Embedded newform 111.2.q.b.50.2

• Label: 111.2.q.b.50.2
• Level: $111$
• Weight: $2$
• Character: 111.50
• 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			50.2
Character	\(\chi\)	\(=\)	111.50
Dual form			111.2.q.b.20.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.81850 + 1.27332i) q^{2} +(-0.547308 - 1.64331i) q^{3} +(1.00153 - 2.75169i) q^{4} +(-0.160411 + 1.83351i) q^{5} +(3.08774 + 2.29144i) q^{6} +(-3.30299 + 2.77153i) q^{7} +(0.533366 + 1.99055i) q^{8} +(-2.40091 + 1.79879i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{3} - 24 q^{4} - 12 q^{6} - 24 q^{7} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(38\)	\(76\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.81850	+	1.27332i	−1.28587	+	0.900377i	−0.998351	−	0.0573982i	\(-0.981720\pi\)
							−0.287520	+	0.957775i	\(0.592831\pi\)
\(3\)	−0.547308	−	1.64331i	−0.315988	−	0.948763i
							\(5\)	−0.160411	+	1.83351i	−0.0717381	+	0.819970i	0.872050	+	0.489417i	\(0.162791\pi\)
−0.943788	+	0.330552i	\(0.892765\pi\)
\(7\)	−3.30299	+	2.77153i	−1.24841	+	1.04754i	−0.251593	+	0.967833i	\(0.580955\pi\)
							−0.996818	+	0.0797087i	\(0.974601\pi\)
\(11\)	−1.51121	+	2.61749i	−0.455647	+	0.789203i	−0.998725	−	0.0504789i	\(-0.983925\pi\)
							0.543079	+	0.839682i	\(0.317259\pi\)
\(13\)	−1.37247	−	2.94327i	−0.380655	−	0.816316i	−0.999525	−	0.0308232i	\(-0.990187\pi\)
							0.618870	−	0.785493i	\(-0.287591\pi\)
\(17\)	−0.223698	+	0.479722i	−0.0542547	+	0.116350i	−0.931566	−	0.363572i	\(-0.881557\pi\)
							0.877311	+	0.479922i	\(0.159335\pi\)
\(19\)	−3.22467	+	4.60530i	−0.739789	+	1.05653i	0.256162	+	0.966634i	\(0.417542\pi\)
							−0.995951	+	0.0898943i	\(0.971347\pi\)
\(23\)	−4.79995	−	1.28614i	−1.00086	−	0.268179i	−0.279054	−	0.960275i	\(-0.590021\pi\)
							−0.721805	+	0.692096i	\(0.756687\pi\)
\(29\)	7.17589	−	1.92277i	1.33253	−	0.357050i	0.478872	−	0.877885i	\(-0.341046\pi\)
							0.853658	+	0.520835i	\(0.174379\pi\)
\(31\)	1.08838	−	1.08838i	0.195478	−	0.195478i	−0.602580	−	0.798058i	\(-0.705861\pi\)
							0.798058	+	0.602580i	\(0.205861\pi\)
\(37\)	4.30127	−	4.30105i	0.707125	−	0.707089i
							\(41\)	−0.674488	−	0.245494i	−0.105337	−	0.0383397i	0.288814	−	0.957385i	\(-0.406739\pi\)
−0.394151	+	0.919046i	\(0.628961\pi\)
\(43\)	−0.280822	−	0.280822i	−0.0428250	−	0.0428250i	0.685370	−	0.728195i	\(-0.259640\pi\)
							−0.728195	+	0.685370i	\(0.759640\pi\)
\(47\)	−2.19370	+	1.26653i	−0.319983	+	0.184742i	−0.651385	−	0.758747i	\(-0.725812\pi\)
							0.331402	+	0.943490i	\(0.392478\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.q.b.50.2	yes	120
3.2	odd	2	inner	111.2.q.b.50.9	yes	120
37.20	odd	36	inner	111.2.q.b.20.9	yes	120
111.20	even	36	inner	111.2.q.b.20.2	✓	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.q.b.20.2	✓	120	111.20	even	36	inner
111.2.q.b.20.9	yes	120	37.20	odd	36	inner
111.2.q.b.50.2	yes	120	1.1	even	1	trivial
111.2.q.b.50.9	yes	120	3.2	odd	2	inner