Embedded newform 230.3.k.b.3.7

• Label: 230.3.k.b.3.7
• 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			3.7
Character	\(\chi\)	\(=\)	230.3
Dual form			230.3.k.b.77.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.13214 + 0.847507i) q^{2} +(0.583215 - 0.217528i) q^{3} +(0.563465 + 1.91899i) q^{4} +(0.599174 + 4.96397i) q^{5} +(0.844635 + 0.248007i) q^{6} +(0.414739 - 1.90653i) q^{7} +(-0.988434 + 2.65009i) q^{8} +(-6.50892 + 5.64002i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	0.583215	−	0.217528i	0.194405	−	0.0725093i	−0.250376	−	0.968149i	\(-0.580554\pi\)
0.444781	+	0.895639i	\(0.353282\pi\)
\(5\)	0.599174	+	4.96397i	0.119835	+	0.992794i
							\(7\)	0.414739	−	1.90653i	0.0592485	−	0.272361i	−0.937987	−	0.346671i	\(-0.887312\pi\)
0.997235	+	0.0743104i	\(0.0236756\pi\)
\(11\)	−1.50877	+	10.4938i	−0.137161	+	0.953978i	0.798730	+	0.601690i	\(0.205506\pi\)
							−0.935891	+	0.352289i	\(0.885404\pi\)
\(13\)	0.618379	+	2.84264i	0.0475676	+	0.218665i	0.994795	−	0.101898i	\(-0.0324916\pi\)
							−0.947227	+	0.320563i	\(0.896128\pi\)
\(17\)	27.9975	−	15.2878i	1.64691	−	0.899283i	0.657619	−	0.753351i	\(-0.271564\pi\)
							0.989295	−	0.145932i	\(-0.0466181\pi\)
\(19\)	−0.595348	−	2.02757i	−0.0313341	−	0.106714i	0.942341	−	0.334654i	\(-0.108619\pi\)
							−0.973675	+	0.227940i	\(0.926801\pi\)
\(23\)	−5.83199	+	22.2483i	−0.253565	+	0.967318i
							\(29\)	5.45243	−	18.5693i	0.188015	−	0.640319i	−0.810495	−	0.585746i	\(-0.800802\pi\)
0.998510	−	0.0545738i	\(-0.0173800\pi\)
\(31\)	2.14261	+	4.69165i	0.0691163	+	0.151344i	0.941037	−	0.338303i	\(-0.109853\pi\)
							−0.871921	+	0.489647i	\(0.837126\pi\)
\(37\)	34.9039	+	2.49638i	0.943350	+	0.0674697i	0.534534	−	0.845147i	\(-0.320487\pi\)
							0.408816	+	0.912617i	\(0.365942\pi\)
\(41\)	32.4497	−	37.4489i	0.791455	−	0.913388i	−0.206425	−	0.978462i	\(-0.566183\pi\)
							0.997880	+	0.0650741i	\(0.0207284\pi\)
\(43\)	−24.8344	+	9.26274i	−0.577543	+	0.215413i	−0.621224	−	0.783633i	\(-0.713364\pi\)
							0.0436806	+	0.999046i	\(0.486092\pi\)
\(47\)	36.0032	−	36.0032i	0.766026	−	0.766026i	−0.211379	−	0.977404i	\(-0.567795\pi\)
							0.977404	+	0.211379i	\(0.0677953\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.3.k.b.3.7	✓	240
5.2	odd	4	inner	230.3.k.b.187.6	yes	240
23.8	even	11	inner	230.3.k.b.123.6	yes	240
115.77	odd	44	inner	230.3.k.b.77.7	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.k.b.3.7	✓	240	1.1	even	1	trivial
230.3.k.b.77.7	yes	240	115.77	odd	44	inner
230.3.k.b.123.6	yes	240	23.8	even	11	inner
230.3.k.b.187.6	yes	240	5.2	odd	4	inner