Embedded newform 126.3.r.a.11.10

• Label: 126.3.r.a.11.10
• 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			11.10
Character	\(\chi\)	\(=\)	126.11
Dual form			126.3.r.a.23.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(-2.24880 - 1.98568i) q^{3} -2.00000 q^{4} +(1.84316 + 1.06415i) q^{5} +(2.80817 - 3.18028i) q^{6} +(6.16730 + 3.31125i) q^{7} -2.82843i q^{8} +(1.11417 + 8.93077i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 64 q^{4} + 8 q^{6} + 2 q^{7} - 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.24880	−	1.98568i	−0.749599	−	0.661892i
\(5\)	1.84316	+	1.06415i	0.368631	+	0.212829i								0.672860	−	0.739770i	\(-0.265065\pi\)
							−0.304229	+	0.952599i	\(0.598399\pi\)
\(7\)	6.16730	+	3.31125i	0.881043	+	0.473036i
							\(11\)	7.75793	−	4.47904i	0.705266	−	0.407185i	−0.104040	−	0.994573i	\(-0.533177\pi\)
0.809306	+	0.587388i	\(0.199844\pi\)
\(13\)	10.4189	+	18.0461i	0.801454	+	1.38816i	0.918659	+	0.395051i	\(0.129273\pi\)
							−0.117205	+	0.993108i	\(0.537393\pi\)
\(17\)	9.01197	+	5.20307i	0.530116	+	0.306063i	0.741064	−	0.671435i	\(-0.234322\pi\)
							−0.210948	+	0.977497i	\(0.567655\pi\)
\(19\)	6.15121	+	10.6542i	0.323748	+	0.560748i	0.981258	−	0.192698i	\(-0.0617237\pi\)
							−0.657510	+	0.753446i	\(0.728390\pi\)
\(23\)	−16.5455	−	9.55254i	−0.719369	−	0.415328i	0.0951515	−	0.995463i	\(-0.469666\pi\)
							−0.814520	+	0.580135i	\(0.803000\pi\)
\(29\)	−28.3968	−	16.3949i	−0.979199	−	0.565341i	−0.0771708	−	0.997018i	\(-0.524589\pi\)
							−0.902028	+	0.431677i	\(0.857922\pi\)
\(31\)	37.8877			1.22218			0.611092	−	0.791560i	\(-0.290731\pi\)
							0.611092	+	0.791560i	\(0.290731\pi\)
\(37\)	16.9479	+	29.3546i	0.458051	+	0.793367i	0.998858	−	0.0477797i	\(-0.0152145\pi\)
							−0.540807	+	0.841146i	\(0.681881\pi\)
\(41\)	−28.7819	+	16.6172i	−0.701997	+	0.405298i	−0.808091	−	0.589058i	\(-0.799499\pi\)
							0.106094	+	0.994356i	\(0.466166\pi\)
\(43\)	10.3473	−	17.9220i	0.240634	−	0.416791i	−0.720261	−	0.693703i	\(-0.755978\pi\)
							0.960895	+	0.276912i	\(0.0893112\pi\)
\(47\)		−	66.2205i		−	1.40895i	−0.709730	−	0.704474i	\(-0.751183\pi\)
							0.709730	−	0.704474i	\(-0.248817\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.3.r.a.11.10	yes	32
3.2	odd	2		378.3.r.a.305.4		32
7.2	even	3		126.3.i.a.65.15	✓	32
9.4	even	3		378.3.i.a.179.4		32
9.5	odd	6		126.3.i.a.95.15	yes	32
21.2	odd	6		378.3.i.a.359.5		32
63.23	odd	6	inner	126.3.r.a.23.2	yes	32
63.58	even	3		378.3.r.a.233.12		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.i.a.65.15	✓	32	7.2	even	3
126.3.i.a.95.15	yes	32	9.5	odd	6
126.3.r.a.11.10	yes	32	1.1	even	1	trivial
126.3.r.a.23.2	yes	32	63.23	odd	6	inner
378.3.i.a.179.4		32	9.4	even	3
378.3.i.a.359.5		32	21.2	odd	6
378.3.r.a.233.12		32	63.58	even	3
378.3.r.a.305.4		32	3.2	odd	2