Embedded newform 126.3.r.a.11.5

• Label: 126.3.r.a.11.5
• Level: $126$
• Weight: $3$
• Character: 126.11
• Analytic conductor: $3.433$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	126.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43325133094\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.5
Character	\(\chi\)	\(=\)	126.11
Dual form			126.3.r.a.23.13

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(0.146807 + 2.99641i) q^{3} -2.00000 q^{4} +(2.36201 + 1.36371i) q^{5} +(4.23756 - 0.207617i) q^{6} +(-2.54299 + 6.52175i) q^{7} +2.82843i q^{8} +(-8.95690 + 0.879787i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 64 q^{4} + 8 q^{6} + 2 q^{7} - 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(73\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.41421i		−	0.707107i
							\(3\)	0.146807	+	2.99641i	0.0489357	+	0.998802i
\(5\)	2.36201	+	1.36371i	0.472402	+	0.272742i								0.717245	−	0.696821i	\(-0.245403\pi\)
							−0.244842	+	0.969563i	\(0.578736\pi\)
\(7\)	−2.54299	+	6.52175i	−0.363285	+	0.931678i
							\(11\)	−2.52158	+	1.45583i	−0.229234	+	0.132348i	−0.610219	−	0.792233i	\(-0.708918\pi\)
0.380984	+	0.924581i	\(0.375585\pi\)
\(13\)	10.4745	+	18.1424i	0.805731	+	1.39557i	0.915796	+	0.401643i	\(0.131561\pi\)
							−0.110065	+	0.993924i	\(0.535106\pi\)
\(17\)	24.7647	+	14.2979i	1.45675	+	0.841053i	0.998850	−	0.0479514i	\(-0.0152693\pi\)
							0.457898	+	0.889005i	\(0.348603\pi\)
\(19\)	−17.7418	−	30.7298i	−0.933782	−	1.61736i	−0.776792	−	0.629757i	\(-0.783155\pi\)
							−0.156989	−	0.987600i	\(-0.550179\pi\)
\(23\)	13.6186	+	7.86269i	0.592112	+	0.341856i	0.765932	−	0.642921i	\(-0.222278\pi\)
							−0.173820	+	0.984777i	\(0.555611\pi\)
\(29\)	18.1582	+	10.4836i	0.626144	+	0.361505i	0.779257	−	0.626704i	\(-0.215597\pi\)
							−0.153113	+	0.988209i	\(0.548930\pi\)
\(31\)	−12.4651			−0.402101			−0.201050	−	0.979581i	\(-0.564435\pi\)
							−0.201050	+	0.979581i	\(0.564435\pi\)
\(37\)	−5.80585	−	10.0560i	−0.156915	−	0.271785i	0.776840	−	0.629698i	\(-0.216821\pi\)
							−0.933755	+	0.357914i	\(0.883488\pi\)
\(41\)	26.4059	−	15.2455i	0.644047	−	0.371841i	−0.142125	−	0.989849i	\(-0.545393\pi\)
							0.786172	+	0.618008i	\(0.212060\pi\)
\(43\)	−12.6505	+	21.9113i	−0.294197	+	0.509564i	−0.974798	−	0.223090i	\(-0.928386\pi\)
							0.680601	+	0.732655i	\(0.261719\pi\)
\(47\)		−	84.6070i		−	1.80015i	−0.435736	−	0.900075i	\(-0.643512\pi\)
							0.435736	−	0.900075i	\(-0.356488\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.3.r.a.11.5	yes	32
3.2	odd	2		378.3.r.a.305.11		32
7.2	even	3		126.3.i.a.65.2	✓	32
9.4	even	3		378.3.i.a.179.11		32
9.5	odd	6		126.3.i.a.95.2	yes	32
21.2	odd	6		378.3.i.a.359.14		32
63.23	odd	6	inner	126.3.r.a.23.13	yes	32
63.58	even	3		378.3.r.a.233.3		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.i.a.65.2	✓	32	7.2	even	3
126.3.i.a.95.2	yes	32	9.5	odd	6
126.3.r.a.11.5	yes	32	1.1	even	1	trivial
126.3.r.a.23.13	yes	32	63.23	odd	6	inner
378.3.i.a.179.11		32	9.4	even	3
378.3.i.a.359.14		32	21.2	odd	6
378.3.r.a.233.3		32	63.58	even	3
378.3.r.a.305.11		32	3.2	odd	2