Embedded newform 588.2.ba.a.187.3

• Label: 588.2.ba.a.187.3
• 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			187.3
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.a.283.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37584 + 0.327226i) q^{2} +(0.988831 + 0.149042i) q^{3} +(1.78585 - 0.900417i) q^{4} +(3.55278 - 1.39436i) q^{5} +(-1.40924 + 0.118513i) q^{6} +(-2.21601 - 1.44544i) q^{7} +(-2.16239 + 1.82320i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37584	+	0.327226i	−0.972863	+	0.231384i
							\(3\)	0.988831	+	0.149042i	0.570902	+	0.0860496i
\(5\)	3.55278	−	1.39436i	1.58885	−	0.623578i								0.604001	−	0.796984i	\(-0.293572\pi\)
							0.984850	+	0.173406i	\(0.0554772\pi\)
\(7\)	−2.21601	−	1.44544i	−0.837574	−	0.546324i
							\(11\)	−0.595096	−	1.92925i	−0.179428	−	0.581692i	−0.999949	−	0.0100757i	\(-0.996793\pi\)
0.820521	−	0.571616i	\(-0.193683\pi\)
\(13\)	5.27543	+	1.20408i	1.46314	+	0.333953i	0.878658	−	0.477453i	\(-0.158440\pi\)
							0.584484	+	0.811405i	\(0.301297\pi\)
\(17\)	−0.391577	+	0.574338i	−0.0949713	+	0.139297i	−0.870784	−	0.491666i	\(-0.836388\pi\)
							0.775812	+	0.630964i	\(0.217340\pi\)
\(19\)	0.251951	−	0.436392i	0.0578015	−	0.100115i	−0.835677	−	0.549222i	\(-0.814924\pi\)
							0.893478	+	0.449106i	\(0.148258\pi\)
\(23\)	−1.78826	−	2.62290i	−0.372878	−	0.546912i	0.593235	−	0.805029i	\(-0.297850\pi\)
							−0.966114	+	0.258117i	\(0.916898\pi\)
\(29\)	−7.98019	+	3.84306i	−1.48188	+	0.713638i	−0.987793	−	0.155772i	\(-0.950213\pi\)
							−0.494091	+	0.869410i	\(0.664499\pi\)
\(31\)	1.38686	+	2.40212i	0.249088	+	0.431433i	0.963273	−	0.268524i	\(-0.0865359\pi\)
							−0.714185	+	0.699957i	\(0.753203\pi\)
\(37\)	0.466966	−	6.23123i	0.0767688	−	1.02441i	−0.817222	−	0.576323i	\(-0.804487\pi\)
							0.893991	−	0.448085i	\(-0.147894\pi\)
\(41\)	−5.04383	−	4.02232i	−0.787714	−	0.628181i	0.144740	−	0.989470i	\(-0.453765\pi\)
							−0.932454	+	0.361289i	\(0.882337\pi\)
\(43\)	3.66094	−	2.91950i	0.558288	−	0.445220i	−0.303251	−	0.952911i	\(-0.598072\pi\)
							0.861539	+	0.507691i	\(0.169501\pi\)
\(47\)	8.78299	+	8.14942i	1.28113	+	1.18872i	0.971283	+	0.237929i	\(0.0764686\pi\)
							0.309848	+	0.950786i	\(0.399722\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.187.3	✓	336
4.3	odd	2		588.2.ba.b.187.17	yes	336
49.38	odd	42		588.2.ba.b.283.17	yes	336
196.87	even	42	inner	588.2.ba.a.283.3	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.3	✓	336	1.1	even	1	trivial
588.2.ba.a.283.3	yes	336	196.87	even	42	inner
588.2.ba.b.187.17	yes	336	4.3	odd	2
588.2.ba.b.283.17	yes	336	49.38	odd	42