Embedded newform 896.2.bt.a.59.44

• Label: 896.2.bt.a.59.44
• Level: $896$
• Weight: $2$
• Character: 896.59
• Analytic conductor: $7.155$
• Analytic rank: $0$
• Dimension: $4032$
• 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			59.44
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.44

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.670857 + 1.24497i) q^{2} +(1.16189 + 1.24054i) q^{3} +(-1.09990 - 1.67039i) q^{4} +(-0.922351 - 2.03567i) q^{5} +(-2.32390 + 0.614296i) q^{6} +(-2.59529 - 0.514266i) q^{7} +(2.81747 - 0.248749i) q^{8} +(0.00726440 - 0.110833i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.670857	+	1.24497i	−0.474368	+	0.880327i
							\(3\)	1.16189	+	1.24054i	0.670818	+	0.716226i	0.971378	−	0.237537i	\(-0.0763402\pi\)
−0.300560	+	0.953763i	\(0.597174\pi\)
\(5\)	−0.922351	−	2.03567i	−0.412488	−	0.910380i	−0.995215	−	0.0977120i	\(-0.968848\pi\)
							0.582727	−	0.812668i	\(-0.301986\pi\)
\(7\)	−2.59529	−	0.514266i	−0.980927	−	0.194374i
							\(11\)	0.0927529	+	0.149159i	0.0279661	+	0.0449731i	0.862887	−	0.505396i	\(-0.168654\pi\)
−0.834921	+	0.550369i	\(0.814487\pi\)
\(13\)	0.575771	+	5.84591i	0.159690	+	1.62136i	0.657070	+	0.753830i	\(0.271796\pi\)
							−0.497380	+	0.867533i	\(0.665704\pi\)
\(17\)	1.81110	+	0.238436i	0.439257	+	0.0578292i	0.346912	−	0.937898i	\(-0.387230\pi\)
							0.0923451	+	0.995727i	\(0.470564\pi\)
\(19\)	−0.984832	+	5.96504i	−0.225936	+	1.36847i	0.597866	+	0.801596i	\(0.296015\pi\)
							−0.823802	+	0.566877i	\(0.808151\pi\)
\(23\)	2.21088	+	2.52103i	0.461000	+	0.525670i	0.934901	−	0.354908i	\(-0.115488\pi\)
							−0.473901	+	0.880578i	\(0.657154\pi\)
\(29\)	4.61400	+	8.63220i	0.856799	+	1.60296i	0.796389	+	0.604785i	\(0.206741\pi\)
							0.0604103	+	0.998174i	\(0.480759\pi\)
\(31\)	−10.3241	+	2.76634i	−1.85427	+	0.496849i	−0.999745	−	0.0225662i	\(-0.992816\pi\)
							−0.854522	+	0.519416i	\(0.826150\pi\)
\(37\)	−0.0775414	+	0.206023i	−0.0127477	+	0.0338699i	−0.942127	−	0.335257i	\(-0.891177\pi\)
							0.929379	+	0.369127i	\(0.120343\pi\)
\(41\)	1.42858	+	7.18194i	0.223106	+	1.12163i	0.916181	+	0.400765i	\(0.131256\pi\)
							−0.693075	+	0.720866i	\(0.743744\pi\)
\(43\)	0.314111	−	0.0952845i	0.0479014	−	0.0145307i	−0.266243	−	0.963906i	\(-0.585782\pi\)
							0.314144	+	0.949375i	\(0.398282\pi\)
\(47\)	−2.05677	+	2.68044i	−0.300011	+	0.390982i	−0.918896	−	0.394499i	\(-0.870918\pi\)
							0.618885	+	0.785482i	\(0.287585\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.44	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.126	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.126	yes	4032
896.243	even	96	inner	896.2.bt.a.243.44	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.44	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.126	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.126	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.44	yes	4032	896.243	even	96	inner