Embedded newform 603.2.k.a.38.14

• Label: 603.2.k.a.38.14
• 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			38.14
Character	\(\chi\)	\(=\)	603.38
Dual form			603.2.k.a.365.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.912112 - 1.57982i) q^{2} +(1.09608 - 1.34112i) q^{3} +(-0.663895 + 1.14990i) q^{4} +(-0.255149 - 0.441931i) q^{5} +(-3.11848 - 0.508369i) q^{6} -3.55097i q^{7} -1.22626 q^{8} +(-0.597201 - 2.93996i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 63 q^{4} + 3 q^{5} + 7 q^{6} + 12 q^{8} - 7 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.912112	−	1.57982i	−0.644960	−	1.11710i	−0.984311	−	0.176444i	\(-0.943541\pi\)
							0.339350	−	0.940660i	\(-0.389793\pi\)
\(3\)	1.09608	−	1.34112i	0.632824	−	0.774295i
							\(5\)	−0.255149	−	0.441931i	−0.114106	−	0.197638i	0.803316	−	0.595553i	\(-0.203067\pi\)
−0.917422	+	0.397915i	\(0.869734\pi\)
\(7\)		−	3.55097i		−	1.34214i	−0.741394	−	0.671070i	\(-0.765835\pi\)
							0.741394	−	0.671070i	\(-0.234165\pi\)
\(11\)	−1.16662			−0.351750			−0.175875	−	0.984412i	\(-0.556276\pi\)
							−0.175875	+	0.984412i	\(0.556276\pi\)
\(13\)		−	1.62982i		−	0.452032i	−0.974124	−	0.226016i	\(-0.927430\pi\)
							0.974124	−	0.226016i	\(-0.0725701\pi\)
\(17\)	4.83825	+	2.79337i	1.17345	+	0.677491i	0.954490	−	0.298243i	\(-0.0964005\pi\)
							0.218959	+	0.975734i	\(0.429734\pi\)
\(19\)	1.00058	−	1.73306i	0.229549	−	0.397591i	−0.728125	−	0.685444i	\(-0.759608\pi\)
							0.957675	+	0.287853i	\(0.0929415\pi\)
\(23\)			7.36612i			1.53594i	0.640484	+	0.767971i	\(0.278734\pi\)
							−0.640484	+	0.767971i	\(0.721266\pi\)
\(29\)		−	0.194041i		−	0.0360326i	−0.999838	−	0.0180163i	\(-0.994265\pi\)
							0.999838	−	0.0180163i	\(-0.00573507\pi\)
\(31\)	−6.71378	−	3.87620i	−1.20583	−	0.696187i	−0.243985	−	0.969779i	\(-0.578455\pi\)
							−0.961846	+	0.273592i	\(0.911788\pi\)
\(37\)	−2.71090	+	4.69542i	−0.445670	+	0.771923i	−0.998099	−	0.0616373i	\(-0.980368\pi\)
							0.552429	+	0.833560i	\(0.313701\pi\)
\(41\)	−1.61652	+	2.79990i	−0.252458	+	0.437271i	−0.964202	−	0.265169i	\(-0.914572\pi\)
							0.711744	+	0.702439i	\(0.247906\pi\)
\(43\)	2.13818	+	1.23448i	0.326070	+	0.188257i	0.654095	−	0.756412i	\(-0.273050\pi\)
							−0.328025	+	0.944669i	\(0.606383\pi\)
\(47\)		−	4.05030i		−	0.590797i	−0.955374	−	0.295398i	\(-0.904548\pi\)
							0.955374	−	0.295398i	\(-0.0954523\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.k.a.38.14	✓	132
9.5	odd	6		603.2.t.a.239.14	yes	132
67.30	odd	6		603.2.t.a.164.14	yes	132
603.365	even	6	inner	603.2.k.a.365.14	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.14	✓	132	1.1	even	1	trivial
603.2.k.a.365.14	yes	132	603.365	even	6	inner
603.2.t.a.164.14	yes	132	67.30	odd	6
603.2.t.a.239.14	yes	132	9.5	odd	6