LMFDB - Embedded newform 867.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [867,2,Mod(1,867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(867, 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("867.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	867.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:	$6.92302985525$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	867.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{16} -1.00000 q^{18} +4.00000 q^{19} +4.00000 q^{21} -4.00000 q^{22} -4.00000 q^{23} -3.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +4.00000 q^{31} -5.00000 q^{32} -4.00000 q^{33} -1.00000 q^{36} -8.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} -8.00000 q^{41} -4.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} +4.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +5.00000 q^{50} -2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -12.0000 q^{56} -4.00000 q^{57} +12.0000 q^{59} -8.00000 q^{61} -4.00000 q^{62} -4.00000 q^{63} +7.00000 q^{64} +4.00000 q^{66} +12.0000 q^{67} +4.00000 q^{69} -12.0000 q^{71} +3.00000 q^{72} +8.00000 q^{74} +5.00000 q^{75} -4.00000 q^{76} -16.0000 q^{77} +2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{81} +8.00000 q^{82} -12.0000 q^{83} -4.00000 q^{84} -4.00000 q^{86} +12.0000 q^{88} -10.0000 q^{89} -8.00000 q^{91} +4.00000 q^{92} -4.00000 q^{93} +8.00000 q^{94} +5.00000 q^{96} -16.0000 q^{97} -9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−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
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	−4.00000	−0.852803
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−3.00000	−0.612372
$25$	−5.00000	−1.00000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	4.00000	0.755929
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−5.00000	−0.883883
$33$	−4.00000	−0.696311
$34$	0	0
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−4.00000	−0.648886
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	−4.00000	−0.617213
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	4.00000	0.589768
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	5.00000	0.707107
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−12.0000	−1.60357
$57$	−4.00000	−0.529813
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	−4.00000	−0.508001
$63$	−4.00000	−0.503953
$64$	7.00000	0.875000
$65$	0	0
$66$	4.00000	0.492366
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	3.00000	0.353553
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	8.00000	0.929981
$75$	5.00000	0.577350
$76$	−4.00000	−0.458831
$77$	−16.0000	−1.82337
$78$	2.00000	0.226455
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	8.00000	0.883452
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	12.0000	1.27920
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	4.00000	0.417029
$93$	−4.00000	−0.414781
$94$	8.00000	0.825137
$95$	0	0
$96$	5.00000	0.510310
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	−9.00000	−0.909137
$99$	4.00000	0.402015
$100$	5.00000	0.500000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	6.00000	0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	4.00000	0.377964
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	−12.0000	−1.10469
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	8.00000	0.724286
$123$	8.00000	0.721336
$124$	−4.00000	−0.359211
$125$	0	0
$126$	4.00000	0.356348
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	3.00000	0.265165
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	4.00000	0.348155
$133$	−16.0000	−1.38738
$134$	−12.0000	−1.03664
$135$	0	0
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	−4.00000	−0.340503
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	12.0000	1.00702
$143$	8.00000	0.668994
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	0	0
