Embedded newform 896.2.bt.a.59.12

• Label: 896.2.bt.a.59.12
• 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			59.12
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35833 + 0.393614i) q^{2} +(2.30151 + 2.45730i) q^{3} +(1.69014 - 1.06932i) q^{4} +(-0.422408 - 0.932274i) q^{5} +(-4.09344 - 2.43192i) q^{6} +(-1.83851 - 1.90260i) q^{7} +(-1.87487 + 2.11775i) q^{8} +(-0.545151 + 8.31740i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35833	+	0.393614i	−0.960486	+	0.278327i
							\(3\)	2.30151	+	2.45730i	1.32878	+	1.41872i	0.836172	+	0.548467i	\(0.184788\pi\)
0.492604	+	0.870253i	\(0.336045\pi\)
\(5\)	−0.422408	−	0.932274i	−0.188906	−	0.416925i	0.792235	−	0.610216i	\(-0.208917\pi\)
							−0.981142	+	0.193291i	\(0.938084\pi\)
\(7\)	−1.83851	−	1.90260i	−0.694891	−	0.719115i
							\(11\)	2.50543	+	4.02905i	0.755414	+	1.21480i	0.971177	+	0.238359i	\(0.0766095\pi\)
−0.215763	+	0.976446i	\(0.569224\pi\)
\(13\)	0.241464	+	2.45162i	0.0669700	+	0.679958i	0.968796	+	0.247859i	\(0.0797269\pi\)
							−0.901826	+	0.432099i	\(0.857773\pi\)
\(17\)	−6.48139	−	0.853291i	−1.57197	−	0.206954i	−0.706425	−	0.707788i	\(-0.749693\pi\)
							−0.865543	+	0.500834i	\(0.833027\pi\)
\(19\)	−0.175931	+	1.06560i	−0.0403612	+	0.244464i	−0.999223	−	0.0394084i	\(-0.987453\pi\)
							0.958862	+	0.283873i	\(0.0916193\pi\)
\(23\)	4.54754	+	5.18547i	0.948227	+	1.08125i	0.996598	+	0.0824160i	\(0.0262636\pi\)
							−0.0483713	+	0.998829i	\(0.515403\pi\)
\(29\)	−2.82319	−	5.28182i	−0.524253	−	0.980809i	−0.995126	−	0.0986079i	\(-0.968561\pi\)
							0.470873	−	0.882201i	\(-0.343939\pi\)
\(31\)	4.44572	−	1.19123i	0.798475	−	0.213951i	0.163561	−	0.986533i	\(-0.447702\pi\)
							0.634914	+	0.772582i	\(0.281035\pi\)
\(37\)	−1.93835	+	5.15007i	−0.318662	+	0.846666i	0.675360	+	0.737488i	\(0.263988\pi\)
							−0.994022	+	0.109177i	\(0.965178\pi\)
\(41\)	1.25548	+	6.31173i	0.196073	+	0.985726i	0.945990	+	0.324195i	\(0.105093\pi\)
							−0.749917	+	0.661532i	\(0.769907\pi\)
\(43\)	−2.11908	+	0.642817i	−0.323157	+	0.0980286i	−0.447695	−	0.894186i	\(-0.647755\pi\)
							0.124538	+	0.992215i	\(0.460255\pi\)
\(47\)	−3.92845	+	5.11965i	−0.573023	+	0.746778i	−0.986504	−	0.163734i	\(-0.947646\pi\)
							0.413481	+	0.910513i	\(0.364313\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.12	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.96	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.96	yes	4032
896.243	even	96	inner	896.2.bt.a.243.12	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.12	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.96	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.96	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.12	yes	4032	896.243	even	96	inner