Embedded newform 588.2.ba.a.187.27

• Label: 588.2.ba.a.187.27
• 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			187.27
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.a.283.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40887 - 0.122812i) q^{2} +(0.988831 + 0.149042i) q^{3} +(1.96983 - 0.346052i) q^{4} +(-2.61791 + 1.02745i) q^{5} +(1.41144 + 0.0885414i) q^{6} +(2.63745 + 0.209446i) q^{7} +(2.73274 - 0.729461i) q^{8} +(0.955573 + 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40887	−	0.122812i	0.996222	−	0.0868410i
							\(3\)	0.988831	+	0.149042i	0.570902	+	0.0860496i
\(5\)	−2.61791	+	1.02745i	−1.17076	+	0.459491i								−0.869443	−	0.494033i	\(-0.835522\pi\)
							−0.301319	+	0.953523i	\(0.597427\pi\)
\(7\)	2.63745	+	0.209446i	0.996862	+	0.0791632i
							\(11\)	0.0741409	+	0.240359i	0.0223543	+	0.0724710i	0.966063	−	0.258308i	\(-0.0831648\pi\)
−0.943708	+	0.330779i	\(0.892689\pi\)
\(13\)	1.99817	+	0.456069i	0.554193	+	0.126491i	0.490440	−	0.871475i	\(-0.336836\pi\)
							0.0637526	+	0.997966i	\(0.479693\pi\)
\(17\)	0.553691	−	0.812116i	0.134290	−	0.196967i	−0.753203	−	0.657788i	\(-0.771492\pi\)
							0.887493	+	0.460821i	\(0.152445\pi\)
\(19\)	−2.11045	+	3.65541i	−0.484171	+	0.838609i	−0.999835	−	0.0181822i	\(-0.994212\pi\)
							0.515664	+	0.856791i	\(0.327545\pi\)
\(23\)	−4.14505	−	6.07967i	−0.864302	−	1.26770i	−0.962156	−	0.272498i	\(-0.912150\pi\)
							0.0978540	−	0.995201i	\(-0.468802\pi\)
\(29\)	0.989785	−	0.476655i	0.183798	−	0.0885127i	−0.339722	−	0.940526i	\(-0.610333\pi\)
							0.523520	+	0.852013i	\(0.324619\pi\)
\(31\)	4.36774	+	7.56514i	0.784469	+	1.35874i	0.929316	+	0.369285i	\(0.120398\pi\)
							−0.144847	+	0.989454i	\(0.546269\pi\)
\(37\)	0.541656	−	7.22789i	0.0890476	−	1.18826i	−0.756367	−	0.654147i	\(-0.773028\pi\)
							0.845415	−	0.534111i	\(-0.179353\pi\)
\(41\)	−4.29173	−	3.42254i	−0.670256	−	0.534511i	0.228178	−	0.973619i	\(-0.426723\pi\)
							−0.898434	+	0.439108i	\(0.855294\pi\)
\(43\)	−4.26544	+	3.40157i	−0.650473	+	0.518735i	−0.892218	−	0.451605i	\(-0.850852\pi\)
							0.241745	+	0.970340i	\(0.422280\pi\)
\(47\)	−7.47808	−	6.93865i	−1.09079	−	1.01211i	−0.999868	−	0.0162479i	\(-0.994828\pi\)
							−0.0909227	−	0.995858i	\(-0.528982\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.187.27	✓	336
4.3	odd	2		588.2.ba.b.187.14	yes	336
49.38	odd	42		588.2.ba.b.283.14	yes	336
196.87	even	42	inner	588.2.ba.a.283.27	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.27	✓	336	1.1	even	1	trivial
588.2.ba.a.283.27	yes	336	196.87	even	42	inner
588.2.ba.b.187.14	yes	336	4.3	odd	2
588.2.ba.b.283.14	yes	336	49.38	odd	42