Embedded newform 896.2.bt.a.59.45

• Label: 896.2.bt.a.59.45
• 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			59.45
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.45

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.609118 + 1.27631i) q^{2} +(-1.02363 - 1.09291i) q^{3} +(-1.25795 - 1.55485i) q^{4} +(-1.53360 - 3.38472i) q^{5} +(2.01841 - 0.640753i) q^{6} +(-2.64350 + 0.109222i) q^{7} +(2.75072 - 0.658449i) q^{8} +(0.0495577 - 0.756104i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.609118	+	1.27631i	−0.430712	+	0.902490i
							\(3\)	−1.02363	−	1.09291i	−0.590991	−	0.630995i	0.362656	−	0.931923i	\(-0.381870\pi\)
−0.953647	+	0.300929i	\(0.902703\pi\)
\(5\)	−1.53360	−	3.38472i	−0.685845	−	1.51369i	−0.849733	−	0.527214i	\(-0.823237\pi\)
							0.163887	−	0.986479i	\(-0.447597\pi\)
\(7\)	−2.64350	+	0.109222i	−0.999148	+	0.0412820i
							\(11\)	−1.25699	−	2.02140i	−0.378996	−	0.609475i	0.603143	−	0.797633i	\(-0.293915\pi\)
−0.982139	+	0.188158i	\(0.939748\pi\)
\(13\)	−0.485387	−	4.92822i	−0.134622	−	1.36684i	−0.790725	−	0.612171i	\(-0.790296\pi\)
							0.656103	−	0.754671i	\(-0.272204\pi\)
\(17\)	6.92932	+	0.912263i	1.68061	+	0.221256i	0.909575	−	0.415539i	\(-0.136407\pi\)
							0.771033	+	0.636795i	\(0.219740\pi\)
\(19\)	0.701942	−	4.25160i	0.161037	−	0.975384i	−0.777166	−	0.629296i	\(-0.783343\pi\)
							0.938202	−	0.346088i	\(-0.112490\pi\)
\(23\)	−3.72016	−	4.24203i	−0.775707	−	0.884525i	0.220149	−	0.975466i	\(-0.429346\pi\)
							−0.995856	+	0.0909415i	\(0.971012\pi\)
\(29\)	−1.64810	−	3.08337i	−0.306044	−	0.572568i	0.681140	−	0.732153i	\(-0.261484\pi\)
							−0.987184	+	0.159585i	\(0.948984\pi\)
\(31\)	3.46023	−	0.927166i	0.621475	−	0.166524i	0.0656770	−	0.997841i	\(-0.479079\pi\)
							0.555798	+	0.831317i	\(0.312413\pi\)
\(37\)	0.784341	−	2.08394i	0.128945	−	0.342598i	−0.855624	−	0.517599i	\(-0.826826\pi\)
							0.984568	+	0.175000i	\(0.0559927\pi\)
\(41\)	1.55306	+	7.80776i	0.242547	+	1.21937i	0.889535	+	0.456867i	\(0.151029\pi\)
							−0.646988	+	0.762501i	\(0.723971\pi\)
\(43\)	−6.31223	+	1.91479i	−0.962606	+	0.292003i	−0.732199	−	0.681091i	\(-0.761506\pi\)
							−0.230407	+	0.973094i	\(0.574006\pi\)
\(47\)	−0.720660	+	0.939182i	−0.105119	+	0.136994i	−0.842950	−	0.537991i	\(-0.819183\pi\)
							0.737831	+	0.674985i	\(0.235850\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.45	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.124	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.124	yes	4032
896.243	even	96	inner	896.2.bt.a.243.45	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.45	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.124	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.124	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.45	yes	4032	896.243	even	96	inner