Embedded newform 144.3.w.a.5.20

• Label: 144.3.w.a.5.20
• 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			5.20
Character	\(\chi\)	\(=\)	144.5
Dual form			144.3.w.a.29.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.608739 - 1.90511i) q^{2} +(2.61769 - 1.46550i) q^{3} +(-3.25887 + 2.31943i) q^{4} +(-2.13498 - 7.96785i) q^{5} +(-4.38543 - 4.09487i) q^{6} +(-10.2471 + 5.91616i) q^{7} +(6.40256 + 4.79658i) q^{8} +(4.70460 - 7.67247i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184q - 6q^{2} - 4q^{3} - 2q^{4} - 6q^{5} - 10q^{6} + O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.608739	−	1.90511i	−0.304370	−	0.952554i
							\(3\)	2.61769	−	1.46550i	0.872563	−	0.488501i
\(5\)	−2.13498	−	7.96785i	−0.426996	−	1.59357i								−0.759526	−	0.650477i	\(-0.774569\pi\)
							0.332530	−	0.943093i	\(-0.392098\pi\)
\(7\)	−10.2471	+	5.91616i	−1.46387	+	0.845165i	−0.999187	−	0.0403127i	\(-0.987165\pi\)
							−0.464682	+	0.885478i	\(0.653831\pi\)
\(11\)	−2.67965	−	0.718009i	−0.243604	−	0.0652736i	0.134951	−	0.990852i	\(-0.456912\pi\)
							−0.378555	+	0.925579i	\(0.623579\pi\)
\(13\)	−0.984428	−	3.67394i	−0.0757253	−	0.282611i	0.917672	−	0.397340i	\(-0.130067\pi\)
							−0.993397	+	0.114729i	\(0.963400\pi\)
\(17\)		−	3.92397i		−	0.230822i	−0.993318	−	0.115411i	\(-0.963181\pi\)
							0.993318	−	0.115411i	\(-0.0368185\pi\)
\(19\)	21.0377	−	21.0377i	1.10725	−	1.10725i	0.113736	−	0.993511i	\(-0.463718\pi\)
							0.993511	−	0.113736i	\(-0.0362819\pi\)
\(23\)	−9.61304	+	16.6503i	−0.417958	+	0.723925i	−0.995734	−	0.0922705i	\(-0.970588\pi\)
							0.577776	+	0.816196i	\(0.303921\pi\)
\(29\)	9.36387	−	34.9464i	0.322892	−	1.20505i	−0.593522	−	0.804818i	\(-0.702263\pi\)
							0.916414	−	0.400231i	\(-0.131070\pi\)
\(31\)	−10.4943	+	18.1766i	−0.338525	+	0.586343i	−0.984156	−	0.177307i	\(-0.943261\pi\)
							0.645630	+	0.763650i	\(0.276595\pi\)
\(37\)	7.79895	+	7.79895i	0.210782	+	0.210782i	0.804600	−	0.593817i	\(-0.202380\pi\)
							−0.593817	+	0.804600i	\(0.702380\pi\)
\(41\)	15.2735	−	26.4545i	0.372524	−	0.645231i	−0.617429	−	0.786627i	\(-0.711826\pi\)
							0.989953	+	0.141395i	\(0.0451589\pi\)
\(43\)	7.11563	−	26.5559i	0.165480	−	0.617579i	−0.832499	−	0.554027i	\(-0.813090\pi\)
							0.997979	−	0.0635521i	\(-0.0202429\pi\)
\(47\)	71.8427	−	41.4784i	1.52857	−	0.882519i	0.529146	−	0.848531i	\(-0.322512\pi\)
							0.999422	−	0.0339887i	\(-0.0108210\pi\)