Embedded newform 896.2.bt.a.59.118

• Label: 896.2.bt.a.59.118
• 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			59.118
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.118

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38660 - 0.278097i) q^{2} +(-2.14037 - 2.28525i) q^{3} +(1.84532 - 0.771220i) q^{4} +(-1.12881 - 2.49135i) q^{5} +(-3.60336 - 2.57350i) q^{6} +(2.40444 - 1.10393i) q^{7} +(2.34425 - 1.58255i) q^{8} +(-0.444974 + 6.78900i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38660	−	0.278097i	0.980475	−	0.196644i
							\(3\)	−2.14037	−	2.28525i	−1.23574	−	1.31939i	−0.930128	−	0.367236i	\(-0.880304\pi\)
−0.305616	−	0.952155i	\(-0.598862\pi\)
\(5\)	−1.12881	−	2.49135i	−0.504821	−	1.11416i	−0.973058	−	0.230561i	\(-0.925944\pi\)
							0.468236	−	0.883603i	\(-0.344890\pi\)
\(7\)	2.40444	−	1.10393i	0.908794	−	0.417246i
							\(11\)	−1.32528	−	2.13122i	−0.399586	−	0.642586i	0.586283	−	0.810106i	\(-0.300591\pi\)
−0.985869	+	0.167520i	\(0.946424\pi\)
\(13\)	−0.576747	−	5.85581i	−0.159961	−	1.62411i	−0.655272	−	0.755393i	\(-0.727446\pi\)
							0.495311	−	0.868716i	\(-0.335054\pi\)
\(17\)	5.78851	+	0.762072i	1.40392	+	0.184830i	0.794137	−	0.607739i	\(-0.207923\pi\)
							0.609783	+	0.792568i	\(0.291257\pi\)
\(19\)	−0.615732	+	3.72943i	−0.141259	+	0.855591i	0.818447	+	0.574583i	\(0.194836\pi\)
							−0.959705	+	0.281008i	\(0.909331\pi\)
\(23\)	−0.178155	−	0.203147i	−0.0371478	−	0.0423590i	0.732968	−	0.680264i	\(-0.238135\pi\)
							−0.770115	+	0.637905i	\(0.779801\pi\)
\(29\)	2.96432	+	5.54586i	0.550461	+	1.02984i	0.991221	+	0.132215i	\(0.0422090\pi\)
							−0.440760	+	0.897625i	\(0.645291\pi\)
\(31\)	−9.11731	+	2.44298i	−1.63752	+	0.438772i	−0.956080	−	0.293105i	\(-0.905312\pi\)
							−0.681438	+	0.731876i	\(0.738645\pi\)
\(37\)	−3.07849	+	8.17934i	−0.506100	+	1.34468i	0.398012	+	0.917380i	\(0.369700\pi\)
							−0.904112	+	0.427295i	\(0.859467\pi\)
\(41\)	1.44910	+	7.28510i	0.226311	+	1.13774i	0.912108	+	0.409949i	\(0.134454\pi\)
							−0.685797	+	0.727793i	\(0.740546\pi\)
\(43\)	3.93333	−	1.19316i	0.599828	−	0.181956i	0.0242542	−	0.999706i	\(-0.492279\pi\)
							0.575573	+	0.817750i	\(0.304779\pi\)
\(47\)	2.38950	−	3.11406i	0.348545	−	0.454233i	−0.585877	−	0.810400i	\(-0.699250\pi\)
							0.934422	+	0.356167i	\(0.115917\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.118	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.33	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.33	yes	4032
896.243	even	96	inner	896.2.bt.a.243.118	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.118	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.33	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.33	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.118	yes	4032	896.243	even	96	inner