LMFDB - Embedded newform 384.5.h.d.65.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,5,Mod(65,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("384.65");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	384.h (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:	yes
Analytic conductor:	$39.6940658242$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			65.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	384.65

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} -19.5959 q^{5} -97.9796 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} -19.5959 q^{5} -97.9796 q^{7} +81.0000 q^{9} +142.000 q^{11} -176.363 q^{15} -881.816 q^{21} -241.000 q^{25} +729.000 q^{27} +1469.69 q^{29} +1861.61 q^{31} +1278.00 q^{33} +1920.00 q^{35} -1587.27 q^{45} +7199.00 q^{49} +4605.04 q^{53} -2782.62 q^{55} -6862.00 q^{59} -7936.35 q^{63} +8158.00 q^{73} -2169.00 q^{75} -13913.1 q^{77} +8524.22 q^{79} +6561.00 q^{81} -4178.00 q^{83} +13227.2 q^{87} +16754.5 q^{93} +17282.0 q^{97} +11502.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{3} + 162 q^{9}+O(q^{10}) $ 2 * q + 18 * q^3 + 162 * q^9	$ 2 q + 18 q^{3} + 162 q^{9} + 284 q^{11} - 482 q^{25} + 1458 q^{27} + 2556 q^{33} + 3840 q^{35} + 14398 q^{49} - 13724 q^{59} + 16316 q^{73} - 4338 q^{75} + 13122 q^{81} - 8356 q^{83} + 34564 q^{97} + 23004 q^{99}+O(q^{100}) $ 2 * q + 18 * q^3 + 162 * q^9 + 284 * q^11 - 482 * q^25 + 1458 * q^27 + 2556 * q^33 + 3840 * q^35 + 14398 * q^49 - 13724 * q^59 + 16316 * q^73 - 4338 * q^75 + 13122 * q^81 - 8356 * q^83 + 34564 * q^97 + 23004 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	9.00000	1.00000
$3$	9.00000	1.00000
$4$	0	0
$5$	−19.5959	−0.783837	−0.391918	−	0.920000i	$-0.628188\pi$
$5$	−19.5959	−0.783837	−0.391918	+	0.920000i	$0.628188\pi$
$6$	0	0
$7$	−97.9796	−1.99958	−0.999792	−	0.0204082i	$-0.993503\pi$
$7$	−97.9796	−1.99958	−0.999792	+	0.0204082i	$0.993503\pi$
$8$	0	0
$9$	81.0000	1.00000
$10$	0	0
$11$	142.000	1.17355	0.586777	−	0.809749i	$-0.300397\pi$
$11$	142.000	1.17355	0.586777	+	0.809749i	$0.300397\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−176.363	−0.783837
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	−881.816	−1.99958
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−241.000	−0.385600
$26$	0	0
$27$	729.000	1.00000
$28$	0	0
$29$	1469.69	1.74756	0.873778	−	0.486326i	$-0.161663\pi$
$29$	1469.69	1.74756	0.873778	+	0.486326i	$0.161663\pi$
$30$	0	0
$31$	1861.61	1.93716	0.968581	−	0.248699i	$-0.0800031\pi$
$31$	1861.61	1.93716	0.968581	+	0.248699i	$0.0800031\pi$
$32$	0	0
$33$	1278.00	1.17355
$34$	0	0
$35$	1920.00	1.56735
$36$	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	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	−1587.27	−0.783837
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	7199.00	2.99833
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4605.04	1.63939	0.819694	−	0.572802i	$-0.194143\pi$
$53$	4605.04	1.63939	0.819694	+	0.572802i	$0.194143\pi$
$54$	0	0
$55$	−2782.62	−0.919874
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6862.00	−1.97127	−0.985636	−	0.168882i	$-0.945984\pi$
$59$	−6862.00	−1.97127	−0.985636	+	0.168882i	$0.945984\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−7936.35	−1.99958
$64$	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	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	8158.00	1.53087	0.765434	−	0.643514i	$-0.222524\pi$
$73$	8158.00	1.53087	0.765434	+	0.643514i	$0.222524\pi$
$74$	0	0
