Embedded newform 50.6.d.a.11.4

• Label: 50.6.d.a.11.4
• 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			11.4
Character	\(\chi\)	\(=\)	50.11
Dual form			50.6.d.a.41.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.23607 + 2.35114i) q^{2} +(2.33322 - 7.18091i) q^{3} +(4.94427 - 15.2169i) q^{4} +(-54.1589 + 13.8497i) q^{5} +(9.33287 + 28.7236i) q^{6} +54.5999 q^{7} +(19.7771 + 60.8676i) q^{8} +(150.470 + 109.323i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} - 11 q^{3} - 96 q^{4} - 120 q^{5} - 44 q^{6} + 548 q^{7} - 384 q^{8} - 893 q^{9}+O(q^{10}) \)

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
							\(3\)	2.33322	−	7.18091i	0.149676	−	0.460655i	−0.847907	−	0.530146i	\(-0.822137\pi\)
0.997583	+	0.0694902i	\(0.0221373\pi\)
\(5\)	−54.1589	+	13.8497i	−0.968824	+	0.247752i
							\(7\)	54.5999			0.421159			0.210580	−	0.977577i	\(-0.432465\pi\)
0.210580	+	0.977577i	\(0.432465\pi\)
\(11\)	−49.4604	+	35.9351i	−0.123247	+	0.0895442i	−0.647701	−	0.761895i	\(-0.724269\pi\)
							0.524454	+	0.851439i	\(0.324269\pi\)
\(13\)	436.167	+	316.894i	0.715805	+	0.520063i	0.885041	−	0.465513i	\(-0.154130\pi\)
							−0.169236	+	0.985576i	\(0.554130\pi\)
\(17\)	575.436	+	1771.01i	0.482920	+	1.48627i	0.834971	+	0.550293i	\(0.185484\pi\)
							−0.352052	+	0.935980i	\(0.614516\pi\)
\(19\)	590.724	+	1818.06i	0.375406	+	1.15538i	0.943205	+	0.332212i	\(0.107795\pi\)
							−0.567799	+	0.823167i	\(0.692205\pi\)
\(23\)	−2558.77	+	1859.05i	−1.00858	+	0.732777i	−0.963911	−	0.266224i	\(-0.914224\pi\)
							−0.0446704	+	0.999002i	\(0.514224\pi\)
\(29\)	1058.22	−	3256.87i	0.233658	−	0.719126i	−0.763638	−	0.645644i	\(-0.776589\pi\)
							0.997297	−	0.0734816i	\(-0.0234110\pi\)
\(31\)	−1642.05	−	5053.69i	−0.306889	−	0.944506i	−0.978966	−	0.204025i	\(-0.934598\pi\)
							0.672077	−	0.740481i	\(-0.265402\pi\)
\(37\)	8623.29	+	6265.19i	1.03554	+	0.752367i	0.969411	−	0.245443i	\(-0.0789336\pi\)
							0.0661339	+	0.997811i	\(0.478934\pi\)
\(41\)	−2128.87	−	1546.72i	−0.197783	−	0.143698i	0.484486	−	0.874799i	\(-0.339007\pi\)
							−0.682269	+	0.731101i	\(0.739007\pi\)
\(43\)	1595.86			0.131621			0.0658103	−	0.997832i	\(-0.479037\pi\)
							0.0658103	+	0.997832i	\(0.479037\pi\)
\(47\)	1116.50	−	3436.23i	0.0737249	−	0.226902i	−0.907403	−	0.420261i	\(-0.861938\pi\)
							0.981128	+	0.193360i	\(0.0619384\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.d.a.11.4	✓	24
25.16	even	5	inner	50.6.d.a.41.4	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.d.a.11.4	✓	24	1.1	even	1	trivial
50.6.d.a.41.4	yes	24	25.16	even	5	inner