LMFDB - Embedded newform 441.3.d.c.244.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,3,Mod(244,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("441.244");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	441.d (of order $2$, degree $1$, 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:	$12.0163796583$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 7 $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			244.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	441.244
Dual form			441.3.d.c.244.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{4} +O(q^{10})$	$q-4.00000 q^{4} +12.1244i q^{13} +16.0000 q^{16} -36.3731i q^{19} +25.0000 q^{25} -60.6218i q^{31} +73.0000 q^{37} +61.0000 q^{43} -48.4974i q^{52} -96.9948i q^{61} -64.0000 q^{64} -13.0000 q^{67} +109.119i q^{73} +145.492i q^{76} +11.0000 q^{79} -193.990i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4}+O(q^{10}) $ 2 * q - 8 * q^4	$ 2 q - 8 q^{4} + 32 q^{16} + 50 q^{25} + 146 q^{37} + 122 q^{43} - 128 q^{64} - 26 q^{67} + 22 q^{79}+O(q^{100}) $ 2 * q - 8 * q^4 + 32 * q^16 + 50 * q^25 + 146 * q^37 + 122 * q^43 - 128 * q^64 - 26 * q^67 + 22 * q^79

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$-1$	$1$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−4.00000	−1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	12.1244i	0.932643i	0.884615	+	0.466321i	$0.154421\pi$
$13$	12.1244i	0.932643i	−0.884615	+	0.466321i	$0.845579\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	− 36.3731i	− 1.91437i	−0.289474	−	0.957186i	$-0.593480\pi$
$19$	− 36.3731i	− 1.91437i	0.289474	−	0.957186i	$-0.406520\pi$
$20$	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
$25$	25.0000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	− 60.6218i	− 1.95554i	−0.209677	−	0.977771i	$-0.567241\pi$
$31$	− 60.6218i	− 1.95554i	0.209677	−	0.977771i	$-0.432759\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	73.0000	1.97297	0.986486	−	0.163843i	$-0.0523889\pi$
$37$	73.0000	1.97297	0.986486	+	0.163843i	$0.0523889\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	61.0000	1.41860	0.709302	−	0.704904i	$-0.249010\pi$
$43$	61.0000	1.41860	0.709302	+	0.704904i	$0.249010\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	− 48.4974i	− 0.932643i
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	− 96.9948i	− 1.59008i	−0.606557	−	0.795040i	$-0.707450\pi$
$61$	− 96.9948i	− 1.59008i	0.606557	−	0.795040i	$-0.292550\pi$
$62$	0	0
$63$	0	0
$64$	−64.0000	−1.00000
$65$	0	0
$66$	0	0
$67$	−13.0000	−0.194030	−0.0970149	−	0.995283i	$-0.530929\pi$
$67$	−13.0000	−0.194030	−0.0970149	+	0.995283i	$0.530929\pi$
$68$	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
$73$	109.119i	1.49478i	0.664384	+	0.747392i	$0.268694\pi$
$73$	109.119i	1.49478i	−0.664384	+	0.747392i	$0.731306\pi$
$74$	0	0
$75$	0	0
$76$	145.492i	1.91437i
$77$	0	0
$78$	0	0
$79$	11.0000	0.139241	0.0696203	−	0.997574i	$-0.477821\pi$
$79$	11.0000	0.139241	0.0696203	+	0.997574i	$0.477821\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 193.990i	− 1.99989i	−0.0103093	−	0.999947i	$-0.503282\pi$
$97$	− 193.990i	− 1.99989i	0.0103093	−	0.999947i	$-0.496718\pi$
$98$	0	0
$99$	0	0
$100$	−100.000	−1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	133.368i	1.29483i	0.762136	+	0.647417i	$0.224151\pi$
$103$	133.368i	1.29483i	−0.762136	+	0.647417i	$0.775849\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−71.0000	−0.651376	−0.325688	−	0.945477i	$-0.605596\pi$
