LMFDB - Embedded newform 4560.2.a.o.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 570)
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)

$f(q)$

$=$

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

$q-1.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +4.00000 q^{13} -1.00000 q^{15} -2.00000 q^{17} +1.00000 q^{19} -2.00000 q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.00000 q^{31} +2.00000 q^{33} +2.00000 q^{35} +8.00000 q^{37} -4.00000 q^{39} -8.00000 q^{41} +6.00000 q^{43} +1.00000 q^{45} +12.0000 q^{47} -3.00000 q^{49} +2.00000 q^{51} -6.00000 q^{53} -2.00000 q^{55} -1.00000 q^{57} +2.00000 q^{61} +2.00000 q^{63} +4.00000 q^{65} -8.00000 q^{67} +4.00000 q^{69} +8.00000 q^{71} +14.0000 q^{73} -1.00000 q^{75} -4.00000 q^{77} +1.00000 q^{81} -4.00000 q^{83} -2.00000 q^{85} +8.00000 q^{91} -8.00000 q^{93} +1.00000 q^{95} -12.0000 q^{97} -2.00000 q^{99} +O(q^{100})$

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	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	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
$33$	2.00000	0.348155
$34$	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
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	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
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	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	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−4.00000	−0.455842
$78$	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
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	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
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	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
$121$	−7.00000	−0.636364
$122$	0	0
$123$	8.00000	0.721336
$124$	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
$129$	−6.00000	−0.528271
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	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
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	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
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	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
$205$	−8.00000	−0.558744
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	−14.0000	−0.946032
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	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
$231$	4.00000	0.263181
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.0000	−0.646846	−0.323423	−	0.946254i	$-0.604834\pi$
$239$	−10.0000	−0.646846	−0.323423	+	0.946254i	$0.604834\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	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
$249$	4.00000	0.253490
$250$	0	0
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	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
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	−2.00000	−0.120605
$276$	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
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	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
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	−16.0000	−0.944450
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	12.0000	0.703452
$292$	0	0
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	−16.0000	−0.925304
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	−2.00000	−0.114897
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	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
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−2.00000	−0.111283
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−10.0000	−0.553001
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	0	0
$333$	8.00000	0.438397
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	28.0000	1.52526	0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000	1.52526	0.762629	+	0.646837i	$0.223908\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	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
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	14.0000	0.732793
$366$	0	0
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	0	0
$369$	−8.00000	−0.416463
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	−36.0000	−1.86401	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	−36.0000	−1.86401	−0.932005	+	0.362446i	$0.881942\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	6.00000	0.304997
$388$	0	0
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	−18.0000	−0.907980
$394$	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
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	32.0000	1.59403
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−16.0000	−0.793091
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	22.0000	1.08518
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	12.0000	0.583460
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	4.00000	0.193574
$428$	0	0
$429$	8.00000	0.386244
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.00000	−0.191346
$438$	0	0
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−10.0000	−0.472984
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	0	0
$453$	−8.00000	−0.375873
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	6.00000	0.278844	0.139422	−	0.990233i	$-0.455476\pi$
$463$	6.00000	0.278844	0.139422	+	0.990233i	$0.455476\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	−12.0000	−0.551761
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	32.0000	1.45907
$482$	0	0
$483$	8.00000	0.364013
$484$	0	0
$485$	−12.0000	−0.544892
$486$	0	0
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	0	0
$489$	14.0000	0.633102
$490$	0	0
$491$	−22.0000	−0.992846	−0.496423	−	0.868081i	$-0.665354\pi$
$491$	−22.0000	−0.992846	−0.496423	+	0.868081i	$0.665354\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	16.0000	0.717698
$498$	0	0
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	−3.00000	−0.133235
$508$	0	0
$509$	20.0000	0.886484	0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000	0.886484	0.443242	+	0.896402i	$0.353828\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−4.00000	−0.176261
$516$	0	0
$517$	−24.0000	−1.05552
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	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
$533$	−32.0000	−1.38607
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.00000	0.258438
$540$	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
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−8.00000	−0.339581
$556$	0	0
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	−18.0000	−0.751961
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	0	0
$579$	−24.0000	−0.997406
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	−8.00000	−0.330195	−0.165098	−	0.986277i	$-0.552794\pi$
