LMFDB - Embedded newform 4650.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(1,4650)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$37.1304369399$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	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	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	−8.00000	−1.94029	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−8.00000	−1.94029	−0.970143	+	0.242536i	$0.922021\pi$
$18$	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	$0.796740\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	3.00000	0.639602
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	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	$0.766494\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	3.00000	0.522233
$34$	−8.00000	−1.37199
$35$	0	0
$36$	1.00000	0.166667
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	−7.00000	−1.13555
$39$	2.00000	0.320256
$40$	0	0
$41$	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	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	−7.00000	−1.03209
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	1.00000	0.144338
$49$	2.00000	0.285714
$50$	0	0
$51$	−8.00000	−1.12022
$52$	2.00000	0.277350
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−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	$0.372279\pi$
$60$	0	0
$61$	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
$65$	0	0
$66$	3.00000	0.369274
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−8.00000	−0.970143
$69$	−7.00000	−0.842701
$70$	0	0
$71$	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	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−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	$0.238911\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	−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	$0.550827\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	−7.00000	−0.729800
$93$	−1.00000	−0.103695
$94$	−6.00000	−0.618853
$95$	0	0
$96$	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	$0.598524\pi$
$98$	2.00000	0.202031
$99$	3.00000	0.301511
$100$	0	0
$101$	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	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−5.00000	−0.485643
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	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	$0.438644\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	−3.00000	−0.283473
$113$	−1.00000	−0.0940721	−0.0470360	−	0.998893i	$-0.514978\pi$
$113$	−1.00000	−0.0940721	−0.0470360	+	0.998893i	$0.514978\pi$
$114$	−7.00000	−0.655610
$115$	0	0
$116$	−8.00000	−0.742781
$117$	2.00000	0.184900
$118$	6.00000	0.552345
$119$	24.0000	2.20008
$120$	0	0
$121$	−2.00000	−0.181818
$122$	2.00000	0.181071
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	−3.00000	−0.267261
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000	0.0883883
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	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
$135$	0	0
$136$	−8.00000	−0.685994
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	−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	$0.639408\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	9.00000	0.755263
$143$	6.00000	0.501745
$144$	1.00000	0.0833333
$145$	0	0
$146$	−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	$0.620181\pi$
$150$	0	0
$151$	−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
$155$	0	0
$156$	2.00000	0.160128
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	13.0000	1.03422
$159$	−5.00000	−0.396526
$160$	0	0
$161$	21.0000	1.65503
$162$	1.00000	0.0785674
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	0	0
$165$	0	0
$166$	16.0000	1.24184
$167$	−15.0000	−1.16073	−0.580367	−	0.814355i	$-0.697091\pi$
$167$	−15.0000	−1.16073	−0.580367	+	0.814355i	$0.697091\pi$
$168$	−3.00000	−0.231455
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−7.00000	−0.535303
