Embedded newform 111.2.q.b.59.7

• Label: 111.2.q.b.59.7
• Level: $111$
• Weight: $2$
• Character: 111.59
• Analytic conductor: $0.886$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 111 = 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	111.q (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.886339462436\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(10\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			59.7
Character	\(\chi\)	\(=\)	111.59
Dual form			111.2.q.b.32.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.420790 + 0.902386i) q^{2} +(-1.71042 + 0.272882i) q^{3} +(0.648338 - 0.772659i) q^{4} +(1.24569 + 1.77903i) q^{5} +(-0.965972 - 1.42863i) q^{6} +(-0.867836 + 4.92174i) q^{7} +(2.89354 + 0.775323i) q^{8} +(2.85107 - 0.933487i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{3} - 24 q^{4} - 12 q^{6} - 24 q^{7} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).

\(n\)	\(38\)	\(76\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{31}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.420790	+	0.902386i	0.297543	+	0.638083i	0.997068	−	0.0765252i	\(-0.0243826\pi\)
							−0.699524	+	0.714609i	\(0.746605\pi\)
\(3\)	−1.71042	+	0.272882i	−0.987511	+	0.157549i
							\(5\)	1.24569	+	1.77903i	0.557090	+	0.795607i	0.994619	−	0.103597i	\(-0.0330353\pi\)
−0.437529	+	0.899204i	\(0.644146\pi\)
\(7\)	−0.867836	+	4.92174i	−0.328011	+	1.86024i	0.159597	+	0.987182i	\(0.448980\pi\)
							−0.487608	+	0.873062i	\(0.662131\pi\)
\(11\)	−2.36164	−	4.09049i	−0.712063	−	1.23333i	−0.964082	−	0.265606i	\(-0.914428\pi\)
							0.252019	−	0.967722i	\(-0.418905\pi\)
\(13\)	−0.286332	−	3.27279i	−0.0794142	−	0.907709i	−0.926510	−	0.376269i	\(-0.877207\pi\)
							0.847096	−	0.531440i	\(-0.178349\pi\)
\(17\)	−0.0360699	+	0.412281i	−0.00874823	+	0.0999928i	−0.999313	−	0.0370495i	\(-0.988204\pi\)
							0.990565	+	0.137042i	\(0.0437596\pi\)
\(19\)	1.79512	+	0.837079i	0.411829	+	0.192039i	0.617481	−	0.786586i	\(-0.288153\pi\)
							−0.205651	+	0.978625i	\(0.565931\pi\)
\(23\)	−1.00930	−	3.76676i	−0.210454	−	0.785424i	−0.987718	−	0.156249i	\(-0.950060\pi\)
							0.777264	−	0.629175i	\(-0.216607\pi\)
\(29\)	0.994311	−	3.71082i	0.184639	−	0.689082i	−0.810069	−	0.586335i	\(-0.800570\pi\)
							0.994708	−	0.102747i	\(-0.0327632\pi\)
\(31\)	−0.758750	+	0.758750i	−0.136275	+	0.136275i	−0.771954	−	0.635678i	\(-0.780720\pi\)
							0.635678	+	0.771954i	\(0.280720\pi\)
\(37\)	−3.19979	+	5.17314i	−0.526043	+	0.850458i
							\(41\)	−0.196941	−	0.165253i	−0.0307571	−	0.0258082i	0.627280	−	0.778794i	\(-0.284168\pi\)
−0.658037	+	0.752986i	\(0.728613\pi\)
\(43\)	−4.12406	−	4.12406i	−0.628913	−	0.628913i	0.318881	−	0.947795i	\(-0.396693\pi\)
							−0.947795	+	0.318881i	\(0.896693\pi\)
\(47\)	7.62486	+	4.40221i	1.11220	+	0.642129i	0.939398	−	0.342828i	\(-0.111385\pi\)
							0.172801	+	0.984957i	\(0.444718\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.q.b.59.7	yes	120
3.2	odd	2	inner	111.2.q.b.59.4	yes	120
37.32	odd	36	inner	111.2.q.b.32.4	✓	120
111.32	even	36	inner	111.2.q.b.32.7	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.q.b.32.4	✓	120	37.32	odd	36	inner
111.2.q.b.32.7	yes	120	111.32	even	36	inner
111.2.q.b.59.4	yes	120	3.2	odd	2	inner
111.2.q.b.59.7	yes	120	1.1	even	1	trivial