Embedded newform 242.2.g.b.25.4

• Label: 242.2.g.b.25.4
• Level: $242$
• Weight: $2$
• Character: 242.25
• 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			25.4
Character	\(\chi\)	\(=\)	242.25
Dual form			242.2.g.b.213.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.466667 - 0.884433i) q^{2} +(0.340662 + 1.04845i) q^{3} +(-0.564443 + 0.825472i) q^{4} +(0.456414 - 2.25249i) q^{5} +(0.768307 - 0.790569i) q^{6} +(-0.473473 - 1.80128i) q^{7} +(0.993482 + 0.113991i) q^{8} +(1.44386 - 1.04902i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{2} - 8 q^{3} + 6 q^{4} + 5 q^{5} - 2 q^{6} - 2 q^{7} + 6 q^{8} - 80 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(123\)
\(\chi(n)\)	\(e\left(\frac{19}{55}\right)\)

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
							\(3\)	0.340662	+	1.04845i	0.196681	+	0.605322i	0.999953	+	0.00970945i	\(0.00309066\pi\)
−0.803272	+	0.595613i	\(0.796909\pi\)
\(5\)	0.456414	−	2.25249i	0.204114	−	1.00734i	−0.738702	−	0.674032i	\(-0.764561\pi\)
							0.942817	−	0.333312i	\(-0.108166\pi\)
\(7\)	−0.473473	−	1.80128i	−0.178956	−	0.680820i	−0.994737	−	0.102459i	\(-0.967329\pi\)
							0.815781	−	0.578361i	\(-0.196307\pi\)
\(11\)	−2.99649	+	1.42163i	−0.903477	+	0.428637i
							\(13\)	3.50369	−	1.25011i	0.971748	−	0.346719i	0.197993	−	0.980204i	\(-0.436558\pi\)
0.773755	+	0.633485i	\(0.218376\pi\)
\(17\)	4.96011	−	1.15342i	1.20300	−	0.279746i	0.423355	−	0.905964i	\(-0.360852\pi\)
							0.779648	+	0.626217i	\(0.215398\pi\)
\(19\)	2.49195	−	4.13243i	0.571692	−	0.948045i	−0.427341	−	0.904090i	\(-0.640550\pi\)
							0.999034	−	0.0439549i	\(-0.0139958\pi\)
\(23\)	−5.18618	−	3.33296i	−1.08139	−	0.694969i	−0.126514	−	0.991965i	\(-0.540379\pi\)
							−0.954879	+	0.296995i	\(0.904015\pi\)
\(29\)	−6.36883	+	2.69149i	−1.18266	+	0.499797i	−0.889797	−	0.456357i	\(-0.849154\pi\)
							−0.292865	+	0.956154i	\(0.594609\pi\)
\(31\)	−2.44791	+	3.17451i	−0.439658	+	0.570158i	−0.958638	−	0.284628i	\(-0.908130\pi\)
							0.518980	+	0.854786i	\(0.326312\pi\)
\(37\)	−0.339565	+	11.8863i	−0.0558241	+	1.95410i	0.183181	+	0.983079i	\(0.441360\pi\)
							−0.239005	+	0.971018i	\(0.576821\pi\)
\(41\)	3.62419	−	0.627191i	0.566003	−	0.0979507i	0.119664	−	0.992814i	\(-0.461818\pi\)
							0.446339	+	0.894864i	\(0.352727\pi\)
\(43\)	3.24422	−	7.10384i	0.494738	−	1.08333i	−0.483406	−	0.875396i	\(-0.660600\pi\)
							0.978144	−	0.207929i	\(-0.0666724\pi\)
\(47\)	−9.38215	+	4.61289i	−1.36853	+	0.672860i	−0.969730	−	0.244179i	\(-0.921481\pi\)
							−0.398798	+	0.917039i	\(0.630572\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.2.g.b.25.4	✓	240
121.92	even	55	inner	242.2.g.b.213.4	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.g.b.25.4	✓	240	1.1	even	1	trivial
242.2.g.b.213.4	yes	240	121.92	even	55	inner