Embedded newform 603.2.y.a.4.18

• Label: 603.2.y.a.4.18
• 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			4.18
Character	\(\chi\)	\(=\)	603.4
Dual form			603.2.y.a.151.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.608303 - 1.33200i) q^{2} +(1.29661 + 1.14839i) q^{3} +(-0.0944641 + 0.109017i) q^{4} +(-2.16014 - 0.864788i) q^{5} +(0.740915 - 2.42565i) q^{6} +(5.10130 + 0.487115i) q^{7} +(-2.60735 - 0.765586i) q^{8} +(0.362415 + 2.97803i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 5 q^{2} - 19 q^{3} - 137 q^{4} - 14 q^{5} - 16 q^{6} - 11 q^{7} - 20 q^{8} - 15 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.608303	−	1.33200i	−0.430135	−	0.941865i	−0.993305	−	0.115523i	\(-0.963145\pi\)
							0.563170	−	0.826341i	\(-0.309582\pi\)
\(3\)	1.29661	+	1.14839i	0.748600	+	0.663022i
							\(5\)	−2.16014	−	0.864788i	−0.966042	−	0.386745i	−0.165625	−	0.986189i	\(-0.552964\pi\)
−0.800417	+	0.599444i	\(0.795388\pi\)
\(7\)	5.10130	+	0.487115i	1.92811	+	0.184112i	0.987747	−	0.156063i	\(-0.0498804\pi\)
							0.940362	+	0.340175i	\(0.110486\pi\)
\(11\)	3.81840	+	1.52866i	1.15129	+	0.460907i	0.867234	−	0.497901i	\(-0.165896\pi\)
							0.284056	+	0.958808i	\(0.408320\pi\)
\(13\)	−1.69675	−	1.61784i	−0.470592	−	0.448709i	0.417294	−	0.908772i	\(-0.362979\pi\)
							−0.887887	+	0.460062i	\(0.847827\pi\)
\(17\)	0.706177	−	0.136104i	0.171273	−	0.0330102i	−0.102894	−	0.994692i	\(-0.532810\pi\)
							0.274167	+	0.961682i	\(0.411598\pi\)
\(19\)	4.26077	+	5.98342i	0.977488	+	1.37269i	0.927452	+	0.373943i	\(0.121994\pi\)
							0.0500368	+	0.998747i	\(0.484066\pi\)
\(23\)	0.280301	−	5.88425i	0.0584469	−	1.22695i	−0.759055	−	0.651026i	\(-0.774339\pi\)
							0.817502	−	0.575925i	\(-0.195358\pi\)
\(29\)	2.18481	−	3.78421i	0.405710	−	0.702710i	−0.588694	−	0.808356i	\(-0.700358\pi\)
							0.994404	+	0.105646i	\(0.0336911\pi\)
\(31\)	−7.36895	−	2.16372i	−1.32350	−	0.388616i	−0.457747	−	0.889082i	\(-0.651344\pi\)
							−0.865756	+	0.500467i	\(0.833162\pi\)
\(37\)	5.18888	−	8.98741i	0.853047	−	1.47752i	−0.0253985	−	0.999677i	\(-0.508085\pi\)
							0.878445	−	0.477843i	\(-0.158581\pi\)
\(41\)	6.68986	+	7.72051i	1.04478	+	1.20574i	0.978136	+	0.207964i	\(0.0666838\pi\)
							0.0666444	+	0.997777i	\(0.478771\pi\)
\(43\)	0.228574	−	0.660420i	0.0348572	−	0.100713i	−0.926230	−	0.376960i	\(-0.876969\pi\)
							0.961087	+	0.276247i	\(0.0890907\pi\)
\(47\)	0.658737	−	0.339603i	0.0960867	−	0.0495361i	−0.409513	−	0.912304i	\(-0.634301\pi\)
							0.505600	+	0.862768i	\(0.331271\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.y.a.4.18	✓	1320
9.7	even	3		603.2.ba.a.205.18	yes	1320
67.17	even	33		603.2.ba.a.553.18	yes	1320
603.151	even	33	inner	603.2.y.a.151.18	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.y.a.4.18	✓	1320	1.1	even	1	trivial
603.2.y.a.151.18	yes	1320	603.151	even	33	inner
603.2.ba.a.205.18	yes	1320	9.7	even	3
603.2.ba.a.553.18	yes	1320	67.17	even	33