Embedded newform 504.2.cy.a.347.5

• Label: 504.2.cy.a.347.5
• Level: $504$
• Weight: $2$
• Character: 504.347
• 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.cy (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			347.5
Character	\(\chi\)	\(=\)	504.347
Dual form			504.2.cy.a.443.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40410 - 0.168823i) q^{2} +(1.00207 + 1.41275i) q^{3} +(1.94300 + 0.474089i) q^{4} +3.02374 q^{5} +(-1.16851 - 2.15281i) q^{6} +(-2.60675 - 0.452601i) q^{7} +(-2.64813 - 0.993691i) q^{8} +(-0.991703 + 2.83135i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 3 q^{2} - 2 q^{3} + q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40410	−	0.168823i	−0.992849	−	0.119376i
							\(3\)	1.00207	+	1.41275i	0.578547	+	0.815649i
\(5\)	3.02374			1.35226										0.676128	−	0.736784i	\(-0.263657\pi\)
							0.676128	+	0.736784i	\(0.263657\pi\)
\(7\)	−2.60675	−	0.452601i	−0.985259	−	0.171067i
							\(11\)			3.40788i			1.02751i	0.857936	+	0.513757i	\(0.171747\pi\)
−0.857936	+	0.513757i	\(0.828253\pi\)
\(13\)	3.91055	+	2.25776i	1.08459	+	0.626189i	0.932131	−	0.362121i	\(-0.117947\pi\)
							0.152460	+	0.988310i	\(0.451281\pi\)
\(17\)	−6.56539	−	3.79053i	−1.59234	−	0.919338i	−0.992904	−	0.118915i	\(-0.962058\pi\)
							−0.599436	−	0.800423i	\(-0.704608\pi\)
\(19\)	3.27538	+	5.67312i	0.751423	+	1.30150i	0.947133	+	0.320841i	\(0.103966\pi\)
							−0.195710	+	0.980662i	\(0.562701\pi\)
\(23\)	3.03483			0.632807			0.316403	−	0.948625i	\(-0.397525\pi\)
							0.316403	+	0.948625i	\(0.397525\pi\)
\(29\)	−1.34279	−	2.32579i	−0.249350	−	0.431888i	0.713995	−	0.700150i	\(-0.246884\pi\)
							−0.963346	+	0.268263i	\(0.913550\pi\)
\(31\)	−0.391387	+	0.225967i	−0.0702952	+	0.0405849i	−0.534736	−	0.845019i	\(-0.679589\pi\)
							0.464440	+	0.885604i	\(0.346255\pi\)
\(37\)	7.61727	−	4.39783i	1.25227	−	0.723000i	0.280712	−	0.959792i	\(-0.409430\pi\)
							0.971560	+	0.236793i	\(0.0760962\pi\)
\(41\)	1.93865	+	1.11928i	0.302767	+	0.174802i	0.643685	−	0.765291i	\(-0.277405\pi\)
							−0.340919	+	0.940093i	\(0.610738\pi\)
\(43\)	1.82164	+	3.15518i	0.277798	+	0.481160i	0.970837	−	0.239740i	\(-0.0770622\pi\)
							−0.693039	+	0.720900i	\(0.743729\pi\)
\(47\)	1.19642	−	2.07227i	0.174516	−	0.302271i	−0.765478	−	0.643463i	\(-0.777497\pi\)
							0.939994	+	0.341192i	\(0.110831\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.cy.a.347.5	yes	184
7.2	even	3		504.2.bt.a.275.58	yes	184
8.3	odd	2	inner	504.2.cy.a.347.28	yes	184
9.2	odd	6		504.2.bt.a.11.35	✓	184
56.51	odd	6		504.2.bt.a.275.35	yes	184
63.2	odd	6	inner	504.2.cy.a.443.28	yes	184
72.11	even	6		504.2.bt.a.11.58	yes	184
504.443	even	6	inner	504.2.cy.a.443.5	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.bt.a.11.35	✓	184	9.2	odd	6
504.2.bt.a.11.58	yes	184	72.11	even	6
504.2.bt.a.275.35	yes	184	56.51	odd	6
504.2.bt.a.275.58	yes	184	7.2	even	3
504.2.cy.a.347.5	yes	184	1.1	even	1	trivial
504.2.cy.a.347.28	yes	184	8.3	odd	2	inner
504.2.cy.a.443.5	yes	184	504.443	even	6	inner
504.2.cy.a.443.28	yes	184	63.2	odd	6	inner