Embedded newform 230.4.j.a.9.7

• Label: 230.4.j.a.9.7
• Level: $230$
• Weight: $4$
• Character: 230.9
• Analytic conductor: $13.570$
• Analytic rank: $0$
• Dimension: $360$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	230.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.5704393013\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(36\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			9.7
Character	\(\chi\)	\(=\)	230.9
Dual form			230.4.j.a.179.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.563465 - 1.91899i) q^{2} +(-3.14614 + 2.72615i) q^{3} +(-3.36501 + 2.16256i) q^{4} +(-3.36640 - 10.6615i) q^{5} +(7.00418 + 4.50131i) q^{6} +(17.7210 + 8.09292i) q^{7} +(6.04600 + 5.23889i) q^{8} +(-1.37617 + 9.57148i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q + 144 q^{4} + 32 q^{5} + 8 q^{6} + 224 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.563465	−	1.91899i	−0.199215	−	0.678464i
							\(3\)	−3.14614	+	2.72615i	−0.605475	+	0.524647i	−0.902762	−	0.430141i	\(-0.858464\pi\)
0.297286	+	0.954788i	\(0.403918\pi\)
\(5\)	−3.36640	−	10.6615i	−0.301100	−	0.953592i
							\(7\)	17.7210	+	8.09292i	0.956845	+	0.436977i	0.831739	−	0.555167i	\(-0.187346\pi\)
0.125106	+	0.992143i	\(0.460073\pi\)
\(11\)	−26.0618	−	7.65243i	−0.714357	−	0.209754i	−0.0956942	−	0.995411i	\(-0.530507\pi\)
							−0.618663	+	0.785657i	\(0.712325\pi\)
\(13\)	46.8693	−	21.4045i	0.999940	−	0.456657i	0.152931	−	0.988237i	\(-0.451129\pi\)
							0.847008	+	0.531580i	\(0.178401\pi\)
\(17\)	10.0534	−	15.6435i	0.143431	−	0.223182i	−0.762104	−	0.647454i	\(-0.775834\pi\)
							0.905535	+	0.424272i	\(0.139470\pi\)
\(19\)	−40.2014	+	25.8359i	−0.485413	+	0.311956i	−0.760358	−	0.649504i	\(-0.774977\pi\)
							0.274946	+	0.961460i	\(0.411340\pi\)
\(23\)	32.0242	−	105.553i	0.290327	−	0.956928i
							\(29\)	−223.109	−	143.383i	−1.42863	−	0.918123i	−0.999892	−	0.0146936i	\(-0.995323\pi\)
−0.428736	−	0.903430i	\(-0.641041\pi\)
\(31\)	201.546	−	232.596i	1.16770	−	1.34760i	0.241575	−	0.970382i	\(-0.422336\pi\)
							0.926125	−	0.377216i	\(-0.123119\pi\)
\(37\)	−301.727	−	43.3818i	−1.34064	−	0.192755i	−0.565606	−	0.824676i	\(-0.691358\pi\)
							−0.775033	+	0.631921i	\(0.782267\pi\)
\(41\)	−40.1360	−	279.152i	−0.152883	−	1.06332i	−0.911356	−	0.411619i	\(-0.864964\pi\)
							0.758473	−	0.651704i	\(-0.225946\pi\)
\(43\)	135.544	−	117.450i	0.480705	−	0.416533i	−0.380508	−	0.924778i	\(-0.624251\pi\)
							0.861213	+	0.508245i	\(0.169705\pi\)
\(47\)			325.865i			1.01133i	0.862731	+	0.505663i	\(0.168752\pi\)
							−0.862731	+	0.505663i	\(0.831248\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.j.a.9.7	✓	360
5.4	even	2	inner	230.4.j.a.9.30	yes	360
23.18	even	11	inner	230.4.j.a.179.30	yes	360
115.64	even	22	inner	230.4.j.a.179.7	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.j.a.9.7	✓	360	1.1	even	1	trivial
230.4.j.a.9.30	yes	360	5.4	even	2	inner
230.4.j.a.179.7	yes	360	115.64	even	22	inner
230.4.j.a.179.30	yes	360	23.18	even	11	inner