Embedded newform 50.6.e.a.9.1

• Label: 50.6.e.a.9.1
• Level: $50$
• Weight: $6$
• Character: 50.9
• 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			9.1
Character	\(\chi\)	\(=\)	50.9
Dual form			50.6.e.a.39.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.35114 + 3.23607i) q^{2} +(-19.7385 + 6.41342i) q^{3} +(-4.94427 - 15.2169i) q^{4} +(-52.1632 - 20.0999i) q^{5} +(25.6537 - 78.9540i) q^{6} +5.87904i q^{7} +(60.8676 + 19.7771i) q^{8} +(151.885 - 110.351i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 192 q^{4} + 180 q^{5} + 72 q^{6} + 686 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)

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
							\(3\)	−19.7385	+	6.41342i	−1.26622	+	0.411421i	−0.863709	−	0.503991i	\(-0.831864\pi\)
−0.402516	+	0.915413i	\(0.631864\pi\)
\(5\)	−52.1632	−	20.0999i	−0.933123	−	0.359557i
							\(7\)			5.87904i			0.0453484i	0.999743	+	0.0226742i	\(0.00721804\pi\)
−0.999743	+	0.0226742i	\(0.992782\pi\)
\(11\)	−9.61629	−	6.98664i	−0.0239621	−	0.0174095i	0.575740	−	0.817633i	\(-0.304714\pi\)
							−0.599702	+	0.800224i	\(0.704714\pi\)
\(13\)	365.888	+	503.601i	0.600468	+	0.826473i	0.995751	−	0.0920865i	\(-0.0293536\pi\)
							−0.395283	+	0.918559i	\(0.629354\pi\)
\(17\)	773.507	+	251.328i	0.649145	+	0.210920i	0.615037	−	0.788498i	\(-0.289141\pi\)
							0.0341080	+	0.999418i	\(0.489141\pi\)
\(19\)	721.421	−	2220.30i	0.458463	−	1.41100i	−0.408557	−	0.912733i	\(-0.633968\pi\)
							0.867021	−	0.498272i	\(-0.166032\pi\)
\(23\)	−359.922	+	495.390i	−0.141869	+	0.195267i	−0.874039	−	0.485856i	\(-0.838508\pi\)
							0.732169	+	0.681123i	\(0.238508\pi\)
\(29\)	−2357.33	−	7255.11i	−0.520505	−	1.60195i	−0.773037	−	0.634361i	\(-0.781263\pi\)
							0.252532	−	0.967588i	\(-0.418737\pi\)
\(31\)	522.411	−	1607.81i	0.0976355	−	0.300491i	−0.890296	−	0.455382i	\(-0.849503\pi\)
							0.987932	+	0.154891i	\(0.0495026\pi\)
\(37\)	3767.99	+	5186.19i	0.452486	+	0.622793i	0.972929	−	0.231103i	\(-0.0742334\pi\)
							−0.520444	+	0.853896i	\(0.674233\pi\)
\(41\)	−3257.86	+	2366.97i	−0.302672	+	0.219904i	−0.728746	−	0.684784i	\(-0.759896\pi\)
							0.426074	+	0.904689i	\(0.359896\pi\)
\(43\)		−	8270.82i		−	0.682146i	−0.940037	−	0.341073i	\(-0.889210\pi\)
							0.940037	−	0.341073i	\(-0.110790\pi\)
\(47\)	−23877.5	+	7758.28i	−1.57668	+	0.512296i	−0.961200	−	0.275853i	\(-0.911040\pi\)
							−0.615484	+	0.788149i	\(0.711040\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.e.a.9.1	✓	48
25.14	even	10	inner	50.6.e.a.39.1	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.e.a.9.1	✓	48	1.1	even	1	trivial
50.6.e.a.39.1	yes	48	25.14	even	10	inner