Embedded newform 29.8.e.a.4.1

• Label: 29.8.e.a.4.1
• Level: $29$
• Weight: $8$
• Character: 29.4
• 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			4.1
Character	\(\chi\)	\(=\)	29.4
Dual form			29.8.e.a.22.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-19.9614 - 4.55605i) q^{2} +(16.7140 + 34.7070i) q^{3} +(262.375 + 126.353i) q^{4} +(46.2709 - 202.726i) q^{5} +(-175.507 - 768.948i) q^{6} +(-856.134 + 412.292i) q^{7} +(-2612.69 - 2083.55i) q^{8} +(438.356 - 549.681i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{2} - 7 q^{3} + 709 q^{4} + 191 q^{5} + 1801 q^{6} - 667 q^{7} + 9086 q^{8} + 9103 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−19.9614	−	4.55605i	−1.76435	−	0.402702i	−0.787448	−	0.616381i	\(-0.788598\pi\)
							−0.976904	+	0.213679i	\(0.931455\pi\)
\(3\)	16.7140	+	34.7070i	0.357401	+	0.742151i	0.999705	−	0.0242992i	\(-0.00773545\pi\)
							−0.642304	+	0.766450i	\(0.722021\pi\)
\(5\)	46.2709	−	202.726i	0.165544	−	0.725295i	−0.822198	−	0.569201i	\(-0.807253\pi\)
							0.987742	−	0.156094i	\(-0.0498903\pi\)
\(7\)	−856.134	+	412.292i	−0.943406	+	0.454320i	−0.841369	−	0.540461i	\(-0.818250\pi\)
							−0.102036	+	0.994781i	\(0.532536\pi\)
\(11\)	2675.91	−	2133.97i	0.606174	−	0.483408i	−0.271647	−	0.962397i	\(-0.587568\pi\)
							0.877821	+	0.478989i	\(0.158997\pi\)
\(13\)	3839.16	+	4814.16i	0.484657	+	0.607741i	0.962692	−	0.270600i	\(-0.0872221\pi\)
							−0.478035	+	0.878341i	\(0.658651\pi\)
\(17\)			18540.8i			0.915287i	0.889136	+	0.457644i	\(0.151306\pi\)
							−0.889136	+	0.457644i	\(0.848694\pi\)
\(19\)	−10949.4	+	22736.7i	−0.366229	+	0.760483i	−0.999914	−	0.0131083i	\(-0.995827\pi\)
							0.633685	+	0.773591i	\(0.281542\pi\)
\(23\)	17481.6	+	76591.7i	0.299594	+	1.31261i	0.870735	+	0.491753i	\(0.163644\pi\)
							−0.571141	+	0.820852i	\(0.693499\pi\)
\(29\)	131331.	+	1460.89i	0.999938	+	0.0111231i
							\(31\)	−241449.	−	55109.2i	−1.45566	−	0.332245i	−0.579779	−	0.814774i	\(-0.696861\pi\)
−0.875880	+	0.482529i	\(0.839718\pi\)
\(37\)	392585.	+	313076.i	1.27417	+	1.01612i	0.998494	+	0.0548655i	\(0.0174730\pi\)
							0.275676	+	0.961251i	\(0.411098\pi\)
\(41\)			338398.i			0.766804i	0.923581	+	0.383402i	\(0.125248\pi\)
							−0.923581	+	0.383402i	\(0.874752\pi\)
\(43\)	−817807.	+	186659.i	−1.56860	+	0.358022i	−0.916475	−	0.400093i	\(-0.868978\pi\)
							−0.652121	+	0.758115i	\(0.726121\pi\)
\(47\)	198726.	−	158479.i	0.279199	−	0.222653i	−0.473871	−	0.880594i	\(-0.657144\pi\)
							0.753070	+	0.657941i	\(0.228572\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.8.e.a.4.1	✓	96
29.22	even	14	inner	29.8.e.a.22.1	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.e.a.4.1	✓	96	1.1	even	1	trivial
29.8.e.a.22.1	yes	96	29.22	even	14	inner