$75$	−2169.00	−0.385600
$76$	0	0
$77$	−13913.1	−2.34662
$78$	0	0
$79$	8524.22	1.36584	0.682921	−	0.730492i	$-0.260709\pi$
$79$	8524.22	1.36584	0.682921	+	0.730492i	$0.260709\pi$
$80$	0	0
$81$	6561.00	1.00000
$82$	0	0
$83$	−4178.00	−0.606474	−0.303237	−	0.952915i	$-0.598067\pi$
$83$	−4178.00	−0.606474	−0.303237	+	0.952915i	$0.598067\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	13227.2	1.74756
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	16754.5	1.93716
$94$	0	0
$95$	0	0
$96$	0	0
$97$	17282.0	1.83675	0.918376	−	0.395709i	$-0.129501\pi$
$97$	17282.0	1.83675	0.918376	+	0.395709i	$0.129501\pi$
$98$	0	0
$99$	11502.0	1.17355
$100$	0	0
$101$	−13031.3	−1.27745	−0.638726	−	0.769434i	$-0.720538\pi$
$101$	−13031.3	−1.27745	−0.638726	+	0.769434i	$0.720538\pi$
$102$	0	0
$103$	−2057.57	−0.193946	−0.0969729	−	0.995287i	$-0.530916\pi$
$103$	−2057.57	−0.193946	−0.0969729	+	0.995287i	$0.530916\pi$
$104$	0	0
$105$	17280.0	1.56735
$106$	0	0
$107$	−15502.0	−1.35400	−0.677002	−	0.735981i	$-0.736721\pi$
$107$	−15502.0	−1.35400	−0.677002	+	0.735981i	$0.736721\pi$
$108$	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	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5523.00	0.377228
$122$	0	0
$123$	0	0
$124$	0	0
$125$	16970.1	1.08608
$126$	0	0
$127$	−24788.8	−1.53691	−0.768455	−	0.639903i	$-0.778974\pi$
$127$	−24788.8	−1.53691	−0.768455	+	0.639903i	$0.778974\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	28178.0	1.64198	0.820989	−	0.570943i	$-0.193422\pi$
$131$	28178.0	1.64198	0.820989	+	0.570943i	$0.193422\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−14285.4	−0.783837
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	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
$145$	−28800.0	−1.36980
$146$	0	0
$147$	64791.0	2.99833
$148$	0	0
$149$	19889.9	0.895899	0.447950	−	0.894059i	$-0.352154\pi$
$149$	19889.9	0.895899	0.447950	+	0.894059i	$0.352154\pi$
$150$	0	0
$151$	−14599.0	−0.640277	−0.320139	−	0.947371i	$-0.603729\pi$
$151$	−14599.0	−0.640277	−0.320139	+	0.947371i	$0.603729\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−36480.0	−1.51842
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	41445.4	1.63939
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−25043.6	−0.919874
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	28561.0	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−15774.7	−0.527071	−0.263536	−	0.964650i	$-0.584889\pi$
$173$	−15774.7	−0.527071	−0.263536	+	0.964650i	$0.584889\pi$
$174$	0	0
$175$	23613.1	0.771039
$176$	0	0
$177$	−61758.0	−1.97127
$178$	0	0
$179$	−11182.0	−0.348990	−0.174495	−	0.984658i	$-0.555829\pi$
$179$	−11182.0	−0.348990	−0.174495	+	0.984658i	$0.555829\pi$
$180$	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
$189$	−71427.1	−1.99958
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	−9602.00	−0.257779	−0.128889	−	0.991659i	$-0.541141\pi$
$193$	−9602.00	−0.257779	−0.128889	+	0.991659i	$0.541141\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	66528.1	1.71425	0.857123	−	0.515112i	$-0.172250\pi$
$197$	66528.1	1.71425	0.857123	+	0.515112i	$0.172250\pi$
$198$	0	0
$199$	69271.6	1.74924	0.874619	−	0.484811i	$-0.161112\pi$
$199$	69271.6	1.74924	0.874619	+	0.484811i	$0.161112\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−144000.	−3.49438
$204$	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
$217$	−182400.	−3.87352
$218$	0	0
$219$	73422.0	1.53087