$147$	−9.00000	−0.742307
$148$	8.00000	0.657596
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−5.00000	−0.408248
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	12.0000	0.973329
$153$	0	0
$154$	16.0000	1.28932
$155$	0	0
$156$	2.00000	0.160128
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	4.00000	0.318223
$159$	6.00000	0.475831
$160$	0	0
$161$	16.0000	1.26098
$162$	−1.00000	−0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	12.0000	0.931381
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	12.0000	0.925820
$169$	−9.00000	−0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	−4.00000	−0.304997
$173$	16.0000	1.21646	0.608229	−	0.793762i	$-0.291880\pi$
$173$	16.0000	1.21646	0.608229	+	0.793762i	$0.291880\pi$
$174$	0	0
$175$	20.0000	1.51186
$176$	−4.00000	−0.301511
$177$	−12.0000	−0.901975
$178$	10.0000	0.749532
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	8.00000	0.592999
$183$	8.00000	0.591377
$184$	−12.0000	−0.884652
$185$	0	0
$186$	4.00000	0.293294
$187$	0	0
$188$	8.00000	0.583460
$189$	4.00000	0.290957
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−7.00000	−0.505181
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	−9.00000	−0.642857
$197$	16.0000	1.13995	0.569976	−	0.821661i	$-0.306952\pi$
$197$	16.0000	1.13995	0.569976	+	0.821661i	$0.306952\pi$
$198$	−4.00000	−0.284268
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−15.0000	−1.06066
$201$	−12.0000	−0.846415
$202$	6.00000	0.422159
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.00000	−0.278019
$208$	−2.00000	−0.138675
$209$	16.0000	1.10674
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	6.00000	0.412082
$213$	12.0000	0.822226
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−3.00000	−0.204124
$217$	−16.0000	−1.08615
$218$	−8.00000	−0.541828
$219$	0	0
$220$	0	0
$221$	0	0
$222$	−8.00000	−0.536925
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	20.0000	1.33631
$225$	−5.00000	−0.333333
$226$	8.00000	0.532152
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	4.00000	0.264906
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	0	0
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−12.0000	−0.781133
$237$	4.00000	0.259828
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	16.0000	1.03065	0.515325	−	0.856995i	$-0.327671\pi$
$241$	16.0000	1.03065	0.515325	+	0.856995i	$0.327671\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	8.00000	0.512148
$245$	0	0
$246$	−8.00000	−0.510061
$247$	8.00000	0.509028
$248$	12.0000	0.762001
$249$	12.0000	0.760469
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	4.00000	0.251976
$253$	−16.0000	−1.00591
$254$	8.00000	0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	4.00000	0.249029
$259$	32.0000	1.98838
$260$	0	0
$261$	0	0
$262$	4.00000	0.247121
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	16.0000	0.981023
$267$	10.0000	0.611990
$268$	−12.0000	−0.733017
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	10.0000	0.604122
$275$	−20.0000	−1.20605
$276$	−4.00000	−0.240772
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	4.00000	0.239904
$279$	4.00000	0.239474
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	−8.00000	−0.476393
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−8.00000	−0.473050
$287$	32.0000	1.88890
$288$	−5.00000	−0.294628
$289$	0	0
$290$	0	0
$291$	16.0000	0.937937
$292$	0	0
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	−24.0000	−1.39497
$297$	−4.00000	−0.232104
$298$	−6.00000	−0.347571
$299$	−8.00000	−0.462652
$300$	−5.00000	−0.288675
$301$	−16.0000	−0.922225
$302$	0	0
$303$	6.00000	0.344691
$304$	−4.00000	−0.229416
$305$	0	0
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	16.0000	0.911685
$309$	0	0
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	−6.00000	−0.339683
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	4.00000	0.225018
