Embedded newform 144.3.w.a.5.5

• Label: 144.3.w.a.5.5
• 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			5.5
Character	\(\chi\)	\(=\)	144.5
Dual form			144.3.w.a.29.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.88357 + 0.672432i) q^{2} +(2.93495 + 0.621365i) q^{3} +(3.09567 - 2.53315i) q^{4} +(1.03307 + 3.85549i) q^{5} +(-5.94600 + 0.803167i) q^{6} +(-11.2715 + 6.50760i) q^{7} +(-4.12754 + 6.85298i) q^{8} +(8.22781 + 3.64735i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184q - 6q^{2} - 4q^{3} - 2q^{4} - 6q^{5} - 10q^{6} + O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.88357	+	0.672432i	−0.941785	+	0.336216i
							\(3\)	2.93495	+	0.621365i	0.978315	+	0.207122i
\(5\)	1.03307	+	3.85549i	0.206615	+	0.771097i								0.988951	+	0.148241i	\(0.0473611\pi\)
							−0.782336	+	0.622856i	\(0.785972\pi\)
\(7\)	−11.2715	+	6.50760i	−1.61021	+	0.929658i	−0.620895	+	0.783894i	\(0.713230\pi\)
							−0.989319	+	0.145763i	\(0.953436\pi\)
\(11\)	9.91527	+	2.65679i	0.901388	+	0.241526i	0.679612	−	0.733572i	\(-0.262148\pi\)
							0.221776	+	0.975098i	\(0.428815\pi\)
\(13\)	−3.10413	−	11.5848i	−0.238780	−	0.891138i	−0.976409	−	0.215931i	\(-0.930721\pi\)
							0.737629	−	0.675206i	\(-0.235945\pi\)
\(17\)			31.4441i			1.84966i	0.380386	+	0.924828i	\(0.375791\pi\)
							−0.380386	+	0.924828i	\(0.624209\pi\)
\(19\)	−8.43842	+	8.43842i	−0.444127	+	0.444127i	−0.893396	−	0.449269i	\(-0.851684\pi\)
							0.449269	+	0.893396i	\(0.351684\pi\)
\(23\)	0.489846	−	0.848438i	0.0212976	−	0.0368886i	−0.855180	−	0.518331i	\(-0.826554\pi\)
							0.876478	+	0.481442i	\(0.159887\pi\)
\(29\)	5.25776	−	19.6222i	0.181302	−	0.676629i	−0.814090	−	0.580739i	\(-0.802764\pi\)
							0.995392	−	0.0958899i	\(-0.0305697\pi\)
\(31\)	0.00519834	−	0.00900379i	0.000167688	−	0.000290445i	−0.865942	−	0.500145i	\(-0.833280\pi\)
							0.866109	+	0.499855i	\(0.166613\pi\)
\(37\)	14.4004	+	14.4004i	0.389201	+	0.389201i	0.874402	−	0.485202i	\(-0.161254\pi\)
							−0.485202	+	0.874402i	\(0.661254\pi\)
\(41\)	13.2394	−	22.9314i	0.322913	−	0.559302i	−0.658175	−	0.752865i	\(-0.728671\pi\)
							0.981088	+	0.193563i	\(0.0620045\pi\)
\(43\)	0.237711	−	0.887151i	0.00552817	−	0.0206314i	−0.963107	−	0.269120i	\(-0.913267\pi\)
							0.968635	+	0.248489i	\(0.0799338\pi\)
\(47\)	−30.9514	+	17.8698i	−0.658541	+	0.380209i	−0.791721	−	0.610883i	\(-0.790815\pi\)
							0.133180	+	0.991092i	\(0.457481\pi\)