LMFDB - Embedded newform 4020.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.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:	$32.0998616126$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +2.00000 q^{13} -1.00000 q^{15} +5.00000 q^{17} +1.00000 q^{19} -2.00000 q^{21} -3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -9.00000 q^{29} -4.00000 q^{31} -2.00000 q^{33} +2.00000 q^{35} -11.0000 q^{37} +2.00000 q^{39} +6.00000 q^{41} +10.0000 q^{43} -1.00000 q^{45} -5.00000 q^{47} -3.00000 q^{49} +5.00000 q^{51} -12.0000 q^{53} +2.00000 q^{55} +1.00000 q^{57} +7.00000 q^{59} +8.00000 q^{61} -2.00000 q^{63} -2.00000 q^{65} -1.00000 q^{67} -3.00000 q^{69} +4.00000 q^{71} -5.00000 q^{73} +1.00000 q^{75} +4.00000 q^{77} +4.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} -5.00000 q^{85} -9.00000 q^{87} -9.00000 q^{89} -4.00000 q^{91} -4.00000 q^{93} -1.00000 q^{95} +2.00000 q^{97} -2.00000 q^{99} +O(q^{100})$

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	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−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
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	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
$33$	−2.00000	−0.348155
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−5.00000	−0.729325	−0.364662	−	0.931140i	$-0.618816\pi$
$47$	−5.00000	−0.729325	−0.364662	+	0.931140i	$0.618816\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	5.00000	0.700140
$52$	0	0
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	7.00000	0.911322	0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000	0.911322	0.455661	+	0.890153i	$0.349403\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	−5.00000	−0.585206	−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000	−0.585206	−0.292603	+	0.956234i	$0.594521\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	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
$85$	−5.00000	−0.542326
$86$	0	0
$87$	−9.00000	−0.964901
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	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
$107$	17.0000	1.64345	0.821726	−	0.569883i	$-0.193011\pi$
$107$	17.0000	1.64345	0.821726	+	0.569883i	$0.193011\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−11.0000	−1.04407
$112$	0	0
$113$	−20.0000	−1.88144	−0.940721	−	0.339182i	$-0.889850\pi$
$113$	−20.0000	−1.88144	−0.940721	+	0.339182i	$0.889850\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−10.0000	−0.916698
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	0	0
$129$	10.0000	0.880451
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	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
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	−5.00000	−0.421076
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	9.00000	0.747409
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	5.00000	0.404226
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	−12.0000	−0.951662
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	−5.00000	−0.391630	−0.195815	−	0.980641i	$-0.562735\pi$
$163$	−5.00000	−0.391630	−0.195815	+	0.980641i	$0.562735\pi$
$164$	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
$169$	−9.00000	−0.692308
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	25.0000	1.90071	0.950357	−	0.311160i	$-0.100718\pi$
$173$	25.0000	1.90071	0.950357	+	0.311160i	$0.100718\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	7.00000	0.526152
$178$	0	0
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	0	0
$185$	11.0000	0.808736
$186$	0	0
$187$	−10.0000	−0.731272
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	19.0000	1.36765	0.683825	−	0.729646i	$-0.260315\pi$
$193$	19.0000	1.36765	0.683825	+	0.729646i	$0.260315\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	20.0000	1.42494	0.712470	−	0.701702i	$-0.247576\pi$
$197$	20.0000	1.42494	0.712470	+	0.701702i	$0.247576\pi$
$198$	0	0
$199$	−15.0000	−1.06332	−0.531661	−	0.846957i	$-0.678432\pi$
$199$	−15.0000	−1.06332	−0.531661	+	0.846957i	$0.678432\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	18.0000	1.26335
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	−3.00000	−0.208514
