Embedded newform 144.3.w.a.5.12

• Label: 144.3.w.a.5.12
• 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			5.12
Character	\(\chi\)	\(=\)	144.5
Dual form			144.3.w.a.29.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.51447 + 1.30628i) q^{2} +(-2.22089 - 2.01684i) q^{3} +(0.587245 - 3.95666i) q^{4} +(-1.88738 - 7.04379i) q^{5} +(5.99804 + 0.153334i) q^{6} +(-8.89672 + 5.13652i) q^{7} +(4.27915 + 6.75935i) q^{8} +(0.864706 + 8.95836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184q - 6q^{2} - 4q^{3} - 2q^{4} - 6q^{5} - 10q^{6} + O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.51447	+	1.30628i	−0.757235	+	0.653142i
							\(3\)	−2.22089	−	2.01684i	−0.740297	−	0.672280i
\(5\)	−1.88738	−	7.04379i	−0.377476	−	1.40876i								−0.849694	−	0.527276i	\(-0.823213\pi\)
							0.472218	−	0.881482i	\(-0.343453\pi\)
\(7\)	−8.89672	+	5.13652i	−1.27096	+	0.733789i	−0.975169	−	0.221464i	\(-0.928917\pi\)
							−0.295791	+	0.955253i	\(0.595583\pi\)
\(11\)	8.94555	+	2.39695i	0.813232	+	0.217905i	0.641385	−	0.767219i	\(-0.278360\pi\)
							0.171847	+	0.985124i	\(0.445027\pi\)
\(13\)	4.68710	+	17.4925i	0.360546	+	1.34558i	0.873359	+	0.487076i	\(0.161937\pi\)
							−0.512813	+	0.858500i	\(0.671397\pi\)
\(17\)		−	16.2105i		−	0.953558i	−0.879023	−	0.476779i	\(-0.841804\pi\)
							0.879023	−	0.476779i	\(-0.158196\pi\)
\(19\)	−10.1733	+	10.1733i	−0.535435	+	0.535435i	−0.922185	−	0.386750i	\(-0.873598\pi\)
							0.386750	+	0.922185i	\(0.373598\pi\)
\(23\)	−15.7930	+	27.3544i	−0.686654	+	1.18932i	0.286260	+	0.958152i	\(0.407588\pi\)
							−0.972914	+	0.231168i	\(0.925745\pi\)
\(29\)	5.89303	−	21.9931i	0.203208	−	0.758383i	−0.786780	−	0.617233i	\(-0.788253\pi\)
							0.989988	−	0.141150i	\(-0.0450799\pi\)
\(31\)	−4.70872	+	8.15575i	−0.151894	+	0.263089i	−0.931924	−	0.362654i	\(-0.881871\pi\)
							0.780030	+	0.625743i	\(0.215204\pi\)
\(37\)	−9.86943	−	9.86943i	−0.266741	−	0.266741i	0.561044	−	0.827786i	\(-0.310400\pi\)
							−0.827786	+	0.561044i	\(0.810400\pi\)
\(41\)	−3.91236	+	6.77640i	−0.0954234	+	0.165278i	−0.909785	−	0.415079i	\(-0.863754\pi\)
							0.814362	+	0.580357i	\(0.197087\pi\)
\(43\)	−17.7682	+	66.3117i	−0.413213	+	1.54213i	0.375175	+	0.926954i	\(0.377582\pi\)
							−0.788388	+	0.615178i	\(0.789084\pi\)
\(47\)	−47.8768	+	27.6417i	−1.01866	+	0.588121i	−0.913714	−	0.406357i	\(-0.866799\pi\)
							−0.104941	+	0.994478i	\(0.533465\pi\)