LMFDB - Embedded newform 3024.2.b.q.1567.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(1567,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3024.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.b (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:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-6}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 24x^{2} + 81 $ x^4 + 24*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1567.1
Root			$-4.46512i$ of defining polynomial
Character	$\chi$	$=$	3024.1567
Dual form			3024.2.b.q.1567.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.46512i q^{5} +2.64575 q^{7} +O(q^{10})$	$q-4.46512i q^{5} +2.64575 q^{7} +4.46512i q^{11} -7.34847i q^{17} -8.64575 q^{19} +2.88335i q^{23} -14.9373 q^{25} -8.29150 q^{31} -11.8136i q^{35} -3.93725 q^{37} -2.88335i q^{41} +7.00000 q^{49} +19.9373 q^{55} -10.2318i q^{71} +11.8136i q^{77} -32.8118 q^{85} -14.9771i q^{89} +38.6043i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 24 q^{19} - 28 q^{25} - 12 q^{31} + 16 q^{37} + 28 q^{49} + 48 q^{55} - 36 q^{85}+O(q^{100}) $ 4 * q - 24 * q^19 - 28 * q^25 - 12 * q^31 + 16 * q^37 + 28 * q^49 + 48 * q^55 - 36 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 4.46512i	− 1.99686i	−0.0560116	−	0.998430i	$-0.517838\pi$
$5$	− 4.46512i	− 1.99686i	0.0560116	−	0.998430i	$-0.482162\pi$
$6$	0	0
$7$	2.64575	1.00000
$7$	2.64575	1.00000
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.46512i	1.34628i	0.739514	+	0.673141i	$0.235055\pi$
$11$	4.46512i	1.34628i	−0.739514	+	0.673141i	$0.764945\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.34847i	− 1.78227i	−0.453743	−	0.891133i	$-0.649911\pi$
$17$	− 7.34847i	− 1.78227i	0.453743	−	0.891133i	$-0.350089\pi$
$18$	0	0
$19$	−8.64575	−1.98347	−0.991736	−	0.128298i	$-0.959049\pi$
$19$	−8.64575	−1.98347	−0.991736	+	0.128298i	$0.959049\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.88335i	0.601221i	0.953747	+	0.300610i	$0.0971904\pi$
$23$	2.88335i	0.601221i	−0.953747	+	0.300610i	$0.902810\pi$
$24$	0	0
$25$	−14.9373	−2.98745
$26$	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
$31$	−8.29150	−1.48920	−0.744599	−	0.667512i	$-0.767359\pi$
$31$	−8.29150	−1.48920	−0.744599	+	0.667512i	$0.767359\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 11.8136i	− 1.99686i
$36$	0	0
$37$	−3.93725	−0.647281	−0.323640	−	0.946180i	$-0.604907\pi$
$37$	−3.93725	−0.647281	−0.323640	+	0.946180i	$0.604907\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.88335i	− 0.450304i	−0.974324	−	0.225152i	$-0.927712\pi$
$41$	− 2.88335i	− 0.450304i	0.974324	−	0.225152i	$-0.0722879\pi$
$42$	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
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	19.9373	2.68834
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	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
$71$	− 10.2318i	− 1.21429i	−0.794590	−	0.607147i	$-0.792314\pi$
$71$	− 10.2318i	− 1.21429i	0.794590	−	0.607147i	$-0.207686\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.8136i	1.34628i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−32.8118	−3.55894
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 14.9771i	− 1.58757i	−0.608198	−	0.793785i	$-0.708107\pi$
$89$	− 14.9771i	− 1.58757i	0.608198	−	0.793785i	$-0.291893\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	38.6043i	3.96072i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.34847i	0.731200i	0.930772	+	0.365600i	$0.119136\pi$
$101$	7.34847i	0.731200i	−0.930772	+	0.365600i	$0.880864\pi$
$102$	0	0
$103$	9.70850	0.956607	0.478303	−	0.878195i	$-0.341252\pi$
$103$	9.70850	0.956607	0.478303	+	0.878195i	$0.341252\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 19.4422i	− 1.87955i	−0.341793	−	0.939775i	$-0.611034\pi$
$107$	− 19.4422i	− 1.87955i	0.341793	−	0.939775i	$-0.388966\pi$
$108$	0	0
$109$	−10.8745	−1.04159	−0.520794	−	0.853682i	$-0.674364\pi$
$109$	−10.8745	−1.04159	−0.520794	+	0.853682i	$0.674364\pi$
$110$	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
$115$	12.8745	1.20055
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 19.4422i	− 1.78227i
$120$	0	0
$121$	−8.93725	−0.812478
$122$	0	0
$123$	0	0
$124$	0	0
