Embedded newform 896.2.bt.a.59.28

• Label: 896.2.bt.a.59.28
• 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			59.28
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08905 - 0.902198i) q^{2} +(0.240168 + 0.256425i) q^{3} +(0.372077 + 1.96508i) q^{4} +(-1.62052 - 3.57656i) q^{5} +(-0.0302101 - 0.495940i) q^{6} +(-2.00714 - 1.72377i) q^{7} +(1.36768 - 2.47577i) q^{8} +(0.188136 - 2.87041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08905	−	0.902198i	−0.770077	−	0.637950i
							\(3\)	0.240168	+	0.256425i	0.138661	+	0.148047i	0.795431	−	0.606044i	\(-0.207244\pi\)
−0.656770	+	0.754091i	\(0.728078\pi\)
\(5\)	−1.62052	−	3.57656i	−0.724718	−	1.59949i	−0.798344	−	0.602201i	\(-0.794291\pi\)
							0.0736263	−	0.997286i	\(-0.476543\pi\)
\(7\)	−2.00714	−	1.72377i	−0.758628	−	0.651524i
							\(11\)	−2.09737	−	3.37284i	−0.632381	−	1.01695i	−0.996335	−	0.0855340i	\(-0.972740\pi\)
0.363955	−	0.931417i	\(-0.381426\pi\)
\(13\)	0.370736	+	3.76414i	0.102824	+	1.04399i	0.898711	+	0.438541i	\(0.144505\pi\)
							−0.795888	+	0.605444i	\(0.792995\pi\)
\(17\)	−1.82830	−	0.240700i	−0.443427	−	0.0583783i	−0.0944928	−	0.995526i	\(-0.530123\pi\)
							−0.348934	+	0.937147i	\(0.613456\pi\)
\(19\)	0.203202	−	1.23078i	0.0466178	−	0.282360i	−0.953181	−	0.302399i	\(-0.902213\pi\)
							0.999799	+	0.0200394i	\(0.00637917\pi\)
\(23\)	2.48434	+	2.83284i	0.518020	+	0.590689i	0.950411	−	0.310998i	\(-0.100663\pi\)
							−0.432390	+	0.901686i	\(0.642330\pi\)
\(29\)	0.205316	+	0.384119i	0.0381262	+	0.0713292i	0.900263	−	0.435347i	\(-0.143374\pi\)
							−0.862137	+	0.506676i	\(0.830874\pi\)
\(31\)	4.53524	−	1.21521i	0.814553	−	0.218259i	0.172589	−	0.984994i	\(-0.444787\pi\)
							0.641964	+	0.766735i	\(0.278120\pi\)
\(37\)	−2.86369	+	7.60865i	−0.470788	+	1.25085i	0.460854	+	0.887476i	\(0.347543\pi\)
							−0.931642	+	0.363378i	\(0.881623\pi\)
\(41\)	−0.563012	−	2.83045i	−0.0879276	−	0.442042i	−0.999521	−	0.0309489i	\(-0.990147\pi\)
							0.911593	−	0.411093i	\(-0.134853\pi\)
\(43\)	8.05464	−	2.44335i	1.22832	−	0.372607i	0.391549	−	0.920157i	\(-0.371939\pi\)
							0.836773	+	0.547550i	\(0.184439\pi\)
\(47\)	−1.06111	+	1.38287i	−0.154779	+	0.201712i	−0.864306	−	0.502966i	\(-0.832242\pi\)
							0.709527	+	0.704678i	\(0.248909\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.28	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.57	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.57	yes	4032
896.243	even	96	inner	896.2.bt.a.243.28	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.28	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.57	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.57	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.28	yes	4032	896.243	even	96	inner