Embedded newform 144.2.u.a.131.1

• Label: 144.2.u.a.131.1
• Level: $144$
• Weight: $2$
• Character: 144.131
• Analytic conductor: $1.150$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.14984578911\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			131.1
Character	\(\chi\)	\(=\)	144.131
Dual form			144.2.u.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40455 + 0.165045i) q^{2} +(1.03912 - 1.38573i) q^{3} +(1.94552 - 0.463628i) q^{4} +(1.15827 - 0.310357i) q^{5} +(-1.23078 + 2.11782i) q^{6} +(0.356047 + 0.616691i) q^{7} +(-2.65606 + 0.972288i) q^{8} +(-0.840471 - 2.87986i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 6 q^{2} - 4 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} - 4 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40455	+	0.165045i	−0.993167	+	0.116705i
							\(3\)	1.03912	−	1.38573i	0.599935	−	0.800049i
\(5\)	1.15827	−	0.310357i	0.517994	−	0.138796i								0.00965542	−	0.999953i	\(-0.496927\pi\)
							0.508339	+	0.861157i	\(0.330260\pi\)
\(7\)	0.356047	+	0.616691i	0.134573	+	0.233087i	0.925434	−	0.378908i	\(-0.123700\pi\)
							−0.790861	+	0.611996i	\(0.790367\pi\)
\(11\)	−2.28146	−	0.611314i	−0.687885	−	0.184318i	−0.102087	−	0.994775i	\(-0.532552\pi\)
							−0.585798	+	0.810457i	\(0.699219\pi\)
\(13\)	4.90394	−	1.31401i	1.36011	−	0.364440i	0.496253	−	0.868178i	\(-0.334709\pi\)
							0.863857	+	0.503738i	\(0.168042\pi\)
\(17\)			0.863180i			0.209352i	0.994506	+	0.104676i	\(0.0333806\pi\)
							−0.994506	+	0.104676i	\(0.966619\pi\)
\(19\)	−0.539682	+	0.539682i	−0.123811	+	0.123811i	−0.766297	−	0.642486i	\(-0.777903\pi\)
							0.642486	+	0.766297i	\(0.277903\pi\)
\(23\)	−0.689728	−	0.398215i	−0.143818	−	0.0830335i	0.426364	−	0.904552i	\(-0.359794\pi\)
							−0.570182	+	0.821518i	\(0.693128\pi\)
\(29\)	8.31534	+	2.22809i	1.54412	+	0.413746i	0.927594	−	0.373589i	\(-0.121873\pi\)
							0.616526	+	0.787335i	\(0.288540\pi\)
\(31\)	−4.18508	−	2.41626i	−0.751663	−	0.433973i	0.0746314	−	0.997211i	\(-0.476222\pi\)
							−0.826295	+	0.563238i	\(0.809555\pi\)
\(37\)	−6.95600	+	6.95600i	−1.14356	+	1.14356i	−0.155765	+	0.987794i	\(0.549784\pi\)
							−0.987794	+	0.155765i	\(0.950216\pi\)
\(41\)	−3.17027	+	5.49108i	−0.495114	+	0.857562i	−0.999984	−	0.00563304i	\(-0.998207\pi\)
							0.504870	+	0.863195i	\(0.331540\pi\)
\(43\)	−3.22509	+	12.0362i	−0.491822	+	1.83550i	0.0553283	+	0.998468i	\(0.482379\pi\)
							−0.547150	+	0.837035i	\(0.684287\pi\)
\(47\)	1.31575	+	2.27895i	0.191923	+	0.332420i	0.945887	−	0.324495i	\(-0.105194\pi\)
							−0.753965	+	0.656915i	\(0.771861\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.u.a.131.1	yes	88
3.2	odd	2		432.2.v.a.179.22		88
4.3	odd	2		576.2.y.a.239.8		88
9.2	odd	6	inner	144.2.u.a.83.7	yes	88
9.7	even	3		432.2.v.a.35.16		88
12.11	even	2		1728.2.z.a.1583.8		88
16.5	even	4		576.2.y.a.527.4		88
16.11	odd	4	inner	144.2.u.a.59.7	yes	88
36.7	odd	6		1728.2.z.a.1007.8		88
36.11	even	6		576.2.y.a.47.4		88
48.5	odd	4		1728.2.z.a.719.8		88
48.11	even	4		432.2.v.a.395.16		88
144.11	even	12	inner	144.2.u.a.11.1	✓	88
144.43	odd	12		432.2.v.a.251.22		88
144.101	odd	12		576.2.y.a.335.8		88
144.133	even	12		1728.2.z.a.143.8		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.1	✓	88	144.11	even	12	inner
144.2.u.a.59.7	yes	88	16.11	odd	4	inner
144.2.u.a.83.7	yes	88	9.2	odd	6	inner
144.2.u.a.131.1	yes	88	1.1	even	1	trivial
432.2.v.a.35.16		88	9.7	even	3
432.2.v.a.179.22		88	3.2	odd	2
432.2.v.a.251.22		88	144.43	odd	12
432.2.v.a.395.16		88	48.11	even	4
576.2.y.a.47.4		88	36.11	even	6
576.2.y.a.239.8		88	4.3	odd	2
576.2.y.a.335.8		88	144.101	odd	12
576.2.y.a.527.4		88	16.5	even	4
1728.2.z.a.143.8		88	144.133	even	12
1728.2.z.a.719.8		88	48.5	odd	4
1728.2.z.a.1007.8		88	36.7	odd	6
1728.2.z.a.1583.8		88	12.11	even	2