Embedded newform 588.2.ba.a.103.3

• Label: 588.2.ba.a.103.3
• 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			103.3
Character	\(\chi\)	\(=\)	588.103
Dual form			588.2.ba.a.451.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33039 - 0.479649i) q^{2} +(-0.826239 - 0.563320i) q^{3} +(1.53987 + 1.27624i) q^{4} +(-1.47341 - 0.110417i) q^{5} +(0.829024 + 1.14574i) q^{6} +(2.51658 + 0.816579i) q^{7} +(-1.43648 - 2.43650i) q^{8} +(0.365341 + 0.930874i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33039	−	0.479649i	−0.940728	−	0.339163i
							\(3\)	−0.826239	−	0.563320i	−0.477029	−	0.325233i
\(5\)	−1.47341	−	0.110417i	−0.658928	−	0.0493798i								−0.258930	−	0.965896i	\(-0.583370\pi\)
							−0.399997	+	0.916516i	\(0.630989\pi\)
\(7\)	2.51658	+	0.816579i	0.951180	+	0.308638i
							\(11\)	−4.52475	−	1.77583i	−1.36426	−	0.535434i	−0.433532	−	0.901138i	\(-0.642733\pi\)
−0.930731	+	0.365704i	\(0.880828\pi\)
\(13\)	−0.493917	+	0.393885i	−0.136988	+	0.109244i	−0.689586	−	0.724203i	\(-0.742208\pi\)
							0.552599	+	0.833447i	\(0.313636\pi\)
\(17\)	1.54551	+	1.66566i	0.374841	+	0.403982i	0.892094	−	0.451850i	\(-0.149236\pi\)
							−0.517253	+	0.855832i	\(0.673045\pi\)
\(19\)	1.95455	−	3.38538i	0.448405	−	0.776660i	−0.549878	−	0.835245i	\(-0.685326\pi\)
							0.998282	+	0.0585856i	\(0.0186591\pi\)
\(23\)	3.99742	−	4.30820i	0.833520	−	0.898321i	−0.162567	−	0.986698i	\(-0.551977\pi\)
							0.996087	+	0.0883764i	\(0.0281678\pi\)
\(29\)	0.615243	+	2.69556i	0.114248	+	0.500552i	0.999380	+	0.0352124i	\(0.0112108\pi\)
							−0.885132	+	0.465340i	\(0.845932\pi\)
\(31\)	−5.33859	−	9.24671i	−0.958840	−	1.66076i	−0.725326	−	0.688405i	\(-0.758311\pi\)
							−0.233513	−	0.972354i	\(-0.575022\pi\)
\(37\)	−11.4735	−	3.53911i	−1.88624	−	0.581826i	−0.990553	−	0.137132i	\(-0.956212\pi\)
							−0.895682	−	0.444695i	\(-0.853312\pi\)
\(41\)	0.556524	+	1.15563i	0.0869145	+	0.180480i	0.939896	−	0.341462i	\(-0.110922\pi\)
							−0.852981	+	0.521942i	\(0.825208\pi\)
\(43\)	0.349636	−	0.726027i	0.0533190	−	0.110718i	−0.872607	−	0.488423i	\(-0.837572\pi\)
							0.925926	+	0.377705i	\(0.123287\pi\)
\(47\)	−10.2573	+	1.54603i	−1.49617	+	0.225512i	−0.845553	−	0.533891i	\(-0.820729\pi\)
							−0.650620	+	0.759403i	\(0.725491\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.ba.a.103.3	✓	336
4.3	odd	2		588.2.ba.b.103.6	yes	336
49.10	odd	42		588.2.ba.b.451.6	yes	336
196.59	even	42	inner	588.2.ba.a.451.3	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.ba.a.103.3	✓	336	1.1	even	1	trivial
588.2.ba.a.451.3	yes	336	196.59	even	42	inner
588.2.ba.b.103.6	yes	336	4.3	odd	2
588.2.ba.b.451.6	yes	336	49.10	odd	42