Embedded newform 603.2.f.c.238.18

• Label: 603.2.f.c.238.18
• Level: $603$
• Weight: $2$
• Character: 603.238
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $128$
• 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			238.18
Character	\(\chi\)	\(=\)	603.238
Dual form			603.2.f.c.565.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.762492 + 1.32067i) q^{2} +(-0.244454 - 1.71471i) q^{3} +(-0.162787 - 0.281956i) q^{4} +(-1.67796 + 2.90631i) q^{5} +(2.45097 + 0.984611i) q^{6} +1.28044 q^{7} -2.55347 q^{8} +(-2.88048 + 0.838336i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + q^{2} - 3 q^{3} - 65 q^{4} + 5 q^{5} + 3 q^{6} - 8 q^{7} - 36 q^{8} + 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.762492	+	1.32067i	−0.539163	+	0.933858i	0.459786	+	0.888030i	\(0.347926\pi\)
							−0.998949	+	0.0458283i	\(0.985407\pi\)
\(3\)	−0.244454	−	1.71471i	−0.141135	−	0.989990i
							\(5\)	−1.67796	+	2.90631i	−0.750406	+	1.29974i	0.197220	+	0.980359i	\(0.436809\pi\)
−0.947626	+	0.319382i	\(0.896525\pi\)
\(7\)	1.28044			0.483961			0.241980	−	0.970281i	\(-0.422203\pi\)
							0.241980	+	0.970281i	\(0.422203\pi\)
\(11\)	0.498811			0.150397			0.0751986	−	0.997169i	\(-0.476041\pi\)
							0.0751986	+	0.997169i	\(0.476041\pi\)
\(13\)	−2.43972			−0.676657			−0.338329	−	0.941028i	\(-0.609862\pi\)
							−0.338329	+	0.941028i	\(0.609862\pi\)
\(17\)	0.414143	+	0.717317i	0.100445	+	0.173975i	0.911868	−	0.410484i	\(-0.134640\pi\)
							−0.811423	+	0.584459i	\(0.801307\pi\)
\(19\)	−3.54174	−	6.13448i	−0.812532	−	1.40735i	−0.911087	−	0.412214i	\(-0.864755\pi\)
							0.0985553	−	0.995132i	\(-0.468578\pi\)
\(23\)	0.804873			0.167828			0.0839138	−	0.996473i	\(-0.473258\pi\)
							0.0839138	+	0.996473i	\(0.473258\pi\)
\(29\)	−4.21308			−0.782349			−0.391174	−	0.920317i	\(-0.627931\pi\)
							−0.391174	+	0.920317i	\(0.627931\pi\)
\(31\)	2.46644	+	4.27199i	0.442985	+	0.767272i	0.997909	−	0.0646283i	\(-0.0205862\pi\)
							−0.554924	+	0.831901i	\(0.687253\pi\)
\(37\)	−1.81101	−	3.13677i	−0.297729	−	0.515681i	0.677887	−	0.735166i	\(-0.262896\pi\)
							−0.975616	+	0.219485i	\(0.929562\pi\)
\(41\)	−3.84178	−	6.65417i	−0.599986	−	1.03921i	−0.992822	−	0.119598i	\(-0.961839\pi\)
							0.392836	−	0.919608i	\(-0.371494\pi\)
\(43\)	−3.39333	−	5.87742i	−0.517478	−	0.896298i	−0.999794	−	0.0203010i	\(-0.993538\pi\)
							0.482316	−	0.875997i	\(-0.339796\pi\)
\(47\)	−7.04116			−1.02706			−0.513529	−	0.858072i	\(-0.671662\pi\)
							−0.513529	+	0.858072i	\(0.671662\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.f.c.238.18	✓	128
9.7	even	3		603.2.h.c.439.47	yes	128
67.29	even	3		603.2.h.c.364.47	yes	128
603.565	even	3	inner	603.2.f.c.565.18	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.f.c.238.18	✓	128	1.1	even	1	trivial
603.2.f.c.565.18	yes	128	603.565	even	3	inner
603.2.h.c.364.47	yes	128	67.29	even	3
603.2.h.c.439.47	yes	128	9.7	even	3