$172$	−1.00000	−0.0762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−8.00000	−0.606478
$175$	0	0
$176$	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	$0.452238\pi$
$180$	0	0
$181$	−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
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	−24.0000	−1.75505
$188$	−6.00000	−0.437595
$189$	−3.00000	−0.218218
$190$	0	0
$191$	−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	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	2.00000	0.142857
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	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	$0.533911\pi$
$200$	0	0
$201$	−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
$210$	0	0
$211$	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
$215$	0	0
$216$	1.00000	0.0680414
$217$	3.00000	0.203653
$218$	4.00000	0.270914
$219$	−1.00000	−0.0675737
$220$	0	0
$221$	−16.0000	−1.07628
$222$	4.00000	0.268462
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	−1.00000	−0.0665190
$227$	−25.0000	−1.65931	−0.829654	−	0.558278i	$-0.811462\pi$
$227$	−25.0000	−1.65931	−0.829654	+	0.558278i	$0.811462\pi$
$228$	−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	$0.641321\pi$
$230$	0	0
$231$	−9.00000	−0.592157
$232$	−8.00000	−0.525226
$233$	25.0000	1.63780	0.818902	−	0.573933i	$-0.194583\pi$
$233$	25.0000	1.63780	0.818902	+	0.573933i	$0.194583\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	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	$0.416693\pi$
$240$	0	0
$241$	−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
$245$	0	0
$246$	0	0
$247$	−14.0000	−0.890799
$248$	−1.00000	−0.0635001
$249$	16.0000	1.01396
$250$	0	0
$251$	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
$255$	0	0
$256$	1.00000	0.0625000
$257$	23.0000	1.43470	0.717350	−	0.696713i	$-0.245355\pi$
$257$	23.0000	1.43470	0.717350	+	0.696713i	$0.245355\pi$
$258$	−1.00000	−0.0622573
$259$	−12.0000	−0.745644
$260$	0	0
$261$	−8.00000	−0.495188
$262$	6.00000	0.370681
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	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	$0.684892\pi$
$270$	0	0
$271$	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
$275$	0	0
$276$	−7.00000	−0.421350
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−10.0000	−0.599760
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−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	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	9.00000	0.534052
$285$	0	0
$286$	6.00000	0.354787
$287$	0	0
$288$	1.00000	0.0589256
$289$	47.0000	2.76471
$290$	0	0
$291$	−6.00000	−0.351726
$292$	−1.00000	−0.0585206
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	4.00000	0.232495
$297$	3.00000	0.174078
$298$	−9.00000	−0.521356
$299$	−14.0000	−0.809641
$300$	0	0
$301$	3.00000	0.172917
$302$	−8.00000	−0.460348
$303$	5.00000	0.287242
$304$	−7.00000	−0.401478
$305$	0	0
$306$	−8.00000	−0.457330
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	−9.00000	−0.512823
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−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	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	7.00000	0.395033
$315$	0	0
$316$	13.0000	0.731307
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	−5.00000	−0.280386
$319$	−24.0000	−1.34374
$320$	0	0
$321$	3.00000	0.167444
$322$	21.0000	1.17028
$323$	56.0000	3.11592
$324$	1.00000	0.0555556
$325$	0	0
$326$	−18.0000	−0.996928
$327$	4.00000	0.221201
$328$	0	0
$329$	18.0000	0.992372
$330$	0	0
$331$	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
$335$	0	0
$336$	−3.00000	−0.163663
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	−9.00000	−0.489535
$339$	−1.00000	−0.0543125
$340$	0	0
$341$	−3.00000	−0.162459
$342$	−7.00000	−0.378517
$343$	15.0000	0.809924
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	6.00000	0.322562
