Embedded newform 896.2.bt.a.59.99

• Label: 896.2.bt.a.59.99
• 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			59.99
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.99

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.09538 - 0.894509i) q^{2} +(1.24638 + 1.33074i) q^{3} +(0.399708 - 1.95965i) q^{4} +(-1.36081 - 3.00338i) q^{5} +(2.55562 + 0.342772i) q^{6} +(-0.257072 + 2.63323i) q^{7} +(-1.31509 - 2.50410i) q^{8} +(-0.0212123 + 0.323637i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.09538	−	0.894509i	0.774550	−	0.632513i
							\(3\)	1.24638	+	1.33074i	0.719596	+	0.768305i	0.980385	−	0.197091i	\(-0.0631494\pi\)
−0.260789	+	0.965396i	\(0.583983\pi\)
\(5\)	−1.36081	−	3.00338i	−0.608574	−	1.34315i	−0.919640	−	0.392762i	\(-0.871519\pi\)
							0.311066	−	0.950388i	\(-0.399314\pi\)
\(7\)	−0.257072	+	2.63323i	−0.0971641	+	0.995268i
							\(11\)	−1.13779	−	1.82972i	−0.343057	−	0.551681i	0.631625	−	0.775274i	\(-0.282388\pi\)
−0.974683	+	0.223593i	\(0.928221\pi\)
\(13\)	−0.322782	−	3.27726i	−0.0895237	−	0.908949i	−0.930112	−	0.367275i	\(-0.880291\pi\)
							0.840589	−	0.541674i	\(-0.182209\pi\)
\(17\)	2.68712	+	0.353766i	0.651722	+	0.0858009i	0.449136	−	0.893464i	\(-0.351732\pi\)
							0.202586	+	0.979264i	\(0.435065\pi\)
\(19\)	0.843338	−	5.10802i	0.193475	−	1.17186i	−0.697083	−	0.716991i	\(-0.745519\pi\)
							0.890558	−	0.454870i	\(-0.150314\pi\)
\(23\)	5.30872	+	6.05343i	1.10694	+	1.26223i	0.962078	+	0.272776i	\(0.0879418\pi\)
							0.144866	+	0.989451i	\(0.453725\pi\)
\(29\)	−3.59358	−	6.72312i	−0.667312	−	1.24845i	−0.956654	−	0.291227i	\(-0.905937\pi\)
							0.289343	−	0.957226i	\(-0.406563\pi\)
\(31\)	−4.03382	+	1.08086i	−0.724496	+	0.194128i	−0.602177	−	0.798363i	\(-0.705700\pi\)
							−0.122319	+	0.992491i	\(0.539033\pi\)
\(37\)	0.394460	−	1.04805i	0.0648488	−	0.172299i	−0.899508	−	0.436904i	\(-0.856075\pi\)
							0.964357	+	0.264605i	\(0.0852416\pi\)
\(41\)	1.50409	+	7.56157i	0.234899	+	1.18092i	0.900583	+	0.434685i	\(0.143140\pi\)
							−0.665683	+	0.746234i	\(0.731860\pi\)
\(43\)	9.37721	−	2.84454i	1.43001	−	0.433789i	0.522075	−	0.852899i	\(-0.325158\pi\)
							0.907935	+	0.419111i	\(0.137658\pi\)
\(47\)	−0.676815	+	0.882042i	−0.0987236	+	0.128659i	−0.840103	−	0.542427i	\(-0.817505\pi\)
							0.741379	+	0.671086i	\(0.234172\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.99	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.15	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.15	yes	4032
896.243	even	96	inner	896.2.bt.a.243.99	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.99	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.15	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.15	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.99	yes	4032	896.243	even	96	inner