Embedded newform 672.2.cj.a.445.43

• Label: 672.2.cj.a.445.43
• Level: $672$
• Weight: $2$
• Character: 672.445
• 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.cj (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			445.43
Character	\(\chi\)	\(=\)	672.445
Dual form			672.2.cj.a.373.43

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.704167 - 1.22644i) q^{2} +(0.793353 - 0.608761i) q^{3} +(-1.00830 - 1.72723i) q^{4} +(-2.36095 + 3.07685i) q^{5} +(-0.187954 - 1.40167i) q^{6} +(0.609575 + 2.57457i) q^{7} +(-2.82835 + 0.0203503i) q^{8} +(0.258819 - 0.965926i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(421\)	\(449\)	\(577\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.704167	−	1.22644i	0.497922	−	0.867222i
							\(3\)	0.793353	−	0.608761i	0.458043	−	0.351469i
\(5\)	−2.36095	+	3.07685i	−1.05585	+	1.37601i								−0.132882	+	0.991132i	\(0.542423\pi\)
							−0.922968	+	0.384878i	\(0.874244\pi\)
\(7\)	0.609575	+	2.57457i	0.230398	+	0.973097i
							\(11\)	−0.0301406	+	0.228941i	−0.00908773	+	0.0690282i	−0.995388	−	0.0959330i	\(-0.969417\pi\)
0.986300	+	0.164961i	\(0.0527499\pi\)
\(13\)	−2.01602	+	4.86709i	−0.559142	+	1.34989i	0.351304	+	0.936261i	\(0.385738\pi\)
							−0.910446	+	0.413627i	\(0.864262\pi\)
\(17\)	6.56572	+	3.79072i	1.59242	+	0.919385i	0.992890	+	0.119034i	\(0.0379799\pi\)
							0.599532	+	0.800351i	\(0.295353\pi\)
\(19\)	−4.91688	+	0.647320i	−1.12801	+	0.148505i	−0.671353	−	0.741138i	\(-0.734287\pi\)
							−0.456657	+	0.889643i	\(0.650953\pi\)
\(23\)	1.54685	−	5.77293i	0.322541	−	1.20374i	−0.594220	−	0.804302i	\(-0.702539\pi\)
							0.916761	−	0.399436i	\(-0.130794\pi\)
\(29\)	4.95485	+	2.05237i	0.920093	+	0.381115i	0.791912	−	0.610636i	\(-0.209086\pi\)
							0.128181	+	0.991751i	\(0.459086\pi\)
\(31\)	−1.38077	+	2.39156i	−0.247994	+	0.429537i	−0.962969	−	0.269612i	\(-0.913105\pi\)
							0.714975	+	0.699150i	\(0.246438\pi\)
\(37\)	−7.03560	+	9.16898i	−1.15665	+	1.50737i	−0.330597	+	0.943772i	\(0.607250\pi\)
							−0.826049	+	0.563599i	\(0.809416\pi\)
\(41\)	−1.67990	−	1.67990i	−0.262356	−	0.262356i	0.563654	−	0.826011i	\(-0.309395\pi\)
							−0.826011	+	0.563654i	\(0.809395\pi\)
\(43\)	2.81160	−	1.16460i	0.428765	−	0.177600i	−0.157856	−	0.987462i	\(-0.550458\pi\)
							0.586620	+	0.809862i	\(0.300458\pi\)
\(47\)	0.600038	−	0.346432i	0.0875245	−	0.0505323i	−0.455599	−	0.890185i	\(-0.650575\pi\)
							0.543123	+	0.839653i	\(0.317242\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.cj.a.445.43	yes	512
7.2	even	3	inner	672.2.cj.a.541.1	yes	512
32.21	even	8	inner	672.2.cj.a.277.1	✓	512
224.149	even	24	inner	672.2.cj.a.373.43	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.cj.a.277.1	✓	512	32.21	even	8	inner
672.2.cj.a.373.43	yes	512	224.149	even	24	inner
672.2.cj.a.445.43	yes	512	1.1	even	1	trivial
672.2.cj.a.541.1	yes	512	7.2	even	3	inner