Embedded newform 121.2.e.a.12.5

• Label: 121.2.e.a.12.5
• Level: $121$
• Weight: $2$
• Character: 121.12
• Analytic conductor: $0.966$
• Analytic rank: $0$
• Dimension: $100$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	121.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.966189864457\)
Analytic rank:	\(0\)
Dimension:	\(100\)
Relative dimension:	\(10\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			12.5
Character	\(\chi\)	\(=\)	121.12
Dual form			121.2.e.a.111.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.322674 + 0.207370i) q^{2} +1.45814 q^{3} +(-0.769714 + 1.68544i) q^{4} +(-0.307743 + 2.14040i) q^{5} +(-0.470506 + 0.302376i) q^{6} +(-0.298898 + 0.344946i) q^{7} +(-0.210316 - 1.46278i) q^{8} -0.873817 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 6 q^{2} - 18 q^{3} - 16 q^{4} - 7 q^{5} - 23 q^{6} - q^{7} + 4 q^{8} + 70 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.322674	+	0.207370i	−0.228165	+	0.146633i	−0.649729	−	0.760166i	\(-0.725118\pi\)
							0.421564	+	0.906799i	\(0.361481\pi\)
\(3\)	1.45814			0.841860			0.420930	−	0.907093i	\(-0.361704\pi\)
							0.420930	+	0.907093i	\(0.361704\pi\)
\(5\)	−0.307743	+	2.14040i	−0.137627	+	0.957216i	0.797605	+	0.603180i	\(0.206100\pi\)
							−0.935232	+	0.354036i	\(0.884809\pi\)
\(7\)	−0.298898	+	0.344946i	−0.112973	+	0.130377i	−0.809419	−	0.587231i	\(-0.800218\pi\)
							0.696447	+	0.717609i	\(0.254763\pi\)
\(11\)	3.31661	−	0.00975866i	0.999996	−	0.00294235i
							\(13\)	2.34464	−	5.13404i	0.650285	−	1.42393i	−0.241019	−	0.970521i	\(-0.577481\pi\)
0.891304	−	0.453406i	\(-0.149791\pi\)
\(17\)	4.21708	+	1.23825i	1.02279	+	0.300319i	0.749777	−	0.661691i	\(-0.230161\pi\)
							0.273015	+	0.962010i	\(0.411979\pi\)
\(19\)	−1.55562	+	0.456771i	−0.356884	+	0.104791i	−0.455260	−	0.890359i	\(-0.650454\pi\)
							0.0983756	+	0.995149i	\(0.468635\pi\)
\(23\)	−0.712520	−	0.822292i	−0.148571	−	0.171460i	0.676586	−	0.736364i	\(-0.263459\pi\)
							−0.825157	+	0.564904i	\(0.808913\pi\)
\(29\)	0.497638	−	0.146120i	0.0924090	−	0.0271337i	−0.235201	−	0.971947i	\(-0.575575\pi\)
							0.327610	+	0.944813i	\(0.393757\pi\)
\(31\)	−3.30307	−	7.23272i	−0.593249	−	1.29903i	−0.933459	−	0.358685i	\(-0.883225\pi\)
							0.340209	−	0.940350i	\(-0.389502\pi\)
\(37\)	−0.851706	−	1.86498i	−0.140020	−	0.306600i	0.826611	−	0.562773i	\(-0.190266\pi\)
							−0.966631	+	0.256173i	\(0.917538\pi\)
\(41\)	−8.03453	+	5.16348i	−1.25478	+	0.806400i	−0.987561	−	0.157236i	\(-0.949742\pi\)
							−0.267221	+	0.963635i	\(0.586105\pi\)
\(43\)	−0.901491	−	6.27001i	−0.137476	−	0.956167i	−0.935446	−	0.353470i	\(-0.885002\pi\)
							0.797970	−	0.602697i	\(-0.205907\pi\)
\(47\)	3.21551	+	2.06649i	0.469031	+	0.301428i	0.753724	−	0.657191i	\(-0.228255\pi\)
							−0.284693	+	0.958619i	\(0.591892\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.2.e.a.12.5	✓	100
121.111	even	11	inner	121.2.e.a.111.5	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.2.e.a.12.5	✓	100	1.1	even	1	trivial
121.2.e.a.111.5	yes	100	121.111	even	11	inner