Embedded newform 672.2.cj.a.613.11

• Label: 672.2.cj.a.613.11
• 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			613.11
Character	\(\chi\)	\(=\)	672.613
Dual form			672.2.cj.a.205.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24675 - 0.667536i) q^{2} +(0.991445 + 0.130526i) q^{3} +(1.10879 + 1.66451i) q^{4} +(0.0406300 + 0.308616i) q^{5} +(-1.14896 - 0.824559i) q^{6} +(-2.20129 - 1.46775i) q^{7} +(-0.271274 - 2.81539i) q^{8} +(0.965926 + 0.258819i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24675	−	0.667536i	−0.881588	−	0.472019i
							\(3\)	0.991445	+	0.130526i	0.572411	+	0.0753593i
\(5\)	0.0406300	+	0.308616i	0.0181703	+	0.138017i								0.998125	−	0.0612011i	\(-0.0194931\pi\)
							−0.979955	+	0.199218i	\(0.936160\pi\)
\(7\)	−2.20129	−	1.46775i	−0.832011	−	0.554759i
							\(11\)	−1.82454	−	2.37778i	−0.550119	−	0.716929i	0.432692	−	0.901542i	\(-0.357564\pi\)
−0.982811	+	0.184613i	\(0.940897\pi\)
\(13\)	3.46803	−	1.43651i	0.961859	−	0.398415i	0.154184	−	0.988042i	\(-0.450725\pi\)
							0.807676	+	0.589627i	\(0.200725\pi\)
\(17\)	−3.10055	−	1.79010i	−0.751994	−	0.434164i	0.0744197	−	0.997227i	\(-0.476290\pi\)
							−0.826414	+	0.563063i	\(0.809623\pi\)
\(19\)	−2.83303	−	2.17386i	−0.649941	−	0.498717i	0.230377	−	0.973101i	\(-0.426004\pi\)
							−0.880318	+	0.474384i	\(0.842671\pi\)
\(23\)	−2.54404	−	0.681672i	−0.530468	−	0.142139i	−0.0163637	−	0.999866i	\(-0.505209\pi\)
							−0.514104	+	0.857728i	\(0.671876\pi\)
\(29\)	−2.49387	−	6.02073i	−0.463100	−	1.11802i	−0.967118	−	0.254328i	\(-0.918146\pi\)
							0.504018	−	0.863693i	\(-0.331854\pi\)
\(31\)	−0.213193	+	0.369260i	−0.0382905	+	0.0663211i	−0.884536	−	0.466473i	\(-0.845525\pi\)
							0.846245	+	0.532794i	\(0.178858\pi\)
\(37\)	0.683796	+	5.19395i	0.112415	+	0.853879i	0.951018	+	0.309136i	\(0.100040\pi\)
							−0.838603	+	0.544744i	\(0.816627\pi\)
\(41\)	5.82871	−	5.82871i	0.910292	−	0.910292i	−0.0860030	−	0.996295i	\(-0.527409\pi\)
							0.996295	+	0.0860030i	\(0.0274095\pi\)
\(43\)	3.41160	−	8.23634i	0.520265	−	1.25603i	−0.417474	−	0.908689i	\(-0.637085\pi\)
							0.937739	−	0.347341i	\(-0.112915\pi\)
\(47\)	1.55873	−	0.899935i	0.227365	−	0.131269i	−0.381991	−	0.924166i	\(-0.624761\pi\)
							0.609356	+	0.792897i	\(0.291428\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.cj.a.613.11	yes	512
7.2	even	3	inner	672.2.cj.a.37.34	✓	512
32.13	even	8	inner	672.2.cj.a.109.34	yes	512
224.205	even	24	inner	672.2.cj.a.205.11	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.cj.a.37.34	✓	512	7.2	even	3	inner
672.2.cj.a.109.34	yes	512	32.13	even	8	inner
672.2.cj.a.205.11	yes	512	224.205	even	24	inner
672.2.cj.a.613.11	yes	512	1.1	even	1	trivial