Embedded newform 588.2.ba.b.283.3

• Label: 588.2.ba.b.283.3
• Level: $588$
• Weight: $2$
• Character: 588.283
• 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			283.3
Character	\(\chi\)	\(=\)	588.283
Dual form			588.2.ba.b.187.3

$q$-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 \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(493\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{19}{42}\right)\)

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								0.864229	−	0.503099i	\(-0.167807\pi\)
							0.291330	+	0.956623i	\(0.405902\pi\)
\(7\)	1.51456	+	2.16936i	0.572449	+	0.819940i
							\(11\)	−1.46471	+	4.74848i	−0.441627	+	1.43172i	0.412166	+	0.911109i	\(0.364772\pi\)
−0.853793	+	0.520612i	\(0.825704\pi\)
\(13\)	1.62731	−	0.371422i	0.451334	−	0.103014i	0.00918920	−	0.999958i	\(-0.497075\pi\)
							0.442145	+	0.896944i	\(0.354218\pi\)
\(17\)	−1.65529	−	2.42786i	−0.401467	−	0.588843i	0.571268	−	0.820763i	\(-0.306452\pi\)
							−0.972735	+	0.231920i	\(0.925499\pi\)
\(19\)	0.522408	+	0.904838i	0.119849	+	0.207584i	0.919708	−	0.392604i	\(-0.128426\pi\)
							−0.799859	+	0.600188i	\(0.795092\pi\)
\(23\)	−0.413634	+	0.606690i	−0.0862487	+	0.126504i	−0.866939	−	0.498415i	\(-0.833916\pi\)
							0.780690	+	0.624919i	\(0.214868\pi\)
\(29\)	3.85385	+	1.85591i	0.715641	+	0.344635i	0.756019	−	0.654550i	\(-0.227142\pi\)
							−0.0403776	+	0.999184i	\(0.512856\pi\)
\(31\)	0.702049	−	1.21599i	0.126092	−	0.218397i	−0.796067	−	0.605208i	\(-0.793090\pi\)
							0.922159	+	0.386811i	\(0.126423\pi\)
\(37\)	−0.131088	−	1.74925i	−0.0215507	−	0.287575i	−0.997558	−	0.0698381i	\(-0.977752\pi\)
							0.976008	−	0.217736i	\(-0.0698673\pi\)
\(41\)	−4.25079	+	3.38989i	−0.663862	+	0.529412i	−0.896440	−	0.443166i	\(-0.853855\pi\)
							0.232578	+	0.972578i	\(0.425284\pi\)
\(43\)	−7.54761	−	6.01902i	−1.15100	−	0.917892i	−0.153471	−	0.988153i	\(-0.549045\pi\)
							−0.997529	+	0.0702613i	\(0.977617\pi\)
\(47\)	−1.55491	+	1.44274i	−0.226807	+	0.210446i	−0.785388	−	0.619004i	\(-0.787536\pi\)
							0.558581	+	0.829450i	\(0.311346\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.11	✓	336	49.40	odd	42
588.2.ba.a.283.11	yes	336	4.3	odd	2
588.2.ba.b.187.3	yes	336	196.187	even	42	inner
588.2.ba.b.283.3	yes	336	1.1	even	1	trivial