Embedded newform 896.2.bk.b.587.68

• Label: 896.2.bk.b.587.68
• Level: $896$
• Weight: $2$
• Character: 896.587
• Analytic conductor: $7.155$
• Analytic rank: $0$
• Dimension: $1984$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.bk (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.15459602111\)
Analytic rank:	\(0\)
Dimension:	\(1984\)
Relative dimension:	\(124\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			587.68
Character	\(\chi\)	\(=\)	896.587
Dual form			896.2.bk.b.867.68

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.121820 - 1.40896i) q^{2} +(0.241376 + 2.45073i) q^{3} +(-1.97032 - 0.343278i) q^{4} +(1.79105 - 3.35081i) q^{5} +(3.48238 - 0.0415403i) q^{6} +(2.63458 - 0.242921i) q^{7} +(-0.723689 + 2.73428i) q^{8} +(-3.00545 + 0.597822i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1984 q - 32 q^{2} - 32 q^{4} - 16 q^{7} - 32 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(129\)	\(645\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{32}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.121820	−	1.40896i	0.0861397	−	0.996283i
							\(3\)	0.241376	+	2.45073i	0.139358	+	1.41493i	0.769598	+	0.638528i	\(0.220457\pi\)
−0.630240	+	0.776400i	\(0.717043\pi\)
\(5\)	1.79105	−	3.35081i	0.800981	−	1.49853i	−0.0646839	−	0.997906i	\(-0.520604\pi\)
							0.865665	−	0.500624i	\(-0.166896\pi\)
\(7\)	2.63458	−	0.242921i	0.995776	−	0.0918156i
							\(11\)	4.32475	−	3.54923i	1.30396	−	1.07013i	0.311162	−	0.950357i	\(-0.399282\pi\)
0.992801	−	0.119777i	\(-0.0382181\pi\)
\(13\)	−3.48969	+	1.86528i	−0.967864	+	0.517334i	−0.878021	−	0.478622i	\(-0.841137\pi\)
							−0.0898432	+	0.995956i	\(0.528637\pi\)
\(17\)	−1.52974	+	0.633641i	−0.371018	+	0.153680i	−0.560398	−	0.828223i	\(-0.689352\pi\)
							0.189381	+	0.981904i	\(0.439352\pi\)
\(19\)	1.59036	−	5.24273i	0.364854	−	1.20276i	−0.561935	−	0.827182i	\(-0.689943\pi\)
							0.926789	−	0.375582i	\(-0.122557\pi\)
\(23\)	−2.89879	+	4.33834i	−0.604439	+	0.904607i	−0.999904	−	0.0138743i	\(-0.995584\pi\)
							0.395465	+	0.918481i	\(0.370584\pi\)
\(29\)	−2.58130	−	2.11842i	−0.479336	−	0.393381i	0.363464	−	0.931608i	\(-0.381594\pi\)
							−0.842800	+	0.538227i	\(0.819094\pi\)
\(31\)	−3.24340	−	3.24340i	−0.582531	−	0.582531i	0.353067	−	0.935598i	\(-0.385139\pi\)
							−0.935598	+	0.353067i	\(0.885139\pi\)
\(37\)	9.25237	−	2.80668i	1.52108	−	0.461415i	0.584360	−	0.811494i	\(-0.301345\pi\)
							0.936720	+	0.350080i	\(0.113845\pi\)
\(41\)	1.74679	+	1.16717i	0.272803	+	0.182281i	0.684445	−	0.729064i	\(-0.260044\pi\)
							−0.411642	+	0.911346i	\(0.635044\pi\)
\(43\)	6.82316	+	0.672022i	1.04052	+	0.102482i	0.603817	−	0.797123i	\(-0.293646\pi\)
							0.436704	+	0.899605i	\(0.356146\pi\)
\(47\)	−2.72779	+	1.12989i	−0.397889	+	0.164811i	−0.572650	−	0.819800i	\(-0.694085\pi\)
							0.174761	+	0.984611i	\(0.444085\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bk.b.587.68	yes	1984
7.6	odd	2	inner	896.2.bk.b.587.67	✓	1984
128.99	odd	32	inner	896.2.bk.b.867.67	yes	1984
896.867	even	32	inner	896.2.bk.b.867.68	yes	1984

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bk.b.587.67	✓	1984	7.6	odd	2	inner
896.2.bk.b.587.68	yes	1984	1.1	even	1	trivial
896.2.bk.b.867.67	yes	1984	128.99	odd	32	inner
896.2.bk.b.867.68	yes	1984	896.867	even	32	inner