Embedded newform 126.3.i.a.65.6

• Label: 126.3.i.a.65.6
• Level: $126$
• Weight: $3$
• Character: 126.65
• Analytic conductor: $3.433$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	126.i (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			65.6
Character	\(\chi\)	\(=\)	126.65
Dual form			126.3.i.a.95.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 + 0.707107i) q^{2} +(0.894073 - 2.86367i) q^{3} +(1.00000 - 1.73205i) q^{4} -3.35218i q^{5} +(0.929913 + 4.13948i) q^{6} +(-6.40395 - 2.82655i) q^{7} +2.82843i q^{8} +(-7.40127 - 5.12067i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 32 q^{4} + 8 q^{6} + 2 q^{7} + 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22474	+	0.707107i	−0.612372	+	0.353553i
							\(3\)	0.894073	−	2.86367i	0.298024	−	0.954558i
\(5\)		−	3.35218i		−	0.670435i								−0.942141	−	0.335218i	\(-0.891190\pi\)
							0.942141	−	0.335218i	\(-0.108810\pi\)
\(7\)	−6.40395	−	2.82655i	−0.914850	−	0.403793i
							\(11\)			14.1890i			1.28991i	0.764222	+	0.644953i	\(0.223123\pi\)
−0.764222	+	0.644953i	\(0.776877\pi\)
\(13\)	−8.31908	−	14.4091i	−0.639930	−	1.10839i	−0.985448	−	0.169978i	\(-0.945630\pi\)
							0.345518	−	0.938412i	\(-0.387703\pi\)
\(17\)	−13.5532	+	7.82495i	−0.797248	+	0.460291i	−0.842508	−	0.538684i	\(-0.818922\pi\)
							0.0452601	+	0.998975i	\(0.485588\pi\)
\(19\)	17.3162	−	29.9925i	0.911378	−	1.57855i	0.0992583	−	0.995062i	\(-0.468353\pi\)
							0.812120	−	0.583491i	\(-0.198314\pi\)
\(23\)		−	26.2640i		−	1.14191i	−0.820980	−	0.570956i	\(-0.806572\pi\)
							0.820980	−	0.570956i	\(-0.193428\pi\)
\(29\)	18.2368	+	10.5290i	0.628855	+	0.363069i	0.780308	−	0.625395i	\(-0.215062\pi\)
							−0.151454	+	0.988464i	\(0.548395\pi\)
\(31\)	16.8738	−	29.2264i	0.544317	−	0.942785i	−0.454332	−	0.890832i	\(-0.650122\pi\)
							0.998650	−	0.0519531i	\(-0.0165446\pi\)
\(37\)	−31.9815	+	55.3936i	−0.864365	+	1.49712i	0.00331203	+	0.999995i	\(0.498946\pi\)
							−0.867677	+	0.497129i	\(0.834388\pi\)
\(41\)	5.44200	−	3.14194i	0.132732	−	0.0766328i	−0.432164	−	0.901795i	\(-0.642250\pi\)
							0.564896	+	0.825162i	\(0.308916\pi\)
\(43\)	0.0742627	−	0.128627i	0.00172704	−	0.00299132i	−0.865161	−	0.501495i	\(-0.832784\pi\)
							0.866888	+	0.498504i	\(0.166117\pi\)
\(47\)	36.3552	−	20.9897i	0.773516	−	0.446590i	−0.0606115	−	0.998161i	\(-0.519305\pi\)
							0.834127	+	0.551572i	\(0.185972\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.3.i.a.65.6	✓	32
3.2	odd	2		378.3.i.a.359.15		32
7.4	even	3		126.3.r.a.11.1	yes	32
9.4	even	3		378.3.r.a.233.2		32
9.5	odd	6		126.3.r.a.23.9	yes	32
21.11	odd	6		378.3.r.a.305.10		32
63.4	even	3		378.3.i.a.179.10		32
63.32	odd	6	inner	126.3.i.a.95.6	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.i.a.65.6	✓	32	1.1	even	1	trivial
126.3.i.a.95.6	yes	32	63.32	odd	6	inner
126.3.r.a.11.1	yes	32	7.4	even	3
126.3.r.a.23.9	yes	32	9.5	odd	6
378.3.i.a.179.10		32	63.4	even	3
378.3.i.a.359.15		32	3.2	odd	2
378.3.r.a.233.2		32	9.4	even	3
378.3.r.a.305.10		32	21.11	odd	6