Embedded newform 230.4.j.a.9.20

• Label: 230.4.j.a.9.20
• 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			9.20
Character	\(\chi\)	\(=\)	230.9
Dual form			230.4.j.a.179.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.563465 + 1.91899i) q^{2} +(-6.71164 + 5.81567i) q^{3} +(-3.36501 + 2.16256i) q^{4} +(5.63759 + 9.65492i) q^{5} +(-14.9420 - 9.60261i) q^{6} +(-25.7501 - 11.7597i) q^{7} +(-6.04600 - 5.23889i) q^{8} +(7.38159 - 51.3401i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q + 144 q^{4} + 32 q^{5} + 8 q^{6} + 224 q^{9}+O(q^{10}) \)

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
							\(3\)	−6.71164	+	5.81567i	−1.29165	+	1.11923i	−0.305715	+	0.952123i	\(0.598896\pi\)
−0.985940	+	0.167102i	\(0.946559\pi\)
\(5\)	5.63759	+	9.65492i	0.504242	+	0.863563i
							\(7\)	−25.7501	−	11.7597i	−1.39037	−	0.634962i	−0.427277	−	0.904121i	\(-0.640527\pi\)
−0.963096	+	0.269159i	\(0.913254\pi\)
\(11\)	0.379343	+	0.111385i	0.0103978	+	0.00305308i	0.286927	−	0.957952i	\(-0.407366\pi\)
							−0.276530	+	0.961005i	\(0.589184\pi\)
\(13\)	9.29694	−	4.24577i	0.198347	−	0.0905819i	−0.313765	−	0.949501i	\(-0.601591\pi\)
							0.512112	+	0.858919i	\(0.328863\pi\)
\(17\)	−53.1143	+	82.6475i	−0.757772	+	1.17912i	0.221222	+	0.975224i	\(0.428996\pi\)
							−0.978993	+	0.203892i	\(0.934641\pi\)
\(19\)	72.1254	−	46.3522i	0.870879	−	0.559680i	−0.0271423	−	0.999632i	\(-0.508641\pi\)
							0.898022	+	0.439951i	\(0.145004\pi\)
\(23\)	−104.570	−	35.1008i	−0.948017	−	0.318219i
							\(29\)	110.207	+	70.8256i	0.705686	+	0.453517i	0.843631	−	0.536923i	\(-0.180413\pi\)
−0.137945	+	0.990440i	\(0.544050\pi\)
\(31\)	−64.5471	+	74.4913i	−0.373968	+	0.431582i	−0.911271	−	0.411808i	\(-0.864898\pi\)
							0.537303	+	0.843389i	\(0.319443\pi\)
\(37\)	351.410	+	50.5252i	1.56139	+	0.224494i	0.868184	−	0.496243i	\(-0.165287\pi\)
							0.693208	+	0.720737i	\(0.256197\pi\)
\(41\)	−53.2055	−	370.052i	−0.202666	−	1.40957i	−0.796332	−	0.604860i	\(-0.793229\pi\)
							0.593666	−	0.804712i	\(-0.297680\pi\)
\(43\)	143.179	−	124.066i	0.507783	−	0.439996i	−0.362910	−	0.931824i	\(-0.618217\pi\)
							0.870692	+	0.491828i	\(0.163671\pi\)
\(47\)		−	411.144i		−	1.27599i	−0.770041	−	0.637994i	\(-0.779764\pi\)
							0.770041	−	0.637994i	\(-0.220236\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.j.a.9.20	yes	360
5.4	even	2	inner	230.4.j.a.9.17	✓	360
23.18	even	11	inner	230.4.j.a.179.17	yes	360
115.64	even	22	inner	230.4.j.a.179.20	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.j.a.9.17	✓	360	5.4	even	2	inner
230.4.j.a.9.20	yes	360	1.1	even	1	trivial
230.4.j.a.179.17	yes	360	23.18	even	11	inner
230.4.j.a.179.20	yes	360	115.64	even	22	inner