Embedded newform 144.3.w.a.5.11

• Label: 144.3.w.a.5.11
• 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			5.11
Character	\(\chi\)	\(=\)	144.5
Dual form			144.3.w.a.29.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.58134 + 1.22449i) q^{2} +(1.74774 + 2.43832i) q^{3} +(1.00126 - 3.87266i) q^{4} +(0.973349 + 3.63259i) q^{5} +(-5.74946 - 1.71573i) q^{6} +(11.7048 - 6.75777i) q^{7} +(3.15868 + 7.35002i) q^{8} +(-2.89081 + 8.52310i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184q - 6q^{2} - 4q^{3} - 2q^{4} - 6q^{5} - 10q^{6} + O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.58134	+	1.22449i	−0.790669	+	0.612243i
							\(3\)	1.74774	+	2.43832i	0.582580	+	0.812773i
\(5\)	0.973349	+	3.63259i	0.194670	+	0.726517i								0.992352	+	0.123440i	\(0.0393925\pi\)
							−0.797682	+	0.603078i	\(0.793941\pi\)
\(7\)	11.7048	−	6.75777i	1.67211	−	0.965395i	0.705661	−	0.708549i	\(-0.250650\pi\)
							0.966452	−	0.256846i	\(-0.0826833\pi\)
\(11\)	4.44465	+	1.19094i	0.404059	+	0.108267i	0.455124	−	0.890428i	\(-0.349595\pi\)
							−0.0510651	+	0.998695i	\(0.516262\pi\)
\(13\)	1.01931	+	3.80413i	0.0784087	+	0.292625i	0.993985	−	0.109520i	\(-0.0349314\pi\)
							−0.915576	+	0.402145i	\(0.868265\pi\)
\(17\)		−	20.6241i		−	1.21318i	−0.795013	−	0.606592i	\(-0.792536\pi\)
							0.795013	−	0.606592i	\(-0.207464\pi\)
\(19\)	−19.5324	+	19.5324i	−1.02802	+	1.02802i	−0.0284249	+	0.999596i	\(0.509049\pi\)
							−0.999596	+	0.0284249i	\(0.990951\pi\)
\(23\)	−14.7668	+	25.5769i	−0.642035	+	1.11204i	0.342943	+	0.939356i	\(0.388576\pi\)
							−0.984978	+	0.172681i	\(0.944757\pi\)
\(29\)	−5.65607	+	21.1088i	−0.195037	+	0.727888i	0.797220	+	0.603689i	\(0.206303\pi\)
							−0.992257	+	0.124199i	\(0.960364\pi\)
\(31\)	6.01256	−	10.4141i	0.193954	−	0.335938i	−0.752603	−	0.658474i	\(-0.771202\pi\)
							0.946557	+	0.322537i	\(0.104536\pi\)
\(37\)	−20.6078	−	20.6078i	−0.556968	−	0.556968i	0.371475	−	0.928443i	\(-0.378852\pi\)
							−0.928443	+	0.371475i	\(0.878852\pi\)
\(41\)	21.7630	−	37.6946i	0.530804	−	0.919379i	−0.468550	−	0.883437i	\(-0.655223\pi\)
							0.999354	−	0.0359424i	\(-0.0114433\pi\)
\(43\)	5.80291	−	21.6568i	0.134952	−	0.503646i	−0.865047	−	0.501692i	\(-0.832711\pi\)
							0.999998	−	0.00195433i	\(-0.000622083\pi\)
\(47\)	9.68895	−	5.59392i	0.206148	−	0.119020i	−0.393372	−	0.919379i	\(-0.628692\pi\)
							0.599520	+	0.800360i	\(0.295358\pi\)