Embedded newform 603.2.y.a.4.20

• Label: 603.2.y.a.4.20
• Level: $603$
• Weight: $2$
• Character: 603.4
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $1320$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.y (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.81497924188\)
Analytic rank:	\(0\)
Dimension:	\(1320\)
Relative dimension:	\(66\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			4.20
Character	\(\chi\)	\(=\)	603.4
Dual form			603.2.y.a.151.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.558230 - 1.22235i) q^{2} +(1.26728 - 1.18067i) q^{3} +(0.127194 - 0.146789i) q^{4} +(0.953627 + 0.381775i) q^{5} +(-2.15064 - 0.889983i) q^{6} +(0.642878 + 0.0613874i) q^{7} +(-2.82914 - 0.830712i) q^{8} +(0.212021 - 2.99250i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 5 q^{2} - 19 q^{3} - 137 q^{4} - 14 q^{5} - 16 q^{6} - 11 q^{7} - 20 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(e\left(\frac{1}{33}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.558230	−	1.22235i	−0.394728	−	0.864335i	−0.997778	−	0.0666308i	\(-0.978775\pi\)
							0.603049	−	0.797704i	\(-0.293952\pi\)
\(3\)	1.26728	−	1.18067i	0.731667	−	0.681662i
							\(5\)	0.953627	+	0.381775i	0.426475	+	0.170735i	0.574963	−	0.818179i	\(-0.305016\pi\)
−0.148488	+	0.988914i	\(0.547441\pi\)
\(7\)	0.642878	+	0.0613874i	0.242985	+	0.0232023i	0.215838	−	0.976429i	\(-0.430752\pi\)
							0.0271470	+	0.999631i	\(0.491358\pi\)
\(11\)	5.13858	+	2.05718i	1.54934	+	0.620263i	0.979491	−	0.201486i	\(-0.0645772\pi\)
							0.569850	+	0.821749i	\(0.307001\pi\)
\(13\)	−2.24874	−	2.14417i	−0.623687	−	0.594685i	0.310889	−	0.950446i	\(-0.399373\pi\)
							−0.934577	+	0.355761i	\(0.884222\pi\)
\(17\)	0.864967	−	0.166709i	0.209785	−	0.0404328i	−0.0832757	−	0.996527i	\(-0.526538\pi\)
							0.293061	+	0.956094i	\(0.405326\pi\)
\(19\)	−2.74209	−	3.85072i	−0.629078	−	0.883416i	0.369879	−	0.929080i	\(-0.379399\pi\)
							−0.998957	+	0.0456636i	\(0.985460\pi\)
\(23\)	−0.125211	+	2.62851i	−0.0261084	+	0.548082i	0.947486	+	0.319797i	\(0.103615\pi\)
							−0.973594	+	0.228285i	\(0.926688\pi\)
\(29\)	2.06763	−	3.58124i	0.383949	−	0.665020i	−0.607674	−	0.794187i	\(-0.707897\pi\)
							0.991623	+	0.129167i	\(0.0412304\pi\)
\(31\)	5.78033	+	1.69726i	1.03818	+	0.304837i	0.756032	−	0.654535i	\(-0.227136\pi\)
							0.282146	+	0.959371i	\(0.408954\pi\)
\(37\)	−1.58209	+	2.74027i	−0.260095	+	0.450497i	−0.966267	−	0.257543i	\(-0.917087\pi\)
							0.706172	+	0.708040i	\(0.250420\pi\)
\(41\)	8.01155	+	9.24583i	1.25119	+	1.44396i	0.848993	+	0.528404i	\(0.177209\pi\)
							0.402202	+	0.915551i	\(0.368245\pi\)
\(43\)	−0.819011	+	2.36638i	−0.124898	+	0.360869i	−0.990064	−	0.140616i	\(-0.955092\pi\)
							0.865166	+	0.501485i	\(0.167213\pi\)
\(47\)	−0.485555	+	0.250321i	−0.0708255	+	0.0365131i	−0.493275	−	0.869873i	\(-0.664201\pi\)
							0.422450	+	0.906386i	\(0.361170\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.y.a.4.20	✓	1320
9.7	even	3		603.2.ba.a.205.20	yes	1320
67.17	even	33		603.2.ba.a.553.20	yes	1320
603.151	even	33	inner	603.2.y.a.151.20	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.y.a.4.20	✓	1320	1.1	even	1	trivial
603.2.y.a.151.20	yes	1320	603.151	even	33	inner
603.2.ba.a.205.20	yes	1320	9.7	even	3
603.2.ba.a.553.20	yes	1320	67.17	even	33