Embedded newform 22.9.d.a.17.4

• Label: 22.9.d.a.17.4
• Level: $22$
• Weight: $9$
• Character: 22.17
• 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			17.4
Character	\(\chi\)	\(=\)	22.17
Dual form			22.9.d.a.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.7600 + 3.49613i) q^{2} +(67.1008 + 48.7516i) q^{3} +(103.554 - 75.2365i) q^{4} +(202.501 - 623.233i) q^{5} +(-892.445 - 289.973i) q^{6} +(-159.403 - 219.399i) q^{7} +(-851.204 + 1171.58i) q^{8} +(98.3408 + 302.662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 182 q^{3} + 1024 q^{4} + 1410 q^{5} + 4480 q^{6} - 10950 q^{7} - 1402 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)

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
							\(3\)	67.1008	+	48.7516i	0.828405	+	0.601872i	0.919108	−	0.394006i	\(-0.128911\pi\)
−0.0907025	+	0.995878i	\(0.528911\pi\)
\(5\)	202.501	−	623.233i	0.324001	−	0.997172i	−0.647889	−	0.761735i	\(-0.724348\pi\)
							0.971890	−	0.235437i	\(-0.0756522\pi\)
\(7\)	−159.403	−	219.399i	−0.0663900	−	0.0913781i	0.774531	−	0.632536i	\(-0.217986\pi\)
							−0.840921	+	0.541158i	\(0.817986\pi\)
\(11\)	13100.4	−	6537.54i	0.894772	−	0.446523i
							\(13\)	26853.1	−	8725.11i	0.940203	−	0.305490i	0.201474	−	0.979494i	\(-0.435427\pi\)
0.738728	+	0.674003i	\(0.235427\pi\)
\(17\)	129962.	+	42227.3i	1.55604	+	0.505590i	0.955748	−	0.294188i	\(-0.0950491\pi\)
							0.600297	+	0.799777i	\(0.295049\pi\)
\(19\)	−35767.8	+	49230.2i	−0.274459	+	0.377761i	−0.923889	−	0.382661i	\(-0.875008\pi\)
							0.649430	+	0.760422i	\(0.275008\pi\)
\(23\)	−271641.			−0.970697			−0.485349	−	0.874321i	\(-0.661307\pi\)
							−0.485349	+	0.874321i	\(0.661307\pi\)
\(29\)	174326.	+	239940.i	0.246474	+	0.339242i	0.914272	−	0.405100i	\(-0.132763\pi\)
							−0.667799	+	0.744342i	\(0.732763\pi\)
\(31\)	−258483.	−	795528.i	−0.279888	−	0.861407i	−0.987884	−	0.155191i	\(-0.950401\pi\)
							0.707996	−	0.706216i	\(-0.249599\pi\)
\(37\)	−1.00508e6	+	730232.i	−0.536281	+	0.389631i	−0.822702	−	0.568473i	\(-0.807534\pi\)
							0.286421	+	0.958104i	\(0.407534\pi\)
\(41\)	−764101.	+	1.05170e6i	−0.270406	+	0.372181i	−0.922527	−	0.385934i	\(-0.873879\pi\)
							0.652121	+	0.758115i	\(0.273879\pi\)
\(43\)		−	3.02970e6i		−	0.886189i	−0.896475	−	0.443094i	\(-0.853881\pi\)
							0.896475	−	0.443094i	\(-0.146119\pi\)
\(47\)	−1.52747e6	−	1.10977e6i	−0.313027	−	0.227428i	0.420167	−	0.907447i	\(-0.361971\pi\)
							−0.733194	+	0.680019i	\(0.761971\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.9.d.a.17.4	yes	32
11.2	odd	10	inner	22.9.d.a.13.4	✓	32
11.3	even	5		242.9.b.e.241.18		32
11.8	odd	10		242.9.b.e.241.17		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.9.d.a.13.4	✓	32	11.2	odd	10	inner
22.9.d.a.17.4	yes	32	1.1	even	1	trivial
242.9.b.e.241.17		32	11.8	odd	10
242.9.b.e.241.18		32	11.3	even	5