Embedded newform 50.6.d.b.11.5

• Label: 50.6.d.b.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	50.11
Dual form			50.6.d.b.41.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.23607 - 2.35114i) q^{2} +(1.57970 - 4.86182i) q^{3} +(4.94427 - 15.2169i) q^{4} +(-52.9218 - 18.0078i) q^{5} +(-6.31881 - 19.4473i) q^{6} -251.763 q^{7} +(-19.7771 - 60.8676i) q^{8} +(175.449 + 127.471i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 28 q^{2} + 7 q^{3} - 112 q^{4} - 145 q^{5} - 28 q^{6} - 236 q^{7} + 448 q^{8} - 446 q^{9}+O(q^{10}) \)

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
							\(3\)	1.57970	−	4.86182i	0.101338	−	0.311886i	−0.887516	−	0.460778i	\(-0.847571\pi\)
0.988854	+	0.148892i	\(0.0475706\pi\)
\(5\)	−52.9218	−	18.0078i	−0.946695	−	0.322132i
							\(7\)	−251.763			−1.94199			−0.970995	−	0.239100i	\(-0.923148\pi\)
−0.970995	+	0.239100i	\(0.923148\pi\)
\(11\)	261.800	−	190.209i	0.652361	−	0.473968i	−0.211714	−	0.977332i	\(-0.567905\pi\)
							0.864074	+	0.503364i	\(0.167905\pi\)
\(13\)	−631.790	−	459.022i	−1.03685	−	0.753313i	−0.0671790	−	0.997741i	\(-0.521400\pi\)
							−0.969667	+	0.244428i	\(0.921400\pi\)
\(17\)	190.896	+	587.519i	0.160205	+	0.493060i	0.998651	−	0.0519247i	\(-0.0165356\pi\)
							−0.838446	+	0.544984i	\(0.816536\pi\)
\(19\)	−623.259	−	1918.19i	−0.396081	−	1.21901i	−0.928116	−	0.372290i	\(-0.878573\pi\)
							0.532035	−	0.846722i	\(-0.321427\pi\)
\(23\)	−999.815	+	726.408i	−0.394094	+	0.286326i	−0.767131	−	0.641490i	\(-0.778316\pi\)
							0.373037	+	0.927816i	\(0.378316\pi\)
\(29\)	2627.86	−	8087.71i	0.580239	−	1.78579i	−0.0373651	−	0.999302i	\(-0.511896\pi\)
							0.617604	−	0.786489i	\(-0.288104\pi\)
\(31\)	63.5037	+	195.444i	0.0118685	+	0.0365274i	0.956815	−	0.290696i	\(-0.0938869\pi\)
							−0.944947	+	0.327224i	\(0.893887\pi\)
\(37\)	−1116.40	−	811.111i	−0.134065	−	0.0974038i	0.518732	−	0.854937i	\(-0.326404\pi\)
							−0.652797	+	0.757533i	\(0.726404\pi\)
\(41\)	−708.695	−	514.897i	−0.0658415	−	0.0478366i	0.554378	−	0.832265i	\(-0.312956\pi\)
							−0.620219	+	0.784429i	\(0.712956\pi\)
\(43\)	10928.3			0.901323			0.450661	−	0.892695i	\(-0.351188\pi\)
							0.450661	+	0.892695i	\(0.351188\pi\)
\(47\)	−2492.49	+	7671.09i	−0.164584	+	0.506538i	−0.999005	−	0.0445890i	\(-0.985802\pi\)
							0.834421	+	0.551127i	\(0.185802\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.d.b.11.5	✓	28
25.16	even	5	inner	50.6.d.b.41.5	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.d.b.11.5	✓	28	1.1	even	1	trivial
50.6.d.b.41.5	yes	28	25.16	even	5	inner