Embedded newform 588.2.ba.a.187.21

• Label: 588.2.ba.a.187.21
• 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			187.21
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.a.283.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.789543 - 1.17330i) q^{2} +(0.988831 + 0.149042i) q^{3} +(-0.753245 - 1.85273i) q^{4} +(2.68885 - 1.05530i) q^{5} +(0.955595 - 1.04252i) q^{6} +(0.891718 - 2.49095i) q^{7} +(-2.76852 - 0.579034i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.789543	−	1.17330i	0.558291	−	0.829645i
							\(3\)	0.988831	+	0.149042i	0.570902	+	0.0860496i
\(5\)	2.68885	−	1.05530i	1.20249	−	0.471943i								0.322334	−	0.946626i	\(-0.395532\pi\)
							0.880157	+	0.474683i	\(0.157437\pi\)
\(7\)	0.891718	−	2.49095i	0.337038	−	0.941491i
							\(11\)	0.268235	+	0.869598i	0.0808760	+	0.262194i	0.986975	−	0.160871i	\(-0.0514304\pi\)
−0.906099	+	0.423065i	\(0.860954\pi\)
\(13\)	2.28247	+	0.520959i	0.633044	+	0.144488i	0.526989	−	0.849872i	\(-0.323321\pi\)
							0.106055	+	0.994360i	\(0.466178\pi\)
\(17\)	−3.81661	+	5.59793i	−0.925663	+	1.35770i	0.00816129	+	0.999967i	\(0.497402\pi\)
							−0.933824	+	0.357732i	\(0.883550\pi\)
\(19\)	−3.81043	+	6.59986i	−0.874173	+	1.51411i	−0.0165316	+	0.999863i	\(0.505262\pi\)
							−0.857641	+	0.514248i	\(0.828071\pi\)
\(23\)	−2.79569	−	4.10052i	−0.582941	−	0.855017i	0.415571	−	0.909561i	\(-0.363582\pi\)
							−0.998511	+	0.0545438i	\(0.982630\pi\)
\(29\)	5.68903	−	2.73969i	1.05643	−	0.508748i	0.176717	−	0.984262i	\(-0.443452\pi\)
							0.879708	+	0.475514i	\(0.157738\pi\)
\(31\)	−1.62430	−	2.81336i	−0.291732	−	0.505295i	0.682487	−	0.730897i	\(-0.260898\pi\)
							−0.974219	+	0.225603i	\(0.927565\pi\)
\(37\)	0.408644	−	5.45298i	0.0671807	−	0.896464i	−0.857198	−	0.514988i	\(-0.827797\pi\)
							0.924378	−	0.381477i	\(-0.124584\pi\)
\(41\)	5.53532	+	4.41427i	0.864471	+	0.689393i	0.951778	−	0.306789i	\(-0.0992545\pi\)
							−0.0873066	+	0.996181i	\(0.527826\pi\)
\(43\)	−0.978269	+	0.780143i	−0.149185	+	0.118971i	−0.695238	−	0.718780i	\(-0.744701\pi\)
							0.546053	+	0.837750i	\(0.316130\pi\)
\(47\)	−2.33568	−	2.16719i	−0.340693	−	0.316117i	0.491206	−	0.871043i	\(-0.336556\pi\)
							−0.831900	+	0.554926i	\(0.812747\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.187.21	✓	336
4.3	odd	2		588.2.ba.b.187.7	yes	336
49.38	odd	42		588.2.ba.b.283.7	yes	336
196.87	even	42	inner	588.2.ba.a.283.21	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.21	✓	336	1.1	even	1	trivial
588.2.ba.a.283.21	yes	336	196.87	even	42	inner
588.2.ba.b.187.7	yes	336	4.3	odd	2
588.2.ba.b.283.7	yes	336	49.38	odd	42