Embedded newform 230.4.j.a.9.16

• Label: 230.4.j.a.9.16
• 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			9.16
Character	\(\chi\)	\(=\)	230.9
Dual form			230.4.j.a.179.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.563465 - 1.91899i) q^{2} +(6.02364 - 5.21951i) q^{3} +(-3.36501 + 2.16256i) q^{4} +(10.3489 - 4.23080i) q^{5} +(-13.4103 - 8.61826i) q^{6} +(-6.46040 - 2.95037i) q^{7} +(6.04600 + 5.23889i) q^{8} +(5.19840 - 36.1557i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q + 144 q^{4} + 32 q^{5} + 8 q^{6} + 224 q^{9}+O(q^{10}) \)

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
							\(3\)	6.02364	−	5.21951i	1.15925	−	1.00450i	0.159399	−	0.987214i	\(-0.449044\pi\)
0.999851	−	0.0172811i	\(-0.00550101\pi\)
\(5\)	10.3489	−	4.23080i	0.925636	−	0.378414i
							\(7\)	−6.46040	−	2.95037i	−0.348829	−	0.159305i	0.233293	−	0.972407i	\(-0.425050\pi\)
−0.582122	+	0.813102i	\(0.697777\pi\)
\(11\)	−49.6274	−	14.5719i	−1.36029	−	0.399418i	−0.481427	−	0.876486i	\(-0.659881\pi\)
							−0.878866	+	0.477068i	\(0.841700\pi\)
\(13\)	−20.7954	+	9.49696i	−0.443663	+	0.202614i	−0.624704	−	0.780861i	\(-0.714780\pi\)
							0.181041	+	0.983475i	\(0.442053\pi\)
\(17\)	43.6265	−	67.8842i	0.622411	−	0.968490i	−0.376700	−	0.926335i	\(-0.622941\pi\)
							0.999111	−	0.0421551i	\(-0.0134223\pi\)
\(19\)	19.3287	−	12.4218i	0.233384	−	0.149987i	−0.418722	−	0.908114i	\(-0.637522\pi\)
							0.652106	+	0.758127i	\(0.273886\pi\)
\(23\)	−74.1257	−	81.6846i	−0.672012	−	0.740540i
							\(29\)	26.8035	+	17.2255i	0.171630	+	0.110300i	0.623636	−	0.781715i	\(-0.285655\pi\)
−0.452005	+	0.892015i	\(0.649291\pi\)
\(31\)	150.475	−	173.658i	0.871811	−	1.00612i	−0.128086	−	0.991763i	\(-0.540883\pi\)
							0.999897	−	0.0143608i	\(-0.00457133\pi\)
\(37\)	194.688	+	27.9919i	0.865041	+	0.124374i	0.560529	−	0.828134i	\(-0.310598\pi\)
							0.304512	+	0.952509i	\(0.401507\pi\)
\(41\)	70.7550	+	492.112i	0.269514	+	1.87451i	0.453029	+	0.891496i	\(0.350343\pi\)
							−0.183515	+	0.983017i	\(0.558748\pi\)
\(43\)	−32.3299	+	28.0140i	−0.114657	+	0.0993512i	−0.710303	−	0.703896i	\(-0.751442\pi\)
							0.595645	+	0.803248i	\(0.296896\pi\)
\(47\)			472.776i			1.46726i	0.679546	+	0.733632i	\(0.262177\pi\)
							−0.679546	+	0.733632i	\(0.737823\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.j.a.9.16	✓	360
5.4	even	2	inner	230.4.j.a.9.21	yes	360
23.18	even	11	inner	230.4.j.a.179.21	yes	360
115.64	even	22	inner	230.4.j.a.179.16	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.j.a.9.16	✓	360	1.1	even	1	trivial
230.4.j.a.9.21	yes	360	5.4	even	2	inner
230.4.j.a.179.16	yes	360	115.64	even	22	inner
230.4.j.a.179.21	yes	360	23.18	even	11	inner