Embedded newform 333.3.bg.a.88.9

• Label: 333.3.bg.a.88.9
• Level: $333$
• Weight: $3$
• Character: 333.88
• Analytic conductor: $9.074$
• Analytic rank: $0$
• Dimension: $296$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 333 = 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	333.bg (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.07359280320\)
Analytic rank:	\(0\)
Dimension:	\(296\)
Relative dimension:	\(74\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			88.9
Character	\(\chi\)	\(=\)	333.88
Dual form			333.3.bg.a.193.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.832081 + 3.10537i) q^{2} +(-0.901375 - 2.86138i) q^{3} +(-5.48686 - 3.16784i) q^{4} +(0.442510 + 1.65147i) q^{5} +(9.63567 - 0.418199i) q^{6} +4.49844 q^{7} +(5.30967 - 5.30967i) q^{8} +(-7.37505 + 5.15836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 296 q - 2 q^{2} - 6 q^{3} - 6 q^{4} + 4 q^{5} + 12 q^{6} - 4 q^{7} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(38\)	\(298\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.832081	+	3.10537i	−0.416041	+	1.55268i	0.366702	+	0.930339i	\(0.380487\pi\)
							−0.782742	+	0.622346i	\(0.786180\pi\)
\(3\)	−0.901375	−	2.86138i	−0.300458	−	0.953795i
							\(5\)	0.442510	+	1.65147i	0.0885021	+	0.330294i	0.995954	−	0.0898610i	\(-0.0286423\pi\)
−0.907452	+	0.420155i	\(0.861976\pi\)
\(7\)	4.49844			0.642634			0.321317	−	0.946972i	\(-0.395875\pi\)
							0.321317	+	0.946972i	\(0.395875\pi\)
\(11\)	−2.53571	−	1.46399i	−0.230519	−	0.133090i	0.380293	−	0.924866i	\(-0.375823\pi\)
							−0.610811	+	0.791776i	\(0.709157\pi\)
\(13\)	−2.14782	−	8.01578i	−0.165217	−	0.616599i	−0.998012	−	0.0630176i	\(-0.979928\pi\)
							0.832795	−	0.553581i	\(-0.186739\pi\)
\(17\)	−1.35716	−	5.06501i	−0.0798332	−	0.297942i	0.914452	−	0.404693i	\(-0.132622\pi\)
							−0.994286	+	0.106752i	\(0.965955\pi\)
\(19\)	−29.4706	−	7.89662i	−1.55108	−	0.415612i	−0.621256	−	0.783607i	\(-0.713377\pi\)
							−0.929828	+	0.367996i	\(0.880044\pi\)
\(23\)	1.03084	+	0.276212i	0.0448190	+	0.0120092i	0.281159	−	0.959661i	\(-0.409281\pi\)
							−0.236340	+	0.971670i	\(0.575948\pi\)
\(29\)	20.7792	−	5.56777i	0.716524	−	0.191992i	0.117904	−	0.993025i	\(-0.462383\pi\)
							0.598620	+	0.801033i	\(0.295716\pi\)
\(31\)	12.1220	−	45.2400i	0.391033	−	1.45935i	−0.437401	−	0.899267i	\(-0.644101\pi\)
							0.828434	−	0.560087i	\(-0.189232\pi\)
\(37\)	−33.5989	−	15.4957i	−0.908078	−	0.418801i
							\(41\)	−1.44583	−	0.834748i	−0.0352640	−	0.0203597i	0.482264	−	0.876026i	\(-0.339814\pi\)
−0.517528	+	0.855666i	\(0.673148\pi\)
\(43\)	−9.95360	−	37.1473i	−0.231479	−	0.863891i	−0.979705	−	0.200447i	\(-0.935761\pi\)
							0.748226	−	0.663444i	\(-0.230906\pi\)
\(47\)	30.5353	−	52.8887i	0.649688	−	1.12529i	−0.333509	−	0.942747i	\(-0.608233\pi\)
							0.983197	−	0.182546i	\(-0.0584338\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	333.3.bg.a.88.9	yes	296
9.4	even	3		333.3.ba.a.310.9	yes	296
37.8	odd	12		333.3.ba.a.304.9	✓	296
333.193	odd	12	inner	333.3.bg.a.193.9	yes	296

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
333.3.ba.a.304.9	✓	296	37.8	odd	12
333.3.ba.a.310.9	yes	296	9.4	even	3
333.3.bg.a.88.9	yes	296	1.1	even	1	trivial
333.3.bg.a.193.9	yes	296	333.193	odd	12	inner