Embedded newform 268.2.h.a.231.20

• Label: 268.2.h.a.231.20
• Level: $268$
• Weight: $2$
• Character: 268.231
• Analytic conductor: $2.140$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 268 = 2^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	268.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.13999077417\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			231.20
Character	\(\chi\)	\(=\)	268.231
Dual form			268.2.h.a.239.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.425784 + 1.34859i) q^{2} -1.04429 q^{3} +(-1.63742 + 1.14842i) q^{4} +3.14744i q^{5} +(-0.444640 - 1.40832i) q^{6} +(0.0992949 - 0.171984i) q^{7} +(-2.24594 - 1.71923i) q^{8} -1.90946 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 3 q^{2} + q^{4} - 3 q^{6} + 44 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/268\mathbb{Z}\right)^\times\).

\(n\)	\(69\)	\(135\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.425784	+	1.34859i	0.301074	+	0.953601i
							\(3\)	−1.04429			−0.602919			−0.301460	−	0.953479i	\(-0.597474\pi\)
−0.301460	+	0.953479i	\(0.597474\pi\)
\(5\)			3.14744i			1.40758i	0.710410	+	0.703788i	\(0.248509\pi\)
							−0.710410	+	0.703788i	\(0.751491\pi\)
\(7\)	0.0992949	−	0.171984i	0.0375300	−	0.0650038i	−0.846650	−	0.532150i	\(-0.821384\pi\)
							0.884180	+	0.467146i	\(0.154718\pi\)
\(11\)	1.91375	−	3.31472i	0.577018	−	0.999425i	−0.418801	−	0.908078i	\(-0.637550\pi\)
							0.995819	−	0.0913467i	\(-0.0291171\pi\)
\(13\)	−3.87764	+	2.23876i	−1.07546	+	0.620920i	−0.929669	−	0.368395i	\(-0.879907\pi\)
							−0.145795	+	0.989315i	\(0.546574\pi\)
\(17\)	2.56578	+	4.44405i	0.622292	+	1.07784i	0.989058	+	0.147528i	\(0.0471317\pi\)
							−0.366766	+	0.930313i	\(0.619535\pi\)
\(19\)	−0.782712	+	0.451899i	−0.179566	+	0.103673i	−0.587089	−	0.809522i	\(-0.699726\pi\)
							0.407523	+	0.913195i	\(0.366393\pi\)
\(23\)	−7.68851	+	4.43896i	−1.60317	+	0.925588i	−0.612316	+	0.790613i	\(0.709762\pi\)
							−0.990849	+	0.134975i	\(0.956905\pi\)
\(29\)	3.83368	−	6.64014i	0.711897	−	1.23304i	−0.252247	−	0.967663i	\(-0.581169\pi\)
							0.964144	−	0.265379i	\(-0.0854972\pi\)
\(31\)	−3.73149	+	6.46314i	−0.670196	+	1.16081i	0.307652	+	0.951499i	\(0.400457\pi\)
							−0.977848	+	0.209315i	\(0.932877\pi\)
\(37\)	3.26583	+	5.65658i	0.536899	+	0.929937i	0.999069	+	0.0431451i	\(0.0137378\pi\)
							−0.462170	+	0.886792i	\(0.652929\pi\)
\(41\)	8.03670	+	4.63999i	1.25512	+	0.724645i	0.972122	−	0.234475i	\(-0.0753371\pi\)
							0.283000	+	0.959120i	\(0.408670\pi\)
\(43\)	11.1467			1.69985			0.849927	−	0.526901i	\(-0.176646\pi\)
							0.849927	+	0.526901i	\(0.176646\pi\)
\(47\)	−3.47530	−	2.00646i	−0.506924	−	0.292673i	0.224644	−	0.974441i	\(-0.427878\pi\)
							−0.731568	+	0.681768i	\(0.761211\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	268.2.h.a.231.20	yes	64
4.3	odd	2	inner	268.2.h.a.231.9	✓	64
67.38	odd	6	inner	268.2.h.a.239.9	yes	64
268.239	even	6	inner	268.2.h.a.239.20	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
268.2.h.a.231.9	✓	64	4.3	odd	2	inner
268.2.h.a.231.20	yes	64	1.1	even	1	trivial
268.2.h.a.239.9	yes	64	67.38	odd	6	inner
268.2.h.a.239.20	yes	64	268.239	even	6	inner