LMFDB - Embedded newform 8464.2.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,1,0,-2,0,2,0,-2,0,-6,0,-1] 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:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -2.00000 q^{5} +2.00000 q^{7} -2.00000 q^{9} -6.00000 q^{11} -1.00000 q^{13} -2.00000 q^{15} -6.00000 q^{17} -4.00000 q^{19} +2.00000 q^{21} -1.00000 q^{25} -5.00000 q^{27} +9.00000 q^{29} -3.00000 q^{31} -6.00000 q^{33} -4.00000 q^{35} +2.00000 q^{37} -1.00000 q^{39} -1.00000 q^{41} +4.00000 q^{45} +1.00000 q^{47} -3.00000 q^{49} -6.00000 q^{51} +12.0000 q^{53} +12.0000 q^{55} -4.00000 q^{57} +12.0000 q^{59} -8.00000 q^{61} -4.00000 q^{63} +2.00000 q^{65} -10.0000 q^{67} +13.0000 q^{71} +5.00000 q^{73} -1.00000 q^{75} -12.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +2.00000 q^{83} +12.0000 q^{85} +9.00000 q^{87} +12.0000 q^{89} -2.00000 q^{91} -3.00000 q^{93} +8.00000 q^{95} +2.00000 q^{97} +12.0000 q^{99} +O(q^{100})$

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	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	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	−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
$19$	−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
$21$	2.00000	0.436436
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	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
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−1.00000	−0.156174	−0.0780869	−	0.996947i	$-0.524881\pi$
$41$	−1.00000	−0.156174	−0.0780869	+	0.996947i	$0.524881\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	0	0
$47$	1.00000	0.145865	0.0729325	−	0.997337i	$-0.476764\pi$
$47$	1.00000	0.145865	0.0729325	+	0.997337i	$0.476764\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	12.0000	1.61808
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.0000	1.54282	0.771408	−	0.636341i	$-0.219553\pi$
$71$	13.0000	1.54282	0.771408	+	0.636341i	$0.219553\pi$
$72$	0	0
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−12.0000	−1.36753
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	0	0
$87$	9.00000	0.964901
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−3.00000	−0.311086
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	12.0000	1.20605
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	−1.00000	−0.0901670
$124$	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
$129$	0	0
$130$	0	0
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	10.0000	0.860663
$136$	0	0
$137$	−16.0000	−1.36697	−0.683486	−	0.729964i	$-0.739537\pi$
$137$	−16.0000	−1.36697	−0.683486	+	0.729964i	$0.739537\pi$
$138$	0	0
$139$	19.0000	1.61156	0.805779	−	0.592216i	$-0.201747\pi$
$139$	19.0000	1.61156	0.805779	+	0.592216i	$0.201747\pi$
$140$	0	0
$141$	1.00000	0.0842152
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	−18.0000	−1.49482
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	0	0
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	0	0
$153$	12.0000	0.970143
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	−12.0000	−0.957704	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−12.0000	−0.957704	−0.478852	+	0.877896i	$0.658947\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	0	0
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	0	0
$165$	12.0000	0.934199
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	8.00000	0.611775
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	36.0000	2.63258
$188$	0	0
$189$	−10.0000	−0.727393
$190$	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
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	1.00000	0.0712470	0.0356235	−	0.999365i	$-0.488658\pi$
$197$	1.00000	0.0712470	0.0356235	+	0.999365i	$0.488658\pi$
$198$	0	0
$199$	−22.0000	−1.55954	−0.779769	−	0.626067i	$-0.784664\pi$
$199$	−22.0000	−1.55954	−0.779769	+	0.626067i	$0.784664\pi$
$200$	0	0
$201$	−10.0000	−0.705346
$202$	0	0
$203$	18.0000	1.26335
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	24.0000	1.66011
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	13.0000	0.890745
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	5.00000	0.337869
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	−17.0000	−1.11371	−0.556854	−	0.830611i	$-0.687992\pi$
