Embedded newform 121.2.g.a.4.7

• Label: 121.2.g.a.4.7
• 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			4.7
Character	\(\chi\)	\(=\)	121.4
Dual form			121.2.g.a.91.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.609785 + 0.0348688i) q^{2} +(-0.704377 + 2.16785i) q^{3} +(-1.61634 - 0.185458i) q^{4} +(0.903893 + 1.71307i) q^{5} +(-0.505109 + 1.29736i) q^{6} +(-0.168606 - 0.0712535i) q^{7} +(-2.18283 - 0.377753i) q^{8} +(-1.77637 - 1.29061i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 44 q^{2} - 32 q^{3} - 34 q^{4} - 43 q^{5} - 22 q^{6} - 44 q^{7} - 44 q^{8} - 110 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.609785	+	0.0348688i	0.431183	+	0.0246559i	0.271333	−	0.962485i	\(-0.412535\pi\)
							0.159850	+	0.987141i	\(0.448899\pi\)
\(3\)	−0.704377	+	2.16785i	−0.406672	+	1.25161i	0.512819	+	0.858497i	\(0.328601\pi\)
							−0.919491	+	0.393111i	\(0.871399\pi\)
\(5\)	0.903893	+	1.71307i	0.404233	+	0.766107i	0.999294	−	0.0375739i	\(-0.0119630\pi\)
							−0.595061	+	0.803681i	\(0.702872\pi\)
\(7\)	−0.168606	−	0.0712535i	−0.0637271	−	0.0269313i	0.357174	−	0.934038i	\(-0.383740\pi\)
							−0.420901	+	0.907107i	\(0.638286\pi\)
\(11\)	1.89661	+	2.72082i	0.571848	+	0.820360i
							\(13\)	1.01899	−	0.575448i	0.282616	−	0.159600i	−0.343907	−	0.939004i	\(-0.611750\pi\)
0.626522	+	0.779403i	\(0.284478\pi\)
\(17\)	4.97934	−	1.77662i	1.20767	−	0.430895i	0.345935	−	0.938259i	\(-0.387562\pi\)
							0.861732	+	0.507364i	\(0.169380\pi\)
\(19\)	0.128546	−	4.49970i	0.0294905	−	1.03230i	−0.841276	−	0.540606i	\(-0.818195\pi\)
							0.870767	−	0.491697i	\(-0.163623\pi\)
\(23\)	−1.71364	−	1.97764i	−0.357318	−	0.412367i	0.548421	−	0.836202i	\(-0.315229\pi\)
							−0.905739	+	0.423835i	\(0.860684\pi\)
\(29\)	1.34184	+	1.96238i	0.249174	+	0.364406i	0.929699	−	0.368319i	\(-0.120067\pi\)
							−0.680525	+	0.732725i	\(0.738248\pi\)
\(31\)	−0.765537	+	3.77808i	−0.137495	+	0.678562i	0.850736	+	0.525593i	\(0.176156\pi\)
							−0.988231	+	0.152970i	\(0.951116\pi\)
\(37\)	−3.51933	+	3.23001i	−0.578575	+	0.531011i	−0.911155	−	0.412064i	\(-0.864808\pi\)
							0.332580	+	0.943075i	\(0.392081\pi\)
\(41\)	−2.49940	−	9.50872i	−0.390341	−	1.48501i	−0.814895	−	0.579609i	\(-0.803205\pi\)
							0.424553	−	0.905403i	\(-0.360431\pi\)
\(43\)	0.706987	+	4.91720i	0.107814	+	0.749866i	0.969970	+	0.243223i	\(0.0782047\pi\)
							−0.862156	+	0.506643i	\(0.830886\pi\)
\(47\)	−5.12156	+	4.18788i	−0.747057	+	0.610865i	−0.927979	−	0.372632i	\(-0.878455\pi\)
							0.180922	+	0.983497i	\(0.442092\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.2.g.a.4.7	✓	400
121.91	even	55	inner	121.2.g.a.91.7	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.2.g.a.4.7	✓	400	1.1	even	1	trivial
121.2.g.a.91.7	yes	400	121.91	even	55	inner