Embedded newform 143.4.q.a.3.10

• Label: 143.4.q.a.3.10
• 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			3.10
Character	\(\chi\)	\(=\)	143.3
Dual form			143.4.q.a.48.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.20952 - 1.42897i) q^{2} +(-1.99079 + 2.21099i) q^{3} +(2.90603 + 3.22747i) q^{4} +(4.33588 + 3.15020i) q^{5} +(9.54892 - 4.25145i) q^{6} +(-17.9899 - 19.9798i) q^{7} +(3.97026 + 12.2192i) q^{8} +(1.89701 + 18.0489i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 3 q^{2} - 3 q^{3} + 149 q^{4} - 12 q^{5} - 35 q^{6} - 3 q^{7} - 44 q^{8} + 369 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.20952	−	1.42897i	−1.13474	−	0.505218i	−0.248584	−	0.968610i	\(-0.579965\pi\)
							−0.886153	+	0.463393i	\(0.846632\pi\)
\(3\)	−1.99079	+	2.21099i	−0.383127	+	0.425506i	−0.903603	−	0.428371i	\(-0.859088\pi\)
							0.520476	+	0.853876i	\(0.325754\pi\)
\(5\)	4.33588	+	3.15020i	0.387813	+	0.281762i	0.764559	−	0.644554i	\(-0.222957\pi\)
							−0.376746	+	0.926317i	\(0.622957\pi\)
\(7\)	−17.9899	−	19.9798i	−0.971364	−	1.07881i	−0.996864	−	0.0791340i	\(-0.974785\pi\)
							0.0255002	−	0.999675i	\(-0.491882\pi\)
\(11\)	−31.3951	+	18.5835i	−0.860544	+	0.509377i
							\(13\)	46.2995	+	7.30477i	0.987782	+	0.155845i
\(17\)	63.2487	−	28.1601i	0.902357	−	0.401755i								0.0975077	−	0.995235i	\(-0.468913\pi\)
							0.804849	+	0.593480i	\(0.202246\pi\)
\(19\)	−4.24444	−	0.902183i	−0.0512495	−	0.0108934i	0.182216	−	0.983259i	\(-0.441673\pi\)
							−0.233465	+	0.972365i	\(0.575006\pi\)
\(23\)	−72.3808	−	125.367i	−0.656193	−	1.13656i	−0.981593	−	0.190984i	\(-0.938832\pi\)
							0.325400	−	0.945576i	\(-0.394501\pi\)
\(29\)	222.373	−	47.2669i	1.42392	−	0.302664i	0.569391	−	0.822067i	\(-0.307179\pi\)
							0.854529	+	0.519403i	\(0.173846\pi\)
\(31\)	103.732	−	75.3657i	0.600994	−	0.436648i	−0.245237	−	0.969463i	\(-0.578866\pi\)
							0.846232	+	0.532815i	\(0.178866\pi\)
\(37\)	−52.0411	+	11.0617i	−0.231230	+	0.0491494i	−0.322069	−	0.946716i	\(-0.604378\pi\)
							0.0908394	+	0.995866i	\(0.471045\pi\)
\(41\)	267.523	−	297.114i	1.01903	−	1.13174i	0.0277937	−	0.999614i	\(-0.491152\pi\)
							0.991232	−	0.132130i	\(-0.0421815\pi\)
\(43\)	138.257	−	239.467i	0.490324	−	0.849266i	−0.509614	−	0.860403i	\(-0.670212\pi\)
							0.999938	+	0.0111371i	\(0.00354511\pi\)
\(47\)	117.913	+	362.899i	0.365945	+	1.12626i	0.949387	+	0.314108i	\(0.101705\pi\)
							−0.583443	+	0.812154i	\(0.698295\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	143.4.q.a.3.10	✓	320
11.4	even	5	inner	143.4.q.a.81.31	yes	320
13.9	even	3	inner	143.4.q.a.113.31	yes	320
143.48	even	15	inner	143.4.q.a.48.10	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.q.a.3.10	✓	320	1.1	even	1	trivial
143.4.q.a.48.10	yes	320	143.48	even	15	inner
143.4.q.a.81.31	yes	320	11.4	even	5	inner
143.4.q.a.113.31	yes	320	13.9	even	3	inner