Embedded newform 896.2.bt.a.243.44

• Label: 896.2.bt.a.243.44
• Level: $896$
• Weight: $2$
• Character: 896.243
• 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			243.44
Character	\(\chi\)	\(=\)	896.243
Dual form			896.2.bt.a.59.44

$q$-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 \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(645\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{15}{32}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.44	✓	4032	896.59	even	96	inner
896.2.bt.a.115.126	yes	4032	7.3	odd	6	inner
896.2.bt.a.187.126	yes	4032	128.59	odd	32	inner
896.2.bt.a.243.44	yes	4032	1.1	even	1	trivial