$317$	−32.0000	−1.79730	−0.898650	−	0.438667i	$-0.855451\pi$
$317$	−32.0000	−1.79730	−0.898650	+	0.438667i	$0.855451\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	−16.0000	−0.891645
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	−10.0000	−0.554700
$326$	−20.0000	−1.10770
$327$	−8.00000	−0.442401
$328$	−24.0000	−1.32518
$329$	32.0000	1.76422
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	12.0000	0.658586
$333$	−8.00000	−0.438397
$334$	12.0000	0.656611
$335$	0	0
$336$	−4.00000	−0.218218
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	9.00000	0.489535
$339$	8.00000	0.434500
$340$	0	0
$341$	16.0000	0.866449
$342$	−4.00000	−0.216295
$343$	−8.00000	−0.431959
$344$	12.0000	0.646997
$345$	0	0
$346$	−16.0000	−0.860165
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−20.0000	−1.06904
$351$	−2.00000	−0.106752
$352$	−20.0000	−1.06600
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	4.00000	0.211407
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	8.00000	0.420471
$363$	−5.00000	−0.262432
$364$	8.00000	0.419314
$365$	0	0
$366$	−8.00000	−0.418167
$367$	12.0000	0.626395	0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000	0.626395	0.313197	+	0.949688i	$0.398600\pi$
$368$	4.00000	0.208514
$369$	−8.00000	−0.416463
$370$	0	0
$371$	24.0000	1.24602
$372$	4.00000	0.207390
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	0	0
$376$	−24.0000	−1.23771
$377$	0	0
$378$	−4.00000	−0.205738
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	8.00000	0.409316
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	0	0
$387$	4.00000	0.203331
$388$	16.0000	0.812277
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	27.0000	1.36371
$393$	4.00000	0.201773
$394$	−16.0000	−0.806068
$395$	0	0
$396$	−4.00000	−0.201008
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	−20.0000	−1.00251
$399$	16.0000	0.801002
$400$	5.00000	0.250000
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	12.0000	0.598506
$403$	8.00000	0.398508
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	−32.0000	−1.58618
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	0	0
$413$	−48.0000	−2.36193
$414$	4.00000	0.196589
$415$	0	0
$416$	−10.0000	−0.490290
$417$	4.00000	0.195881
$418$	−16.0000	−0.782586
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−4.00000	−0.194717
$423$	−8.00000	−0.388973
$424$	−18.0000	−0.874157
$425$	0	0
$426$	−12.0000	−0.581402
$427$	32.0000	1.54859
$428$	−12.0000	−0.580042
$429$	−8.00000	−0.386244
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	1.00000	0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	−8.00000	−0.383131
$437$	−16.0000	−0.765384
$438$	0	0
$439$	36.0000	1.71819	0.859093	−	0.511819i	$-0.171028\pi$
$439$	36.0000	1.71819	0.859093	+	0.511819i	$0.171028\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	16.0000	0.757622
$447$	−6.00000	−0.283790
$448$	−28.0000	−1.32288
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	5.00000	0.235702
$451$	−32.0000	−1.50682
$452$	8.00000	0.376288
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	−12.0000	−0.561951
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	0	0
$461$	34.0000	1.58354	0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000	1.58354	0.791769	+	0.610821i	$0.209160\pi$
$462$	−16.0000	−0.744387
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	0	0
$465$	0	0
$466$	8.00000	0.370593
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	−2.00000	−0.0924500
$469$	−48.0000	−2.21643
$470$	0	0
$471$	2.00000	0.0921551
$472$	36.0000	1.65703
$473$	16.0000	0.735681
$474$	−4.00000	−0.183726
$475$	−20.0000	−0.917663
$476$	0	0
$477$	−6.00000	−0.274721
$478$	8.00000	0.365911
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	−16.0000	−0.728780
$483$	−16.0000	−0.728025
$484$	−5.00000	−0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−24.0000	−1.08643
