Embedded newform 144.2.x.e.133.4

• Label: 144.2.x.e.133.4
• 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			133.4
Character	\(\chi\)	\(=\)	144.133
Dual form			144.2.x.e.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.709450 + 1.22339i) q^{2} +(-0.705067 + 1.58205i) q^{3} +(-0.993361 - 1.73587i) q^{4} +(-0.679606 + 2.53632i) q^{5} +(-1.43525 - 1.98496i) q^{6} +(-0.614293 - 0.354662i) q^{7} +(2.82838 + 0.0162452i) q^{8} +(-2.00576 - 2.23090i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 4 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 28 q^{6} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.709450	+	1.22339i	−0.501657	+	0.865067i
							\(3\)	−0.705067	+	1.58205i	−0.407071	+	0.913397i
\(5\)	−0.679606	+	2.53632i	−0.303929	+	1.13428i								0.629934	+	0.776648i	\(0.283082\pi\)
							−0.933863	+	0.357630i	\(0.883585\pi\)
\(7\)	−0.614293	−	0.354662i	−0.232181	−	0.134050i	0.379397	−	0.925234i	\(-0.376132\pi\)
							−0.611578	+	0.791184i	\(0.709465\pi\)
\(11\)	−3.63167	+	0.973103i	−1.09499	+	0.293402i	−0.760723	−	0.649077i	\(-0.775155\pi\)
							−0.334267	+	0.942478i	\(0.608489\pi\)
\(13\)	−0.519100	−	0.139092i	−0.143972	−	0.0385773i	0.186113	−	0.982528i	\(-0.440411\pi\)
							−0.330085	+	0.943951i	\(0.607078\pi\)
\(17\)	6.08347			1.47546			0.737729	−	0.675096i	\(-0.235898\pi\)
							0.737729	+	0.675096i	\(0.235898\pi\)
\(19\)	−1.86732	+	1.86732i	−0.428392	+	0.428392i	−0.888080	−	0.459688i	\(-0.847961\pi\)
							0.459688	+	0.888080i	\(0.347961\pi\)
\(23\)	−4.94479	+	2.85488i	−1.03106	+	0.595283i	−0.917288	−	0.398224i	\(-0.869627\pi\)
							−0.113772	+	0.993507i	\(0.536293\pi\)
\(29\)	2.59481	+	9.68396i	0.481844	+	1.79827i	0.593870	+	0.804561i	\(0.297599\pi\)
							−0.112026	+	0.993705i	\(0.535734\pi\)
\(31\)	2.14190	+	3.70987i	0.384696	+	0.666313i	0.991727	−	0.128365i	\(-0.0409730\pi\)
							−0.607031	+	0.794678i	\(0.707640\pi\)
\(37\)	3.75493	+	3.75493i	0.617307	+	0.617307i	0.944840	−	0.327533i	\(-0.106217\pi\)
							−0.327533	+	0.944840i	\(0.606217\pi\)
\(41\)	1.57053	−	0.906743i	0.245275	−	0.141610i	−0.372324	−	0.928103i	\(-0.621439\pi\)
							0.617599	+	0.786493i	\(0.288106\pi\)
\(43\)	8.62809	−	2.31189i	1.31577	−	0.352560i	0.468379	−	0.883528i	\(-0.344838\pi\)
							0.847392	+	0.530968i	\(0.178172\pi\)
\(47\)	−2.95451	+	5.11737i	−0.430960	+	0.746445i	−0.996956	−	0.0779629i	\(-0.975158\pi\)
							0.565996	+	0.824408i	\(0.308492\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.x.e.133.4	yes	72
3.2	odd	2		432.2.y.e.181.15		72
4.3	odd	2		576.2.bb.e.241.12		72
9.4	even	3	inner	144.2.x.e.85.8	yes	72
9.5	odd	6		432.2.y.e.37.11		72
12.11	even	2		1728.2.bc.e.1585.13		72
16.3	odd	4		576.2.bb.e.529.15		72
16.13	even	4	inner	144.2.x.e.61.8	yes	72
36.23	even	6		1728.2.bc.e.1009.6		72
36.31	odd	6		576.2.bb.e.49.15		72
48.29	odd	4		432.2.y.e.397.11		72
48.35	even	4		1728.2.bc.e.721.6		72
144.13	even	12	inner	144.2.x.e.13.4	✓	72
144.67	odd	12		576.2.bb.e.337.12		72
144.77	odd	12		432.2.y.e.253.15		72
144.131	even	12		1728.2.bc.e.145.13		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.e.13.4	✓	72	144.13	even	12	inner
144.2.x.e.61.8	yes	72	16.13	even	4	inner
144.2.x.e.85.8	yes	72	9.4	even	3	inner
144.2.x.e.133.4	yes	72	1.1	even	1	trivial
432.2.y.e.37.11		72	9.5	odd	6
432.2.y.e.181.15		72	3.2	odd	2
432.2.y.e.253.15		72	144.77	odd	12
432.2.y.e.397.11		72	48.29	odd	4
576.2.bb.e.49.15		72	36.31	odd	6
576.2.bb.e.241.12		72	4.3	odd	2
576.2.bb.e.337.12		72	144.67	odd	12
576.2.bb.e.529.15		72	16.3	odd	4
1728.2.bc.e.145.13		72	144.131	even	12
1728.2.bc.e.721.6		72	48.35	even	4
1728.2.bc.e.1009.6		72	36.23	even	6
1728.2.bc.e.1585.13		72	12.11	even	2