Embedded newform 29.6.d.a.16.5

• Label: 29.6.d.a.16.5
• 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			16.5
Character	\(\chi\)	\(=\)	29.16
Dual form			29.6.d.a.20.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.07802 - 1.48230i) q^{2} +(18.8759 - 23.6696i) q^{3} +(-12.6747 - 15.8935i) q^{4} +(15.8039 + 7.61077i) q^{5} +(-93.1857 + 44.8759i) q^{6} +(-39.8314 + 49.9470i) q^{7} +(39.7806 + 174.290i) q^{8} +(-149.879 - 656.661i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.07802	−	1.48230i	−0.544123	−	0.262036i	0.141568	−	0.989928i	\(-0.454786\pi\)
							−0.685691	+	0.727893i	\(0.740500\pi\)
\(3\)	18.8759	−	23.6696i	1.21089	−	1.51841i	0.418704	−	0.908123i	\(-0.362484\pi\)
							0.792184	−	0.610283i	\(-0.208944\pi\)
\(5\)	15.8039	+	7.61077i	0.282709	+	0.136146i	0.569865	−	0.821738i	\(-0.306995\pi\)
							−0.287156	+	0.957884i	\(0.592710\pi\)
\(7\)	−39.8314	+	49.9470i	−0.307242	+	0.385269i	−0.911349	−	0.411634i	\(-0.864958\pi\)
							0.604107	+	0.796903i	\(0.293530\pi\)
\(11\)	−25.0902	+	109.927i	−0.0625204	+	0.273920i	−0.996520	−	0.0833544i	\(-0.973437\pi\)
							0.934000	+	0.357274i	\(0.116294\pi\)
\(13\)	148.388	−	650.132i	0.243524	−	1.06695i	−0.694259	−	0.719725i	\(-0.744268\pi\)
							0.937783	−	0.347222i	\(-0.112875\pi\)
\(17\)	−137.506			−0.115398			−0.0576990	−	0.998334i	\(-0.518376\pi\)
							−0.0576990	+	0.998334i	\(0.518376\pi\)
\(19\)	1469.67	+	1842.91i	0.933979	+	1.17117i	0.985014	+	0.172475i	\(0.0551765\pi\)
							−0.0510351	+	0.998697i	\(0.516252\pi\)
\(23\)	2898.41	−	1395.80i	1.14246	−	0.550179i	0.235698	−	0.971826i	\(-0.424262\pi\)
							0.906760	+	0.421647i	\(0.138548\pi\)
\(29\)	4343.01	−	1284.30i	0.958950	−	0.283577i
							\(31\)	76.5604	+	36.8695i	0.0143087	+	0.00689070i	0.441024	−	0.897495i	\(-0.354615\pi\)
−0.426716	+	0.904386i	\(0.640330\pi\)
\(37\)	2814.00	+	12328.9i	0.337924	+	1.48054i	0.803377	+	0.595470i	\(0.203034\pi\)
							−0.465453	+	0.885073i	\(0.654109\pi\)
\(41\)	6149.81			0.571350			0.285675	−	0.958327i	\(-0.407782\pi\)
							0.285675	+	0.958327i	\(0.407782\pi\)
\(43\)	6405.94	−	3084.94i	0.528338	−	0.254434i	−0.150645	−	0.988588i	\(-0.548135\pi\)
							0.678983	+	0.734154i	\(0.262421\pi\)
\(47\)	3653.00	−	16004.8i	0.241215	−	1.05683i	−0.698697	−	0.715418i	\(-0.746236\pi\)
							0.939912	−	0.341416i	\(-0.110906\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.16.5	✓	66
29.7	even	7		841.6.a.i.1.21		33
29.20	even	7	inner	29.6.d.a.20.5	yes	66
29.22	even	14		841.6.a.h.1.13		33

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.16.5	✓	66	1.1	even	1	trivial
29.6.d.a.20.5	yes	66	29.20	even	7	inner
841.6.a.h.1.13		33	29.22	even	14
841.6.a.i.1.21		33	29.7	even	7