Embedded newform 672.2.cj.a.613.6

• Label: 672.2.cj.a.613.6
• Level: $672$
• Weight: $2$
• Character: 672.613
• 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			613.6
Character	\(\chi\)	\(=\)	672.613
Dual form			672.2.cj.a.205.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35076 - 0.418857i) q^{2} +(-0.991445 - 0.130526i) q^{3} +(1.64912 + 1.13155i) q^{4} +(-0.308260 - 2.34147i) q^{5} +(1.28453 + 0.591583i) q^{6} +(-2.03564 - 1.69002i) q^{7} +(-1.75361 - 2.21920i) q^{8} +(0.965926 + 0.258819i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35076	−	0.418857i	−0.955133	−	0.296177i
							\(3\)	−0.991445	−	0.130526i	−0.572411	−	0.0753593i
\(5\)	−0.308260	−	2.34147i	−0.137858	−	1.04714i								−0.910855	−	0.412726i	\(-0.864577\pi\)
							0.772997	−	0.634409i	\(-0.218757\pi\)
\(7\)	−2.03564	−	1.69002i	−0.769398	−	0.638769i
							\(11\)	−0.192387	−	0.250724i	−0.0580069	−	0.0755960i	0.763449	−	0.645868i	\(-0.223504\pi\)
−0.821456	+	0.570272i	\(0.806838\pi\)
\(13\)	−2.68240	+	1.11109i	−0.743963	+	0.308160i	−0.722276	−	0.691605i	\(-0.756904\pi\)
							−0.0216876	+	0.999765i	\(0.506904\pi\)
\(17\)	−6.06080	−	3.49921i	−1.46996	−	0.848682i	−0.470529	−	0.882385i	\(-0.655937\pi\)
							−0.999432	+	0.0337022i	\(0.989270\pi\)
\(19\)	1.94566	+	1.49296i	0.446365	+	0.342508i	0.807359	−	0.590060i	\(-0.200896\pi\)
							−0.360994	+	0.932568i	\(0.617563\pi\)
\(23\)	8.20372	+	2.19818i	1.71059	+	0.458352i	0.975570	−	0.219688i	\(-0.0705040\pi\)
							0.735024	+	0.678041i	\(0.237171\pi\)
\(29\)	−1.81484	−	4.38141i	−0.337007	−	0.813607i	−0.998000	−	0.0632150i	\(-0.979865\pi\)
							0.660993	−	0.750392i	\(-0.270135\pi\)
\(31\)	−4.88813	+	8.46649i	−0.877934	+	1.52063i	−0.0243305	+	0.999704i	\(0.507745\pi\)
							−0.853604	+	0.520923i	\(0.825588\pi\)
\(37\)	−0.565519	−	4.29554i	−0.0929707	−	0.706183i	−0.972924	−	0.231123i	\(-0.925760\pi\)
							0.879954	−	0.475059i	\(-0.157573\pi\)
\(41\)	−4.89332	+	4.89332i	−0.764208	+	0.764208i	−0.977080	−	0.212872i	\(-0.931718\pi\)
							0.212872	+	0.977080i	\(0.431718\pi\)
\(43\)	−3.77926	+	9.12395i	−0.576332	+	1.39139i	0.319750	+	0.947502i	\(0.396401\pi\)
							−0.896083	+	0.443887i	\(0.853599\pi\)
\(47\)	2.17463	−	1.25552i	0.317202	−	0.183137i	−0.332943	−	0.942947i	\(-0.608042\pi\)
							0.650145	+	0.759810i	\(0.274708\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.cj.a.613.6	yes	512
7.2	even	3	inner	672.2.cj.a.37.38	✓	512
32.13	even	8	inner	672.2.cj.a.109.38	yes	512
224.205	even	24	inner	672.2.cj.a.205.6	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.cj.a.37.38	✓	512	7.2	even	3	inner
672.2.cj.a.109.38	yes	512	32.13	even	8	inner
672.2.cj.a.205.6	yes	512	224.205	even	24	inner
672.2.cj.a.613.6	yes	512	1.1	even	1	trivial