Embedded newform 603.2.l.a.200.20

• Label: 603.2.l.a.200.20
• 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			200.20
Character	\(\chi\)	\(=\)	603.200
Dual form			603.2.l.a.401.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.716678 + 1.24132i) q^{2} +(0.0154551 - 1.73198i) q^{3} +(-0.0272550 - 0.0472070i) q^{4} +(-2.10023 - 3.63771i) q^{5} +(2.13887 + 1.26046i) q^{6} +(0.623932 + 0.360227i) q^{7} -2.78858 q^{8} +(-2.99952 - 0.0535358i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 66 q^{4} + 10 q^{6} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.716678	+	1.24132i	−0.506768	+	0.877748i	0.493201	+	0.869915i	\(0.335827\pi\)
							−0.999969	+	0.00783265i	\(0.997507\pi\)
\(3\)	0.0154551	−	1.73198i	0.00892299	−	0.999960i
							\(5\)	−2.10023	−	3.63771i	−0.939254	−	1.62684i	−0.766867	−	0.641806i	\(-0.778186\pi\)
−0.172386	−	0.985029i	\(-0.555148\pi\)
\(7\)	0.623932	+	0.360227i	0.235824	+	0.136153i	0.613256	−	0.789884i	\(-0.289859\pi\)
							−0.377432	+	0.926037i	\(0.623193\pi\)
\(11\)	−2.31102	+	4.00280i	−0.696798	+	1.20689i	0.272773	+	0.962078i	\(0.412059\pi\)
							−0.969571	+	0.244811i	\(0.921274\pi\)
\(13\)	2.54791	−	1.47104i	0.706664	−	0.407993i	−0.103160	−	0.994665i	\(-0.532895\pi\)
							0.809825	+	0.586672i	\(0.199562\pi\)
\(17\)		−	1.78021i		−	0.431765i	−0.976419	−	0.215882i	\(-0.930737\pi\)
							0.976419	−	0.215882i	\(-0.0692627\pi\)
\(19\)	1.44122			0.330638			0.165319	−	0.986240i	\(-0.447135\pi\)
							0.165319	+	0.986240i	\(0.447135\pi\)
\(23\)	−6.61637	+	3.81997i	−1.37961	+	0.796518i	−0.992112	−	0.125353i	\(-0.959994\pi\)
							−0.387497	+	0.921871i	\(0.626660\pi\)
\(29\)	7.18245	+	4.14679i	1.33375	+	0.770039i	0.985872	−	0.167502i	\(-0.0535700\pi\)
							0.347875	+	0.937541i	\(0.386903\pi\)
\(31\)	−7.83707	+	4.52473i	−1.40758	+	0.812666i	−0.995154	−	0.0983259i	\(-0.968651\pi\)
							−0.412424	+	0.910992i	\(0.635318\pi\)
\(37\)	−6.79136			−1.11649			−0.558246	−	0.829675i	\(-0.688526\pi\)
							−0.558246	+	0.829675i	\(0.688526\pi\)
\(41\)	2.54725	+	4.41197i	0.397814	+	0.689033i	0.993456	−	0.114217i	\(-0.0364358\pi\)
							−0.595642	+	0.803250i	\(0.703102\pi\)
\(43\)	−1.72605	−	0.996535i	−0.263220	−	0.151970i	0.362583	−	0.931952i	\(-0.381895\pi\)
							−0.625803	+	0.779982i	\(0.715228\pi\)
\(47\)	−8.39328	−	4.84586i	−1.22429	−	0.706842i	−0.258457	−	0.966023i	\(-0.583214\pi\)
							−0.965829	+	0.259181i	\(0.916547\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.l.a.200.20	✓	132
9.5	odd	6	inner	603.2.l.a.401.47	yes	132
67.66	odd	2	inner	603.2.l.a.200.47	yes	132
603.401	even	6	inner	603.2.l.a.401.20	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.l.a.200.20	✓	132	1.1	even	1	trivial
603.2.l.a.200.47	yes	132	67.66	odd	2	inner
603.2.l.a.401.20	yes	132	603.401	even	6	inner
603.2.l.a.401.47	yes	132	9.5	odd	6	inner