Embedded newform 672.2.bs.a.155.12

• Label: 672.2.bs.a.155.12
• 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			155.12
Character	\(\chi\)	\(=\)	672.155
Dual form			672.2.bs.a.659.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.938001 - 1.05837i) q^{2} +(0.196518 - 1.72087i) q^{3} +(-0.240309 + 1.98551i) q^{4} +(-0.272376 + 0.657573i) q^{5} +(-2.00565 + 1.40618i) q^{6} +(-0.707107 + 0.707107i) q^{7} +(2.32682 - 1.60807i) q^{8} +(-2.92276 - 0.676364i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 16 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.938001	−	1.05837i	−0.663267	−	0.748383i
							\(3\)	0.196518	−	1.72087i	0.113460	−	0.993543i
\(5\)	−0.272376	+	0.657573i	−0.121810	+	0.294075i								−0.973009	−	0.230768i	\(-0.925876\pi\)
							0.851199	+	0.524843i	\(0.175876\pi\)
\(7\)	−0.707107	+	0.707107i	−0.267261	+	0.267261i
							\(11\)	0.333911	−	0.806133i	0.100678	−	0.243058i	−0.865513	−	0.500887i	\(-0.833007\pi\)
0.966191	+	0.257829i	\(0.0830071\pi\)
\(13\)	−3.61811	+	1.49867i	−1.00348	+	0.415656i	−0.823073	−	0.567936i	\(-0.807742\pi\)
							−0.180409	+	0.983592i	\(0.557742\pi\)
\(17\)	−2.19624			−0.532666			−0.266333	−	0.963881i	\(-0.585812\pi\)
							−0.266333	+	0.963881i	\(0.585812\pi\)
\(19\)	−0.129495	−	0.312630i	−0.0297083	−	0.0717222i	0.908329	−	0.418256i	\(-0.137359\pi\)
							−0.938037	+	0.346534i	\(0.887359\pi\)
\(23\)	−3.23540	+	3.23540i	−0.674627	+	0.674627i	−0.958779	−	0.284152i	\(-0.908288\pi\)
							0.284152	+	0.958779i	\(0.408288\pi\)
\(29\)	−8.23604	+	3.41148i	−1.52939	+	0.633496i	−0.979447	−	0.201703i	\(-0.935352\pi\)
							−0.549948	+	0.835199i	\(0.685352\pi\)
\(31\)			2.20582i			0.396176i	0.980184	+	0.198088i	\(0.0634733\pi\)
							−0.980184	+	0.198088i	\(0.936527\pi\)
\(37\)	0.747413	+	0.309589i	0.122874	+	0.0508961i	0.443273	−	0.896386i	\(-0.353817\pi\)
							−0.320399	+	0.947283i	\(0.603817\pi\)
\(41\)	−3.44795	−	3.44795i	−0.538480	−	0.538480i	0.384602	−	0.923082i	\(-0.374339\pi\)
							−0.923082	+	0.384602i	\(0.874339\pi\)
\(43\)	−0.261172	−	0.108181i	−0.0398284	−	0.0164975i	0.362680	−	0.931914i	\(-0.381862\pi\)
							−0.402509	+	0.915416i	\(0.631862\pi\)
\(47\)			6.37201i			0.929453i	0.885454	+	0.464727i	\(0.153847\pi\)
							−0.885454	+	0.464727i	\(0.846153\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.bs.a.155.12	✓	192
3.2	odd	2	inner	672.2.bs.a.155.37	yes	192
32.19	odd	8	inner	672.2.bs.a.659.37	yes	192
96.83	even	8	inner	672.2.bs.a.659.12	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.bs.a.155.12	✓	192	1.1	even	1	trivial
672.2.bs.a.155.37	yes	192	3.2	odd	2	inner
672.2.bs.a.659.12	yes	192	96.83	even	8	inner
672.2.bs.a.659.37	yes	192	32.19	odd	8	inner