LMFDB - Embedded newform 882.2.a.h.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [882,2,Mod(1,882)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("882.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	882.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,-3,0,0,1,0,-3,-3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.04280545828$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	882.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{8} -3.00000 q^{10} -3.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} -6.00000 q^{17} -2.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} -6.00000 q^{23} +4.00000 q^{25} -2.00000 q^{26} +9.00000 q^{29} +7.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} -10.0000 q^{37} -2.00000 q^{38} -3.00000 q^{40} -4.00000 q^{43} -3.00000 q^{44} -6.00000 q^{46} -12.0000 q^{47} +4.00000 q^{50} -2.00000 q^{52} -3.00000 q^{53} +9.00000 q^{55} +9.00000 q^{58} +3.00000 q^{59} +4.00000 q^{61} +7.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +2.00000 q^{67} -6.00000 q^{68} -2.00000 q^{73} -10.0000 q^{74} -2.00000 q^{76} +5.00000 q^{79} -3.00000 q^{80} -9.00000 q^{83} +18.0000 q^{85} -4.00000 q^{86} -3.00000 q^{88} +6.00000 q^{89} -6.00000 q^{92} -12.0000 q^{94} +6.00000 q^{95} +13.0000 q^{97} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.00000	−0.948683
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	−3.00000	−0.639602
$23$	−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
$25$	4.00000	0.800000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	−3.00000	−0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	0	0
$50$	4.00000	0.565685
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	9.00000	1.21356
$56$	0	0
$57$	0	0
$58$	9.00000	1.18176
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	7.00000	0.889001
$63$	0	0
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	−2.00000	−0.229416
$77$	0	0
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	0	0
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	18.0000	1.95237
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−3.00000	−0.319801
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	0	0
$94$	−12.0000	−1.23771
$95$	6.00000	0.615587
$96$	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
$99$	0	0
$100$	4.00000	0.400000
$101$	−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
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−3.00000	−0.291386
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	9.00000	0.858116
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	18.0000	1.67851
$116$	9.00000	0.835629
$117$	0	0
$118$	3.00000	0.276172
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	4.00000	0.362143
$123$	0	0
$124$	7.00000	0.628619
$125$	3.00000	0.268328
$126$	0	0
$127$	−1.00000	−0.0887357	−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000	−0.0887357	−0.0443678	+	0.999015i	$0.514127\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	6.00000	0.526235
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	0	0
$134$	2.00000	0.172774
$135$	0	0
$136$	−6.00000	−0.514496
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	−27.0000	−2.24223
$146$	−2.00000	−0.165521
$147$	0	0
$148$	−10.0000	−0.821995
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	0	0
$155$	−21.0000	−1.68676
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	5.00000	0.397779
$159$	0	0
$160$	−3.00000	−0.237171
$161$	0	0
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	0	0