$489$	−20.0000	−0.904431
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−8.00000	−0.360668
$493$	0	0
$494$	−8.00000	−0.359937
$495$	0	0
$496$	−4.00000	−0.179605
$497$	48.0000	2.15309
$498$	−12.0000	−0.537733
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	−12.0000	−0.535586
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	−12.0000	−0.534522
$505$	0	0
$506$	16.0000	0.711287
$507$	9.00000	0.399704
$508$	8.00000	0.354943
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	0	0
$512$	11.0000	0.486136
$513$	−4.00000	−0.176604
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	4.00000	0.176090
$517$	−32.0000	−1.40736
$518$	−32.0000	−1.40600
$519$	−16.0000	−0.702322
$520$	0	0
$521$	24.0000	1.05146	0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000	1.05146	0.525730	+	0.850652i	$0.323792\pi$
$522$	0	0
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	4.00000	0.174741
$525$	−20.0000	−0.872872
$526$	24.0000	1.04645
$527$	0	0
$528$	4.00000	0.174078
$529$	−7.00000	−0.304348
$530$	0	0
$531$	12.0000	0.520756
$532$	16.0000	0.693688
$533$	−16.0000	−0.693037
$534$	−10.0000	−0.432742
$535$	0	0
$536$	36.0000	1.55496
$537$	4.00000	0.172613
$538$	0	0
$539$	36.0000	1.55063
$540$	0	0
$541$	−40.0000	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$541$	−40.0000	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$542$	0	0
$543$	8.00000	0.343313
$544$	0	0
$545$	0	0
$546$	−8.00000	−0.342368
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	10.0000	0.427179
$549$	−8.00000	−0.341432
$550$	20.0000	0.852803
$551$	0	0
$552$	12.0000	0.510754
$553$	16.0000	0.680389
$554$	−8.00000	−0.339887
$555$	0	0
$556$	4.00000	0.169638
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	−4.00000	−0.169334
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	10.0000	0.421825
$563$	−44.0000	−1.85438	−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000	−1.85438	−0.927189	+	0.374593i	$0.877783\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−4.00000	−0.168133
$567$	−4.00000	−0.167984
$568$	−36.0000	−1.51053
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	−8.00000	−0.334497
$573$	8.00000	0.334205
$574$	−32.0000	−1.33565
$575$	20.0000	0.834058
$576$	7.00000	0.291667
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	48.0000	1.99138
$582$	−16.0000	−0.663221
$583$	−24.0000	−0.993978
$584$	0	0
$585$	0	0
$586$	10.0000	0.413096
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	9.00000	0.371154
$589$	16.0000	0.659269
$590$	0	0
$591$	−16.0000	−0.658152
$592$	8.00000	0.328798
$593$	46.0000	1.88899	0.944497	−	0.328521i	$-0.106550\pi$
$593$	46.0000	1.88899	0.944497	+	0.328521i	$0.106550\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−20.0000	−0.818546
$598$	8.00000	0.327144
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	15.0000	0.612372
$601$	16.0000	0.652654	0.326327	−	0.945257i	$-0.394189\pi$
$601$	16.0000	0.652654	0.326327	+	0.945257i	$0.394189\pi$
$602$	16.0000	0.652111
$603$	12.0000	0.488678
$604$	0	0
$605$	0	0
$606$	−6.00000	−0.243733
$607$	4.00000	0.162355	0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000	0.162355	0.0811775	+	0.996700i	$0.474132\pi$
$608$	−20.0000	−0.811107
$609$	0	0
$610$	0	0
$611$	−16.0000	−0.647291
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	−48.0000	−1.93398
$617$	24.0000	0.966204	0.483102	−	0.875564i	$-0.339510\pi$
$617$	24.0000	0.966204	0.483102	+	0.875564i	$0.339510\pi$
$618$	0	0
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	12.0000	0.481156
$623$	40.0000	1.60257
$624$	2.00000	0.0800641
$625$	25.0000	1.00000
$626$	16.0000	0.639489
$627$	−16.0000	−0.638978
$628$	2.00000	0.0798087
$629$	0	0
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−12.0000	−0.477334
$633$	−4.00000	−0.158986
$634$	32.0000	1.27088
$635$	0	0
$636$	−6.00000	−0.237915
$637$	18.0000	0.713186
