Embedded newform 144.2.x.e.133.9

• Label: 144.2.x.e.133.9
• 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			133.9
Character	\(\chi\)	\(=\)	144.133
Dual form			144.2.x.e.13.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.282685 - 1.38567i) q^{2} +(0.944122 - 1.45211i) q^{3} +(-1.84018 - 0.783419i) q^{4} +(0.131764 - 0.491749i) q^{5} +(-1.74526 - 1.71874i) q^{6} +(2.40518 + 1.38863i) q^{7} +(-1.60575 + 2.32842i) q^{8} +(-1.21727 - 2.74194i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 4 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 28 q^{6} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.282685	−	1.38567i	0.199889	−	0.979819i
							\(3\)	0.944122	−	1.45211i	0.545089	−	0.838378i
\(5\)	0.131764	−	0.491749i	0.0589266	−	0.219917i								−0.930183	−	0.367095i	\(-0.880352\pi\)
							0.989110	+	0.147178i	\(0.0470190\pi\)
\(7\)	2.40518	+	1.38863i	0.909072	+	0.524853i	0.880132	−	0.474728i	\(-0.157454\pi\)
							0.0289393	+	0.999581i	\(0.490787\pi\)
\(11\)	−3.96934	+	1.06358i	−1.19680	+	0.320682i	−0.801570	−	0.597900i	\(-0.796002\pi\)
							−0.395230	+	0.918582i	\(0.629335\pi\)
\(13\)	2.22553	+	0.596329i	0.617251	+	0.165392i	0.553878	−	0.832598i	\(-0.313147\pi\)
							0.0633733	+	0.997990i	\(0.479814\pi\)
\(17\)	2.87908			0.698279			0.349139	−	0.937071i	\(-0.386474\pi\)
							0.349139	+	0.937071i	\(0.386474\pi\)
\(19\)	−3.48018	+	3.48018i	−0.798409	+	0.798409i	−0.982845	−	0.184436i	\(-0.940954\pi\)
							0.184436	+	0.982845i	\(0.440954\pi\)
\(23\)	3.85945	−	2.22826i	0.804752	−	0.464624i	−0.0403783	−	0.999184i	\(-0.512856\pi\)
							0.845130	+	0.534561i	\(0.179523\pi\)
\(29\)	−1.57657	−	5.88383i	−0.292761	−	1.09260i	−0.942979	−	0.332852i	\(-0.891989\pi\)
							0.650218	−	0.759748i	\(-0.274677\pi\)
\(31\)	−1.28296	−	2.22216i	−0.230427	−	0.399112i	0.727507	−	0.686101i	\(-0.240679\pi\)
							−0.957934	+	0.286989i	\(0.907346\pi\)
\(37\)	7.64112	+	7.64112i	1.25619	+	1.25619i	0.952896	+	0.303297i	\(0.0980873\pi\)
							0.303297	+	0.952896i	\(0.401913\pi\)
\(41\)	−4.84731	+	2.79860i	−0.757023	+	0.437067i	−0.828226	−	0.560394i	\(-0.810650\pi\)
							0.0712028	+	0.997462i	\(0.477316\pi\)
\(43\)	−3.40160	+	0.911456i	−0.518739	+	0.138996i	−0.508683	−	0.860954i	\(-0.669868\pi\)
							−0.0100559	+	0.999949i	\(0.503201\pi\)
\(47\)	−4.94233	+	8.56037i	−0.720913	+	1.24866i	0.239721	+	0.970842i	\(0.422944\pi\)
							−0.960634	+	0.277817i	\(0.910389\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.x.e.133.9	yes	72
3.2	odd	2		432.2.y.e.181.10		72
4.3	odd	2		576.2.bb.e.241.7		72
9.4	even	3	inner	144.2.x.e.85.14	yes	72
9.5	odd	6		432.2.y.e.37.5		72
12.11	even	2		1728.2.bc.e.1585.8		72
16.3	odd	4		576.2.bb.e.529.3		72
16.13	even	4	inner	144.2.x.e.61.14	yes	72
36.23	even	6		1728.2.bc.e.1009.11		72
36.31	odd	6		576.2.bb.e.49.3		72
48.29	odd	4		432.2.y.e.397.5		72
48.35	even	4		1728.2.bc.e.721.11		72
144.13	even	12	inner	144.2.x.e.13.9	✓	72
144.67	odd	12		576.2.bb.e.337.7		72
144.77	odd	12		432.2.y.e.253.10		72
144.131	even	12		1728.2.bc.e.145.8		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.e.13.9	✓	72	144.13	even	12	inner
144.2.x.e.61.14	yes	72	16.13	even	4	inner
144.2.x.e.85.14	yes	72	9.4	even	3	inner
144.2.x.e.133.9	yes	72	1.1	even	1	trivial
432.2.y.e.37.5		72	9.5	odd	6
432.2.y.e.181.10		72	3.2	odd	2
432.2.y.e.253.10		72	144.77	odd	12
432.2.y.e.397.5		72	48.29	odd	4
576.2.bb.e.49.3		72	36.31	odd	6
576.2.bb.e.241.7		72	4.3	odd	2
576.2.bb.e.337.7		72	144.67	odd	12
576.2.bb.e.529.3		72	16.3	odd	4
1728.2.bc.e.145.8		72	144.131	even	12
1728.2.bc.e.721.11		72	48.35	even	4
1728.2.bc.e.1009.11		72	36.23	even	6
1728.2.bc.e.1585.8		72	12.11	even	2