Embedded newform 896.2.bt.a.59.104

• Label: 896.2.bt.a.59.104
• 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			59.104
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.104

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.20526 + 0.739832i) q^{2} +(0.318916 + 0.340503i) q^{3} +(0.905297 + 1.78338i) q^{4} +(-0.790841 - 1.74542i) q^{5} +(0.132461 + 0.646338i) q^{6} +(-0.869456 - 2.49881i) q^{7} +(-0.228283 + 2.81920i) q^{8} +(0.181974 - 2.77639i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.20526	+	0.739832i	0.852247	+	0.523140i
							\(3\)	0.318916	+	0.340503i	0.184126	+	0.196589i	0.815278	−	0.579070i	\(-0.196584\pi\)
−0.631152	+	0.775659i	\(0.717418\pi\)
\(5\)	−0.790841	−	1.74542i	−0.353675	−	0.780577i	−0.999849	−	0.0173868i	\(-0.994465\pi\)
							0.646174	−	0.763190i	\(-0.276368\pi\)
\(7\)	−0.869456	−	2.49881i	−0.328623	−	0.944461i
							\(11\)	−1.95798	−	3.14869i	−0.590353	−	0.949365i	−0.999322	−	0.0368166i	\(-0.988278\pi\)
0.408969	−	0.912548i	\(-0.365888\pi\)
\(13\)	−0.287969	−	2.92380i	−0.0798683	−	0.810917i	−0.948777	−	0.315945i	\(-0.897678\pi\)
							0.868909	−	0.494972i	\(-0.164822\pi\)
\(17\)	4.84837	+	0.638300i	1.17590	+	0.154810i	0.693049	−	0.720891i	\(-0.256267\pi\)
							0.482853	+	0.875701i	\(0.339600\pi\)
\(19\)	0.546483	−	3.31000i	0.125372	−	0.759366i	−0.847976	−	0.530034i	\(-0.822179\pi\)
							0.973348	−	0.229332i	\(-0.0736542\pi\)
\(23\)	5.28297	+	6.02407i	1.10158	+	1.25611i	0.964082	+	0.265604i	\(0.0855713\pi\)
							0.137493	+	0.990503i	\(0.456095\pi\)
\(29\)	−1.14377	−	2.13984i	−0.212393	−	0.397358i	0.753054	−	0.657958i	\(-0.228580\pi\)
							−0.965447	+	0.260600i	\(0.916080\pi\)
\(31\)	−4.68884	+	1.25637i	−0.842141	+	0.225651i	−0.654003	−	0.756492i	\(-0.726912\pi\)
							−0.188138	+	0.982143i	\(0.560245\pi\)
\(37\)	−2.36696	+	6.28887i	−0.389127	+	1.03388i	0.585623	+	0.810584i	\(0.300850\pi\)
							−0.974750	+	0.223301i	\(0.928317\pi\)
\(41\)	−0.0955212	−	0.480217i	−0.0149179	−	0.0749973i	0.972609	−	0.232446i	\(-0.0746730\pi\)
							−0.987527	+	0.157449i	\(0.949673\pi\)
\(43\)	−6.00645	+	1.82204i	−0.915976	+	0.277858i	−0.712868	−	0.701298i	\(-0.752604\pi\)
							−0.203108	+	0.979156i	\(0.565104\pi\)
\(47\)	−1.82918	+	2.38383i	−0.266813	+	0.347717i	−0.907475	−	0.420106i	\(-0.861993\pi\)
							0.640663	+	0.767822i	\(0.278660\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.104	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.65	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.65	yes	4032
896.243	even	96	inner	896.2.bt.a.243.104	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.104	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.65	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.65	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.104	yes	4032	896.243	even	96	inner