LMFDB - Embedded newform 3840.2.k.z.1921.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3840,2,Mod(1921,3840)] 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("3840.1921"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,8,0,-2,0,0,0,0,0,2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$30.6625543762$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1921.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3840.1921
Dual form			3840.2.k.z.1921.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} -6.00000i q^{13} +1.00000 q^{15} -2.00000 q^{17} -4.00000i q^{19} -4.00000i q^{21} -8.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} -6.00000i q^{29} +4.00000i q^{35} +6.00000i q^{37} -6.00000 q^{39} -10.0000 q^{41} -4.00000i q^{43} -1.00000i q^{45} -8.00000 q^{47} +9.00000 q^{49} +2.00000i q^{51} -10.0000i q^{53} -4.00000 q^{57} +6.00000i q^{61} -4.00000 q^{63} +6.00000 q^{65} +4.00000i q^{67} +8.00000i q^{69} +14.0000 q^{73} +1.00000i q^{75} -16.0000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -2.00000i q^{85} -6.00000 q^{87} -2.00000 q^{89} -24.0000i q^{91} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{7} - 2 q^{9} + 2 q^{15} - 4 q^{17} - 16 q^{23} - 2 q^{25} - 12 q^{39} - 20 q^{41} - 16 q^{47} + 18 q^{49} - 8 q^{57} - 8 q^{63} + 12 q^{65} + 28 q^{73} - 32 q^{79} + 2 q^{81} - 12 q^{87} - 4 q^{89}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q + 8 * q^7 - 2 * q^9 + 2 * q^15 - 4 * q^17 - 16 * q^23 - 2 * q^25 - 12 * q^39 - 20 * q^41 - 16 * q^47 + 18 * q^49 - 8 * q^57 - 8 * q^63 + 12 * q^65 + 28 * q^73 - 32 * q^79 + 2 * q^81 - 12 * q^87 - 4 * q^89 + 8 * q^95 + 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$.

$n$	$511$	$1537$	$2561$	$2821$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	− 4.00000i	− 0.872872i
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 6.00000i	− 1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$	− 6.00000i	− 1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.00000i	0.676123i
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	2.00000i	0.280056i
$52$	0	0
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	6.00000i	0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$	6.00000i	0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	0	0
$65$	6.00000	0.744208
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	8.00000i	0.963087i
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	− 2.00000i	− 0.216930i
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	− 24.0000i	− 2.51588i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.00000	0.410391
$96$	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
$99$	0	0
$100$	0	0
$101$	14.0000i	1.39305i	0.717532	+	0.696526i	$0.245272\pi$
$101$	14.0000i	1.39305i	−0.717532	+	0.696526i	$0.754728\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	− 10.0000i	− 0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$	− 10.0000i	− 0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	− 8.00000i	− 0.746004i
$116$	0	0
$117$	6.00000i	0.554700i
$118$	0	0
$119$	−8.00000	−0.733359
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	10.0000i	0.901670i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	− 16.0000i	− 1.39793i	−0.715158	−	0.698963i	$-0.753645\pi$
$131$	− 16.0000i	− 1.39793i	0.715158	−	0.698963i	$-0.246355\pi$
$132$	0	0
$133$	− 16.0000i	− 1.38738i
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	− 12.0000i	− 1.01783i	−0.860818	−	0.508913i	$-0.830047\pi$
$139$	− 12.0000i	− 1.01783i	0.860818	−	0.508913i	$-0.169953\pi$
$140$	0	0
$141$	8.00000i	0.673722i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	− 9.00000i	− 0.742307i
$148$	0	0
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	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
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	−32.0000	−2.52195
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	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
$169$	−23.0000	−1.76923
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 8.00000i	− 0.597948i	−0.954261	−	0.298974i	$-0.903356\pi$
