Embedded newform 603.2.l.a.200.10

• Label: 603.2.l.a.200.10
• Level: $603$
• Weight: $2$
• Character: 603.200
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $132$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.l (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			200.10
Character	\(\chi\)	\(=\)	603.200
Dual form			603.2.l.a.401.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13151 + 1.95983i) q^{2} +(1.69797 - 0.341928i) q^{3} +(-1.56063 - 2.70309i) q^{4} +(-1.14681 - 1.98634i) q^{5} +(-1.25114 + 3.71462i) q^{6} +(0.398259 + 0.229935i) q^{7} +2.53742 q^{8} +(2.76617 - 1.16116i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 66 q^{4} + 10 q^{6} + 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/603\mathbb{Z}\right)^\times\).

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13151	+	1.95983i	−0.800098	+	1.38581i	0.119453	+	0.992840i	\(0.461886\pi\)
							−0.919551	+	0.392971i	\(0.871447\pi\)
\(3\)	1.69797	−	0.341928i	0.980321	−	0.197412i
							\(5\)	−1.14681	−	1.98634i	−0.512871	−	0.888318i	−0.999889	−	0.0149262i	\(-0.995249\pi\)
0.487018	−	0.873392i	\(-0.338085\pi\)
\(7\)	0.398259	+	0.229935i	0.150528	+	0.0869073i	0.573372	−	0.819295i	\(-0.305635\pi\)
							−0.422844	+	0.906202i	\(0.638968\pi\)
\(11\)	−1.08598	+	1.88096i	−0.327434	+	0.567132i	−0.982002	−	0.188871i	\(-0.939517\pi\)
							0.654568	+	0.756003i	\(0.272851\pi\)
\(13\)	−0.247499	+	0.142894i	−0.0686440	+	0.0396316i	−0.533929	−	0.845529i	\(-0.679285\pi\)
							0.465285	+	0.885161i	\(0.345952\pi\)
\(17\)		−	6.86526i		−	1.66507i	−0.553973	−	0.832535i	\(-0.686889\pi\)
							0.553973	−	0.832535i	\(-0.313111\pi\)
\(19\)	3.54773			0.813904			0.406952	−	0.913450i	\(-0.366592\pi\)
							0.406952	+	0.913450i	\(0.366592\pi\)
\(23\)	0.878941	−	0.507457i	0.183272	−	0.105812i	−0.405557	−	0.914070i	\(-0.632922\pi\)
							0.588829	+	0.808258i	\(0.299589\pi\)
\(29\)	1.46765	+	0.847346i	0.272535	+	0.157348i	0.630039	−	0.776563i	\(-0.283039\pi\)
							−0.357504	+	0.933912i	\(0.616372\pi\)
\(31\)	5.28930	−	3.05378i	0.949987	−	0.548475i	0.0569100	−	0.998379i	\(-0.481875\pi\)
							0.893077	+	0.449904i	\(0.148542\pi\)
\(37\)	9.03606			1.48552			0.742760	−	0.669558i	\(-0.233516\pi\)
							0.742760	+	0.669558i	\(0.233516\pi\)
\(41\)	−4.23884	−	7.34188i	−0.661995	−	1.14661i	−0.980091	−	0.198551i	\(-0.936377\pi\)
							0.318095	−	0.948059i	\(-0.396957\pi\)
\(43\)	3.36614	+	1.94344i	0.513332	+	0.296372i	0.734202	−	0.678931i	\(-0.237556\pi\)
							−0.220870	+	0.975303i	\(0.570890\pi\)
\(47\)	1.15091	+	0.664479i	0.167878	+	0.0969243i	0.581585	−	0.813486i	\(-0.302433\pi\)
							−0.413707	+	0.910410i	\(0.635766\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.l.a.200.10	✓	132
9.5	odd	6	inner	603.2.l.a.401.57	yes	132
67.66	odd	2	inner	603.2.l.a.200.57	yes	132
603.401	even	6	inner	603.2.l.a.401.10	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.l.a.200.10	✓	132	1.1	even	1	trivial
603.2.l.a.200.57	yes	132	67.66	odd	2	inner
603.2.l.a.401.10	yes	132	603.401	even	6	inner
603.2.l.a.401.57	yes	132	9.5	odd	6	inner