LMFDB - Embedded newform 4864.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4864,2,Mod(1,4864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4864, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.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:	$38.8392355432$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +4.00000 q^{11} +1.00000 q^{13} +2.00000 q^{15} -5.00000 q^{17} +1.00000 q^{19} -1.00000 q^{21} -9.00000 q^{23} -1.00000 q^{25} -5.00000 q^{27} +5.00000 q^{29} +2.00000 q^{31} +4.00000 q^{33} -2.00000 q^{35} +10.0000 q^{37} +1.00000 q^{39} +12.0000 q^{41} +10.0000 q^{43} -4.00000 q^{45} +12.0000 q^{47} -6.00000 q^{49} -5.00000 q^{51} +9.00000 q^{53} +8.00000 q^{55} +1.00000 q^{57} +9.00000 q^{59} -2.00000 q^{61} +2.00000 q^{63} +2.00000 q^{65} +13.0000 q^{67} -9.00000 q^{69} +10.0000 q^{71} -7.00000 q^{73} -1.00000 q^{75} -4.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} -8.00000 q^{83} -10.0000 q^{85} +5.00000 q^{87} -6.00000 q^{89} -1.00000 q^{91} +2.00000 q^{93} +2.00000 q^{95} -6.00000 q^{97} -8.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	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
$9$	−2.00000	−0.666667
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	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
$39$	1.00000	0.160128
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	−4.00000	−0.596285
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−5.00000	−0.700140
$52$	0	0
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	13.0000	1.58820	0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000	1.58820	0.794101	+	0.607785i	$0.207942\pi$
$68$	0	0
$69$	−9.00000	−1.08347
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	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
$77$	−4.00000	−0.455842
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	−10.0000	−1.08465
$86$	0	0
$87$	5.00000	0.536056
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	2.00000	0.207390
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	−8.00000	−0.804030
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	11.0000	1.06341	0.531705	−	0.846930i	$-0.321551\pi$
$107$	11.0000	1.06341	0.531705	+	0.846930i	$0.321551\pi$
$108$	0	0
$109$	15.0000	1.43674	0.718370	−	0.695662i	$-0.244889\pi$
$109$	15.0000	1.43674	0.718370	+	0.695662i	$0.244889\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−18.0000	−1.67851
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	5.00000	0.458349
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	12.0000	1.08200
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	0	0
$129$	10.0000	0.880451
$130$	0	0
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−10.0000	−0.860663
$136$	0	0
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	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
$141$	12.0000	1.01058
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	10.0000	0.830455
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	10.0000	0.808452
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	9.00000	0.713746
$160$	0	0
$161$	9.00000	0.709299
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	0	0
$165$	8.00000	0.622799
$166$	0	0
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	9.00000	0.676481
$178$	0	0
$179$	−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
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	20.0000	1.47043
$186$	0	0
$187$	−20.0000	−1.46254
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	23.0000	1.66422	0.832111	−	0.554609i	$-0.187132\pi$
$191$	23.0000	1.66422	0.832111	+	0.554609i	$0.187132\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	0	0
