0.451359	+	0.892343i	$0.350940\pi$
$444$	4.00000	0.189832
$445$	−3.00000	−0.142214
$446$	10.0000	0.473514
$447$	9.00000	0.425685
$448$	3.00000	0.141737
$449$	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
$451$	0	0
$452$	1.00000	0.0470360
$453$	8.00000	0.375873
$454$	−25.0000	−1.17331
$455$	−6.00000	−0.281284
$456$	−7.00000	−0.327805
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	13.0000	0.607450
$459$	−8.00000	−0.373408
$460$	7.00000	0.326377
$461$	−20.0000	−0.931493	−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000	−0.931493	−0.465746	+	0.884918i	$0.654214\pi$
$462$	9.00000	0.418718
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	−8.00000	−0.371391
$465$	1.00000	0.0463739
$466$	25.0000	1.15810
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	−2.00000	−0.0924500
$469$	30.0000	1.38527
$470$	−6.00000	−0.276759
$471$	7.00000	0.322543
$472$	−6.00000	−0.276172
$473$	3.00000	0.137940
$474$	13.0000	0.597110
$475$	−7.00000	−0.321182
$476$	24.0000	1.10004
$477$	5.00000	0.228934
$478$	−8.00000	−0.365911
$479$	−27.0000	−1.23366	−0.616831	−	0.787096i	$-0.711584\pi$
$479$	−27.0000	−1.23366	−0.616831	+	0.787096i	$0.711584\pi$
$480$	1.00000	0.0456435
$481$	8.00000	0.364769
$482$	22.0000	1.00207
$483$	−21.0000	−0.955533
$484$	−2.00000	−0.0909091
$485$	6.00000	0.272446
$486$	1.00000	0.0453609
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	−2.00000	−0.0905357
$489$	−18.0000	−0.813988
$490$	−2.00000	−0.0903508
$491$	−29.0000	−1.30875	−0.654376	−	0.756169i	$-0.727069\pi$
$491$	−29.0000	−1.30875	−0.654376	+	0.756169i	$0.727069\pi$
$492$	0	0
$493$	−64.0000	−2.88242
$494$	−14.0000	−0.629890
$495$	3.00000	0.134840
$496$	−1.00000	−0.0449013
$497$	27.0000	1.21112
$498$	−16.0000	−0.716977
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	1.00000	0.0447214
$501$	−15.0000	−0.670151
$502$	−20.0000	−0.892644
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	−3.00000	−0.133631
$505$	5.00000	0.222497
$506$	−21.0000	−0.933564
$507$	9.00000	0.399704
$508$	−12.0000	−0.532414
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	8.00000	0.354246
$511$	3.00000	0.132712
$512$	−1.00000	−0.0441942
$513$	7.00000	0.309058
$514$	23.0000	1.01449
$515$	8.00000	0.352522
$516$	−1.00000	−0.0440225
$517$	18.0000	0.791639
$518$	12.0000	0.527250
$519$	6.00000	0.263371
$520$	2.00000	0.0877058
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	8.00000	0.350150
$523$	29.0000	1.26808	0.634041	−	0.773300i	$-0.281395\pi$
$523$	29.0000	1.26808	0.634041	+	0.773300i	$0.281395\pi$
$524$	6.00000	0.262111
$525$	−3.00000	−0.130931
$526$	−24.0000	−1.04645
$527$	−8.00000	−0.348485
$528$	−3.00000	−0.130558
$529$	26.0000	1.13043
$530$	−5.00000	−0.217186
$531$	6.00000	0.260378
$532$	−21.0000	−0.910465
$533$	0	0
$534$	−3.00000	−0.129823
$535$	−3.00000	−0.129701
$536$	−10.0000	−0.431934
$537$	−4.00000	−0.172613
$538$	18.0000	0.776035
$539$	6.00000	0.258438
$540$	−1.00000	−0.0430331
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	−7.00000	−0.300676
$543$	21.0000	0.901196
$544$	−8.00000	−0.342997
$545$	4.00000	0.171341
$546$	−6.00000	−0.256776
$547$	−14.0000	−0.598597	−0.299298	−	0.954160i	$-0.596753\pi$
