Embedded newform 672.2.ch.a.11.3

• Label: 672.2.ch.a.11.3
• Level: $672$
• Weight: $2$
• Character: 672.11
• Analytic conductor: $5.366$
• Analytic rank: $0$
• Dimension: $992$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.ch (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.36594701583\)
Analytic rank:	\(0\)
Dimension:	\(992\)
Relative dimension:	\(124\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			11.3
Character	\(\chi\)	\(=\)	672.11
Dual form			672.2.ch.a.611.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41292 + 0.0604964i) q^{2} +(0.856964 - 1.50520i) q^{3} +(1.99268 - 0.170953i) q^{4} +(0.139662 + 1.06084i) q^{5} +(-1.11976 + 2.17856i) q^{6} +(-1.22300 + 2.34612i) q^{7} +(-2.80515 + 0.362093i) q^{8} +(-1.53123 - 2.57980i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 4 q^{3} - 8 q^{4} - 16 q^{6} - 16 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(421\)	\(449\)	\(577\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41292	+	0.0604964i	−0.999085	+	0.0427774i
							\(3\)	0.856964	−	1.50520i	0.494768	−	0.869025i
\(5\)	0.139662	+	1.06084i	0.0624586	+	0.474420i								0.993879	+	0.110475i	\(0.0352372\pi\)
							−0.931420	+	0.363945i	\(0.881429\pi\)
\(7\)	−1.22300	+	2.34612i	−0.462250	+	0.886750i
							\(11\)	0.413520	−	0.317305i	0.124681	−	0.0956711i	−0.544539	−	0.838736i	\(-0.683295\pi\)
0.669220	+	0.743065i	\(0.266629\pi\)
\(13\)	1.25306	+	3.02514i	0.347535	+	0.839024i	0.996910	+	0.0785551i	\(0.0250307\pi\)
							−0.649375	+	0.760469i	\(0.724969\pi\)
\(17\)	0.0157601	−	0.0272973i	0.00382238	−	0.00662056i	−0.864108	−	0.503307i	\(-0.832117\pi\)
							0.867930	+	0.496686i	\(0.165450\pi\)
\(19\)	5.07010	+	3.89043i	1.16316	+	0.892525i	0.995634	−	0.0933427i	\(-0.0297552\pi\)
							0.167527	+	0.985868i	\(0.446422\pi\)
\(23\)	0.253476	+	0.0679187i	0.0528534	+	0.0141620i	0.285149	−	0.958483i	\(-0.407957\pi\)
							−0.232295	+	0.972645i	\(0.574624\pi\)
\(29\)	2.46066	+	5.94056i	0.456934	+	1.10314i	0.969632	+	0.244567i	\(0.0786457\pi\)
							−0.512699	+	0.858568i	\(0.671354\pi\)
\(31\)	1.33288	+	0.769539i	0.239392	+	0.138213i	0.614897	−	0.788607i	\(-0.289197\pi\)
							−0.375505	+	0.926820i	\(0.622531\pi\)
\(37\)	−4.35538	+	0.573396i	−0.716020	+	0.0942658i	−0.479729	−	0.877417i	\(-0.659265\pi\)
							−0.236291	+	0.971682i	\(0.575932\pi\)
\(41\)	7.44578	+	7.44578i	1.16284	+	1.16284i	0.983852	+	0.178983i	\(0.0572808\pi\)
							0.178983	+	0.983852i	\(0.442719\pi\)
\(43\)	4.07053	−	9.82714i	0.620750	−	1.49862i	−0.230073	−	0.973173i	\(-0.573897\pi\)
							0.850824	−	0.525451i	\(-0.176103\pi\)
\(47\)	−1.61842	+	0.934396i	−0.236071	+	0.136296i	−0.613370	−	0.789796i	\(-0.710186\pi\)
							0.377299	+	0.926092i	\(0.376853\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.ch.a.11.3	✓	992
3.2	odd	2	inner	672.2.ch.a.11.122	yes	992
7.2	even	3	inner	672.2.ch.a.107.85	yes	992
21.2	odd	6	inner	672.2.ch.a.107.40	yes	992
32.3	odd	8	inner	672.2.ch.a.515.40	yes	992
96.35	even	8	inner	672.2.ch.a.515.85	yes	992
224.163	odd	24	inner	672.2.ch.a.611.122	yes	992
672.611	even	24	inner	672.2.ch.a.611.3	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.ch.a.11.3	✓	992	1.1	even	1	trivial
672.2.ch.a.11.122	yes	992	3.2	odd	2	inner
672.2.ch.a.107.40	yes	992	21.2	odd	6	inner
672.2.ch.a.107.85	yes	992	7.2	even	3	inner
672.2.ch.a.515.40	yes	992	32.3	odd	8	inner
672.2.ch.a.515.85	yes	992	96.35	even	8	inner
672.2.ch.a.611.3	yes	992	672.611	even	24	inner
672.2.ch.a.611.122	yes	992	224.163	odd	24	inner