Embedded newform 230.3.k.b.13.8

• Label: 230.3.k.b.13.8
• Level: $230$
• Weight: $3$
• Character: 230.13
• 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.k (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26704608029\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(12\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			13.8
Character	\(\chi\)	\(=\)	230.13
Dual form			230.3.k.b.177.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41061 + 0.100889i) q^{2} +(0.302791 + 1.39191i) q^{3} +(1.97964 + 0.284630i) q^{4} +(-2.63202 + 4.25117i) q^{5} +(0.286693 + 1.99399i) q^{6} +(-5.20021 + 2.83953i) q^{7} +(2.76379 + 0.601225i) q^{8} +(6.34096 - 2.89582i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{7}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41061	+	0.100889i	0.705305	+	0.0504444i
							\(3\)	0.302791	+	1.39191i	0.100930	+	0.463970i	0.999675	+	0.0254971i	\(0.00811685\pi\)
−0.898744	+	0.438473i	\(0.855520\pi\)
\(5\)	−2.63202	+	4.25117i	−0.526404	+	0.850235i
							\(7\)	−5.20021	+	2.83953i	−0.742887	+	0.405647i	−0.805658	−	0.592381i	\(-0.798188\pi\)
0.0627710	+	0.998028i	\(0.480006\pi\)
\(11\)	−12.2025	+	14.0824i	−1.10932	+	1.28022i	−0.152889	+	0.988243i	\(0.548858\pi\)
							−0.956426	+	0.291975i	\(0.905688\pi\)
\(13\)	4.43840	+	2.42355i	0.341415	+	0.186427i	0.640804	−	0.767704i	\(-0.278601\pi\)
							−0.299389	+	0.954131i	\(0.596783\pi\)
\(17\)	3.49894	+	2.61927i	0.205820	+	0.154075i	0.697215	−	0.716862i	\(-0.254422\pi\)
							−0.491396	+	0.870937i	\(0.663513\pi\)
\(19\)	7.14287	+	1.02699i	0.375940	+	0.0540521i	0.327695	−	0.944783i	\(-0.393728\pi\)
							0.0482451	+	0.998836i	\(0.484637\pi\)
\(23\)	−20.7644	+	9.89133i	−0.902801	+	0.430058i
							\(29\)	29.0844	−	4.18171i	1.00291	−	0.144197i	0.378755	−	0.925497i	\(-0.376352\pi\)
0.624155	+	0.781300i	\(0.285443\pi\)
\(31\)	33.9744	−	21.8340i	1.09595	−	0.704323i	0.137761	−	0.990466i	\(-0.456010\pi\)
							0.958187	+	0.286143i	\(0.0923732\pi\)
\(37\)	24.1808	+	9.01895i	0.653534	+	0.243756i	0.654289	−	0.756244i	\(-0.272968\pi\)
							−0.000755643		1.00000i	\(0.500241\pi\)
\(41\)	−0.860004	+	1.88315i	−0.0209757	+	0.0459304i	−0.919830	−	0.392317i	\(-0.871673\pi\)
							0.898854	+	0.438248i	\(0.144401\pi\)
\(43\)	1.03806	+	4.77191i	0.0241410	+	0.110975i	0.987641	−	0.156735i	\(-0.0500969\pi\)
							−0.963500	+	0.267710i	\(0.913733\pi\)
\(47\)	21.6665	−	21.6665i	0.460990	−	0.460990i	−0.437990	−	0.898980i	\(-0.644309\pi\)
							0.898980	+	0.437990i	\(0.144309\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.13.8	✓	240
5.2	odd	4	inner	230.3.k.b.197.8	yes	240
23.16	even	11	inner	230.3.k.b.223.8	yes	240
115.62	odd	44	inner	230.3.k.b.177.8	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.13.8	✓	240	1.1	even	1	trivial
230.3.k.b.177.8	yes	240	115.62	odd	44	inner
230.3.k.b.197.8	yes	240	5.2	odd	4	inner
230.3.k.b.223.8	yes	240	23.16	even	11	inner