Embedded newform 143.4.s.a.96.32

• Label: 143.4.s.a.96.32
• Level: $143$
• Weight: $4$
• Character: 143.96
• Analytic conductor: $8.437$
• Analytic rank: $0$
• Dimension: $320$
• 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			96.32
Character	\(\chi\)	\(=\)	143.96
Dual form			143.4.s.a.73.32

$q$-expansion

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

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.77994	+	0.598684i	1.33641	+	0.211667i	0.783406	−	0.621510i	\(-0.213481\pi\)
							0.553006	+	0.833177i	\(0.313481\pi\)
\(3\)	−0.225426	−	0.693791i	−0.0433833	−	0.133520i	0.927019	−	0.375015i	\(-0.122362\pi\)
							−0.970402	+	0.241495i	\(0.922362\pi\)
\(5\)	0.325977	−	0.0516297i	0.0291563	−	0.00461790i	−0.141839	−	0.989890i	\(-0.545302\pi\)
							0.170996	+	0.985272i	\(0.445302\pi\)
\(7\)	15.8324	+	31.0728i	0.854869	+	1.67778i	0.727702	+	0.685894i	\(0.240589\pi\)
							0.127168	+	0.991881i	\(0.459411\pi\)
\(11\)	36.4495	−	1.56072i	0.999085	−	0.0427795i
							\(13\)	31.8383	+	34.3995i	0.679258	+	0.733900i
\(17\)	−25.1624	−	18.2815i	−0.358986	−	0.260819i								0.393643	−	0.919263i	\(-0.371215\pi\)
							−0.752629	+	0.658445i	\(0.771215\pi\)
\(19\)	3.72002	−	7.30095i	0.0449174	−	0.0881554i	−0.867457	−	0.497512i	\(-0.834247\pi\)
							0.912374	+	0.409357i	\(0.134247\pi\)
\(23\)		−	135.009i		−	1.22397i	−0.790871	−	0.611983i	\(-0.790372\pi\)
							0.790871	−	0.611983i	\(-0.209628\pi\)
\(29\)	−97.7741	−	31.7687i	−0.626076	−	0.203424i	−0.0212398	−	0.999774i	\(-0.506761\pi\)
							−0.604836	+	0.796350i	\(0.706761\pi\)
\(31\)	−28.2951	+	178.648i	−0.163934	+	1.03504i	0.759283	+	0.650760i	\(0.225550\pi\)
							−0.923217	+	0.384278i	\(0.874450\pi\)
\(37\)	−179.194	+	91.3039i	−0.796198	+	0.405683i	−0.804255	−	0.594285i	\(-0.797435\pi\)
							0.00805720	+	0.999968i	\(0.497435\pi\)
\(41\)	149.690	−	293.783i	0.570187	−	1.11905i	−0.408320	−	0.912839i	\(-0.633885\pi\)
							0.978507	−	0.206215i	\(-0.0661148\pi\)
\(43\)	−113.926			−0.404037			−0.202019	−	0.979382i	\(-0.564750\pi\)
							−0.202019	+	0.979382i	\(0.564750\pi\)
\(47\)	133.101	−	261.226i	0.413081	−	0.810718i	−0.586918	−	0.809646i	\(-0.699659\pi\)
							0.999999	−	0.00107164i	\(-0.000341115\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.s.a.8.9	✓	320	13.8	odd	4	inner
143.4.s.a.18.9	yes	320	11.7	odd	10	inner
143.4.s.a.73.32	yes	320	143.73	even	20	inner
143.4.s.a.96.32	yes	320	1.1	even	1	trivial