Embedded newform 296.2.q.a.101.14

• Label: 296.2.q.a.101.14
• Level: $296$
• Weight: $2$
• Character: 296.101
• Analytic conductor: $2.364$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 296 = 2^{3} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	296.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.36357189983\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			101.14
Character	\(\chi\)	\(=\)	296.101
Dual form			296.2.q.a.85.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.484093 - 1.32878i) q^{2} +(0.0574215 - 0.0331523i) q^{3} +(-1.53131 + 1.28651i) q^{4} +(0.0588225 + 0.101884i) q^{5} +(-0.0718494 - 0.0602516i) q^{6} +(1.40284 + 2.42979i) q^{7} +(2.45078 + 1.41198i) q^{8} +(-1.49780 + 2.59427i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 6 q^{2} - 2 q^{4} + 2 q^{7} + 30 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/296\mathbb{Z}\right)^\times\).

\(n\)	\(113\)	\(149\)	\(223\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.484093	−	1.32878i	−0.342306	−	0.939589i
							\(3\)	0.0574215	−	0.0331523i	0.0331523	−	0.0191405i	−0.483332	−	0.875437i	\(-0.660574\pi\)
0.516485	+	0.856297i	\(0.327240\pi\)
\(5\)	0.0588225	+	0.101884i	0.0263062	+	0.0455638i	0.878879	−	0.477045i	\(-0.158292\pi\)
							−0.852573	+	0.522609i	\(0.824959\pi\)
\(7\)	1.40284	+	2.42979i	0.530223	+	0.918373i	0.999378	+	0.0352572i	\(0.0112250\pi\)
							−0.469156	+	0.883116i	\(0.655442\pi\)
\(11\)			1.77206i			0.534297i	0.963655	+	0.267149i	\(0.0860815\pi\)
							−0.963655	+	0.267149i	\(0.913919\pi\)
\(13\)	0.666698	+	1.15475i	0.184909	+	0.320271i	0.943546	−	0.331242i	\(-0.107468\pi\)
							−0.758637	+	0.651513i	\(0.774134\pi\)
\(17\)	3.86149	+	2.22943i	0.936549	+	0.540717i	0.888877	−	0.458146i	\(-0.151486\pi\)
							0.0476722	+	0.998863i	\(0.484820\pi\)
\(19\)	−3.11573	−	5.39660i	−0.714797	−	1.23806i	−0.963038	−	0.269366i	\(-0.913186\pi\)
							0.248241	−	0.968698i	\(-0.420147\pi\)
\(23\)			6.03806i			1.25902i	0.776992	+	0.629511i	\(0.216745\pi\)
							−0.776992	+	0.629511i	\(0.783255\pi\)
\(29\)	8.43088			1.56558			0.782788	−	0.622289i	\(-0.213797\pi\)
							0.782788	+	0.622289i	\(0.213797\pi\)
\(31\)			7.13978i			1.28234i	0.767398	+	0.641171i	\(0.221551\pi\)
							−0.767398	+	0.641171i	\(0.778449\pi\)
\(37\)	−2.72190	−	5.43979i	−0.447477	−	0.894295i
							\(41\)	−2.42678	−	4.20331i	−0.379000	−	0.656447i	0.611917	−	0.790922i	\(-0.290399\pi\)
−0.990917	+	0.134475i	\(0.957065\pi\)
\(43\)	−11.6336			−1.77410			−0.887051	−	0.461671i	\(-0.847250\pi\)
							−0.887051	+	0.461671i	\(0.847250\pi\)
\(47\)	6.45491			0.941546			0.470773	−	0.882254i	\(-0.343975\pi\)
							0.470773	+	0.882254i	\(0.343975\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	296.2.q.a.101.14	yes	72
4.3	odd	2		1184.2.y.a.1137.18		72
8.3	odd	2		1184.2.y.a.1137.19		72
8.5	even	2	inner	296.2.q.a.101.1	yes	72
37.11	even	6	inner	296.2.q.a.85.1	✓	72
148.11	odd	6		1184.2.y.a.529.19		72
296.11	odd	6		1184.2.y.a.529.18		72
296.85	even	6	inner	296.2.q.a.85.14	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.q.a.85.1	✓	72	37.11	even	6	inner
296.2.q.a.85.14	yes	72	296.85	even	6	inner
296.2.q.a.101.1	yes	72	8.5	even	2	inner
296.2.q.a.101.14	yes	72	1.1	even	1	trivial
1184.2.y.a.529.18		72	296.11	odd	6
1184.2.y.a.529.19		72	148.11	odd	6
1184.2.y.a.1137.18		72	4.3	odd	2
1184.2.y.a.1137.19		72	8.3	odd	2