Embedded newform 588.2.ba.b.283.28

• Label: 588.2.ba.b.283.28
• 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			283.28
Character	\(\chi\)	\(=\)	588.283
Dual form			588.2.ba.b.187.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39803 - 0.213330i) q^{2} +(-0.988831 + 0.149042i) q^{3} +(1.90898 - 0.596484i) q^{4} +(3.55694 + 1.39600i) q^{5} +(-1.35062 + 0.419313i) q^{6} +(0.339306 - 2.62390i) q^{7} +(2.54157 - 1.24115i) q^{8} +(0.955573 - 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 28 q^{3} - 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39803	−	0.213330i	0.988557	−	0.150847i
							\(3\)	−0.988831	+	0.149042i	−0.570902	+	0.0860496i
\(5\)	3.55694	+	1.39600i	1.59071	+	0.624309i								0.985182	−	0.171512i	\(-0.0548651\pi\)
							0.605532	+	0.795821i	\(0.292960\pi\)
\(7\)	0.339306	−	2.62390i	0.128246	−	0.991742i
							\(11\)	−1.15615	+	3.74814i	−0.348592	+	1.13011i	0.596310	+	0.802754i	\(0.296633\pi\)
−0.944902	+	0.327354i	\(0.893843\pi\)
\(13\)	−1.58868	+	0.362606i	−0.440621	+	0.100569i	−0.437075	−	0.899425i	\(-0.643986\pi\)
							−0.00354549	+	0.999994i	\(0.501129\pi\)
\(17\)	−1.04348	−	1.53051i	−0.253081	−	0.371202i	0.678582	−	0.734524i	\(-0.262595\pi\)
							−0.931663	+	0.363322i	\(0.881642\pi\)
\(19\)	−3.08266	−	5.33932i	−0.707211	−	1.22493i	−0.965888	−	0.258961i	\(-0.916620\pi\)
							0.258677	−	0.965964i	\(-0.416713\pi\)
\(23\)	−3.78511	+	5.55173i	−0.789250	+	1.15762i	0.194995	+	0.980804i	\(0.437531\pi\)
							−0.984245	+	0.176812i	\(0.943421\pi\)
\(29\)	4.41883	+	2.12800i	0.820556	+	0.395159i	0.796565	−	0.604553i	\(-0.206648\pi\)
							0.0239910	+	0.999712i	\(0.492363\pi\)
\(31\)	0.0510275	−	0.0883823i	0.00916482	−	0.0158739i	−0.861407	−	0.507916i	\(-0.830416\pi\)
							0.870571	+	0.492042i	\(0.163749\pi\)
\(37\)	−0.846858	−	11.3005i	−0.139223	−	1.85780i	−0.432329	−	0.901716i	\(-0.642308\pi\)
							0.293106	−	0.956080i	\(-0.405311\pi\)
\(41\)	−9.06047	+	7.22548i	−1.41501	+	1.12843i	−0.442173	+	0.896930i	\(0.645792\pi\)
							−0.972835	+	0.231502i	\(0.925636\pi\)
\(43\)	9.59021	+	7.64793i	1.46249	+	1.16630i	0.951873	+	0.306492i	\(0.0991554\pi\)
							0.510620	+	0.859807i	\(0.329416\pi\)
\(47\)	5.13225	−	4.76203i	0.748616	−	0.694614i	−0.210511	−	0.977592i	\(-0.567513\pi\)
							0.959126	+	0.282978i	\(0.0913222\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.b.283.28	yes	336
4.3	odd	2		588.2.ba.a.283.16	yes	336
49.40	odd	42		588.2.ba.a.187.16	✓	336
196.187	even	42	inner	588.2.ba.b.187.28	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.16	✓	336	49.40	odd	42
588.2.ba.a.283.16	yes	336	4.3	odd	2
588.2.ba.b.187.28	yes	336	196.187	even	42	inner
588.2.ba.b.283.28	yes	336	1.1	even	1	trivial