Embedded newform 672.2.bs.a.659.40

• Label: 672.2.bs.a.659.40
• Level: $672$
• Weight: $2$
• Character: 672.659
• Analytic conductor: $5.366$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.bs (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.36594701583\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(48\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			659.40
Character	\(\chi\)	\(=\)	672.659
Dual form			672.2.bs.a.155.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.21431 - 0.724880i) q^{2} +(-0.132387 - 1.72698i) q^{3} +(0.949098 - 1.76046i) q^{4} +(-0.733476 - 1.77077i) q^{5} +(-1.41262 - 2.00113i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(-0.123621 - 2.82572i) q^{8} +(-2.96495 + 0.457262i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 16 q^{4}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(421\)	\(449\)	\(577\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.21431	−	0.724880i	0.858647	−	0.512567i
							\(3\)	−0.132387	−	1.72698i	−0.0764339	−	0.997075i
\(5\)	−0.733476	−	1.77077i	−0.328020	−	0.791911i								−0.998739	−	0.0501996i	\(-0.984014\pi\)
							0.670719	−	0.741712i	\(-0.265986\pi\)
\(7\)	−0.707107	−	0.707107i	−0.267261	−	0.267261i
							\(11\)	−0.219221	−	0.529247i	−0.0660977	−	0.159574i	0.887379	−	0.461041i	\(-0.152524\pi\)
−0.953477	+	0.301467i	\(0.902524\pi\)
\(13\)	5.66926	+	2.34828i	1.57237	+	0.651296i	0.987181	−	0.159602i	\(-0.0510211\pi\)
							0.585187	+	0.810898i	\(0.301021\pi\)
\(17\)	−0.0922529			−0.0223746			−0.0111873	−	0.999937i	\(-0.503561\pi\)
							−0.0111873	+	0.999937i	\(0.503561\pi\)
\(19\)	−1.63700	+	3.95206i	−0.375553	+	0.906665i	0.617235	+	0.786779i	\(0.288253\pi\)
							−0.992788	+	0.119886i	\(0.961747\pi\)
\(23\)	−1.32228	−	1.32228i	−0.275714	−	0.275714i	0.555681	−	0.831395i	\(-0.312457\pi\)
							−0.831395	+	0.555681i	\(0.812457\pi\)
\(29\)	−2.81672	−	1.16673i	−0.523053	−	0.216655i	0.105505	−	0.994419i	\(-0.466354\pi\)
							−0.628557	+	0.777763i	\(0.716354\pi\)
\(31\)			0.629905i			0.113134i	0.998399	+	0.0565672i	\(0.0180155\pi\)
							−0.998399	+	0.0565672i	\(0.981985\pi\)
\(37\)	−0.995577	+	0.412381i	−0.163672	+	0.0677951i	−0.463015	−	0.886350i	\(-0.653232\pi\)
							0.299343	+	0.954145i	\(0.403232\pi\)
\(41\)	8.34848	−	8.34848i	1.30381	−	1.30381i	0.378014	−	0.925800i	\(-0.376607\pi\)
							0.925800	−	0.378014i	\(-0.123393\pi\)
\(43\)	2.69663	−	1.11698i	0.411232	−	0.170338i	−0.167470	−	0.985877i	\(-0.553560\pi\)
							0.578702	+	0.815539i	\(0.303560\pi\)
\(47\)			3.09832i			0.451937i	0.974135	+	0.225968i	\(0.0725546\pi\)
							−0.974135	+	0.225968i	\(0.927445\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.bs.a.659.40	yes	192
3.2	odd	2	inner	672.2.bs.a.659.9	yes	192
32.27	odd	8	inner	672.2.bs.a.155.9	✓	192
96.59	even	8	inner	672.2.bs.a.155.40	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.bs.a.155.9	✓	192	32.27	odd	8	inner
672.2.bs.a.155.40	yes	192	96.59	even	8	inner
672.2.bs.a.659.9	yes	192	3.2	odd	2	inner
672.2.bs.a.659.40	yes	192	1.1	even	1	trivial