$179$	− 8.00000i	− 0.597948i	0.954261	−	0.298974i	$-0.0966444\pi$
$180$	0	0
$181$	− 14.0000i	− 1.04061i	−0.853980	−	0.520306i	$-0.825818\pi$
$181$	− 14.0000i	− 1.04061i	0.853980	−	0.520306i	$-0.174182\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	0	0
$188$	0	0
$189$	4.00000i	0.290957i
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	− 6.00000i	− 0.429669i
$196$	0	0
$197$	− 10.0000i	− 0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$	− 10.0000i	− 0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	− 24.0000i	− 1.68447i
$204$	0	0
$205$	− 10.0000i	− 0.698430i
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 12.0000i	− 0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$	− 12.0000i	− 0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 14.0000i	− 0.946032i
$220$	0	0
$221$	12.0000i	0.807207i
$222$	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
$225$	1.00000	0.0666667
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	− 6.00000i	− 0.396491i	−0.980152	−	0.198246i	$-0.936476\pi$
$229$	− 6.00000i	− 0.396491i	0.980152	−	0.198246i	$-0.0635244\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	− 8.00000i	− 0.521862i
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	9.00000i	0.574989i
$246$	0	0
$247$	−24.0000	−1.52708
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	− 8.00000i	− 0.504956i	−0.967603	−	0.252478i	$-0.918755\pi$
$251$	− 8.00000i	− 0.504956i	0.967603	−	0.252478i	$-0.0812455\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−2.00000	−0.125245
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	24.0000i	1.49129i
$260$	0	0
$261$	6.00000i	0.371391i
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	2.00000i	0.122398i
$268$	0	0
$269$	− 6.00000i	− 0.365826i	−0.983129	−	0.182913i	$-0.941447\pi$
$269$	− 6.00000i	− 0.365826i	0.983129	−	0.182913i	$-0.0585527\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	−24.0000	−1.45255
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	− 4.00000i	− 0.236940i
$286$	0	0
$287$	−40.0000	−2.36113
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	− 2.00000i	− 0.117242i
$292$	0	0
$293$	− 10.0000i	− 0.584206i	−0.956387	−	0.292103i	$-0.905645\pi$
$293$	− 10.0000i	− 0.584206i	0.956387	−	0.292103i	$-0.0943550\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	48.0000i	2.77591i
$300$	0	0
$301$	− 16.0000i	− 0.922225i
$302$	0	0
$303$	14.0000	0.804279
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	− 4.00000i	− 0.227552i
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	− 4.00000i	− 0.225374i
$316$	0	0
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	6.00000i	0.332820i
$326$	0	0
$327$	−10.0000	−0.553001
$328$	0	0
$329$	−32.0000	−1.76422
$330$	0	0
$331$	28.0000i	1.53902i	0.638635	+	0.769510i	$0.279499\pi$
$331$	28.0000i	1.53902i	−0.638635	+	0.769510i	$0.720501\pi$
$332$	0	0
$333$	− 6.00000i	− 0.328798i
$334$	0	0
$335$	−4.00000	−0.218543
$336$	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
$339$	− 6.00000i	− 0.325875i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	−8.00000	−0.430706
$346$	0	0
$347$	− 36.0000i	− 1.93258i	−0.257454	−	0.966291i	$-0.582883\pi$
$347$	− 36.0000i	− 1.93258i	0.257454	−	0.966291i	$-0.417117\pi$
$348$	0	0
$349$	− 26.0000i	− 1.39175i	−0.718164	−	0.695874i	$-0.755017\pi$
$349$	− 26.0000i	− 1.39175i	0.718164	−	0.695874i	$-0.244983\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	8.00000i	0.423405i
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	− 11.0000i	− 0.577350i
$364$	0	0
$365$	14.0000i	0.732793i
$366$	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
$369$	10.0000	0.520579
$370$	0	0
$371$	− 40.0000i	− 2.07670i
$372$	0	0
$373$	− 2.00000i	− 0.103556i	−0.998659	−	0.0517780i	$-0.983511\pi$
$373$	− 2.00000i	− 0.103556i	0.998659	−	0.0517780i	$-0.0164888\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−36.0000	−1.85409
$378$	0	0
$379$	4.00000i	0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$	4.00000i	0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	0	0
$381$	− 4.00000i	− 0.204926i
