Embedded newform 504.2.cy.a.347.6

• Label: 504.2.cy.a.347.6
• 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			347.6
Character	\(\chi\)	\(=\)	504.347
Dual form			504.2.cy.a.443.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39790 + 0.214156i) q^{2} +(1.53386 - 0.804539i) q^{3} +(1.90827 - 0.598739i) q^{4} +2.18793 q^{5} +(-1.97189 + 1.45315i) q^{6} +(-2.62554 - 0.326387i) q^{7} +(-2.53936 + 1.24565i) q^{8} +(1.70544 - 2.46809i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 3 q^{2} - 2 q^{3} + q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39790	+	0.214156i	−0.988468	+	0.151431i
							\(3\)	1.53386	−	0.804539i	0.885573	−	0.464501i
\(5\)	2.18793			0.978472										0.489236	−	0.872152i	\(-0.337276\pi\)
							0.489236	+	0.872152i	\(0.337276\pi\)
\(7\)	−2.62554	−	0.326387i	−0.992362	−	0.123363i
							\(11\)		−	5.32343i		−	1.60507i	−0.596602	−	0.802537i	\(-0.703483\pi\)
0.596602	−	0.802537i	\(-0.296517\pi\)
\(13\)	−2.92370	−	1.68800i	−0.810888	−	0.468167i	0.0363760	−	0.999338i	\(-0.488419\pi\)
							−0.847264	+	0.531172i	\(0.821752\pi\)
\(17\)	2.32284	+	1.34110i	0.563373	+	0.325263i	0.754498	−	0.656302i	\(-0.227880\pi\)
							−0.191125	+	0.981566i	\(0.561214\pi\)
\(19\)	−2.09752	−	3.63301i	−0.481204	−	0.833470i	0.518563	−	0.855039i	\(-0.326467\pi\)
							−0.999767	+	0.0215693i	\(0.993134\pi\)
\(23\)	4.62259			0.963877			0.481938	−	0.876205i	\(-0.339933\pi\)
							0.481938	+	0.876205i	\(0.339933\pi\)
\(29\)	3.36913	+	5.83550i	0.625631	+	1.08363i	0.988418	+	0.151753i	\(0.0484919\pi\)
							−0.362787	+	0.931872i	\(0.618175\pi\)
\(31\)	0.0488117	−	0.0281815i	0.00876684	−	0.00506154i	−0.495610	−	0.868545i	\(-0.665056\pi\)
							0.504377	+	0.863484i	\(0.331722\pi\)
\(37\)	−4.10814	+	2.37184i	−0.675375	+	0.389928i	−0.798110	−	0.602512i	\(-0.794167\pi\)
							0.122735	+	0.992439i	\(0.460833\pi\)
\(41\)	4.69162	+	2.70871i	0.732708	+	0.423029i	0.819412	−	0.573205i	\(-0.194300\pi\)
							−0.0867040	+	0.996234i	\(0.527633\pi\)
\(43\)	3.27297	+	5.66896i	0.499124	+	0.864508i	0.999999	−	0.00101150i	\(-0.000321971\pi\)
							−0.500876	+	0.865519i	\(0.666989\pi\)
\(47\)	−2.66424	+	4.61460i	−0.388620	+	0.673109i	−0.992264	−	0.124145i	\(-0.960381\pi\)
							0.603645	+	0.797254i	\(0.293715\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.cy.a.347.6	yes	184
7.2	even	3		504.2.bt.a.275.67	yes	184
8.3	odd	2	inner	504.2.cy.a.347.38	yes	184
9.2	odd	6		504.2.bt.a.11.26	✓	184
56.51	odd	6		504.2.bt.a.275.26	yes	184
63.2	odd	6	inner	504.2.cy.a.443.38	yes	184
72.11	even	6		504.2.bt.a.11.67	yes	184
504.443	even	6	inner	504.2.cy.a.443.6	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.bt.a.11.26	✓	184	9.2	odd	6
504.2.bt.a.11.67	yes	184	72.11	even	6
504.2.bt.a.275.26	yes	184	56.51	odd	6
504.2.bt.a.275.67	yes	184	7.2	even	3
504.2.cy.a.347.6	yes	184	1.1	even	1	trivial
504.2.cy.a.347.38	yes	184	8.3	odd	2	inner
504.2.cy.a.443.6	yes	184	504.443	even	6	inner
504.2.cy.a.443.38	yes	184	63.2	odd	6	inner