Embedded newform 507.2.m.b.40.17

• Label: 507.2.m.b.40.17
• 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			40.17
Character	\(\chi\)	\(=\)	507.40
Dual form			507.2.m.b.469.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.66897 + 0.657842i) q^{2} +(0.120537 - 0.992709i) q^{3} +(4.91973 + 2.58207i) q^{4} +(0.113107 - 0.163863i) q^{5} +(0.974754 - 2.57022i) q^{6} +(-1.95577 - 1.73266i) q^{7} +(7.31692 + 6.48223i) q^{8} +(-0.970942 - 0.239316i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - q^{2} - 17 q^{3} - 21 q^{4} - 6 q^{5} - q^{6} - 8 q^{7} - 9 q^{8} - 17 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.66897	+	0.657842i	1.88725	+	0.465164i	0.999992	+	0.00388245i	\(0.00123583\pi\)
							0.887254	+	0.461282i	\(0.152610\pi\)
\(3\)	0.120537	−	0.992709i	0.0695919	−	0.573141i
							\(5\)	0.113107	−	0.163863i	0.0505828	−	0.0732818i	−0.796876	−	0.604142i	\(-0.793516\pi\)
0.847459	+	0.530861i	\(0.178131\pi\)
\(7\)	−1.95577	−	1.73266i	−0.739213	−	0.654885i	0.207033	−	0.978334i	\(-0.433619\pi\)
							−0.946246	+	0.323449i	\(0.895158\pi\)
\(11\)	2.78789	−	0.687153i	0.840580	−	0.207184i	0.204557	−	0.978855i	\(-0.434425\pi\)
							0.636023	+	0.771670i	\(0.280578\pi\)
\(13\)	−3.28223	+	1.49231i	−0.910326	+	0.413893i
							\(17\)	−2.02855	−	1.79714i	−0.491995	−	0.435870i	0.380290	−	0.924867i	\(-0.375824\pi\)
−0.872285	+	0.488998i	\(0.837363\pi\)
\(19\)	−1.21614			−0.279002			−0.139501	−	0.990222i	\(-0.544550\pi\)
							−0.139501	+	0.990222i	\(0.544550\pi\)
\(23\)	−6.60947			−1.37817			−0.689085	−	0.724681i	\(-0.741987\pi\)
							−0.689085	+	0.724681i	\(0.741987\pi\)
\(29\)	4.36264	+	1.07529i	0.810122	+	0.199677i	0.622557	−	0.782575i	\(-0.286094\pi\)
							0.187566	+	0.982252i	\(0.439940\pi\)
\(31\)	−2.66381	+	7.02388i	−0.478434	+	1.26153i	0.450996	+	0.892526i	\(0.351069\pi\)
							−0.929429	+	0.369000i	\(0.879700\pi\)
\(37\)	3.75089	−	9.89028i	0.616642	−	1.62595i	−0.153550	−	0.988141i	\(-0.549071\pi\)
							0.770193	−	0.637811i	\(-0.220160\pi\)
\(41\)	0.662454	−	5.45580i	0.103458	−	0.852053i	−0.844916	−	0.534899i	\(-0.820350\pi\)
							0.948374	−	0.317154i	\(-0.102727\pi\)
\(43\)	−2.31766	−	6.11117i	−0.353440	−	0.931945i	−0.987353	−	0.158540i	\(-0.949321\pi\)
							0.633913	−	0.773405i	\(-0.281448\pi\)
\(47\)	−8.48580	+	4.45369i	−1.23778	+	0.649638i	−0.951246	−	0.308432i	\(-0.900196\pi\)
							−0.286535	+	0.958070i	\(0.592503\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.m.b.40.17	✓	204
169.131	even	13	inner	507.2.m.b.469.17	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.m.b.40.17	✓	204	1.1	even	1	trivial
507.2.m.b.469.17	yes	204	169.131	even	13	inner