$165$	0	0
$166$	−9.00000	−0.698535
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	18.0000	1.38054
$171$	0	0
$172$	−4.00000	−0.304997
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	6.00000	0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	0	0
$184$	−6.00000	−0.442326
$185$	30.0000	2.20564
$186$	0	0
$187$	18.0000	1.31629
$188$	−12.0000	−0.875190
$189$	0	0
$190$	6.00000	0.435286
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−7.00000	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$194$	13.0000	0.933346
$195$	0	0
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	−6.00000	−0.422159
$203$	0	0
$204$	0	0
$205$	0	0
$206$	16.0000	1.11477
$207$	0	0
$208$	−2.00000	−0.138675
$209$	6.00000	0.415029
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	−3.00000	−0.206041
$213$	0	0
$214$	3.00000	0.205076
$215$	12.0000	0.818393
$216$	0	0
$217$	0	0
$218$	−10.0000	−0.677285
$219$	0	0
$220$	9.00000	0.606780
$221$	12.0000	0.807207
$222$	0	0
$223$	1.00000	0.0669650	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000	0.0669650	0.0334825	+	0.999439i	$0.489340\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.0000	−0.995585	−0.497792	−	0.867296i	$-0.665856\pi$
$227$	−15.0000	−0.995585	−0.497792	+	0.867296i	$0.665856\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	18.0000	1.18688
$231$	0	0
$232$	9.00000	0.590879
$233$	−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
$235$	36.0000	2.34838
$236$	3.00000	0.195283
$237$	0	0
$238$	0	0
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	0	0
$241$	−23.0000	−1.48156	−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000	−1.48156	−0.740780	+	0.671748i	$0.765544\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	4.00000	0.256074
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	7.00000	0.444500
$249$	0	0
$250$	3.00000	0.189737
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	18.0000	1.13165
$254$	−1.00000	−0.0627456
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	6.00000	0.372104
$261$	0	0
$262$	15.0000	0.926703
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	9.00000	0.552866
$266$	0	0
$267$	0	0
$268$	2.00000	0.122169
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	19.0000	1.15417	0.577084	−	0.816685i	$-0.304191\pi$
$271$	19.0000	1.15417	0.577084	+	0.816685i	$0.304191\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	6.00000	0.362473
$275$	−12.0000	−0.723627
$276$	0	0
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	−2.00000	−0.119952
$279$	0	0
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	0	0
$286$	6.00000	0.354787
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	−27.0000	−1.58549
$291$	0	0
$292$	−2.00000	−0.117041
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	−9.00000	−0.524000
$296$	−10.0000	−0.581238
$297$	0	0
$298$	−6.00000	−0.347571
$299$	12.0000	0.693978
$300$	0	0
$301$	0	0
$302$	5.00000	0.287718
$303$	0	0
$304$	−2.00000	−0.114708
$305$	−12.0000	−0.687118
$306$	0	0
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	0	0
$309$	0	0
$310$	−21.0000	−1.19272
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−17.0000	−0.960897	−0.480448	−	0.877023i	$-0.659526\pi$
$313$	−17.0000	−0.960897	−0.480448	+	0.877023i	$0.659526\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	5.00000	0.281272
$317$	21.0000	1.17948	0.589739	−	0.807594i	$-0.299231\pi$
$317$	21.0000	1.17948	0.589739	+	0.807594i	$0.299231\pi$
$318$	0	0
$319$	−27.0000	−1.51171
$320$	−3.00000	−0.167705