$201$	13.0000	0.916949
$202$	0	0
$203$	−5.00000	−0.350931
$204$	0	0
$205$	24.0000	1.67623
$206$	0	0
$207$	18.0000	1.25109
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	0	0
$213$	10.0000	0.685189
$214$	0	0
$215$	20.0000	1.36399
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	−7.00000	−0.473016
$220$	0	0
$221$	−5.00000	−0.336336
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	1.00000	0.0663723	0.0331862	−	0.999449i	$-0.489435\pi$
$227$	1.00000	0.0663723	0.0331862	+	0.999449i	$0.489435\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	19.0000	1.22901	0.614504	−	0.788914i	$-0.289356\pi$
$239$	19.0000	1.22901	0.614504	+	0.788914i	$0.289356\pi$
$240$	0	0
$241$	−16.0000	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−12.0000	−0.766652
$246$	0	0
$247$	1.00000	0.0636285
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	−36.0000	−2.26330
$254$	0	0
$255$	−10.0000	−0.626224
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	0	0
$261$	−10.0000	−0.618984
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	3.00000	0.182237	0.0911185	−	0.995840i	$-0.470956\pi$
$271$	3.00000	0.182237	0.0911185	+	0.995840i	$0.470956\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	0	0
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−6.00000	−0.351726
$292$	0	0
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	0	0
$295$	18.0000	1.04800
$296$	0	0
$297$	−20.0000	−1.16052
$298$	0	0
$299$	−9.00000	−0.520483
$300$	0	0
$301$	−10.0000	−0.576390
$302$	0	0
$303$	−4.00000	−0.229794
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−33.0000	−1.87126	−0.935629	−	0.352985i	$-0.885167\pi$
$311$	−33.0000	−1.87126	−0.935629	+	0.352985i	$0.885167\pi$
$312$	0	0
$313$	21.0000	1.18699	0.593495	−	0.804838i	$-0.297748\pi$
$313$	21.0000	1.18699	0.593495	+	0.804838i	$0.297748\pi$
$314$	0	0
$315$	4.00000	0.225374
$316$	0	0
$317$	31.0000	1.74113	0.870567	−	0.492050i	$-0.163752\pi$
$317$	31.0000	1.74113	0.870567	+	0.492050i	$0.163752\pi$
$318$	0	0
$319$	20.0000	1.11979
$320$	0	0
$321$	11.0000	0.613960
$322$	0	0
$323$	−5.00000	−0.278207
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	15.0000	0.829502
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−1.00000	−0.0549650	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−1.00000	−0.0549650	−0.0274825	+	0.999622i	$0.508749\pi$
$332$	0	0
$333$	−20.0000	−1.09599
$334$	0	0
$335$	26.0000	1.42053
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	−18.0000	−0.969087
$346$	0	0
$347$	−22.0000	−1.18102	−0.590511	−	0.807030i	$-0.701074\pi$
$347$	−22.0000	−1.18102	−0.590511	+	0.807030i	$0.701074\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	−35.0000	−1.86286	−0.931431	−	0.363918i	$-0.881439\pi$
$353$	−35.0000	−1.86286	−0.931431	+	0.363918i	$0.881439\pi$
$354$	0	0
$355$	20.0000	1.06149
$356$	0	0
$357$	5.00000	0.264628
$358$	0	0
$359$	−3.00000	−0.158334	−0.0791670	−	0.996861i	$-0.525226\pi$
$359$	−3.00000	−0.158334	−0.0791670	+	0.996861i	$0.525226\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	−14.0000	−0.732793
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	−24.0000	−1.24939
$370$	0	0
$371$	−9.00000	−0.467257
$372$	0	0
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	5.00000	0.257513
$378$	0	0
$379$	17.0000	0.873231	0.436616	−	0.899648i	$-0.356177\pi$
$379$	17.0000	0.873231	0.436616	+	0.899648i	$0.356177\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	−20.0000	−1.01666
$388$	0	0
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	45.0000	2.27575
$392$	0	0
$393$	−14.0000	−0.706207
$394$	0	0
$395$	20.0000	1.00631
$396$	0	0
