Embedded newform 29.15.c.a.12.16

• Label: 29.15.c.a.12.16
• Level: $29$
• Weight: $15$
• Character: 29.12
• Analytic conductor: $36.055$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	29.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(36.0554007641\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			12.16
Character	\(\chi\)	\(=\)	29.12
Dual form			29.15.c.a.17.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-34.5470 - 34.5470i) q^{2} +(-634.913 - 634.913i) q^{3} -13997.0i q^{4} +29356.7i q^{5} +43868.6i q^{6} +160670. q^{7} +(-1.04957e6 + 1.04957e6i) q^{8} -3.97674e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 312 q^{2} - 2 q^{3} - 4 q^{7} - 689310 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−34.5470	−	34.5470i	−0.269898	−	0.269898i	0.559161	−	0.829059i	\(-0.311123\pi\)
							−0.829059	+	0.559161i	\(0.811123\pi\)
\(3\)	−634.913	−	634.913i	−0.290312	−	0.290312i	0.546891	−	0.837204i	\(-0.315811\pi\)
							−0.837204	+	0.546891i	\(0.815811\pi\)
\(5\)			29356.7i			0.375766i	0.982191	+	0.187883i	\(0.0601625\pi\)
							−0.982191	+	0.187883i	\(0.939837\pi\)
\(7\)	160670.			0.195096			0.0975481	−	0.995231i	\(-0.468900\pi\)
							0.0975481	+	0.995231i	\(0.468900\pi\)
\(11\)	1.06941e7	+	1.06941e7i	0.548779	+	0.548779i	0.926088	−	0.377309i	\(-0.123150\pi\)
							−0.377309	+	0.926088i	\(0.623150\pi\)
\(13\)			9.62712e7i			1.53424i	0.641505	+	0.767119i	\(0.278311\pi\)
							−0.641505	+	0.767119i	\(0.721689\pi\)
\(17\)	7.55411e7	+	7.55411e7i	0.184095	+	0.184095i	0.793137	−	0.609043i	\(-0.208446\pi\)
							−0.609043	+	0.793137i	\(0.708446\pi\)
\(19\)	1.97113e8	+	1.97113e8i	0.220516	+	0.220516i	0.808716	−	0.588199i	\(-0.200163\pi\)
							−0.588199	+	0.808716i	\(0.700163\pi\)
\(23\)	−5.72486e9			−1.68139			−0.840697	−	0.541505i	\(-0.817855\pi\)
							−0.840697	+	0.541505i	\(0.817855\pi\)
\(29\)	1.50011e10	+	8.51613e9i	0.869637	+	0.493692i
							\(31\)	1.02407e9	+	1.02407e9i	0.0372218	+	0.0372218i	0.725473	−	0.688251i	\(-0.241621\pi\)
−0.688251	+	0.725473i	\(0.741621\pi\)
\(37\)	1.20893e11	−	1.20893e11i	1.27347	−	1.27347i	0.329218	−	0.944254i	\(-0.393215\pi\)
							0.944254	−	0.329218i	\(-0.106785\pi\)
\(41\)	3.51588e10	−	3.51588e10i	0.180529	−	0.180529i	−0.611057	−	0.791586i	\(-0.709256\pi\)
							0.791586	+	0.611057i	\(0.209256\pi\)
\(43\)	2.35663e11	+	2.35663e11i	0.866985	+	0.866985i	0.992137	−	0.125153i	\(-0.0399421\pi\)
							−0.125153	+	0.992137i	\(0.539942\pi\)
\(47\)	6.42547e11	−	6.42547e11i	1.26829	−	1.26829i	0.321325	−	0.946969i	\(-0.395872\pi\)
							0.946969	−	0.321325i	\(-0.104128\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.15.c.a.12.16	✓	68
29.17	odd	4	inner	29.15.c.a.17.16	yes	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.15.c.a.12.16	✓	68	1.1	even	1	trivial
29.15.c.a.17.16	yes	68	29.17	odd	4	inner