Embedded newform 504.2.y.a.173.8

• Label: 504.2.y.a.173.8
• Level: $504$
• Weight: $2$
• Character: 504.173
• 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.y (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			173.8
Character	\(\chi\)	\(=\)	504.173
Dual form			504.2.y.a.437.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38113 + 0.304086i) q^{2} +(1.31500 + 1.12729i) q^{3} +(1.81506 - 0.839967i) q^{4} +3.74037i q^{5} +(-2.15898 - 1.15707i) q^{6} +(0.706907 - 2.54957i) q^{7} +(-2.25142 + 1.71204i) q^{8} +(0.458435 + 2.96477i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 3 q^{2} + q^{4} + 6 q^{6} - 2 q^{7} - 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38113	+	0.304086i	−0.976609	+	0.215021i
							\(3\)	1.31500	+	1.12729i	0.759214	+	0.650841i
\(5\)			3.74037i			1.67274i								0.548163	+	0.836372i	\(0.315327\pi\)
							−0.548163	+	0.836372i	\(0.684673\pi\)
\(7\)	0.706907	−	2.54957i	0.267186	−	0.963645i
							\(11\)	−4.90753			−1.47968			−0.739838	−	0.672785i	\(-0.765098\pi\)
−0.739838	+	0.672785i	\(0.765098\pi\)
\(13\)	2.14654	+	3.71792i	0.595344	+	1.03117i	0.993498	+	0.113848i	\(0.0363177\pi\)
							−0.398154	+	0.917319i	\(0.630349\pi\)
\(17\)	1.12692	+	1.95188i	0.273317	+	0.473399i	0.969709	−	0.244262i	\(-0.0785458\pi\)
							−0.696392	+	0.717662i	\(0.745212\pi\)
\(19\)	−1.73128	+	2.99867i	−0.397183	+	0.687941i	−0.993377	−	0.114899i	\(-0.963346\pi\)
							0.596194	+	0.802840i	\(0.296679\pi\)
\(23\)		−	5.07719i		−	1.05867i	−0.848413	−	0.529334i	\(-0.822442\pi\)
							0.848413	−	0.529334i	\(-0.177558\pi\)
\(29\)	3.39458	−	5.87959i	0.630358	−	1.09181i	−0.357120	−	0.934058i	\(-0.616241\pi\)
							0.987478	−	0.157754i	\(-0.0504253\pi\)
\(31\)	0.0893329	+	0.0515764i	0.0160447	+	0.00926339i	0.508001	−	0.861357i	\(-0.330385\pi\)
							−0.491956	+	0.870620i	\(0.663718\pi\)
\(37\)	2.98913	+	1.72577i	0.491410	+	0.283716i	0.725159	−	0.688581i	\(-0.241766\pi\)
							−0.233749	+	0.972297i	\(0.575100\pi\)
\(41\)	5.15402	+	8.92702i	0.804922	+	1.39417i	0.916344	+	0.400393i	\(0.131126\pi\)
							−0.111422	+	0.993773i	\(0.535540\pi\)
\(43\)	−1.43927	−	0.830965i	−0.219487	−	0.126721i	0.386226	−	0.922404i	\(-0.373779\pi\)
							−0.605713	+	0.795683i	\(0.707112\pi\)
\(47\)	0.464462	+	0.804471i	0.0677487	+	0.117344i	0.897910	−	0.440179i	\(-0.145085\pi\)
							−0.830161	+	0.557523i	\(0.811752\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.y.a.173.8	✓	184
7.3	odd	6		504.2.ca.a.101.55	yes	184
8.5	even	2	inner	504.2.y.a.173.26	yes	184
9.5	odd	6		504.2.ca.a.5.38	yes	184
56.45	odd	6		504.2.ca.a.101.38	yes	184
63.59	even	6	inner	504.2.y.a.437.26	yes	184
72.5	odd	6		504.2.ca.a.5.55	yes	184
504.437	even	6	inner	504.2.y.a.437.8	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.y.a.173.8	✓	184	1.1	even	1	trivial
504.2.y.a.173.26	yes	184	8.5	even	2	inner
504.2.y.a.437.8	yes	184	504.437	even	6	inner
504.2.y.a.437.26	yes	184	63.59	even	6	inner
504.2.ca.a.5.38	yes	184	9.5	odd	6
504.2.ca.a.5.55	yes	184	72.5	odd	6
504.2.ca.a.101.38	yes	184	56.45	odd	6
504.2.ca.a.101.55	yes	184	7.3	odd	6