LMFDB - Embedded newform 320.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,2,Mod(1,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, 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("320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	320.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:	$2.55521286468$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-2.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} +6.00000 q^{13} -2.00000 q^{15} +2.00000 q^{17} +8.00000 q^{19} -4.00000 q^{21} +6.00000 q^{23} +1.00000 q^{25} +4.00000 q^{27} +2.00000 q^{29} -4.00000 q^{31} +8.00000 q^{33} +2.00000 q^{35} -2.00000 q^{37} -12.0000 q^{39} -10.0000 q^{41} -2.00000 q^{43} +1.00000 q^{45} +2.00000 q^{47} -3.00000 q^{49} -4.00000 q^{51} -2.00000 q^{53} -4.00000 q^{55} -16.0000 q^{57} -2.00000 q^{61} +2.00000 q^{63} +6.00000 q^{65} -6.00000 q^{67} -12.0000 q^{69} +12.0000 q^{71} +10.0000 q^{73} -2.00000 q^{75} -8.00000 q^{77} +8.00000 q^{79} -11.0000 q^{81} -10.0000 q^{83} +2.00000 q^{85} -4.00000 q^{87} -6.00000 q^{89} +12.0000 q^{91} +8.00000 q^{93} +8.00000 q^{95} +10.0000 q^{97} -4.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	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
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	8.00000	1.39262
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−12.0000	−1.92154
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	−16.0000	−2.11925
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	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
$65$	6.00000	0.744208
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	0	0
$69$	−12.0000	−1.44463
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	−2.00000	−0.230940
$76$	0	0
$77$	−8.00000	−0.911685
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	−4.00000	−0.428845
$88$	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
$91$	12.0000	1.25794
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	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
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	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
$115$	6.00000	0.559503
$116$	0	0
$117$	6.00000	0.554700
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	20.0000	1.80334
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	−24.0000	−2.00698
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	8.00000	0.622799
$166$	0	0
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	8.00000	0.611775
$172$	0	0
$173$	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
$175$	2.00000	0.151186
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−8.00000	−0.585018
$188$	0	0
$189$	8.00000	0.581914
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	−12.0000	−0.859338
$196$	0	0
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	4.00000	0.280745
$204$	0	0
$205$	−10.0000	−0.698430
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	−32.0000	−2.21349
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	−24.0000	−1.64445
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	−20.0000	−1.35147
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	0	0
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	48.0000	3.05417
$248$	0	0
$249$	20.0000	1.26745
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	−24.0000	−1.50887
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	26.0000	1.62184	0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000	1.62184	0.810918	+	0.585160i	$0.198968\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−2.00000	−0.123325	−0.0616626	−	0.998097i	$-0.519640\pi$
$263$	−2.00000	−0.123325	−0.0616626	+	0.998097i	$0.519640\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	12.0000	0.734388
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	−24.0000	−1.45255
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−26.0000	−1.54554	−0.772770	−	0.634686i	$-0.781129\pi$
$283$	−26.0000	−1.54554	−0.772770	+	0.634686i	$0.781129\pi$
$284$	0	0
$285$	−16.0000	−0.947758
$286$	0	0
$287$	−20.0000	−1.18056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−20.0000	−1.17242
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−16.0000	−0.928414
$298$	0	0
$299$	36.0000	2.08193
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	28.0000	1.60856
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−22.0000	−1.25561	−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000	−1.25561	−0.627803	+	0.778372i	$0.716046\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	−28.0000	−1.58773	−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000	−1.58773	−0.793867	+	0.608091i	$0.791935\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	16.0000	0.890264
