Embedded newform 230.3.k.a.223.1

• Label: 230.3.k.a.223.1
• Level: $230$
• Weight: $3$
• Character: 230.223
• 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.k (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26704608029\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(12\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			223.1
Character	\(\chi\)	\(=\)	230.223
Dual form			230.3.k.a.197.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.100889 + 1.41061i) q^{2} +(-5.03168 - 1.09457i) q^{3} +(-1.97964 + 0.284630i) q^{4} +(-4.91446 + 0.920912i) q^{5} +(1.03638 - 7.20817i) q^{6} +(-2.44662 + 4.48066i) q^{7} +(-0.601225 - 2.76379i) q^{8} +(15.9330 + 7.27636i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 24 q^{2} - 4 q^{5} - 74 q^{7} + 48 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{4}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.100889	+	1.41061i	0.0504444	+	0.705305i
							\(3\)	−5.03168	−	1.09457i	−1.67723	−	0.364858i	−0.729509	−	0.683971i	\(-0.760251\pi\)
−0.947717	+	0.319113i	\(0.896615\pi\)
\(5\)	−4.91446	+	0.920912i	−0.982892	+	0.184182i
							\(7\)	−2.44662	+	4.48066i	−0.349518	+	0.640094i	−0.992004	−	0.126203i	\(-0.959721\pi\)
0.642487	+	0.766297i	\(0.277903\pi\)
\(11\)	−14.1111	−	16.2851i	−1.28283	−	1.48047i	−0.793770	−	0.608217i	\(-0.791885\pi\)
							−0.489061	−	0.872249i	\(-0.662661\pi\)
\(13\)	3.78190	+	6.92603i	0.290915	+	0.532772i	0.981735	−	0.190256i	\(-0.0609318\pi\)
							−0.690819	+	0.723027i	\(0.742750\pi\)
\(17\)	8.79921	+	11.7544i	0.517600	+	0.691433i	0.980843	−	0.194798i	\(-0.0624052\pi\)
							−0.463243	+	0.886231i	\(0.653314\pi\)
\(19\)	22.2902	−	3.20484i	1.17317	−	0.168676i	0.471951	−	0.881625i	\(-0.343550\pi\)
							0.701216	+	0.712949i	\(0.252641\pi\)
\(23\)	22.5306	+	4.62308i	0.979591	+	0.201003i
							\(29\)	−14.9423	−	2.14837i	−0.515250	−	0.0740818i	−0.120215	−	0.992748i	\(-0.538358\pi\)
−0.395035	+	0.918666i	\(0.629268\pi\)
\(31\)	−12.2723	−	7.88691i	−0.395880	−	0.254416i	0.327522	−	0.944843i	\(-0.393786\pi\)
							−0.723402	+	0.690427i	\(0.757423\pi\)
\(37\)	−3.26443	−	8.75228i	−0.0882279	−	0.236548i	0.885341	−	0.464943i	\(-0.153925\pi\)
							−0.973569	+	0.228395i	\(0.926652\pi\)
\(41\)	−19.3991	−	42.4781i	−0.473149	−	1.03605i	−0.984291	−	0.176556i	\(-0.943504\pi\)
							0.511141	−	0.859497i	\(-0.329223\pi\)
\(43\)	−60.3734	−	13.1334i	−1.40403	−	0.305428i	−0.554157	−	0.832412i	\(-0.686959\pi\)
							−0.849875	+	0.526984i	\(0.823323\pi\)
\(47\)	32.8275	−	32.8275i	0.698458	−	0.698458i	−0.265620	−	0.964078i	\(-0.585577\pi\)
							0.964078	+	0.265620i	\(0.0855767\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.a.223.1	yes	240
5.2	odd	4	inner	230.3.k.a.177.1	yes	240
23.13	even	11	inner	230.3.k.a.13.1	✓	240
115.82	odd	44	inner	230.3.k.a.197.1	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.a.13.1	✓	240	23.13	even	11	inner
230.3.k.a.177.1	yes	240	5.2	odd	4	inner
230.3.k.a.197.1	yes	240	115.82	odd	44	inner
230.3.k.a.223.1	yes	240	1.1	even	1	trivial