Embedded newform 29.13.c.a.12.17

• Label: 29.13.c.a.12.17
• 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			12.17
Character	\(\chi\)	\(=\)	29.12
Dual form			29.13.c.a.17.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(18.1237 + 18.1237i) q^{2} +(-105.386 - 105.386i) q^{3} -3439.06i q^{4} -24861.6i q^{5} -3819.95i q^{6} +4681.32 q^{7} +(136563. - 136563. i) q^{8} -509229. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q + 88 q^{2} - 2 q^{3} - 4 q^{7} + 79650 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	18.1237	+	18.1237i	0.283183	+	0.283183i	0.834377	−	0.551194i	\(-0.185828\pi\)
							−0.551194	+	0.834377i	\(0.685828\pi\)
\(3\)	−105.386	−	105.386i	−0.144562	−	0.144562i	0.631122	−	0.775684i	\(-0.282595\pi\)
							−0.775684	+	0.631122i	\(0.782595\pi\)
\(5\)		−	24861.6i		−	1.59114i	−0.605861	−	0.795570i	\(-0.707171\pi\)
							0.605861	−	0.795570i	\(-0.292829\pi\)
\(7\)	4681.32			0.0397906			0.0198953	−	0.999802i	\(-0.493667\pi\)
							0.0198953	+	0.999802i	\(0.493667\pi\)
\(11\)	−920806.	−	920806.i	−0.519771	−	0.519771i	0.397731	−	0.917502i	\(-0.369798\pi\)
							−0.917502	+	0.397731i	\(0.869798\pi\)
\(13\)			8.01379e6i			1.66027i	0.557565	+	0.830133i	\(0.311736\pi\)
							−0.557565	+	0.830133i	\(0.688264\pi\)
\(17\)	8.76150e6	+	8.76150e6i	0.362982	+	0.362982i	0.864910	−	0.501928i	\(-0.167376\pi\)
							−0.501928	+	0.864910i	\(0.667376\pi\)
\(19\)	−1.61358e7	−	1.61358e7i	−0.342980	−	0.342980i	0.514506	−	0.857487i	\(-0.327975\pi\)
							−0.857487	+	0.514506i	\(0.827975\pi\)
\(23\)	1.37276e8			0.927315			0.463657	−	0.886015i	\(-0.346537\pi\)
							0.463657	+	0.886015i	\(0.346537\pi\)
\(29\)	5.94823e8	+	663558.i	0.999999	+	0.00111555i
							\(31\)	−8.47123e8	−	8.47123e8i	−0.954501	−	0.954501i	0.0445079	−	0.999009i	\(-0.485828\pi\)
−0.999009	+	0.0445079i	\(0.985828\pi\)
\(37\)	−5.97077e8	+	5.97077e8i	−0.232713	+	0.232713i	−0.813824	−	0.581111i	\(-0.802618\pi\)
							0.581111	+	0.813824i	\(0.302618\pi\)
\(41\)	−2.87521e9	+	2.87521e9i	−0.605294	+	0.605294i	−0.941713	−	0.336419i	\(-0.890784\pi\)
							0.336419	+	0.941713i	\(0.390784\pi\)
\(43\)	−7.62128e7	−	7.62128e7i	−0.0120564	−	0.0120564i	0.701053	−	0.713109i	\(-0.252714\pi\)
							−0.713109	+	0.701053i	\(0.752714\pi\)
\(47\)	5.73565e9	−	5.73565e9i	0.532103	−	0.532103i	−0.389095	−	0.921198i	\(-0.627212\pi\)
							0.921198	+	0.389095i	\(0.127212\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.13.c.a.12.17	✓	58
29.17	odd	4	inner	29.13.c.a.17.17	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.13.c.a.12.17	✓	58	1.1	even	1	trivial
29.13.c.a.17.17	yes	58	29.17	odd	4	inner