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−87887.7	−1.76733	−0.883666	−	0.468117i	$-0.844933\pi$
$223$	−87887.7	−1.76733	−0.883666	+	0.468117i	$0.844933\pi$
$224$	0	0
$225$	−19521.0	−0.385600
$226$	0	0
$227$	−16658.0	−0.323274	−0.161637	−	0.986850i	$-0.551677\pi$
$227$	−16658.0	−0.323274	−0.161637	+	0.986850i	$0.551677\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	−125218.	−2.34662
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	76718.0	1.36584
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−29762.0	−0.512422	−0.256211	−	0.966621i	$-0.582474\pi$
$241$	−29762.0	−0.512422	−0.256211	+	0.966621i	$0.582474\pi$
$242$	0	0
$243$	59049.0	1.00000
$244$	0	0
$245$	−141071.	−2.35020
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−37602.0	−0.606474
$250$	0	0
$251$	−94898.0	−1.50629	−0.753147	−	0.657853i	$-0.771465\pi$
$251$	−94898.0	−1.50629	−0.753147	+	0.657853i	$0.771465\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	119045.	1.74756
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−90240.0	−1.28501
$266$	0	0
$267$	0	0
$268$	0	0
$269$	139033.	1.92138	0.960690	−	0.277622i	$-0.0895462\pi$
$269$	139033.	1.92138	0.960690	+	0.277622i	$0.0895462\pi$
$270$	0	0
$271$	31255.5	0.425586	0.212793	−	0.977097i	$-0.431744\pi$
$271$	31255.5	0.425586	0.212793	+	0.977097i	$0.431744\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−34222.0	−0.452522
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	150791.	1.93716
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	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
$289$	83521.0	1.00000
$290$	0	0
$291$	155538.	1.83675
$292$	0	0
$293$	−170191.	−1.98244	−0.991220	−	0.132221i	$-0.957789\pi$
$293$	−170191.	−1.98244	−0.991220	+	0.132221i	$0.957789\pi$
$294$	0	0
$295$	134467.	1.54516
$296$	0	0
$297$	103518.	1.17355
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−117282.	−1.27745
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	−18518.1	−0.193946
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−84962.0	−0.867234	−0.433617	−	0.901097i	$-0.642763\pi$
$313$	−84962.0	−0.867234	−0.433617	+	0.901097i	$0.642763\pi$
$314$	0	0
$315$	155520.	1.56735
$316$	0	0
$317$	−179989.	−1.79113	−0.895563	−	0.444934i	$-0.853227\pi$
$317$	−179989.	−1.79113	−0.895563	+	0.444934i	$0.853227\pi$
$318$	0	0
$319$	208697.	2.05085
$320$	0	0
$321$	−139518.	−1.35400
$322$	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
$337$	191038.	1.68213	0.841066	−	0.540933i	$-0.181929\pi$
$337$	191038.	1.68213	0.841066	+	0.540933i	$0.181929\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	264349.	2.27336
$342$	0	0
$343$	−470106.	−3.99584
$344$	0	0
$345$	0	0
$346$	0	0
$347$	229582.	1.90668	0.953342	−	0.301891i	$-0.0976180\pi$
$347$	229582.	1.90668	0.953342	+	0.301891i	$0.0976180\pi$
$348$	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	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	130321.	1.00000
$362$	0	0
$363$	49707.0	0.377228
$364$	0	0
$365$	−159863.	−1.19995
$366$	0	0
$367$	−132174.	−0.981331	−0.490665	−	0.871348i	$-0.663246\pi$
$367$	−132174.	−0.981331	−0.490665	+	0.871348i	$0.663246\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−451200.	−3.27809
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	152731.	1.08608
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−223100.	−1.53691
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	272640.	1.83937
$386$	0	0
$387$	0	0
$388$	0	0
$389$	143344.	0.947285	0.473643	−	0.880717i	$-0.342939\pi$
