Embedded newform 504.2.cq.a.277.3

• Label: 504.2.cq.a.277.3
• 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			277.3
Character	\(\chi\)	\(=\)	504.277
Dual form			504.2.cq.a.373.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40384 - 0.171013i) q^{2} +(-1.37821 + 1.04907i) q^{3} +(1.94151 + 0.480149i) q^{4} +(0.213745 + 0.123406i) q^{5} +(2.11418 - 1.23703i) q^{6} +(-2.18219 + 1.49601i) q^{7} +(-2.64345 - 1.00607i) q^{8} +(0.798909 - 2.89167i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 2 q^{2} - 2 q^{4} - 2 q^{6} - 2 q^{7} - 8 q^{8} - 2 q^{9}+O(q^{10}) \)

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								0.547033	−	0.837111i	\(-0.315757\pi\)
							−0.451443	+	0.892300i	\(0.649091\pi\)
\(7\)	−2.18219	+	1.49601i	−0.824792	+	0.565437i
							\(11\)	1.61868	−	0.934548i	0.488052	−	0.281777i	−0.235714	−	0.971822i	\(-0.575743\pi\)
0.723766	+	0.690046i	\(0.242410\pi\)
\(13\)	−4.72805	+	2.72974i	−1.31132	+	0.757093i	−0.982315	−	0.187234i	\(-0.940048\pi\)
							−0.329009	+	0.944327i	\(0.606715\pi\)
\(17\)	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	−0.955625	−	0.294586i	\(-0.904818\pi\)
							−0.222693	+	0.974889i	\(0.571485\pi\)
\(23\)	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	0.501151	−	0.865360i	\(-0.332910\pi\)
							−0.498848	+	0.866689i	\(0.666244\pi\)
\(31\)	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	\(-0.328370\pi\)
							0.999878	−	0.0155929i	\(-0.00496358\pi\)
\(41\)	−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	−0.408987	−	0.912540i	\(-0.634118\pi\)
							−0.994777	+	0.102077i	\(0.967451\pi\)
\(47\)	−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.277.3	yes	184
7.2	even	3		504.2.w.a.205.58	✓	184
8.5	even	2	inner	504.2.cq.a.277.4	yes	184
9.4	even	3		504.2.w.a.445.65	yes	184
56.37	even	6		504.2.w.a.205.65	yes	184
63.58	even	3	inner	504.2.cq.a.373.3	yes	184
72.13	even	6		504.2.w.a.445.58	yes	184
504.373	even	6	inner	504.2.cq.a.373.4	yes	184

    

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