Embedded newform 603.2.f.c.238.1

• Label: 603.2.f.c.238.1
• Level: $603$
• Weight: $2$
• Character: 603.238
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.81497924188\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			238.1
Character	\(\chi\)	\(=\)	603.238
Dual form			603.2.f.c.565.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31801 + 2.28286i) q^{2} +(-0.497255 - 1.65914i) q^{3} +(-2.47429 - 4.28560i) q^{4} +(-0.680757 + 1.17911i) q^{5} +(4.44296 + 1.05160i) q^{6} -3.81969 q^{7} +7.77252 q^{8} +(-2.50548 + 1.65003i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + q^{2} - 3 q^{3} - 65 q^{4} + 5 q^{5} + 3 q^{6} - 8 q^{7} - 36 q^{8} + 5 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/603\mathbb{Z}\right)^\times\).

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31801	+	2.28286i	−0.931973	+	1.61422i	−0.152027	+	0.988376i	\(0.548580\pi\)
							−0.779945	+	0.625848i	\(0.784753\pi\)
\(3\)	−0.497255	−	1.65914i	−0.287090	−	0.957904i
							\(5\)	−0.680757	+	1.17911i	−0.304444	+	0.527312i	−0.977137	−	0.212609i	\(-0.931804\pi\)
0.672694	+	0.739921i	\(0.265137\pi\)
\(7\)	−3.81969			−1.44371			−0.721854	−	0.692045i	\(-0.756710\pi\)
							−0.721854	+	0.692045i	\(0.756710\pi\)
\(11\)	0.885646			0.267032			0.133516	−	0.991047i	\(-0.457373\pi\)
							0.133516	+	0.991047i	\(0.457373\pi\)
\(13\)	0.522563			0.144933			0.0724664	−	0.997371i	\(-0.476913\pi\)
							0.0724664	+	0.997371i	\(0.476913\pi\)
\(17\)	−1.75089	−	3.03264i	−0.424654	−	0.735523i	0.571734	−	0.820439i	\(-0.306271\pi\)
							−0.996388	+	0.0849165i	\(0.972938\pi\)
\(19\)	−1.56747	−	2.71494i	−0.359603	−	0.622850i	0.628292	−	0.777978i	\(-0.283754\pi\)
							−0.987894	+	0.155128i	\(0.950421\pi\)
\(23\)	6.48293			1.35178			0.675892	−	0.737001i	\(-0.263759\pi\)
							0.675892	+	0.737001i	\(0.263759\pi\)
\(29\)	−0.367156			−0.0681791			−0.0340895	−	0.999419i	\(-0.510853\pi\)
							−0.0340895	+	0.999419i	\(0.510853\pi\)
\(31\)	−2.90704	−	5.03514i	−0.522119	−	0.904337i	−0.999669	−	0.0257324i	\(-0.991808\pi\)
							0.477550	−	0.878605i	\(-0.341525\pi\)
\(37\)	3.82368	+	6.62281i	0.628609	+	1.08878i	0.987831	+	0.155531i	\(0.0497087\pi\)
							−0.359222	+	0.933252i	\(0.616958\pi\)
\(41\)	5.09488	+	8.82460i	0.795687	+	1.37817i	0.922402	+	0.386231i	\(0.126223\pi\)
							−0.126715	+	0.991939i	\(0.540443\pi\)
\(43\)	5.82606	+	10.0910i	0.888466	+	1.53887i	0.841689	+	0.539963i	\(0.181562\pi\)
							0.0467777	+	0.998905i	\(0.485105\pi\)
\(47\)	10.6155			1.54843			0.774217	−	0.632920i	\(-0.218144\pi\)
							0.774217	+	0.632920i	\(0.218144\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.f.c.238.1	✓	128
9.7	even	3		603.2.h.c.439.64	yes	128
67.29	even	3		603.2.h.c.364.64	yes	128
603.565	even	3	inner	603.2.f.c.565.1	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.f.c.238.1	✓	128	1.1	even	1	trivial
603.2.f.c.565.1	yes	128	603.565	even	3	inner
603.2.h.c.364.64	yes	128	67.29	even	3
603.2.h.c.439.64	yes	128	9.7	even	3