Embedded newform 29.6.d.a.16.8

• Label: 29.6.d.a.16.8
• 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			16.8
Character	\(\chi\)	\(=\)	29.16
Dual form			29.6.d.a.20.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.74798 + 1.80493i) q^{2} +(-13.9558 + 17.5000i) q^{3} +(-9.16211 - 11.4889i) q^{4} +(-40.2504 - 19.3836i) q^{5} +(-83.8924 + 40.4005i) q^{6} +(-30.7029 + 38.5002i) q^{7} +(-43.2242 - 189.378i) q^{8} +(-57.4139 - 251.547i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.74798	+	1.80493i	0.662555	+	0.319070i	0.734767	−	0.678320i	\(-0.237291\pi\)
							−0.0722119	+	0.997389i	\(0.523006\pi\)
\(3\)	−13.9558	+	17.5000i	−0.895266	+	1.12263i	0.0965979	+	0.995323i	\(0.469204\pi\)
							−0.991864	+	0.127304i	\(0.959368\pi\)
\(5\)	−40.2504	−	19.3836i	−0.720021	−	0.346744i	0.0377297	−	0.999288i	\(-0.487987\pi\)
							−0.757751	+	0.652544i	\(0.773702\pi\)
\(7\)	−30.7029	+	38.5002i	−0.236829	+	0.296974i	−0.886016	−	0.463655i	\(-0.846538\pi\)
							0.649187	+	0.760629i	\(0.275109\pi\)
\(11\)	−124.623	+	546.010i	−0.310540	+	1.36056i	0.543086	+	0.839677i	\(0.317256\pi\)
							−0.853626	+	0.520886i	\(0.825602\pi\)
\(13\)	−134.054	+	587.331i	−0.220000	+	0.963883i	0.737477	+	0.675373i	\(0.236017\pi\)
							−0.957477	+	0.288511i	\(0.906840\pi\)
\(17\)	−1323.91			−1.11106			−0.555530	−	0.831496i	\(-0.687485\pi\)
							−0.555530	+	0.831496i	\(0.687485\pi\)
\(19\)	375.194	+	470.479i	0.238436	+	0.298990i	0.886624	−	0.462491i	\(-0.153044\pi\)
							−0.648188	+	0.761481i	\(0.724473\pi\)
\(23\)	3630.88	−	1748.54i	1.43118	−	0.689218i	0.451959	−	0.892039i	\(-0.350725\pi\)
							0.979216	+	0.202821i	\(0.0650110\pi\)
\(29\)	4180.32	+	1742.42i	0.923028	+	0.384732i
							\(31\)	−2253.73	−	1085.34i	−0.421208	−	0.202843i	0.211259	−	0.977430i	\(-0.432244\pi\)
−0.632468	+	0.774587i	\(0.717958\pi\)
\(37\)	−217.466	−	952.783i	−0.0261149	−	0.114417i	0.960191	−	0.279346i	\(-0.0901176\pi\)
							−0.986305	+	0.164929i	\(0.947260\pi\)
\(41\)	−11735.4			−1.09028			−0.545142	−	0.838343i	\(-0.683524\pi\)
							−0.545142	+	0.838343i	\(0.683524\pi\)
\(43\)	−8700.04	+	4189.72i	−0.717547	+	0.345552i	−0.756773	−	0.653678i	\(-0.773225\pi\)
							0.0392261	+	0.999230i	\(0.487511\pi\)
\(47\)	−2269.26	+	9942.28i	−0.149844	+	0.656510i	0.843083	+	0.537784i	\(0.180738\pi\)
							−0.992927	+	0.118727i	\(0.962119\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.16.8	✓	66
29.7	even	7		841.6.a.i.1.11		33
29.20	even	7	inner	29.6.d.a.20.8	yes	66
29.22	even	14		841.6.a.h.1.23		33

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.16.8	✓	66	1.1	even	1	trivial
29.6.d.a.20.8	yes	66	29.20	even	7	inner
841.6.a.h.1.23		33	29.22	even	14
841.6.a.i.1.11		33	29.7	even	7