Embedded newform 126.3.r.a.11.2

• Label: 126.3.r.a.11.2
• 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			11.2
Character	\(\chi\)	\(=\)	126.11
Dual form			126.3.r.a.23.10

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(-2.82827 - 1.00044i) q^{3} -2.00000 q^{4} +(0.857116 + 0.494856i) q^{5} +(-1.41483 + 3.99978i) q^{6} +(-6.76024 - 1.81637i) q^{7} +2.82843i q^{8} +(6.99824 + 5.65903i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 64 q^{4} + 8 q^{6} + 2 q^{7} - 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.82827	−	1.00044i	−0.942757	−	0.333480i
\(5\)	0.857116	+	0.494856i	0.171423	+	0.0989713i								0.583257	−	0.812288i	\(-0.301778\pi\)
							−0.411834	+	0.911259i	\(0.635111\pi\)
\(7\)	−6.76024	−	1.81637i	−0.965748	−	0.259481i
							\(11\)	−17.4109	+	10.0522i	−1.58281	+	0.913837i	−0.588365	+	0.808595i	\(0.700228\pi\)
−0.994447	+	0.105242i	\(0.966438\pi\)
\(13\)	5.16190	+	8.94067i	0.397069	+	0.687744i	0.993363	−	0.115023i	\(-0.0366940\pi\)
							−0.596294	+	0.802766i	\(0.703361\pi\)
\(17\)	−5.20329	−	3.00412i	−0.306076	−	0.176713i	0.339093	−	0.940753i	\(-0.389880\pi\)
							−0.645169	+	0.764040i	\(0.723213\pi\)
\(19\)	8.72476	+	15.1117i	0.459198	+	0.795354i	0.998919	−	0.0464902i	\(-0.0148036\pi\)
							−0.539721	+	0.841844i	\(0.681470\pi\)
\(23\)	−28.6166	−	16.5218i	−1.24420	−	0.718340i	−0.274255	−	0.961657i	\(-0.588431\pi\)
							−0.969947	+	0.243317i	\(0.921765\pi\)
\(29\)	−21.5690	−	12.4528i	−0.743757	−	0.429409i	0.0796765	−	0.996821i	\(-0.474611\pi\)
							−0.823434	+	0.567412i	\(0.807945\pi\)
\(31\)	15.9782			0.515426			0.257713	−	0.966222i	\(-0.417031\pi\)
							0.257713	+	0.966222i	\(0.417031\pi\)
\(37\)	−20.2862	−	35.1368i	−0.548277	−	0.949643i	−0.998393	−	0.0566731i	\(-0.981951\pi\)
							0.450116	−	0.892970i	\(-0.351383\pi\)
\(41\)	−51.3151	+	29.6268i	−1.25159	+	0.722604i	−0.971425	−	0.237349i	\(-0.923722\pi\)
							−0.280162	+	0.959953i	\(0.590388\pi\)
\(43\)	−10.4214	+	18.0505i	−0.242359	+	0.419778i	−0.961386	−	0.275204i	\(-0.911255\pi\)
							0.719027	+	0.694982i	\(0.244588\pi\)
\(47\)		−	0.295274i		−	0.00628242i	−0.999995	−	0.00314121i	\(-0.999000\pi\)
							0.999995	−	0.00314121i	\(-0.000999880\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.3.r.a.11.2	yes	32
3.2	odd	2		378.3.r.a.305.13		32
7.2	even	3		126.3.i.a.65.7	✓	32
9.4	even	3		378.3.i.a.179.13		32
9.5	odd	6		126.3.i.a.95.7	yes	32
21.2	odd	6		378.3.i.a.359.12		32
63.23	odd	6	inner	126.3.r.a.23.10	yes	32
63.58	even	3		378.3.r.a.233.5		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.i.a.65.7	✓	32	7.2	even	3
126.3.i.a.95.7	yes	32	9.5	odd	6
126.3.r.a.11.2	yes	32	1.1	even	1	trivial
126.3.r.a.23.10	yes	32	63.23	odd	6	inner
378.3.i.a.179.13		32	9.4	even	3
378.3.i.a.359.12		32	21.2	odd	6
378.3.r.a.233.5		32	63.58	even	3
378.3.r.a.305.13		32	3.2	odd	2