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−8.00000	−0.315981	−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000	−0.315981	−0.157991	+	0.987441i	$0.550502\pi$
$642$	12.0000	0.473602
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	3.00000	0.117851
$649$	48.0000	1.88416
$650$	10.0000	0.392232
$651$	16.0000	0.627089
$652$	−20.0000	−0.783260
$653$	−32.0000	−1.25226	−0.626128	−	0.779720i	$-0.715361\pi$
$653$	−32.0000	−1.25226	−0.626128	+	0.779720i	$0.715361\pi$
$654$	8.00000	0.312825
$655$	0	0
$656$	8.00000	0.312348
$657$	0	0
$658$	−32.0000	−1.24749
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	−36.0000	−1.39707
$665$	0	0
$666$	8.00000	0.309994
$667$	0	0
$668$	12.0000	0.464294
$669$	16.0000	0.618596
$670$	0	0
$671$	−32.0000	−1.23535
$672$	−20.0000	−0.771517
$673$	32.0000	1.23351	0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000	1.23351	0.616755	+	0.787155i	$0.288447\pi$
$674$	−16.0000	−0.616297
$675$	5.00000	0.192450
$676$	9.00000	0.346154
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	−8.00000	−0.307238
$679$	64.0000	2.45609
$680$	0	0
$681$	12.0000	0.459841
$682$	−16.0000	−0.612672
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	8.00000	0.305441
$687$	10.0000	0.381524
$688$	−4.00000	−0.152499
$689$	−12.0000	−0.457164
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−16.0000	−0.608229
$693$	−16.0000	−0.607790
$694$	−4.00000	−0.151838
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−2.00000	−0.0757011
$699$	8.00000	0.302588
$700$	−20.0000	−0.755929
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	2.00000	0.0754851
$703$	−32.0000	−1.20690
$704$	28.0000	1.05529
$705$	0	0
$706$	−18.0000	−0.677439
$707$	24.0000	0.902613
$708$	12.0000	0.450988
$709$	40.0000	1.50223	0.751116	−	0.660171i	$-0.229516\pi$
$709$	40.0000	1.50223	0.751116	+	0.660171i	$0.229516\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	−30.0000	−1.12430
$713$	−16.0000	−0.599205
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	8.00000	0.298765
$718$	32.0000	1.19423
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	0	0
$722$	3.00000	0.111648
$723$	−16.0000	−0.595046
$724$	8.00000	0.297318
$725$	0	0
$726$	5.00000	0.185567
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	−24.0000	−0.889499
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−8.00000	−0.295689
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−12.0000	−0.442928
$735$	0	0
$736$	20.0000	0.737210
$737$	48.0000	1.76810
$738$	8.00000	0.294484
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	−24.0000	−0.881068
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	−12.0000	−0.439941
$745$	0	0
$746$	−26.0000	−0.951928
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−48.0000	−1.75388
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	8.00000	0.291730
$753$	−12.0000	−0.437304
$754$	0	0
$755$	0	0
$756$	−4.00000	−0.145479
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	20.0000	0.726433
$759$	16.0000	0.580763
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−8.00000	−0.289809
$763$	−32.0000	−1.15848
$764$	8.00000	0.289430
$765$	0	0
$766$	−16.0000	−0.578103
$767$	24.0000	0.866590
$768$	17.0000	0.613435
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	0	0
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−4.00000	−0.143777
$775$	−20.0000	−0.718421
$776$	−48.0000	−1.72310
$777$	−32.0000	−1.14799
$778$	−6.00000	−0.215110
$779$	−32.0000	−1.14652
$780$	0	0
$781$	−48.0000	−1.71758
$782$	0	0
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	−4.00000	−0.142675
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	−16.0000	−0.569976
$789$	24.0000	0.854423
$790$	0	0
$791$	32.0000	1.13779
$792$	12.0000	0.426401
$793$	−16.0000	−0.568177
$794$	8.00000	0.283909
$795$	0	0
$796$	−20.0000	−0.708881
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	−16.0000	−0.566394
