Embedded newform 242.2.g.a.15.5

• Label: 242.2.g.a.15.5
• Level: $242$
• Weight: $2$
• Character: 242.15
• Analytic conductor: $1.932$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	242.g (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.93237972891\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(5\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label			15.5
Character	\(\chi\)	\(=\)	242.15
Dual form			242.2.g.a.113.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0855750 - 0.996332i) q^{2} +(0.751841 - 2.31393i) q^{3} +(-0.985354 + 0.170522i) q^{4} +(-1.44457 + 0.0826036i) q^{5} +(-2.36978 - 0.551069i) q^{6} +(-0.642307 - 0.939344i) q^{7} +(0.254218 + 0.967147i) q^{8} +(-2.36195 - 1.71606i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 5 q^{2} + 10 q^{3} + 5 q^{4} - q^{5} + q^{6} - 5 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/242\mathbb{Z}\right)^\times\).

\(n\)	\(123\)
\(\chi(n)\)	\(e\left(\frac{26}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0855750	−	0.996332i	−0.0605107	−	0.704513i
							\(3\)	0.751841	−	2.31393i	0.434076	−	1.33595i	−0.459955	−	0.887942i	\(-0.652135\pi\)
0.894031	−	0.448005i	\(-0.147865\pi\)
\(5\)	−1.44457	+	0.0826036i	−0.646032	+	0.0369415i	−0.377034	−	0.926199i	\(-0.623056\pi\)
							−0.268998	+	0.963141i	\(0.586692\pi\)
\(7\)	−0.642307	−	0.939344i	−0.242769	−	0.355039i	0.684773	−	0.728756i	\(-0.259901\pi\)
							−0.927543	+	0.373717i	\(0.878083\pi\)
\(11\)	−2.54440	−	2.12745i	−0.767165	−	0.641450i
							\(13\)	1.93278	−	1.98878i	0.536056	−	0.551588i	−0.393951	−	0.919132i	\(-0.628892\pi\)
0.930006	+	0.367543i	\(0.119801\pi\)
\(17\)	−2.76382	−	1.56080i	−0.670325	−	0.378550i	0.118683	−	0.992932i	\(-0.462133\pi\)
							−0.789009	+	0.614382i	\(0.789405\pi\)
\(19\)	3.15471	+	2.89537i	0.723740	+	0.664243i	0.950859	−	0.309624i	\(-0.100203\pi\)
							−0.227119	+	0.973867i	\(0.572931\pi\)
\(23\)	8.02260	+	2.35565i	1.67283	+	0.491186i	0.974461	−	0.224555i	\(-0.0720929\pi\)
							0.698365	+	0.715741i	\(0.253911\pi\)
\(29\)	−1.39000	−	0.159488i	−0.258117	−	0.0296161i	−0.0160329	−	0.999871i	\(-0.505104\pi\)
							−0.242084	+	0.970255i	\(0.577831\pi\)
\(31\)	2.15064	−	4.07591i	0.386266	−	0.732055i	−0.612004	−	0.790855i	\(-0.709636\pi\)
							0.998270	+	0.0587998i	\(0.0187273\pi\)
\(37\)	7.16255	−	3.52159i	1.17752	−	0.578946i	0.255770	−	0.966738i	\(-0.417671\pi\)
							0.921747	+	0.387792i	\(0.126762\pi\)
\(41\)	−4.14136	+	1.75015i	−0.646771	+	0.273328i	−0.687877	−	0.725827i	\(-0.741457\pi\)
							0.0411062	+	0.999155i	\(0.486912\pi\)
\(43\)	2.21912	−	1.42614i	0.338412	−	0.217484i	−0.360386	−	0.932803i	\(-0.617355\pi\)
							0.698798	+	0.715319i	\(0.253719\pi\)
\(47\)	−2.86050	−	4.74360i	−0.417246	−	0.691925i	0.574731	−	0.818343i	\(-0.305107\pi\)
							−0.991977	+	0.126417i	\(0.959652\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.2.g.a.15.5	✓	200
121.113	even	55	inner	242.2.g.a.113.5	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.g.a.15.5	✓	200	1.1	even	1	trivial
242.2.g.a.113.5	yes	200	121.113	even	55	inner