Embedded newform 603.2.l.a.200.9

• Label: 603.2.l.a.200.9
• 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			200.9
Character	\(\chi\)	\(=\)	603.200
Dual form			603.2.l.a.401.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16336 + 2.01499i) q^{2} +(-1.59193 + 0.682477i) q^{3} +(-1.70679 - 2.95625i) q^{4} +(0.719334 + 1.24592i) q^{5} +(0.476791 - 4.00168i) q^{6} +(1.79079 + 1.03391i) q^{7} +3.28900 q^{8} +(2.06845 - 2.17290i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 66 q^{4} + 10 q^{6} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16336	+	2.01499i	−0.822617	+	1.42481i	0.0811106	+	0.996705i	\(0.474153\pi\)
							−0.903727	+	0.428109i	\(0.859180\pi\)
\(3\)	−1.59193	+	0.682477i	−0.919098	+	0.394028i
							\(5\)	0.719334	+	1.24592i	0.321696	+	0.557194i	0.980838	−	0.194825i	\(-0.0624138\pi\)
−0.659142	+	0.752018i	\(0.729080\pi\)
\(7\)	1.79079	+	1.03391i	0.676856	+	0.390783i	0.798669	−	0.601770i	\(-0.205538\pi\)
							−0.121813	+	0.992553i	\(0.538871\pi\)
\(11\)	0.393539	−	0.681630i	0.118656	−	0.205519i	−0.800579	−	0.599227i	\(-0.795475\pi\)
							0.919235	+	0.393708i	\(0.128808\pi\)
\(13\)	4.96935	−	2.86905i	1.37825	−	0.795732i	0.386300	−	0.922373i	\(-0.373753\pi\)
							0.991949	+	0.126641i	\(0.0404196\pi\)
\(17\)			4.16350i			1.00980i	0.863179	+	0.504898i	\(0.168470\pi\)
							−0.863179	+	0.504898i	\(0.831530\pi\)
\(19\)	7.83603			1.79771			0.898855	−	0.438247i	\(-0.144401\pi\)
							0.898855	+	0.438247i	\(0.144401\pi\)
\(23\)	5.07255	−	2.92864i	1.05770	−	0.610663i	0.132905	−	0.991129i	\(-0.457569\pi\)
							0.924795	+	0.380465i	\(0.124236\pi\)
\(29\)	−0.235138	−	0.135757i	−0.0436641	−	0.0252095i	0.478009	−	0.878355i	\(-0.341359\pi\)
							−0.521673	+	0.853145i	\(0.674692\pi\)
\(31\)	−3.64760	+	2.10594i	−0.655128	+	0.378238i	−0.790418	−	0.612568i	\(-0.790137\pi\)
							0.135290	+	0.990806i	\(0.456803\pi\)
\(37\)	−5.57288			−0.916177			−0.458088	−	0.888907i	\(-0.651466\pi\)
							−0.458088	+	0.888907i	\(0.651466\pi\)
\(41\)	−3.76167	−	6.51540i	−0.587474	−	1.01753i	−0.994562	−	0.104146i	\(-0.966789\pi\)
							0.407088	−	0.913389i	\(-0.366544\pi\)
\(43\)	0.641037	+	0.370103i	0.0977572	+	0.0564402i	0.548082	−	0.836425i	\(-0.315358\pi\)
							−0.450324	+	0.892865i	\(0.648692\pi\)
\(47\)	−1.85069	−	1.06850i	−0.269951	−	0.155856i	0.358914	−	0.933371i	\(-0.383147\pi\)
							−0.628865	+	0.777514i	\(0.716480\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.l.a.200.9	✓	132
9.5	odd	6	inner	603.2.l.a.401.58	yes	132
67.66	odd	2	inner	603.2.l.a.200.58	yes	132
603.401	even	6	inner	603.2.l.a.401.9	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.l.a.200.9	✓	132	1.1	even	1	trivial
603.2.l.a.200.58	yes	132	67.66	odd	2	inner
603.2.l.a.401.9	yes	132	603.401	even	6	inner
603.2.l.a.401.58	yes	132	9.5	odd	6	inner