Embedded newform 121.4.c.g.81.1

• Label: 121.4.c.g.81.1
• Level: $121$
• Weight: $4$
• Character: 121.81
• Analytic conductor: $7.139$
• Analytic rank: $0$
• Dimension: $8$
• 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:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			81.1
Root			\(0.535233 - 1.64728i\) of defining polynomial
Character	\(\chi\)	\(=\)	121.81
Dual form			121.4.c.g.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99350 - 1.44836i) q^{2} +(-0.165602 + 0.509670i) q^{3} +(-0.595848 - 1.83383i) q^{4} +(1.24257 - 0.902778i) q^{5} +(1.06831 - 0.776175i) q^{6} +(-8.72933 - 26.8661i) q^{7} +(-7.55982 + 23.2667i) q^{8} +(21.6111 + 15.7014i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} + 8 q^{3} - 10 q^{4} + 10 q^{5} - 32 q^{6} + 8 q^{7} + 42 q^{8} - 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99350	−	1.44836i	−0.704809	−	0.512074i	0.176686	−	0.984267i	\(-0.443462\pi\)
							−0.881495	+	0.472194i	\(0.843462\pi\)
\(3\)	−0.165602	+	0.509670i	−0.0318701	+	0.0980860i	−0.965726	−	0.259563i	\(-0.916422\pi\)
							0.933856	+	0.357649i	\(0.116422\pi\)
\(5\)	1.24257	−	0.902778i	0.111139	−	0.0807470i	−0.530828	−	0.847480i	\(-0.678119\pi\)
							0.641966	+	0.766733i	\(0.278119\pi\)
\(7\)	−8.72933	−	26.8661i	−0.471340	−	1.45063i	−0.850831	−	0.525439i	\(-0.823901\pi\)
							0.379492	−	0.925195i	\(-0.376099\pi\)
\(11\)		0			0
							\(13\)	−55.3886	−	40.2422i	−1.18170	−	0.858552i	−0.189333	−	0.981913i	\(-0.560633\pi\)
−0.992362	+	0.123361i	\(0.960633\pi\)
\(17\)	−44.7822	+	32.5362i	−0.638899	+	0.464187i	−0.859472	−	0.511183i	\(-0.829207\pi\)
							0.220573	+	0.975370i	\(0.429207\pi\)
\(19\)	−17.0506	+	52.4764i	−0.205878	+	0.633627i	0.793798	+	0.608181i	\(0.208100\pi\)
							−0.999676	+	0.0254456i	\(0.991900\pi\)
\(23\)	−178.315			−1.61658			−0.808290	−	0.588785i	\(-0.799606\pi\)
							−0.808290	+	0.588785i	\(0.799606\pi\)
\(29\)	34.9720	+	107.633i	0.223936	+	0.689203i	0.998398	+	0.0565831i	\(0.0180206\pi\)
							−0.774462	+	0.632620i	\(0.781979\pi\)
\(31\)	−57.2887	−	41.6227i	−0.331915	−	0.241150i	0.409328	−	0.912387i	\(-0.365763\pi\)
							−0.741243	+	0.671237i	\(0.765763\pi\)
\(37\)	−65.0988	−	200.353i	−0.289248	−	0.890213i	−0.985093	−	0.172022i	\(-0.944970\pi\)
							0.695845	−	0.718192i	\(-0.255030\pi\)
\(41\)	59.3091	−	182.535i	0.225915	−	0.695295i	−0.772282	−	0.635280i	\(-0.780885\pi\)
							0.998197	−	0.0600158i	\(-0.0191151\pi\)
\(43\)	−208.210			−0.738413			−0.369207	−	0.929347i	\(-0.620371\pi\)
							−0.369207	+	0.929347i	\(0.620371\pi\)
\(47\)	158.376	−	487.431i	0.491521	−	1.51275i	−0.330787	−	0.943705i	\(-0.607314\pi\)
							0.822309	−	0.569042i	\(-0.192686\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.4.c.g.81.1		8
11.2	odd	10		121.4.c.d.27.1		8
11.3	even	5	inner	121.4.c.g.3.1		8
11.4	even	5	inner	121.4.c.g.9.2		8
11.5	even	5		121.4.a.b.1.2	✓	2
11.6	odd	10		121.4.a.e.1.1	yes	2
11.7	odd	10		121.4.c.d.9.1		8
11.8	odd	10		121.4.c.d.3.2		8
11.9	even	5	inner	121.4.c.g.27.2		8
11.10	odd	2		121.4.c.d.81.2		8
33.5	odd	10		1089.4.a.x.1.1		2
33.17	even	10		1089.4.a.k.1.2		2
44.27	odd	10		1936.4.a.z.1.1		2
44.39	even	10		1936.4.a.y.1.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.4.a.b.1.2	✓	2	11.5	even	5
121.4.a.e.1.1	yes	2	11.6	odd	10
121.4.c.d.3.2		8	11.8	odd	10
121.4.c.d.9.1		8	11.7	odd	10
121.4.c.d.27.1		8	11.2	odd	10
121.4.c.d.81.2		8	11.10	odd	2
121.4.c.g.3.1		8	11.3	even	5	inner
121.4.c.g.9.2		8	11.4	even	5	inner
121.4.c.g.27.2		8	11.9	even	5	inner
121.4.c.g.81.1		8	1.1	even	1	trivial
1089.4.a.k.1.2		2	33.17	even	10
1089.4.a.x.1.1		2	33.5	odd	10
1936.4.a.y.1.1		2	44.39	even	10
1936.4.a.z.1.1		2	44.27	odd	10