LMFDB - Embedded newform 8112.2.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8112,2,Mod(1,8112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8112.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.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:	$64.7746461197$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +2.00000 q^{15} -4.00000 q^{17} -4.00000 q^{19} -1.00000 q^{21} -6.00000 q^{23} -1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +1.00000 q^{31} -2.00000 q^{33} -2.00000 q^{35} +10.0000 q^{37} -4.00000 q^{41} -1.00000 q^{43} +2.00000 q^{45} +10.0000 q^{47} -6.00000 q^{49} -4.00000 q^{51} +8.00000 q^{53} -4.00000 q^{55} -4.00000 q^{57} +2.00000 q^{59} -5.00000 q^{61} -1.00000 q^{63} +7.00000 q^{67} -6.00000 q^{69} -10.0000 q^{71} -7.00000 q^{73} -1.00000 q^{75} +2.00000 q^{77} -17.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -8.00000 q^{85} +6.00000 q^{87} -16.0000 q^{89} +1.00000 q^{93} -8.00000 q^{95} +13.0000 q^{97} -2.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	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	$0.597491\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	−17.0000	−1.91265	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−17.0000	−1.91265	−0.956325	+	0.292306i	$0.905577\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	−12.0000	−1.11901
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−9.00000	−0.798621	−0.399310	−	0.916816i	$-0.630750\pi$
$127$	−9.00000	−0.798621	−0.399310	+	0.916816i	$0.630750\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	2.00000	0.172133
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	0	0
$139$	−3.00000	−0.254457	−0.127228	−	0.991873i	$-0.540608\pi$
$139$	−3.00000	−0.254457	−0.127228	+	0.991873i	$0.540608\pi$
$140$	0	0
$141$	10.0000	0.842152
$142$	0	0
$143$	0	0
$144$	0	0
$145$	12.0000	0.996546
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	−1.00000	−0.0798087	−0.0399043	−	0.999204i	$-0.512705\pi$
$157$	−1.00000	−0.0798087	−0.0399043	+	0.999204i	$0.512705\pi$
$158$	0	0
$159$	8.00000	0.634441
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	7.00000	0.548282	0.274141	−	0.961689i	$-0.411606\pi$
$163$	7.00000	0.548282	0.274141	+	0.961689i	$0.411606\pi$
$164$	0	0
$165$	−4.00000	−0.311400
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	2.00000	0.150329
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−5.00000	−0.369611
$184$	0	0
$185$	20.0000	1.47043
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−22.0000	−1.59186	−0.795932	−	0.605386i	$-0.793019\pi$
$191$	−22.0000	−1.59186	−0.795932	+	0.605386i	$0.793019\pi$
$192$	0	0
$193$	−17.0000	−1.22369	−0.611843	−	0.790979i	$-0.709572\pi$
$193$	−17.0000	−1.22369	−0.611843	+	0.790979i	$0.709572\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	0	0
$201$	7.00000	0.493742
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	−8.00000	−0.558744
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	27.0000	1.85876	0.929378	−	0.369129i	$-0.120344\pi$
$211$	27.0000	1.85876	0.929378	+	0.369129i	$0.120344\pi$
$212$	0	0
$213$	−10.0000	−0.685189
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	−1.00000	−0.0678844
$218$	0	0
$219$	−7.00000	−0.473016
$220$	0	0
$221$	0	0
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	20.0000	1.30466
$236$	0	0
$237$	−17.0000	−1.10427
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−12.0000	−0.766652
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	0	0
$255$	−8.00000	−0.500979
$256$	0	0
$257$	24.0000	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$257$	24.0000	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	16.0000	0.982872
