Embedded newform 22.9.d.a.17.3

• Label: 22.9.d.a.17.3
• 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			17.3
Character	\(\chi\)	\(=\)	22.17
Dual form			22.9.d.a.13.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.7600 + 3.49613i) q^{2} +(-9.77853 - 7.10452i) q^{3} +(103.554 - 75.2365i) q^{4} +(24.4502 - 75.2499i) q^{5} +(130.055 + 42.2574i) q^{6} +(1395.71 + 1921.02i) q^{7} +(-851.204 + 1171.58i) q^{8} +(-1982.32 - 6100.94i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 182 q^{3} + 1024 q^{4} + 1410 q^{5} + 4480 q^{6} - 10950 q^{7} - 1402 q^{9}+O(q^{10}) \)

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
							\(3\)	−9.77853	−	7.10452i	−0.120723	−	0.0877101i	0.525786	−	0.850617i	\(-0.323771\pi\)
−0.646508	+	0.762907i	\(0.723771\pi\)
\(5\)	24.4502	−	75.2499i	0.0391203	−	0.120400i	−0.929589	−	0.368597i	\(-0.879838\pi\)
							0.968709	+	0.248197i	\(0.0798382\pi\)
\(7\)	1395.71	+	1921.02i	0.581302	+	0.800093i	0.993837	−	0.110848i	\(-0.0353567\pi\)
							−0.412536	+	0.910942i	\(0.635357\pi\)
\(11\)	−13065.8	+	6606.32i	−0.892413	+	0.451220i
							\(13\)	−43063.3	+	13992.1i	−1.50777	+	0.489903i	−0.942272	−	0.334849i	\(-0.891315\pi\)
−0.565495	+	0.824752i	\(0.691315\pi\)
\(17\)	−94403.2	−	30673.5i	−1.13029	−	0.367254i	−0.316606	−	0.948557i	\(-0.602543\pi\)
							−0.813687	+	0.581303i	\(0.802543\pi\)
\(19\)	−60823.2	+	83715.9i	−0.466718	+	0.642382i	−0.975885	−	0.218285i	\(-0.929954\pi\)
							0.509167	+	0.860668i	\(0.329954\pi\)
\(23\)	−46.8726			−0.000167497			−8.37486e−5	−	1.00000i	\(-0.500027\pi\)
							−8.37486e−5		1.00000i	\(0.500027\pi\)
\(29\)	81169.4	+	111720.i	0.114763	+	0.157957i	0.862534	−	0.505999i	\(-0.168876\pi\)
							−0.747771	+	0.663956i	\(0.768876\pi\)
\(31\)	−298179.	−	917702.i	−0.322872	−	0.993699i	−0.972392	−	0.233354i	\(-0.925030\pi\)
							0.649519	−	0.760345i	\(-0.274970\pi\)
\(37\)	−2.76189e6	+	2.00663e6i	−1.47366	+	1.07068i	−0.494134	+	0.869386i	\(0.664515\pi\)
							−0.979531	+	0.201294i	\(0.935485\pi\)
\(41\)	1.08886e6	−	1.49868e6i	0.385332	−	0.530364i	−0.571655	−	0.820494i	\(-0.693699\pi\)
							0.956987	+	0.290130i	\(0.0936985\pi\)
\(43\)			2.14432e6i			0.627213i	0.949553	+	0.313606i	\(0.101537\pi\)
							−0.949553	+	0.313606i	\(0.898463\pi\)
\(47\)	−3.87516e6	−	2.81547e6i	−0.794142	−	0.576978i	0.115048	−	0.993360i	\(-0.463298\pi\)
							−0.909190	+	0.416382i	\(0.863298\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.9.d.a.17.3	yes	32
11.2	odd	10	inner	22.9.d.a.13.3	✓	32
11.3	even	5		242.9.b.e.241.28		32
11.8	odd	10		242.9.b.e.241.27		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.9.d.a.13.3	✓	32	11.2	odd	10	inner
22.9.d.a.17.3	yes	32	1.1	even	1	trivial
242.9.b.e.241.27		32	11.8	odd	10
242.9.b.e.241.28		32	11.3	even	5