Embedded newform 588.2.ba.a.283.16

• Label: 588.2.ba.a.283.16
• 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			283.16
Character	\(\chi\)	\(=\)	588.283
Dual form			588.2.ba.a.187.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.317209 - 1.37818i) q^{2} +(0.988831 - 0.149042i) q^{3} +(-1.79876 - 0.874341i) q^{4} +(3.55694 + 1.39600i) q^{5} +(0.108259 - 1.41006i) q^{6} +(-0.339306 + 2.62390i) q^{7} +(-1.77558 + 2.20166i) q^{8} +(0.955573 - 0.294755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.317209	−	1.37818i	0.224300	−	0.974520i
							\(3\)	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	\(-0.703367\pi\)
0.944902	−	0.327354i	\(-0.106157\pi\)
\(13\)	−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	\(0.0833802\pi\)
							−0.258677	+	0.965964i	\(0.583287\pi\)
\(23\)	3.78511	−	5.55173i	0.789250	−	1.15762i	−0.194995	−	0.980804i	\(-0.562469\pi\)
							0.984245	−	0.176812i	\(-0.0565786\pi\)
\(29\)	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	−0.870571	−	0.492042i	\(-0.836251\pi\)
							0.861407	+	0.507916i	\(0.169584\pi\)
\(37\)	−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	\(-0.900845\pi\)
							−0.510620	−	0.859807i	\(-0.670584\pi\)
\(47\)	−5.13225	+	4.76203i	−0.748616	+	0.694614i	−0.959126	−	0.282978i	\(-0.908678\pi\)
							0.210511	+	0.977592i	\(0.432487\pi\)

(See \(a_n\) instead)

Twists

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

    

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