Embedded newform 121.2.g.a.4.6

• Label: 121.2.g.a.4.6
• Level: $121$
• Weight: $2$
• Character: 121.4
• Analytic conductor: $0.966$
• Analytic rank: $0$
• Dimension: $400$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	121.g (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.966189864457\)
Analytic rank:	\(0\)
Dimension:	\(400\)
Relative dimension:	\(10\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label			4.6
Character	\(\chi\)	\(=\)	121.4
Dual form			121.2.g.a.91.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.282035 - 0.0161273i) q^{2} +(0.138004 - 0.424731i) q^{3} +(-1.90768 - 0.218886i) q^{4} +(-1.62748 - 3.08441i) q^{5} +(-0.0457716 + 0.117563i) q^{6} +(-3.79767 - 1.60491i) q^{7} +(1.09122 + 0.188843i) q^{8} +(2.26570 + 1.64613i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 44 q^{2} - 32 q^{3} - 34 q^{4} - 43 q^{5} - 22 q^{6} - 44 q^{7} - 44 q^{8} - 110 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.282035	−	0.0161273i	−0.199429	−	0.0114038i	−0.0429111	−	0.999079i	\(-0.513663\pi\)
							−0.156518	+	0.987675i	\(0.550027\pi\)
\(3\)	0.138004	−	0.424731i	0.0796764	−	0.245219i	−0.903282	−	0.429048i	\(-0.858849\pi\)
							0.982958	+	0.183829i	\(0.0588492\pi\)
\(5\)	−1.62748	−	3.08441i	−0.727829	−	1.37939i	−0.917578	−	0.397557i	\(-0.869858\pi\)
							0.189748	−	0.981833i	\(-0.439233\pi\)
\(7\)	−3.79767	−	1.60491i	−1.43538	−	0.606598i	−0.473682	−	0.880696i	\(-0.657075\pi\)
							−0.961702	+	0.274098i	\(0.911621\pi\)
\(11\)	3.15431	−	1.02486i	0.951060	−	0.309005i
							\(13\)	0.594366	−	0.335654i	0.164847	−	0.0930936i	−0.407101	−	0.913383i	\(-0.633460\pi\)
0.571949	+	0.820289i	\(0.306188\pi\)
\(17\)	−4.45649	+	1.59007i	−1.08086	+	0.385649i	−0.815652	−	0.578543i	\(-0.803622\pi\)
							−0.265205	+	0.964192i	\(0.585440\pi\)
\(19\)	0.194787	−	6.81845i	0.0446873	−	1.56426i	−0.600117	−	0.799912i	\(-0.704879\pi\)
							0.644805	−	0.764347i	\(-0.276939\pi\)
\(23\)	0.828934	+	0.956641i	0.172845	+	0.199473i	0.835562	−	0.549396i	\(-0.185142\pi\)
							−0.662717	+	0.748870i	\(0.730597\pi\)
\(29\)	1.43783	+	2.10276i	0.266998	+	0.390472i	0.935551	−	0.353193i	\(-0.114904\pi\)
							−0.668553	+	0.743665i	\(0.733086\pi\)
\(31\)	1.51489	−	7.47628i	0.272083	−	1.34278i	−0.577960	−	0.816065i	\(-0.696151\pi\)
							0.850042	−	0.526715i	\(-0.176576\pi\)
\(37\)	3.75058	−	3.44225i	0.616591	−	0.565902i	−0.305831	−	0.952086i	\(-0.598934\pi\)
							0.922422	+	0.386184i	\(0.126207\pi\)
\(41\)	0.627430	+	2.38699i	0.0979881	+	0.372786i	0.998010	−	0.0630597i	\(-0.0200859\pi\)
							−0.900022	+	0.435845i	\(0.856449\pi\)
\(43\)	0.389041	+	2.70584i	0.0593283	+	0.412637i	0.997744	+	0.0671297i	\(0.0213841\pi\)
							−0.938416	+	0.345507i	\(0.887707\pi\)
\(47\)	−4.08196	+	3.33780i	−0.595414	+	0.486868i	−0.882040	−	0.471175i	\(-0.843830\pi\)
							0.286625	+	0.958043i	\(0.407466\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.2.g.a.4.6	✓	400
121.91	even	55	inner	121.2.g.a.91.6	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.2.g.a.4.6	✓	400	1.1	even	1	trivial
121.2.g.a.91.6	yes	400	121.91	even	55	inner