Embedded newform 29.13.c.a.12.5

• Label: 29.13.c.a.12.5
• 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			12.5
Character	\(\chi\)	\(=\)	29.12
Dual form			29.13.c.a.17.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-69.9680 - 69.9680i) q^{2} +(814.355 + 814.355i) q^{3} +5695.05i q^{4} -19209.4i q^{5} -113958. i q^{6} -6147.85 q^{7} +(111882. - 111882. i) q^{8} +794908. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q + 88 q^{2} - 2 q^{3} - 4 q^{7} + 79650 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−69.9680	−	69.9680i	−1.09325	−	1.09325i	−0.995179	−	0.0980709i	\(-0.968733\pi\)
							−0.0980709	−	0.995179i	\(-0.531267\pi\)
\(3\)	814.355	+	814.355i	1.11709	+	1.11709i	0.992167	+	0.124918i	\(0.0398669\pi\)
							0.124918	+	0.992167i	\(0.460133\pi\)
\(5\)		−	19209.4i		−	1.22940i	−0.788760	−	0.614702i	\(-0.789276\pi\)
							0.788760	−	0.614702i	\(-0.210724\pi\)
\(7\)	−6147.85			−0.0522559			−0.0261280	−	0.999659i	\(-0.508318\pi\)
							−0.0261280	+	0.999659i	\(0.508318\pi\)
\(11\)	−44580.0	−	44580.0i	−0.0251643	−	0.0251643i	0.694413	−	0.719577i	\(-0.255664\pi\)
							−0.719577	+	0.694413i	\(0.755664\pi\)
\(13\)		−	1.32454e6i		−	0.274412i	−0.990543	−	0.137206i	\(-0.956188\pi\)
							0.990543	−	0.137206i	\(-0.0438123\pi\)
\(17\)	−1.55900e7	−	1.55900e7i	−0.645881	−	0.645881i	0.306114	−	0.951995i	\(-0.400971\pi\)
							−0.951995	+	0.306114i	\(0.900971\pi\)
\(19\)	−6.20809e6	−	6.20809e6i	−0.131958	−	0.131958i	0.638043	−	0.770001i	\(-0.279744\pi\)
							−0.770001	+	0.638043i	\(0.779744\pi\)
\(23\)	−6.23321e6			−0.0421061			−0.0210530	−	0.999778i	\(-0.506702\pi\)
							−0.0210530	+	0.999778i	\(0.506702\pi\)
\(29\)	5.86494e8	−	9.91967e7i	0.985996	−	0.166767i
							\(31\)	−1.14357e9	−	1.14357e9i	−1.28852	−	1.28852i	−0.935687	−	0.352832i	\(-0.885219\pi\)
−0.352832	−	0.935687i	\(-0.614781\pi\)
\(37\)	−3.05748e9	+	3.05748e9i	−1.19166	+	1.19166i	−0.215060	+	0.976601i	\(0.568995\pi\)
							−0.976601	+	0.215060i	\(0.931005\pi\)
\(41\)	4.02383e9	−	4.02383e9i	0.847103	−	0.847103i	−0.142668	−	0.989771i	\(-0.545568\pi\)
							0.989771	+	0.142668i	\(0.0455681\pi\)
\(43\)	−7.00332e9	−	7.00332e9i	−1.10788	−	1.10788i	−0.993429	−	0.114453i	\(-0.963488\pi\)
							−0.114453	−	0.993429i	\(-0.536512\pi\)
\(47\)	5.84296e9	−	5.84296e9i	0.542058	−	0.542058i	−0.382074	−	0.924132i	\(-0.624790\pi\)
							0.924132	+	0.382074i	\(0.124790\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.13.c.a.12.5	✓	58
29.17	odd	4	inner	29.13.c.a.17.5	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.13.c.a.12.5	✓	58	1.1	even	1	trivial
29.13.c.a.17.5	yes	58	29.17	odd	4	inner