Embedded newform 896.2.bk.b.587.119

• Label: 896.2.bk.b.587.119
• 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			587.119
Character	\(\chi\)	\(=\)	896.587
Dual form			896.2.bk.b.867.119

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41075 + 0.0989659i) q^{2} +(-0.214830 - 2.18121i) q^{3} +(1.98041 + 0.279232i) q^{4} +(-0.223923 + 0.418930i) q^{5} +(-0.0872059 - 3.09839i) q^{6} +(1.27402 + 2.31881i) q^{7} +(2.76622 + 0.589918i) q^{8} +(-1.76917 + 0.351909i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1984 q - 32 q^{2} - 32 q^{4} - 16 q^{7} - 32 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41075	+	0.0989659i	0.997548	+	0.0699794i
							\(3\)	−0.214830	−	2.18121i	−0.124032	−	1.25932i	−0.832678	−	0.553757i	\(-0.813194\pi\)
0.708646	−	0.705564i	\(-0.249306\pi\)
\(5\)	−0.223923	+	0.418930i	−0.100141	+	0.187351i	−0.927003	−	0.375054i	\(-0.877624\pi\)
							0.826861	+	0.562406i	\(0.190124\pi\)
\(7\)	1.27402	+	2.31881i	0.481535	+	0.876427i
							\(11\)	0.770242	−	0.632121i	0.232237	−	0.190592i	−0.511077	−	0.859535i	\(-0.670753\pi\)
0.743313	+	0.668943i	\(0.233253\pi\)
\(13\)	1.79555	−	0.959739i	0.497995	−	0.266184i	−0.203221	−	0.979133i	\(-0.565141\pi\)
							0.701216	+	0.712949i	\(0.252641\pi\)
\(17\)	−0.649265	+	0.268934i	−0.157470	+	0.0652261i	−0.460026	−	0.887905i	\(-0.652160\pi\)
							0.302556	+	0.953132i	\(0.402160\pi\)
\(19\)	0.434895	−	1.43366i	0.0997717	−	0.328903i	−0.892818	−	0.450418i	\(-0.851275\pi\)
							0.992590	+	0.121514i	\(0.0387751\pi\)
\(23\)	1.89526	−	2.83645i	0.395188	−	0.591441i	−0.579510	−	0.814965i	\(-0.696756\pi\)
							0.974698	+	0.223524i	\(0.0717560\pi\)
\(29\)	−5.31288	−	4.36017i	−0.986577	−	0.809663i	−0.00485615	−	0.999988i	\(-0.501546\pi\)
							−0.981721	+	0.190325i	\(0.939046\pi\)
\(31\)	4.48427	+	4.48427i	0.805398	+	0.805398i	0.983934	−	0.178535i	\(-0.0571358\pi\)
							−0.178535	+	0.983934i	\(0.557136\pi\)
\(37\)	−7.92596	+	2.40431i	−1.30302	+	0.395267i	−0.864127	−	0.503273i	\(-0.832129\pi\)
							−0.438892	+	0.898540i	\(0.644629\pi\)
\(41\)	−4.22201	−	2.82105i	−0.659367	−	0.440575i	0.180348	−	0.983603i	\(-0.442278\pi\)
							−0.839714	+	0.543028i	\(0.817278\pi\)
\(43\)	−10.5202	−	1.03615i	−1.60431	−	0.158011i	−0.744142	−	0.668021i	\(-0.767142\pi\)
							−0.860168	+	0.510010i	\(0.829642\pi\)
\(47\)	8.64505	−	3.58090i	1.26101	−	0.522327i	0.350790	−	0.936454i	\(-0.385913\pi\)
							0.910219	+	0.414127i	\(0.135913\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bk.b.587.119	✓	1984
7.6	odd	2	inner	896.2.bk.b.587.120	yes	1984
128.99	odd	32	inner	896.2.bk.b.867.120	yes	1984
896.867	even	32	inner	896.2.bk.b.867.119	yes	1984

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bk.b.587.119	✓	1984	1.1	even	1	trivial
896.2.bk.b.587.120	yes	1984	7.6	odd	2	inner
896.2.bk.b.867.119	yes	1984	896.867	even	32	inner
896.2.bk.b.867.120	yes	1984	128.99	odd	32	inner