Embedded newform 896.2.bt.a.59.17

• Label: 896.2.bt.a.59.17
• 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			59.17
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30753 + 0.538848i) q^{2} +(-0.653059 - 0.697264i) q^{3} +(1.41929 - 1.40912i) q^{4} +(-0.673592 - 1.48665i) q^{5} +(1.22962 + 0.559796i) q^{6} +(-1.12023 - 2.39689i) q^{7} +(-1.09646 + 2.60726i) q^{8} +(0.136518 - 2.08287i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30753	+	0.538848i	−0.924565	+	0.381023i
							\(3\)	−0.653059	−	0.697264i	−0.377044	−	0.402566i	0.513038	−	0.858366i	\(-0.328520\pi\)
−0.890082	+	0.455800i	\(0.849353\pi\)
\(5\)	−0.673592	−	1.48665i	−0.301240	−	0.664850i	0.697417	−	0.716665i	\(-0.254333\pi\)
							−0.998657	+	0.0518156i	\(0.983499\pi\)
\(7\)	−1.12023	−	2.39689i	−0.423408	−	0.905939i
							\(11\)	−1.17039	−	1.88213i	−0.352885	−	0.567485i	0.623954	−	0.781461i	\(-0.285525\pi\)
−0.976839	+	0.213976i	\(0.931359\pi\)
\(13\)	−0.0286843	−	0.291237i	−0.00795560	−	0.0807746i	0.990387	−	0.138324i	\(-0.0441714\pi\)
							−0.998343	+	0.0575491i	\(0.981671\pi\)
\(17\)	−1.92833	−	0.253870i	−0.467689	−	0.0615725i	−0.107001	−	0.994259i	\(-0.534125\pi\)
							−0.360688	+	0.932686i	\(0.617458\pi\)
\(19\)	0.252238	−	1.52778i	0.0578673	−	0.350497i	−0.942021	−	0.335554i	\(-0.891077\pi\)
							0.999888	−	0.0149432i	\(-0.00475676\pi\)
\(23\)	−1.00813	−	1.14955i	−0.210209	−	0.239698i	0.637121	−	0.770764i	\(-0.280125\pi\)
							−0.847331	+	0.531066i	\(0.821792\pi\)
\(29\)	2.84742	+	5.32715i	0.528753	+	0.989227i	0.994543	+	0.104331i	\(0.0332702\pi\)
							−0.465790	+	0.884895i	\(0.654230\pi\)
\(31\)	0.568188	−	0.152245i	0.102049	−	0.0273441i	−0.207433	−	0.978249i	\(-0.566511\pi\)
							0.309482	+	0.950905i	\(0.399844\pi\)
\(37\)	0.272765	−	0.724718i	0.0448422	−	0.119143i	−0.911587	−	0.411107i	\(-0.865142\pi\)
							0.956429	+	0.291964i	\(0.0943088\pi\)
\(41\)	0.326625	+	1.64205i	0.0510102	+	0.256446i	0.997873	−	0.0651931i	\(-0.0207663\pi\)
							−0.946862	+	0.321639i	\(0.895766\pi\)
\(43\)	−2.02671	+	0.614796i	−0.309070	+	0.0937555i	−0.441010	−	0.897502i	\(-0.645380\pi\)
							0.131939	+	0.991258i	\(0.457880\pi\)
\(47\)	−0.854147	+	1.11315i	−0.124590	+	0.162369i	−0.851484	−	0.524381i	\(-0.824297\pi\)
							0.726893	+	0.686750i	\(0.240963\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.17	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.102	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.102	yes	4032
896.243	even	96	inner	896.2.bt.a.243.17	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.17	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.102	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.102	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.17	yes	4032	896.243	even	96	inner