Embedded newform 50.6.d.a.41.6

• Label: 50.6.d.a.41.6
• Level: $50$
• Weight: $6$
• Character: 50.41
• 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			41.6
Character	\(\chi\)	\(=\)	50.41
Dual form			50.6.d.a.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.23607 - 2.35114i) q^{2} +(9.12748 + 28.0915i) q^{3} +(4.94427 + 15.2169i) q^{4} +(30.2654 - 47.0001i) q^{5} +(36.5099 - 112.366i) q^{6} +204.125 q^{7} +(19.7771 - 60.8676i) q^{8} +(-509.231 + 369.978i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} - 11 q^{3} - 96 q^{4} - 120 q^{5} - 44 q^{6} + 548 q^{7} - 384 q^{8} - 893 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)

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
							\(3\)	9.12748	+	28.0915i	0.585528	+	1.80207i	0.597138	+	0.802139i	\(0.296304\pi\)
−0.0116096	+	0.999933i	\(0.503696\pi\)
\(5\)	30.2654	−	47.0001i	0.541403	−	0.840763i
							\(7\)	204.125			1.57453			0.787266	−	0.616614i	\(-0.211496\pi\)
0.787266	+	0.616614i	\(0.211496\pi\)
\(11\)	329.087	+	239.095i	0.820028	+	0.595785i	0.916720	−	0.399529i	\(-0.130826\pi\)
							−0.0966928	+	0.995314i	\(0.530826\pi\)
\(13\)	−478.940	+	347.971i	−0.786001	+	0.571063i	−0.906774	−	0.421617i	\(-0.861463\pi\)
							0.120773	+	0.992680i	\(0.461463\pi\)
\(17\)	107.089	−	329.587i	0.0898718	−	0.276597i	−0.896012	−	0.444031i	\(-0.853548\pi\)
							0.985883	+	0.167434i	\(0.0535481\pi\)
\(19\)	−286.576	+	881.989i	−0.182119	+	0.560505i	−0.999887	−	0.0150436i	\(-0.995211\pi\)
							0.817768	+	0.575548i	\(0.195211\pi\)
\(23\)	251.599	+	182.797i	0.0991720	+	0.0720527i	0.636266	−	0.771470i	\(-0.280478\pi\)
							−0.537094	+	0.843522i	\(0.680478\pi\)
\(29\)	147.765	+	454.772i	0.0326269	+	0.100415i	0.966044	−	0.258379i	\(-0.0831882\pi\)
							−0.933417	+	0.358794i	\(0.883188\pi\)
\(31\)	154.172	−	474.491i	0.0288138	−	0.0886796i	−0.935615	−	0.353021i	\(-0.885154\pi\)
							0.964429	+	0.264341i	\(0.0851545\pi\)
\(37\)	−4475.50	+	3251.64i	−0.537449	+	0.390479i	−0.823137	−	0.567843i	\(-0.807778\pi\)
							0.285688	+	0.958323i	\(0.407778\pi\)
\(41\)	13046.6	−	9478.92i	1.21210	−	0.880641i	0.216679	−	0.976243i	\(-0.430477\pi\)
							0.995420	+	0.0956015i	\(0.0304774\pi\)
\(43\)	4942.22			0.407616			0.203808	−	0.979011i	\(-0.434668\pi\)
							0.203808	+	0.979011i	\(0.434668\pi\)
\(47\)	4706.86	+	14486.2i	0.310804	+	0.956557i	0.977447	+	0.211180i	\(0.0677305\pi\)
							−0.666643	+	0.745377i	\(0.732269\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.d.a.41.6	yes	24
25.11	even	5	inner	50.6.d.a.11.6	✓	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.d.a.11.6	✓	24	25.11	even	5	inner
50.6.d.a.41.6	yes	24	1.1	even	1	trivial