LMFDB - Embedded newform 1920.2.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1920,2,Mod(1,1920)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1920.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-1,0,-1,0,-2,0,1,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$15.3312771881$
Analytic rank:	$0$
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	$\chi$	$=$	1920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} -6.00000 q^{11} -2.00000 q^{13} +1.00000 q^{15} -2.00000 q^{17} -2.00000 q^{19} +2.00000 q^{21} +6.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +10.0000 q^{29} -4.00000 q^{31} +6.00000 q^{33} +2.00000 q^{35} +2.00000 q^{37} +2.00000 q^{39} +6.00000 q^{41} -1.00000 q^{45} +6.00000 q^{47} -3.00000 q^{49} +2.00000 q^{51} -10.0000 q^{53} +6.00000 q^{55} +2.00000 q^{57} +6.00000 q^{59} +6.00000 q^{61} -2.00000 q^{63} +2.00000 q^{65} -16.0000 q^{67} -6.00000 q^{69} +8.00000 q^{71} +10.0000 q^{73} -1.00000 q^{75} +12.0000 q^{77} +1.00000 q^{81} -12.0000 q^{83} +2.00000 q^{85} -10.0000 q^{87} +6.00000 q^{89} +4.00000 q^{91} +4.00000 q^{93} +2.00000 q^{95} +10.0000 q^{97} -6.00000 q^{99} +O(q^{100})$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−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
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	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
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	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
$33$	6.00000	1.04447
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	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
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	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
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	−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
$87$	−10.0000	−1.07211
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	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
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	22.0000	1.92215	0.961074	−	0.276289i	$-0.0891049\pi$
$131$	22.0000	1.92215	0.961074	+	0.276289i	$0.0891049\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	12.0000	1.00349
$144$	0	0
$145$	−10.0000	−0.830455
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	4.00000	0.321288
$156$	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
$159$	10.0000	0.793052
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	0	0
$165$	−6.00000	−0.467099
$166$	0	0
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	12.0000	0.877527
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	−20.0000	−1.40372
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	−10.0000	−0.675737
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	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
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	−36.0000	−2.26330
$254$	0	0
$255$	−2.00000	−0.125245
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	10.0000	0.618984
$262$	0	0
$263$	14.0000	0.863277	0.431638	−	0.902047i	$-0.357936\pi$
$263$	14.0000	0.863277	0.431638	+	0.902047i	$0.357936\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	−6.00000	−0.361814
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	6.00000	0.348155
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−10.0000	−0.574485
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	0	0
$309$	−10.0000	−0.568880
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	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
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	0	0
$319$	−60.0000	−3.35936
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	−6.00000	−0.331801
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	30.0000	1.64895	0.824475	−	0.565899i	$-0.191471\pi$
$331$	30.0000	1.64895	0.824475	+	0.565899i	$0.191471\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	16.0000	0.874173
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	6.00000	0.323029
$346$	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
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−25.0000	−1.31216
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	20.0000	1.03835
$372$	0	0
$373$	−38.0000	−1.96757	−0.983783	−	0.179364i	$-0.942596\pi$
$373$	−38.0000	−1.96757	−0.983783	+	0.179364i	$0.942596\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−20.0000	−1.03005
$378$	0	0
$379$	18.0000	0.924598	0.462299	−	0.886724i	$-0.347025\pi$
$379$	18.0000	0.924598	0.462299	+	0.886724i	$0.347025\pi$
$380$	0	0
$381$	−2.00000	−0.102463
$382$	0	0
$383$	34.0000	1.73732	0.868659	−	0.495410i	$-0.164982\pi$
$383$	34.0000	1.73732	0.868659	+	0.495410i	$0.164982\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	−22.0000	−1.10975
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	−34.0000	−1.69788	−0.848939	−	0.528490i	$-0.822758\pi$
$401$	−34.0000	−1.69788	−0.848939	+	0.528490i	$0.822758\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	−12.0000	−0.590481
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	14.0000	0.685583
$418$	0	0
$419$	10.0000	0.488532	0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000	0.488532	0.244266	+	0.969708i	$0.421453\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	−12.0000	−0.580721
$428$	0	0
$429$	−12.0000	−0.579365
