Embedded newform 603.2.k.a.38.16

• Label: 603.2.k.a.38.16
• 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			38.16
Character	\(\chi\)	\(=\)	603.38
Dual form			603.2.k.a.365.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.880451 - 1.52499i) q^{2} +(1.37106 - 1.05839i) q^{3} +(-0.550389 + 0.953302i) q^{4} +(0.675144 + 1.16938i) q^{5} +(-2.82118 - 1.15899i) q^{6} +1.51810i q^{7} -1.58344 q^{8} +(0.759619 - 2.90224i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 63 q^{4} + 3 q^{5} + 7 q^{6} + 12 q^{8} - 7 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.880451	−	1.52499i	−0.622573	−	1.07833i	−0.989005	−	0.147883i	\(-0.952754\pi\)
							0.366432	−	0.930445i	\(-0.380579\pi\)
\(3\)	1.37106	−	1.05839i	0.791583	−	0.611062i
							\(5\)	0.675144	+	1.16938i	0.301933	+	0.522964i	0.976574	−	0.215183i	\(-0.0690346\pi\)
−0.674640	+	0.738146i	\(0.735701\pi\)
\(7\)			1.51810i			0.573787i	0.957962	+	0.286894i	\(0.0926226\pi\)
							−0.957962	+	0.286894i	\(0.907377\pi\)
\(11\)	3.80018			1.14580			0.572899	−	0.819626i	\(-0.305819\pi\)
							0.572899	+	0.819626i	\(0.305819\pi\)
\(13\)		−	3.80468i		−	1.05523i	−0.849484	−	0.527615i	\(-0.823086\pi\)
							0.849484	−	0.527615i	\(-0.176914\pi\)
\(17\)	−5.37833	−	3.10518i	−1.30444	−	0.753116i	−0.323274	−	0.946305i	\(-0.604784\pi\)
							−0.981162	+	0.193189i	\(0.938117\pi\)
\(19\)	0.0902351	−	0.156292i	0.0207013	−	0.0358558i	−0.855489	−	0.517821i	\(-0.826743\pi\)
							0.876190	+	0.481965i	\(0.160077\pi\)
\(23\)		−	7.11300i		−	1.48316i	−0.670863	−	0.741581i	\(-0.734076\pi\)
							0.670863	−	0.741581i	\(-0.265924\pi\)
\(29\)		−	8.20901i		−	1.52437i	−0.647356	−	0.762187i	\(-0.724125\pi\)
							0.647356	−	0.762187i	\(-0.275875\pi\)
\(31\)	5.32116	+	3.07217i	0.955708	+	0.551778i	0.894849	−	0.446368i	\(-0.147283\pi\)
							0.0608586	+	0.998146i	\(0.480616\pi\)
\(37\)	−0.845167	+	1.46387i	−0.138945	+	0.240659i	−0.927097	−	0.374821i	\(-0.877704\pi\)
							0.788153	+	0.615480i	\(0.211038\pi\)
\(41\)	−4.47216	+	7.74602i	−0.698435	+	1.20972i	0.270574	+	0.962699i	\(0.412786\pi\)
							−0.969009	+	0.247025i	\(0.920547\pi\)
\(43\)	10.8666	+	6.27385i	1.65715	+	0.956753i	0.974023	+	0.226450i	\(0.0727119\pi\)
							0.683123	+	0.730304i	\(0.260621\pi\)
\(47\)			3.33926i			0.487082i	0.969891	+	0.243541i	\(0.0783090\pi\)
							−0.969891	+	0.243541i	\(0.921691\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.16	✓	132	1.1	even	1	trivial
603.2.k.a.365.16	yes	132	603.365	even	6	inner
603.2.t.a.164.16	yes	132	67.30	odd	6
603.2.t.a.239.16	yes	132	9.5	odd	6