Embedded newform 143.4.w.a.2.1

• Label: 143.4.w.a.2.1
• Level: $143$
• Weight: $4$
• Character: 143.2
• Analytic conductor: $8.437$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 143 = 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	143.w (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.43727313082\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(40\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			2.1
Character	\(\chi\)	\(=\)	143.2
Dual form			143.4.w.a.72.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.61575 - 2.99751i) q^{2} +(-0.866584 - 0.385828i) q^{3} +(9.06625 + 20.3631i) q^{4} +(-16.7403 - 8.52960i) q^{5} +(2.84342 + 4.37848i) q^{6} +(9.35497 - 24.3705i) q^{7} +(12.3033 - 77.6801i) q^{8} +(-17.4644 - 19.3962i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 20 q^{2} - 6 q^{3} - 18 q^{4} - 12 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 642 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.61575	−	2.99751i	−1.63192	−	1.05978i	−0.941852	−	0.336029i	\(-0.890916\pi\)
							−0.690064	−	0.723749i	\(-0.742417\pi\)
\(3\)	−0.866584	−	0.385828i	−0.166774	−	0.0742527i	0.321652	−	0.946858i	\(-0.395762\pi\)
							−0.488426	+	0.872605i	\(0.662429\pi\)
\(5\)	−16.7403	−	8.52960i	−1.49730	−	0.762910i	−0.502479	−	0.864590i	\(-0.667578\pi\)
							−0.994817	+	0.101679i	\(0.967578\pi\)
\(7\)	9.35497	−	24.3705i	0.505121	−	1.31588i	−0.409772	−	0.912188i	\(-0.634392\pi\)
							0.914893	−	0.403697i	\(-0.132275\pi\)
\(11\)	−32.0018	+	17.5181i	−0.877174	+	0.480173i
							\(13\)	16.2780	−	43.9548i	0.347284	−	0.937760i
\(17\)	−30.1223	−	6.40268i	−0.429748	−	0.0913458i								−0.0120410	−	0.999928i	\(-0.503833\pi\)
							−0.417707	+	0.908582i	\(0.637166\pi\)
\(19\)	−2.71729	+	3.35557i	−0.0328100	+	0.0405169i	−0.793287	−	0.608848i	\(-0.791632\pi\)
							0.760477	+	0.649365i	\(0.224965\pi\)
\(23\)	−1.06285	+	0.613638i	−0.00963565	+	0.00556315i	−0.504810	−	0.863230i	\(-0.668437\pi\)
							0.495174	+	0.868794i	\(0.335104\pi\)
\(29\)	91.6875	+	9.63674i	0.587101	+	0.0617068i	0.393422	−	0.919358i	\(-0.371291\pi\)
							0.193679	+	0.981065i	\(0.437958\pi\)
\(31\)	−16.8201	−	33.0112i	−0.0974507	−	0.191258i	0.837135	−	0.546997i	\(-0.184229\pi\)
							−0.934585	+	0.355739i	\(0.884229\pi\)
\(37\)	88.6636	−	71.7984i	0.393952	−	0.319016i	−0.411795	−	0.911277i	\(-0.635098\pi\)
							0.805746	+	0.592261i	\(0.201765\pi\)
\(41\)	51.6346	+	134.513i	0.196682	+	0.512375i	0.996083	−	0.0884234i	\(-0.0281829\pi\)
							−0.799401	+	0.600798i	\(0.794850\pi\)
\(43\)	−204.071	+	353.462i	−0.723735	+	1.25355i	0.235758	+	0.971812i	\(0.424243\pi\)
							−0.959493	+	0.281734i	\(0.909091\pi\)
\(47\)	−134.159	−	21.2487i	−0.416364	−	0.0659456i	−0.0552621	−	0.998472i	\(-0.517599\pi\)
							−0.361102	+	0.932526i	\(0.617599\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	143.4.w.a.2.1	✓	640
11.6	odd	10	inner	143.4.w.a.28.1	yes	640
13.7	odd	12	inner	143.4.w.a.46.1	yes	640
143.72	even	60	inner	143.4.w.a.72.1	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.w.a.2.1	✓	640	1.1	even	1	trivial
143.4.w.a.28.1	yes	640	11.6	odd	10	inner
143.4.w.a.46.1	yes	640	13.7	odd	12	inner
143.4.w.a.72.1	yes	640	143.72	even	60	inner