Embedded newform 143.4.q.a.3.16

• Label: 143.4.q.a.3.16
• Level: $143$
• Weight: $4$
• Character: 143.3
• 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.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.43727313082\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(40\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			3.16
Character	\(\chi\)	\(=\)	143.3
Dual form			143.4.q.a.48.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34373 - 0.598266i) q^{2} +(-0.820388 + 0.911134i) q^{3} +(-3.90536 - 4.33735i) q^{4} +(-16.3994 - 11.9149i) q^{5} +(1.64748 - 0.733505i) q^{6} +(-14.7799 - 16.4148i) q^{7} +(6.28910 + 19.3559i) q^{8} +(2.66514 + 25.3571i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 3 q^{2} - 3 q^{3} + 149 q^{4} - 12 q^{5} - 35 q^{6} - 3 q^{7} - 44 q^{8} + 369 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34373	−	0.598266i	−0.475079	−	0.211519i	0.155207	−	0.987882i	\(-0.450396\pi\)
							−0.630286	+	0.776363i	\(0.717062\pi\)
\(3\)	−0.820388	+	0.911134i	−0.157884	+	0.175348i	−0.816897	−	0.576784i	\(-0.804308\pi\)
							0.659013	+	0.752132i	\(0.270974\pi\)
\(5\)	−16.3994	−	11.9149i	−1.46681	−	1.06570i	−0.981523	−	0.191345i	\(-0.938715\pi\)
							−0.485287	−	0.874355i	\(-0.661285\pi\)
\(7\)	−14.7799	−	16.4148i	−0.798041	−	0.886315i	0.197533	−	0.980296i	\(-0.436707\pi\)
							−0.995574	+	0.0939816i	\(0.970041\pi\)
\(11\)	35.0585	−	10.0947i	0.960957	−	0.276696i
							\(13\)	40.5424	+	23.5226i	0.864957	+	0.501846i
\(17\)	−24.4658	+	10.8929i	−0.349049	+	0.155407i								−0.573770	−	0.819017i	\(-0.694520\pi\)
							0.224721	+	0.974423i	\(0.427853\pi\)
\(19\)	−76.7546	−	16.3147i	−0.926774	−	0.196992i	−0.280287	−	0.959916i	\(-0.590430\pi\)
							−0.646488	+	0.762924i	\(0.723763\pi\)
\(23\)	−22.8550	−	39.5860i	−0.207200	−	0.358880i	0.743632	−	0.668589i	\(-0.233102\pi\)
							−0.950831	+	0.309709i	\(0.899768\pi\)
\(29\)	−91.5377	+	19.4569i	−0.586142	+	0.124588i	−0.491431	−	0.870916i	\(-0.663526\pi\)
							−0.0947109	+	0.995505i	\(0.530193\pi\)
\(31\)	187.160	−	135.980i	1.08435	−	0.787827i	0.105915	−	0.994375i	\(-0.466223\pi\)
							0.978436	+	0.206548i	\(0.0662229\pi\)
\(37\)	171.190	−	36.3875i	0.760634	−	0.161678i	0.188764	−	0.982022i	\(-0.439552\pi\)
							0.571869	+	0.820345i	\(0.306218\pi\)
\(41\)	−189.832	+	210.830i	−0.723094	+	0.803077i	−0.986872	−	0.161506i	\(-0.948365\pi\)
							0.263778	+	0.964584i	\(0.415031\pi\)
\(43\)	−200.125	+	346.626i	−0.709738	+	1.22930i	0.255217	+	0.966884i	\(0.417853\pi\)
							−0.964954	+	0.262418i	\(0.915480\pi\)
\(47\)	101.530	+	312.477i	0.315099	+	0.969776i	0.975713	+	0.219051i	\(0.0702960\pi\)
							−0.660614	+	0.750726i	\(0.729704\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	143.4.q.a.3.16	✓	320
11.4	even	5	inner	143.4.q.a.81.25	yes	320
13.9	even	3	inner	143.4.q.a.113.25	yes	320
143.48	even	15	inner	143.4.q.a.48.16	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.q.a.3.16	✓	320	1.1	even	1	trivial
143.4.q.a.48.16	yes	320	143.48	even	15	inner
143.4.q.a.81.25	yes	320	11.4	even	5	inner
143.4.q.a.113.25	yes	320	13.9	even	3	inner