Embedded newform 22.9.d.a.7.3

• Label: 22.9.d.a.7.3
• Level: $22$
• Weight: $9$
• Character: 22.7
• 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			7.3
Character	\(\chi\)	\(=\)	22.7
Dual form			22.9.d.a.19.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.65003 + 9.15298i) q^{2} +(15.7928 - 48.6053i) q^{3} +(-39.5542 - 121.735i) q^{4} +(-394.920 + 286.926i) q^{5} +(339.861 + 467.778i) q^{6} +(1393.15 - 452.662i) q^{7} +(1377.28 + 447.504i) q^{8} +(3194.90 + 2321.23i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 182 q^{3} + 1024 q^{4} + 1410 q^{5} + 4480 q^{6} - 10950 q^{7} - 1402 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.65003	+	9.15298i	−0.415627	+	0.572061i
							\(3\)	15.7928	−	48.6053i	0.194973	−	0.600065i	−0.805004	−	0.593270i	\(-0.797837\pi\)
0.999977	−	0.00679568i	\(-0.00216315\pi\)
\(5\)	−394.920	+	286.926i	−0.631872	+	0.459082i	−0.857048	−	0.515236i	\(-0.827704\pi\)
							0.225176	+	0.974318i	\(0.427704\pi\)
\(7\)	1393.15	−	452.662i	0.580238	−	0.188531i	−0.00416910	−	0.999991i	\(-0.501327\pi\)
							0.584407	+	0.811460i	\(0.301327\pi\)
\(11\)	2032.01	+	14499.3i	0.138789	+	0.990322i
							\(13\)	−30249.9	+	41635.5i	−1.05913	+	1.45777i	−0.178523	+	0.983936i	\(0.557132\pi\)
−0.880612	+	0.473838i	\(0.842868\pi\)
\(17\)	27631.5	+	38031.5i	0.330833	+	0.455353i	0.941736	−	0.336353i	\(-0.109193\pi\)
							−0.610903	+	0.791705i	\(0.709193\pi\)
\(19\)	16249.3	+	5279.72i	0.124687	+	0.0405132i	0.370696	−	0.928754i	\(-0.379119\pi\)
							−0.246009	+	0.969268i	\(0.579119\pi\)
\(23\)	376904.			1.34685			0.673425	−	0.739256i	\(-0.264822\pi\)
							0.673425	+	0.739256i	\(0.264822\pi\)
\(29\)	−438275.	+	142404.i	−0.619662	+	0.201340i	−0.601990	−	0.798504i	\(-0.705625\pi\)
							−0.0176715	+	0.999844i	\(0.505625\pi\)
\(31\)	−266217.	−	193418.i	−0.288263	−	0.209435i	0.434251	−	0.900792i	\(-0.357013\pi\)
							−0.722513	+	0.691357i	\(0.757013\pi\)
\(37\)	−569199.	−	1.75181e6i	−0.303709	−	0.934719i	−0.980156	−	0.198228i	\(-0.936481\pi\)
							0.676447	−	0.736491i	\(-0.263519\pi\)
\(41\)	2.78797e6	+	905867.i	0.986627	+	0.320575i	0.757510	−	0.652824i	\(-0.226416\pi\)
							0.229118	+	0.973399i	\(0.426416\pi\)
\(43\)		−	568638.i		−	0.166327i	−0.996536	−	0.0831634i	\(-0.973498\pi\)
							0.996536	−	0.0831634i	\(-0.0265023\pi\)
\(47\)	−1.51495e6	+	4.66255e6i	−0.310462	+	0.955503i	0.667121	+	0.744950i	\(0.267527\pi\)
							−0.977582	+	0.210553i	\(0.932473\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.9.d.a.7.3	✓	32
11.5	even	5		242.9.b.e.241.7		32
11.6	odd	10		242.9.b.e.241.8		32
11.8	odd	10	inner	22.9.d.a.19.3	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.9.d.a.7.3	✓	32	1.1	even	1	trivial
22.9.d.a.19.3	yes	32	11.8	odd	10	inner
242.9.b.e.241.7		32	11.5	even	5
242.9.b.e.241.8		32	11.6	odd	10