Embedded newform 22.14.c.a.3.6

• Label: 22.14.c.a.3.6
• 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			3.6
Character	\(\chi\)	\(=\)	22.3
Dual form			22.14.c.a.15.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(51.7771 - 37.6183i) q^{2} +(461.283 + 1419.68i) q^{3} +(1265.73 - 3895.53i) q^{4} +(-3259.92 - 2368.47i) q^{5} +(77289.8 + 56154.3i) q^{6} +(80603.6 - 248072. i) q^{7} +(-81007.0 - 249314. i) q^{8} +(-512880. + 372629. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 384 q^{2} + 2708 q^{3} - 24576 q^{4} + 32920 q^{5} + 47488 q^{6} - 509572 q^{7} + 1572864 q^{8} - 3517894 q^{9}+O(q^{10}) \)

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
							\(3\)	461.283	+	1419.68i	0.365325	+	1.12435i	0.949778	+	0.312926i	\(0.101309\pi\)
−0.584453	+	0.811428i	\(0.698691\pi\)
\(5\)	−3259.92	−	2368.47i	−0.0933042	−	0.0677895i	0.540155	−	0.841566i	\(-0.318366\pi\)
							−0.633459	+	0.773776i	\(0.718366\pi\)
\(7\)	80603.6	−	248072.i	0.258951	−	0.796968i	−0.734075	−	0.679068i	\(-0.762384\pi\)
							0.993026	−	0.117900i	\(-0.0376161\pi\)
\(11\)	3.32709e6	−	4.84285e6i	0.566254	−	0.824231i
							\(13\)	6.77386e6	−	4.92149e6i	0.389228	−	0.282791i	−0.375911	−	0.926656i	\(-0.622670\pi\)
0.765139	+	0.643865i	\(0.222670\pi\)
\(17\)	1.21674e8	+	8.84014e7i	1.22259	+	0.888262i	0.996312	−	0.0858015i	\(-0.0273451\pi\)
							0.226275	+	0.974063i	\(0.427345\pi\)
\(19\)	9.67914e7	+	2.97893e8i	0.471996	+	1.45265i	0.849966	+	0.526837i	\(0.176622\pi\)
							−0.377970	+	0.925818i	\(0.623378\pi\)
\(23\)	7.57302e7			0.106669			0.0533345	−	0.998577i	\(-0.483015\pi\)
							0.0533345	+	0.998577i	\(0.483015\pi\)
\(29\)	1.04802e9	−	3.22547e9i	0.327176	−	1.00694i	−0.643272	−	0.765637i	\(-0.722424\pi\)
							0.970449	−	0.241308i	\(-0.0775763\pi\)
\(31\)	1.46592e9	−	1.06505e9i	0.296660	−	0.215536i	−0.429492	−	0.903071i	\(-0.641307\pi\)
							0.726151	+	0.687535i	\(0.241307\pi\)
\(37\)	1.30847e9	−	4.02706e9i	0.0838402	−	0.258034i	−0.900345	−	0.435177i	\(-0.856686\pi\)
							0.984185	+	0.177144i	\(0.0566857\pi\)
\(41\)	−2.03211e9	−	6.25418e9i	−0.0668115	−	0.205625i	0.912077	−	0.410018i	\(-0.134478\pi\)
							−0.978889	+	0.204394i	\(0.934478\pi\)
\(43\)	1.06426e10			0.256746			0.128373	−	0.991726i	\(-0.459024\pi\)
							0.128373	+	0.991726i	\(0.459024\pi\)
\(47\)	−2.71772e10	−	8.36429e10i	−0.367764	−	1.13186i	−0.948232	−	0.317578i	\(-0.897130\pi\)
							0.580468	−	0.814283i	\(-0.302870\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.14.c.a.3.6	✓	24
11.4	even	5	inner	22.14.c.a.15.6	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.14.c.a.3.6	✓	24	1.1	even	1	trivial
22.14.c.a.15.6	yes	24	11.4	even	5	inner