Embedded newform 230.3.k.a.13.3

• Label: 230.3.k.a.13.3
• 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			13.3
Character	\(\chi\)	\(=\)	230.13
Dual form			230.3.k.a.177.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41061 - 0.100889i) q^{2} +(-0.745890 - 3.42880i) q^{3} +(1.97964 + 0.284630i) q^{4} +(1.45154 - 4.78467i) q^{5} +(0.706232 + 4.91195i) q^{6} +(10.4743 - 5.71940i) q^{7} +(-2.76379 - 0.601225i) q^{8} +(-3.01364 + 1.37628i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 24 q^{2} - 4 q^{5} - 74 q^{7} + 48 q^{8}+O(q^{10}) \)

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
							\(3\)	−0.745890	−	3.42880i	−0.248630	−	1.14293i	−0.916932	−	0.399044i	\(-0.869342\pi\)
0.668302	−	0.743890i	\(-0.267021\pi\)
\(5\)	1.45154	−	4.78467i	0.290307	−	0.956933i
							\(7\)	10.4743	−	5.71940i	1.49633	−	0.817057i	0.497597	−	0.867408i	\(-0.334216\pi\)
0.998732	+	0.0503507i	\(0.0160339\pi\)
\(11\)	1.39769	−	1.61302i	0.127062	−	0.146638i	−0.688653	−	0.725091i	\(-0.741798\pi\)
							0.815716	+	0.578453i	\(0.196343\pi\)
\(13\)	17.9186	+	9.78430i	1.37835	+	0.752638i	0.985932	−	0.167147i	\(-0.0534553\pi\)
							0.392422	+	0.919785i	\(0.371637\pi\)
\(17\)	−6.76426	−	5.06366i	−0.397898	−	0.297863i	0.381528	−	0.924357i	\(-0.375398\pi\)
							−0.779426	+	0.626495i	\(0.784489\pi\)
\(19\)	4.21099	+	0.605449i	0.221631	+	0.0318657i	0.252236	−	0.967666i	\(-0.418834\pi\)
							−0.0306047	+	0.999532i	\(0.509743\pi\)
\(23\)	19.1176	+	12.7873i	0.831201	+	0.555972i
							\(29\)	−43.4590	+	6.24847i	−1.49859	+	0.215464i	−0.842284	−	0.539033i	\(-0.818790\pi\)
−0.656303	+	0.754498i	\(0.727881\pi\)
\(31\)	−8.88408	+	5.70945i	−0.286583	+	0.184176i	−0.676032	−	0.736872i	\(-0.736302\pi\)
							0.389449	+	0.921048i	\(0.372666\pi\)
\(37\)	3.28252	+	1.22431i	0.0887166	+	0.0330896i	0.393431	−	0.919354i	\(-0.371288\pi\)
							−0.304714	+	0.952444i	\(0.598561\pi\)
\(41\)	−4.59571	+	10.0632i	−0.112090	+	0.245444i	−0.957361	−	0.288895i	\(-0.906712\pi\)
							0.845270	+	0.534339i	\(0.179439\pi\)
\(43\)	9.37392	+	43.0912i	0.217998	+	1.00212i	0.947900	+	0.318569i	\(0.103202\pi\)
							−0.729901	+	0.683553i	\(0.760434\pi\)
\(47\)	−18.1342	+	18.1342i	−0.385834	+	0.385834i	−0.873198	−	0.487365i	\(-0.837958\pi\)
							0.487365	+	0.873198i	\(0.337958\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.a.13.3	✓	240
5.2	odd	4	inner	230.3.k.a.197.3	yes	240
23.16	even	11	inner	230.3.k.a.223.3	yes	240
115.62	odd	44	inner	230.3.k.a.177.3	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.a.13.3	✓	240	1.1	even	1	trivial
230.3.k.a.177.3	yes	240	115.62	odd	44	inner
230.3.k.a.197.3	yes	240	5.2	odd	4	inner
230.3.k.a.223.3	yes	240	23.16	even	11	inner