Embedded newform 504.2.w.a.205.7

• Label: 504.2.w.a.205.7
• 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			205.7
Character	\(\chi\)	\(=\)	504.205
Dual form			504.2.w.a.445.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40063 - 0.195508i) q^{2} +(1.23756 + 1.21180i) q^{3} +(1.92355 + 0.547670i) q^{4} -3.52202i q^{5} +(-1.49645 - 1.93924i) q^{6} +(2.62220 - 0.352244i) q^{7} +(-2.58712 - 1.14316i) q^{8} +(0.0630863 + 2.99934i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q + q^{2} + q^{4} - 2 q^{6} - 2 q^{7} - 8 q^{8} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40063	−	0.195508i	−0.990398	−	0.138245i
							\(3\)	1.23756	+	1.21180i	0.714503	+	0.699632i
\(5\)		−	3.52202i		−	1.57509i								−0.616254	−	0.787547i	\(-0.711351\pi\)
							0.616254	−	0.787547i	\(-0.288649\pi\)
\(7\)	2.62220	−	0.352244i	0.991098	−	0.133136i
							\(11\)		−	4.09072i		−	1.23340i	−0.787199	−	0.616699i	\(-0.788470\pi\)
0.787199	−	0.616699i	\(-0.211530\pi\)
\(13\)	1.60261	−	0.925269i	0.444485	−	0.256623i	−0.261013	−	0.965335i	\(-0.584057\pi\)
							0.705498	+	0.708712i	\(0.250723\pi\)
\(17\)	−0.221083	−	0.382927i	−0.0536205	−	0.0928734i	0.837969	−	0.545717i	\(-0.183743\pi\)
							−0.891590	+	0.452844i	\(0.850409\pi\)
\(19\)	−7.14962	−	4.12784i	−1.64024	−	0.946991i	−0.980748	−	0.195276i	\(-0.937440\pi\)
							−0.659488	−	0.751715i	\(-0.729227\pi\)
\(23\)	−2.69299			−0.561527			−0.280763	−	0.959777i	\(-0.590588\pi\)
							−0.280763	+	0.959777i	\(0.590588\pi\)
\(29\)	5.10311	+	2.94628i	0.947624	+	0.547111i	0.892342	−	0.451360i	\(-0.149061\pi\)
							0.0552819	+	0.998471i	\(0.482394\pi\)
\(31\)	2.24444	−	3.88748i	0.403113	−	0.698213i	−0.590987	−	0.806681i	\(-0.701261\pi\)
							0.994100	+	0.108469i	\(0.0345947\pi\)
\(37\)	−0.903318	−	0.521531i	−0.148505	−	0.0857392i	0.423907	−	0.905706i	\(-0.360659\pi\)
							−0.572411	+	0.819967i	\(0.693992\pi\)
\(41\)	4.58739	+	7.94559i	0.716430	+	1.24089i	0.962405	+	0.271617i	\(0.0875584\pi\)
							−0.245975	+	0.969276i	\(0.579108\pi\)
\(43\)	6.26129	+	3.61496i	0.954838	+	0.551276i	0.894580	−	0.446907i	\(-0.147475\pi\)
							0.0602573	+	0.998183i	\(0.480808\pi\)
\(47\)	1.97129	+	3.41438i	0.287543	+	0.498038i	0.973223	−	0.229865i	\(-0.0738284\pi\)
							−0.685680	+	0.727903i	\(0.740495\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.w.a.205.7	✓	184
7.4	even	3		504.2.cq.a.277.69	yes	184
8.5	even	2	inner	504.2.w.a.205.56	yes	184
9.4	even	3		504.2.cq.a.373.69	yes	184
56.53	even	6		504.2.cq.a.277.70	yes	184
63.4	even	3	inner	504.2.w.a.445.56	yes	184
72.13	even	6		504.2.cq.a.373.70	yes	184
504.445	even	6	inner	504.2.w.a.445.7	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.w.a.205.7	✓	184	1.1	even	1	trivial
504.2.w.a.205.56	yes	184	8.5	even	2	inner
504.2.w.a.445.7	yes	184	504.445	even	6	inner
504.2.w.a.445.56	yes	184	63.4	even	3	inner
504.2.cq.a.277.69	yes	184	7.4	even	3
504.2.cq.a.277.70	yes	184	56.53	even	6
504.2.cq.a.373.69	yes	184	9.4	even	3
504.2.cq.a.373.70	yes	184	72.13	even	6