LMFDB - Embedded newform 4560.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(1,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.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:	$36.4117833217$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{3} +1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} -6.00000 q^{11} +4.00000 q^{13} +1.00000 q^{15} -2.00000 q^{17} -1.00000 q^{19} -2.00000 q^{21} -4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.00000 q^{31} -6.00000 q^{33} -2.00000 q^{35} +8.00000 q^{37} +4.00000 q^{39} +2.00000 q^{43} +1.00000 q^{45} -4.00000 q^{47} -3.00000 q^{49} -2.00000 q^{51} -14.0000 q^{53} -6.00000 q^{55} -1.00000 q^{57} -14.0000 q^{61} -2.00000 q^{63} +4.00000 q^{65} -8.00000 q^{67} -4.00000 q^{69} -8.00000 q^{71} -2.00000 q^{73} +1.00000 q^{75} +12.0000 q^{77} +1.00000 q^{81} -12.0000 q^{83} -2.00000 q^{85} -8.00000 q^{89} -8.00000 q^{91} +8.00000 q^{93} -1.00000 q^{95} -4.00000 q^{97} -6.00000 q^{99} +O(q^{100})$

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
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	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
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	12.0000	1.36753
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	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
$115$	−4.00000	−0.373002
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	−24.0000	−2.00698
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−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
$153$	−2.00000	−0.161690
$154$	0	0
$155$	8.00000	0.642575
$156$	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
$159$	−14.0000	−1.11027
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	0	0
$165$	−6.00000	−0.467099
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	0	0
$175$	−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
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	−14.0000	−1.03491
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	12.0000	0.877527
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	6.00000	0.415029
$210$	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
$213$	−8.00000	−0.548151
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	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
$231$	12.0000	0.789542
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	0	0
$255$	−2.00000	−0.125245
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−14.0000	−0.860013
$266$	0	0
$267$	−8.00000	−0.489592
$268$	0	0
$269$	−16.0000	−0.975537	−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000	−0.975537	−0.487769	+	0.872973i	$0.662189\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	−6.00000	−0.361814
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	0	0
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	0	0
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−4.00000	−0.234484
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−6.00000	−0.348155
$298$	0	0
$299$	−16.0000	−0.925304
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	18.0000	1.03407
$304$	0	0
$305$	−14.0000	−0.801638
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−2.00000	−0.113410	−0.0567048	−	0.998391i	$-0.518059\pi$
$311$	−2.00000	−0.113410	−0.0567048	+	0.998391i	$0.518059\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−14.0000	−0.774202
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	0	0
$333$	8.00000	0.438397
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−48.0000	−2.59935
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	25.0000	1.31216
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	−30.0000	−1.56599	−0.782994	−	0.622030i	$-0.786308\pi$
$367$	−30.0000	−1.56599	−0.782994	+	0.622030i	$0.786308\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	28.0000	1.45369
$372$	0	0
$373$	−20.0000	−1.03556	−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000	−1.03556	−0.517780	+	0.855514i	$0.673242\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	2.00000	0.101666
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	−10.0000	−0.504433
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	0	0
$401$	4.00000	0.199750	0.0998752	−	0.995000i	$-0.468156\pi$
$401$	4.00000	0.199750	0.0998752	+	0.995000i	$0.468156\pi$
$402$	0	0
$403$	32.0000	1.59403
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−48.0000	−2.37927
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	28.0000	1.35501
$428$	0	0
$429$	−24.0000	−1.15873
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	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
$435$	0	0
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	40.0000	1.90046	0.950229	−	0.311553i	$-0.100849\pi$
$443$	40.0000	1.90046	0.950229	+	0.311553i	$0.100849\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	0	0
$447$	−6.00000	−0.283790
$448$	0	0