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000i	0.203331i
$388$	0	0
$389$	6.00000i	0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$	6.00000i	0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	−16.0000	−0.807093
$394$	0	0
$395$	− 16.0000i	− 0.805047i
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	0	0
$399$	−16.0000	−0.801002
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	0	0
$411$	− 18.0000i	− 0.887875i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	16.0000i	0.781651i	0.920465	+	0.390826i	$0.127810\pi$
$419$	16.0000i	0.781651i	−0.920465	+	0.390826i	$0.872190\pi$
$420$	0	0
$421$	− 6.00000i	− 0.292422i	−0.989253	−	0.146211i	$-0.953292\pi$
$421$	− 6.00000i	− 0.292422i	0.989253	−	0.146211i	$-0.0467079\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	24.0000i	1.16144i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	0	0
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	0	0
$435$	− 6.00000i	− 0.287678i
$436$	0	0
$437$	32.0000i	1.53077i
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	− 2.00000i	− 0.0948091i
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	24.0000	1.12514
$456$	0	0
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	0	0
$459$	− 2.00000i	− 0.0933520i
$460$	0	0
$461$	2.00000i	0.0931493i	0.998915	+	0.0465746i	$0.0148305\pi$
$461$	2.00000i	0.0931493i	−0.998915	+	0.0465746i	$0.985169\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.0000i	0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$	20.0000i	0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$	0	0
$469$	16.0000i	0.738811i
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.00000i	0.183533i
$476$	0	0
$477$	10.0000i	0.457869i
$478$	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
$481$	36.0000	1.64146
$482$	0	0
$483$	32.0000i	1.45605i
$484$	0	0
$485$	2.00000i	0.0908153i
$486$	0	0
$487$	−36.0000	−1.63132	−0.815658	−	0.578535i	$-0.803625\pi$
$487$	−36.0000	−1.63132	−0.815658	+	0.578535i	$0.803625\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	− 40.0000i	− 1.80517i	−0.430507	−	0.902587i	$-0.641665\pi$
$491$	− 40.0000i	− 1.80517i	0.430507	−	0.902587i	$-0.358335\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 20.0000i	− 0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$	− 20.0000i	− 0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$	0	0
$501$	8.00000i	0.357414i
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	23.0000i	1.02147i
$508$	0	0
$509$	− 6.00000i	− 0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$	− 6.00000i	− 0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$	0	0
$511$	56.0000	2.47729
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	4.00000i	0.176261i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	0	0
$523$	36.0000i	1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$	36.0000i	1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	0	0
$525$	4.00000i	0.174574i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	60.0000i	2.59889i
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	−8.00000	−0.345225
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 10.0000i	− 0.429934i	−0.976621	−	0.214967i	$-0.931036\pi$
$541$	− 10.0000i	− 0.429934i	0.976621	−	0.214967i	$-0.0689643\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	− 4.00000i	− 0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$	− 4.00000i	− 0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$	0	0
$549$	− 6.00000i	− 0.256074i
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	−64.0000	−2.72156
$554$	0	0
$555$	6.00000i	0.254686i
$556$	0	0
$557$	− 14.0000i	− 0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$	− 14.0000i	− 0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	6.00000i	0.252422i
$566$	0	0
$567$	4.00000	0.167984
$568$	0	0
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	0	0
$571$	− 20.0000i	− 0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$	− 20.0000i	− 0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$	0	0
$573$	8.00000i	0.334205i
