Embedded newform 143.4.q.a.3.20

• Label: 143.4.q.a.3.20
• 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			3.20
Character	\(\chi\)	\(=\)	143.3
Dual form			143.4.q.a.48.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.257195 - 0.114511i) q^{2} +(6.30663 - 7.00422i) q^{3} +(-5.30001 - 5.88626i) q^{4} +(-15.3363 - 11.1425i) q^{5} +(-2.42409 + 1.07927i) q^{6} +(15.1685 + 16.8463i) q^{7} +(1.38509 + 4.26287i) q^{8} +(-6.46326 - 61.4939i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 3 q^{2} - 3 q^{3} + 149 q^{4} - 12 q^{5} - 35 q^{6} - 3 q^{7} - 44 q^{8} + 369 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.257195	−	0.114511i	−0.0909322	−	0.0404856i	0.360767	−	0.932656i	\(-0.382515\pi\)
							−0.451699	+	0.892171i	\(0.649182\pi\)
\(3\)	6.30663	−	7.00422i	1.21371	−	1.34796i	0.293784	−	0.955872i	\(-0.405085\pi\)
							0.919927	−	0.392090i	\(-0.128248\pi\)
\(5\)	−15.3363	−	11.1425i	−1.37172	−	0.996615i	−0.997600	−	0.0692355i	\(-0.977944\pi\)
							−0.374122	−	0.927379i	\(-0.622056\pi\)
\(7\)	15.1685	+	16.8463i	0.819022	+	0.909616i	0.997230	−	0.0743764i	\(-0.0236966\pi\)
							−0.178208	+	0.983993i	\(0.557030\pi\)
\(11\)	28.5111	+	22.7622i	0.781493	+	0.623914i
							\(13\)	−35.2254	−	30.9221i	−0.751520	−	0.659711i
\(17\)	11.4971	−	5.11884i	0.164027	−	0.0730295i								−0.323081	−	0.946371i	\(-0.604719\pi\)
							0.487108	+	0.873342i	\(0.338052\pi\)
\(19\)	26.0516	+	5.53743i	0.314560	+	0.0668618i	0.362486	−	0.931989i	\(-0.381928\pi\)
							−0.0479264	+	0.998851i	\(0.515261\pi\)
\(23\)	−58.3550	−	101.074i	−0.529037	−	0.916319i	−0.999427	−	0.0338604i	\(-0.989220\pi\)
							0.470389	−	0.882459i	\(-0.344113\pi\)
\(29\)	−143.047	+	30.4055i	−0.915968	+	0.194695i	−0.641698	−	0.766958i	\(-0.721770\pi\)
							−0.274270	+	0.961653i	\(0.588436\pi\)
\(31\)	83.9028	−	60.9590i	0.486110	−	0.353179i	−0.317577	−	0.948233i	\(-0.602869\pi\)
							0.803686	+	0.595053i	\(0.202869\pi\)
\(37\)	−65.0128	+	13.8189i	−0.288866	+	0.0614004i	−0.350066	−	0.936725i	\(-0.613841\pi\)
							0.0611996	+	0.998126i	\(0.480507\pi\)
\(41\)	216.812	−	240.794i	0.825863	−	0.917213i	−0.171829	−	0.985127i	\(-0.554968\pi\)
							0.997691	+	0.0679135i	\(0.0216342\pi\)
\(43\)	31.1640	−	53.9777i	0.110523	−	0.191431i	−0.805459	−	0.592652i	\(-0.798081\pi\)
							0.915981	+	0.401221i	\(0.131414\pi\)
\(47\)	−142.232	−	437.745i	−0.441418	−	1.35855i	−0.886364	−	0.462988i	\(-0.846777\pi\)
							0.444946	−	0.895557i	\(-0.353223\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	143.4.q.a.3.20	✓	320
11.4	even	5	inner	143.4.q.a.81.21	yes	320
13.9	even	3	inner	143.4.q.a.113.21	yes	320
143.48	even	15	inner	143.4.q.a.48.20	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.q.a.3.20	✓	320	1.1	even	1	trivial
143.4.q.a.48.20	yes	320	143.48	even	15	inner
143.4.q.a.81.21	yes	320	11.4	even	5	inner
143.4.q.a.113.21	yes	320	13.9	even	3	inner