Embedded newform 50.6.d.a.31.4

• Label: 50.6.d.a.31.4
• Level: $50$
• Weight: $6$
• Character: 50.31
• 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			31.4
Character	\(\chi\)	\(=\)	50.31
Dual form			50.6.d.a.21.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.23607 + 3.80423i) q^{2} +(1.10044 - 0.799520i) q^{3} +(-12.9443 + 9.40456i) q^{4} +(-0.782112 - 55.8962i) q^{5} +(4.40178 + 3.19808i) q^{6} -204.325 q^{7} +(-51.7771 - 37.6183i) q^{8} +(-74.5194 + 229.347i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} - 11 q^{3} - 96 q^{4} - 120 q^{5} - 44 q^{6} + 548 q^{7} - 384 q^{8} - 893 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.23607	+	3.80423i	0.218508	+	0.672499i
							\(3\)	1.10044	−	0.799520i	0.0705935	−	0.0512892i	−0.551929	−	0.833891i	\(-0.686108\pi\)
0.622522	+	0.782602i	\(0.286108\pi\)
\(5\)	−0.782112	−	55.8962i	−0.0139909	−	0.999902i
							\(7\)	−204.325			−1.57607			−0.788035	−	0.615631i	\(-0.788901\pi\)
−0.788035	+	0.615631i	\(0.788901\pi\)
\(11\)	−216.464	−	666.208i	−0.539391	−	1.66008i	−0.733964	−	0.679188i	\(-0.762332\pi\)
							0.194573	−	0.980888i	\(-0.437668\pi\)
\(13\)	9.36033	−	28.8081i	0.0153615	−	0.0472777i	−0.943082	−	0.332560i	\(-0.892087\pi\)
							0.958443	+	0.285283i	\(0.0920875\pi\)
\(17\)	−102.245	−	74.2855i	−0.0858067	−	0.0623422i	0.544055	−	0.839050i	\(-0.316888\pi\)
							−0.629861	+	0.776707i	\(0.716888\pi\)
\(19\)	601.013	+	436.661i	0.381944	+	0.277498i	0.762146	−	0.647405i	\(-0.224146\pi\)
							−0.380202	+	0.924903i	\(0.624146\pi\)
\(23\)	−919.400	−	2829.62i	−0.362397	−	1.11534i	−0.951595	−	0.307355i	\(-0.900556\pi\)
							0.589198	−	0.807989i	\(-0.299444\pi\)
\(29\)	−4760.24	+	3458.52i	−1.05108	+	0.763652i	−0.972417	−	0.233250i	\(-0.925064\pi\)
							−0.0786593	+	0.996902i	\(0.525064\pi\)
\(31\)	6899.05	+	5012.45i	1.28939	+	0.936798i	0.999793	−	0.0203496i	\(-0.00647793\pi\)
							0.289599	+	0.957148i	\(0.406478\pi\)
\(37\)	−2989.75	+	9201.51i	−0.359030	+	1.10498i	0.594606	+	0.804017i	\(0.297308\pi\)
							−0.953636	+	0.300963i	\(0.902692\pi\)
\(41\)	2789.60	−	8585.49i	0.259168	−	0.797638i	−0.733812	−	0.679353i	\(-0.762260\pi\)
							0.992980	−	0.118285i	\(-0.0377395\pi\)
\(43\)	−16381.7			−1.35110			−0.675551	−	0.737313i	\(-0.736094\pi\)
							−0.675551	+	0.737313i	\(0.736094\pi\)
\(47\)	14076.4	−	10227.1i	0.929493	−	0.675316i	−0.0163758	−	0.999866i	\(-0.505213\pi\)
							0.945869	+	0.324550i	\(0.105213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.d.a.31.4	yes	24
25.21	even	5	inner	50.6.d.a.21.4	✓	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.d.a.21.4	✓	24	25.21	even	5	inner
50.6.d.a.31.4	yes	24	1.1	even	1	trivial