$109$	−71.0000	−0.651376	−0.325688	+	0.945477i	$0.605596\pi$
$110$	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
$121$	−121.000	−1.00000
$122$	0	0
$123$	0	0
$124$	242.487i	1.95554i
$125$	0	0
$126$	0	0
$127$	107.000	0.842520	0.421260	−	0.906940i	$-0.361588\pi$
$127$	107.000	0.842520	0.421260	+	0.906940i	$0.361588\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	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
$139$	− 157.617i	− 1.13393i	−0.823741	−	0.566966i	$-0.808117\pi$
$139$	− 157.617i	− 1.13393i	0.823741	−	0.566966i	$-0.191883\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−292.000	−1.97297
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	286.000	1.89404	0.947020	−	0.321175i	$-0.104078\pi$
$151$	286.000	1.89404	0.947020	+	0.321175i	$0.104078\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	290.985i	1.85340i	0.375796	+	0.926702i	$0.377369\pi$
$157$	290.985i	1.85340i	−0.375796	+	0.926702i	$0.622631\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−262.000	−1.60736	−0.803681	−	0.595060i	$-0.797128\pi$
$163$	−262.000	−1.60736	−0.803681	+	0.595060i	$0.797128\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	22.0000	0.130178
$170$	0	0
$171$	0	0
$172$	−244.000	−1.41860
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	181.865i	1.00478i	0.864641	+	0.502390i	$0.167546\pi$
$181$	181.865i	1.00478i	−0.864641	+	0.502390i	$0.832454\pi$
$182$	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
$193$	−239.000	−1.23834	−0.619171	−	0.785256i	$-0.712531\pi$
$193$	−239.000	−1.23834	−0.619171	+	0.785256i	$0.712531\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	96.9948i	0.487411i	0.969849	+	0.243706i	$0.0783631\pi$
$199$	96.9948i	0.487411i	−0.969849	+	0.243706i	$0.921637\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	193.990i	0.932643i
$209$	0	0
$210$	0	0
$211$	166.000	0.786730	0.393365	−	0.919382i	$-0.371311\pi$
$211$	166.000	0.786730	0.393365	+	0.919382i	$0.371311\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 290.985i	− 1.30486i	−0.757848	−	0.652432i	$-0.773749\pi$
$223$	− 290.985i	− 1.30486i	0.757848	−	0.652432i	$-0.226251\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 206.114i	− 0.900061i	−0.893013	−	0.450031i	$-0.851413\pi$
$229$	− 206.114i	− 0.900061i	0.893013	−	0.450031i	$-0.148587\pi$
$230$	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
$241$	− 387.979i	− 1.60987i	−0.593361	−	0.804936i	$-0.702199\pi$
$241$	− 387.979i	− 1.60987i	0.593361	−	0.804936i	$-0.297801\pi$
$242$	0	0
$243$	0	0
$244$	387.979i	1.59008i
$245$	0	0
$246$	0	0
$247$	441.000	1.78543
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	256.000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	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
$265$	0	0
$266$	0	0
$267$	0	0
$268$	52.0000	0.194030
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	− 484.974i	− 1.78957i	−0.446494	−	0.894786i	$-0.647328\pi$
$271$	− 484.974i	− 1.78957i	0.446494	−	0.894786i	$-0.352672\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	407.000	1.46931	0.734657	−	0.678439i	$-0.237343\pi$
$277$	407.000	1.46931	0.734657	+	0.678439i	$0.237343\pi$
$278$	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
$283$	− 230.363i	− 0.814003i	−0.913428	−	0.407001i	$-0.866574\pi$
$283$	− 230.363i	− 0.814003i	0.913428	−	0.407001i	$-0.133426\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	− 436.477i	− 1.49478i
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	− 581.969i	− 1.91437i