$547$	−14.0000	−0.598597	−0.299298	+	0.954160i	$0.596753\pi$
$548$	10.0000	0.427179
$549$	2.00000	0.0853579
$550$	−3.00000	−0.127920
$551$	56.0000	2.38568
$552$	7.00000	0.297940
$553$	39.0000	1.65845
$554$	8.00000	0.339887
$555$	4.00000	0.169791
$556$	−10.0000	−0.424094
$557$	17.0000	0.720313	0.360157	−	0.932892i	$-0.382723\pi$
$557$	17.0000	0.720313	0.360157	+	0.932892i	$0.382723\pi$
$558$	1.00000	0.0423334
$559$	−2.00000	−0.0845910
$560$	3.00000	0.126773
$561$	−24.0000	−1.01328
$562$	28.0000	1.18111
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	−6.00000	−0.252646
$565$	1.00000	0.0420703
$566$	16.0000	0.672530
$567$	3.00000	0.125988
$568$	−9.00000	−0.377632
$569$	37.0000	1.55112	0.775560	−	0.631273i	$-0.217467\pi$
$569$	37.0000	1.55112	0.775560	+	0.631273i	$0.217467\pi$
$570$	−7.00000	−0.293198
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	−6.00000	−0.250873
$573$	24.0000	1.00261
$574$	0	0
$575$	7.00000	0.291920
$576$	1.00000	0.0416667
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	−47.0000	−1.95494
$579$	14.0000	0.581820
$580$	−8.00000	−0.332182
$581$	−48.0000	−1.99138
$582$	6.00000	0.248708
$583$	15.0000	0.621237
$584$	−1.00000	−0.0413803
$585$	−2.00000	−0.0826898
$586$	−14.0000	−0.578335
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	−2.00000	−0.0824786
$589$	7.00000	0.288430
$590$	−6.00000	−0.247016
$591$	−6.00000	−0.246807
$592$	−4.00000	−0.164399
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	3.00000	0.123091
$595$	24.0000	0.983904
$596$	−9.00000	−0.368654
$597$	3.00000	0.122782
$598$	14.0000	0.572503
$599$	35.0000	1.43006	0.715031	−	0.699093i	$-0.246413\pi$
$599$	35.0000	1.43006	0.715031	+	0.699093i	$0.246413\pi$
$600$	1.00000	0.0408248
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	−3.00000	−0.122271
$603$	10.0000	0.407231
$604$	−8.00000	−0.325515
$605$	−2.00000	−0.0813116
$606$	5.00000	0.203111
$607$	−13.0000	−0.527654	−0.263827	−	0.964570i	$-0.584985\pi$
$607$	−13.0000	−0.527654	−0.263827	+	0.964570i	$0.584985\pi$
$608$	7.00000	0.283887
$609$	24.0000	0.972529
$610$	−2.00000	−0.0809776
$611$	−12.0000	−0.485468
$612$	8.00000	0.323381
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	−9.00000	−0.362620
$617$	−3.00000	−0.120775	−0.0603877	−	0.998175i	$-0.519234\pi$
$617$	−3.00000	−0.120775	−0.0603877	+	0.998175i	$0.519234\pi$
$618$	8.00000	0.321807
$619$	−18.0000	−0.723481	−0.361741	−	0.932279i	$-0.617817\pi$
$619$	−18.0000	−0.723481	−0.361741	+	0.932279i	$0.617817\pi$
$620$	−1.00000	−0.0401610
$621$	−7.00000	−0.280900
$622$	16.0000	0.641542
$623$	−9.00000	−0.360577
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	10.0000	0.399680
$627$	21.0000	0.838659
$628$	−7.00000	−0.279330
$629$	−32.0000	−1.27592
$630$	−3.00000	−0.119523
$631$	−5.00000	−0.199047	−0.0995234	−	0.995035i	$-0.531732\pi$
$631$	−5.00000	−0.199047	−0.0995234	+	0.995035i	$0.531732\pi$
$632$	−13.0000	−0.517112
$633$	−17.0000	−0.675689
$634$	−24.0000	−0.953162
$635$	−12.0000	−0.476205
$636$	−5.00000	−0.198263
$637$	−4.00000	−0.158486
$638$	24.0000	0.950169
$639$	9.00000	0.356034
$640$	−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	$0.701845\pi$
