Embedded newform 504.2.cq.a.277.13

• Label: 504.2.cq.a.277.13
• 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			277.13
Character	\(\chi\)	\(=\)	504.277
Dual form			504.2.cq.a.373.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27676 - 0.608187i) q^{2} +(-1.55038 - 0.772217i) q^{3} +(1.26022 + 1.55302i) q^{4} +(2.14506 + 1.23845i) q^{5} +(1.50981 + 1.92886i) q^{6} +(-2.51672 - 0.816158i) q^{7} +(-0.664465 - 2.74927i) q^{8} +(1.80736 + 2.39446i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 2 q^{2} - 2 q^{4} - 2 q^{6} - 2 q^{7} - 8 q^{8} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27676	−	0.608187i	−0.902803	−	0.430053i
							\(3\)	−1.55038	−	0.772217i	−0.895113	−	0.445840i
\(5\)	2.14506	+	1.23845i	0.959300	+	0.553852i								0.895957	−	0.444140i	\(-0.146491\pi\)
							0.0633423	+	0.997992i	\(0.479824\pi\)
\(7\)	−2.51672	−	0.816158i	−0.951231	−	0.308479i
							\(11\)	0.934456	−	0.539508i	0.281749	−	0.162668i	−0.352466	−	0.935825i	\(-0.614657\pi\)
0.634215	+	0.773157i	\(0.281323\pi\)
\(13\)	1.48995	−	0.860221i	0.413237	−	0.238582i	−0.278943	−	0.960308i	\(-0.589984\pi\)
							0.692180	+	0.721725i	\(0.256651\pi\)
\(17\)	−3.35590	+	5.81260i	−0.813926	+	1.40976i	0.0961701	+	0.995365i	\(0.469341\pi\)
							−0.910096	+	0.414397i	\(0.863993\pi\)
\(19\)	2.03637	−	1.17570i	0.467175	−	0.269724i	−0.247881	−	0.968790i	\(-0.579734\pi\)
							0.715056	+	0.699067i	\(0.246401\pi\)
\(23\)	−4.27597	+	7.40619i	−0.891601	+	1.54430i	−0.0536448	+	0.998560i	\(0.517084\pi\)
							−0.837956	+	0.545738i	\(0.816249\pi\)
\(29\)	5.53580	+	3.19610i	1.02797	+	0.593501i	0.916403	−	0.400256i	\(-0.131079\pi\)
							0.111570	+	0.993757i	\(0.464412\pi\)
\(31\)	8.77458			1.57596			0.787980	−	0.615701i	\(-0.211127\pi\)
							0.787980	+	0.615701i	\(0.211127\pi\)
\(37\)	0.442028	−	0.255205i	0.0726689	−	0.0419554i	−0.463225	−	0.886241i	\(-0.653308\pi\)
							0.535894	+	0.844285i	\(0.319975\pi\)
\(41\)	−0.0556860	−	0.0964510i	−0.00869669	−	0.0150631i	0.861644	−	0.507513i	\(-0.169435\pi\)
							−0.870341	+	0.492450i	\(0.836102\pi\)
\(43\)	4.83529	+	2.79166i	0.737375	+	0.425724i	0.821114	−	0.570764i	\(-0.193353\pi\)
							−0.0837389	+	0.996488i	\(0.526686\pi\)
\(47\)	3.75253			0.547363			0.273681	−	0.961820i	\(-0.411759\pi\)
							0.273681	+	0.961820i	\(0.411759\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.cq.a.277.13	yes	184
7.2	even	3		504.2.w.a.205.48	✓	184
8.5	even	2	inner	504.2.cq.a.277.14	yes	184
9.4	even	3		504.2.w.a.445.75	yes	184
56.37	even	6		504.2.w.a.205.75	yes	184
63.58	even	3	inner	504.2.cq.a.373.13	yes	184
72.13	even	6		504.2.w.a.445.48	yes	184
504.373	even	6	inner	504.2.cq.a.373.14	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.w.a.205.48	✓	184	7.2	even	3
504.2.w.a.205.75	yes	184	56.37	even	6
504.2.w.a.445.48	yes	184	72.13	even	6
504.2.w.a.445.75	yes	184	9.4	even	3
504.2.cq.a.277.13	yes	184	1.1	even	1	trivial
504.2.cq.a.277.14	yes	184	8.5	even	2	inner
504.2.cq.a.373.13	yes	184	63.58	even	3	inner
504.2.cq.a.373.14	yes	184	504.373	even	6	inner