$266$	0	0
$267$	−16.0000	−0.979184
$268$	0	0
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	0	0
$271$	−11.0000	−0.668202	−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000	−0.668202	−0.334101	+	0.942537i	$0.608433\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.00000	0.120605
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	−5.00000	−0.297219	−0.148610	−	0.988896i	$-0.547480\pi$
$283$	−5.00000	−0.297219	−0.148610	+	0.988896i	$0.547480\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	13.0000	0.762073
$292$	0	0
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	−2.00000	−0.114897
$304$	0	0
$305$	−10.0000	−0.572598
$306$	0	0
$307$	11.0000	0.627803	0.313902	−	0.949456i	$-0.398364\pi$
$307$	11.0000	0.627803	0.313902	+	0.949456i	$0.398364\pi$
$308$	0	0
$309$	−13.0000	−0.739544
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−21.0000	−1.18699	−0.593495	−	0.804838i	$-0.702252\pi$
$313$	−21.0000	−1.18699	−0.593495	+	0.804838i	$0.702252\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	−18.0000	−1.00466
$322$	0	0
$323$	16.0000	0.890264
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−11.0000	−0.608301
$328$	0	0
$329$	−10.0000	−0.551318
$330$	0	0
$331$	25.0000	1.37412	0.687062	−	0.726599i	$-0.258900\pi$
$331$	25.0000	1.37412	0.687062	+	0.726599i	$0.258900\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	0	0
$335$	14.0000	0.764902
$336$	0	0
$337$	−3.00000	−0.163420	−0.0817102	−	0.996656i	$-0.526038\pi$
$337$	−3.00000	−0.163420	−0.0817102	+	0.996656i	$0.526038\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	−12.0000	−0.646058
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−9.00000	−0.481759	−0.240879	−	0.970555i	$-0.577436\pi$
$349$	−9.00000	−0.481759	−0.240879	+	0.970555i	$0.577436\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	−20.0000	−1.06149
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	−14.0000	−0.732793
$366$	0	0
$367$	5.00000	0.260998	0.130499	−	0.991448i	$-0.458342\pi$
$367$	5.00000	0.260998	0.130499	+	0.991448i	$0.458342\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	−8.00000	−0.415339
$372$	0	0
$373$	−33.0000	−1.70868	−0.854338	−	0.519718i	$-0.826037\pi$
$373$	−33.0000	−1.70868	−0.854338	+	0.519718i	$0.826037\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−9.00000	−0.462299	−0.231149	−	0.972918i	$-0.574249\pi$
$379$	−9.00000	−0.462299	−0.231149	+	0.972918i	$0.574249\pi$
$380$	0	0
$381$	−9.00000	−0.461084
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	−1.00000	−0.0508329
$388$	0	0
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	0	0
$393$	−10.0000	−0.504433
$394$	0	0
$395$	−34.0000	−1.71073
$396$	0	0
$397$	3.00000	0.150566	0.0752828	−	0.997162i	$-0.476014\pi$
$397$	3.00000	0.150566	0.0752828	+	0.997162i	$0.476014\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.00000	0.0993808
$406$	0	0
$407$	−20.0000	−0.991363
$408$	0	0
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	0	0
$413$	−2.00000	−0.0984136
$414$	0	0
$415$	−24.0000	−1.17811
$416$	0	0
$417$	−3.00000	−0.146911
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	−11.0000	−0.536107	−0.268054	−	0.963404i	$-0.586380\pi$
$421$	−11.0000	−0.536107	−0.268054	+	0.963404i	$0.586380\pi$
$422$	0	0
$423$	10.0000	0.486217
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	5.00000	0.241967
$428$	0	0
$429$	0	0
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	0	0
$433$	−9.00000	−0.432512	−0.216256	−	0.976337i	$-0.569385\pi$
$433$	−9.00000	−0.432512	−0.216256	+	0.976337i	$0.569385\pi$
$434$	0	0
$435$	12.0000	0.575356
