Embedded newform 896.2.bk.b.587.38

• Label: 896.2.bk.b.587.38
• 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			587.38
Character	\(\chi\)	\(=\)	896.587
Dual form			896.2.bk.b.867.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.798810 - 1.16701i) q^{2} +(0.0362197 + 0.367745i) q^{3} +(-0.723806 + 1.86443i) q^{4} +(1.76828 - 3.30822i) q^{5} +(0.400227 - 0.336026i) q^{6} +(1.64322 - 2.07360i) q^{7} +(2.75399 - 0.644639i) q^{8} +(2.80843 - 0.558632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1984 q - 32 q^{2} - 32 q^{4} - 16 q^{7} - 32 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.798810	−	1.16701i	−0.564844	−	0.825198i
							\(3\)	0.0362197	+	0.367745i	0.0209114	+	0.212317i	0.999962	+	0.00870083i	\(0.00276959\pi\)
−0.979051	+	0.203617i	\(0.934730\pi\)
\(5\)	1.76828	−	3.30822i	0.790800	−	1.47948i	−0.0847757	−	0.996400i	\(-0.527017\pi\)
							0.875575	−	0.483082i	\(-0.160483\pi\)
\(7\)	1.64322	−	2.07360i	0.621078	−	0.783748i
							\(11\)	−1.05525	+	0.866019i	−0.318169	+	0.261115i	−0.779898	−	0.625906i	\(-0.784729\pi\)
0.461729	+	0.887021i	\(0.347229\pi\)
\(13\)	3.10481	−	1.65955i	0.861118	−	0.460277i	0.0191579	−	0.999816i	\(-0.493901\pi\)
							0.841961	+	0.539539i	\(0.181401\pi\)
\(17\)	0.263941	−	0.109328i	0.0640152	−	0.0265159i	−0.350446	−	0.936583i	\(-0.613970\pi\)
							0.414461	+	0.910067i	\(0.363970\pi\)
\(19\)	−0.133873	+	0.441321i	−0.0307126	+	0.101246i	−0.970957	−	0.239255i	\(-0.923097\pi\)
							0.940244	+	0.340501i	\(0.110597\pi\)
\(23\)	−1.20550	+	1.80416i	−0.251364	+	0.376192i	−0.935596	−	0.353071i	\(-0.885137\pi\)
							0.684233	+	0.729264i	\(0.260137\pi\)
\(29\)	4.29084	+	3.52140i	0.796790	+	0.653908i	0.942045	−	0.335486i	\(-0.108900\pi\)
							−0.145256	+	0.989394i	\(0.546400\pi\)
\(31\)	5.81890	+	5.81890i	1.04511	+	1.04511i	0.998933	+	0.0461724i	\(0.0147024\pi\)
							0.0461724	+	0.998933i	\(0.485298\pi\)
\(37\)	0.243354	−	0.0738205i	0.0400071	−	0.0121360i	−0.270218	−	0.962799i	\(-0.587096\pi\)
							0.310225	+	0.950663i	\(0.399596\pi\)
\(41\)	3.01243	+	2.01284i	0.470462	+	0.314353i	0.768104	−	0.640325i	\(-0.221200\pi\)
							−0.297642	+	0.954677i	\(0.596200\pi\)
\(43\)	−6.42392	−	0.632700i	−0.979638	−	0.0964859i	−0.404496	−	0.914540i	\(-0.632553\pi\)
							−0.575142	+	0.818054i	\(0.695053\pi\)
\(47\)	−7.34400	+	3.04198i	−1.07123	+	0.443719i	−0.847425	−	0.530915i	\(-0.821848\pi\)
							−0.223807	+	0.974634i	\(0.571848\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bk.b.587.38	yes	1984
7.6	odd	2	inner	896.2.bk.b.587.37	✓	1984
128.99	odd	32	inner	896.2.bk.b.867.37	yes	1984
896.867	even	32	inner	896.2.bk.b.867.38	yes	1984

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bk.b.587.37	✓	1984	7.6	odd	2	inner
896.2.bk.b.587.38	yes	1984	1.1	even	1	trivial
896.2.bk.b.867.37	yes	1984	128.99	odd	32	inner
896.2.bk.b.867.38	yes	1984	896.867	even	32	inner