Embedded newform 896.2.bt.a.59.121

• Label: 896.2.bt.a.59.121
• 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			59.121
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.121

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40728 - 0.139875i) q^{2} +(-0.883425 - 0.943223i) q^{3} +(1.96087 - 0.393687i) q^{4} +(-1.64535 - 3.63137i) q^{5} +(-1.37516 - 1.20381i) q^{6} +(-0.408569 - 2.61401i) q^{7} +(2.70442 - 0.828305i) q^{8} +(0.0869790 - 1.32704i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40728	−	0.139875i	0.995097	−	0.0989068i
							\(3\)	−0.883425	−	0.943223i	−0.510046	−	0.544570i	0.422150	−	0.906526i	\(-0.361276\pi\)
−0.932195	+	0.361956i	\(0.882109\pi\)
\(5\)	−1.64535	−	3.63137i	−0.735824	−	1.62400i	−0.780623	−	0.625002i	\(-0.785098\pi\)
							0.0447993	−	0.998996i	\(-0.485735\pi\)
\(7\)	−0.408569	−	2.61401i	−0.154425	−	0.988005i
							\(11\)	2.82700	+	4.54618i	0.852372	+	1.37073i	0.927606	+	0.373559i	\(0.121863\pi\)
−0.0752345	+	0.997166i	\(0.523971\pi\)
\(13\)	0.402488	+	4.08653i	0.111630	+	1.13340i	0.873888	+	0.486127i	\(0.161591\pi\)
							−0.762258	+	0.647273i	\(0.775909\pi\)
\(17\)	−2.97594	−	0.391789i	−0.721770	−	0.0950229i	−0.239313	−	0.970942i	\(-0.576922\pi\)
							−0.482457	+	0.875920i	\(0.660256\pi\)
\(19\)	1.07068	−	6.48504i	0.245632	−	1.48777i	−0.525335	−	0.850895i	\(-0.676060\pi\)
							0.770967	−	0.636875i	\(-0.219773\pi\)
\(23\)	−0.168676	−	0.192338i	−0.0351713	−	0.0401052i	0.733987	−	0.679164i	\(-0.237657\pi\)
							−0.769158	+	0.639059i	\(0.779324\pi\)
\(29\)	−2.80403	−	5.24597i	−0.520696	−	0.974153i	−0.995563	−	0.0940957i	\(-0.970004\pi\)
							0.474868	−	0.880057i	\(-0.342496\pi\)
\(31\)	−3.32957	+	0.892156i	−0.598009	+	0.160236i	−0.545111	−	0.838364i	\(-0.683512\pi\)
							−0.0528979	+	0.998600i	\(0.516846\pi\)
\(37\)	1.08007	−	2.86968i	0.177563	−	0.471773i	−0.816910	−	0.576765i	\(-0.804315\pi\)
							0.994473	+	0.104992i	\(0.0334816\pi\)
\(41\)	2.14988	+	10.8082i	0.335754	+	1.68795i	0.667522	+	0.744590i	\(0.267355\pi\)
							−0.331768	+	0.943361i	\(0.607645\pi\)
\(43\)	3.40535	−	1.03300i	0.519310	−	0.157531i	−0.0197391	−	0.999805i	\(-0.506284\pi\)
							0.539050	+	0.842274i	\(0.318784\pi\)
\(47\)	−0.282378	+	0.368002i	−0.0411890	+	0.0536786i	−0.813494	−	0.581574i	\(-0.802437\pi\)
							0.772305	+	0.635252i	\(0.219104\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.121	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.39	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.39	yes	4032
896.243	even	96	inner	896.2.bt.a.243.121	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.121	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.39	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.39	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.121	yes	4032	896.243	even	96	inner