Embedded newform 111.2.m.a.29.1

• Label: 111.2.m.a.29.1
• Level: $111$
• Weight: $2$
• Character: 111.29
• 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			29.1
Character	\(\chi\)	\(=\)	111.29
Dual form			111.2.m.a.23.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.10243 - 0.563345i) q^{2} +(0.615620 - 1.61895i) q^{3} +(2.37081 + 1.36879i) q^{4} +(1.20031 - 0.321622i) q^{5} +(-2.20633 + 3.05693i) q^{6} +(0.940614 - 1.62919i) q^{7} +(-1.13520 - 1.13520i) q^{8} +(-2.24202 - 1.99332i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 6 q^{3} - 12 q^{4} - 4 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(38\)	\(76\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.10243	−	0.563345i	−1.48664	−	0.398345i	−0.578041	−	0.816007i	\(-0.696183\pi\)
							−0.908602	+	0.417662i	\(0.862849\pi\)
\(3\)	0.615620	−	1.61895i	0.355428	−	0.934703i
							\(5\)	1.20031	−	0.321622i	0.536795	−	0.143834i	0.0197710	−	0.999805i	\(-0.493706\pi\)
0.517024	+	0.855971i	\(0.327040\pi\)
\(7\)	0.940614	−	1.62919i	0.355519	−	0.615777i	−0.631688	−	0.775223i	\(-0.717638\pi\)
							0.987207	+	0.159446i	\(0.0509709\pi\)
\(11\)	−2.84343			−0.857327			−0.428663	−	0.903464i	\(-0.641015\pi\)
							−0.428663	+	0.903464i	\(0.641015\pi\)
\(13\)	−0.693631	−	2.58866i	−0.192379	−	0.717966i	−0.992930	−	0.118702i	\(-0.962127\pi\)
							0.800551	−	0.599264i	\(-0.204540\pi\)
\(17\)	1.15779	−	4.32092i	0.280804	−	1.04798i	−0.671046	−	0.741415i	\(-0.734155\pi\)
							0.951851	−	0.306561i	\(-0.0991784\pi\)
\(19\)	1.28770	+	4.80576i	0.295418	+	1.10252i	0.940884	+	0.338728i	\(0.109997\pi\)
							−0.645466	+	0.763789i	\(0.723337\pi\)
\(23\)	6.37655	+	6.37655i	1.32960	+	1.32960i	0.905714	+	0.423889i	\(0.139335\pi\)
							0.423889	+	0.905714i	\(0.360665\pi\)
\(29\)	2.64247	−	2.64247i	0.490694	−	0.490694i	−0.417831	−	0.908525i	\(-0.637209\pi\)
							0.908525	+	0.417831i	\(0.137209\pi\)
\(31\)	0.558014	+	0.558014i	0.100222	+	0.100222i	0.755440	−	0.655218i	\(-0.227423\pi\)
							−0.655218	+	0.755440i	\(0.727423\pi\)
\(37\)	5.38855	+	2.82197i	0.885872	+	0.463929i
							\(41\)	3.42355	−	5.92977i	0.534669	−	0.926075i	−0.464510	−	0.885568i	\(-0.653769\pi\)
0.999179	−	0.0405067i	\(-0.0128972\pi\)
\(43\)	4.64169	−	4.64169i	0.707851	−	0.707851i	−0.258232	−	0.966083i	\(-0.583140\pi\)
							0.966083	+	0.258232i	\(0.0831398\pi\)
\(47\)			2.51610i			0.367011i	0.983019	+	0.183506i	\(0.0587446\pi\)
							−0.983019	+	0.183506i	\(0.941255\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.m.a.29.1	yes	40
3.2	odd	2	inner	111.2.m.a.29.10	yes	40
37.23	odd	12	inner	111.2.m.a.23.10	yes	40
111.23	even	12	inner	111.2.m.a.23.1	✓	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.m.a.23.1	✓	40	111.23	even	12	inner
111.2.m.a.23.10	yes	40	37.23	odd	12	inner
111.2.m.a.29.1	yes	40	1.1	even	1	trivial
111.2.m.a.29.10	yes	40	3.2	odd	2	inner