Embedded newform 121.2.e.a.12.9

• Label: 121.2.e.a.12.9
• 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			12.9
Character	\(\chi\)	\(=\)	121.12
Dual form			121.2.e.a.111.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.77043 - 1.13779i) q^{2} +0.298538 q^{3} +(1.00904 - 2.20948i) q^{4} +(-0.0612837 + 0.426238i) q^{5} +(0.528540 - 0.339672i) q^{6} +(-1.45441 + 1.67848i) q^{7} +(-0.128482 - 0.893611i) q^{8} -2.91088 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 6 q^{2} - 18 q^{3} - 16 q^{4} - 7 q^{5} - 23 q^{6} - q^{7} + 4 q^{8} + 70 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.77043	−	1.13779i	1.25188	−	0.804537i	0.264732	−	0.964322i	\(-0.414717\pi\)
							0.987152	+	0.159785i	\(0.0510803\pi\)
\(3\)	0.298538			0.172361			0.0861804	−	0.996280i	\(-0.472534\pi\)
							0.0861804	+	0.996280i	\(0.472534\pi\)
\(5\)	−0.0612837	+	0.426238i	−0.0274069	+	0.190619i	−0.998926	−	0.0463439i	\(-0.985243\pi\)
							0.971519	+	0.236963i	\(0.0761521\pi\)
\(7\)	−1.45441	+	1.67848i	−0.549716	+	0.634407i	−0.960817	−	0.277183i	\(-0.910599\pi\)
							0.411101	+	0.911590i	\(0.365145\pi\)
\(11\)	−1.93156	−	2.69612i	−0.582387	−	0.812912i
							\(13\)	0.747300	−	1.63636i	0.207264	−	0.453844i	−0.777241	−	0.629203i	\(-0.783381\pi\)
0.984505	+	0.175359i	\(0.0561086\pi\)
\(17\)	1.88649	+	0.553925i	0.457542	+	0.134346i	0.502378	−	0.864648i	\(-0.332458\pi\)
							−0.0448363	+	0.998994i	\(0.514277\pi\)
\(19\)	0.595639	−	0.174896i	0.136649	−	0.0401238i	−0.212693	−	0.977119i	\(-0.568223\pi\)
							0.349342	+	0.936995i	\(0.386405\pi\)
\(23\)	0.369012	+	0.425863i	0.0769444	+	0.0887986i	0.792916	−	0.609330i	\(-0.208562\pi\)
							−0.715972	+	0.698129i	\(0.754016\pi\)
\(29\)	−4.36677	+	1.28220i	−0.810888	+	0.238098i	−0.660788	−	0.750573i	\(-0.729778\pi\)
							−0.150100	+	0.988671i	\(0.547960\pi\)
\(31\)	−1.78921	−	3.91782i	−0.321351	−	0.703661i	0.678160	−	0.734914i	\(-0.262777\pi\)
							−0.999512	+	0.0312531i	\(0.990050\pi\)
\(37\)	−1.33057	−	2.91354i	−0.218744	−	0.478983i	0.768166	−	0.640250i	\(-0.221169\pi\)
							−0.986911	+	0.161267i	\(0.948442\pi\)
\(41\)	8.31561	−	5.34412i	1.29868	−	0.834611i	0.305613	−	0.952156i	\(-0.401139\pi\)
							0.993068	+	0.117545i	\(0.0375024\pi\)
\(43\)	1.10162	+	7.66196i	0.167996	+	1.16844i	0.883019	+	0.469337i	\(0.155507\pi\)
							−0.715023	+	0.699101i	\(0.753584\pi\)
\(47\)	−4.83800	−	3.10920i	−0.705695	−	0.453523i	0.137939	−	0.990441i	\(-0.455952\pi\)
							−0.843635	+	0.536918i	\(0.819589\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.2.e.a.12.9	✓	100
121.111	even	11	inner	121.2.e.a.111.9	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.2.e.a.12.9	✓	100	1.1	even	1	trivial
121.2.e.a.111.9	yes	100	121.111	even	11	inner