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	0	0
$405$	2.00000	0.0993808
$406$	0	0
$407$	40.0000	1.98273
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	−13.0000	−0.641243
$412$	0	0
$413$	−9.00000	−0.442861
$414$	0	0
$415$	−16.0000	−0.785409
$416$	0	0
$417$	−2.00000	−0.0979404
$418$	0	0
$419$	34.0000	1.66101	0.830504	−	0.557012i	$-0.188052\pi$
$419$	34.0000	1.66101	0.830504	+	0.557012i	$0.188052\pi$
$420$	0	0
$421$	−15.0000	−0.731055	−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000	−0.731055	−0.365528	+	0.930800i	$0.619111\pi$
$422$	0	0
$423$	−24.0000	−1.16692
$424$	0	0
$425$	5.00000	0.242536
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	0	0
$435$	10.0000	0.479463
$436$	0	0
$437$	−9.00000	−0.430528
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	0	0
$447$	8.00000	0.378387
$448$	0	0
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	48.0000	2.26023
$452$	0	0
$453$	−8.00000	−0.375873
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	0	0
$459$	25.0000	1.16690
$460$	0	0
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	0	0
$463$	−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
$465$	4.00000	0.185496
$466$	0	0
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	−13.0000	−0.600284
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	40.0000	1.83920
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−18.0000	−0.824163
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	0	0
$483$	9.00000	0.409514
$484$	0	0
$485$	−12.0000	−0.544892
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	0	0
$489$	10.0000	0.452216
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	−25.0000	−1.12594
$494$	0	0
$495$	−16.0000	−0.719147
$496$	0	0
$497$	−10.0000	−0.448561
$498$	0	0
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	0	0
$503$	11.0000	0.490466	0.245233	−	0.969464i	$-0.421136\pi$
$503$	11.0000	0.490466	0.245233	+	0.969464i	$0.421136\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	−16.0000	−0.705044
$516$	0	0
$517$	48.0000	2.11104
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−32.0000	−1.40195	−0.700973	−	0.713188i	$-0.747251\pi$
$521$	−32.0000	−1.40195	−0.700973	+	0.713188i	$0.747251\pi$
$522$	0	0
$523$	−17.0000	−0.743358	−0.371679	−	0.928361i	$-0.621218\pi$
$523$	−17.0000	−0.743358	−0.371679	+	0.928361i	$0.621218\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	−10.0000	−0.435607
$528$	0	0
$529$	58.0000	2.52174
$530$	0	0
$531$	−18.0000	−0.781133
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	22.0000	0.951143
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−24.0000	−1.03375
$540$	0	0
$541$	4.00000	0.171973	0.0859867	−	0.996296i	$-0.472596\pi$
$541$	4.00000	0.171973	0.0859867	+	0.996296i	$0.472596\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	30.0000	1.28506
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	4.00000	0.170716
$550$	0	0
$551$	5.00000	0.213007
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	20.0000	0.848953
$556$	0	0
$557$	22.0000	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$558$	0	0
$559$	10.0000	0.422955
$560$	0	0
$561$	−20.0000	−0.844401
$562$	0	0
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	23.0000	0.960839
$574$	0	0
$575$	9.00000	0.375326
$576$	0	0
$577$	3.00000	0.124892	0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000	0.124892	0.0624458	+	0.998048i	$0.480110\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	36.0000	1.49097
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	48.0000	1.98117	0.990586	−	0.136892i	$-0.0437113\pi$
$587$	48.0000	1.98117	0.990586	+	0.136892i	$0.0437113\pi$
$588$	0	0
