Embedded newform 507.2.m.a.40.9

• Label: 507.2.m.a.40.9
• Level: $507$
• Weight: $2$
• Character: 507.40
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $180$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.m (of order \(13\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(180\)
Relative dimension:	\(15\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label			40.9
Character	\(\chi\)	\(=\)	507.40
Dual form			507.2.m.a.469.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.00839886 - 0.00207013i) q^{2} +(-0.120537 + 0.992709i) q^{3} +(-1.77085 - 0.929412i) q^{4} +(0.646404 - 0.936478i) q^{5} +(0.00306741 - 0.00808810i) q^{6} +(2.26833 + 2.00956i) q^{7} +(0.0258987 + 0.0229442i) q^{8} +(-0.970942 - 0.239316i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 180 q - q^{2} + 15 q^{3} - 15 q^{4} - 2 q^{5} + q^{6} + 4 q^{7} + 3 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{13}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.00839886	−	0.00207013i	−0.00593889	−	0.00146381i	0.236345	−	0.971669i	\(-0.424050\pi\)
							−0.242284	+	0.970205i	\(0.577897\pi\)
\(3\)	−0.120537	+	0.992709i	−0.0695919	+	0.573141i
							\(5\)	0.646404	−	0.936478i	0.289081	−	0.418806i	−0.651357	−	0.758771i	\(-0.725800\pi\)
0.940438	+	0.339966i	\(0.110415\pi\)
\(7\)	2.26833	+	2.00956i	0.857348	+	0.759544i	0.972278	−	0.233830i	\(-0.0751258\pi\)
							−0.114930	+	0.993374i	\(0.536664\pi\)
\(11\)	1.24237	−	0.306218i	0.374590	−	0.0923281i	−0.0475261	−	0.998870i	\(-0.515134\pi\)
							0.422116	+	0.906542i	\(0.361288\pi\)
\(13\)	−1.43459	−	3.30786i	−0.397885	−	0.917435i
							\(17\)	0.224663	+	0.199034i	0.0544888	+	0.0482728i	0.689928	−	0.723878i	\(-0.257642\pi\)
−0.635439	+	0.772151i	\(0.719181\pi\)
\(19\)	5.75144			1.31947			0.659736	−	0.751498i	\(-0.270668\pi\)
							0.659736	+	0.751498i	\(0.270668\pi\)
\(23\)	3.25661			0.679051			0.339526	−	0.940597i	\(-0.389733\pi\)
							0.339526	+	0.940597i	\(0.389733\pi\)
\(29\)	7.91325	+	1.95044i	1.46945	+	0.362188i	0.891178	−	0.453654i	\(-0.149880\pi\)
							0.578275	+	0.815842i	\(0.303726\pi\)
\(31\)	0.00169805	−	0.00447740i	0.000304980	−	0.000804165i	−0.934864	−	0.355007i	\(-0.884479\pi\)
							0.935169	+	0.354203i	\(0.115248\pi\)
\(37\)	−0.732724	+	1.93204i	−0.120459	+	0.317625i	−0.981660	−	0.190640i	\(-0.938944\pi\)
							0.861201	+	0.508265i	\(0.169713\pi\)
\(41\)	1.34365	−	11.0659i	0.209843	−	1.72821i	−0.383743	−	0.923440i	\(-0.625365\pi\)
							0.593585	−	0.804771i	\(-0.297712\pi\)
\(43\)	2.10600	+	5.55306i	0.321162	+	0.846834i	0.994232	+	0.107255i	\(0.0342061\pi\)
							−0.673070	+	0.739579i	\(0.735025\pi\)
\(47\)	6.41912	−	3.36901i	0.936324	−	0.491421i	0.0737082	−	0.997280i	\(-0.476517\pi\)
							0.862616	+	0.505859i	\(0.168824\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.m.a.40.9	✓	180
169.131	even	13	inner	507.2.m.a.469.9	yes	180

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.m.a.40.9	✓	180	1.1	even	1	trivial
507.2.m.a.469.9	yes	180	169.131	even	13	inner