Embedded newform 603.2.y.a.4.13

• Label: 603.2.y.a.4.13
• 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			4.13
Character	\(\chi\)	\(=\)	603.4
Dual form			603.2.y.a.151.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.848158 - 1.85721i) q^{2} +(-1.73202 + 0.00952723i) q^{3} +(-1.42012 + 1.63891i) q^{4} +(-0.374098 - 0.149766i) q^{5} +(1.48672 + 3.20865i) q^{6} +(-0.762785 - 0.0728371i) q^{7} +(0.330270 + 0.0969759i) q^{8} +(2.99982 - 0.0330028i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 5 q^{2} - 19 q^{3} - 137 q^{4} - 14 q^{5} - 16 q^{6} - 11 q^{7} - 20 q^{8} - 15 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.848158	−	1.85721i	−0.599738	−	1.31324i	−0.929375	−	0.369136i	\(-0.879654\pi\)
							0.329637	−	0.944108i	\(-0.393074\pi\)
\(3\)	−1.73202	+	0.00952723i	−0.999985	+	0.00550055i
							\(5\)	−0.374098	−	0.149766i	−0.167302	−	0.0669774i	0.286500	−	0.958080i	\(-0.407508\pi\)
−0.453801	+	0.891103i	\(0.649932\pi\)
\(7\)	−0.762785	−	0.0728371i	−0.288306	−	0.0275299i	−0.0500987	−	0.998744i	\(-0.515954\pi\)
							−0.238207	+	0.971214i	\(0.576560\pi\)
\(11\)	3.47554	+	1.39140i	1.04791	+	0.419522i	0.830769	−	0.556617i	\(-0.187901\pi\)
							0.217146	+	0.976139i	\(0.430325\pi\)
\(13\)	1.64449	+	1.56802i	0.456100	+	0.434891i	0.882975	−	0.469420i	\(-0.155537\pi\)
							−0.426875	+	0.904311i	\(0.640385\pi\)
\(17\)	−1.96782	+	0.379266i	−0.477266	+	0.0919854i	−0.422210	−	0.906498i	\(-0.638746\pi\)
							−0.0550553	+	0.998483i	\(0.517534\pi\)
\(19\)	4.00508	+	5.62434i	0.918827	+	1.29031i	0.956714	+	0.291031i	\(0.0939981\pi\)
							−0.0378863	+	0.999282i	\(0.512062\pi\)
\(23\)	0.175648	−	3.68731i	0.0366252	−	0.768858i	−0.903615	−	0.428346i	\(-0.859097\pi\)
							0.940240	−	0.340512i	\(-0.110600\pi\)
\(29\)	5.02878	−	8.71009i	0.933820	−	1.61742i	0.157096	−	0.987583i	\(-0.449787\pi\)
							0.776724	−	0.629841i	\(-0.216880\pi\)
\(31\)	4.18715	+	1.22946i	0.752034	+	0.220817i	0.635214	−	0.772336i	\(-0.280912\pi\)
							0.116820	+	0.993153i	\(0.462730\pi\)
\(37\)	−1.43917	+	2.49272i	−0.236599	+	0.409801i	−0.959736	−	0.280903i	\(-0.909366\pi\)
							0.723137	+	0.690704i	\(0.242699\pi\)
\(41\)	−0.273617	−	0.315770i	−0.0427317	−	0.0493151i	0.733981	−	0.679170i	\(-0.237660\pi\)
							−0.776713	+	0.629855i	\(0.783114\pi\)
\(43\)	1.89252	−	5.46807i	0.288606	−	0.833873i	−0.703748	−	0.710450i	\(-0.748492\pi\)
							0.992354	−	0.123423i	\(-0.0393873\pi\)
\(47\)	7.90635	−	4.07600i	1.15326	−	0.594546i	0.227917	−	0.973680i	\(-0.426808\pi\)
							0.925342	+	0.379134i	\(0.123778\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.y.a.4.13	✓	1320
9.7	even	3		603.2.ba.a.205.13	yes	1320
67.17	even	33		603.2.ba.a.553.13	yes	1320
603.151	even	33	inner	603.2.y.a.151.13	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.y.a.4.13	✓	1320	1.1	even	1	trivial
603.2.y.a.151.13	yes	1320	603.151	even	33	inner
603.2.ba.a.205.13	yes	1320	9.7	even	3
603.2.ba.a.553.13	yes	1320	67.17	even	33