Embedded newform 896.2.bt.a.59.113

• Label: 896.2.bt.a.59.113
• 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			59.113
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.113

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.32831 - 0.485370i) q^{2} +(1.78779 + 1.90880i) q^{3} +(1.52883 - 1.28945i) q^{4} +(-1.07735 - 2.37777i) q^{5} +(3.30122 + 1.66775i) q^{6} +(-0.653719 - 2.56372i) q^{7} +(1.40491 - 2.45484i) q^{8} +(-0.251129 + 3.83149i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.32831	−	0.485370i	0.939259	−	0.343208i
							\(3\)	1.78779	+	1.90880i	1.03218	+	1.10205i	0.994707	+	0.102750i	\(0.0327642\pi\)
0.0374730	+	0.999298i	\(0.488069\pi\)
\(5\)	−1.07735	−	2.37777i	−0.481807	−	1.06337i	−0.980487	−	0.196584i	\(-0.937015\pi\)
							0.498680	−	0.866786i	\(-0.333818\pi\)
\(7\)	−0.653719	−	2.56372i	−0.247083	−	0.968994i
							\(11\)	0.100553	+	0.161702i	0.0303178	+	0.0487549i	0.864011	−	0.503473i	\(-0.167945\pi\)
−0.833693	+	0.552228i	\(0.813778\pi\)
\(13\)	−0.00670542	−	0.0680813i	−0.00185975	−	0.0188824i	0.994210	−	0.107454i	\(-0.0342698\pi\)
							−0.996070	+	0.0885716i	\(0.971770\pi\)
\(17\)	4.34152	+	0.571571i	1.05297	+	0.138626i	0.637088	−	0.770791i	\(-0.280139\pi\)
							0.415884	+	0.909418i	\(0.363472\pi\)
\(19\)	−0.664261	+	4.02337i	−0.152392	+	0.923024i	0.795855	+	0.605488i	\(0.207022\pi\)
							−0.948246	+	0.317536i	\(0.897145\pi\)
\(23\)	−0.179439	−	0.204611i	−0.0374157	−	0.0426644i	0.732829	−	0.680413i	\(-0.238199\pi\)
							−0.770245	+	0.637748i	\(0.779866\pi\)
\(29\)	1.40481	+	2.62821i	0.260867	+	0.488047i	0.977922	−	0.208969i	\(-0.0670107\pi\)
							−0.717056	+	0.697016i	\(0.754511\pi\)
\(31\)	3.86760	−	1.03632i	0.694642	−	0.186129i	0.105812	−	0.994386i	\(-0.466256\pi\)
							0.588829	+	0.808257i	\(0.299589\pi\)
\(37\)	−0.129333	+	0.343630i	−0.0212623	+	0.0564925i	−0.946111	−	0.323843i	\(-0.895025\pi\)
							0.924848	+	0.380336i	\(0.124192\pi\)
\(41\)	0.119385	+	0.600189i	0.0186448	+	0.0937338i	0.988987	−	0.148001i	\(-0.0472837\pi\)
							−0.970342	+	0.241734i	\(0.922284\pi\)
\(43\)	−6.32818	+	1.91963i	−0.965039	+	0.292741i	−0.733200	−	0.680013i	\(-0.761974\pi\)
							−0.231839	+	0.972754i	\(0.574474\pi\)
\(47\)	−6.22028	+	8.10642i	−0.907321	+	1.18244i	0.0753635	+	0.997156i	\(0.475988\pi\)
							−0.982684	+	0.185287i	\(0.940678\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.113	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.29	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.29	yes	4032
896.243	even	96	inner	896.2.bt.a.243.113	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.113	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.29	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.29	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.113	yes	4032	896.243	even	96	inner