Embedded newform 896.2.bt.a.59.89

• Label: 896.2.bt.a.59.89
• 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			59.89
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.89

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.837509 - 1.13955i) q^{2} +(0.318559 + 0.340122i) q^{3} +(-0.597157 - 1.90877i) q^{4} +(1.12874 + 2.49117i) q^{5} +(0.654383 - 0.0781592i) q^{6} +(-1.93051 - 1.80918i) q^{7} +(-2.67527 - 0.918122i) q^{8} +(0.182006 - 2.77688i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.837509	−	1.13955i	0.592208	−	0.805785i
							\(3\)	0.318559	+	0.340122i	0.183920	+	0.196370i	0.815190	−	0.579194i	\(-0.196633\pi\)
−0.631270	+	0.775563i	\(0.717466\pi\)
\(5\)	1.12874	+	2.49117i	0.504786	+	1.11409i	0.973070	+	0.230509i	\(0.0740390\pi\)
							−0.468284	+	0.883578i	\(0.655128\pi\)
\(7\)	−1.93051	−	1.80918i	−0.729664	−	0.683806i
							\(11\)	−2.46306	−	3.96093i	−0.742641	−	1.19426i	−0.975123	−	0.221663i	\(-0.928851\pi\)
0.232482	−	0.972601i	\(-0.425315\pi\)
\(13\)	−0.210323	−	2.13544i	−0.0583330	−	0.592265i	−0.979271	−	0.202555i	\(-0.935075\pi\)
							0.920938	−	0.389710i	\(-0.127425\pi\)
\(17\)	7.16447	+	0.943220i	1.73764	+	0.228764i	0.931698	−	0.363233i	\(-0.118327\pi\)
							0.805940	+	0.591997i	\(0.201660\pi\)
\(19\)	−0.478581	+	2.89872i	−0.109794	+	0.665012i	0.874030	+	0.485872i	\(0.161498\pi\)
							−0.983824	+	0.179140i	\(0.942669\pi\)
\(23\)	−2.89729	−	3.30373i	−0.604127	−	0.688875i	0.366237	−	0.930521i	\(-0.380646\pi\)
							−0.970364	+	0.241647i	\(0.922312\pi\)
\(29\)	−1.37740	−	2.57693i	−0.255776	−	0.478523i	0.720958	−	0.692979i	\(-0.243702\pi\)
							−0.976734	+	0.214456i	\(0.931202\pi\)
\(31\)	−5.03240	+	1.34843i	−0.903846	+	0.242185i	−0.680667	−	0.732593i	\(-0.738310\pi\)
							−0.223179	+	0.974778i	\(0.571643\pi\)
\(37\)	4.09872	−	10.8900i	0.673825	−	1.79031i	0.0637496	−	0.997966i	\(-0.479694\pi\)
							0.610075	−	0.792344i	\(-0.291139\pi\)
\(41\)	−0.254893	−	1.28143i	−0.0398076	−	0.200126i	0.955764	−	0.294136i	\(-0.0950317\pi\)
							−0.995571	+	0.0940091i	\(0.970032\pi\)
\(43\)	4.80613	−	1.45792i	0.732928	−	0.222331i	0.0983087	−	0.995156i	\(-0.468657\pi\)
							0.634620	+	0.772825i	\(0.281157\pi\)
\(47\)	−2.01167	+	2.62165i	−0.293432	+	0.382407i	−0.916684	−	0.399613i	\(-0.869144\pi\)
							0.623252	+	0.782021i	\(0.285811\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.89	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.5	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.5	yes	4032
896.243	even	96	inner	896.2.bt.a.243.89	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.89	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.5	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.5	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.89	yes	4032	896.243	even	96	inner