Embedded newform 143.4.s.a.8.9

• Label: 143.4.s.a.8.9
• Level: $143$
• Weight: $4$
• Character: 143.8
• Analytic conductor: $8.437$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 143 = 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	143.s (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.43727313082\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(40\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			8.9
Character	\(\chi\)	\(=\)	143.8
Dual form			143.4.s.a.18.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.598684 + 3.77994i) q^{2} +(-0.225426 - 0.693791i) q^{3} +(-6.32110 - 2.05385i) q^{4} +(0.0516297 + 0.325977i) q^{5} +(2.75745 - 0.436737i) q^{6} +(31.0728 - 15.8324i) q^{7} +(-2.35181 + 4.61569i) q^{8} +(21.4129 - 15.5574i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 10 q^{2} - 12 q^{3} - 6 q^{5} - 10 q^{6} - 10 q^{7} - 10 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/143\mathbb{Z}\right)^\times\).

\(n\)	\(67\)	\(79\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.598684	+	3.77994i	−0.211667	+	1.33641i	0.621510	+	0.783406i	\(0.286519\pi\)
							−0.833177	+	0.553006i	\(0.813481\pi\)
\(3\)	−0.225426	−	0.693791i	−0.0433833	−	0.133520i	0.927019	−	0.375015i	\(-0.122362\pi\)
							−0.970402	+	0.241495i	\(0.922362\pi\)
\(5\)	0.0516297	+	0.325977i	0.00461790	+	0.0291563i	0.989890	−	0.141839i	\(-0.0453016\pi\)
							−0.985272	+	0.170996i	\(0.945302\pi\)
\(7\)	31.0728	−	15.8324i	1.67778	−	0.854869i	0.685894	−	0.727702i	\(-0.259411\pi\)
							0.991881	−	0.127168i	\(-0.0405886\pi\)
\(11\)	−1.56072	−	36.4495i	−0.0427795	−	0.999085i
							\(13\)	22.8773	+	40.9100i	0.488078	+	0.872800i
\(17\)	25.1624	+	18.2815i	0.358986	+	0.260819i								0.752629	−	0.658445i	\(-0.228785\pi\)
							−0.393643	+	0.919263i	\(0.628785\pi\)
\(19\)	7.30095	+	3.72002i	0.0881554	+	0.0449174i	0.497512	−	0.867457i	\(-0.334247\pi\)
							−0.409357	+	0.912374i	\(0.634247\pi\)
\(23\)			135.009i			1.22397i	0.790871	+	0.611983i	\(0.209628\pi\)
							−0.790871	+	0.611983i	\(0.790372\pi\)
\(29\)	−97.7741	−	31.7687i	−0.626076	−	0.203424i	−0.0212398	−	0.999774i	\(-0.506761\pi\)
							−0.604836	+	0.796350i	\(0.706761\pi\)
\(31\)	−178.648	−	28.2951i	−1.03504	−	0.163934i	−0.384278	−	0.923217i	\(-0.625550\pi\)
							−0.650760	+	0.759283i	\(0.725550\pi\)
\(37\)	91.3039	+	179.194i	0.405683	+	0.796198i	0.999968	−	0.00805720i	\(-0.00256472\pi\)
							−0.594285	+	0.804255i	\(0.702565\pi\)
\(41\)	293.783	+	149.690i	1.11905	+	0.570187i	0.912839	−	0.408320i	\(-0.133885\pi\)
							0.206215	+	0.978507i	\(0.433885\pi\)
\(43\)	113.926			0.404037			0.202019	−	0.979382i	\(-0.435250\pi\)
							0.202019	+	0.979382i	\(0.435250\pi\)
\(47\)	−261.226	−	133.101i	−0.810718	−	0.413081i	−0.00107164	−	0.999999i	\(-0.500341\pi\)
							−0.809646	+	0.586918i	\(0.800341\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	143.4.s.a.8.9	✓	320
11.7	odd	10	inner	143.4.s.a.73.32	yes	320
13.5	odd	4	inner	143.4.s.a.96.32	yes	320
143.18	even	20	inner	143.4.s.a.18.9	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.s.a.8.9	✓	320	1.1	even	1	trivial
143.4.s.a.18.9	yes	320	143.18	even	20	inner
143.4.s.a.73.32	yes	320	11.7	odd	10	inner
143.4.s.a.96.32	yes	320	13.5	odd	4	inner