$233$	−17.0000	−1.11371	−0.556854	+	0.830611i	$0.687992\pi$
$234$	0	0
$235$	−2.00000	−0.130466
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	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
$243$	16.0000	1.02640
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	2.00000	0.126745
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	12.0000	0.751469
$256$	0	0
$257$	27.0000	1.68421	0.842107	−	0.539311i	$-0.181315\pi$
$257$	27.0000	1.68421	0.842107	+	0.539311i	$0.181315\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	−18.0000	−1.11417
$262$	0	0
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	−24.0000	−1.47431
$266$	0	0
$267$	12.0000	0.734388
$268$	0	0
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	6.00000	0.361814
$276$	0	0
$277$	−27.0000	−1.62227	−0.811136	−	0.584857i	$-0.801151\pi$
$277$	−27.0000	−1.62227	−0.811136	+	0.584857i	$0.801151\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	0	0
$297$	30.0000	1.74078
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−14.0000	−0.804279
$304$	0	0
$305$	16.0000	0.916157
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	14.0000	0.796432
$310$	0	0
$311$	−27.0000	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$312$	0	0
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	0	0
$315$	8.00000	0.450749
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	−54.0000	−3.02342
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	24.0000	1.33540
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	16.0000	0.884802
$328$	0	0
$329$	2.00000	0.110264
$330$	0	0
$331$	−13.0000	−0.714545	−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000	−0.714545	−0.357272	+	0.934000i	$0.616293\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	20.0000	1.09272
$336$	0	0
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	0	0
$339$	−14.0000	−0.760376
$340$	0	0
$341$	18.0000	0.974755
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	0	0
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	−15.0000	−0.798369	−0.399185	−	0.916871i	$-0.630707\pi$
$353$	−15.0000	−0.798369	−0.399185	+	0.916871i	$0.630707\pi$
$354$	0	0
$355$	−26.0000	−1.37994
$356$	0	0
$357$	−12.0000	−0.635107
$358$	0	0
$359$	−28.0000	−1.47778	−0.738892	−	0.673824i	$-0.764651\pi$
$359$	−28.0000	−1.47778	−0.738892	+	0.673824i	$0.764651\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	25.0000	1.31216
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	24.0000	1.24602
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	−9.00000	−0.463524
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−9.00000	−0.461084
$382$	0	0
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	0	0
$385$	24.0000	1.22315
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.00000	−0.202808	−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000	−0.202808	−0.101404	+	0.994845i	$0.532333\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	1.00000	0.0504433
$394$	0	0
$395$	8.00000	0.402524
$396$	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
$399$	−8.00000	−0.400501
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	3.00000	0.149441
$404$	0	0
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	17.0000	0.840596	0.420298	−	0.907386i	$-0.361926\pi$
$409$	17.0000	0.840596	0.420298	+	0.907386i	$0.361926\pi$
$410$	0	0
$411$	−16.0000	−0.789222
$412$	0	0
$413$	24.0000	1.18096
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	19.0000	0.930434
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	−16.0000	−0.774294
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	−18.0000	−0.863034
$436$	0	0
$437$	0	0
$438$	0	0
$439$	19.0000	0.906821	0.453410	−	0.891302i	$-0.350207\pi$
$439$	19.0000	0.906821	0.453410	+	0.891302i	$0.350207\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	41.0000	1.94797	0.973984	−	0.226615i	$-0.0727659\pi$
$443$	41.0000	1.94797	0.973984	+	0.226615i	$0.0727659\pi$
$444$	0	0
$445$	−24.0000	−1.13771
$446$	0	0
$447$	16.0000	0.756774
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	0	0
$453$	17.0000	0.798730
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	0	0
$459$	30.0000	1.40028
$460$	0	0
$461$	5.00000	0.232873	0.116437	−	0.993198i	$-0.462853\pi$
