Embedded newform 588.2.ba.b.187.3

• Label: 588.2.ba.b.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.b.283.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33511 - 0.466348i) q^{2} +(-0.988831 - 0.149042i) q^{3} +(1.56504 + 1.24525i) q^{4} +(2.58391 - 1.01411i) q^{5} +(1.25069 + 0.660127i) q^{6} +(1.51456 - 2.16936i) q^{7} +(-1.50878 - 2.39240i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{3} - 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33511	−	0.466348i	−0.944066	−	0.329758i
							\(3\)	−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	\(-0.825704\pi\)
0.412166	−	0.911109i	\(-0.364772\pi\)
\(13\)	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	−0.799859	−	0.600188i	\(-0.795092\pi\)
							0.919708	+	0.392604i	\(0.128426\pi\)
\(23\)	−0.413634	−	0.606690i	−0.0862487	−	0.126504i	0.780690	−	0.624919i	\(-0.214868\pi\)
							−0.866939	+	0.498415i	\(0.833916\pi\)
\(29\)	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	0.922159	−	0.386811i	\(-0.126423\pi\)
							−0.796067	+	0.605208i	\(0.793090\pi\)
\(37\)	−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	−0.997529	−	0.0702613i	\(-0.977617\pi\)
							−0.153471	+	0.988153i	\(0.549045\pi\)
\(47\)	−1.55491	−	1.44274i	−0.226807	−	0.210446i	0.558581	−	0.829450i	\(-0.311346\pi\)
							−0.785388	+	0.619004i	\(0.787536\pi\)

(See \(a_n\) instead)

Twists

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

    

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