Embedded newform 22.4.c.b.5.2

• Label: 22.4.c.b.5.2
• Level: $22$
• Weight: $4$
• Character: 22.5
• Analytic conductor: $1.298$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	22.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.29804202013\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	\( x^{8} - x^{7} + 71x^{6} - 141x^{5} + 2911x^{4} + 2710x^{3} + 75340x^{2} + 169400x + 5856400 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			5.2
Root			\(2.22300 + 6.84169i\) of defining polynomial
Character	\(\chi\)	\(=\)	22.5
Dual form			22.4.c.b.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.618034 - 1.90211i) q^{2} +(6.31989 + 4.59167i) q^{3} +(-3.23607 + 2.35114i) q^{4} +(4.60996 - 14.1880i) q^{5} +(4.82797 - 14.8590i) q^{6} +(-17.6106 + 12.7948i) q^{7} +(6.47214 + 4.70228i) q^{8} +(10.5141 + 32.3592i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 3 q^{3} - 8 q^{4} + 5 q^{5} + 14 q^{6} - q^{7} + 16 q^{8} - 21 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.618034	−	1.90211i	−0.218508	−	0.672499i
							\(3\)	6.31989	+	4.59167i	1.21626	+	0.883667i	0.995784	−	0.0917244i	\(-0.0292379\pi\)
0.220479	+	0.975392i	\(0.429238\pi\)
\(5\)	4.60996	−	14.1880i	0.412327	−	1.26901i	−0.502293	−	0.864698i	\(-0.667510\pi\)
							0.914620	−	0.404315i	\(-0.132490\pi\)
\(7\)	−17.6106	+	12.7948i	−0.950881	+	0.690856i	−0.951015	−	0.309144i	\(-0.899957\pi\)
							0.000134039		1.00000i	\(0.499957\pi\)
\(11\)	−29.3767	+	21.6335i	−0.805219	+	0.592978i
							\(13\)	−13.6056	−	41.8736i	−0.290270	−	0.893358i	−0.984769	−	0.173865i	\(-0.944374\pi\)
0.694500	−	0.719493i	\(-0.255626\pi\)
\(17\)	−7.69900	+	23.6951i	−0.109840	+	0.338053i	−0.990836	−	0.135071i	\(-0.956874\pi\)
							0.880996	+	0.473124i	\(0.156874\pi\)
\(19\)	−17.7638	−	12.9061i	−0.214489	−	0.155835i	0.475353	−	0.879795i	\(-0.342320\pi\)
							−0.689842	+	0.723960i	\(0.742320\pi\)
\(23\)	177.749			1.61145			0.805723	−	0.592293i	\(-0.201777\pi\)
							0.805723	+	0.592293i	\(0.201777\pi\)
\(29\)	120.864	−	87.8130i	0.773928	−	0.562292i	−0.129223	−	0.991616i	\(-0.541248\pi\)
							0.903151	+	0.429324i	\(0.141248\pi\)
\(31\)	23.2207	+	71.4658i	0.134534	+	0.414053i	0.995517	−	0.0945803i	\(-0.0301509\pi\)
							−0.860983	+	0.508633i	\(0.830151\pi\)
\(37\)	−179.874	+	130.686i	−0.799219	+	0.580667i	−0.910685	−	0.413102i	\(-0.864445\pi\)
							0.111466	+	0.993768i	\(0.464445\pi\)
\(41\)	204.779	+	148.781i	0.780029	+	0.566724i	0.904988	−	0.425438i	\(-0.139880\pi\)
							−0.124959	+	0.992162i	\(0.539880\pi\)
\(43\)	130.623			0.463253			0.231626	−	0.972805i	\(-0.425595\pi\)
							0.231626	+	0.972805i	\(0.425595\pi\)
\(47\)	−403.775	−	293.360i	−1.25312	−	0.910445i	−0.254722	−	0.967014i	\(-0.581984\pi\)
							−0.998399	+	0.0565693i	\(0.981984\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.4.c.b.5.2	✓	8
3.2	odd	2		198.4.f.d.181.1		8
4.3	odd	2		176.4.m.b.49.1		8
11.2	odd	10		242.4.c.q.9.2		8
11.3	even	5		242.4.a.n.1.1		4
11.4	even	5		242.4.c.r.3.1		8
11.5	even	5		242.4.c.r.81.1		8
11.6	odd	10		242.4.c.n.81.1		8
11.7	odd	10		242.4.c.n.3.1		8
11.8	odd	10		242.4.a.o.1.1		4
11.9	even	5	inner	22.4.c.b.9.2	yes	8
11.10	odd	2		242.4.c.q.27.2		8
33.8	even	10		2178.4.a.bt.1.1		4
33.14	odd	10		2178.4.a.by.1.1		4
33.20	odd	10		198.4.f.d.163.1		8
44.3	odd	10		1936.4.a.bn.1.4		4
44.19	even	10		1936.4.a.bm.1.4		4
44.31	odd	10		176.4.m.b.97.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.b.5.2	✓	8	1.1	even	1	trivial
22.4.c.b.9.2	yes	8	11.9	even	5	inner
176.4.m.b.49.1		8	4.3	odd	2
176.4.m.b.97.1		8	44.31	odd	10
198.4.f.d.163.1		8	33.20	odd	10
198.4.f.d.181.1		8	3.2	odd	2
242.4.a.n.1.1		4	11.3	even	5
242.4.a.o.1.1		4	11.8	odd	10
242.4.c.n.3.1		8	11.7	odd	10
242.4.c.n.81.1		8	11.6	odd	10
242.4.c.q.9.2		8	11.2	odd	10
242.4.c.q.27.2		8	11.10	odd	2
242.4.c.r.3.1		8	11.4	even	5
242.4.c.r.81.1		8	11.5	even	5
1936.4.a.bm.1.4		4	44.19	even	10
1936.4.a.bn.1.4		4	44.3	odd	10
2178.4.a.bt.1.1		4	33.8	even	10
2178.4.a.by.1.1		4	33.14	odd	10