Embedded newform 504.2.cz.b.187.6

• Label: 504.2.cz.b.187.6
• Level: $504$
• Weight: $2$
• Character: 504.187
• Analytic conductor: $4.024$
• Analytic rank: $0$
• Dimension: $180$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.cz (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.02446026187\)
Analytic rank:	\(0\)
Dimension:	\(180\)
Relative dimension:	\(90\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			187.6
Character	\(\chi\)	\(=\)	504.187
Dual form			504.2.cz.b.283.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37793 + 0.318310i) q^{2} +(-1.42741 - 0.981074i) q^{3} +(1.79736 - 0.877216i) q^{4} +2.04097 q^{5} +(2.27915 + 0.897487i) q^{6} +(2.55041 - 0.703859i) q^{7} +(-2.19740 + 1.78086i) q^{8} +(1.07499 + 2.80079i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 180 q + 3 q^{2} + q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\)	\(73\)	\(127\)	\(253\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37793	+	0.318310i	−0.974340	+	0.225079i
							\(3\)	−1.42741	−	0.981074i	−0.824115	−	0.566423i
\(5\)	2.04097			0.912749										0.456375	−	0.889788i	\(-0.349148\pi\)
							0.456375	+	0.889788i	\(0.349148\pi\)
\(7\)	2.55041	−	0.703859i	0.963964	−	0.266034i
							\(11\)	3.42360			1.03226			0.516128	−	0.856512i	\(-0.327373\pi\)
0.516128	+	0.856512i	\(0.327373\pi\)
\(13\)	0.634701	+	1.09934i	0.176035	+	0.304901i	0.940519	−	0.339742i	\(-0.110340\pi\)
							−0.764484	+	0.644642i	\(0.777006\pi\)
\(17\)	−7.04041	+	4.06478i	−1.70755	+	0.985854i	−0.769973	+	0.638077i	\(0.779730\pi\)
							−0.937577	+	0.347777i	\(0.886937\pi\)
\(19\)	4.18816	+	2.41804i	0.960831	+	0.554736i	0.896429	−	0.443188i	\(-0.146153\pi\)
							0.0644021	+	0.997924i	\(0.479486\pi\)
\(23\)			0.714525i			0.148989i	0.997221	+	0.0744944i	\(0.0237343\pi\)
							−0.997221	+	0.0744944i	\(0.976266\pi\)
\(29\)	4.05736	+	2.34252i	0.753432	+	0.434994i	0.826933	−	0.562301i	\(-0.190084\pi\)
							−0.0735006	+	0.997295i	\(0.523417\pi\)
\(31\)	2.62731	−	4.55063i	0.471878	−	0.817317i	−0.527604	−	0.849490i	\(-0.676909\pi\)
							0.999482	+	0.0321732i	\(0.0102428\pi\)
\(37\)	2.21573	+	1.27925i	0.364264	+	0.210308i	0.670950	−	0.741503i	\(-0.265887\pi\)
							−0.306686	+	0.951811i	\(0.599220\pi\)
\(41\)	−2.09421	+	1.20910i	−0.327061	+	0.188829i	−0.654536	−	0.756031i	\(-0.727136\pi\)
							0.327474	+	0.944860i	\(0.393803\pi\)
\(43\)	3.16443	−	5.48096i	0.482572	−	0.835839i	−0.517228	−	0.855848i	\(-0.673036\pi\)
							0.999800	+	0.0200088i	\(0.00636944\pi\)
\(47\)	5.97905	+	10.3560i	0.872134	+	1.51058i	0.859785	+	0.510656i	\(0.170597\pi\)
							0.0123483	+	0.999924i	\(0.496069\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.cz.b.187.6	yes	180
7.3	odd	6		504.2.bf.b.115.57	✓	180
8.3	odd	2	inner	504.2.cz.b.187.66	yes	180
9.4	even	3		504.2.bf.b.355.57	yes	180
56.3	even	6		504.2.bf.b.115.58	yes	180
63.31	odd	6	inner	504.2.cz.b.283.66	yes	180
72.67	odd	6		504.2.bf.b.355.58	yes	180
504.283	even	6	inner	504.2.cz.b.283.6	yes	180

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.bf.b.115.57	✓	180	7.3	odd	6
504.2.bf.b.115.58	yes	180	56.3	even	6
504.2.bf.b.355.57	yes	180	9.4	even	3
504.2.bf.b.355.58	yes	180	72.67	odd	6
504.2.cz.b.187.6	yes	180	1.1	even	1	trivial
504.2.cz.b.187.66	yes	180	8.3	odd	2	inner
504.2.cz.b.283.6	yes	180	504.283	even	6	inner
504.2.cz.b.283.66	yes	180	63.31	odd	6	inner