LMFDB - Embedded newform 4608.2.d.q.2305.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(2305,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4608.2305");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2305.6
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	4608.2305
Dual form			4608.2.d.q.2305.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.449490i q^{5} +2.04989 q^{7} +O(q^{10})$	$q+0.449490i q^{5} +2.04989 q^{7} +3.46410i q^{11} +4.79796 q^{25} +9.34847i q^{29} -3.60697 q^{31} +0.921404i q^{35} -2.79796 q^{49} -7.55051i q^{53} -1.55708 q^{55} +11.3137i q^{59} -9.79796 q^{73} +7.10102i q^{77} +17.4634 q^{79} +17.3205i q^{83} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 40 q^{25} + 56 q^{49} - 16 q^{97}+O(q^{100}) $ 8 * q - 40 * q^25 + 56 * q^49 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$.

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.449490i	0.201018i	0.994936	+	0.100509i	$0.0320471\pi$
$5$	0.449490i	0.201018i	−0.994936	+	0.100509i	$0.967953\pi$
$6$	0	0
$7$	2.04989	0.774785	0.387392	−	0.921915i	$-0.373376\pi$
$7$	2.04989	0.774785	0.387392	+	0.921915i	$0.373376\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.46410i	1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$	3.46410i	1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	4.79796	0.959592
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.34847i	1.73597i	0.496593	+	0.867984i	$0.334584\pi$
$29$	9.34847i	1.73597i	−0.496593	+	0.867984i	$0.665416\pi$
$30$	0	0
$31$	−3.60697	−0.647830	−0.323915	−	0.946086i	$-0.604999\pi$
$31$	−3.60697	−0.647830	−0.323915	+	0.946086i	$0.604999\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.921404i	0.155746i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−2.79796	−0.399708
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.55051i	− 1.03714i	−0.855034	−	0.518571i	$-0.826464\pi$
$53$	− 7.55051i	− 1.03714i	0.855034	−	0.518571i	$-0.173536\pi$
$54$	0	0
$55$	−1.55708	−0.209956
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.3137i	1.47292i	0.676481	+	0.736460i	$0.263504\pi$
$59$	11.3137i	1.47292i	−0.676481	+	0.736460i	$0.736496\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−9.79796	−1.14676	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	−9.79796	−1.14676	−0.573382	+	0.819288i	$0.694369\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.10102i	0.809236i
$78$	0	0
$79$	17.4634	1.96478	0.982391	−	0.186834i	$-0.0598227\pi$
$79$	17.4634	1.96478	0.982391	+	0.186834i	$0.0598227\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	17.3205i	1.90117i	0.310460	+	0.950586i	$0.399517\pi$
$83$	17.3205i	1.90117i	−0.310460	+	0.950586i	$0.600483\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 16.4495i	− 1.63679i	−0.574659	−	0.818393i	$-0.694865\pi$
$101$	− 16.4495i	− 1.63679i	0.574659	−	0.818393i	$-0.305135\pi$
$102$	0	0
$103$	−0.492810	−0.0485580	−0.0242790	−	0.999705i	$-0.507729\pi$
$103$	−0.492810	−0.0485580	−0.0242790	+	0.999705i	$0.507729\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.3137i	1.09374i	0.837218	+	0.546869i	$0.184180\pi$
$107$	11.3137i	1.09374i	−0.837218	+	0.546869i	$0.815820\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.40408i	0.393913i
$126$	0	0
$127$	−19.0205	−1.68779	−0.843896	−	0.536507i	$-0.819744\pi$
$127$	−19.0205	−1.68779	−0.843896	+	0.536507i	$0.819744\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	22.6274i	1.97697i	0.151330	+	0.988483i	$0.451644\pi$
$131$	22.6274i	1.97697i	−0.151330	+	0.988483i	$0.548356\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−4.20204	−0.348961
