LMFDB - Embedded newform 2112.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,2,Mod(1,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, 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("2112.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.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:	$16.8644049069$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1056)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +4.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} +4.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -4.00000 q^{13} -4.00000 q^{15} +2.00000 q^{17} -4.00000 q^{19} +2.00000 q^{21} +6.00000 q^{23} +11.0000 q^{25} -1.00000 q^{27} +6.00000 q^{29} +4.00000 q^{31} -1.00000 q^{33} -8.00000 q^{35} +2.00000 q^{37} +4.00000 q^{39} +6.00000 q^{41} -8.00000 q^{43} +4.00000 q^{45} +10.0000 q^{47} -3.00000 q^{49} -2.00000 q^{51} +12.0000 q^{53} +4.00000 q^{55} +4.00000 q^{57} -8.00000 q^{61} -2.00000 q^{63} -16.0000 q^{65} -4.00000 q^{67} -6.00000 q^{69} +6.00000 q^{71} +10.0000 q^{73} -11.0000 q^{75} -2.00000 q^{77} -6.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} +8.00000 q^{85} -6.00000 q^{87} -14.0000 q^{89} +8.00000 q^{91} -4.00000 q^{93} -16.0000 q^{95} -18.0000 q^{97} +1.00000 q^{99} +O(q^{100})$

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
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	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
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	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
$25$	11.0000	2.20000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−8.00000	−1.35225
$36$	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
$39$	4.00000	0.640513
$40$	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
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	0	0
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−16.0000	−1.98456
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	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
$75$	−11.0000	−1.27017
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−16.0000	−1.64157
$96$	0	0
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	8.00000	0.780720
$106$	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
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	24.0000	2.23801
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	24.0000	2.14663
$126$	0	0
$127$	18.0000	1.59724	0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000	1.59724	0.798621	+	0.601834i	$0.205563\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	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
$133$	8.00000	0.693688
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	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
$141$	−10.0000	−0.842152
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	24.0000	1.99309
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	16.0000	1.28515
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	−12.0000	−0.951662
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	−4.00000	−0.311400
$166$	0	0
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−4.00000	−0.305888
$172$	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
$175$	−22.0000	−1.66304
$176$	0	0
$177$	0	0
$178$	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
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	2.00000	0.146254
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	26.0000	1.88129	0.940647	−	0.339387i	$-0.110219\pi$
$191$	26.0000	1.88129	0.940647	+	0.339387i	$0.110219\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	16.0000	1.14578
$196$	0	0
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	−12.0000	−0.842235
$204$	0	0
$205$	24.0000	1.67623
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	−6.00000	−0.411113
$214$	0	0
$215$	−32.0000	−2.18238
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	40.0000	2.60931
$236$	0	0
$237$	6.00000	0.389742
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−12.0000	−0.766652
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	−8.00000	−0.500979
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−32.0000	−1.97320	−0.986602	−	0.163144i	$-0.947836\pi$
$263$	−32.0000	−1.97320	−0.986602	+	0.163144i	$0.947836\pi$
$264$	0	0
$265$	48.0000	2.94862
$266$	0	0
$267$	14.0000	0.856786
$268$	0	0
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	11.0000	0.663325
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	0	0
$285$	16.0000	0.947758
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	18.0000	1.05518
$292$	0	0
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−24.0000	−1.38796
$300$	0	0
$301$	16.0000	0.922225
$302$	0	0
$303$	10.0000	0.574485
$304$	0	0
$305$	−32.0000	−1.83231
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	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
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	−8.00000	−0.450749
$316$	0	0
$317$	−16.0000	−0.898650	−0.449325	−	0.893368i	$-0.648335\pi$
$317$	−16.0000	−0.898650	−0.449325	+	0.893368i	$0.648335\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	−8.00000	−0.445132
$324$	0	0
$325$	−44.0000	−2.44068
$326$	0	0
$327$	−12.0000	−0.663602
$328$	0	0
$329$	−20.0000	−1.10264
$330$	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
$333$	2.00000	0.109599
$334$	0	0
$335$	−16.0000	−0.874173
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	−24.0000	−1.29212
$346$	0	0
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	0	0
$349$	4.00000	0.214115	0.107058	−	0.994253i	$-0.465857\pi$
$349$	4.00000	0.214115	0.107058	+	0.994253i	$0.465857\pi$