$430$	0	0
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	10.0000	0.479463
$436$	0	0
$437$	−12.0000	−0.574038
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	−18.0000	−0.851371
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	−36.0000	−1.69517
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	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
$465$	−4.00000	−0.185496
$466$	0	0
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	12.0000	0.546019
$484$	0	0
$485$	−10.0000	−0.454077
$486$	0	0
$487$	−26.0000	−1.17817	−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000	−1.17817	−0.589086	+	0.808070i	$0.700512\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	−14.0000	−0.631811	−0.315906	−	0.948791i	$-0.602308\pi$
$491$	−14.0000	−0.631811	−0.315906	+	0.948791i	$0.602308\pi$
$492$	0	0
$493$	−20.0000	−0.900755
$494$	0	0
$495$	6.00000	0.269680
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	0	0
$503$	−34.0000	−1.51599	−0.757993	−	0.652263i	$-0.773820\pi$
$503$	−34.0000	−1.51599	−0.757993	+	0.652263i	$0.773820\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	−10.0000	−0.440653
$516$	0	0
$517$	−36.0000	−1.58328
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	34.0000	1.48957	0.744784	−	0.667306i	$-0.232553\pi$
$521$	34.0000	1.48957	0.744784	+	0.667306i	$0.232553\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	−18.0000	−0.776757
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	0	0
$543$	18.0000	0.772454
$544$	0	0
$545$	−6.00000	−0.257012
$546$	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
$549$	6.00000	0.256074
$550$	0	0
$551$	−20.0000	−0.852029
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.00000	0.0848953
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−12.0000	−0.506640
$562$	0	0
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	46.0000	1.92504	0.962520	−	0.271211i	$-0.0874240\pi$
$571$	46.0000	1.92504	0.962520	+	0.271211i	$0.0874240\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	22.0000	0.914289
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	60.0000	2.48495
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	−10.0000	−0.410651	−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000	−0.410651	−0.205325	+	0.978694i	$0.565825\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	0	0
$597$	4.00000	0.163709
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	−16.0000	−0.651570
$604$	0	0
$605$	−25.0000	−1.01639
$606$	0	0
$607$	26.0000	1.05531	0.527654	−	0.849460i	$-0.323072\pi$
$607$	26.0000	1.05531	0.527654	+	0.849460i	$0.323072\pi$
$608$	0	0
$609$	20.0000	0.810441
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	−6.00000	−0.241160	−0.120580	−	0.992704i	$-0.538475\pi$
$619$	−6.00000	−0.241160	−0.120580	+	0.992704i	$0.538475\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−12.0000	−0.479234
$628$	0	0
$629$	−4.00000	−0.159490
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	−10.0000	−0.397464
$634$	0	0
$635$	−2.00000	−0.0793676
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	22.0000	0.860927	0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000	0.860927	0.430463	+	0.902608i	$0.358350\pi$
$654$	0	0
$655$	−22.0000	−0.859611
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	26.0000	1.01282	0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000	1.01282	0.506408	+	0.862294i	$0.330973\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	−4.00000	−0.155113
$666$	0	0
$667$	60.0000	2.32321
$668$	0	0
$669$	2.00000	0.0773245
$670$	0	0
$671$	−36.0000	−1.38976
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	−8.00000	−0.306561
$682$	0	0
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	26.0000	0.991962
$688$	0	0
$689$	20.0000	0.761939
$690$	0	0
$691$	26.0000	0.989087	0.494543	−	0.869153i	$-0.335335\pi$
$691$	26.0000	0.989087	0.494543	+	0.869153i	$0.335335\pi$
$692$	0	0
$693$	12.0000	0.455842
$694$	0	0
$695$	14.0000	0.531050
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	26.0000	0.983410
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	−4.00000	−0.150863
$704$	0	0
$705$	6.00000	0.225973
$706$	0	0
$707$	−20.0000	−0.752177
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	−12.0000	−0.448775
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	−20.0000	−0.744839
$722$	0	0
$723$	−18.0000	−0.669427
$724$	0	0
$725$	10.0000	0.371391
$726$	0	0
$727$	38.0000	1.40934	0.704671	−	0.709534i	$-0.251095\pi$
$727$	38.0000	1.40934	0.704671	+	0.709534i	$0.251095\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	96.0000	3.53621
$738$	0	0
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	0	0
$743$	−50.0000	−1.83432	−0.917161	−	0.398517i	$-0.869525\pi$
$743$	−50.0000	−1.83432	−0.917161	+	0.398517i	$0.869525\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−8.00000	−0.292314
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	0	0
$753$	−18.0000	−0.655956
$754$	0	0
$755$	0	0
$756$	0	0
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	0	0
$759$	36.0000	1.30672
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	0	0
$763$	−12.0000	−0.434429
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	0	0
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−48.0000	−1.71758
$782$	0	0
$783$	−10.0000	−0.357371
$784$	0	0