$461$	5.00000	0.232873	0.116437	+	0.993198i	$0.462853\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	0	0
$465$	6.00000	0.278243
$466$	0	0
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	0	0
$469$	−20.0000	−0.923514
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	−24.0000	−1.09888
$478$	0	0
$479$	−10.0000	−0.456912	−0.228456	−	0.973554i	$-0.573368\pi$
$479$	−10.0000	−0.456912	−0.228456	+	0.973554i	$0.573368\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.00000	−0.181631
$486$	0	0
$487$	35.0000	1.58600	0.793001	−	0.609221i	$-0.208518\pi$
$487$	35.0000	1.58600	0.793001	+	0.609221i	$0.208518\pi$
$488$	0	0
$489$	11.0000	0.497437
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	0	0
$493$	−54.0000	−2.43204
$494$	0	0
$495$	−24.0000	−1.07872
$496$	0	0
$497$	26.0000	1.16626
$498$	0	0
$499$	−3.00000	−0.134298	−0.0671492	−	0.997743i	$-0.521390\pi$
$499$	−3.00000	−0.134298	−0.0671492	+	0.997743i	$0.521390\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	28.0000	1.24598
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	37.0000	1.64000	0.819998	−	0.572366i	$-0.193974\pi$
$509$	37.0000	1.64000	0.819998	+	0.572366i	$0.193974\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	20.0000	0.883022
$514$	0	0
$515$	−28.0000	−1.23383
$516$	0	0
$517$	−6.00000	−0.263880
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	0	0
$525$	−2.00000	−0.0872872
$526$	0	0
$527$	18.0000	0.784092
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−24.0000	−1.04151
$532$	0	0
$533$	1.00000	0.0433148
$534$	0	0
$535$	−16.0000	−0.691740
$536$	0	0
$537$	15.0000	0.647298
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	−11.0000	−0.472927	−0.236463	−	0.971640i	$-0.575988\pi$
$541$	−11.0000	−0.472927	−0.236463	+	0.971640i	$0.575988\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	−32.0000	−1.37073
$546$	0	0
$547$	19.0000	0.812381	0.406191	−	0.913788i	$-0.366857\pi$
$547$	19.0000	0.812381	0.406191	+	0.913788i	$0.366857\pi$
$548$	0	0
$549$	16.0000	0.682863
$550$	0	0
$551$	−36.0000	−1.53365
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	4.00000	0.169485	0.0847427	−	0.996403i	$-0.472993\pi$
$557$	4.00000	0.169485	0.0847427	+	0.996403i	$0.472993\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	36.0000	1.51992
$562$	0	0
$563$	16.0000	0.674320	0.337160	−	0.941447i	$-0.390534\pi$
$563$	16.0000	0.674320	0.337160	+	0.941447i	$0.390534\pi$
$564$	0	0
$565$	28.0000	1.17797
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	38.0000	1.59025	0.795125	−	0.606445i	$-0.207405\pi$
$571$	38.0000	1.59025	0.795125	+	0.606445i	$0.207405\pi$
$572$	0	0
$573$	−22.0000	−0.919063
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	0	0
$579$	−19.0000	−0.789613
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	−72.0000	−2.98194
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	−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
$589$	12.0000	0.494451
$590$	0	0
$591$	1.00000	0.0411345
$592$	0	0
$593$	46.0000	1.88899	0.944497	−	0.328521i	$-0.106550\pi$
$593$	46.0000	1.88899	0.944497	+	0.328521i	$0.106550\pi$
$594$	0	0
$595$	24.0000	0.983904
$596$	0	0
$597$	−22.0000	−0.900400
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	−3.00000	−0.122373	−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000	−0.122373	−0.0611863	+	0.998126i	$0.519488\pi$
$602$	0	0
$603$	20.0000	0.814463
$604$	0	0
$605$	−50.0000	−2.03279
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	0	0
$609$	18.0000	0.729397
$610$	0	0
$611$	−1.00000	−0.0404557
$612$	0	0
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24.0000	0.961540
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	24.0000	0.958468
$628$	0	0
$629$	−12.0000	−0.478471
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	18.0000	0.714308
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	−26.0000	−1.02854
$640$	0	0
$641$	−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
$643$	−18.0000	−0.709851	−0.354925	−	0.934895i	$-0.615494\pi$
$643$	−18.0000	−0.709851	−0.354925	+	0.934895i	$0.615494\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−39.0000	−1.53325	−0.766624	−	0.642096i	$-0.778065\pi$
$647$	−39.0000	−1.53325	−0.766624	+	0.642096i	$0.778065\pi$
$648$	0	0
