Embedded newform 603.2.k.a.38.20

• Label: 603.2.k.a.38.20
• 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			38.20
Character	\(\chi\)	\(=\)	603.38
Dual form			603.2.k.a.365.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.688777 - 1.19300i) q^{2} +(-1.60949 - 0.639965i) q^{3} +(0.0511733 - 0.0886348i) q^{4} +(-1.73441 - 3.00409i) q^{5} +(0.345100 + 2.36090i) q^{6} +2.96153i q^{7} -2.89609 q^{8} +(2.18089 + 2.06003i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 63 q^{4} + 3 q^{5} + 7 q^{6} + 12 q^{8} - 7 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.688777	−	1.19300i	−0.487039	−	0.843576i	0.512850	−	0.858478i	\(-0.328590\pi\)
							−0.999889	+	0.0149023i	\(0.995256\pi\)
\(3\)	−1.60949	−	0.639965i	−0.929237	−	0.369484i
							\(5\)	−1.73441	−	3.00409i	−0.775654	−	1.34347i	−0.934426	−	0.356156i	\(-0.884087\pi\)
0.158773	−	0.987315i	\(-0.449246\pi\)
\(7\)			2.96153i			1.11935i	0.828711	+	0.559677i	\(0.189075\pi\)
							−0.828711	+	0.559677i	\(0.810925\pi\)
\(11\)	−1.69182			−0.510103			−0.255051	−	0.966927i	\(-0.582092\pi\)
							−0.255051	+	0.966927i	\(0.582092\pi\)
\(13\)			3.48345i			0.966135i	0.875583	+	0.483067i	\(0.160477\pi\)
							−0.875583	+	0.483067i	\(0.839523\pi\)
\(17\)	−0.301985	−	0.174351i	−0.0732421	−	0.0422863i	0.462932	−	0.886394i	\(-0.346798\pi\)
							−0.536174	+	0.844108i	\(0.680131\pi\)
\(19\)	1.37853	−	2.38768i	0.316257	−	0.547772i	−0.663447	−	0.748223i	\(-0.730907\pi\)
							0.979704	+	0.200451i	\(0.0642406\pi\)
\(23\)			5.23917i			1.09244i	0.837641	+	0.546221i	\(0.183934\pi\)
							−0.837641	+	0.546221i	\(0.816066\pi\)
\(29\)		−	3.78627i		−	0.703092i	−0.936171	−	0.351546i	\(-0.885656\pi\)
							0.936171	−	0.351546i	\(-0.114344\pi\)
\(31\)	−4.75324	−	2.74428i	−0.853707	−	0.492888i	0.00819278	−	0.999966i	\(-0.497392\pi\)
							−0.861900	+	0.507078i	\(0.830725\pi\)
\(37\)	2.16132	−	3.74351i	0.355319	−	0.615430i	−0.631854	−	0.775088i	\(-0.717706\pi\)
							0.987172	+	0.159658i	\(0.0510391\pi\)
\(41\)	−1.68558	+	2.91952i	−0.263244	+	0.455952i	−0.967102	−	0.254389i	\(-0.918126\pi\)
							0.703858	+	0.710341i	\(0.251459\pi\)
\(43\)	−1.31074	−	0.756757i	−0.199886	−	0.115404i	0.396716	−	0.917941i	\(-0.370150\pi\)
							−0.596602	+	0.802537i	\(0.703483\pi\)
\(47\)			11.9296i			1.74011i	0.492951	+	0.870057i	\(0.335918\pi\)
							−0.492951	+	0.870057i	\(0.664082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.k.a.38.20	✓	132
9.5	odd	6		603.2.t.a.239.20	yes	132
67.30	odd	6		603.2.t.a.164.20	yes	132
603.365	even	6	inner	603.2.k.a.365.20	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.20	✓	132	1.1	even	1	trivial
603.2.k.a.365.20	yes	132	603.365	even	6	inner
603.2.t.a.164.20	yes	132	67.30	odd	6
603.2.t.a.239.20	yes	132	9.5	odd	6