LMFDB - Embedded newform 2166.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,2,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2166.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2166.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:	$17.2955970778$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} -3.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -1.00000 q^{21} +2.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} +3.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -3.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} -5.00000 q^{37} +3.00000 q^{39} +4.00000 q^{41} +1.00000 q^{42} -9.00000 q^{43} -2.00000 q^{44} -4.00000 q^{46} +10.0000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +5.00000 q^{50} -4.00000 q^{51} -3.00000 q^{52} -4.00000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -14.0000 q^{59} +11.0000 q^{61} +3.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +3.00000 q^{67} +4.00000 q^{68} -4.00000 q^{69} +14.0000 q^{71} -1.00000 q^{72} -11.0000 q^{73} +5.00000 q^{74} +5.00000 q^{75} -2.00000 q^{77} -3.00000 q^{78} +1.00000 q^{79} +1.00000 q^{81} -4.00000 q^{82} +8.00000 q^{83} -1.00000 q^{84} +9.00000 q^{86} +2.00000 q^{88} -14.0000 q^{89} -3.00000 q^{91} +4.00000 q^{92} +3.00000 q^{93} -10.0000 q^{94} +1.00000 q^{96} +2.00000 q^{97} +6.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.00000	0.408248
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.00000	−0.288675
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−1.00000	−0.218218
$22$	2.00000	0.426401
$23$	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
$25$	−5.00000	−1.00000
$26$	3.00000	0.588348
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	0	0
$39$	3.00000	0.480384
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	1.00000	0.154303
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−4.00000	−0.589768
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	5.00000	0.707107
$51$	−4.00000	−0.560112
$52$	−3.00000	−0.416025
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	0	0
$59$	−14.0000	−1.82264	−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000	−1.82264	−0.911322	+	0.411693i	$0.864937\pi$
$60$	0	0
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	3.00000	0.381000
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	4.00000	0.485071
$69$	−4.00000	−0.481543
$70$	0	0
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	−1.00000	−0.117851
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	5.00000	0.581238
$75$	5.00000	0.577350
$76$	0	0
$77$	−2.00000	−0.227921
$78$	−3.00000	−0.339683
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−4.00000	−0.441726
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	9.00000	0.970495
$87$	0	0
$88$	2.00000	0.213201
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	4.00000	0.417029
$93$	3.00000	0.311086
$94$	−10.0000	−1.03142
$95$	0	0
$96$	1.00000	0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	6.00000	0.606092
$99$	−2.00000	−0.201008
$100$	−5.00000	−0.500000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	4.00000	0.396059
$103$	−3.00000	−0.295599	−0.147799	−	0.989017i	$-0.547219\pi$
$103$	−3.00000	−0.295599	−0.147799	+	0.989017i	$0.547219\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	4.00000	0.388514
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	−1.00000	−0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	5.00000	0.474579
$112$	1.00000	0.0944911
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.00000	−0.277350
$118$	14.0000	1.28880
$119$	4.00000	0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−11.0000	−0.995893
