Embedded newform 22.14.c.b.3.6

• Label: 22.14.c.b.3.6
• Level: $22$
• Weight: $14$
• Character: 22.3
• Analytic conductor: $23.591$
• Analytic rank: $0$
• Dimension: $28$
• 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:	\(28\)
Relative dimension:	\(7\) 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.b.15.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-51.7771 + 37.6183i) q^{2} +(412.870 + 1270.68i) q^{3} +(1265.73 - 3895.53i) q^{4} +(-41240.7 - 29963.1i) q^{5} +(-69178.0 - 50260.8i) q^{6} +(25390.0 - 78142.3i) q^{7} +(81007.0 + 249314. i) q^{8} +(-154338. + 112133. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 448 q^{2} - 1248 q^{3} - 28672 q^{4} + 71554 q^{5} + 99008 q^{6} + 361734 q^{7} - 1835008 q^{8} - 8615367 q^{9}+O(q^{10}) \)

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
							\(3\)	412.870	+	1270.68i	0.326983	+	1.00635i	0.970538	+	0.240950i	\(0.0774588\pi\)
−0.643555	+	0.765400i	\(0.722541\pi\)
\(5\)	−41240.7	−	29963.1i	−1.18038	−	0.857594i	−0.188163	−	0.982138i	\(-0.560253\pi\)
							−0.992214	+	0.124544i	\(0.960253\pi\)
\(7\)	25390.0	−	78142.3i	0.0815690	−	0.251044i	−0.901952	−	0.431836i	\(-0.857866\pi\)
							0.983521	+	0.180792i	\(0.0578661\pi\)
\(11\)	5.35158e6	+	2.42556e6i	0.910813	+	0.412818i
							\(13\)	1.23266e6	−	895580.i	0.0708291	−	0.0514603i	−0.551807	−	0.833972i	\(-0.686062\pi\)
0.622636	+	0.782511i	\(0.286062\pi\)
\(17\)	−5.78778e7	−	4.20507e7i	−0.581560	−	0.422528i	0.257726	−	0.966218i	\(-0.417027\pi\)
							−0.839286	+	0.543690i	\(0.817027\pi\)
\(19\)	4.23273e7	+	1.30270e8i	0.206406	+	0.635252i	0.999653	+	0.0263520i	\(0.00838908\pi\)
							−0.793247	+	0.608900i	\(0.791611\pi\)
\(23\)	6.72099e8			0.946678			0.473339	−	0.880880i	\(-0.343049\pi\)
							0.473339	+	0.880880i	\(0.343049\pi\)
\(29\)	−6.42713e8	+	1.97807e9i	−0.200646	+	0.617524i	0.799218	+	0.601041i	\(0.205247\pi\)
							−0.999864	+	0.0164833i	\(0.994753\pi\)
\(31\)	−3.30762e9	+	2.40312e9i	−0.669367	+	0.486323i	−0.869813	−	0.493381i	\(-0.835761\pi\)
							0.200446	+	0.979705i	\(0.435761\pi\)
\(37\)	−5.55906e9	+	1.71090e10i	−0.356197	+	1.09626i	0.599115	+	0.800663i	\(0.295519\pi\)
							−0.955312	+	0.295599i	\(0.904481\pi\)
\(41\)	−4.12052e9	−	1.26817e10i	−0.135474	−	0.416947i	0.860189	−	0.509975i	\(-0.170345\pi\)
							−0.995664	+	0.0930279i	\(0.970345\pi\)
\(43\)	6.18843e10			1.49292			0.746459	−	0.665432i	\(-0.231753\pi\)
							0.746459	+	0.665432i	\(0.231753\pi\)
\(47\)	1.03198e10	+	3.17610e10i	0.139648	+	0.429792i	0.996284	−	0.0861294i	\(-0.0274498\pi\)
							−0.856636	+	0.515921i	\(0.827450\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.14.c.b.3.6	✓	28
11.4	even	5	inner	22.14.c.b.15.6	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.14.c.b.3.6	✓	28	1.1	even	1	trivial
22.14.c.b.15.6	yes	28	11.4	even	5	inner