Embedded newform 672.2.cj.a.445.28

• Label: 672.2.cj.a.445.28
• 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			445.28
Character	\(\chi\)	\(=\)	672.445
Dual form			672.2.cj.a.373.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.265620 + 1.38904i) q^{2} +(-0.793353 + 0.608761i) q^{3} +(-1.85889 - 0.737917i) q^{4} +(1.04358 - 1.36002i) q^{5} +(-0.634866 - 1.26370i) q^{6} +(0.389108 + 2.61698i) q^{7} +(1.51876 - 2.38608i) q^{8} +(0.258819 - 0.965926i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.265620	+	1.38904i	−0.187822	+	0.982203i
							\(3\)	−0.793353	+	0.608761i	−0.458043	+	0.351469i
\(5\)	1.04358	−	1.36002i	0.466702	−	0.608217i								−0.499403	−	0.866370i	\(-0.666447\pi\)
							0.966104	+	0.258153i	\(0.0831138\pi\)
\(7\)	0.389108	+	2.61698i	0.147069	+	0.989126i
							\(11\)	−0.293306	+	2.22788i	−0.0884352	+	0.671732i	0.888604	+	0.458674i	\(0.151676\pi\)
−0.977040	+	0.213058i	\(0.931658\pi\)
\(13\)	1.16149	−	2.80410i	0.322141	−	0.777716i	−0.676989	−	0.735994i	\(-0.736715\pi\)
							0.999129	−	0.0417228i	\(-0.0132846\pi\)
\(17\)	2.71429	+	1.56709i	0.658311	+	0.380076i	0.791633	−	0.610997i	\(-0.209231\pi\)
							−0.133322	+	0.991073i	\(0.542564\pi\)
\(19\)	1.73867	−	0.228900i	0.398878	−	0.0525133i	0.0715809	−	0.997435i	\(-0.477196\pi\)
							0.327297	+	0.944922i	\(0.393862\pi\)
\(23\)	−0.704799	+	2.63035i	−0.146961	+	0.548465i	0.852699	+	0.522402i	\(0.174964\pi\)
							−0.999660	+	0.0260634i	\(0.991703\pi\)
\(29\)	4.78528	+	1.98213i	0.888605	+	0.368072i	0.779828	−	0.625994i	\(-0.215307\pi\)
							0.108777	+	0.994066i	\(0.465307\pi\)
\(31\)	−3.73871	+	6.47563i	−0.671491	+	1.16306i	0.305990	+	0.952035i	\(0.401013\pi\)
							−0.977481	+	0.211022i	\(0.932321\pi\)
\(37\)	−5.10799	+	6.65686i	−0.839748	+	1.09438i	0.154575	+	0.987981i	\(0.450599\pi\)
							−0.994323	+	0.106400i	\(0.966068\pi\)
\(41\)	−2.28668	−	2.28668i	−0.357120	−	0.357120i	0.505630	−	0.862750i	\(-0.331260\pi\)
							−0.862750	+	0.505630i	\(0.831260\pi\)
\(43\)	0.425331	−	0.176178i	0.0648624	−	0.0268669i	−0.350016	−	0.936744i	\(-0.613824\pi\)
							0.414879	+	0.909877i	\(0.363824\pi\)
\(47\)	−2.81102	+	1.62294i	−0.410029	+	0.236730i	−0.690802	−	0.723044i	\(-0.742742\pi\)
							0.280773	+	0.959774i	\(0.409409\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.cj.a.445.28	yes	512
7.2	even	3	inner	672.2.cj.a.541.59	yes	512
32.21	even	8	inner	672.2.cj.a.277.59	✓	512
224.149	even	24	inner	672.2.cj.a.373.28	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.cj.a.277.59	✓	512	32.21	even	8	inner
672.2.cj.a.373.28	yes	512	224.149	even	24	inner
672.2.cj.a.445.28	yes	512	1.1	even	1	trivial
672.2.cj.a.541.59	yes	512	7.2	even	3	inner