Embedded newform 603.2.k.a.38.3

• Label: 603.2.k.a.38.3
• Level: $603$
• Weight: $2$
• Character: 603.38
• 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.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			38.3
Character	\(\chi\)	\(=\)	603.38
Dual form			603.2.k.a.365.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28266 - 2.22163i) q^{2} +(-0.385568 + 1.68859i) q^{3} +(-2.29042 + 3.96712i) q^{4} +(0.714646 + 1.23780i) q^{5} +(4.24597 - 1.30929i) q^{6} -2.23337i q^{7} +6.62064 q^{8} +(-2.70268 - 1.30213i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 63 q^{4} + 3 q^{5} + 7 q^{6} + 12 q^{8} - 7 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28266	−	2.22163i	−0.906975	−	1.57093i	−0.818244	−	0.574872i	\(-0.805052\pi\)
							−0.0887315	−	0.996056i	\(-0.528281\pi\)
\(3\)	−0.385568	+	1.68859i	−0.222608	+	0.974908i
							\(5\)	0.714646	+	1.23780i	0.319599	+	0.553562i	0.980404	−	0.196995i	\(-0.0631183\pi\)
−0.660805	+	0.750558i	\(0.729785\pi\)
\(7\)		−	2.23337i		−	0.844133i	−0.906565	−	0.422067i	\(-0.861305\pi\)
							0.906565	−	0.422067i	\(-0.138695\pi\)
\(11\)	0.689733			0.207962			0.103981	−	0.994579i	\(-0.466842\pi\)
							0.103981	+	0.994579i	\(0.466842\pi\)
\(13\)		−	2.44767i		−	0.678863i	−0.940631	−	0.339431i	\(-0.889765\pi\)
							0.940631	−	0.339431i	\(-0.110235\pi\)
\(17\)	2.80221	+	1.61786i	0.679636	+	0.392388i	0.799718	−	0.600376i	\(-0.204982\pi\)
							−0.120082	+	0.992764i	\(0.538316\pi\)
\(19\)	−1.29170	+	2.23730i	−0.296337	+	0.513271i	−0.975295	−	0.220906i	\(-0.929098\pi\)
							0.678958	+	0.734177i	\(0.262432\pi\)
\(23\)			5.97508i			1.24589i	0.782265	+	0.622945i	\(0.214064\pi\)
							−0.782265	+	0.622945i	\(0.785936\pi\)
\(29\)			7.80001i			1.44843i	0.689577	+	0.724213i	\(0.257797\pi\)
							−0.689577	+	0.724213i	\(0.742203\pi\)
\(31\)	5.31103	+	3.06632i	0.953889	+	0.550728i	0.894287	−	0.447494i	\(-0.147684\pi\)
							0.0596019	+	0.998222i	\(0.481017\pi\)
\(37\)	−0.0936827	+	0.162263i	−0.0154013	+	0.0266759i	−0.873623	−	0.486603i	\(-0.838236\pi\)
							0.858222	+	0.513279i	\(0.171569\pi\)
\(41\)	−1.53776	+	2.66348i	−0.240158	+	0.415966i	−0.960759	−	0.277384i	\(-0.910533\pi\)
							0.720601	+	0.693350i	\(0.243866\pi\)
\(43\)	5.11049	+	2.95055i	0.779343	+	0.449954i	0.836197	−	0.548429i	\(-0.184774\pi\)
							−0.0568543	+	0.998382i	\(0.518107\pi\)
\(47\)			10.6446i			1.55267i	0.630317	+	0.776337i	\(0.282925\pi\)
							−0.630317	+	0.776337i	\(0.717075\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.k.a.38.3	✓	132
9.5	odd	6		603.2.t.a.239.3	yes	132
67.30	odd	6		603.2.t.a.164.3	yes	132
603.365	even	6	inner	603.2.k.a.365.3	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.3	✓	132	1.1	even	1	trivial
603.2.k.a.365.3	yes	132	603.365	even	6	inner
603.2.t.a.164.3	yes	132	67.30	odd	6
603.2.t.a.239.3	yes	132	9.5	odd	6