Embedded newform 29.6.d.a.20.9

• Label: 29.6.d.a.20.9
• 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			20.9
Character	\(\chi\)	\(=\)	29.20
Dual form			29.6.d.a.16.9

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

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.16.9	✓	66	29.16	even	7	inner
29.6.d.a.20.9	yes	66	1.1	even	1	trivial
841.6.a.h.1.25		33	29.4	even	14
841.6.a.i.1.9		33	29.25	even	7