Embedded newform 121.2.g.a.4.10

• Label: 121.2.g.a.4.10
• 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			4.10
Character	\(\chi\)	\(=\)	121.4
Dual form			121.2.g.a.91.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.13575 + 0.122126i) q^{2} +(-0.616726 + 1.89809i) q^{3} +(2.55954 + 0.293680i) q^{4} +(-0.685845 - 1.29982i) q^{5} +(-1.54898 + 3.97852i) q^{6} +(-0.751489 - 0.317582i) q^{7} +(1.21486 + 0.210239i) q^{8} +(-0.795333 - 0.577843i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 44 q^{2} - 32 q^{3} - 34 q^{4} - 43 q^{5} - 22 q^{6} - 44 q^{7} - 44 q^{8} - 110 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.13575	+	0.122126i	1.51020	+	0.0863565i	0.792448	−	0.609940i	\(-0.208806\pi\)
							0.717754	+	0.696297i	\(0.245170\pi\)
\(3\)	−0.616726	+	1.89809i	−0.356067	+	1.09586i	0.599321	+	0.800508i	\(0.295437\pi\)
							−0.955388	+	0.295353i	\(0.904563\pi\)
\(5\)	−0.685845	−	1.29982i	−0.306719	−	0.581297i	0.681952	−	0.731397i	\(-0.261131\pi\)
							−0.988671	+	0.150100i	\(0.952040\pi\)
\(7\)	−0.751489	−	0.317582i	−0.284036	−	0.120035i	0.242597	−	0.970127i	\(-0.422001\pi\)
							−0.526634	+	0.850092i	\(0.676546\pi\)
\(11\)	−0.377985	−	3.29502i	−0.113967	−	0.993485i
							\(13\)	2.44353	−	1.37992i	0.677713	−	0.382722i	−0.114114	−	0.993468i	\(-0.536403\pi\)
0.791827	+	0.610746i	\(0.209130\pi\)
\(17\)	−2.49798	+	0.891278i	−0.605850	+	0.216167i	−0.621118	−	0.783717i	\(-0.713321\pi\)
							0.0152689	+	0.999883i	\(0.495140\pi\)
\(19\)	−0.139274	+	4.87521i	−0.0319515	+	1.11845i	0.812548	+	0.582894i	\(0.198080\pi\)
							−0.844500	+	0.535556i	\(0.820102\pi\)
\(23\)	2.28011	+	2.63139i	0.475436	+	0.548682i	0.941916	−	0.335850i	\(-0.109023\pi\)
							−0.466480	+	0.884532i	\(0.654478\pi\)
\(29\)	−3.89665	−	5.69866i	−0.723589	−	1.05822i	−0.995395	−	0.0958631i	\(-0.969439\pi\)
							0.271805	−	0.962352i	\(-0.412379\pi\)
\(31\)	−0.498456	+	2.45998i	−0.0895254	+	0.441825i	0.910098	+	0.414392i	\(0.136006\pi\)
							−0.999624	+	0.0274327i	\(0.991267\pi\)
\(37\)	0.0992572	−	0.0910974i	0.0163178	−	0.0149763i	−0.667975	−	0.744184i	\(-0.732839\pi\)
							0.684292	+	0.729208i	\(0.260111\pi\)
\(41\)	2.11723	+	8.05478i	0.330656	+	1.25795i	0.900016	+	0.435857i	\(0.143555\pi\)
							−0.569360	+	0.822089i	\(0.692809\pi\)
\(43\)	1.55734	+	10.8315i	0.237491	+	1.65179i	0.664314	+	0.747454i	\(0.268724\pi\)
							−0.426822	+	0.904336i	\(0.640367\pi\)
\(47\)	−1.08028	+	0.883339i	−0.157575	+	0.128848i	−0.708513	−	0.705698i	\(-0.750634\pi\)
							0.550938	+	0.834546i	\(0.314270\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.2.g.a.4.10	✓	400
121.91	even	55	inner	121.2.g.a.91.10	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.2.g.a.4.10	✓	400	1.1	even	1	trivial
121.2.g.a.91.10	yes	400	121.91	even	55	inner