$389$	143344.	0.947285	0.473643	+	0.880717i	$0.342939\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	253602.	1.64198
$394$	0	0
$395$	−167040.	−1.07060
$396$	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	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−128569.	−0.783837
$406$	0	0
$407$	0	0
$408$	0	0
$409$	180962.	1.08178	0.540892	−	0.841092i	$-0.318087\pi$
$409$	180962.	1.08178	0.540892	+	0.841092i	$0.318087\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	672336.	3.94172
$414$	0	0
$415$	81871.7	0.475377
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−181778.	−1.03541	−0.517706	−	0.855559i	$-0.673214\pi$
$419$	−181778.	−1.03541	−0.517706	+	0.855559i	$0.673214\pi$
$420$	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	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	73922.0	0.394274	0.197137	−	0.980376i	$-0.436836\pi$
$433$	73922.0	0.394274	0.197137	+	0.980376i	$0.436836\pi$
$434$	0	0
$435$	−259200.	−1.36980
$436$	0	0
$437$	0	0
$438$	0	0
$439$	231526.	1.20135	0.600676	−	0.799493i	$-0.294898\pi$
$439$	231526.	1.20135	0.600676	+	0.799493i	$0.294898\pi$
$440$	0	0
$441$	583119.	2.99833
$442$	0	0
$443$	385102.	1.96231	0.981157	−	0.193214i	$-0.0618912\pi$
$443$	385102.	1.96231	0.981157	+	0.193214i	$0.0618912\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	179009.	0.895899
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−131391.	−0.640277
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−136798.	−0.655009	−0.327505	−	0.944850i	$-0.606208\pi$
$457$	−136798.	−0.655009	−0.327505	+	0.944850i	$0.606208\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−399855.	−1.88148	−0.940742	−	0.339124i	$-0.889869\pi$
$461$	−399855.	−1.88148	−0.940742	+	0.339124i	$0.889869\pi$
$462$	0	0
$463$	222904.	1.03981	0.519906	−	0.854223i	$-0.325967\pi$
$463$	222904.	1.03981	0.519906	+	0.854223i	$0.325967\pi$
$464$	0	0
$465$	−328320.	−1.51842
$466$	0	0
$467$	34222.0	0.156918	0.0784588	−	0.996917i	$-0.475000\pi$
$467$	34222.0	0.156918	0.0784588	+	0.996917i	$0.475000\pi$
$468$	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
$477$	373008.	1.63939
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−338657.	−1.43971
$486$	0	0
$487$	−85536.2	−0.360655	−0.180327	−	0.983607i	$-0.557716\pi$
$487$	−85536.2	−0.360655	−0.180327	+	0.983607i	$0.557716\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−261262.	−1.08371	−0.541855	−	0.840472i	$-0.682278\pi$
$491$	−261262.	−1.08371	−0.541855	+	0.840472i	$0.682278\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−225392.	−0.919874
$496$	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	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	255360.	1.00131
$506$	0	0
$507$	257049.	1.00000
$508$	0	0
$509$	−128647.	−0.496552	−0.248276	−	0.968689i	$-0.579864\pi$
$509$	−128647.	−0.496552	−0.248276	+	0.968689i	$0.579864\pi$
$510$	0	0
$511$	−799317.	−3.06110
$512$	0	0
$513$	0	0
$514$	0	0
$515$	40320.0	0.152022
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−141972.	−0.527071
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	212518.	0.771039
$526$	0	0
$527$	0	0
$528$	0	0
$529$	279841.	1.00000
$530$	0	0
$531$	−555822.	−1.97127
$532$	0	0
$533$	0	0
$534$	0	0
$535$	303776.	1.06132
$536$	0	0
$537$	−100638.	−0.348990
$538$	0	0
$539$	1.02226e6	3.51871
$540$	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
$553$	−835200.	−2.73112
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−582097.	−1.87622	−0.938112	−	0.346331i	$-0.887427\pi$
