Embedded newform 210.3.w.a.47.4

• Label: 210.3.w.a.47.4
• Level: $210$
• Weight: $3$
• Character: 210.47
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			47.4
Character	\(\chi\)	\(=\)	210.47
Dual form			210.3.w.a.143.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366025 + 1.36603i) q^{2} +(-2.25393 - 1.97985i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(0.974348 - 4.90415i) q^{5} +(1.87953 - 3.80360i) q^{6} +(-4.12022 + 5.65896i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(1.16038 + 8.92488i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 32 q^{2} - 6 q^{3} - 12 q^{5} + 4 q^{7} - 128 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.366025	+	1.36603i	0.183013	+	0.683013i
							\(3\)	−2.25393	−	1.97985i	−0.751309	−	0.659950i
\(5\)	0.974348	−	4.90415i	0.194870	−	0.980829i
							\(7\)	−4.12022	+	5.65896i	−0.588602	+	0.808423i
\(11\)	−10.7431	+	6.20251i	−0.976641	+	0.563864i								−0.901254	−	0.433290i	\(-0.857353\pi\)
							−0.0753870	+	0.997154i	\(0.524019\pi\)
\(13\)	17.5103	+	17.5103i	1.34695	+	1.34695i	0.888958	+	0.457989i	\(0.151430\pi\)
							0.457989	+	0.888958i	\(0.348570\pi\)
\(17\)	−2.57491	+	9.60968i	−0.151465	+	0.565275i	0.847917	+	0.530129i	\(0.177856\pi\)
							−0.999382	+	0.0351465i	\(0.988810\pi\)
\(19\)	−6.00440	+	10.3999i	−0.316021	+	0.547365i	−0.979654	−	0.200694i	\(-0.935680\pi\)
							0.663633	+	0.748058i	\(0.269014\pi\)
\(23\)	−2.19573	+	0.588343i	−0.0954664	+	0.0255801i	−0.306236	−	0.951956i	\(-0.599070\pi\)
							0.210770	+	0.977536i	\(0.432403\pi\)
\(29\)	−19.7542			−0.681181			−0.340590	−	0.940212i	\(-0.610627\pi\)
							−0.340590	+	0.940212i	\(0.610627\pi\)
\(31\)	−45.3975	+	26.2103i	−1.46444	+	0.845493i	−0.999212	−	0.0396998i	\(-0.987360\pi\)
							−0.465225	+	0.885193i	\(0.654026\pi\)
\(37\)	7.35531	−	1.97085i	0.198792	−	0.0532662i	−0.158049	−	0.987431i	\(-0.550520\pi\)
							0.356841	+	0.934165i	\(0.383854\pi\)
\(41\)	18.4715			0.450524			0.225262	−	0.974298i	\(-0.427676\pi\)
							0.225262	+	0.974298i	\(0.427676\pi\)
\(43\)	33.8663	−	33.8663i	0.787588	−	0.787588i	−0.193511	−	0.981098i	\(-0.561987\pi\)
							0.981098	+	0.193511i	\(0.0619874\pi\)
\(47\)	22.7292	−	6.09028i	0.483600	−	0.129580i	−0.00877822	−	0.999961i	\(-0.502794\pi\)
							0.492379	+	0.870381i	\(0.336128\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.a.47.4	yes	64
3.2	odd	2		210.3.w.b.47.16	yes	64
5.3	odd	4		210.3.w.b.173.11	yes	64
7.3	odd	6	inner	210.3.w.a.17.9	✓	64
15.8	even	4	inner	210.3.w.a.173.9	yes	64
21.17	even	6		210.3.w.b.17.11	yes	64
35.3	even	12		210.3.w.b.143.16	yes	64
105.38	odd	12	inner	210.3.w.a.143.4	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.9	✓	64	7.3	odd	6	inner
210.3.w.a.47.4	yes	64	1.1	even	1	trivial
210.3.w.a.143.4	yes	64	105.38	odd	12	inner
210.3.w.a.173.9	yes	64	15.8	even	4	inner
210.3.w.b.17.11	yes	64	21.17	even	6
210.3.w.b.47.16	yes	64	3.2	odd	2
210.3.w.b.143.16	yes	64	35.3	even	12
210.3.w.b.173.11	yes	64	5.3	odd	4