$123$	−4.00000	−0.360668
$124$	−3.00000	−0.269408
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	−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
$129$	9.00000	0.792406
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	−3.00000	−0.259161
$135$	0	0
$136$	−4.00000	−0.342997
$137$	−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
$139$	−19.0000	−1.61156	−0.805779	−	0.592216i	$-0.798253\pi$
$139$	−19.0000	−1.61156	−0.805779	+	0.592216i	$0.798253\pi$
$140$	0	0
$141$	−10.0000	−0.842152
$142$	−14.0000	−1.17485
$143$	6.00000	0.501745
$144$	1.00000	0.0833333
$145$	0	0
$146$	11.0000	0.910366
$147$	6.00000	0.494872
$148$	−5.00000	−0.410997
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	−5.00000	−0.408248
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	2.00000	0.161165
$155$	0	0
$156$	3.00000	0.240192
$157$	−21.0000	−1.67598	−0.837991	−	0.545684i	$-0.816270\pi$
$157$	−21.0000	−1.67598	−0.837991	+	0.545684i	$0.816270\pi$
$158$	−1.00000	−0.0795557
$159$	4.00000	0.317221
$160$	0	0
$161$	4.00000	0.315244
$162$	−1.00000	−0.0785674
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	−8.00000	−0.620920
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	1.00000	0.0771517
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	−9.00000	−0.686244
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	0	0
$175$	−5.00000	−0.377964
$176$	−2.00000	−0.150756
$177$	14.0000	1.05230
$178$	14.0000	1.04934
$179$	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
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	3.00000	0.222375
$183$	−11.0000	−0.813143
$184$	−4.00000	−0.294884
$185$	0	0
$186$	−3.00000	−0.219971
$187$	−8.00000	−0.585018
$188$	10.0000	0.729325
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−1.00000	−0.0721688
$193$	−13.0000	−0.935760	−0.467880	−	0.883792i	$-0.654982\pi$
$193$	−13.0000	−0.935760	−0.467880	+	0.883792i	$0.654982\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	2.00000	0.142134
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	5.00000	0.353553
$201$	−3.00000	−0.211604
$202$	10.0000	0.703598
$203$	0	0
$204$	−4.00000	−0.280056
$205$	0	0
$206$	3.00000	0.209020
$207$	4.00000	0.278019
$208$	−3.00000	−0.208013
$209$	0	0
$210$	0	0
$211$	25.0000	1.72107	0.860535	−	0.509390i	$-0.170129\pi$
$211$	25.0000	1.72107	0.860535	+	0.509390i	$0.170129\pi$
$212$	−4.00000	−0.274721
$213$	−14.0000	−0.959264
$214$	10.0000	0.683586
$215$	0	0
$216$	1.00000	0.0680414
$217$	−3.00000	−0.203653
$218$	−14.0000	−0.948200
$219$	11.0000	0.743311
$220$	0	0
$221$	−12.0000	−0.807207
$222$	−5.00000	−0.335578
$223$	9.00000	0.602685	0.301342	−	0.953516i	$-0.402565\pi$
$223$	9.00000	0.602685	0.301342	+	0.953516i	$0.402565\pi$
$224$	−1.00000	−0.0668153
$225$	−5.00000	−0.333333
$226$	−10.0000	−0.665190
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	−11.0000	−0.726900	−0.363450	−	0.931614i	$-0.618401\pi$
$229$	−11.0000	−0.726900	−0.363450	+	0.931614i	$0.618401\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	3.00000	0.196116
$235$	0	0
$236$	−14.0000	−0.911322
$237$	−1.00000	−0.0649570
$238$	−4.00000	−0.259281
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	25.0000	1.61039	0.805196	−	0.593009i	$-0.202060\pi$
$241$	25.0000	1.61039	0.805196	+	0.593009i	$0.202060\pi$
$242$	7.00000	0.449977
$243$	−1.00000	−0.0641500
$244$	11.0000	0.704203
$245$	0	0
$246$	4.00000	0.255031
$247$	0	0
$248$	3.00000	0.190500
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	1.00000	0.0629941
$253$	−8.00000	−0.502956
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	−9.00000	−0.560316
$259$	−5.00000	−0.310685
$260$	0	0
