Embedded newform 111.2.m.a.29.3

• Label: 111.2.m.a.29.3
• 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			29.3
Character	\(\chi\)	\(=\)	111.29
Dual form			111.2.m.a.23.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28890 - 0.345361i) q^{2} +(1.34079 + 1.09649i) q^{3} +(-0.190054 - 0.109728i) q^{4} +(1.53087 - 0.410196i) q^{5} +(-1.34946 - 1.87632i) q^{6} +(0.317227 - 0.549454i) q^{7} +(2.09415 + 2.09415i) q^{8} +(0.595429 + 2.94032i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 6 q^{3} - 12 q^{4} - 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28890	−	0.345361i	−0.911392	−	0.244207i	−0.227490	−	0.973780i	\(-0.573052\pi\)
							−0.683902	+	0.729574i	\(0.739719\pi\)
\(3\)	1.34079	+	1.09649i	0.774105	+	0.633058i
							\(5\)	1.53087	−	0.410196i	0.684626	−	0.183445i	0.100292	−	0.994958i	\(-0.468022\pi\)
0.584334	+	0.811513i	\(0.301356\pi\)
\(7\)	0.317227	−	0.549454i	0.119901	−	0.207674i	−0.799827	−	0.600230i	\(-0.795076\pi\)
							0.919728	+	0.392556i	\(0.128409\pi\)
\(11\)	2.64142			0.796417			0.398208	−	0.917295i	\(-0.369632\pi\)
							0.398208	+	0.917295i	\(0.369632\pi\)
\(13\)	−0.262748	−	0.980589i	−0.0728732	−	0.271966i	0.919869	−	0.392225i	\(-0.128294\pi\)
							−0.992743	+	0.120258i	\(0.961628\pi\)
\(17\)	−0.0282421	+	0.105401i	−0.00684972	+	0.0255635i	−0.969266	−	0.246015i	\(-0.920879\pi\)
							0.962416	+	0.271578i	\(0.0875455\pi\)
\(19\)	−0.101393	−	0.378405i	−0.0232612	−	0.0868120i	0.953319	−	0.301964i	\(-0.0976421\pi\)
							−0.976581	+	0.215152i	\(0.930975\pi\)
\(23\)	−5.15911	−	5.15911i	−1.07575	−	1.07575i	−0.996885	−	0.0788631i	\(-0.974871\pi\)
							−0.0788631	−	0.996885i	\(-0.525129\pi\)
\(29\)	−6.05381	+	6.05381i	−1.12416	+	1.12416i	−0.133056	+	0.991108i	\(0.542479\pi\)
							−0.991108	+	0.133056i	\(0.957521\pi\)
\(31\)	−2.83991	−	2.83991i	−0.510063	−	0.510063i	0.404483	−	0.914546i	\(-0.367452\pi\)
							−0.914546	+	0.404483i	\(0.867452\pi\)
\(37\)	−5.94793	+	1.27361i	−0.977834	+	0.209380i
							\(41\)	5.64477	−	9.77703i	0.881565	−	1.52691i	0.0319637	−	0.999489i	\(-0.489824\pi\)
0.849601	−	0.527426i	\(-0.176843\pi\)
\(43\)	5.03146	−	5.03146i	0.767290	−	0.767290i	−0.210339	−	0.977629i	\(-0.567457\pi\)
							0.977629	+	0.210339i	\(0.0674566\pi\)
\(47\)			1.67745i			0.244682i	0.992488	+	0.122341i	\(0.0390401\pi\)
							−0.992488	+	0.122341i	\(0.960960\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.m.a.29.3	yes	40
3.2	odd	2	inner	111.2.m.a.29.8	yes	40
37.23	odd	12	inner	111.2.m.a.23.8	yes	40
111.23	even	12	inner	111.2.m.a.23.3	✓	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.m.a.23.3	✓	40	111.23	even	12	inner
111.2.m.a.23.8	yes	40	37.23	odd	12	inner
111.2.m.a.29.3	yes	40	1.1	even	1	trivial
111.2.m.a.29.8	yes	40	3.2	odd	2	inner