Embedded newform 603.2.t.a.164.4

• Label: 603.2.t.a.164.4
• Level: $603$
• Weight: $2$
• Character: 603.164
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $132$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.t (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			164.4
Character	\(\chi\)	\(=\)	603.164
Dual form			603.2.t.a.239.4

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.52152 q^{2} +(-1.57859 + 0.712778i) q^{3} +4.35808 q^{4} +(0.296268 + 0.513151i) q^{5} +(3.98045 - 1.79729i) q^{6} +(-4.22020 + 2.43653i) q^{7} -5.94595 q^{8} +(1.98389 - 2.25037i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 6 q^{2} + 3 q^{3} + 126 q^{4} - 3 q^{5} - 2 q^{6} - 6 q^{7} - 12 q^{8} - 7 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.52152			−1.78299			−0.891493	−	0.453034i	\(-0.850341\pi\)
							−0.891493	+	0.453034i	\(0.850341\pi\)
\(3\)	−1.57859	+	0.712778i	−0.911400	+	0.411523i
							\(5\)	0.296268	+	0.513151i	0.132495	+	0.229488i	0.924638	−	0.380848i	\(-0.124368\pi\)
−0.792143	+	0.610336i	\(0.791034\pi\)
\(7\)	−4.22020	+	2.43653i	−1.59509	+	0.920923i	−0.602670	+	0.797990i	\(0.705897\pi\)
							−0.992415	+	0.122933i	\(0.960770\pi\)
\(11\)	−2.04268	−	3.53803i	−0.615892	−	1.06676i	−0.990227	−	0.139463i	\(-0.955462\pi\)
							0.374335	−	0.927294i	\(-0.377871\pi\)
\(13\)	3.56259	+	2.05686i	0.988084	+	0.570470i	0.904701	−	0.426047i	\(-0.140094\pi\)
							0.0833828	+	0.996518i	\(0.473428\pi\)
\(17\)	−1.79131	+	1.03421i	−0.434457	+	0.250834i	−0.701243	−	0.712922i	\(-0.747371\pi\)
							0.266787	+	0.963756i	\(0.414038\pi\)
\(19\)	−0.728168	−	1.26122i	−0.167053	−	0.289345i	0.770329	−	0.637646i	\(-0.220092\pi\)
							−0.937383	+	0.348302i	\(0.886759\pi\)
\(23\)	−7.72346	−	4.45914i	−1.61045	−	0.929796i	−0.989265	−	0.146135i	\(-0.953317\pi\)
							−0.621189	−	0.783661i	\(-0.713350\pi\)
\(29\)	−3.58163	+	2.06785i	−0.665091	+	0.383991i	−0.794214	−	0.607638i	\(-0.792117\pi\)
							0.129123	+	0.991629i	\(0.458784\pi\)
\(31\)			9.07819i			1.63049i	0.579115	+	0.815246i	\(0.303398\pi\)
							−0.579115	+	0.815246i	\(0.696602\pi\)
\(37\)	−0.575424	−	0.996664i	−0.0945991	−	0.163851i	0.814842	−	0.579683i	\(-0.196824\pi\)
							−0.909441	+	0.415832i	\(0.863490\pi\)
\(41\)	2.70788			0.422900			0.211450	−	0.977389i	\(-0.432181\pi\)
							0.211450	+	0.977389i	\(0.432181\pi\)
\(43\)	−1.50542	−	0.869154i	−0.229574	−	0.132545i	0.380801	−	0.924657i	\(-0.375648\pi\)
							−0.610376	+	0.792112i	\(0.708982\pi\)
\(47\)	10.7494	−	6.20619i	1.56797	−	0.905266i	0.571560	−	0.820560i	\(-0.306338\pi\)
							0.996406	−	0.0847058i	\(-0.0269950\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.t.a.164.4	yes	132
9.5	odd	6		603.2.k.a.365.4	yes	132
67.38	odd	6		603.2.k.a.38.4	✓	132
603.239	even	6	inner	603.2.t.a.239.4	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.4	✓	132	67.38	odd	6
603.2.k.a.365.4	yes	132	9.5	odd	6
603.2.t.a.164.4	yes	132	1.1	even	1	trivial
603.2.t.a.239.4	yes	132	603.239	even	6	inner