Embedded newform 29.6.d.a.7.10

• Label: 29.6.d.a.7.10
• 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			7.10
Character	\(\chi\)	\(=\)	29.7
Dual form			29.6.d.a.25.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.95082 + 8.54710i) q^{2} +(-16.4262 - 7.91043i) q^{3} +(-40.4162 + 19.4634i) q^{4} +(-4.08126 - 17.8812i) q^{5} +(35.5667 - 155.828i) q^{6} +(-139.508 - 67.1835i) q^{7} +(-70.2862 - 88.1361i) q^{8} +(55.7366 + 69.8915i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.95082	+	8.54710i	0.344859	+	1.51093i	0.788675	+	0.614810i	\(0.210767\pi\)
							−0.443816	+	0.896118i	\(0.646376\pi\)
\(3\)	−16.4262	−	7.91043i	−1.05374	−	0.507455i	−0.174908	−	0.984585i	\(-0.555963\pi\)
							−0.878833	+	0.477130i	\(0.841677\pi\)
\(5\)	−4.08126	−	17.8812i	−0.0730079	−	0.319868i	0.925216	−	0.379441i	\(-0.123884\pi\)
							−0.998224	+	0.0595721i	\(0.981026\pi\)
\(7\)	−139.508	−	67.1835i	−1.07610	−	0.518224i	−0.190035	−	0.981777i	\(-0.560860\pi\)
							−0.886069	+	0.463553i	\(0.846574\pi\)
\(11\)	−76.3415	+	95.7293i	−0.190230	+	0.238541i	−0.867795	−	0.496922i	\(-0.834464\pi\)
							0.677565	+	0.735463i	\(0.263035\pi\)
\(13\)	−60.7414	+	76.1674i	−0.0996843	+	0.125000i	−0.829171	−	0.558994i	\(-0.811187\pi\)
							0.729487	+	0.683994i	\(0.239759\pi\)
\(17\)	−974.249			−0.817613			−0.408807	−	0.912621i	\(-0.634055\pi\)
							−0.408807	+	0.912621i	\(0.634055\pi\)
\(19\)	−2635.62	+	1269.25i	−1.67494	+	0.806607i	−0.677463	+	0.735557i	\(0.736921\pi\)
							−0.997473	+	0.0710496i	\(0.977365\pi\)
\(23\)	477.879	−	2093.73i	0.188364	−	0.825278i	−0.789115	−	0.614246i	\(-0.789460\pi\)
							0.977479	−	0.211032i	\(-0.0676825\pi\)
\(29\)	4507.83	+	436.612i	0.995342	+	0.0964053i
							\(31\)	342.476	+	1500.48i	0.0640067	+	0.280432i	0.996796	−	0.0799916i	\(-0.0254893\pi\)
−0.932789	+	0.360423i	\(0.882632\pi\)
\(37\)	6030.88	+	7562.48i	0.724229	+	0.908155i	0.998569	−	0.0534780i	\(-0.0170307\pi\)
							−0.274340	+	0.961633i	\(0.588459\pi\)
\(41\)	−6672.69			−0.619928			−0.309964	−	0.950748i	\(-0.600317\pi\)
							−0.309964	+	0.950748i	\(0.600317\pi\)
\(43\)	4491.60	−	19679.0i	0.370450	−	1.62305i	−0.355064	−	0.934842i	\(-0.615541\pi\)
							0.725514	−	0.688207i	\(-0.241602\pi\)
\(47\)	−8370.53	+	10496.3i	−0.552724	+	0.693094i	−0.977194	−	0.212348i	\(-0.931889\pi\)
							0.424470	+	0.905442i	\(0.360460\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.7.10	✓	66
29.5	even	14		841.6.a.h.1.28		33
29.24	even	7		841.6.a.i.1.6		33
29.25	even	7	inner	29.6.d.a.25.10	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.7.10	✓	66	1.1	even	1	trivial
29.6.d.a.25.10	yes	66	29.25	even	7	inner
841.6.a.h.1.28		33	29.5	even	14
841.6.a.i.1.6		33	29.24	even	7