Embedded newform 144.3.w.a.5.9

• Label: 144.3.w.a.5.9
• Level: $144$
• Weight: $3$
• Character: 144.5
• Analytic conductor: $3.924$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	144.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.92371580679\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(46\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			5.9
Character	\(\chi\)	\(=\)	144.5
Dual form			144.3.w.a.29.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.61016 - 1.18633i) q^{2} +(-2.90827 + 0.736170i) q^{3} +(1.18523 + 3.82037i) q^{4} +(-0.295811 - 1.10398i) q^{5} +(5.55613 + 2.26483i) q^{6} +(-7.30360 + 4.21674i) q^{7} +(2.62382 - 7.55748i) q^{8} +(7.91611 - 4.28197i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184q - 6q^{2} - 4q^{3} - 2q^{4} - 6q^{5} - 10q^{6} + O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.61016	−	1.18633i	−0.805080	−	0.593166i
							\(3\)	−2.90827	+	0.736170i	−0.969424	+	0.245390i
\(5\)	−0.295811	−	1.10398i	−0.0591623	−	0.220797i								0.930015	−	0.367521i	\(-0.119794\pi\)
							−0.989177	+	0.146725i	\(0.953127\pi\)
\(7\)	−7.30360	+	4.21674i	−1.04337	+	0.602391i	−0.920786	−	0.390067i	\(-0.872452\pi\)
							−0.122585	+	0.992458i	\(0.539118\pi\)
\(11\)	16.9180	+	4.53318i	1.53800	+	0.412107i	0.925620	−	0.378454i	\(-0.123544\pi\)
							0.612384	+	0.790561i	\(0.290211\pi\)
\(13\)	−3.32968	−	12.4265i	−0.256129	−	0.955886i	−0.967459	−	0.253028i	\(-0.918574\pi\)
							0.711330	−	0.702858i	\(-0.248093\pi\)
\(17\)		−	24.5673i		−	1.44514i	−0.691299	−	0.722569i	\(-0.742961\pi\)
							0.691299	−	0.722569i	\(-0.257039\pi\)
\(19\)	5.17188	−	5.17188i	0.272204	−	0.272204i	−0.557783	−	0.829987i	\(-0.688348\pi\)
							0.829987	+	0.557783i	\(0.188348\pi\)
\(23\)	13.7010	−	23.7307i	0.595694	−	1.03177i	−0.397755	−	0.917492i	\(-0.630211\pi\)
							0.993449	−	0.114280i	\(-0.0364560\pi\)
\(29\)	−6.54235	+	24.4164i	−0.225598	+	0.841944i	0.756566	+	0.653917i	\(0.226876\pi\)
							−0.982164	+	0.188026i	\(0.939791\pi\)
\(31\)	5.21707	−	9.03623i	0.168293	−	0.291491i	−0.769527	−	0.638614i	\(-0.779508\pi\)
							0.937820	+	0.347123i	\(0.112841\pi\)
\(37\)	35.3961	+	35.3961i	0.956652	+	0.956652i	0.999099	−	0.0424465i	\(-0.0135152\pi\)
							−0.0424465	+	0.999099i	\(0.513515\pi\)
\(41\)	30.0752	−	52.0917i	0.733541	−	1.27053i	−0.221820	−	0.975088i	\(-0.571200\pi\)
							0.955361	−	0.295442i	\(-0.0954669\pi\)
\(43\)	−10.6707	+	39.8237i	−0.248157	+	0.926133i	0.723614	+	0.690205i	\(0.242480\pi\)
							−0.971771	+	0.235928i	\(0.924187\pi\)
\(47\)	16.9574	−	9.79036i	0.360796	−	0.208306i	−0.308634	−	0.951181i	\(-0.599872\pi\)
							0.669430	+	0.742875i	\(0.266538\pi\)