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 19.1464i	− 1.56854i	−0.620422	−	0.784268i	$-0.713039\pi$
$149$	− 19.1464i	− 1.56854i	0.620422	−	0.784268i	$-0.286961\pi$
$150$	0	0
$151$	−15.9063	−1.29444	−0.647218	−	0.762305i	$-0.724068\pi$
$151$	−15.9063	−1.29444	−0.647218	+	0.762305i	$0.724068\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.62129i	− 0.130225i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	26.2474i	1.99556i	0.0666220	+	0.997778i	$0.478778\pi$
$173$	26.2474i	1.99556i	−0.0666220	+	0.997778i	$0.521222\pi$
$174$	0	0
$175$	9.83528	0.743477
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.3137i	0.845626i	0.906217	+	0.422813i	$0.138957\pi$
$179$	11.3137i	0.845626i	−0.906217	+	0.422813i	$0.861043\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	9.79796	0.705273	0.352636	−	0.935760i	$-0.385285\pi$
$193$	9.79796	0.705273	0.352636	+	0.935760i	$0.385285\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 27.1464i	− 1.93410i	−0.254581	−	0.967051i	$-0.581938\pi$
$197$	− 27.1464i	− 1.93410i	0.254581	−	0.967051i	$-0.418062\pi$
$198$	0	0
$199$	27.2200	1.92957	0.964787	−	0.263031i	$-0.0847221\pi$
$199$	27.2200	1.92957	0.964787	+	0.263031i	$0.0847221\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	19.1633i	1.34500i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−7.39388	−0.501929
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−28.7771	−1.92706	−0.963528	−	0.267608i	$-0.913767\pi$
$223$	−28.7771	−1.92706	−0.963528	+	0.267608i	$0.913767\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.3923i	0.689761i	0.938647	+	0.344881i	$0.112081\pi$
$227$	10.3923i	0.689761i	−0.938647	+	0.344881i	$0.887919\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	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
$241$	29.3939	1.89343	0.946713	−	0.322078i	$-0.104381\pi$
$241$	29.3939	1.89343	0.946713	+	0.322078i	$0.104381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.25765i	− 0.0803486i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	31.1769i	1.96787i	0.178529	+	0.983935i	$0.442866\pi$
$251$	31.1769i	1.96787i	−0.178529	+	0.983935i	$0.557134\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	3.39388	0.208484
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.6515i	0.893320i	0.894704	+	0.446660i	$0.147387\pi$
$269$	14.6515i	0.893320i	−0.894704	+	0.446660i	$0.852613\pi$
$270$	0	0
$271$	32.8769	1.99713	0.998563	−	0.0535825i	$-0.0170640\pi$
$271$	32.8769	1.99713	0.998563	+	0.0535825i	$0.0170640\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	16.6206i	1.00226i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.0454i	0.703700i	0.936056	+	0.351850i	$0.114447\pi$
$293$	12.0454i	0.703700i	−0.936056	+	0.351850i	$0.885553\pi$
$294$	0	0
$295$	−5.08540	−0.296083
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−9.79796	−0.553813	−0.276907	−	0.960897i	$-0.589309\pi$
$313$	−9.79796	−0.553813	−0.276907	+	0.960897i	$0.589309\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	34.2474i	1.92353i	0.273879	+	0.961764i	$0.411693\pi$
$317$	34.2474i	1.92353i	−0.273879	+	0.961764i	$0.588307\pi$
$318$	0	0
$319$	−32.3840	−1.81316
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−29.3939	−1.60119	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	−29.3939	−1.60119	−0.800593	+	0.599208i	$0.795482\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 12.4949i	− 0.676637i
$342$	0	0
$343$	−20.0847	−1.08447
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.2487i	1.30174i	0.759190	+	0.650870i	$0.225596\pi$