$799$	0	0
$800$	25.0000	0.883883
$801$	−10.0000	−0.353333
$802$	24.0000	0.847469
$803$	0	0
$804$	12.0000	0.423207
$805$	0	0
$806$	−8.00000	−0.281788
$807$	0	0
$808$	−18.0000	−0.633238
$809$	−40.0000	−1.40633	−0.703163	−	0.711029i	$-0.748229\pi$
$809$	−40.0000	−1.40633	−0.703163	+	0.711029i	$0.748229\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	0	0
$813$	0	0
$814$	32.0000	1.12160
$815$	0	0
$816$	0	0
$817$	16.0000	0.559769
$818$	−10.0000	−0.349642
$819$	−8.00000	−0.279543
$820$	0	0
$821$	−32.0000	−1.11681	−0.558404	−	0.829569i	$-0.688586\pi$
$821$	−32.0000	−1.11681	−0.558404	+	0.829569i	$0.688586\pi$
$822$	−10.0000	−0.348790
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	0	0
$825$	20.0000	0.696311
$826$	48.0000	1.67013
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	4.00000	0.139010
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	14.0000	0.485363
$833$	0	0
$834$	−4.00000	−0.138509
$835$	0	0
$836$	−16.0000	−0.553372
$837$	−4.00000	−0.138260
$838$	−4.00000	−0.138178
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	22.0000	0.758170
$843$	10.0000	0.344418
$844$	−4.00000	−0.137686
$845$	0	0
$846$	8.00000	0.275046
$847$	−20.0000	−0.687208
$848$	6.00000	0.206041
$849$	−4.00000	−0.137280
$850$	0	0
$851$	32.0000	1.09695
$852$	−12.0000	−0.411113
$853$	24.0000	0.821744	0.410872	−	0.911693i	$-0.365224\pi$
$853$	24.0000	0.821744	0.410872	+	0.911693i	$0.365224\pi$
$854$	−32.0000	−1.09502
$855$	0	0
$856$	36.0000	1.23045
$857$	8.00000	0.273275	0.136637	−	0.990621i	$-0.456370\pi$
$857$	8.00000	0.273275	0.136637	+	0.990621i	$0.456370\pi$
$858$	8.00000	0.273115
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	−32.0000	−1.09056
$862$	20.0000	0.681203
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	16.0000	0.543075
$869$	−16.0000	−0.542763
$870$	0	0
$871$	24.0000	0.813209
$872$	24.0000	0.812743
$873$	−16.0000	−0.541518
$874$	16.0000	0.541208
$875$	0	0
$876$	0	0
$877$	−40.0000	−1.35070	−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000	−1.35070	−0.675352	+	0.737496i	$0.736008\pi$
$878$	−36.0000	−1.21494
$879$	10.0000	0.337292
$880$	0	0
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	−9.00000	−0.303046
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	0	0
$886$	20.0000	0.671913
$887$	4.00000	0.134307	0.0671534	−	0.997743i	$-0.478608\pi$
$887$	4.00000	0.134307	0.0671534	+	0.997743i	$0.478608\pi$
$888$	24.0000	0.805387
$889$	32.0000	1.07325
$890$	0	0
$891$	4.00000	0.134005
$892$	16.0000	0.535720
$893$	−32.0000	−1.07084
$894$	6.00000	0.200670
$895$	0	0
$896$	−12.0000	−0.400892
$897$	8.00000	0.267112
$898$	−8.00000	−0.266963
$899$	0	0
$900$	5.00000	0.166667
$901$	0	0
$902$	32.0000	1.06548
$903$	16.0000	0.532447
$904$	−24.0000	−0.798228
$905$	0	0
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	12.0000	0.398234
$909$	−6.00000	−0.199007
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	4.00000	0.132453
$913$	−48.0000	−1.58857
$914$	38.0000	1.25693
$915$	0	0
$916$	10.0000	0.330409
$917$	16.0000	0.528367
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	−34.0000	−1.11973
$923$	−24.0000	−0.789970
$924$	−16.0000	−0.526361
$925$	40.0000	1.31519
$926$	40.0000	1.31448
$927$	0	0
$928$	0	0
$929$	24.0000	0.787414	0.393707	−	0.919236i	$-0.371192\pi$
$929$	24.0000	0.787414	0.393707	+	0.919236i	$0.371192\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	8.00000	0.262049
$933$	12.0000	0.392862
$934$	4.00000	0.130884
$935$	0	0
$936$	6.00000	0.196116
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	48.0000	1.56726
$939$	16.0000	0.522140
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	−2.00000	−0.0651635
$943$	32.0000	1.04206
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−16.0000	−0.520205
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−4.00000	−0.129914
$949$	0	0
$950$	20.0000	0.648886
$951$	32.0000	1.03767
