Embedded newform 111.2.q.b.5.2

• Label: 111.2.q.b.5.2
• Level: $111$
• Weight: $2$
• Character: 111.5
• 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			5.2
Character	\(\chi\)	\(=\)	111.5
Dual form			111.2.q.b.89.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.72972 - 0.806582i) q^{2} +(-1.63427 + 0.573716i) q^{3} +(1.05578 + 1.25823i) q^{4} +(2.08232 + 1.45806i) q^{5} +(3.28958 + 0.325808i) q^{6} +(-0.592946 - 3.36277i) q^{7} +(0.176590 + 0.659045i) q^{8} +(2.34170 - 1.87522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{3} - 24 q^{4} - 12 q^{6} - 24 q^{7} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(38\)	\(76\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{23}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.72972	−	0.806582i	−1.22310	−	0.570339i	−0.299644	−	0.954051i	\(-0.596868\pi\)
							−0.923453	+	0.383712i	\(0.874646\pi\)
\(3\)	−1.63427	+	0.573716i	−0.943548	+	0.331235i
							\(5\)	2.08232	+	1.45806i	0.931244	+	0.652064i	0.937526	−	0.347916i	\(-0.113111\pi\)
−0.00628171	+	0.999980i	\(0.502000\pi\)
\(7\)	−0.592946	−	3.36277i	−0.224113	−	1.27101i	−0.864374	−	0.502849i	\(-0.832285\pi\)
							0.640262	−	0.768157i	\(-0.278826\pi\)
\(11\)	2.44341	−	4.23210i	0.736715	−	1.27603i	−0.217252	−	0.976116i	\(-0.569709\pi\)
							0.953967	−	0.299912i	\(-0.0969573\pi\)
\(13\)	2.77277	+	0.242586i	0.769028	+	0.0672812i	0.464916	−	0.885355i	\(-0.346085\pi\)
							0.304112	+	0.952636i	\(0.401640\pi\)
\(17\)	0.565906	−	0.0495103i	0.137252	−	0.0120080i	−0.0183226	−	0.999832i	\(-0.505833\pi\)
							0.155575	+	0.987824i	\(0.450277\pi\)
\(19\)	2.85708	+	6.12702i	0.655458	+	1.40563i	0.899954	+	0.435985i	\(0.143600\pi\)
							−0.244496	+	0.969650i	\(0.578622\pi\)
\(23\)	−2.45038	−	0.656577i	−0.510940	−	0.136906i	−0.00586709	−	0.999983i	\(-0.501868\pi\)
							−0.505072	+	0.863077i	\(0.668534\pi\)
\(29\)	3.84122	−	1.02925i	0.713298	−	0.191128i	0.116118	−	0.993235i	\(-0.462955\pi\)
							0.597179	+	0.802108i	\(0.296288\pi\)
\(31\)	3.46971	−	3.46971i	0.623179	−	0.623179i	−0.323164	−	0.946343i	\(-0.604747\pi\)
							0.946343	+	0.323164i	\(0.104747\pi\)
\(37\)	−6.03926	−	0.726204i	−0.992848	−	0.119387i
							\(41\)	−4.68284	+	3.92937i	−0.731336	+	0.613664i	−0.930496	−	0.366303i	\(-0.880623\pi\)
0.199159	+	0.979967i	\(0.436179\pi\)
\(43\)	−3.97119	−	3.97119i	−0.605601	−	0.605601i	0.336193	−	0.941793i	\(-0.390861\pi\)
							−0.941793	+	0.336193i	\(0.890861\pi\)
\(47\)	−0.645666	+	0.372775i	−0.0941800	+	0.0543749i	−0.546350	−	0.837557i	\(-0.683983\pi\)
							0.452170	+	0.891932i	\(0.350650\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.q.b.5.2	✓	120
3.2	odd	2	inner	111.2.q.b.5.9	yes	120
37.15	odd	36	inner	111.2.q.b.89.9	yes	120
111.89	even	36	inner	111.2.q.b.89.2	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.q.b.5.2	✓	120	1.1	even	1	trivial
111.2.q.b.5.9	yes	120	3.2	odd	2	inner
111.2.q.b.89.2	yes	120	111.89	even	36	inner
111.2.q.b.89.9	yes	120	37.15	odd	36	inner