Embedded newform 144.2.u.a.131.20

• Label: 144.2.u.a.131.20
• 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			131.20
Character	\(\chi\)	\(=\)	144.131
Dual form			144.2.u.a.11.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.29365 - 0.571371i) q^{2} +(1.61259 - 0.632098i) q^{3} +(1.34707 - 1.47831i) q^{4} +(-3.81396 + 1.02195i) q^{5} +(1.72497 - 1.73910i) q^{6} +(1.46715 + 2.54117i) q^{7} +(0.897978 - 2.68210i) q^{8} +(2.20090 - 2.03863i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 6 q^{2} - 4 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} - 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.29365	−	0.571371i	0.914750	−	0.404020i
							\(3\)	1.61259	−	0.632098i	0.931030	−	0.364942i
\(5\)	−3.81396	+	1.02195i	−1.70566	+	0.457029i								−0.974353	−	0.225024i	\(-0.927754\pi\)
							−0.731302	+	0.682053i	\(0.761087\pi\)
\(7\)	1.46715	+	2.54117i	0.554529	+	0.960473i	0.997940	+	0.0641541i	\(0.0204349\pi\)
							−0.443411	+	0.896318i	\(0.646232\pi\)
\(11\)	−2.65006	−	0.710081i	−0.799022	−	0.214097i	−0.163868	−	0.986482i	\(-0.552397\pi\)
							−0.635155	+	0.772385i	\(0.719064\pi\)
\(13\)	−2.34729	+	0.628955i	−0.651021	+	0.174441i	−0.569191	−	0.822206i	\(-0.692743\pi\)
							−0.0818307	+	0.996646i	\(0.526077\pi\)
\(17\)			2.89808i			0.702889i	0.936209	+	0.351444i	\(0.114309\pi\)
							−0.936209	+	0.351444i	\(0.885691\pi\)
\(19\)	−1.99906	+	1.99906i	−0.458615	+	0.458615i	−0.898201	−	0.439586i	\(-0.855125\pi\)
							0.439586	+	0.898201i	\(0.355125\pi\)
\(23\)	−2.07141	−	1.19593i	−0.431918	−	0.249368i	0.268245	−	0.963351i	\(-0.413556\pi\)
							−0.700163	+	0.713983i	\(0.746890\pi\)
\(29\)	8.46218	+	2.26743i	1.57139	+	0.421052i	0.936246	−	0.351344i	\(-0.114275\pi\)
							0.635141	+	0.772396i	\(0.280942\pi\)
\(31\)	−0.439075	−	0.253500i	−0.0788602	−	0.0455300i	0.460051	−	0.887892i	\(-0.347831\pi\)
							−0.538912	+	0.842362i	\(0.681164\pi\)
\(37\)	1.36407	−	1.36407i	0.224251	−	0.224251i	−0.586035	−	0.810286i	\(-0.699312\pi\)
							0.810286	+	0.586035i	\(0.199312\pi\)
\(41\)	0.745739	−	1.29166i	0.116465	−	0.201723i	−0.801899	−	0.597459i	\(-0.796177\pi\)
							0.918364	+	0.395736i	\(0.129510\pi\)
\(43\)	−1.27136	+	4.74478i	−0.193881	+	0.723572i	0.798673	+	0.601765i	\(0.205536\pi\)
							−0.992554	+	0.121807i	\(0.961131\pi\)
\(47\)	−3.25802	−	5.64306i	−0.475231	−	0.823124i	0.524367	−	0.851493i	\(-0.324302\pi\)
							−0.999598	+	0.0283684i	\(0.990969\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.u.a.131.20	yes	88
3.2	odd	2		432.2.v.a.179.3		88
4.3	odd	2		576.2.y.a.239.3		88
9.2	odd	6	inner	144.2.u.a.83.18	yes	88
9.7	even	3		432.2.v.a.35.5		88
12.11	even	2		1728.2.z.a.1583.22		88
16.5	even	4		576.2.y.a.527.9		88
16.11	odd	4	inner	144.2.u.a.59.18	yes	88
36.7	odd	6		1728.2.z.a.1007.22		88
36.11	even	6		576.2.y.a.47.9		88
48.5	odd	4		1728.2.z.a.719.22		88
48.11	even	4		432.2.v.a.395.5		88
144.11	even	12	inner	144.2.u.a.11.20	✓	88
144.43	odd	12		432.2.v.a.251.3		88
144.101	odd	12		576.2.y.a.335.3		88
144.133	even	12		1728.2.z.a.143.22		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.20	✓	88	144.11	even	12	inner
144.2.u.a.59.18	yes	88	16.11	odd	4	inner
144.2.u.a.83.18	yes	88	9.2	odd	6	inner
144.2.u.a.131.20	yes	88	1.1	even	1	trivial
432.2.v.a.35.5		88	9.7	even	3
432.2.v.a.179.3		88	3.2	odd	2
432.2.v.a.251.3		88	144.43	odd	12
432.2.v.a.395.5		88	48.11	even	4
576.2.y.a.47.9		88	36.11	even	6
576.2.y.a.239.3		88	4.3	odd	2
576.2.y.a.335.3		88	144.101	odd	12
576.2.y.a.527.9		88	16.5	even	4
1728.2.z.a.143.22		88	144.133	even	12
1728.2.z.a.719.22		88	48.5	odd	4
1728.2.z.a.1007.22		88	36.7	odd	6
1728.2.z.a.1583.22		88	12.11	even	2