Embedded newform 29.6.d.a.20.8

• Label: 29.6.d.a.20.8
• Level: $29$
• Weight: $6$
• Character: 29.20
• 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			20.8
Character	\(\chi\)	\(=\)	29.20
Dual form			29.6.d.a.16.8

$q$-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 \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{6}{7}\right)\)

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	−0.0722119	−	0.997389i	\(-0.523006\pi\)
							0.734767	+	0.678320i	\(0.237291\pi\)
\(3\)	−13.9558	−	17.5000i	−0.895266	−	1.12263i	−0.991864	−	0.127304i	\(-0.959368\pi\)
							0.0965979	−	0.995323i	\(-0.469204\pi\)
\(5\)	−40.2504	+	19.3836i	−0.720021	+	0.346744i	−0.757751	−	0.652544i	\(-0.773702\pi\)
							0.0377297	+	0.999288i	\(0.487987\pi\)
\(7\)	−30.7029	−	38.5002i	−0.236829	−	0.296974i	0.649187	−	0.760629i	\(-0.275109\pi\)
							−0.886016	+	0.463655i	\(0.846538\pi\)
\(11\)	−124.623	−	546.010i	−0.310540	−	1.36056i	−0.853626	−	0.520886i	\(-0.825602\pi\)
							0.543086	−	0.839677i	\(-0.317256\pi\)
\(13\)	−134.054	−	587.331i	−0.220000	−	0.963883i	−0.957477	−	0.288511i	\(-0.906840\pi\)
							0.737477	−	0.675373i	\(-0.236017\pi\)
\(17\)	−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	−0.648188	−	0.761481i	\(-0.724473\pi\)
							0.886624	+	0.462491i	\(0.153044\pi\)
\(23\)	3630.88	+	1748.54i	1.43118	+	0.689218i	0.979216	−	0.202821i	\(-0.0650110\pi\)
							0.451959	+	0.892039i	\(0.350725\pi\)
\(29\)	4180.32	−	1742.42i	0.923028	−	0.384732i
							\(31\)	−2253.73	+	1085.34i	−0.421208	+	0.202843i	−0.632468	−	0.774587i	\(-0.717958\pi\)
0.211259	+	0.977430i	\(0.432244\pi\)
\(37\)	−217.466	+	952.783i	−0.0261149	+	0.114417i	−0.986305	−	0.164929i	\(-0.947260\pi\)
							0.960191	+	0.279346i	\(0.0901176\pi\)
\(41\)	−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	0.0392261	−	0.999230i	\(-0.487511\pi\)
							−0.756773	+	0.653678i	\(0.773225\pi\)
\(47\)	−2269.26	−	9942.28i	−0.149844	−	0.656510i	−0.992927	−	0.118727i	\(-0.962119\pi\)
							0.843083	−	0.537784i	\(-0.180738\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.16.8	✓	66	29.16	even	7	inner
29.6.d.a.20.8	yes	66	1.1	even	1	trivial
841.6.a.h.1.23		33	29.4	even	14
841.6.a.i.1.11		33	29.25	even	7