Embedded newform 50.6.e.a.19.4

• Label: 50.6.e.a.19.4
• Level: $50$
• Weight: $6$
• Character: 50.19
• 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			19.4
Character	\(\chi\)	\(=\)	50.19
Dual form			50.6.e.a.29.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.80423 + 1.23607i) q^{2} +(4.87367 + 6.70803i) q^{3} +(12.9443 - 9.40456i) q^{4} +(44.0032 + 34.4778i) q^{5} +(-26.8321 - 19.4947i) q^{6} -63.1969i q^{7} +(-37.6183 + 51.7771i) q^{8} +(53.8461 - 165.721i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 192 q^{4} + 180 q^{5} + 72 q^{6} + 686 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.80423	+	1.23607i	−0.672499	+	0.218508i
							\(3\)	4.87367	+	6.70803i	0.312646	+	0.430320i	0.936204	−	0.351457i	\(-0.114314\pi\)
−0.623558	+	0.781777i	\(0.714314\pi\)
\(5\)	44.0032	+	34.4778i	0.787154	+	0.616757i
							\(7\)		−	63.1969i		−	0.487473i	−0.969841	−	0.243737i	\(-0.921627\pi\)
0.969841	−	0.243737i	\(-0.0783733\pi\)
\(11\)	180.403	+	555.222i	0.449533	+	1.38352i	0.877436	+	0.479695i	\(0.159253\pi\)
							−0.427903	+	0.903825i	\(0.640747\pi\)
\(13\)	358.115	+	116.359i	0.587711	+	0.190959i	0.587752	−	0.809041i	\(-0.300013\pi\)
							−4.10061e−5		1.00000i	\(0.500013\pi\)
\(17\)	−1324.69	+	1823.28i	−1.11171	+	1.53014i	−0.292840	+	0.956161i	\(0.594600\pi\)
							−0.818871	+	0.573978i	\(0.805400\pi\)
\(19\)	1683.26	+	1222.96i	1.06971	+	0.777190i	0.975860	−	0.218396i	\(-0.0700824\pi\)
							0.0938507	+	0.995586i	\(0.470082\pi\)
\(23\)	−118.364	+	38.4587i	−0.0466551	+	0.0151592i	−0.332252	−	0.943191i	\(-0.607808\pi\)
							0.285596	+	0.958350i	\(0.407808\pi\)
\(29\)	3336.00	−	2423.75i	0.736600	−	0.535171i	−0.155045	−	0.987907i	\(-0.549552\pi\)
							0.891644	+	0.452736i	\(0.149552\pi\)
\(31\)	−5279.17	−	3835.54i	−0.986645	−	0.716840i	−0.0274614	−	0.999623i	\(-0.508742\pi\)
							−0.959184	+	0.282783i	\(0.908742\pi\)
\(37\)	−10424.0	−	3386.97i	−1.25179	−	0.406731i	−0.393228	−	0.919441i	\(-0.628642\pi\)
							−0.858562	+	0.512710i	\(0.828642\pi\)
\(41\)	3834.39	−	11801.0i	0.356235	−	1.09638i	−0.599055	−	0.800708i	\(-0.704457\pi\)
							0.955290	−	0.295670i	\(-0.0955430\pi\)
\(43\)		−	1571.18i		−	0.129585i	−0.997899	−	0.0647924i	\(-0.979361\pi\)
							0.997899	−	0.0647924i	\(-0.0206385\pi\)
\(47\)	15224.4	+	20954.6i	1.00530	+	1.38368i	0.922016	+	0.387152i	\(0.126541\pi\)
							0.0832848	+	0.996526i	\(0.473459\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.e.a.19.4	✓	48
25.4	even	10	inner	50.6.e.a.29.4	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.e.a.19.4	✓	48	1.1	even	1	trivial
50.6.e.a.29.4	yes	48	25.4	even	10	inner