Embedded newform 22.14.c.a.3.5

• Label: 22.14.c.a.3.5
• Level: $22$
• Weight: $14$
• Character: 22.3
• Analytic conductor: $23.591$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	22.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.5908043694\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			3.5
Character	\(\chi\)	\(=\)	22.3
Dual form			22.14.c.a.15.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(51.7771 - 37.6183i) q^{2} +(460.202 + 1416.35i) q^{3} +(1265.73 - 3895.53i) q^{4} +(46818.1 + 34015.4i) q^{5} +(77108.7 + 56022.7i) q^{6} +(-157481. + 484676. i) q^{7} +(-81007.0 - 249314. i) q^{8} +(-504441. + 366498. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 384 q^{2} + 2708 q^{3} - 24576 q^{4} + 32920 q^{5} + 47488 q^{6} - 509572 q^{7} + 1572864 q^{8} - 3517894 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	51.7771	−	37.6183i	0.572061	−	0.415627i
							\(3\)	460.202	+	1416.35i	0.364468	+	1.12172i	0.950313	+	0.311295i	\(0.100763\pi\)
−0.585845	+	0.810423i	\(0.699237\pi\)
\(5\)	46818.1	+	34015.4i	1.34001	+	0.973576i	0.999443	+	0.0333571i	\(0.0106199\pi\)
							0.340570	+	0.940219i	\(0.389380\pi\)
\(7\)	−157481.	+	484676.i	−0.505930	+	1.55709i	0.293271	+	0.956029i	\(0.405256\pi\)
							−0.799201	+	0.601063i	\(0.794744\pi\)
\(11\)	−5.82529e6	−	767286.i	−0.991437	−	0.130589i
							\(13\)	1.21836e7	−	8.85193e6i	0.700076	−	0.508635i	−0.179881	−	0.983688i	\(-0.557571\pi\)
0.879957	+	0.475054i	\(0.157571\pi\)
\(17\)	−1.40380e8	−	1.01992e8i	−1.41055	−	1.02482i	−0.993243	−	0.116056i	\(-0.962975\pi\)
							−0.417305	−	0.908767i	\(-0.637025\pi\)
\(19\)	5.80124e7	+	1.78544e8i	0.282893	+	0.870656i	0.987022	+	0.160584i	\(0.0513376\pi\)
							−0.704129	+	0.710072i	\(0.748662\pi\)
\(23\)	2.30725e8			0.324985			0.162492	−	0.986710i	\(-0.448047\pi\)
							0.162492	+	0.986710i	\(0.448047\pi\)
\(29\)	−7.50321e8	+	2.30925e9i	−0.234240	+	0.720915i	0.762982	+	0.646420i	\(0.223735\pi\)
							−0.997221	+	0.0744953i	\(0.976265\pi\)
\(31\)	3.92143e9	−	2.84909e9i	0.793585	−	0.576573i	−0.115440	−	0.993314i	\(-0.536828\pi\)
							0.909025	+	0.416741i	\(0.136828\pi\)
\(37\)	−3.31997e9	+	1.02178e10i	−0.212727	+	0.654707i	0.786580	+	0.617488i	\(0.211850\pi\)
							−0.999307	+	0.0372185i	\(0.988150\pi\)
\(41\)	9.68509e8	+	2.98076e9i	0.0318426	+	0.0980015i	0.965715	−	0.259606i	\(-0.0835926\pi\)
							−0.933872	+	0.357607i	\(0.883593\pi\)
\(43\)	6.46817e10			1.56040			0.780201	−	0.625529i	\(-0.215117\pi\)
							0.780201	+	0.625529i	\(0.215117\pi\)
\(47\)	1.54062e10	+	4.74154e10i	0.208478	+	0.641628i	0.999553	+	0.0299091i	\(0.00952178\pi\)
							−0.791075	+	0.611719i	\(0.790478\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.14.c.a.3.5	✓	24
11.4	even	5	inner	22.14.c.a.15.5	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.14.c.a.3.5	✓	24	1.1	even	1	trivial
22.14.c.a.15.5	yes	24	11.4	even	5	inner