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−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	$0.320203\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	3.00000	0.159901
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	−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	$0.491599\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	−21.0000	−1.10374
$363$	−2.00000	−0.104973
$364$	−6.00000	−0.314485
$365$	0	0
$366$	2.00000	0.104542
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	−7.00000	−0.364900
$369$	0	0
$370$	0	0
$371$	15.0000	0.778761
$372$	−1.00000	−0.0518476
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	−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	$0.337733\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	−24.0000	−1.22795
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	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	$0.688325\pi$
$390$	0	0
$391$	56.0000	2.83204
$392$	2.00000	0.101015
$393$	6.00000	0.302660
$394$	−6.00000	−0.302276
$395$	0	0
$396$	3.00000	0.150756
$397$	−25.0000	−1.25471	−0.627357	−	0.778732i	$-0.715863\pi$
$397$	−25.0000	−1.25471	−0.627357	+	0.778732i	$0.715863\pi$
$398$	−3.00000	−0.150376
$399$	21.0000	1.05131
$400$	0	0
$401$	−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
$405$	0	0
$406$	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
$415$	0	0
$416$	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	$0.372190\pi$
$420$	0	0
$421$	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
$425$	0	0
$426$	9.00000	0.436051
$427$	−6.00000	−0.290360
$428$	3.00000	0.145010
$429$	6.00000	0.289683
$430$	0	0
$431$	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	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	3.00000	0.144005
$435$	0	0
$436$	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	$0.713053\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	−16.0000	−0.761042
$443$	−19.0000	−0.902717	−0.451359	−	0.892343i	$-0.649060\pi$
$443$	−19.0000	−0.902717	−0.451359	+	0.892343i	$0.649060\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	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	$0.203620\pi$
$450$	0	0
$451$	0	0
$452$	−1.00000	−0.0470360
$453$	−8.00000	−0.375873
$454$	−25.0000	−1.17331
$455$	0	0
$456$	−7.00000	−0.327805
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	−13.0000	−0.607450
$459$	−8.00000	−0.373408
$460$	0	0
$461$	−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	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	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
$470$	0	0
$471$	7.00000	0.322543
$472$	6.00000	0.276172
$473$	−3.00000	−0.137940
$474$	13.0000	0.597110
$475$	0	0
$476$	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	$0.711584\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	−22.0000	−1.00207
$483$	21.0000	0.955533
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	1.00000	0.0453609
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	2.00000	0.0905357
$489$	−18.0000	−0.813988
$490$	0	0
$491$	−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
$495$	0	0
$496$	−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	$0.647743\pi$
$500$	0	0
$501$	−15.0000	−0.670151
$502$	20.0000	0.892644
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	−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	$0.678518\pi$
$510$	0	0
$511$	3.00000	0.132712
$512$	1.00000	0.0441942
$513$	−7.00000	−0.309058
$514$	23.0000	1.01449
$515$	0	0
$516$	−1.00000	−0.0440225
$517$	−18.0000	−0.791639
$518$	−12.0000	−0.527250
$519$	6.00000	0.263371
$520$	0	0
$521$	−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	$-0.718605\pi$
$523$	−29.0000	−1.26808	−0.634041	+	0.773300i	$0.718605\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	−24.0000	−1.04645
$527$	8.00000	0.348485
$528$	3.00000	0.130558
$529$	26.0000	1.13043
$530$	0	0
$531$	6.00000	0.260378
$532$	21.0000	0.910465
$533$	0	0
$534$	−3.00000	−0.129823