$436$	0	0
$437$	24.0000	1.14808
$438$	0	0
$439$	−1.00000	−0.0477274	−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000	−0.0477274	−0.0238637	+	0.999715i	$0.507597\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	0	0
$445$	−32.0000	−1.51695
$446$	0	0
$447$	0	0
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.00000	−0.140334	−0.0701670	−	0.997535i	$-0.522353\pi$
$457$	−3.00000	−0.140334	−0.0701670	+	0.997535i	$0.522353\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	−19.0000	−0.883005	−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000	−0.883005	−0.441502	+	0.897260i	$0.645554\pi$
$464$	0	0
$465$	2.00000	0.0927478
$466$	0	0
$467$	−26.0000	−1.20314	−0.601568	−	0.798821i	$-0.705457\pi$
$467$	−26.0000	−1.20314	−0.601568	+	0.798821i	$0.705457\pi$
$468$	0	0
$469$	−7.00000	−0.323230
$470$	0	0
$471$	−1.00000	−0.0460776
$472$	0	0
$473$	2.00000	0.0919601
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	8.00000	0.366295
$478$	0	0
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	6.00000	0.273009
$484$	0	0
$485$	26.0000	1.18060
$486$	0	0
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	0	0
$489$	7.00000	0.316551
$490$	0	0
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	0	0
$493$	−24.0000	−1.08091
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	10.0000	0.448561
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	0	0
$508$	0	0
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	−26.0000	−1.14570
$516$	0	0
$517$	−20.0000	−0.879599
$518$	0	0
$519$	−10.0000	−0.438951
$520$	0	0
$521$	−8.00000	−0.350486	−0.175243	−	0.984525i	$-0.556071\pi$
$521$	−8.00000	−0.350486	−0.175243	+	0.984525i	$0.556071\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	−4.00000	−0.174243
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	2.00000	0.0867926
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−36.0000	−1.55642
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	29.0000	1.24681	0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000	1.24681	0.623404	+	0.781900i	$0.285749\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	−22.0000	−0.942376
$546$	0	0
$547$	−35.0000	−1.49649	−0.748246	−	0.663421i	$-0.769104\pi$
$547$	−35.0000	−1.49649	−0.748246	+	0.663421i	$0.769104\pi$
$548$	0	0
$549$	−5.00000	−0.213395
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	17.0000	0.722914
$554$	0	0
$555$	20.0000	0.848953
$556$	0	0
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	26.0000	1.09577	0.547885	−	0.836554i	$-0.315433\pi$
$563$	26.0000	1.09577	0.547885	+	0.836554i	$0.315433\pi$
$564$	0	0
$565$	36.0000	1.51453
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	−22.0000	−0.919063
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	0	0
$579$	−17.0000	−0.706496
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	−16.0000	−0.662652
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.0000	−0.412744	−0.206372	−	0.978474i	$-0.566166\pi$
$587$	−10.0000	−0.412744	−0.206372	+	0.978474i	$0.566166\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	0	0
$597$	5.00000	0.204636
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	7.00000	0.285062
$604$	0	0
$605$	−14.0000	−0.569181
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−17.0000	−0.686624	−0.343312	−	0.939222i	$-0.611549\pi$
$613$	−17.0000	−0.686624	−0.343312	+	0.939222i	$0.611549\pi$
$614$	0	0
$615$	−8.00000	−0.322591
$616$	0	0
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	0	0
$619$	13.0000	0.522514	0.261257	−	0.965269i	$-0.415863\pi$
$619$	13.0000	0.522514	0.261257	+	0.965269i	$0.415863\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	16.0000	0.641026
