Embedded newform 603.2.l.a.200.19

• Label: 603.2.l.a.200.19
• 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			200.19
Character	\(\chi\)	\(=\)	603.200
Dual form			603.2.l.a.401.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.746535 + 1.29304i) q^{2} +(-0.997353 + 1.41608i) q^{3} +(-0.114628 - 0.198542i) q^{4} +(1.00458 + 1.73999i) q^{5} +(-1.08649 - 2.34677i) q^{6} +(1.99677 + 1.15284i) q^{7} -2.64384 q^{8} +(-1.01057 - 2.82467i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 66 q^{4} + 10 q^{6} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.746535	+	1.29304i	−0.527880	+	0.914315i	0.471592	+	0.881817i	\(0.343680\pi\)
							−0.999472	+	0.0324978i	\(0.989654\pi\)
\(3\)	−0.997353	+	1.41608i	−0.575822	+	0.817575i
							\(5\)	1.00458	+	1.73999i	0.449262	+	0.778145i	0.998338	−	0.0576274i	\(-0.0183535\pi\)
−0.549076	+	0.835773i	\(0.685020\pi\)
\(7\)	1.99677	+	1.15284i	0.754710	+	0.435732i	0.827393	−	0.561623i	\(-0.189823\pi\)
							−0.0726833	+	0.997355i	\(0.523156\pi\)
\(11\)	0.289154	−	0.500829i	0.0871832	−	0.151006i	−0.819136	−	0.573599i	\(-0.805547\pi\)
							0.906319	+	0.422593i	\(0.138880\pi\)
\(13\)	−3.55503	+	2.05250i	−0.985987	+	0.569260i	−0.904072	−	0.427379i	\(-0.859437\pi\)
							−0.0819150	+	0.996639i	\(0.526104\pi\)
\(17\)			3.61168i			0.875961i	0.898984	+	0.437980i	\(0.144306\pi\)
							−0.898984	+	0.437980i	\(0.855694\pi\)
\(19\)	−0.193624			−0.0444204			−0.0222102	−	0.999753i	\(-0.507070\pi\)
							−0.0222102	+	0.999753i	\(0.507070\pi\)
\(23\)	−7.46137	+	4.30783i	−1.55580	+	0.898244i	−0.558153	+	0.829738i	\(0.688490\pi\)
							−0.997651	+	0.0685062i	\(0.978177\pi\)
\(29\)	−1.06231	−	0.613322i	−0.197265	−	0.113891i	0.398114	−	0.917336i	\(-0.369665\pi\)
							−0.595379	+	0.803445i	\(0.702998\pi\)
\(31\)	1.39004	−	0.802539i	0.249658	−	0.144140i	−0.369950	−	0.929052i	\(-0.620625\pi\)
							0.619608	+	0.784912i	\(0.287292\pi\)
\(37\)	9.93300			1.63298			0.816488	−	0.577363i	\(-0.195918\pi\)
							0.816488	+	0.577363i	\(0.195918\pi\)
\(41\)	2.39023	+	4.14000i	0.373291	+	0.646559i	0.990070	−	0.140577i	\(-0.0448959\pi\)
							−0.616779	+	0.787137i	\(0.711563\pi\)
\(43\)	−5.15784	−	2.97788i	−0.786563	−	0.454122i	0.0521882	−	0.998637i	\(-0.483380\pi\)
							−0.838751	+	0.544515i	\(0.816714\pi\)
\(47\)	−2.46116	−	1.42095i	−0.358998	−	0.207267i	0.309643	−	0.950853i	\(-0.399790\pi\)
							−0.668641	+	0.743585i	\(0.733124\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.l.a.200.19	✓	132
9.5	odd	6	inner	603.2.l.a.401.48	yes	132
67.66	odd	2	inner	603.2.l.a.200.48	yes	132
603.401	even	6	inner	603.2.l.a.401.19	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
603.2.l.a.200.19	✓	132	1.1	even	1	trivial
603.2.l.a.200.48	yes	132	67.66	odd	2	inner
603.2.l.a.401.19	yes	132	603.401	even	6	inner
603.2.l.a.401.48	yes	132	9.5	odd	6	inner