Embedded newform 144.2.x.e.133.15

• Label: 144.2.x.e.133.15
• 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			133.15
Character	\(\chi\)	\(=\)	144.133
Dual form			144.2.x.e.13.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.24122 + 0.677764i) q^{2} +(-1.37530 - 1.05287i) q^{3} +(1.08127 + 1.68251i) q^{4} +(1.10094 - 4.10876i) q^{5} +(-0.993456 - 2.23898i) q^{6} +(1.63313 + 0.942891i) q^{7} +(0.201751 + 2.82122i) q^{8} +(0.782911 + 2.89604i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 4 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 28 q^{6} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.24122	+	0.677764i	0.877677	+	0.479252i
							\(3\)	−1.37530	−	1.05287i	−0.794031	−	0.607877i
\(5\)	1.10094	−	4.10876i	0.492355	−	1.83750i								−0.0520092	−	0.998647i	\(-0.516563\pi\)
							0.544365	−	0.838849i	\(-0.316771\pi\)
\(7\)	1.63313	+	0.942891i	0.617267	+	0.356379i	0.775804	−	0.630974i	\(-0.217344\pi\)
							−0.158537	+	0.987353i	\(0.550678\pi\)
\(11\)	−1.18389	+	0.317223i	−0.356957	+	0.0956464i	−0.432841	−	0.901470i	\(-0.642489\pi\)
							0.0758838	+	0.997117i	\(0.475822\pi\)
\(13\)	−2.31428	−	0.620109i	−0.641865	−	0.171987i	−0.0768171	−	0.997045i	\(-0.524476\pi\)
							−0.565048	+	0.825058i	\(0.691142\pi\)
\(17\)	3.44622			0.835831			0.417915	−	0.908486i	\(-0.362761\pi\)
							0.417915	+	0.908486i	\(0.362761\pi\)
\(19\)	−4.17726	+	4.17726i	−0.958328	+	0.958328i	−0.999166	−	0.0408377i	\(-0.986997\pi\)
							0.0408377	+	0.999166i	\(0.486997\pi\)
\(23\)	−1.34867	+	0.778655i	−0.281217	+	0.162361i	−0.633974	−	0.773354i	\(-0.718578\pi\)
							0.352757	+	0.935715i	\(0.385244\pi\)
\(29\)	−0.343336	−	1.28135i	−0.0637558	−	0.237940i	0.926693	−	0.375818i	\(-0.122638\pi\)
							−0.990449	+	0.137878i	\(0.955972\pi\)
\(31\)	1.25840	+	2.17961i	0.226015	+	0.391469i	0.956623	−	0.291327i	\(-0.0940969\pi\)
							−0.730609	+	0.682796i	\(0.760764\pi\)
\(37\)	5.77154	+	5.77154i	0.948836	+	0.948836i	0.998753	−	0.0499178i	\(-0.0158959\pi\)
							−0.0499178	+	0.998753i	\(0.515896\pi\)
\(41\)	−3.43245	+	1.98173i	−0.536058	+	0.309493i	−0.743480	−	0.668758i	\(-0.766826\pi\)
							0.207422	+	0.978252i	\(0.433493\pi\)
\(43\)	−2.96053	+	0.793272i	−0.451477	+	0.120973i	−0.477390	−	0.878691i	\(-0.658417\pi\)
							0.0259136	+	0.999664i	\(0.491751\pi\)
\(47\)	0.230692	−	0.399570i	0.0336498	−	0.0582832i	−0.848710	−	0.528858i	\(-0.822620\pi\)
							0.882360	+	0.470575i	\(0.155954\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.x.e.133.15	yes	72
3.2	odd	2		432.2.y.e.181.4		72
4.3	odd	2		576.2.bb.e.241.16		72
9.4	even	3	inner	144.2.x.e.85.4	yes	72
9.5	odd	6		432.2.y.e.37.15		72
12.11	even	2		1728.2.bc.e.1585.1		72
16.3	odd	4		576.2.bb.e.529.6		72
16.13	even	4	inner	144.2.x.e.61.4	yes	72
36.23	even	6		1728.2.bc.e.1009.18		72
36.31	odd	6		576.2.bb.e.49.6		72
48.29	odd	4		432.2.y.e.397.15		72
48.35	even	4		1728.2.bc.e.721.18		72
144.13	even	12	inner	144.2.x.e.13.15	✓	72
144.67	odd	12		576.2.bb.e.337.16		72
144.77	odd	12		432.2.y.e.253.4		72
144.131	even	12		1728.2.bc.e.145.1		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.e.13.15	✓	72	144.13	even	12	inner
144.2.x.e.61.4	yes	72	16.13	even	4	inner
144.2.x.e.85.4	yes	72	9.4	even	3	inner
144.2.x.e.133.15	yes	72	1.1	even	1	trivial
432.2.y.e.37.15		72	9.5	odd	6
432.2.y.e.181.4		72	3.2	odd	2
432.2.y.e.253.4		72	144.77	odd	12
432.2.y.e.397.15		72	48.29	odd	4
576.2.bb.e.49.6		72	36.31	odd	6
576.2.bb.e.241.16		72	4.3	odd	2
576.2.bb.e.337.16		72	144.67	odd	12
576.2.bb.e.529.6		72	16.3	odd	4
1728.2.bc.e.145.1		72	144.131	even	12
1728.2.bc.e.721.18		72	48.35	even	4
1728.2.bc.e.1009.18		72	36.23	even	6
1728.2.bc.e.1585.1		72	12.11	even	2