Embedded newform 507.2.x.b.2.18

• Label: 507.2.x.b.2.18
• Level: $507$
• Weight: $2$
• Character: 507.2
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $2784$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.x (of order \(156\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(2784\)
Relative dimension:	\(58\) over \(\Q(\zeta_{156})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{156}]$

Embedding invariants

Embedding label			2.18
Character	\(\chi\)	\(=\)	507.2
Dual form			507.2.x.b.254.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0282490 + 1.40255i) q^{2} +(-1.72318 - 0.175026i) q^{3} +(0.0320220 + 0.00129044i) q^{4} +(-0.476786 + 2.60174i) q^{5} +(0.294162 - 2.41191i) q^{6} +(1.45655 - 2.20382i) q^{7} +(-0.172117 + 2.84543i) q^{8} +(2.93873 + 0.603205i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2784 q - 50 q^{3} - 92 q^{4} - 50 q^{6} - 100 q^{7} - 56 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{156}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0282490	+	1.40255i	−0.0199751	+	0.991754i	0.858172	+	0.513363i	\(0.171601\pi\)
							−0.878147	+	0.478392i	\(0.841220\pi\)
\(3\)	−1.72318	−	0.175026i	−0.994881	−	0.101051i
							\(5\)	−0.476786	+	2.60174i	−0.213225	+	1.16353i	0.686518	+	0.727113i	\(0.259138\pi\)
−0.899743	+	0.436419i	\(0.856246\pi\)
\(7\)	1.45655	−	2.20382i	0.550523	−	0.832964i	−0.447598	−	0.894235i	\(-0.647721\pi\)
							0.998121	+	0.0612710i	\(0.0195154\pi\)
\(11\)	0.631824	+	0.348011i	0.190502	+	0.104929i	0.574849	−	0.818259i	\(-0.305061\pi\)
							−0.384347	+	0.923189i	\(0.625573\pi\)
\(13\)	−1.24826	+	3.38258i	−0.346204	+	0.938159i
							\(17\)	0.501268	+	2.45537i	0.121575	+	0.595515i	0.994238	+	0.107192i	\(0.0341861\pi\)
−0.872663	+	0.488323i	\(0.837609\pi\)
\(19\)	0.0321338	+	0.119925i	0.00737201	+	0.0275127i	0.969514	−	0.245036i	\(-0.0787999\pi\)
							−0.962142	+	0.272549i	\(0.912133\pi\)
\(23\)	0.107219	+	0.185710i	0.0223568	+	0.0387231i	0.876987	−	0.480513i	\(-0.159550\pi\)
							−0.854631	+	0.519237i	\(0.826216\pi\)
\(29\)	−2.55719	+	2.45621i	−0.474857	+	0.456107i	−0.891538	−	0.452946i	\(-0.850373\pi\)
							0.416680	+	0.909053i	\(0.363193\pi\)
\(31\)	−7.38394	−	3.32324i	−1.32619	−	0.596872i	−0.381362	−	0.924426i	\(-0.624545\pi\)
							−0.944832	+	0.327554i	\(0.893776\pi\)
\(37\)	6.18499	−	0.624893i	1.01681	−	0.102732i	0.421996	−	0.906598i	\(-0.361330\pi\)
							0.594810	+	0.803866i	\(0.297227\pi\)
\(41\)	−11.5473	−	1.63868i	−1.80338	−	0.255919i	−0.843663	−	0.536873i	\(-0.819605\pi\)
							−0.959721	+	0.280955i	\(0.909349\pi\)
\(43\)	−1.83568	+	1.49879i	−0.279939	+	0.228563i	−0.762004	−	0.647572i	\(-0.775784\pi\)
							0.482065	+	0.876136i	\(0.339887\pi\)
\(47\)	−5.46175	+	1.70195i	−0.796679	+	0.248255i	−0.668710	−	0.743523i	\(-0.733153\pi\)
							−0.127968	+	0.991778i	\(0.540846\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.b.2.18	✓	2784
3.2	odd	2	inner	507.2.x.b.2.41	yes	2784
169.85	odd	156	inner	507.2.x.b.254.41	yes	2784
507.254	even	156	inner	507.2.x.b.254.18	yes	2784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.b.2.18	✓	2784	1.1	even	1	trivial
507.2.x.b.2.41	yes	2784	3.2	odd	2	inner
507.2.x.b.254.18	yes	2784	507.254	even	156	inner
507.2.x.b.254.41	yes	2784	169.85	odd	156	inner