Embedded newform 29.6.d.a.16.2

• Label: 29.6.d.a.16.2
• 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			16.2
Character	\(\chi\)	\(=\)	29.16
Dual form			29.6.d.a.20.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.01987 - 4.34374i) q^{2} +(7.86720 - 9.86516i) q^{3} +(42.5383 + 53.3413i) q^{4} +(61.9634 + 29.8400i) q^{5} +(-113.813 + 54.8094i) q^{6} +(56.6689 - 71.0605i) q^{7} +(-80.7019 - 353.578i) q^{8} +(18.6441 + 81.6852i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.01987	−	4.34374i	−1.59450	−	0.767872i	−0.595142	−	0.803621i	\(-0.702904\pi\)
							−0.999361	+	0.0357484i	\(0.988619\pi\)
\(3\)	7.86720	−	9.86516i	0.504681	−	0.632850i	−0.462597	−	0.886569i	\(-0.653082\pi\)
							0.967278	+	0.253719i	\(0.0816537\pi\)
\(5\)	61.9634	+	29.8400i	1.10843	+	0.533794i	0.896301	−	0.443447i	\(-0.146245\pi\)
							0.212134	+	0.977241i	\(0.431959\pi\)
\(7\)	56.6689	−	71.0605i	0.437119	−	0.548130i	−0.513662	−	0.857992i	\(-0.671712\pi\)
							0.950781	+	0.309863i	\(0.100283\pi\)
\(11\)	170.139	−	745.426i	0.423956	−	1.85747i	−0.0845541	−	0.996419i	\(-0.526947\pi\)
							0.508511	−	0.861056i	\(-0.330196\pi\)
\(13\)	−123.327	+	540.332i	−0.202395	+	0.886752i	0.767078	+	0.641554i	\(0.221710\pi\)
							−0.969473	+	0.245198i	\(0.921147\pi\)
\(17\)	273.387			0.229433			0.114717	−	0.993398i	\(-0.463404\pi\)
							0.114717	+	0.993398i	\(0.463404\pi\)
\(19\)	−450.072	−	564.372i	−0.286021	−	0.358659i	0.617977	−	0.786197i	\(-0.287953\pi\)
							−0.903998	+	0.427537i	\(0.859381\pi\)
\(23\)	2891.61	−	1392.53i	1.13978	−	0.548889i	0.233831	−	0.972277i	\(-0.424874\pi\)
							0.905948	+	0.423388i	\(0.139159\pi\)
\(29\)	1718.54	−	4190.20i	0.379458	−	0.925209i
							\(31\)	−6246.17	−	3008.00i	−1.16737	−	0.562177i	−0.253166	−	0.967423i	\(-0.581472\pi\)
−0.914208	+	0.405246i	\(0.867186\pi\)
\(37\)	3167.26	+	13876.7i	0.380347	+	1.66641i	0.696389	+	0.717664i	\(0.254789\pi\)
							−0.316042	+	0.948745i	\(0.602354\pi\)
\(41\)	−4706.98			−0.437303			−0.218652	−	0.975803i	\(-0.570166\pi\)
							−0.218652	+	0.975803i	\(0.570166\pi\)
\(43\)	−5933.92	+	2857.63i	−0.489408	+	0.235686i	−0.662278	−	0.749258i	\(-0.730410\pi\)
							0.172870	+	0.984945i	\(0.444696\pi\)
\(47\)	−2001.91	+	8770.94i	−0.132190	+	0.579164i	0.864833	+	0.502060i	\(0.167424\pi\)
							−0.997023	+	0.0771038i	\(0.975433\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.16.2	✓	66
29.7	even	7		841.6.a.i.1.31		33
29.20	even	7	inner	29.6.d.a.20.2	yes	66
29.22	even	14		841.6.a.h.1.3		33

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.16.2	✓	66	1.1	even	1	trivial
29.6.d.a.20.2	yes	66	29.20	even	7	inner
841.6.a.h.1.3		33	29.22	even	14
841.6.a.i.1.31		33	29.7	even	7