Embedded newform 603.2.l.a.200.5

• Label: 603.2.l.a.200.5
• Level: $603$
• Weight: $2$
• Character: 603.200
• Analytic conductor: $4.815$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• 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			200.5
Character	\(\chi\)	\(=\)	603.200
Dual form			603.2.l.a.401.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23668 + 2.14199i) q^{2} +(-1.34361 + 1.09302i) q^{3} +(-2.05875 - 3.56586i) q^{4} +(-1.63023 - 2.82365i) q^{5} +(-0.679631 - 4.22973i) q^{6} +(-2.34312 - 1.35280i) q^{7} +5.23732 q^{8} +(0.610593 - 2.93721i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 66 q^{4} + 10 q^{6} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23668	+	2.14199i	−0.874464	+	1.51462i	−0.0171309	+	0.999853i	\(0.505453\pi\)
							−0.857333	+	0.514762i	\(0.827880\pi\)
\(3\)	−1.34361	+	1.09302i	−0.775735	+	0.631058i
							\(5\)	−1.63023	−	2.82365i	−0.729063	−	1.26277i	−0.957280	−	0.289164i	\(-0.906623\pi\)
0.228217	−	0.973610i	\(-0.426711\pi\)
\(7\)	−2.34312	−	1.35280i	−0.885615	−	0.511310i	−0.0131090	−	0.999914i	\(-0.504173\pi\)
							−0.872506	+	0.488604i	\(0.837506\pi\)
\(11\)	−1.37897	+	2.38845i	−0.415776	+	0.720146i	−0.995510	−	0.0946605i	\(-0.969823\pi\)
							0.579733	+	0.814806i	\(0.303157\pi\)
\(13\)	2.01042	−	1.16072i	0.557591	−	0.321925i	−0.194587	−	0.980885i	\(-0.562337\pi\)
							0.752178	+	0.658960i	\(0.229003\pi\)
\(17\)			2.78333i			0.675057i	0.941315	+	0.337528i	\(0.109591\pi\)
							−0.941315	+	0.337528i	\(0.890409\pi\)
\(19\)	−6.42144			−1.47318			−0.736589	−	0.676340i	\(-0.763565\pi\)
							−0.736589	+	0.676340i	\(0.763565\pi\)
\(23\)	−3.99140	+	2.30444i	−0.832265	+	0.480508i	−0.854627	−	0.519242i	\(-0.826214\pi\)
							0.0223627	+	0.999750i	\(0.492881\pi\)
\(29\)	0.00781770	+	0.00451355i	0.00145171	+	0.000838145i	0.500726	−	0.865606i	\(-0.333067\pi\)
							−0.499274	+	0.866444i	\(0.666400\pi\)
\(31\)	6.69126	−	3.86320i	1.20179	−	0.693851i	0.240833	−	0.970567i	\(-0.422579\pi\)
							0.960952	+	0.276716i	\(0.0892460\pi\)
\(37\)	8.14751			1.33944			0.669721	−	0.742612i	\(-0.266414\pi\)
							0.669721	+	0.742612i	\(0.266414\pi\)
\(41\)	3.20104	+	5.54437i	0.499919	+	0.865885i	1.00000		9.33930e-5i	\(-2.97279e-5\pi\)
							−0.500081	+	0.865979i	\(0.666696\pi\)
\(43\)	−5.30678	−	3.06387i	−0.809276	−	0.467236i	0.0374285	−	0.999299i	\(-0.488083\pi\)
							−0.846704	+	0.532064i	\(0.821417\pi\)
\(47\)	6.37112	+	3.67837i	0.929323	+	0.536545i	0.886597	−	0.462542i	\(-0.153062\pi\)
							0.0427258	+	0.999087i	\(0.486396\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.l.a.200.5	✓	132
9.5	odd	6	inner	603.2.l.a.401.62	yes	132
67.66	odd	2	inner	603.2.l.a.200.62	yes	132
603.401	even	6	inner	603.2.l.a.401.5	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.l.a.200.5	✓	132	1.1	even	1	trivial
603.2.l.a.200.62	yes	132	67.66	odd	2	inner
603.2.l.a.401.5	yes	132	603.401	even	6	inner
603.2.l.a.401.62	yes	132	9.5	odd	6	inner