$642$	−3.00000	−0.118401
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	21.0000	0.827516
$645$	−1.00000	−0.0393750
$646$	56.0000	2.20329
$647$	49.0000	1.92639	0.963194	−	0.268806i	$-0.0866290\pi$
$647$	49.0000	1.92639	0.963194	+	0.268806i	$0.0866290\pi$
$648$	−1.00000	−0.0392837
$649$	18.0000	0.706562
$650$	2.00000	0.0784465
$651$	3.00000	0.117579
$652$	18.0000	0.704934
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	4.00000	0.156412
$655$	6.00000	0.234439
$656$	0	0
$657$	1.00000	0.0390137
$658$	−18.0000	−0.701713
$659$	−32.0000	−1.24654	−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000	−1.24654	−0.623272	+	0.782006i	$0.714197\pi$
$660$	−3.00000	−0.116775
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	−32.0000	−1.24372
$663$	16.0000	0.621389
$664$	16.0000	0.620920
$665$	−21.0000	−0.814345
$666$	4.00000	0.154997
$667$	−56.0000	−2.16833
$668$	15.0000	0.580367
$669$	10.0000	0.386622
$670$	−10.0000	−0.386334
$671$	6.00000	0.231627
$672$	3.00000	0.115728
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	10.0000	0.385186
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	−43.0000	−1.65262	−0.826312	−	0.563212i	$-0.809565\pi$
$677$	−43.0000	−1.65262	−0.826312	+	0.563212i	$0.809565\pi$
$678$	1.00000	0.0384048
$679$	18.0000	0.690777
$680$	−8.00000	−0.306786
$681$	−25.0000	−0.958002
$682$	3.00000	0.114876
$683$	−15.0000	−0.573959	−0.286980	−	0.957937i	$-0.592651\pi$
$683$	−15.0000	−0.573959	−0.286980	+	0.957937i	$0.592651\pi$
$684$	−7.00000	−0.267652
$685$	10.0000	0.382080
$686$	15.0000	0.572703
$687$	13.0000	0.495981
$688$	1.00000	0.0381246
$689$	−10.0000	−0.380970
$690$	7.00000	0.266485
$691$	5.00000	0.190209	0.0951045	−	0.995467i	$-0.469681\pi$
$691$	5.00000	0.190209	0.0951045	+	0.995467i	$0.469681\pi$
$692$	−6.00000	−0.228086
$693$	9.00000	0.341882
$694$	12.0000	0.455514
$695$	−10.0000	−0.379322
$696$	−8.00000	−0.303239
$697$	0	0
$698$	−20.0000	−0.757011
$699$	25.0000	0.945587
$700$	3.00000	0.113389
$701$	33.0000	1.24639	0.623196	−	0.782065i	$-0.285834\pi$
$701$	33.0000	1.24639	0.623196	+	0.782065i	$0.285834\pi$
$702$	−2.00000	−0.0754851
$703$	28.0000	1.05604
$704$	3.00000	0.113067
$705$	−6.00000	−0.225973
$706$	18.0000	0.677439
$707$	15.0000	0.564133
$708$	−6.00000	−0.225494
$709$	−15.0000	−0.563337	−0.281668	−	0.959512i	$-0.590888\pi$
$709$	−15.0000	−0.563337	−0.281668	+	0.959512i	$0.590888\pi$
$710$	−9.00000	−0.337764
$711$	13.0000	0.487538
$712$	3.00000	0.112430
$713$	−7.00000	−0.262152
$714$	24.0000	0.898177
$715$	−6.00000	−0.224387
$716$	4.00000	0.149487
$717$	−8.00000	−0.298765
$718$	−1.00000	−0.0373197
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	1.00000	0.0372678
$721$	24.0000	0.893807
$722$	−30.0000	−1.11648
$723$	22.0000	0.818189
$724$	−21.0000	−0.780459
$725$	−8.00000	−0.297113
$726$	−2.00000	−0.0742270
$727$	−29.0000	−1.07555	−0.537775	−	0.843088i	$-0.680735\pi$
$727$	−29.0000	−1.07555	−0.537775	+	0.843088i	$0.680735\pi$
$728$	6.00000	0.222375
$729$	1.00000	0.0370370
$730$	−1.00000	−0.0370117
$731$	8.00000	0.295891
$732$	−2.00000	−0.0739221
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	18.0000	0.664392
