Embedded newform 22.9.d.a.13.6

• Label: 22.9.d.a.13.6
• 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			13.6
Character	\(\chi\)	\(=\)	22.13
Dual form			22.9.d.a.17.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.7600 + 3.49613i) q^{2} +(-23.2078 + 16.8615i) q^{3} +(103.554 + 75.2365i) q^{4} +(88.9820 + 273.858i) q^{5} +(-308.665 + 100.291i) q^{6} +(-550.185 + 757.265i) q^{7} +(851.204 + 1171.58i) q^{8} +(-1773.17 + 5457.25i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 182 q^{3} + 1024 q^{4} + 1410 q^{5} + 4480 q^{6} - 10950 q^{7} - 1402 q^{9}+O(q^{10}) \)

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
							\(3\)	−23.2078	+	16.8615i	−0.286516	+	0.208166i	−0.721755	−	0.692149i	\(-0.756664\pi\)
0.435238	+	0.900315i	\(0.356664\pi\)
\(5\)	88.9820	+	273.858i	0.142371	+	0.438173i	0.996664	−	0.0816193i	\(-0.0260092\pi\)
							−0.854292	+	0.519793i	\(0.826009\pi\)
\(7\)	−550.185	+	757.265i	−0.229148	+	0.315395i	−0.908073	−	0.418813i	\(-0.862446\pi\)
							0.678924	+	0.734208i	\(0.262446\pi\)
\(11\)	−4522.48	+	13925.0i	−0.308891	+	0.951097i
							\(13\)	2680.40	+	870.913i	0.0938481	+	0.0304931i	0.355565	−	0.934652i	\(-0.384289\pi\)
−0.261717	+	0.965145i	\(0.584289\pi\)
\(17\)	758.471	−	246.442i	0.00908121	−	0.00295066i	−0.304473	−	0.952521i	\(-0.598480\pi\)
							0.313554	+	0.949570i	\(0.398480\pi\)
\(19\)	35124.3	+	48344.5i	0.269522	+	0.370965i	0.922228	−	0.386646i	\(-0.126366\pi\)
							−0.652706	+	0.757611i	\(0.726366\pi\)
\(23\)	38167.1			0.136388			0.0681942	−	0.997672i	\(-0.478276\pi\)
							0.0681942	+	0.997672i	\(0.478276\pi\)
\(29\)	498127.	−	685614.i	0.704285	−	0.969365i	−0.295616	−	0.955307i	\(-0.595525\pi\)
							0.999901	−	0.0140583i	\(-0.00447505\pi\)
\(31\)	376417.	−	1.15849e6i	0.407589	−	1.25443i	−0.511125	−	0.859507i	\(-0.670771\pi\)
							0.918714	−	0.394924i	\(-0.129229\pi\)
\(37\)	19897.8	+	14456.6i	0.0106169	+	0.00771362i	0.593081	−	0.805143i	\(-0.297911\pi\)
							−0.582464	+	0.812856i	\(0.697911\pi\)
\(41\)	1.17973e6	+	1.62375e6i	0.417489	+	0.574625i	0.965025	−	0.262157i	\(-0.0844340\pi\)
							−0.547536	+	0.836782i	\(0.684434\pi\)
\(43\)			3.69440e6i			1.08061i	0.841469	+	0.540306i	\(0.181691\pi\)
							−0.841469	+	0.540306i	\(0.818309\pi\)
\(47\)	−2.22400e6	+	1.61583e6i	−0.455767	+	0.331134i	−0.791868	−	0.610692i	\(-0.790891\pi\)
							0.336102	+	0.941826i	\(0.390891\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.9.d.a.13.6	✓	32
11.4	even	5		242.9.b.e.241.22		32
11.6	odd	10	inner	22.9.d.a.17.6	yes	32
11.7	odd	10		242.9.b.e.241.21		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.9.d.a.13.6	✓	32	1.1	even	1	trivial
22.9.d.a.17.6	yes	32	11.6	odd	10	inner
242.9.b.e.241.21		32	11.7	odd	10
242.9.b.e.241.22		32	11.4	even	5