Embedded newform 507.2.x.b.2.20

• Label: 507.2.x.b.2.20
• 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			2.20
Character	\(\chi\)	\(=\)	507.2
Dual form			507.2.x.b.254.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0219978 + 1.09218i) q^{2} +(1.01159 + 1.40595i) q^{3} +(0.805998 + 0.0324806i) q^{4} +(0.345605 - 1.88591i) q^{5} +(-1.55781 + 1.07391i) q^{6} +(2.55979 - 3.87306i) q^{7} +(-0.185120 + 3.06041i) q^{8} +(-0.953385 + 2.84448i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2784 q - 50 q^{3} - 92 q^{4} - 50 q^{6} - 100 q^{7} - 56 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0219978	+	1.09218i	−0.0155548	+	0.772290i	0.914441	+	0.404719i	\(0.132631\pi\)
							−0.929996	+	0.367570i	\(0.880190\pi\)
\(3\)	1.01159	+	1.40595i	0.584040	+	0.811725i
							\(5\)	0.345605	−	1.88591i	0.154559	−	0.843402i	−0.811358	−	0.584550i	\(-0.801271\pi\)
0.965917	−	0.258852i	\(-0.0833443\pi\)
\(7\)	2.55979	−	3.87306i	0.967509	−	1.46388i	0.0834458	−	0.996512i	\(-0.473407\pi\)
							0.884063	−	0.467368i	\(-0.154798\pi\)
\(11\)	−3.13027	−	1.72417i	−0.943812	−	0.519856i	−0.0654524	−	0.997856i	\(-0.520849\pi\)
							−0.878360	+	0.478000i	\(0.841362\pi\)
\(13\)	2.53999	+	2.55900i	0.704465	+	0.709738i
							\(17\)	0.458755	+	2.24713i	0.111265	+	0.545010i	0.996765	+	0.0803712i	\(0.0256106\pi\)
−0.885500	+	0.464638i	\(0.846184\pi\)
\(19\)	−0.948964	−	3.54158i	−0.217707	−	0.812494i	−0.985196	−	0.171432i	\(-0.945161\pi\)
							0.767489	−	0.641062i	\(-0.221506\pi\)
\(23\)	3.01771	+	5.22682i	0.629235	+	1.08987i	0.987706	+	0.156326i	\(0.0499651\pi\)
							−0.358470	+	0.933541i	\(0.616702\pi\)
\(29\)	5.22121	−	5.01504i	0.969555	−	0.931270i	−0.0281076	−	0.999605i	\(-0.508948\pi\)
							0.997662	+	0.0683349i	\(0.0217686\pi\)
\(31\)	−0.202654	−	0.0912073i	−0.0363978	−	0.0163813i	0.392132	−	0.919909i	\(-0.371738\pi\)
							−0.428530	+	0.903528i	\(0.640968\pi\)
\(37\)	−9.27189	+	0.936773i	−1.52429	+	0.154005i	−0.826761	−	0.562553i	\(-0.809819\pi\)
							−0.697528	+	0.716558i	\(0.745717\pi\)
\(41\)	−2.24090	−	0.318007i	−0.349970	−	0.0496643i	−0.0366943	−	0.999327i	\(-0.511683\pi\)
							−0.313276	+	0.949662i	\(0.601426\pi\)
\(43\)	−2.91816	+	2.38260i	−0.445015	+	0.363343i	−0.828298	−	0.560287i	\(-0.810691\pi\)
							0.383283	+	0.923631i	\(0.374793\pi\)
\(47\)	−11.2841	+	3.51626i	−1.64595	+	0.512899i	−0.973779	−	0.227495i	\(-0.926947\pi\)
							−0.672173	+	0.740394i	\(0.734639\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.b.2.20	✓	2784
3.2	odd	2	inner	507.2.x.b.2.39	yes	2784
169.85	odd	156	inner	507.2.x.b.254.39	yes	2784
507.254	even	156	inner	507.2.x.b.254.20	yes	2784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.b.2.20	✓	2784	1.1	even	1	trivial
507.2.x.b.2.39	yes	2784	3.2	odd	2	inner
507.2.x.b.254.20	yes	2784	507.254	even	156	inner
507.2.x.b.254.39	yes	2784	169.85	odd	156	inner