Embedded newform 144.2.x.e.133.1

• Label: 144.2.x.e.133.1
• Level: $144$
• Weight: $2$
• Character: 144.133
• Analytic conductor: $1.150$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	144.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.14984578911\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			133.1
Character	\(\chi\)	\(=\)	144.133
Dual form			144.2.x.e.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39409 - 0.237718i) q^{2} +(1.28937 + 1.15652i) q^{3} +(1.88698 + 0.662802i) q^{4} +(-0.326078 + 1.21694i) q^{5} +(-1.52257 - 1.91880i) q^{6} +(0.707732 + 0.408609i) q^{7} +(-2.47306 - 1.37258i) q^{8} +(0.324925 + 2.98235i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 4 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 28 q^{6} - 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39409	−	0.237718i	−0.985771	−	0.168092i
							\(3\)	1.28937	+	1.15652i	0.744415	+	0.667717i
\(5\)	−0.326078	+	1.21694i	−0.145827	+	0.544232i								0.853891	+	0.520452i	\(0.174237\pi\)
							−0.999717	+	0.0237796i	\(0.992430\pi\)
\(7\)	0.707732	+	0.408609i	0.267497	+	0.154440i	0.627750	−	0.778415i	\(-0.283976\pi\)
							−0.360252	+	0.932855i	\(0.617309\pi\)
\(11\)	−1.85518	+	0.497094i	−0.559357	+	0.149879i	−0.527411	−	0.849610i	\(-0.676837\pi\)
							−0.0319463	+	0.999490i	\(0.510171\pi\)
\(13\)	−0.434170	−	0.116336i	−0.120417	−	0.0322657i	0.198107	−	0.980180i	\(-0.436521\pi\)
							−0.318524	+	0.947915i	\(0.603187\pi\)
\(17\)	6.62002			1.60559			0.802795	−	0.596255i	\(-0.203345\pi\)
							0.802795	+	0.596255i	\(0.203345\pi\)
\(19\)	1.18421	−	1.18421i	0.271677	−	0.271677i	−0.558098	−	0.829775i	\(-0.688469\pi\)
							0.829775	+	0.558098i	\(0.188469\pi\)
\(23\)	−2.66201	+	1.53691i	−0.555067	+	0.320468i	−0.751163	−	0.660117i	\(-0.770507\pi\)
							0.196096	+	0.980585i	\(0.437174\pi\)
\(29\)	−2.31943	−	8.65622i	−0.430707	−	1.60742i	−0.751136	−	0.660147i	\(-0.770494\pi\)
							0.320430	−	0.947272i	\(-0.396173\pi\)
\(31\)	−4.61440	−	7.99238i	−0.828771	−	1.43547i	−0.899003	−	0.437943i	\(-0.855707\pi\)
							0.0702316	−	0.997531i	\(-0.477626\pi\)
\(37\)	−2.14134	−	2.14134i	−0.352035	−	0.352035i	0.508831	−	0.860866i	\(-0.330078\pi\)
							−0.860866	+	0.508831i	\(0.830078\pi\)
\(41\)	9.15868	−	5.28777i	1.43035	−	0.825810i	0.433199	−	0.901299i	\(-0.357385\pi\)
							0.997147	+	0.0754884i	\(0.0240516\pi\)
\(43\)	−6.19791	+	1.66072i	−0.945172	+	0.253258i	−0.698312	−	0.715793i	\(-0.746065\pi\)
							−0.246860	+	0.969051i	\(0.579399\pi\)
\(47\)	−0.140916	+	0.244074i	−0.0205547	+	0.0356018i	−0.876120	−	0.482093i	\(-0.839877\pi\)
							0.855565	+	0.517695i	\(0.173210\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.x.e.133.1	yes	72
3.2	odd	2		432.2.y.e.181.18		72
4.3	odd	2		576.2.bb.e.241.6		72
9.4	even	3	inner	144.2.x.e.85.13	yes	72
9.5	odd	6		432.2.y.e.37.6		72
12.11	even	2		1728.2.bc.e.1585.12		72
16.3	odd	4		576.2.bb.e.529.13		72
16.13	even	4	inner	144.2.x.e.61.13	yes	72
36.23	even	6		1728.2.bc.e.1009.7		72
36.31	odd	6		576.2.bb.e.49.13		72
48.29	odd	4		432.2.y.e.397.6		72
48.35	even	4		1728.2.bc.e.721.7		72
144.13	even	12	inner	144.2.x.e.13.1	✓	72
144.67	odd	12		576.2.bb.e.337.6		72
144.77	odd	12		432.2.y.e.253.18		72
144.131	even	12		1728.2.bc.e.145.12		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.e.13.1	✓	72	144.13	even	12	inner
144.2.x.e.61.13	yes	72	16.13	even	4	inner
144.2.x.e.85.13	yes	72	9.4	even	3	inner
144.2.x.e.133.1	yes	72	1.1	even	1	trivial
432.2.y.e.37.6		72	9.5	odd	6
432.2.y.e.181.18		72	3.2	odd	2
432.2.y.e.253.18		72	144.77	odd	12
432.2.y.e.397.6		72	48.29	odd	4
576.2.bb.e.49.13		72	36.31	odd	6
576.2.bb.e.241.6		72	4.3	odd	2
576.2.bb.e.337.6		72	144.67	odd	12
576.2.bb.e.529.13		72	16.3	odd	4
1728.2.bc.e.145.12		72	144.131	even	12
1728.2.bc.e.721.7		72	48.35	even	4
1728.2.bc.e.1009.7		72	36.23	even	6
1728.2.bc.e.1585.12		72	12.11	even	2