Embedded newform 22.9.d.a.13.8

• Label: 22.9.d.a.13.8
• Level: $22$
• Weight: $9$
• Character: 22.13
• 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			13.8
Character	\(\chi\)	\(=\)	22.13
Dual form			22.9.d.a.17.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.7600 + 3.49613i) q^{2} +(123.182 - 89.4972i) q^{3} +(103.554 + 75.2365i) q^{4} +(310.287 + 954.964i) q^{5} +(1638.33 - 532.327i) q^{6} +(-525.022 + 722.631i) q^{7} +(851.204 + 1171.58i) q^{8} +(5136.68 - 15809.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 182 q^{3} + 1024 q^{4} + 1410 q^{5} + 4480 q^{6} - 10950 q^{7} - 1402 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.7600	+	3.49613i	0.672499	+	0.218508i
							\(3\)	123.182	−	89.4972i	1.52077	−	1.10490i	0.559665	−	0.828719i	\(-0.310930\pi\)
0.961104	−	0.276185i	\(-0.0890703\pi\)
\(5\)	310.287	+	954.964i	0.496459	+	1.52794i	0.814671	+	0.579923i	\(0.196917\pi\)
							−0.318213	+	0.948019i	\(0.603083\pi\)
\(7\)	−525.022	+	722.631i	−0.218668	+	0.300971i	−0.904232	−	0.427042i	\(-0.859556\pi\)
							0.685564	+	0.728013i	\(0.259556\pi\)
\(11\)	−12034.4	−	8338.62i	−0.821965	−	0.569539i
							\(13\)	4985.66	+	1619.94i	0.174562	+	0.0567186i	0.394994	−	0.918684i	\(-0.370747\pi\)
−0.220432	+	0.975402i	\(0.570747\pi\)
\(17\)	−35490.8	+	11531.7i	−0.424933	+	0.138069i	−0.513673	−	0.857986i	\(-0.671716\pi\)
							0.0887408	+	0.996055i	\(0.471716\pi\)
\(19\)	−130435.	−	179529.i	−1.00088	−	1.37759i	−0.924785	−	0.380491i	\(-0.875755\pi\)
							−0.0760938	−	0.997101i	\(-0.524245\pi\)
\(23\)	273286.			0.976577			0.488288	−	0.872682i	\(-0.337621\pi\)
							0.488288	+	0.872682i	\(0.337621\pi\)
\(29\)	−429891.	+	591695.i	−0.607809	+	0.836577i	−0.996395	−	0.0848355i	\(-0.972964\pi\)
							0.388586	+	0.921412i	\(0.372964\pi\)
\(31\)	−39533.0	+	121670.i	−0.0428068	+	0.131746i	−0.970176	−	0.242402i	\(-0.922065\pi\)
							0.927369	+	0.374148i	\(0.122065\pi\)
\(37\)	−631073.	−	458501.i	−0.336723	−	0.244643i	0.406555	−	0.913626i	\(-0.366730\pi\)
							−0.743278	+	0.668983i	\(0.766730\pi\)
\(41\)	1.53276e6	+	2.10966e6i	0.542424	+	0.746583i	0.988960	−	0.148183i	\(-0.0473424\pi\)
							−0.446536	+	0.894766i	\(0.647342\pi\)
\(43\)		−	1.91135e6i		−	0.559070i	−0.960136	−	0.279535i	\(-0.909820\pi\)
							0.960136	−	0.279535i	\(-0.0901803\pi\)
\(47\)	−1.60258e6	+	1.16434e6i	−0.328419	+	0.238610i	−0.739759	−	0.672871i	\(-0.765061\pi\)
							0.411341	+	0.911482i	\(0.365061\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.9.d.a.13.8	✓	32
11.4	even	5		242.9.b.e.241.10		32
11.6	odd	10	inner	22.9.d.a.17.8	yes	32
11.7	odd	10		242.9.b.e.241.9		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.9.d.a.13.8	✓	32	1.1	even	1	trivial
22.9.d.a.17.8	yes	32	11.6	odd	10	inner
242.9.b.e.241.9		32	11.7	odd	10
242.9.b.e.241.10		32	11.4	even	5