Embedded newform 896.2.bt.a.59.37

• Label: 896.2.bt.a.59.37
• 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			59.37
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.37

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.848709 + 1.13124i) q^{2} +(-2.14450 - 2.28966i) q^{3} +(-0.559386 - 1.92018i) q^{4} +(-0.811370 - 1.79073i) q^{5} +(4.41020 - 0.482679i) q^{6} +(0.0143955 - 2.64571i) q^{7} +(2.64693 + 0.996877i) q^{8} +(-0.447450 + 6.82676i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.848709	+	1.13124i	−0.600128	+	0.799904i
							\(3\)	−2.14450	−	2.28966i	−1.23813	−	1.32193i	−0.928617	−	0.371039i	\(-0.879002\pi\)
−0.309510	−	0.950896i	\(-0.600165\pi\)
\(5\)	−0.811370	−	1.79073i	−0.362856	−	0.800839i	−0.999558	−	0.0297297i	\(-0.990535\pi\)
							0.636702	−	0.771110i	\(-0.280298\pi\)
\(7\)	0.0143955	−	2.64571i	0.00544099	−	0.999985i
							\(11\)	2.17090	+	3.49109i	0.654551	+	1.05260i	0.993722	+	0.111878i	\(0.0356867\pi\)
−0.339171	+	0.940725i	\(0.610147\pi\)
\(13\)	0.461008	+	4.68070i	0.127861	+	1.29819i	0.818296	+	0.574797i	\(0.194919\pi\)
							−0.690435	+	0.723394i	\(0.742581\pi\)
\(17\)	2.40611	+	0.316770i	0.583567	+	0.0768280i	0.416529	−	0.909122i	\(-0.363246\pi\)
							0.167038	+	0.985950i	\(0.446580\pi\)
\(19\)	0.0519758	−	0.314813i	0.0119241	−	0.0722231i	−0.980020	−	0.198898i	\(-0.936264\pi\)
							0.991944	+	0.126675i	\(0.0404305\pi\)
\(23\)	1.92052	+	2.18993i	0.400456	+	0.456632i	0.916562	−	0.399892i	\(-0.130952\pi\)
							−0.516107	+	0.856524i	\(0.672619\pi\)
\(29\)	2.62732	+	4.91537i	0.487881	+	0.912760i	0.998594	+	0.0530094i	\(0.0168813\pi\)
							−0.510713	+	0.859751i	\(0.670619\pi\)
\(31\)	6.35572	−	1.70301i	1.14152	−	0.305870i	0.361958	−	0.932194i	\(-0.382108\pi\)
							0.779562	+	0.626325i	\(0.215442\pi\)
\(37\)	−3.01003	+	7.99747i	−0.494847	+	1.31478i	0.418731	+	0.908110i	\(0.362475\pi\)
							−0.913577	+	0.406665i	\(0.866692\pi\)
\(41\)	0.201838	+	1.01471i	0.0315218	+	0.158471i	0.993340	−	0.115217i	\(-0.0367564\pi\)
							−0.961819	+	0.273688i	\(0.911756\pi\)
\(43\)	0.267291	−	0.0810819i	0.0407615	−	0.0123649i	−0.269838	−	0.962906i	\(-0.586970\pi\)
							0.310600	+	0.950541i	\(0.399470\pi\)
\(47\)	4.73812	−	6.17484i	0.691126	−	0.900693i	−0.307581	−	0.951522i	\(-0.599520\pi\)
							0.998707	+	0.0508286i	\(0.0161862\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.37	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.123	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.123	yes	4032
896.243	even	96	inner	896.2.bt.a.243.37	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.37	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.123	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.123	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.37	yes	4032	896.243	even	96	inner