Embedded newform 896.2.bt.a.59.96

• Label: 896.2.bt.a.59.96
• 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			59.96
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.96

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.967544 - 1.03144i) q^{2} +(-1.33933 - 1.42999i) q^{3} +(-0.127718 - 1.99592i) q^{4} +(-0.237330 - 0.523798i) q^{5} +(-2.77081 - 0.00214356i) q^{6} +(-0.0581699 - 2.64511i) q^{7} +(-2.18223 - 1.79940i) q^{8} +(-0.0548529 + 0.836894i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.967544	−	1.03144i	0.684157	−	0.729335i
							\(3\)	−1.33933	−	1.42999i	−0.773264	−	0.825606i	0.215277	−	0.976553i	\(-0.430934\pi\)
−0.988542	+	0.150947i	\(0.951768\pi\)
\(5\)	−0.237330	−	0.523798i	−0.106137	−	0.234250i	0.850234	−	0.526406i	\(-0.176461\pi\)
							−0.956371	+	0.292156i	\(0.905627\pi\)
\(7\)	−0.0581699	−	2.64511i	−0.0219862	−	0.999758i
							\(11\)	0.946359	+	1.52187i	0.285338	+	0.458861i	0.960291	−	0.279000i	\(-0.0900029\pi\)
−0.674953	+	0.737861i	\(0.735836\pi\)
\(13\)	−0.344136	−	3.49407i	−0.0954462	−	0.969081i	−0.916983	−	0.398926i	\(-0.869383\pi\)
							0.821537	−	0.570155i	\(-0.193117\pi\)
\(17\)	−2.14037	−	0.281785i	−0.519116	−	0.0683430i	−0.133584	−	0.991037i	\(-0.542649\pi\)
							−0.385532	+	0.922694i	\(0.625982\pi\)
\(19\)	−0.205585	+	1.24521i	−0.0471644	+	0.285671i	−0.999832	−	0.0183441i	\(-0.994161\pi\)
							0.952667	+	0.304015i	\(0.0983272\pi\)
\(23\)	4.70340	+	5.36320i	0.980726	+	1.11830i	0.992968	+	0.118379i	\(0.0377698\pi\)
							−0.0122422	+	0.999925i	\(0.503897\pi\)
\(29\)	0.167349	+	0.313089i	0.0310760	+	0.0581391i	0.896980	−	0.442071i	\(-0.145756\pi\)
							−0.865904	+	0.500210i	\(0.833256\pi\)
\(31\)	3.62869	−	0.972304i	0.651732	−	0.174631i	0.0822200	−	0.996614i	\(-0.473799\pi\)
							0.569512	+	0.821983i	\(0.307132\pi\)
\(37\)	1.55064	−	4.11996i	0.254924	−	0.677317i	−0.745023	−	0.667038i	\(-0.767562\pi\)
							0.999947	−	0.0102781i	\(-0.00327168\pi\)
\(41\)	−1.08763	−	5.46788i	−0.169859	−	0.853939i	−0.967900	−	0.251336i	\(-0.919130\pi\)
							0.798041	−	0.602603i	\(-0.205870\pi\)
\(43\)	−7.14488	+	2.16738i	−1.08958	+	0.330522i	−0.783442	−	0.621464i	\(-0.786538\pi\)
							−0.306141	+	0.951986i	\(0.599038\pi\)
\(47\)	−2.60657	+	3.39695i	−0.380208	+	0.495497i	−0.943782	−	0.330569i	\(-0.892759\pi\)
							0.563574	+	0.826066i	\(0.309426\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.96	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.12	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.12	yes	4032
896.243	even	96	inner	896.2.bt.a.243.96	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.96	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.12	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.12	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.96	yes	4032	896.243	even	96	inner