$261$	0	0
$262$	6.00000	0.370681
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	0	0
$267$	14.0000	0.856786
$268$	3.00000	0.183254
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	4.00000	0.242536
$273$	3.00000	0.181568
$274$	6.00000	0.362473
$275$	10.0000	0.603023
$276$	−4.00000	−0.240772
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	19.0000	1.13954
$279$	−3.00000	−0.179605
$280$	0	0
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	10.0000	0.595491
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	14.0000	0.830747
$285$	0	0
$286$	−6.00000	−0.354787
$287$	4.00000	0.236113
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−2.00000	−0.117242
$292$	−11.0000	−0.643726
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	5.00000	0.290619
$297$	2.00000	0.116052
$298$	22.0000	1.27443
$299$	−12.0000	−0.693978
$300$	5.00000	0.288675
$301$	−9.00000	−0.518751
$302$	20.0000	1.15087
$303$	10.0000	0.574485
$304$	0	0
$305$	0	0
$306$	−4.00000	−0.228665
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	−2.00000	−0.113961
$309$	3.00000	0.170664
$310$	0	0
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	−3.00000	−0.169842
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	21.0000	1.18510
$315$	0	0
$316$	1.00000	0.0562544
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	−4.00000	−0.224309
$319$	0	0
$320$	0	0
$321$	10.0000	0.558146
$322$	−4.00000	−0.222911
$323$	0	0
$324$	1.00000	0.0555556
$325$	15.0000	0.832050
$326$	−11.0000	−0.609234
$327$	−14.0000	−0.774202
$328$	−4.00000	−0.220863
$329$	10.0000	0.551318
$330$	0	0
$331$	15.0000	0.824475	0.412237	−	0.911077i	$-0.364747\pi$
$331$	15.0000	0.824475	0.412237	+	0.911077i	$0.364747\pi$
$332$	8.00000	0.439057
$333$	−5.00000	−0.273998
$334$	−6.00000	−0.328305
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	−19.0000	−1.03500	−0.517498	−	0.855684i	$-0.673136\pi$
$337$	−19.0000	−1.03500	−0.517498	+	0.855684i	$0.673136\pi$
$338$	4.00000	0.217571
$339$	−10.0000	−0.543125
$340$	0	0
$341$	6.00000	0.324918
$342$	0	0
$343$	−13.0000	−0.701934
$344$	9.00000	0.485247
$345$	0	0
$346$	24.0000	1.29025
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	0	0
$349$	−29.0000	−1.55233	−0.776167	−	0.630527i	$-0.782839\pi$
$349$	−29.0000	−1.55233	−0.776167	+	0.630527i	$0.782839\pi$
$350$	5.00000	0.267261
$351$	3.00000	0.160128
$352$	2.00000	0.106600
$353$	−12.0000	−0.638696	−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000	−0.638696	−0.319348	+	0.947638i	$0.603464\pi$
$354$	−14.0000	−0.744092
$355$	0	0
$356$	−14.0000	−0.741999
$357$	−4.00000	−0.211702
$358$	−4.00000	−0.211407
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	0	0
$362$	18.0000	0.946059
$363$	7.00000	0.367405
$364$	−3.00000	−0.157243
$365$	0	0
$366$	11.0000	0.574979
$367$	23.0000	1.20059	0.600295	−	0.799779i	$-0.295050\pi$
$367$	23.0000	1.20059	0.600295	+	0.799779i	$0.295050\pi$
$368$	4.00000	0.208514
$369$	4.00000	0.208232
$370$	0	0
$371$	−4.00000	−0.207670
$372$	3.00000	0.155543
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	−10.0000	−0.515711
$377$	0	0
$378$	1.00000	0.0514344
$379$	−13.0000	−0.667765	−0.333883	−	0.942615i	$-0.608359\pi$
$379$	−13.0000	−0.667765	−0.333883	+	0.942615i	$0.608359\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	8.00000	0.409316
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	13.0000	0.661683
$387$	−9.00000	−0.457496
$388$	2.00000	0.101535
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	6.00000	0.303046
$393$	6.00000	0.302660
$394$	2.00000	0.100759
$395$	0	0
$396$	−2.00000	−0.100504
$397$	15.0000	0.752828	0.376414	−	0.926451i	$-0.377157\pi$
