Embedded newform 588.2.ba.a.187.18

• Label: 588.2.ba.a.187.18
• Level: $588$
• Weight: $2$
• Character: 588.187
• Analytic conductor: $4.695$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	588.ba (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.69520363885\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			187.18
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.a.283.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.546668 - 1.30428i) q^{2} +(0.988831 + 0.149042i) q^{3} +(-1.40231 - 1.42602i) q^{4} +(2.80567 - 1.10114i) q^{5} +(0.734956 - 1.20824i) q^{6} +(-1.52632 + 2.16110i) q^{7} +(-2.62653 + 1.04945i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(493\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{23}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.546668	−	1.30428i	0.386553	−	0.922267i
							\(3\)	0.988831	+	0.149042i	0.570902	+	0.0860496i
\(5\)	2.80567	−	1.10114i	1.25473	−	0.492446i								0.357462	−	0.933928i	\(-0.383642\pi\)
							0.897271	+	0.441481i	\(0.145547\pi\)
\(7\)	−1.52632	+	2.16110i	−0.576896	+	0.816817i
							\(11\)	−1.71367	−	5.55558i	−0.516691	−	1.67507i	−0.718404	−	0.695626i	\(-0.755127\pi\)
0.201713	−	0.979445i	\(-0.435349\pi\)
\(13\)	0.363826	+	0.0830410i	0.100907	+	0.0230314i	0.272676	−	0.962106i	\(-0.412091\pi\)
							−0.171769	+	0.985137i	\(0.554948\pi\)
\(17\)	3.94001	−	5.77893i	0.955592	−	1.40160i	0.0401294	−	0.999194i	\(-0.487223\pi\)
							0.915463	−	0.402402i	\(-0.131825\pi\)
\(19\)	1.74600	−	3.02417i	0.400561	−	0.693791i	−0.593233	−	0.805031i	\(-0.702149\pi\)
							0.993794	+	0.111239i	\(0.0354820\pi\)
\(23\)	0.103798	+	0.152244i	0.0216434	+	0.0317451i	0.836908	−	0.547344i	\(-0.184361\pi\)
							−0.815265	+	0.579089i	\(0.803409\pi\)
\(29\)	−0.320214	+	0.154207i	−0.0594623	+	0.0286355i	−0.463379	−	0.886160i	\(-0.653363\pi\)
							0.403916	+	0.914796i	\(0.367649\pi\)
\(31\)	4.31962	+	7.48180i	0.775827	+	1.34377i	0.934329	+	0.356413i	\(0.116000\pi\)
							−0.158502	+	0.987359i	\(0.550666\pi\)
\(37\)	−0.593000	+	7.91303i	−0.0974885	+	1.30089i	0.707122	+	0.707091i	\(0.249993\pi\)
							−0.804611	+	0.593803i	\(0.797626\pi\)
\(41\)	5.81173	+	4.63470i	0.907640	+	0.723819i	0.961522	−	0.274728i	\(-0.0885879\pi\)
							−0.0538814	+	0.998547i	\(0.517159\pi\)
\(43\)	−4.58736	+	3.65830i	−0.699566	+	0.557885i	−0.907394	−	0.420280i	\(-0.861932\pi\)
							0.207829	+	0.978165i	\(0.433360\pi\)
\(47\)	−5.05565	−	4.69095i	−0.737442	−	0.684246i	0.219145	−	0.975692i	\(-0.429673\pi\)
							−0.956587	+	0.291446i	\(0.905864\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.187.18	✓	336
4.3	odd	2		588.2.ba.b.187.5	yes	336
49.38	odd	42		588.2.ba.b.283.5	yes	336
196.87	even	42	inner	588.2.ba.a.283.18	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.18	✓	336	1.1	even	1	trivial
588.2.ba.a.283.18	yes	336	196.87	even	42	inner
588.2.ba.b.187.5	yes	336	4.3	odd	2
588.2.ba.b.283.5	yes	336	49.38	odd	42