Embedded newform 121.4.c.d.9.1

• Label: 121.4.c.d.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:	\( 2^{4} \)
Twist minimal:	yes
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.d.27.1

$q$-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	\(0.243462\pi\)
							−0.990693	+	0.136117i	\(0.956538\pi\)
\(3\)	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	−0.236801	−	0.971558i	\(-0.576099\pi\)
							−0.997182	+	0.0750170i	\(0.976099\pi\)
\(11\)		0			0
							\(13\)	−21.1566	+	65.1132i	−0.451367	+	1.38917i	0.423980	+	0.905672i	\(0.360633\pi\)
−0.875347	+	0.483495i	\(0.839367\pi\)
\(17\)	−17.1053	−	52.6446i	−0.244038	−	0.751070i	−0.995793	−	0.0916286i	\(-0.970793\pi\)
							0.751756	−	0.659442i	\(-0.229207\pi\)
\(19\)	−44.6391	+	32.4322i	−0.538995	+	0.391603i	−0.823712	−	0.567009i	\(-0.808100\pi\)
							0.284717	+	0.958612i	\(0.408100\pi\)
\(23\)	−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	0.840980	−	0.541067i	\(-0.181979\pi\)
							−0.254708	+	0.967018i	\(0.581979\pi\)
\(31\)	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	−0.251382	−	0.967888i	\(-0.580885\pi\)
							0.842835	+	0.538172i	\(0.180885\pi\)
\(43\)	208.210			0.738413			0.369207	−	0.929347i	\(-0.379629\pi\)
							0.369207	+	0.929347i	\(0.379629\pi\)
\(47\)	−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.d.9.1		8
11.2	odd	10		121.4.c.g.3.1		8
11.3	even	5	inner	121.4.c.d.81.2		8
11.4	even	5		121.4.a.e.1.1	yes	2
11.5	even	5	inner	121.4.c.d.27.1		8
11.6	odd	10		121.4.c.g.27.2		8
11.7	odd	10		121.4.a.b.1.2	✓	2
11.8	odd	10		121.4.c.g.81.1		8
11.9	even	5	inner	121.4.c.d.3.2		8
11.10	odd	2		121.4.c.g.9.2		8
33.26	odd	10		1089.4.a.k.1.2		2
33.29	even	10		1089.4.a.x.1.1		2
44.7	even	10		1936.4.a.z.1.1		2
44.15	odd	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	11.7	odd	10
121.4.a.e.1.1	yes	2	11.4	even	5
121.4.c.d.3.2		8	11.9	even	5	inner
121.4.c.d.9.1		8	1.1	even	1	trivial
121.4.c.d.27.1		8	11.5	even	5	inner
121.4.c.d.81.2		8	11.3	even	5	inner
121.4.c.g.3.1		8	11.2	odd	10
121.4.c.g.9.2		8	11.10	odd	2
121.4.c.g.27.2		8	11.6	odd	10
121.4.c.g.81.1		8	11.8	odd	10
1089.4.a.k.1.2		2	33.26	odd	10
1089.4.a.x.1.1		2	33.29	even	10
1936.4.a.y.1.1		2	44.15	odd	10
1936.4.a.z.1.1		2	44.7	even	10