$589$	2.00000	0.0824086
$590$	0	0
$591$	−12.0000	−0.493614
$592$	0	0
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	10.0000	0.409960
$596$	0	0
$597$	−5.00000	−0.204636
$598$	0	0
$599$	28.0000	1.14405	0.572024	−	0.820237i	$-0.306158\pi$
$599$	28.0000	1.14405	0.572024	+	0.820237i	$0.306158\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	0	0
$603$	−26.0000	−1.05880
$604$	0	0
$605$	10.0000	0.406558
$606$	0	0
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	0	0
$609$	−5.00000	−0.202610
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	−12.0000	−0.484675	−0.242338	−	0.970192i	$-0.577914\pi$
$613$	−12.0000	−0.484675	−0.242338	+	0.970192i	$0.577914\pi$
$614$	0	0
$615$	24.0000	0.967773
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	45.0000	1.80579
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	4.00000	0.159745
$628$	0	0
$629$	−50.0000	−1.99363
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	13.0000	0.516704
$634$	0	0
$635$	12.0000	0.476205
$636$	0	0
$637$	−6.00000	−0.237729
$638$	0	0
$639$	−20.0000	−0.791188
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	0	0
$645$	20.0000	0.787499
$646$	0	0
$647$	−37.0000	−1.45462	−0.727310	−	0.686309i	$-0.759230\pi$
$647$	−37.0000	−1.45462	−0.727310	+	0.686309i	$0.759230\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	0	0
$653$	46.0000	1.80012	0.900060	−	0.435767i	$-0.143523\pi$
$653$	46.0000	1.80012	0.900060	+	0.435767i	$0.143523\pi$
$654$	0	0
$655$	−28.0000	−1.09405
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	19.0000	0.740135	0.370067	−	0.929005i	$-0.379335\pi$
$659$	19.0000	0.740135	0.370067	+	0.929005i	$0.379335\pi$
$660$	0	0
$661$	15.0000	0.583432	0.291716	−	0.956505i	$-0.405774\pi$
$661$	15.0000	0.583432	0.291716	+	0.956505i	$0.405774\pi$
$662$	0	0
$663$	−5.00000	−0.194184
$664$	0	0
$665$	−2.00000	−0.0775567
$666$	0	0
$667$	−45.0000	−1.74241
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	−20.0000	−0.770943	−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000	−0.770943	−0.385472	+	0.922720i	$0.625961\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	15.0000	0.576497	0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000	0.576497	0.288248	+	0.957556i	$0.406927\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	0	0
$681$	1.00000	0.0383201
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	−26.0000	−0.993409
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	9.00000	0.342873
$690$	0	0
$691$	30.0000	1.14125	0.570627	−	0.821209i	$-0.306700\pi$
$691$	30.0000	1.14125	0.570627	+	0.821209i	$0.306700\pi$
$692$	0	0
$693$	8.00000	0.303895
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	−60.0000	−2.27266
$698$	0	0
$699$	−10.0000	−0.378235
$700$	0	0
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	0	0
$705$	24.0000	0.903892
$706$	0	0
$707$	4.00000	0.150435
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	−20.0000	−0.750059
$712$	0	0
$713$	−18.0000	−0.674105
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	19.0000	0.709568
$718$	0	0
$719$	3.00000	0.111881	0.0559406	−	0.998434i	$-0.482184\pi$
$719$	3.00000	0.111881	0.0559406	+	0.998434i	$0.482184\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	−16.0000	−0.595046
$724$	0	0
$725$	−5.00000	−0.185695
$726$	0	0
$727$	−29.0000	−1.07555	−0.537775	−	0.843088i	$-0.680735\pi$
$727$	−29.0000	−1.07555	−0.537775	+	0.843088i	$0.680735\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−50.0000	−1.84932
$732$	0	0
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	0	0
$735$	−12.0000	−0.442627
$736$	0	0
$737$	52.0000	1.91544
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	1.00000	0.0367359
