Embedded newform 896.2.bt.a.115.2

• Label: 896.2.bt.a.115.2
• Level: $896$
• Weight: $2$
• Character: 896.115
• Analytic conductor: $7.155$
• Analytic rank: $0$
• Dimension: $4032$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.bt (of order \(96\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.15459602111\)
Analytic rank:	\(0\)
Dimension:	\(4032\)
Relative dimension:	\(126\) over \(\Q(\zeta_{96})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{96}]$

Embedding invariants

Embedding label			115.2
Character	\(\chi\)	\(=\)	896.115
Dual form			896.2.bt.a.187.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41387 + 0.0313007i) q^{2} +(0.101527 + 0.0236727i) q^{3} +(1.99804 - 0.0885099i) q^{4} +(2.39383 + 3.34059i) q^{5} +(-0.144286 - 0.0302922i) q^{6} +(2.22505 + 1.43148i) q^{7} +(-2.82219 + 0.187681i) q^{8} +(-2.68087 - 1.32206i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(645\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{15}{32}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41387	+	0.0313007i	−0.999755	+	0.0221329i
							\(3\)	0.101527	+	0.0236727i	0.0586165	+	0.0136674i	0.256282	−	0.966602i	\(-0.417503\pi\)
−0.197665	+	0.980270i	\(0.563336\pi\)
\(5\)	2.39383	+	3.34059i	1.07055	+	1.49396i	0.855812	+	0.517287i	\(0.173058\pi\)
							0.214742	+	0.976671i	\(0.431109\pi\)
\(7\)	2.22505	+	1.43148i	0.840991	+	0.541049i
							\(11\)	1.92588	+	0.0630468i	0.580675	+	0.0190093i	0.321646	−	0.946860i	\(-0.395764\pi\)
0.259029	+	0.965869i	\(0.416597\pi\)
\(13\)	−0.353391	+	3.58804i	−0.0980130	+	0.995143i	0.812861	+	0.582458i	\(0.197909\pi\)
							−0.910874	+	0.412685i	\(0.864591\pi\)
\(17\)	3.64537	+	4.75074i	0.884132	+	1.15222i	0.987506	+	0.157580i	\(0.0503693\pi\)
							−0.103375	+	0.994642i	\(0.532964\pi\)
\(19\)	−0.998177	−	0.375687i	−0.228997	−	0.0861885i	0.235105	−	0.971970i	\(-0.424456\pi\)
							−0.464103	+	0.885781i	\(0.653623\pi\)
\(23\)	−0.513699	−	1.51331i	−0.107114	−	0.315547i	0.880137	−	0.474720i	\(-0.157451\pi\)
							−0.987251	+	0.159173i	\(0.949117\pi\)
\(29\)	3.87241	−	7.24477i	0.719089	−	1.34532i	−0.210818	−	0.977525i	\(-0.567613\pi\)
							0.929907	−	0.367795i	\(-0.119887\pi\)
\(31\)	−0.206219	−	0.769621i	−0.0370381	−	0.138228i	0.944931	−	0.327270i	\(-0.106129\pi\)
							−0.981969	+	0.189042i	\(0.939462\pi\)
\(37\)	−6.84150	−	1.12954i	−1.12474	−	0.185695i	−0.428524	−	0.903530i	\(-0.640966\pi\)
							−0.696213	+	0.717836i	\(0.745133\pi\)
\(41\)	1.38004	−	6.93792i	0.215526	−	1.08352i	−0.709816	−	0.704387i	\(-0.751222\pi\)
							0.925342	−	0.379134i	\(-0.123778\pi\)
\(43\)	−10.9011	−	3.30682i	−1.66240	−	0.504285i	−0.687364	−	0.726314i	\(-0.741232\pi\)
							−0.975040	+	0.222029i	\(0.928732\pi\)
\(47\)	11.5301	−	1.51796i	1.68183	−	0.221418i	0.771775	−	0.635896i	\(-0.219369\pi\)
							0.910059	+	0.414479i	\(0.136036\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.115.2	yes	4032
7.5	odd	6	inner	896.2.bt.a.243.84	yes	4032
128.59	odd	32	inner	896.2.bt.a.59.84	✓	4032
896.187	even	96	inner	896.2.bt.a.187.2	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.84	✓	4032	128.59	odd	32	inner
896.2.bt.a.115.2	yes	4032	1.1	even	1	trivial
896.2.bt.a.187.2	yes	4032	896.187	even	96	inner
896.2.bt.a.243.84	yes	4032	7.5	odd	6	inner