Embedded newform 333.3.r.a.38.14

• Label: 333.3.r.a.38.14
• Level: $333$
• Weight: $3$
• Character: 333.38
• Analytic conductor: $9.074$
• Analytic rank: $0$
• Dimension: $144$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.07359280320\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(72\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			38.14
Character	\(\chi\)	\(=\)	333.38
Dual form			333.3.r.a.149.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.48524 - 1.43485i) q^{2} +(2.59682 - 1.50217i) q^{3} +(2.11760 + 3.66779i) q^{4} +(0.333280 - 0.192419i) q^{5} +(-8.60911 + 0.00718270i) q^{6} +(-4.73790 + 8.20628i) q^{7} -0.674958i q^{8} +(4.48699 - 7.80173i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 4 q^{3} + 144 q^{4} - 30 q^{6} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.48524	−	1.43485i	−1.24262	−	0.717426i	−0.272992	−	0.962016i	\(-0.588013\pi\)
							−0.969627	+	0.244590i	\(0.921347\pi\)
\(3\)	2.59682	−	1.50217i	0.865608	−	0.500722i
							\(5\)	0.333280	−	0.192419i	0.0666560	−	0.0384839i	−0.466302	−	0.884626i	\(-0.654414\pi\)
0.532958	+	0.846142i	\(0.321081\pi\)
\(7\)	−4.73790	+	8.20628i	−0.676843	+	1.17233i	0.299084	+	0.954227i	\(0.403319\pi\)
							−0.975927	+	0.218099i	\(0.930014\pi\)
\(11\)	1.68563	+	0.973198i	0.153239	+	0.0884725i	0.574659	−	0.818393i	\(-0.305135\pi\)
							−0.421420	+	0.906866i	\(0.638468\pi\)
\(13\)	2.04023	+	3.53379i	0.156941	+	0.271830i	0.933764	−	0.357889i	\(-0.116503\pi\)
							−0.776823	+	0.629719i	\(0.783170\pi\)
\(17\)		−	26.0044i		−	1.52967i	−0.644227	−	0.764834i	\(-0.722821\pi\)
							0.644227	−	0.764834i	\(-0.277179\pi\)
\(19\)	22.8899			1.20473			0.602365	−	0.798220i	\(-0.294225\pi\)
							0.602365	+	0.798220i	\(0.294225\pi\)
\(23\)	30.8401	−	17.8055i	1.34087	−	0.774154i	0.353938	−	0.935269i	\(-0.384842\pi\)
							0.986936	+	0.161115i	\(0.0515091\pi\)
\(29\)	46.8825	+	27.0676i	1.61664	+	0.933367i	0.987781	+	0.155846i	\(0.0498103\pi\)
							0.628857	+	0.777521i	\(0.283523\pi\)
\(31\)	−8.37153	−	14.4999i	−0.270049	−	0.467739i	0.698825	−	0.715293i	\(-0.253707\pi\)
							−0.968874	+	0.247554i	\(0.920373\pi\)
\(37\)	−6.08276			−0.164399
							\(41\)	62.9563	−	36.3479i	1.53552	−	0.886533i	0.536428	−	0.843946i	\(-0.319773\pi\)
0.999093	−	0.0425869i	\(-0.0135599\pi\)
\(43\)	−12.2310	+	21.1847i	−0.284441	+	0.492667i	−0.972474	−	0.233013i	\(-0.925141\pi\)
							0.688032	+	0.725680i	\(0.258475\pi\)
\(47\)	29.1172	+	16.8108i	0.619515	+	0.357677i	0.776680	−	0.629895i	\(-0.216902\pi\)
							−0.157165	+	0.987572i	\(0.550235\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	333.3.r.a.38.14	✓	144
9.5	odd	6	inner	333.3.r.a.149.14	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
333.3.r.a.38.14	✓	144	1.1	even	1	trivial
333.3.r.a.149.14	yes	144	9.5	odd	6	inner