$587$	−8.00000	−0.330195	−0.165098	+	0.986277i	$0.552794\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	−46.0000	−1.88899	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−46.0000	−1.88899	−0.944497	+	0.328521i	$0.893450\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	0	0
$597$	−20.0000	−0.818546
$598$	0	0
$599$	−20.0000	−0.817178	−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000	−0.817178	−0.408589	+	0.912719i	$0.633979\pi$
$600$	0	0
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	−7.00000	−0.284590
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	48.0000	1.94187
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	0	0
$615$	8.00000	0.322591
$616$	0	0
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	0	0
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	−16.0000	−0.637962
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	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
$641$	32.0000	1.26392	0.631962	−	0.774999i	$-0.282250\pi$
$641$	32.0000	1.26392	0.631962	+	0.774999i	$0.282250\pi$
$642$	0	0
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	0	0
$645$	−6.00000	−0.236250
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−16.0000	−0.627089
$652$	0	0
$653$	−46.0000	−1.80012	−0.900060	−	0.435767i	$-0.856477\pi$
$653$	−46.0000	−1.80012	−0.900060	+	0.435767i	$0.856477\pi$
$654$	0	0
$655$	18.0000	0.703318
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	0	0
$668$	0	0
$669$	4.00000	0.154649
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	4.00000	0.154189	0.0770943	−	0.997024i	$-0.475436\pi$
$673$	4.00000	0.154189	0.0770943	+	0.997024i	$0.475436\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	−24.0000	−0.921035
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	−22.0000	−0.840577
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16.0000	0.606043
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	−12.0000	−0.451946
$706$	0	0
$707$	4.00000	0.150435
$708$	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
$715$	−8.00000	−0.299183
$716$	0	0
$717$	10.0000	0.373457
$718$	0	0
$719$	−50.0000	−1.86469	−0.932343	−	0.361576i	$-0.882239\pi$
$719$	−50.0000	−1.86469	−0.932343	+	0.361576i	$0.882239\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−18.0000	−0.667583	−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000	−0.667583	−0.333792	+	0.942647i	$0.608328\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−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
$745$	10.0000	0.366372
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	24.0000	0.876941
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	2.00000	0.0728841
$754$	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
$759$	−8.00000	−0.290382
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	0	0
$765$	−2.00000	−0.0723102
$766$	0	0
$767$	0	0
$768$	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
$771$	−18.0000	−0.648254
$772$	0	0
$773$	34.0000	1.22290	0.611448	−	0.791285i	$-0.290588\pi$
$773$	34.0000	1.22290	0.611448	+	0.791285i	$0.290588\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	−16.0000	−0.573997
$778$	0	0
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0	0
$783$	0	0
$784$	0	0
$785$	18.0000	0.642448
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	0	0
$789$	24.0000	0.854423
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	6.00000	0.212798
$796$	0	0
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−28.0000	−0.988099
$804$	0	0
$805$	−8.00000	−0.281963
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	0	0
$813$	12.0000	0.420858
$814$	0	0
$815$	−14.0000	−0.490399
$816$	0	0
$817$	6.00000	0.209913
$818$	0	0
$819$	8.00000	0.279543
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	0	0
$823$	−34.0000	−1.18517	−0.592583	−	0.805510i	$-0.701892\pi$
$823$	−34.0000	−1.18517	−0.592583	+	0.805510i	$0.701892\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	0	0
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	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	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	−12.0000	−0.413302
$844$	0	0
$845$	3.00000	0.103203
$846$	0	0
$847$	−14.0000	−0.481046
$848$	0	0
$849$	14.0000	0.480479
$850$	0	0
$851$	−32.0000	−1.09695
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	16.0000	0.545279
$862$	0	0
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−32.0000	−1.08428
$872$	0	0
$873$	−12.0000	−0.406138
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	−12.0000	−0.405211	−0.202606	−	0.979260i	$-0.564941\pi$
$877$	−12.0000	−0.405211	−0.202606	+	0.979260i	$0.564941\pi$
$878$	0	0
$879$	−14.0000	−0.472208
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	12.0000	0.401565
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16.0000	0.534224
$898$	0	0
$899$	0	0
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	−12.0000	−0.399335
$904$	0	0
$905$	2.00000	0.0664822
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	2.00000	0.0663358
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	0	0
$915$	−2.00000	−0.0661180
$916$	0	0
$917$	36.0000	1.18882
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−32.0000	−1.05444
$922$	0	0
$923$	32.0000	1.05329
$924$	0	0
$925$	8.00000	0.263038
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	0	0
$933$	−18.0000	−0.589294
$934$	0	0
$935$	4.00000	0.130814
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	0	0
$941$	−8.00000	−0.260793	−0.130396	−	0.991462i	$-0.541625\pi$
$941$	−8.00000	−0.260793	−0.130396	+	0.991462i	$0.541625\pi$
$942$	0	0
$943$	32.0000	1.04206
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	0	0
$949$	56.0000	1.81784
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	0	0
$955$	18.0000	0.582466
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−44.0000	−1.42083
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	24.0000	0.772587
$966$	0	0
$967$	−58.0000	−1.86515	−0.932577	−	0.360971i	$-0.882445\pi$
$967$	−58.0000	−1.86515	−0.932577	+	0.360971i	$0.882445\pi$
$968$	0	0
$969$	2.00000	0.0642493
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	−24.0000	−0.763928
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	12.0000	0.380808
$994$	0	0
$995$	20.0000	0.634043
$996$	0	0
$997$	−62.0000	−1.96356	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−62.0000	−1.96356	−0.981780	+	0.190022i	$0.939144\pi$
$998$	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.o.1.1		1
4.3	odd	2		570.2.a.l.1.1	✓	1
12.11	even	2		1710.2.a.b.1.1		1
20.3	even	4		2850.2.d.q.799.1		2
20.7	even	4		2850.2.d.q.799.2		2
20.19	odd	2		2850.2.a.e.1.1		1
60.59	even	2		8550.2.a.bf.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.l.1.1	✓	1	4.3	odd	2
1710.2.a.b.1.1		1	12.11	even	2
2850.2.a.e.1.1		1	20.19	odd	2
2850.2.d.q.799.1		2	20.3	even	4
2850.2.d.q.799.2		2	20.7	even	4
4560.2.a.o.1.1		1	1.1	even	1	trivial
8550.2.a.bf.1.1		1	60.59	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

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