Embedded newform 144.2.x.e.85.13

• Label: 144.2.x.e.85.13
• Level: $144$
• Weight: $2$
• Character: 144.85
• 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			85.13
Character	\(\chi\)	\(=\)	144.85
Dual form			144.2.x.e.61.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.902916 - 1.08846i) q^{2} +(-1.15652 - 1.28937i) q^{3} +(-0.369486 - 1.96557i) q^{4} +(1.21694 - 0.326078i) q^{5} +(-2.44766 + 0.0946360i) q^{6} +(-0.707732 + 0.408609i) q^{7} +(-2.47306 - 1.37258i) q^{8} +(-0.324925 + 2.98235i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 4 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 28 q^{6} - 8 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.902916	−	1.08846i	0.638458	−	0.769657i
							\(3\)	−1.15652	−	1.28937i	−0.667717	−	0.744415i
\(5\)	1.21694	−	0.326078i	0.544232	−	0.145827i								0.0237796	−	0.999717i	\(-0.492430\pi\)
							0.520452	+	0.853891i	\(0.325763\pi\)
\(7\)	−0.707732	+	0.408609i	−0.267497	+	0.154440i	−0.627750	−	0.778415i	\(-0.716024\pi\)
							0.360252	+	0.932855i	\(0.382691\pi\)
\(11\)	0.497094	−	1.85518i	0.149879	−	0.559357i	−0.849610	−	0.527411i	\(-0.823163\pi\)
							0.999490	−	0.0319463i	\(-0.0101706\pi\)
\(13\)	0.116336	+	0.434170i	0.0322657	+	0.120417i	0.980180	−	0.198107i	\(-0.0634794\pi\)
							−0.947915	+	0.318524i	\(0.896813\pi\)
\(17\)	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	\(-0.229493\pi\)
							−0.196096	+	0.980585i	\(0.562826\pi\)
\(29\)	8.65622	+	2.31943i	1.60742	+	0.430707i	0.947272	−	0.320430i	\(-0.103827\pi\)
							0.660147	+	0.751136i	\(0.270494\pi\)
\(31\)	−4.61440	+	7.99238i	−0.828771	+	1.43547i	0.0702316	+	0.997531i	\(0.477626\pi\)
							−0.899003	+	0.437943i	\(0.855707\pi\)
\(37\)	−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	\(-0.642615\pi\)
							−0.997147	+	0.0754884i	\(0.975948\pi\)
\(43\)	1.66072	−	6.19791i	0.253258	−	0.945172i	−0.715793	−	0.698312i	\(-0.753935\pi\)
							0.969051	−	0.246860i	\(-0.0793987\pi\)
\(47\)	−0.140916	−	0.244074i	−0.0205547	−	0.0356018i	0.855565	−	0.517695i	\(-0.173210\pi\)
							−0.876120	+	0.482093i	\(0.839877\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.x.e.85.13	yes	72
3.2	odd	2		432.2.y.e.37.6		72
4.3	odd	2		576.2.bb.e.49.13		72
9.2	odd	6		432.2.y.e.181.18		72
9.7	even	3	inner	144.2.x.e.133.1	yes	72
12.11	even	2		1728.2.bc.e.1009.7		72
16.3	odd	4		576.2.bb.e.337.6		72
16.13	even	4	inner	144.2.x.e.13.1	✓	72
36.7	odd	6		576.2.bb.e.241.6		72
36.11	even	6		1728.2.bc.e.1585.12		72
48.29	odd	4		432.2.y.e.253.18		72
48.35	even	4		1728.2.bc.e.145.12		72
144.29	odd	12		432.2.y.e.397.6		72
144.61	even	12	inner	144.2.x.e.61.13	yes	72
144.83	even	12		1728.2.bc.e.721.7		72
144.115	odd	12		576.2.bb.e.529.13		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.e.13.1	✓	72	16.13	even	4	inner
144.2.x.e.61.13	yes	72	144.61	even	12	inner
144.2.x.e.85.13	yes	72	1.1	even	1	trivial
144.2.x.e.133.1	yes	72	9.7	even	3	inner
432.2.y.e.37.6		72	3.2	odd	2
432.2.y.e.181.18		72	9.2	odd	6
432.2.y.e.253.18		72	48.29	odd	4
432.2.y.e.397.6		72	144.29	odd	12
576.2.bb.e.49.13		72	4.3	odd	2
576.2.bb.e.241.6		72	36.7	odd	6
576.2.bb.e.337.6		72	16.3	odd	4
576.2.bb.e.529.13		72	144.115	odd	12
1728.2.bc.e.145.12		72	48.35	even	4
1728.2.bc.e.721.7		72	144.83	even	12
1728.2.bc.e.1009.7		72	12.11	even	2
1728.2.bc.e.1585.12		72	36.11	even	6