Embedded newform 50.3.f.a.47.1

• Label: 50.3.f.a.47.1
• Level: $50$
• Weight: $3$
• Character: 50.47
• Analytic conductor: $1.362$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	50.f (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.36240132180\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(2\) over \(\Q(\zeta_{20})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial:	\( x^{16} + 30x^{14} + 351x^{12} + 2130x^{10} + 7341x^{8} + 14480x^{6} + 15196x^{4} + 6560x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 5^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			47.1
Root			\(-1.84816i\) of defining polynomial
Character	\(\chi\)	\(=\)	50.47
Dual form			50.3.f.a.33.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.26007 - 0.642040i) q^{2} +(-0.742607 + 0.117617i) q^{3} +(1.17557 - 1.61803i) q^{4} +(4.02861 - 2.96147i) q^{5} +(-0.860225 + 0.624990i) q^{6} +(2.71368 + 2.71368i) q^{7} +(0.442463 - 2.79360i) q^{8} +(-8.02188 + 2.60647i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 2 q^{3} + 4 q^{6} - 2 q^{7} + 8 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{17}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.26007	−	0.642040i	0.630037	−	0.321020i
							\(3\)	−0.742607	+	0.117617i	−0.247536	+	0.0392058i	−0.278969	−	0.960300i	\(-0.589993\pi\)
0.0314335	+	0.999506i	\(0.489993\pi\)
\(5\)	4.02861	−	2.96147i	0.805722	−	0.592293i
							\(7\)	2.71368	+	2.71368i	0.387668	+	0.387668i	0.873855	−	0.486187i	\(-0.161613\pi\)
−0.486187	+	0.873855i	\(0.661613\pi\)
\(11\)	−4.47760	+	13.7806i	−0.407054	+	1.25278i	0.512114	+	0.858918i	\(0.328863\pi\)
							−0.919168	+	0.393866i	\(0.871137\pi\)
\(13\)	−16.6235	−	8.47008i	−1.27873	−	0.651545i	−0.323169	−	0.946341i	\(-0.604748\pi\)
							−0.955560	+	0.294797i	\(0.904748\pi\)
\(17\)	10.5727	+	1.67455i	0.621925	+	0.0985032i	0.459441	−	0.888208i	\(-0.348050\pi\)
							0.162483	+	0.986711i	\(0.448050\pi\)
\(19\)	11.3496	+	15.6214i	0.597346	+	0.822176i	0.995462	−	0.0951586i	\(-0.0303358\pi\)
							−0.398116	+	0.917335i	\(0.630336\pi\)
\(23\)	−8.15541	−	16.0059i	−0.354583	−	0.695908i	0.642965	−	0.765896i	\(-0.277704\pi\)
							−0.997548	+	0.0699875i	\(0.977704\pi\)
\(29\)	−15.1793	+	20.8925i	−0.523423	+	0.720430i	−0.986110	−	0.166091i	\(-0.946885\pi\)
							0.462687	+	0.886522i	\(0.346885\pi\)
\(31\)	6.46961	−	4.70045i	0.208697	−	0.151627i	−0.478527	−	0.878073i	\(-0.658829\pi\)
							0.687224	+	0.726446i	\(0.258829\pi\)
\(37\)	31.9799	−	62.7641i	0.864322	−	1.69633i	0.159193	−	0.987247i	\(-0.449111\pi\)
							0.705128	−	0.709080i	\(-0.250889\pi\)
\(41\)	1.20943	+	3.72225i	0.0294984	+	0.0907867i	0.964722	−	0.263271i	\(-0.0848015\pi\)
							−0.935223	+	0.354058i	\(0.884801\pi\)
\(43\)	9.83401	−	9.83401i	0.228698	−	0.228698i	−0.583451	−	0.812149i	\(-0.698298\pi\)
							0.812149	+	0.583451i	\(0.198298\pi\)
\(47\)	5.34000	+	33.7155i	0.113617	+	0.717350i	0.977070	+	0.212920i	\(0.0682974\pi\)
							−0.863453	+	0.504430i	\(0.831703\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.3.f.a.47.1	yes	16
4.3	odd	2		400.3.bg.a.97.2		16
5.2	odd	4		250.3.f.c.243.2		16
5.3	odd	4		250.3.f.a.243.1		16
5.4	even	2		250.3.f.b.7.2		16
25.6	even	5		250.3.f.a.107.1		16
25.8	odd	20	inner	50.3.f.a.33.1	✓	16
25.17	odd	20		250.3.f.b.143.2		16
25.19	even	10		250.3.f.c.107.2		16
100.83	even	20		400.3.bg.a.33.2		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.3.f.a.33.1	✓	16	25.8	odd	20	inner
50.3.f.a.47.1	yes	16	1.1	even	1	trivial
250.3.f.a.107.1		16	25.6	even	5
250.3.f.a.243.1		16	5.3	odd	4
250.3.f.b.7.2		16	5.4	even	2
250.3.f.b.143.2		16	25.17	odd	20
250.3.f.c.107.2		16	25.19	even	10
250.3.f.c.243.2		16	5.2	odd	4
400.3.bg.a.33.2		16	100.83	even	20
400.3.bg.a.97.2		16	4.3	odd	2