$535$	0	0
$536$	−10.0000	−0.431934
$537$	4.00000	0.172613
$538$	−18.0000	−0.776035
$539$	6.00000	0.258438
$540$	0	0
$541$	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
$545$	0	0
$546$	−6.00000	−0.256776
$547$	14.0000	0.598597	0.299298	−	0.954160i	$-0.403247\pi$
$547$	14.0000	0.598597	0.299298	+	0.954160i	$0.403247\pi$
$548$	−10.0000	−0.427179
$549$	2.00000	0.0853579
$550$	0	0
$551$	56.0000	2.38568
$552$	−7.00000	−0.297940
$553$	−39.0000	−1.65845
$554$	8.00000	0.339887
$555$	0	0
$556$	−10.0000	−0.424094
$557$	−17.0000	−0.720313	−0.360157	−	0.932892i	$-0.617277\pi$
$557$	−17.0000	−0.720313	−0.360157	+	0.932892i	$0.617277\pi$
$558$	−1.00000	−0.0423334
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	−24.0000	−1.01328
$562$	−28.0000	−1.18111
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	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	$0.217467\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	6.00000	0.250873
$573$	−24.0000	−1.00261
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	47.0000	1.95494
$579$	14.0000	0.581820
$580$	0	0
$581$	−48.0000	−1.99138
$582$	−6.00000	−0.248708
$583$	−15.0000	−0.621237
$584$	−1.00000	−0.0413803
$585$	0	0
$586$	−14.0000	−0.578335
$587$	−16.0000	−0.660391	−0.330195	−	0.943913i	$-0.607115\pi$
$587$	−16.0000	−0.660391	−0.330195	+	0.943913i	$0.607115\pi$
$588$	2.00000	0.0824786
$589$	7.00000	0.288430
$590$	0	0
$591$	−6.00000	−0.246807
$592$	4.00000	0.164399
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	−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	$0.246413\pi$
$600$	0	0
$601$	−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
$605$	0	0
$606$	5.00000	0.203111
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	−7.00000	−0.283887
$609$	24.0000	0.972529
$610$	0	0
$611$	−12.0000	−0.485468
$612$	−8.00000	−0.323381
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	−9.00000	−0.362620
$617$	3.00000	0.120775	0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000	0.120775	0.0603877	+	0.998175i	$0.480766\pi$
$618$	−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	$0.617817\pi$
$620$	0	0
$621$	−7.00000	−0.280900
$622$	−16.0000	−0.641542
$623$	9.00000	0.360577
$624$	2.00000	0.0800641
$625$	0	0
$626$	10.0000	0.399680
$627$	−21.0000	−0.838659
$628$	7.00000	0.279330
$629$	−32.0000	−1.27592
$630$	0	0
$631$	−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
$635$	0	0
$636$	−5.00000	−0.198263
$637$	4.00000	0.158486
$638$	−24.0000	−0.950169
$639$	9.00000	0.356034
$640$	0	0
$641$	−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	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	21.0000	0.827516
$645$	0	0
$646$	56.0000	2.20329
$647$	−49.0000	−1.92639	−0.963194	−	0.268806i	$-0.913371\pi$
$647$	−49.0000	−1.92639	−0.963194	+	0.268806i	$0.913371\pi$
$648$	1.00000	0.0392837
$649$	18.0000	0.706562
$650$	0	0
$651$	3.00000	0.117579
$652$	−18.0000	−0.704934
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	4.00000	0.156412
$655$	0	0
$656$	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	$0.714197\pi$
$660$	0	0
$661$	−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
$665$	0	0
$666$	4.00000	0.154997
$667$	56.0000	2.16833
$668$	−15.0000	−0.580367
$669$	10.0000	0.386622
$670$	0	0
$671$	6.00000	0.231627
$672$	−3.00000	−0.115728
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	43.0000	1.65262	0.826312	−	0.563212i	$-0.190435\pi$
$677$	43.0000	1.65262	0.826312	+	0.563212i	$0.190435\pi$
$678$	−1.00000	−0.0384048
$679$	18.0000	0.690777
$680$	0	0
$681$	−25.0000	−0.958002
$682$	−3.00000	−0.114876
$683$	15.0000	0.573959	0.286980	−	0.957937i	$-0.407349\pi$
$683$	15.0000	0.573959	0.286980	+	0.957937i	$0.407349\pi$
$684$	−7.00000	−0.267652
$685$	0	0
$686$	15.0000	0.572703
$687$	−13.0000	−0.495981
$688$	−1.00000	−0.0381246
$689$	−10.0000	−0.380970
$690$	0	0
$691$	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
$695$	0	0
$696$	−8.00000	−0.303239
