Embedded newform 230.3.i.a.109.15

• Label: 230.3.i.a.109.15
• 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			109.15
Character	\(\chi\)	\(=\)	230.109
Dual form			230.3.i.a.19.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28641 + 0.587486i) q^{2} +(-0.725559 + 2.47103i) q^{3} +(1.30972 + 1.51150i) q^{4} +(4.93077 - 0.829144i) q^{5} +(-2.38506 + 2.75251i) q^{6} +(-0.102488 + 0.712820i) q^{7} +(0.796860 + 2.71386i) q^{8} +(1.99174 + 1.28002i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 48 q^{4} - 8 q^{6} + 96 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.725559	+	2.47103i	−0.241853	+	0.823676i	0.745686	+	0.666298i	\(0.232122\pi\)
−0.987539	+	0.157378i	\(0.949696\pi\)
\(5\)	4.93077	−	0.829144i	0.986155	−	0.165829i
							\(7\)	−0.102488	+	0.712820i	−0.0146412	+	0.101831i	−0.995832	−	0.0912107i	\(-0.970926\pi\)
0.981190	+	0.193042i	\(0.0618354\pi\)
\(11\)	10.3444	−	4.72412i	0.940397	−	0.429465i	0.114587	−	0.993413i	\(-0.463445\pi\)
							0.825810	+	0.563948i	\(0.190718\pi\)
\(13\)	−4.37372	+	0.628847i	−0.336440	+	0.0483728i	−0.308465	−	0.951236i	\(-0.599815\pi\)
							−0.0279756	+	0.999609i	\(0.508906\pi\)
\(17\)	−21.2072	+	24.4744i	−1.24748	+	1.43967i	−0.393533	+	0.919310i	\(0.628747\pi\)
							−0.853948	+	0.520359i	\(0.825798\pi\)
\(19\)	−3.60054	+	3.11988i	−0.189502	+	0.164204i	−0.744445	−	0.667684i	\(-0.767286\pi\)
							0.554943	+	0.831889i	\(0.312740\pi\)
\(23\)	0.814663	−	22.9856i	0.0354201	−	0.999373i
							\(29\)	24.5625	−	28.3467i	0.846983	−	0.977471i	−0.152959	−	0.988233i	\(-0.548880\pi\)
0.999942	+	0.0107619i	\(0.00342567\pi\)
\(31\)	−32.2132	+	9.45864i	−1.03913	+	0.305117i	−0.756419	−	0.654087i	\(-0.773053\pi\)
							−0.282715	+	0.959204i	\(0.591235\pi\)
\(37\)	19.5470	+	12.5621i	0.528297	+	0.339516i	0.777446	−	0.628949i	\(-0.216515\pi\)
							−0.249150	+	0.968465i	\(0.580151\pi\)
\(41\)	51.2789	−	32.9549i	1.25070	−	0.803779i	0.263721	−	0.964599i	\(-0.415050\pi\)
							0.986984	+	0.160820i	\(0.0514138\pi\)
\(43\)	−57.0767	−	16.7592i	−1.32736	−	0.389749i	−0.460220	−	0.887805i	\(-0.652229\pi\)
							−0.867145	+	0.498056i	\(0.834047\pi\)
\(47\)			11.8605i			0.252352i	0.992008	+	0.126176i	\(0.0402704\pi\)
							−0.992008	+	0.126176i	\(0.959730\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.i.a.109.15	yes	240
5.4	even	2	inner	230.3.i.a.109.10	yes	240
23.19	odd	22	inner	230.3.i.a.19.10	✓	240
115.19	odd	22	inner	230.3.i.a.19.15	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.i.a.19.10	✓	240	23.19	odd	22	inner
230.3.i.a.19.15	yes	240	115.19	odd	22	inner
230.3.i.a.109.10	yes	240	5.4	even	2	inner
230.3.i.a.109.15	yes	240	1.1	even	1	trivial