$324$	0	0
$325$	6.00000	0.332820
$326$	0	0
$327$	−28.0000	−1.54840
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	−6.00000	−0.327815
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	16.0000	0.866449
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	−12.0000	−0.646058
$346$	0	0
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	24.0000	1.28103
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	0	0
$357$	−8.00000	−0.423405
$358$	0	0
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	0	0
$363$	−10.0000	−0.524864
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	−4.00000	−0.207670
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	−2.00000	−0.103280
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	0	0
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	−8.00000	−0.403547
$394$	0	0
$395$	8.00000	0.402524
$396$	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
$399$	−32.0000	−1.60200
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	−24.0000	−1.19553
$404$	0	0
$405$	−11.0000	−0.546594
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−10.0000	−0.490881
$416$	0	0
$417$	32.0000	1.56705
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	2.00000	0.0972433
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	−4.00000	−0.193574
$428$	0	0
$429$	48.0000	2.31746
$430$	0	0
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	0	0
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	0	0
$437$	48.0000	2.29615
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	20.0000	0.945968
$448$	0	0
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	40.0000	1.88353
$452$	0	0
$453$	40.0000	1.87936
$454$	0	0
$455$	12.0000	0.562569
$456$	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
$459$	8.00000	0.373408
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	38.0000	1.76601	0.883005	−	0.469364i	$-0.155517\pi$
$463$	38.0000	1.76601	0.883005	+	0.469364i	$0.155517\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	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
$469$	−12.0000	−0.554109
$470$	0	0
$471$	20.0000	0.921551
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	8.00000	0.367065
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−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
$481$	−12.0000	−0.547153
$482$	0	0
$483$	−24.0000	−1.09204
$484$	0	0
$485$	10.0000	0.454077
$486$	0	0
$487$	−38.0000	−1.72194	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000	−1.72194	−0.860972	+	0.508652i	$0.830144\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	0	0
$501$	28.0000	1.25095
$502$	0	0
$503$	14.0000	0.624229	0.312115	−	0.950044i	$-0.398963\pi$
$503$	14.0000	0.624229	0.312115	+	0.950044i	$0.398963\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	−46.0000	−2.04293
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	20.0000	0.884748
$512$	0	0
$513$	32.0000	1.41283
$514$	0	0
$515$	−2.00000	−0.0881305
$516$	0	0
$517$	−8.00000	−0.351840
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−60.0000	−2.59889
$534$	0	0
$535$	−6.00000	−0.259403
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	−36.0000	−1.54491
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	−14.0000	−0.598597	−0.299298	−	0.954160i	$-0.596753\pi$
$547$	−14.0000	−0.598597	−0.299298	+	0.954160i	$0.596753\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	16.0000	0.681623
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	46.0000	1.94908	0.974541	−	0.224208i	$-0.0719796\pi$
$557$	46.0000	1.94908	0.974541	+	0.224208i	$0.0719796\pi$
$558$	0	0
$559$	−12.0000	−0.507546
$560$	0	0
$561$	16.0000	0.675521
$562$	0	0
$563$	30.0000	1.26435	0.632175	−	0.774826i	$-0.282163\pi$
$563$	30.0000	1.26435	0.632175	+	0.774826i	$0.282163\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	−22.0000	−0.923913
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	−20.0000	−0.829740
$582$	0	0
$583$	8.00000	0.331326
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	10.0000	0.412744	0.206372	−	0.978474i	$-0.433834\pi$
$587$	10.0000	0.412744	0.206372	+	0.978474i	$0.433834\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	−44.0000	−1.80992
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	0	0
$597$	−32.0000	−1.30967
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	−6.00000	−0.244339
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	0	0
$615$	20.0000	0.806478
