Embedded newform 143.4.o.a.32.8

• Label: 143.4.o.a.32.8
• Level: $143$
• Weight: $4$
• Character: 143.32
• 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			32.8
Character	\(\chi\)	\(=\)	143.32
Dual form			143.4.o.a.76.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.61125 + 0.967632i) q^{2} +(3.54274 - 6.13621i) q^{3} +(5.17664 - 2.98873i) q^{4} +(3.98133 - 3.98133i) q^{5} +(-6.85614 + 25.5875i) q^{6} +(-33.6422 - 9.01441i) q^{7} +(5.34682 - 5.34682i) q^{8} +(-11.6020 - 20.0953i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} - 12 q^{4} - 8 q^{5} - 652 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.61125	+	0.967632i	−1.27677	+	0.342110i	−0.832621	−	0.553843i	\(-0.813161\pi\)
							−0.444150	+	0.895953i	\(0.646494\pi\)
\(3\)	3.54274	−	6.13621i	0.681801	−	1.18091i	−0.292630	−	0.956226i	\(-0.594530\pi\)
							0.974431	−	0.224688i	\(-0.0721363\pi\)
\(5\)	3.98133	−	3.98133i	0.356101	−	0.356101i	−0.506273	−	0.862373i	\(-0.668977\pi\)
							0.862373	+	0.506273i	\(0.168977\pi\)
\(7\)	−33.6422	−	9.01441i	−1.81651	−	0.486732i	−0.820163	−	0.572131i	\(-0.806117\pi\)
							−0.996347	+	0.0853983i	\(0.972784\pi\)
\(11\)	−28.4641	+	22.8210i	−0.780203	+	0.625526i
							\(13\)	26.0134	−	38.9911i	0.554986	−	0.831860i
\(17\)	40.5399	+	70.2172i	0.578375	+	1.00178i								0.995666	+	0.0930023i	\(0.0296464\pi\)
							−0.417291	+	0.908773i	\(0.637020\pi\)
\(19\)	−4.85115	+	18.1047i	−0.0585752	+	0.218606i	−0.989009	−	0.147854i	\(-0.952764\pi\)
							0.930434	+	0.366459i	\(0.119430\pi\)
\(23\)	−89.4399	−	51.6381i	−0.810848	−	0.468143i	0.0364024	−	0.999337i	\(-0.488410\pi\)
							−0.847250	+	0.531194i	\(0.821744\pi\)
\(29\)	−173.839	−	100.366i	−1.11314	−	0.642673i	−0.173500	−	0.984834i	\(-0.555508\pi\)
							−0.939641	+	0.342161i	\(0.888841\pi\)
\(31\)	−57.0064	+	57.0064i	−0.330279	+	0.330279i	−0.852692	−	0.522413i	\(-0.825032\pi\)
							0.522413	+	0.852692i	\(0.325032\pi\)
\(37\)	−288.648	+	77.3431i	−1.28253	+	0.343652i	−0.834817	−	0.550527i	\(-0.814427\pi\)
							−0.447709	+	0.894179i	\(0.647760\pi\)
\(41\)	−360.067	+	96.4797i	−1.37154	+	0.367502i	−0.868040	−	0.496494i	\(-0.834620\pi\)
							−0.503498	+	0.863996i	\(0.667954\pi\)
\(43\)	65.0919	+	112.742i	0.230847	+	0.399839i	0.958058	−	0.286576i	\(-0.0925170\pi\)
							−0.727211	+	0.686414i	\(0.759184\pi\)
\(47\)	15.8700	+	15.8700i	0.0492528	+	0.0492528i	0.731304	−	0.682051i	\(-0.238912\pi\)
							−0.682051	+	0.731304i	\(0.738912\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	143.4.o.a.32.8	✓	160
11.10	odd	2	inner	143.4.o.a.32.33	yes	160
13.11	odd	12	inner	143.4.o.a.76.33	yes	160
143.76	even	12	inner	143.4.o.a.76.8	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.o.a.32.8	✓	160	1.1	even	1	trivial
143.4.o.a.32.33	yes	160	11.10	odd	2	inner
143.4.o.a.76.8	yes	160	143.76	even	12	inner
143.4.o.a.76.33	yes	160	13.11	odd	12	inner