Embedded newform 672.2.cf.a.187.59

• Label: 672.2.cf.a.187.59
• Level: $672$
• Weight: $2$
• Character: 672.187
• Analytic conductor: $5.366$
• Analytic rank: $0$
• Dimension: $512$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.cf (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.36594701583\)
Analytic rank:	\(0\)
Dimension:	\(512\)
Relative dimension:	\(64\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			187.59
Character	\(\chi\)	\(=\)	672.187
Dual form			672.2.cf.a.115.59

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36521 - 0.369061i) q^{2} +(0.608761 - 0.793353i) q^{3} +(1.72759 - 1.00769i) q^{4} +(2.34252 + 3.05283i) q^{5} +(0.538291 - 1.30776i) q^{6} +(-1.86291 + 1.87872i) q^{7} +(1.98662 - 2.01329i) q^{8} +(-0.258819 - 0.965926i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(421\)	\(449\)	\(577\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{8}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36521	−	0.369061i	0.965348	−	0.260965i
							\(3\)	0.608761	−	0.793353i	0.351469	−	0.458043i
\(5\)	2.34252	+	3.05283i	1.04761	+	1.36527i								0.928057	+	0.372437i	\(0.121478\pi\)
							0.119548	+	0.992828i	\(0.461856\pi\)
\(7\)	−1.86291	+	1.87872i	−0.704112	+	0.710089i
							\(11\)	0.395187	+	3.00175i	0.119154	+	0.905061i	0.941721	+	0.336394i	\(0.109207\pi\)
−0.822568	+	0.568667i	\(0.807459\pi\)
\(13\)	−0.873213	−	2.10812i	−0.242186	−	0.584688i	0.755314	−	0.655363i	\(-0.227485\pi\)
							−0.997499	+	0.0706754i	\(0.977485\pi\)
\(17\)	−0.223893	−	0.387794i	−0.0543020	−	0.0940538i	0.837597	−	0.546289i	\(-0.183960\pi\)
							−0.891899	+	0.452235i	\(0.850627\pi\)
\(19\)	0.424929	−	3.22766i	0.0974855	−	0.740476i	−0.870978	−	0.491321i	\(-0.836514\pi\)
							0.968464	−	0.249154i	\(-0.0801526\pi\)
\(23\)	−0.709634	+	0.190146i	−0.147969	+	0.0396482i	−0.332043	−	0.943264i	\(-0.607738\pi\)
							0.184074	+	0.982912i	\(0.441071\pi\)
\(29\)	2.38739	+	5.76366i	0.443326	+	1.07028i	0.974774	+	0.223193i	\(0.0716482\pi\)
							−0.531448	+	0.847091i	\(0.678352\pi\)
\(31\)	−3.58756	−	6.21384i	−0.644345	−	1.11604i	−0.984452	−	0.175652i	\(-0.943797\pi\)
							0.340107	−	0.940387i	\(-0.389537\pi\)
\(37\)	5.46588	−	4.19412i	0.898585	−	0.689509i	−0.0524392	−	0.998624i	\(-0.516700\pi\)
							0.951025	+	0.309115i	\(0.100033\pi\)
\(41\)	−7.46503	−	7.46503i	−1.16584	−	1.16584i	−0.983175	−	0.182666i	\(-0.941527\pi\)
							−0.182666	−	0.983175i	\(-0.558473\pi\)
\(43\)	−8.18619	−	3.39083i	−1.24838	−	0.517097i	−0.342058	−	0.939679i	\(-0.611124\pi\)
							−0.906325	+	0.422582i	\(0.861124\pi\)
\(47\)	−2.17444	−	1.25541i	−0.317175	−	0.183121i	0.332958	−	0.942942i	\(-0.391953\pi\)
							−0.650133	+	0.759821i	\(0.725287\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.cf.a.187.59	yes	512
7.3	odd	6	inner	672.2.cf.a.283.28	yes	512
32.19	odd	8	inner	672.2.cf.a.19.28	✓	512
224.115	even	24	inner	672.2.cf.a.115.59	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.cf.a.19.28	✓	512	32.19	odd	8	inner
672.2.cf.a.115.59	yes	512	224.115	even	24	inner
672.2.cf.a.187.59	yes	512	1.1	even	1	trivial
672.2.cf.a.283.28	yes	512	7.3	odd	6	inner