$125$	44.3710i	3.96866i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−22.8745	−1.98347
$134$	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
$139$	−21.1660	−1.79528	−0.897639	−	0.440732i	$-0.854719\pi$
$139$	−21.1660	−1.79528	−0.897639	+	0.440732i	$0.854719\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	37.0225i	2.97372i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.62864i	0.601221i
$162$	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
$173$	23.9073i	1.81764i	0.417187	+	0.908821i	$0.363016\pi$
$173$	23.9073i	1.81764i	−0.417187	+	0.908821i	$0.636984\pi$
$174$	0	0
$175$	−39.5203	−2.98745
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.4422i	1.45318i	0.687071	+	0.726590i	$0.258896\pi$
$179$	19.4422i	1.45318i	−0.687071	+	0.726590i	$0.741104\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	17.5803i	1.29253i
$186$	0	0
$187$	32.8118	2.39943
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 26.5105i	− 1.91823i	−0.283010	−	0.959117i	$-0.591333\pi$
$191$	− 26.5105i	− 1.91823i	0.283010	−	0.959117i	$-0.408667\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	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
$199$	−26.6458	−1.88887	−0.944434	−	0.328702i	$-0.893389\pi$
$199$	−26.6458	−1.88887	−0.944434	+	0.328702i	$0.893389\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−12.8745	−0.899195
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 38.6043i	− 2.67031i
$210$	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
$217$	−21.9373	−1.48920
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	25.2288	1.68944	0.844721	−	0.535207i	$-0.179766\pi$
$223$	25.2288	1.68944	0.844721	+	0.535207i	$0.179766\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	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
$239$	− 19.4422i	− 1.25761i	−0.777562	−	0.628806i	$-0.783544\pi$
$239$	− 19.4422i	− 1.25761i	0.777562	−	0.628806i	$-0.216456\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 31.2558i	− 1.99686i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−12.8745	−0.809413
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 19.1621i	− 1.19530i	−0.801759	−	0.597648i	$-0.796102\pi$
$257$	− 19.1621i	− 1.19530i	0.801759	−	0.597648i	$-0.203898\pi$
$258$	0	0
$259$	−10.4170	−0.647281
$260$	0	0
$261$	0	0
$262$	0	0
$263$	14.9771i	0.923528i	0.887003	+	0.461764i	$0.152783\pi$
$263$	14.9771i	0.923528i	−0.887003	+	0.461764i	$0.847217\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 29.6740i	− 1.80926i	−0.426199	−	0.904629i	$-0.640148\pi$
$269$	− 29.6740i	− 1.80926i	0.426199	−	0.904629i	$-0.359852\pi$
$270$	0	0
$271$	10.5830	0.642872	0.321436	−	0.946931i	$-0.395835\pi$
$271$	10.5830	0.642872	0.321436	+	0.946931i	$0.395835\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 66.6966i	− 4.02195i
$276$	0	0
$277$	8.06275	0.484443	0.242222	−	0.970221i	$-0.422124\pi$
$277$	8.06275	0.484443	0.242222	+	0.970221i	$0.422124\pi$
$278$	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
$283$	−26.4575	−1.57274	−0.786368	−	0.617758i	$-0.788041\pi$
$283$	−26.4575	−1.57274	−0.786368	+	0.617758i	$0.788041\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 7.62864i	− 0.450304i
$288$	0	0
$289$	−37.0000	−2.17647
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 22.0454i	− 1.28791i	−0.765065	−	0.643953i	$-0.777293\pi$
$293$	− 22.0454i	− 1.28791i	0.765065	−	0.643953i	$-0.222707\pi$
$294$	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
$304$	0	0
$305$	0	0
$306$	0	0
$307$	27.3542	1.56119	0.780595	−	0.625038i	$-0.214916\pi$
$307$	27.3542	1.56119	0.780595	+	0.625038i	$0.214916\pi$
$308$	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
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	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
$323$	63.5330i	3.53507i
$324$	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
$337$	30.7490	1.67501	0.837503	−	0.546433i	$-0.184015\pi$
$337$	30.7490	1.67501	0.837503	+	0.546433i	$0.184015\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 37.0225i	− 2.00488i
$342$	0	0
$343$	18.5203	1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.280168i	0.0150402i	0.999972	+	0.00752011i	$0.00239375\pi$
