Embedded newform 22.9.d.a.13.7

• Label: 22.9.d.a.13.7
• 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			13.7
Character	\(\chi\)	\(=\)	22.13
Dual form			22.9.d.a.17.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.7600 + 3.49613i) q^{2} +(43.1191 - 31.3278i) q^{3} +(103.554 + 75.2365i) q^{4} +(-227.453 - 700.028i) q^{5} +(573.486 - 186.337i) q^{6} +(1739.62 - 2394.38i) q^{7} +(851.204 + 1171.58i) q^{8} +(-1149.64 + 3538.23i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 182 q^{3} + 1024 q^{4} + 1410 q^{5} + 4480 q^{6} - 10950 q^{7} - 1402 q^{9}+O(q^{10}) \)

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
							\(3\)	43.1191	−	31.3278i	0.532334	−	0.386763i	−0.288896	−	0.957360i	\(-0.593288\pi\)
0.821230	+	0.570597i	\(0.193288\pi\)
\(5\)	−227.453	−	700.028i	−0.363925	−	1.12004i	−0.950652	−	0.310260i	\(-0.899584\pi\)
							0.586727	−	0.809785i	\(-0.300416\pi\)
\(7\)	1739.62	−	2394.38i	0.724539	−	0.997242i	−0.274822	−	0.961495i	\(-0.588619\pi\)
							0.999361	−	0.0357468i	\(-0.0113810\pi\)
\(11\)	−1531.32	−	14560.7i	−0.104591	−	0.994515i
							\(13\)	22015.2	+	7153.19i	0.770815	+	0.250453i	0.667914	−	0.744239i	\(-0.267188\pi\)
0.102901	+	0.994692i	\(0.467188\pi\)
\(17\)	72333.3	−	23502.5i	0.866049	−	0.281397i	0.157897	−	0.987456i	\(-0.449529\pi\)
							0.708153	+	0.706059i	\(0.249529\pi\)
\(19\)	59650.8	+	82102.3i	0.457722	+	0.630000i	0.974034	−	0.226400i	\(-0.0726957\pi\)
							−0.516312	+	0.856400i	\(0.672696\pi\)
\(23\)	−465898.			−1.66487			−0.832433	−	0.554126i	\(-0.813052\pi\)
							−0.832433	+	0.554126i	\(0.813052\pi\)
\(29\)	−619281.	+	852367.i	−0.875580	+	1.20513i	0.102046	+	0.994780i	\(0.467461\pi\)
							−0.977626	+	0.210352i	\(0.932539\pi\)
\(31\)	−280480.	+	863227.i	−0.303707	+	0.934713i	0.676450	+	0.736489i	\(0.263518\pi\)
							−0.980157	+	0.198225i	\(0.936482\pi\)
\(37\)	336886.	+	244762.i	0.179753	+	0.130598i	0.674023	−	0.738710i	\(-0.264565\pi\)
							−0.494270	+	0.869308i	\(0.664565\pi\)
\(41\)	1.14673e6	+	1.57834e6i	0.405814	+	0.558554i	0.962191	−	0.272375i	\(-0.0878092\pi\)
							−0.556378	+	0.830930i	\(0.687809\pi\)
\(43\)		−	2.26980e6i		−	0.663917i	−0.943294	−	0.331958i	\(-0.892291\pi\)
							0.943294	−	0.331958i	\(-0.107709\pi\)
\(47\)	7.70667e6	−	5.59922e6i	1.57934	−	1.14746i	0.661890	−	0.749601i	\(-0.269755\pi\)
							0.917448	−	0.397855i	\(-0.130245\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.9.d.a.13.7	✓	32
11.4	even	5		242.9.b.e.241.30		32
11.6	odd	10	inner	22.9.d.a.17.7	yes	32
11.7	odd	10		242.9.b.e.241.29		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.9.d.a.13.7	✓	32	1.1	even	1	trivial
22.9.d.a.17.7	yes	32	11.6	odd	10	inner
242.9.b.e.241.29		32	11.7	odd	10
242.9.b.e.241.30		32	11.4	even	5