$449$	−24.0000	−1.13263	−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000	−1.13263	−0.566315	+	0.824189i	$0.691631\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−8.00000	−0.375873
$454$	0	0
$455$	−8.00000	−0.375046
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−6.00000	−0.278844	−0.139422	−	0.990233i	$-0.544524\pi$
$463$	−6.00000	−0.278844	−0.139422	+	0.990233i	$0.544524\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	−12.0000	−0.551761
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−14.0000	−0.641016
$478$	0	0
$479$	−26.0000	−1.18797	−0.593985	−	0.804476i	$-0.702446\pi$
$479$	−26.0000	−1.18797	−0.593985	+	0.804476i	$0.702446\pi$
$480$	0	0
$481$	32.0000	1.45907
$482$	0	0
$483$	8.00000	0.364013
$484$	0	0
$485$	−4.00000	−0.181631
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	−10.0000	−0.452216
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	16.0000	0.717698
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−12.0000	−0.528783
$516$	0	0
$517$	24.0000	1.05552
$518$	0	0
$519$	−10.0000	−0.438951
$520$	0	0
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	−2.00000	−0.0872872
$526$	0	0
$527$	−16.0000	−0.696971
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	4.00000	0.172935
$536$	0	0
$537$	0	0
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	−22.0000	−0.944110
$544$	0	0
$545$	−14.0000	−0.599694
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	0	0
$549$	−14.0000	−0.597505
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.00000	0.339581
$556$	0	0
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	8.00000	0.335377	0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000	0.335377	0.167689	+	0.985840i	$0.446370\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	−18.0000	−0.751961
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	0	0
$579$	−16.0000	−0.664937
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	84.0000	3.47892
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	2.00000	0.0822690
$592$	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
$597$	20.0000	0.818546
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	25.0000	1.01639
$606$	0	0
$607$	4.00000	0.162355	0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000	0.162355	0.0811775	+	0.996700i	$0.474132\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16.0000	−0.647291
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	16.0000	0.641026
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.00000	0.239617
$628$	0	0
$629$	−16.0000	−0.637962
$630$	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
$633$	−4.00000	−0.158986
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	−12.0000	−0.475457
$638$	0	0
$639$	−8.00000	−0.316475
$640$	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
$643$	−10.0000	−0.394362	−0.197181	−	0.980367i	$-0.563179\pi$
$643$	−10.0000	−0.394362	−0.197181	+	0.980367i	$0.563179\pi$
$644$	0	0
$645$	2.00000	0.0787499
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−16.0000	−0.627089
$652$	0	0
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	−10.0000	−0.390732
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	50.0000	1.94477	0.972387	−	0.233373i	$-0.0749763\pi$
$661$	50.0000	1.94477	0.972387	+	0.233373i	$0.0749763\pi$
$662$	0	0
$663$	−8.00000	−0.310694
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	0	0
$668$	0	0
$669$	20.0000	0.773245
$670$	0	0
$671$	84.0000	3.24278
$672$	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
$675$	1.00000	0.0384900
$676$	0	0
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	20.0000	0.766402
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	10.0000	0.382080
$686$	0	0
$687$	10.0000	0.381524
$688$	0	0
$689$	−56.0000	−2.13343
$690$	0	0
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	0	0
$693$	12.0000	0.455842
$694$	0	0
$695$	16.0000	0.606915
$696$	0	0
$697$	0	0
$698$	0	0
$699$	26.0000	0.983410
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	−4.00000	−0.150649
$706$	0	0
$707$	−36.0000	−1.35392
$708$	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
$711$	0	0
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	−24.0000	−0.897549
$716$	0	0
$717$	−22.0000	−0.821605
$718$	0	0
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	0	0
$723$	18.0000	0.669427
$724$	0	0
$725$	0	0
$726$	0	0
$727$	50.0000	1.85440	0.927199	−	0.374570i	$-0.122210\pi$
$727$	50.0000	1.85440	0.927199	+	0.374570i	$0.122210\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.00000	−0.147945
$732$	0	0
$733$	38.0000	1.40356	0.701781	−	0.712393i	$-0.252388\pi$
$733$	38.0000	1.40356	0.701781	+	0.712393i	$0.252388\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	48.0000	1.76810
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−8.00000	−0.292314
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	10.0000	0.364420
