Embedded newform 896.2.bt.a.59.70

• Label: 896.2.bt.a.59.70
• 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			59.70
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.70

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.223307 - 1.39647i) q^{2} +(-2.05738 - 2.19664i) q^{3} +(-1.90027 - 0.623684i) q^{4} +(-0.906651 - 2.00102i) q^{5} +(-3.52697 + 2.38254i) q^{6} +(-0.667156 - 2.56025i) q^{7} +(-1.29530 + 2.51440i) q^{8} +(-0.396213 + 6.04505i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.223307	−	1.39647i	0.157902	−	0.987455i
							\(3\)	−2.05738	−	2.19664i	−1.18783	−	1.26823i	−0.955299	−	0.295640i	\(-0.904467\pi\)
−0.232527	−	0.972590i	\(-0.574700\pi\)
\(5\)	−0.906651	−	2.00102i	−0.405466	−	0.894884i	−0.996124	−	0.0879611i	\(-0.971965\pi\)
							0.590657	−	0.806922i	\(-0.298868\pi\)
\(7\)	−0.667156	−	2.56025i	−0.252161	−	0.967685i
							\(11\)	−2.85743	−	4.59512i	−0.861548	−	1.38548i	−0.921951	−	0.387308i	\(-0.873405\pi\)
0.0604026	−	0.998174i	\(-0.480762\pi\)
\(13\)	0.268283	+	2.72392i	0.0744082	+	0.755480i	0.957913	+	0.287058i	\(0.0926772\pi\)
							−0.883505	+	0.468422i	\(0.844823\pi\)
\(17\)	0.579952	+	0.0763521i	0.140659	+	0.0185181i	0.200527	−	0.979688i	\(-0.435735\pi\)
							−0.0598679	+	0.998206i	\(0.519068\pi\)
\(19\)	0.0684435	−	0.414556i	0.0157020	−	0.0951058i	−0.977646	−	0.210259i	\(-0.932569\pi\)
							0.993348	+	0.115153i	\(0.0367359\pi\)
\(23\)	−3.84198	−	4.38094i	−0.801109	−	0.913490i	0.196645	−	0.980475i	\(-0.436995\pi\)
							−0.997754	+	0.0669852i	\(0.978662\pi\)
\(29\)	−4.07500	−	7.62379i	−0.756709	−	1.41570i	−0.904407	−	0.426670i	\(-0.859687\pi\)
							0.147699	−	0.989032i	\(-0.452813\pi\)
\(31\)	7.12321	−	1.90866i	1.27937	−	0.342805i	0.445755	−	0.895155i	\(-0.352935\pi\)
							0.833612	+	0.552350i	\(0.186269\pi\)
\(37\)	1.05134	−	2.79335i	0.172840	−	0.459224i	−0.820883	−	0.571097i	\(-0.806518\pi\)
							0.993723	+	0.111872i	\(0.0356848\pi\)
\(41\)	0.0934480	+	0.469795i	0.0145941	+	0.0733696i	0.987396	−	0.158269i	\(-0.0505913\pi\)
							−0.972802	+	0.231639i	\(0.925591\pi\)
\(43\)	7.19770	−	2.18340i	1.09764	−	0.332965i	0.311019	−	0.950404i	\(-0.399330\pi\)
							0.786619	+	0.617438i	\(0.211830\pi\)
\(47\)	−5.83222	+	7.60070i	−0.850717	+	1.10868i	0.142197	+	0.989838i	\(0.454583\pi\)
							−0.992914	+	0.118837i	\(0.962083\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.70	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.16	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.16	yes	4032
896.243	even	96	inner	896.2.bt.a.243.70	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.70	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.16	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.16	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.70	yes	4032	896.243	even	96	inner