$208$	0	0
$209$	−2.00000	−0.138343
$210$	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
$213$	4.00000	0.274075
$214$	0	0
$215$	−10.0000	−0.681994
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	−5.00000	−0.337869
$220$	0	0
$221$	10.0000	0.672673
$222$	0	0
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	11.0000	0.730096	0.365048	−	0.930989i	$-0.381053\pi$
$227$	11.0000	0.730096	0.365048	+	0.930989i	$0.381053\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	5.00000	0.326164
$236$	0	0
$237$	4.00000	0.259828
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	−4.00000	−0.253490
$250$	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
$253$	6.00000	0.377217
$254$	0	0
$255$	−5.00000	−0.313112
$256$	0	0
$257$	−23.0000	−1.43470	−0.717350	−	0.696713i	$-0.754645\pi$
$257$	−23.0000	−1.43470	−0.717350	+	0.696713i	$0.754645\pi$
$258$	0	0
$259$	22.0000	1.36701
$260$	0	0
$261$	−9.00000	−0.557086
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	−9.00000	−0.550791
$268$	0	0
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	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
$273$	−4.00000	−0.242091
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−19.0000	−1.12943	−0.564716	−	0.825285i	$-0.691014\pi$
$283$	−19.0000	−1.12943	−0.564716	+	0.825285i	$0.691014\pi$
$284$	0	0
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	2.00000	0.117242
$292$	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
$295$	−7.00000	−0.407556
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	−6.00000	−0.346989
$300$	0	0
$301$	−20.0000	−1.15278
$302$	0	0
$303$	−8.00000	−0.459588
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	0	0
$321$	17.0000	0.948847
$322$	0	0
$323$	5.00000	0.278207
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	−10.0000	−0.553001
$328$	0	0
$329$	10.0000	0.551318
$330$	0	0
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	0	0
$333$	−11.0000	−0.602796
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	−20.0000	−1.08625
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	0	0
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	0	0
$357$	−10.0000	−0.529256
$358$	0	0
$359$	−19.0000	−1.00278	−0.501391	−	0.865221i	$-0.667178\pi$
$359$	−19.0000	−1.00278	−0.501391	+	0.865221i	$0.667178\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	5.00000	0.261712
$366$	0	0
$367$	34.0000	1.77479	0.887393	−	0.461014i	$-0.152514\pi$
$367$	34.0000	1.77479	0.887393	+	0.461014i	$0.152514\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	24.0000	1.24602
$372$	0	0
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−18.0000	−0.927047
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	0	0
$381$	−13.0000	−0.666010
$382$	0	0
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	10.0000	0.508329
$388$	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
$391$	−15.0000	−0.758583
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	15.0000	0.752828	0.376414	−	0.926451i	$-0.377157\pi$
$397$	15.0000	0.752828	0.376414	+	0.926451i	$0.377157\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	22.0000	1.09050
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	−22.0000	−1.08518
$412$	0	0
$413$	−14.0000	−0.688895
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	5.00000	0.244266	0.122133	−	0.992514i	$-0.461027\pi$
$419$	5.00000	0.244266	0.122133	+	0.992514i	$0.461027\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	0	0
$423$	−5.00000	−0.243108
$424$	0	0
$425$	5.00000	0.242536
$426$	0	0
$427$	−16.0000	−0.774294
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	0	0
$433$	36.0000	1.73005	0.865025	−	0.501729i	$-0.167303\pi$
$433$	36.0000	1.73005	0.865025	+	0.501729i	$0.167303\pi$
$434$	0	0
$435$	9.00000	0.431517
$436$	0	0
$437$	−3.00000	−0.143509
$438$	0	0
$439$	21.0000	1.00228	0.501138	−	0.865368i	$-0.332915\pi$
$439$	21.0000	1.00228	0.501138	+	0.865368i	$0.332915\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	0	0
$445$	9.00000	0.426641
$446$	0	0
$447$	−15.0000	−0.709476
$448$	0	0
$449$	−5.00000	−0.235965	−0.117982	−	0.993016i	$-0.537643\pi$