$305$	0	0
$306$	0	0
$307$	− 60.6218i	− 0.197465i	−0.995114	−	0.0987325i	$-0.968521\pi$
$307$	− 60.6218i	− 0.197465i	0.995114	−	0.0987325i	$-0.0314788\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	181.865i	0.581039i	0.956869	+	0.290520i	$0.0938282\pi$
$313$	181.865i	0.581039i	−0.956869	+	0.290520i	$0.906172\pi$
$314$	0	0
$315$	0	0
$316$	−44.0000	−0.139241
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	303.109i	0.932643i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	661.000	1.99698	0.998489	−	0.0549442i	$-0.0174981\pi$
$331$	661.000	1.99698	0.998489	+	0.0549442i	$0.0174981\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	649.000	1.92582	0.962908	−	0.269830i	$-0.0869675\pi$
$337$	649.000	1.92582	0.962908	+	0.269830i	$0.0869675\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	484.974i	1.38961i	0.719198	+	0.694805i	$0.244510\pi$
$349$	484.974i	1.38961i	−0.719198	+	0.694805i	$0.755490\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	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
$361$	−962.000	−2.66482
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 545.596i	− 1.48664i	−0.668937	−	0.743319i	$-0.733251\pi$
$367$	− 545.596i	− 1.48664i	0.668937	−	0.743319i	$-0.266749\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−577.000	−1.54692	−0.773458	−	0.633847i	$-0.781475\pi$
$373$	−577.000	−1.54692	−0.773458	+	0.633847i	$0.781475\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−611.000	−1.61214	−0.806069	−	0.591822i	$-0.798409\pi$
$379$	−611.000	−1.61214	−0.806069	+	0.591822i	$0.798409\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	775.959i	1.99989i
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 666.840i	− 1.67970i	−0.542821	−	0.839848i	$-0.682644\pi$
$397$	− 666.840i	− 1.67970i	0.542821	−	0.839848i	$-0.317356\pi$
$398$	0	0
$399$	0	0
$400$	400.000	1.00000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	735.000	1.82382
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	278.860i	0.681810i	0.940098	+	0.340905i	$0.110733\pi$
$409$	278.860i	0.681810i	−0.940098	+	0.340905i	$0.889267\pi$
$410$	0	0
$411$	0	0
$412$	− 533.472i	− 1.29483i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−481.000	−1.14252	−0.571259	−	0.820770i	$-0.693545\pi$
$421$	−481.000	−1.14252	−0.571259	+	0.820770i	$0.693545\pi$
$422$	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
$433$	788.083i	1.82005i	0.414550	+	0.910027i	$0.363939\pi$
$433$	788.083i	1.82005i	−0.414550	+	0.910027i	$0.636061\pi$
$434$	0	0
$435$	0	0
$436$	284.000	0.651376
$437$	0	0
$438$	0	0
$439$	872.954i	1.98850i	0.107062	+	0.994252i	$0.465856\pi$
$439$	872.954i	1.98850i	−0.107062	+	0.994252i	$0.534144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	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
$457$	−767.000	−1.67834	−0.839168	−	0.543872i	$-0.816958\pi$
$457$	−767.000	−1.67834	−0.839168	+	0.543872i	$0.816958\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	−397.000	−0.857451	−0.428726	−	0.903435i	$-0.641037\pi$
$463$	−397.000	−0.857451	−0.428726	+	0.903435i	$0.641037\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 909.327i	− 1.91437i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	885.078i	1.84008i
$482$	0	0
$483$	0	0
$484$	484.000	1.00000
$485$	0	0
$486$	0	0
$487$	349.000	0.716632	0.358316	−	0.933600i	$-0.383351\pi$
$487$	349.000	0.716632	0.358316	+	0.933600i	$0.383351\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	− 969.948i	− 1.95554i
