Embedded newform 896.2.bt.a.59.61

• Label: 896.2.bt.a.59.61
• 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			59.61
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.61

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.117481 + 1.40933i) q^{2} +(-2.14119 - 2.28612i) q^{3} +(-1.97240 - 0.331138i) q^{4} +(0.173797 + 0.383578i) q^{5} +(3.47344 - 2.74905i) q^{6} +(-2.08094 + 1.63392i) q^{7} +(0.698401 - 2.74085i) q^{8} +(-0.445462 + 6.79644i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.117481	+	1.40933i	−0.0830717	+	0.996544i
							\(3\)	−2.14119	−	2.28612i	−1.23621	−	1.31989i	−0.929832	−	0.367984i	\(-0.880048\pi\)
−0.306382	−	0.951909i	\(-0.599119\pi\)
\(5\)	0.173797	+	0.383578i	0.0777243	+	0.171541i	0.945679	−	0.325103i	\(-0.105399\pi\)
							−0.867954	+	0.496644i	\(0.834566\pi\)
\(7\)	−2.08094	+	1.63392i	−0.786522	+	0.617563i
							\(11\)	−1.02033	−	1.64083i	−0.307641	−	0.494727i	0.658556	−	0.752532i	\(-0.271168\pi\)
−0.966197	+	0.257804i	\(0.917001\pi\)
\(13\)	0.154480	+	1.56846i	0.0428450	+	0.435013i	0.992547	+	0.121859i	\(0.0388855\pi\)
							−0.949702	+	0.313154i	\(0.898614\pi\)
\(17\)	0.0405464	+	0.00533804i	0.00983395	+	0.00129466i	0.135442	−	0.990785i	\(-0.456755\pi\)
							−0.125608	+	0.992080i	\(0.540088\pi\)
\(19\)	−1.03939	+	6.29550i	−0.238453	+	1.44429i	0.553063	+	0.833139i	\(0.313459\pi\)
							−0.791516	+	0.611148i	\(0.790708\pi\)
\(23\)	−5.40270	−	6.16060i	−1.12654	−	1.28457i	−0.954078	−	0.299559i	\(-0.903160\pi\)
							−0.172464	−	0.985016i	\(-0.555173\pi\)
\(29\)	3.61705	+	6.76703i	0.671670	+	1.25661i	0.954721	+	0.297501i	\(0.0961533\pi\)
							−0.283052	+	0.959105i	\(0.591347\pi\)
\(31\)	−4.31568	+	1.15638i	−0.775119	+	0.207693i	−0.624632	−	0.780919i	\(-0.714751\pi\)
							−0.150488	+	0.988612i	\(0.548084\pi\)
\(37\)	4.11191	−	10.9251i	0.675994	−	1.79607i	0.0744617	−	0.997224i	\(-0.476276\pi\)
							0.601532	−	0.798849i	\(-0.294557\pi\)
\(41\)	−1.12769	−	5.66928i	−0.176116	−	0.885394i	−0.963249	−	0.268610i	\(-0.913436\pi\)
							0.787133	−	0.616783i	\(-0.211564\pi\)
\(43\)	9.81769	−	2.97817i	1.49718	−	0.454166i	0.567765	−	0.823191i	\(-0.307808\pi\)
							0.929420	+	0.369025i	\(0.120308\pi\)
\(47\)	5.35236	−	6.97534i	0.780722	−	1.01746i	−0.218412	−	0.975857i	\(-0.570088\pi\)
							0.999134	−	0.0416005i	\(-0.0132457\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.61	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.108	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.108	yes	4032
896.243	even	96	inner	896.2.bt.a.243.61	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.61	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.108	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.108	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.61	yes	4032	896.243	even	96	inner