Embedded newform 588.2.ba.b.187.5

• Label: 588.2.ba.b.187.5
• 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			187.5
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.b.283.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.25978 + 0.642609i) q^{2} +(-0.988831 - 0.149042i) q^{3} +(1.17411 - 1.61910i) q^{4} +(2.80567 - 1.10114i) q^{5} +(1.34149 - 0.447671i) q^{6} +(1.52632 - 2.16110i) q^{7} +(-0.438676 + 2.79420i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{3} - 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.25978	+	0.642609i	−0.890801	+	0.454393i
							\(3\)	−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	\(0.244873\pi\)
−0.201713	+	0.979445i	\(0.564651\pi\)
\(13\)	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	−0.993794	−	0.111239i	\(-0.964518\pi\)
							0.593233	+	0.805031i	\(0.297851\pi\)
\(23\)	−0.103798	−	0.152244i	−0.0216434	−	0.0317451i	0.815265	−	0.579089i	\(-0.196591\pi\)
							−0.836908	+	0.547344i	\(0.815639\pi\)
\(29\)	−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	\(-0.884000\pi\)
							0.158502	−	0.987359i	\(-0.449334\pi\)
\(37\)	−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	−0.207829	−	0.978165i	\(-0.566640\pi\)
							0.907394	+	0.420280i	\(0.138068\pi\)
\(47\)	5.05565	+	4.69095i	0.737442	+	0.684246i	0.956587	−	0.291446i	\(-0.0941364\pi\)
							−0.219145	+	0.975692i	\(0.570327\pi\)

(See \(a_n\) instead)

Twists

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

    

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