$397$	15.0000	0.752828	0.376414	+	0.926451i	$0.377157\pi$
$398$	−5.00000	−0.250627
$399$	0	0
$400$	−5.00000	−0.250000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	3.00000	0.149626
$403$	9.00000	0.448322
$404$	−10.0000	−0.497519
$405$	0	0
$406$	0	0
$407$	10.0000	0.495682
$408$	4.00000	0.198030
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	−3.00000	−0.147799
$413$	−14.0000	−0.688895
$414$	−4.00000	−0.196589
$415$	0	0
$416$	3.00000	0.147087
$417$	19.0000	0.930434
$418$	0	0
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−25.0000	−1.21698
$423$	10.0000	0.486217
$424$	4.00000	0.194257
$425$	−20.0000	−0.970143
$426$	14.0000	0.678302
$427$	11.0000	0.532327
$428$	−10.0000	−0.483368
$429$	−6.00000	−0.289683
$430$	0	0
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	−1.00000	−0.0481125
$433$	−9.00000	−0.432512	−0.216256	−	0.976337i	$-0.569385\pi$
$433$	−9.00000	−0.432512	−0.216256	+	0.976337i	$0.569385\pi$
$434$	3.00000	0.144005
$435$	0	0
$436$	14.0000	0.670478
$437$	0	0
$438$	−11.0000	−0.525600
$439$	19.0000	0.906821	0.453410	−	0.891302i	$-0.350207\pi$
$439$	19.0000	0.906821	0.453410	+	0.891302i	$0.350207\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	12.0000	0.570782
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	5.00000	0.237289
$445$	0	0
$446$	−9.00000	−0.426162
$447$	22.0000	1.04056
$448$	1.00000	0.0472456
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	5.00000	0.235702
$451$	−8.00000	−0.376705
$452$	10.0000	0.470360
$453$	20.0000	0.939682
$454$	−8.00000	−0.375459
$455$	0	0
$456$	0	0
$457$	5.00000	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$458$	11.0000	0.513996
$459$	−4.00000	−0.186704
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	−2.00000	−0.0930484
$463$	−9.00000	−0.418265	−0.209133	−	0.977887i	$-0.567064\pi$
$463$	−9.00000	−0.418265	−0.209133	+	0.977887i	$0.567064\pi$
$464$	0	0
$465$	0	0
$466$	−18.0000	−0.833834
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	−3.00000	−0.138675
$469$	3.00000	0.138527
$470$	0	0
$471$	21.0000	0.967629
$472$	14.0000	0.644402
$473$	18.0000	0.827641
$474$	1.00000	0.0459315
$475$	0	0
$476$	4.00000	0.183340
$477$	−4.00000	−0.183147
$478$	12.0000	0.548867
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	15.0000	0.683941
$482$	−25.0000	−1.13872
$483$	−4.00000	−0.182006
$484$	−7.00000	−0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	−11.0000	−0.497947
$489$	−11.0000	−0.497437
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	−4.00000	−0.180334
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−3.00000	−0.134704
$497$	14.0000	0.627986
$498$	8.00000	0.358489
$499$	5.00000	0.223831	0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000	0.223831	0.111915	+	0.993718i	$0.464301\pi$
$500$	0	0
$501$	−6.00000	−0.268060
$502$	18.0000	0.803379
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	8.00000	0.355643
$507$	4.00000	0.177646
$508$	−8.00000	−0.354943
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	−11.0000	−0.486611
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	12.0000	0.529297
$515$	0	0
$516$	9.00000	0.396203
$517$	−20.0000	−0.879599
$518$	5.00000	0.219687
$519$	24.0000	1.05348
$520$	0	0
$521$	34.0000	1.48957	0.744784	−	0.667306i	$-0.232553\pi$
$521$	34.0000	1.48957	0.744784	+	0.667306i	$0.232553\pi$
$522$	0	0
$523$	37.0000	1.61790	0.808949	−	0.587879i	$-0.200037\pi$
$523$	37.0000	1.61790	0.808949	+	0.587879i	$0.200037\pi$
$524$	−6.00000	−0.262111
$525$	5.00000	0.218218
$526$	−18.0000	−0.784837
$527$	−12.0000	−0.522728
$528$	2.00000	0.0870388
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−14.0000	−0.607548
$532$	0	0
