Embedded newform 333.3.n.a.110.17

• Label: 333.3.n.a.110.17
• Level: $333$
• Weight: $3$
• Character: 333.110
• Analytic conductor: $9.074$
• Analytic rank: $0$
• Dimension: $148$
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.07359280320\)
Analytic rank:	\(0\)
Dimension:	\(148\)
Relative dimension:	\(74\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			110.17
Character	\(\chi\)	\(=\)	333.110
Dual form			333.3.n.a.221.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.21885 + 2.11111i) q^{2} +(2.94107 + 0.591713i) q^{3} +(-0.971192 - 1.68215i) q^{4} +(4.22130 + 7.31151i) q^{5} +(-4.83389 + 5.48771i) q^{6} +(-0.385107 + 0.667025i) q^{7} -5.01585 q^{8} +(8.29975 + 3.48053i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 148 q - 6 q^{3} - 146 q^{4} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(38\)	\(298\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.21885	+	2.11111i	−0.609425	+	1.05556i	0.381910	+	0.924199i	\(0.375266\pi\)
							−0.991335	+	0.131356i	\(0.958067\pi\)
\(3\)	2.94107	+	0.591713i	0.980356	+	0.197238i
							\(5\)	4.22130	+	7.31151i	0.844261	+	1.46230i	0.886262	+	0.463185i	\(0.153293\pi\)
−0.0420013	+	0.999118i	\(0.513373\pi\)
\(7\)	−0.385107	+	0.667025i	−0.0550153	+	0.0952893i	−0.892221	−	0.451598i	\(-0.850854\pi\)
							0.837206	+	0.546887i	\(0.184187\pi\)
\(11\)	−9.18278	−	5.30168i	−0.834799	−	0.481971i	0.0206943	−	0.999786i	\(-0.493412\pi\)
							−0.855493	+	0.517815i	\(0.826746\pi\)
\(13\)	−20.0911	+	11.5996i	−1.54547	+	0.892277i	−0.546990	+	0.837139i	\(0.684227\pi\)
							−0.998479	+	0.0551380i	\(0.982440\pi\)
\(17\)	20.0892			1.18172			0.590859	−	0.806775i	\(-0.298789\pi\)
							0.590859	+	0.806775i	\(0.298789\pi\)
\(19\)		−	24.1782i		−	1.27254i	−0.771468	−	0.636268i	\(-0.780477\pi\)
							0.771468	−	0.636268i	\(-0.219523\pi\)
\(23\)	17.4808	+	30.2777i	0.760035	+	1.31642i	0.942832	+	0.333269i	\(0.108152\pi\)
							−0.182796	+	0.983151i	\(0.558515\pi\)
\(29\)	16.3122	−	28.2536i	0.562490	−	0.974262i	−0.434788	−	0.900533i	\(-0.643177\pi\)
							0.997278	−	0.0737288i	\(-0.0234899\pi\)
\(31\)	−20.6515	+	11.9231i	−0.666176	+	0.384617i	−0.794626	−	0.607099i	\(-0.792333\pi\)
							0.128450	+	0.991716i	\(0.459000\pi\)
\(37\)	32.2507	+	18.1354i	0.871640	+	0.490146i
							\(41\)	38.8186	−	22.4119i	0.946796	−	0.546633i	0.0547118	−	0.998502i	\(-0.482576\pi\)
0.892084	+	0.451869i	\(0.149243\pi\)
\(43\)	−23.6969	−	13.6814i	−0.551091	−	0.318172i	0.198471	−	0.980107i	\(-0.436402\pi\)
							−0.749562	+	0.661934i	\(0.769736\pi\)
\(47\)	−5.44989	−	3.14650i	−0.115955	−	0.0669467i	0.440901	−	0.897556i	\(-0.354659\pi\)
							−0.556856	+	0.830609i	\(0.687992\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	333.3.n.a.110.17	✓	148
9.5	odd	6	inner	333.3.n.a.221.58	yes	148
37.36	even	2	inner	333.3.n.a.110.58	yes	148
333.221	odd	6	inner	333.3.n.a.221.17	yes	148

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
333.3.n.a.110.17	✓	148	1.1	even	1	trivial
333.3.n.a.110.58	yes	148	37.36	even	2	inner
333.3.n.a.221.17	yes	148	333.221	odd	6	inner
333.3.n.a.221.58	yes	148	9.5	odd	6	inner