Embedded newform 59.2.c.a.3.3

• Label: 59.2.c.a.3.3
• Level: $59$
• Weight: $2$
• Character: 59.3
• Analytic conductor: $0.471$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	59.c (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.471117371926\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(4\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			3.3
Character	\(\chi\)	\(=\)	59.3
Dual form			59.2.c.a.20.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.554859 + 0.256705i) q^{2} +(2.89290 + 0.974731i) q^{3} +(-1.05280 + 1.23945i) q^{4} +(-2.81882 - 1.69603i) q^{5} +(-1.85537 + 0.201783i) q^{6} +(0.0467597 - 0.862432i) q^{7} +(0.593097 - 2.13614i) q^{8} +(5.03048 + 3.82407i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 26 q^{2} - 23 q^{3} - 30 q^{4} - 25 q^{5} - 13 q^{6} - 23 q^{7} - 8 q^{8} - 21 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/59\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{25}{29}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.554859	+	0.256705i	−0.392344	+	0.181518i	−0.606135	−	0.795362i	\(-0.707281\pi\)
							0.213791	+	0.976879i	\(0.431419\pi\)
\(3\)	2.89290	+	0.974731i	1.67022	+	0.562761i	0.986033	−	0.166550i	\(-0.0532627\pi\)
							0.684182	+	0.729311i	\(0.260159\pi\)
\(5\)	−2.81882	−	1.69603i	−1.26062	−	0.758488i	−0.281066	−	0.959688i	\(-0.590688\pi\)
							−0.979550	+	0.201201i	\(0.935516\pi\)
\(7\)	0.0467597	−	0.862432i	0.0176735	−	0.325969i	−0.976305	−	0.216397i	\(-0.930569\pi\)
							0.993979	−	0.109571i	\(-0.0349478\pi\)
\(11\)	0.631059	−	3.84929i	0.190272	−	1.16061i	−0.703034	−	0.711156i	\(-0.748172\pi\)
							0.893306	−	0.449449i	\(-0.148380\pi\)
\(13\)	−1.31325	+	0.998305i	−0.364230	+	0.276880i	−0.771241	−	0.636543i	\(-0.780364\pi\)
							0.407012	+	0.913423i	\(0.366571\pi\)
\(17\)	0.182290	+	3.36214i	0.0442117	+	0.815438i	0.933607	+	0.358299i	\(0.116643\pi\)
							−0.889395	+	0.457139i	\(0.848874\pi\)
\(19\)	−3.16614	+	2.99913i	−0.726362	+	0.688047i	−0.958646	−	0.284601i	\(-0.908139\pi\)
							0.232284	+	0.972648i	\(0.425380\pi\)
\(23\)	−2.04440	+	0.450007i	−0.426287	+	0.0938330i	−0.422932	−	0.906161i	\(-0.638999\pi\)
							−0.00335541	+	0.999994i	\(0.501068\pi\)
\(29\)	−3.93910	−	1.82242i	−0.731472	−	0.338415i	0.0185498	−	0.999828i	\(-0.494095\pi\)
							−0.750022	+	0.661413i	\(0.769957\pi\)
\(31\)	5.88624	+	5.57574i	1.05720	+	1.00143i	0.999995	+	0.00307034i	\(0.000977320\pi\)
							0.0572045	+	0.998362i	\(0.481781\pi\)
\(37\)	−0.331878	−	1.19532i	−0.0545604	−	0.196509i	0.931247	−	0.364388i	\(-0.118722\pi\)
							−0.985808	+	0.167879i	\(0.946308\pi\)
\(41\)	−3.25469	−	0.716412i	−0.508297	−	0.111885i	−0.0465849	−	0.998914i	\(-0.514834\pi\)
							−0.461713	+	0.887030i	\(0.652765\pi\)
\(43\)	−0.471222	−	2.87433i	−0.0718607	−	0.438331i	−0.998112	−	0.0614239i	\(-0.980436\pi\)
							0.926251	−	0.376907i	\(-0.123012\pi\)
\(47\)	5.86181	−	3.52693i	0.855032	−	0.514456i	−0.0192832	−	0.999814i	\(-0.506138\pi\)
							0.874315	+	0.485358i	\(0.161311\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	59.2.c.a.3.3	✓	112
3.2	odd	2		531.2.i.a.298.2		112
4.3	odd	2		944.2.m.c.593.1		112
59.16	even	29		3481.2.a.p.1.27		56
59.20	even	29	inner	59.2.c.a.20.3	yes	112
59.43	odd	58		3481.2.a.q.1.30		56
177.20	odd	58		531.2.i.a.433.2		112
236.79	odd	58		944.2.m.c.433.1		112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
59.2.c.a.3.3	✓	112	1.1	even	1	trivial
59.2.c.a.20.3	yes	112	59.20	even	29	inner
531.2.i.a.298.2		112	3.2	odd	2
531.2.i.a.433.2		112	177.20	odd	58
944.2.m.c.433.1		112	236.79	odd	58
944.2.m.c.593.1		112	4.3	odd	2
3481.2.a.p.1.27		56	59.16	even	29
3481.2.a.q.1.30		56	59.43	odd	58