$347$	24.2487i	1.30174i	−0.759190	+	0.650870i	$0.774404\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 4.40408i	− 0.230520i
$366$	0	0
$367$	30.3342	1.58343	0.791715	−	0.610890i	$-0.209188\pi$
$367$	30.3342	1.58343	0.791715	+	0.610890i	$0.209188\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 15.4777i	− 0.803562i
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−3.19184	−0.162671
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 4.85357i	− 0.246086i	−0.992401	−	0.123043i	$-0.960735\pi$
$389$	− 4.85357i	− 0.246086i	0.992401	−	0.123043i	$-0.0392653\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.84961i	0.394957i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	23.1918i	1.14120i
$414$	0	0
$415$	−7.78539	−0.382170
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.3923i	0.507697i	0.967244	+	0.253849i	$0.0816965\pi$
$419$	10.3923i	0.507697i	−0.967244	+	0.253849i	$0.918303\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−6.72112	−0.320782	−0.160391	−	0.987054i	$-0.551275\pi$
$439$	−6.72112	−0.320782	−0.160391	+	0.987054i	$0.551275\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 31.1769i	− 1.48126i	−0.671913	−	0.740630i	$-0.734527\pi$
$443$	− 31.1769i	− 1.48126i	0.671913	−	0.740630i	$-0.265473\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 12.9444i	− 0.602880i	−0.953485	−	0.301440i	$-0.902533\pi$
$461$	− 12.9444i	− 0.602880i	0.953485	−	0.301440i	$-0.0974673\pi$
$462$	0	0
$463$	42.6335	1.98135	0.990673	−	0.136260i	$-0.0435083\pi$
$463$	42.6335	1.98135	0.990673	+	0.136260i	$0.0435083\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 17.3205i	− 0.801498i	−0.916188	−	0.400749i	$-0.868750\pi$
$467$	− 17.3205i	− 0.801498i	0.916188	−	0.400749i	$-0.131250\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 0.898979i	− 0.0408206i
$486$	0	0
$487$	29.7627	1.34868	0.674338	−	0.738422i	$-0.264429\pi$
$487$	29.7627	1.34868	0.674338	+	0.738422i	$0.264429\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	22.6274i	1.02116i	0.859830	+	0.510581i	$0.170569\pi$
$491$	22.6274i	1.02116i	−0.859830	+	0.510581i	$0.829431\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	7.39388	0.329023
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 29.8434i	− 1.32278i	−0.750040	−	0.661392i	$-0.769966\pi$
$509$	− 29.8434i	− 1.32278i	0.750040	−	0.661392i	$-0.230034\pi$
$510$	0	0
$511$	−20.0847	−0.888496
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 0.221513i	− 0.00976103i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−5.08540	−0.219861
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 9.69241i	− 0.417482i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	35.7980	1.52228
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 21.8434i	− 0.925533i	−0.886480	−	0.462767i	$-0.846857\pi$
$557$	− 21.8434i	− 0.925533i	0.886480	−	0.462767i	$-0.153143\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	38.1051i	1.60594i	0.596020	+	0.802970i	$0.296748\pi$
$563$	38.1051i	1.60594i	−0.596020	+	0.802970i	$0.703252\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	29.3939	1.22368	0.611842	−	0.790980i	$-0.290429\pi$
$577$	29.3939	1.22368	0.611842	+	0.790980i	$0.290429\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	35.5051i	1.47300i
$582$	0	0
$583$	26.1557	1.08326
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 45.2548i	− 1.86787i	−0.357447	−	0.933933i	$-0.616353\pi$
$587$	− 45.2548i	− 1.86787i	0.357447	−	0.933933i	$-0.383647\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	48.9898	1.99834	0.999168	−	0.0407909i	$-0.0129877\pi$