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	8.00000	0.258738
$957$	0	0
$958$	12.0000	0.387702
$959$	40.0000	1.29167
$960$	0	0
$961$	−15.0000	−0.483871
$962$	16.0000	0.515861
$963$	12.0000	0.386695
$964$	−16.0000	−0.515325
$965$	0	0
$966$	16.0000	0.514792
$967$	−24.0000	−0.771788	−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000	−0.771788	−0.385894	+	0.922543i	$0.626107\pi$
$968$	15.0000	0.482118
$969$	0	0
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	1.00000	0.0320750
$973$	16.0000	0.512936
$974$	4.00000	0.128168
$975$	10.0000	0.320256
$976$	8.00000	0.256074
$977$	46.0000	1.47167	0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000	1.47167	0.735835	+	0.677161i	$0.236790\pi$
$978$	20.0000	0.639529
$979$	−40.0000	−1.27841
$980$	0	0
$981$	8.00000	0.255420
$982$	−20.0000	−0.638226
$983$	20.0000	0.637901	0.318950	−	0.947771i	$-0.396670\pi$
$983$	20.0000	0.637901	0.318950	+	0.947771i	$0.396670\pi$
$984$	24.0000	0.765092
$985$	0	0
$986$	0	0
$987$	−32.0000	−1.01857
$988$	−8.00000	−0.254514
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−12.0000	−0.381193	−0.190596	−	0.981669i	$-0.561042\pi$
$991$	−12.0000	−0.381193	−0.190596	+	0.981669i	$0.561042\pi$
$992$	−20.0000	−0.635001
$993$	−20.0000	−0.634681
$994$	−48.0000	−1.52247
$995$	0	0
$996$	−12.0000	−0.380235
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	−36.0000	−1.13956
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.2.a.a.1.1		1
3.2	odd	2		2601.2.a.i.1.1		1
17.2	even	8		867.2.e.d.616.1		4
17.3	odd	16		867.2.h.d.757.1		8
17.4	even	4		51.2.d.b.16.2	yes	2
17.5	odd	16		867.2.h.d.688.2		8
17.6	odd	16		867.2.h.d.733.1		8
17.7	odd	16		867.2.h.d.712.2		8
17.8	even	8		867.2.e.d.829.2		4
17.9	even	8		867.2.e.d.829.1		4
17.10	odd	16		867.2.h.d.712.1		8
17.11	odd	16		867.2.h.d.733.2		8
17.12	odd	16		867.2.h.d.688.1		8
17.13	even	4		51.2.d.b.16.1	✓	2
17.14	odd	16		867.2.h.d.757.2		8
17.15	even	8		867.2.e.d.616.2		4
17.16	even	2		867.2.a.b.1.1		1
51.38	odd	4		153.2.d.a.118.1		2
51.47	odd	4		153.2.d.a.118.2		2
51.50	odd	2		2601.2.a.j.1.1		1
68.47	odd	4		816.2.c.c.577.2		2
68.55	odd	4		816.2.c.c.577.1		2
85.4	even	4		1275.2.g.a.526.1		2
85.13	odd	4		1275.2.d.d.424.1		2
85.38	odd	4		1275.2.d.b.424.1		2
85.47	odd	4		1275.2.d.b.424.2		2
85.64	even	4		1275.2.g.a.526.2		2
85.72	odd	4		1275.2.d.d.424.2		2
136.13	even	4		3264.2.c.e.577.2		2
136.21	even	4		3264.2.c.e.577.1		2
136.115	odd	4		3264.2.c.d.577.1		2
136.123	odd	4		3264.2.c.d.577.2		2
204.47	even	4		2448.2.c.j.577.1		2
204.191	even	4		2448.2.c.j.577.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.d.b.16.1	✓	2	17.13	even	4
51.2.d.b.16.2	yes	2	17.4	even	4
153.2.d.a.118.1		2	51.38	odd	4
153.2.d.a.118.2		2	51.47	odd	4
816.2.c.c.577.1		2	68.55	odd	4
816.2.c.c.577.2		2	68.47	odd	4
867.2.a.a.1.1		1	1.1	even	1	trivial
867.2.a.b.1.1		1	17.16	even	2
867.2.e.d.616.1		4	17.2	even	8
867.2.e.d.616.2		4	17.15	even	8
867.2.e.d.829.1		4	17.9	even	8
867.2.e.d.829.2		4	17.8	even	8
867.2.h.d.688.1		8	17.12	odd	16
867.2.h.d.688.2		8	17.5	odd	16
867.2.h.d.712.1		8	17.10	odd	16
867.2.h.d.712.2		8	17.7	odd	16
867.2.h.d.733.1		8	17.6	odd	16
867.2.h.d.733.2		8	17.11	odd	16
867.2.h.d.757.1		8	17.3	odd	16
867.2.h.d.757.2		8	17.14	odd	16
1275.2.d.b.424.1		2	85.38	odd	4
1275.2.d.b.424.2		2	85.47	odd	4
1275.2.d.d.424.1		2	85.13	odd	4
1275.2.d.d.424.2		2	85.72	odd	4
1275.2.g.a.526.1		2	85.4	even	4
1275.2.g.a.526.2		2	85.64	even	4
2448.2.c.j.577.1		2	204.47	even	4
2448.2.c.j.577.2		2	204.191	even	4
2601.2.a.i.1.1		1	3.2	odd	2
2601.2.a.j.1.1		1	51.50	odd	2
3264.2.c.d.577.1		2	136.115	odd	4
3264.2.c.d.577.2		2	136.123	odd	4
3264.2.c.e.577.1		2	136.21	even	4
3264.2.c.e.577.2		2	136.13	even	4

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:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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