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	8.00000	0.319489
$628$	0	0
$629$	−40.0000	−1.59490
$630$	0	0
$631$	−9.00000	−0.358284	−0.179142	−	0.983823i	$-0.557332\pi$
$631$	−9.00000	−0.358284	−0.179142	+	0.983823i	$0.557332\pi$
$632$	0	0
$633$	27.0000	1.07315
$634$	0	0
$635$	−18.0000	−0.714308
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10.0000	−0.395594
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	31.0000	1.22252	0.611260	−	0.791430i	$-0.290663\pi$
$643$	31.0000	1.22252	0.611260	+	0.791430i	$0.290663\pi$
$644$	0	0
$645$	−2.00000	−0.0787499
$646$	0	0
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	−1.00000	−0.0391931
$652$	0	0
$653$	−4.00000	−0.156532	−0.0782660	−	0.996933i	$-0.524938\pi$
$653$	−4.00000	−0.156532	−0.0782660	+	0.996933i	$0.524938\pi$
$654$	0	0
$655$	−20.0000	−0.781465
$656$	0	0
$657$	−7.00000	−0.273096
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	19.0000	0.739014	0.369507	−	0.929228i	$-0.379527\pi$
$661$	19.0000	0.739014	0.369507	+	0.929228i	$0.379527\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	−36.0000	−1.39393
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	10.0000	0.386046
$672$	0	0
$673$	7.00000	0.269830	0.134915	−	0.990857i	$-0.456924\pi$
$673$	7.00000	0.269830	0.134915	+	0.990857i	$0.456924\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	8.00000	0.307465	0.153732	−	0.988113i	$-0.450871\pi$
$677$	8.00000	0.307465	0.153732	+	0.988113i	$0.450871\pi$
$678$	0	0
$679$	−13.0000	−0.498894
$680$	0	0
$681$	28.0000	1.07296
$682$	0	0
$683$	46.0000	1.76014	0.880071	−	0.474843i	$-0.157495\pi$
$683$	46.0000	1.76014	0.880071	+	0.474843i	$0.157495\pi$
$684$	0	0
$685$	16.0000	0.611329
$686$	0	0
$687$	22.0000	0.839352
$688$	0	0
$689$	0	0
$690$	0	0
$691$	5.00000	0.190209	0.0951045	−	0.995467i	$-0.469681\pi$
$691$	5.00000	0.190209	0.0951045	+	0.995467i	$0.469681\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	−6.00000	−0.227593
$696$	0	0
$697$	16.0000	0.606043
$698$	0	0
$699$	−6.00000	−0.226941
$700$	0	0
$701$	28.0000	1.05755	0.528773	−	0.848763i	$-0.322652\pi$
$701$	28.0000	1.05755	0.528773	+	0.848763i	$0.322652\pi$
$702$	0	0
$703$	−40.0000	−1.50863
$704$	0	0
$705$	20.0000	0.753244
$706$	0	0
$707$	2.00000	0.0752177
$708$	0	0
$709$	−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
$711$	−17.0000	−0.637550
$712$	0	0
$713$	−6.00000	−0.224702
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	13.0000	0.484145
$722$	0	0
$723$	−14.0000	−0.520666
$724$	0	0
$725$	−6.00000	−0.222834
$726$	0	0
$727$	−41.0000	−1.52061	−0.760303	−	0.649569i	$-0.774949\pi$
$727$	−41.0000	−1.52061	−0.760303	+	0.649569i	$0.774949\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.00000	0.147945
$732$	0	0
$733$	−35.0000	−1.29275	−0.646377	−	0.763018i	$-0.723717\pi$
$733$	−35.0000	−1.29275	−0.646377	+	0.763018i	$0.723717\pi$
$734$	0	0
$735$	−12.0000	−0.442627
$736$	0	0
$737$	−14.0000	−0.515697
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	18.0000	0.657706
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	−16.0000	−0.583072
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	0	0
$759$	12.0000	0.435572
$760$	0	0
$761$	−8.00000	−0.290000	−0.145000	−	0.989432i	$-0.546318\pi$
$761$	−8.00000	−0.290000	−0.145000	+	0.989432i	$0.546318\pi$
$762$	0	0
$763$	11.0000	0.398227
$764$	0	0
$765$	−8.00000	−0.289241
$766$	0	0
$767$	0	0
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	0	0
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−10.0000	−0.358748
$778$	0	0
$779$	16.0000	0.573259
$780$	0	0
