Embedded newform 230.3.i.a.19.5

• Label: 230.3.i.a.19.5
• Level: $230$
• Weight: $3$
• Character: 230.19
• Analytic conductor: $6.267$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	230.i (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26704608029\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			19.5
Character	\(\chi\)	\(=\)	230.19
Dual form			230.3.i.a.109.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28641 + 0.587486i) q^{2} +(-0.424421 - 1.44544i) q^{3} +(1.30972 - 1.51150i) q^{4} +(3.70278 + 3.35997i) q^{5} +(1.39516 + 1.61010i) q^{6} +(1.33136 + 9.25983i) q^{7} +(-0.796860 + 2.71386i) q^{8} +(5.66211 - 3.63882i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 48 q^{4} - 8 q^{6} + 96 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{15}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28641	+	0.587486i	−0.643207	+	0.293743i
							\(3\)	−0.424421	−	1.44544i	−0.141474	−	0.481815i	0.858021	−	0.513615i	\(-0.171694\pi\)
−0.999494	+	0.0318005i	\(0.989876\pi\)
\(5\)	3.70278	+	3.35997i	0.740557	+	0.671994i
							\(7\)	1.33136	+	9.25983i	0.190195	+	1.32283i	0.831490	+	0.555539i	\(0.187488\pi\)
−0.641296	+	0.767294i	\(0.721603\pi\)
\(11\)	−11.1673	−	5.09994i	−1.01521	−	0.463631i	−0.162886	−	0.986645i	\(-0.552080\pi\)
							−0.852324	+	0.523014i	\(0.824807\pi\)
\(13\)	−16.2447	−	2.33563i	−1.24959	−	0.179664i	−0.514415	−	0.857541i	\(-0.671991\pi\)
							−0.735175	+	0.677878i	\(0.762900\pi\)
\(17\)	16.9201	+	19.5268i	0.995301	+	1.14864i	0.988889	+	0.148659i	\(0.0474956\pi\)
							0.00641209	+	0.999979i	\(0.497959\pi\)
\(19\)	22.5651	+	19.5528i	1.18764	+	1.02909i	0.998892	+	0.0470684i	\(0.0149879\pi\)
							0.188746	+	0.982026i	\(0.439558\pi\)
\(23\)	2.98448	+	22.8055i	0.129760	+	0.991545i
							\(29\)	9.51860	+	10.9851i	0.328228	+	0.378795i	0.895746	−	0.444566i	\(-0.146642\pi\)
−0.567519	+	0.823361i	\(0.692096\pi\)
\(31\)	−22.7320	−	6.67472i	−0.733290	−	0.215314i	−0.106293	−	0.994335i	\(-0.533898\pi\)
							−0.626997	+	0.779021i	\(0.715716\pi\)
\(37\)	37.0350	−	23.8009i	1.00094	−	0.643268i	0.0659105	−	0.997826i	\(-0.479005\pi\)
							0.935034	+	0.354557i	\(0.115368\pi\)
\(41\)	32.0940	+	20.6255i	0.782780	+	0.503062i	0.869955	−	0.493131i	\(-0.164148\pi\)
							−0.0871753	+	0.996193i	\(0.527784\pi\)
\(43\)	5.57263	−	1.63627i	0.129596	−	0.0380528i	−0.216291	−	0.976329i	\(-0.569396\pi\)
							0.345887	+	0.938276i	\(0.387578\pi\)
\(47\)			52.2193i			1.11105i	0.831500	+	0.555525i	\(0.187483\pi\)
							−0.831500	+	0.555525i	\(0.812517\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.i.a.19.5	✓	240
5.4	even	2	inner	230.3.i.a.19.20	yes	240
23.17	odd	22	inner	230.3.i.a.109.20	yes	240
115.109	odd	22	inner	230.3.i.a.109.5	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.i.a.19.5	✓	240	1.1	even	1	trivial
230.3.i.a.19.20	yes	240	5.4	even	2	inner
230.3.i.a.109.5	yes	240	115.109	odd	22	inner
230.3.i.a.109.20	yes	240	23.17	odd	22	inner