$601$	48.9898	1.99834	0.999168	+	0.0407909i	$0.0129877\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 0.449490i	− 0.0182744i
$606$	0	0
$607$	−46.7333	−1.89684	−0.948422	−	0.317010i	$-0.897321\pi$
$607$	−46.7333	−1.89684	−0.948422	+	0.317010i	$0.897321\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	22.0102	0.880408
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−31.8912	−1.26957	−0.634785	−	0.772689i	$-0.718911\pi$
$631$	−31.8912	−1.26957	−0.634785	+	0.772689i	$0.718911\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 8.54950i	− 0.339276i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−39.1918	−1.53841
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 21.7526i	− 0.851243i	−0.904901	−	0.425622i	$-0.860055\pi$
$653$	− 21.7526i	− 0.851243i	0.904901	−	0.425622i	$-0.139945\pi$
$654$	0	0
$655$	−10.1708	−0.397406
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 45.2548i	− 1.76288i	−0.472298	−	0.881439i	$-0.656575\pi$
$659$	− 45.2548i	− 1.76288i	0.472298	−	0.881439i	$-0.343425\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 38.7423i	− 1.48899i	−0.667628	−	0.744495i	$-0.732690\pi$
$677$	− 38.7423i	− 1.48899i	0.667628	−	0.744495i	$-0.267310\pi$
$678$	0	0
$679$	−4.09978	−0.157335
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 51.9615i	− 1.98825i	−0.108227	−	0.994126i	$-0.534517\pi$
$683$	− 51.9615i	− 1.98825i	0.108227	−	0.994126i	$-0.465483\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 52.9444i	− 1.99968i	−0.0178345	−	0.999841i	$-0.505677\pi$
$701$	− 52.9444i	− 1.99968i	0.0178345	−	0.999841i	$-0.494323\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 33.7196i	− 1.26816i
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−1.01021	−0.0376220
$722$	0	0
$723$	0	0
$724$	0	0
$725$	44.8536i	1.66582i
$726$	0	0
$727$	39.5193	1.46569	0.732845	−	0.680395i	$-0.238192\pi$
$727$	39.5193	1.46569	0.732845	+	0.680395i	$0.238192\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	8.60612	0.315304
$746$	0	0
$747$	0	0
$748$	0	0
$749$	23.1918i	0.847411i
$750$	0	0
$751$	45.1762	1.64850	0.824251	−	0.566224i	$-0.191596\pi$
$751$	45.1762	1.64850	0.824251	+	0.566224i	$0.191596\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 7.14972i	− 0.260205i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	48.9898	1.76662	0.883309	−	0.468792i	$-0.155311\pi$
$769$	48.9898	1.76662	0.883309	+	0.468792i	$0.155311\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.7423i	0.817985i	0.912538	+	0.408993i	$0.134120\pi$
$773$	22.7423i	0.817985i	−0.912538	+	0.408993i	$0.865880\pi$
$774$	0	0
$775$	−17.3061	−0.621653
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 50.2474i	− 1.77986i	−0.456101	−	0.889928i	$-0.650754\pi$
$797$	− 50.2474i	− 1.77986i	0.456101	−	0.889928i	$-0.349246\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 33.9411i	− 1.19776i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11.9546i	0.417218i	0.977999	+	0.208609i	$0.0668936\pi$
$821$	11.9546i	0.417218i	−0.977999	+	0.208609i	$0.933106\pi$
$822$	0	0
$823$	8.27820	0.288560	0.144280	−	0.989537i	$-0.453913\pi$
$823$	8.27820	0.288560	0.144280	+	0.989537i	$0.453913\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 56.5685i	− 1.96708i	−0.180688	−	0.983540i	$-0.557832\pi$
$827$	− 56.5685i	− 1.96708i	0.180688	−	0.983540i	$-0.442168\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−58.3939	−2.01358
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.84337i	0.201018i
$846$	0	0
$847$	−2.04989	−0.0704350
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−11.7980	−0.401143
