Embedded newform 29.6.e.a.13.4

• Label: 29.6.e.a.13.4
• Level: $29$
• Weight: $6$
• Character: 29.13
• Analytic conductor: $4.651$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65113077458\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			13.4
Character	\(\chi\)	\(=\)	29.13
Dual form			29.6.e.a.9.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.27278 + 4.71948i) q^{2} +(-0.126978 - 0.101261i) q^{3} +(2.84371 + 3.56589i) q^{4} +(36.9063 + 17.7731i) q^{5} +(0.766495 - 0.369124i) q^{6} +(-124.573 + 156.209i) q^{7} +(-186.713 + 42.6160i) q^{8} +(-54.0667 - 236.882i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 7 q^{2} - 7 q^{3} + 165 q^{4} - 53 q^{5} - 311 q^{6} - 27 q^{7} - 1498 q^{8} + 1567 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.27278	+	4.71948i	−0.401775	+	0.834294i	0.597694	+	0.801724i	\(0.296084\pi\)
							−0.999469	+	0.0325704i	\(0.989631\pi\)
\(3\)	−0.126978	−	0.101261i	−0.00814563	−	0.00649593i	0.619409	−	0.785069i	\(-0.287372\pi\)
							−0.627554	+	0.778573i	\(0.715944\pi\)
\(5\)	36.9063	+	17.7731i	0.660199	+	0.317935i	0.733813	−	0.679352i	\(-0.237739\pi\)
							−0.0736137	+	0.997287i	\(0.523453\pi\)
\(7\)	−124.573	+	156.209i	−0.960898	+	1.20493i	0.0178460	+	0.999841i	\(0.494319\pi\)
							−0.978744	+	0.205087i	\(0.934252\pi\)
\(11\)	15.6712	+	3.57686i	0.0390500	+	0.00891291i	0.242001	−	0.970276i	\(-0.422196\pi\)
							−0.202951	+	0.979189i	\(0.565053\pi\)
\(13\)	−150.197	+	658.055i	−0.246492	+	1.07995i	0.688488	+	0.725248i	\(0.258275\pi\)
							−0.934979	+	0.354702i	\(0.884582\pi\)
\(17\)			2192.25i			1.83979i	0.392170	+	0.919893i	\(0.371724\pi\)
							−0.392170	+	0.919893i	\(0.628276\pi\)
\(19\)	2266.18	−	1807.21i	1.44016	−	1.14849i	0.477211	−	0.878789i	\(-0.341648\pi\)
							0.962945	−	0.269697i	\(-0.0869237\pi\)
\(23\)	2187.43	−	1053.41i	0.862213	−	0.415220i	0.0501170	−	0.998743i	\(-0.484041\pi\)
							0.812096	+	0.583523i	\(0.198326\pi\)
\(29\)	3228.52	+	3176.13i	0.712866	+	0.701300i
							\(31\)	−607.991	+	1262.51i	−0.113630	+	0.235955i	−0.950026	−	0.312170i	\(-0.898944\pi\)
0.836396	+	0.548125i	\(0.184658\pi\)
\(37\)	12851.0	−	2933.15i	1.54323	−	0.352233i	0.635606	−	0.772014i	\(-0.280750\pi\)
							0.907625	+	0.419781i	\(0.137893\pi\)
\(41\)			14625.9i			1.35882i	0.733759	+	0.679410i	\(0.237764\pi\)
							−0.733759	+	0.679410i	\(0.762236\pi\)
\(43\)	−2762.03	−	5735.40i	−0.227801	−	0.473035i	0.755469	−	0.655184i	\(-0.227409\pi\)
							−0.983271	+	0.182149i	\(0.941695\pi\)
\(47\)	−767.691	−	175.221i	−0.0506923	−	0.0115702i	0.197100	−	0.980383i	\(-0.436848\pi\)
							−0.247792	+	0.968813i	\(0.579705\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.e.a.13.4	yes	72
29.3	odd	28		841.6.a.l.1.22		72
29.9	even	14	inner	29.6.e.a.9.4	✓	72
29.26	odd	28		841.6.a.l.1.51		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.e.a.9.4	✓	72	29.9	even	14	inner
29.6.e.a.13.4	yes	72	1.1	even	1	trivial
841.6.a.l.1.22		72	29.3	odd	28
841.6.a.l.1.51		72	29.26	odd	28