Embedded newform 29.8.d.a.16.12

• Label: 29.8.d.a.16.12
• 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			16.12
Character	\(\chi\)	\(=\)	29.16
Dual form			29.8.d.a.20.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.88508 + 3.79725i) q^{2} +(-16.2849 + 20.4207i) q^{3} +(-32.0514 - 40.1912i) q^{4} +(-135.199 - 65.1086i) q^{5} +(-205.950 + 99.1805i) q^{6} +(1044.62 - 1309.91i) q^{7} +(-349.386 - 1530.76i) q^{8} +(334.849 + 1467.07i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q + q^{2} - 5 q^{3} - 1275 q^{4} + 133 q^{5} - 2675 q^{6} + 657 q^{7} - 6562 q^{8} - 15054 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.88508	+	3.79725i	0.696949	+	0.335633i	0.748590	−	0.663033i	\(-0.230731\pi\)
							−0.0516415	+	0.998666i	\(0.516445\pi\)
\(3\)	−16.2849	+	20.4207i	−0.348226	+	0.436662i	−0.924841	−	0.380354i	\(-0.875802\pi\)
							0.576614	+	0.817016i	\(0.304373\pi\)
\(5\)	−135.199	−	65.1086i	−0.483704	−	0.232940i	0.176107	−	0.984371i	\(-0.443649\pi\)
							−0.659811	+	0.751431i	\(0.729364\pi\)
\(7\)	1044.62	−	1309.91i	1.15111	−	1.44344i	0.274919	−	0.961467i	\(-0.411349\pi\)
							0.876186	−	0.481973i	\(-0.160080\pi\)
\(11\)	1118.08	−	4898.62i	0.253278	−	1.10968i	−0.675006	−	0.737812i	\(-0.735859\pi\)
							0.928284	−	0.371871i	\(-0.121284\pi\)
\(13\)	−607.841	+	2663.13i	−0.0767341	+	0.336194i	−0.998694	−	0.0510914i	\(-0.983730\pi\)
							0.921960	+	0.387285i	\(0.126587\pi\)
\(17\)	−12851.6			−0.634432			−0.317216	−	0.948353i	\(-0.602748\pi\)
							−0.317216	+	0.948353i	\(0.602748\pi\)
\(19\)	−28616.2	−	35883.6i	−0.957138	−	1.20021i	−0.979700	−	0.200468i	\(-0.935754\pi\)
							0.0225621	−	0.999745i	\(-0.492818\pi\)
\(23\)	66486.3	−	32018.1i	1.13942	−	0.548717i	0.233583	−	0.972337i	\(-0.424955\pi\)
							0.905840	+	0.423620i	\(0.139241\pi\)
\(29\)	−18452.7	+	130036.i	−0.140497	+	0.990081i
							\(31\)	107855.	+	51940.3i	0.650241	+	0.313140i	0.729770	−	0.683693i	\(-0.239627\pi\)
−0.0795285	+	0.996833i	\(0.525341\pi\)
\(37\)	125515.	+	549916.i	0.407370	+	1.78480i	0.596244	+	0.802803i	\(0.296659\pi\)
							−0.188874	+	0.982001i	\(0.560484\pi\)
\(41\)	−10199.8			−0.0231126			−0.0115563	−	0.999933i	\(-0.503679\pi\)
							−0.0115563	+	0.999933i	\(0.503679\pi\)
\(43\)	440393.	−	212082.i	0.844697	−	0.406785i	0.0390908	−	0.999236i	\(-0.487554\pi\)
							0.805607	+	0.592451i	\(0.201840\pi\)
\(47\)	78463.3	−	343770.i	0.110236	−	0.482976i	−0.889428	−	0.457075i	\(-0.848897\pi\)
							0.999664	−	0.0259016i	\(-0.00824566\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.8.d.a.16.12	✓	102
29.20	even	7	inner	29.8.d.a.20.12	yes	102

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.d.a.16.12	✓	102	1.1	even	1	trivial
29.8.d.a.20.12	yes	102	29.20	even	7	inner