Embedded newform 126.3.r.a.11.15

• Label: 126.3.r.a.11.15
• 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			11.15
Character	\(\chi\)	\(=\)	126.11
Dual form			126.3.r.a.23.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(2.02240 - 2.21583i) q^{3} -2.00000 q^{4} +(-7.20455 - 4.15955i) q^{5} +(3.13365 + 2.86011i) q^{6} +(5.54044 - 4.27827i) q^{7} -2.82843i q^{8} +(-0.819791 - 8.96259i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 64 q^{4} + 8 q^{6} + 2 q^{7} - 20 q^{9}+O(q^{10}) \)

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
							\(3\)	2.02240	−	2.21583i	0.674134	−	0.738609i
\(5\)	−7.20455	−	4.15955i	−1.44091	−	0.831909i								−0.442999	−	0.896522i	\(-0.646085\pi\)
							−0.997910	+	0.0646127i	\(0.979419\pi\)
\(7\)	5.54044	−	4.27827i	0.791491	−	0.611181i
							\(11\)	9.13192	−	5.27232i	0.830174	−	0.479301i	−0.0237379	−	0.999718i	\(-0.507557\pi\)
0.853912	+	0.520417i	\(0.174223\pi\)
\(13\)	1.24066	+	2.14889i	0.0954357	+	0.165300i	0.909790	−	0.415068i	\(-0.136242\pi\)
							−0.814355	+	0.580368i	\(0.802909\pi\)
\(17\)	−27.5318	−	15.8955i	−1.61952	−	0.935029i	−0.987045	−	0.160446i	\(-0.948707\pi\)
							−0.632473	−	0.774582i	\(-0.717960\pi\)
\(19\)	6.85989	+	11.8817i	0.361047	+	0.625351i	0.988133	−	0.153598i	\(-0.0490862\pi\)
							−0.627087	+	0.778949i	\(0.715753\pi\)
\(23\)	27.8304	+	16.0679i	1.21002	+	0.698603i	0.962763	−	0.270348i	\(-0.0871387\pi\)
							0.247253	+	0.968951i	\(0.420472\pi\)
\(29\)	2.05673	+	1.18745i	0.0709216	+	0.0409466i	0.535041	−	0.844826i	\(-0.320296\pi\)
							−0.464120	+	0.885772i	\(0.653629\pi\)
\(31\)	20.7685			0.669952			0.334976	−	0.942227i	\(-0.391272\pi\)
							0.334976	+	0.942227i	\(0.391272\pi\)
\(37\)	5.23692	+	9.07061i	0.141538	+	0.245152i	0.928076	−	0.372391i	\(-0.121462\pi\)
							−0.786538	+	0.617542i	\(0.788128\pi\)
\(41\)	−43.1999	+	24.9415i	−1.05366	+	0.608329i	−0.923671	−	0.383187i	\(-0.874827\pi\)
							−0.129986	+	0.991516i	\(0.541493\pi\)
\(43\)	16.3930	−	28.3936i	0.381234	−	0.660316i	−0.610005	−	0.792397i	\(-0.708833\pi\)
							0.991239	+	0.132081i	\(0.0421660\pi\)
\(47\)			41.5702i			0.884473i	0.896899	+	0.442236i	\(0.145815\pi\)
							−0.896899	+	0.442236i	\(0.854185\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.3.r.a.11.15	yes	32
3.2	odd	2		378.3.r.a.305.7		32
7.2	even	3		126.3.i.a.65.13	✓	32
9.4	even	3		378.3.i.a.179.7		32
9.5	odd	6		126.3.i.a.95.13	yes	32
21.2	odd	6		378.3.i.a.359.2		32
63.23	odd	6	inner	126.3.r.a.23.7	yes	32
63.58	even	3		378.3.r.a.233.15		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.i.a.65.13	✓	32	7.2	even	3
126.3.i.a.95.13	yes	32	9.5	odd	6
126.3.r.a.11.15	yes	32	1.1	even	1	trivial
126.3.r.a.23.7	yes	32	63.23	odd	6	inner
378.3.i.a.179.7		32	9.4	even	3
378.3.i.a.359.2		32	21.2	odd	6
378.3.r.a.233.15		32	63.58	even	3
378.3.r.a.305.7		32	3.2	odd	2