Embedded newform 50.6.d.a.31.6

• Label: 50.6.d.a.31.6
• 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			31.6
Character	\(\chi\)	\(=\)	50.31
Dual form			50.6.d.a.21.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.23607 + 3.80423i) q^{2} +(19.6428 - 14.2713i) q^{3} +(-12.9443 + 9.40456i) q^{4} +(-42.2315 - 36.6265i) q^{5} +(78.5712 + 57.0853i) q^{6} +208.145 q^{7} +(-51.7771 - 37.6183i) q^{8} +(107.078 - 329.551i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} - 11 q^{3} - 96 q^{4} - 120 q^{5} - 44 q^{6} + 548 q^{7} - 384 q^{8} - 893 q^{9}+O(q^{10}) \)

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
							\(3\)	19.6428	−	14.2713i	1.26009	−	0.915506i	0.261324	−	0.965251i	\(-0.415841\pi\)
0.998762	+	0.0497448i	\(0.0158408\pi\)
\(5\)	−42.2315	−	36.6265i	−0.755460	−	0.655195i
							\(7\)	208.145			1.60554			0.802768	−	0.596292i	\(-0.203360\pi\)
0.802768	+	0.596292i	\(0.203360\pi\)
\(11\)	−78.0392	−	240.180i	−0.194460	−	0.598487i	−0.999982	−	0.00592157i	\(-0.998115\pi\)
							0.805522	−	0.592566i	\(-0.201885\pi\)
\(13\)	146.158	−	449.827i	0.239863	−	0.738223i	−0.756576	−	0.653906i	\(-0.773129\pi\)
							0.996439	−	0.0843168i	\(-0.0268708\pi\)
\(17\)	1742.17	+	1265.76i	1.46207	+	1.06226i	0.982817	+	0.184584i	\(0.0590939\pi\)
							0.479257	+	0.877675i	\(0.340906\pi\)
\(19\)	−1523.40	−	1106.82i	−0.968123	−	0.703382i	−0.0130999	−	0.999914i	\(-0.504170\pi\)
							−0.955023	+	0.296532i	\(0.904170\pi\)
\(23\)	935.235	+	2878.36i	0.368639	+	1.13455i	0.947671	+	0.319249i	\(0.103431\pi\)
							−0.579032	+	0.815305i	\(0.696569\pi\)
\(29\)	−6038.94	+	4387.55i	−1.33342	+	0.968784i	−0.333758	+	0.942659i	\(0.608317\pi\)
							−0.999659	+	0.0261255i	\(0.991683\pi\)
\(31\)	−2495.39	−	1813.01i	−0.466374	−	0.338841i	0.329652	−	0.944102i	\(-0.393069\pi\)
							−0.796027	+	0.605262i	\(0.793069\pi\)
\(37\)	1100.98	−	3388.45i	0.132213	−	0.406909i	−0.862933	−	0.505318i	\(-0.831375\pi\)
							0.995146	+	0.0984088i	\(0.0313753\pi\)
\(41\)	−2421.84	+	7453.67i	−0.225002	+	0.692485i	0.773289	+	0.634053i	\(0.218610\pi\)
							−0.998291	+	0.0584320i	\(0.981390\pi\)
\(43\)	−1729.68			−0.142658			−0.0713289	−	0.997453i	\(-0.522724\pi\)
							−0.0713289	+	0.997453i	\(0.522724\pi\)
\(47\)	−14675.1	+	10662.1i	−0.969030	+	0.704041i	−0.955230	−	0.295863i	\(-0.904393\pi\)
							−0.0137995	+	0.999905i	\(0.504393\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.d.a.31.6	yes	24
25.21	even	5	inner	50.6.d.a.21.6	✓	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.d.a.21.6	✓	24	25.21	even	5	inner
50.6.d.a.31.6	yes	24	1.1	even	1	trivial