Embedded newform 22.14.c.a.3.3

• Label: 22.14.c.a.3.3
• 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			3.3
Character	\(\chi\)	\(=\)	22.3
Dual form			22.14.c.a.15.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(51.7771 - 37.6183i) q^{2} +(-252.702 - 777.736i) q^{3} +(1265.73 - 3895.53i) q^{4} +(25133.6 + 18260.6i) q^{5} +(-42341.2 - 30762.7i) q^{6} +(29224.9 - 89945.0i) q^{7} +(-81007.0 - 249314. i) q^{8} +(748819. - 544049. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 384 q^{2} + 2708 q^{3} - 24576 q^{4} + 32920 q^{5} + 47488 q^{6} - 509572 q^{7} + 1572864 q^{8} - 3517894 q^{9}+O(q^{10}) \)

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
							\(3\)	−252.702	−	777.736i	−0.200134	−	0.615948i	−0.999878	−	0.0156075i	\(-0.995032\pi\)
0.799745	−	0.600340i	\(-0.204968\pi\)
\(5\)	25133.6	+	18260.6i	0.719366	+	0.522650i	0.886182	−	0.463338i	\(-0.153348\pi\)
							−0.166815	+	0.985988i	\(0.553348\pi\)
\(7\)	29224.9	−	89945.0i	0.0938893	−	0.288961i	−0.893073	−	0.449911i	\(-0.851456\pi\)
							0.986963	+	0.160950i	\(0.0514557\pi\)
\(11\)	5.31376e6	+	2.50731e6i	0.904378	+	0.426733i
							\(13\)	6.79286e6	−	4.93530e6i	0.390320	−	0.283584i	−0.375267	−	0.926917i	\(-0.622449\pi\)
0.765587	+	0.643333i	\(0.222449\pi\)
\(17\)	−4.48186e7	−	3.25626e7i	−0.450340	−	0.327191i	0.339390	−	0.940646i	\(-0.389779\pi\)
							−0.789730	+	0.613455i	\(0.789779\pi\)
\(19\)	−5.25782e7	−	1.61819e8i	−0.256394	−	0.789099i	−0.993552	−	0.113379i	\(-0.963833\pi\)
							0.737158	−	0.675720i	\(-0.236167\pi\)
\(23\)	−8.42275e7			−0.118638			−0.0593189	−	0.998239i	\(-0.518893\pi\)
							−0.0593189	+	0.998239i	\(0.518893\pi\)
\(29\)	4.54247e7	−	1.39803e8i	0.0141809	−	0.0436444i	−0.943716	−	0.330758i	\(-0.892696\pi\)
							0.957897	+	0.287113i	\(0.0926957\pi\)
\(31\)	1.82496e9	−	1.32591e9i	0.369319	−	0.268326i	−0.387609	−	0.921824i	\(-0.626699\pi\)
							0.756929	+	0.653498i	\(0.226699\pi\)
\(37\)	5.46734e9	−	1.68268e10i	0.350320	−	1.07817i	−0.608353	−	0.793666i	\(-0.708170\pi\)
							0.958674	−	0.284508i	\(-0.0918303\pi\)
\(41\)	−9.26760e9	−	2.85227e10i	−0.304700	−	0.937770i	−0.979789	−	0.200034i	\(-0.935895\pi\)
							0.675089	−	0.737736i	\(-0.264105\pi\)
\(43\)	−2.30048e9			−0.0554975			−0.0277487	−	0.999615i	\(-0.508834\pi\)
							−0.0277487	+	0.999615i	\(0.508834\pi\)
\(47\)	3.16247e10	+	9.73309e10i	0.427948	+	1.31709i	0.900143	+	0.435594i	\(0.143462\pi\)
							−0.472195	+	0.881494i	\(0.656538\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.14.c.a.3.3	✓	24
11.4	even	5	inner	22.14.c.a.15.3	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.14.c.a.3.3	✓	24	1.1	even	1	trivial
22.14.c.a.15.3	yes	24	11.4	even	5	inner