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	24.0000	0.963087
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	64.0000	2.55591
$628$	0	0
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	−6.00000	−0.238103
$636$	0	0
$637$	−18.0000	−0.713186
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	30.0000	1.18308	0.591542	−	0.806274i	$-0.298519\pi$
$643$	30.0000	1.18308	0.591542	+	0.806274i	$0.298519\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	0	0
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	16.0000	0.627089
$652$	0	0
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	12.0000	0.464642
$668$	0	0
$669$	4.00000	0.154649
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	−36.0000	−1.37952
$682$	0	0
$683$	−34.0000	−1.30097	−0.650487	−	0.759517i	$-0.725435\pi$
$683$	−34.0000	−1.30097	−0.650487	+	0.759517i	$0.725435\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	−20.0000	−0.763048
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	0	0
$693$	−8.00000	−0.303895
$694$	0	0
$695$	−16.0000	−0.606915
$696$	0	0
$697$	−20.0000	−0.757554
$698$	0	0
$699$	−4.00000	−0.151294
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	−16.0000	−0.603451
$704$	0	0
$705$	−4.00000	−0.150649
$706$	0	0
$707$	−28.0000	−1.05305
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	−24.0000	−0.897549
$716$	0	0
$717$	−16.0000	−0.597531
$718$	0	0
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	36.0000	1.33885
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−4.00000	−0.147945
$732$	0	0
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	0	0
$737$	24.0000	0.884051
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	−96.0000	−3.52665
$742$	0	0
$743$	−42.0000	−1.54083	−0.770415	−	0.637542i	$-0.779951\pi$
$743$	−42.0000	−1.54083	−0.770415	+	0.637542i	$0.779951\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	0	0
$747$	−10.0000	−0.365881
$748$	0	0
$749$	−12.0000	−0.438470
$750$	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
$753$	40.0000	1.45768
$754$	0	0
$755$	−20.0000	−0.727875
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	0	0
$759$	48.0000	1.74229
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	28.0000	1.01367
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	−52.0000	−1.87273
$772$	0	0
$773$	38.0000	1.36677	0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000	1.36677	0.683383	+	0.730061i	$0.260508\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	8.00000	0.286998
$778$	0	0
$779$	−80.0000	−2.86630
$780$	0	0
$781$	−48.0000	−1.71758
$782$	0	0
$783$	8.00000	0.285897
$784$	0	0
$785$	−10.0000	−0.356915
$786$	0	0
$787$	−14.0000	−0.499046	−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000	−0.499046	−0.249523	+	0.968369i	$0.580274\pi$
$788$	0	0
$789$	4.00000	0.142404
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	4.00000	0.141865
$796$	0	0
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	−40.0000	−1.41157
$804$	0	0
$805$	12.0000	0.422944
$806$	0	0
$807$	36.0000	1.26726
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	56.0000	1.96401
$814$	0	0
$815$	6.00000	0.210171
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	12.0000	0.419314
$820$	0	0
$821$	38.0000	1.32621	0.663105	−	0.748527i	$-0.269238\pi$
$821$	38.0000	1.32621	0.663105	+	0.748527i	$0.269238\pi$
$822$	0	0
$823$	6.00000	0.209147	0.104573	−	0.994517i	$-0.466652\pi$
$823$	6.00000	0.209147	0.104573	+	0.994517i	$0.466652\pi$
$824$	0	0
$825$	8.00000	0.278524
$826$	0	0
$827$	50.0000	1.73867	0.869335	−	0.494223i	$-0.164547\pi$
$827$	50.0000	1.73867	0.869335	+	0.494223i	$0.164547\pi$
$828$	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
$831$	−44.0000	−1.52634
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−14.0000	−0.484490
$836$	0	0
$837$	−16.0000	−0.553041
$838$	0	0
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	20.0000	0.688837
$844$	0	0
$845$	23.0000	0.791224
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	52.0000	1.78464
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	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
$861$	40.0000	1.36320
$862$	0	0
$863$	46.0000	1.56586	0.782929	−	0.622111i	$-0.213725\pi$
$863$	46.0000	1.56586	0.782929	+	0.622111i	$0.213725\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	26.0000	0.883006
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−36.0000	−1.21981
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	0	0
$879$	36.0000	1.21425