$347$	0.280168i	0.0150402i	−0.999972	+	0.00752011i	$0.997606\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.21040i	0.490220i	0.969495	+	0.245110i	$0.0788241\pi$
$353$	9.21040i	0.490220i	−0.969495	+	0.245110i	$0.921176\pi$
$354$	0	0
$355$	−45.6863	−2.42478
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.4422i	1.02612i	0.858352	+	0.513061i	$0.171488\pi$
$359$	19.4422i	1.02612i	−0.858352	+	0.513061i	$0.828512\pi$
$360$	0	0
$361$	55.7490	2.93416
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−10.7712	−0.562254	−0.281127	−	0.959671i	$-0.590708\pi$
$367$	−10.7712	−0.562254	−0.281127	+	0.959671i	$0.590708\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.12549	0.0582758	0.0291379	−	0.999575i	$-0.490724\pi$
$373$	1.12549	0.0582758	0.0291379	+	0.999575i	$0.490724\pi$
$374$	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
$385$	52.7490	2.68834
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	21.1882	1.07154
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	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
$407$	− 17.5803i	− 0.871423i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−27.9373	−1.36158	−0.680789	−	0.732479i	$-0.738363\pi$
$421$	−27.9373	−1.36158	−0.680789	+	0.732479i	$0.738363\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	109.766i	5.32443i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	32.2772i	1.55474i	0.629044	+	0.777370i	$0.283447\pi$
$431$	32.2772i	1.55474i	−0.629044	+	0.777370i	$0.716553\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 24.9288i	− 1.19250i
$438$	0	0
$439$	5.29150	0.252550	0.126275	−	0.991995i	$-0.459698\pi$
$439$	5.29150	0.252550	0.126275	+	0.991995i	$0.459698\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.06830i	0.335825i	0.985802	+	0.167913i	$0.0537026\pi$
$443$	7.06830i	0.335825i	−0.985802	+	0.167913i	$0.946297\pi$
$444$	0	0
$445$	−66.8745	−3.17016
$446$	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
$451$	12.8745	0.606237
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	42.7490	1.99971	0.999857	−	0.0168929i	$-0.00537742\pi$
$457$	42.7490	1.99971	0.999857	+	0.0168929i	$0.00537742\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 41.7678i	− 1.94532i	−0.232233	−	0.972660i	$-0.574603\pi$
$461$	− 41.7678i	− 1.94532i	0.232233	−	0.972660i	$-0.425397\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	129.144	5.92552
$476$	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	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 41.2075i	− 1.85967i	−0.367981	−	0.929833i	$-0.619951\pi$
$491$	− 41.2075i	− 1.85967i	0.367981	−	0.929833i	$-0.380049\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 27.0709i	− 1.21429i
$498$	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
$505$	32.8118	1.46010
$506$	0	0
$507$	0	0
$508$	0	0
$509$	36.7423i	1.62858i	0.580461	+	0.814288i	$0.302872\pi$
$509$	36.7423i	1.62858i	−0.580461	+	0.814288i	$0.697128\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 43.3496i	− 1.91021i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	37.0225i	1.62199i	0.585056	+	0.810993i	$0.301073\pi$
$521$	37.0225i	1.62199i	−0.585056	+	0.810993i	$0.698927\pi$
$522$	0	0
$523$	−6.16601	−0.269621	−0.134810	−	0.990871i	$-0.543043\pi$
$523$	−6.16601	−0.269621	−0.134810	+	0.990871i	$0.543043\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	60.9299i	2.65415i
$528$	0	0
$529$	14.6863	0.638533
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−86.8118	−3.75320
$536$	0	0
$537$	0	0
$538$	0	0
$539$	31.2558i	1.34628i
$540$	0	0
$541$	43.6863	1.87822	0.939110	−	0.343617i	$-0.111652\pi$
$541$	43.6863	1.87822	0.939110	+	0.343617i	$0.111652\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	48.5559i	2.07991i
$546$	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
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	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
$575$	− 43.0694i	− 1.79612i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	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		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	71.6863	2.95378
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 43.3496i	− 1.78015i	−0.455811	−	0.890077i	$-0.650651\pi$
$593$	− 43.3496i	− 1.78015i	0.455811	−	0.890077i	$-0.349349\pi$