$350$	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
$355$	24.0000	1.27379
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	40.0000	2.09370
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	−24.0000	−1.24602
$372$	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
$375$	−24.0000	−1.23935
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	−18.0000	−0.922168
$382$	0	0
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	−8.00000	−0.406663
$388$	0	0
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	−24.0000	−1.20757
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	−8.00000	−0.400501
$400$	0	0
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	0	0
$405$	4.00000	0.198762
$406$	0	0
$407$	2.00000	0.0991363
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	0	0
$413$	0	0
$414$	0	0
$415$	48.0000	2.35623
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	10.0000	0.486217
$424$	0	0
$425$	22.0000	1.06716
$426$	0	0
$427$	16.0000	0.774294
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	0	0
$435$	−24.0000	−1.15071
$436$	0	0
$437$	−24.0000	−1.14808
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	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
$445$	−56.0000	−2.65465
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	0	0
$453$	−2.00000	−0.0939682
$454$	0	0
$455$	32.0000	1.50018
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	−16.0000	−0.741982
$466$	0	0
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	−44.0000	−2.01886
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	12.0000	0.546019
$484$	0	0
$485$	−72.0000	−3.26935
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	20.0000	0.893534
$502$	0	0
$503$	4.00000	0.178351	0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000	0.178351	0.0891756	+	0.996016i	$0.471577\pi$
$504$	0	0
$505$	−40.0000	−1.77998
$506$	0	0
$507$	−3.00000	−0.133235
$508$	0	0
$509$	−28.0000	−1.24108	−0.620539	−	0.784176i	$-0.713086\pi$
$509$	−28.0000	−1.24108	−0.620539	+	0.784176i	$0.713086\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	64.0000	2.82018
$516$	0	0
$517$	10.0000	0.439799
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	22.0000	0.960159
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−24.0000	−1.03956
$534$	0	0
$535$	16.0000	0.691740
$536$	0	0
$537$	24.0000	1.03568
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	36.0000	1.54776	0.773880	−	0.633332i	$-0.218313\pi$
$541$	36.0000	1.54776	0.773880	+	0.633332i	$0.218313\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	48.0000	2.05609
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	−8.00000	−0.341432
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	12.0000	0.510292
$554$	0	0
$555$	−8.00000	−0.339581
$556$	0	0
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	0	0
$559$	32.0000	1.35346
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	0	0
$563$	44.0000	1.85438	0.927189	−	0.374593i	$-0.122217\pi$
$563$	44.0000	1.85438	0.927189	+	0.374593i	$0.122217\pi$
$564$	0	0
$565$	40.0000	1.68281
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	−26.0000	−1.08617
$574$	0	0
$575$	66.0000	2.75239
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	−6.00000	−0.249351
$580$	0	0
$581$	−24.0000	−0.995688
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	−16.0000	−0.661519
$586$	0	0
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−22.0000	−0.904959
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	0	0
$597$	−16.0000	−0.654836
$598$	0	0
$599$	46.0000	1.87951	0.939755	−	0.341850i	$-0.111053\pi$
$599$	46.0000	1.87951	0.939755	+	0.341850i	$0.111053\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	4.00000	0.162623
$606$	0	0
$607$	−30.0000	−1.21766	−0.608831	−	0.793300i	$-0.708361\pi$
$607$	−30.0000	−1.21766	−0.608831	+	0.793300i	$0.708361\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	−40.0000	−1.61823
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	0	0
$615$	−24.0000	−0.967773
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	28.0000	1.12180
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	4.00000	0.159745
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	0	0
$633$	8.00000	0.317971
$634$	0	0
$635$	72.0000	2.85723
$636$	0	0
$637$	12.0000	0.475457
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	0	0
$645$	32.0000	1.26000
$646$	0	0
$647$	6.00000	0.235884	0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000	0.235884	0.117942	+	0.993020i	$0.462370\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	8.00000	0.313064	0.156532	−	0.987673i	$-0.449969\pi$
$653$	8.00000	0.313064	0.156532	+	0.987673i	$0.449969\pi$
$654$	0	0
$655$	−48.0000	−1.87552
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	32.0000	1.24091
$666$	0	0
$667$	36.0000	1.39393
$668$	0	0
$669$	12.0000	0.463947
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	0	0
$675$	−11.0000	−0.423390
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	36.0000	1.38155
$680$	0	0
$681$	4.00000	0.153280
$682$	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
$685$	56.0000	2.13965
$686$	0	0
$687$	10.0000	0.381524
$688$	0	0
$689$	−48.0000	−1.82865
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	48.0000	1.82074
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−14.0000	−0.528773	−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000	−0.528773	−0.264386	+	0.964417i	$0.585169\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	−40.0000	−1.50649
$706$	0	0
$707$	20.0000	0.752177
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	−16.0000	−0.598366
$716$	0	0
$717$	−20.0000	−0.746914
$718$	0	0
