Embedded newform 50.6.d.a.11.1

• Label: 50.6.d.a.11.1
• 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			11.1
Character	\(\chi\)	\(=\)	50.11
Dual form			50.6.d.a.41.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.23607 + 2.35114i) q^{2} +(-6.86375 + 21.1244i) q^{3} +(4.94427 - 15.2169i) q^{4} +(-38.8214 - 40.2232i) q^{5} +(-27.4550 - 84.4978i) q^{6} -78.8464 q^{7} +(19.7771 + 60.8676i) q^{8} +(-202.540 - 147.154i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} - 11 q^{3} - 96 q^{4} - 120 q^{5} - 44 q^{6} + 548 q^{7} - 384 q^{8} - 893 q^{9}+O(q^{10}) \)

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
							\(3\)	−6.86375	+	21.1244i	−0.440310	+	1.35513i	0.447237	+	0.894416i	\(0.352408\pi\)
−0.887546	+	0.460718i	\(0.847592\pi\)
\(5\)	−38.8214	−	40.2232i	−0.694458	−	0.719534i
							\(7\)	−78.8464			−0.608187			−0.304093	−	0.952642i	\(-0.598353\pi\)
−0.304093	+	0.952642i	\(0.598353\pi\)
\(11\)	267.704	−	194.498i	0.667072	−	0.484656i	−0.201972	−	0.979391i	\(-0.564735\pi\)
							0.869044	+	0.494735i	\(0.164735\pi\)
\(13\)	374.407	+	272.023i	0.614449	+	0.446423i	0.850978	−	0.525201i	\(-0.176010\pi\)
							−0.236529	+	0.971624i	\(0.576010\pi\)
\(17\)	−93.8680	−	288.896i	−0.0787762	−	0.242448i	0.903911	−	0.427721i	\(-0.140683\pi\)
							−0.982687	+	0.185272i	\(0.940683\pi\)
\(19\)	−790.654	−	2433.38i	−0.502461	−	1.54642i	−0.804998	−	0.593278i	\(-0.797833\pi\)
							0.302537	−	0.953138i	\(-0.402167\pi\)
\(23\)	2310.13	−	1678.41i	0.910576	−	0.661572i	−0.0305842	−	0.999532i	\(-0.509737\pi\)
							0.941161	+	0.337960i	\(0.109737\pi\)
\(29\)	−432.234	+	1330.28i	−0.0954385	+	0.293730i	−0.987368	−	0.158446i	\(-0.949352\pi\)
							0.891929	+	0.452175i	\(0.149352\pi\)
\(31\)	−2398.43	−	7381.61i	−0.448252	−	1.37958i	−0.878877	−	0.477048i	\(-0.841707\pi\)
							0.430625	−	0.902531i	\(-0.358293\pi\)
\(37\)	−5281.02	−	3836.89i	−0.634181	−	0.460760i	0.223665	−	0.974666i	\(-0.428198\pi\)
							−0.857846	+	0.513906i	\(0.828198\pi\)
\(41\)	−7082.66	−	5145.85i	−0.658016	−	0.478077i	0.207976	−	0.978134i	\(-0.433312\pi\)
							−0.865992	+	0.500057i	\(0.833312\pi\)
\(43\)	−13310.9			−1.09783			−0.548915	−	0.835878i	\(-0.684959\pi\)
							−0.548915	+	0.835878i	\(0.684959\pi\)
\(47\)	9122.07	−	28074.8i	0.602350	−	1.85384i	0.0882758	−	0.996096i	\(-0.471864\pi\)
							0.514074	−	0.857746i	\(-0.328136\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.d.a.11.1	✓	24
25.16	even	5	inner	50.6.d.a.41.1	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.d.a.11.1	✓	24	1.1	even	1	trivial
50.6.d.a.41.1	yes	24	25.16	even	5	inner