Embedded newform 603.2.h.c.439.64

• Label: 603.2.h.c.439.64
• Level: $603$
• Weight: $2$
• Character: 603.439
• 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.h (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			439.64
Character	\(\chi\)	\(=\)	603.439
Dual form			603.2.h.c.364.64

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.63602 q^{2} +(-0.497255 + 1.65914i) q^{3} +4.94858 q^{4} +(-0.680757 - 1.17911i) q^{5} +(-1.31077 + 4.37351i) q^{6} +(1.90985 + 3.30795i) q^{7} +7.77252 q^{8} +(-2.50548 - 1.65003i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 2 q^{2} - 3 q^{3} + 130 q^{4} + 5 q^{5} - 6 q^{6} + 4 q^{7} - 36 q^{8} + 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.63602			1.86395			0.931973	−	0.362528i	\(-0.118086\pi\)
							0.931973	+	0.362528i	\(0.118086\pi\)
\(3\)	−0.497255	+	1.65914i	−0.287090	+	0.957904i
							\(5\)	−0.680757	−	1.17911i	−0.304444	−	0.527312i	0.672694	−	0.739921i	\(-0.265137\pi\)
−0.977137	+	0.212609i	\(0.931804\pi\)
\(7\)	1.90985	+	3.30795i	0.721854	+	1.25029i	0.960256	+	0.279121i	\(0.0900431\pi\)
							−0.238402	+	0.971167i	\(0.576624\pi\)
\(11\)	−0.442823	−	0.766992i	−0.133516	−	0.231257i	0.791513	−	0.611152i	\(-0.209293\pi\)
							−0.925030	+	0.379895i	\(0.875960\pi\)
\(13\)	−0.261281	+	0.452553i	−0.0724664	+	0.125516i	−0.899982	−	0.435928i	\(-0.856420\pi\)
							0.827515	+	0.561443i	\(0.189754\pi\)
\(17\)	−1.75089	−	3.03264i	−0.424654	−	0.735523i	0.571734	−	0.820439i	\(-0.306271\pi\)
							−0.996388	+	0.0849165i	\(0.972938\pi\)
\(19\)	−1.56747	−	2.71494i	−0.359603	−	0.622850i	0.628292	−	0.777978i	\(-0.283754\pi\)
							−0.987894	+	0.155128i	\(0.950421\pi\)
\(23\)	−3.24146	+	5.61438i	−0.675892	+	1.17068i	0.300315	+	0.953840i	\(0.402908\pi\)
							−0.976207	+	0.216839i	\(0.930425\pi\)
\(29\)	0.183578	+	0.317966i	0.0340895	+	0.0590448i	0.882567	−	0.470187i	\(-0.155814\pi\)
							−0.848477	+	0.529232i	\(0.822480\pi\)
\(31\)	5.81408			1.04424			0.522119	−	0.852872i	\(-0.325142\pi\)
							0.522119	+	0.852872i	\(0.325142\pi\)
\(37\)	3.82368	+	6.62281i	0.628609	+	1.08878i	0.987831	+	0.155531i	\(0.0497087\pi\)
							−0.359222	+	0.933252i	\(0.616958\pi\)
\(41\)	−10.1898			−1.59137			−0.795687	−	0.605708i	\(-0.792890\pi\)
							−0.795687	+	0.605708i	\(0.792890\pi\)
\(43\)	5.82606	−	10.0910i	0.888466	−	1.53887i	0.0467777	−	0.998905i	\(-0.485105\pi\)
							0.841689	−	0.539963i	\(-0.181562\pi\)
\(47\)	−5.30777	−	9.19332i	−0.774217	−	1.34098i	−0.935233	−	0.354032i	\(-0.884810\pi\)
							0.161016	−	0.986952i	\(-0.448523\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.h.c.439.64	yes	128
9.4	even	3		603.2.f.c.238.1	✓	128
67.29	even	3		603.2.f.c.565.1	yes	128
603.364	even	3	inner	603.2.h.c.364.64	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.f.c.238.1	✓	128	9.4	even	3
603.2.f.c.565.1	yes	128	67.29	even	3
603.2.h.c.364.64	yes	128	603.364	even	3	inner
603.2.h.c.439.64	yes	128	1.1	even	1	trivial