Embedded newform 507.2.x.b.2.1

• Label: 507.2.x.b.2.1
• 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			2.1
Character	\(\chi\)	\(=\)	507.2
Dual form			507.2.x.b.254.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0548695 + 2.72425i) q^{2} +(0.0796955 + 1.73022i) q^{3} +(-5.42016 - 0.218425i) q^{4} +(0.698419 - 3.81115i) q^{5} +(-4.71792 + 0.122175i) q^{6} +(1.76691 - 2.67341i) q^{7} +(0.563407 - 9.31422i) q^{8} +(-2.98730 + 0.275781i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2784 q - 50 q^{3} - 92 q^{4} - 50 q^{6} - 100 q^{7} - 56 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0548695	+	2.72425i	−0.0387986	+	1.92634i	0.248745	+	0.968569i	\(0.419982\pi\)
							−0.287543	+	0.957768i	\(0.592839\pi\)
\(3\)	0.0796955	+	1.73022i	0.0460122	+	0.998941i
							\(5\)	0.698419	−	3.81115i	0.312343	−	1.70440i	−0.335016	−	0.942213i	\(-0.608742\pi\)
0.647358	−	0.762186i	\(-0.275874\pi\)
\(7\)	1.76691	−	2.67341i	0.667830	−	1.01045i	−0.329959	−	0.943995i	\(-0.607035\pi\)
							0.997789	−	0.0664589i	\(-0.0211701\pi\)
\(11\)	−1.61808	−	0.891248i	−0.487871	−	0.268721i	0.219437	−	0.975627i	\(-0.429578\pi\)
							−0.707308	+	0.706905i	\(0.750091\pi\)
\(13\)	−2.52324	−	2.57551i	−0.699822	−	0.714318i
							\(17\)	−0.284641	−	1.39426i	−0.0690355	−	0.338158i	0.930574	−	0.366104i	\(-0.119309\pi\)
−0.999609	+	0.0279457i	\(0.991103\pi\)
\(19\)	0.769835	+	2.87306i	0.176612	+	0.659126i	0.996271	+	0.0862743i	\(0.0274961\pi\)
							−0.819659	+	0.572851i	\(0.805837\pi\)
\(23\)	−2.92698	−	5.06968i	−0.610318	−	1.05710i	−0.991187	−	0.132473i	\(-0.957708\pi\)
							0.380868	−	0.924629i	\(-0.375625\pi\)
\(29\)	−1.00933	+	0.969477i	−0.187428	+	0.180027i	−0.780565	−	0.625074i	\(-0.785069\pi\)
							0.593137	+	0.805102i	\(0.297889\pi\)
\(31\)	3.74456	+	1.68529i	0.672543	+	0.302687i	0.717761	−	0.696290i	\(-0.245167\pi\)
							−0.0452171	+	0.998977i	\(0.514398\pi\)
\(37\)	5.00085	−	0.505254i	0.822134	−	0.0830633i	0.319526	−	0.947578i	\(-0.396476\pi\)
							0.502608	+	0.864514i	\(0.332374\pi\)
\(41\)	−10.7285	−	1.52248i	−1.67551	−	0.237771i	−0.762872	−	0.646550i	\(-0.776211\pi\)
							−0.912634	+	0.408779i	\(0.865955\pi\)
\(43\)	2.95576	−	2.41331i	0.450750	−	0.368026i	−0.379709	−	0.925106i	\(-0.623976\pi\)
							0.830459	+	0.557080i	\(0.188078\pi\)
\(47\)	−4.32908	+	1.34899i	−0.631461	+	0.196771i	−0.596494	−	0.802617i	\(-0.703440\pi\)
							−0.0349664	+	0.999388i	\(0.511132\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.b.2.1	✓	2784
3.2	odd	2	inner	507.2.x.b.2.58	yes	2784
169.85	odd	156	inner	507.2.x.b.254.58	yes	2784
507.254	even	156	inner	507.2.x.b.254.1	yes	2784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.b.2.1	✓	2784	1.1	even	1	trivial
507.2.x.b.2.58	yes	2784	3.2	odd	2	inner
507.2.x.b.254.1	yes	2784	507.254	even	156	inner
507.2.x.b.254.58	yes	2784	169.85	odd	156	inner