$321$	0	0
$322$	0	0
$323$	12.0000	0.667698
$324$	0	0
$325$	−8.00000	−0.443760
$326$	−10.0000	−0.553849
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−9.00000	−0.493939
$333$	0	0
$334$	18.0000	0.984916
$335$	−6.00000	−0.327815
$336$	0	0
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	18.0000	0.976187
$341$	−21.0000	−1.13721
$342$	0	0
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	6.00000	0.322562
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	−3.00000	−0.159901
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−20.0000	−1.05118
$363$	0	0
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	37.0000	1.93138	0.965692	−	0.259690i	$-0.0836203\pi$
$367$	37.0000	1.93138	0.965692	+	0.259690i	$0.0836203\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	30.0000	1.55963
$371$	0	0
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	18.0000	0.930758
$375$	0	0
$376$	−12.0000	−0.618853
$377$	−18.0000	−0.927047
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	12.0000	0.613973
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	0	0
$385$	0	0
$386$	−7.00000	−0.356291
$387$	0	0
$388$	13.0000	0.659975
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	36.0000	1.82060
$392$	0	0
$393$	0	0
$394$	−18.0000	−0.906827
$395$	−15.0000	−0.754732
$396$	0	0
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	4.00000	0.200000
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	−14.0000	−0.697390
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	30.0000	1.48704
$408$	0	0
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	0	0
$411$	0	0
$412$	16.0000	0.788263
$413$	0	0
$414$	0	0
$415$	27.0000	1.32538
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	6.00000	0.293470
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	−16.0000	−0.778868
$423$	0	0
$424$	−3.00000	−0.145693
$425$	−24.0000	−1.16417
$426$	0	0
$427$	0	0
$428$	3.00000	0.145010
$429$	0	0
$430$	12.0000	0.578691
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	12.0000	0.574038
$438$	0	0
$439$	19.0000	0.906821	0.453410	−	0.891302i	$-0.350207\pi$
$439$	19.0000	0.906821	0.453410	+	0.891302i	$0.350207\pi$
$440$	9.00000	0.429058
$441$	0	0
$442$	12.0000	0.570782
$443$	−15.0000	−0.712672	−0.356336	−	0.934358i	$-0.615974\pi$
$443$	−15.0000	−0.712672	−0.356336	+	0.934358i	$0.615974\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	1.00000	0.0473514
$447$	0	0
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	−15.0000	−0.703985
$455$	0	0
$456$	0	0
$457$	−13.0000	−0.608114	−0.304057	−	0.952654i	$-0.598341\pi$
$457$	−13.0000	−0.608114	−0.304057	+	0.952654i	$0.598341\pi$
$458$	−20.0000	−0.934539
$459$	0	0
$460$	18.0000	0.839254
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	9.00000	0.417815
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	0	0
$470$	36.0000	1.66056
$471$	0	0
$472$	3.00000	0.138086
$473$	12.0000	0.551761
$474$	0	0
$475$	−8.00000	−0.367065
$476$	0	0
$477$	0	0
$478$	18.0000	0.823301
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	−23.0000	−1.04762
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−39.0000	−1.77090
$486$	0	0
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	4.00000	0.181071
$489$	0	0
$490$	0	0
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	0	0
$493$	−54.0000	−2.43204
$494$	4.00000	0.179969
$495$	0	0
$496$	7.00000	0.314309
$497$	0	0
$498$	0	0
$499$	38.0000	1.70111	0.850557	−	0.525883i	$-0.176265\pi$
$499$	38.0000	1.70111	0.850557	+	0.525883i	$0.176265\pi$