$533$	−12.0000	−0.519778
$534$	−14.0000	−0.605839
$535$	0	0
$536$	−3.00000	−0.129580
$537$	−4.00000	−0.172613
$538$	−2.00000	−0.0862261
$539$	12.0000	0.516877
$540$	0	0
$541$	−31.0000	−1.33279	−0.666397	−	0.745597i	$-0.732164\pi$
$541$	−31.0000	−1.33279	−0.666397	+	0.745597i	$0.732164\pi$
$542$	16.0000	0.687259
$543$	18.0000	0.772454
$544$	−4.00000	−0.171499
$545$	0	0
$546$	−3.00000	−0.128388
$547$	11.0000	0.470326	0.235163	−	0.971956i	$-0.424438\pi$
$547$	11.0000	0.470326	0.235163	+	0.971956i	$0.424438\pi$
$548$	−6.00000	−0.256307
$549$	11.0000	0.469469
$550$	−10.0000	−0.426401
$551$	0	0
$552$	4.00000	0.170251
$553$	1.00000	0.0425243
$554$	2.00000	0.0849719
$555$	0	0
$556$	−19.0000	−0.805779
$557$	34.0000	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$558$	3.00000	0.127000
$559$	27.0000	1.14198
$560$	0	0
$561$	8.00000	0.337760
$562$	8.00000	0.337460
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	−10.0000	−0.421076
$565$	0	0
$566$	20.0000	0.840663
$567$	1.00000	0.0419961
$568$	−14.0000	−0.587427
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−7.00000	−0.292941	−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000	−0.292941	−0.146470	+	0.989215i	$0.546791\pi$
$572$	6.00000	0.250873
$573$	8.00000	0.334205
$574$	−4.00000	−0.166957
$575$	−20.0000	−0.834058
$576$	1.00000	0.0416667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	1.00000	0.0415945
$579$	13.0000	0.540262
$580$	0	0
$581$	8.00000	0.331896
$582$	2.00000	0.0829027
$583$	8.00000	0.331326
$584$	11.0000	0.455183
$585$	0	0
$586$	−14.0000	−0.578335
$587$	−2.00000	−0.0825488	−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000	−0.0825488	−0.0412744	+	0.999148i	$0.513142\pi$
$588$	6.00000	0.247436
$589$	0	0
$590$	0	0
$591$	2.00000	0.0822690
$592$	−5.00000	−0.205499
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	−22.0000	−0.901155
$597$	−5.00000	−0.204636
$598$	12.0000	0.490716
$599$	−46.0000	−1.87951	−0.939755	−	0.341850i	$-0.888947\pi$
$599$	−46.0000	−1.87951	−0.939755	+	0.341850i	$0.888947\pi$
$600$	−5.00000	−0.204124
$601$	27.0000	1.10135	0.550676	−	0.834719i	$-0.314370\pi$
$601$	27.0000	1.10135	0.550676	+	0.834719i	$0.314370\pi$
$602$	9.00000	0.366813
$603$	3.00000	0.122169
$604$	−20.0000	−0.813788
$605$	0	0
$606$	−10.0000	−0.406222
$607$	5.00000	0.202944	0.101472	−	0.994838i	$-0.467645\pi$
$607$	5.00000	0.202944	0.101472	+	0.994838i	$0.467645\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−30.0000	−1.21367
$612$	4.00000	0.161690
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	2.00000	0.0805823
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	−3.00000	−0.120678
$619$	1.00000	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	−10.0000	−0.400963
$623$	−14.0000	−0.560898
$624$	3.00000	0.120096
$625$	25.0000	1.00000
$626$	−6.00000	−0.239808
$627$	0	0
$628$	−21.0000	−0.837991
$629$	−20.0000	−0.797452
$630$	0	0
$631$	17.0000	0.676759	0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000	0.676759	0.338380	+	0.941010i	$0.390121\pi$
$632$	−1.00000	−0.0397779
$633$	−25.0000	−0.993661
$634$	−12.0000	−0.476581
$635$	0	0
$636$	4.00000	0.158610
$637$	18.0000	0.713186
$638$	0	0
$639$	14.0000	0.553831
$640$	0	0
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	−10.0000	−0.394669
$643$	37.0000	1.45914	0.729569	−	0.683907i	$-0.239721\pi$
$643$	37.0000	1.45914	0.729569	+	0.683907i	$0.239721\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	0	0
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	−1.00000	−0.0392837
$649$	28.0000	1.09910
$650$	−15.0000	−0.588348
$651$	3.00000	0.117579
$652$	11.0000	0.430793
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	4.00000	0.156174
