Embedded newform 29.13.c.a.12.6

• Label: 29.13.c.a.12.6
• Level: $29$
• Weight: $13$
• Character: 29.12
• Analytic conductor: $26.506$
• Analytic rank: $0$
• Dimension: $58$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	29.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(26.5058207010\)
Analytic rank:	\(0\)
Dimension:	\(58\)
Relative dimension:	\(29\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			12.6
Character	\(\chi\)	\(=\)	29.12
Dual form			29.13.c.a.17.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-58.0743 - 58.0743i) q^{2} +(433.237 + 433.237i) q^{3} +2649.26i q^{4} +10278.3i q^{5} -50319.9i q^{6} -72219.2 q^{7} +(-84018.7 + 84018.7i) q^{8} -156053. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q + 88 q^{2} - 2 q^{3} - 4 q^{7} + 79650 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−58.0743	−	58.0743i	−0.907411	−	0.907411i	0.0886513	−	0.996063i	\(-0.471744\pi\)
							−0.996063	+	0.0886513i	\(0.971744\pi\)
\(3\)	433.237	+	433.237i	0.594289	+	0.594289i	0.938787	−	0.344498i	\(-0.111951\pi\)
							−0.344498	+	0.938787i	\(0.611951\pi\)
\(5\)			10278.3i			0.657811i	0.944363	+	0.328906i	\(0.106680\pi\)
							−0.944363	+	0.328906i	\(0.893320\pi\)
\(7\)	−72219.2			−0.613853			−0.306927	−	0.951733i	\(-0.599301\pi\)
							−0.306927	+	0.951733i	\(0.599301\pi\)
\(11\)	−1.84490e6	−	1.84490e6i	−1.04140	−	1.04140i	−0.999105	−	0.0422925i	\(-0.986534\pi\)
							−0.0422925	−	0.999105i	\(-0.513466\pi\)
\(13\)			1.12272e6i			0.232601i	0.993214	+	0.116301i	\(0.0371036\pi\)
							−0.993214	+	0.116301i	\(0.962896\pi\)
\(17\)	2.49019e7	+	2.49019e7i	1.03167	+	1.03167i	0.999482	+	0.0321857i	\(0.0102468\pi\)
							0.0321857	+	0.999482i	\(0.489753\pi\)
\(19\)	707970.	+	707970.i	0.0150485	+	0.0150485i	0.714591	−	0.699542i	\(-0.246613\pi\)
							−0.699542	+	0.714591i	\(0.746613\pi\)
\(23\)	7.96747e7			0.538212			0.269106	−	0.963111i	\(-0.413272\pi\)
							0.269106	+	0.963111i	\(0.413272\pi\)
\(29\)	−2.10846e8	−	5.56200e8i	−0.354469	−	0.935068i
							\(31\)	1.83986e8	+	1.83986e8i	0.207307	+	0.207307i	0.803122	−	0.595815i	\(-0.203171\pi\)
−0.595815	+	0.803122i	\(0.703171\pi\)
\(37\)	2.00336e9	−	2.00336e9i	0.780817	−	0.780817i	−0.199152	−	0.979969i	\(-0.563819\pi\)
							0.979969	+	0.199152i	\(0.0638187\pi\)
\(41\)	2.40230e9	−	2.40230e9i	0.505735	−	0.505735i	−0.407479	−	0.913215i	\(-0.633592\pi\)
							0.913215	+	0.407479i	\(0.133592\pi\)
\(43\)	6.06919e9	+	6.06919e9i	0.960108	+	0.960108i	0.999234	−	0.0391263i	\(-0.0124575\pi\)
							−0.0391263	+	0.999234i	\(0.512457\pi\)
\(47\)	−2.78165e9	+	2.78165e9i	−0.258057	+	0.258057i	−0.824263	−	0.566207i	\(-0.808410\pi\)
							0.566207	+	0.824263i	\(0.308410\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.13.c.a.12.6	✓	58
29.17	odd	4	inner	29.13.c.a.17.6	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.13.c.a.12.6	✓	58	1.1	even	1	trivial
29.13.c.a.17.6	yes	58	29.17	odd	4	inner