Embedded newform 242.2.g.a.25.3

• Label: 242.2.g.a.25.3
• Level: $242$
• Weight: $2$
• Character: 242.25
• 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			25.3
Character	\(\chi\)	\(=\)	242.25
Dual form			242.2.g.a.213.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.466667 + 0.884433i) q^{2} +(-0.0758337 - 0.233392i) q^{3} +(-0.564443 + 0.825472i) q^{4} +(0.588996 - 2.90681i) q^{5} +(0.171031 - 0.175986i) q^{6} +(1.08715 + 4.13596i) q^{7} +(-0.993482 - 0.113991i) q^{8} +(2.37833 - 1.72796i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 5 q^{2} + 10 q^{3} + 5 q^{4} - q^{5} + q^{6} - 5 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(123\)
\(\chi(n)\)	\(e\left(\frac{19}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.466667	+	0.884433i	0.329984	+	0.625389i
							\(3\)	−0.0758337	−	0.233392i	−0.0437826	−	0.134749i	0.926776	−	0.375615i	\(-0.122569\pi\)
−0.970558	+	0.240866i	\(0.922569\pi\)
\(5\)	0.588996	−	2.90681i	0.263407	−	1.29996i	−0.601839	−	0.798617i	\(-0.705565\pi\)
							0.865246	−	0.501347i	\(-0.167162\pi\)
\(7\)	1.08715	+	4.13596i	0.410905	+	1.56324i	0.775049	+	0.631902i	\(0.217725\pi\)
							−0.364144	+	0.931343i	\(0.618638\pi\)
\(11\)	3.22702	+	0.765747i	0.972982	+	0.230882i
							\(13\)	−2.04627	+	0.730108i	−0.567533	+	0.202496i	−0.604201	−	0.796832i	\(-0.706508\pi\)
0.0366675	+	0.999328i	\(0.488326\pi\)
\(17\)	6.24610	−	1.45247i	1.51490	−	0.352275i	0.615056	−	0.788484i	\(-0.289134\pi\)
							0.899846	+	0.436209i	\(0.143679\pi\)
\(19\)	−1.03571	+	1.71754i	−0.237609	+	0.394030i	−0.952182	−	0.305530i	\(-0.901166\pi\)
							0.714573	+	0.699560i	\(0.246621\pi\)
\(23\)	−7.17717	−	4.61249i	−1.49654	−	0.961770i	−0.995339	−	0.0964421i	\(-0.969254\pi\)
							−0.501205	−	0.865328i	\(-0.667110\pi\)
\(29\)	−5.70140	+	2.40943i	−1.05872	+	0.447420i	−0.847928	−	0.530111i	\(-0.822150\pi\)
							−0.210794	+	0.977530i	\(0.567605\pi\)
\(31\)	−4.93708	+	6.40252i	−0.886726	+	1.14993i	0.101179	+	0.994868i	\(0.467738\pi\)
							−0.987905	+	0.155059i	\(0.950443\pi\)
\(37\)	0.213597	−	7.47686i	0.0351151	−	1.22919i	−0.770769	−	0.637115i	\(-0.780128\pi\)
							0.805884	−	0.592074i	\(-0.201691\pi\)
\(41\)	−8.41380	+	1.45607i	−1.31402	+	0.227399i	−0.784091	−	0.620646i	\(-0.786871\pi\)
							−0.529924	+	0.848045i	\(0.677780\pi\)
\(43\)	1.60345	−	3.51107i	0.244524	−	0.535433i	−0.747081	−	0.664733i	\(-0.768545\pi\)
							0.991606	+	0.129299i	\(0.0412727\pi\)
\(47\)	4.92539	−	2.42165i	0.718442	−	0.353234i	−0.0451086	−	0.998982i	\(-0.514363\pi\)
							0.763551	+	0.645748i	\(0.223454\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.2.g.a.25.3	✓	200
121.92	even	55	inner	242.2.g.a.213.3	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.g.a.25.3	✓	200	1.1	even	1	trivial
242.2.g.a.213.3	yes	200	121.92	even	55	inner