Embedded newform 111.2.q.b.89.1

• Label: 111.2.q.b.89.1
• Level: $111$
• Weight: $2$
• Character: 111.89
• 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			89.1
Character	\(\chi\)	\(=\)	111.89
Dual form			111.2.q.b.5.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.22435 + 1.03723i) q^{2} +(1.24469 + 1.20447i) q^{3} +(2.58629 - 3.08222i) q^{4} +(2.78522 - 1.95024i) q^{5} +(-4.01793 - 1.38814i) q^{6} +(0.634954 - 3.60100i) q^{7} +(-1.28540 + 4.79718i) q^{8} +(0.0984907 + 2.99838i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{3} - 24 q^{4} - 12 q^{6} - 24 q^{7} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(38\)	\(76\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.22435	+	1.03723i	−1.57285	+	0.733432i	−0.996486	−	0.0837612i	\(-0.973307\pi\)
							−0.576364	+	0.817193i	\(0.695529\pi\)
\(3\)	1.24469	+	1.20447i	0.718620	+	0.695403i
							\(5\)	2.78522	−	1.95024i	1.24559	−	0.872172i	0.250211	−	0.968191i	\(-0.419500\pi\)
0.995379	+	0.0960197i	\(0.0306112\pi\)
\(7\)	0.634954	−	3.60100i	0.239990	−	1.36105i	−0.591855	−	0.806044i	\(-0.701604\pi\)
							0.831845	−	0.555007i	\(-0.187285\pi\)
\(11\)	0.183466	+	0.317773i	0.0553171	+	0.0958120i	0.892358	−	0.451328i	\(-0.149050\pi\)
							−0.837041	+	0.547140i	\(0.815716\pi\)
\(13\)	−1.83786	+	0.160792i	−0.509730	+	0.0445956i	−0.339120	−	0.940743i	\(-0.610129\pi\)
							−0.170610	+	0.985339i	\(0.554574\pi\)
\(17\)	−2.38397	−	0.208571i	−0.578198	−	0.0505858i	−0.205694	−	0.978616i	\(-0.565945\pi\)
							−0.372504	+	0.928031i	\(0.621501\pi\)
\(19\)	−2.29817	+	4.92843i	−0.527235	+	1.13066i	0.444758	+	0.895651i	\(0.353290\pi\)
							−0.971993	+	0.235009i	\(0.924488\pi\)
\(23\)	−3.68249	+	0.986720i	−0.767852	+	0.205745i	−0.621422	−	0.783476i	\(-0.713445\pi\)
							−0.146430	+	0.989221i	\(0.546778\pi\)
\(29\)	−1.48710	−	0.398468i	−0.276148	−	0.0739936i	0.118087	−	0.993003i	\(-0.462324\pi\)
							−0.394235	+	0.919010i	\(0.628990\pi\)
\(31\)	2.30320	+	2.30320i	0.413668	+	0.413668i	0.883014	−	0.469347i	\(-0.155510\pi\)
							−0.469347	+	0.883014i	\(0.655510\pi\)
\(37\)	5.77564	−	1.90841i	0.949509	−	0.313741i
							\(41\)	8.83113	+	7.41020i	1.37919	+	1.15728i	0.969511	+	0.245049i	\(0.0788039\pi\)
0.409679	+	0.912230i	\(0.365641\pi\)
\(43\)	−7.26963	+	7.26963i	−1.10861	+	1.10861i	−0.115275	+	0.993334i	\(0.536775\pi\)
							−0.993334	+	0.115275i	\(0.963225\pi\)
\(47\)	−2.81626	−	1.62597i	−0.410793	−	0.237171i	0.280337	−	0.959902i	\(-0.409554\pi\)
							−0.691130	+	0.722730i	\(0.742887\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.q.b.89.1	yes	120
3.2	odd	2	inner	111.2.q.b.89.10	yes	120
37.5	odd	36	inner	111.2.q.b.5.10	yes	120
111.5	even	36	inner	111.2.q.b.5.1	✓	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.q.b.5.1	✓	120	111.5	even	36	inner
111.2.q.b.5.10	yes	120	37.5	odd	36	inner
111.2.q.b.89.1	yes	120	1.1	even	1	trivial
111.2.q.b.89.10	yes	120	3.2	odd	2	inner