Embedded newform 272.2.j.a.13.25

• Label: 272.2.j.a.13.25
• Level: $272$
• Weight: $2$
• Character: 272.13
• Analytic conductor: $2.172$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	272.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17193093498\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.25
Character	\(\chi\)	\(=\)	272.13
Dual form			272.2.j.a.21.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.930565 + 1.06492i) q^{2} +1.70026i q^{3} +(-0.268097 + 1.98195i) q^{4} +2.90401 q^{5} +(-1.81063 + 1.58220i) q^{6} +(-0.453147 - 0.453147i) q^{7} +(-2.36009 + 1.55883i) q^{8} +0.109126 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 4 q^{4} - 4 q^{5} + 6 q^{6} - 60 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/272\mathbb{Z}\right)^\times\).

\(n\)	\(69\)	\(239\)	\(241\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.930565	+	1.06492i	0.658009	+	0.753010i
							\(3\)			1.70026i			0.981644i	0.871260	+	0.490822i	\(0.163303\pi\)
−0.871260	+	0.490822i	\(0.836697\pi\)
\(5\)	2.90401			1.29871			0.649356	−	0.760484i	\(-0.275038\pi\)
							0.649356	+	0.760484i	\(0.275038\pi\)
\(7\)	−0.453147	−	0.453147i	−0.171274	−	0.171274i	0.616265	−	0.787539i	\(-0.288645\pi\)
							−0.787539	+	0.616265i	\(0.788645\pi\)
\(11\)		−	2.30130i		−	0.693867i	−0.937890	−	0.346934i	\(-0.887223\pi\)
							0.937890	−	0.346934i	\(-0.112777\pi\)
\(13\)	−2.83013	−	2.83013i	−0.784937	−	0.784937i	0.195722	−	0.980659i	\(-0.437295\pi\)
							−0.980659	+	0.195722i	\(0.937295\pi\)
\(17\)	2.55560	−	3.23557i	0.619823	−	0.784741i
							\(19\)	−2.28566	+	2.28566i	−0.524367	+	0.524367i	−0.918887	−	0.394520i	\(-0.870911\pi\)
0.394520	+	0.918887i	\(0.370911\pi\)
\(23\)	−2.63418	−	2.63418i	−0.549264	−	0.549264i	0.376964	−	0.926228i	\(-0.376968\pi\)
							−0.926228	+	0.376964i	\(0.876968\pi\)
\(29\)			3.29992i			0.612779i	0.951906	+	0.306390i	\(0.0991210\pi\)
							−0.951906	+	0.306390i	\(0.900879\pi\)
\(31\)	−3.92140	−	3.92140i	−0.704305	−	0.704305i	0.261027	−	0.965332i	\(-0.415939\pi\)
							−0.965332	+	0.261027i	\(0.915939\pi\)
\(37\)	1.27321			0.209314			0.104657	−	0.994508i	\(-0.466625\pi\)
							0.104657	+	0.994508i	\(0.466625\pi\)
\(41\)	4.47191	+	4.47191i	0.698395	+	0.698395i	0.964064	−	0.265669i	\(-0.0855929\pi\)
							−0.265669	+	0.964064i	\(0.585593\pi\)
\(43\)	4.30802	+	4.30802i	0.656967	+	0.656967i	0.954661	−	0.297694i	\(-0.0962176\pi\)
							−0.297694	+	0.954661i	\(0.596218\pi\)
\(47\)	5.63952			0.822609			0.411305	−	0.911498i	\(-0.365073\pi\)
							0.411305	+	0.911498i	\(0.365073\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.2.j.a.13.25	✓	68
4.3	odd	2		1088.2.j.a.81.10		68
16.5	even	4		272.2.s.a.149.10	yes	68
16.11	odd	4		1088.2.s.a.625.10		68
17.4	even	4		272.2.s.a.157.10	yes	68
68.55	odd	4		1088.2.s.a.1041.10		68
272.21	even	4	inner	272.2.j.a.21.25	yes	68
272.123	odd	4		1088.2.j.a.497.25		68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
272.2.j.a.13.25	✓	68	1.1	even	1	trivial
272.2.j.a.21.25	yes	68	272.21	even	4	inner
272.2.s.a.149.10	yes	68	16.5	even	4
272.2.s.a.157.10	yes	68	17.4	even	4
1088.2.j.a.81.10		68	4.3	odd	2
1088.2.j.a.497.25		68	272.123	odd	4
1088.2.s.a.625.10		68	16.11	odd	4
1088.2.s.a.1041.10		68	68.55	odd	4