Embedded newform 29.6.d.a.16.9

• Label: 29.6.d.a.16.9
• 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			16.9
Character	\(\chi\)	\(=\)	29.16
Dual form			29.6.d.a.20.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.98347 + 1.91834i) q^{2} +(6.52648 - 8.18394i) q^{3} +(-7.76363 - 9.73529i) q^{4} +(64.0881 + 30.8632i) q^{5} +(41.6976 - 20.0805i) q^{6} +(85.0547 - 106.655i) q^{7} +(-43.7334 - 191.609i) q^{8} +(29.6906 + 130.083i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.98347	+	1.91834i	0.704185	+	0.339118i	0.751473	−	0.659763i	\(-0.229344\pi\)
							−0.0472879	+	0.998881i	\(0.515058\pi\)
\(3\)	6.52648	−	8.18394i	0.418674	−	0.525000i	−0.527110	−	0.849797i	\(-0.676724\pi\)
							0.945784	+	0.324797i	\(0.105296\pi\)
\(5\)	64.0881	+	30.8632i	1.14644	+	0.552098i	0.907963	−	0.419050i	\(-0.137637\pi\)
							0.238480	+	0.971148i	\(0.423351\pi\)
\(7\)	85.0547	−	106.655i	0.656074	−	0.822691i	−0.336835	−	0.941564i	\(-0.609357\pi\)
							0.992910	+	0.118872i	\(0.0379279\pi\)
\(11\)	−58.7994	+	257.617i	−0.146518	+	0.641938i	0.847319	+	0.531085i	\(0.178215\pi\)
							−0.993837	+	0.110853i	\(0.964642\pi\)
\(13\)	2.84949	−	12.4844i	0.00467638	−	0.0204885i	−0.972536	−	0.232753i	\(-0.925227\pi\)
							0.977212	+	0.212264i	\(0.0680838\pi\)
\(17\)	−2149.82			−1.80418			−0.902089	−	0.431551i	\(-0.857967\pi\)
							−0.902089	+	0.431551i	\(0.857967\pi\)
\(19\)	721.037	+	904.152i	0.458219	+	0.574589i	0.956243	−	0.292575i	\(-0.0945120\pi\)
							−0.498023	+	0.867164i	\(0.665941\pi\)
\(23\)	−2674.06	+	1287.76i	−1.05403	+	0.507592i	−0.878926	−	0.476959i	\(-0.841739\pi\)
							−0.175100	+	0.984551i	\(0.556025\pi\)
\(29\)	−3250.21	−	3153.93i	−0.717657	−	0.696397i
							\(31\)	2734.08	+	1316.66i	0.510983	+	0.246076i	0.671564	−	0.740946i	\(-0.265623\pi\)
−0.160582	+	0.987023i	\(0.551337\pi\)
\(37\)	755.116	+	3308.38i	0.0906796	+	0.397293i	0.999816	−	0.0192062i	\(-0.00611391\pi\)
							−0.909136	+	0.416499i	\(0.863257\pi\)
\(41\)	17005.6			1.57991			0.789955	−	0.613165i	\(-0.210104\pi\)
							0.789955	+	0.613165i	\(0.210104\pi\)
\(43\)	12222.6	−	5886.08i	1.00807	−	0.485462i	0.144401	−	0.989519i	\(-0.453874\pi\)
							0.863670	+	0.504058i	\(0.168160\pi\)
\(47\)	−2581.38	+	11309.8i	−0.170454	+	0.746807i	0.815359	+	0.578956i	\(0.196540\pi\)
							−0.985812	+	0.167851i	\(0.946317\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.16.9	✓	66
29.7	even	7		841.6.a.i.1.9		33
29.20	even	7	inner	29.6.d.a.20.9	yes	66
29.22	even	14		841.6.a.h.1.25		33

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.16.9	✓	66	1.1	even	1	trivial
29.6.d.a.20.9	yes	66	29.20	even	7	inner
841.6.a.h.1.25		33	29.22	even	14
841.6.a.i.1.9		33	29.7	even	7