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	0	0
$579$	6.00000i	0.249351i
$580$	0	0
$581$	− 48.0000i	− 1.99138i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	36.0000i	1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$	36.0000i	1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−10.0000	−0.411345
$592$	0	0
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	0	0
$595$	− 8.00000i	− 0.327968i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	11.0000i	0.447214i
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	48.0000i	1.94187i
$612$	0	0
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	0	0
$615$	−10.0000	−0.403239
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	0	0
$619$	− 28.0000i	− 1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$	− 28.0000i	− 1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$	0	0
$621$	− 8.00000i	− 0.321029i
$622$	0	0
$623$	−8.00000	−0.320513
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 12.0000i	− 0.478471i
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	4.00000i	0.158735i
$636$	0	0
$637$	− 54.0000i	− 2.13956i
$638$	0	0
$639$	0	0
$640$	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
$643$	36.0000i	1.41970i	0.704352	+	0.709851i	$0.251238\pi$
$643$	36.0000i	1.41970i	−0.704352	+	0.709851i	$0.748762\pi$
$644$	0	0
$645$	− 4.00000i	− 0.157500i
$646$	0	0
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	0	0
$657$	−14.0000	−0.546192
$658$	0	0
$659$	16.0000i	0.623272i	0.950202	+	0.311636i	$0.100877\pi$
$659$	16.0000i	0.623272i	−0.950202	+	0.311636i	$0.899123\pi$
$660$	0	0
$661$	2.00000i	0.0777910i	0.999243	+	0.0388955i	$0.0123839\pi$
$661$	2.00000i	0.0777910i	−0.999243	+	0.0388955i	$0.987616\pi$
$662$	0	0
$663$	12.0000	0.466041
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	48.0000i	1.85857i
$668$	0	0
$669$	12.0000i	0.463947i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	0	0
$685$	18.0000i	0.687745i
$686$	0	0
$687$	−6.00000	−0.228914
$688$	0	0
$689$	−60.0000	−2.28582
$690$	0	0
$691$	− 4.00000i	− 0.152167i	−0.997101	−	0.0760836i	$-0.975758\pi$
$691$	− 4.00000i	− 0.152167i	0.997101	−	0.0760836i	$-0.0242416\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	− 10.0000i	− 0.378235i
$700$	0	0
$701$	− 38.0000i	− 1.43524i	−0.696435	−	0.717620i	$-0.745231\pi$
$701$	− 38.0000i	− 1.43524i	0.696435	−	0.717620i	$-0.254769\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	56.0000i	2.10610i
$708$	0	0
$709$	− 22.0000i	− 0.826227i	−0.910679	−	0.413114i	$-0.864441\pi$
$709$	− 22.0000i	− 0.826227i	0.910679	−	0.413114i	$-0.135559\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 24.0000i	− 0.896296i
$718$	0	0
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	− 2.00000i	− 0.0743808i
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	8.00000i	0.295891i
$732$	0	0
$733$	− 46.0000i	− 1.69905i	−0.527549	−	0.849524i	$-0.676889\pi$
$733$	− 46.0000i	− 1.69905i	0.527549	−	0.849524i	$-0.323111\pi$
$734$	0	0
$735$	9.00000	0.331970
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 4.00000i	− 0.147142i	−0.997290	−	0.0735712i	$-0.976560\pi$
$739$	− 4.00000i	− 0.147142i	0.997290	−	0.0735712i	$-0.0234396\pi$
$740$	0	0
$741$	24.0000i	0.881662i
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	16.0000i	0.584627i
$750$	0	0
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	0	0
$753$	−8.00000	−0.291536
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 42.0000i	− 1.52652i	−0.646094	−	0.763258i	$-0.723599\pi$
$757$	− 42.0000i	− 1.52652i	0.646094	−	0.763258i	$-0.276401\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−50.0000	−1.81250	−0.906249	−	0.422744i	$-0.861067\pi$
$761$	−50.0000	−1.81250	−0.906249	+	0.422744i	$0.861067\pi$
$762$	0	0
$763$	− 40.0000i	− 1.44810i
$764$	0	0
$765$	2.00000i	0.0723102i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	− 6.00000i	− 0.216085i
$772$	0	0
$773$	38.0000i	1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$	38.0000i	1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	24.0000	0.860995
$778$	0	0
$779$	40.0000i	1.43315i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	−10.0000	−0.356915
