Embedded newform 672.2.bs.b.155.13

• Label: 672.2.bs.b.155.13
• 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			155.13
Character	\(\chi\)	\(=\)	672.155
Dual form			672.2.bs.b.659.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09166 + 0.899047i) q^{2} +(0.650725 + 1.60517i) q^{3} +(0.383428 - 1.96290i) q^{4} +(-0.365005 + 0.881200i) q^{5} +(-2.15349 - 1.16726i) q^{6} +(0.707107 - 0.707107i) q^{7} +(1.34617 + 2.48753i) q^{8} +(-2.15311 + 2.08904i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q + 16 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09166	+	0.899047i	−0.771918	+	0.635722i
							\(3\)	0.650725	+	1.60517i	0.375696	+	0.926743i
\(5\)	−0.365005	+	0.881200i	−0.163235	+	0.394085i								−0.984240	−	0.176836i	\(-0.943414\pi\)
							0.821005	+	0.570921i	\(0.193414\pi\)
\(7\)	0.707107	−	0.707107i	0.267261	−	0.267261i
							\(11\)	−0.792129	+	1.91237i	−0.238836	+	0.576601i	−0.997163	−	0.0752733i	\(-0.976017\pi\)
0.758327	+	0.651874i	\(0.226017\pi\)
\(13\)	−1.79007	+	0.741471i	−0.496476	+	0.205647i	−0.616849	−	0.787082i	\(-0.711591\pi\)
							0.120373	+	0.992729i	\(0.461591\pi\)
\(17\)	1.29907			0.315071			0.157536	−	0.987513i	\(-0.449645\pi\)
							0.157536	+	0.987513i	\(0.449645\pi\)
\(19\)	−0.650712	−	1.57096i	−0.149284	−	0.360402i	0.831493	−	0.555535i	\(-0.187486\pi\)
							−0.980777	+	0.195132i	\(0.937486\pi\)
\(23\)	−4.07699	+	4.07699i	−0.850112	+	0.850112i	−0.990147	−	0.140035i	\(-0.955279\pi\)
							0.140035	+	0.990147i	\(0.455279\pi\)
\(29\)	−2.99231	+	1.23946i	−0.555659	+	0.230161i	−0.642799	−	0.766035i	\(-0.722227\pi\)
							0.0871407	+	0.996196i	\(0.472227\pi\)
\(31\)			8.87104i			1.59329i	0.604450	+	0.796643i	\(0.293393\pi\)
							−0.604450	+	0.796643i	\(0.706607\pi\)
\(37\)	−7.81407	−	3.23669i	−1.28463	−	0.532109i	−0.367246	−	0.930124i	\(-0.619699\pi\)
							−0.917379	+	0.398014i	\(0.869699\pi\)
\(41\)	1.90450	+	1.90450i	0.297433	+	0.297433i	0.840007	−	0.542575i	\(-0.182551\pi\)
							−0.542575	+	0.840007i	\(0.682551\pi\)
\(43\)	0.541023	+	0.224099i	0.0825053	+	0.0341748i	0.423554	−	0.905871i	\(-0.360782\pi\)
							−0.341049	+	0.940045i	\(0.610782\pi\)
\(47\)			0.552208i			0.0805478i	0.999189	+	0.0402739i	\(0.0128230\pi\)
							−0.999189	+	0.0402739i	\(0.987177\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.bs.b.155.13	✓	192
3.2	odd	2	inner	672.2.bs.b.155.36	yes	192
32.19	odd	8	inner	672.2.bs.b.659.36	yes	192
96.83	even	8	inner	672.2.bs.b.659.13	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.bs.b.155.13	✓	192	1.1	even	1	trivial
672.2.bs.b.155.36	yes	192	3.2	odd	2	inner
672.2.bs.b.659.13	yes	192	96.83	even	8	inner
672.2.bs.b.659.36	yes	192	32.19	odd	8	inner