Embedded newform 242.2.g.b.31.2

• Label: 242.2.g.b.31.2
• Level: $242$
• Weight: $2$
• Character: 242.31
• 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			31.2
Character	\(\chi\)	\(=\)	242.31
Dual form			242.2.g.b.203.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.774142 - 0.633012i) q^{2} +(-2.15202 + 1.56354i) q^{3} +(0.198590 - 0.980083i) q^{4} +(2.23021 - 1.09652i) q^{5} +(-0.676233 + 2.57266i) q^{6} +(0.232047 + 2.70167i) q^{7} +(-0.466667 - 0.884433i) q^{8} +(1.25951 - 3.87637i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{2} - 8 q^{3} + 6 q^{4} + 5 q^{5} - 2 q^{6} - 2 q^{7} + 6 q^{8} - 80 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(123\)
\(\chi(n)\)	\(e\left(\frac{43}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.774142	−	0.633012i	0.547401	−	0.447607i
							\(3\)	−2.15202	+	1.56354i	−1.24247	+	0.902709i	−0.997761	−	0.0668872i	\(-0.978693\pi\)
−0.244711	+	0.969596i	\(0.578693\pi\)
\(5\)	2.23021	−	1.09652i	0.997382	−	0.490380i	0.131869	−	0.991267i	\(-0.457902\pi\)
							0.865512	+	0.500887i	\(0.166993\pi\)
\(7\)	0.232047	+	2.70167i	0.0877054	+	1.02113i	0.899421	+	0.437083i	\(0.143988\pi\)
							−0.811716	+	0.584052i	\(0.801466\pi\)
\(11\)	1.32743	+	3.03940i	0.400235	+	0.916412i
							\(13\)	4.92173	−	0.564716i	1.36504	−	0.156624i	0.599689	−	0.800233i	\(-0.295291\pi\)
0.765354	+	0.643609i	\(0.222564\pi\)
\(17\)	1.77531	+	2.59631i	0.430576	+	0.629697i	0.978357	−	0.206925i	\(-0.0663455\pi\)
							−0.547781	+	0.836622i	\(0.684527\pi\)
\(19\)	−1.44925	+	0.517091i	−0.332480	+	0.118629i	−0.497013	−	0.867743i	\(-0.665570\pi\)
							0.164533	+	0.986372i	\(0.447388\pi\)
\(23\)	2.74196	−	3.16440i	0.571739	−	0.659822i	−0.394069	−	0.919081i	\(-0.628933\pi\)
							0.965808	+	0.259259i	\(0.0834783\pi\)
\(29\)	0.0248064	+	0.0321695i	0.00460644	+	0.00597374i	0.794200	−	0.607657i	\(-0.207890\pi\)
							−0.789593	+	0.613630i	\(0.789708\pi\)
\(31\)	−6.06826	−	5.56940i	−1.08989	−	1.00029i	−0.999991	−	0.00434372i	\(-0.998617\pi\)
							−0.0899020	−	0.995951i	\(-0.528655\pi\)
\(37\)	−7.18996	+	4.06036i	−1.18202	+	0.667519i	−0.952129	−	0.305697i	\(-0.901111\pi\)
							−0.229894	+	0.973216i	\(0.573838\pi\)
\(41\)	−3.82996	+	0.219005i	−0.598139	+	0.0342028i	−0.353537	−	0.935421i	\(-0.615021\pi\)
							−0.244602	+	0.969623i	\(0.578657\pi\)
\(43\)	0.934994	−	6.50302i	0.142585	−	0.991702i	−0.785374	−	0.619021i	\(-0.787530\pi\)
							0.927960	−	0.372681i	\(-0.121561\pi\)
\(47\)	2.48120	−	6.37291i	0.361920	−	0.929585i	−0.626715	−	0.779249i	\(-0.715601\pi\)
							0.988635	−	0.150336i	\(-0.0480355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.2.g.b.31.2	✓	240
121.82	even	55	inner	242.2.g.b.203.2	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.g.b.31.2	✓	240	1.1	even	1	trivial
242.2.g.b.203.2	yes	240	121.82	even	55	inner