$866$	0	0
$867$	0	0
$868$	0	0
$869$	60.4949i	2.05215i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.02788i	0.305198i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−38.9898	−1.30768
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−5.08540	−0.169986
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 33.7196i	− 1.12461i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−60.0000	−1.98571
$914$	0	0
$915$	0	0
$916$	0	0
$917$	46.3837i	1.53172i
$918$	0	0
$919$	−59.6041	−1.96616	−0.983078	−	0.183187i	$-0.941359\pi$
$919$	−59.6041	−1.96616	−0.983078	+	0.183187i	$0.941359\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 60.9444i	− 1.98673i	−0.115003	−	0.993365i	$-0.536688\pi$
$941$	− 60.9444i	− 1.98673i	0.115003	−	0.993365i	$-0.463312\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 56.5685i	− 1.83823i	−0.393989	−	0.919115i	$-0.628905\pi$
$947$	− 56.5685i	− 1.83823i	0.393989	−	0.919115i	$-0.371095\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−17.9898	−0.580316
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.40408i	0.141772i
$966$	0	0
$967$	−47.3047	−1.52122	−0.760609	−	0.649211i	$-0.775099\pi$
$967$	−47.3047	−1.52122	−0.760609	+	0.649211i	$0.775099\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.46410i	0.111168i	0.998454	+	0.0555842i	$0.0177021\pi$
$971$	3.46410i	0.111168i	−0.998454	+	0.0555842i	$0.982298\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	12.2020	0.388789
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	23.6917	0.752591	0.376296	−	0.926500i	$-0.377198\pi$
$991$	23.6917	0.752591	0.376296	+	0.926500i	$0.377198\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	12.2351i	0.387879i
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.d.q.2305.6		8
3.2	odd	2	inner	4608.2.d.q.2305.4		8
4.3	odd	2	inner	4608.2.d.q.2305.5		8
8.3	odd	2	inner	4608.2.d.q.2305.3		8
8.5	even	2	inner	4608.2.d.q.2305.4		8
12.11	even	2	inner	4608.2.d.q.2305.3		8
16.3	odd	4		4608.2.a.s.1.4	yes	4
16.5	even	4		4608.2.a.bc.1.1	yes	4
16.11	odd	4		4608.2.a.bc.1.2	yes	4
16.13	even	4		4608.2.a.s.1.3	✓	4
24.5	odd	2	CM	4608.2.d.q.2305.6		8
24.11	even	2	inner	4608.2.d.q.2305.5		8
48.5	odd	4		4608.2.a.s.1.3	✓	4
48.11	even	4		4608.2.a.s.1.4	yes	4
48.29	odd	4		4608.2.a.bc.1.1	yes	4
48.35	even	4		4608.2.a.bc.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.a.s.1.3	✓	4	16.13	even	4
4608.2.a.s.1.3	✓	4	48.5	odd	4
4608.2.a.s.1.4	yes	4	16.3	odd	4
4608.2.a.s.1.4	yes	4	48.11	even	4
4608.2.a.bc.1.1	yes	4	16.5	even	4
4608.2.a.bc.1.1	yes	4	48.29	odd	4
4608.2.a.bc.1.2	yes	4	16.11	odd	4
4608.2.a.bc.1.2	yes	4	48.35	even	4
4608.2.d.q.2305.3		8	8.3	odd	2	inner
4608.2.d.q.2305.3		8	12.11	even	2	inner
4608.2.d.q.2305.4		8	3.2	odd	2	inner
4608.2.d.q.2305.4		8	8.5	even	2	inner
4608.2.d.q.2305.5		8	4.3	odd	2	inner
4608.2.d.q.2305.5		8	24.11	even	2	inner
4608.2.d.q.2305.6		8	1.1	even	1	trivial
4608.2.d.q.2305.6		8	24.5	odd	2	CM

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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.449490i q^{5} +2.04989 q^{7} +O(q^{10})\)	\(q+0.449490i q^{5} +2.04989 q^{7} +3.46410i q^{11} +4.79796 q^{25} +9.34847i q^{29} -3.60697 q^{31} +0.921404i q^{35} -2.79796 q^{49} -7.55051i q^{53} -1.55708 q^{55} +11.3137i q^{59} -9.79796 q^{73} +7.10102i q^{77} +17.4634 q^{79} +17.3205i q^{83} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 40 q^{25} + 56 q^{49} - 16 q^{97}+O(q^{100}) \) 8 * q - 40 * q^25 + 56 * q^49 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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