Embedded newform 507.2.x.b.2.17

• Label: 507.2.x.b.2.17
• 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			2.17
Character	\(\chi\)	\(=\)	507.2
Dual form			507.2.x.b.254.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0287183 + 1.42585i) q^{2} +(-0.617987 - 1.61805i) q^{3} +(-0.0338548 - 0.00136430i) q^{4} +(-0.204323 + 1.11496i) q^{5} +(2.32485 - 0.834692i) q^{6} +(0.633950 - 0.959192i) q^{7} +(-0.169299 + 2.79885i) q^{8} +(-2.23618 + 1.99987i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2784 q - 50 q^{3} - 92 q^{4} - 50 q^{6} - 100 q^{7} - 56 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0287183	+	1.42585i	−0.0203069	+	1.00823i	0.853248	+	0.521505i	\(0.174629\pi\)
							−0.873555	+	0.486725i	\(0.838191\pi\)
\(3\)	−0.617987	−	1.61805i	−0.356795	−	0.934183i
							\(5\)	−0.204323	+	1.11496i	−0.0913761	+	0.498623i	0.905812	+	0.423680i	\(0.139262\pi\)
−0.997188	+	0.0749428i	\(0.976123\pi\)
\(7\)	0.633950	−	0.959192i	0.239611	−	0.362541i	−0.694519	−	0.719474i	\(-0.744383\pi\)
							0.934130	+	0.356934i	\(0.116178\pi\)
\(11\)	2.88618	+	1.58972i	0.870216	+	0.479319i	0.853833	−	0.520547i	\(-0.174272\pi\)
							0.0163832	+	0.999866i	\(0.494785\pi\)
\(13\)	2.12246	−	2.91465i	0.588665	−	0.808377i
							\(17\)	0.0539382	+	0.264207i	0.0130819	+	0.0640796i	0.985808	−	0.167879i	\(-0.0536918\pi\)
−0.972726	+	0.231958i	\(0.925487\pi\)
\(19\)	1.17860	+	4.39860i	0.270390	+	1.00911i	0.958868	+	0.283851i	\(0.0916122\pi\)
							−0.688479	+	0.725256i	\(0.741721\pi\)
\(23\)	0.118383	+	0.205046i	0.0246846	+	0.0427550i	0.878104	−	0.478470i	\(-0.158809\pi\)
							−0.853419	+	0.521225i	\(0.825475\pi\)
\(29\)	1.06432	−	1.02229i	0.197639	−	0.189835i	−0.587371	−	0.809318i	\(-0.699837\pi\)
							0.785010	+	0.619483i	\(0.212658\pi\)
\(31\)	1.73487	+	0.780799i	0.311591	+	0.140236i	0.560174	−	0.828375i	\(-0.310734\pi\)
							−0.248583	+	0.968611i	\(0.579965\pi\)
\(37\)	5.86965	−	0.593032i	0.964964	−	0.0974939i	0.394571	−	0.918866i	\(-0.370893\pi\)
							0.570393	+	0.821372i	\(0.306791\pi\)
\(41\)	7.54291	+	1.07042i	1.17800	+	0.167171i	0.701938	−	0.712238i	\(-0.252318\pi\)
							0.476067	+	0.879409i	\(0.342062\pi\)
\(43\)	−4.81896	+	3.93456i	−0.734885	+	0.600015i	−0.924203	−	0.381901i	\(-0.875270\pi\)
							0.189319	+	0.981916i	\(0.439372\pi\)
\(47\)	−4.84017	+	1.50826i	−0.706012	+	0.220002i	−0.629425	−	0.777061i	\(-0.716710\pi\)
							−0.0765861	+	0.997063i	\(0.524402\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.b.2.17	✓	2784
3.2	odd	2	inner	507.2.x.b.2.42	yes	2784
169.85	odd	156	inner	507.2.x.b.254.42	yes	2784
507.254	even	156	inner	507.2.x.b.254.17	yes	2784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.b.2.17	✓	2784	1.1	even	1	trivial
507.2.x.b.2.42	yes	2784	3.2	odd	2	inner
507.2.x.b.254.17	yes	2784	507.254	even	156	inner
507.2.x.b.254.42	yes	2784	169.85	odd	156	inner