$785$	−18.0000	−0.642448
$786$	0	0
$787$	−48.0000	−1.71102	−0.855508	−	0.517790i	$-0.826755\pi$
$787$	−48.0000	−1.71102	−0.855508	+	0.517790i	$0.826755\pi$
$788$	0	0
$789$	−14.0000	−0.498413
$790$	0	0
$791$	36.0000	1.28001
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	−10.0000	−0.354663
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	−60.0000	−2.11735
$804$	0	0
$805$	12.0000	0.422944
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	−26.0000	−0.912983	−0.456492	−	0.889728i	$-0.650894\pi$
$811$	−26.0000	−0.912983	−0.456492	+	0.889728i	$0.650894\pi$
$812$	0	0
$813$	4.00000	0.140286
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	0	0
$818$	0	0
$819$	4.00000	0.139771
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	2.00000	0.0697156	0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000	0.0697156	0.0348578	+	0.999392i	$0.488902\pi$
$824$	0	0
$825$	6.00000	0.208893
$826$	0	0
$827$	−4.00000	−0.139094	−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000	−0.139094	−0.0695468	+	0.997579i	$0.522155\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	6.00000	0.207639
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	−18.0000	−0.619953
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	−50.0000	−1.71802
$848$	0	0
$849$	−8.00000	−0.274559
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	0	0
$859$	2.00000	0.0682391	0.0341196	−	0.999418i	$-0.489137\pi$
$859$	2.00000	0.0682391	0.0341196	+	0.999418i	$0.489137\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	0	0
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	32.0000	1.08428
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	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
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	6.00000	0.201688
$886$	0	0
$887$	−46.0000	−1.54453	−0.772264	−	0.635301i	$-0.780876\pi$
$887$	−46.0000	−1.54453	−0.772264	+	0.635301i	$0.780876\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	−6.00000	−0.201008
$892$	0	0
$893$	−12.0000	−0.401565
$894$	0	0
$895$	−18.0000	−0.601674
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	−40.0000	−1.33407
$900$	0	0
$901$	20.0000	0.666297
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	0	0
$909$	10.0000	0.331679
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	72.0000	2.38285
$914$	0	0
$915$	6.00000	0.198354
$916$	0	0
$917$	−44.0000	−1.45301
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	0	0
$923$	−16.0000	−0.526646
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	10.0000	0.328443
$928$	0	0
$929$	58.0000	1.90292	0.951459	−	0.307775i	$-0.0995844\pi$
$929$	58.0000	1.90292	0.951459	+	0.307775i	$0.0995844\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	−12.0000	−0.392442
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	0	0
$943$	36.0000	1.17232
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	32.0000	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$947$	32.0000	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	−14.0000	−0.453981
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	0	0
$957$	60.0000	1.93952
$958$	0	0
$959$	4.00000	0.129167
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	22.0000	0.708205
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−54.0000	−1.73294	−0.866471	−	0.499227i	$-0.833617\pi$
$971$	−54.0000	−1.73294	−0.866471	+	0.499227i	$0.833617\pi$
$972$	0	0
$973$	28.0000	0.897639
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	46.0000	1.47167	0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000	1.47167	0.735835	+	0.677161i	$0.236790\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	0	0
$981$	6.00000	0.191565
$982$	0	0
$983$	−26.0000	−0.829271	−0.414636	−	0.909988i	$-0.636091\pi$
$983$	−26.0000	−0.829271	−0.414636	+	0.909988i	$0.636091\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	12.0000	0.381964
$988$	0	0
$989$	0	0
$990$	0	0
$991$	60.0000	1.90596	0.952981	−	0.303029i	$-0.0979978\pi$
$991$	60.0000	1.90596	0.952981	+	0.303029i	$0.0979978\pi$
$992$	0	0
$993$	−30.0000	−0.952021
$994$	0	0
$995$	4.00000	0.126809
$996$	0	0
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.a.b.1.1	✓	1
3.2	odd	2		5760.2.a.bf.1.1		1
4.3	odd	2		1920.2.a.q.1.1	yes	1
5.4	even	2		9600.2.a.bv.1.1		1
8.3	odd	2		1920.2.a.j.1.1	yes	1
8.5	even	2		1920.2.a.u.1.1	yes	1
12.11	even	2		5760.2.a.bo.1.1		1
16.3	odd	4		3840.2.k.h.1921.2		2
16.5	even	4		3840.2.k.v.1921.2		2
16.11	odd	4		3840.2.k.h.1921.1		2
16.13	even	4		3840.2.k.v.1921.1		2
20.19	odd	2		9600.2.a.i.1.1		1
24.5	odd	2		5760.2.a.c.1.1		1
24.11	even	2		5760.2.a.v.1.1		1
40.19	odd	2		9600.2.a.bg.1.1		1
40.29	even	2		9600.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.a.b.1.1	✓	1	1.1	even	1	trivial
1920.2.a.j.1.1	yes	1	8.3	odd	2
1920.2.a.q.1.1	yes	1	4.3	odd	2
1920.2.a.u.1.1	yes	1	8.5	even	2
3840.2.k.h.1921.1		2	16.11	odd	4
3840.2.k.h.1921.2		2	16.3	odd	4
3840.2.k.v.1921.1		2	16.13	even	4
3840.2.k.v.1921.2		2	16.5	even	4
5760.2.a.c.1.1		1	24.5	odd	2
5760.2.a.v.1.1		1	24.11	even	2
5760.2.a.bf.1.1		1	3.2	odd	2
5760.2.a.bo.1.1		1	12.11	even	2
9600.2.a.i.1.1		1	20.19	odd	2
9600.2.a.x.1.1		1	40.29	even	2
9600.2.a.bg.1.1		1	40.19	odd	2
9600.2.a.bv.1.1		1	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.a (trivial)

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