Embedded newform 144.2.u.a.131.11

• Label: 144.2.u.a.131.11
• 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			131.11
Character	\(\chi\)	\(=\)	144.131
Dual form			144.2.u.a.11.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.311077 + 1.37958i) q^{2} +(-1.36635 - 1.06447i) q^{3} +(-1.80646 - 0.858309i) q^{4} +(2.31044 - 0.619079i) q^{5} +(1.89356 - 1.55385i) q^{6} +(2.51270 + 4.35213i) q^{7} +(1.74605 - 2.22515i) q^{8} +(0.733803 + 2.90887i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 6 q^{2} - 4 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} - 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.311077	+	1.37958i	−0.219965	+	0.975508i
							\(3\)	−1.36635	−	1.06447i	−0.788860	−	0.614573i
\(5\)	2.31044	−	0.619079i	1.03326	−	0.276861i								0.297941	−	0.954584i	\(-0.403700\pi\)
							0.735317	+	0.677724i	\(0.237033\pi\)
\(7\)	2.51270	+	4.35213i	0.949712	+	1.64495i	0.746030	+	0.665912i	\(0.231957\pi\)
							0.203681	+	0.979037i	\(0.434709\pi\)
\(11\)	−1.03121	−	0.276313i	−0.310922	−	0.0833114i	0.0999837	−	0.994989i	\(-0.468121\pi\)
							−0.410906	+	0.911678i	\(0.634788\pi\)
\(13\)	4.00543	−	1.07325i	1.11091	−	0.297666i	0.343708	−	0.939077i	\(-0.388317\pi\)
							0.767198	+	0.641411i	\(0.221650\pi\)
\(17\)		−	2.22468i		−	0.539564i	−0.962921	−	0.269782i	\(-0.913048\pi\)
							0.962921	−	0.269782i	\(-0.0869517\pi\)
\(19\)	−0.697254	+	0.697254i	−0.159961	+	0.159961i	−0.782549	−	0.622588i	\(-0.786081\pi\)
							0.622588	+	0.782549i	\(0.286081\pi\)
\(23\)	−2.20833	−	1.27498i	−0.460469	−	0.265852i	0.251773	−	0.967786i	\(-0.418987\pi\)
							−0.712241	+	0.701935i	\(0.752320\pi\)
\(29\)	0.589549	+	0.157969i	0.109477	+	0.0293342i	0.313141	−	0.949707i	\(-0.398619\pi\)
							−0.203665	+	0.979041i	\(0.565285\pi\)
\(31\)	0.190501	+	0.109986i	0.0342150	+	0.0197541i	0.517010	−	0.855979i	\(-0.327045\pi\)
							−0.482795	+	0.875733i	\(0.660378\pi\)
\(37\)	−5.16341	+	5.16341i	−0.848859	+	0.848859i	−0.989991	−	0.141132i	\(-0.954926\pi\)
							0.141132	+	0.989991i	\(0.454926\pi\)
\(41\)	0.828296	−	1.43465i	0.129358	−	0.224055i	−0.794070	−	0.607826i	\(-0.792042\pi\)
							0.923428	+	0.383772i	\(0.125375\pi\)
\(43\)	1.33457	−	4.98067i	0.203519	−	0.759545i	−0.786376	−	0.617748i	\(-0.788045\pi\)
							0.989896	−	0.141797i	\(-0.0452881\pi\)
\(47\)	−5.76715	−	9.98900i	−0.841226	−	1.45705i	−0.888859	−	0.458181i	\(-0.848501\pi\)
							0.0476333	−	0.998865i	\(-0.484832\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.u.a.131.11	yes	88
3.2	odd	2		432.2.v.a.179.12		88
4.3	odd	2		576.2.y.a.239.17		88
9.2	odd	6	inner	144.2.u.a.83.3	yes	88
9.7	even	3		432.2.v.a.35.20		88
12.11	even	2		1728.2.z.a.1583.4		88
16.5	even	4		576.2.y.a.527.6		88
16.11	odd	4	inner	144.2.u.a.59.3	yes	88
36.7	odd	6		1728.2.z.a.1007.4		88
36.11	even	6		576.2.y.a.47.6		88
48.5	odd	4		1728.2.z.a.719.4		88
48.11	even	4		432.2.v.a.395.20		88
144.11	even	12	inner	144.2.u.a.11.11	✓	88
144.43	odd	12		432.2.v.a.251.12		88
144.101	odd	12		576.2.y.a.335.17		88
144.133	even	12		1728.2.z.a.143.4		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.11	✓	88	144.11	even	12	inner
144.2.u.a.59.3	yes	88	16.11	odd	4	inner
144.2.u.a.83.3	yes	88	9.2	odd	6	inner
144.2.u.a.131.11	yes	88	1.1	even	1	trivial
432.2.v.a.35.20		88	9.7	even	3
432.2.v.a.179.12		88	3.2	odd	2
432.2.v.a.251.12		88	144.43	odd	12
432.2.v.a.395.20		88	48.11	even	4
576.2.y.a.47.6		88	36.11	even	6
576.2.y.a.239.17		88	4.3	odd	2
576.2.y.a.335.17		88	144.101	odd	12
576.2.y.a.527.6		88	16.5	even	4
1728.2.z.a.143.4		88	144.133	even	12
1728.2.z.a.719.4		88	48.5	odd	4
1728.2.z.a.1007.4		88	36.7	odd	6
1728.2.z.a.1583.4		88	12.11	even	2