Embedded newform 603.2.l.a.200.7

• Label: 603.2.l.a.200.7
• Level: $603$
• Weight: $2$
• Character: 603.200
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $132$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.81497924188\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(66\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			200.7
Character	\(\chi\)	\(=\)	603.200
Dual form			603.2.l.a.401.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18642 + 2.05493i) q^{2} +(0.117735 - 1.72804i) q^{3} +(-1.81517 - 3.14397i) q^{4} +(0.408528 + 0.707591i) q^{5} +(3.41134 + 2.29212i) q^{6} +(4.18823 + 2.41807i) q^{7} +3.86854 q^{8} +(-2.97228 - 0.406902i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 66 q^{4} + 10 q^{6} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(136\)	\(470\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18642	+	2.05493i	−0.838924	+	1.45306i	0.0518717	+	0.998654i	\(0.483481\pi\)
							−0.890795	+	0.454405i	\(0.849852\pi\)
\(3\)	0.117735	−	1.72804i	0.0679742	−	0.997687i
							\(5\)	0.408528	+	0.707591i	0.182699	+	0.316444i	0.942799	−	0.333362i	\(-0.108183\pi\)
−0.760100	+	0.649807i	\(0.774850\pi\)
\(7\)	4.18823	+	2.41807i	1.58300	+	0.913946i	0.994418	+	0.105510i	\(0.0336474\pi\)
							0.588583	+	0.808437i	\(0.299686\pi\)
\(11\)	0.881900	−	1.52750i	0.265903	−	0.460557i	−0.701897	−	0.712279i	\(-0.747663\pi\)
							0.967800	+	0.251721i	\(0.0809966\pi\)
\(13\)	2.13475	−	1.23250i	0.592073	−	0.341833i	−0.173844	−	0.984773i	\(-0.555619\pi\)
							0.765917	+	0.642940i	\(0.222286\pi\)
\(17\)		−	2.34117i		−	0.567818i	−0.958851	−	0.283909i	\(-0.908369\pi\)
							0.958851	−	0.283909i	\(-0.0916313\pi\)
\(19\)	−1.02477			−0.235099			−0.117550	−	0.993067i	\(-0.537504\pi\)
							−0.117550	+	0.993067i	\(0.537504\pi\)
\(23\)	−3.12005	+	1.80136i	−0.650576	+	0.375610i	−0.788677	−	0.614808i	\(-0.789234\pi\)
							0.138101	+	0.990418i	\(0.455900\pi\)
\(29\)	−1.03785	−	0.599200i	−0.192723	−	0.111269i	0.400534	−	0.916282i	\(-0.368825\pi\)
							−0.593257	+	0.805013i	\(0.702158\pi\)
\(31\)	1.36679	−	0.789118i	0.245483	−	0.141730i	−0.372211	−	0.928148i	\(-0.621400\pi\)
							0.617694	+	0.786418i	\(0.288067\pi\)
\(37\)	7.34466			1.20745			0.603727	−	0.797191i	\(-0.293682\pi\)
							0.603727	+	0.797191i	\(0.293682\pi\)
\(41\)	3.97263	+	6.88080i	0.620421	+	1.07460i	0.989407	+	0.145166i	\(0.0463715\pi\)
							−0.368987	+	0.929435i	\(0.620295\pi\)
\(43\)	8.20067	+	4.73466i	1.25059	+	0.722029i	0.971227	−	0.238157i	\(-0.0765432\pi\)
							0.279364	+	0.960185i	\(0.409877\pi\)
\(47\)	8.44193	+	4.87395i	1.23138	+	0.710939i	0.967318	−	0.253566i	\(-0.0816035\pi\)
							0.264065	+	0.964505i	\(0.414937\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.l.a.200.7	✓	132
9.5	odd	6	inner	603.2.l.a.401.60	yes	132
67.66	odd	2	inner	603.2.l.a.200.60	yes	132
603.401	even	6	inner	603.2.l.a.401.7	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.l.a.200.7	✓	132	1.1	even	1	trivial
603.2.l.a.200.60	yes	132	67.66	odd	2	inner
603.2.l.a.401.7	yes	132	603.401	even	6	inner
603.2.l.a.401.60	yes	132	9.5	odd	6	inner