Embedded newform 507.2.q.a.55.5

• Label: 507.2.q.a.55.5
• Level: $507$
• Weight: $2$
• Character: 507.55
• 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.q (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(15\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			55.5
Character	\(\chi\)	\(=\)	507.55
Dual form			507.2.q.a.295.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.826374 + 1.01212i) q^{2} +(0.996757 + 0.0804666i) q^{3} +(0.0585493 + 0.286793i) q^{4} +(-0.0644718 - 0.530973i) q^{5} +(-0.905137 + 0.942347i) q^{6} +(0.386707 - 0.244538i) q^{7} +(-2.65259 - 1.39219i) q^{8} +(0.987050 + 0.160411i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q + q^{2} - 15 q^{3} + 15 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} + 6 q^{8} + 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{28}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.826374	+	1.01212i	−0.584335	+	0.715680i	−0.979163	−	0.203075i	\(-0.934907\pi\)
							0.394828	+	0.918755i	\(0.370804\pi\)
\(3\)	0.996757	+	0.0804666i	0.575478	+	0.0464574i
							\(5\)	−0.0644718	−	0.530973i	−0.0288326	−	0.237458i	0.971167	−	0.238401i	\(-0.0766231\pi\)
−1.00000		0.000942372i	\(0.999700\pi\)
\(7\)	0.386707	−	0.244538i	0.146161	−	0.0924268i	−0.459383	−	0.888238i	\(-0.651930\pi\)
							0.605545	+	0.795811i	\(0.292955\pi\)
\(11\)	6.45072	−	1.04834i	1.94496	−	0.316088i	0.945024	−	0.327000i	\(-0.106038\pi\)
							0.999940	+	0.0109122i	\(0.00347354\pi\)
\(13\)	−0.596867	+	3.55581i	−0.165541	+	0.986203i
							\(17\)	2.55335	−	1.61464i	0.619278	−	0.391608i	−0.187668	−	0.982233i	\(-0.560093\pi\)
0.806947	+	0.590625i	\(0.201119\pi\)
\(19\)	−4.18596	+	7.25030i	−0.960326	+	1.66333i	−0.238646	+	0.971107i	\(0.576704\pi\)
							−0.721680	+	0.692227i	\(0.756630\pi\)
\(23\)	2.46142	+	4.26330i	0.513241	+	0.888960i	0.999882	+	0.0153579i	\(0.00488878\pi\)
							−0.486641	+	0.873602i	\(0.661778\pi\)
\(29\)	−0.532692	+	0.652429i	−0.0989185	+	0.121153i	−0.821692	−	0.569932i	\(-0.806970\pi\)
							0.722773	+	0.691085i	\(0.242867\pi\)
\(31\)	−6.84146	−	1.68627i	−1.22876	−	0.302863i	−0.429069	−	0.903272i	\(-0.641158\pi\)
							−0.799693	+	0.600409i	\(0.795005\pi\)
\(37\)	0.0619163	+	0.213760i	0.0101790	+	0.0351419i	0.965447	−	0.260600i	\(-0.0839204\pi\)
							−0.955268	+	0.295742i	\(0.904433\pi\)
\(41\)	1.42987	+	0.115431i	0.223308	+	0.0180273i	0.191614	−	0.981470i	\(-0.438628\pi\)
							0.0316938	+	0.999498i	\(0.489910\pi\)
\(43\)	2.22271	−	7.67368i	0.338960	−	1.17023i	−0.592254	−	0.805752i	\(-0.701762\pi\)
							0.931214	−	0.364474i	\(-0.118751\pi\)
\(47\)	−0.0989089	−	0.0876257i	−0.0144274	−	0.0127815i	0.655878	−	0.754867i	\(-0.272298\pi\)
							−0.670306	+	0.742085i	\(0.733837\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.q.a.55.5	✓	360
169.126	even	39	inner	507.2.q.a.295.5	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.q.a.55.5	✓	360	1.1	even	1	trivial
507.2.q.a.295.5	yes	360	169.126	even	39	inner