$449$	−5.00000	−0.235965	−0.117982	+	0.993016i	$0.537643\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	0	0
$453$	5.00000	0.234920
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	0	0
$459$	5.00000	0.233380
$460$	0	0
$461$	9.00000	0.419172	0.209586	−	0.977790i	$-0.432788\pi$
$461$	9.00000	0.419172	0.209586	+	0.977790i	$0.432788\pi$
$462$	0	0
$463$	−18.0000	−0.836531	−0.418265	−	0.908325i	$-0.637362\pi$
$463$	−18.0000	−0.836531	−0.418265	+	0.908325i	$0.637362\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	0	0
$471$	−13.0000	−0.599008
$472$	0	0
$473$	−20.0000	−0.919601
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−12.0000	−0.549442
$478$	0	0
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	0	0
$481$	−22.0000	−1.00311
$482$	0	0
$483$	6.00000	0.273009
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	0	0
$489$	−5.00000	−0.226108
$490$	0	0
$491$	−31.0000	−1.39901	−0.699505	−	0.714628i	$-0.746596\pi$
$491$	−31.0000	−1.39901	−0.699505	+	0.714628i	$0.746596\pi$
$492$	0	0
$493$	−45.0000	−2.02670
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	29.0000	1.28540	0.642701	−	0.766117i	$-0.277814\pi$
$509$	29.0000	1.28540	0.642701	+	0.766117i	$0.277814\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	4.00000	0.176261
$516$	0	0
$517$	10.0000	0.439799
$518$	0	0
$519$	25.0000	1.09738
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	−7.00000	−0.306089	−0.153044	−	0.988219i	$-0.548908\pi$
$523$	−7.00000	−0.306089	−0.153044	+	0.988219i	$0.548908\pi$
$524$	0	0
$525$	−2.00000	−0.0872872
$526$	0	0
$527$	−20.0000	−0.871214
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	7.00000	0.303774
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	−17.0000	−0.734974
$536$	0	0
$537$	−24.0000	−1.03568
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	−5.00000	−0.214571
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	8.00000	0.341432
$550$	0	0
$551$	−9.00000	−0.383413
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	11.0000	0.466924
$556$	0	0
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	0	0
$559$	20.0000	0.845910
$560$	0	0
$561$	−10.0000	−0.422200
$562$	0	0
$563$	−10.0000	−0.421450	−0.210725	−	0.977545i	$-0.567582\pi$
$563$	−10.0000	−0.421450	−0.210725	+	0.977545i	$0.567582\pi$
$564$	0	0
$565$	20.0000	0.841406
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	−3.00000	−0.125767	−0.0628833	−	0.998021i	$-0.520030\pi$
$569$	−3.00000	−0.125767	−0.0628833	+	0.998021i	$0.520030\pi$
$570$	0	0
$571$	−37.0000	−1.54840	−0.774201	−	0.632940i	$-0.781848\pi$
$571$	−37.0000	−1.54840	−0.774201	+	0.632940i	$0.781848\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	0	0
$575$	−3.00000	−0.125109
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	19.0000	0.789613
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	24.0000	0.993978
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	20.0000	0.822690
$592$	0	0
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	0	0
$595$	10.0000	0.409960
$596$	0	0
$597$	−15.0000	−0.613909
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	0	0
$609$	18.0000	0.729397
$610$	0	0
$611$	−10.0000	−0.404557
$612$	0	0
$613$	29.0000	1.17130	0.585649	−	0.810564i	$-0.300840\pi$
$613$	29.0000	1.17130	0.585649	+	0.810564i	$0.300840\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−15.0000	−0.603877	−0.301939	−	0.953327i	$-0.597634\pi$
$617$	−15.0000	−0.603877	−0.301939	+	0.953327i	$0.597634\pi$
$618$	0	0
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	0	0
$621$	−3.00000	−0.120386
$622$	0	0
$623$	18.0000	0.721155
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.00000	−0.0798723
$628$	0	0
$629$	−55.0000	−2.19299
$630$	0	0
$631$	18.0000	0.716569	0.358284	−	0.933613i	$-0.383362\pi$
$631$	18.0000	0.716569	0.358284	+	0.933613i	$0.383362\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	13.0000	0.515889
$636$	0	0
$637$	−6.00000	−0.237729
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	0	0
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	0	0
$645$	−10.0000	−0.393750
$646$	0	0
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	0	0
$649$	−14.0000	−0.549548
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	−5.00000	−0.195069
