Embedded newform 507.2.t.b.322.4

• Label: 507.2.t.b.322.4
• Level: $507$
• Weight: $2$
• Character: 507.322
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $360$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.t (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(15\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			322.4
Character	\(\chi\)	\(=\)	507.322
Dual form			507.2.t.b.433.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.123498 + 1.52980i) q^{2} +(0.845190 + 0.534466i) q^{3} +(-0.350944 - 0.0570339i) q^{4} +(-0.731801 + 0.826032i) q^{5} +(-0.922007 + 1.22697i) q^{6} +(-0.602375 - 0.578589i) q^{7} +(-0.604002 + 2.45053i) q^{8} +(0.428693 + 0.903450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 15 q^{3} - 14 q^{4} + 3 q^{7} + 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{41}{78}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.123498	+	1.52980i	−0.0873266	+	1.08173i	0.793587	+	0.608457i	\(0.208211\pi\)
							−0.880914	+	0.473277i	\(0.843071\pi\)
\(3\)	0.845190	+	0.534466i	0.487971	+	0.308574i
							\(5\)	−0.731801	+	0.826032i	−0.327271	+	0.369413i	−0.888999	−	0.457909i	\(-0.848598\pi\)
0.561728	+	0.827322i	\(0.310137\pi\)
\(7\)	−0.602375	−	0.578589i	−0.227676	−	0.218686i	0.570202	−	0.821504i	\(-0.306865\pi\)
							−0.797878	+	0.602818i	\(0.794044\pi\)
\(11\)	5.24852	+	2.49045i	1.58249	+	0.750900i	0.998161	−	0.0606186i	\(-0.0193073\pi\)
							0.584327	+	0.811518i	\(0.301359\pi\)
\(13\)	3.38608	−	1.23872i	0.939131	−	0.343560i
							\(17\)	−4.41268	+	4.59409i	−1.07023	+	1.11423i	−0.0771261	+	0.997021i	\(0.524574\pi\)
−0.993107	+	0.117210i	\(0.962605\pi\)
\(19\)	−3.82210	−	2.20669i	−0.876849	−	0.506249i	−0.00723079	−	0.999974i	\(-0.502302\pi\)
							−0.869618	+	0.493725i	\(0.835635\pi\)
\(23\)	−3.20742	−	5.55542i	−0.668794	−	1.15838i	−0.978242	−	0.207469i	\(-0.933478\pi\)
							0.309448	−	0.950916i	\(-0.399856\pi\)
\(29\)	1.71982	+	0.138839i	0.319363	+	0.0257817i	0.239109	−	0.970993i	\(-0.423145\pi\)
							0.0802544	+	0.996774i	\(0.474427\pi\)
\(31\)	−0.273356	−	0.0331915i	−0.0490963	−	0.00596137i	0.0959517	−	0.995386i	\(-0.469411\pi\)
							−0.145048	+	0.989425i	\(0.546334\pi\)
\(37\)	1.31056	+	3.07600i	0.215455	+	0.505692i	0.992279	−	0.124026i	\(-0.0395807\pi\)
							−0.776824	+	0.629718i	\(0.783170\pi\)
\(41\)	3.85712	−	6.09955i	0.602381	−	0.952590i	−0.397014	−	0.917813i	\(-0.629954\pi\)
							0.999395	−	0.0347771i	\(-0.0110721\pi\)
\(43\)	−2.75176	−	1.17242i	−0.419639	−	0.178792i	0.171816	−	0.985129i	\(-0.445036\pi\)
							−0.591456	+	0.806337i	\(0.701447\pi\)
\(47\)	10.0129	+	3.79741i	1.46054	+	0.553909i	0.951720	−	0.306967i	\(-0.0993140\pi\)
							0.508816	+	0.860875i	\(0.330083\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.t.b.322.4	✓	360
169.95	even	78	inner	507.2.t.b.433.4	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.t.b.322.4	✓	360	1.1	even	1	trivial
507.2.t.b.433.4	yes	360	169.95	even	78	inner