Embedded newform 22.4.c.b.9.1

• Label: 22.4.c.b.9.1
• Level: $22$
• Weight: $4$
• Character: 22.9
• 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			9.1
Root			\(-2.53202 + 7.79275i\) of defining polynomial
Character	\(\chi\)	\(=\)	22.9
Dual form			22.4.c.b.5.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.618034 + 1.90211i) q^{2} +(-6.12891 + 4.45291i) q^{3} +(-3.23607 - 2.35114i) q^{4} +(1.67119 + 5.14341i) q^{5} +(-4.68207 - 14.4099i) q^{6} +(17.9196 + 13.0193i) q^{7} +(6.47214 - 4.70228i) q^{8} +(9.39163 - 28.9045i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 3 q^{3} - 8 q^{4} + 5 q^{5} + 14 q^{6} - q^{7} + 16 q^{8} - 21 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)

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
							\(3\)	−6.12891	+	4.45291i	−1.17951	+	0.856963i	−0.992116	−	0.125322i	\(-0.960004\pi\)
−0.187393	+	0.982285i	\(0.560004\pi\)
\(5\)	1.67119	+	5.14341i	0.149476	+	0.460041i	0.997559	−	0.0698226i	\(-0.0222433\pi\)
							−0.848083	+	0.529863i	\(0.822243\pi\)
\(7\)	17.9196	+	13.0193i	0.967566	+	0.702978i	0.954896	−	0.296942i	\(-0.0959667\pi\)
							0.0126708	+	0.999920i	\(0.495967\pi\)
\(11\)	−11.0504	+	34.7691i	−0.302892	+	0.953025i
							\(13\)	23.7408	−	73.0668i	0.506502	−	1.55885i	−0.291730	−	0.956501i	\(-0.594231\pi\)
0.798231	−	0.602351i	\(-0.205769\pi\)
\(17\)	18.3211	+	56.3866i	0.261384	+	0.804456i	0.992505	+	0.122208i	\(0.0389975\pi\)
							−0.731121	+	0.682248i	\(0.761003\pi\)
\(19\)	−77.0690	+	55.9939i	−0.930571	+	0.676099i	−0.946133	−	0.323780i	\(-0.895046\pi\)
							0.0155616	+	0.999879i	\(0.495046\pi\)
\(23\)	142.484			1.29174			0.645868	−	0.763449i	\(-0.276495\pi\)
							0.645868	+	0.763449i	\(0.276495\pi\)
\(29\)	16.5188	+	12.0016i	0.105774	+	0.0768496i	0.639415	−	0.768862i	\(-0.279177\pi\)
							−0.533641	+	0.845711i	\(0.679177\pi\)
\(31\)	65.9146	−	202.864i	0.381891	−	1.17534i	−0.556820	−	0.830633i	\(-0.687979\pi\)
							0.938711	−	0.344706i	\(-0.112021\pi\)
\(37\)	117.775	+	85.5683i	0.523298	+	0.380199i	0.817845	−	0.575439i	\(-0.195169\pi\)
							−0.294547	+	0.955637i	\(0.595169\pi\)
\(41\)	−67.0130	+	48.6878i	−0.255260	+	0.185457i	−0.708055	−	0.706157i	\(-0.750427\pi\)
							0.452795	+	0.891615i	\(0.350427\pi\)
\(43\)	−151.373			−0.536841			−0.268420	−	0.963302i	\(-0.586502\pi\)
							−0.268420	+	0.963302i	\(0.586502\pi\)
\(47\)	−73.1063	+	53.1148i	−0.226886	+	0.164842i	−0.695421	−	0.718603i	\(-0.744782\pi\)
							0.468535	+	0.883445i	\(0.344782\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.4.c.b.9.1	yes	8
3.2	odd	2		198.4.f.d.163.2		8
4.3	odd	2		176.4.m.b.97.2		8
11.2	odd	10		242.4.c.n.3.2		8
11.3	even	5		242.4.c.r.81.2		8
11.4	even	5		242.4.a.n.1.3		4
11.5	even	5	inner	22.4.c.b.5.1	✓	8
11.6	odd	10		242.4.c.q.27.1		8
11.7	odd	10		242.4.a.o.1.3		4
11.8	odd	10		242.4.c.n.81.2		8
11.9	even	5		242.4.c.r.3.2		8
11.10	odd	2		242.4.c.q.9.1		8
33.5	odd	10		198.4.f.d.181.2		8
33.26	odd	10		2178.4.a.by.1.3		4
33.29	even	10		2178.4.a.bt.1.3		4
44.7	even	10		1936.4.a.bm.1.2		4
44.15	odd	10		1936.4.a.bn.1.2		4
44.27	odd	10		176.4.m.b.49.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.b.5.1	✓	8	11.5	even	5	inner
22.4.c.b.9.1	yes	8	1.1	even	1	trivial
176.4.m.b.49.2		8	44.27	odd	10
176.4.m.b.97.2		8	4.3	odd	2
198.4.f.d.163.2		8	3.2	odd	2
198.4.f.d.181.2		8	33.5	odd	10
242.4.a.n.1.3		4	11.4	even	5
242.4.a.o.1.3		4	11.7	odd	10
242.4.c.n.3.2		8	11.2	odd	10
242.4.c.n.81.2		8	11.8	odd	10
242.4.c.q.9.1		8	11.10	odd	2
242.4.c.q.27.1		8	11.6	odd	10
242.4.c.r.3.2		8	11.9	even	5
242.4.c.r.81.2		8	11.3	even	5
1936.4.a.bm.1.2		4	44.7	even	10
1936.4.a.bn.1.2		4	44.15	odd	10
2178.4.a.bt.1.3		4	33.29	even	10
2178.4.a.by.1.3		4	33.26	odd	10