Embedded newform 242.2.g.a.5.3

• Label: 242.2.g.a.5.3
• Level: $242$
• Weight: $2$
• Character: 242.5
• 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			5.3
Character	\(\chi\)	\(=\)	242.5
Dual form			242.2.g.a.97.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.516397 + 0.856349i) q^{2} +(-0.324490 - 0.235756i) q^{3} +(-0.466667 - 0.884433i) q^{4} +(1.42498 - 1.16520i) q^{5} +(0.369456 - 0.156133i) q^{6} +(0.331985 - 0.430526i) q^{7} +(0.998369 + 0.0570888i) q^{8} +(-0.877338 - 2.70017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 5 q^{2} + 10 q^{3} + 5 q^{4} - q^{5} + q^{6} - 5 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	−0.324490	−	0.235756i	−0.187345	−	0.136114i	0.490160	−	0.871633i	\(-0.336938\pi\)
−0.677504	+	0.735519i	\(0.736938\pi\)
\(5\)	1.42498	−	1.16520i	0.637270	−	0.521093i	−0.258271	−	0.966073i	\(-0.583153\pi\)
							0.895541	+	0.444979i	\(0.146789\pi\)
\(7\)	0.331985	−	0.430526i	0.125479	−	0.162724i	−0.724972	−	0.688778i	\(-0.758147\pi\)
							0.850451	+	0.526055i	\(0.176329\pi\)
\(11\)	−2.46329	−	2.22086i	−0.742709	−	0.669614i
							\(13\)	1.21254	−	0.209839i	0.336298	−	0.0581987i	0.000128245	−	1.00000i	\(-0.499959\pi\)
0.336170	+	0.941801i	\(0.390868\pi\)
\(17\)	6.49772	−	0.745544i	1.57593	−	0.180821i	0.718538	−	0.695488i	\(-0.244812\pi\)
							0.857390	+	0.514667i	\(0.172084\pi\)
\(19\)	3.59636	−	2.03096i	0.825061	−	0.465933i	−0.0205144	−	0.999790i	\(-0.506530\pi\)
							0.845575	+	0.533856i	\(0.179258\pi\)
\(23\)	0.512172	+	0.150387i	0.106795	+	0.0313579i	0.334694	−	0.942327i	\(-0.391367\pi\)
							−0.227898	+	0.973685i	\(0.573185\pi\)
\(29\)	0.229561	+	1.13293i	0.0426283	+	0.210379i	0.995736	−	0.0922538i	\(-0.0294071\pi\)
							−0.953107	+	0.302633i	\(0.902134\pi\)
\(31\)	−3.34940	+	1.64679i	−0.601570	+	0.295772i	−0.716484	−	0.697603i	\(-0.754250\pi\)
							0.114914	+	0.993375i	\(0.463341\pi\)
\(37\)	1.18977	+	1.22424i	0.195596	+	0.201264i	0.807834	−	0.589410i	\(-0.200640\pi\)
							−0.612238	+	0.790674i	\(0.709731\pi\)
\(41\)	−0.792233	−	9.22380i	−0.123726	−	1.44052i	−0.750271	−	0.661130i	\(-0.770077\pi\)
							0.626545	−	0.779385i	\(-0.284468\pi\)
\(43\)	−6.08038	+	3.90762i	−0.927250	+	0.595907i	−0.914752	−	0.404015i	\(-0.867614\pi\)
							−0.0124972	+	0.999922i	\(0.503978\pi\)
\(47\)	−0.0409518	+	0.00952293i	−0.00597343	+	0.00138906i	−0.229482	−	0.973313i	\(-0.573703\pi\)
							0.223509	+	0.974702i	\(0.428249\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.2.g.a.5.3	✓	200
121.97	even	55	inner	242.2.g.a.97.3	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.g.a.5.3	✓	200	1.1	even	1	trivial
242.2.g.a.97.3	yes	200	121.97	even	55	inner