Embedded newform 603.2.k.a.38.19

• Label: 603.2.k.a.38.19
• Level: $603$
• Weight: $2$
• Character: 603.38
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.81497924188\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(66\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			38.19
Character	\(\chi\)	\(=\)	603.38
Dual form			603.2.k.a.365.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.714188 - 1.23701i) q^{2} +(1.73068 - 0.0688806i) q^{3} +(-0.0201279 + 0.0348626i) q^{4} +(-1.51065 - 2.61653i) q^{5} +(-1.32124 - 2.09167i) q^{6} -0.535303i q^{7} -2.79925 q^{8} +(2.99051 - 0.238421i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 63 q^{4} + 3 q^{5} + 7 q^{6} + 12 q^{8} - 7 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/603\mathbb{Z}\right)^\times\).

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.714188	−	1.23701i	−0.505007	−	0.874698i	−0.999983	−	0.00579116i	\(-0.998157\pi\)
							0.494976	−	0.868906i	\(-0.335177\pi\)
\(3\)	1.73068	−	0.0688806i	0.999209	−	0.0397683i
							\(5\)	−1.51065	−	2.61653i	−0.675585	−	1.17015i	−0.976298	−	0.216432i	\(-0.930558\pi\)
0.300713	−	0.953715i	\(-0.402775\pi\)
\(7\)		−	0.535303i		−	0.202325i	−0.994870	−	0.101163i	\(-0.967744\pi\)
							0.994870	−	0.101163i	\(-0.0322563\pi\)
\(11\)	−1.78908			−0.539428			−0.269714	−	0.962941i	\(-0.586929\pi\)
							−0.269714	+	0.962941i	\(0.586929\pi\)
\(13\)		−	1.04905i		−	0.290954i	−0.989362	−	0.145477i	\(-0.953528\pi\)
							0.989362	−	0.145477i	\(-0.0464716\pi\)
\(17\)	−3.23605	−	1.86833i	−0.784856	−	0.453137i	0.0532923	−	0.998579i	\(-0.483028\pi\)
							−0.838149	+	0.545442i	\(0.816362\pi\)
\(19\)	0.335495	−	0.581095i	0.0769679	−	0.133312i	−0.824972	−	0.565173i	\(-0.808809\pi\)
							0.901940	+	0.431861i	\(0.142143\pi\)
\(23\)			0.271857i			0.0566861i	0.999598	+	0.0283430i	\(0.00902308\pi\)
							−0.999598	+	0.0283430i	\(0.990977\pi\)
\(29\)			6.13641i			1.13950i	0.821817	+	0.569751i	\(0.192961\pi\)
							−0.821817	+	0.569751i	\(0.807039\pi\)
\(31\)	1.36890	+	0.790333i	0.245861	+	0.141948i	0.617868	−	0.786282i	\(-0.287997\pi\)
							−0.372006	+	0.928230i	\(0.621330\pi\)
\(37\)	1.65647	−	2.86910i	0.272323	−	0.471676i	−0.697134	−	0.716941i	\(-0.745542\pi\)
							0.969456	+	0.245265i	\(0.0788749\pi\)
\(41\)	5.93039	−	10.2717i	0.926172	−	1.60418i	0.136506	−	0.990639i	\(-0.456413\pi\)
							0.789666	−	0.613537i	\(-0.210254\pi\)
\(43\)	−5.72612	−	3.30597i	−0.873225	−	0.504156i	−0.00480609	−	0.999988i	\(-0.501530\pi\)
							−0.868418	+	0.495832i	\(0.834863\pi\)
\(47\)		−	3.47005i		−	0.506158i	−0.967446	−	0.253079i	\(-0.918557\pi\)
							0.967446	−	0.253079i	\(-0.0814433\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.k.a.38.19	✓	132
9.5	odd	6		603.2.t.a.239.19	yes	132
67.30	odd	6		603.2.t.a.164.19	yes	132
603.365	even	6	inner	603.2.k.a.365.19	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.19	✓	132	1.1	even	1	trivial
603.2.k.a.365.19	yes	132	603.365	even	6	inner
603.2.t.a.164.19	yes	132	67.30	odd	6
603.2.t.a.239.19	yes	132	9.5	odd	6