Embedded newform 603.2.k.a.38.7

• Label: 603.2.k.a.38.7
• 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			38.7
Character	\(\chi\)	\(=\)	603.38
Dual form			603.2.k.a.365.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17719 - 2.03895i) q^{2} +(1.35469 + 1.07927i) q^{3} +(-1.77155 + 3.06842i) q^{4} +(-1.13402 - 1.96417i) q^{5} +(0.605848 - 4.03265i) q^{6} +0.964702i q^{7} +3.63306 q^{8} +(0.670366 + 2.92414i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 63 q^{4} + 3 q^{5} + 7 q^{6} + 12 q^{8} - 7 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17719	−	2.03895i	−0.832399	−	1.44176i	−0.896131	−	0.443791i	\(-0.853633\pi\)
							0.0637314	−	0.997967i	\(-0.479700\pi\)
\(3\)	1.35469	+	1.07927i	0.782130	+	0.623115i
							\(5\)	−1.13402	−	1.96417i	−0.507147	−	0.878405i	−0.999966	−	0.00827280i	\(-0.997367\pi\)
0.492818	−	0.870132i	\(-0.335967\pi\)
\(7\)			0.964702i			0.364623i	0.983241	+	0.182311i	\(0.0583579\pi\)
							−0.983241	+	0.182311i	\(0.941642\pi\)
\(11\)	5.79371			1.74687			0.873434	−	0.486942i	\(-0.161888\pi\)
							0.873434	+	0.486942i	\(0.161888\pi\)
\(13\)		−	0.156245i		−	0.0433346i	−0.999765	−	0.0216673i	\(-0.993103\pi\)
							0.999765	−	0.0216673i	\(-0.00689745\pi\)
\(17\)	3.10656	+	1.79358i	0.753452	+	0.435006i	0.826940	−	0.562290i	\(-0.190080\pi\)
							−0.0734876	+	0.997296i	\(0.523413\pi\)
\(19\)	−1.25784	+	2.17864i	−0.288568	+	0.499814i	−0.973468	−	0.228823i	\(-0.926512\pi\)
							0.684900	+	0.728637i	\(0.259846\pi\)
\(23\)			1.99422i			0.415824i	0.978147	+	0.207912i	\(0.0666668\pi\)
							−0.978147	+	0.207912i	\(0.933333\pi\)
\(29\)		−	3.24232i		−	0.602083i	−0.953611	−	0.301041i	\(-0.902666\pi\)
							0.953611	−	0.301041i	\(-0.0973342\pi\)
\(31\)	−3.63537	−	2.09888i	−0.652932	−	0.376970i	0.136647	−	0.990620i	\(-0.456367\pi\)
							−0.789579	+	0.613649i	\(0.789701\pi\)
\(37\)	2.42142	−	4.19402i	0.398079	−	0.689492i	−0.595410	−	0.803422i	\(-0.703010\pi\)
							0.993489	+	0.113930i	\(0.0363438\pi\)
\(41\)	3.37425	−	5.84437i	0.526969	−	0.912737i	−0.472537	−	0.881311i	\(-0.656662\pi\)
							0.999506	−	0.0314264i	\(-0.0100050\pi\)
\(43\)	9.45567	+	5.45923i	1.44198	+	0.832525i	0.997982	−	0.0635047i	\(-0.0202278\pi\)
							0.443994	+	0.896030i	\(0.353561\pi\)
\(47\)		−	10.6826i		−	1.55822i	−0.626889	−	0.779108i	\(-0.715672\pi\)
							0.626889	−	0.779108i	\(-0.284328\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.k.a.38.7	✓	132
9.5	odd	6		603.2.t.a.239.7	yes	132
67.30	odd	6		603.2.t.a.164.7	yes	132
603.365	even	6	inner	603.2.k.a.365.7	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.7	✓	132	1.1	even	1	trivial
603.2.k.a.365.7	yes	132	603.365	even	6	inner
603.2.t.a.164.7	yes	132	67.30	odd	6
603.2.t.a.239.7	yes	132	9.5	odd	6