Embedded newform 121.2.g.a.4.8

• Label: 121.2.g.a.4.8
• 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			4.8
Character	\(\chi\)	\(=\)	121.4
Dual form			121.2.g.a.91.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.43269 + 0.0819240i) q^{2} +(0.403457 - 1.24171i) q^{3} +(0.0589155 + 0.00675993i) q^{4} +(-0.621601 - 1.17807i) q^{5} +(0.679753 - 1.74593i) q^{6} +(2.51104 + 1.06117i) q^{7} +(-2.74416 - 0.474896i) q^{8} +(1.04798 + 0.761401i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 44 q^{2} - 32 q^{3} - 34 q^{4} - 43 q^{5} - 22 q^{6} - 44 q^{7} - 44 q^{8} - 110 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.43269	+	0.0819240i	1.01306	+	0.0579290i	0.555727	−	0.831365i	\(-0.312440\pi\)
							0.457336	+	0.889294i	\(0.348804\pi\)
\(3\)	0.403457	−	1.24171i	0.232936	−	0.716903i	−0.764453	−	0.644680i	\(-0.776991\pi\)
							0.997388	−	0.0722231i	\(-0.0230094\pi\)
\(5\)	−0.621601	−	1.17807i	−0.277989	−	0.526847i	0.705298	−	0.708911i	\(-0.250813\pi\)
							−0.983287	+	0.182064i	\(0.941722\pi\)
\(7\)	2.51104	+	1.06117i	0.949083	+	0.401086i	0.808179	−	0.588937i	\(-0.200454\pi\)
							0.140904	+	0.990023i	\(0.454999\pi\)
\(11\)	−1.54552	+	2.93451i	−0.465993	+	0.884789i
							\(13\)	−0.264287	+	0.149250i	−0.0733000	+	0.0413944i	−0.527947	−	0.849277i	\(-0.677038\pi\)
0.454647	+	0.890672i	\(0.349765\pi\)
\(17\)	−4.21636	+	1.50439i	−1.02262	+	0.364869i	−0.793526	−	0.608536i	\(-0.791757\pi\)
							−0.229091	+	0.973405i	\(0.573575\pi\)
\(19\)	−0.0547469	+	1.91639i	−0.0125598	+	0.439650i	0.968841	+	0.247682i	\(0.0796689\pi\)
							−0.981401	+	0.191968i	\(0.938513\pi\)
\(23\)	−2.11764	−	2.44389i	−0.441559	−	0.509586i	0.490725	−	0.871315i	\(-0.336732\pi\)
							−0.932283	+	0.361729i	\(0.882187\pi\)
\(29\)	3.02941	+	4.43037i	0.562547	+	0.822699i	0.996952	−	0.0780186i	\(-0.0248594\pi\)
							−0.434405	+	0.900718i	\(0.643041\pi\)
\(31\)	1.41302	−	6.97352i	0.253786	−	1.25248i	−0.627020	−	0.779003i	\(-0.715726\pi\)
							0.880806	−	0.473478i	\(-0.157002\pi\)
\(37\)	−6.05298	+	5.55537i	−0.995103	+	0.913297i	−0.996234	−	0.0867021i	\(-0.972367\pi\)
							0.00113129	+	0.999999i	\(0.499640\pi\)
\(41\)	−1.64107	−	6.24327i	−0.256292	−	0.975035i	−0.963430	−	0.267960i	\(-0.913651\pi\)
							0.707138	−	0.707075i	\(-0.249986\pi\)
\(43\)	0.0340136	+	0.236570i	0.00518703	+	0.0360766i	0.992251	−	0.124253i	\(-0.0396534\pi\)
							−0.987064	+	0.160329i	\(0.948744\pi\)
\(47\)	7.16023	−	5.85489i	1.04443	−	0.854024i	0.0549176	−	0.998491i	\(-0.482510\pi\)
							0.989510	+	0.144467i	\(0.0461468\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.2.g.a.4.8	✓	400
121.91	even	55	inner	121.2.g.a.91.8	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.2.g.a.4.8	✓	400	1.1	even	1	trivial
121.2.g.a.91.8	yes	400	121.91	even	55	inner