Embedded newform 882.2.y.a.571.26

• Label: 882.2.y.a.571.26
• Level: $882$
• Weight: $2$
• Character: 882.571
• Analytic conductor: $7.043$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.y (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.04280545828\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			571.26
Character	\(\chi\)	\(=\)	882.571
Dual form			882.2.y.a.709.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.826239 + 0.563320i) q^{2} +(1.57735 + 0.715520i) q^{3} +(0.365341 - 0.930874i) q^{4} +(-0.148541 - 0.650800i) q^{5} +(-1.70633 + 0.297362i) q^{6} +(2.62085 + 0.362116i) q^{7} +(0.222521 + 0.974928i) q^{8} +(1.97606 + 2.25725i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{2} - 13 q^{3} + 28 q^{4} - 4 q^{5} - 5 q^{6} - q^{7} + 56 q^{8} + 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.826239	+	0.563320i	−0.584239	+	0.398327i
							\(3\)	1.57735	+	0.715520i	0.910683	+	0.413106i
\(5\)	−0.148541	−	0.650800i	−0.0664294	−	0.291046i								0.930791	−	0.365551i	\(-0.119120\pi\)
							−0.997221	+	0.0745049i	\(0.976262\pi\)
\(7\)	2.62085	+	0.362116i	0.990589	+	0.136867i
							\(11\)	−2.04145	−	0.983113i	−0.615522	−	0.296420i	0.100025	−	0.994985i	\(-0.468108\pi\)
−0.715546	+	0.698565i	\(0.753822\pi\)
\(13\)	−5.34073	+	3.64125i	−1.48125	+	1.00990i	−0.490918	+	0.871206i	\(0.663339\pi\)
							−0.990335	+	0.138695i	\(0.955709\pi\)
\(17\)	1.98282	+	5.05215i	0.480905	+	1.22533i	0.941288	+	0.337605i	\(0.109617\pi\)
							−0.460383	+	0.887720i	\(0.652288\pi\)
\(19\)	3.77562	+	6.53957i	0.866187	+	1.50028i	0.865863	+	0.500281i	\(0.166770\pi\)
							0.000323945		1.00000i	\(0.499897\pi\)
\(23\)	−5.24690	−	6.57940i	−1.09405	−	1.37190i	−0.922171	−	0.386782i	\(-0.873587\pi\)
							−0.171882	−	0.985117i	\(-0.554985\pi\)
\(29\)	3.30759	+	0.498539i	0.614204	+	0.0925763i	0.448774	−	0.893645i	\(-0.351861\pi\)
							0.165430	+	0.986222i	\(0.447099\pi\)
\(31\)	2.25489	+	3.90558i	0.404990	+	0.701463i	0.994320	−	0.106429i	\(-0.0339417\pi\)
							−0.589330	+	0.807892i	\(0.700608\pi\)
\(37\)	1.29323	+	0.194923i	0.212606	+	0.0320452i	0.254482	−	0.967078i	\(-0.418095\pi\)
							−0.0418754	+	0.999123i	\(0.513333\pi\)
\(41\)	6.92553	+	2.13624i	1.08159	+	0.333625i	0.783804	−	0.621009i	\(-0.213277\pi\)
							0.297782	+	0.954634i	\(0.403753\pi\)
\(43\)	3.50865	−	1.08228i	0.535065	−	0.165046i	−0.0154352	−	0.999881i	\(-0.504913\pi\)
							0.550500	+	0.834835i	\(0.314437\pi\)
\(47\)	7.65525	−	5.21926i	1.11663	−	0.761308i	0.143067	−	0.989713i	\(-0.454303\pi\)
							0.973566	+	0.228405i	\(0.0733511\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.y.a.571.26	✓	336
9.7	even	3		882.2.bb.b.277.26	yes	336
49.23	even	21		882.2.bb.b.121.26	yes	336
441.268	even	21	inner	882.2.y.a.709.26	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.y.a.571.26	✓	336	1.1	even	1	trivial
882.2.y.a.709.26	yes	336	441.268	even	21	inner
882.2.bb.b.121.26	yes	336	49.23	even	21
882.2.bb.b.277.26	yes	336	9.7	even	3