$781$	20.0000	0.715656
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−49.0000	−1.74666	−0.873331	−	0.487128i	$-0.838045\pi$
$787$	−49.0000	−1.74666	−0.873331	+	0.487128i	$0.838045\pi$
$788$	0	0
$789$	−8.00000	−0.284808
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	0	0
$794$	0	0
$795$	16.0000	0.567462
$796$	0	0
$797$	−8.00000	−0.283375	−0.141687	−	0.989911i	$-0.545253\pi$
$797$	−8.00000	−0.283375	−0.141687	+	0.989911i	$0.545253\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	0	0
$801$	−16.0000	−0.565332
$802$	0	0
$803$	14.0000	0.494049
$804$	0	0
$805$	12.0000	0.422944
$806$	0	0
$807$	−20.0000	−0.704033
$808$	0	0
$809$	32.0000	1.12506	0.562530	−	0.826777i	$-0.309828\pi$
$809$	32.0000	1.12506	0.562530	+	0.826777i	$0.309828\pi$
$810$	0	0
$811$	−51.0000	−1.79085	−0.895426	−	0.445210i	$-0.853129\pi$
$811$	−51.0000	−1.79085	−0.895426	+	0.445210i	$0.853129\pi$
$812$	0	0
$813$	−11.0000	−0.385787
$814$	0	0
$815$	14.0000	0.490399
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	−11.0000	−0.382046	−0.191023	−	0.981586i	$-0.561180\pi$
$829$	−11.0000	−0.382046	−0.191023	+	0.981586i	$0.561180\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	24.0000	0.831551
$834$	0	0
$835$	−24.0000	−0.830554
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	22.0000	0.759524	0.379762	−	0.925084i	$-0.376006\pi$
$839$	22.0000	0.759524	0.379762	+	0.925084i	$0.376006\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−2.00000	−0.0688837
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	−5.00000	−0.171600
$850$	0	0
$851$	−60.0000	−2.05677
$852$	0	0
$853$	−53.0000	−1.81469	−0.907343	−	0.420392i	$-0.861893\pi$
$853$	−53.0000	−1.81469	−0.907343	+	0.420392i	$0.861893\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	54.0000	1.84460	0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000	1.84460	0.922302	+	0.386469i	$0.126305\pi$
$858$	0	0
$859$	35.0000	1.19418	0.597092	−	0.802173i	$-0.296323\pi$
$859$	35.0000	1.19418	0.597092	+	0.802173i	$0.296323\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	0	0
$863$	26.0000	0.885050	0.442525	−	0.896756i	$-0.354083\pi$
$863$	26.0000	0.885050	0.442525	+	0.896756i	$0.354083\pi$
$864$	0	0
$865$	−20.0000	−0.680020
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	34.0000	1.15337
$870$	0	0
$871$	0	0
$872$	0	0
$873$	13.0000	0.439983
$874$	0	0
$875$	12.0000	0.405674
$876$	0	0
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	0	0
$879$	26.0000	0.876958
$880$	0	0
$881$	−40.0000	−1.34763	−0.673817	−	0.738898i	$-0.735346\pi$
$881$	−40.0000	−1.34763	−0.673817	+	0.738898i	$0.735346\pi$
$882$	0	0
$883$	−19.0000	−0.639401	−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000	−0.639401	−0.319700	+	0.947519i	$0.603582\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	0	0
$889$	9.00000	0.301850
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	−40.0000	−1.33855
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.00000	0.200111
$900$	0	0
$901$	−32.0000	−1.06607
$902$	0	0
$903$	1.00000	0.0332779
$904$	0	0
$905$	−20.0000	−0.664822
$906$	0	0
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	0	0
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	0	0
$913$	24.0000	0.794284
$914$	0	0
$915$	−10.0000	−0.330590
$916$	0	0
$917$	10.0000	0.330229
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	11.0000	0.362462
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−10.0000	−0.328798
$926$	0	0
$927$	−13.0000	−0.426976
$928$	0	0
$929$	28.0000	0.918650	0.459325	−	0.888268i	$-0.348091\pi$
$929$	28.0000	0.918650	0.459325	+	0.888268i	$0.348091\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	16.0000	0.523256