$786$	0	0
$787$	− 12.0000i	− 0.427754i	−0.976861	−	0.213877i	$-0.931391\pi$
$787$	− 12.0000i	− 0.427754i	0.976861	−	0.213877i	$-0.0686091\pi$
$788$	0	0
$789$	24.0000i	0.854423i
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	36.0000	1.27840
$794$	0	0
$795$	− 10.0000i	− 0.354663i
$796$	0	0
$797$	34.0000i	1.20434i	0.798367	+	0.602171i	$0.205697\pi$
$797$	34.0000i	1.20434i	−0.798367	+	0.602171i	$0.794303\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 32.0000i	− 1.12785i
$806$	0	0
$807$	−6.00000	−0.211210
$808$	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
$811$	− 12.0000i	− 0.421377i	−0.977553	−	0.210688i	$-0.932429\pi$
$811$	− 12.0000i	− 0.421377i	0.977553	−	0.210688i	$-0.0675706\pi$
$812$	0	0
$813$	8.00000i	0.280572i
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	24.0000i	0.838628i
$820$	0	0
$821$	− 50.0000i	− 1.74501i	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	− 50.0000i	− 1.74501i	0.488603	−	0.872506i	$-0.337507\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 20.0000i	− 0.695468i	−0.937593	−	0.347734i	$-0.886951\pi$
$827$	− 20.0000i	− 0.695468i	0.937593	−	0.347734i	$-0.113049\pi$
$828$	0	0
$829$	38.0000i	1.31979i	0.751356	+	0.659897i	$0.229400\pi$
$829$	38.0000i	1.31979i	−0.751356	+	0.659897i	$0.770600\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	− 8.00000i	− 0.276851i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	2.00000i	0.0688837i
$844$	0	0
$845$	− 23.0000i	− 0.791224i
$846$	0	0
$847$	44.0000	1.51186
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	− 48.0000i	− 1.64542i
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	−38.0000	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$858$	0	0
$859$	36.0000i	1.22830i	0.789188	+	0.614152i	$0.210502\pi$
$859$	36.0000i	1.22830i	−0.789188	+	0.614152i	$0.789498\pi$
$860$	0	0
$861$	40.0000i	1.36320i
$862$	0	0
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	24.0000	0.813209
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	− 4.00000i	− 0.135225i
$876$	0	0
$877$	10.0000i	0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$	10.0000i	0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$	0	0
$879$	−10.0000	−0.337292
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	− 28.0000i	− 0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$	− 28.0000i	− 0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	0	0
$893$	32.0000i	1.07084i
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	48.0000	1.60267
$898$	0	0
$899$	0	0
$900$	0	0
$901$	20.0000i	0.666297i
$902$	0	0
$903$	−16.0000	−0.532447
$904$	0	0
$905$	14.0000	0.465376
$906$	0	0
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	0	0
$909$	− 14.0000i	− 0.464351i
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	6.00000i	0.198354i
$916$	0	0
$917$	− 64.0000i	− 2.11347i
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 6.00000i	− 0.197279i
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	− 36.0000i	− 1.17985i
$932$	0	0
$933$	− 16.0000i	− 0.523816i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	− 14.0000i	− 0.456873i
$940$	0	0
$941$	− 14.0000i	− 0.456387i	−0.973616	−	0.228193i	$-0.926718\pi$
$941$	− 14.0000i	− 0.456387i	0.973616	−	0.228193i	$-0.0732819\pi$
$942$	0	0
$943$	80.0000	2.60516
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	44.0000i	1.42981i	0.699223	+	0.714904i	$0.253530\pi$
$947$	44.0000i	1.42981i	−0.699223	+	0.714904i	$0.746470\pi$
$948$	0	0
$949$	− 84.0000i	− 2.72676i
$950$	0	0
$951$	2.00000	0.0648544
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	0	0
$955$	− 8.00000i	− 0.258874i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	72.0000	2.32500
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 4.00000i	− 0.128898i
$964$	0	0
$965$	− 6.00000i	− 0.193147i
$966$	0	0
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	− 56.0000i	− 1.79713i	−0.438845	−	0.898563i	$-0.644612\pi$
$971$	− 56.0000i	− 1.79713i	0.438845	−	0.898563i	$-0.355388\pi$
$972$	0	0
$973$	− 48.0000i	− 1.53881i
$974$	0	0
$975$	6.00000	0.192154
