Embedded newform 896.2.bt.a.59.102

• Label: 896.2.bt.a.59.102
• 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			59.102
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.102

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.18717 - 0.768529i) q^{2} +(-1.11193 - 1.18720i) q^{3} +(0.818728 - 1.82474i) q^{4} +(-0.808349 - 1.78406i) q^{5} +(-2.23244 - 0.554850i) q^{6} +(-2.60572 + 0.458517i) q^{7} +(-0.430400 - 2.79549i) q^{8} +(0.0231637 - 0.353409i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.18717	−	0.768529i	0.839453	−	0.543432i
							\(3\)	−1.11193	−	1.18720i	−0.641974	−	0.685429i	0.323410	−	0.946259i	\(-0.395171\pi\)
−0.965385	+	0.260830i	\(0.916004\pi\)
\(5\)	−0.808349	−	1.78406i	−0.361505	−	0.797858i	−0.999610	−	0.0279089i	\(-0.991115\pi\)
							0.638106	−	0.769949i	\(-0.279718\pi\)
\(7\)	−2.60572	+	0.458517i	−0.984869	+	0.173303i
							\(11\)	−0.848008	−	1.36371i	−0.255684	−	0.411173i	0.696099	−	0.717946i	\(-0.254918\pi\)
−0.951783	+	0.306773i	\(0.900751\pi\)
\(13\)	0.233356	+	2.36930i	0.0647212	+	0.657126i	0.971743	+	0.236042i	\(0.0758502\pi\)
							−0.907022	+	0.421084i	\(0.861650\pi\)
\(17\)	3.23212	+	0.425517i	0.783904	+	0.103203i	0.511845	−	0.859078i	\(-0.328962\pi\)
							0.272059	+	0.962281i	\(0.412295\pi\)
\(19\)	−0.651948	+	3.94879i	−0.149567	+	0.905914i	0.801741	+	0.597672i	\(0.203907\pi\)
							−0.951308	+	0.308242i	\(0.900259\pi\)
\(23\)	−0.132069	−	0.150596i	−0.0275383	−	0.0314014i	0.737907	−	0.674903i	\(-0.235814\pi\)
							−0.765445	+	0.643501i	\(0.777481\pi\)
\(29\)	−1.84314	−	3.44828i	−0.342263	−	0.640329i	0.650622	−	0.759401i	\(-0.274508\pi\)
							−0.992886	+	0.119072i	\(0.962008\pi\)
\(31\)	−2.19557	+	0.588301i	−0.394336	+	0.105662i	−0.450538	−	0.892757i	\(-0.648768\pi\)
							0.0562016	+	0.998419i	\(0.482101\pi\)
\(37\)	−1.69390	+	4.50059i	−0.278476	+	0.739893i	0.720448	+	0.693509i	\(0.243936\pi\)
							−0.998924	+	0.0463834i	\(0.985230\pi\)
\(41\)	−0.483590	−	2.43117i	−0.0755240	−	0.379685i	0.924475	−	0.381243i	\(-0.124504\pi\)
							−0.999999	+	0.00155842i	\(0.999504\pi\)
\(43\)	3.05002	−	0.925214i	0.465124	−	0.141094i	−0.0490198	−	0.998798i	\(-0.515610\pi\)
							0.514144	+	0.857704i	\(0.328110\pi\)
\(47\)	2.15066	−	2.80279i	0.313705	−	0.408829i	−0.609714	−	0.792622i	\(-0.708716\pi\)
							0.923419	+	0.383793i	\(0.125382\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.102	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.19	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.19	yes	4032
896.243	even	96	inner	896.2.bt.a.243.102	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.102	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.19	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.19	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.102	yes	4032	896.243	even	96	inner