$557$	−582097.	−1.87622	−0.938112	+	0.346331i	$0.887427\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	527662.	1.66471	0.832356	−	0.554242i	$-0.186992\pi$
$563$	527662.	1.66471	0.832356	+	0.554242i	$0.186992\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−642844.	−1.99958
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	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
$577$	581758.	1.74739	0.873697	−	0.486471i	$-0.161716\pi$
$577$	581758.	1.74739	0.873697	+	0.486471i	$0.161716\pi$
$578$	0	0
$579$	−86418.0	−0.257779
$580$	0	0
$581$	409359.	1.21270
$582$	0	0
$583$	653916.	1.92391
$584$	0	0
$585$	0	0
$586$	0	0
$587$	74738.0	0.216903	0.108451	−	0.994102i	$-0.465411\pi$
$587$	74738.0	0.216903	0.108451	+	0.994102i	$0.465411\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	598753.	1.71425
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	623444.	1.74924
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−712802.	−1.97342	−0.986711	−	0.162485i	$-0.948049\pi$
$601$	−712802.	−1.97342	−0.986711	+	0.162485i	$0.948049\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−108228.	−0.295685
$606$	0	0
$607$	708098.	1.92184	0.960918	−	0.276833i	$-0.0892850\pi$
$607$	708098.	1.92184	0.960918	+	0.276833i	$0.0892850\pi$
$608$	0	0
$609$	−1.29600e6	−3.49438
$610$	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	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	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
$625$	−181919.	−0.465713
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	303247.	0.761619	0.380809	−	0.924654i	$-0.375645\pi$
$631$	303247.	0.761619	0.380809	+	0.924654i	$0.375645\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	485760.	1.20469
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	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	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	−974404.	−2.31339
$650$	0	0
$651$	−1.64160e6	−3.87352
$652$	0	0
$653$	837823.	1.96484	0.982418	−	0.186696i	$-0.0597780\pi$
$653$	837823.	1.96484	0.982418	+	0.186696i	$0.0597780\pi$
$654$	0	0
$655$	−552174.	−1.28704
$656$	0	0
$657$	660798.	1.53087
$658$	0	0
$659$	−335662.	−0.772914	−0.386457	−	0.922307i	$-0.626301\pi$
$659$	−335662.	−0.772914	−0.386457	+	0.922307i	$0.626301\pi$
$660$	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
$669$	−790989.	−1.76733
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−869758.	−1.92030	−0.960148	−	0.279491i	$-0.909834\pi$
$673$	−869758.	−1.92030	−0.960148	+	0.279491i	$0.909834\pi$
$674$	0	0
$675$	−175689.	−0.385600
$676$	0	0
$677$	197821.	0.431613	0.215807	−	0.976436i	$-0.430762\pi$
$677$	197821.	0.431613	0.215807	+	0.976436i	$0.430762\pi$
$678$	0	0
$679$	−1.69328e6	−3.67274
$680$	0	0
$681$	−149922.	−0.323274
$682$	0	0
$683$	−846578.	−1.81479	−0.907393	−	0.420282i	$-0.861931\pi$
$683$	−846578.	−1.81479	−0.907393	+	0.420282i	$0.861931\pi$
$684$	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
$693$	−1.12696e6	−2.34662
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	70055.4	0.142563	0.0712813	−	0.997456i	$-0.477291\pi$
$701$	70055.4	0.142563	0.0712813	+	0.997456i	$0.477291\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.27680e6	2.55437
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	690462.	1.36584
$712$	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
$721$	201600.	0.387811
$722$	0	0
$723$	−267858.	−0.512422
$724$	0	0
$725$	−354196.	−0.673857
$726$	0	0
$727$	156277.	0.295684	0.147842	−	0.989011i	$-0.452767\pi$
