Embedded newform 22.9.d.a.7.2

• Label: 22.9.d.a.7.2
• Level: $22$
• Weight: $9$
• Character: 22.7
• Analytic conductor: $8.962$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	22.d (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.96232942134\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			7.2
Character	\(\chi\)	\(=\)	22.7
Dual form			22.9.d.a.19.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.65003 + 9.15298i) q^{2} +(-20.1856 + 62.1248i) q^{3} +(-39.5542 - 121.735i) q^{4} +(927.050 - 673.541i) q^{5} +(-434.393 - 597.890i) q^{6} +(320.984 - 104.294i) q^{7} +(1377.28 + 447.504i) q^{8} +(1855.92 + 1348.41i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 182 q^{3} + 1024 q^{4} + 1410 q^{5} + 4480 q^{6} - 10950 q^{7} - 1402 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.65003	+	9.15298i	−0.415627	+	0.572061i
							\(3\)	−20.1856	+	62.1248i	−0.249205	+	0.766973i	0.745712	+	0.666269i	\(0.232110\pi\)
−0.994916	+	0.100704i	\(0.967890\pi\)
\(5\)	927.050	−	673.541i	1.48328	−	1.07767i	0.506798	−	0.862065i	\(-0.330829\pi\)
							0.976481	−	0.215601i	\(-0.0691711\pi\)
\(7\)	320.984	−	104.294i	0.133687	−	0.0434377i	−0.241409	−	0.970423i	\(-0.577610\pi\)
							0.375097	+	0.926986i	\(0.377610\pi\)
\(11\)	−6886.83	+	12920.2i	−0.470380	+	0.882464i
							\(13\)	15745.6	−	21672.0i	0.551298	−	0.758796i	−0.438890	−	0.898541i	\(-0.644628\pi\)
0.990188	+	0.139745i	\(0.0446282\pi\)
\(17\)	59602.7	+	82036.1i	0.713626	+	0.982221i	0.999712	+	0.0240174i	\(0.00764573\pi\)
							−0.286086	+	0.958204i	\(0.592354\pi\)
\(19\)	133297.	+	43310.7i	1.02283	+	0.332339i	0.771953	−	0.635680i	\(-0.219280\pi\)
							0.250880	+	0.968018i	\(0.419280\pi\)
\(23\)	−168451.			−0.601952			−0.300976	−	0.953632i	\(-0.597312\pi\)
							−0.300976	+	0.953632i	\(0.597312\pi\)
\(29\)	904383.	−	293852.i	1.27868	−	0.415467i	0.410562	−	0.911833i	\(-0.365333\pi\)
							0.868113	+	0.496366i	\(0.165333\pi\)
\(31\)	−67386.9	−	48959.5i	−0.0729674	−	0.0530139i	0.550704	−	0.834701i	\(-0.314359\pi\)
							−0.623671	+	0.781687i	\(0.714359\pi\)
\(37\)	584965.	+	1.80034e6i	0.312121	+	0.960610i	0.976923	+	0.213590i	\(0.0685158\pi\)
							−0.664802	+	0.747019i	\(0.731484\pi\)
\(41\)	196917.	+	63982.1i	0.0696862	+	0.0226424i	0.343653	−	0.939097i	\(-0.388336\pi\)
							−0.273966	+	0.961739i	\(0.588336\pi\)
\(43\)			1.81948e6i			0.532200i	0.963945	+	0.266100i	\(0.0857351\pi\)
							−0.963945	+	0.266100i	\(0.914265\pi\)
\(47\)	2.51259e6	−	7.73295e6i	0.514908	−	1.58472i	−0.268541	−	0.963268i	\(-0.586541\pi\)
							0.783449	−	0.621456i	\(-0.213459\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.9.d.a.7.2	✓	32
11.5	even	5		242.9.b.e.241.13		32
11.6	odd	10		242.9.b.e.241.14		32
11.8	odd	10	inner	22.9.d.a.19.2	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.9.d.a.7.2	✓	32	1.1	even	1	trivial
22.9.d.a.19.2	yes	32	11.8	odd	10	inner
242.9.b.e.241.13		32	11.5	even	5
242.9.b.e.241.14		32	11.6	odd	10