$649$	−72.0000	−2.82625
$650$	0	0
$651$	−6.00000	−0.235159
$652$	0	0
$653$	11.0000	0.430463	0.215232	−	0.976563i	$-0.430949\pi$
$653$	11.0000	0.430463	0.215232	+	0.976563i	$0.430949\pi$
$654$	0	0
$655$	−2.00000	−0.0781465
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	−14.0000	−0.545363	−0.272681	−	0.962104i	$-0.587910\pi$
$659$	−14.0000	−0.545363	−0.272681	+	0.962104i	$0.587910\pi$
$660$	0	0
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	0	0
$663$	6.00000	0.233021
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	0	0
$668$	0	0
$669$	12.0000	0.463947
$670$	0	0
$671$	48.0000	1.85302
$672$	0	0
$673$	−29.0000	−1.11787	−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000	−1.11787	−0.558934	+	0.829212i	$0.688789\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	40.0000	1.53732	0.768662	−	0.639655i	$-0.220923\pi$
$677$	40.0000	1.53732	0.768662	+	0.639655i	$0.220923\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	14.0000	0.536481
$682$	0	0
$683$	27.0000	1.03313	0.516563	−	0.856249i	$-0.327211\pi$
$683$	27.0000	1.03313	0.516563	+	0.856249i	$0.327211\pi$
$684$	0	0
$685$	32.0000	1.22266
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	0	0
$693$	24.0000	0.911685
$694$	0	0
$695$	−38.0000	−1.44142
$696$	0	0
$697$	6.00000	0.227266
$698$	0	0
$699$	−17.0000	−0.642999
$700$	0	0
$701$	−26.0000	−0.982006	−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000	−0.982006	−0.491003	+	0.871158i	$0.663370\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	−2.00000	−0.0753244
$706$	0	0
$707$	−28.0000	−1.05305
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−12.0000	−0.448775
$716$	0	0
$717$	9.00000	0.336111
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	28.0000	1.04277
$722$	0	0
$723$	−14.0000	−0.520666
$724$	0	0
$725$	−9.00000	−0.334252
$726$	0	0
$727$	30.0000	1.11264	0.556319	−	0.830969i	$-0.312213\pi$
$727$	30.0000	1.11264	0.556319	+	0.830969i	$0.312213\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	42.0000	1.55131	0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000	1.55131	0.775653	+	0.631160i	$0.217421\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	0	0
$737$	60.0000	2.21013
$738$	0	0
$739$	11.0000	0.404642	0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000	0.404642	0.202321	+	0.979319i	$0.435152\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	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
$745$	−32.0000	−1.17239
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	16.0000	0.584627
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	−34.0000	−1.23739
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	13.0000	0.471250	0.235625	−	0.971844i	$-0.424286\pi$
$761$	13.0000	0.471250	0.235625	+	0.971844i	$0.424286\pi$
$762$	0	0
$763$	32.0000	1.15848
$764$	0	0
$765$	−24.0000	−0.867722
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	27.0000	0.972381
$772$	0	0
$773$	−12.0000	−0.431610	−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000	−0.431610	−0.215805	+	0.976436i	$0.569238\pi$
$774$	0	0
$775$	3.00000	0.107763
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	4.00000	0.143315
$780$	0	0
$781$	−78.0000	−2.79106
$782$	0	0
$783$	−45.0000	−1.60817
$784$	0	0
$785$	24.0000	0.856597
$786$	0	0
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	0	0
$789$	4.00000	0.142404
$790$	0	0
$791$	−28.0000	−0.995565
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	−24.0000	−0.851192
$796$	0	0
$797$	−4.00000	−0.141687	−0.0708436	−	0.997487i	$-0.522569\pi$
$797$	−4.00000	−0.141687	−0.0708436	+	0.997487i	$0.522569\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	0	0
$801$	−24.0000	−0.847998
$802$	0	0
$803$	−30.0000	−1.05868
$804$	0	0
$805$	0	0
$806$	0	0
$807$	9.00000	0.316815
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	21.0000	0.737410	0.368705	−	0.929547i	$-0.379801\pi$
$811$	21.0000	0.737410	0.368705	+	0.929547i	$0.379801\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	0	0
$815$	−22.0000	−0.770626
$816$	0	0
$817$	0	0
$818$	0	0
$819$	4.00000	0.139771
$820$	0	0
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	0	0
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	0	0
$825$	6.00000	0.208893
