Embedded newform 896.2.bt.a.59.126

• Label: 896.2.bt.a.59.126
• 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			59.126
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.126

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41419 - 0.00839265i) q^{2} +(-1.76078 - 1.87997i) q^{3} +(1.99986 - 0.0237376i) q^{4} +(0.730890 + 1.61311i) q^{5} +(-2.50585 - 2.64385i) q^{6} +(-2.25264 - 1.38767i) q^{7} +(2.82798 - 0.0503535i) q^{8} +(-0.237715 + 3.62684i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41419	−	0.00839265i	0.999982	−	0.00593450i
							\(3\)	−1.76078	−	1.87997i	−1.01659	−	1.08540i	−0.996329	−	0.0856040i	\(-0.972718\pi\)
−0.0202576	−	0.999795i	\(-0.506449\pi\)
\(5\)	0.730890	+	1.61311i	0.326864	+	0.721404i	0.999833	−	0.0182662i	\(-0.00581465\pi\)
							−0.672969	+	0.739671i	\(0.734981\pi\)
\(7\)	−2.25264	−	1.38767i	−0.851417	−	0.524490i
							\(11\)	−1.86186	−	2.99412i	−0.561373	−	0.902761i	−0.999992	−	0.00408773i	\(-0.998699\pi\)
0.438619	−	0.898673i	\(-0.355468\pi\)
\(13\)	−0.112563	−	1.14287i	−0.0312194	−	0.316976i	−0.998093	−	0.0617217i	\(-0.980341\pi\)
							0.966874	−	0.255254i	\(-0.0821591\pi\)
\(17\)	−3.98296	−	0.524366i	−0.966009	−	0.127177i	−0.369015	−	0.929423i	\(-0.620305\pi\)
							−0.596994	+	0.802246i	\(0.703638\pi\)
\(19\)	1.07732	−	6.52520i	0.247153	−	1.49698i	−0.519249	−	0.854623i	\(-0.673788\pi\)
							0.766402	−	0.642361i	\(-0.222045\pi\)
\(23\)	−0.872315	−	0.994684i	−0.181890	−	0.207406i	0.653708	−	0.756747i	\(-0.273212\pi\)
							−0.835599	+	0.549341i	\(0.814879\pi\)
\(29\)	−0.651972	−	1.21975i	−0.121068	−	0.226503i	0.814085	−	0.580745i	\(-0.197239\pi\)
							−0.935154	+	0.354243i	\(0.884739\pi\)
\(31\)	−2.24712	+	0.602113i	−0.403594	+	0.108143i	−0.454905	−	0.890540i	\(-0.650327\pi\)
							0.0513109	+	0.998683i	\(0.483660\pi\)
\(37\)	−0.849901	+	2.25813i	−0.139723	+	0.371235i	−0.987176	−	0.159635i	\(-0.948968\pi\)
							0.847453	+	0.530870i	\(0.178135\pi\)
\(41\)	−2.41955	−	12.1639i	−0.377870	−	1.89968i	−0.433359	−	0.901222i	\(-0.642672\pi\)
							0.0554887	−	0.998459i	\(-0.482328\pi\)
\(43\)	8.77525	−	2.66194i	1.33821	−	0.405943i	0.461579	−	0.887099i	\(-0.347283\pi\)
							0.876635	+	0.481156i	\(0.159783\pi\)
\(47\)	−2.40933	+	3.13991i	−0.351438	+	0.458002i	−0.935303	−	0.353848i	\(-0.884873\pi\)
							0.583865	+	0.811851i	\(0.301540\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.126	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.43	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.43	yes	4032
896.243	even	96	inner	896.2.bt.a.243.126	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.126	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.43	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.43	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.126	yes	4032	896.243	even	96	inner