$594$	0	0
$595$	−86.8118	−3.55894
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 38.6043i	− 1.57733i	−0.614824	−	0.788664i	$-0.710773\pi$
$599$	− 38.6043i	− 1.57733i	0.614824	−	0.788664i	$-0.289227\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	39.9059i	1.62240i
$606$	0	0
$607$	−26.4575	−1.07388	−0.536939	−	0.843621i	$-0.680419\pi$
$607$	−26.4575	−1.07388	−0.536939	+	0.843621i	$0.680419\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	7.12549	0.287796	0.143898	−	0.989593i	$-0.454036\pi$
$613$	7.12549	0.287796	0.143898	+	0.989593i	$0.454036\pi$
$614$	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
$619$	−28.4170	−1.14218	−0.571088	−	0.820889i	$-0.693478\pi$
$619$	−28.4170	−1.14218	−0.571088	+	0.820889i	$0.693478\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 39.6257i	− 1.58757i
$624$	0	0
$625$	123.435	4.93741
$626$	0	0
$627$	0	0
$628$	0	0
$629$	28.9328i	1.15363i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	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
$643$	11.4797	0.452717	0.226358	−	0.974044i	$-0.427318\pi$
$643$	11.4797	0.452717	0.226358	+	0.974044i	$0.427318\pi$
$644$	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
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	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
$659$	1.86193i	0.0725305i	0.999342	+	0.0362652i	$0.0115461\pi$
$659$	1.86193i	0.0725305i	−0.999342	+	0.0362652i	$0.988454\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	102.137i	3.96072i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19.7224i	0.757993i	0.925398	+	0.378997i	$0.123731\pi$
$677$	19.7224i	0.757993i	−0.925398	+	0.378997i	$0.876269\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 51.7195i	− 1.97899i	−0.144568	−	0.989495i	$-0.546179\pi$
$683$	− 51.7195i	− 1.97899i	0.144568	−	0.989495i	$-0.453821\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	42.3320	1.61039	0.805193	−	0.593013i	$-0.202062\pi$
$691$	42.3320	1.61039	0.805193	+	0.593013i	$0.202062\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	94.5087i	3.58492i
$696$	0	0
$697$	−21.1882	−0.802562
$698$	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
$703$	34.0405	1.28386
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19.4422i	0.731200i
$708$	0	0
$709$	−40.8745	−1.53507	−0.767537	−	0.641004i	$-0.778518\pi$
$709$	−40.8745	−1.53507	−0.767537	+	0.641004i	$0.778518\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 23.9073i	− 0.895337i
$714$	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
$721$	25.6863	0.956607
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−52.9150	−1.96251	−0.981255	−	0.192715i	$-0.938271\pi$
$727$	−52.9150	−1.96251	−0.981255	+	0.192715i	$0.938271\pi$
$728$	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
$743$	21.7652i	0.798489i	0.916844	+	0.399245i	$0.130728\pi$
$743$	21.7652i	0.798489i	−0.916844	+	0.399245i	$0.869272\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 51.4393i	− 1.87955i
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 36.7423i	− 1.33191i	−0.745992	−	0.665955i	$-0.768024\pi$
$761$	− 36.7423i	− 1.33191i	0.745992	−	0.665955i	$-0.231976\pi$
$762$	0	0
$763$	−28.7712	−1.04159
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 28.6526i	− 1.03056i	−0.857021	−	0.515282i	$-0.827687\pi$
$773$	− 28.6526i	− 1.03056i	0.857021	−	0.515282i	$-0.172313\pi$
$774$	0	0
$775$	123.852	4.44891
$776$	0	0
$777$	0	0
$778$	0	0
$779$	24.9288i	0.893166i
$780$	0	0
$781$	45.6863	1.63478
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−21.1660	−0.754487	−0.377243	−	0.926114i	$-0.623128\pi$
$787$	−21.1660	−0.754487	−0.377243	+	0.926114i	$0.623128\pi$
$788$	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
$797$	24.9288i	0.883022i	0.897256	+	0.441511i	$0.145557\pi$
$797$	24.9288i	0.883022i	−0.897256	+	0.441511i	$0.854443\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	34.0627	1.20055
$806$	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
$811$	11.8340	0.415548	0.207774	−	0.978177i	$-0.433378\pi$
$811$	11.8340	0.415548	0.207774	+	0.978177i	$0.433378\pi$
$812$	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
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 53.3012i	− 1.85346i	−0.375722	−	0.926732i	$-0.622605\pi$
