Embedded newform 603.2.l.a.200.12

• Label: 603.2.l.a.200.12
• 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			200.12
Character	\(\chi\)	\(=\)	603.200
Dual form			603.2.l.a.401.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00027 + 1.73252i) q^{2} +(-1.71459 - 0.245331i) q^{3} +(-1.00108 - 1.73392i) q^{4} +(-1.52156 - 2.63541i) q^{5} +(2.14009 - 2.72516i) q^{6} +(3.85817 + 2.22752i) q^{7} +0.00431800 q^{8} +(2.87962 + 0.841285i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 66 q^{4} + 10 q^{6} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00027	+	1.73252i	−0.707298	+	1.22508i	0.258558	+	0.965996i	\(0.416753\pi\)
							−0.965856	+	0.259080i	\(0.916581\pi\)
\(3\)	−1.71459	−	0.245331i	−0.989918	−	0.141642i
							\(5\)	−1.52156	−	2.63541i	−0.680460	−	1.17859i	−0.974840	−	0.222904i	\(-0.928446\pi\)
0.294380	−	0.955688i	\(-0.404887\pi\)
\(7\)	3.85817	+	2.22752i	1.45825	+	0.841923i	0.998926	−	0.0463446i	\(-0.0147572\pi\)
							0.459327	+	0.888267i	\(0.348091\pi\)
\(11\)	0.709185	−	1.22834i	0.213827	−	0.370360i	−0.739082	−	0.673616i	\(-0.764740\pi\)
							0.952909	+	0.303256i	\(0.0980737\pi\)
\(13\)	−5.59099	+	3.22796i	−1.55066	+	0.895274i	−0.552573	+	0.833465i	\(0.686354\pi\)
							−0.998088	+	0.0618096i	\(0.980313\pi\)
\(17\)			1.07720i			0.261259i	0.991431	+	0.130630i	\(0.0416999\pi\)
							−0.991431	+	0.130630i	\(0.958300\pi\)
\(19\)	−1.95221			−0.447869			−0.223934	−	0.974604i	\(-0.571890\pi\)
							−0.223934	+	0.974604i	\(0.571890\pi\)
\(23\)	−1.10687	+	0.639051i	−0.230798	+	0.133251i	−0.610940	−	0.791677i	\(-0.709208\pi\)
							0.380142	+	0.924928i	\(0.375875\pi\)
\(29\)	−0.597111	−	0.344742i	−0.110881	−	0.0640170i	0.443534	−	0.896258i	\(-0.353725\pi\)
							−0.554415	+	0.832240i	\(0.687058\pi\)
\(31\)	3.08813	−	1.78293i	0.554644	−	0.320224i	−0.196349	−	0.980534i	\(-0.562908\pi\)
							0.750993	+	0.660310i	\(0.229575\pi\)
\(37\)	−8.82360			−1.45059			−0.725295	−	0.688438i	\(-0.758297\pi\)
							−0.725295	+	0.688438i	\(0.758297\pi\)
\(41\)	−5.55545	−	9.62233i	−0.867616	−	1.50275i	−0.864426	−	0.502760i	\(-0.832318\pi\)
							−0.00318969	−	0.999995i	\(-0.501015\pi\)
\(43\)	−0.111123	−	0.0641570i	−0.0169461	−	0.00978386i	0.491503	−	0.870876i	\(-0.336448\pi\)
							−0.508449	+	0.861092i	\(0.669781\pi\)
\(47\)	−5.08344	−	2.93492i	−0.741495	−	0.428102i	0.0811175	−	0.996705i	\(-0.474151\pi\)
							−0.822613	+	0.568602i	\(0.807484\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.l.a.200.12	✓	132
9.5	odd	6	inner	603.2.l.a.401.55	yes	132
67.66	odd	2	inner	603.2.l.a.200.55	yes	132
603.401	even	6	inner	603.2.l.a.401.12	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.l.a.200.12	✓	132	1.1	even	1	trivial
603.2.l.a.200.55	yes	132	67.66	odd	2	inner
603.2.l.a.401.12	yes	132	603.401	even	6	inner
603.2.l.a.401.55	yes	132	9.5	odd	6	inner