$500$	3.00000	0.134164
$501$	0	0
$502$	−9.00000	−0.401690
$503$	−18.0000	−0.802580	−0.401290	−	0.915951i	$-0.631438\pi$
$503$	−18.0000	−0.802580	−0.401290	+	0.915951i	$0.631438\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	18.0000	0.800198
$507$	0	0
$508$	−1.00000	−0.0443678
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	−48.0000	−2.11513
$516$	0	0
$517$	36.0000	1.58328
$518$	0	0
$519$	0	0
$520$	6.00000	0.263117
$521$	−24.0000	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$522$	0	0
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	15.0000	0.655278
$525$	0	0
$526$	24.0000	1.04645
$527$	−42.0000	−1.82955
$528$	0	0
$529$	13.0000	0.565217
$530$	9.00000	0.390935
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−9.00000	−0.389104
$536$	2.00000	0.0863868
$537$	0	0
$538$	3.00000	0.129339
$539$	0	0
$540$	0	0
$541$	−16.0000	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$542$	19.0000	0.816120
$543$	0	0
$544$	−6.00000	−0.257248
$545$	30.0000	1.28506
$546$	0	0
$547$	14.0000	0.598597	0.299298	−	0.954160i	$-0.403247\pi$
$547$	14.0000	0.598597	0.299298	+	0.954160i	$0.403247\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	−12.0000	−0.511682
$551$	−18.0000	−0.766826
$552$	0	0
$553$	0	0
$554$	−4.00000	−0.169944
$555$	0	0
$556$	−2.00000	−0.0848189
$557$	15.0000	0.635570	0.317785	−	0.948163i	$-0.397061\pi$
$557$	15.0000	0.635570	0.317785	+	0.948163i	$0.397061\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	−18.0000	−0.759284
$563$	3.00000	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$563$	3.00000	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$564$	0	0
$565$	0	0
$566$	−20.0000	−0.840663
$567$	0	0
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	6.00000	0.250873
$573$	0	0
$574$	0	0
$575$	−24.0000	−1.00087
$576$	0	0
$577$	1.00000	0.0416305	0.0208153	−	0.999783i	$-0.493374\pi$
$577$	1.00000	0.0416305	0.0208153	+	0.999783i	$0.493374\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	−27.0000	−1.12111
$581$	0	0
$582$	0	0
$583$	9.00000	0.372742
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−9.00000	−0.371787
$587$	−27.0000	−1.11441	−0.557205	−	0.830375i	$-0.688126\pi$
$587$	−27.0000	−1.11441	−0.557205	+	0.830375i	$0.688126\pi$
$588$	0	0
$589$	−14.0000	−0.576860
$590$	−9.00000	−0.370524
$591$	0	0
$592$	−10.0000	−0.410997
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	12.0000	0.490716
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	−23.0000	−0.938190	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000	−0.938190	−0.469095	+	0.883148i	$0.655420\pi$
$602$	0	0
$603$	0	0
$604$	5.00000	0.203447
$605$	6.00000	0.243935
$606$	0	0
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	−12.0000	−0.485866
$611$	24.0000	0.970936
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	−26.0000	−1.04927
$615$	0	0
$616$	0	0
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	−21.0000	−0.843380
$621$	0	0
$622$	12.0000	0.481156
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−17.0000	−0.679457
$627$	0	0
$628$	4.00000	0.159617
$629$	60.0000	2.39236
$630$	0	0
$631$	−37.0000	−1.47295	−0.736473	−	0.676467i	$-0.763510\pi$
$631$	−37.0000	−1.47295	−0.736473	+	0.676467i	$0.763510\pi$
$632$	5.00000	0.198889
$633$	0	0
$634$	21.0000	0.834017
$635$	3.00000	0.119051
$636$	0	0
$637$	0	0
$638$	−27.0000	−1.06894
$639$	0	0
$640$	−3.00000	−0.118585
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	−38.0000	−1.49857	−0.749287	−	0.662246i	$-0.769604\pi$
