Embedded newform 504.2.cq.a.277.4

• Label: 504.2.cq.a.277.4
• Level: $504$
• Weight: $2$
• Character: 504.277
• Analytic conductor: $4.024$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.02446026187\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(92\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			277.4
Character	\(\chi\)	\(=\)	504.277
Dual form			504.2.cq.a.373.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40384 + 0.171013i) q^{2} +(1.37821 - 1.04907i) q^{3} +(1.94151 - 0.480149i) q^{4} +(-0.213745 - 0.123406i) q^{5} +(-1.75537 + 1.70841i) q^{6} +(-2.18219 + 1.49601i) q^{7} +(-2.64345 + 1.00607i) q^{8} +(0.798909 - 2.89167i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 2 q^{2} - 2 q^{4} - 2 q^{6} - 2 q^{7} - 8 q^{8} - 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40384	+	0.171013i	−0.992662	+	0.120925i
							\(3\)	1.37821	−	1.04907i	0.795708	−	0.605680i
\(5\)	−0.213745	−	0.123406i	−0.0955898	−	0.0551888i								0.451443	−	0.892300i	\(-0.350909\pi\)
							−0.547033	+	0.837111i	\(0.684243\pi\)
\(7\)	−2.18219	+	1.49601i	−0.824792	+	0.565437i
							\(11\)	−1.61868	+	0.934548i	−0.488052	+	0.281777i	−0.723766	−	0.690046i	\(-0.757590\pi\)
0.235714	+	0.971822i	\(0.424257\pi\)
\(13\)	4.72805	−	2.72974i	1.31132	−	0.757093i	0.329009	−	0.944327i	\(-0.393285\pi\)
							0.982315	+	0.187234i	\(0.0599521\pi\)
\(17\)	2.66475	−	4.61548i	0.646297	−	1.11942i	−0.337704	−	0.941253i	\(-0.609650\pi\)
							0.984000	−	0.178166i	\(-0.0570165\pi\)
\(19\)	5.13617	−	2.96537i	1.17832	−	0.680302i	0.222693	−	0.974889i	\(-0.428515\pi\)
							0.955625	+	0.294586i	\(0.0951818\pi\)
\(23\)	0.586230	−	1.01538i	0.122237	−	0.211721i	−0.798412	−	0.602111i	\(-0.794326\pi\)
							0.920650	+	0.390390i	\(0.127660\pi\)
\(29\)	−0.0123976	−	0.00715774i	−0.00230217	−	0.00132916i	0.498848	−	0.866689i	\(-0.333756\pi\)
							−0.501151	+	0.865360i	\(0.667090\pi\)
\(31\)	6.29225			1.13012			0.565060	−	0.825050i	\(-0.308853\pi\)
							0.565060	+	0.825050i	\(0.308853\pi\)
\(37\)	−9.20518	+	5.31461i	−1.51332	+	0.873717i	−0.513443	+	0.858124i	\(0.671630\pi\)
							−0.999878	+	0.0155929i	\(0.995036\pi\)
\(41\)	−5.17669	−	8.96629i	−0.808463	−	1.40030i	−0.913928	−	0.405877i	\(-0.866966\pi\)
							0.105464	−	0.994423i	\(-0.466367\pi\)
\(43\)	9.20509	+	5.31456i	1.40376	+	0.810463i	0.994777	−	0.102077i	\(-0.0325487\pi\)
							0.408987	+	0.912540i	\(0.365882\pi\)
\(47\)	−9.22012			−1.34489			−0.672446	−	0.740146i	\(-0.734756\pi\)
							−0.672446	+	0.740146i	\(0.734756\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.cq.a.277.4	yes	184
7.2	even	3		504.2.w.a.205.65	yes	184
8.5	even	2	inner	504.2.cq.a.277.3	yes	184
9.4	even	3		504.2.w.a.445.58	yes	184
56.37	even	6		504.2.w.a.205.58	✓	184
63.58	even	3	inner	504.2.cq.a.373.4	yes	184
72.13	even	6		504.2.w.a.445.65	yes	184
504.373	even	6	inner	504.2.cq.a.373.3	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.w.a.205.58	✓	184	56.37	even	6
504.2.w.a.205.65	yes	184	7.2	even	3
504.2.w.a.445.58	yes	184	9.4	even	3
504.2.w.a.445.65	yes	184	72.13	even	6
504.2.cq.a.277.3	yes	184	8.5	even	2	inner
504.2.cq.a.277.4	yes	184	1.1	even	1	trivial
504.2.cq.a.373.3	yes	184	504.373	even	6	inner
504.2.cq.a.373.4	yes	184	63.58	even	3	inner