$976$	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
$979$	0	0
$980$	0	0
$981$	10.0000i	0.319275i
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	32.0000i	1.01857i
$988$	0	0
$989$	32.0000i	1.01754i
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	0	0
$996$	0	0
$997$	30.0000i	0.950110i	0.879956	+	0.475055i	$0.157572\pi$
$997$	30.0000i	0.950110i	−0.879956	+	0.475055i	$0.842428\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.k.z.1921.1		2
4.3	odd	2		3840.2.k.a.1921.2		2
8.3	odd	2		3840.2.k.a.1921.1		2
8.5	even	2	inner	3840.2.k.z.1921.2		2
16.3	odd	4		960.2.a.g.1.1		1
16.5	even	4		240.2.a.a.1.1		1
16.11	odd	4		120.2.a.a.1.1	✓	1
16.13	even	4		960.2.a.n.1.1		1
48.5	odd	4		720.2.a.f.1.1		1
48.11	even	4		360.2.a.e.1.1		1
48.29	odd	4		2880.2.a.b.1.1		1
48.35	even	4		2880.2.a.r.1.1		1
80.3	even	4		4800.2.f.u.3649.1		2
80.13	odd	4		4800.2.f.n.3649.2		2
80.19	odd	4		4800.2.a.bl.1.1		1
80.27	even	4		600.2.f.c.49.1		2
80.29	even	4		4800.2.a.bh.1.1		1
80.37	odd	4		1200.2.f.f.49.2		2
80.43	even	4		600.2.f.c.49.2		2
80.53	odd	4		1200.2.f.f.49.1		2
80.59	odd	4		600.2.a.a.1.1		1
80.67	even	4		4800.2.f.u.3649.2		2
80.69	even	4		1200.2.a.r.1.1		1
80.77	odd	4		4800.2.f.n.3649.1		2
112.27	even	4		5880.2.a.p.1.1		1
144.11	even	12		3240.2.q.a.1081.1		2
144.43	odd	12		3240.2.q.m.1081.1		2
144.59	even	12		3240.2.q.a.2161.1		2
144.139	odd	12		3240.2.q.m.2161.1		2
240.53	even	4		3600.2.f.l.2449.2		2
240.59	even	4		1800.2.a.c.1.1		1
240.107	odd	4		1800.2.f.g.649.2		2
240.149	odd	4		3600.2.a.bo.1.1		1
240.197	even	4		3600.2.f.l.2449.1		2
240.203	odd	4		1800.2.f.g.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.a.a.1.1	✓	1	16.11	odd	4
240.2.a.a.1.1		1	16.5	even	4
360.2.a.e.1.1		1	48.11	even	4
600.2.a.a.1.1		1	80.59	odd	4
600.2.f.c.49.1		2	80.27	even	4
600.2.f.c.49.2		2	80.43	even	4
720.2.a.f.1.1		1	48.5	odd	4
960.2.a.g.1.1		1	16.3	odd	4
960.2.a.n.1.1		1	16.13	even	4
1200.2.a.r.1.1		1	80.69	even	4
1200.2.f.f.49.1		2	80.53	odd	4
1200.2.f.f.49.2		2	80.37	odd	4
1800.2.a.c.1.1		1	240.59	even	4
1800.2.f.g.649.1		2	240.203	odd	4
1800.2.f.g.649.2		2	240.107	odd	4
2880.2.a.b.1.1		1	48.29	odd	4
2880.2.a.r.1.1		1	48.35	even	4
3240.2.q.a.1081.1		2	144.11	even	12
3240.2.q.a.2161.1		2	144.59	even	12
3240.2.q.m.1081.1		2	144.43	odd	12
3240.2.q.m.2161.1		2	144.139	odd	12
3600.2.a.bo.1.1		1	240.149	odd	4
3600.2.f.l.2449.1		2	240.197	even	4
3600.2.f.l.2449.2		2	240.53	even	4
3840.2.k.a.1921.1		2	8.3	odd	2
3840.2.k.a.1921.2		2	4.3	odd	2
3840.2.k.z.1921.1		2	1.1	even	1	trivial
3840.2.k.z.1921.2		2	8.5	even	2	inner
4800.2.a.bh.1.1		1	80.29	even	4
4800.2.a.bl.1.1		1	80.19	odd	4
4800.2.f.n.3649.1		2	80.77	odd	4
4800.2.f.n.3649.2		2	80.13	odd	4
4800.2.f.u.3649.1		2	80.3	even	4
4800.2.f.u.3649.2		2	80.67	even	4
5880.2.a.p.1.1		1	112.27	even	4

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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} -6.00000i q^{13} +1.00000 q^{15} -2.00000 q^{17} -4.00000i q^{19} -4.00000i q^{21} -8.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} -6.00000i q^{29} +4.00000i q^{35} +6.00000i q^{37} -6.00000 q^{39} -10.0000 q^{41} -4.00000i q^{43} -1.00000i q^{45} -8.00000 q^{47} +9.00000 q^{49} +2.00000i q^{51} -10.0000i q^{53} -4.00000 q^{57} +6.00000i q^{61} -4.00000 q^{63} +6.00000 q^{65} +4.00000i q^{67} +8.00000i q^{69} +14.0000 q^{73} +1.00000i q^{75} -16.0000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -2.00000i q^{85} -6.00000 q^{87} -2.00000 q^{89} -24.0000i q^{91} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{7} - 2 q^{9} + 2 q^{15} - 4 q^{17} - 16 q^{23} - 2 q^{25} - 12 q^{39} - 20 q^{41} - 16 q^{47} + 18 q^{49} - 8 q^{57} - 8 q^{63} + 12 q^{65} + 28 q^{73} - 32 q^{79} + 2 q^{81} - 12 q^{87} - 4 q^{89}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q + 8 * q^7 - 2 * q^9 + 2 * q^15 - 4 * q^17 - 16 * q^23 - 2 * q^25 - 12 * q^39 - 20 * q^41 - 16 * q^47 + 18 * q^49 - 8 * q^57 - 8 * q^63 + 12 * q^65 + 28 * q^73 - 32 * q^79 + 2 * q^81 - 12 * q^87 - 4 * q^89 + 8 * q^95 + 4 * q^97

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