Embedded newform 896.2.bt.a.59.94

• Label: 896.2.bt.a.59.94
• 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			59.94
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.94

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.947996 - 1.04943i) q^{2} +(0.853150 + 0.910898i) q^{3} +(-0.202607 - 1.98971i) q^{4} +(-0.393787 - 0.869106i) q^{5} +(1.76471 - 0.0317925i) q^{6} +(-2.56041 - 0.666576i) q^{7} +(-2.28013 - 1.67362i) q^{8} +(0.0943375 - 1.43931i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.947996	−	1.04943i	0.670334	−	0.742059i
							\(3\)	0.853150	+	0.910898i	0.492566	+	0.525907i	0.927133	−	0.374732i	\(-0.122265\pi\)
−0.434567	+	0.900639i	\(0.643099\pi\)
\(5\)	−0.393787	−	0.869106i	−0.176107	−	0.388676i	0.801837	−	0.597542i	\(-0.203856\pi\)
							−0.977944	+	0.208866i	\(0.933023\pi\)
\(7\)	−2.56041	−	0.666576i	−0.967742	−	0.251942i
							\(11\)	0.934764	+	1.50322i	0.281842	+	0.453239i	0.959327	−	0.282298i	\(-0.0910967\pi\)
−0.677485	+	0.735537i	\(0.736930\pi\)
\(13\)	0.0202423	+	0.205524i	0.00561421	+	0.0570020i	0.997584	−	0.0694759i	\(-0.0221327\pi\)
							−0.991969	+	0.126478i	\(0.959633\pi\)
\(17\)	−3.88190	−	0.511062i	−0.941499	−	0.123951i	−0.355875	−	0.934534i	\(-0.615817\pi\)
							−0.585624	+	0.810583i	\(0.699150\pi\)
\(19\)	1.30794	−	7.92206i	0.300061	−	1.81745i	−0.234272	−	0.972171i	\(-0.575271\pi\)
							0.534333	−	0.845274i	\(-0.320563\pi\)
\(23\)	−5.43699	−	6.19969i	−1.13369	−	1.29273i	−0.950874	−	0.309578i	\(-0.899812\pi\)
							−0.182816	−	0.983147i	\(-0.558521\pi\)
\(29\)	3.72291	+	6.96507i	0.691326	+	1.29338i	0.945310	+	0.326172i	\(0.105759\pi\)
							−0.253984	+	0.967208i	\(0.581741\pi\)
\(31\)	4.12243	−	1.10460i	0.740409	−	0.198392i	0.131149	−	0.991363i	\(-0.458133\pi\)
							0.609260	+	0.792971i	\(0.291467\pi\)
\(37\)	−3.22325	+	8.56397i	−0.529899	+	1.40791i	0.351790	+	0.936079i	\(0.385573\pi\)
							−0.881690	+	0.471830i	\(0.843594\pi\)
\(41\)	−0.686167	−	3.44960i	−0.107161	−	0.538736i	−0.996650	−	0.0817799i	\(-0.973940\pi\)
							0.889489	−	0.456956i	\(-0.151060\pi\)
\(43\)	−2.17244	+	0.659003i	−0.331294	+	0.100497i	−0.451550	−	0.892246i	\(-0.649129\pi\)
							0.120255	+	0.992743i	\(0.461629\pi\)
\(47\)	−2.33081	+	3.03757i	−0.339984	+	0.443075i	−0.931786	−	0.363008i	\(-0.881750\pi\)
							0.591802	+	0.806083i	\(0.298417\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.94	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.11	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.11	yes	4032
896.243	even	96	inner	896.2.bt.a.243.94	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.94	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.11	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.11	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.94	yes	4032	896.243	even	96	inner