Embedded newform 29.6.d.a.16.6

• Label: 29.6.d.a.16.6
• 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			16.6
Character	\(\chi\)	\(=\)	29.16
Dual form			29.6.d.a.20.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37713 - 0.663191i) q^{2} +(-4.45689 + 5.58876i) q^{3} +(-18.4950 - 23.1920i) q^{4} +(71.8910 + 34.6209i) q^{5} +(9.84412 - 4.74068i) q^{6} +(-153.020 + 191.881i) q^{7} +(20.9732 + 91.8897i) q^{8} +(42.7022 + 187.091i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37713	−	0.663191i	−0.243444	−	0.117237i	0.308184	−	0.951327i	\(-0.400279\pi\)
							−0.551628	+	0.834090i	\(0.685993\pi\)
\(3\)	−4.45689	+	5.58876i	−0.285909	+	0.358519i	−0.903958	−	0.427621i	\(-0.859352\pi\)
							0.618049	+	0.786140i	\(0.287923\pi\)
\(5\)	71.8910	+	34.6209i	1.28602	+	0.619317i	0.946931	−	0.321436i	\(-0.104166\pi\)
							0.339093	+	0.940753i	\(0.389880\pi\)
\(7\)	−153.020	+	191.881i	−1.18033	+	1.48008i	−0.337974	+	0.941156i	\(0.609741\pi\)
							−0.842353	+	0.538927i	\(0.818830\pi\)
\(11\)	22.8891	−	100.283i	0.0570356	−	0.249889i	−0.938371	−	0.345631i	\(-0.887665\pi\)
							0.995406	+	0.0957413i	\(0.0305222\pi\)
\(13\)	−70.1262	+	307.243i	−0.115086	+	0.504224i	0.884223	+	0.467064i	\(0.154688\pi\)
							−0.999309	+	0.0371602i	\(0.988169\pi\)
\(17\)	1227.02			1.02974			0.514872	−	0.857267i	\(-0.327839\pi\)
							0.514872	+	0.857267i	\(0.327839\pi\)
\(19\)	−1328.12	−	1665.41i	−0.844023	−	1.05837i	−0.997531	−	0.0702238i	\(-0.977629\pi\)
							0.153508	−	0.988147i	\(-0.450943\pi\)
\(23\)	355.718	−	171.305i	0.140212	−	0.0675228i	−0.362462	−	0.931999i	\(-0.618064\pi\)
							0.502674	+	0.864476i	\(0.332349\pi\)
\(29\)	−2177.00	+	3971.37i	−0.480689	+	0.876891i
							\(31\)	−928.579	−	447.180i	−0.173546	−	0.0835753i	0.345094	−	0.938568i	\(-0.387847\pi\)
−0.518640	+	0.854993i	\(0.673561\pi\)
\(37\)	1644.24	+	7203.88i	0.197452	+	0.865092i	0.972447	+	0.233125i	\(0.0748950\pi\)
							−0.774995	+	0.631967i	\(0.782248\pi\)
\(41\)	8189.96			0.760890			0.380445	−	0.924803i	\(-0.375771\pi\)
							0.380445	+	0.924803i	\(0.375771\pi\)
\(43\)	20917.5	−	10073.4i	1.72520	−	0.830812i	0.737328	−	0.675535i	\(-0.236087\pi\)
							0.987871	−	0.155278i	\(-0.0496272\pi\)
\(47\)	−3570.07	+	15641.5i	−0.235739	+	1.03284i	0.709049	+	0.705160i	\(0.249125\pi\)
							−0.944788	+	0.327682i	\(0.893733\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.16.6	✓	66
29.7	even	7		841.6.a.i.1.18		33
29.20	even	7	inner	29.6.d.a.20.6	yes	66
29.22	even	14		841.6.a.h.1.16		33

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.16.6	✓	66	1.1	even	1	trivial
29.6.d.a.20.6	yes	66	29.20	even	7	inner
841.6.a.h.1.16		33	29.22	even	14
841.6.a.i.1.18		33	29.7	even	7