Embedded newform 333.3.bn.a.86.13

• Label: 333.3.bn.a.86.13
• Level: $333$
• Weight: $3$
• Character: 333.86
• Analytic conductor: $9.074$
• Analytic rank: $0$
• Dimension: $444$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 333 = 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	333.bn (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.07359280320\)
Analytic rank:	\(0\)
Dimension:	\(444\)
Relative dimension:	\(74\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			86.13
Character	\(\chi\)	\(=\)	333.86
Dual form			333.3.bn.a.182.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.961211 + 2.64090i) q^{2} +(1.31409 + 2.69688i) q^{3} +(-2.98627 - 2.50578i) q^{4} +(-1.65563 - 0.291932i) q^{5} +(-8.38532 + 0.878108i) q^{6} +(-8.10155 - 6.79801i) q^{7} +(-0.247518 + 0.142905i) q^{8} +(-5.54635 + 7.08788i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 444 q - 9 q^{2} - 12 q^{3} - 3 q^{4} - 9 q^{5} - 12 q^{6} - 6 q^{7} + 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/333\mathbb{Z}\right)^\times\).

\(n\)	\(38\)	\(298\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{9}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.961211	+	2.64090i	−0.480605	+	1.32045i	0.428370	+	0.903603i	\(0.359088\pi\)
							−0.908976	+	0.416849i	\(0.863134\pi\)
\(3\)	1.31409	+	2.69688i	0.438029	+	0.898961i
							\(5\)	−1.65563	−	0.291932i	−0.331126	−	0.0583864i	0.00561382	−	0.999984i	\(-0.498213\pi\)
−0.336739	+	0.941598i	\(0.609324\pi\)
\(7\)	−8.10155	−	6.79801i	−1.15736	−	0.971144i	−0.157498	−	0.987519i	\(-0.550343\pi\)
							−0.999866	+	0.0163757i	\(0.994787\pi\)
\(11\)	0.355868	+	0.205460i	0.0323516	+	0.0186782i	0.516089	−	0.856535i	\(-0.327388\pi\)
							−0.483737	+	0.875213i	\(0.660721\pi\)
\(13\)	1.22762	+	1.03009i	0.0944319	+	0.0792378i	0.688781	−	0.724969i	\(-0.258146\pi\)
							−0.594349	+	0.804207i	\(0.702590\pi\)
\(17\)	9.03279	−	1.59272i	0.531341	−	0.0936897i	0.0984602	−	0.995141i	\(-0.468608\pi\)
							0.432881	+	0.901451i	\(0.357497\pi\)
\(19\)	0.545253	+	0.457522i	0.0286975	+	0.0240801i	0.657024	−	0.753870i	\(-0.271815\pi\)
							−0.628326	+	0.777950i	\(0.716260\pi\)
\(23\)	15.4222	−	8.90402i	0.670531	−	0.387131i	−0.125747	−	0.992062i	\(-0.540133\pi\)
							0.796278	+	0.604931i	\(0.206799\pi\)
\(29\)	−17.2954	−	9.98548i	−0.596392	−	0.344327i	0.171229	−	0.985231i	\(-0.445226\pi\)
							−0.767621	+	0.640904i	\(0.778560\pi\)
\(31\)	4.37966	−	7.58579i	0.141279	−	0.244703i	−0.786699	−	0.617336i	\(-0.788212\pi\)
							0.927979	+	0.372633i	\(0.121545\pi\)
\(37\)	−35.3911	+	10.7922i	−0.956516	+	0.291681i
							\(41\)	0.513167	−	0.611568i	0.0125163	−	0.0149163i	−0.759750	−	0.650215i	\(-0.774679\pi\)
0.772266	+	0.635299i	\(0.219123\pi\)
\(43\)	−26.3563	−	45.6504i	−0.612936	−	1.06164i	−0.990743	−	0.135752i	\(-0.956655\pi\)
							0.377806	−	0.925885i	\(-0.376679\pi\)
\(47\)			45.0084i			0.957625i	0.877917	+	0.478812i	\(0.158933\pi\)
							−0.877917	+	0.478812i	\(0.841067\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	333.3.bn.a.86.13	yes	444
9.2	odd	6		333.3.bk.a.308.13	yes	444
37.34	even	9		333.3.bk.a.293.13	✓	444
333.182	odd	18	inner	333.3.bn.a.182.13	yes	444

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
333.3.bk.a.293.13	✓	444	37.34	even	9
333.3.bk.a.308.13	yes	444	9.2	odd	6
333.3.bn.a.86.13	yes	444	1.1	even	1	trivial
333.3.bn.a.182.13	yes	444	333.182	odd	18	inner