Embedded newform 121.2.g.a.4.4

• Label: 121.2.g.a.4.4
• 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			4.4
Character	\(\chi\)	\(=\)	121.4
Dual form			121.2.g.a.91.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.828838 - 0.0473947i) q^{2} +(1.01247 - 3.11606i) q^{3} +(-1.30224 - 0.149418i) q^{4} +(0.0511865 + 0.0970092i) q^{5} +(-0.986857 + 2.53472i) q^{6} +(-0.0339504 - 0.0143476i) q^{7} +(2.70833 + 0.468694i) q^{8} +(-6.25768 - 4.54647i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 44 q^{2} - 32 q^{3} - 34 q^{4} - 43 q^{5} - 22 q^{6} - 44 q^{7} - 44 q^{8} - 110 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.828838	−	0.0473947i	−0.586077	−	0.0335131i	−0.238464	−	0.971151i	\(-0.576644\pi\)
							−0.347613	+	0.937638i	\(0.613008\pi\)
\(3\)	1.01247	−	3.11606i	0.584549	−	1.79906i	−0.0165245	−	0.999863i	\(-0.505260\pi\)
							0.601074	−	0.799194i	\(-0.294740\pi\)
\(5\)	0.0511865	+	0.0970092i	0.0228913	+	0.0433838i	0.895613	−	0.444835i	\(-0.146738\pi\)
							−0.872721	+	0.488219i	\(0.837647\pi\)
\(7\)	−0.0339504	−	0.0143476i	−0.0128320	−	0.00542287i	0.382845	−	0.923813i	\(-0.374944\pi\)
							−0.395677	+	0.918390i	\(0.629490\pi\)
\(11\)	−3.15538	+	1.02157i	−0.951382	+	0.308015i
							\(13\)	2.41843	−	1.36575i	0.670752	−	0.378791i	−0.118420	−	0.992964i	\(-0.537783\pi\)
0.789172	+	0.614172i	\(0.210510\pi\)
\(17\)	4.98785	−	1.77966i	1.20973	−	0.431631i	0.347275	−	0.937763i	\(-0.387107\pi\)
							0.862456	+	0.506132i	\(0.168925\pi\)
\(19\)	0.0703444	−	2.46237i	0.0161381	−	0.564907i	−0.950787	−	0.309846i	\(-0.899723\pi\)
							0.966925	−	0.255061i	\(-0.0820957\pi\)
\(23\)	2.57372	+	2.97024i	0.536658	+	0.619337i	0.957722	−	0.287694i	\(-0.0928886\pi\)
							−0.421064	+	0.907031i	\(0.638343\pi\)
\(29\)	4.38533	+	6.41335i	0.814336	+	1.19093i	0.978855	+	0.204555i	\(0.0655746\pi\)
							−0.164519	+	0.986374i	\(0.552607\pi\)
\(31\)	−0.222975	+	1.10043i	−0.0400476	+	0.197643i	−0.995112	−	0.0987571i	\(-0.968513\pi\)
							0.955064	+	0.296400i	\(0.0957860\pi\)
\(37\)	3.94536	−	3.62101i	0.648612	−	0.595291i	−0.282834	−	0.959169i	\(-0.591275\pi\)
							0.931447	+	0.363878i	\(0.118547\pi\)
\(41\)	1.70211	+	6.47549i	0.265824	+	1.01130i	0.957346	+	0.288945i	\(0.0933045\pi\)
							−0.691521	+	0.722356i	\(0.743059\pi\)
\(43\)	−0.380986	−	2.64982i	−0.0580999	−	0.404094i	−0.998031	−	0.0627284i	\(-0.980020\pi\)
							0.939931	−	0.341365i	\(-0.110889\pi\)
\(47\)	−6.96344	+	5.69398i	−1.01572	+	0.830552i	−0.985617	−	0.168995i	\(-0.945948\pi\)
							−0.0301052	+	0.999547i	\(0.509584\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.2.g.a.4.4	✓	400
121.91	even	55	inner	121.2.g.a.91.4	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.2.g.a.4.4	✓	400	1.1	even	1	trivial
121.2.g.a.91.4	yes	400	121.91	even	55	inner