Embedded newform 242.2.g.b.15.2

• Label: 242.2.g.b.15.2
• Level: $242$
• Weight: $2$
• Character: 242.15
• Analytic conductor: $1.932$
• Analytic rank: $0$
• Dimension: $240$
• 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:	\(240\)
Relative dimension:	\(6\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label			15.2
Character	\(\chi\)	\(=\)	242.15
Dual form			242.2.g.b.113.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0855750 + 0.996332i) q^{2} +(-0.398506 + 1.22647i) q^{3} +(-0.985354 + 0.170522i) q^{4} +(3.76248 - 0.215146i) q^{5} +(-1.25608 - 0.292088i) q^{6} +(-0.343343 - 0.502123i) q^{7} +(-0.254218 - 0.967147i) q^{8} +(1.08162 + 0.785842i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{2} - 8 q^{3} + 6 q^{4} + 5 q^{5} - 2 q^{6} - 2 q^{7} + 6 q^{8} - 80 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.398506	+	1.22647i	−0.230077	+	0.708105i	0.767659	+	0.640858i	\(0.221421\pi\)
−0.997736	+	0.0672467i	\(0.978579\pi\)
\(5\)	3.76248	−	0.215146i	1.68263	−	0.0962163i	0.810579	−	0.585630i	\(-0.199153\pi\)
							0.872052	+	0.489413i	\(0.162789\pi\)
\(7\)	−0.343343	−	0.502123i	−0.129771	−	0.189785i	0.755114	−	0.655594i	\(-0.227582\pi\)
							−0.884885	+	0.465809i	\(0.845763\pi\)
\(11\)	3.31622	+	0.0517716i	0.999878	+	0.0156097i
							\(13\)	−3.72360	+	3.83149i	−1.03274	+	1.06266i	−0.0347434	+	0.999396i	\(0.511061\pi\)
−0.997997	+	0.0632680i	\(0.979848\pi\)
\(17\)	−3.10440	−	1.75313i	−0.752927	−	0.425197i	0.0669360	−	0.997757i	\(-0.478678\pi\)
							−0.819863	+	0.572560i	\(0.805950\pi\)
\(19\)	−5.83774	−	5.35783i	−1.33927	−	1.22917i	−0.951588	−	0.307375i	\(-0.900549\pi\)
							−0.387680	−	0.921794i	\(-0.626723\pi\)
\(23\)	−1.18986	−	0.349374i	−0.248103	−	0.0728496i	0.155317	−	0.987865i	\(-0.450360\pi\)
							−0.403419	+	0.915015i	\(0.632178\pi\)
\(29\)	7.77624	+	0.892241i	1.44401	+	0.165685i	0.800308	−	0.599590i	\(-0.204670\pi\)
							0.643704	+	0.765275i	\(0.277397\pi\)
\(31\)	−2.42014	+	4.58667i	−0.434670	+	0.823791i	−0.999999	−	0.00107410i	\(-0.999658\pi\)
							0.565330	+	0.824865i	\(0.308749\pi\)
\(37\)	9.55393	−	4.69735i	1.57066	−	0.772240i	0.571817	−	0.820381i	\(-0.306239\pi\)
							0.998841	+	0.0481413i	\(0.0153298\pi\)
\(41\)	0.546939	−	0.231139i	0.0854176	−	0.0360978i	−0.346143	−	0.938182i	\(-0.612509\pi\)
							0.431560	+	0.902084i	\(0.357963\pi\)
\(43\)	0.542751	−	0.348805i	0.0827688	−	0.0531923i	−0.498602	−	0.866831i	\(-0.666153\pi\)
							0.581371	+	0.813639i	\(0.302517\pi\)
\(47\)	−4.75269	−	7.88145i	−0.693251	−	1.14963i	−0.981387	−	0.192038i	\(-0.938490\pi\)
							0.288137	−	0.957589i	\(-0.406964\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.2.g.b.15.2	✓	240
121.113	even	55	inner	242.2.g.b.113.2	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.g.b.15.2	✓	240	1.1	even	1	trivial
242.2.g.b.113.2	yes	240	121.113	even	55	inner