Embedded newform 603.2.t.a.164.19

• Label: 603.2.t.a.164.19
• 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			164.19
Character	\(\chi\)	\(=\)	603.164
Dual form			603.2.t.a.239.19

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.42838 q^{2} +(-1.73068 + 0.0688806i) q^{3} +0.0402558 q^{4} +(1.51065 + 2.61653i) q^{5} +(2.47206 - 0.0983874i) q^{6} +(0.463586 - 0.267651i) q^{7} +2.79925 q^{8} +(2.99051 - 0.238421i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 6 q^{2} + 3 q^{3} + 126 q^{4} - 3 q^{5} - 2 q^{6} - 6 q^{7} - 12 q^{8} - 7 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.42838			−1.01001			−0.505007	−	0.863115i	\(-0.668510\pi\)
							−0.505007	+	0.863115i	\(0.668510\pi\)
\(3\)	−1.73068	+	0.0688806i	−0.999209	+	0.0397683i
							\(5\)	1.51065	+	2.61653i	0.675585	+	1.17015i	0.976298	+	0.216432i	\(0.0694421\pi\)
−0.300713	+	0.953715i	\(0.597225\pi\)
\(7\)	0.463586	−	0.267651i	0.175219	−	0.101163i	−0.409825	−	0.912164i	\(-0.634410\pi\)
							0.585044	+	0.811001i	\(0.301077\pi\)
\(11\)	−0.894540	−	1.54939i	−0.269714	−	0.467158i	0.699074	−	0.715049i	\(-0.253596\pi\)
							−0.968788	+	0.247891i	\(0.920262\pi\)
\(13\)	−0.908502	−	0.524524i	−0.251973	−	0.145477i	0.368694	−	0.929551i	\(-0.379805\pi\)
							−0.620667	+	0.784074i	\(0.713138\pi\)
\(17\)	3.23605	−	1.86833i	0.784856	−	0.453137i	−0.0532923	−	0.998579i	\(-0.516972\pi\)
							0.838149	+	0.545442i	\(0.183638\pi\)
\(19\)	0.335495	+	0.581095i	0.0769679	+	0.133312i	0.901940	−	0.431861i	\(-0.142143\pi\)
							−0.824972	+	0.565173i	\(0.808809\pi\)
\(23\)	−0.235435	−	0.135928i	−0.0490916	−	0.0283430i	0.475253	−	0.879849i	\(-0.342356\pi\)
							−0.524345	+	0.851506i	\(0.675690\pi\)
\(29\)	5.31429	−	3.06821i	0.986838	−	0.569751i	0.0825106	−	0.996590i	\(-0.473706\pi\)
							0.904328	+	0.426839i	\(0.140373\pi\)
\(31\)			1.58067i			0.283896i	0.989874	+	0.141948i	\(0.0453366\pi\)
							−0.989874	+	0.141948i	\(0.954663\pi\)
\(37\)	1.65647	+	2.86910i	0.272323	+	0.471676i	0.969456	−	0.245265i	\(-0.0788749\pi\)
							−0.697134	+	0.716941i	\(0.745542\pi\)
\(41\)	11.8608			1.85234			0.926172	−	0.377102i	\(-0.123079\pi\)
							0.926172	+	0.377102i	\(0.123079\pi\)
\(43\)	5.72612	+	3.30597i	0.873225	+	0.504156i	0.868418	−	0.495832i	\(-0.165137\pi\)
							0.00480609	+	0.999988i	\(0.498470\pi\)
\(47\)	−3.00515	+	1.73502i	−0.438346	+	0.253079i	−0.702896	−	0.711293i	\(-0.748110\pi\)
							0.264550	+	0.964372i	\(0.414777\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.19	✓	132	67.38	odd	6
603.2.k.a.365.19	yes	132	9.5	odd	6
603.2.t.a.164.19	yes	132	1.1	even	1	trivial
603.2.t.a.239.19	yes	132	603.239	even	6	inner