Embedded newform 230.3.i.a.109.2

• Label: 230.3.i.a.109.2
• 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			109.2
Character	\(\chi\)	\(=\)	230.109
Dual form			230.3.i.a.19.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28641 - 0.587486i) q^{2} +(-1.02272 + 3.48306i) q^{3} +(1.30972 + 1.51150i) q^{4} +(-2.37251 + 4.40127i) q^{5} +(3.36189 - 3.87983i) q^{6} +(-1.08475 + 7.54462i) q^{7} +(-0.796860 - 2.71386i) q^{8} +(-3.51448 - 2.25862i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 48 q^{4} - 8 q^{6} + 96 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.02272	+	3.48306i	−0.340906	+	1.16102i	0.593507	+	0.804829i	\(0.297743\pi\)
−0.934413	+	0.356191i	\(0.884075\pi\)
\(5\)	−2.37251	+	4.40127i	−0.474502	+	0.880255i
							\(7\)	−1.08475	+	7.54462i	−0.154965	+	1.07780i	0.752778	+	0.658275i	\(0.228714\pi\)
−0.907742	+	0.419528i	\(0.862196\pi\)
\(11\)	−6.50838	+	2.97228i	−0.591671	+	0.270207i	−0.688667	−	0.725078i	\(-0.741804\pi\)
							0.0969960	+	0.995285i	\(0.469077\pi\)
\(13\)	7.13042	−	1.02520i	0.548494	−	0.0788615i	0.137504	−	0.990501i	\(-0.456092\pi\)
							0.410990	+	0.911640i	\(0.365183\pi\)
\(17\)	5.52068	−	6.37120i	0.324746	−	0.374777i	−0.569777	−	0.821800i	\(-0.692970\pi\)
							0.894523	+	0.447023i	\(0.147516\pi\)
\(19\)	5.13612	−	4.45047i	0.270322	−	0.234235i	−0.509142	−	0.860683i	\(-0.670037\pi\)
							0.779464	+	0.626447i	\(0.215492\pi\)
\(23\)	−19.1315	−	12.7666i	−0.831804	−	0.555069i
							\(29\)	−0.932091	+	1.07569i	−0.0321411	+	0.0370928i	−0.771593	−	0.636117i	\(-0.780540\pi\)
0.739451	+	0.673210i	\(0.235085\pi\)
\(31\)	−6.94594	+	2.03951i	−0.224063	+	0.0657907i	−0.391837	−	0.920035i	\(-0.628160\pi\)
							0.167774	+	0.985825i	\(0.446342\pi\)
\(37\)	36.5317	+	23.4775i	0.987344	+	0.634528i	0.931435	−	0.363908i	\(-0.118558\pi\)
							0.0559094	+	0.998436i	\(0.482194\pi\)
\(41\)	−39.5895	+	25.4426i	−0.965596	+	0.620551i	−0.925541	−	0.378646i	\(-0.876390\pi\)
							−0.0400550	+	0.999197i	\(0.512753\pi\)
\(43\)	−29.5620	−	8.68018i	−0.687488	−	0.201865i	−0.0807118	−	0.996737i	\(-0.525719\pi\)
							−0.606776	+	0.794873i	\(0.707538\pi\)
\(47\)		−	9.78818i		−	0.208259i	−0.994564	−	0.104130i	\(-0.966794\pi\)
							0.994564	−	0.104130i	\(-0.0332057\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.i.a.109.2	yes	240
5.4	even	2	inner	230.3.i.a.109.23	yes	240
23.19	odd	22	inner	230.3.i.a.19.23	yes	240
115.19	odd	22	inner	230.3.i.a.19.2	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.i.a.19.2	✓	240	115.19	odd	22	inner
230.3.i.a.19.23	yes	240	23.19	odd	22	inner
230.3.i.a.109.2	yes	240	1.1	even	1	trivial
230.3.i.a.109.23	yes	240	5.4	even	2	inner