Embedded newform 29.6.d.a.16.1

• Label: 29.6.d.a.16.1
• Level: $29$
• Weight: $6$
• Character: 29.16
• Analytic conductor: $4.651$
• Analytic rank: $0$
• Dimension: $66$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	29.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65113077458\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			16.1
Character	\(\chi\)	\(=\)	29.16
Dual form			29.6.d.a.20.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.29727 - 4.47733i) q^{2} +(-16.9028 + 21.1954i) q^{3} +(46.4410 + 58.2352i) q^{4} +(-29.3057 - 14.1129i) q^{5} +(252.049 - 121.380i) q^{6} +(-125.979 + 157.972i) q^{7} +(-97.5569 - 427.425i) q^{8} +(-109.469 - 479.616i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.29727	−	4.47733i	−1.64354	−	0.791487i	−0.999652	−	0.0263670i	\(-0.991606\pi\)
							−0.643888	−	0.765120i	\(-0.722680\pi\)
\(3\)	−16.9028	+	21.1954i	−1.08431	+	1.35969i	−0.156057	+	0.987748i	\(0.549878\pi\)
							−0.928257	+	0.371939i	\(0.878693\pi\)
\(5\)	−29.3057	−	14.1129i	−0.524236	−	0.252459i	0.152998	−	0.988227i	\(-0.451107\pi\)
							−0.677234	+	0.735768i	\(0.736822\pi\)
\(7\)	−125.979	+	157.972i	−0.971744	+	1.21853i	0.00408480	+	0.999992i	\(0.498700\pi\)
							−0.975829	+	0.218537i	\(0.929872\pi\)
\(11\)	28.6749	−	125.633i	0.0714530	−	0.313056i	−0.926554	−	0.376162i	\(-0.877244\pi\)
							0.998007	+	0.0631063i	\(0.0201007\pi\)
\(13\)	52.5736	−	230.340i	0.0862799	−	0.378017i	−0.913291	−	0.407308i	\(-0.866468\pi\)
							0.999571	+	0.0292907i	\(0.00932486\pi\)
\(17\)	1590.68			1.33494			0.667470	−	0.744637i	\(-0.267377\pi\)
							0.667470	+	0.744637i	\(0.267377\pi\)
\(19\)	455.054	+	570.620i	0.289187	+	0.362629i	0.905110	−	0.425177i	\(-0.139788\pi\)
							−0.615923	+	0.787806i	\(0.711217\pi\)
\(23\)	−3244.29	+	1562.37i	−1.27879	+	0.615833i	−0.945079	−	0.326842i	\(-0.894016\pi\)
							−0.333712	+	0.942675i	\(0.608301\pi\)
\(29\)	2542.38	−	3747.99i	0.561365	−	0.827568i
							\(31\)	2326.29	+	1120.28i	0.434771	+	0.209374i	0.638450	−	0.769663i	\(-0.279576\pi\)
−0.203680	+	0.979038i	\(0.565290\pi\)
\(37\)	920.767	+	4034.14i	0.110572	+	0.484448i	0.999644	+	0.0266804i	\(0.00849363\pi\)
							−0.889072	+	0.457767i	\(0.848649\pi\)
\(41\)	−6002.77			−0.557689			−0.278845	−	0.960336i	\(-0.589951\pi\)
							−0.278845	+	0.960336i	\(0.589951\pi\)
\(43\)	−754.262	+	363.234i	−0.0622087	+	0.0299581i	−0.464729	−	0.885453i	\(-0.653848\pi\)
							0.402521	+	0.915411i	\(0.368134\pi\)
\(47\)	6459.90	−	28302.7i	0.426561	−	1.86889i	−0.0647494	−	0.997902i	\(-0.520625\pi\)
							0.491310	−	0.870985i	\(-0.336518\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.16.1	✓	66
29.7	even	7		841.6.a.i.1.32		33
29.20	even	7	inner	29.6.d.a.20.1	yes	66
29.22	even	14		841.6.a.h.1.2		33

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.16.1	✓	66	1.1	even	1	trivial
29.6.d.a.20.1	yes	66	29.20	even	7	inner
841.6.a.h.1.2		33	29.22	even	14
841.6.a.i.1.32		33	29.7	even	7