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	−21.0000	−0.685309
$940$	0	0
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	−20.0000	−0.649913	−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000	−0.649913	−0.324956	+	0.945729i	$0.605350\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	−44.0000	−1.42381
$956$	0	0
$957$	−12.0000	−0.387905
$958$	0	0
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−18.0000	−0.580042
$964$	0	0
$965$	−34.0000	−1.09450
$966$	0	0
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	0	0
$969$	16.0000	0.513994
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	3.00000	0.0961756
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24.0000	−0.767828	−0.383914	−	0.923369i	$-0.625424\pi$
$977$	−24.0000	−0.767828	−0.383914	+	0.923369i	$0.625424\pi$
$978$	0	0
$979$	32.0000	1.02272
$980$	0	0
$981$	−11.0000	−0.351203
$982$	0	0
$983$	30.0000	0.956851	0.478426	−	0.878128i	$-0.341208\pi$
$983$	30.0000	0.956851	0.478426	+	0.878128i	$0.341208\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	0	0
$987$	−10.0000	−0.318304
$988$	0	0
$989$	6.00000	0.190789
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	0	0
$993$	25.0000	0.793351
$994$	0	0
$995$	10.0000	0.317021
$996$	0	0
$997$	7.00000	0.221692	0.110846	−	0.993838i	$-0.464644\pi$
$997$	7.00000	0.221692	0.110846	+	0.993838i	$0.464644\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.bd.1.1		1
4.3	odd	2		2028.2.a.b.1.1		1
12.11	even	2		6084.2.a.e.1.1		1
13.3	even	3		624.2.q.d.529.1		2
13.9	even	3		624.2.q.d.289.1		2
13.12	even	2		8112.2.a.u.1.1		1
39.29	odd	6		1872.2.t.e.1153.1		2
39.35	odd	6		1872.2.t.e.289.1		2
52.3	odd	6		156.2.i.a.61.1	✓	2
52.7	even	12		2028.2.q.g.361.2		4
52.11	even	12		2028.2.q.g.1837.2		4
52.15	even	12		2028.2.q.g.1837.1		4
52.19	even	12		2028.2.q.g.361.1		4
52.23	odd	6		2028.2.i.f.529.1		2
52.31	even	4		2028.2.b.c.337.1		2
52.35	odd	6		156.2.i.a.133.1	yes	2
52.43	odd	6		2028.2.i.f.2005.1		2
52.47	even	4		2028.2.b.c.337.2		2
52.51	odd	2		2028.2.a.a.1.1		1
156.35	even	6		468.2.l.b.289.1		2
156.47	odd	4		6084.2.b.b.4393.1		2
156.83	odd	4		6084.2.b.b.4393.2		2
156.107	even	6		468.2.l.b.217.1		2
156.155	even	2		6084.2.a.l.1.1		1
260.3	even	12		3900.2.by.c.3649.2		4
260.87	even	12		3900.2.by.c.1849.2		4
260.107	even	12		3900.2.by.c.3649.1		4
260.139	odd	6		3900.2.q.e.601.1		2
260.159	odd	6		3900.2.q.e.2401.1		2
260.243	even	12		3900.2.by.c.1849.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.i.a.61.1	✓	2	52.3	odd	6
156.2.i.a.133.1	yes	2	52.35	odd	6
468.2.l.b.217.1		2	156.107	even	6
468.2.l.b.289.1		2	156.35	even	6
624.2.q.d.289.1		2	13.9	even	3
624.2.q.d.529.1		2	13.3	even	3
1872.2.t.e.289.1		2	39.35	odd	6
1872.2.t.e.1153.1		2	39.29	odd	6
2028.2.a.a.1.1		1	52.51	odd	2
2028.2.a.b.1.1		1	4.3	odd	2
2028.2.b.c.337.1		2	52.31	even	4
2028.2.b.c.337.2		2	52.47	even	4
2028.2.i.f.529.1		2	52.23	odd	6
2028.2.i.f.2005.1		2	52.43	odd	6
2028.2.q.g.361.1		4	52.19	even	12
2028.2.q.g.361.2		4	52.7	even	12
2028.2.q.g.1837.1		4	52.15	even	12
2028.2.q.g.1837.2		4	52.11	even	12
3900.2.q.e.601.1		2	260.139	odd	6
3900.2.q.e.2401.1		2	260.159	odd	6
3900.2.by.c.1849.1		4	260.243	even	12
3900.2.by.c.1849.2		4	260.87	even	12
3900.2.by.c.3649.1		4	260.107	even	12
3900.2.by.c.3649.2		4	260.3	even	12
6084.2.a.e.1.1		1	12.11	even	2
6084.2.a.l.1.1		1	156.155	even	2
6084.2.b.b.4393.1		2	156.47	odd	4
6084.2.b.b.4393.2		2	156.83	odd	4
8112.2.a.u.1.1		1	13.12	even	2
8112.2.a.bd.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

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