$735$	−2.00000	−0.0737711
$736$	−7.00000	−0.258023
$737$	30.0000	1.10506
$738$	0	0
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	−4.00000	−0.147043
$741$	−14.0000	−0.514303
$742$	−15.0000	−0.550667
$743$	1.00000	0.0366864	0.0183432	−	0.999832i	$-0.494161\pi$
$743$	1.00000	0.0366864	0.0183432	+	0.999832i	$0.494161\pi$
$744$	−1.00000	−0.0366618
$745$	−9.00000	−0.329734
$746$	−13.0000	−0.475964
$747$	−16.0000	−0.585409
$748$	24.0000	0.877527
$749$	−9.00000	−0.328853
$750$	1.00000	0.0365148
$751$	6.00000	0.218943	0.109472	−	0.993990i	$-0.465084\pi$
$751$	6.00000	0.218943	0.109472	+	0.993990i	$0.465084\pi$
$752$	6.00000	0.218797
$753$	−20.0000	−0.728841
$754$	−16.0000	−0.582686
$755$	−8.00000	−0.291150
$756$	−3.00000	−0.109109
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−19.0000	−0.690111
$759$	−21.0000	−0.762252
$760$	7.00000	0.253917
$761$	49.0000	1.77625	0.888124	−	0.459603i	$-0.152008\pi$
$761$	49.0000	1.77625	0.888124	+	0.459603i	$0.152008\pi$
$762$	−12.0000	−0.434714
$763$	12.0000	0.434429
$764$	−24.0000	−0.868290
$765$	8.00000	0.289241
$766$	36.0000	1.30073
$767$	−12.0000	−0.433295
$768$	−1.00000	−0.0360844
$769$	9.00000	0.324548	0.162274	−	0.986746i	$-0.448117\pi$
$769$	9.00000	0.324548	0.162274	+	0.986746i	$0.448117\pi$
$770$	−9.00000	−0.324337
$771$	23.0000	0.828325
$772$	−14.0000	−0.503871
$773$	−1.00000	−0.0359675	−0.0179838	−	0.999838i	$-0.505725\pi$
$773$	−1.00000	−0.0359675	−0.0179838	+	0.999838i	$0.505725\pi$
$774$	−1.00000	−0.0359443
$775$	−1.00000	−0.0359211
$776$	−6.00000	−0.215387
$777$	12.0000	0.430498
$778$	22.0000	0.788738
$779$	0	0
$780$	2.00000	0.0716115
$781$	27.0000	0.966136
$782$	−56.0000	−2.00256
$783$	8.00000	0.285897
$784$	2.00000	0.0714286
$785$	−7.00000	−0.249841
$786$	6.00000	0.214013
$787$	−27.0000	−0.962446	−0.481223	−	0.876598i	$-0.659807\pi$
$787$	−27.0000	−0.962446	−0.481223	+	0.876598i	$0.659807\pi$
$788$	6.00000	0.213741
$789$	−24.0000	−0.854423
$790$	−13.0000	−0.462519
$791$	3.00000	0.106668
$792$	−3.00000	−0.106600
$793$	−4.00000	−0.142044
$794$	−25.0000	−0.887217
$795$	−5.00000	−0.177332
$796$	−3.00000	−0.106332
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	−21.0000	−0.743392
$799$	48.0000	1.69812
$800$	−1.00000	−0.0353553
$801$	−3.00000	−0.106000
$802$	35.0000	1.23589
$803$	3.00000	0.105868
$804$	−10.0000	−0.352673
$805$	21.0000	0.740153
$806$	−2.00000	−0.0704470
$807$	18.0000	0.633630
$808$	−5.00000	−0.175899
$809$	13.0000	0.457056	0.228528	−	0.973537i	$-0.426609\pi$
$809$	13.0000	0.457056	0.228528	+	0.973537i	$0.426609\pi$
$810$	−1.00000	−0.0351364
$811$	−23.0000	−0.807639	−0.403820	−	0.914839i	$-0.632318\pi$
$811$	−23.0000	−0.807639	−0.403820	+	0.914839i	$0.632318\pi$
$812$	−24.0000	−0.842235
$813$	−7.00000	−0.245501
$814$	12.0000	0.420600
$815$	18.0000	0.630512
$816$	−8.00000	−0.280056
$817$	−7.00000	−0.244899
$818$	−12.0000	−0.419570
$819$	−6.00000	−0.209657
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	10.0000	0.348790
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	−8.00000	−0.278693
$825$	−3.00000	−0.104447
$826$	−18.0000	−0.626300
