Embedded newform 50.6.d.b.11.4

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

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.23607 - 2.35114i) q^{2} +(-0.839907 + 2.58497i) q^{3} +(4.94427 - 15.2169i) q^{4} +(-19.5514 + 52.3712i) q^{5} +(3.35963 + 10.3399i) q^{6} +200.961 q^{7} +(-19.7771 - 60.8676i) q^{8} +(190.615 + 138.490i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 28 q^{2} + 7 q^{3} - 112 q^{4} - 145 q^{5} - 28 q^{6} - 236 q^{7} + 448 q^{8} - 446 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.839907	+	2.58497i	−0.0538800	+	0.165826i	−0.974375	−	0.224928i	\(-0.927785\pi\)
0.920495	+	0.390753i	\(0.127785\pi\)
\(5\)	−19.5514	+	52.3712i	−0.349745	+	0.936845i
							\(7\)	200.961			1.55013			0.775063	−	0.631884i	\(-0.217718\pi\)
0.775063	+	0.631884i	\(0.217718\pi\)
\(11\)	273.600	−	198.782i	0.681764	−	0.495331i	−0.192178	−	0.981360i	\(-0.561555\pi\)
							0.873943	+	0.486029i	\(0.161555\pi\)
\(13\)	−211.332	−	153.542i	−0.346822	−	0.251981i	0.400712	−	0.916204i	\(-0.368763\pi\)
							−0.747535	+	0.664223i	\(0.768763\pi\)
\(17\)	477.780	+	1470.46i	0.400964	+	1.23404i	0.924218	+	0.381865i	\(0.124718\pi\)
							−0.523254	+	0.852177i	\(0.675282\pi\)
\(19\)	−201.008	−	618.639i	−0.127741	−	0.393146i	0.866650	−	0.498917i	\(-0.166269\pi\)
							−0.994390	+	0.105771i	\(0.966269\pi\)
\(23\)	994.898	−	722.836i	0.392156	−	0.284918i	−0.374182	−	0.927355i	\(-0.622077\pi\)
							0.766338	+	0.642437i	\(0.222077\pi\)
\(29\)	−1356.48	+	4174.83i	−0.299516	+	0.921814i	0.682152	+	0.731211i	\(0.261044\pi\)
							−0.981667	+	0.190603i	\(0.938956\pi\)
\(31\)	−123.412	−	379.824i	−0.0230650	−	0.0709868i	0.938861	−	0.344295i	\(-0.111882\pi\)
							−0.961926	+	0.273308i	\(0.911882\pi\)
\(37\)	−2501.75	−	1817.63i	−0.300427	−	0.218273i	0.427351	−	0.904086i	\(-0.359447\pi\)
							−0.727778	+	0.685813i	\(0.759447\pi\)
\(41\)	−12359.3	−	8979.57i	−1.14825	−	0.834249i	−0.159999	−	0.987117i	\(-0.551149\pi\)
							−0.988247	+	0.152868i	\(0.951149\pi\)
\(43\)	−18820.9			−1.55228			−0.776140	−	0.630561i	\(-0.782825\pi\)
							−0.776140	+	0.630561i	\(0.782825\pi\)
\(47\)	2166.83	−	6668.80i	0.143080	−	0.440355i	−0.853679	−	0.520800i	\(-0.825634\pi\)
							0.996759	+	0.0804442i	\(0.0256339\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.d.b.11.4	✓	28
25.16	even	5	inner	50.6.d.b.41.4	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.d.b.11.4	✓	28	1.1	even	1	trivial
50.6.d.b.41.4	yes	28	25.16	even	5	inner