Embedded newform 29.13.c.a.12.15

• Label: 29.13.c.a.12.15
• 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			12.15
Character	\(\chi\)	\(=\)	29.12
Dual form			29.13.c.a.17.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.50807 + 2.50807i) q^{2} +(773.536 + 773.536i) q^{3} -4083.42i q^{4} -10269.1i q^{5} +3880.17i q^{6} +192386. q^{7} +(20514.6 - 20514.6i) q^{8} +665275. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q + 88 q^{2} - 2 q^{3} - 4 q^{7} + 79650 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.50807	+	2.50807i	0.0391886	+	0.0391886i	0.726430	−	0.687241i	\(-0.241178\pi\)
							−0.687241	+	0.726430i	\(0.741178\pi\)
\(3\)	773.536	+	773.536i	1.06109	+	1.06109i	0.998008	+	0.0630839i	\(0.0200936\pi\)
							0.0630839	+	0.998008i	\(0.479906\pi\)
\(5\)		−	10269.1i		−	0.657219i	−0.944466	−	0.328610i	\(-0.893420\pi\)
							0.944466	−	0.328610i	\(-0.106580\pi\)
\(7\)	192386.			1.63525			0.817626	−	0.575750i	\(-0.195290\pi\)
							0.817626	+	0.575750i	\(0.195290\pi\)
\(11\)	−2.03959e6	−	2.03959e6i	−1.15129	−	1.15129i	−0.986293	−	0.165001i	\(-0.947237\pi\)
							−0.165001	−	0.986293i	\(-0.552763\pi\)
\(13\)		−	3.65691e6i		−	0.757626i	−0.925473	−	0.378813i	\(-0.876332\pi\)
							0.925473	−	0.378813i	\(-0.123668\pi\)
\(17\)	−1.31395e7	−	1.31395e7i	−0.544359	−	0.544359i	0.380445	−	0.924804i	\(-0.375771\pi\)
							−0.924804	+	0.380445i	\(0.875771\pi\)
\(19\)	861415.	+	861415.i	0.0183101	+	0.0183101i	0.716203	−	0.697892i	\(-0.245879\pi\)
							−0.697892	+	0.716203i	\(0.745879\pi\)
\(23\)	4.78047e7			0.322926			0.161463	−	0.986879i	\(-0.448379\pi\)
							0.161463	+	0.986879i	\(0.448379\pi\)
\(29\)	−3.11603e8	−	5.06674e8i	−0.523858	−	0.851805i
							\(31\)	1.10962e9	+	1.10962e9i	1.25028	+	1.25028i	0.955596	+	0.294679i	\(0.0952128\pi\)
0.294679	+	0.955596i	\(0.404787\pi\)
\(37\)	2.12904e8	−	2.12904e8i	0.0829800	−	0.0829800i	−0.664398	−	0.747379i	\(-0.731312\pi\)
							0.747379	+	0.664398i	\(0.231312\pi\)
\(41\)	−1.88223e9	+	1.88223e9i	−0.396250	+	0.396250i	−0.876908	−	0.480658i	\(-0.840398\pi\)
							0.480658	+	0.876908i	\(0.340398\pi\)
\(43\)	6.39640e9	+	6.39640e9i	1.01187	+	1.01187i	0.999929	+	0.0119422i	\(0.00380140\pi\)
							0.0119422	+	0.999929i	\(0.496199\pi\)
\(47\)	1.07810e10	−	1.07810e10i	1.00017	−	1.00017i	0.000165120	−	1.00000i	\(-0.499947\pi\)
							1.00000		0.000165120i	\(-5.25594e-5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.13.c.a.12.15	✓	58
29.17	odd	4	inner	29.13.c.a.17.15	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.13.c.a.12.15	✓	58	1.1	even	1	trivial
29.13.c.a.17.15	yes	58	29.17	odd	4	inner