Embedded newform 603.2.f.c.238.11

• Label: 603.2.f.c.238.11
• 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			238.11
Character	\(\chi\)	\(=\)	603.238
Dual form			603.2.f.c.565.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.03546 + 1.79347i) q^{2} +(1.65146 - 0.522197i) q^{3} +(-1.14435 - 1.98207i) q^{4} +(0.450767 - 0.780752i) q^{5} +(-0.773473 + 3.50255i) q^{6} -0.961690 q^{7} +0.597864 q^{8} +(2.45462 - 1.72477i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + q^{2} - 3 q^{3} - 65 q^{4} + 5 q^{5} + 3 q^{6} - 8 q^{7} - 36 q^{8} + 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.03546	+	1.79347i	−0.732180	+	1.26817i	0.223770	+	0.974642i	\(0.428163\pi\)
							−0.955950	+	0.293530i	\(0.905170\pi\)
\(3\)	1.65146	−	0.522197i	0.953469	−	0.301490i
							\(5\)	0.450767	−	0.780752i	0.201589	−	0.349163i	−0.747451	−	0.664317i	\(-0.768723\pi\)
0.949041	+	0.315154i	\(0.102056\pi\)
\(7\)	−0.961690			−0.363485			−0.181742	−	0.983346i	\(-0.558174\pi\)
							−0.181742	+	0.983346i	\(0.558174\pi\)
\(11\)	4.29125			1.29386			0.646930	−	0.762549i	\(-0.276052\pi\)
							0.646930	+	0.762549i	\(0.276052\pi\)
\(13\)	−0.0410203			−0.0113770			−0.00568849	−	0.999984i	\(-0.501811\pi\)
							−0.00568849	+	0.999984i	\(0.501811\pi\)
\(17\)	1.56395	+	2.70883i	0.379313	+	0.656989i	0.990962	−	0.134140i	\(-0.0428272\pi\)
							−0.611650	+	0.791129i	\(0.709494\pi\)
\(19\)	−2.74665	−	4.75734i	−0.630125	−	1.09141i	−0.987526	−	0.157457i	\(-0.949670\pi\)
							0.357401	−	0.933951i	\(-0.383663\pi\)
\(23\)	4.35013			0.907065			0.453532	−	0.891240i	\(-0.350164\pi\)
							0.453532	+	0.891240i	\(0.350164\pi\)
\(29\)	2.14657			0.398608			0.199304	−	0.979938i	\(-0.436132\pi\)
							0.199304	+	0.979938i	\(0.436132\pi\)
\(31\)	0.234083	+	0.405443i	0.0420425	+	0.0728198i	0.886281	−	0.463148i	\(-0.153280\pi\)
							−0.844238	+	0.535968i	\(0.819947\pi\)
\(37\)	1.63675	+	2.83493i	0.269079	+	0.466059i	0.968624	−	0.248529i	\(-0.0799473\pi\)
							−0.699545	+	0.714589i	\(0.746614\pi\)
\(41\)	−0.607797	−	1.05274i	−0.0949220	−	0.164410i	0.814654	−	0.579947i	\(-0.196927\pi\)
							−0.909576	+	0.415538i	\(0.863593\pi\)
\(43\)	−2.51013	−	4.34767i	−0.382791	−	0.663013i	0.608669	−	0.793424i	\(-0.291704\pi\)
							−0.991460	+	0.130411i	\(0.958370\pi\)
\(47\)	10.0391			1.46435			0.732176	−	0.681116i	\(-0.238505\pi\)
							0.732176	+	0.681116i	\(0.238505\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.f.c.238.11	✓	128
9.7	even	3		603.2.h.c.439.54	yes	128
67.29	even	3		603.2.h.c.364.54	yes	128
603.565	even	3	inner	603.2.f.c.565.11	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.f.c.238.11	✓	128	1.1	even	1	trivial
603.2.f.c.565.11	yes	128	603.565	even	3	inner
603.2.h.c.364.54	yes	128	67.29	even	3
603.2.h.c.439.54	yes	128	9.7	even	3