$658$	0	0
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	0	0
$663$	10.0000	0.388368
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	27.0000	1.04544
$668$	0	0
$669$	19.0000	0.734582
$670$	0	0
$671$	−16.0000	−0.617673
$672$	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
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	11.0000	0.421521
$682$	0	0
$683$	−30.0000	−1.14792	−0.573959	−	0.818884i	$-0.694593\pi$
$683$	−30.0000	−1.14792	−0.573959	+	0.818884i	$0.694593\pi$
$684$	0	0
$685$	22.0000	0.840577
$686$	0	0
$687$	−6.00000	−0.228914
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−43.0000	−1.63580	−0.817899	−	0.575362i	$-0.804861\pi$
$691$	−43.0000	−1.63580	−0.817899	+	0.575362i	$0.804861\pi$
$692$	0	0
$693$	4.00000	0.151947
$694$	0	0
$695$	−12.0000	−0.455186
$696$	0	0
$697$	30.0000	1.13633
$698$	0	0
$699$	−10.0000	−0.378235
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	−11.0000	−0.414873
$704$	0	0
$705$	5.00000	0.188311
$706$	0	0
$707$	16.0000	0.601742
$708$	0	0
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	12.0000	0.449404
$714$	0	0
$715$	4.00000	0.149592
$716$	0	0
$717$	−12.0000	−0.448148
$718$	0	0
$719$	−53.0000	−1.97657	−0.988283	−	0.152631i	$-0.951225\pi$
$719$	−53.0000	−1.97657	−0.988283	+	0.152631i	$0.951225\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	17.0000	0.632237
$724$	0	0
$725$	−9.00000	−0.334252
$726$	0	0
$727$	−46.0000	−1.70605	−0.853023	−	0.521874i	$-0.825233\pi$
$727$	−46.0000	−1.70605	−0.853023	+	0.521874i	$0.825233\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	50.0000	1.84932
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	2.00000	0.0736709
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	0	0
$745$	15.0000	0.549557
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	−34.0000	−1.24233
$750$	0	0
$751$	41.0000	1.49611	0.748056	−	0.663636i	$-0.230988\pi$
$751$	41.0000	1.49611	0.748056	+	0.663636i	$0.230988\pi$
$752$	0	0
$753$	−20.0000	−0.728841
$754$	0	0
$755$	−5.00000	−0.181969
$756$	0	0
$757$	−44.0000	−1.59921	−0.799604	−	0.600528i	$-0.794957\pi$
$757$	−44.0000	−1.59921	−0.799604	+	0.600528i	$0.794957\pi$
$758$	0	0
$759$	6.00000	0.217786
$760$	0	0
$761$	11.0000	0.398750	0.199375	−	0.979923i	$-0.436109\pi$
$761$	11.0000	0.398750	0.199375	+	0.979923i	$0.436109\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	0	0
$765$	−5.00000	−0.180775
$766$	0	0
$767$	14.0000	0.505511
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	−23.0000	−0.828325
$772$	0	0
$773$	39.0000	1.40273	0.701366	−	0.712801i	$-0.252574\pi$
$773$	39.0000	1.40273	0.701366	+	0.712801i	$0.252574\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	22.0000	0.789246
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	−9.00000	−0.321634
$784$	0	0
$785$	13.0000	0.463990
$786$	0	0
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	0	0
$789$	12.0000	0.427211
$790$	0	0
$791$	40.0000	1.42224
$792$	0	0
$793$	16.0000	0.568177
$794$	0	0
$795$	12.0000	0.425596
$796$	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
$799$	−25.0000	−0.884436
$800$	0	0
$801$	−9.00000	−0.317999
$802$	0	0
$803$	10.0000	0.352892
$804$	0	0
$805$	−6.00000	−0.211472
$806$	0	0
$807$	−2.00000	−0.0704033
$808$	0	0
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	0	0
$811$	14.0000	0.491606	0.245803	−	0.969320i	$-0.420948\pi$
$811$	14.0000	0.491606	0.245803	+	0.969320i	$0.420948\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	0	0
$815$	5.00000	0.175142
$816$	0	0
$817$	10.0000	0.349856
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	15.0000	0.523504	0.261752	−	0.965135i	$-0.415700\pi$
$821$	15.0000	0.523504	0.261752	+	0.965135i	$0.415700\pi$
$822$	0	0
$823$	39.0000	1.35945	0.679727	−	0.733465i	$-0.262098\pi$
$823$	39.0000	1.35945	0.679727	+	0.733465i	$0.262098\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	−31.0000	−1.07798	−0.538988	−	0.842314i	$-0.681193\pi$
$827$	−31.0000	−1.07798	−0.538988	+	0.842314i	$0.681193\pi$
$828$	0	0
$829$	−19.0000	−0.659897	−0.329949	−	0.943999i	$-0.607031\pi$
$829$	−19.0000	−0.659897	−0.329949	+	0.943999i	$0.607031\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	0	0
