Embedded newform 603.2.t.a.164.10

• Label: 603.2.t.a.164.10
• Level: $603$
• Weight: $2$
• Character: 603.164
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $132$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.81497924188\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(66\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			164.10
Character	\(\chi\)	\(=\)	603.164
Dual form			603.2.t.a.239.10

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.25415 q^{2} +(0.276897 + 1.70977i) q^{3} +3.08119 q^{4} +(1.58205 + 2.74019i) q^{5} +(-0.624168 - 3.85409i) q^{6} +(-1.26597 + 0.730907i) q^{7} -2.43718 q^{8} +(-2.84666 + 0.946863i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 6 q^{2} + 3 q^{3} + 126 q^{4} - 3 q^{5} - 2 q^{6} - 6 q^{7} - 12 q^{8} - 7 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.25415			−1.59393			−0.796963	−	0.604029i	\(-0.793561\pi\)
							−0.796963	+	0.604029i	\(0.793561\pi\)
\(3\)	0.276897	+	1.70977i	0.159867	+	0.987139i
							\(5\)	1.58205	+	2.74019i	0.707513	+	1.22545i	0.965777	+	0.259374i	\(0.0835164\pi\)
−0.258264	+	0.966074i	\(0.583150\pi\)
\(7\)	−1.26597	+	0.730907i	−0.478491	+	0.276257i	−0.719788	−	0.694194i	\(-0.755761\pi\)
							0.241296	+	0.970452i	\(0.422427\pi\)
\(11\)	−0.837756	−	1.45104i	−0.252593	−	0.437504i	0.711646	−	0.702538i	\(-0.247950\pi\)
							−0.964239	+	0.265034i	\(0.914617\pi\)
\(13\)	−4.66664	−	2.69428i	−1.29429	−	0.747260i	−0.314880	−	0.949131i	\(-0.601964\pi\)
							−0.979412	+	0.201871i	\(0.935298\pi\)
\(17\)	−2.28074	+	1.31678i	−0.553160	+	0.319367i	−0.750395	−	0.660989i	\(-0.770137\pi\)
							0.197236	+	0.980356i	\(0.436804\pi\)
\(19\)	−3.98417	−	6.90078i	−0.914030	−	1.58315i	−0.808315	−	0.588751i	\(-0.799620\pi\)
							−0.105716	−	0.994396i	\(-0.533713\pi\)
\(23\)	3.90789	+	2.25622i	0.814850	+	0.470454i	0.848637	−	0.528975i	\(-0.177423\pi\)
							−0.0337870	+	0.999429i	\(0.510757\pi\)
\(29\)	−2.07503	+	1.19802i	−0.385323	+	0.222466i	−0.680132	−	0.733090i	\(-0.738077\pi\)
							0.294809	+	0.955556i	\(0.404744\pi\)
\(31\)			2.86768i			0.515051i	0.966271	+	0.257526i	\(0.0829072\pi\)
							−0.966271	+	0.257526i	\(0.917093\pi\)
\(37\)	−0.470709	−	0.815292i	−0.0773841	−	0.134033i	0.824736	−	0.565517i	\(-0.191323\pi\)
							−0.902121	+	0.431484i	\(0.857990\pi\)
\(41\)	−2.50503			−0.391221			−0.195610	−	0.980682i	\(-0.562669\pi\)
							−0.195610	+	0.980682i	\(0.562669\pi\)
\(43\)	10.4956	+	6.05962i	1.60056	+	0.924083i	0.991375	+	0.131054i	\(0.0418362\pi\)
							0.609184	+	0.793029i	\(0.291497\pi\)
\(47\)	−0.0198672	+	0.0114703i	−0.00289793	+	0.00167312i	−0.501448	−	0.865188i	\(-0.667199\pi\)
							0.498550	+	0.866861i	\(0.333866\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.t.a.164.10	yes	132
9.5	odd	6		603.2.k.a.365.10	yes	132
67.38	odd	6		603.2.k.a.38.10	✓	132
603.239	even	6	inner	603.2.t.a.239.10	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.k.a.38.10	✓	132	67.38	odd	6
603.2.k.a.365.10	yes	132	9.5	odd	6
603.2.t.a.164.10	yes	132	1.1	even	1	trivial
603.2.t.a.239.10	yes	132	603.239	even	6	inner