Embedded newform 504.2.bt.a.11.9

• Label: 504.2.bt.a.11.9
• Level: $504$
• Weight: $2$
• Character: 504.11
• 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.bt (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			11.9
Character	\(\chi\)	\(=\)	504.11
Dual form			504.2.bt.a.275.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33451 - 0.468067i) q^{2} +(-1.72686 - 0.133969i) q^{3} +(1.56183 + 1.24928i) q^{4} +(-0.658616 + 1.14076i) q^{5} +(2.24181 + 0.987070i) q^{6} +(1.31090 + 2.29816i) q^{7} +(-1.49952 - 2.39821i) q^{8} +(2.96410 + 0.462692i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 2 q^{3} - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33451	−	0.468067i	−0.943640	−	0.330973i
							\(3\)	−1.72686	−	0.133969i	−0.997004	−	0.0773470i
\(5\)	−0.658616	+	1.14076i	−0.294542	+	0.510162i								−0.974878	−	0.222738i	\(-0.928500\pi\)
							0.680336	+	0.732900i	\(0.261834\pi\)
\(7\)	1.31090	+	2.29816i	0.495475	+	0.868622i
							\(11\)	−1.37368	+	0.793096i	−0.414181	+	0.239128i	−0.692585	−	0.721337i	\(-0.743528\pi\)
0.278404	+	0.960464i	\(0.410195\pi\)
\(13\)	1.42320	−	0.821683i	0.394724	−	0.227894i	−0.289481	−	0.957184i	\(-0.593483\pi\)
							0.684205	+	0.729290i	\(0.260149\pi\)
\(17\)	−1.53075	−	0.883781i	−0.371262	−	0.214348i	0.302748	−	0.953071i	\(-0.402096\pi\)
							−0.674010	+	0.738722i	\(0.735429\pi\)
\(19\)	−2.52349	−	4.37081i	−0.578928	−	1.00273i	−0.995603	−	0.0936767i	\(-0.970138\pi\)
							0.416675	−	0.909056i	\(-0.363195\pi\)
\(23\)	−2.79893	+	4.84789i	−0.583617	+	1.01085i	0.411429	+	0.911442i	\(0.365030\pi\)
							−0.995046	+	0.0994126i	\(0.968304\pi\)
\(29\)	0.477339	−	0.826775i	0.0886396	−	0.153528i	−0.818297	−	0.574796i	\(-0.805081\pi\)
							0.906936	+	0.421268i	\(0.138415\pi\)
\(31\)			2.16446i			0.388748i	0.980927	+	0.194374i	\(0.0622675\pi\)
							−0.980927	+	0.194374i	\(0.937732\pi\)
\(37\)	−6.65295	+	3.84108i	−1.09374	+	0.631470i	−0.934569	−	0.355782i	\(-0.884215\pi\)
							−0.159168	+	0.987251i	\(0.550881\pi\)
\(41\)	5.12334	−	2.95796i	0.800132	−	0.461956i	−0.0433856	−	0.999058i	\(-0.513814\pi\)
							0.843517	+	0.537102i	\(0.180481\pi\)
\(43\)	−4.98920	+	8.64154i	−0.760845	+	1.31782i	0.181570	+	0.983378i	\(0.441882\pi\)
							−0.942415	+	0.334445i	\(0.891451\pi\)
\(47\)	−9.19303			−1.34094			−0.670470	−	0.741936i	\(-0.733908\pi\)
							−0.670470	+	0.741936i	\(0.733908\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.bt.a.11.9	✓	184
7.2	even	3		504.2.cy.a.443.54	yes	184
8.3	odd	2	inner	504.2.bt.a.11.84	yes	184
9.5	odd	6		504.2.cy.a.347.23	yes	184
56.51	odd	6		504.2.cy.a.443.23	yes	184
63.23	odd	6	inner	504.2.bt.a.275.84	yes	184
72.59	even	6		504.2.cy.a.347.54	yes	184
504.275	even	6	inner	504.2.bt.a.275.9	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.bt.a.11.9	✓	184	1.1	even	1	trivial
504.2.bt.a.11.84	yes	184	8.3	odd	2	inner
504.2.bt.a.275.9	yes	184	504.275	even	6	inner
504.2.bt.a.275.84	yes	184	63.23	odd	6	inner
504.2.cy.a.347.23	yes	184	9.5	odd	6
504.2.cy.a.347.54	yes	184	72.59	even	6
504.2.cy.a.443.23	yes	184	56.51	odd	6
504.2.cy.a.443.54	yes	184	7.2	even	3