Embedded newform 50.3.f.a.13.1

• Label: 50.3.f.a.13.1
• Level: $50$
• Weight: $3$
• Character: 50.13
• 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			13.1
Root			\(1.78563i\) of defining polynomial
Character	\(\chi\)	\(=\)	50.13
Dual form			50.3.f.a.27.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39680 + 0.221232i) q^{2} +(-1.14941 + 2.25584i) q^{3} +(1.90211 - 0.618034i) q^{4} +(-4.68874 + 1.73657i) q^{5} +(1.10643 - 3.40526i) q^{6} +(-6.58346 + 6.58346i) q^{7} +(-2.52015 + 1.28408i) q^{8} +(1.52238 + 2.09537i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 2 q^{3} + 4 q^{6} - 2 q^{7} + 8 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{19}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39680	+	0.221232i	−0.698401	+	0.110616i
							\(3\)	−1.14941	+	2.25584i	−0.383137	+	0.751948i	−0.999365	−	0.0356190i	\(-0.988660\pi\)
0.616229	+	0.787567i	\(0.288660\pi\)
\(5\)	−4.68874	+	1.73657i	−0.937749	+	0.347314i
							\(7\)	−6.58346	+	6.58346i	−0.940495	+	0.940495i	−0.998326	−	0.0578317i	\(-0.981581\pi\)
0.0578317	+	0.998326i	\(0.481581\pi\)
\(11\)	−3.81477	−	2.77159i	−0.346797	−	0.251963i	0.400727	−	0.916198i	\(-0.368758\pi\)
							−0.747524	+	0.664235i	\(0.768758\pi\)
\(13\)	17.5244	+	2.77559i	1.34803	+	0.213507i	0.788363	−	0.615210i	\(-0.210929\pi\)
							0.559668	+	0.828717i	\(0.310929\pi\)
\(17\)	−7.44532	−	14.6123i	−0.437960	−	0.859545i	−0.999485	−	0.0320852i	\(-0.989785\pi\)
							0.561525	−	0.827460i	\(-0.310215\pi\)
\(19\)	16.0908	+	5.22822i	0.846885	+	0.275170i	0.700141	−	0.714005i	\(-0.253121\pi\)
							0.146744	+	0.989174i	\(0.453121\pi\)
\(23\)	5.91002	+	37.3144i	0.256957	+	1.62237i	0.691960	+	0.721936i	\(0.256747\pi\)
							−0.435003	+	0.900429i	\(0.643253\pi\)
\(29\)	−1.46851	+	0.477149i	−0.0506384	+	0.0164534i	−0.334227	−	0.942493i	\(-0.608475\pi\)
							0.283588	+	0.958946i	\(0.408475\pi\)
\(31\)	−9.29144	+	28.5961i	−0.299724	+	0.922456i	0.681870	+	0.731474i	\(0.261167\pi\)
							−0.981594	+	0.190982i	\(0.938833\pi\)
\(37\)	1.26903	−	8.01234i	0.0342981	−	0.216550i	−0.964587	−	0.263766i	\(-0.915035\pi\)
							0.998885	+	0.0472168i	\(0.0150352\pi\)
\(41\)	−30.0830	+	21.8566i	−0.733733	+	0.533088i	−0.890742	−	0.454509i	\(-0.849815\pi\)
							0.157009	+	0.987597i	\(0.449815\pi\)
\(43\)	−25.9880	−	25.9880i	−0.604371	−	0.604371i	0.337098	−	0.941470i	\(-0.390555\pi\)
							−0.941470	+	0.337098i	\(0.890555\pi\)
\(47\)	41.1918	+	20.9883i	0.876421	+	0.446559i	0.833500	−	0.552520i	\(-0.186334\pi\)
							0.0429212	+	0.999078i	\(0.486334\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.3.f.a.13.1	✓	16
4.3	odd	2		400.3.bg.a.113.2		16
5.2	odd	4		250.3.f.a.207.2		16
5.3	odd	4		250.3.f.c.207.1		16
5.4	even	2		250.3.f.b.43.2		16
25.2	odd	20	inner	50.3.f.a.27.1	yes	16
25.11	even	5		250.3.f.a.93.2		16
25.14	even	10		250.3.f.c.93.1		16
25.23	odd	20		250.3.f.b.157.2		16
100.27	even	20		400.3.bg.a.177.2		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.3.f.a.13.1	✓	16	1.1	even	1	trivial
50.3.f.a.27.1	yes	16	25.2	odd	20	inner
250.3.f.a.93.2		16	25.11	even	5
250.3.f.a.207.2		16	5.2	odd	4
250.3.f.b.43.2		16	5.4	even	2
250.3.f.b.157.2		16	25.23	odd	20
250.3.f.c.93.1		16	25.14	even	10
250.3.f.c.207.1		16	5.3	odd	4
400.3.bg.a.113.2		16	4.3	odd	2
400.3.bg.a.177.2		16	100.27	even	20