Embedded newform 588.2.ba.a.187.11

• Label: 588.2.ba.a.187.11
• 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			187.11
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.a.283.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.564817 - 1.29653i) q^{2} +(0.988831 + 0.149042i) q^{3} +(-1.36196 + 1.46460i) q^{4} +(2.58391 - 1.01411i) q^{5} +(-0.365271 - 1.36623i) q^{6} +(-1.51456 + 2.16936i) q^{7} +(2.66815 + 0.938592i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.564817	−	1.29653i	−0.399386	−	0.916783i
							\(3\)	0.988831	+	0.149042i	0.570902	+	0.0860496i
\(5\)	2.58391	−	1.01411i	1.15556	−	0.453523i								0.291330	−	0.956623i	\(-0.405902\pi\)
							0.864229	+	0.503099i	\(0.167807\pi\)
\(7\)	−1.51456	+	2.16936i	−0.572449	+	0.819940i
							\(11\)	1.46471	+	4.74848i	0.441627	+	1.43172i	0.853793	+	0.520612i	\(0.174296\pi\)
−0.412166	+	0.911109i	\(0.635228\pi\)
\(13\)	1.62731	+	0.371422i	0.451334	+	0.103014i	0.442145	−	0.896944i	\(-0.354218\pi\)
							0.00918920	+	0.999958i	\(0.497075\pi\)
\(17\)	−1.65529	+	2.42786i	−0.401467	+	0.588843i	−0.972735	−	0.231920i	\(-0.925499\pi\)
							0.571268	+	0.820763i	\(0.306452\pi\)
\(19\)	−0.522408	+	0.904838i	−0.119849	+	0.207584i	−0.919708	−	0.392604i	\(-0.871574\pi\)
							0.799859	+	0.600188i	\(0.204908\pi\)
\(23\)	0.413634	+	0.606690i	0.0862487	+	0.126504i	0.866939	−	0.498415i	\(-0.166084\pi\)
							−0.780690	+	0.624919i	\(0.785132\pi\)
\(29\)	3.85385	−	1.85591i	0.715641	−	0.344635i	−0.0403776	−	0.999184i	\(-0.512856\pi\)
							0.756019	+	0.654550i	\(0.227142\pi\)
\(31\)	−0.702049	−	1.21599i	−0.126092	−	0.218397i	0.796067	−	0.605208i	\(-0.206910\pi\)
							−0.922159	+	0.386811i	\(0.873577\pi\)
\(37\)	−0.131088	+	1.74925i	−0.0215507	+	0.287575i	0.976008	+	0.217736i	\(0.0698673\pi\)
							−0.997558	+	0.0698381i	\(0.977752\pi\)
\(41\)	−4.25079	−	3.38989i	−0.663862	−	0.529412i	0.232578	−	0.972578i	\(-0.425284\pi\)
							−0.896440	+	0.443166i	\(0.853855\pi\)
\(43\)	7.54761	−	6.01902i	1.15100	−	0.917892i	0.153471	−	0.988153i	\(-0.450955\pi\)
							0.997529	+	0.0702613i	\(0.0223833\pi\)
\(47\)	1.55491	+	1.44274i	0.226807	+	0.210446i	0.785388	−	0.619004i	\(-0.212464\pi\)
							−0.558581	+	0.829450i	\(0.688654\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.187.11	✓	336
4.3	odd	2		588.2.ba.b.187.3	yes	336
49.38	odd	42		588.2.ba.b.283.3	yes	336
196.87	even	42	inner	588.2.ba.a.283.11	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.11	✓	336	1.1	even	1	trivial
588.2.ba.a.283.11	yes	336	196.87	even	42	inner
588.2.ba.b.187.3	yes	336	4.3	odd	2
588.2.ba.b.283.3	yes	336	49.38	odd	42