Embedded newform 896.2.bt.a.59.19

• Label: 896.2.bt.a.59.19
• 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			59.19
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28681 + 0.586611i) q^{2} +(-1.14015 - 1.21732i) q^{3} +(1.31178 - 1.50972i) q^{4} +(-1.45265 - 3.20606i) q^{5} +(2.18125 + 0.897645i) q^{6} +(1.23437 + 2.34016i) q^{7} +(-0.802393 + 2.71223i) q^{8} +(0.0142699 - 0.217717i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28681	+	0.586611i	−0.909914	+	0.414797i
							\(3\)	−1.14015	−	1.21732i	−0.658265	−	0.702822i	0.310564	−	0.950552i	\(-0.399482\pi\)
−0.968829	+	0.247730i	\(0.920315\pi\)
\(5\)	−1.45265	−	3.20606i	−0.649644	−	1.43379i	−0.886722	−	0.462304i	\(-0.847023\pi\)
							0.237078	−	0.971491i	\(-0.423810\pi\)
\(7\)	1.23437	+	2.34016i	0.466547	+	0.884496i
							\(11\)	0.141805	+	0.228041i	0.0427558	+	0.0687569i	0.869884	−	0.493257i	\(-0.164194\pi\)
−0.827128	+	0.562013i	\(0.810027\pi\)
\(13\)	0.462877	+	4.69967i	0.128379	+	1.30345i	0.816283	+	0.577652i	\(0.196031\pi\)
							−0.687904	+	0.725802i	\(0.741469\pi\)
\(17\)	5.63810	+	0.742270i	1.36744	+	0.180027i	0.778243	−	0.627963i	\(-0.216111\pi\)
							0.589197	+	0.807990i	\(0.299444\pi\)
\(19\)	−1.00576	+	6.09181i	−0.230737	+	1.39756i	0.581176	+	0.813778i	\(0.302593\pi\)
							−0.811913	+	0.583778i	\(0.801574\pi\)
\(23\)	3.03603	+	3.46193i	0.633057	+	0.721863i	0.976067	−	0.217469i	\(-0.0697800\pi\)
							−0.343011	+	0.939332i	\(0.611447\pi\)
\(29\)	−1.06035	−	1.98378i	−0.196902	−	0.368378i	0.764022	−	0.645191i	\(-0.223222\pi\)
							−0.960924	+	0.276812i	\(0.910722\pi\)
\(31\)	−1.92956	+	0.517024i	−0.346559	+	0.0928603i	−0.427900	−	0.903826i	\(-0.640746\pi\)
							0.0813408	+	0.996686i	\(0.474080\pi\)
\(37\)	1.82642	−	4.85267i	0.300261	−	0.797774i	−0.696515	−	0.717542i	\(-0.745267\pi\)
							0.996776	−	0.0802320i	\(-0.0255661\pi\)
\(41\)	1.38829	+	6.97942i	0.216815	+	1.09000i	0.923830	+	0.382804i	\(0.125041\pi\)
							−0.707015	+	0.707199i	\(0.749959\pi\)
\(43\)	6.50639	−	1.97369i	0.992215	−	0.300985i	0.247879	−	0.968791i	\(-0.420266\pi\)
							0.744336	+	0.667806i	\(0.232766\pi\)
\(47\)	3.53650	−	4.60886i	0.515852	−	0.672271i	−0.460699	−	0.887556i	\(-0.652401\pi\)
							0.976551	+	0.215285i	\(0.0690681\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.19	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.104	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.104	yes	4032
896.243	even	96	inner	896.2.bt.a.243.19	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.19	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.104	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.104	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.19	yes	4032	896.243	even	96	inner