Embedded newform 144.3.w.a.5.10

• Label: 144.3.w.a.5.10
• 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			5.10
Character	\(\chi\)	\(=\)	144.5
Dual form			144.3.w.a.29.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59704 - 1.20393i) q^{2} +(-1.75135 - 2.43573i) q^{3} +(1.10109 + 3.84547i) q^{4} +(-2.49590 - 9.31484i) q^{5} +(-0.135487 + 5.99847i) q^{6} +(7.55275 - 4.36058i) q^{7} +(2.87120 - 7.46701i) q^{8} +(-2.86558 + 8.53161i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184q - 6q^{2} - 4q^{3} - 2q^{4} - 6q^{5} - 10q^{6} + O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59704	−	1.20393i	−0.798521	−	0.601967i
							\(3\)	−1.75135	−	2.43573i	−0.583782	−	0.811911i
\(5\)	−2.49590	−	9.31484i	−0.499181	−	1.86297i								−0.505228	−	0.862986i	\(-0.668592\pi\)
							0.00604760	−	0.999982i	\(-0.498075\pi\)
\(7\)	7.55275	−	4.36058i	1.07896	−	0.622941i	0.148348	−	0.988935i	\(-0.452605\pi\)
							0.930617	+	0.365995i	\(0.119271\pi\)
\(11\)	−8.59716	−	2.30360i	−0.781560	−	0.209418i	−0.154088	−	0.988057i	\(-0.549244\pi\)
							−0.627473	+	0.778639i	\(0.715911\pi\)
\(13\)	−1.01775	−	3.79829i	−0.0782884	−	0.292176i	0.915670	−	0.401930i	\(-0.131660\pi\)
							−0.993959	+	0.109754i	\(0.964994\pi\)
\(17\)			20.4214i			1.20126i	0.799527	+	0.600630i	\(0.205083\pi\)
							−0.799527	+	0.600630i	\(0.794917\pi\)
\(19\)	1.44592	−	1.44592i	0.0761009	−	0.0761009i	−0.668032	−	0.744133i	\(-0.732863\pi\)
							0.744133	+	0.668032i	\(0.232863\pi\)
\(23\)	6.97317	−	12.0779i	0.303181	−	0.525125i	−0.673673	−	0.739029i	\(-0.735285\pi\)
							0.976855	+	0.213904i	\(0.0686179\pi\)
\(29\)	−8.45427	+	31.5518i	−0.291527	+	1.08799i	0.652410	+	0.757866i	\(0.273758\pi\)
							−0.943937	+	0.330126i	\(0.892909\pi\)
\(31\)	12.0086	−	20.7996i	0.387375	−	0.670954i	−0.604720	−	0.796438i	\(-0.706715\pi\)
							0.992096	+	0.125484i	\(0.0400484\pi\)
\(37\)	−14.6915	−	14.6915i	−0.397067	−	0.397067i	0.480130	−	0.877197i	\(-0.340589\pi\)
							−0.877197	+	0.480130i	\(0.840589\pi\)
\(41\)	29.9984	−	51.9587i	0.731667	−	1.26729i	−0.224503	−	0.974473i	\(-0.572076\pi\)
							0.956170	−	0.292812i	\(-0.0945909\pi\)
\(43\)	9.90936	−	36.9823i	0.230450	−	0.860052i	−0.749697	−	0.661781i	\(-0.769801\pi\)
							0.980147	−	0.198271i	\(-0.0635326\pi\)
\(47\)	31.8352	−	18.3801i	0.677344	−	0.391065i	−0.121509	−	0.992590i	\(-0.538773\pi\)
							0.798854	+	0.601525i	\(0.205440\pi\)