$727$	156277.	0.295684	0.147842	+	0.989011i	$0.452767\pi$
$728$	0	0
$729$	531441.	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	−1.26964e6	−2.35020
$736$	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	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−389760.	−0.702239
$746$	0	0
$747$	−338418.	−0.606474
$748$	0	0
$749$	1.51888e6	2.70745
$750$	0	0
$751$	755521.	1.33957	0.669787	−	0.742554i	$-0.266386\pi$
$751$	755521.	1.33957	0.669787	+	0.742554i	$0.266386\pi$
$752$	0	0
$753$	−854082.	−1.50629
$754$	0	0
$755$	286080.	0.501873
$756$	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	1.00000			$0$
$761$	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
$769$	439678.	0.743502	0.371751	−	0.928333i	$-0.378758\pi$
$769$	439678.	0.743502	0.371751	+	0.928333i	$0.378758\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.18722e6	−1.98688	−0.993440	−	0.114353i	$-0.963521\pi$
$773$	−1.18722e6	−1.98688	−0.993440	+	0.114353i	$0.963521\pi$
$774$	0	0
$775$	−448649.	−0.746969
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1.07141e6	1.74756
$784$	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
$795$	−812160.	−1.28501
$796$	0	0
$797$	−1.20446e6	−1.89617	−0.948084	−	0.318020i	$-0.896982\pi$
$797$	−1.20446e6	−1.89617	−0.948084	+	0.318020i	$0.896982\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.15844e6	1.79656
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.25130e6	1.92138
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	281299.	0.425586
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.00439e6	1.49010	0.745050	−	0.667008i	$-0.232425\pi$
$821$	1.00439e6	1.49010	0.745050	+	0.667008i	$0.232425\pi$
$822$	0	0
$823$	741412.	1.09461	0.547305	−	0.836933i	$-0.315654\pi$
$823$	741412.	1.09461	0.547305	+	0.836933i	$0.315654\pi$
$824$	0	0
$825$	−307998.	−0.452522
$826$	0	0
$827$	1.02226e6	1.49468	0.747342	−	0.664439i	$-0.231330\pi$
$827$	1.02226e6	1.49468	0.747342	+	0.664439i	$0.231330\pi$
$828$	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
$837$	1.35712e6	1.93716
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.45272e6	2.05395
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−559679.	−0.783837
$846$	0	0
$847$	−541141.	−0.754300
$848$	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	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	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	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	309120.	0.413138
$866$	0	0
$867$	751689.	1.00000
$868$	0	0
$869$	1.21044e6	1.60289
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.39984e6	1.83675
$874$	0	0
$875$	−1.66272e6	−2.17172
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	−1.53171e6	−1.98244
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	1.21020e6	1.54516
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	2.42880e6	3.07318
$890$	0	0
$891$	931662.	1.17355
$892$	0	0
$893$	0	0
$894$	0	0
$895$	219122.	0.273551
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.73600e6	3.38530
$900$	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
$909$	−1.05553e6	−1.27745
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−593276.	−0.711730
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.76087e6	−3.28327
$918$	0	0
$919$	−1.13509e6	−1.34400	−0.672002	−	0.740549i	$-0.734565\pi$
$919$	−1.13509e6	−1.34400	−0.672002	+	0.740549i	$0.734565\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−166663.	−0.193946
