Embedded newform 672.2.bs.b.155.7

• Label: 672.2.bs.b.155.7
• 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			155.7
Character	\(\chi\)	\(=\)	672.155
Dual form			672.2.bs.b.659.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31896 + 0.510228i) q^{2} +(0.712074 - 1.57891i) q^{3} +(1.47933 - 1.34595i) q^{4} +(-0.745289 + 1.79929i) q^{5} +(-0.133596 + 2.44584i) q^{6} +(0.707107 - 0.707107i) q^{7} +(-1.26445 + 2.53005i) q^{8} +(-1.98590 - 2.24860i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q + 16 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31896	+	0.510228i	−0.932649	+	0.360786i
							\(3\)	0.712074	−	1.57891i	0.411116	−	0.911583i
\(5\)	−0.745289	+	1.79929i	−0.333303	+	0.804665i								0.665022	+	0.746823i	\(0.268422\pi\)
							−0.998326	+	0.0578420i	\(0.981578\pi\)
\(7\)	0.707107	−	0.707107i	0.267261	−	0.267261i
							\(11\)	1.52280	−	3.67637i	0.459142	−	1.10847i	−0.509604	−	0.860409i	\(-0.670208\pi\)
0.968745	−	0.248057i	\(-0.0797921\pi\)
\(13\)	−5.68137	+	2.35330i	−1.57573	+	0.652688i	−0.987730	−	0.156174i	\(-0.950084\pi\)
							−0.587998	+	0.808862i	\(0.700084\pi\)
\(17\)	−4.91779			−1.19274			−0.596369	−	0.802710i	\(-0.703391\pi\)
							−0.596369	+	0.802710i	\(0.703391\pi\)
\(19\)	−2.28461	−	5.51554i	−0.524126	−	1.26535i	−0.935319	−	0.353804i	\(-0.884888\pi\)
							0.411193	−	0.911548i	\(-0.365112\pi\)
\(23\)	1.63764	−	1.63764i	0.341472	−	0.341472i	−0.515448	−	0.856921i	\(-0.672375\pi\)
							0.856921	+	0.515448i	\(0.172375\pi\)
\(29\)	8.40043	−	3.47957i	1.55992	−	0.646140i	0.574845	−	0.818262i	\(-0.305062\pi\)
							0.985076	+	0.172122i	\(0.0550622\pi\)
\(31\)		−	2.41600i		−	0.433926i	−0.976180	−	0.216963i	\(-0.930385\pi\)
							0.976180	−	0.216963i	\(-0.0696150\pi\)
\(37\)	−3.10392	−	1.28569i	−0.510281	−	0.211365i	0.112661	−	0.993634i	\(-0.464063\pi\)
							−0.622942	+	0.782268i	\(0.714063\pi\)
\(41\)	−4.31669	−	4.31669i	−0.674153	−	0.674153i	0.284518	−	0.958671i	\(-0.408167\pi\)
							−0.958671	+	0.284518i	\(0.908167\pi\)
\(43\)	−4.41153	−	1.82732i	−0.672752	−	0.278663i	0.0200413	−	0.999799i	\(-0.493620\pi\)
							−0.692793	+	0.721136i	\(0.743620\pi\)
\(47\)			2.45389i			0.357937i	0.983855	+	0.178968i	\(0.0572760\pi\)
							−0.983855	+	0.178968i	\(0.942724\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.bs.b.155.7	✓	192
3.2	odd	2	inner	672.2.bs.b.155.42	yes	192
32.19	odd	8	inner	672.2.bs.b.659.42	yes	192
96.83	even	8	inner	672.2.bs.b.659.7	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.bs.b.155.7	✓	192	1.1	even	1	trivial
672.2.bs.b.155.42	yes	192	3.2	odd	2	inner
672.2.bs.b.659.7	yes	192	96.83	even	8	inner
672.2.bs.b.659.42	yes	192	32.19	odd	8	inner