Embedded newform 121.4.c.c.9.1

• Label: 121.4.c.c.9.1
• Level: $121$
• Weight: $4$
• Character: 121.9
• Analytic conductor: $7.139$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	121.c (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.13923111069\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	8.0.324000000.3
Defining polynomial:	\( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 11)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			9.1
Root			\(1.40126 - 1.01807i\) of defining polynomial
Character	\(\chi\)	\(=\)	121.9
Dual form			121.4.c.c.27.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.226216 + 0.696222i) q^{2} +(-4.79602 + 3.48451i) q^{3} +(6.03859 + 4.38729i) q^{4} +(-3.97285 - 12.2272i) q^{5} +(-1.34106 - 4.12734i) q^{6} +(-13.6952 - 9.95015i) q^{7} +(-9.15848 + 6.65403i) q^{8} +(2.51651 - 7.74502i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} - 44 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.226216	+	0.696222i	−0.0799795	+	0.246152i	−0.983049	−	0.183343i	\(-0.941308\pi\)
							0.903070	+	0.429494i	\(0.141308\pi\)
\(3\)	−4.79602	+	3.48451i	−0.922994	+	0.670594i	−0.944267	−	0.329179i	\(-0.893228\pi\)
							0.0212735	+	0.999774i	\(0.493228\pi\)
\(5\)	−3.97285	−	12.2272i	−0.355342	−	1.09363i	−0.955811	−	0.293983i	\(-0.905019\pi\)
							0.600468	−	0.799648i	\(-0.294981\pi\)
\(7\)	−13.6952	−	9.95015i	−0.739472	−	0.537258i	0.153074	−	0.988215i	\(-0.451083\pi\)
							−0.892546	+	0.450957i	\(0.851083\pi\)
\(11\)		0			0
							\(13\)	23.0653	−	70.9878i	0.492090	−	1.51450i	−0.329352	−	0.944207i	\(-0.606830\pi\)
0.821443	−	0.570291i	\(-0.193170\pi\)
\(17\)	−25.5819	−	78.7328i	−0.364971	−	1.12327i	−0.949999	−	0.312253i	\(-0.898916\pi\)
							0.585028	−	0.811013i	\(-0.301084\pi\)
\(19\)	54.9509	−	39.9242i	0.663505	−	0.482065i	−0.204340	−	0.978900i	\(-0.565505\pi\)
							0.867845	+	0.496835i	\(0.165505\pi\)
\(23\)	13.3538			0.121064			0.0605319	−	0.998166i	\(-0.480720\pi\)
							0.0605319	+	0.998166i	\(0.480720\pi\)
\(29\)	−136.720	−	99.3327i	−0.875456	−	0.636056i	0.0565897	−	0.998398i	\(-0.481977\pi\)
							−0.932045	+	0.362342i	\(0.881977\pi\)
\(31\)	−20.2398	+	62.2918i	−0.117264	+	0.360901i	−0.992412	−	0.122953i	\(-0.960763\pi\)
							0.875149	+	0.483854i	\(0.160763\pi\)
\(37\)	−33.0535	−	24.0148i	−0.146864	−	0.106703i	0.511927	−	0.859029i	\(-0.328932\pi\)
							−0.658791	+	0.752326i	\(0.728932\pi\)
\(41\)	−222.422	+	161.599i	−0.847230	+	0.615548i	−0.924381	−	0.381471i	\(-0.875418\pi\)
							0.0771511	+	0.997019i	\(0.475418\pi\)
\(43\)	−2.28719			−0.00811146			−0.00405573	−	0.999992i	\(-0.501291\pi\)
							−0.00405573	+	0.999992i	\(0.501291\pi\)
\(47\)	−58.1247	+	42.2301i	−0.180391	+	0.131061i	−0.674316	−	0.738443i	\(-0.735562\pi\)
							0.493926	+	0.869504i	\(0.335562\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.4.c.c.9.1		8
11.2	odd	10		121.4.c.f.3.1		8
11.3	even	5	inner	121.4.c.c.81.2		8
11.4	even	5		11.4.a.a.1.1	✓	2
11.5	even	5	inner	121.4.c.c.27.1		8
11.6	odd	10		121.4.c.f.27.2		8
11.7	odd	10		121.4.a.c.1.2		2
11.8	odd	10		121.4.c.f.81.1		8
11.9	even	5	inner	121.4.c.c.3.2		8
11.10	odd	2		121.4.c.f.9.2		8
33.26	odd	10		99.4.a.c.1.2		2
33.29	even	10		1089.4.a.v.1.1		2
44.7	even	10		1936.4.a.w.1.1		2
44.15	odd	10		176.4.a.i.1.1		2
55.4	even	10		275.4.a.b.1.2		2
55.37	odd	20		275.4.b.c.199.2		4
55.48	odd	20		275.4.b.c.199.3		4
77.48	odd	10		539.4.a.e.1.1		2
88.37	even	10		704.4.a.p.1.1		2
88.59	odd	10		704.4.a.n.1.2		2
132.59	even	10		1584.4.a.bc.1.2		2
143.103	even	10		1859.4.a.a.1.2		2
165.59	odd	10		2475.4.a.q.1.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.1	✓	2	11.4	even	5
99.4.a.c.1.2		2	33.26	odd	10
121.4.a.c.1.2		2	11.7	odd	10
121.4.c.c.3.2		8	11.9	even	5	inner
121.4.c.c.9.1		8	1.1	even	1	trivial
121.4.c.c.27.1		8	11.5	even	5	inner
121.4.c.c.81.2		8	11.3	even	5	inner
121.4.c.f.3.1		8	11.2	odd	10
121.4.c.f.9.2		8	11.10	odd	2
121.4.c.f.27.2		8	11.6	odd	10
121.4.c.f.81.1		8	11.8	odd	10
176.4.a.i.1.1		2	44.15	odd	10
275.4.a.b.1.2		2	55.4	even	10
275.4.b.c.199.2		4	55.37	odd	20
275.4.b.c.199.3		4	55.48	odd	20
539.4.a.e.1.1		2	77.48	odd	10
704.4.a.n.1.2		2	88.59	odd	10
704.4.a.p.1.1		2	88.37	even	10
1089.4.a.v.1.1		2	33.29	even	10
1584.4.a.bc.1.2		2	132.59	even	10
1859.4.a.a.1.2		2	143.103	even	10
1936.4.a.w.1.1		2	44.7	even	10
2475.4.a.q.1.1		2	165.59	odd	10