Embedded newform 230.3.k.a.3.10

• Label: 230.3.k.a.3.10
• Level: $230$
• Weight: $3$
• Character: 230.3
• 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			3.10
Character	\(\chi\)	\(=\)	230.3
Dual form			230.3.k.a.77.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13214 - 0.847507i) q^{2} +(3.55876 - 1.32735i) q^{3} +(0.563465 + 1.91899i) q^{4} +(-2.88474 + 4.08390i) q^{5} +(-5.15393 - 1.51333i) q^{6} +(-1.27831 + 5.87629i) q^{7} +(0.988434 - 2.65009i) q^{8} +(4.10115 - 3.55367i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 24 q^{2} - 4 q^{5} - 74 q^{7} + 48 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{8}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13214	−	0.847507i	−0.566068	−	0.423753i
							\(3\)	3.55876	−	1.32735i	1.18625	−	0.442449i	0.322599	−	0.946536i	\(-0.395443\pi\)
0.863653	+	0.504086i	\(0.168171\pi\)
\(5\)	−2.88474	+	4.08390i	−0.576949	+	0.816780i
							\(7\)	−1.27831	+	5.87629i	−0.182616	+	0.839470i	0.791097	+	0.611691i	\(0.209510\pi\)
−0.973713	+	0.227780i	\(0.926853\pi\)
\(11\)	−2.41594	+	16.8032i	−0.219631	+	1.52757i	0.519772	+	0.854305i	\(0.326017\pi\)
							−0.739403	+	0.673263i	\(0.764892\pi\)
\(13\)	0.608823	+	2.79871i	0.0468325	+	0.215286i	0.994616	−	0.103629i	\(-0.0330454\pi\)
							−0.947784	+	0.318914i	\(0.896682\pi\)
\(17\)	−5.78171	+	3.15705i	−0.340101	+	0.185709i	−0.640218	−	0.768193i	\(-0.721156\pi\)
							0.300117	+	0.953902i	\(0.402974\pi\)
\(19\)	3.50127	+	11.9242i	0.184277	+	0.627590i	0.998869	+	0.0475552i	\(0.0151430\pi\)
							−0.814591	+	0.580035i	\(0.803039\pi\)
\(23\)	6.97338	−	21.9174i	0.303190	−	0.952930i
							\(29\)	6.93720	−	23.6259i	0.239214	−	0.814688i	−0.749126	−	0.662427i	\(-0.769526\pi\)
0.988340	−	0.152261i	\(-0.0486553\pi\)
\(31\)	−1.29562	−	2.83702i	−0.0417943	−	0.0915168i	0.887583	−	0.460647i	\(-0.152383\pi\)
							−0.929378	+	0.369131i	\(0.879655\pi\)
\(37\)	70.4872	+	5.04134i	1.90506	+	0.136252i	0.973698	−	0.227844i	\(-0.0731677\pi\)
							0.931361	+	0.364097i	\(0.118622\pi\)
\(41\)	−30.8628	+	35.6176i	−0.752752	+	0.868722i	−0.994832	−	0.101534i	\(-0.967625\pi\)
							0.242080	+	0.970256i	\(0.422170\pi\)
\(43\)	64.2533	−	23.9653i	1.49426	−	0.557332i	0.536252	−	0.844058i	\(-0.319840\pi\)
							0.958012	+	0.286727i	\(0.0925671\pi\)
\(47\)	−36.1592	+	36.1592i	−0.769345	+	0.769345i	−0.977991	−	0.208646i	\(-0.933094\pi\)
							0.208646	+	0.977991i	\(0.433094\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.a.3.10	✓	240
5.2	odd	4	inner	230.3.k.a.187.3	yes	240
23.8	even	11	inner	230.3.k.a.123.3	yes	240
115.77	odd	44	inner	230.3.k.a.77.10	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.a.3.10	✓	240	1.1	even	1	trivial
230.3.k.a.77.10	yes	240	115.77	odd	44	inner
230.3.k.a.123.3	yes	240	23.8	even	11	inner
230.3.k.a.187.3	yes	240	5.2	odd	4	inner