Embedded newform 29.8.e.a.5.4

• Label: 29.8.e.a.5.4
• Level: $29$
• Weight: $8$
• Character: 29.5
• Analytic conductor: $9.059$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.05916573904\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			5.4
Character	\(\chi\)	\(=\)	29.5
Dual form			29.8.e.a.6.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.9983 + 8.77088i) q^{2} +(-44.4854 + 10.1535i) q^{3} +(15.5523 - 68.1393i) q^{4} +(-208.135 - 260.993i) q^{5} +(400.210 - 501.847i) q^{6} +(-9.74527 - 42.6968i) q^{7} +(-354.671 - 736.482i) q^{8} +(-94.5645 + 45.5399i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{2} - 7 q^{3} + 709 q^{4} + 191 q^{5} + 1801 q^{6} - 667 q^{7} + 9086 q^{8} + 9103 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{11}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.9983	+	8.77088i	−0.972124	+	0.775243i	−0.974416	−	0.224754i	\(-0.927842\pi\)
							0.00229107	+	0.999997i	\(0.499271\pi\)
\(3\)	−44.4854	+	10.1535i	−0.951246	+	0.217116i	−0.669867	−	0.742481i	\(-0.733649\pi\)
							−0.281379	+	0.959597i	\(0.590792\pi\)
\(5\)	−208.135	−	260.993i	−0.744648	−	0.933759i	0.254800	−	0.966994i	\(-0.417990\pi\)
							−0.999448	+	0.0332351i	\(0.989419\pi\)
\(7\)	−9.74527	−	42.6968i	−0.0107387	−	0.0470492i	0.969275	−	0.245981i	\(-0.0791100\pi\)
							−0.980013	+	0.198932i	\(0.936253\pi\)
\(11\)	−1433.59	+	2976.88i	−0.324751	+	0.674353i	−0.997874	−	0.0651782i	\(-0.979238\pi\)
							0.673122	+	0.739531i	\(0.264953\pi\)
\(13\)	2750.36	+	1324.50i	0.347207	+	0.167206i	0.599356	−	0.800482i	\(-0.295423\pi\)
							−0.252150	+	0.967688i	\(0.581138\pi\)
\(17\)			16836.7i			0.831161i	0.909557	+	0.415580i	\(0.136421\pi\)
							−0.909557	+	0.415580i	\(0.863579\pi\)
\(19\)	−2081.62	−	475.116i	−0.0696248	−	0.0158914i	0.187567	−	0.982252i	\(-0.439940\pi\)
							−0.257191	+	0.966360i	\(0.582797\pi\)
\(23\)	26497.5	−	33226.9i	0.454107	−	0.569432i	−0.501093	−	0.865394i	\(-0.667068\pi\)
							0.955200	+	0.295961i	\(0.0956399\pi\)
\(29\)	−38992.2	−	125417.i	−0.296883	−	0.954914i
							\(31\)	42984.0	−	34278.6i	0.259144	−	0.206660i	−0.485296	−	0.874350i	\(-0.661288\pi\)
0.744440	+	0.667690i	\(0.232717\pi\)
\(37\)	125897.	+	261429.i	0.408612	+	0.848491i	0.999140	+	0.0414736i	\(0.0132052\pi\)
							−0.590528	+	0.807017i	\(0.701080\pi\)
\(41\)			47131.8i			0.106800i	0.998573	+	0.0533999i	\(0.0170058\pi\)
							−0.998573	+	0.0533999i	\(0.982994\pi\)
\(43\)	767319.	+	611917.i	1.47176	+	1.17369i	0.946482	+	0.322756i	\(0.104609\pi\)
							0.525276	+	0.850932i	\(0.323962\pi\)
\(47\)	338441.	−	702781.i	0.475490	−	0.987364i	−0.515928	−	0.856632i	\(-0.672553\pi\)
							0.991418	−	0.130732i	\(-0.0417328\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.8.e.a.5.4	✓	96
29.6	even	14	inner	29.8.e.a.6.4	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.e.a.5.4	✓	96	1.1	even	1	trivial
29.8.e.a.6.4	yes	96	29.6	even	14	inner