Embedded newform 29.6.d.a.16.3

• Label: 29.6.d.a.16.3
• Level: $29$
• Weight: $6$
• Character: 29.16
• Analytic conductor: $4.651$
• Analytic rank: $0$
• Dimension: $66$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	29.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65113077458\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			16.3
Character	\(\chi\)	\(=\)	29.16
Dual form			29.6.d.a.20.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.13336 - 2.95367i) q^{2} +(1.76173 - 2.20914i) q^{3} +(8.94225 + 11.2132i) q^{4} +(-38.8583 - 18.7131i) q^{5} +(-17.3304 + 8.34587i) q^{6} +(0.981087 - 1.23024i) q^{7} +(26.7482 + 117.192i) q^{8} +(52.2960 + 229.124i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.13336	−	2.95367i	−1.08423	−	0.522140i	−0.195567	−	0.980690i	\(-0.562655\pi\)
							−0.888668	+	0.458550i	\(0.848369\pi\)
\(3\)	1.76173	−	2.20914i	0.113015	−	0.141716i	−0.722106	−	0.691782i	\(-0.756826\pi\)
							0.835121	+	0.550066i	\(0.185397\pi\)
\(5\)	−38.8583	−	18.7131i	−0.695118	−	0.334751i	0.0527418	−	0.998608i	\(-0.483204\pi\)
							−0.747859	+	0.663857i	\(0.768918\pi\)
\(7\)	0.981087	−	1.23024i	0.00756767	−	0.00948956i	−0.778033	−	0.628223i	\(-0.783783\pi\)
							0.785601	+	0.618734i	\(0.212354\pi\)
\(11\)	−96.3835	+	422.284i	−0.240171	+	1.05226i	0.700690	+	0.713466i	\(0.252876\pi\)
							−0.940861	+	0.338793i	\(0.889981\pi\)
\(13\)	−44.0450	+	192.974i	−0.0722833	+	0.316694i	−0.998124	−	0.0612194i	\(-0.980501\pi\)
							0.925841	+	0.377913i	\(0.123358\pi\)
\(17\)	−978.583			−0.821250			−0.410625	−	0.911804i	\(-0.634689\pi\)
							−0.410625	+	0.911804i	\(0.634689\pi\)
\(19\)	−85.9478	−	107.775i	−0.0546199	−	0.0684912i	0.753771	−	0.657137i	\(-0.228233\pi\)
							−0.808391	+	0.588646i	\(0.799661\pi\)
\(23\)	−1793.27	+	863.593i	−0.706848	+	0.340400i	−0.752532	−	0.658556i	\(-0.771168\pi\)
							0.0456838	+	0.998956i	\(0.485453\pi\)
\(29\)	−2270.74	+	3918.53i	−0.501387	+	0.865223i
							\(31\)	2200.70	+	1059.80i	0.411298	+	0.198071i	0.628079	−	0.778149i	\(-0.283841\pi\)
−0.216781	+	0.976220i	\(0.569556\pi\)
\(37\)	−390.052	−	1708.93i	−0.0468401	−	0.205220i	0.946093	−	0.323895i	\(-0.104992\pi\)
							−0.992933	+	0.118675i	\(0.962135\pi\)
\(41\)	−12150.5			−1.12884			−0.564422	−	0.825486i	\(-0.690901\pi\)
							−0.564422	+	0.825486i	\(0.690901\pi\)
\(43\)	−5640.93	+	2716.53i	−0.465243	+	0.224049i	−0.651794	−	0.758396i	\(-0.725983\pi\)
							0.186551	+	0.982445i	\(0.440269\pi\)
\(47\)	1219.69	−	5343.80i	0.0805385	−	0.352862i	−0.918562	−	0.395278i	\(-0.870648\pi\)
							0.999100	+	0.0424157i	\(0.0135054\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.16.3	✓	66
29.7	even	7		841.6.a.i.1.26		33
29.20	even	7	inner	29.6.d.a.20.3	yes	66
29.22	even	14		841.6.a.h.1.8		33

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.16.3	✓	66	1.1	even	1	trivial
29.6.d.a.20.3	yes	66	29.20	even	7	inner
841.6.a.h.1.8		33	29.22	even	14
841.6.a.i.1.26		33	29.7	even	7