Embedded newform 882.2.bc.a.563.52

• Label: 882.2.bc.a.563.52
• Level: $882$
• Weight: $2$
• Character: 882.563
• Analytic conductor: $7.043$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.bc (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.04280545828\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(56\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			563.52
Character	\(\chi\)	\(=\)	882.563
Dual form			882.2.bc.a.47.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.680173 - 0.733052i) q^{2} +(1.33939 - 1.09820i) q^{3} +(-0.0747301 - 0.997204i) q^{4} +(-0.764115 + 0.958170i) q^{5} +(0.105978 - 1.72881i) q^{6} +(-2.62352 - 0.342277i) q^{7} +(-0.781831 - 0.623490i) q^{8} +(0.587918 - 2.94183i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q - 56 q^{4} + 14 q^{6} - 2 q^{7} - 26 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.680173	−	0.733052i	0.480955	−	0.518346i
							\(3\)	1.33939	−	1.09820i	0.773296	−	0.634046i
\(5\)	−0.764115	+	0.958170i	−0.341723	+	0.428507i								−0.922763	−	0.385368i	\(-0.874074\pi\)
							0.581040	+	0.813875i	\(0.302646\pi\)
\(7\)	−2.62352	−	0.342277i	−0.991597	−	0.129369i
							\(11\)	−5.88408	−	1.34300i	−1.77412	−	0.404931i	−0.794737	−	0.606953i	\(-0.792391\pi\)
−0.979381	+	0.202022i	\(0.935249\pi\)
\(13\)	2.69317	−	2.90255i	0.746951	−	0.805021i	−0.239183	−	0.970974i	\(-0.576880\pi\)
							0.986134	+	0.165953i	\(0.0530701\pi\)
\(17\)	0.143807	−	1.91898i	0.0348784	−	0.465420i	−0.952232	−	0.305377i	\(-0.901218\pi\)
							0.987110	−	0.160043i	\(-0.0511634\pi\)
\(19\)	−4.52761	+	2.61402i	−1.03870	+	0.599696i	−0.919466	−	0.393169i	\(-0.871379\pi\)
							−0.119239	+	0.992866i	\(0.538045\pi\)
\(23\)	3.23815	−	6.72409i	0.675201	−	1.40207i	−0.228346	−	0.973580i	\(-0.573332\pi\)
							0.903547	−	0.428489i	\(-0.140954\pi\)
\(29\)	1.11162	−	1.63045i	0.206422	−	0.302766i	−0.709041	−	0.705167i	\(-0.750872\pi\)
							0.915463	+	0.402401i	\(0.131824\pi\)
\(31\)	−0.950942	+	0.549026i	−0.170794	+	0.0986081i	−0.582960	−	0.812501i	\(-0.698106\pi\)
							0.412166	+	0.911109i	\(0.364772\pi\)
\(37\)	−0.411677	−	0.280677i	−0.0676794	−	0.0461430i	0.529008	−	0.848617i	\(-0.322564\pi\)
							−0.596687	+	0.802474i	\(0.703517\pi\)
\(41\)	2.14231	+	5.45851i	0.334572	+	0.852475i	0.994841	+	0.101446i	\(0.0323470\pi\)
							−0.660269	+	0.751029i	\(0.729558\pi\)
\(43\)	3.81558	−	9.72193i	0.581870	−	1.48258i	−0.272105	−	0.962268i	\(-0.587720\pi\)
							0.853975	−	0.520314i	\(-0.174185\pi\)
\(47\)	−1.39211	−	1.29169i	−0.203060	−	0.188412i	0.572087	−	0.820193i	\(-0.306134\pi\)
							−0.775147	+	0.631781i	\(0.782324\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.bc.a.563.52	yes	672
9.2	odd	6		882.2.bk.a.857.56	yes	672
49.47	odd	42		882.2.bk.a.635.56	yes	672
441.47	even	42	inner	882.2.bc.a.47.52	✓	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.bc.a.47.52	✓	672	441.47	even	42	inner
882.2.bc.a.563.52	yes	672	1.1	even	1	trivial
882.2.bk.a.635.56	yes	672	49.47	odd	42
882.2.bk.a.857.56	yes	672	9.2	odd	6