Embedded newform 22.9.d.a.13.2

• Label: 22.9.d.a.13.2
• 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			13.2
Character	\(\chi\)	\(=\)	22.13
Dual form			22.9.d.a.17.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.7600 - 3.49613i) q^{2} +(-42.1296 + 30.6089i) q^{3} +(103.554 + 75.2365i) q^{4} +(-302.083 - 929.717i) q^{5} +(560.326 - 182.061i) q^{6} +(-997.523 + 1372.97i) q^{7} +(-851.204 - 1171.58i) q^{8} +(-1189.47 + 3660.80i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 182 q^{3} + 1024 q^{4} + 1410 q^{5} + 4480 q^{6} - 10950 q^{7} - 1402 q^{9}+O(q^{10}) \)

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
							\(3\)	−42.1296	+	30.6089i	−0.520118	+	0.377888i	−0.816648	−	0.577136i	\(-0.804170\pi\)
0.296530	+	0.955023i	\(0.404170\pi\)
\(5\)	−302.083	−	929.717i	−0.483333	−	1.48755i	−0.834380	−	0.551189i	\(-0.814174\pi\)
							0.351047	−	0.936358i	\(-0.385826\pi\)
\(7\)	−997.523	+	1372.97i	−0.415462	+	0.571834i	−0.964540	−	0.263937i	\(-0.914979\pi\)
							0.549078	+	0.835771i	\(0.314979\pi\)
\(11\)	14478.5	−	2175.51i	0.988899	−	0.148590i
							\(13\)	50392.3	+	16373.5i	1.76438	+	0.573280i	0.997639	−	0.0686816i	\(-0.0218793\pi\)
0.766737	+	0.641962i	\(0.221879\pi\)
\(17\)	−133152.	+	43263.8i	−1.59424	+	0.517999i	−0.965673	−	0.259760i	\(-0.916357\pi\)
							−0.628563	+	0.777758i	\(0.716357\pi\)
\(19\)	120130.	+	165345.i	0.921803	+	1.26875i	0.962972	+	0.269600i	\(0.0868916\pi\)
							−0.0411697	+	0.999152i	\(0.513108\pi\)
\(23\)	254681.			0.910091			0.455046	−	0.890468i	\(-0.349623\pi\)
							0.455046	+	0.890468i	\(0.349623\pi\)
\(29\)	277655.	−	382160.i	0.392567	−	0.540322i	−0.566292	−	0.824205i	\(-0.691623\pi\)
							0.958859	+	0.283883i	\(0.0916226\pi\)
\(31\)	−28832.1	+	88736.0i	−0.0312197	+	0.0960845i	−0.965452	−	0.260580i	\(-0.916086\pi\)
							0.934232	+	0.356665i	\(0.116086\pi\)
\(37\)	415572.	+	301931.i	0.221738	+	0.161102i	0.693109	−	0.720833i	\(-0.256241\pi\)
							−0.471371	+	0.881935i	\(0.656241\pi\)
\(41\)	814750.	+	1.12141e6i	0.288329	+	0.396851i	0.928471	−	0.371406i	\(-0.121124\pi\)
							−0.640141	+	0.768257i	\(0.721124\pi\)
\(43\)		−	418471.i		−	0.122403i	−0.998125	−	0.0612015i	\(-0.980507\pi\)
							0.998125	−	0.0612015i	\(-0.0194932\pi\)
\(47\)	1.08989e6	−	791854.i	0.223354	−	0.162276i	−0.470481	−	0.882410i	\(-0.655920\pi\)
							0.693835	+	0.720134i	\(0.255920\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.9.d.a.13.2	✓	32
11.4	even	5		242.9.b.e.241.25		32
11.6	odd	10	inner	22.9.d.a.17.2	yes	32
11.7	odd	10		242.9.b.e.241.26		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.9.d.a.13.2	✓	32	1.1	even	1	trivial
22.9.d.a.17.2	yes	32	11.6	odd	10	inner
242.9.b.e.241.25		32	11.4	even	5
242.9.b.e.241.26		32	11.7	odd	10