Embedded newform 896.2.bt.a.59.34

• Label: 896.2.bt.a.59.34
• Level: $896$
• Weight: $2$
• Character: 896.59
• 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			59.34
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.907648 + 1.08452i) q^{2} +(-0.200094 - 0.213638i) q^{3} +(-0.352350 - 1.96872i) q^{4} +(1.10648 + 2.44205i) q^{5} +(0.413309 - 0.0230968i) q^{6} +(-0.304978 + 2.62811i) q^{7} +(2.45492 + 1.40477i) q^{8} +(0.190606 - 2.90808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(129\)	\(645\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{17}{32}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.907648	+	1.08452i	−0.641804	+	0.766869i
							\(3\)	−0.200094	−	0.213638i	−0.115524	−	0.123344i	0.669492	−	0.742820i	\(-0.266512\pi\)
−0.785016	+	0.619475i	\(0.787345\pi\)
\(5\)	1.10648	+	2.44205i	0.494832	+	1.09212i	0.976452	+	0.215737i	\(0.0692153\pi\)
							−0.481620	+	0.876380i	\(0.659951\pi\)
\(7\)	−0.304978	+	2.62811i	−0.115271	+	0.993334i
							\(11\)	0.447706	+	0.719970i	0.134989	+	0.217079i	0.909731	−	0.415197i	\(-0.136287\pi\)
−0.774743	+	0.632276i	\(0.782121\pi\)
\(13\)	0.195782	+	1.98780i	0.0543000	+	0.551317i	0.983395	+	0.181477i	\(0.0580877\pi\)
							−0.929095	+	0.369841i	\(0.879412\pi\)
\(17\)	−0.168214	−	0.0221457i	−0.0407978	−	0.00537113i	0.110100	−	0.993921i	\(-0.464883\pi\)
							−0.150897	+	0.988549i	\(0.548216\pi\)
\(19\)	−1.13709	+	6.88727i	−0.260867	+	1.58005i	0.460606	+	0.887605i	\(0.347632\pi\)
							−0.721473	+	0.692443i	\(0.756534\pi\)
\(23\)	−3.64963	−	4.16161i	−0.761001	−	0.867756i	0.233513	−	0.972354i	\(-0.424978\pi\)
							−0.994515	+	0.104598i	\(0.966644\pi\)
\(29\)	3.05990	+	5.72466i	0.568208	+	1.06304i	0.987858	+	0.155357i	\(0.0496528\pi\)
							−0.419650	+	0.907686i	\(0.637847\pi\)
\(31\)	8.77998	−	2.35259i	1.57693	−	0.422537i	0.638956	−	0.769243i	\(-0.279366\pi\)
							0.937974	+	0.346706i	\(0.112700\pi\)
\(37\)	0.932385	−	2.47729i	0.153283	−	0.407263i	−0.836843	−	0.547443i	\(-0.815601\pi\)
							0.990126	+	0.140180i	\(0.0447680\pi\)
\(41\)	0.238220	+	1.19761i	0.0372037	+	0.187036i	0.994918	−	0.100684i	\(-0.0321030\pi\)
							−0.957715	+	0.287719i	\(0.907103\pi\)
\(43\)	−3.09297	+	0.938241i	−0.471673	+	0.143080i	−0.517168	−	0.855884i	\(-0.673014\pi\)
							0.0454946	+	0.998965i	\(0.485514\pi\)
\(47\)	−7.79276	+	10.1557i	−1.13669	+	1.48137i	−0.284094	+	0.958796i	\(0.591693\pi\)
							−0.852597	+	0.522569i	\(0.824974\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.34	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.119	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.119	yes	4032
896.243	even	96	inner	896.2.bt.a.243.34	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.34	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.119	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.119	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.34	yes	4032	896.243	even	96	inner