$827$	−44.0000	−1.53003	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000	−1.53003	−0.765015	+	0.644013i	$0.777268\pi$
$828$	7.00000	0.243267
$829$	−21.0000	−0.729360	−0.364680	−	0.931133i	$-0.618822\pi$
$829$	−21.0000	−0.729360	−0.364680	+	0.931133i	$0.618822\pi$
$830$	16.0000	0.555368
$831$	8.00000	0.277517
$832$	−2.00000	−0.0693375
$833$	16.0000	0.554367
$834$	−10.0000	−0.346272
$835$	15.0000	0.519096
$836$	−21.0000	−0.726300
$837$	1.00000	0.0345651
$838$	−16.0000	−0.552711
$839$	−27.0000	−0.932144	−0.466072	−	0.884747i	$-0.654331\pi$
$839$	−27.0000	−0.932144	−0.466072	+	0.884747i	$0.654331\pi$
$840$	3.00000	0.103510
$841$	35.0000	1.20690
$842$	−34.0000	−1.17172
$843$	28.0000	0.964371
$844$	17.0000	0.585164
$845$	−9.00000	−0.309609
$846$	−6.00000	−0.206284
$847$	−6.00000	−0.206162
$848$	5.00000	0.171701
$849$	16.0000	0.549119
$850$	−8.00000	−0.274398
$851$	−28.0000	−0.959828
$852$	−9.00000	−0.308335
$853$	45.0000	1.54077	0.770385	−	0.637579i	$-0.220064\pi$
$853$	45.0000	1.54077	0.770385	+	0.637579i	$0.220064\pi$
$854$	−6.00000	−0.205316
$855$	−7.00000	−0.239395
$856$	3.00000	0.102538
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	−6.00000	−0.204837
$859$	−24.0000	−0.818869	−0.409435	−	0.912339i	$-0.634274\pi$
$859$	−24.0000	−0.818869	−0.409435	+	0.912339i	$0.634274\pi$
$860$	1.00000	0.0340997
$861$	0	0
$862$	−32.0000	−1.08992
$863$	−31.0000	−1.05525	−0.527626	−	0.849477i	$-0.676918\pi$
$863$	−31.0000	−1.05525	−0.527626	+	0.849477i	$0.676918\pi$
$864$	1.00000	0.0340207
$865$	−6.00000	−0.204006
$866$	−19.0000	−0.645646
$867$	−47.0000	−1.59620
$868$	−3.00000	−0.101827
$869$	39.0000	1.32298
$870$	−8.00000	−0.271225
$871$	−20.0000	−0.677674
$872$	−4.00000	−0.135457
$873$	6.00000	0.203069
$874$	49.0000	1.65745
$875$	3.00000	0.101419
$876$	−1.00000	−0.0337869
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	26.0000	0.877457
$879$	−14.0000	−0.472208
$880$	3.00000	0.101130
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	−2.00000	−0.0673435
$883$	−13.0000	−0.437485	−0.218742	−	0.975783i	$-0.570195\pi$
$883$	−13.0000	−0.437485	−0.218742	+	0.975783i	$0.570195\pi$
$884$	−16.0000	−0.538138
$885$	−6.00000	−0.201688
$886$	−19.0000	−0.638317
$887$	22.0000	0.738688	0.369344	−	0.929293i	$-0.379582\pi$
$887$	22.0000	0.738688	0.369344	+	0.929293i	$0.379582\pi$
$888$	−4.00000	−0.134231
$889$	−36.0000	−1.20740
$890$	3.00000	0.100560
$891$	3.00000	0.100504
$892$	−10.0000	−0.334825
$893$	−42.0000	−1.40548
$894$	−9.00000	−0.301005
$895$	4.00000	0.133705
$896$	−3.00000	−0.100223
$897$	14.0000	0.467446
$898$	−34.0000	−1.13459
$899$	8.00000	0.266815
$900$	1.00000	0.0333333
$901$	40.0000	1.33259
$902$	0	0
$903$	−3.00000	−0.0998337
$904$	−1.00000	−0.0332595
$905$	−21.0000	−0.698064
$906$	−8.00000	−0.265782
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	25.0000	0.829654
$909$	5.00000	0.165840
$910$	6.00000	0.198898
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	7.00000	0.231793
$913$	−48.0000	−1.58857
$914$	38.0000	1.25693
$915$	−2.00000	−0.0661180
$916$	−13.0000	−0.429532
$917$	18.0000	0.594412
$918$	8.00000	0.264039
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	−7.00000	−0.230783
