Embedded newform 143.4.o.a.32.9

• Label: 143.4.o.a.32.9
• 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			32.9
Character	\(\chi\)	\(=\)	143.32
Dual form			143.4.o.a.76.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.51428 + 0.941650i) q^{2} +(2.24550 - 3.88932i) q^{3} +(4.53529 - 2.61845i) q^{4} +(14.9878 - 14.9878i) q^{5} +(-4.22895 + 15.7827i) q^{6} +(11.9754 + 3.20880i) q^{7} +(7.10844 - 7.10844i) q^{8} +(3.41546 + 5.91574i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} - 12 q^{4} - 8 q^{5} - 652 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.51428	+	0.941650i	−1.24249	+	0.332923i	−0.819430	−	0.573179i	\(-0.805710\pi\)
							−0.423057	+	0.906103i	\(0.639043\pi\)
\(3\)	2.24550	−	3.88932i	0.432147	−	0.748500i	−0.564911	−	0.825152i	\(-0.691090\pi\)
							0.997058	+	0.0766516i	\(0.0244229\pi\)
\(5\)	14.9878	−	14.9878i	1.34055	−	1.34055i	0.445039	−	0.895511i	\(-0.353190\pi\)
							0.895511	−	0.445039i	\(-0.146810\pi\)
\(7\)	11.9754	+	3.20880i	0.646611	+	0.173259i	0.567196	−	0.823583i	\(-0.308028\pi\)
							0.0794151	+	0.996842i	\(0.474695\pi\)
\(11\)	−14.5776	−	33.4439i	−0.399574	−	0.916701i
							\(13\)	24.7955	+	39.7767i	0.529003	+	0.848620i
\(17\)	−16.0862	−	27.8622i	−0.229499	−	0.397504i								0.728161	−	0.685407i	\(-0.240375\pi\)
							−0.957660	+	0.287902i	\(0.907042\pi\)
\(19\)	−2.93244	+	10.9440i	−0.0354077	+	0.132144i	−0.981367	−	0.192141i	\(-0.938457\pi\)
							0.945960	+	0.324285i	\(0.105124\pi\)
\(23\)	93.7675	+	54.1367i	0.850082	+	0.490795i	0.860678	−	0.509149i	\(-0.170040\pi\)
							−0.0105968	+	0.999944i	\(0.503373\pi\)
\(29\)	−97.5935	−	56.3456i	−0.624919	−	0.360797i	0.153863	−	0.988092i	\(-0.450829\pi\)
							−0.778782	+	0.627295i	\(0.784162\pi\)
\(31\)	59.6798	−	59.6798i	0.345768	−	0.345768i	−0.512762	−	0.858531i	\(-0.671378\pi\)
							0.858531	+	0.512762i	\(0.171378\pi\)
\(37\)	−161.865	+	43.3716i	−0.719200	+	0.192709i	−0.599815	−	0.800139i	\(-0.704759\pi\)
							−0.119386	+	0.992848i	\(0.538092\pi\)
\(41\)	179.980	−	48.2255i	0.685565	−	0.183697i	0.100809	−	0.994906i	\(-0.467857\pi\)
							0.584756	+	0.811209i	\(0.301190\pi\)
\(43\)	−89.1911	−	154.484i	−0.316314	−	0.547873i	0.663402	−	0.748264i	\(-0.269112\pi\)
							−0.979716	+	0.200391i	\(0.935779\pi\)
\(47\)	73.9184	+	73.9184i	0.229406	+	0.229406i	0.812445	−	0.583038i	\(-0.198136\pi\)
							−0.583038	+	0.812445i	\(0.698136\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	143.4.o.a.32.9	✓	160
11.10	odd	2	inner	143.4.o.a.32.32	yes	160
13.11	odd	12	inner	143.4.o.a.76.32	yes	160
143.76	even	12	inner	143.4.o.a.76.9	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.o.a.32.9	✓	160	1.1	even	1	trivial
143.4.o.a.32.32	yes	160	11.10	odd	2	inner
143.4.o.a.76.9	yes	160	143.76	even	12	inner
143.4.o.a.76.32	yes	160	13.11	odd	12	inner