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	−27.0000	−0.936620
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	15.0000	0.518476
$838$	0	0
$839$	38.0000	1.31191	0.655953	−	0.754802i	$-0.272267\pi$
$839$	38.0000	1.31191	0.655953	+	0.754802i	$0.272267\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	6.00000	0.206651
$844$	0	0
$845$	24.0000	0.825625
$846$	0	0
$847$	50.0000	1.71802
$848$	0	0
$849$	−14.0000	−0.480479
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	−16.0000	−0.547188
$856$	0	0
$857$	−9.00000	−0.307434	−0.153717	−	0.988115i	$-0.549124\pi$
$857$	−9.00000	−0.307434	−0.153717	+	0.988115i	$0.549124\pi$
$858$	0	0
$859$	1.00000	0.0341196	0.0170598	−	0.999854i	$-0.494569\pi$
$859$	1.00000	0.0341196	0.0170598	+	0.999854i	$0.494569\pi$
$860$	0	0
$861$	−2.00000	−0.0681598
$862$	0	0
$863$	−49.0000	−1.66798	−0.833990	−	0.551780i	$-0.813949\pi$
$863$	−49.0000	−1.66798	−0.833990	+	0.551780i	$0.813949\pi$
$864$	0	0
$865$	4.00000	0.136004
$866$	0	0
$867$	19.0000	0.645274
$868$	0	0
$869$	24.0000	0.814144
$870$	0	0
$871$	10.0000	0.338837
$872$	0	0
$873$	−4.00000	−0.135379
$874$	0	0
$875$	24.0000	0.811348
$876$	0	0
$877$	46.0000	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$878$	0	0
$879$	−12.0000	−0.404750
$880$	0	0
$881$	38.0000	1.28025	0.640126	−	0.768270i	$-0.278882\pi$
$881$	38.0000	1.28025	0.640126	+	0.768270i	$0.278882\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	−24.0000	−0.806751
$886$	0	0
$887$	21.0000	0.705111	0.352555	−	0.935791i	$-0.385313\pi$
$887$	21.0000	0.705111	0.352555	+	0.935791i	$0.385313\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	−6.00000	−0.201008
$892$	0	0
$893$	−4.00000	−0.133855
$894$	0	0
$895$	−30.0000	−1.00279
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−27.0000	−0.900500
$900$	0	0
$901$	−72.0000	−2.39867
$902$	0	0
$903$	0	0
$904$	0	0
$905$	28.0000	0.930751
$906$	0	0
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	0	0
$909$	28.0000	0.928701
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	16.0000	0.528944
$916$	0	0
$917$	2.00000	0.0660458
$918$	0	0
$919$	18.0000	0.593765	0.296883	−	0.954914i	$-0.404053\pi$
$919$	18.0000	0.593765	0.296883	+	0.954914i	$0.404053\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	−13.0000	−0.427900
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	−28.0000	−0.919641
$928$	0	0
$929$	15.0000	0.492134	0.246067	−	0.969253i	$-0.420862\pi$
$929$	15.0000	0.492134	0.246067	+	0.969253i	$0.420862\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	0	0
$933$	−27.0000	−0.883940
$934$	0	0
$935$	−72.0000	−2.35465
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	−28.0000	−0.913745
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	20.0000	0.650600
$946$	0	0
$947$	29.0000	0.942373	0.471187	−	0.882034i	$-0.343826\pi$
$947$	29.0000	0.942373	0.471187	+	0.882034i	$0.343826\pi$
$948$	0	0
$949$	−5.00000	−0.162307
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	−4.00000	−0.129573	−0.0647864	−	0.997899i	$-0.520637\pi$
$953$	−4.00000	−0.129573	−0.0647864	+	0.997899i	$0.520637\pi$
$954$	0	0
$955$	44.0000	1.42381
$956$	0	0
$957$	−54.0000	−1.74557
$958$	0	0
$959$	−32.0000	−1.03333
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−16.0000	−0.515593
$964$	0	0
$965$	38.0000	1.22326
$966$	0	0
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	0	0
$973$	38.0000	1.21822
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	−72.0000	−2.30113
$980$	0	0
$981$	−32.0000	−1.02168
$982$	0	0
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	2.00000	0.0636607
$988$	0	0
$989$	0	0
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	−13.0000	−0.412543
$994$	0	0
$995$	44.0000	1.39489
$996$	0	0
$997$	30.0000	0.950110	0.475055	−	0.879956i	$-0.342428\pi$
$997$	30.0000	0.950110	0.475055	+	0.879956i	$0.342428\pi$
$998$	0	0
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.l.1.1		1
4.3	odd	2		4232.2.a.a.1.1	✓	1
23.22	odd	2		8464.2.a.n.1.1		1
92.91	even	2		4232.2.a.d.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.a.1.1	✓	1	4.3	odd	2
4232.2.a.d.1.1	yes	1	92.91	even	2
8464.2.a.l.1.1		1	1.1	even	1	trivial
8464.2.a.n.1.1		1	23.22	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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