$643$	−38.0000	−1.49857	−0.749287	+	0.662246i	$0.769604\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	−9.00000	−0.353281
$650$	−8.00000	−0.313786
$651$	0	0
$652$	−10.0000	−0.391630
$653$	39.0000	1.52619	0.763094	−	0.646288i	$-0.223679\pi$
$653$	39.0000	1.52619	0.763094	+	0.646288i	$0.223679\pi$
$654$	0	0
$655$	−45.0000	−1.75830
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	40.0000	1.55582	0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	−9.00000	−0.349268
$665$	0	0
$666$	0	0
$667$	−54.0000	−2.09089
$668$	18.0000	0.696441
$669$	0	0
$670$	−6.00000	−0.231800
$671$	−12.0000	−0.463255
$672$	0	0
$673$	17.0000	0.655302	0.327651	−	0.944799i	$-0.393743\pi$
$673$	17.0000	0.655302	0.327651	+	0.944799i	$0.393743\pi$
$674$	5.00000	0.192593
$675$	0	0
$676$	−9.00000	−0.346154
$677$	33.0000	1.26829	0.634147	−	0.773213i	$-0.281352\pi$
$677$	33.0000	1.26829	0.634147	+	0.773213i	$0.281352\pi$
$678$	0	0
$679$	0	0
$680$	18.0000	0.690268
$681$	0	0
$682$	−21.0000	−0.804132
$683$	51.0000	1.95146	0.975730	−	0.218975i	$-0.0702714\pi$
$683$	51.0000	1.95146	0.975730	+	0.218975i	$0.0702714\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	0	0
$688$	−4.00000	−0.152499
$689$	6.00000	0.228582
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−12.0000	−0.455514
$695$	6.00000	0.227593
$696$	0	0
$697$	0	0
$698$	−14.0000	−0.529908
$699$	0	0
$700$	0	0
$701$	9.00000	0.339925	0.169963	−	0.985451i	$-0.445635\pi$
$701$	9.00000	0.339925	0.169963	+	0.985451i	$0.445635\pi$
$702$	0	0
$703$	20.0000	0.754314
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−24.0000	−0.903252
$707$	0	0
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	0	0
$712$	6.00000	0.224860
$713$	−42.0000	−1.57291
$714$	0	0
$715$	−18.0000	−0.673162
$716$	−12.0000	−0.448461
$717$	0	0
$718$	30.0000	1.11959
$719$	42.0000	1.56634	0.783168	−	0.621810i	$-0.213603\pi$
$719$	42.0000	1.56634	0.783168	+	0.621810i	$0.213603\pi$
$720$	0	0
$721$	0	0
$722$	−15.0000	−0.558242
$723$	0	0
$724$	−20.0000	−0.743294
$725$	36.0000	1.33701
$726$	0	0
$727$	31.0000	1.14973	0.574863	−	0.818250i	$-0.305055\pi$
$727$	31.0000	1.14973	0.574863	+	0.818250i	$0.305055\pi$
$728$	0	0
$729$	0	0
$730$	6.00000	0.222070
$731$	24.0000	0.887672
$732$	0	0
$733$	−20.0000	−0.738717	−0.369358	−	0.929287i	$-0.620423\pi$
$733$	−20.0000	−0.738717	−0.369358	+	0.929287i	$0.620423\pi$
$734$	37.0000	1.36569
$735$	0	0
$736$	−6.00000	−0.221163
$737$	−6.00000	−0.221013
$738$	0	0
$739$	26.0000	0.956425	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956425	0.478213	+	0.878244i	$0.341285\pi$
$740$	30.0000	1.10282
$741$	0	0
$742$	0	0
$743$	−54.0000	−1.98107	−0.990534	−	0.137268i	$-0.956168\pi$
$743$	−54.0000	−1.98107	−0.990534	+	0.137268i	$0.956168\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	−4.00000	−0.146450
$747$	0	0
$748$	18.0000	0.658145
$749$	0	0
$750$	0	0
$751$	23.0000	0.839282	0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000	0.839282	0.419641	+	0.907690i	$0.362156\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	−18.0000	−0.655521
$755$	−15.0000	−0.545906
$756$	0	0
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	−28.0000	−1.01701
$759$	0	0
$760$	6.00000	0.217643
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	0	0
$764$	12.0000	0.434145
$765$	0	0
$766$	−30.0000	−1.08394
$767$	−6.00000	−0.216647
$768$	0	0
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	0	0
$772$	−7.00000	−0.251936
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	28.0000	1.00579
$776$	13.0000	0.466673
$777$	0	0
