Embedded newform 111.2.m.a.14.1

• Label: 111.2.m.a.14.1
• Level: $111$
• Weight: $2$
• Character: 111.14
• Analytic conductor: $0.886$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 111 = 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	111.m (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.886339462436\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			14.1
Character	\(\chi\)	\(=\)	111.14
Dual form			111.2.m.a.8.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.683439 - 2.55063i) q^{2} +(0.839686 - 1.51490i) q^{3} +(-4.30657 + 2.48640i) q^{4} +(-0.283720 + 1.05886i) q^{5} +(-4.43783 - 1.10639i) q^{6} +(-1.64962 - 2.85722i) q^{7} +(5.55078 + 5.55078i) q^{8} +(-1.58985 - 2.54408i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 6 q^{3} - 12 q^{4} - 4 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(38\)	\(76\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.683439	−	2.55063i	−0.483264	−	1.80357i	−0.587752	−	0.809041i	\(-0.699987\pi\)
							0.104488	−	0.994526i	\(-0.466680\pi\)
\(3\)	0.839686	−	1.51490i	0.484793	−	0.874629i
							\(5\)	−0.283720	+	1.05886i	−0.126884	+	0.473536i	−0.999900	−	0.0141553i	\(-0.995494\pi\)
0.873016	+	0.487691i	\(0.162161\pi\)
\(7\)	−1.64962	−	2.85722i	−0.623497	−	1.07993i	−0.988830	−	0.149051i	\(-0.952378\pi\)
							0.365333	−	0.930877i	\(-0.380955\pi\)
\(11\)	1.47499			0.444726			0.222363	−	0.974964i	\(-0.428623\pi\)
							0.222363	+	0.974964i	\(0.428623\pi\)
\(13\)	3.81992	+	1.02354i	1.05945	+	0.283880i	0.746154	−	0.665774i	\(-0.231898\pi\)
							0.313301	+	0.949654i	\(0.398565\pi\)
\(17\)	6.35827	−	1.70369i	1.54211	−	0.413206i	0.615160	−	0.788402i	\(-0.289091\pi\)
							0.926946	+	0.375196i	\(0.122425\pi\)
\(19\)	−4.74934	−	1.27258i	−1.08957	−	0.291950i	−0.331061	−	0.943609i	\(-0.607407\pi\)
							−0.758512	+	0.651659i	\(0.774073\pi\)
\(23\)	−2.17312	−	2.17312i	−0.453127	−	0.453127i	0.443264	−	0.896391i	\(-0.353820\pi\)
							−0.896391	+	0.443264i	\(0.853820\pi\)
\(29\)	−0.685058	+	0.685058i	−0.127212	+	0.127212i	−0.767846	−	0.640634i	\(-0.778672\pi\)
							0.640634	+	0.767846i	\(0.278672\pi\)
\(31\)	1.98007	+	1.98007i	0.355631	+	0.355631i	0.862200	−	0.506569i	\(-0.169086\pi\)
							−0.506569	+	0.862200i	\(0.669086\pi\)
\(37\)	−2.43252	+	5.57520i	−0.399904	+	0.916557i
							\(41\)	2.55696	+	4.42878i	0.399330	+	0.691660i	0.993643	−	0.112574i	\(-0.0359094\pi\)
−0.594313	+	0.804234i	\(0.702576\pi\)
\(43\)	2.78848	−	2.78848i	0.425239	−	0.425239i	−0.461764	−	0.887003i	\(-0.652783\pi\)
							0.887003	+	0.461764i	\(0.152783\pi\)
\(47\)			6.97869i			1.01795i	0.860782	+	0.508973i	\(0.169975\pi\)
							−0.860782	+	0.508973i	\(0.830025\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.m.a.14.1	yes	40
3.2	odd	2	inner	111.2.m.a.14.10	yes	40
37.8	odd	12	inner	111.2.m.a.8.10	yes	40
111.8	even	12	inner	111.2.m.a.8.1	✓	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.m.a.8.1	✓	40	111.8	even	12	inner
111.2.m.a.8.10	yes	40	37.8	odd	12	inner
111.2.m.a.14.1	yes	40	1.1	even	1	trivial
111.2.m.a.14.10	yes	40	3.2	odd	2	inner