Embedded newform 50.4.e.a.9.5

• Label: 50.4.e.a.9.5
• Level: $50$
• Weight: $4$
• Character: 50.9
• Analytic conductor: $2.950$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	50.e (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.95009550029\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			9.5
Character	\(\chi\)	\(=\)	50.9
Dual form			50.4.e.a.39.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.17557 - 1.61803i) q^{2} +(-6.75819 + 2.19587i) q^{3} +(-1.23607 - 3.80423i) q^{4} +(-8.51031 + 7.25084i) q^{5} +(-4.39174 + 13.5164i) q^{6} +23.3487i q^{7} +(-7.60845 - 2.47214i) q^{8} +(19.0078 - 13.8100i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 32 q^{4} - 30 q^{5} - 12 q^{6} + 26 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.17557	−	1.61803i	0.415627	−	0.572061i
							\(3\)	−6.75819	+	2.19587i	−1.30061	+	0.422595i	−0.875795	−	0.482683i	\(-0.839662\pi\)
−0.424819	+	0.905278i	\(0.639662\pi\)
\(5\)	−8.51031	+	7.25084i	−0.761185	+	0.648535i
							\(7\)			23.3487i			1.26071i	0.776307	+	0.630355i	\(0.217091\pi\)
−0.776307	+	0.630355i	\(0.782909\pi\)
\(11\)	−51.2554	−	37.2392i	−1.40492	−	1.02073i	−0.994037	−	0.109043i	\(-0.965221\pi\)
							−0.410880	−	0.911689i	\(-0.634779\pi\)
\(13\)	−3.44706	−	4.74447i	−0.0735418	−	0.101222i	0.770661	−	0.637245i	\(-0.219926\pi\)
							−0.844203	+	0.536023i	\(0.819926\pi\)
\(17\)	70.6774	+	22.9645i	1.00834	+	0.327630i	0.766193	−	0.642610i	\(-0.222148\pi\)
							0.242147	+	0.970240i	\(0.422148\pi\)
\(19\)	−3.29893	+	10.1531i	−0.0398330	+	0.122593i	−0.968996	−	0.247078i	\(-0.920530\pi\)
							0.929163	+	0.369671i	\(0.120530\pi\)
\(23\)	−77.5886	+	106.792i	−0.703406	+	0.968156i	0.296508	+	0.955030i	\(0.404178\pi\)
							−0.999914	+	0.0131252i	\(0.995822\pi\)
\(29\)	−13.5338	−	41.6528i	−0.0866609	−	0.266715i	0.898330	−	0.439321i	\(-0.144781\pi\)
							−0.984991	+	0.172607i	\(0.944781\pi\)
\(31\)	−97.2735	+	299.377i	−0.563575	+	1.73451i	0.108572	+	0.994089i	\(0.465372\pi\)
							−0.672147	+	0.740418i	\(0.734628\pi\)
\(37\)	84.1012	+	115.755i	0.373680	+	0.514326i	0.953896	−	0.300136i	\(-0.0970321\pi\)
							−0.580217	+	0.814462i	\(0.697032\pi\)
\(41\)	105.776	−	76.8508i	0.402913	−	0.292733i	−0.367813	−	0.929900i	\(-0.619893\pi\)
							0.770726	+	0.637166i	\(0.219893\pi\)
\(43\)			223.280i			0.791858i	0.918281	+	0.395929i	\(0.129577\pi\)
							−0.918281	+	0.395929i	\(0.870423\pi\)
\(47\)	−510.915	+	166.006i	−1.58563	+	0.515202i	−0.963499	−	0.267711i	\(-0.913733\pi\)
							−0.622131	+	0.782913i	\(0.713733\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.4.e.a.9.5	✓	32
5.2	odd	4		250.4.d.d.201.8		32
5.3	odd	4		250.4.d.c.201.1		32
5.4	even	2		250.4.e.b.49.4		32
25.2	odd	20		250.4.d.d.51.8		32
25.8	odd	20		1250.4.a.n.1.3		16
25.11	even	5		250.4.e.b.199.4		32
25.14	even	10	inner	50.4.e.a.39.5	yes	32
25.17	odd	20		1250.4.a.m.1.14		16
25.23	odd	20		250.4.d.c.51.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.4.e.a.9.5	✓	32	1.1	even	1	trivial
50.4.e.a.39.5	yes	32	25.14	even	10	inner
250.4.d.c.51.1		32	25.23	odd	20
250.4.d.c.201.1		32	5.3	odd	4
250.4.d.d.51.8		32	25.2	odd	20
250.4.d.d.201.8		32	5.2	odd	4
250.4.e.b.49.4		32	5.4	even	2
250.4.e.b.199.4		32	25.11	even	5
1250.4.a.m.1.14		16	25.17	odd	20
1250.4.a.n.1.3		16	25.8	odd	20