Embedded newform 121.2.g.a.5.6

• Label: 121.2.g.a.5.6
• Level: $121$
• Weight: $2$
• Character: 121.5
• 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			5.6
Character	\(\chi\)	\(=\)	121.5
Dual form			121.2.g.a.97.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.215633 + 0.357587i) q^{2} +(0.525350 + 0.381689i) q^{3} +(0.851964 + 1.61465i) q^{4} +(-2.63241 + 2.15251i) q^{5} +(-0.249770 + 0.105554i) q^{6} +(-0.169441 + 0.219735i) q^{7} +(-1.59487 - 0.0911981i) q^{8} +(-0.796745 - 2.45213i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 44 q^{2} - 32 q^{3} - 34 q^{4} - 43 q^{5} - 22 q^{6} - 44 q^{7} - 44 q^{8} - 110 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{37}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.215633	+	0.357587i	−0.152476	+	0.252853i	−0.923203	−	0.384313i	\(-0.874439\pi\)
							0.770727	+	0.637165i	\(0.219893\pi\)
\(3\)	0.525350	+	0.381689i	0.303311	+	0.220368i	0.729021	−	0.684491i	\(-0.239976\pi\)
							−0.425710	+	0.904860i	\(0.639976\pi\)
\(5\)	−2.63241	+	2.15251i	−1.17725	+	0.962632i	−0.999769	−	0.0214926i	\(-0.993158\pi\)
							−0.177480	+	0.984124i	\(0.556795\pi\)
\(7\)	−0.169441	+	0.219735i	−0.0640427	+	0.0830521i	−0.822834	−	0.568282i	\(-0.807608\pi\)
							0.758791	+	0.651334i	\(0.225790\pi\)
\(11\)	3.19528	+	0.888907i	0.963415	+	0.268016i
							\(13\)	4.83082	−	0.836007i	1.33983	−	0.231867i	0.544865	−	0.838524i	\(-0.316581\pi\)
0.794964	+	0.606657i	\(0.207490\pi\)
\(17\)	0.172074	−	0.0197436i	0.0417340	−	0.00478854i	−0.0930992	−	0.995657i	\(-0.529677\pi\)
							0.134833	+	0.990868i	\(0.456950\pi\)
\(19\)	3.86342	−	2.18178i	0.886330	−	0.500534i	0.0198828	−	0.999802i	\(-0.493671\pi\)
							0.866447	+	0.499269i	\(0.166398\pi\)
\(23\)	−1.45447	−	0.427072i	−0.303279	−	0.0890507i	0.126553	−	0.991960i	\(-0.459609\pi\)
							−0.429831	+	0.902909i	\(0.641427\pi\)
\(29\)	0.731887	+	3.61200i	0.135908	+	0.670732i	0.988873	+	0.148764i	\(0.0475294\pi\)
							−0.852965	+	0.521968i	\(0.825198\pi\)
\(31\)	1.39438	−	0.685572i	0.250439	−	0.123132i	−0.311685	−	0.950185i	\(-0.600893\pi\)
							0.562124	+	0.827053i	\(0.309984\pi\)
\(37\)	−5.80410	−	5.97228i	−0.954188	−	0.981836i	0.0456655	−	0.998957i	\(-0.485459\pi\)
							−0.999853	+	0.0171207i	\(0.994550\pi\)
\(41\)	−0.756493	−	8.80769i	−0.118144	−	1.37553i	−0.779944	−	0.625849i	\(-0.784752\pi\)
							0.661800	−	0.749681i	\(-0.269793\pi\)
\(43\)	−7.12111	+	4.57646i	−1.08596	+	0.697903i	−0.955926	−	0.293606i	\(-0.905145\pi\)
							−0.130033	+	0.991510i	\(0.541508\pi\)
\(47\)	−7.67697	+	1.78520i	−1.11980	+	0.260399i	−0.745235	−	0.666801i	\(-0.767663\pi\)
							−0.374566	+	0.927200i	\(0.622208\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.2.g.a.5.6	✓	400
121.97	even	55	inner	121.2.g.a.97.6	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.2.g.a.5.6	✓	400	1.1	even	1	trivial
121.2.g.a.97.6	yes	400	121.97	even	55	inner