Embedded newform 603.2.f.c.238.2

• Label: 603.2.f.c.238.2
• 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			238.2
Character	\(\chi\)	\(=\)	603.238
Dual form			603.2.f.c.565.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29651 + 2.24561i) q^{2} +(-1.64424 - 0.544495i) q^{3} +(-2.36185 - 4.09085i) q^{4} +(0.741692 - 1.28465i) q^{5} +(3.35449 - 2.98639i) q^{6} +1.85123 q^{7} +7.06261 q^{8} +(2.40705 + 1.79056i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + q^{2} - 3 q^{3} - 65 q^{4} + 5 q^{5} + 3 q^{6} - 8 q^{7} - 36 q^{8} + 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29651	+	2.24561i	−0.916768	+	1.58789i	−0.112476	+	0.993654i	\(0.535878\pi\)
							−0.804292	+	0.594234i	\(0.797455\pi\)
\(3\)	−1.64424	−	0.544495i	−0.949302	−	0.314364i
							\(5\)	0.741692	−	1.28465i	0.331695	−	0.574512i	−0.651150	−	0.758949i	\(-0.725713\pi\)
0.982844	+	0.184437i	\(0.0590463\pi\)
\(7\)	1.85123			0.699698			0.349849	−	0.936806i	\(-0.386233\pi\)
							0.349849	+	0.936806i	\(0.386233\pi\)
\(11\)	−5.57555			−1.68109			−0.840546	−	0.541740i	\(-0.817766\pi\)
							−0.840546	+	0.541740i	\(0.817766\pi\)
\(13\)	2.03633			0.564775			0.282388	−	0.959300i	\(-0.408874\pi\)
							0.282388	+	0.959300i	\(0.408874\pi\)
\(17\)	1.16284	+	2.01409i	0.282029	+	0.488489i	0.971884	−	0.235459i	\(-0.0756592\pi\)
							−0.689855	+	0.723947i	\(0.742326\pi\)
\(19\)	−3.18210	−	5.51156i	−0.730024	−	1.26444i	−0.956872	−	0.290508i	\(-0.906176\pi\)
							0.226849	−	0.973930i	\(-0.427158\pi\)
\(23\)	−4.51950			−0.942381			−0.471190	−	0.882032i	\(-0.656176\pi\)
							−0.471190	+	0.882032i	\(0.656176\pi\)
\(29\)	−2.39675			−0.445065			−0.222532	−	0.974925i	\(-0.571432\pi\)
							−0.222532	+	0.974925i	\(0.571432\pi\)
\(31\)	1.39779	+	2.42104i	0.251050	+	0.434832i	0.963815	−	0.266571i	\(-0.0858907\pi\)
							−0.712765	+	0.701403i	\(0.752557\pi\)
\(37\)	−0.814018	−	1.40992i	−0.133824	−	0.231790i	0.791324	−	0.611397i	\(-0.209392\pi\)
							−0.925148	+	0.379608i	\(0.876059\pi\)
\(41\)	−1.24066	−	2.14889i	−0.193759	−	0.335601i	0.752734	−	0.658325i	\(-0.228735\pi\)
							−0.946493	+	0.322724i	\(0.895401\pi\)
\(43\)	−5.99031	−	10.3755i	−0.913514	−	1.58225i	−0.809063	−	0.587722i	\(-0.800025\pi\)
							−0.104451	−	0.994530i	\(-0.533308\pi\)
\(47\)	3.25138			0.474262			0.237131	−	0.971478i	\(-0.423793\pi\)
							0.237131	+	0.971478i	\(0.423793\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.f.c.238.2	✓	128
9.7	even	3		603.2.h.c.439.63	yes	128
67.29	even	3		603.2.h.c.364.63	yes	128
603.565	even	3	inner	603.2.f.c.565.2	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.f.c.238.2	✓	128	1.1	even	1	trivial
603.2.f.c.565.2	yes	128	603.565	even	3	inner
603.2.h.c.364.63	yes	128	67.29	even	3
603.2.h.c.439.63	yes	128	9.7	even	3