Embedded newform 143.4.o.a.76.2

• Label: 143.4.o.a.76.2
• Level: $143$
• Weight: $4$
• Character: 143.76
• Analytic conductor: $8.437$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 143 = 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	143.o (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.43727313082\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(40\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			76.2
Character	\(\chi\)	\(=\)	143.76
Dual form			143.4.o.a.32.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.87783 - 1.30701i) q^{2} +(-4.77756 - 8.27498i) q^{3} +(15.1567 + 8.75074i) q^{4} +(-11.5708 - 11.5708i) q^{5} +(12.4886 + 46.6082i) q^{6} +(20.5354 - 5.50244i) q^{7} +(-33.9281 - 33.9281i) q^{8} +(-32.1502 + 55.6857i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} - 12 q^{4} - 8 q^{5} - 652 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.87783	−	1.30701i	−1.72457	−	0.462098i	−0.745651	−	0.666336i	\(-0.767862\pi\)
							−0.978921	+	0.204238i	\(0.934528\pi\)
\(3\)	−4.77756	−	8.27498i	−0.919442	−	1.59252i	−0.800264	−	0.599647i	\(-0.795308\pi\)
							−0.119178	−	0.992873i	\(-0.538026\pi\)
\(5\)	−11.5708	−	11.5708i	−1.03492	−	1.03492i	−0.999368	−	0.0355515i	\(-0.988681\pi\)
							−0.0355515	−	0.999368i	\(-0.511319\pi\)
\(7\)	20.5354	−	5.50244i	1.10881	−	0.297104i	0.342461	−	0.939532i	\(-0.388739\pi\)
							0.766346	+	0.642428i	\(0.222073\pi\)
\(11\)	19.7391	+	30.6817i	0.541052	+	0.840989i
							\(13\)	−4.84102	+	46.6215i	−0.103281	+	0.994652i
\(17\)	0.695475	−	1.20460i	0.00992220	−	0.0171858i								−0.861022	−	0.508568i	\(-0.830175\pi\)
							0.870944	+	0.491383i	\(0.163508\pi\)
\(19\)	19.0088	+	70.9417i	0.229522	+	0.856587i	0.980542	+	0.196308i	\(0.0628952\pi\)
							−0.751021	+	0.660279i	\(0.770438\pi\)
\(23\)	90.2819	−	52.1243i	0.818481	−	0.472550i	−0.0314112	−	0.999507i	\(-0.510000\pi\)
							0.849892	+	0.526956i	\(0.176667\pi\)
\(29\)	−102.356	+	59.0954i	−0.655416	+	0.378405i	−0.790528	−	0.612426i	\(-0.790194\pi\)
							0.135112	+	0.990830i	\(0.456861\pi\)
\(31\)	5.98325	+	5.98325i	0.0346653	+	0.0346653i	0.724227	−	0.689562i	\(-0.242197\pi\)
							−0.689562	+	0.724227i	\(0.742197\pi\)
\(37\)	86.8630	+	23.2749i	0.385951	+	0.103415i	0.446577	−	0.894745i	\(-0.352643\pi\)
							−0.0606257	+	0.998161i	\(0.519310\pi\)
\(41\)	−40.9487	−	10.9722i	−0.155979	−	0.0417943i	0.179985	−	0.983669i	\(-0.442395\pi\)
							−0.335963	+	0.941875i	\(0.609062\pi\)
\(43\)	−118.603	+	205.427i	−0.420624	+	0.728542i	−0.996001	−	0.0893468i	\(-0.971522\pi\)
							0.575377	+	0.817888i	\(0.304855\pi\)
\(47\)	−175.113	+	175.113i	−0.543464	+	0.543464i	−0.924543	−	0.381078i	\(-0.875553\pi\)
							0.381078	+	0.924543i	\(0.375553\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	143.4.o.a.76.2	yes	160
11.10	odd	2	inner	143.4.o.a.76.39	yes	160
13.6	odd	12	inner	143.4.o.a.32.39	yes	160
143.32	even	12	inner	143.4.o.a.32.2	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.o.a.32.2	✓	160	143.32	even	12	inner
143.4.o.a.32.39	yes	160	13.6	odd	12	inner
143.4.o.a.76.2	yes	160	1.1	even	1	trivial
143.4.o.a.76.39	yes	160	11.10	odd	2	inner