Embedded newform 896.2.bt.a.59.86

• Label: 896.2.bt.a.59.86
• 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			59.86
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.86

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.761624 - 1.19161i) q^{2} +(-2.27004 - 2.42370i) q^{3} +(-0.839859 - 1.81511i) q^{4} +(1.10515 + 2.43913i) q^{5} +(-4.61701 + 0.859052i) q^{6} +(-2.12033 + 1.58247i) q^{7} +(-2.80256 - 0.381652i) q^{8} +(-0.525017 + 8.01020i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.761624	−	1.19161i	0.538549	−	0.842594i
							\(3\)	−2.27004	−	2.42370i	−1.31061	−	1.39932i	−0.863092	−	0.505047i	\(-0.831475\pi\)
−0.447516	−	0.894276i	\(-0.647691\pi\)
\(5\)	1.10515	+	2.43913i	0.494240	+	1.09081i	0.976645	+	0.214862i	\(0.0689300\pi\)
							−0.482405	+	0.875948i	\(0.660237\pi\)
\(7\)	−2.12033	+	1.58247i	−0.801409	+	0.598117i
							\(11\)	−0.263460	−	0.423678i	−0.0794361	−	0.127744i	0.807079	−	0.590444i	\(-0.201047\pi\)
−0.886515	+	0.462700i	\(0.846881\pi\)
\(13\)	−0.0923431	−	0.937575i	−0.0256114	−	0.260037i	−0.999461	−	0.0328412i	\(-0.989544\pi\)
							0.973849	−	0.227195i	\(-0.0729556\pi\)
\(17\)	3.79970	+	0.500239i	0.921562	+	0.121326i	0.576354	−	0.817200i	\(-0.304475\pi\)
							0.345208	+	0.938526i	\(0.387808\pi\)
\(19\)	0.311227	−	1.88507i	0.0714003	−	0.432466i	−0.926957	−	0.375168i	\(-0.877585\pi\)
							0.998357	−	0.0572975i	\(-0.0182483\pi\)
\(23\)	4.68413	+	5.34123i	0.976710	+	1.11372i	0.993496	+	0.113865i	\(0.0363230\pi\)
							−0.0167867	+	0.999859i	\(0.505344\pi\)
\(29\)	−0.586291	−	1.09687i	−0.108871	−	0.203684i	0.821586	−	0.570085i	\(-0.193090\pi\)
							−0.930457	+	0.366401i	\(0.880590\pi\)
\(31\)	−2.85333	+	0.764547i	−0.512473	+	0.137317i	−0.505783	−	0.862661i	\(-0.668796\pi\)
							−0.00669028	+	0.999978i	\(0.502130\pi\)
\(37\)	−3.89636	+	10.3524i	−0.640558	+	1.70192i	0.0693185	+	0.997595i	\(0.477918\pi\)
							−0.709876	+	0.704326i	\(0.751249\pi\)
\(41\)	1.25084	+	6.28841i	0.195349	+	0.982085i	0.946684	+	0.322164i	\(0.104410\pi\)
							−0.751335	+	0.659921i	\(0.770590\pi\)
\(43\)	−5.81772	+	1.76479i	−0.887194	+	0.269127i	−0.700809	−	0.713349i	\(-0.747177\pi\)
							−0.186385	+	0.982477i	\(0.559677\pi\)
\(47\)	1.20523	−	1.57068i	0.175800	−	0.229107i	−0.697139	−	0.716936i	\(-0.745544\pi\)
							0.872940	+	0.487828i	\(0.162211\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.86	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.3	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.3	yes	4032
896.243	even	96	inner	896.2.bt.a.243.86	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.86	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.3	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.3	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.86	yes	4032	896.243	even	96	inner