Embedded newform 507.2.m.b.40.7

• Label: 507.2.m.b.40.7
• 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			40.7
Character	\(\chi\)	\(=\)	507.40
Dual form			507.2.m.b.469.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09223 - 0.269211i) q^{2} +(0.120537 - 0.992709i) q^{3} +(-0.650419 - 0.341366i) q^{4} +(-2.21140 + 3.20377i) q^{5} +(-0.398902 + 1.05182i) q^{6} +(-0.833262 - 0.738206i) q^{7} +(2.30254 + 2.03987i) q^{8} +(-0.970942 - 0.239316i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - q^{2} - 17 q^{3} - 21 q^{4} - 6 q^{5} - q^{6} - 8 q^{7} - 9 q^{8} - 17 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09223	−	0.269211i	−0.772324	−	0.190361i	−0.166588	−	0.986027i	\(-0.553275\pi\)
							−0.605736	+	0.795666i	\(0.707121\pi\)
\(3\)	0.120537	−	0.992709i	0.0695919	−	0.573141i
							\(5\)	−2.21140	+	3.20377i	−0.988969	+	1.43277i	−0.0893968	+	0.995996i	\(0.528494\pi\)
−0.899572	+	0.436773i	\(0.856121\pi\)
\(7\)	−0.833262	−	0.738206i	−0.314944	−	0.279016i	0.490810	−	0.871267i	\(-0.336701\pi\)
							−0.805754	+	0.592251i	\(0.798239\pi\)
\(11\)	3.28661	−	0.810076i	0.990949	−	0.244247i	0.289664	−	0.957128i	\(-0.406456\pi\)
							0.701285	+	0.712881i	\(0.252610\pi\)
\(13\)	−3.49660	−	0.879659i	−0.969782	−	0.243974i
							\(17\)	−2.91197	−	2.57978i	−0.706257	−	0.625689i	0.231560	−	0.972821i	\(-0.425617\pi\)
−0.937816	+	0.347132i	\(0.887156\pi\)
\(19\)	6.43900			1.47721			0.738604	−	0.674139i	\(-0.235485\pi\)
							0.738604	+	0.674139i	\(0.235485\pi\)
\(23\)	6.02110			1.25549			0.627743	−	0.778420i	\(-0.283979\pi\)
							0.627743	+	0.778420i	\(0.283979\pi\)
\(29\)	4.72501	+	1.16461i	0.877412	+	0.216263i	0.652200	−	0.758047i	\(-0.273846\pi\)
							0.225212	+	0.974310i	\(0.427692\pi\)
\(31\)	3.08619	−	8.13762i	0.554297	−	1.46156i	−0.306132	−	0.951989i	\(-0.599035\pi\)
							0.860429	−	0.509571i	\(-0.170196\pi\)
\(37\)	−3.20298	+	8.44555i	−0.526566	+	1.38844i	0.363067	+	0.931763i	\(0.381730\pi\)
							−0.889633	+	0.456677i	\(0.849039\pi\)
\(41\)	0.725596	−	5.97582i	0.113319	−	0.933267i	−0.819539	−	0.573023i	\(-0.805771\pi\)
							0.932858	−	0.360243i	\(-0.117306\pi\)
\(43\)	−3.27217	−	8.62801i	−0.499002	−	1.31576i	−0.913918	−	0.405900i	\(-0.866958\pi\)
							0.414916	−	0.909860i	\(-0.363811\pi\)
\(47\)	1.47579	−	0.774556i	0.215266	−	0.112981i	−0.353644	−	0.935380i	\(-0.615057\pi\)
							0.568910	+	0.822400i	\(0.307365\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.m.b.40.7	✓	204
169.131	even	13	inner	507.2.m.b.469.7	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.m.b.40.7	✓	204	1.1	even	1	trivial
507.2.m.b.469.7	yes	204	169.131	even	13	inner