Embedded newform 603.2.k.a.38.6

• Label: 603.2.k.a.38.6
• Level: $603$
• Weight: $2$
• Character: 603.38
• 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.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			38.6
Character	\(\chi\)	\(=\)	603.38
Dual form			603.2.k.a.365.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20278 - 2.08327i) q^{2} +(-1.17583 - 1.27178i) q^{3} +(-1.89335 + 3.27938i) q^{4} +(-0.366678 - 0.635105i) q^{5} +(-1.23520 + 3.97925i) q^{6} -4.48576i q^{7} +4.29802 q^{8} +(-0.234838 + 2.99079i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 63 q^{4} + 3 q^{5} + 7 q^{6} + 12 q^{8} - 7 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20278	−	2.08327i	−0.850493	−	1.47310i	−0.880764	−	0.473555i	\(-0.842971\pi\)
							0.0302716	−	0.999542i	\(-0.490363\pi\)
\(3\)	−1.17583	−	1.27178i	−0.678867	−	0.734261i
							\(5\)	−0.366678	−	0.635105i	−0.163983	−	0.284028i	0.772310	−	0.635245i	\(-0.219101\pi\)
−0.936294	+	0.351218i	\(0.885768\pi\)
\(7\)		−	4.48576i		−	1.69546i	−0.530430	−	0.847729i	\(-0.677970\pi\)
							0.530430	−	0.847729i	\(-0.322030\pi\)
\(11\)	−1.09923			−0.331432			−0.165716	−	0.986174i	\(-0.552993\pi\)
							−0.165716	+	0.986174i	\(0.552993\pi\)
\(13\)			5.07106i			1.40646i	0.710963	+	0.703229i	\(0.248259\pi\)
							−0.710963	+	0.703229i	\(0.751741\pi\)
\(17\)	−6.89379	−	3.98013i	−1.67199	−	0.965324i	−0.966523	−	0.256581i	\(-0.917404\pi\)
							−0.705467	−	0.708742i	\(-0.749263\pi\)
\(19\)	−2.60366	+	4.50967i	−0.597321	+	1.03459i	0.395894	+	0.918296i	\(0.370435\pi\)
							−0.993215	+	0.116294i	\(0.962899\pi\)
\(23\)			4.91873i			1.02563i	0.858500	+	0.512813i	\(0.171397\pi\)
							−0.858500	+	0.512813i	\(0.828603\pi\)
\(29\)		−	1.40905i		−	0.261654i	−0.991405	−	0.130827i	\(-0.958237\pi\)
							0.991405	−	0.130827i	\(-0.0417633\pi\)
\(31\)	1.20627	+	0.696441i	0.216653	+	0.125085i	0.604399	−	0.796681i	\(-0.293413\pi\)
							−0.387747	+	0.921766i	\(0.626746\pi\)
\(37\)	−1.22896	+	2.12863i	−0.202040	+	0.349944i	−0.949186	−	0.314716i	\(-0.898091\pi\)
							0.747145	+	0.664661i	\(0.231424\pi\)
\(41\)	−0.323077	+	0.559586i	−0.0504562	+	0.0873927i	−0.890150	−	0.455667i	\(-0.849401\pi\)
							0.839694	+	0.543059i	\(0.182734\pi\)
\(43\)	6.25086	+	3.60893i	0.953247	+	0.550357i	0.894088	−	0.447891i	\(-0.147825\pi\)
							0.0591588	+	0.998249i	\(0.481158\pi\)
\(47\)		−	8.57220i		−	1.25038i	−0.780471	−	0.625192i	\(-0.785021\pi\)
							0.780471	−	0.625192i	\(-0.214979\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.k.a.38.6	✓	132
9.5	odd	6		603.2.t.a.239.6	yes	132
67.30	odd	6		603.2.t.a.164.6	yes	132
603.365	even	6	inner	603.2.k.a.365.6	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.6	✓	132	1.1	even	1	trivial
603.2.k.a.365.6	yes	132	603.365	even	6	inner
603.2.t.a.164.6	yes	132	67.30	odd	6
603.2.t.a.239.6	yes	132	9.5	odd	6