Embedded newform 672.2.bs.a.155.7

• Label: 672.2.bs.a.155.7
• 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			155.7
Character	\(\chi\)	\(=\)	672.155
Dual form			672.2.bs.a.659.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26204 - 0.638158i) q^{2} +(1.63116 + 0.582516i) q^{3} +(1.18551 + 1.61077i) q^{4} +(-0.102034 + 0.246332i) q^{5} +(-1.68685 - 1.77610i) q^{6} +(-0.707107 + 0.707107i) q^{7} +(-0.468240 - 2.78940i) q^{8} +(2.32135 + 1.90035i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 16 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26204	−	0.638158i	−0.892400	−	0.451246i
							\(3\)	1.63116	+	0.582516i	0.941749	+	0.336316i
\(5\)	−0.102034	+	0.246332i	−0.0456310	+	0.110163i								−0.945052	−	0.326921i	\(-0.893989\pi\)
							0.899421	+	0.437084i	\(0.143989\pi\)
\(7\)	−0.707107	+	0.707107i	−0.267261	+	0.267261i
							\(11\)	−2.42681	+	5.85884i	−0.731711	+	1.76651i	−0.0949010	+	0.995487i	\(0.530253\pi\)
−0.636810	+	0.771021i	\(0.719747\pi\)
\(13\)	0.975149	−	0.403920i	0.270458	−	0.112027i	−0.243333	−	0.969943i	\(-0.578241\pi\)
							0.513791	+	0.857916i	\(0.328241\pi\)
\(17\)	−7.05903			−1.71207			−0.856033	−	0.516921i	\(-0.827078\pi\)
							−0.856033	+	0.516921i	\(0.827078\pi\)
\(19\)	0.190280	+	0.459377i	0.0436533	+	0.105388i	0.944202	−	0.329366i	\(-0.106835\pi\)
							−0.900549	+	0.434755i	\(0.856835\pi\)
\(23\)	−0.00755555	+	0.00755555i	−0.00157544	+	0.00157544i	−0.707894	−	0.706319i	\(-0.750355\pi\)
							0.706319	+	0.707894i	\(0.250355\pi\)
\(29\)	5.48638	−	2.27253i	1.01879	−	0.421999i	0.190139	−	0.981757i	\(-0.439106\pi\)
							0.828656	+	0.559759i	\(0.189106\pi\)
\(31\)			4.77761i			0.858084i	0.903285	+	0.429042i	\(0.141149\pi\)
							−0.903285	+	0.429042i	\(0.858851\pi\)
\(37\)	−5.18118	−	2.14612i	−0.851781	−	0.352819i	−0.0862935	−	0.996270i	\(-0.527502\pi\)
							−0.765488	+	0.643450i	\(0.777502\pi\)
\(41\)	−3.41445	−	3.41445i	−0.533248	−	0.533248i	0.388290	−	0.921537i	\(-0.373066\pi\)
							−0.921537	+	0.388290i	\(0.873066\pi\)
\(43\)	−7.15962	−	2.96561i	−1.09183	−	0.452252i	−0.237187	−	0.971464i	\(-0.576225\pi\)
							−0.854645	+	0.519213i	\(0.826225\pi\)
\(47\)		−	2.51273i		−	0.366520i	−0.983064	−	0.183260i	\(-0.941335\pi\)
							0.983064	−	0.183260i	\(-0.0586650\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.bs.a.155.7	✓	192
3.2	odd	2	inner	672.2.bs.a.155.42	yes	192
32.19	odd	8	inner	672.2.bs.a.659.42	yes	192
96.83	even	8	inner	672.2.bs.a.659.7	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.bs.a.155.7	✓	192	1.1	even	1	trivial
672.2.bs.a.155.42	yes	192	3.2	odd	2	inner
672.2.bs.a.659.7	yes	192	96.83	even	8	inner
672.2.bs.a.659.42	yes	192	32.19	odd	8	inner