$742$	0	0
$743$	−50.0000	−1.83432	−0.917161	−	0.398517i	$-0.869525\pi$
$743$	−50.0000	−1.83432	−0.917161	+	0.398517i	$0.869525\pi$
$744$	0	0
$745$	16.0000	0.586195
$746$	0	0
$747$	16.0000	0.585409
$748$	0	0
$749$	−11.0000	−0.401931
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	−36.0000	−1.30672
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	−15.0000	−0.543036
$764$	0	0
$765$	20.0000	0.723102
$766$	0	0
$767$	9.00000	0.324971
$768$	0	0
$769$	1.00000	0.0360609	0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000	0.0360609	0.0180305	+	0.999837i	$0.494260\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	0	0
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	−10.0000	−0.358748
$778$	0	0
$779$	12.0000	0.429945
$780$	0	0
$781$	40.0000	1.43131
$782$	0	0
$783$	−25.0000	−0.893427
$784$	0	0
$785$	4.00000	0.142766
$786$	0	0
$787$	−11.0000	−0.392108	−0.196054	−	0.980593i	$-0.562813\pi$
$787$	−11.0000	−0.392108	−0.196054	+	0.980593i	$0.562813\pi$
$788$	0	0
$789$	16.0000	0.569615
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	0	0
$795$	18.0000	0.638394
$796$	0	0
$797$	5.00000	0.177109	0.0885545	−	0.996071i	$-0.471775\pi$
$797$	5.00000	0.177109	0.0885545	+	0.996071i	$0.471775\pi$
$798$	0	0
$799$	−60.0000	−2.12265
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	−28.0000	−0.988099
$804$	0	0
$805$	18.0000	0.634417
$806$	0	0
$807$	−6.00000	−0.211210
$808$	0	0
$809$	5.00000	0.175791	0.0878953	−	0.996130i	$-0.471986\pi$
$809$	5.00000	0.175791	0.0878953	+	0.996130i	$0.471986\pi$
$810$	0	0
$811$	−9.00000	−0.316033	−0.158016	−	0.987436i	$-0.550510\pi$
$811$	−9.00000	−0.316033	−0.158016	+	0.987436i	$0.550510\pi$
$812$	0	0
$813$	3.00000	0.105215
$814$	0	0
$815$	20.0000	0.700569
$816$	0	0
$817$	10.0000	0.349856
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	50.0000	1.74501	0.872506	−	0.488603i	$-0.162493\pi$
$821$	50.0000	1.74501	0.872506	+	0.488603i	$0.162493\pi$
$822$	0	0
$823$	−3.00000	−0.104573	−0.0522867	−	0.998632i	$-0.516651\pi$
$823$	−3.00000	−0.104573	−0.0522867	+	0.998632i	$0.516651\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−19.0000	−0.660695	−0.330347	−	0.943859i	$-0.607166\pi$
$827$	−19.0000	−0.660695	−0.330347	+	0.943859i	$0.607166\pi$
$828$	0	0
$829$	7.00000	0.243120	0.121560	−	0.992584i	$-0.461210\pi$
$829$	7.00000	0.243120	0.121560	+	0.992584i	$0.461210\pi$
$830$	0	0
$831$	6.00000	0.208138
$832$	0	0
$833$	30.0000	1.03944
$834$	0	0
$835$	48.0000	1.66111
$836$	0	0
$837$	−10.0000	−0.345651
$838$	0	0
$839$	−22.0000	−0.759524	−0.379762	−	0.925084i	$-0.623994\pi$
$839$	−22.0000	−0.759524	−0.379762	+	0.925084i	$0.623994\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	−20.0000	−0.688837
$844$	0	0
$845$	−24.0000	−0.825625
$846$	0	0
$847$	−5.00000	−0.171802
$848$	0	0
$849$	−24.0000	−0.823678
$850$	0	0
$851$	−90.0000	−3.08516
$852$	0	0
$853$	−40.0000	−1.36957	−0.684787	−	0.728743i	$-0.740105\pi$
$853$	−40.0000	−1.36957	−0.684787	+	0.728743i	$0.740105\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	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	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	0	0
$863$	30.0000	1.02121	0.510606	−	0.859815i	$-0.329421\pi$
$863$	30.0000	1.02121	0.510606	+	0.859815i	$0.329421\pi$
$864$	0	0
$865$	−28.0000	−0.952029
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	40.0000	1.35691
$870$	0	0
$871$	13.0000	0.440488
$872$	0	0
$873$	12.0000	0.406138
$874$	0	0
$875$	12.0000	0.405674
$876$	0	0
$877$	−41.0000	−1.38447	−0.692236	−	0.721671i	$-0.743374\pi$
$877$	−41.0000	−1.38447	−0.692236	+	0.721671i	$0.743374\pi$
$878$	0	0