$833$	−15.0000	−0.519719
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	51.0000	1.76072	0.880358	−	0.474310i	$-0.157302\pi$
$839$	51.0000	1.76072	0.880358	+	0.474310i	$0.157302\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	14.0000	0.481046
$848$	0	0
$849$	−19.0000	−0.652078
$850$	0	0
$851$	33.0000	1.13123
$852$	0	0
$853$	−9.00000	−0.308154	−0.154077	−	0.988059i	$-0.549240\pi$
$853$	−9.00000	−0.308154	−0.154077	+	0.988059i	$0.549240\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	26.0000	0.888143	0.444072	−	0.895991i	$-0.353534\pi$
$857$	26.0000	0.888143	0.444072	+	0.895991i	$0.353534\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	0	0
$863$	37.0000	1.25949	0.629747	−	0.776800i	$-0.283158\pi$
$863$	37.0000	1.25949	0.629747	+	0.776800i	$0.283158\pi$
$864$	0	0
$865$	−25.0000	−0.850026
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	−13.0000	−0.438979	−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000	−0.438979	−0.219489	+	0.975615i	$0.570439\pi$
$878$	0	0
$879$	14.0000	0.472208
$880$	0	0
$881$	−17.0000	−0.572745	−0.286372	−	0.958118i	$-0.592449\pi$
$881$	−17.0000	−0.572745	−0.286372	+	0.958118i	$0.592449\pi$
$882$	0	0
$883$	10.0000	0.336527	0.168263	−	0.985742i	$-0.446184\pi$
$883$	10.0000	0.336527	0.168263	+	0.985742i	$0.446184\pi$
$884$	0	0
$885$	−7.00000	−0.235302
$886$	0	0
$887$	−27.0000	−0.906571	−0.453286	−	0.891365i	$-0.649748\pi$
$887$	−27.0000	−0.906571	−0.453286	+	0.891365i	$0.649748\pi$
$888$	0	0
$889$	26.0000	0.872012
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	−5.00000	−0.167319
$894$	0	0
$895$	24.0000	0.802232
$896$	0	0
$897$	−6.00000	−0.200334
$898$	0	0
$899$	36.0000	1.20067
$900$	0	0
$901$	−60.0000	−1.99889
$902$	0	0
$903$	−20.0000	−0.665558
$904$	0	0
$905$	5.00000	0.166206
$906$	0	0
$907$	−39.0000	−1.29497	−0.647487	−	0.762077i	$-0.724180\pi$
$907$	−39.0000	−1.29497	−0.647487	+	0.762077i	$0.724180\pi$
$908$	0	0
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−57.0000	−1.88849	−0.944247	−	0.329238i	$-0.893208\pi$
$911$	−57.0000	−1.88849	−0.944247	+	0.329238i	$0.893208\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	0	0
$915$	−8.00000	−0.264472
$916$	0	0
$917$	24.0000	0.792550
$918$	0	0
$919$	52.0000	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	0	0
$923$	8.00000	0.263323
$924$	0	0
$925$	−11.0000	−0.361678
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	0	0
$933$	−10.0000	−0.327385
$934$	0	0
$935$	10.0000	0.327035
$936$	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
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	0	0
$943$	−18.0000	−0.586161
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	21.0000	0.682408	0.341204	−	0.939989i	$-0.389165\pi$
$947$	21.0000	0.682408	0.341204	+	0.939989i	$0.389165\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	10.0000	0.324272
$952$	0	0
$953$	35.0000	1.13376	0.566881	−	0.823800i	$-0.308150\pi$
$953$	35.0000	1.13376	0.566881	+	0.823800i	$0.308150\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	0	0
$957$	18.0000	0.581857
$958$	0	0
$959$	44.0000	1.42083
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	17.0000	0.547817
$964$	0	0
$965$	−19.0000	−0.611632
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	0	0
$969$	5.00000	0.160623
$970$	0	0
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	0	0
$973$	−24.0000	−0.769405
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	41.0000	1.31171	0.655853	−	0.754889i	$-0.272309\pi$
$977$	41.0000	1.31171	0.655853	+	0.754889i	$0.272309\pi$
$978$	0	0
$979$	18.0000	0.575282
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	0	0
$985$	−20.0000	−0.637253
$986$	0	0
$987$	10.0000	0.318304
$988$	0	0
$989$	−30.0000	−0.953945
$990$	0	0
$991$	30.0000	0.952981	0.476491	−	0.879180i	$-0.341909\pi$
$991$	30.0000	0.952981	0.476491	+	0.879180i	$0.341909\pi$
$992$	0	0
$993$	14.0000	0.444277
$994$	0	0
$995$	15.0000	0.475532
$996$	0	0
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	0	0
$999$	−11.0000	−0.348025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.a.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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