Embedded newform 230.4.j.a.9.10

• Label: 230.4.j.a.9.10
• 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			9.10
Character	\(\chi\)	\(=\)	230.9
Dual form			230.4.j.a.179.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.563465 - 1.91899i) q^{2} +(0.729490 - 0.632107i) q^{3} +(-3.36501 + 2.16256i) q^{4} +(-6.36706 - 9.19024i) q^{5} +(-1.62405 - 1.04371i) q^{6} +(6.54228 + 2.98776i) q^{7} +(6.04600 + 5.23889i) q^{8} +(-3.70990 + 25.8029i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q + 144 q^{4} + 32 q^{5} + 8 q^{6} + 224 q^{9}+O(q^{10}) \)

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
							\(3\)	0.729490	−	0.632107i	0.140390	−	0.121649i	−0.581835	−	0.813307i	\(-0.697665\pi\)
0.722225	+	0.691658i	\(0.243119\pi\)
\(5\)	−6.36706	−	9.19024i	−0.569487	−	0.822000i
							\(7\)	6.54228	+	2.98776i	0.353250	+	0.161324i	0.584134	−	0.811658i	\(-0.301434\pi\)
−0.230884	+	0.972981i	\(0.574162\pi\)
\(11\)	−10.0488	−	2.95059i	−0.275439	−	0.0808760i	0.141096	−	0.989996i	\(-0.454937\pi\)
							−0.416535	+	0.909120i	\(0.636756\pi\)
\(13\)	−58.4477	+	26.6922i	−1.24696	+	0.569467i	−0.925963	−	0.377613i	\(-0.876745\pi\)
							−0.320996	+	0.947081i	\(0.604018\pi\)
\(17\)	−25.4001	+	39.5233i	−0.362378	+	0.563871i	−0.973792	−	0.227440i	\(-0.926965\pi\)
							0.611414	+	0.791311i	\(0.290601\pi\)
\(19\)	120.848	−	77.6644i	1.45918	−	0.937759i	0.460436	−	0.887693i	\(-0.347693\pi\)
							0.998746	−	0.0500663i	\(-0.0159433\pi\)
\(23\)	−50.0130	+	98.3143i	−0.453410	+	0.891302i
							\(29\)	219.111	+	140.814i	1.40303	+	0.901672i	0.999909	−	0.0134754i	\(-0.00428948\pi\)
0.403120	+	0.915147i	\(0.367926\pi\)
\(31\)	−31.1059	+	35.8981i	−0.180219	+	0.207983i	−0.838670	−	0.544640i	\(-0.816666\pi\)
							0.658451	+	0.752623i	\(0.271212\pi\)
\(37\)	−140.655	−	20.2232i	−0.624961	−	0.0898559i	−0.177444	−	0.984131i	\(-0.556783\pi\)
							−0.447518	+	0.894275i	\(0.647692\pi\)
\(41\)	54.4481	+	378.695i	0.207399	+	1.44249i	0.781601	+	0.623778i	\(0.214403\pi\)
							−0.574202	+	0.818713i	\(0.694688\pi\)
\(43\)	−7.29633	+	6.32231i	−0.0258763	+	0.0224219i	−0.667703	−	0.744428i	\(-0.732722\pi\)
							0.641827	+	0.766850i	\(0.278177\pi\)
\(47\)			450.261i			1.39739i	0.715420	+	0.698694i	\(0.246235\pi\)
							−0.715420	+	0.698694i	\(0.753765\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.4.j.a.9.10	✓	360
5.4	even	2	inner	230.4.j.a.9.27	yes	360
23.18	even	11	inner	230.4.j.a.179.27	yes	360
115.64	even	22	inner	230.4.j.a.179.10	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.j.a.9.10	✓	360	1.1	even	1	trivial
230.4.j.a.9.27	yes	360	5.4	even	2	inner
230.4.j.a.179.10	yes	360	115.64	even	22	inner
230.4.j.a.179.27	yes	360	23.18	even	11	inner