$879$	−21.0000	−0.708312
$880$	0	0
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	0	0
$883$	−26.0000	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$884$	0	0
$885$	18.0000	0.605063
$886$	0	0
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	0	0
$889$	−6.00000	−0.201234
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	12.0000	0.401565
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	−9.00000	−0.300501
$898$	0	0
$899$	10.0000	0.333519
$900$	0	0
$901$	−45.0000	−1.49917
$902$	0	0
$903$	−10.0000	−0.332779
$904$	0	0
$905$	4.00000	0.132964
$906$	0	0
$907$	51.0000	1.69343	0.846714	−	0.532049i	$-0.178578\pi$
$907$	51.0000	1.69343	0.846714	+	0.532049i	$0.178578\pi$
$908$	0	0
$909$	8.00000	0.265343
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	0	0
$913$	−32.0000	−1.05905
$914$	0	0
$915$	−4.00000	−0.132236
$916$	0	0
$917$	14.0000	0.462321
$918$	0	0
$919$	−43.0000	−1.41844	−0.709220	−	0.704988i	$-0.750953\pi$
$919$	−43.0000	−1.41844	−0.709220	+	0.704988i	$0.750953\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	10.0000	0.329154
$924$	0	0
$925$	−10.0000	−0.328798
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	13.0000	0.426516	0.213258	−	0.976996i	$-0.431592\pi$
$929$	13.0000	0.426516	0.213258	+	0.976996i	$0.431592\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	−33.0000	−1.08037
$934$	0	0
$935$	−40.0000	−1.30814
$936$	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
$939$	21.0000	0.685309
$940$	0	0
$941$	−3.00000	−0.0977972	−0.0488986	−	0.998804i	$-0.515571\pi$
$941$	−3.00000	−0.0977972	−0.0488986	+	0.998804i	$0.515571\pi$
$942$	0	0
$943$	−108.000	−3.51696
$944$	0	0
$945$	10.0000	0.325300
$946$	0	0
$947$	38.0000	1.23483	0.617417	−	0.786636i	$-0.288179\pi$
$947$	38.0000	1.23483	0.617417	+	0.786636i	$0.288179\pi$
$948$	0	0
$949$	−7.00000	−0.227230
$950$	0	0
$951$	31.0000	1.00524
$952$	0	0
$953$	−8.00000	−0.259145	−0.129573	−	0.991570i	$-0.541361\pi$
$953$	−8.00000	−0.259145	−0.129573	+	0.991570i	$0.541361\pi$
$954$	0	0
$955$	46.0000	1.48853
$956$	0	0
$957$	20.0000	0.646508
$958$	0	0
$959$	13.0000	0.419792
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−22.0000	−0.708940
$964$	0	0
$965$	−8.00000	−0.257529
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	−5.00000	−0.160623
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	2.00000	0.0641171
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	−30.0000	−0.957826
$982$	0	0
$983$	2.00000	0.0637901	0.0318950	−	0.999491i	$-0.489846\pi$
$983$	2.00000	0.0637901	0.0318950	+	0.999491i	$0.489846\pi$
$984$	0	0
$985$	−24.0000	−0.764704
$986$	0	0
$987$	−12.0000	−0.381964
$988$	0	0
$989$	−90.0000	−2.86183
$990$	0	0
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	0	0
$993$	−1.00000	−0.0317340
$994$	0	0
$995$	−10.0000	−0.317021
$996$	0	0
$997$	−50.0000	−1.58352	−0.791758	−	0.610835i	$-0.790834\pi$
$997$	−50.0000	−1.58352	−0.791758	+	0.610835i	$0.790834\pi$
$998$	0	0
$999$	−50.0000	−1.58193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.n.1.1		1
4.3	odd	2		4864.2.a.f.1.1		1
8.3	odd	2		4864.2.a.k.1.1		1
8.5	even	2		4864.2.a.c.1.1		1
16.3	odd	4		2432.2.c.a.1217.1	✓	2
16.5	even	4		2432.2.c.b.1217.1	yes	2
16.11	odd	4		2432.2.c.a.1217.2	yes	2
16.13	even	4		2432.2.c.b.1217.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.a.1217.1	✓	2	16.3	odd	4
2432.2.c.a.1217.2	yes	2	16.11	odd	4
2432.2.c.b.1217.1	yes	2	16.5	even	4
2432.2.c.b.1217.2	yes	2	16.13	even	4
4864.2.a.c.1.1		1	8.5	even	2
4864.2.a.f.1.1		1	4.3	odd	2
4864.2.a.k.1.1		1	8.3	odd	2
4864.2.a.n.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

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