Embedded newform 29.15.c.a.12.10

• Label: 29.15.c.a.12.10
• 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			12.10
Character	\(\chi\)	\(=\)	29.12
Dual form			29.15.c.a.17.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-94.2182 - 94.2182i) q^{2} +(-1537.46 - 1537.46i) q^{3} +1370.15i q^{4} -119266. i q^{5} +289713. i q^{6} +1.40104e6 q^{7} +(-1.41458e6 + 1.41458e6i) q^{8} -55424.1i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 312 q^{2} - 2 q^{3} - 4 q^{7} - 689310 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−94.2182	−	94.2182i	−0.736080	−	0.736080i	0.235737	−	0.971817i	\(-0.424250\pi\)
							−0.971817	+	0.235737i	\(0.924250\pi\)
\(3\)	−1537.46	−	1537.46i	−0.702998	−	0.702998i	0.262055	−	0.965053i	\(-0.415600\pi\)
							−0.965053	+	0.262055i	\(0.915600\pi\)
\(5\)		−	119266.i		−	1.52661i	−0.646038	−	0.763305i	\(-0.723575\pi\)
							0.646038	−	0.763305i	\(-0.276425\pi\)
\(7\)	1.40104e6			1.70123			0.850617	−	0.525786i	\(-0.176229\pi\)
							0.850617	+	0.525786i	\(0.176229\pi\)
\(11\)	−1.90295e7	−	1.90295e7i	−0.976513	−	0.976513i	0.0232176	−	0.999730i	\(-0.492609\pi\)
							−0.999730	+	0.0232176i	\(0.992609\pi\)
\(13\)		−	4.65120e6i		−	0.0741245i	−0.999313	−	0.0370622i	\(-0.988200\pi\)
							0.999313	−	0.0370622i	\(-0.0118000\pi\)
\(17\)	1.14820e8	+	1.14820e8i	0.279817	+	0.279817i	0.833036	−	0.553219i	\(-0.186601\pi\)
							−0.553219	+	0.833036i	\(0.686601\pi\)
\(19\)	−1.04300e9	−	1.04300e9i	−1.16684	−	1.16684i	−0.982947	−	0.183889i	\(-0.941131\pi\)
							−0.183889	−	0.982947i	\(-0.558869\pi\)
\(23\)	−3.85162e9			−1.13122			−0.565612	−	0.824671i	\(-0.691360\pi\)
							−0.565612	+	0.824671i	\(0.691360\pi\)
\(29\)	1.18944e10	−	1.24932e10i	0.689538	−	0.724250i
							\(31\)	1.46959e10	+	1.46959e10i	0.534150	+	0.534150i	0.921805	−	0.387655i	\(-0.126715\pi\)
−0.387655	+	0.921805i	\(0.626715\pi\)
\(37\)	−1.16835e11	+	1.16835e11i	−1.23072	+	1.23072i	−0.267033	+	0.963687i	\(0.586043\pi\)
							−0.963687	+	0.267033i	\(0.913957\pi\)
\(41\)	1.57242e11	−	1.57242e11i	0.807389	−	0.807389i	−0.176849	−	0.984238i	\(-0.556591\pi\)
							0.984238	+	0.176849i	\(0.0565905\pi\)
\(43\)	7.07368e10	+	7.07368e10i	0.260235	+	0.260235i	0.825150	−	0.564914i	\(-0.191091\pi\)
							−0.564914	+	0.825150i	\(0.691091\pi\)
\(47\)	2.97411e11	−	2.97411e11i	0.587046	−	0.587046i	−0.349785	−	0.936830i	\(-0.613745\pi\)
							0.936830	+	0.349785i	\(0.113745\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.15.c.a.12.10	✓	68
29.17	odd	4	inner	29.15.c.a.17.10	yes	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.15.c.a.12.10	✓	68	1.1	even	1	trivial
29.15.c.a.17.10	yes	68	29.17	odd	4	inner