Embedded newform 242.2.g.b.5.6

• Label: 242.2.g.b.5.6
• Level: $242$
• Weight: $2$
• Character: 242.5
• 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			5.6
Character	\(\chi\)	\(=\)	242.5
Dual form			242.2.g.b.97.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.516397 - 0.856349i) q^{2} +(2.41632 + 1.75556i) q^{3} +(-0.466667 - 0.884433i) q^{4} +(-1.13886 + 0.931243i) q^{5} +(2.75115 - 1.16265i) q^{6} +(0.346651 - 0.449545i) q^{7} +(-0.998369 - 0.0570888i) q^{8} +(1.82956 + 5.63081i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{2} - 8 q^{3} + 6 q^{4} + 5 q^{5} - 2 q^{6} - 2 q^{7} + 6 q^{8} - 80 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(123\)
\(\chi(n)\)	\(e\left(\frac{37}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.516397	−	0.856349i	0.365148	−	0.605530i
							\(3\)	2.41632	+	1.75556i	1.39506	+	1.01357i	0.995288	+	0.0969583i	\(0.0309113\pi\)
0.399774	+	0.916614i	\(0.369089\pi\)
\(5\)	−1.13886	+	0.931243i	−0.509315	+	0.416465i	−0.852440	−	0.522825i	\(-0.824878\pi\)
							0.343125	+	0.939290i	\(0.388514\pi\)
\(7\)	0.346651	−	0.449545i	0.131022	−	0.169912i	−0.721821	−	0.692080i	\(-0.756695\pi\)
							0.852843	+	0.522168i	\(0.174877\pi\)
\(11\)	1.46516	−	2.97545i	0.441761	−	0.897133i
							\(13\)	1.44921	−	0.250796i	0.401938	−	0.0695582i	0.0340313	−	0.999421i	\(-0.489165\pi\)
0.367907	+	0.929863i	\(0.380074\pi\)
\(17\)	−1.00526	+	0.115343i	−0.243812	+	0.0279749i	−0.235036	−	0.971987i	\(-0.575521\pi\)
							−0.00877616	+	0.999961i	\(0.502794\pi\)
\(19\)	−2.11090	+	1.19208i	−0.484275	+	0.273483i	−0.714475	−	0.699661i	\(-0.753335\pi\)
							0.230201	+	0.973143i	\(0.426062\pi\)
\(23\)	−4.75835	−	1.39718i	−0.992184	−	0.291331i	−0.254939	−	0.966957i	\(-0.582055\pi\)
							−0.737245	+	0.675626i	\(0.763874\pi\)
\(29\)	−1.05096	−	5.18669i	−0.195158	−	0.963145i	−0.951189	−	0.308608i	\(-0.900137\pi\)
							0.756031	−	0.654536i	\(-0.227136\pi\)
\(31\)	−2.68314	+	1.31921i	−0.481906	+	0.236937i	−0.665972	−	0.745977i	\(-0.731983\pi\)
							0.184066	+	0.982914i	\(0.441074\pi\)
\(37\)	−8.03172	−	8.26444i	−1.32041	−	1.35867i	−0.889952	−	0.456053i	\(-0.849263\pi\)
							−0.430454	−	0.902613i	\(-0.641646\pi\)
\(41\)	−0.820311	−	9.55071i	−0.128111	−	1.49157i	−0.724723	−	0.689040i	\(-0.758032\pi\)
							0.596612	−	0.802530i	\(-0.296513\pi\)
\(43\)	9.78140	−	6.28612i	1.49165	−	0.958625i	0.495721	−	0.868482i	\(-0.334904\pi\)
							0.995929	−	0.0901429i	\(-0.0287324\pi\)
\(47\)	−3.09663	+	0.720091i	−0.451690	+	0.105036i	−0.446169	−	0.894949i	\(-0.647212\pi\)
							−0.00552114	+	0.999985i	\(0.501757\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.2.g.b.5.6	✓	240
121.97	even	55	inner	242.2.g.b.97.6	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.g.b.5.6	✓	240	1.1	even	1	trivial
242.2.g.b.97.6	yes	240	121.97	even	55	inner