$497$	0	0
$498$	0	0
$499$	−851.000	−1.70541	−0.852705	−	0.522392i	$-0.825040\pi$
$499$	−851.000	−1.70541	−0.852705	+	0.522392i	$0.825040\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−428.000	−0.842520
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	1030.57i	1.97050i	0.171128	+	0.985249i	$0.445259\pi$
$523$	1030.57i	1.97050i	−0.171128	+	0.985249i	$0.554741\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−529.000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−241.000	−0.445471	−0.222736	−	0.974879i	$-0.571499\pi$
$541$	−241.000	−0.445471	−0.222736	+	0.974879i	$0.571499\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	506.000	0.925046	0.462523	−	0.886607i	$-0.346944\pi$
$547$	506.000	0.925046	0.462523	+	0.886607i	$0.346944\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	630.466i	1.13393i
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	739.586i	1.32305i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	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
$571$	−181.000	−0.316988	−0.158494	−	0.987360i	$-0.550664\pi$
$571$	−181.000	−0.316988	−0.158494	+	0.987360i	$0.550664\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1151.81i	1.99621i	0.0615251	+	0.998106i	$0.480404\pi$
$577$	1151.81i	1.99621i	−0.0615251	+	0.998106i	$0.519596\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	−2205.00	−3.74363
$590$	0	0
$591$	0	0
$592$	1168.00	1.97297
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	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
$601$	84.8705i	0.141215i	0.997504	+	0.0706077i	$0.0224939\pi$
$601$	84.8705i	0.141215i	−0.997504	+	0.0706077i	$0.977506\pi$
$602$	0	0
$603$	0	0
$604$	−1144.00	−1.89404
$605$	0	0
$606$	0	0
$607$	− 254.611i	− 0.419459i	−0.977759	−	0.209729i	$-0.932742\pi$
$607$	− 254.611i	− 0.419459i	0.977759	−	0.209729i	$-0.0672583\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1126.00	−1.83687	−0.918434	−	0.395574i	$-0.870546\pi$
$613$	−1126.00	−1.83687	−0.918434	+	0.395574i	$0.870546\pi$
$614$	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
$619$	− 424.352i	− 0.685545i	−0.939418	−	0.342773i	$-0.888634\pi$
$619$	− 424.352i	− 0.685545i	0.939418	−	0.342773i	$-0.111366\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	0	0
$628$	− 1163.94i	− 1.85340i
$629$	0	0
$630$	0	0
$631$	674.000	1.06815	0.534073	−	0.845438i	$-0.320661\pi$
$631$	674.000	1.06815	0.534073	+	0.845438i	$0.320661\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	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
$643$	351.606i	0.546822i	0.961897	+	0.273411i	$0.0881518\pi$
$643$	351.606i	0.546822i	−0.961897	+	0.273411i	$0.911848\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	1048.00	1.60736
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	763.834i	1.15557i	0.816188	+	0.577787i	$0.196084\pi$
$661$	763.834i	1.15557i	−0.816188	+	0.577787i	$0.803916\pi$
$662$	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
$673$	−23.0000	−0.0341753	−0.0170877	−	0.999854i	$-0.505439\pi$
$673$	−23.0000	−0.0341753	−0.0170877	+	0.999854i	$0.505439\pi$
$674$	0	0
$675$	0	0
$676$	−88.0000	−0.130178
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	976.000	1.41860
$689$	0	0
$690$	0	0
$691$	933.575i	1.35105i	0.737337	+	0.675525i	$0.236083\pi$
$691$	933.575i	1.35105i	−0.737337	+	0.675525i	$0.763917\pi$
$692$	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