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	−2.00000	−0.0673054	−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000	−0.0673054	−0.0336527	+	0.999434i	$0.510714\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.00000	0.0671534	0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000	0.0671534	0.0335767	+	0.999436i	$0.489310\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	44.0000	1.47406
$892$	0	0
$893$	16.0000	0.535420
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−72.0000	−2.40401
$898$	0	0
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	0	0
$909$	−14.0000	−0.464351
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	40.0000	1.32381
$914$	0	0
$915$	4.00000	0.132236
$916$	0	0
$917$	8.00000	0.264183
$918$	0	0
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	0	0
$921$	44.0000	1.44985
$922$	0	0
$923$	72.0000	2.36991
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	−2.00000	−0.0656886
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−24.0000	−0.786568
$932$	0	0
$933$	56.0000	1.83336
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	34.0000	1.11073	0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000	1.11073	0.555366	+	0.831606i	$0.312578\pi$
$938$	0	0
$939$	12.0000	0.391605
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	0	0
$943$	−60.0000	−1.95387
$944$	0	0
$945$	8.00000	0.260240
$946$	0	0
$947$	−38.0000	−1.23483	−0.617417	−	0.786636i	$-0.711821\pi$
$947$	−38.0000	−1.23483	−0.617417	+	0.786636i	$0.711821\pi$
$948$	0	0
$949$	60.0000	1.94768
$950$	0	0
$951$	4.00000	0.129709
$952$	0	0
$953$	58.0000	1.87880	0.939402	−	0.342817i	$-0.111381\pi$
$953$	58.0000	1.87880	0.939402	+	0.342817i	$0.111381\pi$
$954$	0	0
$955$	4.00000	0.129437
$956$	0	0
$957$	16.0000	0.517207
$958$	0	0
$959$	4.00000	0.129167
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−6.00000	−0.193347
$964$	0	0
$965$	2.00000	0.0643823
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	0	0
$969$	−32.0000	−1.02799
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	−32.0000	−1.02587
$974$	0	0
$975$	−12.0000	−0.384308
$976$	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
$979$	24.0000	0.767043
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	46.0000	1.46717	0.733586	−	0.679597i	$-0.237845\pi$
$983$	46.0000	1.46717	0.733586	+	0.679597i	$0.237845\pi$
$984$	0	0
$985$	22.0000	0.700978
$986$	0	0
$987$	−8.00000	−0.254643
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	0	0
$993$	−8.00000	−0.253872
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.a.b.1.1		1
3.2	odd	2		2880.2.a.o.1.1		1
4.3	odd	2		320.2.a.e.1.1		1
5.2	odd	4		1600.2.c.c.449.2		2
5.3	odd	4		1600.2.c.c.449.1		2
5.4	even	2		1600.2.a.t.1.1		1
8.3	odd	2		160.2.a.a.1.1	✓	1
8.5	even	2		160.2.a.b.1.1	yes	1
12.11	even	2		2880.2.a.d.1.1		1
16.3	odd	4		1280.2.d.h.641.2		2
16.5	even	4		1280.2.d.b.641.2		2
16.11	odd	4		1280.2.d.h.641.1		2
16.13	even	4		1280.2.d.b.641.1		2
20.3	even	4		1600.2.c.f.449.2		2
20.7	even	4		1600.2.c.f.449.1		2
20.19	odd	2		1600.2.a.e.1.1		1
24.5	odd	2		1440.2.a.l.1.1		1
24.11	even	2		1440.2.a.i.1.1		1
40.3	even	4		800.2.c.a.449.1		2
40.13	odd	4		800.2.c.b.449.2		2
40.19	odd	2		800.2.a.i.1.1		1
40.27	even	4		800.2.c.a.449.2		2
40.29	even	2		800.2.a.a.1.1		1
40.37	odd	4		800.2.c.b.449.1		2
56.13	odd	2		7840.2.a.e.1.1		1
56.27	even	2		7840.2.a.w.1.1		1
120.29	odd	2		7200.2.a.l.1.1		1
120.53	even	4		7200.2.f.g.6049.1		2
120.59	even	2		7200.2.a.bp.1.1		1
120.77	even	4		7200.2.f.g.6049.2		2
120.83	odd	4		7200.2.f.w.6049.2		2
120.107	odd	4		7200.2.f.w.6049.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.2.a.a.1.1	✓	1	8.3	odd	2
160.2.a.b.1.1	yes	1	8.5	even	2
320.2.a.b.1.1		1	1.1	even	1	trivial
320.2.a.e.1.1		1	4.3	odd	2
800.2.a.a.1.1		1	40.29	even	2
800.2.a.i.1.1		1	40.19	odd	2
800.2.c.a.449.1		2	40.3	even	4
800.2.c.a.449.2		2	40.27	even	4
800.2.c.b.449.1		2	40.37	odd	4
800.2.c.b.449.2		2	40.13	odd	4
1280.2.d.b.641.1		2	16.13	even	4
1280.2.d.b.641.2		2	16.5	even	4
1280.2.d.h.641.1		2	16.11	odd	4
1280.2.d.h.641.2		2	16.3	odd	4
1440.2.a.i.1.1		1	24.11	even	2
1440.2.a.l.1.1		1	24.5	odd	2
1600.2.a.e.1.1		1	20.19	odd	2
1600.2.a.t.1.1		1	5.4	even	2
1600.2.c.c.449.1		2	5.3	odd	4
1600.2.c.c.449.2		2	5.2	odd	4
1600.2.c.f.449.1		2	20.7	even	4
1600.2.c.f.449.2		2	20.3	even	4
2880.2.a.d.1.1		1	12.11	even	2
2880.2.a.o.1.1		1	3.2	odd	2
7200.2.a.l.1.1		1	120.29	odd	2
7200.2.a.bp.1.1		1	120.59	even	2
7200.2.f.g.6049.1		2	120.53	even	4
7200.2.f.g.6049.2		2	120.77	even	4
7200.2.f.w.6049.1		2	120.107	odd	4
7200.2.f.w.6049.2		2	120.83	odd	4
7840.2.a.e.1.1		1	56.13	odd	2
7840.2.a.w.1.1		1	56.27	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more