Embedded newform 896.2.bk.b.587.56

• Label: 896.2.bk.b.587.56
• 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			587.56
Character	\(\chi\)	\(=\)	896.587
Dual form			896.2.bk.b.867.56

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.196978 - 1.40043i) q^{2} +(0.0178932 + 0.181673i) q^{3} +(-1.92240 + 0.551708i) q^{4} +(-1.42659 + 2.66896i) q^{5} +(0.250896 - 0.0608439i) q^{6} +(-1.80680 - 1.93274i) q^{7} +(1.15130 + 2.58351i) q^{8} +(2.90967 - 0.578770i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1984 q - 32 q^{2} - 32 q^{4} - 16 q^{7} - 32 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.196978	−	1.40043i	−0.139285	−	0.990252i
							\(3\)	0.0178932	+	0.181673i	0.0103307	+	0.104889i	0.998967	−	0.0454364i	\(-0.0144678\pi\)
−0.988637	+	0.150325i	\(0.951968\pi\)
\(5\)	−1.42659	+	2.66896i	−0.637991	+	1.19360i	0.330325	+	0.943867i	\(0.392842\pi\)
							−0.968316	+	0.249729i	\(0.919658\pi\)
\(7\)	−1.80680	−	1.93274i	−0.682906	−	0.730506i
							\(11\)	2.82044	−	2.31468i	0.850395	−	0.697901i	−0.104602	−	0.994514i	\(-0.533357\pi\)
0.954998	+	0.296613i	\(0.0958570\pi\)
\(13\)	−1.74000	+	0.930050i	−0.482589	+	0.257949i	−0.694704	−	0.719295i	\(-0.744465\pi\)
							0.212115	+	0.977245i	\(0.431965\pi\)
\(17\)	−6.82627	+	2.82753i	−1.65561	+	0.685778i	−0.997730	−	0.0673416i	\(-0.978548\pi\)
							−0.657884	+	0.753119i	\(0.728548\pi\)
\(19\)	0.999467	−	3.29480i	0.229293	−	0.755879i	−0.764499	−	0.644625i	\(-0.777013\pi\)
							0.993792	−	0.111254i	\(-0.0354866\pi\)
\(23\)	−1.40759	+	2.10661i	−0.293503	+	0.439259i	−0.948690	−	0.316209i	\(-0.897590\pi\)
							0.655186	+	0.755467i	\(0.272590\pi\)
\(29\)	−7.43460	−	6.10142i	−1.38057	−	1.13300i	−0.975820	−	0.218577i	\(-0.929858\pi\)
							−0.404750	−	0.914427i	\(-0.632642\pi\)
\(31\)	3.15473	+	3.15473i	0.566606	+	0.566606i	0.931176	−	0.364570i	\(-0.118784\pi\)
							−0.364570	+	0.931176i	\(0.618784\pi\)
\(37\)	−6.75940	+	2.05044i	−1.11124	+	0.337090i	−0.791961	−	0.610571i	\(-0.790940\pi\)
							−0.319277	+	0.947662i	\(0.603440\pi\)
\(41\)	−6.27371	−	4.19196i	−0.979789	−	0.654674i	−0.0409971	−	0.999159i	\(-0.513053\pi\)
							−0.938792	+	0.344485i	\(0.888053\pi\)
\(43\)	−8.66549	−	0.853476i	−1.32147	−	0.130154i	−0.587441	−	0.809267i	\(-0.699864\pi\)
							−0.734033	+	0.679113i	\(0.762364\pi\)
\(47\)	−11.1483	+	4.61777i	−1.62614	+	0.673571i	−0.994792	−	0.101925i	\(-0.967500\pi\)
							−0.631353	+	0.775496i	\(0.717500\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bk.b.587.56	yes	1984
7.6	odd	2	inner	896.2.bk.b.587.55	✓	1984
128.99	odd	32	inner	896.2.bk.b.867.55	yes	1984
896.867	even	32	inner	896.2.bk.b.867.56	yes	1984

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bk.b.587.55	✓	1984	7.6	odd	2	inner
896.2.bk.b.587.56	yes	1984	1.1	even	1	trivial
896.2.bk.b.867.55	yes	1984	128.99	odd	32	inner
896.2.bk.b.867.56	yes	1984	896.867	even	32	inner