Embedded newform 588.2.ba.a.103.9

• Label: 588.2.ba.a.103.9
• Level: $588$
• Weight: $2$
• Character: 588.103
• 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			103.9
Character	\(\chi\)	\(=\)	588.103
Dual form			588.2.ba.a.451.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.706021 - 1.22537i) q^{2} +(-0.826239 - 0.563320i) q^{3} +(-1.00307 + 1.73028i) q^{4} +(-0.481855 - 0.0361100i) q^{5} +(-0.106934 + 1.41016i) q^{6} +(-1.51904 - 2.16623i) q^{7} +(2.82842 + 0.00752107i) q^{8} +(0.365341 + 0.930874i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(493\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{29}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.706021	−	1.22537i	−0.499232	−	0.866468i
							\(3\)	−0.826239	−	0.563320i	−0.477029	−	0.325233i
\(5\)	−0.481855	−	0.0361100i	−0.215492	−	0.0161489i								−0.0334534	−	0.999440i	\(-0.510651\pi\)
							−0.182039	+	0.983291i	\(0.558270\pi\)
\(7\)	−1.51904	−	2.16623i	−0.574141	−	0.818756i
							\(11\)	−1.90604	−	0.748066i	−0.574693	−	0.225550i	0.0601530	−	0.998189i	\(-0.480841\pi\)
−0.634846	+	0.772639i	\(0.718936\pi\)
\(13\)	−2.42641	+	1.93500i	−0.672965	+	0.536672i	−0.899275	−	0.437384i	\(-0.855905\pi\)
							0.226310	+	0.974055i	\(0.427334\pi\)
\(17\)	2.61776	+	2.82127i	0.634899	+	0.684258i	0.965765	−	0.259419i	\(-0.0835312\pi\)
							−0.330866	+	0.943678i	\(0.607341\pi\)
\(19\)	−1.11411	+	1.92970i	−0.255595	+	0.442703i	−0.965057	−	0.262041i	\(-0.915605\pi\)
							0.709462	+	0.704743i	\(0.248938\pi\)
\(23\)	2.12464	−	2.28981i	0.443017	−	0.477459i	−0.471581	−	0.881823i	\(-0.656316\pi\)
							0.914598	+	0.404364i	\(0.132507\pi\)
\(29\)	2.12663	+	9.31737i	0.394905	+	1.73019i	0.647000	+	0.762490i	\(0.276024\pi\)
							−0.252094	+	0.967703i	\(0.581119\pi\)
\(31\)	3.08596	+	5.34504i	0.554255	+	0.959997i	0.997961	+	0.0638250i	\(0.0203299\pi\)
							−0.443706	+	0.896172i	\(0.646337\pi\)
\(37\)	−3.27251	−	1.00944i	−0.537998	−	0.165950i	0.0138361	−	0.999904i	\(-0.495596\pi\)
							−0.551834	+	0.833954i	\(0.686072\pi\)
\(41\)	4.28859	+	8.90536i	0.669766	+	1.39078i	0.907749	+	0.419514i	\(0.137799\pi\)
							−0.237983	+	0.971269i	\(0.576486\pi\)
\(43\)	−0.903128	+	1.87537i	−0.137726	+	0.285991i	−0.958412	−	0.285389i	\(-0.907877\pi\)
							0.820686	+	0.571379i	\(0.193592\pi\)
\(47\)	4.21028	−	0.634598i	0.614133	−	0.0925656i	0.165392	−	0.986228i	\(-0.447111\pi\)
							0.448740	+	0.893662i	\(0.351873\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.103.9	✓	336
4.3	odd	2		588.2.ba.b.103.13	yes	336
49.10	odd	42		588.2.ba.b.451.13	yes	336
196.59	even	42	inner	588.2.ba.a.451.9	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.103.9	✓	336	1.1	even	1	trivial
588.2.ba.a.451.9	yes	336	196.59	even	42	inner
588.2.ba.b.103.13	yes	336	4.3	odd	2
588.2.ba.b.451.13	yes	336	49.10	odd	42