Embedded newform 603.2.k.a.38.10

• Label: 603.2.k.a.38.10
• 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			38.10
Character	\(\chi\)	\(=\)	603.38
Dual form			603.2.k.a.365.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12708 - 1.95215i) q^{2} +(-0.276897 - 1.70977i) q^{3} +(-1.54060 + 2.66839i) q^{4} +(-1.58205 - 2.74019i) q^{5} +(-3.02565 + 2.46759i) q^{6} +1.46181i q^{7} +2.43718 q^{8} +(-2.84666 + 0.946863i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 63 q^{4} + 3 q^{5} + 7 q^{6} + 12 q^{8} - 7 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12708	−	1.95215i	−0.796963	−	1.38038i	−0.921585	−	0.388175i	\(-0.873105\pi\)
							0.124623	−	0.992204i	\(-0.460228\pi\)
\(3\)	−0.276897	−	1.70977i	−0.159867	−	0.987139i
							\(5\)	−1.58205	−	2.74019i	−0.707513	−	1.22545i	−0.965777	−	0.259374i	\(-0.916484\pi\)
0.258264	−	0.966074i	\(-0.416850\pi\)
\(7\)			1.46181i			0.552514i	0.961084	+	0.276257i	\(0.0890941\pi\)
							−0.961084	+	0.276257i	\(0.910906\pi\)
\(11\)	−1.67551			−0.505186			−0.252593	−	0.967573i	\(-0.581283\pi\)
							−0.252593	+	0.967573i	\(0.581283\pi\)
\(13\)		−	5.38857i		−	1.49452i	−0.664532	−	0.747260i	\(-0.731369\pi\)
							0.664532	−	0.747260i	\(-0.268631\pi\)
\(17\)	2.28074	+	1.31678i	0.553160	+	0.319367i	0.750395	−	0.660989i	\(-0.229863\pi\)
							−0.197236	+	0.980356i	\(0.563196\pi\)
\(19\)	−3.98417	+	6.90078i	−0.914030	+	1.58315i	−0.105716	+	0.994396i	\(0.533713\pi\)
							−0.808315	+	0.588751i	\(0.799620\pi\)
\(23\)		−	4.51244i		−	0.940908i	−0.882424	−	0.470454i	\(-0.844090\pi\)
							0.882424	−	0.470454i	\(-0.155910\pi\)
\(29\)		−	2.39603i		−	0.444932i	−0.974940	−	0.222466i	\(-0.928589\pi\)
							0.974940	−	0.222466i	\(-0.0714107\pi\)
\(31\)	2.48349	+	1.43384i	0.446047	+	0.257526i	0.706159	−	0.708053i	\(-0.250426\pi\)
							−0.260112	+	0.965578i	\(0.583759\pi\)
\(37\)	−0.470709	+	0.815292i	−0.0773841	+	0.134033i	−0.902121	−	0.431484i	\(-0.857990\pi\)
							0.824736	+	0.565517i	\(0.191323\pi\)
\(41\)	−1.25252	+	2.16942i	−0.195610	+	0.338807i	−0.947100	−	0.320937i	\(-0.896002\pi\)
							0.751490	+	0.659744i	\(0.229335\pi\)
\(43\)	−10.4956	−	6.05962i	−1.60056	−	0.924083i	−0.991375	−	0.131054i	\(-0.958164\pi\)
							−0.609184	−	0.793029i	\(-0.708503\pi\)
\(47\)		−	0.0229407i		−	0.00334624i	−0.999999	−	0.00167312i	\(-0.999467\pi\)
							0.999999	−	0.00167312i	\(-0.000532571\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.k.a.38.10	✓	132
9.5	odd	6		603.2.t.a.239.10	yes	132
67.30	odd	6		603.2.t.a.164.10	yes	132
603.365	even	6	inner	603.2.k.a.365.10	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.10	✓	132	1.1	even	1	trivial
603.2.k.a.365.10	yes	132	603.365	even	6	inner
603.2.t.a.164.10	yes	132	67.30	odd	6
603.2.t.a.239.10	yes	132	9.5	odd	6