Embedded newform 50.6.d.b.11.3

• Label: 50.6.d.b.11.3
• Level: $50$
• Weight: $6$
• Character: 50.11
• Analytic conductor: $8.019$
• Analytic rank: $0$
• Dimension: $28$
• 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:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			11.3
Character	\(\chi\)	\(=\)	50.11
Dual form			50.6.d.b.41.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.23607 - 2.35114i) q^{2} +(-4.00985 + 12.3411i) q^{3} +(4.94427 - 15.2169i) q^{4} +(32.6779 - 45.3558i) q^{5} +(16.0394 + 49.3642i) q^{6} +24.1971 q^{7} +(-19.7771 - 60.8676i) q^{8} +(60.3684 + 43.8602i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 28 q^{2} + 7 q^{3} - 112 q^{4} - 145 q^{5} - 28 q^{6} - 236 q^{7} + 448 q^{8} - 446 q^{9}+O(q^{10}) \)

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
							\(3\)	−4.00985	+	12.3411i	−0.257232	+	0.791679i	0.736150	+	0.676819i	\(0.236642\pi\)
−0.993382	+	0.114860i	\(0.963358\pi\)
\(5\)	32.6779	−	45.3558i	0.584561	−	0.811350i
							\(7\)	24.1971			0.186646			0.0933228	−	0.995636i	\(-0.470251\pi\)
0.0933228	+	0.995636i	\(0.470251\pi\)
\(11\)	478.475	−	347.632i	1.19228	−	0.866240i	0.198774	−	0.980045i	\(-0.436304\pi\)
							0.993503	+	0.113806i	\(0.0363041\pi\)
\(13\)	715.822	+	520.075i	1.17475	+	0.853509i	0.991570	−	0.129570i	\(-0.0413597\pi\)
							0.183184	+	0.983079i	\(0.441360\pi\)
\(17\)	−489.226	−	1505.68i	−0.410570	−	1.26361i	−0.916154	−	0.400827i	\(-0.868723\pi\)
							0.505583	−	0.862778i	\(-0.331277\pi\)
\(19\)	−42.6095	−	131.139i	−0.0270784	−	0.0833387i	0.936604	−	0.350390i	\(-0.113951\pi\)
							−0.963682	+	0.267051i	\(0.913951\pi\)
\(23\)	−2640.44	+	1918.39i	−1.04078	+	0.756168i	−0.970437	−	0.241355i	\(-0.922408\pi\)
							−0.0703388	+	0.997523i	\(0.522408\pi\)
\(29\)	614.681	−	1891.79i	0.135723	−	0.417714i	−0.859978	−	0.510330i	\(-0.829523\pi\)
							0.995702	+	0.0926167i	\(0.0295231\pi\)
\(31\)	2970.61	+	9142.61i	0.555191	+	1.70870i	0.695440	+	0.718585i	\(0.255210\pi\)
							−0.140249	+	0.990116i	\(0.544790\pi\)
\(37\)	−139.046	−	101.023i	−0.0166976	−	0.0121315i	0.579405	−	0.815040i	\(-0.303285\pi\)
							−0.596103	+	0.802908i	\(0.703285\pi\)
\(41\)	−10640.1	−	7730.52i	−0.988526	−	0.718206i	−0.0289280	−	0.999581i	\(-0.509209\pi\)
							−0.959598	+	0.281375i	\(0.909209\pi\)
\(43\)	118.681			0.00978835			0.00489417	−	0.999988i	\(-0.498442\pi\)
							0.00489417	+	0.999988i	\(0.498442\pi\)
\(47\)	−2355.28	+	7248.79i	−0.155524	+	0.478653i	−0.998214	−	0.0597462i	\(-0.980971\pi\)
							0.842690	+	0.538400i	\(0.180971\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.d.b.11.3	✓	28
25.16	even	5	inner	50.6.d.b.41.3	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.d.b.11.3	✓	28	1.1	even	1	trivial
50.6.d.b.41.3	yes	28	25.16	even	5	inner