Embedded newform 59.2.c.a.15.3

• Label: 59.2.c.a.15.3
• Level: $59$
• Weight: $2$
• Character: 59.15
• Analytic conductor: $0.471$
• Analytic rank: $0$
• Dimension: $112$
• 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			15.3
Character	\(\chi\)	\(=\)	59.15
Dual form			59.2.c.a.4.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.670646 - 0.0729372i) q^{2} +(0.865386 + 1.01881i) q^{3} +(-1.50879 + 0.332111i) q^{4} +(0.331060 - 0.251665i) q^{5} +(0.654677 + 0.620143i) q^{6} +(-0.928345 - 2.32997i) q^{7} +(-2.26622 + 0.763578i) q^{8} +(0.196262 - 1.19715i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 26 q^{2} - 23 q^{3} - 30 q^{4} - 25 q^{5} - 13 q^{6} - 23 q^{7} - 8 q^{8} - 21 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{28}{29}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.670646	−	0.0729372i	0.474219	−	0.0515744i	0.132110	−	0.991235i	\(-0.457825\pi\)
							0.342108	+	0.939661i	\(0.388859\pi\)
\(3\)	0.865386	+	1.01881i	0.499631	+	0.588211i	0.952946	−	0.303139i	\(-0.0980347\pi\)
							−0.453315	+	0.891350i	\(0.649759\pi\)
\(5\)	0.331060	−	0.251665i	0.148055	−	0.112548i	−0.528525	−	0.848918i	\(-0.677255\pi\)
							0.676579	+	0.736370i	\(0.263462\pi\)
\(7\)	−0.928345	−	2.32997i	−0.350881	−	0.880646i	−0.993434	−	0.114405i	\(-0.963504\pi\)
							0.642553	−	0.766241i	\(-0.277875\pi\)
\(11\)	−0.551218	−	0.255021i	−0.166199	−	0.0768917i	0.335026	−	0.942209i	\(-0.391255\pi\)
							−0.501225	+	0.865317i	\(0.667117\pi\)
\(13\)	0.568579	+	3.46818i	0.157695	+	0.961900i	0.940540	+	0.339683i	\(0.110320\pi\)
							−0.782845	+	0.622217i	\(0.786232\pi\)
\(17\)	−0.0439170	+	0.110223i	−0.0106514	+	0.0267331i	−0.934206	−	0.356733i	\(-0.883890\pi\)
							0.923555	+	0.383466i	\(0.125270\pi\)
\(19\)	−3.52708	+	5.20206i	−0.809168	+	1.19343i	0.168999	+	0.985616i	\(0.445946\pi\)
							−0.978168	+	0.207818i	\(0.933364\pi\)
\(23\)	−0.233431	−	4.30537i	−0.0486736	−	0.897732i	−0.916294	−	0.400506i	\(-0.868834\pi\)
							0.867621	−	0.497227i	\(-0.165648\pi\)
\(29\)	7.91458	+	0.860763i	1.46970	+	0.159840i	0.807669	−	0.589636i	\(-0.200729\pi\)
							0.662032	+	0.749476i	\(0.269694\pi\)
\(31\)	−1.03526	−	1.52689i	−0.185937	−	0.274237i	0.723283	−	0.690551i	\(-0.242632\pi\)
							−0.909221	+	0.416314i	\(0.863322\pi\)
\(37\)	2.60938	+	0.879204i	0.428980	+	0.144540i	0.525505	−	0.850790i	\(-0.323876\pi\)
							−0.0965252	+	0.995331i	\(0.530773\pi\)
\(41\)	0.288828	−	5.32712i	0.0451074	−	0.831956i	−0.885208	−	0.465196i	\(-0.845984\pi\)
							0.930315	−	0.366761i	\(-0.119533\pi\)
\(43\)	−9.32450	+	4.31397i	−1.42197	+	0.657875i	−0.971960	−	0.235147i	\(-0.924443\pi\)
							−0.450013	+	0.893022i	\(0.648581\pi\)
\(47\)	1.73583	+	1.31954i	0.253197	+	0.192475i	0.724072	−	0.689725i	\(-0.242268\pi\)
							−0.470875	+	0.882200i	\(0.656062\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	59.2.c.a.15.3	yes	112
3.2	odd	2		531.2.i.a.487.2		112
4.3	odd	2		944.2.m.c.369.1		112
59.2	odd	58		3481.2.a.q.1.32		56
59.4	even	29	inner	59.2.c.a.4.3	✓	112
59.57	even	29		3481.2.a.p.1.25		56
177.122	odd	58		531.2.i.a.181.2		112
236.63	odd	58		944.2.m.c.417.1		112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
59.2.c.a.4.3	✓	112	59.4	even	29	inner
59.2.c.a.15.3	yes	112	1.1	even	1	trivial
531.2.i.a.181.2		112	177.122	odd	58
531.2.i.a.487.2		112	3.2	odd	2
944.2.m.c.369.1		112	4.3	odd	2
944.2.m.c.417.1		112	236.63	odd	58
3481.2.a.p.1.25		56	59.57	even	29
3481.2.a.q.1.32		56	59.2	odd	58