Embedded newform 507.2.m.b.40.3

• Label: 507.2.m.b.40.3
• Level: $507$
• Weight: $2$
• Character: 507.40
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $204$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.m (of order \(13\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(204\)
Relative dimension:	\(17\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label			40.3
Character	\(\chi\)	\(=\)	507.40
Dual form			507.2.m.b.469.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.29214 - 0.564962i) q^{2} +(0.120537 - 0.992709i) q^{3} +(3.16382 + 1.66050i) q^{4} +(-1.62559 + 2.35507i) q^{5} +(-0.837130 + 2.20733i) q^{6} +(3.18940 + 2.82556i) q^{7} +(-2.77972 - 2.46262i) q^{8} +(-0.970942 - 0.239316i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - q^{2} - 17 q^{3} - 21 q^{4} - 6 q^{5} - q^{6} - 8 q^{7} - 9 q^{8} - 17 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{13}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.29214	−	0.564962i	−1.62079	−	0.399489i	−0.678583	−	0.734524i	\(-0.737406\pi\)
							−0.942206	+	0.335035i	\(0.891252\pi\)
\(3\)	0.120537	−	0.992709i	0.0695919	−	0.573141i
							\(5\)	−1.62559	+	2.35507i	−0.726984	+	1.05322i	0.268952	+	0.963154i	\(0.413323\pi\)
−0.995936	+	0.0900646i	\(0.971293\pi\)
\(7\)	3.18940	+	2.82556i	1.20548	+	1.06796i	0.995927	+	0.0901634i	\(0.0287389\pi\)
							0.209552	+	0.977798i	\(0.432800\pi\)
\(11\)	1.08698	−	0.267916i	0.327736	−	0.0807797i	−0.0720146	−	0.997404i	\(-0.522943\pi\)
							0.399751	+	0.916624i	\(0.369097\pi\)
\(13\)	−0.861346	+	3.50115i	−0.238894	+	0.971046i
							\(17\)	−1.95440	−	1.73145i	−0.474011	−	0.419937i	0.391997	−	0.919966i	\(-0.371784\pi\)
−0.866009	+	0.500029i	\(0.833323\pi\)
\(19\)	−7.80578			−1.79077			−0.895385	−	0.445294i	\(-0.853099\pi\)
							−0.895385	+	0.445294i	\(0.853099\pi\)
\(23\)	−5.59591			−1.16683			−0.583414	−	0.812175i	\(-0.698283\pi\)
							−0.583414	+	0.812175i	\(0.698283\pi\)
\(29\)	−0.848092	−	0.209036i	−0.157487	−	0.0388170i	0.159784	−	0.987152i	\(-0.448920\pi\)
							−0.317271	+	0.948335i	\(0.602766\pi\)
\(31\)	0.167191	−	0.440847i	0.0300284	−	0.0791785i	−0.919164	−	0.393876i	\(-0.871134\pi\)
							0.949192	+	0.314698i	\(0.101903\pi\)
\(37\)	2.71345	−	7.15477i	0.446088	−	1.17624i	−0.503848	−	0.863793i	\(-0.668083\pi\)
							0.949936	−	0.312445i	\(-0.101148\pi\)
\(41\)	−1.00198	+	8.25206i	−0.156483	+	1.28876i	0.676906	+	0.736069i	\(0.263320\pi\)
							−0.833389	+	0.552686i	\(0.813603\pi\)
\(43\)	−2.95742	−	7.79807i	−0.451002	−	1.18919i	−0.947103	−	0.320930i	\(-0.896005\pi\)
							0.496101	−	0.868265i	\(-0.334765\pi\)
\(47\)	4.90232	−	2.57294i	0.715077	−	0.375302i	−0.0676140	−	0.997712i	\(-0.521539\pi\)
							0.782691	+	0.622410i	\(0.213846\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.m.b.40.3	✓	204
169.131	even	13	inner	507.2.m.b.469.3	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.m.b.40.3	✓	204	1.1	even	1	trivial
507.2.m.b.469.3	yes	204	169.131	even	13	inner