Embedded newform 296.2.q.a.85.30

• Label: 296.2.q.a.85.30
• Level: $296$
• Weight: $2$
• Character: 296.85
• 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			85.30
Character	\(\chi\)	\(=\)	296.85
Dual form			296.2.q.a.101.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.06765 + 0.927430i) q^{2} +(-0.280448 - 0.161917i) q^{3} +(0.279748 + 1.98034i) q^{4} +(1.57735 - 2.73205i) q^{5} +(-0.149254 - 0.432966i) q^{6} +(2.58572 - 4.47860i) q^{7} +(-1.53795 + 2.37375i) q^{8} +(-1.44757 - 2.50726i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 6 q^{2} - 2 q^{4} + 2 q^{7} + 30 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(113\)	\(149\)	\(223\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.06765	+	0.927430i	0.754942	+	0.655792i
							\(3\)	−0.280448	−	0.161917i	−0.161917	−	0.0934827i	0.416852	−	0.908974i	\(-0.363133\pi\)
−0.578769	+	0.815492i	\(0.696467\pi\)
\(5\)	1.57735	−	2.73205i	0.705413	−	1.22181i	−0.261129	−	0.965304i	\(-0.584095\pi\)
							0.966542	−	0.256507i	\(-0.0825717\pi\)
\(7\)	2.58572	−	4.47860i	0.977311	−	1.69275i	0.305221	−	0.952281i	\(-0.401270\pi\)
							0.672089	−	0.740470i	\(-0.265397\pi\)
\(11\)			2.54946i			0.768691i	0.923189	+	0.384346i	\(0.125573\pi\)
							−0.923189	+	0.384346i	\(0.874427\pi\)
\(13\)	−1.47545	+	2.55556i	−0.409217	+	0.708785i	−0.994802	−	0.101826i	\(-0.967532\pi\)
							0.585585	+	0.810611i	\(0.300865\pi\)
\(17\)	1.46355	−	0.844981i	0.354963	−	0.204938i	−0.311906	−	0.950113i	\(-0.600967\pi\)
							0.666869	+	0.745175i	\(0.267634\pi\)
\(19\)	−3.12153	+	5.40664i	−0.716127	+	1.24037i	0.246396	+	0.969169i	\(0.420754\pi\)
							−0.962523	+	0.271200i	\(0.912580\pi\)
\(23\)			4.40633i			0.918782i	0.888234	+	0.459391i	\(0.151932\pi\)
							−0.888234	+	0.459391i	\(0.848068\pi\)
\(29\)	0.503739			0.0935420			0.0467710	−	0.998906i	\(-0.485107\pi\)
							0.0467710	+	0.998906i	\(0.485107\pi\)
\(31\)			9.67551i			1.73777i	0.495012	+	0.868886i	\(0.335164\pi\)
							−0.495012	+	0.868886i	\(0.664836\pi\)
\(37\)	3.84713	−	4.71164i	0.632464	−	0.774589i
							\(41\)	1.45568	−	2.52132i	0.227340	−	0.393764i	−0.729679	−	0.683790i	\(-0.760331\pi\)
0.957019	+	0.290026i	\(0.0936639\pi\)
\(43\)	4.61165			0.703270			0.351635	−	0.936137i	\(-0.385626\pi\)
							0.351635	+	0.936137i	\(0.385626\pi\)
\(47\)	1.20891			0.176337			0.0881686	−	0.996106i	\(-0.471899\pi\)
							0.0881686	+	0.996106i	\(0.471899\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	296.2.q.a.85.30	yes	72
4.3	odd	2		1184.2.y.a.529.20		72
8.3	odd	2		1184.2.y.a.529.17		72
8.5	even	2	inner	296.2.q.a.85.18	✓	72
37.27	even	6	inner	296.2.q.a.101.18	yes	72
148.27	odd	6		1184.2.y.a.1137.17		72
296.27	odd	6		1184.2.y.a.1137.20		72
296.101	even	6	inner	296.2.q.a.101.30	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.q.a.85.18	✓	72	8.5	even	2	inner
296.2.q.a.85.30	yes	72	1.1	even	1	trivial
296.2.q.a.101.18	yes	72	37.27	even	6	inner
296.2.q.a.101.30	yes	72	296.101	even	6	inner
1184.2.y.a.529.17		72	8.3	odd	2
1184.2.y.a.529.20		72	4.3	odd	2
1184.2.y.a.1137.17		72	148.27	odd	6
1184.2.y.a.1137.20		72	296.27	odd	6