Embedded newform 672.2.bs.a.155.20

• Label: 672.2.bs.a.155.20
• 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			155.20
Character	\(\chi\)	\(=\)	672.155
Dual form			672.2.bs.a.659.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.393886 - 1.35825i) q^{2} +(1.34373 - 1.09288i) q^{3} +(-1.68971 + 1.06999i) q^{4} +(-0.833504 + 2.01226i) q^{5} +(-2.01369 - 1.39465i) q^{6} +(-0.707107 + 0.707107i) q^{7} +(2.11888 + 1.87360i) q^{8} +(0.611211 - 2.93708i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 16 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.393886	−	1.35825i	−0.278519	−	0.960431i
							\(3\)	1.34373	−	1.09288i	0.775802	−	0.630977i
\(5\)	−0.833504	+	2.01226i	−0.372754	+	0.899908i								0.620527	+	0.784185i	\(0.286919\pi\)
							−0.993281	+	0.115724i	\(0.963081\pi\)
\(7\)	−0.707107	+	0.707107i	−0.267261	+	0.267261i
							\(11\)	0.822974	−	1.98684i	0.248136	−	0.599053i	−0.749910	−	0.661540i	\(-0.769903\pi\)
0.998046	+	0.0624867i	\(0.0199031\pi\)
\(13\)	6.21470	−	2.57421i	1.72365	−	0.713958i	0.723938	−	0.689865i	\(-0.242330\pi\)
							0.999710	−	0.0240928i	\(-0.00766972\pi\)
\(17\)	−4.25374			−1.03168			−0.515842	−	0.856684i	\(-0.672521\pi\)
							−0.515842	+	0.856684i	\(0.672521\pi\)
\(19\)	1.26616	+	3.05678i	0.290477	+	0.701273i	0.999994	−	0.00341522i	\(-0.00108710\pi\)
							−0.709518	+	0.704688i	\(0.751087\pi\)
\(23\)	5.56921	−	5.56921i	1.16126	−	1.16126i	0.177062	−	0.984200i	\(-0.443341\pi\)
							0.984200	−	0.177062i	\(-0.0566592\pi\)
\(29\)	4.27957	−	1.77266i	0.794696	−	0.329174i	0.0518661	−	0.998654i	\(-0.483483\pi\)
							0.742830	+	0.669480i	\(0.233483\pi\)
\(31\)		−	7.60002i		−	1.36500i	−0.730884	−	0.682502i	\(-0.760892\pi\)
							0.730884	−	0.682502i	\(-0.239108\pi\)
\(37\)	4.04039	+	1.67359i	0.664237	+	0.275136i	0.689220	−	0.724552i	\(-0.257953\pi\)
							−0.0249835	+	0.999688i	\(0.507953\pi\)
\(41\)	2.32106	+	2.32106i	0.362489	+	0.362489i	0.864729	−	0.502239i	\(-0.167490\pi\)
							−0.502239	+	0.864729i	\(0.667490\pi\)
\(43\)	−6.26435	−	2.59478i	−0.955305	−	0.395700i	−0.150083	−	0.988673i	\(-0.547954\pi\)
							−0.805222	+	0.592973i	\(0.797954\pi\)
\(47\)			4.54348i			0.662735i	0.943502	+	0.331368i	\(0.107510\pi\)
							−0.943502	+	0.331368i	\(0.892490\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.bs.a.155.20	✓	192
3.2	odd	2	inner	672.2.bs.a.155.29	yes	192
32.19	odd	8	inner	672.2.bs.a.659.29	yes	192
96.83	even	8	inner	672.2.bs.a.659.20	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.bs.a.155.20	✓	192	1.1	even	1	trivial
672.2.bs.a.155.29	yes	192	3.2	odd	2	inner
672.2.bs.a.659.20	yes	192	96.83	even	8	inner
672.2.bs.a.659.29	yes	192	32.19	odd	8	inner