$827$	− 53.3012i	− 1.85346i	0.375722	−	0.926732i	$-0.377395\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 51.4393i	− 1.78227i
$834$	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
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 58.0465i	− 1.99686i
$846$	0	0
$847$	−23.6458	−0.812478
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 11.3525i	− 0.389159i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 56.4647i	− 1.92880i	−0.264450	−	0.964399i	$-0.585190\pi$
$857$	− 56.4647i	− 1.92880i	0.264450	−	0.964399i	$-0.414810\pi$
$858$	0	0
$859$	23.1033	0.788273	0.394137	−	0.919052i	$-0.371044\pi$
$859$	23.1033	0.788273	0.394137	+	0.919052i	$0.371044\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	58.3267i	1.98546i	0.120351	+	0.992731i	$0.461598\pi$
$863$	58.3267i	1.98546i	−0.120351	+	0.992731i	$0.538402\pi$
$864$	0	0
$865$	106.749	3.62958
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	117.395i	3.96866i
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 44.3710i	− 1.49490i	−0.664320	−	0.747448i	$-0.731279\pi$
$881$	− 44.3710i	− 1.49490i	0.664320	−	0.747448i	$-0.268721\pi$
$882$	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
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	86.8118	2.90180
$896$	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
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 58.3267i	− 1.93245i	−0.257702	−	0.966224i	$-0.582965\pi$
$911$	− 58.3267i	− 1.93245i	0.257702	−	0.966224i	$-0.417035\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	58.8118	1.93372
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 36.7423i	− 1.20548i	−0.797939	−	0.602739i	$-0.794076\pi$
$929$	− 36.7423i	− 1.20548i	0.797939	−	0.602739i	$-0.205924\pi$
$930$	0	0
$931$	−60.5203	−1.98347
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 146.508i	− 4.79133i
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 33.8590i	− 1.10377i	−0.833920	−	0.551886i	$-0.813908\pi$
$941$	− 33.8590i	− 1.10377i	0.833920	−	0.551886i	$-0.186092\pi$
$942$	0	0
$943$	8.31373	0.270732
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 29.1137i	− 0.946068i	−0.881044	−	0.473034i	$-0.843159\pi$
$947$	− 29.1137i	− 0.946068i	0.881044	−	0.473034i	$-0.156841\pi$
$948$	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
$955$	−118.373	−3.83045
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	37.7490	1.21771
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 17.8605i	− 0.574949i
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−56.0000	−1.79528
$974$	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
$979$	66.8745	2.13732
$980$	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
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	118.976i	3.77180i
$996$	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	3024.2.b.q.1567.1	✓	4
3.2	odd	2	inner	3024.2.b.q.1567.4	yes	4
4.3	odd	2		3024.2.b.t.1567.1	yes	4
7.6	odd	2		3024.2.b.t.1567.4	yes	4
12.11	even	2		3024.2.b.t.1567.4	yes	4
21.20	even	2		3024.2.b.t.1567.1	yes	4
28.27	even	2	inner	3024.2.b.q.1567.4	yes	4
84.83	odd	2	CM	3024.2.b.q.1567.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3024.2.b.q.1567.1	✓	4	1.1	even	1	trivial
3024.2.b.q.1567.1	✓	4	84.83	odd	2	CM
3024.2.b.q.1567.4	yes	4	3.2	odd	2	inner
3024.2.b.q.1567.4	yes	4	28.27	even	2	inner
3024.2.b.t.1567.1	yes	4	4.3	odd	2
3024.2.b.t.1567.1	yes	4	21.20	even	2
3024.2.b.t.1567.4	yes	4	7.6	odd	2
3024.2.b.t.1567.4	yes	4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.46512i q^{5} +2.64575 q^{7} +O(q^{10})\)	\(q-4.46512i q^{5} +2.64575 q^{7} +4.46512i q^{11} -7.34847i q^{17} -8.64575 q^{19} +2.88335i q^{23} -14.9373 q^{25} -8.29150 q^{31} -11.8136i q^{35} -3.93725 q^{37} -2.88335i q^{41} +7.00000 q^{49} +19.9373 q^{55} -10.2318i q^{71} +11.8136i q^{77} -32.8118 q^{85} -14.9771i q^{89} +38.6043i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 24 q^{19} - 28 q^{25} - 12 q^{31} + 16 q^{37} + 28 q^{49} + 48 q^{55} - 36 q^{85}+O(q^{100}) \) 4 * q - 24 * q^19 - 28 * q^25 - 12 * q^31 + 16 * q^37 + 28 * q^49 + 48 * q^55 - 36 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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