Embedded newform 230.3.k.b.13.3

• Label: 230.3.k.b.13.3
• Level: $230$
• Weight: $3$
• Character: 230.13
• 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			13.3
Character	\(\chi\)	\(=\)	230.13
Dual form			230.3.k.b.177.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41061 + 0.100889i) q^{2} +(-0.800118 - 3.67808i) q^{3} +(1.97964 + 0.284630i) q^{4} +(-2.59942 - 4.27118i) q^{5} +(-0.757577 - 5.26906i) q^{6} +(6.22281 - 3.39791i) q^{7} +(2.76379 + 0.601225i) q^{8} +(-4.70140 + 2.14706i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{7}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41061	+	0.100889i	0.705305	+	0.0504444i
							\(3\)	−0.800118	−	3.67808i	−0.266706	−	1.22603i	−0.894293	−	0.447481i	\(-0.852321\pi\)
0.627588	−	0.778546i	\(-0.284042\pi\)
\(5\)	−2.59942	−	4.27118i	−0.519885	−	0.854236i
							\(7\)	6.22281	−	3.39791i	0.888973	−	0.485416i	0.0312401	−	0.999512i	\(-0.490054\pi\)
0.857733	+	0.514096i	\(0.171873\pi\)
\(11\)	−8.60252	+	9.92784i	−0.782047	+	0.902531i	−0.997255	−	0.0740373i	\(-0.976412\pi\)
							0.215208	+	0.976568i	\(0.430957\pi\)
\(13\)	−9.61121	−	5.24812i	−0.739324	−	0.403701i	0.0650057	−	0.997885i	\(-0.479293\pi\)
							−0.804329	+	0.594184i	\(0.797475\pi\)
\(17\)	−19.2691	−	14.4246i	−1.13347	−	0.848508i	−0.143485	−	0.989652i	\(-0.545831\pi\)
							−0.989989	+	0.141144i	\(0.954922\pi\)
\(19\)	27.0293	+	3.88623i	1.42260	+	0.204538i	0.810265	−	0.586063i	\(-0.199323\pi\)
							0.612331	+	0.790602i	\(0.290232\pi\)
\(23\)	−1.55361	−	22.9475i	−0.0675484	−	0.997716i
							\(29\)	52.2909	−	7.51830i	1.80314	−	0.259252i	0.842826	−	0.538187i	\(-0.180890\pi\)
0.960310	+	0.278935i	\(0.0899813\pi\)
\(31\)	−15.0737	+	9.68727i	−0.486248	+	0.312493i	−0.760695	−	0.649110i	\(-0.775142\pi\)
							0.274447	+	0.961602i	\(0.411505\pi\)
\(37\)	61.4888	+	22.9341i	1.66186	+	0.619841i	0.992405	−	0.123012i	\(-0.0392554\pi\)
							0.669454	+	0.742853i	\(0.266528\pi\)
\(41\)	−12.3770	+	27.1019i	−0.301879	+	0.661023i	−0.998402	−	0.0565115i	\(-0.982002\pi\)
							0.696523	+	0.717535i	\(0.254730\pi\)
\(43\)	−0.620439	−	2.85211i	−0.0144288	−	0.0663282i	0.969370	−	0.245607i	\(-0.0789872\pi\)
							−0.983798	+	0.179279i	\(0.942624\pi\)
\(47\)	23.1474	−	23.1474i	0.492497	−	0.492497i	−0.416595	−	0.909092i	\(-0.636777\pi\)
							0.909092	+	0.416595i	\(0.136777\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.13.3	✓	240
5.2	odd	4	inner	230.3.k.b.197.3	yes	240
23.16	even	11	inner	230.3.k.b.223.3	yes	240
115.62	odd	44	inner	230.3.k.b.177.3	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.13.3	✓	240	1.1	even	1	trivial
230.3.k.b.177.3	yes	240	115.62	odd	44	inner
230.3.k.b.197.3	yes	240	5.2	odd	4	inner
230.3.k.b.223.3	yes	240	23.16	even	11	inner