$928$	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
$937$	464162.	0.528677	0.264338	−	0.964430i	$-0.414846\pi$
$937$	464162.	0.528677	0.264338	+	0.964430i	$0.414846\pi$
$938$	0	0
$939$	−764658.	−0.867234
$940$	0	0
$941$	−787854.	−0.889747	−0.444873	−	0.895593i	$-0.646751\pi$
$941$	−787854.	−0.889747	−0.444873	+	0.895593i	$0.646751\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	1.39968e6	1.56735
$946$	0	0
$947$	−87982.0	−0.0981056	−0.0490528	−	0.998796i	$-0.515620\pi$
$947$	−87982.0	−0.0981056	−0.0490528	+	0.998796i	$0.515620\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−1.61990e6	−1.79113
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.87827e6	2.05085
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.54208e6	2.75259
$962$	0	0
$963$	−1.25566e6	−1.35400
$964$	0	0
$965$	188160.	0.202056
$966$	0	0
$967$	−580137.	−0.620408	−0.310204	−	0.950670i	$-0.600397\pi$
$967$	−580137.	−0.620408	−0.310204	+	0.950670i	$0.600397\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.83922e6	−1.95072	−0.975360	−	0.220621i	$-0.929192\pi$
$971$	−1.83922e6	−1.95072	−0.975360	+	0.220621i	$0.929192\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	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
$985$	−1.30368e6	−1.34369
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1.96341e6	1.99924	0.999619	−	0.0276138i	$-0.00879087\pi$
$991$	1.96341e6	1.99924	0.999619	+	0.0276138i	$0.00879087\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.35744e6	−1.37112
$996$	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	384.5.h.d.65.1	yes	2
3.2	odd	2		384.5.h.a.65.2	yes	2
4.3	odd	2		384.5.h.a.65.1	✓	2
8.3	odd	2	inner	384.5.h.d.65.2	yes	2
8.5	even	2		384.5.h.a.65.2	yes	2
12.11	even	2	inner	384.5.h.d.65.2	yes	2
24.5	odd	2	CM	384.5.h.d.65.1	yes	2
24.11	even	2		384.5.h.a.65.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.h.a.65.1	✓	2	4.3	odd	2
384.5.h.a.65.1	✓	2	24.11	even	2
384.5.h.a.65.2	yes	2	3.2	odd	2
384.5.h.a.65.2	yes	2	8.5	even	2
384.5.h.d.65.1	yes	2	1.1	even	1	trivial
384.5.h.d.65.1	yes	2	24.5	odd	2	CM
384.5.h.d.65.2	yes	2	8.3	odd	2	inner
384.5.h.d.65.2	yes	2	12.11	even	2	inner

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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	384.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} -19.5959 q^{5} -97.9796 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} -19.5959 q^{5} -97.9796 q^{7} +81.0000 q^{9} +142.000 q^{11} -176.363 q^{15} -881.816 q^{21} -241.000 q^{25} +729.000 q^{27} +1469.69 q^{29} +1861.61 q^{31} +1278.00 q^{33} +1920.00 q^{35} -1587.27 q^{45} +7199.00 q^{49} +4605.04 q^{53} -2782.62 q^{55} -6862.00 q^{59} -7936.35 q^{63} +8158.00 q^{73} -2169.00 q^{75} -13913.1 q^{77} +8524.22 q^{79} +6561.00 q^{81} -4178.00 q^{83} +13227.2 q^{87} +16754.5 q^{93} +17282.0 q^{97} +11502.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} + 162 q^{9}+O(q^{10}) \) 2 * q + 18 * q^3 + 162 * q^9	\( 2 q + 18 q^{3} + 162 q^{9} + 284 q^{11} - 482 q^{25} + 1458 q^{27} + 2556 q^{33} + 3840 q^{35} + 14398 q^{49} - 13724 q^{59} + 16316 q^{73} - 4338 q^{75} + 13122 q^{81} - 8356 q^{83} + 34564 q^{97} + 23004 q^{99}+O(q^{100}) \) 2 * q + 18 * q^3 + 162 * q^9 + 284 * q^11 - 482 * q^25 + 1458 * q^27 + 2556 * q^33 + 3840 * q^35 + 14398 * q^49 - 13724 * q^59 + 16316 * q^73 - 4338 * q^75 + 13122 * q^81 - 8356 * q^83 + 34564 * q^97 + 23004 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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