Embedded newform 504.2.w.a.205.13

• Label: 504.2.w.a.205.13
• Level: $504$
• Weight: $2$
• Character: 504.205
• 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.w (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			205.13
Character	\(\chi\)	\(=\)	504.205
Dual form			504.2.w.a.445.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28717 - 0.585837i) q^{2} +(-1.65782 + 0.501625i) q^{3} +(1.31359 + 1.50814i) q^{4} +4.18515i q^{5} +(2.42776 + 0.325538i) q^{6} +(1.88927 - 1.85221i) q^{7} +(-0.807285 - 2.71077i) q^{8} +(2.49674 - 1.66321i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q + q^{2} + q^{4} - 2 q^{6} - 2 q^{7} - 8 q^{8} - 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28717	−	0.585837i	−0.910163	−	0.414249i
							\(3\)	−1.65782	+	0.501625i	−0.957144	+	0.289613i
\(5\)			4.18515i			1.87166i								0.352457	+	0.935828i	\(0.385346\pi\)
							−0.352457	+	0.935828i	\(0.614654\pi\)
\(7\)	1.88927	−	1.85221i	0.714076	−	0.700068i
							\(11\)		−	1.71793i		−	0.517975i	−0.965881	−	0.258988i	\(-0.916611\pi\)
0.965881	−	0.258988i	\(-0.0833889\pi\)
\(13\)	3.88784	−	2.24464i	1.07829	−	0.622552i	0.147857	−	0.989009i	\(-0.452763\pi\)
							0.930435	+	0.366457i	\(0.119429\pi\)
\(17\)	3.47739	+	6.02302i	0.843391	+	1.46080i	0.887011	+	0.461748i	\(0.152778\pi\)
							−0.0436195	+	0.999048i	\(0.513889\pi\)
\(19\)	−0.135957	−	0.0784950i	−0.0311908	−	0.0180080i	0.484324	−	0.874889i	\(-0.339066\pi\)
							−0.515514	+	0.856881i	\(0.672399\pi\)
\(23\)	5.22930			1.09038			0.545192	−	0.838311i	\(-0.316457\pi\)
							0.545192	+	0.838311i	\(0.316457\pi\)
\(29\)	−1.17927	−	0.680852i	−0.218985	−	0.126431i	0.386495	−	0.922291i	\(-0.373686\pi\)
							−0.605480	+	0.795860i	\(0.707019\pi\)
\(31\)	−1.57091	+	2.72089i	−0.282143	+	0.488686i	−0.971912	−	0.235343i	\(-0.924379\pi\)
							0.689769	+	0.724029i	\(0.257712\pi\)
\(37\)	2.27910	+	1.31584i	0.374682	+	0.216323i	0.675502	−	0.737358i	\(-0.263927\pi\)
							−0.300820	+	0.953681i	\(0.597260\pi\)
\(41\)	1.52330	+	2.63843i	0.237899	+	0.412053i	0.960111	−	0.279619i	\(-0.0902080\pi\)
							−0.722212	+	0.691671i	\(0.756875\pi\)
\(43\)	6.37569	+	3.68100i	0.972283	+	0.561348i	0.899931	−	0.436031i	\(-0.143616\pi\)
							0.0723515	+	0.997379i	\(0.476950\pi\)
\(47\)	−1.42046	−	2.46031i	−0.207196	−	0.358874i	0.743634	−	0.668587i	\(-0.233100\pi\)
							−0.950830	+	0.309713i	\(0.899767\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.w.a.205.13	✓	184
7.4	even	3		504.2.cq.a.277.73	yes	184
8.5	even	2	inner	504.2.w.a.205.50	yes	184
9.4	even	3		504.2.cq.a.373.73	yes	184
56.53	even	6		504.2.cq.a.277.74	yes	184
63.4	even	3	inner	504.2.w.a.445.50	yes	184
72.13	even	6		504.2.cq.a.373.74	yes	184
504.445	even	6	inner	504.2.w.a.445.13	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.w.a.205.13	✓	184	1.1	even	1	trivial
504.2.w.a.205.50	yes	184	8.5	even	2	inner
504.2.w.a.445.13	yes	184	504.445	even	6	inner
504.2.w.a.445.50	yes	184	63.4	even	3	inner
504.2.cq.a.277.73	yes	184	7.4	even	3
504.2.cq.a.277.74	yes	184	56.53	even	6
504.2.cq.a.373.73	yes	184	9.4	even	3
504.2.cq.a.373.74	yes	184	72.13	even	6