Embedded newform 672.2.bs.a.155.10

• Label: 672.2.bs.a.155.10
• Level: $672$
• Weight: $2$
• Character: 672.155
• 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			155.10
Character	\(\chi\)	\(=\)	672.155
Dual form			672.2.bs.a.659.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07981 + 0.913242i) q^{2} +(1.37333 - 1.05544i) q^{3} +(0.331980 - 1.97225i) q^{4} +(-0.295998 + 0.714603i) q^{5} +(-0.519067 + 2.39386i) q^{6} +(-0.707107 + 0.707107i) q^{7} +(1.44267 + 2.43284i) q^{8} +(0.772089 - 2.89894i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 16 q^{4}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07981	+	0.913242i	−0.763541	+	0.645759i
							\(3\)	1.37333	−	1.05544i	0.792894	−	0.609359i
\(5\)	−0.295998	+	0.714603i	−0.132374	+	0.319580i								−0.976144	−	0.217126i	\(-0.930332\pi\)
							0.843769	+	0.536706i	\(0.180332\pi\)
\(7\)	−0.707107	+	0.707107i	−0.267261	+	0.267261i
							\(11\)	−2.38741	+	5.76372i	−0.719831	+	1.73783i	−0.0459997	+	0.998941i	\(0.514647\pi\)
−0.673832	+	0.738885i	\(0.735353\pi\)
\(13\)	−6.19332	+	2.56536i	−1.71772	+	0.711502i	−0.717833	+	0.696215i	\(0.754866\pi\)
							−0.999883	+	0.0152866i	\(0.995134\pi\)
\(17\)	0.122603			0.0297355			0.0148677	−	0.999889i	\(-0.495267\pi\)
							0.0148677	+	0.999889i	\(0.495267\pi\)
\(19\)	2.83600	+	6.84670i	0.650622	+	1.57074i	0.811878	+	0.583828i	\(0.198446\pi\)
							−0.161256	+	0.986913i	\(0.551554\pi\)
\(23\)	2.39746	−	2.39746i	0.499906	−	0.499906i	−0.411503	−	0.911409i	\(-0.634996\pi\)
							0.911409	+	0.411503i	\(0.134996\pi\)
\(29\)	−6.52614	+	2.70321i	−1.21187	+	0.501974i	−0.894817	−	0.446432i	\(-0.852694\pi\)
							−0.317056	+	0.948407i	\(0.602694\pi\)
\(31\)		−	3.24804i		−	0.583365i	−0.956515	−	0.291683i	\(-0.905785\pi\)
							0.956515	−	0.291683i	\(-0.0942151\pi\)
\(37\)	5.82335	+	2.41211i	0.957353	+	0.396549i	0.805990	−	0.591929i	\(-0.201634\pi\)
							0.151363	+	0.988478i	\(0.451634\pi\)
\(41\)	3.21801	+	3.21801i	0.502568	+	0.502568i	0.912235	−	0.409667i	\(-0.134355\pi\)
							−0.409667	+	0.912235i	\(0.634355\pi\)
\(43\)	1.90336	+	0.788398i	0.290260	+	0.120230i	0.523062	−	0.852294i	\(-0.324790\pi\)
							−0.232802	+	0.972524i	\(0.574790\pi\)
\(47\)		−	7.17054i		−	1.04593i	−0.852354	−	0.522966i	\(-0.824826\pi\)
							0.852354	−	0.522966i	\(-0.175174\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.bs.a.155.10	✓	192
3.2	odd	2	inner	672.2.bs.a.155.39	yes	192
32.19	odd	8	inner	672.2.bs.a.659.39	yes	192
96.83	even	8	inner	672.2.bs.a.659.10	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.bs.a.155.10	✓	192	1.1	even	1	trivial
672.2.bs.a.155.39	yes	192	3.2	odd	2	inner
672.2.bs.a.659.10	yes	192	96.83	even	8	inner
672.2.bs.a.659.39	yes	192	32.19	odd	8	inner