Embedded newform 672.2.bs.a.155.16

• Label: 672.2.bs.a.155.16
• 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			155.16
Character	\(\chi\)	\(=\)	672.155
Dual form			672.2.bs.a.659.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.669636 - 1.24563i) q^{2} +(1.65765 + 0.502191i) q^{3} +(-1.10318 + 1.66823i) q^{4} +(1.25795 - 3.03696i) q^{5} +(-0.484479 - 2.40110i) q^{6} +(-0.707107 + 0.707107i) q^{7} +(2.81672 + 0.257038i) q^{8} +(2.49561 + 1.66491i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 16 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.669636	−	1.24563i	−0.473504	−	0.880792i
							\(3\)	1.65765	+	0.502191i	0.957045	+	0.289940i
\(5\)	1.25795	−	3.03696i	0.562573	−	1.35817i								−0.345129	−	0.938555i	\(-0.612165\pi\)
							0.907702	−	0.419616i	\(-0.137835\pi\)
\(7\)	−0.707107	+	0.707107i	−0.267261	+	0.267261i
							\(11\)	0.937538	−	2.26342i	0.282678	−	0.682446i	−0.717218	−	0.696849i	\(-0.754585\pi\)
0.999896	+	0.0144030i	\(0.00458478\pi\)
\(13\)	−0.722797	+	0.299392i	−0.200468	+	0.0830365i	−0.480658	−	0.876908i	\(-0.659602\pi\)
							0.280191	+	0.959944i	\(0.409602\pi\)
\(17\)	1.62949			0.395209			0.197605	−	0.980282i	\(-0.436684\pi\)
							0.197605	+	0.980282i	\(0.436684\pi\)
\(19\)	−0.439982	−	1.06221i	−0.100939	−	0.243688i	0.865340	−	0.501185i	\(-0.167102\pi\)
							−0.966279	+	0.257497i	\(0.917102\pi\)
\(23\)	5.92550	−	5.92550i	1.23555	−	1.23555i	0.273751	−	0.961801i	\(-0.411735\pi\)
							0.961801	−	0.273751i	\(-0.0882645\pi\)
\(29\)	−5.95296	+	2.46580i	−1.10544	+	0.457887i	−0.859364	−	0.511365i	\(-0.829140\pi\)
							−0.246072	+	0.969251i	\(0.579140\pi\)
\(31\)			3.01611i			0.541709i	0.962620	+	0.270855i	\(0.0873063\pi\)
							−0.962620	+	0.270855i	\(0.912694\pi\)
\(37\)	−1.35853	−	0.562722i	−0.223341	−	0.0925110i	0.268207	−	0.963361i	\(-0.413569\pi\)
							−0.491548	+	0.870850i	\(0.663569\pi\)
\(41\)	−1.93389	−	1.93389i	−0.302023	−	0.302023i	0.539782	−	0.841805i	\(-0.318507\pi\)
							−0.841805	+	0.539782i	\(0.818507\pi\)
\(43\)	10.4900	+	4.34509i	1.59971	+	0.662620i	0.991374	−	0.131063i	\(-0.0418390\pi\)
							0.608332	+	0.793683i	\(0.291839\pi\)
\(47\)		−	11.1075i		−	1.62020i	−0.586291	−	0.810100i	\(-0.699413\pi\)
							0.586291	−	0.810100i	\(-0.300587\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.bs.a.155.16	✓	192
3.2	odd	2	inner	672.2.bs.a.155.33	yes	192
32.19	odd	8	inner	672.2.bs.a.659.33	yes	192
96.83	even	8	inner	672.2.bs.a.659.16	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.bs.a.155.16	✓	192	1.1	even	1	trivial
672.2.bs.a.155.33	yes	192	3.2	odd	2	inner
672.2.bs.a.659.16	yes	192	96.83	even	8	inner
672.2.bs.a.659.33	yes	192	32.19	odd	8	inner