Embedded newform 144.2.x.e.133.14

• Label: 144.2.x.e.133.14
• 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			133.14
Character	\(\chi\)	\(=\)	144.133
Dual form			144.2.x.e.13.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.06791 - 0.927124i) q^{2} +(1.71768 + 0.222657i) q^{3} +(0.280882 - 1.98018i) q^{4} +(-0.798307 + 2.97932i) q^{5} +(2.04077 - 1.35472i) q^{6} +(-1.78208 - 1.02889i) q^{7} +(-1.53591 - 2.37507i) q^{8} +(2.90085 + 0.764906i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 4 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 28 q^{6} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.06791	−	0.927124i	0.755129	−	0.655576i
							\(3\)	1.71768	+	0.222657i	0.991703	+	0.128551i
\(5\)	−0.798307	+	2.97932i	−0.357014	+	1.33239i								0.520918	+	0.853607i	\(0.325590\pi\)
							−0.877932	+	0.478786i	\(0.841077\pi\)
\(7\)	−1.78208	−	1.02889i	−0.673564	−	0.388882i	0.123862	−	0.992299i	\(-0.460472\pi\)
							−0.797426	+	0.603417i	\(0.793805\pi\)
\(11\)	−0.446347	+	0.119598i	−0.134579	+	0.0360602i	−0.325480	−	0.945549i	\(-0.605526\pi\)
							0.190901	+	0.981609i	\(0.438859\pi\)
\(13\)	−5.67550	−	1.52075i	−1.57410	−	0.421779i	−0.637008	−	0.770857i	\(-0.719828\pi\)
							−0.937094	+	0.349078i	\(0.886495\pi\)
\(17\)	0.0443921			0.0107667			0.00538333	−	0.999986i	\(-0.498286\pi\)
							0.00538333	+	0.999986i	\(0.498286\pi\)
\(19\)	−1.10726	+	1.10726i	−0.254023	+	0.254023i	−0.822618	−	0.568595i	\(-0.807487\pi\)
							0.568595	+	0.822618i	\(0.307487\pi\)
\(23\)	7.89263	−	4.55681i	1.64573	−	0.950161i	0.666984	−	0.745072i	\(-0.267585\pi\)
							0.978743	−	0.205089i	\(-0.0657485\pi\)
\(29\)	1.86402	+	6.95662i	0.346140	+	1.29181i	0.891275	+	0.453464i	\(0.149812\pi\)
							−0.545134	+	0.838349i	\(0.683521\pi\)
\(31\)	0.542236	+	0.939180i	0.0973884	+	0.168682i	0.910603	−	0.413282i	\(-0.135618\pi\)
							−0.813214	+	0.581964i	\(0.802284\pi\)
\(37\)	−0.769054	−	0.769054i	−0.126432	−	0.126432i	0.641060	−	0.767491i	\(-0.278495\pi\)
							−0.767491	+	0.641060i	\(0.778495\pi\)
\(41\)	−5.77193	+	3.33242i	−0.901423	+	0.520437i	−0.877662	−	0.479281i	\(-0.840898\pi\)
							−0.0237617	+	0.999718i	\(0.507564\pi\)
\(43\)	11.0600	−	2.96351i	1.68663	−	0.451932i	0.717115	−	0.696955i	\(-0.245462\pi\)
							0.969517	+	0.245023i	\(0.0787956\pi\)
\(47\)	−1.22453	+	2.12095i	−0.178617	+	0.309373i	−0.941407	−	0.337273i	\(-0.890496\pi\)
							0.762790	+	0.646646i	\(0.223829\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.x.e.133.14	yes	72
3.2	odd	2		432.2.y.e.181.5		72
4.3	odd	2		576.2.bb.e.241.2		72
9.4	even	3	inner	144.2.x.e.85.11	yes	72
9.5	odd	6		432.2.y.e.37.8		72
12.11	even	2		1728.2.bc.e.1585.16		72
16.3	odd	4		576.2.bb.e.529.9		72
16.13	even	4	inner	144.2.x.e.61.11	yes	72
36.23	even	6		1728.2.bc.e.1009.3		72
36.31	odd	6		576.2.bb.e.49.9		72
48.29	odd	4		432.2.y.e.397.8		72
48.35	even	4		1728.2.bc.e.721.3		72
144.13	even	12	inner	144.2.x.e.13.14	✓	72
144.67	odd	12		576.2.bb.e.337.2		72
144.77	odd	12		432.2.y.e.253.5		72
144.131	even	12		1728.2.bc.e.145.16		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.e.13.14	✓	72	144.13	even	12	inner
144.2.x.e.61.11	yes	72	16.13	even	4	inner
144.2.x.e.85.11	yes	72	9.4	even	3	inner
144.2.x.e.133.14	yes	72	1.1	even	1	trivial
432.2.y.e.37.8		72	9.5	odd	6
432.2.y.e.181.5		72	3.2	odd	2
432.2.y.e.253.5		72	144.77	odd	12
432.2.y.e.397.8		72	48.29	odd	4
576.2.bb.e.49.9		72	36.31	odd	6
576.2.bb.e.241.2		72	4.3	odd	2
576.2.bb.e.337.2		72	144.67	odd	12
576.2.bb.e.529.9		72	16.3	odd	4
1728.2.bc.e.145.16		72	144.131	even	12
1728.2.bc.e.721.3		72	48.35	even	4
1728.2.bc.e.1009.3		72	36.23	even	6
1728.2.bc.e.1585.16		72	12.11	even	2