Embedded newform 22.3.d.a.13.2

• Label: 22.3.d.a.13.2
• Level: $22$
• Weight: $3$
• Character: 22.13
• Analytic conductor: $0.599$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	22.d (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.599456581593\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{10})\)
Coefficient field:	8.0.64000000.1
Defining polynomial:	\( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			13.2
Root			\(1.34500 + 0.437016i\) of defining polynomial
Character	\(\chi\)	\(=\)	22.13
Dual form			22.3.d.a.17.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.34500 + 0.437016i) q^{2} +(-1.86723 + 1.35662i) q^{3} +(1.61803 + 1.17557i) q^{4} +(-2.45454 - 7.55429i) q^{5} +(-3.10429 + 1.00865i) q^{6} +(-2.13277 + 2.93550i) q^{7} +(1.66251 + 2.28825i) q^{8} +(-1.13502 + 3.49324i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} + 4 q^{4} + 2 q^{5} - 20 q^{6} - 30 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.34500	+	0.437016i	0.672499	+	0.218508i
							\(3\)	−1.86723	+	1.35662i	−0.622411	+	0.452208i	−0.853763	−	0.520662i	\(-0.825685\pi\)
0.231352	+	0.972870i	\(0.425685\pi\)
\(5\)	−2.45454	−	7.55429i	−0.490907	−	1.51086i	−0.823239	−	0.567695i	\(-0.807835\pi\)
							0.332331	−	0.943163i	\(-0.392165\pi\)
\(7\)	−2.13277	+	2.93550i	−0.304681	+	0.419357i	−0.933713	−	0.358022i	\(-0.883451\pi\)
							0.629032	+	0.777379i	\(0.283451\pi\)
\(11\)	10.5316	+	3.17569i	0.957420	+	0.288699i
							\(13\)	−10.5015	−	3.41216i	−0.807811	−	0.262474i	−0.124141	−	0.992265i	\(-0.539617\pi\)
−0.683670	+	0.729791i	\(0.739617\pi\)
\(17\)	20.0856	−	6.52621i	1.18151	−	0.383895i	0.348580	−	0.937279i	\(-0.386664\pi\)
							0.832926	+	0.553384i	\(0.186664\pi\)
\(19\)	4.09361	+	5.63437i	0.215453	+	0.296546i	0.903040	−	0.429556i	\(-0.141330\pi\)
							−0.687587	+	0.726102i	\(0.741330\pi\)
\(23\)	−34.5781			−1.50340			−0.751698	−	0.659507i	\(-0.770765\pi\)
							−0.751698	+	0.659507i	\(0.770765\pi\)
\(29\)	25.8411	−	35.5672i	0.891073	−	1.22646i	−0.0821567	−	0.996619i	\(-0.526181\pi\)
							0.973229	−	0.229837i	\(-0.0738192\pi\)
\(31\)	−5.73331	+	17.6453i	−0.184945	+	0.569203i	−0.999947	−	0.0102547i	\(-0.996736\pi\)
							0.815002	+	0.579458i	\(0.196736\pi\)
\(37\)	5.08796	+	3.69662i	0.137512	+	0.0999086i	0.654415	−	0.756136i	\(-0.272915\pi\)
							−0.516902	+	0.856044i	\(0.672915\pi\)
\(41\)	−0.251778	−	0.346543i	−0.00614092	−	0.00845226i	0.805935	−	0.592003i	\(-0.201663\pi\)
							−0.812076	+	0.583551i	\(0.801663\pi\)
\(43\)		−	7.55230i		−	0.175635i	−0.996137	−	0.0878174i	\(-0.972011\pi\)
							0.996137	−	0.0878174i	\(-0.0279892\pi\)
\(47\)	−2.74350	+	1.99327i	−0.0583724	+	0.0424100i	−0.616589	−	0.787285i	\(-0.711486\pi\)
							0.558216	+	0.829695i	\(0.311486\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.3.d.a.13.2	✓	8
3.2	odd	2		198.3.j.a.145.1		8
4.3	odd	2		176.3.n.b.145.2		8
11.2	odd	10		242.3.d.e.239.2		8
11.3	even	5		242.3.d.e.161.2		8
11.4	even	5		242.3.b.d.241.8		8
11.5	even	5		242.3.d.c.215.1		8
11.6	odd	10	inner	22.3.d.a.17.2	yes	8
11.7	odd	10		242.3.b.d.241.4		8
11.8	odd	10		242.3.d.d.161.1		8
11.9	even	5		242.3.d.d.239.1		8
11.10	odd	2		242.3.d.c.233.1		8
33.17	even	10		198.3.j.a.127.1		8
33.26	odd	10		2178.3.d.l.1693.4		8
33.29	even	10		2178.3.d.l.1693.8		8
44.39	even	10		176.3.n.b.17.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.3.d.a.13.2	✓	8	1.1	even	1	trivial
22.3.d.a.17.2	yes	8	11.6	odd	10	inner
176.3.n.b.17.2		8	44.39	even	10
176.3.n.b.145.2		8	4.3	odd	2
198.3.j.a.127.1		8	33.17	even	10
198.3.j.a.145.1		8	3.2	odd	2
242.3.b.d.241.4		8	11.7	odd	10
242.3.b.d.241.8		8	11.4	even	5
242.3.d.c.215.1		8	11.5	even	5
242.3.d.c.233.1		8	11.10	odd	2
242.3.d.d.161.1		8	11.8	odd	10
242.3.d.d.239.1		8	11.9	even	5
242.3.d.e.161.2		8	11.3	even	5
242.3.d.e.239.2		8	11.2	odd	10
2178.3.d.l.1693.4		8	33.26	odd	10
2178.3.d.l.1693.8		8	33.29	even	10