$778$	−30.0000	−1.07555
$779$	0	0
$780$	0	0
$781$	0	0
$782$	36.0000	1.28736
$783$	0	0
$784$	0	0
$785$	−12.0000	−0.428298
$786$	0	0
$787$	16.0000	0.570338	0.285169	−	0.958477i	$-0.407950\pi$
$787$	16.0000	0.570338	0.285169	+	0.958477i	$0.407950\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	−15.0000	−0.533676
$791$	0	0
$792$	0	0
$793$	−8.00000	−0.284088
$794$	−8.00000	−0.283909
$795$	0	0
$796$	−8.00000	−0.283552
$797$	−27.0000	−0.956389	−0.478195	−	0.878254i	$-0.658709\pi$
$797$	−27.0000	−0.956389	−0.478195	+	0.878254i	$0.658709\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	4.00000	0.141421
$801$	0	0
$802$	−24.0000	−0.847469
$803$	6.00000	0.211735
$804$	0	0
$805$	0	0
$806$	−14.0000	−0.493129
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	0	0
$814$	30.0000	1.05150
$815$	30.0000	1.05085
$816$	0	0
$817$	8.00000	0.279885
$818$	−11.0000	−0.384606
$819$	0	0
$820$	0	0
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	0	0
$827$	9.00000	0.312961	0.156480	−	0.987681i	$-0.449985\pi$
$827$	9.00000	0.312961	0.156480	+	0.987681i	$0.449985\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	27.0000	0.937184
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	0	0
$835$	−54.0000	−1.86875
$836$	6.00000	0.207514
$837$	0	0
$838$	36.0000	1.24360
$839$	54.0000	1.86429	0.932144	−	0.362089i	$-0.117936\pi$
$839$	54.0000	1.86429	0.932144	+	0.362089i	$0.117936\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	8.00000	0.275698
$843$	0	0
$844$	−16.0000	−0.550743
$845$	27.0000	0.928828
$846$	0	0
$847$	0	0
$848$	−3.00000	−0.103020
$849$	0	0
$850$	−24.0000	−0.823193
$851$	60.0000	2.05677
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	0	0
$855$	0	0
$856$	3.00000	0.102538
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	12.0000	0.409197
$861$	0	0
$862$	24.0000	0.817443
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	−26.0000	−0.883516
$867$	0	0
$868$	0	0
$869$	−15.0000	−0.508840
$870$	0	0
$871$	−4.00000	−0.135535
$872$	−10.0000	−0.338643
$873$	0	0
$874$	12.0000	0.405906
$875$	0	0
$876$	0	0
$877$	56.0000	1.89099	0.945493	−	0.325643i	$-0.105581\pi$
$877$	56.0000	1.89099	0.945493	+	0.325643i	$0.105581\pi$
$878$	19.0000	0.641219
$879$	0	0
$880$	9.00000	0.303390
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−15.0000	−0.503935
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	0	0
$889$	0	0
$890$	−18.0000	−0.603361
$891$	0	0
$892$	1.00000	0.0334825
$893$	24.0000	0.803129
$894$	0	0
$895$	36.0000	1.20335
$896$	0	0
$897$	0	0
$898$	18.0000	0.600668
$899$	63.0000	2.10117
$900$	0	0
$901$	18.0000	0.599667
$902$	0	0
$903$	0	0
$904$	0	0
$905$	60.0000	1.99447
$906$	0	0
$907$	50.0000	1.66022	0.830111	−	0.557598i	$-0.188277\pi$
$907$	50.0000	1.66022	0.830111	+	0.557598i	$0.188277\pi$
$908$	−15.0000	−0.497792
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	27.0000	0.893570
$914$	−13.0000	−0.430002
$915$	0	0
$916$	−20.0000	−0.660819
$917$	0	0
$918$	0	0
$919$	−28.0000	−0.923635	−0.461817	−	0.886975i	$-0.652802\pi$
$919$	−28.0000	−0.923635	−0.461817	+	0.886975i	$0.652802\pi$
$920$	18.0000	0.593442
$921$	0	0
$922$	−18.0000	−0.592798
$923$	0	0
$924$	0	0
$925$	−40.0000	−1.31519
$926$	−4.00000	−0.131448
$927$	0	0
$928$	9.00000	0.295439
$929$	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
$931$	0	0
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−12.0000	−0.392652
$935$	−54.0000	−1.76599
$936$	0	0
$937$	37.0000	1.20874	0.604369	−	0.796705i	$-0.293425\pi$
$937$	37.0000	1.20874	0.604369	+	0.796705i	$0.293425\pi$
