Embedded newform 672.2.bs.a.155.18

• Label: 672.2.bs.a.155.18
• 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			155.18
Character	\(\chi\)	\(=\)	672.155
Dual form			672.2.bs.a.659.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.550610 + 1.30262i) q^{2} +(0.864233 - 1.50103i) q^{3} +(-1.39366 - 1.43447i) q^{4} +(1.15904 - 2.79817i) q^{5} +(1.47943 + 1.95225i) q^{6} +(-0.707107 + 0.707107i) q^{7} +(2.63594 - 1.02558i) q^{8} +(-1.50620 - 2.59449i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 16 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.550610	+	1.30262i	−0.389340	+	0.921094i
							\(3\)	0.864233	−	1.50103i	0.498965	−	0.866622i
\(5\)	1.15904	−	2.79817i	0.518338	−	1.25138i								−0.420586	−	0.907253i	\(-0.638176\pi\)
							0.938924	−	0.344125i	\(-0.111824\pi\)
\(7\)	−0.707107	+	0.707107i	−0.267261	+	0.267261i
							\(11\)	1.27611	−	3.08079i	0.384761	−	0.928894i	−0.606270	−	0.795259i	\(-0.707335\pi\)
0.991031	−	0.133635i	\(-0.0426651\pi\)
\(13\)	1.40797	−	0.583201i	0.390501	−	0.161751i	−0.178789	−	0.983887i	\(-0.557218\pi\)
							0.569290	+	0.822137i	\(0.307218\pi\)
\(17\)	−7.42017			−1.79966			−0.899828	−	0.436244i	\(-0.856308\pi\)
							−0.899828	+	0.436244i	\(0.856308\pi\)
\(19\)	0.996362	+	2.40543i	0.228581	+	0.551843i	0.996005	−	0.0892964i	\(-0.0284619\pi\)
							−0.767424	+	0.641140i	\(0.778462\pi\)
\(23\)	−2.38103	+	2.38103i	−0.496480	+	0.496480i	−0.910340	−	0.413860i	\(-0.864180\pi\)
							0.413860	+	0.910340i	\(0.364180\pi\)
\(29\)	5.71257	−	2.36623i	1.06080	−	0.439397i	0.217064	−	0.976157i	\(-0.430352\pi\)
							0.843735	+	0.536760i	\(0.180352\pi\)
\(31\)		−	0.776227i		−	0.139415i	−0.997567	−	0.0697073i	\(-0.977793\pi\)
							0.997567	−	0.0697073i	\(-0.0222065\pi\)
\(37\)	−4.82603	−	1.99901i	−0.793394	−	0.328634i	−0.0510866	−	0.998694i	\(-0.516268\pi\)
							−0.742307	+	0.670060i	\(0.766268\pi\)
\(41\)	1.91757	+	1.91757i	0.299474	+	0.299474i	0.840808	−	0.541334i	\(-0.182081\pi\)
							−0.541334	+	0.840808i	\(0.682081\pi\)
\(43\)	3.35650	+	1.39031i	0.511862	+	0.212020i	0.623638	−	0.781713i	\(-0.285654\pi\)
							−0.111776	+	0.993733i	\(0.535654\pi\)
\(47\)		−	9.07653i		−	1.32395i	−0.749527	−	0.661974i	\(-0.769719\pi\)
							0.749527	−	0.661974i	\(-0.230281\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.bs.a.155.18	✓	192
3.2	odd	2	inner	672.2.bs.a.155.31	yes	192
32.19	odd	8	inner	672.2.bs.a.659.31	yes	192
96.83	even	8	inner	672.2.bs.a.659.18	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.bs.a.155.18	✓	192	1.1	even	1	trivial
672.2.bs.a.155.31	yes	192	3.2	odd	2	inner
672.2.bs.a.659.18	yes	192	96.83	even	8	inner
672.2.bs.a.659.31	yes	192	32.19	odd	8	inner