$657$	−11.0000	−0.429151
$658$	−10.0000	−0.389841
$659$	−14.0000	−0.545363	−0.272681	−	0.962104i	$-0.587910\pi$
$659$	−14.0000	−0.545363	−0.272681	+	0.962104i	$0.587910\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	−15.0000	−0.582992
$663$	12.0000	0.466041
$664$	−8.00000	−0.310460
$665$	0	0
$666$	5.00000	0.193746
$667$	0	0
$668$	6.00000	0.232147
$669$	−9.00000	−0.347960
$670$	0	0
$671$	−22.0000	−0.849301
$672$	1.00000	0.0385758
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	19.0000	0.731853
$675$	5.00000	0.192450
$676$	−4.00000	−0.153846
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	10.0000	0.384048
$679$	2.00000	0.0767530
$680$	0	0
$681$	−8.00000	−0.306561
$682$	−6.00000	−0.229752
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	11.0000	0.419676
$688$	−9.00000	−0.343122
$689$	12.0000	0.457164
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	−24.0000	−0.912343
$693$	−2.00000	−0.0759737
$694$	−16.0000	−0.607352
$695$	0	0
$696$	0	0
$697$	16.0000	0.606043
$698$	29.0000	1.09767
$699$	−18.0000	−0.680823
$700$	−5.00000	−0.188982
$701$	−44.0000	−1.66186	−0.830929	−	0.556379i	$-0.812190\pi$
$701$	−44.0000	−1.66186	−0.830929	+	0.556379i	$0.812190\pi$
$702$	−3.00000	−0.113228
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	12.0000	0.451626
$707$	−10.0000	−0.376089
$708$	14.0000	0.526152
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	0	0
$711$	1.00000	0.0375029
$712$	14.0000	0.524672
$713$	−12.0000	−0.449404
$714$	4.00000	0.149696
$715$	0	0
$716$	4.00000	0.149487
$717$	12.0000	0.448148
$718$	36.0000	1.34351
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	−3.00000	−0.111726
$722$	0	0
$723$	−25.0000	−0.929760
$724$	−18.0000	−0.668965
$725$	0	0
$726$	−7.00000	−0.259794
$727$	17.0000	0.630495	0.315248	−	0.949009i	$-0.397912\pi$
$727$	17.0000	0.630495	0.315248	+	0.949009i	$0.397912\pi$
$728$	3.00000	0.111187
$729$	1.00000	0.0370370
$730$	0	0
$731$	−36.0000	−1.33151
$732$	−11.0000	−0.406572
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	−23.0000	−0.848945
$735$	0	0
$736$	−4.00000	−0.147442
$737$	−6.00000	−0.221013
$738$	−4.00000	−0.147242
$739$	−5.00000	−0.183928	−0.0919640	−	0.995762i	$-0.529314\pi$
$739$	−5.00000	−0.183928	−0.0919640	+	0.995762i	$0.529314\pi$
$740$	0	0
$741$	0	0
$742$	4.00000	0.146845
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	−3.00000	−0.109985
$745$	0	0
$746$	−26.0000	−0.951928
$747$	8.00000	0.292705
$748$	−8.00000	−0.292509
$749$	−10.0000	−0.365392
$750$	0	0
$751$	13.0000	0.474377	0.237188	−	0.971464i	$-0.423774\pi$
$751$	13.0000	0.474377	0.237188	+	0.971464i	$0.423774\pi$
$752$	10.0000	0.364662
$753$	18.0000	0.655956
$754$	0	0
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	5.00000	0.181728	0.0908640	−	0.995863i	$-0.471037\pi$
$757$	5.00000	0.181728	0.0908640	+	0.995863i	$0.471037\pi$
$758$	13.0000	0.472181
$759$	8.00000	0.290382
$760$	0	0
$761$	−34.0000	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$761$	−34.0000	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$762$	−8.00000	−0.289809
$763$	14.0000	0.506834
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−14.0000	−0.505841
$767$	42.0000	1.51653
$768$	−1.00000	−0.0360844
$769$	27.0000	0.973645	0.486822	−	0.873501i	$-0.338156\pi$
$769$	27.0000	0.973645	0.486822	+	0.873501i	$0.338156\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	−13.0000	−0.467880
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	9.00000	0.323498
$775$	15.0000	0.538816
$776$	−2.00000	−0.0717958
$777$	5.00000	0.179374
$778$	16.0000	0.573628
$779$	0	0
$780$	0	0
$781$	−28.0000	−1.00192
