Embedded newform 50.6.d.b.31.1

• Label: 50.6.d.b.31.1
• Level: $50$
• Weight: $6$
• Character: 50.31
• Analytic conductor: $8.019$
• Analytic rank: $0$
• Dimension: $28$
• 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:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			31.1
Character	\(\chi\)	\(=\)	50.31
Dual form			50.6.d.b.21.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23607 - 3.80423i) q^{2} +(-23.0554 + 16.7507i) q^{3} +(-12.9443 + 9.40456i) q^{4} +(-50.3632 + 24.2601i) q^{5} +(92.2216 + 67.0029i) q^{6} +20.3938 q^{7} +(51.7771 + 37.6183i) q^{8} +(175.874 - 541.284i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 28 q^{2} + 7 q^{3} - 112 q^{4} - 145 q^{5} - 28 q^{6} - 236 q^{7} + 448 q^{8} - 446 q^{9}+O(q^{10}) \)

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
							\(3\)	−23.0554	+	16.7507i	−1.47901	+	1.07456i	−0.501130	+	0.865372i	\(0.667082\pi\)
−0.977875	+	0.209188i	\(0.932918\pi\)
\(5\)	−50.3632	+	24.2601i	−0.900923	+	0.433978i
							\(7\)	20.3938			0.157309			0.0786546	−	0.996902i	\(-0.474938\pi\)
0.0786546	+	0.996902i	\(0.474938\pi\)
\(11\)	−57.1025	−	175.743i	−0.142290	−	0.437923i	0.854363	−	0.519677i	\(-0.173948\pi\)
							−0.996653	+	0.0817544i	\(0.973948\pi\)
\(13\)	−42.6701	+	131.325i	−0.0700269	+	0.215521i	−0.979945	−	0.199267i	\(-0.936144\pi\)
							0.909918	+	0.414787i	\(0.136144\pi\)
\(17\)	1758.93	+	1277.94i	1.47614	+	1.07248i	0.978776	+	0.204934i	\(0.0656978\pi\)
							0.497362	+	0.867543i	\(0.334302\pi\)
\(19\)	−1883.38	−	1368.36i	−1.19689	−	0.869591i	−0.202914	−	0.979197i	\(-0.565041\pi\)
							−0.993975	+	0.109606i	\(0.965041\pi\)
\(23\)	−627.573	−	1931.47i	−0.247368	−	0.761322i	−0.995238	−	0.0974760i	\(-0.968923\pi\)
							0.747869	−	0.663846i	\(-0.231077\pi\)
\(29\)	1243.88	−	903.731i	0.274652	−	0.199547i	−0.441929	−	0.897050i	\(-0.645706\pi\)
							0.716582	+	0.697503i	\(0.245706\pi\)
\(31\)	5987.07	+	4349.86i	1.11895	+	0.812964i	0.984049	−	0.177895i	\(-0.0569289\pi\)
							0.134899	+	0.990859i	\(0.456929\pi\)
\(37\)	−268.579	+	826.601i	−0.0322528	+	0.0992639i	−0.965887	−	0.258964i	\(-0.916619\pi\)
							0.933634	+	0.358228i	\(0.116619\pi\)
\(41\)	4845.54	−	14913.0i	0.450176	−	1.38550i	−0.426530	−	0.904474i	\(-0.640264\pi\)
							0.876706	−	0.481027i	\(-0.159736\pi\)
\(43\)	−15841.8			−1.30658			−0.653288	−	0.757110i	\(-0.726611\pi\)
							−0.653288	+	0.757110i	\(0.726611\pi\)
\(47\)	−1359.77	+	987.932i	−0.0897886	+	0.0652353i	−0.631774	−	0.775153i	\(-0.717673\pi\)
							0.541985	+	0.840388i	\(0.317673\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.d.b.31.1	yes	28
25.21	even	5	inner	50.6.d.b.21.1	✓	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.d.b.21.1	✓	28	25.21	even	5	inner
50.6.d.b.31.1	yes	28	1.1	even	1	trivial