Embedded newform 121.4.c.g.9.2

• Label: 121.4.c.g.9.2
• Level: $121$
• Weight: $4$
• Character: 121.9
• 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			9.2
Root			\(-1.40126 + 1.01807i\) of defining polynomial
Character	\(\chi\)	\(=\)	121.9
Dual form			121.4.c.g.27.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.761449 - 2.34350i) q^{2} +(0.433551 - 0.314993i) q^{3} +(1.55995 + 1.13337i) q^{4} +(-0.474619 - 1.46073i) q^{5} +(-0.408059 - 1.25588i) q^{6} +(22.8537 + 16.6042i) q^{7} +(19.7919 - 14.3796i) q^{8} +(-8.25471 + 25.4054i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} + 8 q^{3} - 10 q^{4} + 10 q^{5} - 32 q^{6} + 8 q^{7} + 42 q^{8} - 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.761449	−	2.34350i	0.269213	−	0.828552i	−0.721480	−	0.692435i	\(-0.756538\pi\)
							0.990693	−	0.136117i	\(-0.0434623\pi\)
\(3\)	0.433551	−	0.314993i	0.0834369	−	0.0606205i	−0.545285	−	0.838251i	\(-0.683578\pi\)
							0.628722	+	0.777630i	\(0.283578\pi\)
\(5\)	−0.474619	−	1.46073i	−0.0424512	−	0.130651i	0.927585	−	0.373613i	\(-0.121881\pi\)
							−0.970036	+	0.242962i	\(0.921881\pi\)
\(7\)	22.8537	+	16.6042i	1.23398	+	0.896541i	0.997182	−	0.0750170i	\(-0.0239011\pi\)
							0.236801	+	0.971558i	\(0.423901\pi\)
\(11\)		0			0
							\(13\)	21.1566	−	65.1132i	0.451367	−	1.38917i	−0.423980	−	0.905672i	\(-0.639367\pi\)
0.875347	−	0.483495i	\(-0.160633\pi\)
\(17\)	17.1053	+	52.6446i	0.244038	+	0.751070i	0.995793	+	0.0916286i	\(0.0292073\pi\)
							−0.751756	+	0.659442i	\(0.770793\pi\)
\(19\)	44.6391	−	32.4322i	0.538995	−	0.391603i	−0.284717	−	0.958612i	\(-0.591900\pi\)
							0.823712	+	0.567009i	\(0.191900\pi\)
\(23\)	−178.315			−1.61658			−0.808290	−	0.588785i	\(-0.799606\pi\)
							−0.808290	+	0.588785i	\(0.799606\pi\)
\(29\)	−91.5579	−	66.5207i	−0.586271	−	0.425951i	0.254708	−	0.967018i	\(-0.418021\pi\)
							−0.840980	+	0.541067i	\(0.818021\pi\)
\(31\)	21.8824	−	67.3470i	0.126780	−	0.390189i	−0.867441	−	0.497540i	\(-0.834237\pi\)
							0.994221	+	0.107351i	\(0.0342368\pi\)
\(37\)	170.431	+	123.825i	0.757261	+	0.550182i	0.898069	−	0.439855i	\(-0.144970\pi\)
							−0.140808	+	0.990037i	\(0.544970\pi\)
\(41\)	−155.273	+	112.813i	−0.591454	+	0.429716i	−0.842835	−	0.538172i	\(-0.819115\pi\)
							0.251382	+	0.967888i	\(0.419115\pi\)
\(43\)	−208.210			−0.738413			−0.369207	−	0.929347i	\(-0.620371\pi\)
							−0.369207	+	0.929347i	\(0.620371\pi\)
\(47\)	−414.634	+	301.249i	−1.28682	+	0.934929i	−0.999736	−	0.0229763i	\(-0.992686\pi\)
							−0.287084	+	0.957906i	\(0.592686\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.4.a.b.1.2	✓	2	11.4	even	5
121.4.a.e.1.1	yes	2	11.7	odd	10
121.4.c.d.3.2		8	11.2	odd	10
121.4.c.d.9.1		8	11.10	odd	2
121.4.c.d.27.1		8	11.6	odd	10
121.4.c.d.81.2		8	11.8	odd	10
121.4.c.g.3.1		8	11.9	even	5	inner
121.4.c.g.9.2		8	1.1	even	1	trivial
121.4.c.g.27.2		8	11.5	even	5	inner
121.4.c.g.81.1		8	11.3	even	5	inner
1089.4.a.k.1.2		2	33.29	even	10
1089.4.a.x.1.1		2	33.26	odd	10
1936.4.a.y.1.1		2	44.7	even	10
1936.4.a.z.1.1		2	44.15	odd	10