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	0	0
$723$	18.0000	0.669427
$724$	0	0
$725$	66.0000	2.45118
$726$	0	0
$727$	−48.0000	−1.78022	−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000	−1.78022	−0.890111	+	0.455744i	$0.849373\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.0000	−0.591781
$732$	0	0
$733$	28.0000	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$733$	28.0000	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$734$	0	0
$735$	12.0000	0.442627
$736$	0	0
$737$	−4.00000	−0.147342
$738$	0	0
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	0	0
$741$	−16.0000	−0.587775
$742$	0	0
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	0	0
$745$	−24.0000	−0.879292
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	−8.00000	−0.292314
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	−16.0000	−0.583072
$754$	0	0
$755$	8.00000	0.291150
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	−6.00000	−0.217786
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	−24.0000	−0.868858
$764$	0	0
$765$	8.00000	0.289241
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	0	0
$775$	44.0000	1.58053
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	−40.0000	−1.42766
$786$	0	0
$787$	−16.0000	−0.570338	−0.285169	−	0.958477i	$-0.592050\pi$
$787$	−16.0000	−0.570338	−0.285169	+	0.958477i	$0.592050\pi$
$788$	0	0
$789$	32.0000	1.13923
$790$	0	0
$791$	−20.0000	−0.711118
$792$	0	0
$793$	32.0000	1.13635
$794$	0	0
$795$	−48.0000	−1.70238
$796$	0	0
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	20.0000	0.707549
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	10.0000	0.352892
$804$	0	0
$805$	−48.0000	−1.69178
$806$	0	0
$807$	−4.00000	−0.140807
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	0	0
$813$	10.0000	0.350715
$814$	0	0
$815$	−16.0000	−0.560456
$816$	0	0
$817$	32.0000	1.11954
$818$	0	0
$819$	8.00000	0.279543
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	−11.0000	−0.382971
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−80.0000	−2.76851
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	6.00000	0.206651
$844$	0	0
$845$	12.0000	0.412813
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	−24.0000	−0.823678
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	56.0000	1.91740	0.958702	−	0.284413i	$-0.0917988\pi$
$853$	56.0000	1.91740	0.958702	+	0.284413i	$0.0917988\pi$
$854$	0	0
$855$	−16.0000	−0.547188
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	0	0
$863$	2.00000	0.0680808	0.0340404	−	0.999420i	$-0.489163\pi$
$863$	2.00000	0.0680808	0.0340404	+	0.999420i	$0.489163\pi$
$864$	0	0
$865$	56.0000	1.90406
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	−6.00000	−0.203536
$870$	0	0
$871$	16.0000	0.542139
$872$	0	0
$873$	−18.0000	−0.609208
$874$	0	0
$875$	−48.0000	−1.62270
$876$	0	0
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	0	0
$879$	14.0000	0.472208
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	−36.0000	−1.20740
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−40.0000	−1.33855
$894$	0	0
$895$	−96.0000	−3.20893
$896$	0	0
$897$	24.0000	0.801337
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	24.0000	0.799556
$902$	0	0
$903$	−16.0000	−0.532447
$904$	0	0
$905$	−40.0000	−1.32964
$906$	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
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	32.0000	1.05789
$916$	0	0
$917$	24.0000	0.792550
$918$	0	0
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	−24.0000	−0.789970
$924$	0	0
$925$	22.0000	0.723356
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	0	0
$933$	10.0000	0.327385
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	36.0000	1.17232
$944$	0	0
$945$	8.00000	0.260240
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	−40.0000	−1.29845
$950$	0	0
$951$	16.0000	0.518836
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	104.000	3.36536
$956$	0	0
$957$	−6.00000	−0.193952
$958$	0	0
$959$	−28.0000	−0.904167
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	24.0000	0.772587
$966$	0	0
$967$	−30.0000	−0.964735	−0.482367	−	0.875969i	$-0.660223\pi$
$967$	−30.0000	−0.964735	−0.482367	+	0.875969i	$0.660223\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−24.0000	−0.769405
$974$	0	0
$975$	44.0000	1.40913
$976$	0	0
$977$	−50.0000	−1.59964	−0.799821	−	0.600239i	$-0.795072\pi$
$977$	−50.0000	−1.59964	−0.799821	+	0.600239i	$0.795072\pi$
$978$	0	0
$979$	−14.0000	−0.447442
$980$	0	0
$981$	12.0000	0.383131
$982$	0	0
$983$	−42.0000	−1.33959	−0.669796	−	0.742545i	$-0.733618\pi$
$983$	−42.0000	−1.33959	−0.669796	+	0.742545i	$0.733618\pi$
$984$	0	0
$985$	88.0000	2.80391
$986$	0	0
$987$	20.0000	0.636607
$988$	0	0
$989$	−48.0000	−1.52631
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	0	0
$993$	12.0000	0.380808
$994$	0	0
$995$	64.0000	2.02894
$996$	0	0
$997$	20.0000	0.633406	0.316703	−	0.948525i	$-0.397424\pi$
$997$	20.0000	0.633406	0.316703	+	0.948525i	$0.397424\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.a.o.1.1		1
3.2	odd	2		6336.2.a.a.1.1		1
4.3	odd	2		2112.2.a.bc.1.1		1
8.3	odd	2		1056.2.a.a.1.1	✓	1
8.5	even	2		1056.2.a.f.1.1	yes	1
12.11	even	2		6336.2.a.e.1.1		1
24.5	odd	2		3168.2.a.ba.1.1		1
24.11	even	2		3168.2.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1056.2.a.a.1.1	✓	1	8.3	odd	2
1056.2.a.f.1.1	yes	1	8.5	even	2
2112.2.a.o.1.1		1	1.1	even	1	trivial
2112.2.a.bc.1.1		1	4.3	odd	2
3168.2.a.ba.1.1		1	24.5	odd	2
3168.2.a.bb.1.1		1	24.11	even	2
6336.2.a.a.1.1		1	3.2	odd	2
6336.2.a.e.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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