Embedded newform 50.6.d.a.11.2

• Label: 50.6.d.a.11.2
• Level: $50$
• Weight: $6$
• Character: 50.11
• Analytic conductor: $8.019$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(8.01919099065\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			11.2
Character	\(\chi\)	\(=\)	50.11
Dual form			50.6.d.a.41.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.23607 + 2.35114i) q^{2} +(-4.88978 + 15.0492i) q^{3} +(4.94427 - 15.2169i) q^{4} +(51.0006 - 22.8897i) q^{5} +(-19.5591 - 60.1968i) q^{6} +188.761 q^{7} +(19.7771 + 60.8676i) q^{8} +(-5.97726 - 4.34273i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} - 11 q^{3} - 96 q^{4} - 120 q^{5} - 44 q^{6} + 548 q^{7} - 384 q^{8} - 893 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.23607	+	2.35114i	−0.572061	+	0.415627i
							\(3\)	−4.88978	+	15.0492i	−0.313680	+	0.965406i	0.662615	+	0.748960i	\(0.269447\pi\)
−0.976295	+	0.216446i	\(0.930553\pi\)
\(5\)	51.0006	−	22.8897i	0.912326	−	0.409464i
							\(7\)	188.761			1.45602			0.728010	−	0.685566i	\(-0.240445\pi\)
0.728010	+	0.685566i	\(0.240445\pi\)
\(11\)	−312.265	+	226.874i	−0.778111	+	0.565330i	−0.904411	−	0.426662i	\(-0.859690\pi\)
							0.126301	+	0.991992i	\(0.459690\pi\)
\(13\)	−145.558	−	105.754i	−0.238879	−	0.173556i	0.461905	−	0.886930i	\(-0.347166\pi\)
							−0.700784	+	0.713374i	\(0.747166\pi\)
\(17\)	199.706	+	614.630i	0.167598	+	0.515812i	0.999218	−	0.0395317i	\(-0.0125866\pi\)
							−0.831621	+	0.555344i	\(0.812587\pi\)
\(19\)	632.511	+	1946.67i	0.401961	+	1.23711i	0.923406	+	0.383824i	\(0.125393\pi\)
							−0.521445	+	0.853285i	\(0.674607\pi\)
\(23\)	1864.81	−	1354.86i	0.735045	−	0.534042i	−0.156110	−	0.987740i	\(-0.549895\pi\)
							0.891156	+	0.453698i	\(0.149895\pi\)
\(29\)	−584.904	+	1800.15i	−0.129149	+	0.397478i	−0.994634	−	0.103456i	\(-0.967010\pi\)
							0.865486	+	0.500934i	\(0.167010\pi\)
\(31\)	1341.84	+	4129.77i	0.250783	+	0.771830i	0.994631	+	0.103482i	\(0.0329983\pi\)
							−0.743849	+	0.668348i	\(0.767002\pi\)
\(37\)	−6107.84	−	4437.61i	−0.733472	−	0.532899i	0.157188	−	0.987569i	\(-0.449757\pi\)
							−0.890660	+	0.454670i	\(0.849757\pi\)
\(41\)	−10358.4	−	7525.79i	−0.962347	−	0.699186i	−0.00865227	−	0.999963i	\(-0.502754\pi\)
							−0.953695	+	0.300777i	\(0.902754\pi\)
\(43\)	21746.1			1.79354			0.896769	−	0.442500i	\(-0.145908\pi\)
							0.896769	+	0.442500i	\(0.145908\pi\)
\(47\)	595.550	−	1832.91i	0.0393254	−	0.121031i	−0.929467	−	0.368906i	\(-0.879732\pi\)
							0.968792	+	0.247875i	\(0.0797323\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.d.a.11.2	✓	24
25.16	even	5	inner	50.6.d.a.41.2	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.d.a.11.2	✓	24	1.1	even	1	trivial
50.6.d.a.41.2	yes	24	25.16	even	5	inner