Embedded newform 29.8.d.a.7.10

• Label: 29.8.d.a.7.10
• Level: $29$
• Weight: $8$
• Character: 29.7
• 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			7.10
Character	\(\chi\)	\(=\)	29.7
Dual form			29.8.d.a.25.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.240160 - 1.05221i) q^{2} +(53.2557 + 25.6466i) q^{3} +(114.275 - 55.0317i) q^{4} +(117.842 + 516.297i) q^{5} +(14.1957 - 62.1954i) q^{6} +(-631.923 - 304.318i) q^{7} +(-171.482 - 215.031i) q^{8} +(814.854 + 1021.79i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q + q^{2} - 5 q^{3} - 1275 q^{4} + 133 q^{5} - 2675 q^{6} + 657 q^{7} - 6562 q^{8} - 15054 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.240160	−	1.05221i	−0.0212273	−	0.0930029i	0.963205	−	0.268769i	\(-0.0866169\pi\)
							−0.984432	+	0.175766i	\(0.943760\pi\)
\(3\)	53.2557	+	25.6466i	1.13879	+	0.548410i	0.905647	−	0.424033i	\(-0.139386\pi\)
							0.233139	+	0.972443i	\(0.425100\pi\)
\(5\)	117.842	+	516.297i	0.421603	+	1.84716i	0.523024	+	0.852318i	\(0.324804\pi\)
							−0.101421	+	0.994844i	\(0.532339\pi\)
\(7\)	−631.923	−	304.318i	−0.696340	−	0.335340i	0.0520075	−	0.998647i	\(-0.483438\pi\)
							−0.748347	+	0.663307i	\(0.769152\pi\)
\(11\)	356.396	−	446.907i	0.0807344	−	0.101238i	−0.739824	−	0.672801i	\(-0.765091\pi\)
							0.820558	+	0.571563i	\(0.193663\pi\)
\(13\)	−5190.67	+	6508.89i	−0.655272	+	0.821685i	−0.992819	−	0.119624i	\(-0.961831\pi\)
							0.337548	+	0.941308i	\(0.390403\pi\)
\(17\)	19442.3			0.959791			0.479895	−	0.877326i	\(-0.340675\pi\)
							0.479895	+	0.877326i	\(0.340675\pi\)
\(19\)	21869.5	−	10531.8i	0.731476	−	0.352261i	−0.0307882	−	0.999526i	\(-0.509802\pi\)
							0.762265	+	0.647265i	\(0.224087\pi\)
\(23\)	10914.2	−	47818.1i	0.187044	−	0.819493i	−0.791121	−	0.611660i	\(-0.790502\pi\)
							0.978164	−	0.207832i	\(-0.0666409\pi\)
\(29\)	118743.	+	56124.9i	0.904096	+	0.427329i
							\(31\)	−60381.9	−	264550.i	−0.364033	−	1.59493i	−0.742848	−	0.669460i	\(-0.766526\pi\)
0.378815	−	0.925472i	\(-0.376332\pi\)
\(37\)	133007.	+	166786.i	0.431687	+	0.541318i	0.949331	−	0.314278i	\(-0.101762\pi\)
							−0.517644	+	0.855596i	\(0.673191\pi\)
\(41\)	−367982.			−0.833840			−0.416920	−	0.908943i	\(-0.636891\pi\)
							−0.416920	+	0.908943i	\(0.636891\pi\)
\(43\)	−83229.9	+	364654.i	−0.159639	+	0.699426i	0.830227	+	0.557425i	\(0.188211\pi\)
							−0.989867	+	0.142001i	\(0.954647\pi\)
\(47\)	576234.	−	722574.i	0.809573	−	1.01517i	−0.189870	−	0.981809i	\(-0.560807\pi\)
							0.999443	−	0.0333637i	\(-0.0106220\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.8.d.a.7.10	✓	102
29.25	even	7	inner	29.8.d.a.25.10	yes	102

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.d.a.7.10	✓	102	1.1	even	1	trivial
29.8.d.a.25.10	yes	102	29.25	even	7	inner