Embedded newform 603.2.y.a.4.11

• Label: 603.2.y.a.4.11
• Level: $603$
• Weight: $2$
• Character: 603.4
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $1320$
• CM: no
• 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			4.11
Character	\(\chi\)	\(=\)	603.4
Dual form			603.2.y.a.151.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.918977 - 2.01228i) q^{2} +(-0.842148 - 1.51353i) q^{3} +(-1.89503 + 2.18698i) q^{4} +(1.15998 + 0.464386i) q^{5} +(-2.27174 + 3.08554i) q^{6} +(-2.47192 - 0.236040i) q^{7} +(1.89714 + 0.557052i) q^{8} +(-1.58157 + 2.54924i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 5 q^{2} - 19 q^{3} - 137 q^{4} - 14 q^{5} - 16 q^{6} - 11 q^{7} - 20 q^{8} - 15 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.918977	−	2.01228i	−0.649815	−	1.42290i	−0.891716	−	0.452596i	\(-0.850498\pi\)
							0.241901	−	0.970301i	\(-0.422229\pi\)
\(3\)	−0.842148	−	1.51353i	−0.486215	−	0.873839i
							\(5\)	1.15998	+	0.464386i	0.518759	+	0.207680i	0.616241	−	0.787558i	\(-0.288655\pi\)
−0.0974823	+	0.995237i	\(0.531079\pi\)
\(7\)	−2.47192	−	0.236040i	−0.934298	−	0.0892146i	−0.383209	−	0.923661i	\(-0.625181\pi\)
							−0.551088	+	0.834447i	\(0.685787\pi\)
\(11\)	−1.28336	−	0.513781i	−0.386949	−	0.154911i	0.170027	−	0.985439i	\(-0.445614\pi\)
							−0.556976	+	0.830528i	\(0.688039\pi\)
\(13\)	−4.77296	−	4.55101i	−1.32378	−	1.26222i	−0.941010	−	0.338378i	\(-0.890122\pi\)
							−0.382770	−	0.923844i	\(-0.625030\pi\)
\(17\)	2.54096	−	0.489729i	0.616272	−	0.118777i	0.128445	−	0.991717i	\(-0.459002\pi\)
							0.487828	+	0.872940i	\(0.337789\pi\)
\(19\)	3.95117	+	5.54864i	0.906460	+	1.27294i	0.961597	+	0.274466i	\(0.0885011\pi\)
							−0.0551368	+	0.998479i	\(0.517559\pi\)
\(23\)	−0.343420	+	7.20927i	−0.0716080	+	1.50324i	0.622076	+	0.782957i	\(0.286290\pi\)
							−0.693684	+	0.720280i	\(0.744014\pi\)
\(29\)	−2.43178	+	4.21196i	−0.451569	+	0.782141i	−0.998484	−	0.0550475i	\(-0.982469\pi\)
							0.546914	+	0.837189i	\(0.315802\pi\)
\(31\)	3.23644	+	0.950306i	0.581283	+	0.170680i	0.559138	−	0.829075i	\(-0.311132\pi\)
							0.0221449	+	0.999755i	\(0.492950\pi\)
\(37\)	1.39434	−	2.41507i	0.229229	−	0.397036i	−0.728351	−	0.685204i	\(-0.759713\pi\)
							0.957580	+	0.288168i	\(0.0930463\pi\)
\(41\)	2.33780	+	2.69796i	0.365103	+	0.421351i	0.908343	−	0.418227i	\(-0.137348\pi\)
							−0.543240	+	0.839577i	\(0.682803\pi\)
\(43\)	3.35533	−	9.69458i	0.511683	−	1.47841i	−0.332888	−	0.942966i	\(-0.608023\pi\)
							0.844571	−	0.535444i	\(-0.179856\pi\)
\(47\)	−10.0896	+	5.20157i	−1.47172	+	0.758726i	−0.992874	−	0.119170i	\(-0.961977\pi\)
							−0.478850	+	0.877897i	\(0.658946\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.y.a.4.11	✓	1320
9.7	even	3		603.2.ba.a.205.11	yes	1320
67.17	even	33		603.2.ba.a.553.11	yes	1320
603.151	even	33	inner	603.2.y.a.151.11	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.y.a.4.11	✓	1320	1.1	even	1	trivial
603.2.y.a.151.11	yes	1320	603.151	even	33	inner
603.2.ba.a.205.11	yes	1320	9.7	even	3
603.2.ba.a.553.11	yes	1320	67.17	even	33