Embedded newform 230.3.i.a.109.4

• Label: 230.3.i.a.109.4
• Level: $230$
• Weight: $3$
• Character: 230.109
• 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.i (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26704608029\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(24\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			109.4
Character	\(\chi\)	\(=\)	230.109
Dual form			230.3.i.a.19.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28641 - 0.587486i) q^{2} +(-0.810314 + 2.75968i) q^{3} +(1.30972 + 1.51150i) q^{4} +(4.95827 + 0.644601i) q^{5} +(2.66367 - 3.07404i) q^{6} +(-0.976678 + 6.79294i) q^{7} +(-0.796860 - 2.71386i) q^{8} +(0.612076 + 0.393358i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 48 q^{4} - 8 q^{6} + 96 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28641	−	0.587486i	−0.643207	−	0.293743i
							\(3\)	−0.810314	+	2.75968i	−0.270105	+	0.919892i	0.707014	+	0.707199i	\(0.250042\pi\)
−0.977119	+	0.212693i	\(0.931777\pi\)
\(5\)	4.95827	+	0.644601i	0.991655	+	0.128920i
							\(7\)	−0.976678	+	6.79294i	−0.139525	+	0.970421i	0.792976	+	0.609253i	\(0.208531\pi\)
−0.932501	+	0.361167i	\(0.882378\pi\)
\(11\)	15.1121	−	6.90144i	1.37382	−	0.627404i	0.414587	−	0.910010i	\(-0.363926\pi\)
							0.959236	+	0.282606i	\(0.0911987\pi\)
\(13\)	−16.4246	+	2.36151i	−1.26343	+	0.181654i	−0.741279	−	0.671197i	\(-0.765780\pi\)
							−0.522154	+	0.852851i	\(0.674871\pi\)
\(17\)	5.15630	−	5.95069i	0.303312	−	0.350041i	−0.583548	−	0.812078i	\(-0.698336\pi\)
							0.886860	+	0.462038i	\(0.152882\pi\)
\(19\)	−8.26853	+	7.16472i	−0.435186	+	0.377091i	−0.844724	−	0.535202i	\(-0.820235\pi\)
							0.409538	+	0.912293i	\(0.365690\pi\)
\(23\)	9.37166	+	21.0041i	0.407463	+	0.913222i
							\(29\)	−29.7713	+	34.3579i	−1.02660	+	1.18476i	−0.0439942	+	0.999032i	\(0.514008\pi\)
−0.982602	+	0.185723i	\(0.940537\pi\)
\(31\)	23.0191	−	6.75902i	0.742552	−	0.218033i	0.111490	−	0.993766i	\(-0.464438\pi\)
							0.631062	+	0.775733i	\(0.282619\pi\)
\(37\)	−22.6337	−	14.5458i	−0.611721	−	0.393129i	0.197781	−	0.980246i	\(-0.436626\pi\)
							−0.809502	+	0.587117i	\(0.800263\pi\)
\(41\)	−58.9339	+	37.8745i	−1.43741	+	0.923769i	−0.437716	+	0.899113i	\(0.644212\pi\)
							−0.999696	+	0.0246551i	\(0.992151\pi\)
\(43\)	73.8331	+	21.6794i	1.71705	+	0.504171i	0.984326	−	0.176356i	\(-0.0564310\pi\)
							0.732723	+	0.680527i	\(0.238249\pi\)
\(47\)		−	13.1589i		−	0.279976i	−0.990153	−	0.139988i	\(-0.955294\pi\)
							0.990153	−	0.139988i	\(-0.0447064\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.i.a.109.4	yes	240
5.4	even	2	inner	230.3.i.a.109.21	yes	240
23.19	odd	22	inner	230.3.i.a.19.21	yes	240
115.19	odd	22	inner	230.3.i.a.19.4	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.i.a.19.4	✓	240	115.19	odd	22	inner
230.3.i.a.19.21	yes	240	23.19	odd	22	inner
230.3.i.a.109.4	yes	240	1.1	even	1	trivial
230.3.i.a.109.21	yes	240	5.4	even	2	inner