$697$	0	0
$698$	20.0000	0.757011
$699$	25.0000	0.945587
$700$	0	0
$701$	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
$705$	0	0
$706$	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	$0.590888\pi$
$710$	0	0
$711$	13.0000	0.487538
$712$	−3.00000	−0.112430
$713$	7.00000	0.262152
$714$	24.0000	0.898177
$715$	0	0
$716$	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	$0.688971\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	30.0000	1.11648
$723$	−22.0000	−0.818189
$724$	−21.0000	−0.780459
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	29.0000	1.07555	0.537775	−	0.843088i	$-0.319265\pi$
$727$	29.0000	1.07555	0.537775	+	0.843088i	$0.319265\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	2.00000	0.0739221
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	18.0000	0.664392
$735$	0	0
$736$	−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	$0.763152\pi$
$740$	0	0
$741$	−14.0000	−0.514303
$742$	15.0000	0.550667
$743$	−1.00000	−0.0366864	−0.0183432	−	0.999832i	$-0.505839\pi$
$743$	−1.00000	−0.0366864	−0.0183432	+	0.999832i	$0.505839\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−13.0000	−0.475964
$747$	16.0000	0.585409
$748$	−24.0000	−0.877527
$749$	−9.00000	−0.328853
$750$	0	0
$751$	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
$755$	0	0
$756$	−3.00000	−0.109109
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	19.0000	0.690111
$759$	−21.0000	−0.762252
$760$	0	0
$761$	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
$765$	0	0
$766$	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	$0.448117\pi$
$770$	0	0
$771$	23.0000	0.828325
$772$	14.0000	0.503871
$773$	1.00000	0.0359675	0.0179838	−	0.999838i	$-0.494275\pi$
$773$	1.00000	0.0359675	0.0179838	+	0.999838i	$0.494275\pi$
$774$	−1.00000	−0.0359443
$775$	0	0
$776$	−6.00000	−0.215387
$777$	−12.0000	−0.430498
$778$	−22.0000	−0.788738
$779$	0	0
$780$	0	0
$781$	27.0000	0.966136
$782$	56.0000	2.00256
$783$	−8.00000	−0.285897
$784$	2.00000	0.0714286
$785$	0	0
$786$	6.00000	0.214013
$787$	27.0000	0.962446	0.481223	−	0.876598i	$-0.340193\pi$
$787$	27.0000	0.962446	0.481223	+	0.876598i	$0.340193\pi$
$788$	−6.00000	−0.213741
$789$	−24.0000	−0.854423
$790$	0	0
$791$	3.00000	0.106668
$792$	3.00000	0.106600
$793$	4.00000	0.142044
$794$	−25.0000	−0.887217
$795$	0	0
$796$	−3.00000	−0.106332
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	21.0000	0.743392
$799$	48.0000	1.69812
$800$	0	0
$801$	−3.00000	−0.106000
$802$	−35.0000	−1.23589
$803$	−3.00000	−0.105868
$804$	−10.0000	−0.352673
$805$	0	0
$806$	−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	$0.426609\pi$
$810$	0	0
$811$	−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
$815$	0	0
$816$	−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	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−18.0000	−0.626300
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	−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	$0.618822\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	2.00000	0.0693375
$833$	−16.0000	−0.554367
$834$	−10.0000	−0.346272
$835$	0	0
$836$	−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	$0.654331\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	34.0000	1.17172
$843$	−28.0000	−0.964371
$844$	17.0000	0.585164
$845$	0	0
$846$	−6.00000	−0.206284
$847$	6.00000	0.206162
$848$	−5.00000	−0.171701
$849$	16.0000	0.549119
$850$	0	0
$851$	−28.0000	−0.959828
$852$	9.00000	0.308335
$853$	−45.0000	−1.54077	−0.770385	−	0.637579i	$-0.779936\pi$
$853$	−45.0000	−1.54077	−0.770385	+	0.637579i	$0.779936\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	3.00000	0.102538
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	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	$0.634274\pi$
$860$	0	0
$861$	0	0
$862$	32.0000	1.08992
$863$	31.0000	1.05525	0.527626	−	0.849477i	$-0.323082\pi$