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	− 2655.23i	− 3.77700i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	934.000	1.31735	0.658674	−	0.752428i	$-0.271118\pi$
$709$	934.000	1.31735	0.658674	+	0.752428i	$0.271118\pi$
$710$	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	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	− 727.461i	− 1.00478i
$725$	0	0
$726$	0	0
$727$	− 1103.32i	− 1.51763i	−0.651307	−	0.758815i	$-0.725779\pi$
$727$	− 1103.32i	− 1.51763i	0.651307	−	0.758815i	$-0.274221\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 375.855i	− 0.512763i	−0.966576	−	0.256381i	$-0.917470\pi$
$733$	− 375.855i	− 0.512763i	0.966576	−	0.256381i	$-0.0825303\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1331.00	1.80108	0.900541	−	0.434771i	$-0.143171\pi$
$739$	1331.00	1.80108	0.900541	+	0.434771i	$0.143171\pi$
$740$	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
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−179.000	−0.238349	−0.119174	−	0.992873i	$-0.538025\pi$
$751$	−179.000	−0.238349	−0.119174	+	0.992873i	$0.538025\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	838.000	1.10700	0.553501	−	0.832849i	$-0.313292\pi$
$757$	838.000	1.10700	0.553501	+	0.832849i	$0.313292\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1273.06i	1.65547i	0.561118	+	0.827736i	$0.310371\pi$
$769$	1273.06i	1.65547i	−0.561118	+	0.827736i	$0.689629\pi$
$770$	0	0
$771$	0	0
$772$	956.000	1.23834
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	− 1515.54i	− 1.95554i
$776$	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
$787$	− 193.990i	− 0.246493i	−0.992376	−	0.123246i	$-0.960669\pi$
$787$	− 193.990i	− 0.246493i	0.992376	−	0.123246i	$-0.0393305\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1176.00	1.48298
$794$	0	0
$795$	0	0
$796$	− 387.979i	− 0.487411i
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	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
$811$	581.969i	0.717594i	0.933416	+	0.358797i	$0.116813\pi$
$811$	581.969i	0.717594i	−0.933416	+	0.358797i	$0.883187\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2218.76i	− 2.71574i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	1058.00	1.28554	0.642770	−	0.766059i	$-0.277785\pi$
$823$	1058.00	1.28554	0.642770	+	0.766059i	$0.277785\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	− 400.104i	− 0.482634i	−0.970446	−	0.241317i	$-0.922421\pi$
$829$	− 400.104i	− 0.482634i	0.970446	−	0.241317i	$-0.0775794\pi$
$830$	0	0
$831$	0	0
$832$	− 775.959i	− 0.932643i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−841.000	−1.00000
$842$	0	0
$843$	0	0
$844$	−664.000	−0.786730
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1636.79i	1.91886i	0.281946	+	0.959430i	$0.409020\pi$
$853$	1636.79i	1.91886i	−0.281946	+	0.959430i	$0.590980\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	969.948i	1.12916i	0.825378	+	0.564580i	$0.190962\pi$
$859$	969.948i	1.12916i	−0.825378	+	0.564580i	$0.809038\pi$
$860$	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
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 157.617i	− 0.180961i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−598.000	−0.681870	−0.340935	−	0.940087i	$-0.610744\pi$
$877$	−598.000	−0.681870	−0.340935	+	0.940087i	$0.610744\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	−443.000	−0.501699	−0.250849	−	0.968026i	$-0.580710\pi$
$883$	−443.000	−0.501699	−0.250849	+	0.968026i	$0.580710\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	1163.94i	1.30486i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1667.00	−1.83793	−0.918964	−	0.394342i	$-0.870972\pi$