$921$	12.0000	0.395413
$922$	20.0000	0.658665
$923$	−18.0000	−0.592477
$924$	−9.00000	−0.296078
$925$	−4.00000	−0.131519
$926$	40.0000	1.31448
$927$	8.00000	0.262754
$928$	8.00000	0.262613
$929$	3.00000	0.0984268	0.0492134	−	0.998788i	$-0.484329\pi$
$929$	3.00000	0.0984268	0.0492134	+	0.998788i	$0.484329\pi$
$930$	−1.00000	−0.0327913
$931$	−14.0000	−0.458831
$932$	−25.0000	−0.818902
$933$	16.0000	0.523816
$934$	0	0
$935$	24.0000	0.784884
$936$	2.00000	0.0653720
$937$	−16.0000	−0.522697	−0.261349	−	0.965244i	$-0.584167\pi$
$937$	−16.0000	−0.522697	−0.261349	+	0.965244i	$0.584167\pi$
$938$	−30.0000	−0.979535
$939$	10.0000	0.326338
$940$	6.00000	0.195698
$941$	−28.0000	−0.912774	−0.456387	−	0.889781i	$-0.650857\pi$
$941$	−28.0000	−0.912774	−0.456387	+	0.889781i	$0.650857\pi$
$942$	−7.00000	−0.228072
$943$	0	0
$944$	6.00000	0.195283
$945$	−3.00000	−0.0975900
$946$	−3.00000	−0.0975384
$947$	38.0000	1.23483	0.617417	−	0.786636i	$-0.288179\pi$
$947$	38.0000	1.23483	0.617417	+	0.786636i	$0.288179\pi$
$948$	−13.0000	−0.422220
$949$	−2.00000	−0.0649227
$950$	7.00000	0.227110
$951$	−24.0000	−0.778253
$952$	−24.0000	−0.777844
$953$	12.0000	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$954$	−5.00000	−0.161881
$955$	−24.0000	−0.776622
$956$	8.00000	0.258738
$957$	24.0000	0.775810
$958$	27.0000	0.872330
$959$	30.0000	0.968751
$960$	−1.00000	−0.0322749
$961$	1.00000	0.0322581
$962$	−8.00000	−0.257930
$963$	−3.00000	−0.0966736
$964$	−22.0000	−0.708572
$965$	−14.0000	−0.450676
$966$	21.0000	0.675664
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	2.00000	0.0642824
$969$	56.0000	1.79898
$970$	−6.00000	−0.192648
$971$	−54.0000	−1.73294	−0.866471	−	0.499227i	$-0.833617\pi$
$971$	−54.0000	−1.73294	−0.866471	+	0.499227i	$0.833617\pi$
$972$	−1.00000	−0.0320750
$973$	−30.0000	−0.961756
$974$	20.0000	0.640841
$975$	2.00000	0.0640513
$976$	2.00000	0.0640184
$977$	46.0000	1.47167	0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000	1.47167	0.735835	+	0.677161i	$0.236790\pi$
$978$	18.0000	0.575577
$979$	−9.00000	−0.287641
$980$	2.00000	0.0638877
$981$	4.00000	0.127710
$982$	29.0000	0.925427
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	64.0000	2.03818
$987$	−18.0000	−0.572946
$988$	14.0000	0.445399
$989$	7.00000	0.222587
$990$	−3.00000	−0.0953463
$991$	5.00000	0.158830	0.0794151	−	0.996842i	$-0.474695\pi$
$991$	5.00000	0.158830	0.0794151	+	0.996842i	$0.474695\pi$
$992$	1.00000	0.0317500
$993$	−32.0000	−1.01549
$994$	−27.0000	−0.856388
$995$	−3.00000	−0.0951064
$996$	16.0000	0.506979
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	20.0000	0.633089
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.a.e.1.1	✓	1
3.2	odd	2		2790.2.a.u.1.1		1
4.3	odd	2		7440.2.a.x.1.1		1
5.2	odd	4		4650.2.d.ba.3349.1		2
5.3	odd	4		4650.2.d.ba.3349.2		2
5.4	even	2		4650.2.a.bk.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.e.1.1	✓	1	1.1	even	1	trivial
2790.2.a.u.1.1		1	3.2	odd	2
4650.2.a.bk.1.1		1	5.4	even	2
4650.2.d.ba.3349.1		2	5.2	odd	4
4650.2.d.ba.3349.2		2	5.3	odd	4
7440.2.a.x.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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