Embedded newform 242.2.g.a.25.5

• Label: 242.2.g.a.25.5
• 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			25.5
Character	\(\chi\)	\(=\)	242.25
Dual form			242.2.g.a.213.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.466667 + 0.884433i) q^{2} +(0.902513 + 2.77765i) q^{3} +(-0.564443 + 0.825472i) q^{4} +(-0.0878091 + 0.433355i) q^{5} +(-2.03547 + 2.09445i) q^{6} +(-0.274842 - 1.04561i) q^{7} +(-0.993482 - 0.113991i) q^{8} +(-4.47375 + 3.25037i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 5 q^{2} + 10 q^{3} + 5 q^{4} - q^{5} + q^{6} - 5 q^{8} - 40 q^{9}+O(q^{10}) \)

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
							\(3\)	0.902513	+	2.77765i	0.521066	+	1.60368i	0.771967	+	0.635663i	\(0.219273\pi\)
−0.250901	+	0.968013i	\(0.580727\pi\)
\(5\)	−0.0878091	+	0.433355i	−0.0392694	+	0.193802i	−0.994915	−	0.100716i	\(-0.967887\pi\)
							0.955646	+	0.294519i	\(0.0951593\pi\)
\(7\)	−0.274842	−	1.04561i	−0.103880	−	0.395202i	0.894805	−	0.446457i	\(-0.147314\pi\)
							−0.998686	+	0.0512545i	\(0.983678\pi\)
\(11\)	2.62541	−	2.02662i	0.791592	−	0.611050i
							\(13\)	1.91769	−	0.684229i	0.531870	−	0.189771i	−0.0564403	−	0.998406i	\(-0.517975\pi\)
0.588311	+	0.808635i	\(0.299793\pi\)
\(17\)	−0.889291	+	0.206796i	−0.215685	+	0.0501553i	−0.332947	−	0.942946i	\(-0.608043\pi\)
							0.117262	+	0.993101i	\(0.462588\pi\)
\(19\)	1.26921	−	2.10475i	0.291177	−	0.482862i	−0.676023	−	0.736881i	\(-0.736298\pi\)
							0.967199	+	0.254019i	\(0.0817525\pi\)
\(23\)	−2.04361	−	1.31335i	−0.426122	−	0.273852i	0.309953	−	0.950752i	\(-0.399687\pi\)
							−0.736075	+	0.676900i	\(0.763323\pi\)
\(29\)	−4.11519	+	1.73909i	−0.764172	+	0.322942i	−0.736287	−	0.676669i	\(-0.763423\pi\)
							−0.0278849	+	0.999611i	\(0.508877\pi\)
\(31\)	−4.60078	+	5.96640i	−0.826325	+	1.07160i	0.169958	+	0.985451i	\(0.445637\pi\)
							−0.996283	+	0.0861459i	\(0.972545\pi\)
\(37\)	0.0988480	−	3.46013i	0.0162505	−	0.568842i	−0.950149	−	0.311797i	\(-0.899069\pi\)
							0.966399	−	0.257045i	\(-0.0827489\pi\)
\(41\)	11.7346	−	2.03076i	1.83264	−	0.317151i	0.853617	−	0.520902i	\(-0.174404\pi\)
							0.979023	+	0.203751i	\(0.0653133\pi\)
\(43\)	1.06826	−	2.33916i	0.162908	−	0.356718i	−0.810521	−	0.585710i	\(-0.800816\pi\)
							0.973428	+	0.228992i	\(0.0735430\pi\)
\(47\)	−6.12009	+	3.00905i	−0.892707	+	0.438915i	−0.829126	−	0.559062i	\(-0.811161\pi\)
							−0.0635816	+	0.997977i	\(0.520252\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.2.g.a.25.5	✓	200
121.92	even	55	inner	242.2.g.a.213.5	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.g.a.25.5	✓	200	1.1	even	1	trivial
242.2.g.a.213.5	yes	200	121.92	even	55	inner