Embedded newform 126.3.r.a.11.4

• Label: 126.3.r.a.11.4
• 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			11.4
Character	\(\chi\)	\(=\)	126.11
Dual form			126.3.r.a.23.12

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(-0.677712 + 2.92245i) q^{3} -2.00000 q^{4} +(-5.54397 - 3.20081i) q^{5} +(4.13297 + 0.958429i) q^{6} +(4.41544 - 5.43175i) q^{7} +2.82843i q^{8} +(-8.08141 - 3.96116i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 64 q^{4} + 8 q^{6} + 2 q^{7} - 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.677712	+	2.92245i	−0.225904	+	0.974150i
\(5\)	−5.54397	−	3.20081i	−1.10879	−	0.640162i								−0.170277	−	0.985396i	\(-0.554466\pi\)
							−0.938517	+	0.345234i	\(0.887799\pi\)
\(7\)	4.41544	−	5.43175i	0.630777	−	0.775964i
							\(11\)	−16.5236	+	9.53988i	−1.50214	+	0.867262i	−0.502144	+	0.864784i	\(0.667455\pi\)
−0.999997	+	0.00247766i	\(0.999211\pi\)
\(13\)	−11.8281	−	20.4870i	−0.909857	−	1.57592i	−0.814261	−	0.580499i	\(-0.802857\pi\)
							−0.0955966	−	0.995420i	\(-0.530476\pi\)
\(17\)	−4.77629	−	2.75759i	−0.280958	−	0.162211i	0.352899	−	0.935661i	\(-0.385196\pi\)
							−0.633857	+	0.773450i	\(0.718529\pi\)
\(19\)	2.39342	+	4.14552i	0.125969	+	0.218185i	0.922111	−	0.386924i	\(-0.126463\pi\)
							−0.796142	+	0.605110i	\(0.793129\pi\)
\(23\)	6.64254	+	3.83507i	0.288806	+	0.166742i	0.637403	−	0.770530i	\(-0.280009\pi\)
							−0.348597	+	0.937273i	\(0.613342\pi\)
\(29\)	11.7658	+	6.79297i	0.405716	+	0.234240i	0.688947	−	0.724811i	\(-0.258073\pi\)
							−0.283231	+	0.959052i	\(0.591406\pi\)
\(31\)	13.5975			0.438628			0.219314	−	0.975654i	\(-0.429618\pi\)
							0.219314	+	0.975654i	\(0.429618\pi\)
\(37\)	31.7248	+	54.9490i	0.857428	+	1.48511i	0.874374	+	0.485252i	\(0.161272\pi\)
							−0.0169461	+	0.999856i	\(0.505394\pi\)
\(41\)	−32.6243	+	18.8356i	−0.795714	+	0.459406i	−0.841970	−	0.539524i	\(-0.818604\pi\)
							0.0462560	+	0.998930i	\(0.485271\pi\)
\(43\)	−3.41749	+	5.91926i	−0.0794764	+	0.137657i	−0.903024	−	0.429590i	\(-0.858658\pi\)
							0.823548	+	0.567247i	\(0.191991\pi\)
\(47\)		−	83.5555i		−	1.77778i	−0.458124	−	0.888888i	\(-0.651479\pi\)
							0.458124	−	0.888888i	\(-0.348521\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.3.r.a.11.4	yes	32
3.2	odd	2		378.3.r.a.305.15		32
7.2	even	3		126.3.i.a.65.3	✓	32
9.4	even	3		378.3.i.a.179.15		32
9.5	odd	6		126.3.i.a.95.3	yes	32
21.2	odd	6		378.3.i.a.359.10		32
63.23	odd	6	inner	126.3.r.a.23.12	yes	32
63.58	even	3		378.3.r.a.233.7		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.i.a.65.3	✓	32	7.2	even	3
126.3.i.a.95.3	yes	32	9.5	odd	6
126.3.r.a.11.4	yes	32	1.1	even	1	trivial
126.3.r.a.23.12	yes	32	63.23	odd	6	inner
378.3.i.a.179.15		32	9.4	even	3
378.3.i.a.359.10		32	21.2	odd	6
378.3.r.a.233.7		32	63.58	even	3
378.3.r.a.305.15		32	3.2	odd	2