Embedded newform 588.2.ba.b.187.28

• Label: 588.2.ba.b.187.28
• 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			187.28
Character	\(\chi\)	\(=\)	588.187
Dual form			588.2.ba.b.283.28

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

\(n\)	\(197\)	\(295\)	\(493\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{23}{42}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.187.16	✓	336	4.3	odd	2
588.2.ba.a.283.16	yes	336	49.38	odd	42
588.2.ba.b.187.28	yes	336	1.1	even	1	trivial
588.2.ba.b.283.28	yes	336	196.87	even	42	inner