Embedded newform 22.14.c.a.3.2

• Label: 22.14.c.a.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	22.3
Dual form			22.14.c.a.15.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(51.7771 - 37.6183i) q^{2} +(-417.662 - 1285.43i) q^{3} +(1265.73 - 3895.53i) q^{4} +(-34004.8 - 24705.9i) q^{5} +(-69981.0 - 50844.2i) q^{6} +(96600.4 - 297306. i) q^{7} +(-81007.0 - 249314. i) q^{8} +(-188057. + 136631. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 384 q^{2} + 2708 q^{3} - 24576 q^{4} + 32920 q^{5} + 47488 q^{6} - 509572 q^{7} + 1572864 q^{8} - 3517894 q^{9}+O(q^{10}) \)

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
							\(3\)	−417.662	−	1285.43i	−0.330778	−	1.01803i	−0.968765	−	0.247983i	\(-0.920232\pi\)
0.637986	−	0.770048i	\(-0.279768\pi\)
\(5\)	−34004.8	−	24705.9i	−0.973273	−	0.707124i	−0.0170780	−	0.999854i	\(-0.505436\pi\)
							−0.956195	+	0.292730i	\(0.905436\pi\)
\(7\)	96600.4	−	297306.i	0.310343	−	0.955137i	−0.667286	−	0.744801i	\(-0.732544\pi\)
							0.977629	−	0.210336i	\(-0.0674558\pi\)
\(11\)	−5.59972e6	+	1.77930e6i	−0.953045	+	0.302828i
							\(13\)	1.20489e7	−	8.75407e6i	0.692337	−	0.503012i	−0.185091	−	0.982721i	\(-0.559258\pi\)
0.877427	+	0.479709i	\(0.159258\pi\)
\(17\)	−2.63734e7	−	1.91614e7i	−0.265001	−	0.192535i	0.447348	−	0.894360i	\(-0.352369\pi\)
							−0.712349	+	0.701825i	\(0.752369\pi\)
\(19\)	8.48992e7	+	2.61293e8i	0.414005	+	1.27417i	0.913138	+	0.407650i	\(0.133652\pi\)
							−0.499134	+	0.866525i	\(0.666348\pi\)
\(23\)	8.50387e8			1.19780			0.598902	−	0.800822i	\(-0.295604\pi\)
							0.598902	+	0.800822i	\(0.295604\pi\)
\(29\)	−6.52448e8	+	2.00803e9i	−0.203685	+	0.626877i	0.796080	+	0.605191i	\(0.206903\pi\)
							−0.999765	+	0.0216860i	\(0.993097\pi\)
\(31\)	−1.22356e9	+	8.88971e8i	−0.247614	+	0.179902i	−0.704669	−	0.709536i	\(-0.748904\pi\)
							0.457055	+	0.889439i	\(0.348904\pi\)
\(37\)	−6.59770e9	+	2.03056e10i	−0.422748	+	1.30108i	0.482386	+	0.875959i	\(0.339770\pi\)
							−0.905134	+	0.425126i	\(0.860230\pi\)
\(41\)	−8.97514e9	−	2.76226e10i	−0.295084	−	0.908176i	−0.983193	−	0.182569i	\(-0.941559\pi\)
							0.688109	−	0.725608i	\(-0.258441\pi\)
\(43\)	−1.35369e10			−0.326568			−0.163284	−	0.986579i	\(-0.552209\pi\)
							−0.163284	+	0.986579i	\(0.552209\pi\)
\(47\)	−2.85602e10	−	8.78992e10i	−0.386478	−	1.18946i	−0.935402	−	0.353585i	\(-0.884963\pi\)
							0.548924	−	0.835872i	\(-0.315037\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.14.c.a.3.2	✓	24
11.4	even	5	inner	22.14.c.a.15.2	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.14.c.a.3.2	✓	24	1.1	even	1	trivial
22.14.c.a.15.2	yes	24	11.4	even	5	inner