$754$	0	0
$755$	−8.00000	−0.291150
$756$	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
$759$	24.0000	0.871145
$760$	0	0
$761$	−34.0000	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$761$	−34.0000	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$762$	0	0
$763$	28.0000	1.01367
$764$	0	0
$765$	−2.00000	−0.0723102
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	−54.0000	−1.94225	−0.971123	−	0.238581i	$-0.923318\pi$
$773$	−54.0000	−1.94225	−0.971123	+	0.238581i	$0.923318\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	−16.0000	−0.573997
$778$	0	0
$779$	0	0
$780$	0	0
$781$	48.0000	1.71758
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.00000	0.0713831
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	−56.0000	−1.98862
$794$	0	0
$795$	−14.0000	−0.496529
$796$	0	0
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	−8.00000	−0.282666
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	8.00000	0.281963
$806$	0	0
$807$	−16.0000	−0.563227
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	−36.0000	−1.26413	−0.632065	−	0.774915i	$-0.717793\pi$
$811$	−36.0000	−1.26413	−0.632065	+	0.774915i	$0.717793\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	0	0
$815$	−10.0000	−0.350285
$816$	0	0
$817$	−2.00000	−0.0699711
$818$	0	0
$819$	−8.00000	−0.279543
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	−30.0000	−1.04573	−0.522867	−	0.852414i	$-0.675138\pi$
$823$	−30.0000	−1.04573	−0.522867	+	0.852414i	$0.675138\pi$
$824$	0	0
$825$	−6.00000	−0.208893
$826$	0	0
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	0	0
$829$	50.0000	1.73657	0.868286	−	0.496064i	$-0.165222\pi$
$829$	50.0000	1.73657	0.868286	+	0.496064i	$0.165222\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	8.00000	0.276520
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	20.0000	0.688837
$844$	0	0
$845$	3.00000	0.103203
$846$	0	0
$847$	−50.0000	−1.71802
$848$	0	0
$849$	6.00000	0.205919
$850$	0	0
$851$	−32.0000	−1.09695
$852$	0	0
$853$	−2.00000	−0.0684787	−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000	−0.0684787	−0.0342393	+	0.999414i	$0.510901\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	0	0
$867$	−13.0000	−0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−32.0000	−1.08428
$872$	0	0
$873$	−4.00000	−0.135379
$874$	0	0
$875$	−2.00000	−0.0676123
$876$	0	0
$877$	36.0000	1.21563	0.607817	−	0.794077i	$-0.292045\pi$
$877$	36.0000	1.21563	0.607817	+	0.794077i	$0.292045\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	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
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	−6.00000	−0.201008
$892$	0	0
$893$	4.00000	0.133855
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−16.0000	−0.534224
$898$	0	0
$899$	0	0
$900$	0	0
$901$	28.0000	0.932815
$902$	0	0
$903$	−4.00000	−0.133112
$904$	0	0
$905$	−22.0000	−0.731305
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	18.0000	0.597022
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	0	0
$913$	72.0000	2.38285
$914$	0	0
$915$	−14.0000	−0.462826
$916$	0	0
$917$	20.0000	0.660458
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	0	0
$923$	−32.0000	−1.05329
$924$	0	0
$925$	8.00000	0.263038
$926$	0	0
$927$	−12.0000	−0.394132
$928$	0	0
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	0	0
$933$	−2.00000	−0.0654771
$934$	0	0
$935$	12.0000	0.392442
$936$	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
$939$	26.0000	0.848478
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	0	0
$951$	26.0000	0.843108
$952$	0	0
$953$	50.0000	1.61966	0.809829	−	0.586665i	$-0.199560\pi$
$953$	50.0000	1.61966	0.809829	+	0.586665i	$0.199560\pi$
$954$	0	0
$955$	−18.0000	−0.582466
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−20.0000	−0.645834
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	−16.0000	−0.515058
$966$	0	0
$967$	−6.00000	−0.192947	−0.0964735	−	0.995336i	$-0.530756\pi$
$967$	−6.00000	−0.192947	−0.0964735	+	0.995336i	$0.530756\pi$
$968$	0	0
$969$	2.00000	0.0642493
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	−32.0000	−1.02587
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−26.0000	−0.831814	−0.415907	−	0.909407i	$-0.636536\pi$
$977$	−26.0000	−0.831814	−0.415907	+	0.909407i	$0.636536\pi$
$978$	0	0
$979$	48.0000	1.53409
$980$	0	0
$981$	−14.0000	−0.446986
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	20.0000	0.634043
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	0	0
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.y.1.1		1
4.3	odd	2		2280.2.a.d.1.1	✓	1
12.11	even	2		6840.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.d.1.1	✓	1	4.3	odd	2
4560.2.a.y.1.1		1	1.1	even	1	trivial
6840.2.a.j.1.1		1	12.11	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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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