$782$	−16.0000	−0.572159
$783$	0	0
$784$	−6.00000	−0.214286
$785$	0	0
$786$	−6.00000	−0.214013
$787$	−47.0000	−1.67537	−0.837685	−	0.546154i	$-0.816091\pi$
$787$	−47.0000	−1.67537	−0.837685	+	0.546154i	$0.816091\pi$
$788$	−2.00000	−0.0712470
$789$	−18.0000	−0.640817
$790$	0	0
$791$	10.0000	0.355559
$792$	2.00000	0.0710669
$793$	−33.0000	−1.17186
$794$	−15.0000	−0.532330
$795$	0	0
$796$	5.00000	0.177220
$797$	26.0000	0.920967	0.460484	−	0.887668i	$-0.347676\pi$
$797$	26.0000	0.920967	0.460484	+	0.887668i	$0.347676\pi$
$798$	0	0
$799$	40.0000	1.41510
$800$	5.00000	0.176777
$801$	−14.0000	−0.494666
$802$	−18.0000	−0.635602
$803$	22.0000	0.776363
$804$	−3.00000	−0.105802
$805$	0	0
$806$	−9.00000	−0.317011
$807$	−2.00000	−0.0704033
$808$	10.0000	0.351799
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	−10.0000	−0.350500
$815$	0	0
$816$	−4.00000	−0.140028
$817$	0	0
$818$	30.0000	1.04893
$819$	−3.00000	−0.104828
$820$	0	0
$821$	48.0000	1.67521	0.837606	−	0.546275i	$-0.183955\pi$
$821$	48.0000	1.67521	0.837606	+	0.546275i	$0.183955\pi$
$822$	−6.00000	−0.209274
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	3.00000	0.104510
$825$	−10.0000	−0.348155
$826$	14.0000	0.487122
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	4.00000	0.139010
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	−3.00000	−0.104006
$833$	−24.0000	−0.831551
$834$	−19.0000	−0.657916
$835$	0	0
$836$	0	0
$837$	3.00000	0.103695
$838$	−18.0000	−0.621800
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−10.0000	−0.344623
$843$	8.00000	0.275535
$844$	25.0000	0.860535
$845$	0	0
$846$	−10.0000	−0.343807
$847$	−7.00000	−0.240523
$848$	−4.00000	−0.137361
$849$	20.0000	0.686398
$850$	20.0000	0.685994
$851$	−20.0000	−0.685591
$852$	−14.0000	−0.479632
$853$	3.00000	0.102718	0.0513590	−	0.998680i	$-0.483645\pi$
$853$	3.00000	0.102718	0.0513590	+	0.998680i	$0.483645\pi$
$854$	−11.0000	−0.376412
$855$	0	0
$856$	10.0000	0.341793
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	6.00000	0.204837
$859$	7.00000	0.238837	0.119418	−	0.992844i	$-0.461897\pi$
$859$	7.00000	0.238837	0.119418	+	0.992844i	$0.461897\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	10.0000	0.340601
$863$	−22.0000	−0.748889	−0.374444	−	0.927249i	$-0.622167\pi$
$863$	−22.0000	−0.748889	−0.374444	+	0.927249i	$0.622167\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	9.00000	0.305832
$867$	1.00000	0.0339618
$868$	−3.00000	−0.101827
$869$	−2.00000	−0.0678454
$870$	0	0
$871$	−9.00000	−0.304953
$872$	−14.0000	−0.474100
$873$	2.00000	0.0676897
$874$	0	0
$875$	0	0
$876$	11.0000	0.371656
$877$	−25.0000	−0.844190	−0.422095	−	0.906552i	$-0.638705\pi$
$877$	−25.0000	−0.844190	−0.422095	+	0.906552i	$0.638705\pi$
$878$	−19.0000	−0.641219
$879$	−14.0000	−0.472208
$880$	0	0
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	6.00000	0.202031
$883$	−9.00000	−0.302874	−0.151437	−	0.988467i	$-0.548390\pi$
$883$	−9.00000	−0.302874	−0.151437	+	0.988467i	$0.548390\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−24.0000	−0.806296
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	−5.00000	−0.167789
$889$	−8.00000	−0.268311
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	9.00000	0.301342
$893$	0	0
$894$	−22.0000	−0.735790
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	12.0000	0.400668
$898$	36.0000	1.20134
$899$	0	0
$900$	−5.00000	−0.166667
$901$	−16.0000	−0.533037
$902$	8.00000	0.266371
$903$	9.00000	0.299501
$904$	−10.0000	−0.332595
$905$	0	0
$906$	−20.0000	−0.664455
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	8.00000	0.265489
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−60.0000	−1.98789	−0.993944	−	0.109885i	$-0.964952\pi$
