Embedded newform 29.8.e.a.4.6

• Label: 29.8.e.a.4.6
• 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			4.6
Character	\(\chi\)	\(=\)	29.4
Dual form			29.8.e.a.22.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.62931 - 1.51310i) q^{2} +(34.2559 + 71.1332i) q^{3} +(-73.6657 - 35.4756i) q^{4} +(94.1794 - 412.627i) q^{5} +(-119.462 - 523.396i) q^{6} +(849.727 - 409.207i) q^{7} +(1115.16 + 889.311i) q^{8} +(-2522.89 + 3163.60i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{2} - 7 q^{3} + 709 q^{4} + 191 q^{5} + 1801 q^{6} - 667 q^{7} + 9086 q^{8} + 9103 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.62931	−	1.51310i	−0.585954	−	0.133740i	−0.0807412	−	0.996735i	\(-0.525729\pi\)
							−0.505212	+	0.862995i	\(0.668586\pi\)
\(3\)	34.2559	+	71.1332i	0.732507	+	1.52107i	0.849301	+	0.527909i	\(0.177024\pi\)
							−0.116794	+	0.993156i	\(0.537262\pi\)
\(5\)	94.1794	−	412.627i	0.336946	−	1.47626i	−0.468433	−	0.883499i	\(-0.655181\pi\)
							0.805379	−	0.592760i	\(-0.201962\pi\)
\(7\)	849.727	−	409.207i	0.936346	−	0.450920i	0.0974669	−	0.995239i	\(-0.468926\pi\)
							0.838879	+	0.544319i	\(0.183212\pi\)
\(11\)	1351.27	−	1077.60i	0.306103	−	0.244109i	−0.458375	−	0.888759i	\(-0.651568\pi\)
							0.764478	+	0.644650i	\(0.222997\pi\)
\(13\)	8469.65	+	10620.6i	1.06921	+	1.34075i	0.936922	+	0.349538i	\(0.113662\pi\)
							0.132290	+	0.991211i	\(0.457767\pi\)
\(17\)		−	707.802i		−	0.0349414i	−0.999847	−	0.0174707i	\(-0.994439\pi\)
							0.999847	−	0.0174707i	\(-0.00556138\pi\)
\(19\)	16174.8	−	33587.3i	0.541004	−	1.12341i	−0.433937	−	0.900943i	\(-0.642876\pi\)
							0.974941	−	0.222463i	\(-0.0714098\pi\)
\(23\)	−4263.08	−	18677.8i	−0.0730593	−	0.320094i	0.925172	−	0.379549i	\(-0.123921\pi\)
							−0.998231	+	0.0594552i	\(0.981064\pi\)
\(29\)	94253.5	+	91466.7i	0.717636	+	0.696418i
							\(31\)	272681.	+	62237.6i	1.64395	+	0.375221i	0.941633	−	0.336642i	\(-0.109291\pi\)
0.702318	+	0.711863i	\(0.252148\pi\)
\(37\)	−56902.7	−	45378.4i	−0.184683	−	0.147280i	0.526784	−	0.849999i	\(-0.323398\pi\)
							−0.711467	+	0.702719i	\(0.751969\pi\)
\(41\)		−	652669.i		−	1.47894i	−0.673192	−	0.739468i	\(-0.735077\pi\)
							0.673192	−	0.739468i	\(-0.264923\pi\)
\(43\)	−555905.	+	126882.i	−1.06626	+	0.243366i	−0.719431	−	0.694564i	\(-0.755597\pi\)
							−0.346824	+	0.937930i	\(0.612740\pi\)
\(47\)	−136662.	+	108985.i	−0.192002	+	0.153117i	−0.714773	−	0.699357i	\(-0.753470\pi\)
							0.522771	+	0.852473i	\(0.324898\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.8.e.a.4.6	✓	96
29.22	even	14	inner	29.8.e.a.22.6	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.e.a.4.6	✓	96	1.1	even	1	trivial
29.8.e.a.22.6	yes	96	29.22	even	14	inner