Embedded newform 50.6.e.a.39.7

• Label: 50.6.e.a.39.7
• Level: $50$
• Weight: $6$
• Character: 50.39
• Analytic conductor: $8.019$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	50.e (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.01919099065\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			39.7
Character	\(\chi\)	\(=\)	50.39
Dual form			50.6.e.a.9.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.35114 + 3.23607i) q^{2} +(-27.7680 - 9.02236i) q^{3} +(-4.94427 + 15.2169i) q^{4} +(12.4445 + 54.4989i) q^{5} +(-36.0894 - 111.072i) q^{6} -185.806i q^{7} +(-60.8676 + 19.7771i) q^{8} +(493.066 + 358.234i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 192 q^{4} + 180 q^{5} + 72 q^{6} + 686 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.35114	+	3.23607i	0.415627	+	0.572061i
							\(3\)	−27.7680	−	9.02236i	−1.78132	−	0.578785i	−0.782290	−	0.622914i	\(-0.785949\pi\)
−0.999026	+	0.0441291i	\(0.985949\pi\)
\(5\)	12.4445	+	54.4989i	0.222614	+	0.974907i
							\(7\)		−	185.806i		−	1.43323i	−0.697470	−	0.716614i	\(-0.745691\pi\)
0.697470	−	0.716614i	\(-0.254309\pi\)
\(11\)	455.216	−	330.734i	1.13432	−	0.824133i	0.148003	−	0.988987i	\(-0.452715\pi\)
							0.986318	+	0.164854i	\(0.0527154\pi\)
\(13\)	110.725	−	152.401i	0.181714	−	0.250108i	−0.708436	−	0.705775i	\(-0.750599\pi\)
							0.890151	+	0.455666i	\(0.150599\pi\)
\(17\)	736.538	−	239.316i	0.618120	−	0.200839i	0.0168144	−	0.999859i	\(-0.494648\pi\)
							0.601305	+	0.799019i	\(0.294648\pi\)
\(19\)	18.9785	+	58.4098i	0.0120608	+	0.0371195i	0.956906	−	0.290398i	\(-0.0937878\pi\)
							−0.944845	+	0.327518i	\(0.893788\pi\)
\(23\)	−451.179	−	620.994i	−0.177840	−	0.244775i	0.710786	−	0.703408i	\(-0.248339\pi\)
							−0.888626	+	0.458633i	\(0.848339\pi\)
\(29\)	2228.50	−	6858.61i	0.492059	−	1.51440i	−0.329432	−	0.944179i	\(-0.606857\pi\)
							0.821491	−	0.570222i	\(-0.193143\pi\)
\(31\)	1714.19	+	5275.73i	0.320372	+	0.986003i	0.973487	+	0.228744i	\(0.0734619\pi\)
							−0.653115	+	0.757259i	\(0.726538\pi\)
\(37\)	6181.24	−	8507.74i	0.742286	−	1.02167i	−0.256198	−	0.966624i	\(-0.582470\pi\)
							0.998484	−	0.0550445i	\(-0.0175301\pi\)
\(41\)	684.301	+	497.174i	0.0635752	+	0.0461901i	0.619119	−	0.785297i	\(-0.287490\pi\)
							−0.555544	+	0.831487i	\(0.687490\pi\)
\(43\)		−	14837.7i		−	1.22376i	−0.790950	−	0.611881i	\(-0.790413\pi\)
							0.790950	−	0.611881i	\(-0.209587\pi\)
\(47\)	14319.8	+	4652.79i	0.945567	+	0.307233i	0.740913	−	0.671601i	\(-0.234393\pi\)
							0.204654	+	0.978834i	\(0.434393\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.e.a.39.7	yes	48
25.9	even	10	inner	50.6.e.a.9.7	✓	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.e.a.9.7	✓	48	25.9	even	10	inner
50.6.e.a.39.7	yes	48	1.1	even	1	trivial