$911$	−60.0000	−1.98789	−0.993944	+	0.109885i	$0.964952\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	−5.00000	−0.165385
$915$	0	0
$916$	−11.0000	−0.363450
$917$	−6.00000	−0.198137
$918$	4.00000	0.132020
$919$	55.0000	1.81428	0.907141	−	0.420826i	$-0.138260\pi$
$919$	55.0000	1.81428	0.907141	+	0.420826i	$0.138260\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	−6.00000	−0.197599
$923$	−42.0000	−1.38245
$924$	2.00000	0.0657952
$925$	25.0000	0.821995
$926$	9.00000	0.295758
$927$	−3.00000	−0.0985329
$928$	0	0
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	0	0
$931$	0	0
$932$	18.0000	0.589610
$933$	−10.0000	−0.327385
$934$	−8.00000	−0.261768
$935$	0	0
$936$	3.00000	0.0980581
$937$	−25.0000	−0.816714	−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000	−0.816714	−0.408357	+	0.912822i	$0.633898\pi$
$938$	−3.00000	−0.0979535
$939$	−6.00000	−0.195803
$940$	0	0
$941$	−32.0000	−1.04317	−0.521585	−	0.853199i	$-0.674659\pi$
$941$	−32.0000	−1.04317	−0.521585	+	0.853199i	$0.674659\pi$
$942$	−21.0000	−0.684217
$943$	16.0000	0.521032
$944$	−14.0000	−0.455661
$945$	0	0
$946$	−18.0000	−0.585230
$947$	10.0000	0.324956	0.162478	−	0.986712i	$-0.448051\pi$
$947$	10.0000	0.324956	0.162478	+	0.986712i	$0.448051\pi$
$948$	−1.00000	−0.0324785
$949$	33.0000	1.07123
$950$	0	0
$951$	−12.0000	−0.389127
$952$	−4.00000	−0.129641
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	4.00000	0.129505
$955$	0	0
$956$	−12.0000	−0.388108
$957$	0	0
$958$	−6.00000	−0.193851
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−15.0000	−0.483619
$963$	−10.0000	−0.322245
$964$	25.0000	0.805196
$965$	0	0
$966$	4.00000	0.128698
$967$	51.0000	1.64005	0.820025	−	0.572328i	$-0.193960\pi$
$967$	51.0000	1.64005	0.820025	+	0.572328i	$0.193960\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	0	0
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	−1.00000	−0.0320750
$973$	−19.0000	−0.609112
$974$	−32.0000	−1.02535
$975$	−15.0000	−0.480384
$976$	11.0000	0.352101
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	11.0000	0.351741
$979$	28.0000	0.894884
$980$	0	0
$981$	14.0000	0.446986
$982$	−30.0000	−0.957338
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	4.00000	0.127515
$985$	0	0
$986$	0	0
$987$	−10.0000	−0.318304
$988$	0	0
$989$	−36.0000	−1.14473
$990$	0	0
$991$	−17.0000	−0.540023	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000	−0.540023	−0.270011	+	0.962857i	$0.587027\pi$
$992$	3.00000	0.0952501
$993$	−15.0000	−0.476011
$994$	−14.0000	−0.444053
$995$	0	0
$996$	−8.00000	−0.253490
$997$	−49.0000	−1.55185	−0.775923	−	0.630828i	$-0.782715\pi$
$997$	−49.0000	−1.55185	−0.775923	+	0.630828i	$0.782715\pi$
$998$	−5.00000	−0.158272
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.2.a.b.1.1		1
3.2	odd	2		6498.2.a.u.1.1		1
19.7	even	3		114.2.e.b.49.1	yes	2
19.11	even	3		114.2.e.b.7.1	✓	2
19.18	odd	2		2166.2.a.h.1.1		1
57.11	odd	6		342.2.g.c.235.1		2
57.26	odd	6		342.2.g.c.163.1		2
57.56	even	2		6498.2.a.g.1.1		1
76.7	odd	6		912.2.q.b.49.1		2
76.11	odd	6		912.2.q.b.577.1		2
228.11	even	6		2736.2.s.k.577.1		2
228.83	even	6		2736.2.s.k.1873.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.e.b.7.1	✓	2	19.11	even	3
114.2.e.b.49.1	yes	2	19.7	even	3
342.2.g.c.163.1		2	57.26	odd	6
342.2.g.c.235.1		2	57.11	odd	6
912.2.q.b.49.1		2	76.7	odd	6
912.2.q.b.577.1		2	76.11	odd	6
2166.2.a.b.1.1		1	1.1	even	1	trivial
2166.2.a.h.1.1		1	19.18	odd	2
2736.2.s.k.577.1		2	228.11	even	6
2736.2.s.k.1873.1		2	228.83	even	6
6498.2.a.g.1.1		1	57.56	even	2
6498.2.a.u.1.1		1	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more