$907$	−1667.00	−1.83793	−0.918964	+	0.394342i	$0.870972\pi$
$908$	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
$913$	0	0
$914$	0	0
$915$	0	0
$916$	824.456i	0.900061i
$917$	0	0
$918$	0	0
$919$	1837.00	1.99891	0.999456	−	0.0329825i	$-0.0105006\pi$
$919$	1837.00	1.99891	0.999456	+	0.0329825i	$0.0105006\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	1825.00	1.97297
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1758.03i	− 1.87623i	−0.346318	−	0.938117i	$-0.612568\pi$
$937$	− 1758.03i	− 1.87623i	0.346318	−	0.938117i	$-0.387432\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	−1323.00	−1.39410
$950$	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
$961$	−2714.00	−2.82414
$962$	0	0
$963$	0	0
$964$	1551.92i	1.60987i
$965$	0	0
$966$	0	0
$967$	1787.00	1.84798	0.923992	−	0.382412i	$-0.124907\pi$
$967$	1787.00	1.84798	0.923992	+	0.382412i	$0.124907\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	− 1551.92i	− 1.59008i
$977$	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	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−1764.00	−1.78543
$989$	0	0
$990$	0	0
$991$	1693.00	1.70838	0.854188	−	0.519965i	$-0.174055\pi$
$991$	1693.00	1.70838	0.854188	+	0.519965i	$0.174055\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1952.02i	1.95789i	0.204112	+	0.978947i	$0.434569\pi$
$997$	1952.02i	1.95789i	−0.204112	+	0.978947i	$0.565431\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.3.d.c.244.2		2
3.2	odd	2	CM	441.3.d.c.244.2		2
7.2	even	3		63.3.m.b.10.1	✓	2
7.3	odd	6		63.3.m.b.19.1	yes	2
7.4	even	3		441.3.m.c.19.1		2
7.5	odd	6		441.3.m.c.325.1		2
7.6	odd	2	inner	441.3.d.c.244.1		2
21.2	odd	6		63.3.m.b.10.1	✓	2
21.5	even	6		441.3.m.c.325.1		2
21.11	odd	6		441.3.m.c.19.1		2
21.17	even	6		63.3.m.b.19.1	yes	2
21.20	even	2	inner	441.3.d.c.244.1		2
28.3	even	6		1008.3.cg.d.145.1		2
28.23	odd	6		1008.3.cg.d.577.1		2
84.23	even	6		1008.3.cg.d.577.1		2
84.59	odd	6		1008.3.cg.d.145.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.3.m.b.10.1	✓	2	7.2	even	3
63.3.m.b.10.1	✓	2	21.2	odd	6
63.3.m.b.19.1	yes	2	7.3	odd	6
63.3.m.b.19.1	yes	2	21.17	even	6
441.3.d.c.244.1		2	7.6	odd	2	inner
441.3.d.c.244.1		2	21.20	even	2	inner
441.3.d.c.244.2		2	1.1	even	1	trivial
441.3.d.c.244.2		2	3.2	odd	2	CM
441.3.m.c.19.1		2	7.4	even	3
441.3.m.c.19.1		2	21.11	odd	6
441.3.m.c.325.1		2	7.5	odd	6
441.3.m.c.325.1		2	21.5	even	6
1008.3.cg.d.145.1		2	28.3	even	6
1008.3.cg.d.145.1		2	84.59	odd	6
1008.3.cg.d.577.1		2	28.23	odd	6
1008.3.cg.d.577.1		2	84.23	even	6

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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	441.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.00000 q^{4} +O(q^{10})\)	\(q-4.00000 q^{4} +12.1244i q^{13} +16.0000 q^{16} -36.3731i q^{19} +25.0000 q^{25} -60.6218i q^{31} +73.0000 q^{37} +61.0000 q^{43} -48.4974i q^{52} -96.9948i q^{61} -64.0000 q^{64} -13.0000 q^{67} +109.119i q^{73} +145.492i q^{76} +11.0000 q^{79} -193.990i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4}+O(q^{10}) \) 2 * q - 8 * q^4	\( 2 q - 8 q^{4} + 32 q^{16} + 50 q^{25} + 146 q^{37} + 122 q^{43} - 128 q^{64} - 26 q^{67} + 22 q^{79}+O(q^{100}) \) 2 * q - 8 * q^4 + 32 * q^16 + 50 * q^25 + 146 * q^37 + 122 * q^43 - 128 * q^64 - 26 * q^67 + 22 * q^79

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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