$863$	31.0000	1.05525	0.527626	+	0.849477i	$0.323082\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−19.0000	−0.645646
$867$	47.0000	1.59620
$868$	3.00000	0.101827
$869$	39.0000	1.32298
$870$	0	0
$871$	−20.0000	−0.677674
$872$	4.00000	0.135457
$873$	−6.00000	−0.203069
$874$	49.0000	1.65745
$875$	0	0
$876$	−1.00000	−0.0337869
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−26.0000	−0.877457
$879$	−14.0000	−0.472208
$880$	0	0
$881$	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	$-0.429805\pi$
$883$	13.0000	0.437485	0.218742	+	0.975783i	$0.429805\pi$
$884$	−16.0000	−0.538138
$885$	0	0
$886$	−19.0000	−0.638317
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	4.00000	0.134231
$889$	−36.0000	−1.20740
$890$	0	0
$891$	3.00000	0.100504
$892$	10.0000	0.334825
$893$	42.0000	1.40548
$894$	−9.00000	−0.301005
$895$	0	0
$896$	−3.00000	−0.100223
$897$	−14.0000	−0.467446
$898$	34.0000	1.13459
$899$	8.00000	0.266815
$900$	0	0
$901$	40.0000	1.33259
$902$	0	0
$903$	3.00000	0.0998337
$904$	−1.00000	−0.0332595
$905$	0	0
$906$	−8.00000	−0.265782
$907$	40.0000	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$908$	−25.0000	−0.829654
$909$	5.00000	0.165840
$910$	0	0
$911$	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
$915$	0	0
$916$	−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	$0.552741\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	−20.0000	−0.658665
$923$	18.0000	0.592477
$924$	−9.00000	−0.296078
$925$	0	0
$926$	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	$0.484329\pi$
$930$	0	0
$931$	−14.0000	−0.458831
$932$	25.0000	0.818902
$933$	−16.0000	−0.523816
$934$	0	0
$935$	0	0
$936$	2.00000	0.0653720
$937$	16.0000	0.522697	0.261349	−	0.965244i	$-0.415833\pi$
$937$	16.0000	0.522697	0.261349	+	0.965244i	$0.415833\pi$
$938$	30.0000	0.979535
$939$	10.0000	0.326338
$940$	0	0
$941$	−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
$945$	0	0
$946$	−3.00000	−0.0975384
$947$	−38.0000	−1.23483	−0.617417	−	0.786636i	$-0.711821\pi$
$947$	−38.0000	−1.23483	−0.617417	+	0.786636i	$0.711821\pi$
$948$	13.0000	0.422220
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	−24.0000	−0.778253
$952$	24.0000	0.777844
$953$	−12.0000	−0.388718	−0.194359	−	0.980930i	$-0.562263\pi$
$953$	−12.0000	−0.388718	−0.194359	+	0.980930i	$0.562263\pi$
$954$	−5.00000	−0.161881
$955$	0	0
$956$	8.00000	0.258738
$957$	−24.0000	−0.775810
$958$	−27.0000	−0.872330
$959$	30.0000	0.968751
$960$	0	0
$961$	1.00000	0.0322581
$962$	8.00000	0.257930
$963$	3.00000	0.0966736
$964$	−22.0000	−0.708572
$965$	0	0
$966$	21.0000	0.675664
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	−2.00000	−0.0642824
$969$	56.0000	1.79898
$970$	0	0
$971$	−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
$975$	0	0
$976$	2.00000	0.0640184
$977$	−46.0000	−1.47167	−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000	−1.47167	−0.735835	+	0.677161i	$0.763210\pi$
$978$	−18.0000	−0.575577
$979$	−9.00000	−0.287641
$980$	0	0
$981$	4.00000	0.127710
$982$	−29.0000	−0.925427
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	0	0
$985$	0	0
$986$	64.0000	2.03818
$987$	18.0000	0.572946
$988$	−14.0000	−0.445399
$989$	7.00000	0.222587
$990$	0	0
$991$	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
$995$	0	0
$996$	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	$0.469711\pi$
$998$	−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	4650.2.a.bk.1.1		1
5.2	odd	4		4650.2.d.ba.3349.2		2
5.3	odd	4		4650.2.d.ba.3349.1		2
5.4	even	2		930.2.a.e.1.1	✓	1
15.14	odd	2		2790.2.a.u.1.1		1
20.19	odd	2		7440.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.e.1.1	✓	1	5.4	even	2
2790.2.a.u.1.1		1	15.14	odd	2
4650.2.a.bk.1.1		1	1.1	even	1	trivial
4650.2.d.ba.3349.1		2	5.3	odd	4
4650.2.d.ba.3349.2		2	5.2	odd	4
7440.2.a.x.1.1		1	20.19	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:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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