Embedded newform 603.2.f.c.238.15

• Label: 603.2.f.c.238.15
• 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			238.15
Character	\(\chi\)	\(=\)	603.238
Dual form			603.2.f.c.565.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.772546 + 1.33809i) q^{2} +(-1.60578 - 0.649205i) q^{3} +(-0.193654 - 0.335419i) q^{4} +(-1.83310 + 3.17502i) q^{5} +(2.10923 - 1.64714i) q^{6} -3.81842 q^{7} -2.49176 q^{8} +(2.15706 + 2.08496i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + q^{2} - 3 q^{3} - 65 q^{4} + 5 q^{5} + 3 q^{6} - 8 q^{7} - 36 q^{8} + 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.772546	+	1.33809i	−0.546272	+	0.946172i	0.452253	+	0.891890i	\(0.350620\pi\)
							−0.998526	+	0.0542820i	\(0.982713\pi\)
\(3\)	−1.60578	−	0.649205i	−0.927098	−	0.374819i
							\(5\)	−1.83310	+	3.17502i	−0.819786	+	1.41991i	0.0860532	+	0.996291i	\(0.472574\pi\)
−0.905840	+	0.423621i	\(0.860759\pi\)
\(7\)	−3.81842			−1.44323			−0.721614	−	0.692296i	\(-0.756599\pi\)
							−0.721614	+	0.692296i	\(0.756599\pi\)
\(11\)	−5.08221			−1.53235			−0.766173	−	0.642635i	\(-0.777841\pi\)
							−0.766173	+	0.642635i	\(0.777841\pi\)
\(13\)	3.76654			1.04465			0.522325	−	0.852746i	\(-0.325065\pi\)
							0.522325	+	0.852746i	\(0.325065\pi\)
\(17\)	0.236371	+	0.409407i	0.0573284	+	0.0992958i	0.893265	−	0.449530i	\(-0.148408\pi\)
							−0.835937	+	0.548826i	\(0.815075\pi\)
\(19\)	3.69855	+	6.40607i	0.848505	+	1.46965i	0.882542	+	0.470234i	\(0.155830\pi\)
							−0.0340366	+	0.999421i	\(0.510836\pi\)
\(23\)	−2.48751			−0.518682			−0.259341	−	0.965786i	\(-0.583505\pi\)
							−0.259341	+	0.965786i	\(0.583505\pi\)
\(29\)	−0.175783			−0.0326420			−0.0163210	−	0.999867i	\(-0.505195\pi\)
							−0.0163210	+	0.999867i	\(0.505195\pi\)
\(31\)	0.428282	+	0.741807i	0.0769218	+	0.133232i	0.901920	−	0.431902i	\(-0.142157\pi\)
							−0.824999	+	0.565135i	\(0.808824\pi\)
\(37\)	−0.464197	−	0.804013i	−0.0763135	−	0.132179i	0.825343	−	0.564631i	\(-0.190982\pi\)
							−0.901657	+	0.432452i	\(0.857648\pi\)
\(41\)	3.62723	+	6.28255i	0.566478	+	0.981169i	0.996910	+	0.0785460i	\(0.0250278\pi\)
							−0.430432	+	0.902623i	\(0.641639\pi\)
\(43\)	−3.18589	−	5.51812i	−0.485843	−	0.841505i	0.514024	−	0.857776i	\(-0.328154\pi\)
							−0.999868	+	0.0162702i	\(0.994821\pi\)
\(47\)	3.55006			0.517829			0.258914	−	0.965900i	\(-0.416635\pi\)
							0.258914	+	0.965900i	\(0.416635\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.f.c.238.15	✓	128
9.7	even	3		603.2.h.c.439.50	yes	128
67.29	even	3		603.2.h.c.364.50	yes	128
603.565	even	3	inner	603.2.f.c.565.15	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.f.c.238.15	✓	128	1.1	even	1	trivial
603.2.f.c.565.15	yes	128	603.565	even	3	inner
603.2.h.c.364.50	yes	128	67.29	even	3
603.2.h.c.439.50	yes	128	9.7	even	3