Embedded newform 672.2.cj.a.445.21

• Label: 672.2.cj.a.445.21
• 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			445.21
Character	\(\chi\)	\(=\)	672.445
Dual form			672.2.cj.a.373.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.769477 - 1.18655i) q^{2} +(-0.793353 + 0.608761i) q^{3} +(-0.815810 + 1.82605i) q^{4} +(-0.656043 + 0.854972i) q^{5} +(1.33279 + 0.472927i) q^{6} +(-1.61894 - 2.09261i) q^{7} +(2.79445 - 0.437101i) q^{8} +(0.258819 - 0.965926i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.769477	−	1.18655i	−0.544102	−	0.839019i
							\(3\)	−0.793353	+	0.608761i	−0.458043	+	0.351469i
\(5\)	−0.656043	+	0.854972i	−0.293392	+	0.382355i								−0.916670	−	0.399645i	\(-0.869134\pi\)
							0.623279	+	0.782000i	\(0.285800\pi\)
\(7\)	−1.61894	−	2.09261i	−0.611903	−	0.790933i
							\(11\)	0.0687624	−	0.522302i	0.0207326	−	0.157480i	−0.977947	−	0.208854i	\(-0.933027\pi\)
0.998679	+	0.0513738i	\(0.0163600\pi\)
\(13\)	0.136320	−	0.329106i	0.0378084	−	0.0912776i	−0.903847	−	0.427855i	\(-0.859269\pi\)
							0.941656	+	0.336577i	\(0.109269\pi\)
\(17\)	−0.277999	−	0.160503i	−0.0674248	−	0.0389277i	0.465909	−	0.884833i	\(-0.345728\pi\)
							−0.533333	+	0.845905i	\(0.679061\pi\)
\(19\)	1.79731	−	0.236620i	0.412331	−	0.0542844i	0.0784924	−	0.996915i	\(-0.474989\pi\)
							0.333838	+	0.942630i	\(0.391656\pi\)
\(23\)	−1.81094	+	6.75853i	−0.377608	+	1.40925i	0.471889	+	0.881658i	\(0.343572\pi\)
							−0.849497	+	0.527594i	\(0.823094\pi\)
\(29\)	7.66980	+	3.17694i	1.42425	+	0.589942i	0.955924	−	0.293615i	\(-0.0948585\pi\)
							0.468323	+	0.883558i	\(0.344858\pi\)
\(31\)	0.0700522	−	0.121334i	0.0125818	−	0.0217922i	−0.859666	−	0.510857i	\(-0.829328\pi\)
							0.872248	+	0.489064i	\(0.162662\pi\)
\(37\)	2.91012	−	3.79255i	0.478421	−	0.623491i	−0.490326	−	0.871539i	\(-0.663122\pi\)
							0.968748	+	0.248048i	\(0.0797891\pi\)
\(41\)	7.65548	+	7.65548i	1.19558	+	1.19558i	0.975475	+	0.220110i	\(0.0706416\pi\)
							0.220110	+	0.975475i	\(0.429358\pi\)
\(43\)	9.87475	−	4.09025i	1.50588	−	0.623758i	0.531181	−	0.847259i	\(-0.321749\pi\)
							0.974704	+	0.223501i	\(0.0717485\pi\)
\(47\)	−9.73108	+	5.61824i	−1.41942	+	0.819505i	−0.996248	−	0.0865425i	\(-0.972418\pi\)
							−0.423176	+	0.906047i	\(0.639085\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.cj.a.445.21	yes	512
7.2	even	3	inner	672.2.cj.a.541.21	yes	512
32.21	even	8	inner	672.2.cj.a.277.21	✓	512
224.149	even	24	inner	672.2.cj.a.373.21	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.cj.a.277.21	✓	512	32.21	even	8	inner
672.2.cj.a.373.21	yes	512	224.149	even	24	inner
672.2.cj.a.445.21	yes	512	1.1	even	1	trivial
672.2.cj.a.541.21	yes	512	7.2	even	3	inner