$938$	0	0
$939$	0	0
$940$	36.0000	1.17419
$941$	−15.0000	−0.488986	−0.244493	−	0.969651i	$-0.578622\pi$
$941$	−15.0000	−0.488986	−0.244493	+	0.969651i	$0.578622\pi$
$942$	0	0
$943$	0	0
$944$	3.00000	0.0976417
$945$	0	0
$946$	12.0000	0.390154
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	−8.00000	−0.259554
$951$	0	0
$952$	0	0
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	0	0
$955$	−36.0000	−1.16493
$956$	18.0000	0.582162
$957$	0	0
$958$	−24.0000	−0.775405
$959$	0	0
$960$	0	0
$961$	18.0000	0.580645
$962$	20.0000	0.644826
$963$	0	0
$964$	−23.0000	−0.740780
$965$	21.0000	0.676014
$966$	0	0
$967$	29.0000	0.932577	0.466289	−	0.884633i	$-0.345591\pi$
$967$	29.0000	0.932577	0.466289	+	0.884633i	$0.345591\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	−39.0000	−1.25221
$971$	−39.0000	−1.25157	−0.625785	−	0.779996i	$-0.715221\pi$
$971$	−39.0000	−1.25157	−0.625785	+	0.779996i	$0.715221\pi$
$972$	0	0
$973$	0	0
$974$	11.0000	0.352463
$975$	0	0
$976$	4.00000	0.128037
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	0	0
$982$	9.00000	0.287202
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	54.0000	1.72058
$986$	−54.0000	−1.71971
$987$	0	0
$988$	4.00000	0.127257
$989$	24.0000	0.763156
$990$	0	0
$991$	−19.0000	−0.603555	−0.301777	−	0.953378i	$-0.597580\pi$
$991$	−19.0000	−0.603555	−0.301777	+	0.953378i	$0.597580\pi$
$992$	7.00000	0.222250
$993$	0	0
$994$	0	0
$995$	24.0000	0.760851
$996$	0	0
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	38.0000	1.20287
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.a.h.1.1		1
3.2	odd	2		882.2.a.e.1.1		1
4.3	odd	2		7056.2.a.h.1.1		1
7.2	even	3		882.2.g.e.361.1		2
7.3	odd	6		126.2.g.a.37.1	✓	2
7.4	even	3		882.2.g.e.667.1		2
7.5	odd	6		126.2.g.a.109.1	yes	2
7.6	odd	2		882.2.a.j.1.1		1
12.11	even	2		7056.2.a.bx.1.1		1
21.2	odd	6		882.2.g.g.361.1		2
21.5	even	6		126.2.g.d.109.1	yes	2
21.11	odd	6		882.2.g.g.667.1		2
21.17	even	6		126.2.g.d.37.1	yes	2
21.20	even	2		882.2.a.a.1.1		1
28.3	even	6		1008.2.s.b.289.1		2
28.19	even	6		1008.2.s.b.865.1		2
28.27	even	2		7056.2.a.by.1.1		1
63.5	even	6		1134.2.e.g.865.1		2
63.31	odd	6		1134.2.h.f.541.1		2
63.38	even	6		1134.2.e.g.919.1		2
63.40	odd	6		1134.2.e.k.865.1		2
63.47	even	6		1134.2.h.j.109.1		2
63.52	odd	6		1134.2.e.k.919.1		2
63.59	even	6		1134.2.h.j.541.1		2
63.61	odd	6		1134.2.h.f.109.1		2
84.47	odd	6		1008.2.s.o.865.1		2
84.59	odd	6		1008.2.s.o.289.1		2
84.83	odd	2		7056.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.g.a.37.1	✓	2	7.3	odd	6
126.2.g.a.109.1	yes	2	7.5	odd	6
126.2.g.d.37.1	yes	2	21.17	even	6
126.2.g.d.109.1	yes	2	21.5	even	6
882.2.a.a.1.1		1	21.20	even	2
882.2.a.e.1.1		1	3.2	odd	2
882.2.a.h.1.1		1	1.1	even	1	trivial
882.2.a.j.1.1		1	7.6	odd	2
882.2.g.e.361.1		2	7.2	even	3
882.2.g.e.667.1		2	7.4	even	3
882.2.g.g.361.1		2	21.2	odd	6
882.2.g.g.667.1		2	21.11	odd	6
1008.2.s.b.289.1		2	28.3	even	6
1008.2.s.b.865.1		2	28.19	even	6
1008.2.s.o.289.1		2	84.59	odd	6
1008.2.s.o.865.1		2	84.47	odd	6
1134.2.e.g.865.1		2	63.5	even	6
1134.2.e.g.919.1		2	63.38	even	6
1134.2.e.k.865.1		2	63.40	odd	6
1134.2.e.k.919.1		2	63.52	odd	6
1134.2.h.f.109.1		2	63.61	odd	6
1134.2.h.f.541.1		2	63.31	odd	6
1134.2.h.j.109.1		2	63.47	even	6
1134.2.h.j.541.1		2	63.59	even	6
7056.2.a.e.1.1		1	84.83	odd	2
7056.2.a.h.1.1		1	4.3	odd	2
7056.2.a.bx.1.1		1	12.11	even	2
7056.2.a.by.1.1		1	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

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