Embedded newform 29.8.d.a.16.5

• Label: 29.8.d.a.16.5
• Level: $29$
• Weight: $8$
• Character: 29.16
• Analytic conductor: $9.059$
• Analytic rank: $0$
• Dimension: $102$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	29.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.05916573904\)
Analytic rank:	\(0\)
Dimension:	\(102\)
Relative dimension:	\(17\) 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			29.8.d.a.20.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.9551 - 5.27569i) q^{2} +(45.2250 - 56.7104i) q^{3} +(12.3742 + 15.5167i) q^{4} +(-371.691 - 178.997i) q^{5} +(-794.630 + 382.674i) q^{6} +(914.083 - 1146.22i) q^{7} +(292.629 + 1282.09i) q^{8} +(-684.112 - 2997.29i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q + q^{2} - 5 q^{3} - 1275 q^{4} + 133 q^{5} - 2675 q^{6} + 657 q^{7} - 6562 q^{8} - 15054 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.9551	−	5.27569i	−0.968301	−	0.466309i	−0.118235	−	0.992986i	\(-0.537724\pi\)
							−0.850066	+	0.526676i	\(0.823438\pi\)
\(3\)	45.2250	−	56.7104i	0.967062	−	1.21266i	−0.0100534	−	0.999949i	\(-0.503200\pi\)
							0.977116	−	0.212708i	\(-0.0682284\pi\)
\(5\)	−371.691	−	178.997i	−1.32980	−	0.640399i	−0.372109	−	0.928189i	\(-0.621365\pi\)
							−0.957694	+	0.287790i	\(0.907080\pi\)
\(7\)	914.083	−	1146.22i	1.00726	−	1.26307i	0.0427344	−	0.999086i	\(-0.486393\pi\)
							0.964528	−	0.263981i	\(-0.0850355\pi\)
\(11\)	1019.58	−	4467.08i	0.230966	−	1.01193i	−0.717875	−	0.696172i	\(-0.754885\pi\)
							0.948841	−	0.315755i	\(-0.102258\pi\)
\(13\)	−3076.77	+	13480.2i	−0.388413	+	1.70175i	0.281710	+	0.959500i	\(0.409098\pi\)
							−0.670123	+	0.742250i	\(0.733759\pi\)
\(17\)	−3866.02			−0.190850			−0.0954250	−	0.995437i	\(-0.530421\pi\)
							−0.0954250	+	0.995437i	\(0.530421\pi\)
\(19\)	12508.8	+	15685.6i	0.418388	+	0.524642i	0.945705	−	0.325026i	\(-0.105373\pi\)
							−0.527316	+	0.849669i	\(0.676802\pi\)
\(23\)	−20380.9	+	9814.93i	−0.349282	+	0.168205i	−0.600296	−	0.799778i	\(-0.704950\pi\)
							0.251014	+	0.967984i	\(0.419236\pi\)
\(29\)	−53724.6	−	119848.i	−0.409054	−	0.912510i
							\(31\)	50974.6	+	24548.1i	0.307318	+	0.147997i	0.581186	−	0.813771i	\(-0.302589\pi\)
−0.273868	+	0.961767i	\(0.588303\pi\)
\(37\)	−83866.3	−	367442.i	−0.272196	−	1.19257i	−0.907415	−	0.420236i	\(-0.861947\pi\)
							0.635219	−	0.772332i	\(-0.280910\pi\)
\(41\)	−90158.6			−0.204298			−0.102149	−	0.994769i	\(-0.532572\pi\)
							−0.102149	+	0.994769i	\(0.532572\pi\)
\(43\)	876682.	−	422188.i	1.68152	−	0.809779i	0.684814	−	0.728718i	\(-0.259884\pi\)
							0.996709	−	0.0810603i	\(-0.0258306\pi\)
\(47\)	−14155.4	+	62018.7i	−0.0198874	+	0.0871324i	−0.983898	−	0.178732i	\(-0.942800\pi\)
							0.964010	+	0.265865i	\(0.0856575\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.8.d.a.16.5	✓	102
29.20	even	7	inner	29.8.d.a.20.5	yes	102

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.d.a.16.5	✓	102	1.1	even	1	trivial
29.8.d.a.20.5	yes	102	29.20	even	7	inner