Embedded newform 29.6.d.a.7.11

• Label: 29.6.d.a.7.11
• Level: $29$
• Weight: $6$
• Character: 29.7
• 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			7.11
Character	\(\chi\)	\(=\)	29.7
Dual form			29.6.d.a.25.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.19360 + 9.61081i) q^{2} +(14.0872 + 6.78404i) q^{3} +(-58.7248 + 28.2804i) q^{4} +(0.197313 + 0.864484i) q^{5} +(-34.2984 + 150.271i) q^{6} +(69.1061 + 33.2798i) q^{7} +(-203.933 - 255.724i) q^{8} +(0.918137 + 1.15131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.19360	+	9.61081i	0.387778	+	1.69897i	0.672284	+	0.740293i	\(0.265313\pi\)
							−0.284506	+	0.958674i	\(0.591830\pi\)
\(3\)	14.0872	+	6.78404i	0.903695	+	0.435196i	0.827222	−	0.561875i	\(-0.189920\pi\)
							0.0764727	+	0.997072i	\(0.475634\pi\)
\(5\)	0.197313	+	0.864484i	0.00352964	+	0.0154644i	0.976662	−	0.214782i	\(-0.0689040\pi\)
							−0.973132	+	0.230246i	\(0.926047\pi\)
\(7\)	69.1061	+	33.2798i	0.533054	+	0.256705i	0.680991	−	0.732291i	\(-0.261549\pi\)
							−0.147937	+	0.988997i	\(0.547263\pi\)
\(11\)	398.388	−	499.563i	0.992716	−	1.24483i	0.0232171	−	0.999730i	\(-0.492609\pi\)
							0.969499	−	0.245096i	\(-0.0788195\pi\)
\(13\)	−489.189	+	613.423i	−0.802820	+	1.00670i	0.196835	+	0.980437i	\(0.436934\pi\)
							−0.999655	+	0.0262674i	\(0.991638\pi\)
\(17\)	755.430			0.633975			0.316987	−	0.948430i	\(-0.397329\pi\)
							0.316987	+	0.948430i	\(0.397329\pi\)
\(19\)	1268.69	−	610.970i	0.806254	−	0.388272i	0.0150983	−	0.999886i	\(-0.495194\pi\)
							0.791156	+	0.611614i	\(0.209480\pi\)
\(23\)	−698.797	+	3061.63i	−0.275443	+	1.20679i	0.628044	+	0.778178i	\(0.283856\pi\)
							−0.903487	+	0.428616i	\(0.859001\pi\)
\(29\)	−1097.66	+	4393.89i	−0.242366	+	0.970185i
							\(31\)	−733.942	−	3215.61i	−0.137169	−	0.600979i	−0.996049	−	0.0888015i	\(-0.971696\pi\)
0.858880	−	0.512177i	\(-0.171161\pi\)
\(37\)	−3538.74	−	4437.44i	−0.424956	−	0.532878i	0.522553	−	0.852607i	\(-0.324980\pi\)
							−0.947509	+	0.319729i	\(0.896408\pi\)
\(41\)	−9596.64			−0.891579			−0.445789	−	0.895138i	\(-0.647077\pi\)
							−0.445789	+	0.895138i	\(0.647077\pi\)
\(43\)	3418.04	−	14975.4i	0.281907	−	1.23512i	−0.613438	−	0.789743i	\(-0.710214\pi\)
							0.895345	−	0.445373i	\(-0.146929\pi\)
\(47\)	16988.6	−	21303.0i	1.12179	−	1.40668i	0.219469	−	0.975619i	\(-0.429567\pi\)
							0.902322	−	0.431062i	\(-0.141861\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.7.11	✓	66
29.5	even	14		841.6.a.h.1.32		33
29.24	even	7		841.6.a.i.1.2		33
29.25	even	7	inner	29.6.d.a.25.11	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.7.11	✓	66	1.1	even	1	trivial
29.6.d.a.25.11	yes	66	29.25	even	7	inner
841.6.a.h.1.32		33	29.5	even	14
841.6.a.i.1.2		33	29.24	even	7