Embedded newform 603.2.k.a.38.8

• Label: 603.2.k.a.38.8
• 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			38.8
Character	\(\chi\)	\(=\)	603.38
Dual form			603.2.k.a.365.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17138 - 2.02889i) q^{2} +(-1.66573 + 0.474716i) q^{3} +(-1.74427 + 3.02116i) q^{4} +(1.98526 + 3.43857i) q^{5} +(2.91435 + 2.82350i) q^{6} +2.23626i q^{7} +3.48727 q^{8} +(2.54929 - 1.58149i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 63 q^{4} + 3 q^{5} + 7 q^{6} + 12 q^{8} - 7 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17138	−	2.02889i	−0.828291	−	1.43464i	−0.899378	−	0.437172i	\(-0.855980\pi\)
							0.0710866	−	0.997470i	\(-0.477353\pi\)
\(3\)	−1.66573	+	0.474716i	−0.961708	+	0.274077i
							\(5\)	1.98526	+	3.43857i	0.887836	+	1.53778i	0.842428	+	0.538809i	\(0.181125\pi\)
0.0454077	+	0.998969i	\(0.485541\pi\)
\(7\)			2.23626i			0.845228i	0.906310	+	0.422614i	\(0.138887\pi\)
							−0.906310	+	0.422614i	\(0.861113\pi\)
\(11\)	−1.25881			−0.379545			−0.189773	−	0.981828i	\(-0.560775\pi\)
							−0.189773	+	0.981828i	\(0.560775\pi\)
\(13\)			0.663534i			0.184031i	0.995758	+	0.0920156i	\(0.0293310\pi\)
							−0.995758	+	0.0920156i	\(0.970669\pi\)
\(17\)	−3.65750	−	2.11166i	−0.887074	−	0.512152i	−0.0140898	−	0.999901i	\(-0.504485\pi\)
							−0.872984	+	0.487748i	\(0.837818\pi\)
\(19\)	−1.41410	+	2.44929i	−0.324416	+	0.561905i	−0.981394	−	0.192005i	\(-0.938501\pi\)
							0.656978	+	0.753910i	\(0.271834\pi\)
\(23\)			4.52081i			0.942653i	0.881959	+	0.471327i	\(0.156225\pi\)
							−0.881959	+	0.471327i	\(0.843775\pi\)
\(29\)		−	10.5678i		−	1.96239i	−0.193022	−	0.981194i	\(-0.561829\pi\)
							0.193022	−	0.981194i	\(-0.438171\pi\)
\(31\)	−1.87566	−	1.08292i	−0.336879	−	0.194497i	0.322012	−	0.946736i	\(-0.395641\pi\)
							−0.658891	+	0.752238i	\(0.728974\pi\)
\(37\)	−3.24082	+	5.61326i	−0.532787	+	0.922814i	0.466480	+	0.884532i	\(0.345522\pi\)
							−0.999267	+	0.0382826i	\(0.987811\pi\)
\(41\)	1.79191	−	3.10367i	0.279849	−	0.484713i	−0.691498	−	0.722378i	\(-0.743049\pi\)
							0.971347	+	0.237666i	\(0.0763823\pi\)
\(43\)	−6.19894	−	3.57896i	−0.945329	−	0.545786i	−0.0537021	−	0.998557i	\(-0.517102\pi\)
							−0.891627	+	0.452771i	\(0.850435\pi\)
\(47\)		−	5.63176i		−	0.821476i	−0.911753	−	0.410738i	\(-0.865271\pi\)
							0.911753	−	0.410738i	\(-0.134729\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.k.a.38.8	✓	132
9.5	odd	6		603.2.t.a.239.8	yes	132
67.30	odd	6		603.2.t.a.164.8	yes	132
603.365	even	6	inner	603.2.k.a.365.8	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.8	✓	132	1.1	even	1	trivial
603.2.k.a.365.8	yes	132	603.365	even	6	inner
603.2.t.a.164.8	yes	132	67.30	odd	6
603.2.t.a.239.8	yes	132	9.5	odd	6