Embedded newform 603.2.t.a.164.16

• Label: 603.2.t.a.164.16
• 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			164.16
Character	\(\chi\)	\(=\)	603.164
Dual form			603.2.t.a.239.16

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.76090 q^{2} +(-1.37106 + 1.05839i) q^{3} +1.10078 q^{4} +(-0.675144 - 1.16938i) q^{5} +(2.41431 - 1.86372i) q^{6} +(-1.31471 + 0.759049i) q^{7} +1.58344 q^{8} +(0.759619 - 2.90224i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 6 q^{2} + 3 q^{3} + 126 q^{4} - 3 q^{5} - 2 q^{6} - 6 q^{7} - 12 q^{8} - 7 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.76090			−1.24515			−0.622573	−	0.782562i	\(-0.713913\pi\)
							−0.622573	+	0.782562i	\(0.713913\pi\)
\(3\)	−1.37106	+	1.05839i	−0.791583	+	0.611062i
							\(5\)	−0.675144	−	1.16938i	−0.301933	−	0.522964i	0.674640	−	0.738146i	\(-0.264299\pi\)
−0.976574	+	0.215183i	\(0.930965\pi\)
\(7\)	−1.31471	+	0.759049i	−0.496914	+	0.286894i	−0.727438	−	0.686173i	\(-0.759289\pi\)
							0.230524	+	0.973067i	\(0.425956\pi\)
\(11\)	1.90009	+	3.29105i	0.572899	+	0.992290i	0.996266	+	0.0863319i	\(0.0275145\pi\)
							−0.423368	+	0.905958i	\(0.639152\pi\)
\(13\)	−3.29495	−	1.90234i	−0.913855	−	0.527615i	−0.0321858	−	0.999482i	\(-0.510247\pi\)
							−0.881670	+	0.471867i	\(0.843580\pi\)
\(17\)	5.37833	−	3.10518i	1.30444	−	0.753116i	0.323274	−	0.946305i	\(-0.395216\pi\)
							0.981162	+	0.193189i	\(0.0618830\pi\)
\(19\)	0.0902351	+	0.156292i	0.0207013	+	0.0358558i	0.876190	−	0.481965i	\(-0.160077\pi\)
							−0.855489	+	0.517821i	\(0.826743\pi\)
\(23\)	6.16004	+	3.55650i	1.28446	+	0.741581i	0.977660	−	0.210194i	\(-0.0674095\pi\)
							0.306797	+	0.951775i	\(0.400743\pi\)
\(29\)	−7.10921	+	4.10451i	−1.32015	+	0.762187i	−0.983752	−	0.179533i	\(-0.942541\pi\)
							−0.336396	+	0.941721i	\(0.609208\pi\)
\(31\)			6.14434i			1.10356i	0.833991	+	0.551778i	\(0.186051\pi\)
							−0.833991	+	0.551778i	\(0.813949\pi\)
\(37\)	−0.845167	−	1.46387i	−0.138945	−	0.240659i	0.788153	−	0.615480i	\(-0.211038\pi\)
							−0.927097	+	0.374821i	\(0.877704\pi\)
\(41\)	−8.94433			−1.39687			−0.698435	−	0.715674i	\(-0.746120\pi\)
							−0.698435	+	0.715674i	\(0.746120\pi\)
\(43\)	−10.8666	−	6.27385i	−1.65715	−	0.956753i	−0.974023	−	0.226450i	\(-0.927288\pi\)
							−0.683123	−	0.730304i	\(-0.739379\pi\)
\(47\)	2.89189	−	1.66963i	0.421825	−	0.243541i	−0.274033	−	0.961720i	\(-0.588358\pi\)
							0.695858	+	0.718180i	\(0.255024\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.t.a.164.16	yes	132
9.5	odd	6		603.2.k.a.365.16	yes	132
67.38	odd	6		603.2.k.a.38.16	✓	132
603.239	even	6	inner	603.2.t.a.239.16	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.16	✓	132	67.38	odd	6
603.2.k.a.365.16	yes	132	9.5	odd	6
603.2.t.a.164.16	yes	132	1.1	even	1	trivial
603.2.t.a.239.16	yes	132	603.239	even	6	inner