Embedded newform 230.3.i.a.109.13

• Label: 230.3.i.a.109.13
• 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			109.13
Character	\(\chi\)	\(=\)	230.109
Dual form			230.3.i.a.19.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28641 + 0.587486i) q^{2} +(-1.62857 + 5.54639i) q^{3} +(1.30972 + 1.51150i) q^{4} +(0.750729 + 4.94332i) q^{5} +(-5.35343 + 6.17819i) q^{6} +(-0.543186 + 3.77794i) q^{7} +(0.796860 + 2.71386i) q^{8} +(-20.5389 - 13.1996i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 48 q^{4} - 8 q^{6} + 96 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.62857	+	5.54639i	−0.542856	+	1.84880i	−0.0143001	+	0.999898i	\(0.504552\pi\)
−0.528556	+	0.848899i	\(0.677266\pi\)
\(5\)	0.750729	+	4.94332i	0.150146	+	0.988664i
							\(7\)	−0.543186	+	3.77794i	−0.0775980	+	0.539706i	0.913528	+	0.406777i	\(0.133347\pi\)
−0.991126	+	0.132929i	\(0.957562\pi\)
\(11\)	15.1678	−	6.92691i	1.37889	−	0.629719i	0.418461	−	0.908235i	\(-0.362570\pi\)
							0.960432	+	0.278515i	\(0.0898423\pi\)
\(13\)	6.09510	−	0.876343i	0.468854	−	0.0674110i	0.0961611	−	0.995366i	\(-0.469344\pi\)
							0.372693	+	0.927955i	\(0.378434\pi\)
\(17\)	8.15256	−	9.40856i	0.479563	−	0.553445i	−0.463484	−	0.886105i	\(-0.653401\pi\)
							0.943047	+	0.332661i	\(0.107946\pi\)
\(19\)	18.5815	−	16.1009i	0.977973	−	0.847418i	−0.0103263	−	0.999947i	\(-0.503287\pi\)
							0.988299	+	0.152528i	\(0.0487416\pi\)
\(23\)	1.00299	+	22.9781i	0.0436081	+	0.999049i
							\(29\)	−26.1605	+	30.1908i	−0.902086	+	1.04106i	0.0968666	+	0.995297i	\(0.469118\pi\)
−0.998952	+	0.0457649i	\(0.985427\pi\)
\(31\)	−37.5160	+	11.0157i	−1.21019	+	0.355345i	−0.823742	−	0.566965i	\(-0.808117\pi\)
							−0.386451	+	0.922310i	\(0.626299\pi\)
\(37\)	−17.7676	−	11.4186i	−0.480207	−	0.308610i	0.278050	−	0.960567i	\(-0.410312\pi\)
							−0.758256	+	0.651957i	\(0.773948\pi\)
\(41\)	29.7061	−	19.0910i	0.724539	−	0.465633i	−0.125674	−	0.992072i	\(-0.540109\pi\)
							0.850213	+	0.526438i	\(0.176473\pi\)
\(43\)	−3.53813	−	1.03889i	−0.0822821	−	0.0241602i	0.240332	−	0.970691i	\(-0.422744\pi\)
							−0.322615	+	0.946530i	\(0.604562\pi\)
\(47\)		−	20.2106i		−	0.430012i	−0.976613	−	0.215006i	\(-0.931023\pi\)
							0.976613	−	0.215006i	\(-0.0689771\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.i.a.109.13	yes	240
5.4	even	2	inner	230.3.i.a.109.12	yes	240
23.19	odd	22	inner	230.3.i.a.19.12	✓	240
115.19	odd	22	inner	230.3.i.a.19.13	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.i.a.19.12	✓	240	23.19	odd	22	inner
230.3.i.a.19.13	yes	240	115.19	odd	22	inner
230.3.i.a.109.12	yes	240	5.4	even	2	inner
230.3.i.a.109.13	yes	240	1.1	even	1	trivial