Embedded newform 144.2.x.e.133.17

• Label: 144.2.x.e.133.17
• 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			133.17
Character	\(\chi\)	\(=\)	144.133
Dual form			144.2.x.e.13.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30272 - 0.550382i) q^{2} +(-1.21980 + 1.22967i) q^{3} +(1.39416 - 1.43399i) q^{4} +(-0.0468197 + 0.174734i) q^{5} +(-0.912266 + 2.27327i) q^{6} +(4.04791 + 2.33706i) q^{7} +(1.02696 - 2.63540i) q^{8} +(-0.0241875 - 2.99990i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 4 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 28 q^{6} - 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.30272	−	0.550382i	0.921162	−	0.389179i
							\(3\)	−1.21980	+	1.22967i	−0.704250	+	0.709952i
\(5\)	−0.0468197	+	0.174734i	−0.0209384	+	0.0781432i								−0.975604	−	0.219536i	\(-0.929546\pi\)
							0.954666	+	0.297679i	\(0.0962125\pi\)
\(7\)	4.04791	+	2.33706i	1.52997	+	0.883327i	0.999362	+	0.0357075i	\(0.0113685\pi\)
							0.530605	+	0.847619i	\(0.321965\pi\)
\(11\)	−0.598734	+	0.160430i	−0.180525	+	0.0483716i	−0.347949	−	0.937513i	\(-0.613122\pi\)
							0.167424	+	0.985885i	\(0.446455\pi\)
\(13\)	−4.41237	−	1.18229i	−1.22377	−	0.327909i	−0.411621	−	0.911355i	\(-0.635037\pi\)
							−0.812152	+	0.583446i	\(0.801704\pi\)
\(17\)	−4.34691			−1.05428			−0.527141	−	0.849778i	\(-0.676736\pi\)
							−0.527141	+	0.849778i	\(0.676736\pi\)
\(19\)	−1.23918	+	1.23918i	−0.284287	+	0.284287i	−0.834816	−	0.550529i	\(-0.814426\pi\)
							0.550529	+	0.834816i	\(0.314426\pi\)
\(23\)	−3.86311	+	2.23037i	−0.805515	+	0.465064i	−0.845396	−	0.534140i	\(-0.820635\pi\)
							0.0398812	+	0.999204i	\(0.487302\pi\)
\(29\)	−2.31752	−	8.64910i	−0.430353	−	1.60610i	−0.751949	−	0.659221i	\(-0.770886\pi\)
							0.321597	−	0.946877i	\(-0.395780\pi\)
\(31\)	−2.25376	−	3.90364i	−0.404788	−	0.701114i	0.589509	−	0.807762i	\(-0.299321\pi\)
							−0.994297	+	0.106648i	\(0.965988\pi\)
\(37\)	2.79692	+	2.79692i	0.459811	+	0.459811i	0.898593	−	0.438783i	\(-0.144590\pi\)
							−0.438783	+	0.898593i	\(0.644590\pi\)
\(41\)	3.67211	−	2.12009i	0.573487	−	0.331103i	−0.185054	−	0.982728i	\(-0.559246\pi\)
							0.758541	+	0.651626i	\(0.225913\pi\)
\(43\)	−0.0131254	+	0.00351694i	−0.00200161	+	0.000536329i	−0.259820	−	0.965657i	\(-0.583663\pi\)
							0.257818	+	0.966193i	\(0.416996\pi\)
\(47\)	−1.17465	+	2.03456i	−0.171341	+	0.296771i	−0.938889	−	0.344221i	\(-0.888143\pi\)
							0.767548	+	0.640991i	\(0.221477\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.x.e.133.17	yes	72
3.2	odd	2		432.2.y.e.181.2		72
4.3	odd	2		576.2.bb.e.241.15		72
9.4	even	3	inner	144.2.x.e.85.9	yes	72
9.5	odd	6		432.2.y.e.37.10		72
12.11	even	2		1728.2.bc.e.1585.11		72
16.3	odd	4		576.2.bb.e.529.14		72
16.13	even	4	inner	144.2.x.e.61.9	yes	72
36.23	even	6		1728.2.bc.e.1009.8		72
36.31	odd	6		576.2.bb.e.49.14		72
48.29	odd	4		432.2.y.e.397.10		72
48.35	even	4		1728.2.bc.e.721.8		72
144.13	even	12	inner	144.2.x.e.13.17	✓	72
144.67	odd	12		576.2.bb.e.337.15		72
144.77	odd	12		432.2.y.e.253.2		72
144.131	even	12		1728.2.bc.e.145.11		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.x.e.13.17	✓	72	144.13	even	12	inner
144.2.x.e.61.9	yes	72	16.13	even	4	inner
144.2.x.e.85.9	yes	72	9.4	even	3	inner
144.2.x.e.133.17	yes	72	1.1	even	1	trivial
432.2.y.e.37.10		72	9.5	odd	6
432.2.y.e.181.2		72	3.2	odd	2
432.2.y.e.253.2		72	144.77	odd	12
432.2.y.e.397.10		72	48.29	odd	4
576.2.bb.e.49.14		72	36.31	odd	6
576.2.bb.e.241.15		72	4.3	odd	2
576.2.bb.e.337.15		72	144.67	odd	12
576.2.bb.e.529.14		72	16.3	odd	4
1728.2.bc.e.145.11		72	144.131	even	12
1728.2.bc.e.721.8		72	48.35	even	4
1728.2.bc.e.1009.8		72	36.23	even	6
1728.2.bc.e.1585.11		72	12.11	even	2