Embedded newform 144.2.x.e.133.13

• Label: 144.2.x.e.133.13
• 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			133.13
Character	\(\chi\)	\(=\)	144.133
Dual form			144.2.x.e.13.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.04877 - 0.948722i) q^{2} +(-1.13227 - 1.31071i) q^{3} +(0.199854 - 1.98999i) q^{4} +(0.00302457 - 0.0112878i) q^{5} +(-2.43100 - 0.300433i) q^{6} +(-1.05753 - 0.610563i) q^{7} +(-1.67835 - 2.27665i) q^{8} +(-0.435936 + 2.96816i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 4 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 28 q^{6} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.04877	−	0.948722i	0.741595	−	0.670848i
							\(3\)	−1.13227	−	1.31071i	−0.653716	−	0.756740i
\(5\)	0.00302457	−	0.0112878i	0.00135263	−	0.00504807i								−0.965246	−	0.261342i	\(-0.915835\pi\)
							0.966599	+	0.256294i	\(0.0825016\pi\)
\(7\)	−1.05753	−	0.610563i	−0.399708	−	0.230771i	0.286650	−	0.958035i	\(-0.407458\pi\)
							−0.686358	+	0.727264i	\(0.740792\pi\)
\(11\)	1.83070	−	0.490535i	0.551977	−	0.147902i	0.0279594	−	0.999609i	\(-0.491099\pi\)
							0.524018	+	0.851707i	\(0.324432\pi\)
\(13\)	5.06759	+	1.35786i	1.40550	+	0.376602i	0.880315	−	0.474389i	\(-0.157331\pi\)
							0.525181	+	0.850991i	\(0.323998\pi\)
\(17\)	−1.54238			−0.374082			−0.187041	−	0.982352i	\(-0.559890\pi\)
							−0.187041	+	0.982352i	\(0.559890\pi\)
\(19\)	4.06823	−	4.06823i	0.933317	−	0.933317i	−0.0645950	−	0.997912i	\(-0.520576\pi\)
							0.997912	+	0.0645950i	\(0.0205756\pi\)
\(23\)	−5.20660	+	3.00603i	−1.08565	+	0.626801i	−0.932415	−	0.361388i	\(-0.882303\pi\)
							−0.153236	+	0.988190i	\(0.548970\pi\)
\(29\)	0.798708	+	2.98082i	0.148316	+	0.553524i	0.999585	+	0.0287946i	\(0.00916687\pi\)
							−0.851269	+	0.524730i	\(0.824166\pi\)
\(31\)	2.92831	+	5.07198i	0.525939	+	0.910954i	0.999543	+	0.0302159i	\(0.00961949\pi\)
							−0.473604	+	0.880738i	\(0.657047\pi\)
\(37\)	0.923082	+	0.923082i	0.151754	+	0.151754i	0.778901	−	0.627147i	\(-0.215777\pi\)
							−0.627147	+	0.778901i	\(0.715777\pi\)
\(41\)	−3.20829	+	1.85231i	−0.501051	+	0.289282i	−0.729147	−	0.684357i	\(-0.760083\pi\)
							0.228096	+	0.973639i	\(0.426750\pi\)
\(43\)	−4.84015	+	1.29691i	−0.738115	+	0.197777i	−0.608240	−	0.793753i	\(-0.708124\pi\)
							−0.129875	+	0.991530i	\(0.541458\pi\)
\(47\)	−1.31218	+	2.27277i	−0.191402	+	0.331517i	−0.945715	−	0.324997i	\(-0.894637\pi\)
							0.754313	+	0.656515i	\(0.227970\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.x.e.133.13	yes	72
3.2	odd	2		432.2.y.e.181.6		72
4.3	odd	2		576.2.bb.e.241.14		72
9.4	even	3	inner	144.2.x.e.85.12	yes	72
9.5	odd	6		432.2.y.e.37.7		72
12.11	even	2		1728.2.bc.e.1585.10		72
16.3	odd	4		576.2.bb.e.529.4		72
16.13	even	4	inner	144.2.x.e.61.12	yes	72
36.23	even	6		1728.2.bc.e.1009.9		72
36.31	odd	6		576.2.bb.e.49.4		72
48.29	odd	4		432.2.y.e.397.7		72
48.35	even	4		1728.2.bc.e.721.9		72
144.13	even	12	inner	144.2.x.e.13.13	✓	72
144.67	odd	12		576.2.bb.e.337.14		72
144.77	odd	12		432.2.y.e.253.6		72
144.131	even	12		1728.2.bc.e.145.10		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.e.13.13	✓	72	144.13	even	12	inner
144.2.x.e.61.12	yes	72	16.13	even	4	inner
144.2.x.e.85.12	yes	72	9.4	even	3	inner
144.2.x.e.133.13	yes	72	1.1	even	1	trivial
432.2.y.e.37.7		72	9.5	odd	6
432.2.y.e.181.6		72	3.2	odd	2
432.2.y.e.253.6		72	144.77	odd	12
432.2.y.e.397.7		72	48.29	odd	4
576.2.bb.e.49.4		72	36.31	odd	6
576.2.bb.e.241.14		72	4.3	odd	2
576.2.bb.e.337.14		72	144.67	odd	12
576.2.bb.e.529.4		72	16.3	odd	4
1728.2.bc.e.145.10		72	144.131	even	12
1728.2.bc.e.721.9		72	48.35	even	4
1728.2.bc.e.1009.9		72	36.23	even	6
1728.2.bc.e.1585.10		72	12.11	even	2